Adaptive Conservative Cell Average Spectral Element Methods for Transient Wigner Equation in Quantum Transport

Abstract

A new adaptive cell average spectral element method (SEM) is proposed to solve the time-dependent Wigner equation for transport in quantum devices. The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of Wigner functions while preserving exactly the mass conservation for the numerical solutions. The key feature of the proposed method is an analytical relation between the cell averages of the Wigner function in the k-space (local electron density for finite range velocity) and the point values of the distribution, resulting in fast transforms between the local electron density and local fluxes of the discretized Wigner equation via the fast sine and cosine transforms. Numerical results with the proposed method are provided to demonstrate its high accuracy, conservation, convergence and a reduction of the cost using adaptive meshes.

A benchmark study of the Signed-particle Monte Carlo algorithm for the Wigner equation

“…Much work has been devoted to the direct solution based on finite-difference scheme [3,4]. Many difficulties of the direct solution comes from the discretization of the diffusion term k · ∇ x f w because of the typically rapid variation in the phase space.…”

“…However, this equation has represented a numerically daunting task and it has raised more problems than solutions. A numerical treatment of the Wigner equation can be dealt with deterministic schemes [1][2][3][4], Direct Simulation Monte Carlo [5][6][7][8], and it is often used to derive reduced transport models, such as quantum-hydrodynamic models [9][10][11][12].…”

Effect of boundary treatments on quantum transport current in the Green’s function and Wigner distribution methods for a nano-scale DG-MOSFET

“…(42) is only needed for jqj < Lq 2 [16]. Now, as V w ðx; q À q 0 Þ is calculated by a numerical quadrature in the r variable (see (57) below), Eq.…”

Stability and accuracy of a pseudospectral scheme for the Wigner function equation

“…The evolution of the WF is governed by the WF equation, also known as the quantum Liouville equation. This equation appears in many fields of application including optics , signal processing and pattern recognition , quantum transport , and quantum tunneling phenomena .…”

“…For quantum transport problems Monte Carlo methods have been considered in . Further, cell‐average spectral element schemes are discussed in , and finite‐element WENO schemes are considered in . In a finite‐difference spectral collocation method is analyzed for simulation of quantum tunneling phenomena.…”