2.4.1 Multiple Refresh

**Figure 2.1:** The electron distribution function for n-type MOSFET at V_ds=6.5V and V_gs=2V simulated with and without Multiple Refresh for the same computational time. Graph is taken for room temperature and corresponds to the drain edge of the gate contact.

The statistical enhancement method used in this work falls into the class of population control methods and conserves the total particle charge. It is based on a partitioned phase space, but instead of the particle weight the number of particles is fixed for each region. In general, the MC algorithm is not modified and is identical for all particles regardless of their statistical weight [154]. The particles can move from one region to another and thereby, the number of particles in a region can change during the MC simulation. To maintain the given particle number in a region, the simulation is halted at fixed time intervals. The number of particles is then examined and if this number deviates from the initial number in the region, the particle sub-ensemble of that region is replaced by an ensemble with a desired number of particles. This process is called refresh and because it is continually repeated - multiple refresh (MR) - to maintain the given number of particles in all regions of the phase space during the simulation. The refresh does not produce any new information. The information contained in the sub-ensemble before the refresh is either identical after the refresh or it is reduced. In order to reduce the computational time only one region of interest may be refreshed. In Figure 2.1 the electron distribution function is shown for 5V n-type MOS transistor (for a more device details, see Section 3.1) at room temperature for V_ds = 6.5V and V_gs = 2V simulated with (dotted line) and without (straight line) MR. The depicted DF corresponds to the drain end of the gate contact. The simulation with and without MR were performed for the same computational time and the obtained DFs are identical within the statistical noise. In the case of the simulation performed without MR the statistical noise becomes too large for values of the distribution function below 10^-4. At the same time the simulation with MR demonstrates good results over the presented range of 10 orders of magnitude.