Quantitative analysis of valley–orbit coupling in germanium doped with group-V donors

The wave functions of electrons localized at P, As, and Sb shallow donors in Ge are calculated in the envelope function approximation taking into account valley–orbit coupling induced by the short-range donor potential. An approach is proposed that makes it possible to include intervalley mixing into the equation for the multicomponent envelope function. The effects of valley–orbit coupling are calculated using perturbation theory and the single-valley “bare” functions are determined by the Ritz method. The parameters of the short-range part of the potential and the coefficient of intervalley mixing are found for each donor individually and yield the best agreement with the measured energies of the singlet and triplet states. The envelope functions of the 1$s(A1)$ and 1$s(T2)$ states are calculated. The parameters of the valley–orbit interaction for each donor are obtained. It is shown how the functions of the 2$s$, 2$p_0$, 2$p_\pm$, and 3p$_0$ excited states should be modified to remain orthogonal to the singlet and triplet functions in the framework of a more rigorous multivalley model.