To enable predictive simulation of semiconductor devices proper models
describing carrier transport are required. The drift-diffusion (DD) transport
model  is by now the most popular model used for device simulation.
However, with down-scaling the feature sizes, non-local effects become more
pronounced and must be accounted for by using energy-transport (ET) or
hydrodynamic (HD) model .
During the last decade Monte-Carlo (MC) methods for solving the time-dependent Boltzmann equation have been developed [35,36] and applied for device simulation [37,38,39]. However, the MC algorithms encounter serious difficulties when applied to the extreme conditions occurring in the advanced semiconductor devices. The carrier distribution can vary by several orders of magnitude in the space and energy domain of interest. Since the MC simulation follows the natural carrier histories, the portion of the simulated trajectories in a given region is proportional to the carrier density in this region. As a consequence, the major part of the simulation time is spent by trajectory computation in densely populated regions, while the statistics in the low density regions remain insufficient. A simple increase of the total simulation time cannot solve the problem within reasonable CPU time, if statistics in the rarely visited regions needs to be increased by orders of magnitude. Thus, reduction of computation time is still an issue and, therefore, the MC device simulation is still not feasible for industrial application.