Wave functions and energies of shallow acceptor states in germanium

Abstract

Approximate wave functions and energies of shallow acceptor states in germanium have been obtained by solving the effective mass equations in the limit of very strong spin-orbit coupling. The variational approach used by Schechter was followed, but with more general trial functions that allow a more complete calculation of the energy spectrum of the acceptor. The new trial functions are constructed from terms with the angular dependence derived by Schechter, each multiplied by an arbitrary radial function. The variational procedure leads to systems of differential equations for the radial functions and energies of the states. Degenerate wave functions for a given acceptor level transform as a basis for a Γ 6, Γ 7 or Γ 8 irreducible representation of the double tetrahedral group. Energies and wave functions were obtained for the ground state, which has Γ 8 symmetry and envelope functions with even parity, and for the first excited state having the same symmetry and parity. Energies and wave functions were also obtained for excited states with Γ 6, Γ 7 and Γ 8 symmetries and envelopes with odd parity. States with Γ 6 or Γ 7 symmetry and envelope functions with even parity were not considered. The resulting energies show fair agreement with the experimental energies. As the new trial functions are quite flexible, it appears that the remaining discrepancy may be due to the inadequacy of the effective mass approximation.