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.
Numerical Comparison of the Sigma Equation and the Wigner Equation
The ability to simulate electron transport in a semiconductor nanostructure constitutes a fundamental requirement for nanoelectronics research. Quantum transport models are indispensable to appropriately model modern devices.
A full quantum description is usually based either on the Liouville-von Neumann equation for the density matrix ρ(x1,x2) or on the Wigner equation for the Wigner function w(r,k). There is a third option based on sigma functions, σ(r,s). The sigma function is intermediate between the density matrix and the Wigner function. Introducing characteristic coordinates (r,s), the stationary Liouville-von Neumann equation takes on its characteristic form, which we call the sigma equation. Fourier transformation of σ(r,s) with respect to s then gives the Wigner function w(r,k).
To study the sigma equation, we compare it to Frensley's finite difference Wigner method. For both equations, we have developed a shooting method. No system matrix needs to be stored, and the shooting method can easily be parallelized. All operators in the sigma equation can be sparsely discretized. We impose anti-periodic boundary conditions in coordinate s, which results in an explicit discretization. In contrast, the Wigner equation is not sparse in coordinate k, and the shooting is much slower.
Fig. 1 shows simulation results for a resonant tunneling diode using a fixed coherence length of 30 nm. The upper solid red line in Fig. 1 is the solution from the sigma equation with the number of meshing points Nr = 1600 and Ns = 641. This solution is numerically accurate, it does not noticeably change if the r-mesh is refined. For the Wigner equation, we always use Nk = 640 points in k and only vary the r-mesh. The dashed lines are I-V curves from Frensley's discretization with Nr = 800 (orange), Nr = 1600 (blue), all the way up to Nr = 102,400 (cyan). If we refine the r-mesh, then the current increases, and the solution from the Wigner equation slowly approaches the solution from the sigma equation. However, the current in the numerically accurate result is unphysically high. And only for a coarse r-mesh do we get the typical negative differential resistance expected for resonant tunneling.
Frensley's discretization uses upwinding for the spatial first order derivative. Upwinding on a coarse mesh introduces a lot of artificial diffusion. This actually results in a very large numerical error. But - at least in this application - the introduced artificial diffusion seems to benefit the physics.
It appears that for a physically acceptable solution, the use of Frensley's discretization on a coarse mesh is advised. Future work will attempt to get a better understanding of the observed breakdown of the stationary Wigner equation in the semi-discrete limit (fixed coherence length, very fine r-mesh).
Fig. 1: Numerical convergence of the solution from the Wigner equation using Frensley's discretization to the solution of the sigma equation in the limit of a very fine r-mesh.