Spin-dependent inter- and intra-valley electron-phonon scattering in germanium

TL;DR

This paper develops a detailed theoretical framework for understanding spin-dependent electron-phonon scattering in germanium, including constructing a ${m k} imes{m p}$ Hamiltonian and calculating scattering matrices for various valleys and phonon modes.

Contribution

The study introduces a comprehensive ${m k} imes{m p}$ Hamiltonian for germanium's conduction bands and provides detailed scattering matrices and selection rules for spin-dependent electron-phonon interactions.

Findings

Constructed a 16x16 ${m k} imes{m p}$ Hamiltonian for germanium.

Calculated scattering matrices and selection rules for various electron-phonon processes.

Verified scattering coefficients using pseudo-potential calculations.

Abstract

We investigate the spin-dependent electron-phonon scatterings of the $L$ and $\Gamma$ valleys and the band structure near the conduction band minima in germanium. We first construct a $16\times16$ ${\bm k}\cdot{\bm p}$ Hamiltonian in the vicinity of the $L$ point in germanium, which ensures the correctness of the band structure of the lowest three conduction bands and highest two valence bands. This Hamiltonian facilitates the analysis of the spin-related properties of the conduction electrons. We then demonstrate the phonon-induced electron scatterings of the $L$ and $\Gamma$ valleys, i.e., the intra-$\Gamma$/ $L$ valley, inter--$\Gamma$-$L$ valley and inter--$L$-$L$ valley scatterings in germanium. The selection rules and complete scattering matrices for these scatterings are calculated, where the scattering matrices for the intra-$\Gamma$ valley scattering, inter--$\Gamma$-$L$ valley scattering and the optical-phonon and the separated transverse-acoustic- and longitudinal-acoustic-phonon contributions to the intra-$\Gamma$ valley scattering have not been reported in the literature. The coefficients in these scattering matrices are obtained via the pseudo-potential calculation, which also verifies our selection rules and wave-vector dependence. We further discuss the Elliott-Yafet mechanisms in these electron-phonon scatterings with the ${\bm k}$$\cdot$${\bm p}$ eigenstates at the $L$ and $\Gamma$ valleys. Our investigation of these electron-phonon scatterings are essential for the study of the optical orientation of spin and hot-electron relaxation in germanium.