@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 = {<p style="text-align: justify;">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<span style="text-align: justify;">&nbsp;–&nbsp;</span>characterized by the Landau-Zener probability<span style="text-align: justify;">&nbsp;–&nbsp;</span>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].</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.020515.250716a},
url = {http://global-sci.org/intro/article_detail/cicp/11241.html}
}