Configuration Mixing of Subsidiary Minima: Corrections to the Ground-State Wave Function for Donors in Silicon

Abstract

The introduction of wave-function components from the region of the $L_1$, $K_1$, and $U_1$ points of the lowest conduction band into the ground-state wave function of the shallow donors As, P, and Sb is shown to improve substantially the agreement between the calculated Fermi contact constants for identified ENDOR shells and the experimental Fermi contact constants measured by Hale and Mieher. This wave-function admixture is the band-structure analogy to configuration mixing in atomic physics, and is calculated here employing first-order perturbation theory, the total impurity potential being the perturbing interaction. If one considers the low-energy $L_1$, $K_1$, and $U_1$ regions as subsidiary minima (strictly correct only for the $L_1$ region), this approach represents a logical extension of the Kohn-Luttinger formalism. This admixture of subsidiary minima is donor dependent (largest for As, intermediate for P, smallest for Sb) and is able to explain satisfactorily the numerous observed donor anomalies, even including the inverted-order cases. The calculated results indicate the positive identification of two new ENDOR shells, shell $C$ as site (5, 5, 5) and shell $F$ as site (2, 2, 0), and suggest the tentative identification of nine other ENDOR shells with lattice sites. Matching experimental Fermi contact constants and calculated values versus $\frac{k_0}{k_{max}}$ for positively and tentatively identified ENDOR shells yields $\frac{k_0}{k_{max}}=0.87\mathrm{}\pm 0.01$. A noninversion component of wave function has been introduced, resulting from the tetrahedral potential admixing $4f-\mathrm{nf}$ wave function (satisfying $A_1$ symmetry) into the solution of the single-valley Schrödinger equation. This addition makes only a slight improvement in the over-all agreement. The subsidiary-minima-admixture approach has also been attempted for the deep donor ${\mathrm{S}}^{+}$, yielding an improved qualitative agreement between theory and experiment. The admixture of subsidiary minima has a number of other physical consequences: (1) The donor-nucleus hyperfine interaction can be reasonably accounted for, including the donor dependence, without employing the sharply peaked Whittaker function and a cutoff radius; (2) the "shear" deformation potential ${\Xi }_u$ determined by ESR or optical experiments using the $1s-A_1$ donor ground state may not yield the true "shear" deformation potential of the ${\Delta }_1$ minima; (3) the energy of the $1s-A_1$ state contains an important second-order correction from the subsidiary minima which can account for between 25% and 50% of the energy correction to the effective-mass value. It is shown the valley-valley coupling terms account for nearly all the energy correction of the $1s-A_1$ state, and that the single-valley correction is very small, contrary to previous work. Analysis of the location of the lattice sites positively and tentatively identified with ENDOR shells yields evidence that the three-dimensional appearance of the wave-function density of the $1s-A_1$ state significantly reflects the tetrahedral symmetry of the atoms surrounding the donor.