Enhanced conductance through side-coupled double quantum dots

Conductance, on-site, and intersite charge fluctuations and spin correlations in the system of two side-coupled quantum dots are calculated using Wilson’s numerical renormalization group (NRG) technique. We also show the spectral density calculated using the density-matrix NRG, which for some parameter ranges remedies inconsistencies of the conventional approach. By changing the gate voltage and the interdot tunneling rate, the system can be tuned to a nonconducting spin-singlet state, the usual Kondo regime with an odd number of electrons occupying the dots, the two-stage Kondo regime with two electrons, or a valence-fluctuating state associated with a Fano resonance. Analytical expressions for the width of the Kondo regime and the Kondo temperature are given. We also study the effect of unequal gate voltages and the stability of the two-stage Kondo effect with respect to such perturbations.

Figure 1

(Color online) Side-coupled configuration of quantum dots.

Figure 2

(Color online) Conductance and correlation functions of DQD vs $\delta$. Besides different values of $t_d$, indicated in the figure, other parameters of the model are $\Gamma ∕D=0.03$ and $U∕D=1$. Temperature is $T∕D={10}^{-9}$, which for all parameters used corresponds to a zero-temperature limit. In particular, in the Kondo plateaus, $T⪡T_K$ for all $\delta$.

Figure 3

(Color online) Eigenvalue diagram for the isolated DQD system. The diagram is symmetric, since for $\delta \to -\delta$, $E(Q,S,r)\to E(-Q,S,r)$. Points $A$ and $B$ correspond to valence-fluctuation regions when the charge on the dot changes, while point $C$ corresponds to the center of the Kondo regime, when the Kondo temperature is the lowest. Parameters are $U∕D=1$ and $t_d∕D=0.3$.

Figure 4

(Color online) (a) The width of conductance peak $\Delta$ vs $t_d$ as obtained from NRG calculations (solid circles), compared with the analytical result as given in the text (dashed line). (b) Kondo temperatures $T_K$ vs $\delta$ as obtained from the widths of Kondo peaks (solid circles). The analytical estimate, Eq. (8), is shown using dashed lines. The rest of parameters are identical to those in Fig. 2.

Figure 5

(Color online) Comparison of spectral densities computed by the conventional NRG and by the density-matrix NRG technique for parameters where the conventional NRG approach has difficulties: $t_d∕D=0.3$, $U∕D=1$, $\Gamma ∕D=0.03$, and $\delta ∕D=0.48$. Note the discontinuities in the spectral density calculated using the conventional NRG. At low frequencies both method produce identical results, so we do not show the Kondo peak which exceeds the vertical scale of the plot. In this calculations, the effective temperature is zero.

Figure 6

(Color online) Zero-temperature spectral density ${\rho }_d\left(\omega \right)$ sweeps for $t_d∕D=0.3$. (a) Spectral density calculated using the conventional NRG approach. (b) Spectral density calculated using the density-matrix NRG approach. Note that the vertical line, representing the Kondo peak, has been artificially broadened. Its true width is given by $T_K$.

Figure 7

(Color online) Three zero-temperature spectral densities ${\rho }_d\left(\omega \right)$ for $t_d∕D=0.3$: at the particle-hole symmetric point $\delta ∕D=0$, in the Kondo regime $\delta ∕D=0.62$, and in the empty-orbital regime $\delta ∕D=1$. Inset: scaling of spectral densities ${\rho }_d(\omega ∕T_K)$ in the Kondo region $0.52<\delta <0.76$. Parameters are as in Fig. 2.

Figure 8

(Color online) Conductance as a function of gate-voltage ${\delta }_d$ for different gate-voltage asymmetries $\kappa$. The label SIAM corresponds to results for conductance of the corresponding single-impurity Anderson model. The temperature is $T∕D={10}^{-9}$.

Figure 9

Kondo temperature as a function of $\kappa$ for (a) the Kondo plateau at ${\delta }_d\sim 0$ and (b) the Kondo plateau at ${\delta }_a\sim 0$. In both cases we calculate the Kondo temperature in the middle of the plateau, where $T_K$ is lowest. Parameters are $U∕D=1$, $\Gamma ∕D=0.03$, and $t_d∕D=0.3$.

Figure 10

(Color online) Conductance as a function of gate-voltage ${\delta }_a$ for different gate-voltage asymmetries $\kappa$. The temperature is fixed to $T∕D={10}^{-9}$ for all $\kappa$.

Figure 11

(Color online) Conductance and correlation functions at $t_d∕D=0.001$ (a)–(e), $t_d∕D=0.01$ (f) and (g), and a blow-up of (f) in the inset of (g). Dashed lines in (a) represent $G∕G_0$ and (b) $S$ of a single quantum dot with otherwise identical parameters. Dashed line in (c) represents $⟨n_d⟩$ of DQD, and finally dashed line in the inset of (g) represents the semianalytical model described in the text. In (h) a schematic plot of different temperatures and interactions is presented as explained in the text. NRG values of the gap in ${\rho }_d\left(\omega \right)$ at $\omega =0$ and $T⪡T_K^0$ are presented with open circles and squares for $t_d∕D=0.001$ and 0.01, respectively. Values of $J_{\mathrm{eff}}$ and analytical results of $T_K^0$ are presented as dashed and dot-dashed lines for $t_d∕D=0.001$ and 0.01, respectively. For analytical estimates of $T_K^0$ different values of $\alpha =2.2$ and 1.3, respectively, were used. Other parameters of the model are $\Gamma ∕D=0.03$, $U∕D=1$.

Figure 12

(Color online) Dynamic magnetic susceptibility $k_{B}T\chi \left(T\right)∕{\left(g{\mu }_B\right)}^2$ for different $\kappa$. Parameters are $U∕D=1$, $\Gamma ∕D=0.03$, $t_d∕D=8\times {10}^{-5}$, and $\delta ∕D=0$. These calculations were performed using a different NRG discretization scheme (Ref. 29) with $\Lambda =4$ and keeping 600 states.