The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene (Communications in Computational Physics, 2018)
A Lagrangian surface hopping algorithm is implemented to study the twodimensional massless Dirac equation for Graphene with an electrostatic potential, inthe semiclassical regime. In this problem, the crossing of the energy levels of thesystem at Dirac points requires a particular treatment in the algorithm in order todescribe the quantum transition – characterized by the Landau-Zener probability – between different energy levels. We first derive the Landau-Zener probability for theunderlying problem, then incorporate it into the surface hopping algorithm. We alsoshow 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 Diracequation, the surface hopping algorithm, and the asymptotic models of [O. Morandi,F. Schurrer, J. Phys. A: Math. Theor. 44 (2011) 265301].