Boltzmann Collision Kernels and Velocity Saturation in Semiconductors

TL;DR

This paper analyzes how different electron-phonon interaction models affect the behavior of solutions to the Boltzmann equation under high external fields, revealing conditions for velocity saturation or saturation exclusion.

Contribution

It extends previous results on collision kernels of rank one to finite rank Hilbert-Schmidt kernels and characterizes velocity saturation in singular collision models.

Findings

Velocity saturation is excluded for finite rank Hilbert-Schmidt kernels.

Velocity saturation generally occurs for certain singular collision kernels.

Asymptotic behavior of moments is determined in high external fields.

Abstract

For different models of the electron-phonon interaction, the asymptotic behaviour of the moments of the stationary homogeneous solution of the linear Boltzmann equation is determined in the limit of a high external field. For Hilbert-Schmidt kernels of a finite rank, a result recently proven for kernels of rank one is found generally valid; as a consequence velocity saturation is excluded for these collision models. For a class of singular collision kernels in contrast, velocity saturation is generally obtained.