General Nano-Electromagnetic Quantum Phase Space Model

Modern manufacturing technologies are able to fabricate structures with varying electromagnetic properties on the nanoscale. Due to manifesting quantum effects the current transport within these structures is significantly different to what we are used to in the macroscopic world. Existing approaches to simulate quantum transport effects fail to capture most interesting yet intricate special cases, such as high frequency scenarios: A new approach based on the Wigner formalism is needed. Wigner mechanics is usually formulated for electrostatic conditions. The formalism bearing many classical concepts was used to develop the Wigner signed particle model, providing a computationally efficient heuristic description of quantum phenomena. In contrast, the developed electromagnetic Wigner theories do not favor numerical implementation. We will thus develop a Wigner based electromagnetic model involving the numerical Monte Carlo theory, integrate it into our open source simulator ViennaWD, and use it to simulate and analyze current transport under general, spatio-temporal electromagnetic conditions. Our envisioned model is based on our recently derived and numerically favorable Wigner current transport equation for general electromagnetic fields. Electromagnetic nanostructures, such as magnetic quantum wires, quantum Hall systems, and Aharonov-Bohm rings, will be investigated. The envisioned general electromagnetic signed particle model will provide a novel and unique way for exploring general spatio-temporal electromagnetic processes in open nano- electromagnetic systems. Fundamental questions about the primary role of forces or potentials will be addressed.

This project developed a Monte Carlo Wigner function based signed particle model describing electron transport dynamics in nanostructures for electromagnetic (EM) conditions. The first key theoretical result is a solution method to treat integrals contained in the continuous formulation of the transport equation. The equation was reformulated to be suitable for the application of stochastic methods by analytical approximation of the involved derivatives of the Wigner function. The second key theoretical result was the study and derivation of practically relevant simplifying assumptions of the transport equation for bounded quantum systems (e.g., nanoelectronic structures). These systems imply a finite coherence length, which allowed us to derive a semi-discrete Wigner equation for general EM fields. However, this introduced multi-dimensional integrals, representing a significant solution effort. Therefore, simplifying yet practical assumptions have been devised to gather experience with the developed theory. One assumption is centered on the fact that for homogeneous magnetic fields certain terms of the equation vanish, resulting in the involved integrals to be solved analytically. An additional term was identified which can be neglected for a broad class of physical conditions, further simplifying the solution effort. The third key theoretical result is the analysis of the case of linear magnetic fields and general electric fields, which led to identifying three operators in the equation. The choice of the magnitude of the magnetic gradient allows to toggle one of the operators, called quantum magnetic term, on and off. Algorithms for solving the resulting two equations were implemented. It was shown that the particle trajectories are modified by the magnetic force and that numerical concepts of the well-understood electrostatic model, i.e., generation and annihilation of signed particles, can be generalized for the quantum magnetic term. However, this results in significantly larger numbers of generated particles and the spatial non-locality of the generation process, which occurs along with the momentum non-locality. These three key theoretical results led to several simulation studies allowing to investigate interesting physical setups of EM fields. Particularly noteworthy are our studies of magneto-tunneling effects as well as so-called snake- and edge-states, which led to identifying local EM interplay effects. In particular, magnetic scenarios resulting in snake trajectories are promising for the improvement of the performance of quantum wire based devices by reducing surface roughness resistance. Additionally, a quantum interference logic device was suggested, which uses charge interference for realizing classical logic operations. The project has achieved all its goals and allowed to significantly advance the theory and solution methods, led to the development of a suitable particle model, and improved the understanding of the complicated interplay between EM fields and quantum electron transport dynamics.