Temperature scaling of effective polaron mobility in energetically disordered media

We study effective mobility in 2 dimensional (2D) and 3 dimensional (3D) systems, where hopping transitions of carriers are described by the Marcus equation under a Gaussian density of states in the dilute limit. Using an effective medium approximation (EMA), we determined the coefficient $C_d$ for the effective mobility expressed by $\mu_{\rm eff}\propto\exp\left[-\lambda/\left(4 k_{\rm B} T\right)- C_d\sigma^2/\left(k_{\rm B} T\right)^2 \right]/\left[\sqrt{\lambda} (k_{\rm B} T)^{3/2}\right]$, where $\lambda$ is the reorganization energy, $\sigma$ is the standard deviation of the Gaussian density of states, and $k_{\rm B} T$ takes its usual meaning. We found $C_d=1/2$ for both 2D and 3D. While various estimates of the coefficient $C_d$ for 3D systems are available in the literature, we provide for the first time the expected $C_d$ value for a 2D system. By means of kinetic Monte-Carlo simulations, we show that the effective mobility is well described by the equation shown above under certain conditions on $\lambda$. We also give examples of analysis of experimental data for 2D and 3D systems based on our theoretical results.