Capacitive interactions and Kondo effect tuning in double quantum impurity systems

We present a study of the correlated transport regimes of a double quantum impurity system with mutual capacitive interactions. Such system can be implemented by a double quantum dot arrangement or by a quantum dot and nearby quantum point contact, with independently connected sets of metallic terminals. Many-body spin correlations arising within each dot-lead subsystem give rise to the Kondo effect under appropriate conditions. The otherwise independent Kondo ground states may be modified by the capacitive coupling, decisively modifying the ground state of the double quantum impurity system. We analyze this coupled system through variational methods and the numerical renormalization group technique. Our results reveal a strong dependence of the coupled system ground state on the electron-hole asymmetries of the individual subsystems, as well as on their hybridization strengths to the respective reservoirs. The electrostatic repulsion produced by the capacitive coupling produces an effective shift of the individual energy levels toward higher energies, with a stronger effect on the “shallower” subsystem (that closer to resonance with the Fermi level), potentially pushing it out of the Kondo regime and dramatically changing the transport properties of the system. The effective remote gating that this entails is found to depend nonlinearly on the capacitive coupling strength, as well as on the independent subsystem levels. The analysis we present here of this mutual interaction should be important to fully characterize transport through such coupled systems.

Figure 1

Illustration of the model system studied: two quantum impurities, $d$ and $q$, capacitively coupled with interaction strength $u$. Each impurity $i(=d,q)$ is coupled to its own set of metallic terminals with hybridization energy ${\Gamma }_i/2$.

Figure 2

(a) External gating of the impurity levels ${\Lambda }_d$ and ${\Lambda }_q$ for identical couplings $\left({\Gamma }_d={\Gamma }_q=0.002D\right)$, but asymmetric impurities, with ${\varepsilon }_d=-12.5{\Gamma }_d(=-0.025D)$, ${\varepsilon }_q=-5{\Gamma }_q(=-0.010D)$, and varying $u$, as obtained variationally. (b) Charge and (c) charge fluctuations of both impurities evaluated using the variational method and NRG, showing how impurity $q$ is depleted as the capacitive coupling $u$ is increased, while impurity $d$ remains essentially unchanged. The NRG calculations were carried out with local Coulomb interaction strengths of $U_d=U_q=0.05D(=25{\Gamma }_d)$.

Figure 3

Magnetic moment squared of impurities (a) $d$ and (c) $q$ as functions of the temperature, for different values of the capacitive coupling $u$, as shown in the legend in panel (b); all panels show results for the same set of $u$ values. (b) and (c) show the same quantities as functions of temperature, normalized to the Kondo temperature of each curve. The inset of (c) shows the dependence of the Kondo temperatures on $u$. All results obtained from NRG calculations. Parameters correspond to those of Fig. 2. Notice that for $u\gtrsim 0.01D(=|{\varepsilon }_q\left|\right)$, impurity $q$ is pushed away from the Kondo regime, first into mixed valence, eventually reducing $n_q$ well below unity (see Fig. 2).

Figure 4

Spectral densities of both impurities for varying capacitive coupling $u$, for parameters ${\Gamma }_d={\Gamma }_q=0.002D$, $U_d=U_q=25{\Gamma }_d$ and ${\varepsilon }_d=-U_d/2$, ${\varepsilon }_q=-U_q/5$. (a) Reference curves with $u/D=0$, showing the local levels dressed by their respective electronic reservoirs, and the ASR of each impurity at the Fermi level. Same $u=0$ curves shown in black in corresponding (b) and (c) panels. (b) The ASR of impurity $d$ remains well defined for all values of $u$. The panel focuses on the Hubbard bands, amplifying the lower region of the vertical axis for clarity (notice different scale). (c) In the case of impurity $q$, the disappearance of the ASR and the shift of the left Hubbard peak toward positive values of $\omega$ with increasing $u$ is a clear indication that the Kondo effect is being quenched. Inset in (c) shows detailed behavior of ${\rho }_{\sigma }^q\left(\omega \right)$ for $\omega \approx 0$. For $u\gtrsim |{\varepsilon }_q|$, the $q$ impurity is pushed out of the Kondo regime.

Figure 5

External gating of the impurity levels ${\Lambda }_d$ and ${\Lambda }_q$ as a function of ${\varepsilon }_q$, as obtained from the variational calculation. (a) For impurities with identical couplings $\left({\Gamma }_d={\Gamma }_q=0.002D\right)$, with fixed ${\varepsilon }_d=-12.5{\Gamma }_d$. (b) For different couplings $\left({\Gamma }_q=25{\Gamma }_d=0.05D\right)$, with fixed ${\varepsilon }_d=-10{\Gamma }_d$.

Figure 6

(a) Comparison of the spectral densities of both impurities (in the electron-hole symmetric regime, ${\varepsilon }_d=-U_d/2)$, when ${\Gamma }_q\gg {\Gamma }_d$ and $U_q\gg U_d$. (b) Conductances per spin channel (in units of $e^2/h)$ for impurities $d$ and $q$, and (c) their occupations, as functions of ${\varepsilon }_q$, for different values of ${\varepsilon }_d$. All results from NRG calculations; parameters: $U_q=10U_d=0.5D$, ${\Gamma }_q=25{\Gamma }_d=0.05D$, $u=0.01D(=U_q/50=U_d/5)$. Vertical dashed lines denote the region in the uncoupled regime $\left(u=0\right)$ for impurity $q$ $\left(\langle n_q\rangle =1\right)$ where the high conductance in the Kondo regime is expected.

Figure 7

Kondo temperatures $T_K^d$ and $T_K^q$ of impurities $d$ and $q$, respectively, as functions of ${\varepsilon }_q$, for three fixed values of ${\varepsilon }_d$. The curves terminate, indicating the approximate values of ${\varepsilon }_q$ at which the Kondo effect is quenched in each case. The dashed line on the left indicates the trend of the transition out of the Kondo regime for impurity $d$. All other parameters as in Fig. 6.

Figure 8

NRG calculations of (a) the conductance of impurities $d$ and $q$, and (b) the occupation of both impurities, as functions of ${\varepsilon }_d$, for different values of ${\varepsilon }_q$. Parameters: $U_q=10U_d=0.5D$, ${\Gamma }_q=25{\Gamma }_d=0.05D$, $u=0.01(=U_q/50=U_d/5)D$. Vertical dashed lines indicate the region $\langle n_d\rangle =1$ for the uncoupled case $\left(u=0\right)$. Notice the typical conductance plateau is shifted by $\approx u$ to lower ${\varepsilon }_d$ values due to the remote gating effect.

Figure 9

System regimes represented in the $u$-${\varepsilon }_q$ parameter space for (a) impurity $q$ and (b) impurity $d$, for fixed ${\varepsilon }_d=-U_d/2$, with $U_d=U_q=0.05D$ and ${\Gamma }_d={\Gamma }_q=0.002D(=0.04U_d)$. The Kondo temperatures of both impurities for select values of ${\varepsilon }_q$ are shown in the central panels, as functions of $u$. (c) and (d) show the system regimes in the ${\varepsilon }_d$-${\varepsilon }_q$ plane for impurities $q$ and $d$, respectively, for fixed $u=0.01D(=U_d/5)$, with $U_q=10U_d=0.5D$, and ${\Gamma }_q=25{\Gamma }_d=0.05D$. The shaded areas indicate that the corresponding impurity is in the Kondo regime, and their boundaries indicate steep but continuous crossovers. The Kondo temperatures are higher closer to the boundaries of the shaded regions.

Figure 10

Typical (a) conductance (per spin channel) and (b) charge and charge fluctuation profiles (full and empty circles, respectively) of an interacting quantum impurity, as functions of the level energy $\varepsilon$. (c) Kondo temperature as a function of $\varepsilon$. The shaded region represents the range of $\varepsilon /D$ where the Kondo effect takes place. Curves calculated using NRG. Parameters: $U_q=0.5D$, ${\Gamma }_q=0.05D$.