Statistical enhancement aims at reduction of the time necessary for computation of the desired device characteristics. Enhancement algorithms are especially useful when the device behavior is governed by rare events in the transport process. Such events are inherent for the sub-threshold regime of device operation, simulations of effects due to discrete dopant distribution, as well as tunneling phenomena. Virtually all Monte Carlo device simulators with statistical enhancement use population control techniques. They are based on the heuristic idea for splitting of the particles entering the given phase-space region D of interest. The alternative idea - to enrich the statistic in D by biasing the probabilities associated with the transport of classical carriers - gives rise to the event-biasing approach. Due to the event biasing, the behavior of the simulated Ensemble Monte Carlo (EMC) particles differs from that of Boltzmann carriers. Nevertheless the Boltzmann distribution function f is recovered by using the proper weights associated with the particles. The approach is derived from the integral form of the linear Boltzmann equation (BE), where Coulomb interactions are neglected.
The approach is generalized for Hartree electrons in the presence of both initial and boundary conditions. The BE becomes nonlinear via the Hartree component of the electric field. The steps used to derive event biasing cannot be applied directly to BE, so the solution is sought in the iterative procedure of coupling the EMC with the Poisson equation (PE). The fact that the electrical field is frozen between two successive solutions of PE is employed. The BE is linear, and event biasing can be applied. The BE solution f(t) is then obtained within a weighting EMC scheme from the phase-space position of the numerical particles. f(t) gives the correct carrier density for the next solution of the PE. A key point in the proof is to use the Markovian character of the evolution: f(t) becomes the initial condition f0 of the BE for the next time step. Finally, f0 can be biased by inverting the weighting scheme used for f(t). This means that the same particles (with respect to phase-space location and weight) which reach time t can be used to continue the evolution in the next time interval. This proves that the biasing scheme can be used to provide the self-consistent distribution function at any time. In the thermodynamic limit of an infinite number of particles, both Boltzmann and biased stochastic processes give the same evolution of the physical averages. For a finite particle number, the computational efforts depend on the variance of the chosen stochastic process.