3.5 Comparison between Master Equation and Monte Carlo Method

      The disadvantage of the ME method is that the number of states that have to be considered becomes often very large. Consequently the matrix operations involved in calculating the exponential operator are time consuming and the approximations do not converge quickly. Numerical instabilities can easily appear. By filtering out the largest eigenvalues the Krylov subspace method rectifies much of the numerical problems. However the a priori identification of important states remains. The MC method on the other hand is a very stable and robust method, that has one major drawback which is the resolution of rare tunnel events, for instance co-tunneling, which take place in presence of much more frequent events. In the past variance reducing methods [95] were developed that can help to alleviate this problem. However, in the case of SET circuits the tunnel rates may vary over a range of many magnitudes, so that most variance reducing methods would not trigger, since no rare state is ever reached. We tackled this problem with a new algorithm that combines the MC method with a partial direct calculation of state probabilities (see Section 3.6). Other good features of the MC method are an easy trade-off between accuracy and simulation time. Thus one can achieve quick approximate results of very large circuits, which would otherwise not be feasible to compute. Furthermore the MC method offers a detail richer microscopic model for the tunnel process, since in real SET circuits electrons tunnel from island to island as simulated by the MC method.