Spin-Orbit Interaction in Single Wall Carbon Nanotubes: Symmetry Adapted Tight-Binding Calculation and Effective Model Analysis

Energy band for single wall carbon nanotubes with spin-orbit interaction is calculated using non-orthogonal tight-binding method. A Bloch function with spin degree of freedom is introduced to adapt the screw symmetry of nanotubes. The energy gap opened by spin-orbit interaction for armchair nanotubes, and the energy band splitting for chiral and zigzag nanotubes are evaluated quantitatively. Spin polarization direction for each split band is shown to be parallel to the nanotube axis. The energy gap and the energy splitting depend on the diameter and chirality in an energy scale of sub-milli-electron volt. An effective model for reproducing the low energy band structure shows that the two mechanism of the band modification, shift of the energy band in two dimensional reciprocal lattice space, and, effective Zeeman energy shift, are relevant. The effective model explains well the energy gap and splitting for more than 300 nanotubes within the diameter between 0.7 to 2.5 nm.