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Volume 21, Issue 2
The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene

Ali Faraj & Shi Jin

DOI: 10.4208/cicp.020515.250716a

Commun. Comput. Phys., 21 (2017), pp. 313-357.

Published online: 2018-04

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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].

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@Article{CiCP-21-313, author = {Ali Faraj and Shi Jin}, title = {The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene}, journal = {Communications in Computational Physics}, year = {2018}, volume = {21}, number = {2}, pages = {313--357}, 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].

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.020515.250716a}, url = {http://global-sci.org/intro/article_detail/cicp/11241.html} }
TY - JOUR T1 - The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene AU - Ali Faraj & Shi Jin JO - Communications in Computational Physics VL - 2 SP - 313 EP - 357 PY - 2018 DA - 2018/04 SN - 21 DO - http://doi.org/10.4208/cicp.020515.250716a UR - https://global-sci.org/intro/article_detail/cicp/11241.html KW - AB -

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].

Ali Faraj and Shi Jin. (2018). The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene. Communications in Computational Physics. 21 (2). 313-357. doi:10.4208/cicp.020515.250716a
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