Tetrahedral potentials and lack-of-inversion symmetry for donors in silicon

Abstract

The extensive electron-nuclear double resonance data of Hale and Mieher for P-, As-, and Sb-doped Si and the comprehensive multiband calculation of Ivey and Mieher employing only a spherical potential $U(r)$ suggests four inversion-related pairs [$(\underset{̱}{224})$, (224)], [$(\underset{̱}{444})$, (444)], [$(\underset{̱}{228})$, (228)], and [$(\underset{̱}{2212})$, (2212)] have been associated with the shells $[M,L]$, $[H,O]$, $[I,P]$, and $[S,T]$. An alternative explanation is provided resulting from tetrahedral potentials of the form ${V}_{\text{tet}}(\mathbf{r})={V}_{t}$ $xyx/{({r}^{2}+{b}^{2})}^{7/2}$ and a corresponding wave function ${\ensuremath{\psi}}_{\text{tet}}\ensuremath{\propto}\ensuremath{-}xyz$ $[{C}_{1}\text{ }\text{exp}(\ensuremath{-}r/{\ensuremath{\lambda}}_{1})+{C}_{2}\text{ }\text{exp}(\ensuremath{-}r/{\ensuremath{\lambda}}_{2})]$ with ${C}_{1}⪢{C}_{2}$ and ${\ensuremath{\lambda}}_{2}\ensuremath{\sim}4$ ${\ensuremath{\lambda}}_{1}$ (for P donor). ${\ensuremath{\psi}}_{\text{tet}}$ leads to a giant octupole moment for the donors dominated by ${C}_{2}\text{ }\text{exp}(\ensuremath{-}r/{\ensuremath{\lambda}}_{2})$. Excellent agreement is obtained for P, good agreement for As, and less satisfactory agreement for Sb. ${V}_{\text{tet}}(\mathbf{r})$ originates from the donor $\mathbf{E}$-field-induced electric dipoles ${\mathbf{p}}_{i}$ at the covalent bond sites [first nearest-neighbor (nn), second nn, and third nn] as calculated by Harrison. Reasons are provided why the inversion-related pairs cannot be explained by a spherical potential $U(r)$. These results provide strong support for the assignments by Ivey and Mieher of the four inversion-related pairs. The interest in donors in Si as possible quantum bits has led to interest in the Stark effect from external electric fields. ${V}_{\text{tet}}(\mathbf{r})$ and the associated ${\ensuremath{\psi}}_{\text{tet}}(\mathbf{r})$, resulting from an ``internal Stark effect,'' are a general feature for impurities with ${T}_{d}$ symmetry in semiconductors.