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.