Multivalley Effective-Mass Approximation for Donor States in Silicon. I. Shallow-Level Group-V Impurities

Abstract

Following the procedures of the Kohn-Luttinger one-valley effective-mass approximation (EMA), but without neglecting the intervalley overlap terms, a multivalley EMA is developed within the pseudopotential formalism for shallow-level group-V donors in silicon. A phenomenological two-parameter model impurity potential is proposed which behaves like a square well at small $r$ and a screened Coulomb potential at large $r$. The relative smoothness of the model impurity potential compared with the true impurity potential justifies the use of the EMA. Within the EMA, the valley-orbit interaction can be incorporated completely in the energy calculations, instead of treated as a perturbation. The effects of the higher-energy subsidiary valleys and the $1s\ensuremath{-}2s$ coupling etc. can also be estimated quantitatively in this EMA. The energy levels of the group-$\mathrm{V}$ donor impurities in silicon are calculated using the variation method. The two potential parameters are adjusted to fit the observed $1s({A}_{1})\ensuremath{\rightarrow}2{p}_{\ifmmode\pm\else\textpm\fi{}}$ and $1s({T}_{2})\ensuremath{\rightarrow}2{p}_{\ifmmode\pm\else\textpm\fi{}}$ transition energies. It is shown that the proposed potential describes well the central-cell effects. For the $p$ states, it is found that the usual one-valley approximation is adequate. It is also found that in the multivalley approximation the valley-orbit interaction shifts the $1s({A}_{1})$ level downward and the $1s({T}_{2})$ and $1s(E)$ levels upward relative to the one-valley $1s$ level. The effects of the uncertainty in the position of the ${\ensuremath{\Delta}}_{1}$ conduction-band valleys and of the coupling between the $1s$ and $2s$ states on the ground-state energy are shown to be negligible. The contribution from wave-function components of an ${L}_{1}$ valley of the conduction band to the ground-state energy is computed to be less than 1% of that from wave-function components of $\mathrm{a} {\ensuremath{\Delta}}_{1}$ valley. It indicates that the higher-energy subsidiary conduction-band valleys may be neglected. The ground-state-model wave function is used to calculate the photo-ionization cross section and to predict the Fermi contact hyperfine constants. These results are compared with reported experimental and theoretical results.