Theory of the motion at the band crossing points in bulk semiconductor crystals and in inversion layers

This paper presents an original investigation of the motion at the band crossing points in the energy dispersion of either bulk crystals or inversion layers. In particular, by using a formalism based on the time dependent Schrödinger equation, we address the quite elusive topic of the belonging of the carriers to the bands that are degenerate at the crossing point. This problem is relevant and delicate for the semiclassical transport modeling in numerically calculated band structures; however, its clarification demands a full-quantum transport treatment. We here propose analytical derivations and numerical calculations clearly demonstrating that, in a given band structure, the motion of the carriers at the band crossing points is entirely governed by the overlap integrals between the eigenfunctions of the Hamiltonian that has produced the same band structure. Our formulation of the problem is quite general and we apply it to the silicon conduction band calculated by means of the nonlocal pseudopotential method, to the hole inversion layers described by a quantized $k⋅p$ approach, and to the electron inversion layers described by the effective mass approximation method. In all the physical systems, our results underline the crucial role played by the abovementioned overlap integrals.