Exact Solutions of the Kinetic Equations Governing Thermally Stimulated Luminescence and Conductivity

Past analyses of the kinetic equations governing thermally stimulated luminescence and conductivity have invariably involved certain approximations; however, the validity of these approximations has never been explicitly determined. In this paper we give a procedure for determining the conditions under which the approximations are valid, and show that for a model involving a single trap depth in the presence of other deep traps, and a single type of recombination center, the validity depends critically on $N$, the number of active traps. For $N<\mathrm{}{10}^{15}$ ${\mathrm{cm}}^{-3\mathrm{}}$ the conventional approximations are inadequate, and the kinetic equations must be analyzed exactly through numerical solutions. Although no such solutions have been reported in the literature, we show that they are not only possible, but are in fact readily obtained for certain parametric ranges. Examination of these exact solutions for small $N$ reveals new features; in particular, the dependence of the processes on the density of deep traps, or on initial filling ratios of the active traps, is markedly different from the dependence at large $N$, where the approximations do hold. This invalidates, for these low densities, many approaches to the analysis of the phenomena which have been recommended on the basis of these approximations. The procedures developed here have been applied to one specific model. However, they can be readily generalized to the solutions of the equations for more complex and realistic models of solids.