A charge carrier drifts through a thin organic semiconductor layer by hopping between…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods
Error, Chance and the Machine, Ch Random walks and Markov chains
Problem
A charge carrier drifts through a thin organic semiconductor layer by hopping between five discrete trap sites labelled k=0,1,2,3,4 along the field direction. Site 0 is a recombination centre (the carrier is lost there) and site 4 is the collecting electrode (the carrier is extracted there). Under the applied field, each attempted hop moves the carrier one site towards the electrode with probability p=0.70 and one site back towards the recombination centre with probability q=0.30; the walk stops as soon as either boundary is reached. Each attempt takes τ=2.0ps. The carrier is injected at site k=1.
((a)) Using the absorption-probability result for a biased walk between absorbing walls, find the probability that the carrier is collected rather than lost to recombination.
((b)) Let Dk be the mean number of hops to absorption (either boundary) starting from site k. Condition on the first hop to write the difference equations for D1,D2,D3 (with D0=D4=0), solve the resulting linear system, and convert D1 to a mean physical time.