Zero-field mobility, exact mean dwell times, and disorder-induced steps in a Gaussian energy distribution

Abstract

Hopping transport in molecularly doped polymers (MDPs) is usually modeled in terms of energetic disorder corresponding to a Gaussian distribution of states, g(ε,σ), whose width σ is taken from experiment. Time-of-flight profiles indicate normal transport and finite mean dwell time 〈τ(T,E)〉. At zero field, thermal equilibrium and detailed balance are shown to yield 〈τ(T)〉 exactly for hopping rates that are products of energetic and geometrical factors. Hopping in g(ε,σ) for standard rates such as Miller-Abrahams, Marcus, or symmetric leads to ln〈τ(T)〉 that goes strictly as T−2 only for symmetric rates. Disorder-induced steps dominate the zero-field mobility μ(T). Monte Carlo simulations with Marcus rates show that extra steps depend on the overall disorder. Dilution and orientation increase the slope of ln μ(T) vs T−2. We interpret extra steps using an auxiliary lattice with spacing a′ chosen to give the same μ(T). Evidence for disorder-induced steps comes from previous studies with variable donor concentration in the same polymer, from much larger mobility changes than the concentration ratio, and from modeling of tritolylamine:polystyrene systems. Exact dwell times for hopping in g(ε,σ) show that current estimates of σ are upper bounds that neglect geometrical disorder. They provide direct tests for the convergence of simulations, either in terms of 〈τ(T)〉 or the distribution of visited sites in an infinite random walk.