Fano-Kondo effect in side-coupled double quantum dots at finite temperatures and the importance of two-stage Kondo screening

We study the zero-bias conductance through the system of two quantum dots, one of which is embedded directly between the source and drain electrodes, while the second dot is side coupled to the first one through a tunneling junction. Modeling the system using the two impurity Anderson model, we compute the temperature dependence of the conductance in various parameter regimes using the numerical renormalization group. We consider the noninteracting case, where we study the extent of the departure from the conventional Fano resonance line shape at finite temperatures, and the case where the embedded and/or the side-coupled quantum dot is interacting, where we study the consequences of the coexistence of the Kondo and Fano effects. If the side-coupled dot is very weakly interacting, the occupancy changes by two when the on-site energy crosses the Fermi level and a Fano-resonance-like shape is observed. If the interaction on the side-coupled dot is sizeable, the occupancy changes only by one and a very different line-shape results, which is strongly and characteristically temperature dependent. These results suggest an intriguing alternative interpretation of the recent experimental results study of the transport properties of the side-coupled double quantum dot [Sasaki et al., Phys. Rev. Lett. 103, 266806 (2009)]: the observed Fano-like conductance antiresonance may, in fact, result from the two-stage Kondo effect in the regime where the experimental temperature is between the higher and the lower Kondo temperature.

Figure 1

(Color online) Schematic representation of the double quantum dot nanostructure consisting of a quantum dot (d), embedded between the source and drain electrodes, and a side-coupled quantum dot (a).

Figure 2

(Color online) Noninteracting model with $U_a=U_d=0$, symmetric case, ${ϵ}_d=0$. (a) Conductance curves for a range of temperatures. Fano-resonance-line fits overlap completely with the conductance curves. (b) Fano parameters as a function of the temperature.

Figure 3

(Color online) Noninteracting model with $U_a=U_d=0$, asymmetric case, ${ϵ}_d\ne 0$. (a) Conductance curves for a range of temperatures. Full curves: NRG results. Dashed lines: Fano resonance line fits. (b) Fano parameters as a function of the temperature. Parameters are extracted by curve fitting in the energy window which corresponds to the horizontal axis in the upper subfigure.

Figure 4

(Color online) Resonance curves for noninteracting (full lines) and interacting models (dashed lines) for a range of scaled temperatures $T/{\Gamma }_a$ and for different ${\Gamma }_a/T_K^0$ ratios.

Figure 5

(Color online) Interacting model with $U_d\ne 0$ and $U_a=0$. Symmetric case, ${ϵ}_d=0$. (a) Conductance curves for a range of temperatures. (b) Fano parameters as a function of the temperature. Curve fitting is performed in the energy window, which corresponds to the horizontal axis in the upper subfigure.

Figure 6

(Color online) Temperature dependence of the conductance for ${ϵ}_a=0$, i.e., at the bottom of the conductance antiresonance, for a range of the interdot coupling strengths.

Figure 7

(Color online) Comparison of the temperature dependence of the conductance $G\left(T\right)$ with the energy dependence of the zero-temperature spectral function on the embedded dot $A_d\left(\omega ,T=0\right)$. For completeness, the spectral function on the side-coupled dot $A_a\left(\omega ,T=0\right)$ is also shown. Spectral functions are rescaled in units of the respective effective hybridization strength, i.e., by $1/\pi {\Gamma }_i$, where $i∊\left\{a,d\right\}$. Conductance curves are rescaled in units of the conductance quantum $G_0=2e^2/h$. The horizontal axis is rescaled in units of ${\Gamma }_a=t^2/{\Gamma }_d$ (note that ${\Gamma }_a$ is different in each subfigure).

Figure 8

(Color online) Conductance curves and level occupancies at $T=0$ for a range of electron-electron interaction strengths $U_a$. Left panels: noninteracting embedded impurity. Right panels: interacting embedded impurity. The arrows show the direction of increasing $U_a$.

Figure 9

(Color online) Comparison of the temperature dependence of the conductance $G\left(T\right)$ with the energy dependence of the zero-temperature spectral function on the embedded dot $A_d\left(\omega ,T=0\right)$.

Figure 10

(Color online) (a) Conductance curves for the fully interacting case with $U_d/D=U_a/D=1$, ${\delta }_d=0$, for a range of different temperatures. (b) Temperature dependence of the conductance for a range of energies of the side-coupled dot.

Figure 11

(Color online) Conductance curves for the fully interacting case with $U_d/D=U_a/D=1$, ${\delta }_d=0$, for increasing interdot tunneling $t$. The arrow indicates the direction of the increasing temperature $T$.