3.1 Introduction

Organic semiconductors can be considered as hopping networks and are characterized by strong disorder in both energy and space [8,9]. This makes it very difficult to solve the problem analytically or simulate the carrier transport and recombination in such a system by starting from a one-particle master equation. Consequently, an analytical approach to this problem is normally based on a specific set of assumptions and simplifications [20,68]. The concept of transport energy is a very useful tool for the analysis of charge hopping transport in organic semiconductors. The importance of the transport energy stems from the fact that it maximizes the probability for a carrier to hop upward. It does not depend on the initial energy of the carrier and serves as an analog of the mobility edge [10].

The transport energy concept is based on the Miller-Abrahams expression [7,71]. This equation can be written as

$$\omega_{if}=\nu_0\exp\left(-2\alpha R_{it}-\frac{E_t-E_i+\mid E_t-E_i\mid}{2k_BT}\right)$$ (3.1)

For a particular density of states $g\left(E\right)$, the transport energy can be obtained in the following way [10]. For an electron with energy $E_i$, the median rate of a upward hop to a neighboring localized state with energy $E_f> E_i$ is

$$\omega_\uparrow=\nu_0\exp\left(-2\alpha R\left(E_t\right)-\frac{E_t-E_i}{k_BT}\right).$$ (3.2)

where

$$R\left(E_t\right)=\left[\frac{4\pi}{3}\int_{-\infty}^{E_t} g\left(E\right)dE\right]^{-1/3}$$

The transport energy can be calculated by maximizing the rate (3.2) with respect to the final energy $E_t$

$$\frac{\partial\omega_\uparrow\left(E_i,E_t\right)}{\partial E_t}=0.$$ (3.3)

After some calculation we obtain

$$g\left(E_t\right)\left[\int_\infty^{E_t} g\left(E\right)dE\right]^{-4/3}=\frac{1}{\alpha k_BT}\left(\frac{9\pi}{2}\right)^{1/3}.$$ (3.4)

Here we can see that the transport energy $E_t$ does not depend on the initial energy $E_i$. The transport energy has been extended to an exponential DOS in [10] and later to a Gaussian DOS in [77].