Local Heating in a Normal-Metal–Quantum-Dot–Superconductor System without Electric Voltage Bias

We investigate the heat generation Q in a quantum dot (QD), coupled to a normal metal and a superconductor, without electric bias voltage. It is found that Q is quite sensitive to the lead temperatures T_\rm L,R and the superconductor gap magnitude \it \Delta. At T_\rm L,R\ll \omega_0 (\omega_0 is the phonon frequency), the superconductor affects Q only at \it \Delta < \omega_0, and the maximum magnitude of negative Q appears at some \it \Delta slightly smaller than \omega_0. At elevated lead temperature, contribution to Q from the superconductor arises at \it \Delta, ranging from less than to much larger than \omega_0. However, the peak value of Q is several times smaller than that in the case of T_\rm L,R\ll \omega_0. Interchanging lead temperatures T_\rm L and T_\rm R leads to quite different Q behaviors, while this makes no difference for a normal-metal–quantum-dot–normal-metal system, and the QD can be cooled much more efficiently when the superconductor is colder.