General Einstein relation model in disordered organic semiconductors under quasiequilibrium

In this work, the Einstein relation between the diffusivity and mobility of charge carriers for disordered organic semiconductors is analyzed. We formulate an analytic theory that allows predicting the Einstein relation for charge carrier hopping in disordered organic semiconductors with Gaussian density of states distribution as a function of disorder, temperature, bias field, and Fermi level, i.e., concentration of occupied states of the DOS under the condition of quasiequilibrium. By scanning the Fermi across the DOS, we calculate the charge carrier mobility and diffusivity as well the $qD/\mu k_{B}T$ ratio. We are thus able to identify the role of mobile and localized states on the interplay of diffusion and drift and can determine under which condition Einstein relation is valid or not.

Figure 1

Dependence of $qD/\mu k_{B}T$ as a function of the disorder parametric in the normalized Fermi level $E_F/\sigma$ (carrier concentration).

Figure 2

Dependence of $qD/\mu k_{B}T$ on the electric field. The parameters are $\sigma =0.06\text{eV}$ and $E_F=-0.35\mathrm{eV}$. The other parameters are the same as those in Fig. 1

Figure 3

(Top) Dependence of the charge carrier mobility $\mu$ (bottom), the diffusivity $D$ (middle) and $qD/\mu k_{B}T$ (top) as a function of the relative charge carrier concentration for a 3D system and $F=1\times {10}^7\text{V}/\text{m}$. The arrows indicate the relative positions of the Fermi level at which the $qD/\mu k_{B}T$ begins to fall off.

Figure 4

Dependence of $qD/\mu k_{B}T$ on carrier concentration for different electric fields for 3D system and for $F=1\times {10}^7\text{V}/\text{m}$ for a 1D system. The disorder parameter $\sigma /k_{B}T$ is $\sigma /k_{B}T=2.3$. The inset shows the dependence of $qD/\mu k_{B}T$ for 3D on the carrier concentration at low electric fields.

Figure 5

Dependence of $D/\mu$ on carrier density. The parameters used here are $F=1\times {10}^7\text{V}/\text{m}$, $\sigma /k_{B}T=3$ and $E_a=0.5$ eV. The other parameters are the same as those in Fig. 1.