The Wigner formulation of quantum statistical mechanics provides a convenient kinetic description of carrier transport processes on the nanometer scale, characteristic of novel nanoelectronic devices. The approach, based on the concept of phase-space, considers rigorously the spatially quantum coherence and can account for processes of de-coherence due to phonons and other scattering mechanisms using the models developed for the Boltzmann transport. Almost two decades ago the coherent Wigner equation was utilized in deterministic 1D device simulators. At that time it was recognized that an extension of the deterministic approaches to two dimensions is prohibited by the enormous increase in the memory requirements, a fact which remains true even for today's computers.
A basic property of the stochastic methods is that they turn the memory requirements of the deterministic counterparts into computation time requirements. Accordingly, we focus on the development of a union of theoretical and numerical stochastic approaches, algorithms, and experimental code for 2D Wigner simulations of nanostructures. In contrast to device simulators which, being tools for investigation of novel structures and materials, rely on well-established algorithms, the union is comprised of mutually related elements which must be developed and tested for relevance and viability. Many open problems need to be addressed, such as the choice of the driving force in the Wigner equation, pure quantum versus mixed classical-quantum approaches, the correct formulation of the boundary conditions, appropriate values for the parameters, and a variety of possible algorithms. We present the first results in this direction. A semi-discrete formulation of the Wigner equation for a typical MOSFET structure has been derived.