Modern micro-electronic devices operate on scales of nanometres and femtoseconds, which leads to a pronouncement of effects that, while they defy purely classical descriptions, are also not purely attributable to coherent quantum models. Here, the Wigner function formalism may be used as a bridge between the purely classical and the fully coherent quantum worlds. This allows the treatment of the mixed mode transport, where purely coherent phenomena such as quantization and tunnelling are considered alongside phase breaking processes, such as the interaction with phonons. Furthermore, the similarity of the Wigner function is very convenient due to its similarity with classical distribution functions, which suggests the use of Boltzmann scattering models deployed in classical simulations in the quantum counterpart, thereby easily accounting for, e.g., phonon interactions, due to the phase-space nature of the formalism.
Mixed mode transport is especially interesting in devices, where the dwelling times of carriers are sufficiently prolonged to make the probability of scattering events non negligible, but the number of such events is small, e.g., in the order of magnitude of 1. Under this assumption, which is warranted for short devices, a method for determining an adjustment of a purely coherent quantum solution to account for scattering phenomena is developed. It uses coherent input data obtained by, e.g. a Non-Equilibrium Green's Function (NEGF) formalism, and calculates a correction by reformulating the problem to resemble a Fredholm integral equation of the second kind. The solution to this restated problem is representable as a Neumann series with its terms obtained by iterative application of the kernel to the free term, which is obtained from the initial condition. This series corresponds to a Boltzmann kind of evolution process, which can be treated using Monte Carlo methods.