Landau levels on a surface of weak topological insulators

TL;DR

This paper investigates how Landau levels form for Dirac electrons on the surface of weak topological insulators, revealing the effects of magnetic field strength and layered structure on their quantization.

Contribution

It provides a detailed analysis of Landau quantization in WTIs, highlighting the nontrivial modulation caused by cone mixing due to layered structure effects.

Findings

Landau levels follow Dirac electron quantization when magnetic length is large.

Decreasing magnetic length causes nontrivial modulation of Landau levels.

Layered structure induces mixing of Dirac cones affecting surface states.

Abstract

A three-dimensional weak topological insulator (WTI), being equivalent to stacked layers of two-dimensional quantum spin-Hall insulators, accommodates massless Dirac electrons on its side surface. A notable feature of WTIs is that surface states typically consist of two Dirac cones in the reciprocal space. We study the Landau quantization of Dirac electrons of WTIs in a perpendicular magnetic field. It is shown that when the magnetic length $l_B$ is much larger than the interlayer distance $a$, surface electrons are quantized into Landau levels according to the ordinary quantization rule for Dirac electrons. It is also shown that, with decreasing $l_B$ toward $a$, each Landau level and its spin state become modulated in a nontrivial manner. We demonstrate that this is attributed to the mixing of two Dirac cones induced by the discreteness of the layered structure.