The Landau-Zener Transition and the Surface Hopping Method for the 2D Dirac Equation for Graphene. (7th February 2017)

• Record Type: Journal Article
• Authors: Faraj, Ali; Jin, Shi
• Is Part Of: Communications in computational physics. Volume 21:Number 2(2017:Feb.)
• Page Start: 313
• Page End: 357
• Publication Date: 2017-02-07
• Subjects: 35Q41, -- 37M05, -- 65M70, -- 65Z05, -- 81-08, -- 81Q20, -- 81V99; Dirac equation, -- Wigner transform, -- semiclassicalmodel, -- band crossing, -- Landau-Zener formula, -- surface hopping algorithm, -- spectral methods; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.020515.250716a
• Languages: English
• ISSNs: 1815-2406

Abstract

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].