Thermally Driven Out-of-Equilibrium Two-Impurity Kondo System

The archetypal two-impurity Kondo problem in a serially coupled double quantum dot is investigated in the presence of a thermal bias $\theta$. The slave-boson formulation is employed to obtain the nonlinear thermal and thermoelectrical responses. When the Kondo correlations prevail over the antiferromagnetic coupling $J$ between dot spins, we demonstrate that the setup shows negative differential thermal conductance regions behaving as a thermal diode. In addition, we report a sign reversal of the thermoelectric current $I(\theta )$ controlled by $t/\mathrm{\Gamma }$ ($t$ and $\mathrm{\Gamma }$ denote the interdot tunnel and reservoir-dot tunnel couplings, respectively) and $\theta$. All these features are attributed to the fact that at large $\theta$ both $Q(\theta )$ (heat current) and $I(\theta )$ are suppressed regardless of the value of $t/\mathrm{\Gamma }$ because the double dot decouples at high thermal biases. Finally, for a finite $J$, we investigate how the Kondo-to-antiferromagnetic crossover is altered by $\theta$.

Figure 1

Schematic of a tunnel-coupled DQD system. Each reservoir is coupled to its respective dot with the tunnel hybridization ${\mathrm{\Gamma }}_{L/R}$ and the dots are coupled with a tunnel amplitude $t$. The left reservoir is heated; meanwhile, the right reservoir remains at the background temperature $T_b$ giving rise to a temperature gradient $\theta$ in the system.

Figure 2

${\tilde{\mathrm{\Gamma }}}_{L/R}$ as a function of the thermal bias $\theta$. The left, middle, and right columns correspond to the weak ($t=0.25\mathrm{\Gamma }$), intermediate ($t=\mathrm{\Gamma }$), and the strong coupling regimes ($t=2.5\mathrm{\Gamma }$), respectively. The parameters are normalized by the common Kondo temperature $T_K$ (for the definition of $T_K$, see the main text). Parameters are $k_B=1$, ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }$, ${ϵ}_L={ϵ}_R=-3.5\mathrm{\Gamma }$, $T_b={10}^{-5}\mathrm{\Gamma }$, $D=100\mathrm{\Gamma }$, ${\mu }_{L,R}={ϵ}_F=0$.

Figure 3

(a) Heat current $Q(\theta )$ for different values of the tunnel coupling $t$. (b) Differential thermal conductance $K(\theta )=dQ/d\theta$ for different values of the tunnel coupling $t$. (c) Linear thermal conductance $K_0(t)$ normalized with the thermal conductance quantum $k_0$ as a function of the tunnel amplitude $t$. This result is equivalent to the transmission function at the Fermi level, $K_0(t)=k_0\mathcal{T}(\omega =0,t)$. Same parameters as in Fig. 2.

Figure 4

(a) Thermoelectric current $I(\theta )$ for different values of the interdot tunnel coupling $t$. For clarity, all curves are rescaled with the factors: ${10}^{-6}$, ${10}^{-5}$, $5\times {10}^{-5}$, and ${10}^{-4}$ for $t=0.25\mathrm{\Gamma }$, $\mathrm{\Gamma }$, $1.5\mathrm{\Gamma }$, and $t=2.5\mathrm{\Gamma }$, respectively. Panels (b) and (c) show transmission probabilities $\mathcal{T}(\omega )$ for $t/\mathrm{\Gamma }<1$ and $t/\mathrm{\Gamma }>1$, respectively. Two different $\theta$ values are considered. The dashed line in (c) indicates the position of local minimum showing that the transmission is not symmetric. Same parameters as in Fig. 2.

Figure 5

$J_c/k_B{T}_{KR}$ for several values of the interdot tunnel coupling $t/\mathrm{\Gamma }$. Inset: $J_c/k_B{T}_{KR}$ as a function of $t/\mathrm{\Gamma }$ at $\theta =3.1T_K$. Same parameters as in Fig. 2.