The Landau-Zener transition and the surface hopping method for the 2D Dirac equation for graphene
This work addresses the need for efficient semiclassical simulation of quantum transport in graphene, but the results are incremental as they primarily validate and compare existing methods.
The authors implemented a Lagrangian surface hopping algorithm for the 2D massless Dirac equation in graphene, incorporating the Landau-Zener transition probability at Dirac points. Numerical experiments compared the algorithm's solutions to the full Dirac equation and asymptotic models, revealing discrepancies in transition probabilities among different models.
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. Sch{ü}rrer, J. Phys. A: Math. Theor. 44 (2011)] 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. Sch{ü}rrer, J. Phys. A: Math. Theor. 44 (2011)].