We investigate the spin-dependent electron-phonon scatterings of the L and Γ valleys and the band structure near the conduction band minima in germanium. We first construct a 16 × 16 k · 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 Γ valleys, i.e., the intra-Γ/L valley, inter-Γ-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-Γ valley scattering, inter-Γ-L valley scattering and the optical-phonon and the separated transverse-acoustic-and longitudinal-acoustic-phonon contributions to the intra-Γ 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 k·p eigenstates at the L and Γ 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.

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We study three-electron energy spectra in Si/SiGe single and vertically coupled double quantum dots where all the relevant effects, such as, the Zeeman splitting, spin-orbit coupling, valley coupling and electron-electron Coulomb interaction are explicitly included. In the absence of magnetic field, our results in single quantum dots agree well with the experiment by Borselli et al. [Appl. Phys. Lett. 98, 123118 (2011)]. We identify the spin and valley configurations of the ground state in the experimental cases and give a complete phase-diagram-like picture of the ground state configuration with respect to the dot size and valley splitting. We also explicitly investigate the three-electron energy spectra of the pure and mixed valley configurations with magnetic fields in both Faraday and Voigt configurations. We find that the ground state can be switched between doublet and quartet by tuning the magnetic field and/or dot size. The three-electron energy spectra present many anticrossing points between different spin states due to the spin-orbit coupling, which are expected to benefit the spin manipulation. We show that the negligibly small intervalley Coulomb interaction can result in magnetic-field independent quartet-doublet degeneracy in the three-electron energy spectrum of the mixed valley configuration. Furthermore, we study the barrier-width and barrierheight dependences in vertically coupled double quantum dots with both pure and mixed valley configurations. Similar to the single quantum dot case, anticrossing behavior and quartet-doublet degeneracy are observed.

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