Semiclassical expansion of the Slater sum for position dependent mass distributions in d dimensions

TL;DR

This paper derives a second-order gradient expansion of the Slater sum for quantum systems with position-dependent mass in multiple dimensions, providing analytical formulas validated by numerical examples.

Contribution

It introduces a new analytical gradient expansion of the Slater sum for systems with spatially varying effective mass in multiple dimensions.

Findings

Derived a second-order gradient expansion valid for d=1,2,3,4

Provided analytical expressions for the Slater sum with position-dependent mass

Numerical example demonstrates the impact of spatially varying effective mass

Abstract

We consider hamiltonian systems with spatially varying effective mass and slowly varying local potential in d dimensions. The Slater sum is defined as the diagonal element of the Bloch propagator. We derive a gradient expansion of the Slater sum up to the second order. We will show that the derived analytical expression is valid for d=1,2,3 and 4. A numerical example is shown to highlight the effect of the spatially varying effective mass.