Time dependent transport in 1D micro- and nanostructures: Solving the Boltzmann and Wigner–Boltzmann equations

For many decades the Boltzmann distribution function has been used to calculate the non-equilibrium properties of mobile particles undergoing the combined action of various scattering mechanisms and externally applied force fields. When the latter give rise to the occurrence of inhomogeneous potential profiles across the region through which the particles are moving, the numerical solution of the Boltzmann equation becomes a highly complicated task. In this work we highlight a particular algorithm that can be used to solve the time dependent Boltzmann equation as well as its quantum mechanical extension, the Wigner–Boltzmann equation. As an illustration, we show the calculated distribution function describing electrons propagating under the action of both a uniform and a pronouncedly non-uniform electric field.