Diffusion and carrier recombination by interstitials in silicon

We analyze a model of the electronic structure of the silicon interstitial, consistent with full local-density approximation (LDA) calculations. The model assumes three charge states: neutral (0), singly (+), and doubly ionized (++). The (0) interstitial is stable in a shared site, the (++) stable in a tetrahedral site, and the (+) state has energy nearly independent of position. In thermal equilibrium the relative occupation of the (0) and (++) states, each near its stable position, depends upon the electronic Fermi energy. The (+) state has much lower probability than either the (0) or the (++), making this a negative-$U$ center. Nevertheless, the predicted diffusion constant for dopant atoms is dominated by motion of the interstitial in the (+) state. It has an activation energy of about one-half the band gap and is also proportional to the total interstitial density. If the interstitial density is established at some high annealing temperature, it depends strongly upon the Fermi energy at that temperature, and is much higher for $p$-type silicon. The moving interstitial also provides radiationless recombination of excess carriers, at a rate calculated using matrix elements derived from the full LDA electronic structure. The recombination rate does not contain an important Boltzmann factor, in contrast to a Huang-Rhys mechanism, but is proportional to the interstitial density and, at high carrier densities, to the square root of the product of the electron and hole densities. This recombination causes an enhancement of the diffusion rate, given near equilibrium by a factor $\left[N_e{\tau }_h{/N}_e^0{+N}_h{\tau }_e{/N}_h^0\right]/[{\tau }_e+{\tau }_h],$ with $N_e,$ $N_e^0,$ and ${\tau }_e$ the density of electrons, the equilibrium density, and an electron emission time, and $h$ indicating the corresponding parameters for holes. For high carrier densities, the enhancement can greatly exceed the equilibrium diffusion.