Electron cooling in diffusive normal metal–superconductor tunnel junctions with a spin-valve ferromagnetic interlayer

Abstract

We investigate heat and charge transport through a diffusive SIF${}_{1}$F${}_{2}$N tunnel junction, where N (S) is a normal (superconducting) electrode, I is an insulator layer, and F${}_{1,2}$ are two ferromagnets with arbitrary direction of magnetization. The flow of an electric current in such structures at subgap bias is accompanied by a heat transfer from the normal metal into the superconductor, which enables refrigeration of electrons in the normal metal. We demonstrate that the refrigeration efficiency depends on the strength of the ferromagnetic exchange field $h$ and the angle $\ensuremath{\alpha}$ between the magnetizations of the two F layers. As expected, for values of $h$ much larger than the superconducting order parameter $\ensuremath{\Delta}$, the proximity effect is suppressed and the efficiency of refrigeration increases with respect to a NIS junction. However, for $h\ensuremath{\sim}\ensuremath{\Delta}$ the cooling power (i.e., the heat flow out of the normal metal reservoir) has a nonmonotonic behavior as a function of $h$ showing a minimum at $h\ensuremath{\approx}\ensuremath{\Delta}$. We also determine the dependence of the cooling power on the lengths of the ferromagnetic layers, the bias voltage, the temperature, the transmission of the tunneling barrier, and the magnetization misalignment angle $\ensuremath{\alpha}$.