EA Spectroscopy as a series of sensors: Investigating the Impact of Film-Processing Temperature on Mobility in Organic Diodes

Team members

Li Jinhan A0327554Y

Liu Chenyang A0328377R

Idea

We will use EA spectroscopy, which will include optical sensors, electrical sensors, and lock-in amplifiers, among other components as a highly sensitive, non-destructive optical sensing platform to measure the internal electric field modulation response of organic diodes under operating conditions, and to quantitatively extract carrier mobility based on this measurement. By systematically controlling the thin film preparation temperature and comparing the EA response characteristics of different samples, the project aims to reveal the influence of film preparation temperature on device mobility.

Introduction

The performance of organic semiconductor devices (such as organic diodes) is largely limited by the charge transport processes within the thin film, and carrier mobility is one of the key parameters characterizing charge transport capability. Since organic thin films typically exhibit significant morphological and microstructure sensitivity, the film fabrication temperature affects factors such as molecular packing, crystallinity, phase separation behavior, and trapped state density, thereby altering the internal electric field distribution and charge injection/transport efficiency, ultimately manifesting as differences in mobility and device response. Therefore, establishing a characterization method capable of reliably tracking the "process-structure-transport" relationship is crucial for process optimization and device performance improvement.

EA Theory

Electroabsorption technology is achieved by measuring the change in absorption coefficient after an applied electric field is applied. Under normal incident conditions, the intensity $I$ of light transmitted through the absorbing medium can be calculated using the Lambert-Beer law:

$I=I_0(1-R{)}^2{e}^{-\alpha d}$----------(1)

Where $I_0$ represents the original light intensity of the incident light, $R$ is the reflectivity, $\alpha$ is the absorption coefficient, and $d$ is the thickness of the absorbing medium.

Both $R$ and $\alpha$ are affected by the external electric field, and their $\mathrm{\Delta }I$ changes as follows:

$\mathrm{\Delta }I=-I(\frac{2R}{1-R}\frac{\mathrm{\Delta }R}{R}+d\mathrm{\Delta }\alpha )$----------(2)

Under classical operating conditions, the change in the value of $R$ is negligible. Equation (2) simplifies to:

$\frac{\mathrm{\Delta }I}{I}=-d\mathrm{\Delta }\alpha$----------(3)

From a microscopic perspective, the energy level E(F) of state $⟨i|$ in electric field F is given by the following equation:

$E(F)=E(0)-m_{i}F-\frac{1}{2}(p_{i}F)F$----------(4)

$m_i$ represents the electric dipole moment of this state, and $p_i$ is its polarizability. Therefore, the optical transition energy shift $\mathrm{\Delta }E$ from the initial state to the final state is given by the following equation:

$\mathrm{\Delta }E(F)=-(m_f-m_i)F-\frac{1}{2}(p_f-p_i)F^2$----------(5)

The first term represents the linear energy change caused by the difference in dipole moments between the initial and final states, which cancels out in isotropic solids without permanent dipoles. The second term describes the energy shift caused by the difference in polarizability between the initial and final states, an effect that always exists.

The absorption variation of the external electric field $\mathrm{\Delta }\alpha$ can be represented by the $\mathrm{\Delta }E$ term of the Maclaurin series, while the third and higher order terms can be ignored due to the small spectral changes.

$\mathrm{\Delta }\alpha (h\nu )=\left(\frac{d\alpha }{dE}\mathrm{\Delta }E\right)+\frac{1}{2}\left(\frac{d^2\alpha }{d^{2}E}\mathrm{\Delta }{E}^2\right)$----------(6)

The sole contribution of the first term comes from the second-order Strac effect $\frac{1}{2}\mathrm{\Delta }p{F}^2$, while the isotropic average of $(\mathrm{\Delta }mF{)}^2$ in the second term does not cancel out, but instead produces $\frac{1}{3}(\mathrm{\Delta }mF{)}^2$ in the randomly distributed matrix. The third contribution of $\mathrm{\Delta }\alpha (hv)$ comes from the transition of the oscillator to the previously forbidden state when the electric field is present, and this transition also has a quadratic relationship with F.

Adding up all contributions to $\mathrm{\Delta }\alpha (hv)$, we get:

$\mathrm{\Delta }\alpha =[a\alpha +b\frac{d\alpha }{dE}+C\frac{d^2\alpha }{d^{2}E}]F^2$----------(7)

For a composite electric field with AC component $E_{ac}\sin (\omega t)$ and DC component $E_{d}c$:

$E(t)=E_{ac}\sin (\omega t)+E_{d}c$----------(8)

Substituting formula (8) into (7) yields the modulation of $\mathrm{\Delta }\alpha$ at the fundamental frequencies $1\omega$ and $2\omega$:

$\mathrm{\Delta }\alpha \propto [\frac{1}{2}{E}_{ac}^2(1+\sin (2\omega t-\frac{\pi }{4}))+2E_{ac}(E_{dc}-E_{bi})\sin (\omega t)+(E_{dc}-E_{bi}{)}^2]$----------(9)

$E_{bi}=\frac{({\varphi }_{anode}-{\varphi }_{cathode})}{qd}$ represents the internal electric field generated by the equilibrium state of the two electrodes EF.

A phase-sensitive lock-in amplifier can measure the $2\omega$ component:

$\mathrm{\Delta }\alpha (2\omega )\propto \frac{1}{2}{E}_{ac}^2\sin (2\omega t-\frac{\pi }{4})$----------(10)

and $1\omega$ components:

$\mathrm{\Delta }\alpha (1\omega )\propto 2E_{ac}(E_{dc}-E_{bi})\sin (\omega t)$----------(11)

Since $\mathrm{\Delta }\alpha (1\omega )$ and $E_{dc}-E_{bi}$ are linearly related, the internal field $E_{dc}$ can be determined by measuring $E_{bi}$ required to eliminate the electroabsorption response at $1\omega$.

EA device setup and operation procedures

A schematic diagram and photograph of the homemade electroabsorption device are shown in the figure.

This device is driven by a modulated driving voltage. The DC bias voltage varies from 0V to 3V in 0.5V steps to change the electric field strength, while a constant AC bias voltage is applied simultaneously. The change in electric field strength after applying this driving voltage leads to a change in the absorption bandgap $\mathrm{\Delta }E$.

Monochromatic light is incident on a glass substrate at a 45° angle, and the reflected light illuminates a photodiode. The voltage output of the photodiode is demodulated by a lock-in amplifier (locked to a constant AC signal). This voltage output reflects the change in reflectivity with the excitation bias voltage $V_{dc}$ within a certain photon energy range.

Experimental Principle

Built-in potential is the potential difference formed by the internal charge distribution of a material when no external voltage is applied.

Under the space charge confined current (SCLC) mechanism, the relationship between current density $J$ and carrier mobility $\mu$ follows the Mott-Gurney law:

$J=\frac{9}{8}{\epsilon }_0{\epsilon }_r\mu \frac{(V-V_{bi}{)}^2}{d^3}$----------(12)

Where ${\epsilon }_0$ is the vacuum permittivity, ${\epsilon }_r$ is the relative permittivity of the material, $V_{bi}$ is the built-in potential, and $d$ is the thickness of the organic layer. This relationship shows that the current density $J$ is proportional to the carrier mobility $\mu$, and also strongly depends on the built-in potential $V_{bi}$ and the film thickness $d$.

Mobility is obtained using the Mott-Gurney law. The J-V relationship in the formula can be directly measured, but $V_{bi}$ is missing. Therefore, the Accurate Estimation Method (EA) is used to obtain the accurate $V_{bi}$ for calculating mobility. The working principle of the EA has been explained in detail above. The following section explains how to use the EA to obtain $V_{bi}$ and how to derive the carrier mobility $\mu$.

How to get $V_{bi}$:

Bulit-in potential is $V_{bi}$， Applied Voltage is $V=V_{DC}+V_{AC}\sin (\omega t)$. Internal electric field :

$F(t)=\frac{V_{bi}-V}{d}$

$F_{DC}(t)=\frac{V_{bi}-V_{DC}}{d};F_{AC}(t)=\frac{V_{ac}}{d}$

$F(t)=F_{DC}(t)+F_{AC}(t)\sin \omega t$

Because the device is made of Ag and ITO, the absorption intensity is difficult to detect. The detector detects the reflected light, and since the intensity of the reflected light is very weak, we use phase-locked detection to output $\mathrm{\Delta }R/R$ to enhance the signal strength.

Because:

$\mathrm{\Delta }A\propto F^2$

So:

$F^2(t)=(F_{DC}+F_{AC}\sin \omega t{)}^2$

Then:

$F^2(t)=F_{DC}^2+2F_{DC}{F}_{AC}\sin \omega t+F_{AC}^2{\sin }^{}\omega t$

Using trigonometric formulas:

$F^2(t)=F_{DC}^2+2F_{DC}{F}_{AC}\sin \omega t+\frac{F_{AC}^2}{2}-\frac{F_{AC}^2}{2}\cos 2\omega t$

The resulting absorption change is as follows:

$\frac{\mathrm{\Delta }R}{R}\propto \mathrm{\Delta }A\propto 2F_{DC}{F}_{AC}$----------(13)

For empty current:

$2F_{DC}{F}_{AC}=0;F_{AC}\ne 0$

$F_{DC}=\frac{V_{bi}-V_{DC}}{d}=0\to V_{bi}=V_{DC}$

As shown in the figure below, by changing the input $V_{bi}$ value in the data, when the null current curve flips, the voltage value at the flip point is $V_{bi}$.

Derivation of the calculation of carrier concentration:

By performing EA, we can get $V_{bi}$ of the devices. By applying the equations below, we can calculate carrier mobility.

Gauss’s Law: $\mathrm{\nabla }\cdot E=\rho e/\epsilon$, in 1D: $\frac{dE}{dz}=\frac{\rho e}{\epsilon }$.

Drift-diffusion equation: $J=ep\mu E-De\frac{dp}{dz}=\epsilon \mu E\frac{dE}{dz}-\epsilon D\frac{d^{2}E}{d{z}^2}$.

MG equation: $J=\frac{9}{8}\epsilon \mu \frac{V^2}{L^3}$.

Combining the above three formulas yields:

$Jz=\frac{1}{2}\epsilon \mu E^2-\epsilon D\frac{dE}{dz}$----------(14)

Since we assume the electric field across OSC is constant, we can ignore the $dE/dz$term, and:

$E(z)=\sqrt{\frac{2J}{\epsilon \mu }}\sqrt{z}$----------(15.a)

$⟨E⟩=\frac{1}{Z}\underset{0}{\overset{Z}{\int }}E(z)dz=\frac{1}{Z}\underset{0}{\overset{Z}{\int }}\sqrt{\frac{2J}{\epsilon \mu }}\sqrt{z}dz=\frac{1}{Z}\cdot \frac{3}{2}\cdot \frac{2}{3}{Z}^{\frac{3}{2}}\cdot \sqrt{\frac{V^2}{Z^3}}=\frac{V}{Z}$----------(15.b)

From Gauss’s Law, we can calculate the carrier density:

$p(z)=\frac{d}{dz}\left(\sqrt{\frac{2J}{\epsilon \mu }}\sqrt{z}\right)\cdot \frac{\epsilon }{e}=\frac{1}{e}\sqrt{\frac{\epsilon J}{2\mu }}\frac{1}{\sqrt{z}}$----------(16)

The average carrier density is:

$⟨p⟩=\frac{1}{Z}\underset{0}{\overset{Z}{\int }}p(z)dz=\frac{1}{Z}\underset{0}{\overset{Z}{\int }}\frac{1}{e}\sqrt{\frac{\epsilon J}{2\mu }}\frac{1}{\sqrt{z}}dz=\frac{1}{Z}\cdot \frac{1}{e}\cdot 2\sqrt{Z}\sqrt{\frac{\epsilon J}{2\mu }}=\frac{1}{e}\frac{3}{2}\cdot \left(\frac{\epsilon }{Z}\right)\frac{V}{Z}$----------(17)

In Mott-Gurney regime:

$⟨Ep⟩=\frac{1}{Z}\underset{0}{\overset{Z}{\int }}E(z)p(z)dz=\frac{1}{Ze}\underset{0}{\overset{Z}{\int }}\sqrt{\frac{\epsilon J}{2\mu }}\frac{1}{\sqrt{z}}\cdot \sqrt{\frac{2J}{\epsilon \mu }}\cdot \sqrt{z}dz=\frac{1}{e}\frac{J}{\mu }=\frac{3}{4}⟨E⟩⟨p⟩$----------(18)

The final result is：

$⟨\mu ⟩=\frac{J}{e⟨Ep⟩}=\frac{J}{3/4\cdot ⟨p⟩⟨E⟩e}$----------(19)