Spherical Model of Shallow Acceptor States in Semiconductors

The effective-mass approximation for shallow acceptor states in cubic semiconductors with degenerate valence bands is reformulated. The Hamiltonian is written as the sum of a spherical term and a cubic corection, thus pointing out the relevance of the spherical symmetry in the acceptor problem and the strong similarity to the case of atoms with the spin-orbit interaction. Without the introduction of any explicit representation of the Hamiltonian, the present formulation yields a meaningful classification of the acceptor states and reduces the eigenvalue problem to simple radial Hamiltonians. These radial Hamiltonians are explicitly given for the most improtant acceptor states and are shown to apply also to the description of the exciton problem. The variational method is used in the numerical calculation. The resulting eigenvalues, eigenfunctions, and related quantities are given as functions of the relevant parameters. The theoretical ionization energies are compared with available experimental data.