Curvature-induced anisotropic spin-orbit splitting in carbon nanotubes

Abstract

We have theoretically explored the spin-orbit interaction in carbon nanotubes. We show that, besides the dependence on chirality and diameter, the effects of spin-orbit coupling are anisotropic: spin splitting is larger for the higher valence or the lower electron band depending on the specific tube. Different tube behaviors can be grouped in three families, according to the so-called chiral index. Curvature-induced changes in the orbital hybridization have a crucial role, and they are shown to be family-dependent. Our results explain recent experimental results which have evidenced the importance of spin-orbit effects in carbon nanotubes. PACS numbers: 71.20.Tx, 71.70.Ej Improvements in the quality of carbon nanotubes (CNTs) have enabled the fabrication of quantum dots aiming at the realization of spintronics devices [1,2,3,4]. CNTs present a high Fermi velocity and a twofold orbital degeneracy originating from the topology of the honeycomb lattice. The unique fourfold degeneracy of CNTs energy states (spin plus orbital moment) has been observed in CNT quantum dots (QDs) by magnetic field spectroscopy measurements [5] and makes them particularly interesting since, besides the spin degree of freedom, they present the orbital moment to allow for quantum manipulation. In a recent experiment [6], spin-orbit coupling has been directly observed in CNT as a splitting of the fourfold degeneracy of a single-electron energy level in ultra-clean QDs. This important finding seems to be in contradiction with the interpretation of earlier experiments in defect-free CNTs, from which independent spin and orbital symmetries and electron-hole symmetry have been deduced [7]. Besides showing the importance of spin-orbit effects in carbon nanotubes, Kuemmeth et al.[6] point out an unexplained anisotropic splitting of electron and holes in carbon nanotube quantum dots, which deserves further exploration.On theoretical grounds, spin-orbit interaction (SOI) has been investigated on CNTs by deriving an effective mass Hamiltonian including a weak SOI in carbon orbitals to the lowest order in perturbation theory [8]. Band splitting was found considering surface curvature effects [9], as well as in the electron spin resonance spectra of achiral CNTs derived by low-energy theory [10]. In an earlier work, we showed that the inclusion of the full lattice symmetry is essential for deriving spin-orbit (SO) effects in CNTs [11]. Employing an empirical tightbinding model, we demonstrated an intrinsic symmetry dependence of SOI effects. As confirmed by recent experimental results [6], we showed that, in the absence of a magnetic field, CNTs present spin-orbit split bands at the Fermi level. In addition, SOI induces zero-field spin splitting in chiral CNTs, while Kramers theorem on time-reversal symmetry alongside the inversion symmetry preserve the spin-degeneracy in achiral-i.e., (n, n) armchair and (n, 0) zigzag-nanotubes [12]. More recent works [13] have indicated the importance of curvature in the SOI effects investigated with a continuum ...

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

“…30 and 33 does not account for a possible change of sign of ε SO , as suggested in other works. 60,61 We have nevertheless seen that for the systems studied in this work, the effect of this shift is minimal. Therefore, its exclusion does not alter the conclusions of our study.…”

Spin-orbit effects in carbon-nanotube double quantum dots

“…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.…”

Spin filtering and entanglement detection due to spin-orbit interaction in carbon nanotube cross-junctions

“…While in semiconductors the SOI typically causes the spin to precess during transport, in CNTs, which are hollow cylinders different from filled quantum wires, the SOI has a different impact. Rather than causing spin rotations, it lifts the energy degeneracy of opposite spins and leads to distinct, fully spin polarized bands with the polarization directions parallel to the rotational symmetry axis, [26][27][28][29][30][31][32][33][34][35] an effect whose consequences were, to date, investigated theoretically mainly in quantum dot setups. [36][37][38][39][40] Recently the possibility of using gates to control the spin filtering properties due to SOI in CNT quantum dots has been also exploited.…”