Auger effect in semiconductors

This paper presents a calculation of the lifetimes ($\tau $) of excess electrons and holes in a semiconductor assuming the Auger effect between bands (electron-electron and hole-hole collisions) to be the only recombination mechanism. If pair annihilation, and the corresponding reverse process of pair creation, are counted separately, there are four classes of processes to be considered. The suitably weighted algebraic sum of the rates of these processes yields a net recombination rate R. If N be the non-equilibrium number of pairs, then $\tau $ = N/R. In the calculation the effect of traps is neglected, and the group of electrons in the conduction band and the group in the valence band are each assumed to be in equilibrium among themselves, but not with each other, by the use of quasi-Fermi levels. Bloch functions $\psi _{{\bf k}}$ = u(${\bf k}$, ${\bf r}$) exp (i${\bf k}$. ${\bf r}$) are used. The matrix element of the Coulomb interaction is obtained as a multiple sum over reciprocal lattice vectors. Most of these terms correspond to Umklapp-type processes whose probability of occurrence is shown to be small. The dominant term, after integration over all initial and final states, yields the dependence of lifetime on temperature, carrier concentration, energy gap and other parameters. The absolute value of the lifetime depends also on an overlap intergral of the form $\int $u$^{\ast}$(${\bf k}$, ${\bf r}$) u(${\bf k}^{\prime}$, ${\bf r}$) d${\bf r}$ where k, ${\bf k}^{\prime}$ are in different bands. This integral is estimated on the basis of a one-dimensional model. The theory is compared with experimental lifetimes in InSb, and shows that the mechanism envisaged may dominate radiative recombination above 240 degrees K and accounts for the order of magnitude of the observed lifetimes ($\sim $ 10$^{-8}$ s) in the neighbourhood of the highest temperature (330 degrees K) at which recombination in InSb has so far been studied.