Bound states of a two-dimensional electron gas in inhomogeneous magnetic fields

TL;DR

This paper derives an exact quantization formula for bound states of a 2D electron gas in inhomogeneous magnetic fields, providing new analytical solutions for specific magnetic field configurations and analyzing the effects of fringing fields.

Contribution

It introduces an analytical transfer matrix method to exactly determine Landau levels in inhomogeneous magnetic fields, including previously unexplored scenarios.

Findings

Exact Landau levels for symmetric power law magnetic fields

Analytical solutions for 2DEG under ferromagnetic films

Discussion of fringing magnetic field effects on energy levels

Abstract

We study the bound states of a two dimensional free electron gas (2DEG) subjected to a perpendicular inhomogeneous magnetic field. An analytical transfer matrix (ATM) based exact quantization formula is derived for magnetic fields that vary (arbitrarily) along one spatial direction. As illustrative examples, we consider (1) a class of symmetric power law magnetic fields confined within a strip, followed by the problem of a (2) 2DEG placed under a thin ferromagnetic film, which are hitherto unexplored. The exact Landau levels for either cases are obtained. Also, the role of the fringing magnetic field (present in the second example) on these levels is discussed.