Quantum resonances in the valence bands of germanium. I. Theoretical considerations

Abstract

The structure of the ${\ensuremath{\Gamma}}_{25}^{+}$ degenerate valence band of Ge has been investigated by a comprehensive study of the complex resonance spectra from electric-dipole transitions in the system of anomalously spaced low-lying Landau levels produced by an applied magnetic field. This paper, the first of a series of four, is devoted to the development of a systematic theoretical spectroscopy to serve as a framework for the analysis of our experiments which will be discussed in succeeding papers. Using group-theoretical techniques, following Luttinger, we construct a generalized effective-mass Hamiltonian for holes, including the effects of elastic strain, in the full six-dimensional space of ${\ensuremath{\Gamma}}_{25}^{+}$ to encompass the spin-orbit-split-off band. The formulation in the extended space sheds considerable light on two important consequences of spin-orbit coupling, the anisotropy of the hole $g$ factor (the Luttinger $q$ term) and a new effect, the spin-dependent contribution to the valence-band deformation potentials. From the Hamiltonian for ${\ensuremath{\Gamma}}_{25}^{+}$ we project, making the split appropriate to large spin-orbit interaction, the Hamiltonian belonging to the subspace of the band edge ${\ensuremath{\Gamma}}_{8}^{+}$. We examine the nature of its eigenstates and develop a systematic scheme based on group theory for classifying the magnetic eigenstates, in terms of which selection rules for quantum transitions can be expressed in unusually clear and compact form. A formalism is presented for generating complete, synthesized quantum-resonance spectra starting with the eigenvalues and eigenfunctions of the effective-mass Hamiltonian. A further projection, representing the decoupling of the ${\ensuremath{\Gamma}}_{8}^{+}$ band by large uniaxial stress, is expanded to second order to evaluate the corrections to the hole effective masses and $g$ factors at finite stress. Finally, we consider the interaction between the projected spaces of the stress-decoupled band-edge states and the spin-orbit-split-off states which contributes two important shifts to the quantum-resonance effective masses: the second-order magnetic interaction and the interaction from the cross terms of strain and magnetic field.