General analytic solution to the Shockley-Read-Hall rate equations with a single-level defect

Abstract

The decay of excess carriers in nondegenerate semiconductors generated by a light impulse $\ensuremath{\delta}(t)$ is governed by the differential equations referred to as the Shockley-Read-Hall (SRH) rate equations. In the past, linear approximations were used or restrictive conditions imposed to obtain an analytical solution limited to low or high injection. For defect level parameters of practical interest, the nonlinear differential equations were numerically solved. Whereas carrier decay is often approximated by one time constant $\ensuremath{\tau},$ in the present paper it is shown that recombination occurs with both the minority $({\ensuremath{\tau}}_{1})$ and majority $({\ensuremath{\tau}}_{2})$ time constants present in the decay. Expressions for ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ are derived without an approximation at a given temperature, for arbitrary excess carrier concentration, doping concentration ${N}_{A,D},$ defect level concentration ${N}_{t},$ cross section ${\ensuremath{\sigma}}_{n,p},$ and energy level ${E}_{t}.$ A general analytic solution to the SRH rate equations represented by an infinite series of monoexponential terms, the frequencies or inverse time constants of which are a linear combination of the fundamental frequencies ${\ensuremath{\lambda}}_{1}=1/{\ensuremath{\tau}}_{1}$ and ${\ensuremath{\lambda}}_{2}=1/{\ensuremath{\tau}}_{2},$ is derived without an approximation. The solution is the sum of the responses to an infinite number of linear systems and in this sense represents the impulse response. A critical point representing the transition between the linear and nonlinear variation of fundamental frequency with excess carrier density is identified. The analytic solution is verified by analyzing the numerical solution of the SRH rate equations for the fundamental frequencies using a multitransient technique. The trapping behavior of the minority carrier at a single-level defect, with excess carrier concentration, is examined.