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

Published online by Cambridge University Press:  07 February 2017

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].

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

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