Electron-Hole Recombination Statistics in Semiconductors through Flaws with Many Charge Conditions

A flaw with $s$ electronic units of negative charge makes transitions to charge $s+1\mathrm{}$ by hole emission at rate $e(p, s)$ or by electron capture at rate $nc(n, s)$ and returns to charge $s$ at rates $e(n, s+1)$ and $pc(p, s+1)$. Here $n$ is the electron density in the conduction band and $p$ is the hole density in the valence band. The steady-state ratio of populations $N_{s+1\mathrm{}}$ to $N_s$ is given by $\frac{c(n, s)[n+n^{*}(s+\frac{1}{2}\left)\right]}{c(p, s+1)[p+p^{*}(s+\frac{1}{2}\left)\right]},$ where $n^{*}(s+\frac{1}{2})=\frac{e(p, s)}{c(n, s)}$ and $p^{*}\left(s\frac{1}{2}\right)=\frac{e(n, s+1)}{c(p, s+1)}$. This distribution corresponds to an effective Fermi level for the flaws only for the condition of thermal equilibrium. Expressions for the recombination rate based on the steady-state distribution are derived. For a given transition $s\rightleftarrows s+1\mathrm{}$ the following special cases are defined: (1) denuded: $n<\mathrm{}{n}^{*}$, $p<\mathrm{}{p}^{*}$; (2) $n$-dominated: $n>\mathrm{}{n}^{*}$, $p<\mathrm{}{p}^{*}$; (3) $p$-dominated: $n<\mathrm{}{n}^{*}$, $p>\mathrm{}{p}^{*}$; (4) flooded: $n>\mathrm{}{n}^{*}$, $p>\mathrm{}{p}^{*}$. Diagrams which aid in visualizing the relative importance of the various transitions are presented. Some speculations on the nature of trapping centers are given.