Inconsistency between Linearized Thomas-Fermi Approximation and Electron-Ionized Impurity Scattering Rate in the first Born Approximation

TL;DR

This paper investigates the inconsistency between the linearized Thomas-Fermi approximation and the electron-impurity scattering rate calculated via the first Born approximation, revealing significant differences in wave vector transfer distributions.

Contribution

It demonstrates the inconsistency between the Thomas-Fermi approximation and first Born approximation for impurity scattering in semiconductors, and analyzes the differential cross-sections in detail.

Findings

Wave vector transfer distribution is inconsistent with Thomas-Fermi approximation.

Scattering probabilities differ by at most 1% from RPA estimates.

Nondegenerate carrier dynamics are less affected by this inconsistency.

Abstract

We show that by computing the electron-impurity scattering rate at the first order via Fermi's golden rule, and assuming that the localized impurity potential is of Yukawa form, one obtains a wave vector transfer distribution which is inconsistent with the finite temperature linearized Thomas-Fermi approximation for {\it n}-type semiconductors. Our previous findings show that this is not the case for the carrier nondegenerate dynamics, because the average wave vector transferred being in general negligible in this regime. Moreover, we examine the behavior of the electron-impurity differential cross-sections in the first Born approximation for relevant values of the wave vector transfer. We find that in the majority of collisions, the scattering probabilities differ at the most by $1$ \% from the estimates computed by means of the impurity potential at random phase approximation level.