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 $K\text{−}{K}^{\prime }$ 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.

Figure 1

(Color online) Schematic of a representative mesoscopic system in question. In (a) each of the two leads $L$ and $R$ has two conduction bands (or “modes”) 1 and 2. The model with two leads in (a) is equivalent in equilibrium to the model in (b) with only one lead. The operators $c_{k\mu \sigma }$ ($\mu =1,2$ and $\sigma =\uparrow ,\downarrow$) are related to $a_{Lk\mu \sigma }$ and $a_{Rk\mu \sigma }$ by the canonical tranformation in Eq. (4). The wiggly lines indicate the interorbital Coulomb interaction $U_{12}$ [the intraorbital itneraction $U_{mm}$ $(m=1,2)$ is not shown].

Figure 2

(Color online) Schematics of the SU(4)-symmetric Anderson model.

Figure 3

(Color online) RG flows for different values of ${\Gamma }_2∕{\Gamma }_1$ with ${\Gamma }_1$ fixed.

Figure 4

(Color online) Comparison of the SU(2) and SU(4) Kondo model.

Figure 5

(Color online) NRG results of the total spectral density $A_d\left(E\right)$ for different values of coupling asymmetry ${\Gamma }_2∕{\Gamma }_1$. We took ${ϵ}_0=-0.8D$, ${\Gamma }_1=0.1D$, $U_{m{m}^{\prime }}=8D$, and ${\Delta }_{\mathrm{orb}}={\Delta }_Z=0$.

Figure 6

(Color online) A model with finite mixing between orbital quantum numbers.

Figure 7

(Color online) The RG flow in the case where there is a finite mixing of the orbital quantum numbers $m$.

Figure 8

(Color online) NRG results on the effects of the finite mixing of the orbital quantum number $m$. We took ${ϵ}_0=-0.8D$, ${\Gamma }_0=0.08D$, $U_{m{m}^{\prime }}=8D$, ${\Delta }_{\mathrm{orb}}=32T_K^{\mathrm{SU}\left(4\right)}$, and ${\Delta }_Z=2T_K^{\mathrm{SU}\left(4\right)}$.

Figure 9

(Color online) Schematic of the two-level SU(2)-symmetric Anderson model.

Figure 10

(Color online) Scaling of the single-particle energy level ${ϵ}_d$, to be compared with ${ϵ}_{m\sigma }$ in Eq. (38), of the single-level Anderson model. ${ϵ}_d^{*}={ϵ}_d(D=\Gamma )$ is the scale-invariant quantity.

Figure 11

(Color online) Scaling of the two-level SU(2)-symmetric Anderson model. The arrowed lines indicate the RG flow of the effective single-particle energy levels ${ϵ}_{\pm }$ [see Eq. (43)] and the widths of the shadowed regions around ${ϵ}_{\pm }$ the RG flow of ${\Gamma }_{\pm }$ [see Eq. (46)]. We have defined ${\Gamma }_0\equiv ({\Gamma }_1+{\Gamma }_2)∕2$.

Figure 12

(Color online) Total single-particle excitation spectrum $A_d\left(\omega \right)$ (a) with only the orbital degeneracy lifted $({\Delta }_{\mathrm{orb}}\ne 0$, ${\Delta }_Z=0)$ and (b) both the orbital and spin degeneracies lifted $({\Delta }_{\mathrm{orb}},{\Delta }_Z\ne 0)$. The short vertical arrows indicate the transition from ${ϵ}_{+}$ to ${ϵ}_{-}$, whose excitation energy is given by ${\Delta }_{\mathrm{eff}}$ [see Eq. (52)]. We took ${ϵ}_0=-0.8D$, ${\Gamma }_0=0.1D$, and $U_{m{m}^{\prime }}=8D$.

Figure 13

Transition from SU(4) to SU(2) Kondo physics as obtainded from SBMFT: As the orbital mixing increases, the SU(4) Kondo effect reduces to an SU(2) spin Kondo effect. This is reflected in both the position of the Kondo resonance [Fig. 2a] as well as the reduction of the odd Kondo temperature down to zero [Fig. 2b], see main text.

Figure 14

(Color online) Equilibrium SBMFT result: Transmission, $\mathcal{T}\left(ϵ\right)$ as a function of the frequency for several values of $\beta$. Left inset corresponds to the Kondo temperature vs $\beta$.

Figure 15

NCA results: Density of states around $\epsilon =0$ for $T=0.25T_K$ (left) and $T=T_K$ (right) and different values of $\beta$. The inset shows a close-up for the $\beta =0.5$ curve (thick dashed) together with the individual even and odd channel contributions (thin dashed).