Effects of the field modulation on the Hofstadter's spectrum

TL;DR

This paper investigates how spatially modulated magnetic fields alter the energy spectrum of 2D electrons, revealing complex band structure changes such as gap openings and closings, distinct from the classic Hofstadter's spectrum.

Contribution

It provides a detailed analysis of how different types and strengths of magnetic field modulations affect the energy spectrum and symmetries of 2D electrons, extending the understanding of Hofstadter's spectrum.

Findings

Modulated fields cause gap opening and closing in the spectrum.

Field modulation leads to band crossing and broadening.

Energy band structure significantly depends on modulation parameters.

Abstract

We study the effect of spatially modulated magnetic fields on the energy spectrum of a two-dimensional (2D) Bloch electron. Taking into account four kinds of modulated fields and using the method of direct diagonalization of the Hamiltonian matrix, we calculate energy spectra with varying system parameters (i.e., the kind of the modulation, the relative strength of the modulated field to the uniform background field, and the period of the modulation) to elucidate that the energy band structure sensitively depends on such parameters: Inclusion of spatially modulated fields into a uniform field leads occurrence of gap opening, gap closing, band crossing, and band broadening, resulting distinctive energy band structure from the Hofstadter's spectrum. We also discuss the effect of the field modulation on the symmetries appeared in the Hofstadter's spectrum in detail.