Current density operator in systems with non-parabolic, position-dependent energy bands

TL;DR

This paper derives the correct form of the quantum current density in systems with non-parabolic, position-dependent energy bands, highlighting issues with velocity operators in discrete models.

Contribution

It provides a generalized derivation of the current density for complex semiconductor models and clarifies the limitations of velocity operators in discrete formulations.

Findings

Velocity operator does not satisfy continuity in discrete models.

Textbook current expressions are invalid for non-parabolic, position-dependent bands.

Derived generalized continuity equation for specific semiconductor models.

Abstract

The present manuscript was written in 1994 and was not published. It addresses the form that the quantum-mechanical current density must take in mesoscopic treatments of semiconductor heterostructures, in which the electron dispersion relations are non-parabolic and position dependent, rendering the textbook expressions inapplicable. The approach is to derive the continuity equation for the specific model under consideration, using generalizations of Green's identities to higher-order derivatives and to discrete models of different topological structure. A new addendum addresses two issues of more current interest: the use of irregular meshes in discrete formulations, and the identification of the Heisenberg velocity operator to evaluate current density. It is demonstrated that on discrete domains the velocity operator fails to satisfy a sensible continuity equation, and therefore cannot be identified with the current density.