Coupling of Spin and Orbital Motion of Electrons

The spin-orbit coupling till recently was commonly presumed negligible in carbon nanotubes (CNT) due to the lightness of carbon atom and symmetries of flat graphene. Indeed, before the work of Kuemmeth et al, the interpretation of the majority of experiments with charge carriers in CNTs assumed the independent spin and orbital degeneracies of the electron states in a CNT. The approximate four-fold degeneracy of single-particle states was seen in measurements of the spatial quantization of the conductance of a CNT well-connected to the leads, and was inferred also from the electron addition spectra in the Coulomb blockade regime (typical for samples with poor connection of a CNT to the leads – “closed” quantum dots). Observation of the conductance enhanced by orbital Kondo effect, compatible with the SU(4) symmetry of single-particle states in CNT-based quantum dots, was also reported in the recent years. One should bear in mind though that the addition spectra for quantum dots containing even moderate number of electrons, measure the energy difference between quite complicated many-body states. The enhancement of conductance by Kondo effect may be observed in transport through an “open” quantum dot containing a single electron, but the enhancement actually is not that sensitive to the presence of the exact SU(4) symmetry. [References to the experimental and theoretical works devoted to the addition spectra and Kondo effect may be found, e.g., in V.V. Deshpande and M. Bockrath, Nature Physics 4, 314 (2008) and A. Makarovski et al, Phys. Rev. B 75, R241407 (2007), respectively.] Contrary to the previous measurements, the experiment of Kuemmeth et al targeted directly the spectra of single-particle states (electrons and holes) in closed quantum dots formed in a high-quality CNT. The experimental results clearly show the presence of spin-orbit interaction due to the curvature