3.4 Monte Carlo Method

  To use the MC method for the simulation of SET circuits was first proposed and implemented by N. Bakhvalov et al. [13]. Other groups adopted this method later [18] [61] [97]. From (3.17), the probability that a tunnel event out of state 0 happens at $\tau$ and not earlier is
 $$\begin{gather} P_0(\tau)=e^{-\Gamma\tau}. \end{gather}$$
To construct random numbers which are distributed like (3.28), from evenly distributed random numbers, one has to take the inverse of the distribution function [58].
 $$\begin{gather} \tau=-\ln(r)/\Gamma,% \index{duration to next tunnel event} \end{gather}$$
where r is an evenly distributed random number from the interval ]0,1[. The MC procedure is then as follows. Starting from all possible tunnel events with their particular tunnel rates, concrete tunnel times $\tau_i$ are computed according to (3.29). The event with the smallest $\tau$ will happen first and thus is taken as the winner of the MC method. Node charges are updated according to the so computed tunnel event, which further brings changing node potentials in its train. New tunnel rates are calculated and a new winner is determined through MC simulation. Doing this many times gives the macroscopic behavior of the circuit. Another possibility is to calculate first an exit tunnel time out of the current state k, $\tau_k = -\ln(r)/\sum_i \Gamma_{ik}$, and selecting afterwards with a second random process, which event it was. One or the other scheme will have a shorter run time, depending on how long the evaluation of a logarithm and the generation of an evenly distributed random number takes.