Phase-dependent heat and charge transport through superconductor–quantum dot hybrids

We analyze heat and charge transport through a single-level quantum dot coupled to two BCS superconductors at different temperatures to first order in the tunnel coupling. In order to describe the system theoretically, we extend a real-time diagrammatic technique that allows us to capture the interplay between superconducting correlations, strong Coulomb interactions, and nonequilibrium physics. We find that a thermoelectric effect can arise due to the superconducting proximity effect on the dot. In the nonlinear regime, the thermoelectric current can also flow at the particle-hole symmetric point due to a level renormalization caused by virtual tunneling between the dot and the leads. The heat current through the quantum dot is sensitive to the superconducting phase difference. In the nonlinear regime, the system can act as a thermal diode.

Figure 1

Schematic sketch of our setup. A single-level quantum dot is tunnel coupled to two superconducting electrodes at temperatures $T_{\text{L}}$ and $T_{\text{R}}$.

Figure 2

Linear-response charge current $I^e$ as a function of (a) phase difference $\varphi$ and (b) detuning $\delta$. Parameters are $U=4k_{\text{B}}T$ and $\mathrm{\Delta }=1.75k_{\text{B}}T$.

Figure 3

Linear-response heat current $I^h$ as a function of (a) phase difference $\varphi$ and (b) detuning $\delta$. Parameters as in Fig. 2.

Figure 4

Charge current $I^e$ in units of ${10}^{-3}e\mathrm{\Gamma }/\hslash$ as a function of phase difference $\varphi$ and detuning $\delta$. The red line indicates a vanishing charge current. Parameters are ${\mathrm{\Delta }}_0=2.32k_{\text{B}}{T}_{\text{L}},U=5k_{\text{B}}{T}_{\text{L}},{\mathrm{\Gamma }}_{\text{R}}=4{\mathrm{\Gamma }}_{\text{L}}$, and $T_{\text{R}}=T_{\text{L}}/2$.

Figure 5

Nonlinear heat current as a function of the asymmetry $a$. Parameters are ${\mathrm{\Delta }}_0=2.32k_{\text{B}}{T}_{\text{L}},U=4.64k_{\text{B}}{T}_{\text{L}},\delta =10\mathrm{\Gamma }$, and $T_{\text{R}}=0.1T_{\text{L}}$.

Figure 6

Diagrams corresponding to different transitions in our setup system. Horizontal lines describe the forward and backward propagation of the dot on the Keldysh contour. Dots indicate tunneling vertices. Dashed lines correspond to tunneling lines which arise from Wick contractions of reservoir operators. Due to the presence of superconducting leads there are both normal (a), (b) and anomalous (c), (d) tunneling lines.