数值求解含时Wigner方程的一种高阶算法
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摘要
基于有限差分方法与谱方法,结合显式格式和隐式格式的特点,针对含时Wigner方程设计了一种高阶数值求解算法.并且应用此数值算法模拟了Gauss波包的散射效应.分别设计了单势垒与双势垒对Gauss波包的散射实验,考察了势垒高度和宽度对散射现象的影响以及双势垒高度与Gauss波包半衰期的关系.
Based on the finite difference method and the spectral method, a high-order scheme for the time-dependent Wigner equation is designed, which takes the advantages of the explicit scheme and the implicit scheme. The scattering effects of one single barrier and a double barrier to a Gaussian wave packet are investigated using the proposed scheme. The effects of the barrier height and width to the scattering phenomenon, and the effect of the double barrier height to the half-life of a Gaussian wave packet are investigated by a series of numerical tests.
Based on the finite difference method and the spectral method, a high-order scheme for the time-dependent Wigner equation is designed, which takes the advantages of the explicit scheme and the implicit scheme. The scattering effects of one single barrier and a double barrier to a Gaussian wave packet are investigated using the proposed scheme. The effects of the barrier height and width to the scattering phenomenon, and the effect of the double barrier height to the half-life of a Gaussian wave packet are investigated by a series of numerical tests.
引文
[1] Wigner E. On the quantum correction for thermodynamic equilibrium[J]. Phys. Rev., 1932, 40(5):749-750.
[2] Ferry D K and Goodnick S M. Transport in Nanostructures. Cambridge Univ. Press, Cambridge,U.K, 1997.
[3] Frensley W R. Wigner function model of a resonant-tunneling semiconductor device[J]. Phys.Rev. B, 1987, 36:1570-1580.
[4] Schleich W P. Quantum Optics in Phase Space. Wiley, England, 2001.
[5] Jensen K L and Buot F A. Numerical simulation of transient response and resonant-tunneling characteristics of double-barrier semiconductor structures as a function of experimental parameters[J]. J. Appl. Phys., 1989, 65(12):5248-5250.
[6] Jensen K L and Buot F A. Numerical aspects on the simulation of I-V characteristics and switching times of resonant tunneling diodes[J]. J. Appl. Phys., 1990, 67:2153-2155.
[7] Dorda A and Schiirrer F. A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes[J]. J. Comput.Phys., 2015, 284:95-116.
[8] Ringhofer C. A spectral method for the numerical solution of quantum tunneling phenomena[J].SIAM J. Num. Anal., 1990, 27:32-50.
[9] Ringhofer C. A spectral collocation technique for the solution of the Wigner-Poisson problem[J].SIAM Journal on Numerical Analysis, 1992, 29(3):679-700.
[10] Yin D, Tang M, and Jin S. The Gaussian beam method for the Wigner equation with discontinuous potentials[J]. Inverse Problems&Imaging, 2013, 7(3).
[11] Cai Z, Fan Y, Li R, Lu T and Wang Y. Quantum hydrodynamics models by moment closure of Wigner equation[J]. J. Math. Phys., 2012, 53:103503.
[12] Li R, Lu T, Wang Y and Yao W. Numerical method for high order hyperbolic moment system of Wigner equation[J]. Commun. Comput. Phys., 2014, 9(3):659-698.
[13] Markowich P A and Ringhofer C. An analysis of the quantum liouville equation[J]. Z. angew.Math. Mech., 1989, 69:121-127.
[14] Arnold A, Lange H and Zweifel P F. A discrete-velocity, stationary Wigner equation[J]. J. Math.Phys., 2000, 41(11):7167-7180.
[15] Goudon T. Analysis of a semidiscrete version of the Wigner equation[J]. SIAM J. Numerical Analysis, 2003, 40(6):2007-2025.
[16] Arnold A. Mathematical properties of quantum evolution equations. In N. Ben Abdallah and G. Frosali, editors, Quantum Transport-Modelling, Analysis and Asymptotics, Lecture Notes Math. 1946, pages 45-109. Springer, Berlin, 2008.
[17] Li R, Lu T and Sun Z P. Stationary wigner equation with inflow boundary conditions:Will a symmetric potential yield a symmetric solution?[J]. SIAM J. Appl. Math., 2014, 70(3):885-897.
[18] Li R, Lu T and Sun Z P. Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions[J]. Frontiers of Mathematics in China, 2017, 12(4):907-919.
[19] Jiang H and Cai W. Effect of boundary treatments on quantum transport current in the Green's function and Wigner distribution methods for a nano-scale DG-MOSFET[J]. J. Comput. Phys.,2010, 229(12):4461-4475.
[20] Jiang H, Cai W and Tsu R. Accuracy of the frensley inflow boundary condition for Wigner equations in simulating resonant tunneling diodes[J]. J. Comput. Phys., 2011, 230:2031-2044.
[2] Ferry D K and Goodnick S M. Transport in Nanostructures. Cambridge Univ. Press, Cambridge,U.K, 1997.
[3] Frensley W R. Wigner function model of a resonant-tunneling semiconductor device[J]. Phys.Rev. B, 1987, 36:1570-1580.
[4] Schleich W P. Quantum Optics in Phase Space. Wiley, England, 2001.
[5] Jensen K L and Buot F A. Numerical simulation of transient response and resonant-tunneling characteristics of double-barrier semiconductor structures as a function of experimental parameters[J]. J. Appl. Phys., 1989, 65(12):5248-5250.
[6] Jensen K L and Buot F A. Numerical aspects on the simulation of I-V characteristics and switching times of resonant tunneling diodes[J]. J. Appl. Phys., 1990, 67:2153-2155.
[7] Dorda A and Schiirrer F. A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes[J]. J. Comput.Phys., 2015, 284:95-116.
[8] Ringhofer C. A spectral method for the numerical solution of quantum tunneling phenomena[J].SIAM J. Num. Anal., 1990, 27:32-50.
[9] Ringhofer C. A spectral collocation technique for the solution of the Wigner-Poisson problem[J].SIAM Journal on Numerical Analysis, 1992, 29(3):679-700.
[10] Yin D, Tang M, and Jin S. The Gaussian beam method for the Wigner equation with discontinuous potentials[J]. Inverse Problems&Imaging, 2013, 7(3).
[11] Cai Z, Fan Y, Li R, Lu T and Wang Y. Quantum hydrodynamics models by moment closure of Wigner equation[J]. J. Math. Phys., 2012, 53:103503.
[12] Li R, Lu T, Wang Y and Yao W. Numerical method for high order hyperbolic moment system of Wigner equation[J]. Commun. Comput. Phys., 2014, 9(3):659-698.
[13] Markowich P A and Ringhofer C. An analysis of the quantum liouville equation[J]. Z. angew.Math. Mech., 1989, 69:121-127.
[14] Arnold A, Lange H and Zweifel P F. A discrete-velocity, stationary Wigner equation[J]. J. Math.Phys., 2000, 41(11):7167-7180.
[15] Goudon T. Analysis of a semidiscrete version of the Wigner equation[J]. SIAM J. Numerical Analysis, 2003, 40(6):2007-2025.
[16] Arnold A. Mathematical properties of quantum evolution equations. In N. Ben Abdallah and G. Frosali, editors, Quantum Transport-Modelling, Analysis and Asymptotics, Lecture Notes Math. 1946, pages 45-109. Springer, Berlin, 2008.
[17] Li R, Lu T and Sun Z P. Stationary wigner equation with inflow boundary conditions:Will a symmetric potential yield a symmetric solution?[J]. SIAM J. Appl. Math., 2014, 70(3):885-897.
[18] Li R, Lu T and Sun Z P. Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions[J]. Frontiers of Mathematics in China, 2017, 12(4):907-919.
[19] Jiang H and Cai W. Effect of boundary treatments on quantum transport current in the Green's function and Wigner distribution methods for a nano-scale DG-MOSFET[J]. J. Comput. Phys.,2010, 229(12):4461-4475.
[20] Jiang H, Cai W and Tsu R. Accuracy of the frensley inflow boundary condition for Wigner equations in simulating resonant tunneling diodes[J]. J. Comput. Phys., 2011, 230:2031-2044.