The Thomas-Reiche-Kuhn sum rule as a consequence of a non-singular optical susceptibility in semiconductors

TL;DR

This paper demonstrates that the Thomas-Reiche-Kuhn sum rule for semiconductors can be derived from a non-singular optical susceptibility expression, linking it to effective mass tensors and momentum matrix elements without using the k-representation.

Contribution

It establishes a direct connection between the TRK sum rule and a non-singular optical susceptibility expression involving effective mass tensors.

Findings

TRK sum rule can be derived from non-singular susceptibility expressions

Effective mass tensor plays a key role in the derivation

Momentum matrix elements of Bloch functions are crucial in the analysis

Abstract

The Thomas-Reiche-Kuhn optical (TRK) sum rules for bulk materials have customarily been obtained by combining the Kramers-Kronig relations with the high frequency limit of the optical susceptibility tensor $\chi_{ij}$. Also, a non-singular expression for $\chi_{ij}$ involve the reduction of some its parts to an effective mass tensor. In this paper we show that the latter procedure is intimately connected to the TRK sum rules, and in fact these sum rules can be obtained from it. In reaching this result, we present before a thorough description of the momentum matrix elements of Bloch eigenfunctions bypassing the so-called $\bf{k}-$representation.