Rigorous Theory and Simpli ed Model of theBand - 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 decoupling yields a tractable expression for the diierential tunneling conductivity. Its evaluation is presented explicitly, taking exactly into account the anisotropy of the six conduction band valleys. A simpliied rate formula for the purpose of device simulation is then derived from the general expression, restricting the eld 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-to-band tunneling is compared with trap-assisted tunneling. We conclude that the pre-breakdown range in silicon is dominated by tunneling via traps. 1 Notation a lattice constant a + k , a k creation and annihilation operator of a Bloch state fkg, resp. Airy function, its derivative and its integral b + q , b q creation and annihilation operator of a phonon with wave vector q c s sound velocity