Derivation of phenomenological expressions for transition matrix elements for electron-phonon scattering

TL;DR

This paper provides a more intuitive derivation of the phenomenological transition matrix elements for electron-phonon scattering and extends the expression to include spin dependence.

Contribution

It introduces a new, more intuitive derivation method and generalizes the expression to account for electron spin states.

Findings

Derivation aligns with existing results but offers clearer motivation.

Extension includes spin dependence in the matrix elements.

Provides a basis for more accurate modeling of electron-phonon interactions.

Abstract

In the literature on electron-phonon scatterings very often a phenomenological expression for the transition matrix element is used which was derived in the textbooks of Ashcroft/Mermin and of Czycholl. There are various steps in the derivation of this expression. In the textbooks in part different arguments have been used in these steps, but the final result is the same. In the present paper again slightly different arguments are used which motivate the procedure in a more intuitive way. Furthermore, we generalize the phenomenological expression to describe the dependence of the matrix elements on the spin state of the initial and final electron state.