Classical Approximation of the Scattering Induced Wigner Correction Equation

Quantum simulations basically rely on two kinetic theories which account for the coherent transport at different levels of approximation. These theories have complementary properties with respect to the ability to account for de-coherence processes, and become computationally expensive in describing mixed mode transport, where both, coherent and de-coherent processes must be taken into account. We consider an ap- proach, where the coherent information, as provided by the non- equilibrium Green's function, is used in a kind of Wigner equa- tion for the scattering induced correction to the coherent Wigner function. Here, we address the opportunity to approximate the equation by taking the classical limit in the Wigner potential term. I. INTRODUCTION The nanometer and femtosecond scales of operation of modern devices give rise to a number of phenomena which are beyond purely classical description. These phenomena are classified in the International Technology Road-map for Semi- conductors (ITRS, www.itrs.net) according to their importance to the performance of next generation devices. It is recognized that '... computationally efficient quantum based simulators' are of utmost interest. Quantum models capable of describing mixed mode transport, where purely coherent phenomena such as quantization and tunneling are considered along with phase breaking processes such as interactions with phonons, are especially relevant. The rising computational requirements resulting from the increasing complexity due to the mixed mode phenomena are a major concern for the development and deployment of these models. The harmony between theoretical and numerical aspects of the classical Boltzmann model is no longer among the characteristics of the quantum mechanical counterpart. The two kinetic theories which are the foundations of quantum simulations will be sketched in the following. We first consider coherent processes. The non-equilibrium Green's function (NEGF) approach offers the most comprehensive, self-consistent way to account for correlations in space and time. However, because of numerical issues the applicability is restricted to stationary structures, basically in the ballistic limit (1). The computational burden can be reduced by work- ing in a mode space, obtained by separation of the problem into longitudinal and transverse directions. Furthermore, if the transverse potential profile along the transport direction remains uniform, the modes in these directions can be de- coupled so that the transport becomes quasi-multidimensional.