Robert Kosik was born in Eisenstadt, Austria, in 1969. He studied technical mathematics at the Technische Universität Wien, where he received the degree of Diplomingenieur in 1996. In 1998 he joined the Institute for Microelectronics at the Technische Universität Wien and received his PhD degree in 2004. His scientific interests include partial differential equations for transport modeling and data analysis in reliability.
Deterministic Numerical Methods for the Wigner Equation
The Wigner function formulation of quantum mechanics is an attractive approach to quantum transport simulation because it is formally close to a classical phase space description and allows for a mixed quantum-classical description of the system. The use of Wigner functions to describe electron transport in quantum electronic devices, such as resonant tunneling diodes, was first proposed by W.R. Frensley in the late 1980s.
In spite of its importance for nanoelectronic device modeling, the equation describing the stationary Wigner function in an open quantum system is still not well understood, both analytically and numerically, especially because of its singular behavior at zero velocity. Recently, it has been shown that the traditional deterministic numerical approach for dealing with this singularity according to Frensley has severe shortcomings. The correct numerical treatment of the singularity is a prerequisite for further applications of the Wigner equation using deterministic methods.
We propose a revised approach aimed at achieving two goals. First, we explicitly include the point k = 0 in the grid and derive two equations for that point. The first one is an algebraic constraint that ensures that the solution of the Wigner equation has no singularity at k = 0. The second is a transport equation for k = 0. The resulting system, which we refer to as the constrained Wigner equation, is over-determined. Schroedinger modes with a smooth Wigner transform at k = 0 fulfill both equations. The second goal is to study the dual problem. After an inverse Fourier transform, the treatment of the singularity can be related to boundary conditions for the characteristic von Neumann equation (sigma equation). The constrained Wigner equation can be related to a sigma equation with fully homogeneous boundary conditions in the non-spatial coordinate. With these boundary conditions, the sigma equation is over-determined as well.
Our approach attacks a so far unresolved problem that is important for physics, engineering and mathematics. To test the revised approach, we simulate the current through an energy barrier, varying the barrier height while keeping the bias fixed. In Fig. 1, the constrained solution is compared with results from the quantum transmitting boundary method (QTBM). The fit with the QTBM (solid line) is excellent.
Fig. 1: Simulation of a 1D device with a single energy barrier of variable height at a fixed bias of 0.1 V. The barrier width is 3 nm, the barrier height varies from 0 to 5 eV. The red markers indicate the solution of the constrained equation, which fits well with the solution using the QTBM (solid line).