On the impact ionization rate in direct gap semiconductors

We present quantum-mechanical theory of impact ionization in semiconductors with the direct band gap in $\Gamma$-point. It is shown that energy dependence of the impact ionization rate $\mathcal{W}(E)$ near a threshold $E_{th}$ is given by superposition of the two terms, one of which is strongly anisotropic and quadratic in $E-E_{th}$, while another one is isotropic and cubic in $E-E_{th}$. Explicit form of the coefficients in such representation is derived in the framework of the 14-band ${\bf k\cdot p}$ model, and we claim the room temperature domination of the cubic contribution for most of the direct-gap materials with $E_g$ up to 1.5 eV.