An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

Abstract

Abstract. We study the limiting behavior of a singularly perturbed Schrödinger-Poisson system describing a 3-dimensional electron gas strongly confined in the vicinity of a plane (x, y) and subject to a strong uniform magnetic field in the plane of the gas. The coupled effects of the confinement and of the magnetic field induce fast oscillations in time that need to be averaged out. We obtain at the limit a system of 2-dimensional Schrödinger equations in the plane (x, y), coupled through an effective selfconsistent electrical potential. In the direction perpendicular to the magnetic field, the electron mass is modified by the field, as the result of an averaging of the cyclotron motion. The main tools of the analysis are the adaptation of the second order long-time averaging theory of ODEs to our PDEs context, and the use of a Sobolev scale adapted to the confinement operator.

The Schrödinger–Poisson System on the Sphere

“…Strongly confined Schrödinger-Poisson systems have previously been studied in Euclidean spaces: in [5,16] for the confinement on a plane and in [3] for the confinement on an axis. The idea here is to investigate the influence of the geometry on the confinement procedure.…”

“…The square of ε also denotes the ratio between the kinetic energy of the particles and the confinement energy. One refers to [5] and [16], where similar singularly perturbed systems were derived and put in dimensionless form, starting from systems in physical variables. Our aim is to derive a simpler asymptotic system satisfied by u ε as ε → 0.…”

“…solves the system 16) which is a "mixed-state" version of the scalar equation (1.2). Indeed, let us decompose the function ω ε on the eigenvectors of the confinement operator H r , setting…”

An Asymptotic Model for the Transport of an Electron Gas in a Slab

“…3 and 21 for a model where no Dirichlet boundary condition is imposed and to Ref. 13 where a magnetic potential is added. The main idea is to introduce two characteristic energies E transp and E conf .…”

“…13, the authors give an asymptotic model for the Schr€ odingerÀPoisson system describing a three-dimensional electron gas con¯ned on a plane and subject to a strong uniform magnetic¯eld lying in the transport plane. In order to deal with the fast oscillations due to the magnetic potential, they use second-order long-time averaging techniques and a Sobolev scale adapted to the con¯nement operator.…”