Real-space equation for single-donor impurities and core excitons in many-valley semiconductors

We solve with high accuracy a previously derived equation in real space for single-donor impurities and core excitons in many-valley semiconductors. That includes, beyond the effective-mass theory, both effects due to the intervalley scattering and to the short-range behavior of the impurity potential. As we previously reported, we obtain in Si a shallow-deep instability for the A₁ singlets and the dispersively screened point-charge potential, at both the substitutional and the interstitial sites. Our prediction of such deep levels appears to be substantially confirmed by the most recent experimental data. We also introduce a direct test on the localization and convergence of the envelope function in →k space. We show that in most cases there is no shallow-deep instability and the envelope function involves only a single band. In these cases this method should in fact provide accurate values for the binding energies.