We built a model of charge transport in a single disordered polymer chain starting from a model Hamiltonian of the system. The parameters entering the Hamiltonian determine both the density of states (DOS) and the hopping rate unlike the most common modelling strategies of transport in polymeric materials that parametrize both the DOS and the hopping rate from the outset. This model incorporates the effect of variable delocalization of one-electron states and is designed to link atomistic calculations of polymeric systems with full device models in multiscale modelling protocols. The initial and final states for the hopping process are determined by static disorder and further stabilized by polaronic effects. The coupling between these states is due to the residual (and much smaller) dynamic disorder. We find that, at lower static disorder, long-distance hopping events become more frequent, i.e. the hopping range and disorder are not unrelated parameters, as commonly assumed. The availability of low energy relatively delocalized states promotes long range displacement of charge and it can be at the origin of the high mobility observed in some polymers. The description of the hopping rates from the model Hamiltonian also allows the identification of the breakdown of the incoherent transport limit.

We describe charge transport along a polymer chain with a generic theoretical model depending in principle on tens of parameters, reflecting the chemistry of the material. The charge carrier states are obtained from a model Hamiltonian that incorporates different types of disorder and electronic structure (e.g., the difference between homo- and copolymer). The hopping rate between these states is described with a general rate expression, which contains the rates most used in the literature as special cases. We demonstrate that the steady state charge mobility in the limit of low charge density and low field ultimately depends on only two parameters: an effective structural disorder and an effective electron-phonon coupling, weighted by the size of the monomer. The results support the experimental observation [N. I. Craciun, J. Wildeman, and P. W. M. Blom, Phys. Rev. Lett. 100, 056601 (2008)] that the mobility in a broad range of (polymeric) semiconductors follows a universal behavior, insensitive to the chemical detail.

We propose an expression of the hopping rate between localized states in semiconducting disordered polymers that contain the most used rates in the literature as special cases. We stress that these rates cannot be obtained directly from electron transfer rate theories as it is not possible to define diabatic localized states if the localization is caused by disorder, as in most polymers, rather than nuclear polarization effects. After defining the separate classes of accepting and inducing nuclear modes in the system, we obtain a general expression of the hopping rate. We show that, under the appropriate limits, this expression reduces to (i) a single-phonon rate expression or (ii) the Miller-Abrahams rate or (iii) a multi-phonon expression. The description of these limits from a more general expression is useful to interpolate between them, to validate the assumptions of each limiting case, and to define the simplest rate expression that still captures the main features of the charge transport. When the rate expression is fed with a range of realistic parameters the deviation from the Miller-Abrahams rate is large or extremely large, especially for hopping toward lower energy states, due to the energy gap law.