Spectral diagrams of Hofstadter type for the Bloch electron in three dimensions

TL;DR

This paper analyzes spectral diagrams of a 3D Bloch electron in a magnetic field, revealing complex structures including potential fractals, and examines how lattice geometry and band coupling influence these diagrams.

Contribution

It introduces a detailed analysis of flux-energy and angle-energy diagrams for a 3D Bloch electron, highlighting the effects of lattice geometry and band coupling on their structure.

Findings

Angle-energy diagrams can have fractal structures even in cubic lattices.

Neglecting Landau band coupling significantly alters diagram shapes.

The structure depends on magnetic field direction and lattice geometry.

Abstract

Flux-energy and angle-energy diagrams for an exact three-dimensional Hamiltonian of the Bloch electron in a uniform magnetic field are analyzed. The dependence of the structure of the diagrams on the direction of the field, the geometry of the Bravais lattice and the number of atoms in an elementary cell is considered. Numerical evidence is given that the angle-energy diagram may have a fractal structure even in the case of a cubic lattice. It is shown that neglecting coupling of Landau bands changes considerably the shape of the diagrams.