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

Published online by Cambridge University Press:  20 August 2015

Sihong Shao ,
Tiao Lu  and
Wei Cai
Sihong Shao*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Tiao Lu*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Wei Cai*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
*
Email:shaosihong@gmail.com
Email:tlu@math.pku.edu.cn
Corresponding author.Email:wcai@uncc.edu
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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.

Keywords

65M7065M5042B2045K0581Q05Wigner equationquantum transportspectral element methodsadaptive mesh
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 711 - 739
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Wigner, E., On the quantum corrections for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749–759.Google Scholar
[2]Zachos, C., Deformation quantization: quantum mechanics lives and works in phase-space, Int. J. Mod. Phys. A., 17 (2002), 297–316.CrossRefGoogle Scholar
[3]Hancock, J., Walton, M. A. and Wynder, B., Quantum mechanics another way, Eur. J. Phys., 25 (2004), 525–534.Google Scholar
[4]Jacoboni, C., Brunetti, R., Bordone, P. and Bertoni, A., Quantum transport and its simulation with the Wigner-function approach, Int. J. High. Speed. Electron. Syst., 11 (2001), 387–423.Google Scholar
[5]Jacoboni, C. and Bordone, P., The Wigner-function approach to non-equilibrium electron transport, Rep. Prog. Phys., 67 (2004), 1033–1071.CrossRefGoogle Scholar
[6]Kosina, H., Wigner function approach to nano device simulation, Int. J. Comput. Sci. Eng., 2 (2006), 100–118.Google Scholar
[7]Sverdlov, V., Ungersboeck, E., Kosina, H. and Selberherr, S., Current transport models for nanoscale semiconductor devices, Mat. Sci. Eng. R., 58 (2008), 228–270.Google Scholar
[8]Frensley, W. R., Wigner-function model of a resonant-tunneling semiconductor device, Phys. Rev. B., 36 (1987), 1570–1580.Google Scholar
[9]Frensley, W. R., Boundary conditions for open quantum systems driven far from equilibrium, Rev. Mod. Phys., 62 (1990), 745–791.Google Scholar
[10]Jensen, K. L. and Buot, F. A., The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach, IEEE Trans. Electron. Devices., 38 (1991), 2337–2347.CrossRefGoogle Scholar
[11]Biegel, B. A., Quantum Electronic Device Simulation, Ph.D. Thesis, Stanford University, 1997.Google Scholar
[12]Biegel, B. A., SQUADS Technical Reference, Unpublished, Stanford University, 1996.Google Scholar
[13]Frensley, W. R., Effect of inelastic processes on the self-consistent potential in the resonant-tunneling diode, Solid. State. Electron., 32 (1989), 1235–1239.CrossRefGoogle Scholar
[14]Kluksdahl, N. C., Kriman, A. M., Ferry, D. K. and Ringhofer, C., Self-consistent study of the resonant-tunneling diode, Phys. Rev. B., 39 (1989), 7720–7735.CrossRefGoogle ScholarPubMed
[15]Zhao, P. J., Wigner-Poisson Simulation of Quantum Devices, Ph.D. Thesis, Stevens Institute of Technology, 2000.Google Scholar
[16]Ferrari, G., Effect of Contact Proximity on Quantum Transport in Mesoscopic Semiconductor Systems, Ph.D. Thesis, Universita degli studi di Modena e Reggio Emilia, 2004.Google Scholar
[17]Kosik, R., Numerical Challenges on the Road to NanoTCAD, Ph.D. Thesis, Institute for Microelectronics, TU Vienna, 2004.Google Scholar
[18]Nedjalkov, M., Kosina, H., Selberherr, S., Ringhofer, C. and Ferry, D. K., Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices, Phys. Rev. B., 70 (2004), 115319.Google Scholar
[19]Shifren, L., Ringhofer, C. and Ferry, D. K., A Wigner function-based quantum ensemble Monte Carlo study of a resonant tunneling diode, IEEE Trans. Electron. Devices., 50 (2003), 769–773.Google Scholar
[20]Querlioz, D., Saint-Martin, J., Do, V. N., Bournel, A. and Dollfus, P., A study of quantum transport in end-of-roadmap DG-MOSFETs using a fully self-consistent Wigner Monte Carlo approach, IEEE Trans. Nanotechnol., 5 (2006), 737–744.CrossRefGoogle Scholar
[21]Ringhofer, C., A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM J. Numer. Anal., 27 (1990), 32–50.Google Scholar
[22]Ringhofer, C., A spectral collocation technique for the solution of the Wigner-Poisson problem, SIAM J. Numer. Anal., 29 (1992), 679–700.Google Scholar
[23]Arnold, A. and Ringhofer, C., Operator splitting methods applied to spectral discretizations of quantum transport equations, SIAM J. Numer. Anal., 32 (1995), 1876–1894.Google Scholar
[24]Arnold, A. and Ringhofer, C., A operator splitting method for the Wigner-Poisson problem, SIAM J. Numer. Anal., 33 (1996), 1622–1643.Google Scholar
[25]Goudon, T., Analysis of a semidiscrete versionof the Wigner equation, SIAM J. Numer. Anal., 40 (2002), 2007–2025.Google Scholar
[26]Goudon, T. and Lohrengel, S., On a discrete model for quantum transport in semi-conductor devices, Transport. Theor. Statist. Phys., 31 (2002), 471–490.CrossRefGoogle Scholar
[27]Cai, W., Gottlieb, D. and Shu, C. W., Essentially nonoscillatory spectral Fourier methods for shock wave calculations, Math. Comput., 52 (1989), 389–410.Google Scholar
[28]Cai, W., Gottlieb, D. and Harten, A., Cell averaging Chebyshev methods for hyperbolic problems, Comput. Math. Applic., 24 (1992), 37–49.CrossRefGoogle Scholar
[29]Gottlieb, S. and Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math. Comput. 67 (1998), 73–85.Google Scholar
[30]Pathria, R. K., Statistical Mechanics, 2nd Edition, Butterworth-Heinemann, Oxford, 1996.Google Scholar
[31]Markowich, P. A., Ringhofer, C. A. and Schmeiser, C., Semiconductor Equations, Springer-Verlag, Wien-New York, 1990.Google Scholar
[32]Boyd, J. P., Chebyshev and Fourier Spectral Methods, 2nd Edition, Dover, New York, 2001.Google Scholar
[33]Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd Edition, Cambridge University Press, Cambridge, 1992.Google Scholar
[34]Swarztrauber, P. N., FFTPACK (version 4), 1985, http://www.netlib.org/fftpack/.Google Scholar
[35]Demeio, L., Splitting-scheme solution of the collisionless Wigner equation with non-parabolic band profile, J. Comput. Electron., 2 (2003), 313–316.Google Scholar
[36]Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th Edition, Dover, New York, 1972.Google Scholar
[37]Kythe, P. K. and Schäferkotter, M. R., Handbook of Computational Methods for Integration, Chapman & Hall/CRC, New York, 2005.Google Scholar