Electron Scattering by Pair Production in Silicon

The energy-dependent rate for inelastic scattering of electrons by production of electron-hole pairs is computed by first-order perturbation theory for silicon using a screened Coulomb interaction with a frequency- and momentum-dependent dielectric function calculated for silicon in the random-phase approximation. The threshold for momentum-conserving pair creation is found to be very close to that determined by energy conservation alone. This follows from the silicon band structure. The absolute scattering rate is close to that obtained experimentally by Bartelink, Moll, and Meyer. Pair production dominates the scattering rate for electrons of energy greater than 6.5 eV above the valence-band maximum. For electrons between 4 and 6.5 eV the scattering rate is dominated by phonon scattering, but energy loss is dominated by pair scattering. Below 4 eV phonon processes dominate inelastic processes as well. The momentum space integrals for pair-creation scattering are performed by a Monte Carlo method. It is found that the calculations with momentum conservation can be reproduced surprisingly well by a simple "random-k" approximation, which effectively ignores momentum conservation. This leads to a very simple expression for pair scattering in the form of a two-dimensional energy fold over one-electron state-density functions, a result first obtained by Berglund and Spicer. This simple form facilitates the calculation tremendously and should make calculations for other materials very simple if the state-density function is known. Using this "random-k" method, the scattering rate for primary holes is obtained and found to be almost identical with that for primary electrons of comparable energy. The secondary-particle energy distribution functions are also determined for primary holes and electrons. One-electron state-density structure is prominent in these distributions.