Exchange between deep donors in semiconductors: A quantum defect approach

Exchange interactions among defects in semiconductors are commonly treated within effective-mass theory using a scaled hydrogenic wave function. However, such a wave function is only applicable to shallow impurities; here, we present a simple but robust generalization to treat deep donors, in which we treat the long-range part of the wave function using the well-established quantum defect theory, and include a model central-cell correction to fix the bound-state eigenvalue at the experimentally observed value. This allows us to compute the effect of binding energy on exchange interactions as a function of donor distance; this is a significant quantity given recent proposals to carry out quantum information processing using deep donors. As expected, exchange interactions are suppressed (or increased), compared to the hydrogenic case, by the greater localization (or delocalization) of the wave functions of deep donors (or supershallow donors with binding energy less than the hydrogenic value). The calculated results are compared with a simple scaling of the Heitler-London hydrogenic exchange; the scaled hydrogenic results give the correct order of magnitude but fail to reproduce quantitatively our calculations. We calculate the donor exchange in silicon including intervalley interference terms for donor pairs along the {100} direction, and also show the influence of the donor type on the distribution of nearest-neighbor exchange constants at different concentrations. Our methods can be used to compute the exchange interactions between two donor electrons with arbitrary binding energy.