Quantum Hall effect in carbon nanotubes and curved graphene strips

We develop a long-wavelength approximation in order to describe the low-energy states of carbon nanotubes in a transverse magnetic field. We show that in the limit where the square of the magnetic length $l=\sqrt{\hslash c∕eB}$ is much larger than the C-C distance times the nanotube radius $R$, the low-energy theory is given by the linear coupling of a two-component Dirac spinor to the corresponding vector potential. We investigate in this regime the evolution of the band structure of zigzag nanotubes for values of $R∕l>1$, showing that for radius $R\approx 20\mathrm{nm}$ a clear pattern of Landau levels starts to develop for magnetic field strength $B\gtrsim 10\mathrm{T}$. The levels tend to be fourfold degenerate, and we clarify the transition to the typical twofold degeneracy of graphene as the nanotube is unrolled to form a curved strip. We show that the dynamics of the Dirac fermions leads to states which are localized at the flanks of the nanotube and that carry chiral currents in the longitudinal direction. We discuss the possibility of observing the quantization of the Hall conductivity in thick carbon nanotubes, which should display steps at even multiples of $2e^2∕h$, with values doubled with respect to those in the odd-integer quantization of graphene.

Figure 1

Schematic representation of the lattice of a zigzag nanotube, showing the four different levels of inequivalent atoms in a unit cell.

Figure 2

Sequence of band structures of a zigzag (510,0) nanotube with radius $R\approx 20\mathrm{nm}$ in transverse magnetic field for (a) $B=0\mathrm{T}$, (b) $B=5\mathrm{T}$, (c) $B=10\mathrm{T}$, and (d) $B=20\mathrm{T}$. $B=20\mathrm{T}$ corresponds to $aR∕l^2\approx 0.1$ and $R∕l\approx 3.5$. Energy is in units of $t$ and momentum is in units of ${\AA }^{-1}$.

Figure 3

Sequence of band structures of a zigzag (500,0) nanotube with radius $R\approx 20\mathrm{nm}$ in transverse magnetic field for (a) $B=0\mathrm{T}$, (b) $B=5\mathrm{T}$, (c) $B=10\mathrm{T}$, and (d) $B=20\mathrm{T}$. Energy is in units of $t$ and momentum is in units of ${\AA }^{-1}$.

Figure 4

Band structure corresponding to a zigzag nanotube with a cut in the longitudinal direction, for $R\approx 20\mathrm{nm}$ and $B=20\mathrm{T}$ (in the same units as in Fig. 2).

Figure 5

Plot of the integral of the current $j$ over the angular variable $\theta$ (in units of $v_F$) for states in the lowest Landau subband of Fig. 2d.

Figure 6

Angular distribution of the eigenfunctions in the lowest Landau subband of Fig. 2d. The panels correspond, from top to bottom, to longitudinal momenta $k=0.15,0,-0.15$ (in units of ${\mathrm{\AA }}^{-1}$).

Figure 7

Plot of the Hall conductivity ${\sigma }_{xy}$ (in units of $e^2∕h$) as a function of the position of the Fermi level ${\epsilon }_F$ in the band structure of Fig. 2d. The full line corresponds to a linear Hall voltage drop and the dashed line to a sharp voltage drop in the bulk of the nanotube.