Rigorous Theory and Simplified Model of the Band-to-band Tunneling in Silicon

-The phonon-assisted band-to-band tunneling rate in crystalline silicon is calculated using the equilibrium Green's function formalism. Electron-phonon collisions, that balance the momentum, are included in the perturbation operator. Houston-type solutions are used for the time dependence of the Bloch states. RPA deeoupling yields a tractable expression for the differential tunneling conductivity. Its evaluation is presented explicitly, taking exactly into account the anisotropy of the six conduction band valleys. A simplified rate formula for the purpose of device simulation is then derived from the general expression, restricting the field strength and using reasonable models for the matrix elements. It is shown that indirect, phonon-assisted tunneling largely exceeds direct tunneling at all events. Finally, band-toband tunneling is compared with trap-assisted tunneling. We conclude that the pre-breakdown range in silicon is dominated by tunneling via traps.