Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-03T23:22:04.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Discontinuous Galerkin Methods for the 2D Simulation of Quantum Transport Phenomena on Quantum Directional Coupler

Published online by Cambridge University Press:  03 June 2015

Li Guo  and
Yan Xu
Li Guo*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
Yan Xu*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
*
Email:lili2010@mail.ustc.edu.cn
Corresponding author.Email:yxu@ustc.edu.cn
Get access

Abstract

In this paper, we present local discontinuous Galerkin methods (LDG) to simulate an important application of the 2D stationary Schrödinger equation called quantum transport phenomena on a typical quantum directional coupler, which frequency change mainly reflects in y-direction. We present the minimal dissipation LDG (MD-LDG) method with polynomial basis functions for the 2D stationary Schrödinger equation which can describe quantum transport phenomena. We also give the MD-LDG method with polynomial basis functions in x-direction and exponential basis functions in y-direction for the 2D stationary Schrödinger equation to reduce the computational cost. The numerical results are shown to demonstrate the accuracy and capability of these methods.

Keywords

65N3035Q40Local discontinuous Galerkin method2D stationary Schrödinger equationquantum transport phenomenaquantum directional coupler
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1012 - 1028
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ben, N. Abdallah and Wu, H., A generalized stationary algorithm for resonant tunneling: multi-mode approximation and high dimension, Communications in Computational Physics, 10(2011), no. 4, 882–898.Google Scholar
[2]Antoine, X., Besse, C., Ehrhardt, M. and Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Communications in Computational Physics, 4(2008), no. 4, 729–796.Google Scholar
[3]Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Scientific Computing, 39(2002), no. 5, 1749–1779.Google Scholar
[4]Bao, W.Z., Jin, S. and Markowich, P.A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, Journal of Computational Physics, 175(2002), 487-524.Google Scholar
[5]Bao, W.Z., Jin, S. and Markowich, P.A., Numerical studies of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regime, SIAM Journal on Scientific Computing, 25(2003), no. 1, 27–64.Google Scholar
[6]Cockburn, B. and Dong, B., An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, Journal of Scientific Computing, 32(2007), no. 2, 233–262.CrossRefGoogle Scholar
[7]Fan, K., Cai, W. and Ji, X., A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions, Journal of Computational Physics, 227(2008), no. 4, 2387–2410.Google Scholar
[8]Gagelman, J. and Yserentant, H., A spectral method for Schrödinger equations with smooth confinement potentials, Numerische Mathematik, 122(2012), no. 2, 383–398.Google Scholar
[9]Jiang, H.Y., Shao, S.H., Cai, W. and Zhang, P.W., Boundary treatments in non-equilibrium Green’s function (NEGF) methods for quantum transport in nano-MOSFETs, Journal of Computational Physics, 227(2008), no. 13, 6553–6573.Google Scholar
[10]Kong, L.H., Duan, Y.L., Wang, L., Yin, X.L. and Ma, Y.P., Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations, Mathmatical and Computer Modelling, 55(2012), no. 5-6, 1798–1812.Google Scholar
[11]Li, X., Zhang, L.M. and Wang, S.S., A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator, Applied Mathematics and Computation, 219(2012), no. 6, 3187–3197.CrossRefGoogle Scholar
[12]Lu, T. and Cai, W., A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schröinger-Poisson equations with discontinuous potentials, Journal of Computational and Applied Mathematics, 220(2008), no. 1-2, 588–614.Google Scholar
[13]Lu, T., Cai, W. and Zhang, P.W., Conservative local discontinuous Galerkin methods for time dependent Schrödinger equation, International Journal of Numerical Analysis and Modeling, 2(2005), no. 1, 75–84.Google Scholar
[14]Markowich, P.A., Pietra, P. and Pohl, C., Numerical approximation of quadratic obaservables of Schrödinger-type equations in the semiclassical limit, Numerische Mathematik, 81(1999), no. 4, 595–630.Google Scholar
[15]Markowich, P.A., Pietra, P., Pohl, C. and Stimming, H.P., A wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM Journal on Numerical Analysis, 40(2002), no. 4, 1281–1310.Google Scholar
[16]Polizzi, E. and Ben Abdallah, N., Subband decomposition approach for the simulation of quantum electron transport in nanostructures, Journal of Computational Physics, 202(2005), 150–180.Google Scholar
[17]Shao, S.H., Lu, T. and Cai, W., Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Communications in Computational Physics, 10(2011), no. 4, 882–898.Google Scholar
[18]Soba, A., A finite element method solver for time-dependent and stationary Schrödinger equations with a generic potential, Communications in Computational Physics, 9(2011), no. 3, 711–739.Google Scholar
[19]Wang, W. and Shu, C.W., The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling Diode, Journal of Scientific Computing, 40(2009), 360–374.Google Scholar
[20]Wu, H., Domain and subband decomposition approach for the 2D simulation of quantum transport phenomena, preprint.Google Scholar
[21]Wu, H., High order scheme for Schrödinger equation with discontinuous potential I: immersed interface method, Numerical Mathematics: Theory, Methods and Applications, 4(2011), no. 4, 576–597.Google Scholar
[22]Xu, Y. and Shu, C.W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, Journal of Computational Physics, 205(2005), no. 1, 72–97.Google Scholar
[23]Yuan, L. and Shu, C.W., Discontinuous Galerkin method based on non-polynomial approximation spaces, Journal of Computational Physics, 218(2006), no. 1, 295–323.Google Scholar