Kondo effect in a topological insulator quantum dot

We investigate theoretically the nonequilibrium transport properties of a topological insulator quantum dot (TIQD) in the Coulomb blockade and Kondo regime. An Anderson impurity model is applied to a TIQD system coupled to two external leads, and we show that the model realizes the spin-orbital Kondo effect at the Dirac point where the edge states are not split by a finite-size effect, leading to an additional $SU\left(4\right)$ symmetry because of the presence of strong mixture among four internal degrees of freedom. In a more realistic situation where the degeneracy is lifted due to the finite-size effect, we demonstrate that there is a richer structure in transport measurements. We illustrate a continuous crossover from four (spin and orbital) Coulomb peaks with large interpair spacing and small intrapair spacing to a double-peak structure in the local density of states (LDOS) as increasing the hybridization strength $\Gamma$ within the Coulomb blockade regime. When temperature falls below the Kondo temperature $T_K$, four Kondo peaks show up in the nonequilibrium LDOS. Two of them are located at the chemical potential of each lead, and the other two are shifted away from the chemical potential by an amount proportional to the TIQD's bare energy level, leading to a triple-peak structure in the differential conductance when a bias voltage is applied.

Figure 1

Sketch of a TIQD attached to two external leads. The geometry of the TIQD is a thin circular slab. The gate voltage $V_G$ adjusts the energy levels so that only those levels near the Dirac point contribute to the transport properties of the TIQD. The biased voltage $\text{eV}$ is controlled by the difference between left and right leads' chemical potential: $\text{eV}={\mu }_L-{\mu }_R$.

Figure 2

Density of states $\rho \left(\omega \right)$ for TIQDs symmetrically coupled to two external leads with parameters $U=0.1\Gamma$ and ${\epsilon }_{\alpha }=\pm 2\Gamma$. Two leads' chemical potentials are set to be ${\mu }_L={\mu }_R=8\Gamma$. Under the mean-field approximation, two distinct peaks are shifted away from the original resonances $\pm 2\Gamma$ to the new ones $\pm 2\Gamma \mp U\langle n\rangle$ in the LDOS.

Figure 3

Local density of states $\pi \Gamma \rho \left(\omega \right)$ vs energy $\omega$ in units of $U$ in the Coulomb blockade regime with the hybridization strength (a) $\Gamma <{\Gamma }_c$ and (b) $\Gamma >{\Gamma }_c$. (a) The LDOS exhibits a quadruple-peak structure at $\epsilon =\pm 0.2U$ and $\epsilon =\pm 0.2U+(1+2\langle n\rangle )U$ in the weak coupling limit $\left(\Gamma =0.1U\right)$. The occupations $\langle n\rangle$ converge to $0.3670$ via the self-consistent algorithm. (b) Four Coulomb peaks merge into a double-peak structure when the hybridization strength $\Gamma$ is above the critical value ${\Gamma }_c\approx 0.2U$ in the strong coupling limit $\left(\Gamma =0.3U\right)$.

Figure 4

Local density of states $\pi \Gamma {\rho }_{\alpha }\left(\omega \right)$ for a TIQD symmetrically coupled to two external leads with the chemical potential ${\mu }_L$ and ${\mu }_R$ in the limit of $U\to \infty$. The TIQD is prepared to have four internal states with levels ${\epsilon }_{1,2}=-\Gamma$ and ${\epsilon }_{3,4}=\Gamma$. Temperature is set to be $T=0.01\Gamma$, while the Kondo temperature is roughly estimated to be $T_K\approx 0.08\Gamma$. Panels (a) and (b) display the equilibrium $\left({\mu }_L={\mu }_R=10\Gamma \right)$ LDOS for negative levels ${\epsilon }_{\alpha =1,2}$ (a) and positive levels ${\epsilon }_{\alpha =3,4}$ (b). It exhibits a double-peak structure at $\omega =10\Gamma$ and $\omega =9\Gamma$ for (a) and at $\omega =10\Gamma$ and $\omega =11\Gamma$ for (b). Panels (c) and (d) are the nonequilibrium LDOS for negative energy states (c) and positive energy states (d). Four Kondo peaks emerge in the nonequilibrium LDOS. Two of them are located at each lead's chemical potential: ${\mu }_L=10.5\Gamma ,{\mu }_R=10\Gamma$ (arrows 1 and 2), and the other two peaks are shifted either downwards from the chemical potential by $-2\Gamma$ $\left(=-2|{\epsilon }_{\alpha }|\right)$ [arrows 3 and 4 in (c)] or upwards from the chemical potential by $2\Gamma$ $\left(=2|{\epsilon }_{\alpha }|\right)$ [arrows 5 and 6 in (d)].

Figure 5

Differential conductance as a function of the applied bias voltage $\Delta \mu$ in units of $e^2/h$ via the EOM method for the temperatures above $T_K$ (solid line) and below $T_K$ (dashed line). When $\Delta \mu$ is adjusted so that the Kondo peaks in the left lead's DOS coincides with those in the right lead's DOS, the differential conductance ${\sigma }_d$ is enhanced. It shows a triple-peak structure at $\Delta \mu =0$ and $\Delta \mu =\pm 2|{\epsilon }_{\alpha }|=\pm 2\Gamma$ (solid line). When the temperature is raised above the Kondo temperature $T_K$, the triple-peak structure in the differential conductance disappears, as shown by the dashed line.