Evaluating effective mass models of the phoshorous donor in silicon

Evaluating effective mass models of a phosphorus donor in silicon is made difficult by conflation of mathematical and physical approximations. We propose a scheme to solve a class of effective mass models with high precision. We construct donor electron states using envelope functions expanded in freely extensible basis sets equipped with tunable parameters. With these states, we compute the expectation values of both the donor's energy as well as the energy variance. We variationally optimize the parameters of these basis states to find stationary points of the energy functional, with variance of the expectation energy used to evaluate the precision of our candidate eigenstates. In this manner, we can find exact energy eigenstates of the implied Hamiltonian of an effective mass model.

To improve the physical verisimilitude of a given effective mass model, we evaluate models of the donor atom's Coulomb potential, in particular considering the effects of a dielectric constant with dynamic response. We present a phenomenological psuedopotential for exchange and correlation effects from the donor's valence electrons. Finally, we include symmetry breaking perturbations, in particular the effect of an electric field upon the donor electron.