Electronic states of group V donors in germanium: variational calculation taking into account the short-range potential

In the framework of the envelope function approximation, the wave functions of low-lying $1s(A_{1})$-, $2s$-, $2p_{0}$-, $2p_\pm$-, $3p_{0}$ states of shallow donor centers P, As, Sb in germanium are calculated considering the short-range part of the impurity potential. The latter is constructed individually for each impurity, taking into account the spatial dispersion of the dielectric function and the difference between the ionic cores of germanium and the impurity center. The envelope function equation was solved using the Ritz variational method, and selected trial wave functions of the orbitally non-degenerate $s$-states are characterized by two spatial scales: the first one is of the order of the donor effective Bohr radius and corresponds to the long-range part of the potential, and the second one, which is an order of magnitude less, 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 due to the attractive “central cell” correction. The envelope functions of $p$-states, in turn, are constructed in such a way they are orthogonal to the ground state envelope functions for each impurity center, and, unlike previous works, are different for various donors.