Theory of the Valence Band Structure of Germanium in an External Magnetic Field

Abstract

The energies of the magnetic sub-bands associated with the $V_1$ and $V_2$ valence bands in germanium have been calculated as a function of $k_z$, the propagation constant parallel to the external magnetic field. Warping of the $V_1$ and $V_2$ bands was neglected. Sub-bands belonging to the $1^{+}$ and $2^{+}$ ladders (light holes) have minima at $k_z=0\mathrm{}$ and show quantum effects consisting of a decrease in curvature as the valence band edge is approached. Sub-bands belonging to the $2^{-}$ ladder (heavy holes) also have minima at $k_z=0\mathrm{}$, but the curvatures increase near the valence band edge. The $1^{-}$ heavy hole sub-bands show very marked quantum effects. The sub-band minima occur at values of $k_z$ different from zero, and local maxima appear at $k_z=0\mathrm{}$. The peculiar nature of the $1^{-}$ magnetic sub-bands may lead to observable effects in various magneto-optic phenomena in germanium.