The Landau-Zener transition and the surface hopping method for the 2D Dirac equation for graphene

TL;DR

This paper develops a surface hopping algorithm incorporating Landau-Zener probabilities to simulate quantum transitions in the 2D Dirac equation for graphene, validated through numerical comparisons with direct solutions and asymptotic models.

Contribution

It introduces a novel surface hopping method tailored for the 2D Dirac equation in graphene, accounting for energy level crossings at Dirac points.

Findings

The derived Landau-Zener probability accurately predicts quantum transitions.

Numerical experiments show the method's effectiveness compared to direct Dirac solutions.

Different asymptotic models yield varying transition probabilities.

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. Sch{\"u}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{\"u}rrer, J. Phys. A: Math. Theor. 44 (2011)].