Valence Bands of Germanium and Silicon in an External Magnetic Field

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 V1 and V2 bands from the V3 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 kH≠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 kH=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 kH≠0 show that the ε1- levels possess negative curvatures near kH=0. Gradual "crossing" or reordering of the heavy hole levels is found at relatively high kH

Theory of the Valence Band Energy Level Structures of Germanium and Silicon in an External Magnetic Field

The problem of the valence band structure of Ge and Si 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, in principle, similar to that of W. Shockley and E. O. Kane, but is modified in some important essentials to include the effects of the magnetic field. The analytical results obtained are somewhat more general than those of J. M. 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, of which the most important are the following: 1. Magnetic field ϰ in the [001] direction, kH = 0; nonspherical symmetry character of energy bands and the coupling of V1 and V2 bands to the V3 band included. 2. Magnetic field ϰ in the [001] direction, kH ≠ 0; nonspherical symmetry character of energy bands included, decoupling of V1 and V2 bands from the V3 band assumed. In addition, a set of algebraic equations is derived whose solution should yield the valence band Landau levels for the cases of the magnetic field in the [101] and the [111] directions. However, no numerical calculations are performed for these cases. The results of the calculations indicate the presence of some interesting transitions between the Landau levels of Ge and Si, as well as the possible presence of other interesting effects which may be observable. Certain of these seem to offer potential millimeter-wave applications possibilities, some of which are discussed.</p