Spin Splitting Induced by Spin-Orbit Interaction in Chiral Nanotubes

We show that chiral tubes present spin splitting at the Fermi level in the absence of a magnetic field, whereas achiral tubes preserve spin degeneracy, as evidenced by tight-binding electronic structure calculations with the inclusion of spin-orbit interaction. These remarkably different behaviors of chiral and nonchiral nanotubes have a symmetry origin, which may provide a global explanation to recently reported spin-dependent transport experiments which were in apparent contradiction.

Figure 1

Band structure calculated around the Fermi level of the achiral tubes: (5,5) armchair tube (a) without and (b) with SO interaction; (9,0) zigzag tube (c) without and (d) with SO interaction.

Figure 2

Same as Fig. 1 for the chiral tubes (7,1) (a),(b) and (9,3) (c),(d).

Figure 3

SO spin-split dispersion relation for the chiral tube (9,3), calculated around the Fermi level in the $\Gamma$ region of the Brillouin zone (BZ). Arrows indicate the spin polarization of the bands.

Figure 4

Spin-resolved DOS at ${\mathbf{k}}_F$ for (a) the (5,5) nanotube and (b) the (7,1) nanotube, calculated with (solid line) and without (dotted line) SO interaction.

Figure 5

Spin-resolved DOS for the (9,3) tube at $\mathbf{k}=0.025$ (solid line) and $\mathbf{k}=0.0125$ (dashed line) and schematic representation of the sequential ordering of spin-states, considering either one or both $\mathbf{k}$ vectors (see text). Wave vector values are given with respect to the BZ length.