PC5271 wiki - User contributions [en]

LED based avalanched photodetector

2025-04-29T10:23:58Z

Runzhi: /* Calculation of quenching equivalent capacitance circuit */

==Group members:==

Cai Shijie

Email:E1184418@u.nus.edu.sg

Nie Huanxin

Email: E1352877@u.nus.edu.sg

Yang Runzhi

Email:E1127408@u.nus.edu.sg

== Idea ==

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

==Main==
=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is relatively close to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

Main Page

2025-04-29T10:20:48Z

Runzhi: /* LED based avalanched photodetector */

'''Welcome to the wiki page for the course PC5271: Physics of Sensors!'''

This is the repository where projects are documented. Creation of new accounts have now been blocked,and editing/creating pages is enabled. If you need an account, please contact Christian.

<span style="color: red">'''Deadline for report'''</span>

Deadline for finishing your report on this wiki will be <span style="color: red">'''29 April 2025 23:59:59</span>''' ;) Please be ensured you are happy with your project page at this point, as this will be the basis for our assessment.
Thanks!! Ramanathan Mahendrian and Christian Kurtsiefer

==Projects==
===[[Project 1 (Example)]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join. Once the project has stabilized, keep stuff in the project page linked by the headline.

===[[Laser Gyroscope]]===
Team members: Darren Koh, Chiew Wen Xin

Build a laser interferometer to detect rotation.

===[[Laser Distance Measurer]]===
Team members: Arya Chowdhury, Liu Sijin, Jonathan Wong

This project aims to build a laser interferometer to measure distances.

(CK: We should have fast laser diodes and fast photodiodes, mounted in optics bench kits)

===[[Non-contact Alcohol Concentration Measurement Device At NIR Spectrum]]===
Team members: Lim Gin Joe,Sun Weijia, Yan Chengrui, Zhu Junyi
This project aims to build a sensor to measure the concentration of alcohol by optical method
(CK: you can check Optics Letters <b>47</b>, 5076-5079 (2022) https://doi.org/10.1364/OL.472890 for some info)

===[[Ultrasonic Acoustic Remote Sensing]]===
Team member(s): Chua Rui Ming

How well can we use sound waves to survey the environment?

(CK: we have some ultrasonic transducers around 40kHz, see datasheets below)

===[[Blood Oxygen Sensor]]===
Team members: He Lingzi, Zhao Lubo, Zhang Ruoxi, Xu Yintong

This project aims to build a sensor to detect the oxygen concentration in the blood.

(CK: We have LEDs at 940nm and 660nm peak wavelenth emission, plus some Si photodiodes)

===[[Terahertz Electromagnetic Wave Detection]]===
Team members: Shizhuo Luo, Bohan Zhang

This project aims to detect Terahertz waves, especially terahertz pulses (This is because they are intense and controllable). We may try different ways like electro-optical sampling and thermopile detectors.

===[[Optical measurement of atmospheric carbon dioxide]]===
Team member(s): Ta Na, Cao Yuan, Qi Kaiyi, Gao Yihan, Chen Yiming

This project aims to make use of the optical properties of carbon dioxide gas to create a portable and accurate measurement device of carbon dioxide.

===[[Photodetector with wavelength @ 780nm and 1560nm]]===
Team members: Sunke Lan

To design photodetector as power monitor with power within 10mW.

(CK: Standard problem, we have already the respective photodiodes)

===[[LED based avalanched photodetector]]===

Team members: Cai Shijie, Nie Huanxin, Yang Runzhi

1.Build a single photondetection circuit using reverse-biased LED. The possible LED is gallium compounds based, emitting wavelength around 700nm(red light).

2.Record singal waveforms under different variables: reverse bias and power of light source.

3.Investigate the photodetector's quantum efficiency, stimulation number's accordance with Poisson distribution.

4.Give numerical calculations of quench time of equivalent quenching circuit (as accurate as possible).

Other devices needed: use LED as single photon source (wavelength shorter than the emitting wavelength 700nm).

===[[Motor Rotation Speed Measurement via the Hall Effect Sensor]]===

Team members: Mi Tianshuo

This project implements a Hall effect sensor to measure the rotation speed of a circuit board-controlled rotary plate.

===[[STM32-Based IMU Attitude Estimation]]===

Team members: Li Ding, Fan Xuting

This project utilizes an STM32 microcontroller and an MPU6050 IMU sensor to measure angular velocity and acceleration, enabling real-time attitude angle computation for motion tracking.

===[[Magnetic field sensing using a fluxgate magnetometer]]===
Team members: Ni Xueqi

This project investigates magnetic field sensing using a fluxgate magnetometer (FLC100). A 5 V supply drives the sensor, and the output is monitored with an oscilloscope. A permanent magnet is modeled in COMSOL and experimentally measured, showing expected distance-dependent field decay. A toroidal solenoid is also studied; due to imperfections, a magnetic field decaying as <math> 1/r^3 </math> outside the toroid is observed. Measurements confirm dipole-like behavior and linear current dependence, demonstrating the fluxgate magnetometer's sensitivity and validating magnetic field modeling.

===[[CO2 Concentration Detector]]===
Team members: Xie Zihan，Zhao Yun，Zhang Wenbo

Infrared absorption-based CO₂ gas sensors are developed based on the principle that different substances exhibit different absorption spectra. Because the chemical structures of different gas molecules vary, their degrees of absorption of infrared radiation at various wavelengths also differ. Consequently, when infrared radiation of different wavelengths is directed at the sample in turn, certain wavelengths are selectively absorbed and thus weakened by the sample, generating an infrared absorption spectrum.

Once the infrared absorption spectrum of a particular substance is known, its infrared absorption peaks can be identified. For the same substance, when the concentration changes, the absorption intensity at a given absorption peak also changes, and this intensity is directly proportional to the concentration. Therefore, by detecting how the gas alters the wavelength and intensity of the light, one can determine the gas concentration.

===[[Humidity Sensor of Graphite]]===
Team members: Xu Ruizhe, Wei Heyi, Li Zerui, Ma Shunyu

This project aims to build a humidity sensor with Graphite.

===[[Temperature and humidity sensors]]===
Team members: Chen Andi, Chen Miaoge, Chen Yingnan, Fang Ye

This project aims to design and evaluate a real-time temperature and humidity monitoring system based on Arduino and the DHT11 sensor. The system is low-cost, easy to implement, and suitable for applications such as smart homes, agriculture, and storage environments. In addition to system development, the project compares the performance of the DHT11 and SHT31 sensors in various environments—indoor, outdoor, and rainy conditions—to assess their accuracy, stability, and response time. The results help guide practical sensor selection, especially in scenarios where cost and simplicity are prioritized over high precision.

===[[Ultrasonic Doppler Speedometer]]===
Team members: Yang Yuzhen, Liu Xueyi, Shao Shuai

Design and build an ultrasonic Doppler speedometer to measure the distance and velocity of a moving object.

==Resources==
===Books and links===
* A good textbook on the Physics of Sensors is Jacob Fraden: Handbook of Mondern Sensors, Springer, ISBN 978-3-319-19302-1 or [https://link.springer.com/book/10.1007/978-3-319-19303-8 doi:10.1007/978-3-319-19303-8]. There shoud be an e-book available through the NUS library at https://linc.nus.edu.sg/record=b3554643
* Another good textbook: John B.Bentley: Principles of Measurement Systems, 4th Edition, Pearson, ISBN: 0-13-043028-5 or https://linc.nus.edu.sg/record=b2458243 in our library.

===Software===
* Various Python extensions. [https://www.python.org Python] is a very powerful free programming language that runs on just about any computer platform. It is open source and completely free.
* [https://www.gnuplot.info Gnuplot]: A free and very mature data display tool that works on just about any platform used that produces excellent publication-grade eps and pdf figures. Can be also used in scripts. Open source and completely free.
* Matlab: Very common, good toolset also for formal mathematics, good graphics. Expensive. We may have a site license, but I am not sure how painful it is for us to get a license for this course. Ask if interested.
* Mathematica: More common among theroetical physicists, very good in formal maths, now with better numerics. Graphs are ok but can be a pain to make looking good. As with Matlab, we do have a campus license. Ask if interested.

===Apps===
Common mobile phones these days are equipped with an amazing toolchest of sensors. There are a few apps that allow you to access them directly, and turn your phone into a powerful sensor. Here some suggestions:

* Physics Toolbox sensor suite on [https://play.google.com/store/apps/details?id=com.chrystianvieyra.physicstoolboxsuite&hl=en_SG Google play store] or [https://apps.apple.com/us/app/physics-toolbox-sensor-suite/id1128914250 Apple App store].

===Data sheets===
A number of components might be useful for several groups. Some common data sheets are here:
* Photodiodes:
** Generic Silicon pin Photodiode type [[Media:Bpw34.pdf|BPW34]]
** Fast photodiodes (Silicon PIN, small area): [[Media:S5971_etc_kpin1025e.pdf|S5971/S5972/S5973]]
* PT 100 Temperature sensors based on platinum wire: [[Media:PT100_TABLA_R_T.pdf|Calibration table]]
* Thermistor type [[Media:Thermistor B57861S.pdf|B57861S]] (R0=10kΩ, B=3988Kelvin). Search for [https://en.wikipedia.org/wiki/Steinhart-Hart_equation Steinhart-Hart equation]. See [[Thermistor]] page here as well.
* Humidity sensor
** Sensirion device the reference unit: [[media:Sensirion SHT30-DIS.pdf|SHT30/31]]
* Thermopile detectors:
** [[Media:Thermopile_G-TPCO-035 TS418-1N426.pdf|G-TPCO-035 / TS418-1N426]]: Thermopile detector with a built-in optical bandpass filter for light around 4μm wavelength for CO<sub>2</sub> absorption
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]
* Ultrasonic detectors:
** plastic detctor, 40 kHz, -74dB: [[Media:MCUSD16P40B12RO.pdf|MCUSD16P40B12RO]]
** metal casing/waterproof, 48 kHz, -90dB, [[Media:MCUSD14A48S09RS-30C.pdf|MCUSD14A48S09RS-30C]]
** metal casing, 40 kHz, sensitivity unknown, [[Media:MCUST16A40S12RO.pdf|MCUST16A40S12RO]]
** metal casing/waterproof, 300kHz, may need high voltage: [[Media:MCUSD13A300B09RS.pdf|MCUSD13A300B09RS]]
* Magnetic field sensor
** Fluxgate magnetometer [[media:Data-sheet FLC-100.pdf|FCL100]]
* Lasers
** Red laser diode [[media:HL6501MG.pdf|HL6501MG]]
* Generic amplifiers
** Instrumentation amplifiers: [[media:Ad8221.pdf|AD8221]] or [[media:AD8226.pdf|AD8226]]
** Conventional operational amplifiers: Precision: [[media:OP27.pdf|OP27]], General purpose: [[media:OP07.pdf|OP07]]
** Transimpedance amplifiers for photodetectors: [[media:AD8015.pdf|AD8015]]

===Some code snippets===
* For the [[media:Generic FPGA board version 3 - Quantum Optics Wiki.pdf|pattern generator]], you need to send the following text file to it to generate ultrasonic pulses:

# This pattern is to generate a burst of 10..20 oscillations at 40 kHz
# every 100ms for a sonar test. Pulses are TTL level on the AUX output,
# I/O lane 0 bit 7 is a sync pulse (10ns long), I/O lane 0 bit 0 copies the
# aux line, bit 1 indicates the pause periode between bursts.
# Internal counter 0 is for burst counting, int counter 1 for pause cycles

# Set device to programming mode: reset table, reset RAM, program params
config 13
writew 0, 60571; # basic address is 0, input thres -0.5V (not used)
writew 0,0,0,0; # external counter preload (not used)
writew 9,999,0,0; # internal cnt preload only first one is relevant
# and determines the number of pulses (minus 1) and
# number minus 1 of multiples of 100us for pause
writew 0,0,0,0,0,0,0,0; # DAC preload - not used

config 4; # switch to RAM write

# This is the RAM sequence- starting with 40kHz burst
writew 0x80,0,0,0,0,0, 0,0x1010; # ad 0: sync pulse 10nsec, load cnt 0
writew 0x01 0,0,1,0,0,1248,0xc004; # ad 1: pulse on (12.49us), if done go ad4
writew 0x01 0,0,1,0,0, 0,0x1100; # ad 2: pulse on for 10ns, decr int cnt 0
writew 0x00,0,0,0,0,0,1249,0x0001; # ad 3: pulse off for 12.5us, then go 1

# Waiting time / pause
writew 0x02,0,0,0,0,0, 0,0x1020; # ad 4: preload internal cntr 1 (10ns)
writew 0x02,0,0,0,0,0,9998,0x1200; # ad 5: decr cnt1 (10ns)
writew 0x02,0,0,0,0,0, 0,0xd008; # ad 6: if count is down goto ad 8(10ns)
writew 0x02,0,0,0,0,0, 0,0x0005; # ad 7: goto ad 5(10ns)

writew 0x02,0,0,0,0,0, 0,0x0000; # ad 8: restart (goto ad 0; 10ns)

# start pattern and keep output level on AUX line to TTL level
config 0x400;

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

== Old Wiki ==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2]

LED based avalanched photodetector

2025-04-29T10:05:38Z

Runzhi: /* Group members: */

==Group members:==

Cai Shijie

Email:E1184418@u.nus.edu.sg

Nie Huanxin

Email: E1352877@u.nus.edu.sg

Yang Runzhi

Email:E1127408@u.nus.edu.sg

== Idea ==

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

==Main==
=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:05:25Z

Runzhi: /* Group members: */

=='''Group members:'''==

Cai Shijie

Email:E1184418@u.nus.edu.sg

Nie Huanxin

Email: E1352877@u.nus.edu.sg

Yang Runzhi

Email:E1127408@u.nus.edu.sg

== Idea ==

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

==Main==
=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:04:54Z

Runzhi: /* Group members: */

=='''Group members:'''==

Cai Shijie Email:E1184418@u.nus.edu.sg

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

== Idea ==

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

==Main==
=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:04:28Z

Runzhi: /* Group members: */

=='''Group members:'''==

Cai Shijie Email:E1184418@u.nus.edu.sg

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

== Idea ==

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

==Main==
=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:03:41Z

Runzhi:

='''Group members:'''=

Cai Shijie Email:E1184418@u.nus.edu.sg

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:03:07Z

Runzhi: /* Part 1. Working Principles: */

'''Group members:'''

Cai Shijie Email:

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Author: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:02:51Z

Runzhi: /* Part 2. Experimental Setup */

'''Group members:'''

Cai Shijie Email:

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:02:39Z

Runzhi: /* Part 3. Results and Analysis */

'''Group members:'''

Cai Shijie Email:

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Editor: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:02:22Z

Runzhi: /* Part 2. Experimental Setup */

'''Group members:'''

Cai Shijie Email:

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Editor: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T10:01:54Z

Runzhi:

'''Group members:'''

Cai Shijie Email:

Nie Huanxin Email: E1352877@u.nus.edu.sg

Yang Runzhi Email:E1127408@u.nus.edu.sg

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T09:59:30Z

Runzhi: /* Idea */

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.

In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T09:59:09Z

Runzhi: /* Idea */

=== Idea ===

Our project aims to construct a photo detector to measure some phenomena in optical experiment. We choose Poisson distribution of photons generated by LED as our target phenomena. In the sensor part, we want to detect number of photons, thus we choose reversed LED as the sensor. Using avalanche effect in LED, we could detect relative number of photons with observable quantities.
In the main part, we constructed this LED based avalanched photodetector(APD), explained the working principle and analyzed the result detected by the LED based APD, especially compared the distribution results of photon with theoretical Poisson distribution.

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T09:37:23Z

Runzhi: /* 3. Results and Analysis */

=== Idea ===

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== Part 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T09:37:13Z

Runzhi: /* 2. Experimental Setup */

=== Idea ===

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== Part 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T09:36:57Z

Runzhi: /* 1. Working Principles: */

=== Idea ===

=== Part 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-29T09:36:29Z

Runzhi:

=== Idea ===

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-28T09:57:52Z

Runzhi: /* Photon detection circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:Circuit2.png|600px]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

File:Circuit2.png

2025-04-28T09:56:52Z

Runzhi:

LED based avalanched photodetector

2025-04-25T11:07:35Z

Runzhi: /* Code Listings */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center|MATLAB code of Quenching Circuit differential equation]]
</div>

LED based avalanched photodetector

2025-04-25T09:37:07Z

Runzhi: /* Equipment and the circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]]

[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T09:36:22Z

Runzhi: /* Equipment and the circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up_s.png|400px|frameless|alt=Fig.1]][[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

File:Chain up s.png

2025-04-25T09:35:56Z

Runzhi: the small version to fit.

== Summary ==
the small version to fit.

LED based avalanched photodetector

2025-04-25T09:34:59Z

Runzhi: /* Equipment and the circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.2]][[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

File:Chain up.png

2025-04-25T09:34:13Z

Runzhi: Runzhi uploaded a new version of File:Chain up.png

LED based avalanched photodetector

2025-04-25T09:32:35Z

Runzhi: /* Equipment and the circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain_up.png|400px|frameless|alt=Fig.1]][[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

File:Chain up.png

2025-04-25T09:31:40Z

Runzhi:

LED based avalanched photodetector

2025-04-25T06:21:12Z

Runzhi: /* Calculation of quenching equivalent capacitance circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png|400px]]

''Fig 1.4 Quench equivalent circuit ''
</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result, which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time 2.png|400px]]

''Fig 1.5 Theoretical and Experimental results of Quench time ''
</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T06:17:04Z

Runzhi: /* Photon detection circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]

''Fig 1.3 APD photodetection and light source circuit''
</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T06:14:32Z

Runzhi: /* Avalanche Photon detection of Reverse-biased LED */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig 1.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig 1.2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T06:13:39Z

Runzhi: /* Avalanche Photon detection of Reverse-biased LED */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|400px|Avalanche Mechanism]]

''Fig.1 Avalanche Mechanism''
</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]

''Fig. 2 PD, APD and SPAD region''
</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T06:08:22Z

Runzhi: /* 3. Results and Analysis */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|600px|Avalanche Mechanism]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T06:07:51Z

Runzhi: /* Avalanche Photon detection of Reverse-biased LED */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png|600px|Avalanche Mechanism]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.

====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse 3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
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<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
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LED based avalanched photodetector

2025-04-25T06:06:38Z

Runzhi: /* 3. Results and Analysis */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse 3_1.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:3_2.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases significantly, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
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[[File:Code2.png|600px|thumb|center]]
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[[File:Code3.png|600px|thumb|center]]
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[[File:Code4.png|600px|thumb|center]]
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[[File:Code5.png|600px|thumb|center]]
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[[File:Code6.png|600px|thumb|center]]
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File:3 2.png

2025-04-25T06:03:24Z

Runzhi:

File:3 1.png

2025-04-25T06:03:05Z

Runzhi:

LED based avalanched photodetector

2025-04-25T05:51:37Z

Runzhi: /* Experiment steps */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

'''The main measurement including the following parts:'''

'''1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.'''

'''2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.'''

'''3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution. '''

'''4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.'''

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T05:50:34Z

Runzhi: /* Equipment and the circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE with working wavelength around 450 nm as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

The main measurement including the following parts:

1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.

2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.

3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution,

4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T05:50:00Z

Runzhi: /* Experiment steps */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of pulse peaks and decay time after excitations. Also, the number of pulse peaks in a small period is

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

The main measurement including the following parts:

1.Working voltage sweep: Connect the measure circuit to oscilloscope. Keep the reverse-bias voltage constant at 25.8 V and vary the working voltage from 3 V to 7 V in 0.5 V increments. During each voltage level, use the oscilloscope to measure number of pulses in 1000ms and the height of pulse peaks in 6 times.

2.Reverse-bias voltage sweep: Connect the measure circuit to oscilloscope. Keep the working voltage constant at 5 V and vary the reverse-bias voltage from 25.4 V to 26.5 V in 0.05 V increments. During each voltage level, use the oscilloscope to measure the decay time after excitation.

3.Confirmation the Poisson distribution: Connect the measure circuit to counter. Keep the reverse-bias voltage constant at 25.8 V and the working voltage constant at 5 V. Set the sample time, then sample and compare the sample results with the theoretical Poisson distribution,

4.Calculate quantum efficiency: Connect the measure circuit to oscilloscope, use the oscilloscope to measure number of pulses in 50ms. Use multimeter to measure the working current and the working voltage of blue LED. Calculate the number of photon generated in 50ms and calculate the quantum efficiency.

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

File:Quench time 2.png

2025-04-25T05:39:15Z

Runzhi:

LED based avalanched photodetector

2025-04-25T05:35:59Z

Runzhi: /* Equipment and the circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Arduino Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of signal peaks and decay time after excitations.

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

working

reverse

counter

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T05:24:05Z

Runzhi: /* Photon detection circuit */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>

====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of signal peaks and decay time after excitations.

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

working

reverse

counter

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

LED based avalanched photodetector

2025-04-25T05:21:03Z

Runzhi: /* Code Listings */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.<br/>
:So the photodetection circuit in our experiment consists of: AND113 red LED (wavelength λ=700nm), resistor <math>R_{0}=10k\Omega</math>, capacitor C=0.33μF, 2231A-30-3 triple channel DC power supply, RTB2004 Digital Oscilloscope, wires, circuit experimental bread board. And we use a series circuit of blue LED (λ=450nm) and resistor R=1kΩ as light source.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>
:And the theoretical filtering frequency of RC circuit is: f=1/2\piRC≈48.25Hz. During experiments, the noise amplitude is stabilized under 5mV, so a photodetection signal threshold is set at <math>V_{th}=10mV</math>.
====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of signal peaks and decay time after excitations.

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

working

reverse

counter

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
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[[File:Code2.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code3.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code4.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code5.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code6.png|600px|thumb|center]]
</div>

File:Code6.png

2025-04-25T05:19:46Z

Runzhi:

File:Code5.png

2025-04-25T05:19:30Z

Runzhi:

File:Code4.png

2025-04-25T05:19:10Z

Runzhi:

File:Code3.png

2025-04-25T05:18:48Z

Runzhi:

LED based avalanched photodetector

2025-04-25T05:18:29Z

Runzhi: /* Code Listings */

=== 1. Working Principles: ===
'''Editor: Nie Huanxin Email: E1352877@u.nus.edu.sg'''
==== Avalanche Photon detection of Reverse-biased LED ====
:When a light-emitting diode (LED) light bulb is forward-biased,electrons from the n-region and holes from the p-region migrate toward the depletion zone under the applied electric field. Upon recombination in the active region, energy is released as photons via radiative recombination of charge carriers, converting electrical energy into light—the fundamental principle of LEDs. However, under reverse bias <math>V_{re}</math>, the LED operates as a photodetector: In the absence of incident photons, the depletion region in LED widens and the built-in electric field is strengthened. Although minority carrier diffusion increases, the absence of photogenerated free carriers results in negligible current (limited only by intrinsic thermal generation). When photons with energy exceeding the bandgap (ℎ𝜈 ≥ 𝐸𝑔) are absorbed, electron-hole pairs (EHPs) will be generated within the depletion zone and separated by strong built-in electric field, then the reverse-biased LED light bulb will operates as a photodetector with distinct voltage-dependent response regimes:
<div style="text-align:center;">[[Image:Avalanche.png]]</div>
#At small reverse voltages (below the avalanche threshold <math>V_{th}</math>), the photocurrent exhibits a near-linear relationship with the applied bias. The observed signal is dominated by leakage current, where photogenerated EHPs are separated by built-in electric field but with negligible current gain. Only primary photocurrent generated from adequate incident photon flux can be detected under this voltage region.
#When <math>V_{BD}</math> approaches the avalanche threshold, the LED turns into Avalanche Photodiode mode, or APD mode. High-energy photogenerated charge carriers in the material gain sufficient kinetic energy to create secondary EHPs via collisions with other carriers, leading to a detectable amplification of the total photocurrent depending on the applied voltage.
#Beyond the breakdown voltage (<math>V_{BD}</math>), the LED operates in Single Photon Avalanche Diode (SPAD) mode, even a single photon can trigger avalanche impact ionization, creating a macroscopic current pulse. SPAD mode have infinitely effective gain, however, the detectable gain is limited by noise, electric loss and quenching.
<div style="text-align:center;">[[Image:SPAD.png]]</div>
:In avalanche multiplication regimes, a critical challenge arises when the signal amplification rate exceeds the current decay rate, leading to persistent conduction and preventing subsequent photon detection. Trapped carriers (e.g., at defect sites) during current decay will also trigger false secondary avalanches (afterpulse). To mitigate these effects, quenching is required to reset the diode to its pre-avalanche state. The simplest quenching method employs a series resistor <math>R_{0}</math> to suppress the avalanche: During an avalanche, the stimulated high current induces a voltage drop across <math>R_{0}</math>, reducing the bias voltage across the diode below its breakdown threshold (<math>V_{th}</math>). This terminates the avalanche, allowing the diode to recover. Once the photocurrent ceases, the bias voltage across the diode returns to its original value, ready for the next photondetection. The effectiveness of quenching largely depends on the resistor <math>R_{0}</math>: If <math>R_{0}</math> is too small for a sufficient voltage drop, the detection accuracy will be greatly degraded due to the inefficiency of quenching; If <math>R_{0}</math> is too large, the amplitude as well as time span of signal pulses may be , owning to unqualified signal waves; Only proper <math>R_{0}</math> can optimize the photodetection efficiency.
:During Experiments, we applied R<sub>0=10kΩ empirically for reverse-biased AND 113 LED, to balance quenching efficiency and timing resolution. Also, the LED's reverse-bias range of <math>V_{re}</math>∈[25.5V,26.4V] are discovered experimentally, where:
*Below 25.5 V, the field is too weak for detectable gain;
*Above 26.4 V, passive quenching fails to suppress runaway avalanches.<br/>
:From the derivations mentioned above, the photodetection mechanism sequence in a reverse-biased LED of APD mode are as follows:
# An incident photon (with energy hν≥E<sub>g</sub>) generates an electron-hole pair (EHP) within the depletion region.
# The high electric field accelerates the primary carriers, enabling impact ionization. Secondary EHPs are created through collisions with the lattice, causing an exponential rise in carrier density (n). The resulting current (I) grows exponentially until reaching a peak value (<math>I_{s}</math>).
# The peak current occurs when the avalanche generation rate balances the recombination rate. At this point, the current <math>I_{s}</math> is proportional to the carrier density <math>n_{s}</math> in the photodetector:<br/><div style="text-align:center;"><math>I_s=\frac{V_R}{R_0}\propto{n_s}\bullet\frac{e^2\tau(T)}{m_e}</math></div><br/>Where <math>\tau(T)</math> is temperature-dependent mean free time, <math>m_e</math> and e are the mass and charge of a electron. Under certain temperature T, the <math>I_{s}</math> is determined by <math>n_{s}</math> only.
# The voltage drop across <math>R_{0}</math> reduces the bias below <math>V_{th}</math>, suppressing the avalanche. The diode then behaves as a discharging capacitor, with the current decaying to zero as the system resets to its pre-avalanche state.
====Photon detection circuit====
:Theoretically, every reverse-biased LED have a photodetection range near its breakdown voltage <math>V_{BD}</math>. However, only the AND113 red LED exhibits an ideal photodetection voltage range below 30V, making it suitable for experimental applications. And photoelectric conversion efficiency depends on the strength of the LED’s built-in electric field, which is governed by the carrier concentration in the doped semiconductor material. A higher carrier density enhances the internal electric field in depletion zone, thereby improving the separation and collection efficiency of photogenerated electron-hole pairs. To minimize interference from ambient light, measurements were conducted inside a lightproof enclosure (a cardboard box) within a darkroom. Additionally, a series RC low-pass filter was integrated into the photodetection circuit to suppress high-frequency noise arising from residual transient photocurrent signals.<br/>
:So the photodetection circuit in our experiment consists of: AND113 red LED (wavelength λ=700nm), resistor <math>R_{0}=10k\Omega</math>, capacitor C=0.33μF, 2231A-30-3 triple channel DC power supply, RTB2004 Digital Oscilloscope, wires, circuit experimental bread board. And we use a series circuit of blue LED (λ=450nm) and resistor R=1kΩ as light source.
<div style="text-align:center;">[[Image:APD Circuit.png]]</div>
:And the theoretical filtering frequency of RC circuit is: f=1/2\piRC≈48.25Hz. During experiments, the noise amplitude is stabilized under 5mV, so a photodetection signal threshold is set at <math>V_{th}=10mV</math>.
====Calculation of quenching equivalent capacitance circuit====
:During quenching period, the reverse-biased LED functions as a capacitor:
<div style="text-align:center;">[[Image:Quench circuit.png]]</div>
:So the theoretical relaxation time of such equavalent circuit can be calculated using a second-order RC circuit differential equation:
<div style="text-align:center;"><math>R_{0}^{2}CC_{LED}\frac{d^{2}V_{LED}}{dt^{2}}+R_{0}(C+2C_{LED})\frac{dV_{LED}}{dt}+V_{LED}=V_{re}</math></div>
:where
<div style="text-align:center;"><math>C\frac{dV_{0}}{dt}|_{t=0}=I_{0}</math></div>
:However, the width and charge density of depletion zone is determined by the reverse voltage applied on the PN junction, leading to the variation of <math>C_{LED}</math>. Using parallel plate capacitor model, we can derive <math>C_{LED}</math> as a function of <math>V_{LED}</math>. The width of depletion zone W can be expressed as:
<div style="text-align:center;"><math>W=\sqrt{\frac{2\varepsilon(V_{bi}-V_{LED})}{q}\frac{N_{A}N_{D}}{N_{A}+N_{D}}}</math></div>
:Where ε is the dielectric constant of the material, <math>V_{bi}</math> is the built-in electric field voltage, <math>N_{A}</math> and <math>N_{A}</math> is the charge carrier density in P and N zone. Simplifying the formula with approximate carrier density <math>N_{0}=\frac{N_{A}N_{D}}{N_{A}+N_{D}}</math>, we can obtain:
<div style="text-align:center;"><math>C_{LED}=\frac{\varepsilon S}{W}=S\sqrt{\frac{\varepsilon eN_{0}}{2(V_{bi}-V_{LED})}}</math></div>
:Since the capacitance is a non-linear function of V_LED, it is extremely difficult to solve the circuit equation directly; instead, translating the equation into numerical calculation programme in MATLAB makes it easier to visualize the theoretical calculation result:
:Which is similar to experimental results:
<div style="text-align:center;">[[Image:Quench time.png]]</div>

=== 2. Experimental Setup ===
'''Author: Yang Runzhi Email:e1127408@u.nus.edu.sg'''

==== Equipment and the circuit====

'''Equipment:'''

'''Core: One red LDE , one blue LDE.'''

'''KEITHLEY Triple Channel DC Power Supply, ROHDE&SCHWARZ RBT2004 Digital Oscilloscope, Counter, Multimeter.'''

'''Breadboard, resistors, capacitors, wires in all kinds.'''

We choose a red LED with working wavelength around 650 nm as detector(sensor) and another bule LDE as light source.

<div style="text-align:center;">
[[File:red_LED.png|400px|frameless|alt=Fig.1]]

''Figure 2.1: The red LED used in experiment''
</div>

Connect the circuit as the circuit diagram shown in part1.The working voltage and reversed-bias voltage is generated by DC Power Supply. The measure part is connected to either the Oscilloscope or the Counter. The distance between the light source and the detector is 1.3 cm.

<div style="text-align:center;">
[[File:chain.png|400px|frameless|alt=Fig.1]]

''Figure 2.2: The circuit used in experiment''
</div>

The cutoff frequency <math>f_c</math> of a simple RC low-pass filter is given by:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi R C}</math></div>

In our case:
* <math>R = 10\,\mathrm{k}\Omega = 10^4\,\Omega</math>
* <math>C = 0.33 \times 10^{-6} \,\mathrm{F}</math>

Then:

<div style="text-align:center;"><math>f_c = \frac{1}{2 \pi \times 10^4 \times 0.33 \times 10^{-6}}\mathrm{Hz} \approx 48.25\,\mathrm{Hz}</math></div>

Therefore, the cutoff frequency of the filter is 48.25 Hz. Signals with higher frequency is filtered.

==== Experiment steps====
First step is test the working voltage and reversed-bias voltage. Changing the 2 parameter till we can detect excitation in the oscilloscope.

<div style="text-align:center;">
[[File:wave.png|400px|frameless|alt=Fig.1]]

''Figure 2.3: The excitation figure in the oscilloscope''
</div>

Figure 2.3 is what we got in this step, with working voltage at 5V level and reversed-bias voltage at 25.8V. The figure shows two important parameter which we can detect in the following experiment: height of signal peaks and decay time after excitations.

Then cover the circuit with a box to avoid interference. The effect of this step is that all the photons detector captured are generated by the blue light source.

working

reverse

counter

<div style="text-align:center;">
[[File:Cover.png|400px|frameless|alt=Fig.1]]

''Figure 2.4: Covered circuit connected with counter''
</div>

=== 3. Results and Analysis ===

'''Author: Cai Shijie Email:e1184418@u.nus.edu.sg'''
'''Date: April 2025'''

The avalanche effect can be observed with the power of the light source around 13 μW. This indicates that the detector is a sensitive APD capable of detecting low photon number densities.

<div style="text-align:center;">
[[File:pulse number.png|400px|frameless|alt=Fig.1]]

''Figure 3.1: Pulse number vs voltage of power supply''
</div>

Figure 3.1 measures the average pulse number per 50 ms versus the voltage of the power supply. The linear curve corresponds to the increasing photon number with higher voltage of the light source.

<div style="text-align:center;">
[[File:pulse hight.png|400px|frameless|alt=Fig.2]]

''Figure 3.2: Pulse height vs voltage of power supply''
</div>

Figure 3.2 shows that the pulse height increases with the voltage of the power supply. For a single-photon avalanche photodiode (SAPD), the curve should be flat, meaning each pulse corresponds to one photon. However, when the power supply voltage is doubled, the pulse height increases by 14%, suggesting that each pulse corresponds to several photons.

By setting the photocurrent pulse number per 50 ms as one sample, 1000 or 10,000 samples are used for statistical analysis and compared with the theoretical Poisson distribution, resulting in Figure 3.3.

<div style="text-align:center;">
[[File:Poisson distribution data.png|600px|frameless|alt=Fig.3]]

''Figure 3.3: Pulse number distribution compared to Poisson distribution''
</div>

Several methods are used to analyze how closely the data match the theoretical model. The Kullback–Leibler (KL) divergence (result: 0.0061), Jensen–Shannon (JS) divergence (result: 0.0366), and Bhattacharyya distance (result: 0.0014) all qualitatively estimate the similarity between the real data and the theoretical Poisson distribution. All results are close to 0, indicating a high degree of similarity between the two distributions.

The Kolmogorov–Smirnov (KS) test is used to obtain a p-value, which is more sensitive than the previous methods. The p-value indicates the probability of observing the test statistic under the assumption that the data follow a Poisson distribution. The p-value obtained is 0.0264, which is smaller than 0.05, thus rejecting the Poisson distribution in this test.

Furthermore, the quantum efficiency (QE) is estimated by '''0.245%'''. The Python, Arduino code, and QE calculation are attached in the appendix.

'''In conclusion''', the LED-based APD cannot fully verify the Poisson distribution of the LED source, as it is not a true single-photon detector.

== Appendix ==

=== QE Estimation ===

'''Given Parameters'''
* Blue LED optical power: <math>P_\text{blue} = 1 \, \mu\text{W} = 1 \times 10^{-6} \, \text{W}</math>
* Wavelength of blue light: <math>\lambda_\text{blue} = 450 \, \text{nm}</math>
* Photon energy:
<math>
E_\text{ph} = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \cdot 3.0 \times 10^8}{450 \times 10^{-9}} \approx 4.42 \times 10^{-19} \, \text{J}
</math>
* Photon emission rate:
<math>
N_\text{emit} = \frac{P_\text{blue}}{E_\text{ph}} = \frac{1 \times 10^{-6}}{4.42 \times 10^{-19}} \approx 2.26 \times 10^{12} \, \text{photons/s}
</math>
* Emission duration: <math>\Delta t = 50 \, \text{ms} = 0.05 \, \text{s}</math>
* Distance between LEDs: <math>d = 0.1 \, \text{m}</math>
* Red LED pn-junction radius: <math>r = 17 \, \mu\text{m} = 1.7 \times 10^{-5} \, \text{m}</math>
* Entrance area of the pn-junction:
<math>
A = \pi r^2 = \pi (1.7 \times 10^{-5})^2 \approx 9.08 \times 10^{-10} \, \text{m}^2
</math>
* Solid angle covered by receiving junction:
<math>
\Omega = \frac{A}{d^2} = \frac{9.08 \times 10^{-10}}{(0.1)^2} = 9.08 \times 10^{-8} \, \text{sr}
</math>
* Fraction of photons geometrically intercepted:
<math>
f = \frac{\Omega}{4\pi} = \frac{9.08 \times 10^{-8}}{4\pi} \approx 7.23 \times 10^{-9}
</math>
* Shell transmission rate at 450 nm (approximate): <math>T_\text{shell} = 0.2</math>
* Number of detected photo-pulses: <math>N_\text{detected} = 4</math>

'''Photons Reaching the pn-Junction in 50 ms:'''
<math>
N_\text{incident} = N_\text{emit} \cdot \Delta t \cdot f \cdot T_\text{shell} = 2.26 \times 10^{12} \cdot 0.05 \cdot 7.23 \times 10^{-9} \cdot 0.2 \approx 1.63 \times 10^3
</math>

<math>
\eta = \frac{N_\text{detected}}{N_\text{incident}} = \frac{4}{1.63 \times 10^3} \approx 2.45 \times 10^{-3} = 0.245\%
</math>

'''Conclusion:''' Using a realistic pn-junction area and accounting for geometric and spectral filtering factors, the estimated quantum efficiency of the red LED functioning as a photon detector is approximately '''0.245%'''. This aligns with expectations given that LEDs are not optimized for photodetection, especially under off-band excitation (blue light in a red LED).

=== Code Listings ===

<div style="text-align:center;">
[[File:Arduino code.png|600px|thumb|center|Arduino code]]
</div>
<br/>
<div style="text-align:center;">
[[File:Code1.png|600px|thumb|center]]
</div>
<div style="text-align:center;">
[[File:Code2.png|600px|thumb|center]]
</div>

File:Code2.png

2025-04-25T05:15:26Z

Runzhi: