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

Abstract

Energy band for single wall carbon nanotubes with spin-orbit interaction is calculated using nonorthogonal 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.

Signatures of spin-orbit interaction in transport properties of finite carbon nanotubes in a parallel magnetic field

“…Derivations using degenerate second-order perturbation theory 30,33 show nevertheless the existence of diagonal terms, which lead to a shift of the energy levels. This shift is valley-and spin-dependent, and causes a modification of Eq.…”

“…The SOI-modified band structure and its implications on the electronic properties of CNT's in a magnetic field have been in the focus of extensive theoretical work. 15,[30][31][32][33][34][35][36] However, at present only a few theoretical works exist 16,37 that investigate quantum transport through CNT's including magnetic field, curvature, and SOI effects.…”

Spin-orbit effects in carbon-nanotube double quantum dots

“…10,14 While specific expressions for the gap are model dependent, the order of magnitude estimate for R of a few nanometers is 10,14,31 E curv ͑ ͒ ϳ ͑បva/R 2 ͒cos 3 ϳ 10 meV. ͑7͒…”

“…Combined with the atomic SO coupling which produces transition matrix elements between different quantum states on the same atom, a spin-dependent coupling between the adjacent A and B atoms arises. 27,30 More recent work [31][32][33] has extended this approach and added to the low-energy effective Hamiltonian of electrons in CNTs a term that is diagonal in sublattice ͑A , B͒ space. The generalized SO Hamiltonian near the Dirac points is…”

“…More recent theoretical investigations extended this work by including the and bands in full as well as the curved bonds between neighboring atoms. [30][31][32][33] Lowest-order perturbation theory shows that the SO coupling is inversely proportional to the radius of curvature and originates from the intra-atomic SO coupling in a carbon atom. Even though this is a weak coupling compared to heavier atoms, the combined effect of curvature and intra-atomic SO coupling splits a fourfolddegenerate level into two Kramers doublets by a fraction of a millielectron volt.…”