Kondo temperature and screening extension in a double quantum dot system

In this work we use the slave-boson mean-field approximation at finite $U$ to study the effects of spin-spin correlations in the transport properties of two quantum dots coupled in series to metallic leads. Different quantum regimes of this system are studied in a wide range of the parameter space. The main aspects related to the interplay between the half-filling Kondo effect and the antiferromagnetic correlation between the quantum dots are reviewed. Slave-boson results for conductance, local density of states in the quantum dots, and the renormalized energy parameters are presented. As a different approach to the Kondo physics in a double-dot system, the Kondo cloud extension inside the metallic leads is calculated and its dependence with the interdot coupling is analyzed. In addition, the cloud extension permits the calculation of the Kondo temperature of the double quantum dot. This result is very similar to the corresponding critical temperature $T_c$, as a function of the parameters of the system, as obtained by using the finite-temperature extension of the slave-boson mean-field approximation.

Figure 1

The figure sketches an artificial molecule consisting of two QDs with intradot Coulomb interaction $U$, and tunnel coupled to each other through the interdot matrix element $t_{\alpha \beta }$. In addition each QD is tunnel coupled to its adjacent metallic lead through the matrix element $t_{L\left(R\right)}$. The leads L and R act as charge reservoirs, being in thermodynamic equilibrium with the DQD.

Figure 2

(a) behavior of the renormalized local energy state ${\tilde{\epsilon }}_{\alpha \left(\beta \right)}$ of the QDs as a function of the gate potential $V_g$. The parameters used are $U=0.5$, $t^{\prime }=0.15$, Fermi energy $E_f=0$, and different magnitudes of $t_{\alpha \beta }$ (see legend). As $t_{\alpha \beta }$ increases, the plateau observed at ${\tilde{\epsilon }}_{\alpha \left(\beta \right)}=0$, characteristic of the Kondo regime, is gradually suppressed. Note that for the largest value of $t_{\alpha \beta }=0.45$ one has now two plateaus located at ${\tilde{\epsilon }}_{\alpha \left(\beta \right)}\approx \pm t_{\alpha \beta }$, which (as noted in Appendix ) are associated with the molecular Kondo regimes at quarter filling. (b) QDs LDOS for $V_g=-U/2$, and the same $t_{\alpha \beta }$ values as in panel (a). For the smallest value of $t_{\alpha \beta }=0.025$ (black squares) a splitting of the Kondo resonance is already observed. Note that for $t_{\alpha \beta }=0.125$ (green inverted triangles), the separation between the peaks is already close to $2t_{\alpha \beta }$, which is the separation expected for single-particle molecular orbitals, as mentioned in the Introduction.

Figure 3

Conductance as a function of $V_g$ for $t_{\alpha \beta }=0.025$ (black squares), $t_{\alpha \beta }=0.05$ (red circles), $t_{\alpha \beta }=0.075$ (blue triangles), $t_{\alpha \beta }=0.125$ (green inverted triangles), $t_{\alpha \beta }=0.25$ (magenta diamonds), and $t_{\alpha \beta }=0.45$ (dark yellow left triangles). The other parameters are $t^{\prime }=0.15$ and $U=0.5$.

Figure 4

Renormalization parameter $Z^2$ as a function of $V_g$ for the same $t_{\alpha \beta }$, $U$, and $t^{\prime }$ values as in Fig. 3. The inset shows the small renormalization of $Z$ characteristic of the molecular Kondo regime.

Figure 5

The figure shows the effect produced by the presence of the impurity $\alpha \left(\beta \right)$ in the LDOS calculated at the site $N=50$ [inside the lead L(R)]. The black squares show the LDOS (at site $N=50$) for the isolated lead; the red circles show the LDOS when each lead is connected to its respective QD, for $t_{\alpha \beta }=0$; and the blue triangles show the LDOS for the impurity.

Figure 6

The figure shows the effect produced by the antiferromagnetic interaction $J=4t_{\alpha \beta }^2/U$ between the QDs in the LDOS at site $N=50$ inside the leads. The black inverted triangles show the LDOS calculated at $N=50$ for $t_{\alpha \beta }=0$. The red triangles, blue circles, and cyan inverted triangles are obtained for $t_{\alpha \beta }=0.03$, $t_{\alpha \beta }=0.06$, and $t_{\alpha \beta }=0.09$, respectively. The magenta dashed curve shows the LDOS for the isolated leads ($t^{\prime }=0$). The other parameters are $V_g=-U/2$, $U=0.5$, and $t^{\prime }=0.2$.

Figure 7

Natural logarithm of the cloud extension function $F$ vs. the distance $N$ (from the border of the semi-infinite chain) for $t_{\alpha \beta }=0.0$ (black squares), $t_{\alpha \beta }=0.04$ (blue triangles), and $t_{\alpha \beta }=0.07$ (red circles). The straight lines are tangents to $\ln F$ and correspond to $t_{\alpha \beta }=0.01$ (purple line),$t_{\alpha \beta }=0.02$ (black line), $t_{\alpha \beta }=0.03$ (red line),..., $t_{\alpha \beta }=0.12$ (green line). The other parameters are $V_g=-U/2$, $U=0.5$, and $t^{\prime }=0.2$.

Figure 8

Renormalization parameter $Z^2$ as a function of temperature $T$ for values of $t_{\alpha \beta }$ ranging from $0.00$ (black leftmost curve) to $0.13$ (purple rightmost curve). The other parameters are $t^{\prime }=0.2$, $U=0.5$, $V_g=-U/2$, and $E_f=0.0$.

Figure 9

Kondo temperature $T_K^{KC}$ and $T_K^{\mathrm{Cut}}$ as a function of $t_{\alpha \beta }$. The red circles correspond to $T_K$ obtained from the extension $\xi$ of the Kondo cloud, while the black squares correspond to $T_K$ as obtained through the critical temperature $T_c$ for which the QD's are decoupled (see text), with $Z\to 0$. The insets show the exponential fits obtained for each $T_K$ in the region where the system is expected to be in the Kondo regime. The values for the other parameters are $V_g=-U/2$, $U=0.5$, and $t^{\prime }=0.2$.

Figure 10

Conductance as a function of $t_{\alpha \beta }$ for $V_g/U=-0.5$ (black squares), $V_g/U=-0.14$ (red circles), $V_g/U=0.0$ (blue triangles), $V_g/U=0.08$ (cyan inverted triangles), $V_g/U=0.22$ (magenta diamonds), $V_g/U=0.6$ (gold left-triangles), and $V_g/U=1.2$ (light blue right-triangles). The other parameters are $t^{\prime }=0.15$ and $U=0.5$.