Electron refrigeration in hybrid structures with spin-split superconductors

Electron tunneling between superconductors and normal metals has been used for an efficient refrigeration of electrons in the latter. Such cooling is a nonlinear effect and usually requires a large voltage. Here we study the electron cooling in heterostructures based on superconductors with a spin-splitting field coupled to normal metals via spin-filtering barriers. The cooling power shows a linear term in the applied voltage. This improves the coefficient of performance of electron refrigeration in the normal metal by shifting its optimum cooling to lower voltage, and also allows for cooling the spin-split superconductor by reverting the sign of the voltage. We also show how tunnel coupling spin-split superconductors with regular ones allows for a highly efficient refrigeration of the latter.

Figure 1

Schematic general setup: An island (blue) with quasiparticle temperature $T_{\text{qp}}$ is electrically connected to two electrodes with quasiparticle temperature $T_{\text{R}}$ through a spin-filtering thin layer of a ferromagnetic insulator. We assume that the film phonons strongly thermalize to the substrate and, thus, their temperature equals $T_{\text{bath}}$. $\dot{Q}$ stands for the cooling power of the island and white arrows show the spin polarization at each FI.

Figure 2

(Top) Cooling power ${\dot{Q}}_N$ of the metallic electrode as a function of the bias voltage at $T_N=T_S=0.2{\mathrm{\Delta }}_0$ for different values of the exchange field $h$. (Bottom) Cooling power optimized vs. bias voltage as a function of the temperature of the junction $T=T_N=T_S$. $V_{\text{opt}}$ stands for the applied bias for which ${\dot{Q}}^{\mathrm{opt}}$ is obtained.

Figure 3

(Top) Cooling power ${\dot{Q}}_S$ of the spin-split superconductor as a function of the bias voltage for different values of the exchange field $h$, at $T_S=T_N=0.2{\mathrm{\Delta }}_0$. (Bottom) Cooling power for optimal bias voltage $V_{\mathrm{opt}}$, as a function of the temperature of the junction $T=T_N=T_S$.

Figure 4

(Top) Final temperature of the superconducting island vs. the voltage across the structure for $T_{\text{bath}}=0.15{\mathrm{\Delta }}_0$. (Bottom) Optimum value of the relative refrigeration efficiency, ${\eta }_R\equiv (T_{\mathrm{bath}}-T_{qp})/T_{\mathrm{bath}}$, in terms of phonon temperature. Both values are obtained for different spin-splitting amplitudes and a qp-ph interaction parametrized by $\tilde{\mathrm{\Sigma }}=300$.

Figure 5

Steady-state temperature in the metallic island attached to two SS electrodes as a function of the bias voltage $V$ across the structure for different exchange fields $h$. Here we disregard the heating of the electrodes. The qp-ph interaction in the normal metal is parametrized by ${\tilde{\mathrm{\Sigma }}}_N=300$.

Figure 6

(Top) Steady-state electronic temperature of the metallic island between two finite spin-split superconductors as a function of the bias voltage $V$ across the whole structure. (Bottom) Optimum value of the relative refrigeration efficiency as a function of phonon temperature. Results are obtained for different values of $h$ in a particular case where ${\tilde{\mathrm{\Sigma }}}_{SS}={\tilde{\mathrm{\Sigma }}}_N$.

Figure 7

Cooling power ${\dot{Q}}_{S^{\prime }}$ as a function of the applied voltage for different exchange fields and order parameters ${\mathrm{\Delta }}_0^{\prime }={\mathrm{\Delta }}^{\prime }(T=0)$ of the superconducting island. $T_{\mathrm{bath}}=0.2{\mathrm{\Delta }}_0$ is set in all the plots.

Figure 8

Steady-state temperatures in the island as a function of the bias voltage for different values of exchange fields and superconducting order parameters ${\mathrm{\Delta }}_0^{\prime }$. We set $T_{\mathrm{bath}}=0.2{\mathrm{\Delta }}_0$, disregard heating of the electrodes, and parametrize the qp-ph interaction in the island by ${\mathrm{\Sigma }}_{S^{\prime }}=300$.