Valence Bands of Germanium and Silicon in an External Magnetic Field

Abstract

The problem of the valence band structure of germanium and silicon in the presence of an external magnetic field is considered from a quantum-mechanical point of view. The analysis is carried out using first- and second-order perturbation theory. The approach is an extension of methods of Shockley and Kane to include the effects of the magnetic field. The usual approximation of the decoupling of the ${V}_{1}$ and ${V}_{2}$ bands from the ${V}_{3}$ band is not made, thus making the analysis applicable to Si as well as Ge. Spherical energy bands are not assumed in this calculation and the case of ${k}_{H}\ensuremath{\ne}0$ is considered. Detailed analysis is carried out for the magnetic field $H$ in the [001] direction. The analytical results obtained are more general than those of Luttinger but reduce to the latter if certain approximations are introduced.Numerical calculations of the Landau energy levels are carried out for certain special cases. The results predict an increase of the hole effective mass with the magnetic field. They also indicate mixing of the Landau levels even at ${k}_{H}=0$, which leads to a prediction of new transitions some of which are of "negative mass" type. The mixing is more pronounced in Si than in Ge. Calculations for ${k}_{H}\ensuremath{\ne}0$ show that the ${{\ensuremath{\epsilon}}_{1}}^{\ensuremath{-}}$ levels possess negative curvatures near ${k}_{H}=0$. Gradual "crossing" or reordering of the heavy hole levels is found at relatively high ${k}_{H}$.