Hofstadter butterflies of carbon nanotubes: Pseudofractality of the magnetoelectronic spectrum

Abstract

The electronic spectrum of a two-dimensional square lattice in a perpendicular magnetic field has become known as the Hofstadter butterfly ͓Hofstadter, Phys. Rev. B 14, 2239 ͑1976͒.͔. We have calculated quasi-onedimensional analogs of the Hofstadter butterfly for carbon nanotubes ͑CNTs͒. For the case of single-wall CNTs, it is straightforward to implement magnetic fields parallel to the tube axis by means of zone folding in the graphene reciprocal lattice. We have also studied perpendicular magnetic fields which, in contrast to the parallel case, lead to a much richer, pseudofractal spectrum. Moreover, we have investigated magnetic fields piercing double-wall CNTs and found strong signatures of interwall interaction in the resulting Hofstadter butterfly spectrum, which can be understood with the help of a minimal model. Ubiquitous to all perpendicular magnetic field spectra is the presence of cusp catastrophes at specific values of energy and magnetic field. Resolving the density of states along the tube circumference allows recognition of the snake states already predicted for nonuniform magnetic fields in the two-dimensional electron gas. An analytic model of the magnetic spectrum of electrons on a cylindrical surface is used to explain some of the results.

Charge transport in disordered graphene-based low dimensional materials

“…A prescription for the inclusion of magnetic field effects into the band structure of carbon-based nano-networks can be found in Ref. [87]. An energy structure similar to that of 2-D graphene is also observed for bulk electrons in nanoribbons, provided that the width W of the ribbon is larger than the magnetic length.…”

Hofstadter butterflies of bilayer graphene

“…19 Since its discovery, various aspects of the so-called Hofstadter butterfly have been studied, 20,21 particularly in relation to graphenelike honeycomb structures. 12,22,23 Featuring a large variety of topologies, all these systems have in common that the atoms inside the unit cell are located on discrete coordinates. All closed loops have commensurate areas, and the atomic network is regular enough that the magnetic phases of all links can be determined individually without the need of a continuously defined gauge field.…”

“…While these represent discrete levels in two-dimensional graphene sheets, they are broadened by the finite width of the ribbon to a peak of the same shape as in carbon nanotubes. 23,32 The mesoscopic character of these split SUSYLLs in dependence on the width W of the ribbon is captured by the functional form of the density of states:…”

Signatures of spin-orbit interaction in transport properties of finite carbon nanotubes in a parallel magnetic field

“…[19][20][21][22][23][24][25] Further theoretical studies [26][27][28][29] reveal that the transport properties are directly related to the magnetic-field modulated density of states. The SOI-modified band structure and its implications on the electronic properties of CNT's in a magnetic field have been in the focus of extensive theoretical work.…”