Theory of paramagnetic impurities in semiconductors

In this paper, a model of a paramagnetic impurity in a semiconductor (or of an F' center in an alkali halide) is proposed. It is an exactly soluble form of the quantum-mechanical 3-body problem. Specifically, we deal with 2 interacting particles in any number of dimensions in an attractive external potential, and present the qualitative features of the resulting eigenvalues and eigenfunctions. We find algebraically the conditions for a magnetic moment to appear (e.g., for an F' center to become unstable with respect to an F center) and discover that even a large 2-body electronic repulsion U does not cause a moment to appear when the one-electron bound state orbite about the impurity are sufficiently great. Conversely, in the case of small, tightly bound orbits, beyond a certain value of U, the impurity does in fact become magnetic in the ground state. Using the exact ground-state solution, we show that a perturbation-theoretic expansion in powers of U has a finite radius of convergence.