Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps

A continuum model for the effective spin-orbit interaction in graphene is derived from a tight-binding model which includes the $\pi$ and $\sigma$ bands. We analyze the combined effects of the intra-atomic spin-orbit coupling, curvature, and applied electric field, using perturbation theory. We recover the effective spin-orbit Hamiltonian derived recently from group theoretical arguments by Kane and Mele. We find, for flat graphene, that the intrinsic spin-orbit coupling ${\Delta }_{\mathrm{int}}\propto {\Delta }^2$ and the Rashba coupling due to a perpendicular electric field $\mathcal{E}$, ${\Delta }_{\mathcal{E}}\propto \Delta$, where $\Delta$ is the intra-atomic spin-orbit coupling constant for carbon. Moreover we show that local curvature of the graphene sheet induces an extra spin-orbit coupling term ${\Delta }_{\mathrm{curv}}\propto \Delta$. For the values of $\mathcal{E}$ and curvature profile reported in actual samples of graphene, we find that ${\Delta }_{\mathrm{int}}<{\Delta }_{\mathcal{E}}\lesssim {\Delta }_{\mathrm{curv}}$. The effect of spin-orbit coupling on derived materials of graphenelike fullerenes, nanotubes, and nanotube caps, is also studied. For fullerenes, only ${\Delta }_{\mathrm{int}}$ is important. Both for nanotubes and nanotube caps ${\Delta }_{\mathrm{curv}}$ is in the order of a few Kelvins. We reproduce the known appearance of a gap and spin-splitting in the energy spectrum of nanotubes due to the spin-orbit coupling. For nanotube caps, spin-orbit coupling causes spin-splitting of the localized states at the cap, which could allow spin-dependent field-effect emission.

Figure 1

(Color online) Black (full) curves: $\sigma$ bands. Red (dashed) curves: $\pi$ bands. The dark and light green (gray) arrows give contributions to the up and down spins at the $A$ sublattice, respectively. The opposite contributions can be defined for the $B$ sublattice. These interband transitions are equivalent to the processes depicted in Fig. 3.

Figure 2

(Color online) Sketch of the relevant orbitals, $p_x$ and $p_z$ needed for the analysis of spin-orbit effects in a curved nanotube. The arrows stand for the different hoppings described in the text.

Figure 3

(Color online) Sketch of the processes leading to an effective intrinsic term in the $\pi$ band of graphene. Transitions drawn in red (dark gray), and indicated by SO, are mediated by the intra-atomic spin-orbit coupling.

Figure 4

(Color online) Left, one-fifth of a fullerene cap closing an armchair nanotube. The full cap is obtained by gluing five triangles like the one in the figure together, forming a pyramid. A pentagon is formed at the apex of the pyramid, from the five triangles like the one shaded in green (gray) in the figure. The edges of the cap are given by the thick black line. The cap contains six pentagons and 70 hexagons, and it closes a $25\times 25$ armchair nanotube. Right, sketch of the folding procedure of a flat honeycomb lattice needed to obtain an armchair nanotube capped by a semispherical fullerene.

Figure 5

(Color online) (Up) Sketch of the matching scheme used to build a zero energy state at a fullerene cap. (Down) The wave function is one-half of a zero energy state at the cap, matched to a decaying state towards the bulk of the nanotube. See text for details.