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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:ali.faraj@esisar.grenoble-inp.fr (A. Faraj), jin@math.wisc.edu (S. Jin)
*Corresponding author.Email addresses:ali.faraj@esisar.grenoble-inp.fr (A. Faraj), jin@math.wisc.edu (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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