SU(3) Kondo effect in spinless triple quantum dots

We discuss a device—a purely capacitively coupled interacting triple quantum dot system in an external magnetic field—for the observation of the SU(3) Kondo effect, identified by the conductance being pinned to a characteristic value of $3/4$ of the unitary limit. The Kondo effect occurs in two plateaus where the dot occupancy is pinned to an integer value, either 1 or 2. We discuss the thermodynamic and spectral properties of the corresponding triple-impurity model and establish how the presence of SU(3) Kondo screening can be identified in the temperature dependence of the conductance through one of the dots. We report results about the robustness of the SU(3) Kondo effect against various perturbations present in real experimental setups, namely unequal reservoir-dot tunneling couplings, gating effects, and nonvanishing interdot tunneling rates. Finally, we describe possible mechanisms to restore the SU(3) Kondo physics by properly tuning the on-site dot potentials. We briefly comment on the spinful case (i.e., the same system in the absence of the magnetic field), which has very different behavior and shows Kondo plateaus in conductance for all integer values of the occupancy, including at the particle-hole symmetric point.

Figure 1

Schematic representation of the capacitively coupled triple quantum dot system. Each quantum dot is attached to two electron reservoirs. We assume that a sufficiently large external magnetic field is applied to fully polarize the electrons, so that we may consider the system to be spinless. The only interaction between each pair of quantum dots is purely capacitive. Interdot charging energies (denoted by $V$) are assumed to be the same: they are characterized by the capacitance $C$, $V=e^2/2C$. Dashed lines indicate the electron transport through each dot.

Figure 2

Temperature dependence of the impurity entropy (left panels) and the $T=0$ dot spectral function (right panels) for a range of the on-site energies $\epsilon$. We consider a symmetric triple quantum dot system. The total occupancy of the triple quantum dot, $\langle n\rangle$, and the zero-temperature linear conductance through one dot, $G$, are also shown. The interdot tunneling is zero, $t=0$. Other parameters are $\Gamma =0.01$ and $V=0.2$.

Figure 3

Spectral function in the $\mathrm{SU}\left(3\right)$ Kondo regime. The TQD is symmetric and $\langle n\rangle =2$.

Figure 4

Temperature dependence of the impurity entropy (left panels) and the $T=0$ dot spectral function (right panels) for a range of the on-site energies $\epsilon$. Same parameters as in Fig. 2, but with smaller hybridization, $\Gamma =0.005$. We plot the positive-frequency side of the spectral function on the logarithmic frequency scale. The dashed line corresponds to $G=(3/4)(e^2/h)$.

Figure 5

Thermodynamic properties in the $\mathrm{SU}\left(3\right)$ Kondo regime. The total occupancy is $\langle n\rangle =2$. We compare the cases of a decoupled triple quantum dot system ($\Gamma =0$) and the triple quantum dot connected to the leads ($\Gamma =0.005$) in which the Kondo correlations are present.

Figure 6

Temperature dependence of the entropy at the particle-hole symmetric point for a range of hybridization parameters $\Gamma$. We consider a symmetric triple quantum dot system with parameters $V=0.2$ and $\epsilon =-V=-0.2$.

Figure 7

Occupancy $n$, differential conductance $G$, and charge fluctuations $\delta n^2=\langle n^2\rangle -{\langle n\rangle }^2$ as a function of the on-site energy for a range of hybridizations $\Gamma$.

Figure 8

Top panel: Temperature dependence of the entropy for a symmetric triple quantum dot system for a range of hybridization parameters $\Gamma$. Other parameters are tuned so that the system is in the $\mathrm{SU}\left(3\right)$ Kondo regime, $V=0.2$, $\epsilon =-0.3$. Bottom panel: Relation between the Kondo temperature $T_K$ and the hybridization $\Gamma$. Here we use an arbitrary definition of the Kondo temperature, $S_{\mathrm{imp}}\left(T_K\right)=0.1k_B$.

Figure 9

Temperature dependence of the linear conductance in the $\mathrm{SU}\left(3\right)$ Kondo regime of a symmetric triple quantum dot.

Figure 10

Top panel: Temperature dependence of the impurity entropy for a range of the hybridization parameter ${\Gamma }_1$, while ${\Gamma }_2={\Gamma }_3=\Gamma$ is held fixed. Bottom panel: Impurity entropy for a strong symmetry breaking and a fit of the $\mathrm{SU}\left(2\right)$ Kondo screening crossover with the universal $\mathrm{SU}\left(2\right)$ entropy curve for the spinful single dot case.

Figure 11

Spectral functions in the case of nonequal hybridizations ${\Gamma }_i$. The dashed line corresponds to the fully symmetric case with $\Gamma \equiv {\Gamma }_i=0.005$.

Figure 12

Temperature dependence of the impurity entropy for a range of the parameter $\delta {\epsilon }_1$ defined as ${\epsilon }_1=\epsilon +\delta {\epsilon }_1$, while ${\epsilon }_2={\epsilon }_3=\epsilon$ with $\epsilon =-0.3$. The rest of the parameters correspond to a symmetric triple quantum dot configuration with $V=0.2$ and $\Gamma =0.005$.

Figure 13

Temperature dependence of the triple quantum dot entropy for different values of interdot tunneling.

Figure 14

Restoration of the $\mathrm{SU}\left(3\right)$ Kondo effect in an asymmetrically lead-dot coupled triple quantum dot (${\Gamma }_1=0.004$ and ${\Gamma }_2={\Gamma }_3=0.005$) by tuning the on-site dot potential ${\epsilon }_1=\epsilon +\delta {\epsilon }_1$, with $V=0.2$ and $\epsilon =-0.3$. The $\mathrm{SU}\left(3\right)$ symmetry is effectively restored for $\delta {\epsilon }_1=3.25\times {10}^{-5}$, which is indicated by the impurity curve having the universal $\mathrm{SU}\left(3\right)$ shape. The curves for parameters near the restoration point show some wiggles; these are numerical artifacts due to the $z$-averaging in NRG.

Figure 15

Occupancy $n$, differential conductance $G$, and charge fluctuations $\delta n^2=\langle n^2\rangle -{\langle n\rangle }^2$ as a function of the on-site energy $\epsilon =-U/2-2V+\delta$ for on-site repulsion $U=0.32$ and for a range of interdot repulsions $V$. $\Gamma =0.01$.