Ali Faraj

Naoufel Ben Abdallah, Ali Faraj

The simulation of the time-dependent evolution of the resonant tunneling diode is done by a multiscale algorithm exploiting the existence of resonant states. After revisiting and improving the algorithm developed in [N. Ben Abdallah, O. Pinaud, J. Comp. Phys. 213 (2006) 288-310] for the stationary case, the time-dependent problem is dealt with. The wave function is decomposed into a non resonant part and a resonant one. The projection method to compute the resonant part leads to an accurate algorithm thanks to a suitable interpolation of the non resonant one. The simulation times are largely reduced.

Ali Faraj, Shi Jin

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)].