Helical Modes in Carbon Nanotubes Generated by Strong Electric Fields

Helical modes, conducting opposite spins in opposite directions, are shown to exist in metallic armchair nanotubes in an all-electric setup. This is a consequence of the interplay between spin-orbit interaction and strong electric fields. The helical regime can also be obtained in chiral metallic nanotubes by applying an additional magnetic field. In particular, it is possible to obtain helical modes at one of the two Dirac points only, while the other one remains gapped. Starting from a tight-binding model we derive the effective low-energy Hamiltonian and the resulting spectrum.

Figure 1

Cross section of a CNT in a uniform electric field $E$. The orientation of the orbitals $p_r$, $p_t$ as well as the local coordinate system $\widehat{\mathbf{r}}$, $\widehat{\mathbf{t}}$ depends on the azimuthal angle $\phi$. The $s$ orbital is indicated by the dashed circles. The electric field $E$ is oriented along the $x$ direction of the global coordinate system. The $z$ direction is along the nanotube.

Figure 2

Energy dispersion $\epsilon (k)$ of a (10,10) armchair nanotube (implying $R=0.67\mathrm{nm}$). The solid (red) lines show the analytical results [see Eq. (4) and Table ] and the dots show numerical results obtained from $H$ in Eq. (1). The axial $k$ shift $\Delta k_{\mathrm{cv}}^z$ has been removed in both, the numerical and the analytical spectrum. The arrows correspond to the $S^y$ projections and are reversed for $E\to -E$. The field strength is $E=1\mathrm{V}/\mathrm{nm}$ so that the splitting $2eE\xi \simeq 0.4\mathrm{meV}$ and the gap $2|\alpha |\simeq 0.24\mathrm{meV}$. The dashed lines indicate the spectrum for the case $\alpha =0$ with the spin-degeneracies at $k=0$. The dashed gray (light blue) lines indicate chemical potentials at which only helical modes exist [see Eq. (5)].

Figure 3

Chiral (23,20)-CNT with electric field $E=0.5\mathrm{V}/\mathrm{nm}$ ($eE\xi \simeq 0.1\mathrm{meV}$). (a) The spectrum and (b) the spin expectation values at the $\mathbf{K}/{\mathbf{K}}^{\prime }$ point for the $|2,k⟩$ subband and magnetic field $B_z=0$. For $B_z=0.44\mathrm{T}$, the bands (c) at ${\mathbf{K}}^{\prime }$ (dashed lines) are gapped, while the spectrum at $\mathbf{K}$ (solid lines) has the same form as in the armchair case [Eq. (4)]. The size of the gap is $2|\alpha |=0.11\mathrm{meV}$. The spin expectation values (d) at $\mathbf{K}$ for $|2,k⟩$ follow closely the armchair case [see Eq. (5)].