9. Stochastic Methods and Monte Carlo

For the NANOTCAD project our working group had the task to simulate resonant tunneling diodes by the Monte Carlo method.

The Monte Carlo approach is motivated by the Monte Carlo (MC) method for classical device simulation, where the dissipation operator is treated exactly and the potential operator can be treated by the method of trajectories.

Regarding the simulation of coherent (ballistic) transport, the Wigner method is inferior to the quantum transmitting boundary method in terms of numerical accuracy and feasibility as will be discussed in Section 10.2. Why then should we bother to solve the Wigner equation and go through all the numerical troubles as described in Chapter 8? The reason is: many-particle interactions, i.e., electron-phonon scattering. Even for many devices at the end of the roadmap scattering is far too important to be neglected.

In this chapter we discuss the modeling of dissipation by electron-phonon scattering. Secondly, we show how coherent transport in a quantum potential can be understood as a stochastic process using particles and ``antiparticles''. Finally we discuss the negative sign problem and the inclusion of annihilation into the stochastic process. This allows for stationary solutions of the stochastic process and reduces the variance of the Monte Carlo algorithm.