Electron cooling by diffusive normal metal-superconductor tunnel junctions

We investigate heat and charge transport in NN'IS tunnel junctions in the diffusive limit. Here, N and S are massive normal and superconducting electrodes (reservoirs), N' is a normal metal strip, and I is an insulator. The flow of electric current in such structures at subgap bias is accompanied by heat transfer from the normal metal into the superconductor, which enables refrigeration of electrons in the normal metal. We show that the two-particle current due to Andreev reflection generates Joule heating, which is deposited in the N electrode and dominates over the single-particle cooling at low enough temperatures. This results in the existence of a limiting temperature for refrigeration. We consider different geometries of the contact, one dimensional and planar, which are commonly used in the experiments. We also discuss the applicability of our results to a double-barrier SINIS microcooler.