# Theory of electron tunneling in semiconductors with degenerate band structure

## Abstract

The theory of electron tunneling, in the one electron approximation, is investigated for semiconductors with degenerate band structure. Previous theory has been confined to simple band structure, but the semiconductors of interest actually have a degenerate structure. For the latter, tunneling theory proves to be quite different, on account of the branch points connecting the degenerate bands' energy functions in complex k space. The theory developed is applied to the "diamond group" structure, in which the valence band system consists of heavy hole, light hole, and spin-orbit split-off bands, and the tunneling probabilities are calculated for tunneling between the latter two bands and the conduction band in the approximation of spherical symmetry. It is shown that although for an electron in a constant electric field the wave functions in the k representation, a(k), used by previous authors lead to WKB-like solutions in the r representation for the case of nondegenerate bands, they do not lead to WKB-like solutions for the degenerate band case. We show, however, by solving the coupled equations that are the degenerate band generalization of the effective mass equation, that inside the forbidden region we do obtain the expected WKB solutions. Furthermore, we find more generally that by treating all the bands simultaneously, we can always derive WKB-like solutions for a slowly varying one-dimensional potential that exists in addition to the normal periodic crystal potential. Hence the a(K) functions previously used cannot be applicable here. We derive the analytic properties of the Bloch wave functions and of the matrix elements of x between Bloch states and show that these quantities become singular as we approach an energy branch point. In order to derive the correct a(k) functions for the constant electric field case, these singularities in the interband matrix elements of x cannot be neglected as has been done previously. We obtain the a(k) functions for "small" applied field by a method similar to that used for obtaining the connection formulas for the WKB solutions to the one-dimensional Schrödinger equation. We then show that these functions represent WKB-like solutions far from the classical turning point and the energy branch points as required. Furthermore, we obtain the appropriate connection formulas for various boundary conditions. In this way we find that no matter which valence band the electron is in on the allowed side far from the turning point, the electron always tunnels via the band that has the branch point connection with the conduction band. Using our derived a(k) functions we obtain the transmission probability for a transition from each of the two valence bands considered to the conduction band. The transmission probability for a transition from the light hole valence band to the conduction and is evaluated using Kane's four-band model for the ε{lunate}(k) functions. This model is then generalized to include effects from higher bands via the k·p interaction and the transmission probability is evaluated for this case also. Finally, the results are expressed in terms of energy band gaps and effective masses of the four-band model as generalized. © 1966.