Abstract
The lack of knowledge about phonon spectra and wave functions of electron (hole) states in disordered organic semiconductors lead to the need to use some phenomenological models of hopping rates to calculate the mobility of charge carriers. The literature [1,2,3,4 ] actively discusses the applicability of the common Miller-Abrahams approximation (MA), the semiclassical Marcus theory, and the so-called "empirical" model [1,2]. The latter empirically interpolates, using the auxiliary parameter α, between the MA limit (at α = 1) and multi-phonon quantum mechanical models [2,3,4]. Until recently, it was believed that the transport energy concept, which provides simple analytical expressions for mobility, is applicable only in the case of the MA model. A breakthrough in this area was a recent numerical Monte-Carlo (MC)-based study [1], which demonstrated that the transport level exists in the framework of several popular models of hopping rates. Moreover, the transport energy position is universal and does not depend on specific parameters of different hopping models. In this paper, we demonstrated, using a modified Arkhipov method [5], that the energy for which upward and downward jumps are equally probable is virtually independent of the hopping transition model (MA, "empirical" or Marcus) and agrees well with numerical results [1]. It is precisely this energy that we identified with the transport level in our previous work (for example [6]). In agreement with numerical results of the work [1], specific parameters of a particular model (the reorganization energy in the Marcus model or the parameter α in the “empirical” model) do not significantly influence this energy. We have derived a compact analytical expression that relates the position of the transport level to a single dimensionless parameter which combines the energetic disorder parameter and localization parameter, while in the work [1] the transport energy was presented as a color-map, where the values of these parameters were plotted along the x- and y-axes. The universal transport energy opens the way to the simple analytical expressions for the mobility, hence for the analysis of experimental and computational data in organic photovoltaics and light-emitting devices without the need for resource-intensive numerical modeling.