7.2 The Open Von Neumann-Poisson Equation

In this section we introduce the quantum transmission boundary method (QTBM). First we discuss how the solution of the coherent transport problem for an open system is reduced to the solution of a set of Schrödinger equations with absorbing boundary conditions. This is also known as the ``mixed state Schrödinger approach''

Then we couple the von Neumann equation self-consistently with the Poisson equation to describe interactions between electrons. For this both a Gummel method and a full Newton solver have been implemented. The assembly of the Jacobian matrix needed for the Newton solver has been parallelized using MPI. It has been observed that the system admits multiple solutions which are difficult to handle with stationary simulations alone.

The final section discusses the description of scattering within the QTBM. We use the hydrodynamical formulation to investigate scattering models which are compared with the Wigner relaxation time model.