In the discussion we point out that for an accurate prediction of the I-V characteristics of resonant tunneling devices scattering mechanisms have to be included in the model: scattering that breaks phase-coherence results in a broadening of the transmission peaks at resonance and a dramatic degradation of the peak-to-valley current ratio of RTDs.

A proper theoretical description of the complicated transport properties beyond the global coherent tunneling picture is given in terms of a non-equilibrium quantum transport theory such as the density matrix, Wigner function or Kadanoff-Baym and Keldysh techniques for the non-equilibrium Green's function. The natural frame to extend semiclassical Monte Carlo methods to the quantum regime is the Wigner formulation of quantum mechanics. This formulation was actually adopted for the Monte Carlo simulator. But many equivalent formulations of quantum mechanics are possible and appropriate in other modes of device simulation.

Chapter 6 summarizes a few of these possible formulations, which have been applied to a variety of problems in the literature. We give formulations on configuration space (Schrödinger equation, the hydrodynamical formulation and hybrids thereof like the Riccati and Prüfer equation). Then we introduce phase space distribution functions focusing on the Wigner distribution.

In Chapter 7 we discuss the simplest quantum transport model: the open boundary Schrödinger equation. We give the formulation of absorbing boundary conditions and compare the Schrödinger, Riccati and Prüfer equations. Then the one-particle Schrödinger equation is extended to the von Neumann equation which describes many non-interacting particles. In the coherent case this is an alternative to the Wigner formulation. For open boundary conditions the von Neumann equation can be solved by a separation ansatz. This reduces the problem to a set of one dimensional Schrödinger equations which then can be solved independently. This method is known as quantum transmitting boundary method (QTBM). It is very fast and ideal for TCAD purposes.

Coupling the Poisson equation with the von Neumann
equation we get a nonlinear system. The
calculation of the Jacobian is parallelized
using MPI [GHLL^{+}98].
The main drawback of the QTBM is that it
is difficult to include scattering in the model.
Methods for describing scattering in the density
matrix formulation are discussed.

Chapter 8 introduces the Wigner function method (WFM). It applies the finite difference method to the Wigner equation for the simulation of RTDs. Although the Wigner method has very good theoretical properties its numerical discretization using finite differences proves to be troublesome. We implement the so called ``standard discretization'' (see [Fre90]) and discuss its numerical properties.

Chapter 9 deals with the development of quantum Monte Carlo methods. In the Wigner formalism the potential operator becomes non-local in momentum space with properties similar to the collision operator. We present a Monte Carlo algorithm which treats potential and collision operator in a unified way as a generalized scattering operator which allows for negative transition ``probabilities'' [NKKS02], [KNS03]. The arising negative sign problem is illustrated.

In the previous chapters we presented several numerical approaches to the simulation of quantum transport in mesoscopic device structures. Chapter 10 is devoted to the application of the von Neumann and the Wigner formulation to a test problem. We discuss the discrepancy in the results from a purely numerical point of view and find that the QTBM is superior to the Wigner function method in terms of numerical accuracy and computational costs.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD