Determination of the Donor Pair Exchange Energy in Phosphorus-Doped Silicon

The EPR spectrum for relatively dilute samples of phosphorus-doped silicon (${10}^{16}$ donors/${\mathrm{cm}}^3$) has been calculated in detail for an assumed random distribution of impurities. The system of donor electron spins is treated as a collection of nearest-neighbor donor pairs. An expression is derived for the donor pair exchange energy using Kohn-Luttinger wave functions and a general exchange-energy expression. The resultant relationship contains an adjustable parameter $a^{*}$, the "effective Bohr radius," which is determined from a comparison of the calculated spectrum and the experimental results obtained for the ratio $C$ of the central-pair and hyperfine line intensities. The resulting expression $J\left(\mathrm{R}\right)$, where $J$ represents the exchange energy and R the separation vector connecting the two pair donors, exhibits an oscillatory spatial dependence due to interference from portions of the wave function arising from different conduction-band valleys. The distribution of pair exchange energies is compared with earlier experimental determinations of this distribution.