Kondo effects in carbon nanotubes: From SU(4) to SU(2) symmetry

We study the Kondo effect in a single-electron transistor device realized in a single-wall carbon nanotube (NT). The K- K′ double orbital degeneracy of a NT, which originates from the peculiar two-dimensional band structure of graphene, plays the role of a pseudospin. Screening of this pseudospin, together with the real spin, can result in an SU(4) Kondo effect at low temperatures. In order to have such an exotic Kondo effect it is crucial that this orbital quantum number be conserved during tunneling. Experimentally, this conservation is not obvious and some mixing in the orbital channel may occur. Here we investigate in detail the role of mixing and asymmetry in the tunneling coupling and analyze how different Kondo effects, from an SU(4) symmetry to a two-level SU(2) Kondo effect, emerge depending on the mixing and/or asymmetry. We use four different theoretical approaches to address both the linear and nonlinear conductance for different values of external magnetic field. Our results point out clearly the experimental conditions to observe exclusively SU(4) Kondo physics. Although we focus on NT quantum dots (QDs) our results also apply to vertical quantum dots. We also mention that a finite amount of orbital mixing corresponds, in pseudospin language, to having noncollinear leads with respect to the orbital "magnetization" axis which defines the two pseudospin orientations in the carbon nanotube QD. In this sense, some of our results are also relevant to the problem of a Kondo QD coupled to noncollinear ferromagnetic leads.