Application of the Method of Generating Function to Radiative and Non-Radiative Transitions of a Trapped Electron in a Crystal

Abstract

Using the method of generating function, we have discussed the shape of the absorption band and the probability for non-radiative transition of a trapped electron in insulating or semiconducting crystal, especially their temperature dependence. We have thereby chosen a model for the vibrational motion of the lattice as general as possible, the normal modes for any two electronic states being different as regards not only their equilibrium positions but also the principal axes and frequencies. For non-radiative transition, we have derived, in comparatively general cases, the high and low temperature behaviors of the probability which correspond to the process through activated states and the tunneling of the lattice co-ordinates, respectively. The result is applied to calculations of the probability for thermal ionization of trapped electrons (or holes) and the capture cross-section of impurities for free electrons (or holes) in non-polar semiconductors. Further, the high temperature expansion of a density matrix is used to discuss the transitions on a most general model in which lattice vibration is no longer of harmonic type. Two problems related to the degeneracy of electronic states are discussed briefly.