Multiphonon relaxation of $1s(T_{2})$ states of a single ionized selenium donor in silicon

The rate of multiphonon relaxation of $1s(T_2)$ level in Se$^+$ donors in silicon was estimated. The calculation is an initial approach to the problem, which uses the most simplified form of wave functions. For the probability of transition, we used a well-known expression from the literature by R. Pässler [R. Pässler. Czech. J. Phys. B, $\mathbf{24}$, 322 (1974)], obtained in the framework of the so-called “static coupling scheme”. The deformation potentials of optical and acoustic phonons were determined by a fitting procedure using published data on the luminescence spectrum of Se+ donors at the $1s(T_2)$–$1s(A_1)$ transition and the Franck–Condon principle. The resulting estimate for the relaxation rate, 10$^3$ s$^{-1}$, was five orders of magnitude less than the rate corresponding to the experimentally measured lifetime. The reason for the discrepancy with the experiment is an oversimplified model that does not take into account several factors, the main of which is the presence of quasi-local vibrational modes. Analysis of the luminescence spectrum at this transition leads to the conclusion that the energies of such vibrational modes lie in the range from 26 to 61 meV. For a satisfactory agreement with the experiment, it is necessary to complicate the model, taking into account the interaction with these modes.