A kinetic model for the transport of electrons in a graphene layer

Abstract

Abstract. In this article, we propose a new numerical model for computation of the transport of electrons in a graphene device. The underlying quantum model for graphene is a massless Dirac equation, whose eigenvalues display a conical singularity responsible for non adiabatic transitions between the two modes. We first derive a kinetic model which takes the form of two Boltzmann equations coupled by a collision operator modeling the non-adiabatic transitions. This collision term includes a Landau-Zener transfer term and a jump operator whose presence is essential in order to ensure a good energy conservation during the transitions. We propose an algorithmic realization of the semi-group solving the kinetic model, by a particle method. We give analytic justification of the model and propose a series of numerical experiments studying the influences of the various sources of errors between the quantum and the kinetic models.

Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing

“…Adapting his methods to the present context is the subject of ongoing work. A model of the dynamics at a 'conical' codimension 2 Bloch band degeneracy was derived in [23].…”

Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

“…A self-contained discussion of Wigner measures, including the sense of convergence and the systematic extraction of observables, can be found in Section 3. This technique has been established for a wide variety of wave problems, including Schrödinger [2,3,4,5,7,8,9,10,11,12,26,28,29,38,41,48], Dirac [22,23], and acoustic [9,40], elastic and Maxwell equations with smooth, random or periodic coefficients [13,26,42].…”

“…While for many classes of problems the Wigner measures approach is worked out, key questions are still open in many interesting problems, such as systems with eigenvalue crossings [22,23], nonsmooth [2,3,4,5], and nonlinear problems. In nonlinear problems in particular, the limit Vlasov-type equation (5) is typically not well-posed for measures.…”

“…is to describe the evolution of macroscopic observables, such as mass mpx, tq " |ψ ε px, tq| 2 , momentum jpk, tq " ε n | p ψ ε pεk, tq| 2 , kinetic energy E kin px, tq " |∇ψ ε px, tq| 2 (2) * agis.athanassoulis@le.ac.uk, Department of Mathematics, University of Leicester, UK when ε Ñ 0. Variations of this problem arises in many different physical contexts, including quantum molecular dynamics [2], mean field limits [12,28,29], wave propagation over large (geophysical) distances [42,44], the formation of rogue waves [21] and the study of graphene [22,23]. We will use the term semiclassical to describe this regime [15,31,32]; other terms used in the literature are zero-dispersion limit [46], high frequency limit [26], and geometric optics [16,17].…”

“…In other words, it was non known heretofore whether we can have any control of the observables in the problem described by the scaling (23) for wavepacket initial data; not even for coherent state initial data. To answer this question one observes that Theorem 2.3 implies the following Corollary 2.4 (Squeezed states for defocusing power nonlinearities).…”