Hofstadter butterflies of carbon nanotubes: Pseudofractality of the magnetoelectronic spectrum

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 (1976).]. We have calculated quasi-one-dimensional analogs of the Hofstadter butterfly for carbon nanotubes (CNTs). 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.

Figure 1

(Color online) Density of states of a (6,6) CNT in dependence on an external magnetic field parallel (left) or perpendicular (right) to the tube axis. For every value of the magnetic field, the DOS is unity normalized over energy. The units $B_0^{\Vert }={\Phi }_0∕r^2\pi$ and $B_0^{\perp }={\Phi }_0∕A_{\text{plaquette}}$ (see text) are scaled such that the physical field scale is the same for both segments of the plot.

Figure 2

(Color online) The structure of a CNT [here, a (3,3)-CNT]: the hexagonal lattice of a graphene sheet is rolled up in such a way that the chiral vector $(n,m)$ becomes the circumference of the resulting cylinder. Magnetic fields parallel to the tube axis pierce the tube cross section $r^2\pi$, while perpendicular magnetic fields pierce the wall made up from hexagonal plaquettes.

Figure 3

(Color online) Scheme to illustrate the physical meaning of the butterfly plots. An external magnetic field distorts the band structure of a CNT in an intricate way. For any fixed magnetic field, the DOS and the transmission can be obtained directly from the band structure. (a), (b), and (c) are sections of the two right panels of the DOS and transmission vs $E$ and $B_{\perp }$.

Figure 4

(Color online) DOS in a graphene ribbons of infinite length and various widths and internal orientations, pierced perpendicularly by magnetic fields. Each ribbon can be classified as an unrolled CNT: The “chiral” vectors refer to the SWCNT which, when unrolled, would result in the corresponding planar ribbon. The density of states is normalized to the number of atoms per unit cell to give a comparable visual appearance.

Figure 5

(Color online) Cusp catastrophes are ubiquitous in butterfly plots. The band structure at magnetic fields below, at, and above the critical magnetic field shows the smooth transition from a strictly monotone band into a third-order parabolic band with changing magnetic field.

Figure 6

(Color online) Two semiconducting SWCNTs of similar diameter as the (6,6) CNT in Fig. 1. The band gap oscillates irregularly with increasing perpendicular field. The large unit cell of the the chiral (6,5) tube leads to bands with low dispersion as soon as the rotational symmetry is broken by the perpendicular magnetic field.

Figure 7

(Color online) Comparing different diameters: the large (200,200) CNT bears strong resemblance to the Hofstadter butterfly of graphene (Ref. 25) combined with the curvature effects (details in text). The straight lines at the lower left corner of the graphene butterfly (bottom panel) indicate the Landau states obtained from an effective mass continuum theory [see Eq. (8)]. The parabolic lines near $E_{\mathrm{F}}$ in the same plot indicate the relativistic Landau levels obtained from the Dirac-like dispersion of graphene [see Eq. (9)].

Figure 8

(Color online) Decomposition of the density of states into the contributions of particular atoms (identified by their angle $\Phi$ toward the magnetic field direction). A plaquette at angle $\theta$ captures a flux of $B_{\perp }{A}_{\text{plaquette}}\cos \theta$. The region at $\Phi =0$ experiences a perpendicular field piercing the tube wall, very much as in the plain graphene sheet (Fig. 7). The regions at $\Phi =\pi ∕2$ experiences a field tangential to the tube wall, leading to a much smaller flux per plaquette, resulting in a stretched impression of the butterfly. The DOS butterfly over the whole CNT unit cell is an overlay of these and many intermediate pictures. In the angle-resolved plot for $B=0.2$ $B_0^{\perp }$ one can see a smooth transition between a region with Landau levels and a region with normal band dispersion. For the stronger field, the systems goes through two oscillations along the angle.

Figure 9

(Color online) Analytical solution of the continuum model. Top panel: the $k_z$-dependent effective potential of free electrons confined to a continuum cylinder in a magnetic field perpendicular to the tube. Middle panel: first three eigenstates of the harmonic approximation to the above potential for various $k_z$ (highlighted wave functions correspond to the selected potentials in the first panel). Each wave function is shifted to the corresponding energy. Superimposed are the lines followed by the extrema of the wave functions. The same pattern can be found in the top left panel of Fig. 8, where the maxima of the DOS show the maxima of the various energy eigenstates. Bottom panel: The $k_z$-dependent effective potential of a discretized tube showing a large number of minima. States located in narrow minima have higher energy, so the low-energy spectrum is mainly determined by the widest potential minima.

Figure 10

(Color online) Zoom into the butterfly of two different armchair SWCNTs. The scales, including the color scale, are chosen according to the scaling law given in the text to produce comparable representations of the data. The tubes correspond to diameters of $2.7\mathrm{nm}$ (top) and $27\mathrm{nm}$ (bottom). The inset in the bottom panel illustrates the shape of the peak at the Fermi level. The white lines crossing the plot in the upper panel are caused by small avoided crossings in the band structure.

Figure 11

(Color online) Butterfly plot of a (6,6)@(11,11) double-wall CNT. In the upper panel, the interwall interaction is switched off, resulting in an overlay of the butterflies of two independent SWCNTs. In the lower panel, the interwall interaction gives rise to a number of new features (see text for details).

Figure 12

(Color online) Upper panel: Spectrum of a system of two concentric atomic rings. Atom spacing and coupling inside each ring are taken from graphene. The distance $\delta r$ between the rings as well as the the parametrization of the coupling between the rings follow those given in the text for DWCNTs. The sketch displays the prevalent links between the shells. Even though the geometry is irregular, the area of circular paths is very near to integer multiples of $d_{\mathrm{CC}}\delta r∕2$, leading to a clear periodicity of the modulation in the spectrum. Lower panel: An isolated unit cell of a DWCNT with the same radii as the planar double ring of the upper panel. This system has smallest closed loops at an angle against the magnetic field, resulting in an effective smallest area of $d_{\mathrm{CC}}\delta r∕4$ and leading to a doubling of the period. Furthermore, the system has two atoms in the rotational periodic cell, leading to two interlaced modulations.