Among the advantages of graphene as medium for spin currents is the weak spinorbit coupling in this material, in accordance with the low atomic number of carbon. This feature is beneficial wherever long spin lifetimes are relevant, as in spintronics networks. Assessing and optimizing the efficiency of graphene and other carbon nanostructures as elements of spintronic circuits requires a detailed understanding of spin-orbit coupling mechanisms. Beyond practical considerations, examination of spin-orbit interactions also provides insight into the basic architecture of graphene and nanosystems derived from it. As outlined in Chapter 4, the two Dirac points are each characterized by an electronic state of fourfold degeneracy, corresponding to two spin and two sublattice degrees of freedom. Addition of the spin-orbit effect to the description of graphene lifts the spin degeneracy and thus opens a gap at the points K and, separating two states with twofold degeneracy. Spin-orbit interaction thus modifies the fundamental electronic structure of graphene.

As specified in Section 8.1, the intrinsic spin-orbit gap in graphene is extremely small. Use of the Rashba effect, on the other hand, makes it possible to tune the gap by applying an external electric field to the graphene sheet. This device allows us to close the gap, or to widen it by several orders of magnitude. Further, a physical sheet of carbon atoms is never ideally flat, and curved segments of the graphene layer turn out to be associated with effective electric fields. This feature implies a critical dependence of the spin-orbit effect on the nanostructure geometry. It acquires major importance as one goes from graphene to carbon nanotubes. Section 8.3 is dedicated to spin-orbit coupling in SWCNTs and emphasizes the geometric origin of this effect. In contrast to graphene, this interaction is very pronounced in SWCNTs, inducing a marked energy splitting between two Kramers doublets at the Dirac point. As it is masked by competing effects, its definite experimental characterization was a relatively recent event [288].

Section 8.2 deals with orbital magnetic moments in SWCNTs and an orbital shift undergone by the Dirac cones as a consequence of magnetic interactions involving these moments. This mechanism turns out to be essential for a proper appreciation of spin-orbit processes in nanotubes, as will be outlined in Section 8.3.