An effective mass theorem for the bidimensional electron gas in a strong magnetic field

TL;DR

This paper derives a 2D effective Schr"odinger system for a strongly confined electron gas under a magnetic field, showing how the magnetic field modifies the electron mass through averaging cyclotron motion effects.

Contribution

It introduces a novel averaging approach for PDEs to analyze the limiting behavior of a confined electron gas in a magnetic field, leading to an effective mass theorem.

Findings

Derivation of a 2D Schr"odinger system with an effective potential

Modification of electron mass due to magnetic field effects

Application of long-time averaging theory to PDEs

Abstract

We study the limiting behavior of a singularly perturbed Schr\"odinger-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 Schr\"odinger 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.