The backward Monte Carlo method for semiconductor device simulation

A backward Monte Carlo method for the numerical solution of the semiconductor Boltzmann equation is presented. The method is particularly suited to simulate rare events. The general theory of the backward Monte Carlo method is described, and several estimators for the contact current are derived from that theory. The transition probabilities for the construction of the backward trajectories are chosen so as to satisfy the principle of detailed balance. This property guarantees stability of the numerical method and allows for a clear physical interpretation of the estimators. A symmetric sampling method which generates wave vectors always in pairs symmetric to the origin can be shown to yield zero current exactly as thermal equilibrium is approached. The properties of the different estimators are evaluated by simulation of an n-channel MOSFET. Quantities varying over many orders of magnitude can be resolved with ease. Such quantities are the drain current in the sub-threshold region, the high-energy tail of the carrier distribution function, and the so-called acceleration integral which varies over 30 orders in the example shown.