Derivation of an Analytical Expression for the Einstein Relation in Organic Semiconductor Thin Films Using the Binomial Expansion Theorem

In this study, we introduce a novel analytical approach based on the binomial theorem and power series expansions to evaluate the Einstein relation in organic semiconductor thin films. The Einstein relation, expressing the intrinsic link between the diffusion coefficient and charge carrier mobility, is of fundamental importance for the theoretical analysis and performance evaluation of semiconductor devices. The derived formulas exhibit accuracy across a wide range of parameters with practical significance, whereas the existing methods in the literature have been evaluated solely through numerical calculation techniques. The derived formulas demonstrate superior suitability for analyzing organic light-emitting devices through the generalized Einstein relation under a range of parameter settings.