×

Time-dependent simulations of quantum waveguides using a time-splitting spectral method. (English) Zbl 1205.81091

Summary: The electron flow through quantum waveguides is modeled by the time-dependent Schrödinger equation with absorbing boundary conditions, which are realized by a negative imaginary potential. The Schrödinger equation is discretized by a time-splitting spectral method, and the quantum waveguides are fed by a mono-energetic incoming plane wave pulse. The resulting algorithm is extremely efficient due to the fast Fourier transform implementation of the spectral scheme. Numerical convergence rates for a one-dimensional scattering problem are calculated. The transmission rates of a two-dimensional T-stub quantum waveguide and a single-branch coupler are numerically computed. Moreover, the transient behavior of a three-dimensional T-stub waveguide is simulated.

MSC:

81Q37 Quantum dots, waveguides, ratchets, etc.
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
82D77 Quantum waveguides, quantum wires

Software:

PovRay; FFTW
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antoine, X.; Arnold, A.; Besse, C.; Ehrhardt, M.; Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4, 729-796 (2008) · Zbl 1364.65178
[2] Appenzeller, J.; Schroer, C.; Schäpers, T.; von der Hart, A.; Förster, A.; Lengeler, B.; Lüth, H., Electron interference in a T-shaped quantum transistor based on Schottky-gate technology, Phys. Rev. B, 53, 9959-9963 (1996)
[3] Arnold, A., Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6, 313-319 (1998)
[4] Arnold, A., Mathematical concepts of open quantum boundary conditions, Transp. Theory Stat. Phys., 30, 561-584 (2001) · Zbl 1019.81010
[5] Arnold, A.; Ehrhardt, M.; Sofronov, I., Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability, Commun. Math. Sci., 1, 501-556 (2003) · Zbl 1085.65513
[6] Arnold, A.; Schulte, M., Transparent boundary conditions for quantum-waveguide simulations, Math. Comput. Simul., 79, 898-905 (2008) · Zbl 1159.82326
[7] Bao, W.; Jin, S.; Markowich, P., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487-524 (2002) · Zbl 1006.65112
[8] Berenger, J., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 185-200 (1994) · Zbl 0814.65129
[9] Burgnies, L.; Vanbésien, O.; Lippens, D., An analysis of wave patterns in multiport quantum waveguide structures, J. Phys. D: Appl. Phys., 32, 706-712 (1999)
[10] Cheng, C.; Lee, J.-H.; Lim, K.-H.; Massoud, H.; Liu, Q.-H., 3D quantum transport solver based on perfectly matched layer and spectral element methods for the simulation of semiconductor nanodevices, J. Comput. Phys., 227, 455-471 (2007) · Zbl 1206.82087
[11] Ehrhardt, M.; Arnold, A., Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma., 6, 57-108 (2001) · Zbl 0993.65097
[12] Frigo, M.; Johnson, S., The design and implementation of FFTW3, Proc. IEEE, 93, 216-231 (2005)
[13] Ge, J.-Y.; Zhang, J., Use of negative complex potential as absorbing potential, J. Chem. Phys., 108, 1429-1433 (1998)
[14] Han, H.; Yin, D.; Huang, Z., Numerical solutions of Schrödinger equations in \(R^3\), Numer. Meth. Part. Diff. Eqs., 23, 511-533 (2007) · Zbl 1122.65088
[15] Hussain, A.; Roberts, G., Procedure for absorbing time-dependent wave functions at low kinetic energies and large bandwidths, Phys. Rev. A, 63, 012703 (2000), (11 pp.)
[16] Jahnke, T.; Lubich, C., Error bounds for exponential operator splittings, BIT, 40, 735-744 (2000) · Zbl 0972.65061
[17] Leforestier, C.; Wyatt, R., Optical potential for laser induced dissociation, J. Chem. Phys., 78, 2334-2344 (1983)
[18] Lent, C.; Kirkner, D., The quantum transmitting boundary method, J. Appl. Phys., 67, 6353-6359 (1990)
[19] Lubich, C., On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math., 67, 365-389 (1994) · Zbl 0795.65063
[20] Lubich, C.; Schädle, A., Fast convolution for non? Reflecting boundary conditions, SIAM J. Sci. Comput., 24, 161-182 (2002) · Zbl 1013.65113
[21] Mahapatra, S.; Sathyamurthy, N., Negative imaginary potentials in time-dependent molecular scattering, J. Chem. Soc., Faraday Trans., 93, 773-779 (1997)
[22] Nedjalkov, M.; Vasileska, D.; Atanassov, E.; Palankovski, V., Ultrafast Wigner transport in quantum wires, J. Comput. Electron., 6, 235-238 (2007)
[23] Neuhauser, D.; Baer, M., The time-dependent Schrödinger equation: application of absorbing boundary conditions, J. Chem. Phys., 90, 4351-4355 (1989)
[24] A. Nissen, G. Kreiss, An optimized perfectly matched layer for the Schrödinger equation, Preprint, Uppsala Universitet, Sweden, 2009.; A. Nissen, G. Kreiss, An optimized perfectly matched layer for the Schrödinger equation, Preprint, Uppsala Universitet, Sweden, 2009.
[25] Pathria, D.; Morris, J., Pseudo-spectral solution of nonlinear Schrödinger equation, J. Comput. Phys., 87, 108-125 (1990) · Zbl 0691.65090
[26] Persistence of Vision Raytracer (Version 3.6), http://www.povray.org.; Persistence of Vision Raytracer (Version 3.6), http://www.povray.org.
[27] Polizzi, E.; Ben Abdallah, N., Self-consistent three-dimensional models for quantum ballistic transport in open systems, Phys. Rev. B, 66, 245301 (2002), (9 pp.)
[28] Polizzi, E.; Ben Abdallah, N., Subband decomposition approach for the simulation of quantum electron transport in nanostructures, J. Comput. Phys., 202, 150-180 (2005) · Zbl 1056.81092
[29] A. Schädle, Numerische Behandlung transparenter Randbedingungen für die Schrödinger-Gleichung, Master Thesis, Universität Tübingen, Germany, 1998.; A. Schädle, Numerische Behandlung transparenter Randbedingungen für die Schrödinger-Gleichung, Master Thesis, Universität Tübingen, Germany, 1998.
[30] Schmidt, F.; Deuflhard, P., Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation, Comput. Math. Appl., 29, 53-76 (1995) · Zbl 0821.65078
[31] Shin, M., Three-dimensional quantum simulations of multigate nanowire field effect transistors, Math. Comput. Simul., 79, 1060-1070 (2008) · Zbl 1159.82333
[32] Soffer, A.; Stucchio, C., Open boundaries for the nonlinear Schrödinger equation, J. Comput. Phys., 225, 1218-1232 (2007) · Zbl 1122.65094
[33] F. Sol, M. Macucci, U. Ravaioli, K. Hess, Theory for a quantum modulated transistor, J. Appl. Phys. 66 (2989) 3892-3906.; F. Sol, M. Macucci, U. Ravaioli, K. Hess, Theory for a quantum modulated transistor, J. Appl. Phys. 66 (2989) 3892-3906.
[34] Svizhenko, A.; Anantram, M.; Govindan, T.; Biegel, B.; Venugopal, R., Two-dimensional quantum mechanical modeling of nanotransistors, J. Appl. Phys., 91, 2343-2354 (2002)
[35] Thean, A.; Leburton, J.-P., Stark effect and single-electron charging in silicon nanocrystal quantum dots, J. Appl. Phys., 89, 2808-2815 (2001)
[36] Tsuchiya, H.; Ogawa, M.; Miyoshi, T., Wigner function formulation of quantum transport in electron waveguides and its application, Jpn. J. Appl. Phys., 30, 3853-3858 (1991)
[37] Tsukada, N.; Wieck, A. D.; Ploog, K., Proposal of novel electron wave coupled devices, Appl. Phys. Lett., 56, 2527-2529 (1990)
[38] Venugopal, R.; Ran, Z.; Datta, S.; Lundstrom, M.; Jovanovic, D., Simulating quantum transport in nanoscale transistors: Real versus mode-space approaches, J. Appl. Phys., 92, 3730-3739 (2002)
[39] Zheng, C., A perfectly matched layer approach to the nonlinear Schrödinger wave equation, J. Comput. Phys., 227, 537-556 (2007) · Zbl 1127.65078
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.