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We propose a unified diffusion-mobility relation which quantifies bothquantum and classical levels of understanding on electron dynamics in orderedand disordered materials. This attempt overcomes the inability of classicalEinstein relation (diffusion-mobility ratio) to explain the quantum behaviors,conceptually well-settles the dimensional effect, phase transition andnonlinear behavior of electronic transport. Our proposed theory relies on thechemical potential which provides the coupling mechanism of charge-heatcurrent, due to electron-phonon coupling. We have derived expressions whichexplain charge transport in both degenerate and nondegenerate materials, andalso provide the linear and nonlinear relationship between the charge densityand chemical potential. Theoretically, we find that the symmetrical nature ofelectron-hole transport in strongly correlated two-dimensional semiconductorsindicates linear dispersion. We also observe the broken symmetry in thenonlinear regime. This generalized diffusion-mobility relation explains boththe strongly and weakly correlated systems from low temperature to hightemperature, in both the relativistic as well as nonrelativistic domains. Invanishing charge density limit of nondegenerate cases, the nonlinear transportreduces to linear like transport, which is the classical Einstein relation.

Unified Quantum Classical Theory of Einstein Diffusion-Mobility Relationship for Ordered and Disordered Semiconductors