Curvature-enhanced spin-orbit coupling in a carbon nanotube

Structure of the spin-orbit coupling varies from material to material and thus finding the correct spin-orbit coupling structure is an important step toward advanced spintronic applications. We show theoretically that the curvature in a carbon nanotube generates two types of the spin-orbit coupling, one of which was not recognized before. In addition to the topological phase-related contribution of the spin-orbit coupling, which appears in the off-diagonal part of the effective Dirac Hamiltonian of carbon nanotubes, there is another contribution that appears in the diagonal part. The existence of the diagonal term can modify spin-orbit coupling effects qualitatively, an example of which is the electron-hole asymmetric spin splitting observed recently, and generate four qualitatively different behavior of energy-level dependence on parallel magnetic field. It is demonstrated that the diagonal term applies to a curved graphene as well. This result should be valuable for spintronic applications of graphitic materials.