The preceding chapters have introduced all the aspects relevant to the simulation of time-resolved quantum transport, in two dimensions, using a Wigner Monte Carlo approach. The work has been presented in six chapters.
Chapter 1 motivated the need for utilizing TCAD in the research and design of nanoelectronics. An overview of the most widespread transport models was given with the qualities and limitations of each. It was found that the Wigner-Boltzmann transport model offers some unique qualities in that it makes time-resolved quantum transport with scattering computationally feasible, while providing many classical analogies to be used. This presented the motivation for the development of a solver for the Wigner-Boltzmann transport equation.
Chapter 2 presented the Wigner formalism of quantum mechanics, which gives a formulation in the phase space. From this background the derivation of the Wigner transport equation was presented. Thereafter, the augmentation of the Wigner equation with semi-classical phonon scattering models was shown to be justified by outlining the rigorous derivation, which finally yielded the Wigner-Boltzmann equation. The semi-discrete form of the Wigner equation, which follows from considering a finite spatial domain, was presented. The various transport problems that can be approached with Wigner-based simulations and the handling of boundary conditions was outlined. Finally, an overview of existing solvers for the Wigner-Boltzmann equation, for one-dimensional problems, was given.
Chapter 3 introduced the signed-particle method – the Monte Carlo approach which has made the solution of the Wigner equation in two spatial dimensions computationally feasible. The mathematical foundation of the signed-particle method was presented to be based on the integral formulation of the Wigner-Boltzmann equation, which is developed into a Neumann series that can be evaluated using Monte Carlo integration. Armed with this theoretical background, the basic building blocks and architecture of an algorithmic implementation of the signed-particle method was shown.
Chapter 4 highlighted the improvements and contributions made by the author to the signed-particle method’s algorithms. The contributions encompass optimized algorithms for increased computational efficiency, statistical enhancements and treatment of discretization effects, which can lead to non-physical behaviour. Finally, the considerably improved accuracy of the signed-particle method, using the optimized algorithms, was demonstrated by a comparison of the numerical results to an exact solution of a physical problem.
Chapter 5 treated the parallelization of the Wigner Ensemble Monte Carlo code, which was developed in the scope of this thesis, using a spatial domain-decomposition approach, which is well-positioned for large scale parallelization. The latter was motivated by evaluating the possible parallelization approaches to Wigner Monte Carlo simulations with due consideration to the typical hardware architecture of supercomputers. Finally, the parallel efficiency of the selected approach was demonstrated using one- and two-dimensional examples.
Chapter 6 demonstrated an application of the Wigner Monte Carlo simulator to investigate electrostatic lenses, which were applied to manipulate the dynamics of wavepackets. The latter is of interest in the emerging field of quantum control, but also for actual devices. The improvement of the drive-current in a nano-scaled channel was shown by focussing electrons using a converging lens. The calculation of a steady-state solution was shown for the first time and has been made practical by the parallelization presented in Chapter 5.