Competition between isotropic and strongly anisotropic terms in the impact ionization rate of narrow- and middle-gap cubic semiconductors

We report on the strong anisotropy of the inter-band process of impact ionization in direct-gap cubic semiconductors with either weak or strong spin-orbit coupling at low effective temperatures of electron distribution $T$, and the crossover to isotropic behavior with increasing $T$. Such anisotropy is related to specific mechanism of the impact ionization involving coupling of the electron and heavy hole states via remote bands, which is vanishing for some high-symmetry propagation directions of an initial electron, namely [100] and [111]. At room temperature impact ionization rate in narrow-gap semiconductors InSb, InAs, GaSb and In$_{0.53}$Ga$_{0.47}$As is isotropic while in middle-gap InP, GaAs and CdTe both terms are comparable. We propose simple and justified analytic generalization of Keldysh formula for the impact ionization rate, which is suitable for incorporation into modelling software.