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

We investigate the heat generation.. in a quantum dot (QD), coupled to a normal metal and a superconductor, without electric bias voltage. It is found that.. is quite sensitive to the lead temperatures.. T-L,T- R and the superconductor gap magnitude..Delta At.. T-L,T-R <<.omega(0) (omega(0) is the phonon frequency), the superconductor affects.. only at.Delta < omega(0), and the maximum magnitude of negative.. appears at some Delta slightly smaller than.. 0. At elevated lead temperature, contribution to.. from the superconductor arises at Delta, ranging from less than to much larger than.. 0. However, the peak value of.. is several times smaller than that in the case of.. T-L,T-R <<omega(0). Interchanging lead temperatures.. L and.. R leads to quite different.. 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.