Ground-state wave function for shallow-donor electrons in silicon. I. Isotropic electron-nuclear-double-resonance hyperfine interactions

In a new theoretical investigation of electrons bound to the shallow-donor impurities (P, As, Sb) in silicon we have calculated the Fermi-contact hyperfine-interaction constants for the ${\mathrm{Si}}^{29}$ lattice nuclei surrounding the impurity nucleus. We have used a model potential which represents the impurity potential, a wave-vector-dependent dielectric function which represents the screening of the impurity potential by the silicon lattice, and pseudopotential Bloch functions for the calculation of the wave-function density of the $1s\left(A_1\right)$ ground state at the nuclear sites. The restrictions of the effective-mass theory to a single band and to conduction-band-minima Bloch functions have been removed because we represent the wave function in terms of a Bloch-function expansion throughout the Brillouin zone for several bands. With the use of the Fermi-contact constants calculated from this wave function, we have been able to make definite matchings of ${\mathrm{Si}}^{29}$ lattice sites and the electron-nuclear-double-resonance (ENDOR) shells measured by Hale and Mieher. With these matchings we are able to explain the experimentally observed lack of inversion symmetry of the electronic wave function and to explain most of the donor dependence of the experimental ENDOR data. Tutorial type discussions of the comparison of this calculation with effective-mass calculations are presented. These results contain several features that are due to the complex values of the wave-function expansion coefficients and the complex values of the Bloch functions; these features cannot be obtained from any real-valued effective-mass Hamiltonian.