Spin-dependent intravalley and intervalley electron-phonon scatterings in germanium

The spin-dependent electron-phonon scattering in the $L$ and $\Gamma$ valleys of germanium crystals has been investigated theoretically. For this purpose, the 16 $\times$ 16 $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian correctly describing the electron dispersion in the vicinity of the $L$ point of the Brillouin zone in germanium in the lowest conduction bands and the highest valence bands has been constructed. This Hamiltonian facilitates the analysis of the spin-dependent properties of conduction electrons. Then, the electron scatterings by phonons in the $L$ and $\Gamma$ valleys, i.e., intra-$L$ valley, intra-$\Gamma$ valley, inter-$L$-$\Gamma$ valley, and inter-$L$-$L$ valley scatterings, have been considered successively. The scattering matrix expanded in powers of the electron wave vectors counted from the centers of the valleys has been constructed using the invariant method for each type of processes. The numerical coefficients in these matrices have been found by the pseudopotential method. The partial contributions of the Elliott and Yafet mechanisms to the spin-dependent electron scattering have been analyzed. The obtained results can be used in studying the optical orientation and relaxation of hot electrons in germanium.