Spin-orbit interaction in bent carbon nanotubes: resonant spin transitions

We develop an effective tight-binding Hamiltonian for spin-orbit (SO) interaction in bent carbon nanotubes (CNT) for the electrons forming the $\pi$ bonds between the nearest neighbor atoms. We account for the bend of the CNT and the intrinsic spin-orbit interaction which introduce mixing of $\pi$ and $\sigma$ bonds between the $p_z$ orbitals along the CNT. The effect contributes to the main origin of the SO coupling--the folding of the graphene plane into the nanotube. We discuss the bend-related contribution of the SO coupling for resonant single-electron spin and charge transitions in a double quantum dot. We report that although the effect of the bend-related SO coupling is weak for the energy spectra, it produces a pronounced increase of the spin transition rates driven by an external electric field. We find that spin-flipping transitions driven by alternate electric fields have usually larger rates when accompanied by charge shift from one dot to the other. Spin-flipping transition rates are non-monotonic functions of the driving amplitude since they are masked by stronger spin-conserving charge transitions. We demonstrate that the fractional resonances--counterparts of multiphoton transitions for atoms in strong laser fields--occurring in electrically controlled nanodevices already at moderate ac amplitudes--can be used to maintain the spin-flip transitions.