Impact ionisation rate calculations in wide band gap semiconductors
Calculations of band-to-band impact ionisation rates performed in the semi-classical Fermi’s Golden Rule approximation are presented here for the semiconductors GaAs, In(_0.53)Ga(_0.47)As and Si(_0.5)Ge(_0.5) at 300K. The crystal band structure is calculated using the empirical pseudopotential method. To increase the speed with which band structure data at arbitrary k-vectors can be obtained, an interpolation scheme has been developed. Energies are quadratically interpolated on adapted meshes designed to ensure accuracy is uniform throughout the Brillouin zone, and pseudowavefunctions are quadratically interpolated on a regular mesh. Matrix elements are calculated from the pseudowavefunctions, and include the terms commonly neglected in calculations for narrow band gap materials and an isotropic approximation to the full wavevector and frequency dependent dielectric function. The numerical integration of the rate over all distinct energy and wavevector conserving transitions is performed using two different algorithms. Results from each are compared and found to be in good agreement, indicating that the algorithms are reliable. The rates for electrons and holes in each material are calculated as functions of the k-vector of the impacting carriers, and found to be highly anisotropic. Average rates for impacting carriers at a given energy are calculated and fitted to Keldysh-type expressions with higher than quadratic dependence of the rate on energy above threshold being obtained in all cases. The average rates calculated here are compared to results obtained by other workers, with reasonable agreement being obtained for GaAs, and poorer agreement obtained for InGaAs and SiGe. Possible reasons for the disagreement are investigated. The impact ionisation thresholds are examined and k-space and energy distributions of generated carriers are determined. The role of threshold anisotropy, variation in the matrix elements and the shape of the bands in determining characteristics of the rate, particularly the softness of the rate's threshold behaviour are investigated.