Calculation of Anisotropic Hyperfine Constants for Lattice Nuclei near a Shallow Donor

Abstract

A method is presented for calculating the magnetic anisotropic (dipolar) hyperfine constants for lattice nuclei near a shallow-donor impurity. The method assumes that the wave function of the donor electron can be expressed in an effective-mass form, i.e., a slowly varying envelope function times conduction-band Bloch functions. For each dipolar hyperfine constant, two separate calculations are performed. One calculation is for a local region about the lattice nucleus of interest. The greatest part of the interaction occurs in this region (about 85 ± 10%). The second calculation is for the more distant region. The dipolar constants in the distant region are calculated without considering the details of the Bloch functions and are evaluated by an integration involving only the envelope function. In the local region, the details of the Bloch functions must be considered. The Bloch functions are expressed in terms of equivalent orbitals. Symmetry arguments using the properties of these orbitals simplify the calculation. The final results show that the local contribution can be expressed as products of a few intrinsic lattice parameters, which are the dipolar matrix elements between equivalent orbitals, and a set of coefficients that is not difficult to evaluate. The resulting dipolar constants vary a great deal from one lattice site to another. Numerical results have been computed for the shallow donors — arsenic, phosphorus, and antimony — in silicon. Comparison of theoretical values with experimental values shows qualitative and semiquantitative agreement.