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The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene

Part of: Equations of mathematical physics and other areas of application Quantum theory Applications to specific physical systems Approximation methods and numerical treatment of dynamical systems General mathematical topics and methods in quantum theory Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  07 February 2017

Ali Faraj  and
Shi Jin
Ali Faraj*
Affiliation:
Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China Grenoble INP, ESISAR, 26902 Valence Cedex 9, France
Shi Jin*
Affiliation:
Department of Mathematics, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, P.R. China Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
*
*Corresponding author.Email addresses:[email protected] (A. Faraj), [email protected] (S. Jin)
*Corresponding author.Email addresses:[email protected] (A. Faraj), [email protected] (S. Jin)
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Abstract

A Lagrangian surface hopping algorithm is implemented to study the two dimensional massless Dirac equation for Graphene with an electrostatic potential, in the semiclassical regime. In this problem, the crossing of the energy levels of the system at Dirac points requires a particular treatment in the algorithm in order to describe the quantum transition—characterized by the Landau-Zener probability— between different energy levels. We first derive the Landau-Zener probability for the underlying problem, then incorporate it into the surface hopping algorithm. We also show that different asymptotic models for this problem derived in [O. Morandi, F. Schurrer, J. Phys. A:Math. Theor. 44 (2011) 265301]may give different transition probabilities. We conduct numerical experiments to compare the solutions to the Dirac equation, the surface hopping algorithm, and the asymptotic models of [O. Morandi, F. Schurrer, J. Phys. A: Math. Theor. 44 (2011) 265301].

Keywords

Dirac equationWigner transformsemiclassicalmodelband crossingLandau-Zener formulasurface hopping algorithmspectral methods

MSC classification

Secondary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 37M05: Simulation 65M70: Spectral, collocation and related methods 65Z05: Applications to physics 81-08: Computational methods 81Q20: Semiclassical techniques, including WKB and Maslov methods 81V99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 313 - 357
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Athanassoulis, A., Katsaounis, T., Kyza, I., Regularized semiclassical limits: linear flows with infinite Lyapunov exponents, Commun. Math. Sci. 14 (2016) 1821–1858.Google Scholar
[2] Beenakker, C. W., Colloquium: Andreev reflection and Klein tunneling in graphene, Rev. Mod. Phys. 80 (2008) 1337.Google Scholar
[3] Bonnaillie-Noël, V., Faraj, A., Nier, F., Simulation of resonant tunneling heterostructures: numerical comparison of a complete Schrödinger-Poisson system and a reduced nonlinear model, J. Comput. Electron. 8, 1 (2009) 11–18.Google Scholar
[4] Brinkman, D., Heitzinger, C., Markowich, P. A., A convergent 2D finite-difference scheme for the Dirac-Poisson system with magnetic potential and the simulation of graphene, J. Comput. Phys. 257, A (2014) 318–332.Google Scholar
[5] Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S., Geim, A. K., The electronic properties of graphene, Rev. Mod. Phys. 81, 1 (2009) 109–162.Google Scholar
[6] Chai, L., Jin, S., Li, Q., Morandi, O., A Multiband Semiclassical Model for Surface Hopping Quantum Dynamics, SIAM Multiscale Model. Simul. 13, 1 (2015) 205–230.Google Scholar
[7] Drukker, K., Basics of surface hopping in mixed quantum/classical simulations, J. Comp. Phys. 153 (1999) 225–272.Google Scholar
[8] Faou, E., Gauckler, L. and Lubich, C., Plane Wave Stability of the split-step Fourier method for the nonlinear Schrödinger equation, Forum Math. Sigma 2, e5 (2014) 45 pp.Google Scholar
[9] Fermanian Kammerer, C., Gérard, P., A Landau-Zener formula for non-degenerated involutive codimension three crossings, Ann. Henri Poincaré 4 (2003) 513–552.Google Scholar
[10] Fermanian Kammerer, C., Gérard, P., Mesures semi-classiques et croisements de mode, Bull. S.M.F 130, 1 (2002) 123–168.Google Scholar
[11] Fermanian Kammerer, C., Méhats, F., A kinetic model for the transport of electrons in a graphene layer, hal-01160791.Google Scholar
[12] Fiori, G., Iannaccone, G., Simulation of Graphene Nanoribbon Field-Effect Transistors, IEEE Electron. Device Lett. 28, 8 (2007) 760–762.Google Scholar
[13] Gérard, P., Markowich, P. A., Mauser, N. J., Poupaud, F., Homogenization Limits and Wigner Transforms, Comm. Pure Appl. Math. 50, 4 (1997) 323–379.Google Scholar
[14] Grecchi, V., Martinez, A., Sacchetti, A., Destruction of the beating effect for a non-linear Schrödinger equation, Commun. Math. Phys. 227, 1 (2002) 191–209.Google Scholar
[15] Hagedorn, G. A., Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps, Commun. Math. Phys. 136 (1991) 433–449.Google Scholar
[16] Hammer, R., Pötz, W., Arnold, A., Single-cone real-space finite difference scheme for the time-dependent Dirac equation, J. Comput. Phys. 265 (2014) 50–70.Google Scholar
[17] Hörmander, L., The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] No. 274, Springer-Verlag, Berlin, 1985.Google Scholar
[18] Huang, Z., Jin, S., Markowich, P. A., Sparber, C., Zheng, C., A time-splitting spectral scheme for the Maxwell-Dirac system, J. Comput. Phys. 208 (2005) 761–789.CrossRefGoogle Scholar
[19] Jin, S., Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 122 (1995) 51–67.Google Scholar
[20] Jin, S., Markowich, P. A., Sparber, C., Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numerica 20 (2011) 121–209.Google Scholar
[21] Jin, S., Novak, K., Acoherent semiclassical transport model for pure-state quantum scattering, Comm. Math. Sci. 8 (2010) 253–275.Google Scholar
[22] Jin, S., Qi, P., A Hybrid Schrödinger/Gaussian Beam Solver for Quantum Barriers and Surface Hopping, Kinetic and Related Models 4 (2011) 1097–1120.Google Scholar
[23] Jin, S., Qi, P., Zhang, Z., An Eulerian surface hopping method for the Schrödinger equation with conical crossings, Multiscale Model. Simul. 9, 1 (2011) 258–281.Google Scholar
[24] Lasser, C., Swart, T., Teufel, S., A rigorous surface hopping algorithm for conical crossings, Commun. Math. Sc. 5, 4 (2007) 789–814.Google Scholar
[25] Lasser, C., Teufel, S., Propagation through conical crossings: an asymptotic semigroup, Comm. Pure Appl. Math. 58, 9 (2005) 1188–1230.CrossRefGoogle Scholar
[26] Lemme, M. C., Echtermeyer, T. J., Baus, M., Kurz, H., A Graphene Field-Effect Device, IEEE Electron. Device Lett. 28, 4 (2007) 282–284.Google Scholar
[27] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.Google Scholar
[28] Morandi, O., Multiband Wigner-function formalism applied to the Zener band transition in a semiconductor, Physical Review B 80, 2 (2009) 024301–024312.Google Scholar
[29] Morandi, O., Schürrer, F., Wigner model for Klein tunneling in graphene, Communications in Applied and Industrial Mathematics 2, 1 (2011).Google Scholar
[30] Morandi, O., Schürrer, F., Wigner model for quantum transport in graphene, J. Phys. A:Math. Theor. 44 (2011) 265301.Google Scholar
[31] Novikov, D. S., Elastic scattering theory and transport in graphene, Phys. Rev. B 76 (2007) 245435.Google Scholar
[32] Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Zhang, Y., Dubonos, S. V., Gregorieva, I. V., Firsov, A. A., Electric Field Effect in Atomically Thin Carbon Films, Science 306, 5696 (2004) 666–669.Google Scholar
[33] Pareschi, L., Central Differencing Based Numerical Schemes for Hyperbolic Conservation Laws with Relaxation Terms, SIAM J. Numer. Anal. 39, 4 (2001) 1395–1417.Google Scholar
[34] Rathe, U.W., Keitel, C. H., Protopapas, M., Knight, P. L., Intense laser-atom dynamics with the two-dimensional Dirac equation, J. Phys. B 30, 15 (1997) L531-L539.Google Scholar
[35] Sholl, D. and Tully, J., A generalized surface hopping method, J. Chem. Phys. 109 (1998) 7702–7710.Google Scholar
[36] Sparber, C., Markowich, P. A., Semiclassical asymptotics for the Maxwell-Dirac system, J. Math. Phys. 44, 10 (2003) 4555–4572.Google Scholar
[37] Thaller, B., The Dirac Equation, Springer, New York, 1992.Google Scholar
[38] Tully, J., Molecular dynamics with electronic transitions, J. Chem. Phys. 93 (1990) 1061–1071.Google Scholar
[39] Tully, J., Preston, R., Trajectory Surface Hopping Approach to Nonadiabatic Molecular Collisions: The Reaction of H + with D 2 , J. Chem. Phys. 55 (1971) 562–572.CrossRefGoogle Scholar
[40] Zener, C., Non-adiabatic crossing of energy levels, Proc. Royal. Soc. London Ser. A 137 (1932) 696–702.Google Scholar