Theory of localized states in semiconductors. II. The pseudo impurity theory application to shallow and deep donors in silicon

A new pseudo impurity theory is developed by combining the general theory of pseudopotentials with effective-mass ideas. It is applicable to a general impurity, both shallow and deep levels, and reduces to the Kohn-Luttinger effective-mass theory in the special cases of isocoric impurities, namely impurities which have isoelectronic cores with the host atoms. The resultant impurity-electron wave function $\psi \left(\overrightarrow{\mathrm{r}}\right)$ in the new theory is given by a many-band expansion and has the correct nodal structure in all atomic cells, including the impurity cell. The impurity pseudopotentials are again constructed from fundamental crystal and atomic properties with no adjustable parameters. An application is presented to donors in silicon and excellent agreement is obtained for many shallow and deep, substitutional and interstitial donors. The accuracy of the calculations is investigated. Further conceptual understanding of the effective-mass notion is also attained: it is the "pseudo" electron, described by the smooth pseudo wave function $\phi \left(\overrightarrow{\mathrm{r}}\right)$, that may be described by the band-minimum effective mass $m^{*}$; whereas the "effective" mass of the real electron, which is described by $\psi \left(\overrightarrow{\mathrm{r}}\right)$, is $m^{*}$ only outside the impurity cell. Inside the impurity cell, the kinetic energy of the real electron is a complicated operator, which reduces to the $m^{*}$ form in an asymptotic way.