Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

4.4 The Combined Backward-Forward MC Method

The forward MC method and the backward MC method can be combined. The following considerations can be found in [P4].

In semiconductor devices, there are various processes which are caused by carriers with energies above a certain energy threshold. Such processes are impact ionization, carrier injection into the oxide, and the generation of interface traps due to hot carriers. To model such processes, only carriers with energies above the threshold need to be considered, whereas carriers with lower energies have no effect. Therefore, a reasonable approximation in the modeling of such processes is to consider only those high energetic carriers that are able to surmount the energy barrier and to neglect the vast majority of carriers close to thermal equilibrium that get reflected on either side of the barrier.

This motivates the introduction of a combined backward-forward Monte Carlo method which simulates only those trajectories passing the energy barrier, as shown in Fig. 4.5. In the first step of this method, a backward trajectory is started from the injection plane at with a random initial state . The contribution of this trajectory to the estimator (4.23) determines the statistical weight of this state:

In the second step, a forward trajectory is started from the state . This trajectory is assigned the statistical weight (4.70). The quantities of interest are computed as a weighted average over an ensemble of forward trajectories. Note that the backward trajectories are only needed to determine the weights. No averages are computed from those trajectories.