Impact ionization rate in direct gap semiconductors

In the framework of the $14$-band $\mathbf{k}\cdot\mathbf{ p}$ model, the intensity of the impact ionization processes in direct gap semiconductors is studied and explicit expressions for the impact ionization rate are obtained. It is shown that the rate of the process near the threshold energy is determined by the sum of the isotropic and strongly anisotropic contributions. The former contribution is proportional to the cube of the distance from the threshold, whereas the latter is a quadratic one arising only because of the coupling with remote bands. The comparison of these contributions under averaging over the nondegenerate isotropic distribution of nonequilibrium electrons characterized by some effective temperature $T^*$ demonstrates that the cubic contribution rather than the commonly used quadratic one is dominant in the direct gap semiconductors with $E_g<1-1.5\,$ eV up to $T^*=300$ K. This should be taken into account in the calculations of the operating characteristics of the devices based on the avalanche multiplication of charge carriers.