Theory of Umklapp-assisted recombination of bound excitons in Si:P

We present calculations for the oscillator strength of the recombination of excitons bound to phosphorus donors in silicon. We show that the direct recombination of the bound exciton cannot account for the experimentally measured oscillator strength of the no-phonon line. Instead, the recombination process is assisted by an Umklapp process of the donor electron state. We make use of the empirical pseudopotential method to evaluate the Umklapp-assisted recombination matrix element in second-order perturbation theory. Our result is in good agreement with experiment. Being potentially useful for quantum computing, the process of Umklapp-assisted recombination can be used to detect optically the spin state of the nucleus of a phosphorus donor, which requires that the energy levels of the nuclear spin are optically resolvable. We therefore present two methods to improve the optical resolution of the optical detection of the spin state of a single nucleus in Si:P.