Higher-order Einstein relations for nonlinear charge transport

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Abstract
Nonlinear terms in relations for current densities are treated macroscopically, semimicroscopically and microscopically. In the macroscopic treatment, terms in phi 2, E2, (grad n)2,(del)2n, and E.grad n are included, where phi is the electrostatic potential, n is the carrier concentration and E is the electric field. The power series expansion of the current density is valid for equilibrium and yields conductivity-diffusion type Einstein relations. In the semimicroscopic approach, a perturbation theory for the density matrix is used, and Einstein relations are then derived by equating the average of the current density operator to zero. In the microscopic approach a Kubo formalism is developed, based on a local nonequilibrium distribution function due to Mori (1958). This leads to Einstein relations via correlation functions and Liouville's equation. A set of such relations which emerge consistently from such a treatment is given.