Electron States of Group-V Donors in Germanium: Variational Calculation Taking into Account the Short-Range Potential

Abstract

The wave functions of low-lying 1s (A1), 2s, 2p0, 2p±, and 3p0 states of P, As, and Sb shallow donor centers in germanium are calculated in the scope of the envelope-function approximation taking into account the short-range impurity potential. The latter is constructed individually for each impurity allowing for the spatial permittivity dispersion and the difference between the ion cores of germanium and the impurity center. The envelope-function equation is solved using the Ritz variational method; herewith, the selected test functions of orbitally nondegenerate s states are characterized by two spatial scales. The first scale, on the order of the donor effective Bohr radius, corresponds to the long-range part of the potential, and the second scale, which is smaller by an order of magnitude, simulates the electron response to the short-range part of the donor potential. The electron density in the donor ground state is shifted to the nucleus, which is due to allowance made for the attracting “central cell” potential. The envelope functions of p states are in turn constructed so that they are orthogonal to envelopes in the ground states for each impurity centers and are different for various donors in contrast with previous works.