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

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 $\alpha$ between the magnetizations of the two F layers. As expected, for values of $h$ much larger than the superconducting order parameter $\Delta$, the proximity effect is suppressed and the efficiency of refrigeration increases with respect to a NIS junction. However, for $h\sim \Delta$ 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 $h\approx \Delta$. 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 $\alpha$.

Figure 1

The SIF${}_1$F${}_2$N junction. The interface at $x=0$ corresponds to the insulating barrier (thick black line). Interfaces at $x=l_1$ and $x=l_{12}$ are fully transparent. $\alpha$ is the relative angle between the magnetization directions of F${}_1$ and F${}_2$.

Figure 2

Cooling power versus exchange field for different orientations of the exchange field vector in the second ferromagnetic layer F${}_2$: $\alpha =0$ (black solid line), $\alpha =\pi /2$ (blue dashed line), and $\alpha =\pi$ (red dash-dotted line), calculated at optimum bias; $W=7\times {10}^{-3}$, $T=0.25{\Delta }_0$, $l_1=\xi$, and $l_2=6\xi$. We have defined $\tilde{P}={10}^{2}P\left(V_{\mathrm{opt}}\right)e^2{R}_0/{\Delta }_0^2$.

Figure 3

The Andreev current as a function of (a) the exchange field for $l_2=6\xi$ and as a function of (b) the F${}_2$ length for $h=0.7\Delta \left(T\right)$. Different magnetic configurations are chosen: $\alpha =0$ (solid black line), $\alpha =\pi /2$ (dashed blue line), $\alpha =\pi$ (dash-dotted red line). The Andreev current is calculated at optimal bias; $W=7\times {10}^{-3}$, $T=0.25{\Delta }_0$, $l_1=\xi$. We have defined ${\tilde{I}}_A=I_A\left(V_{\mathrm{opt}}\right)e{R}_0/{\Delta }_0$.

Figure 4

Cooling power versus length $l_2$ of the F${}_2$ layer for (a) $h=0.7\Delta \left(T\right)$ and (b) $h=1.7\Delta \left(T\right)$. We consider different orientations of the exchange field vector in the second ferromagnetic layer F${}_2$ with respect to the one in F${}_1$: $\alpha =0$ (solid black line), $\alpha =\pi /2$ (dashed blue line), $\alpha =\pi$ (dash-dotted red line), and calculate the cooling power at optimal bias; $W=7\times {10}^{-3}$, $T=0.25{\Delta }_0$, and $l_1=\xi$. $\tilde{P}$ is defined in Fig. 2.

Figure 5

Cooling power versus bias voltage for $h=\Delta \left(T\right)$ (a), $h=1.7\Delta \left(T\right)$ (b), and $h=8\Delta \left(T\right)$ (c) for different orientations of the exchange field vector in the second ferromagnetic layer F${}_2$: $\alpha =0$ (black solid line), $\alpha =\pi /2$ (blue dashed line), and $\alpha =\pi$ (red dash-dotted line); $W=7\times {10}^{-3}$, $T=0.25{\Delta }_0$, $l_1=\xi$, and $l_2=6\xi$. $\tilde{P}$ is defined in Fig. 2.

Figure 6

Dependence of the cooling power on the tunneling parameter $W$ for $h=\Delta \left(T\right)$ (a), $h=1.7\Delta \left(T\right)$ (b), and $h=8\Delta \left(T\right)$ (c) and for different orientations of the exchange field vector in the second ferromagnetic layer F${}_2$: $\alpha =0$ (black solid line), $\alpha =\pi /2$ (blue dashed line), and $\alpha =\pi$ (red dash-dotted line). $P$ is calculated at optimum bias; $T=0.25{\Delta }_0$, $l_1=\xi$, and $l_2=6\xi$. We have defined $\overline{P}(W,V_{\mathrm{opt}})=P(W,V_{\mathrm{opt}})e^2{R}_0/{\Delta }_0^2$. Note the logarithmic scale.

Figure 7

Temperature dependence of the cooling power for $h={\Delta }_0$ (a), $h=1.7{\Delta }_0$ (b), and $h=8{\Delta }_0$ (c), and for different orientations of the exchange field vector in the second ferromagnetic layer F${}_2$: $\alpha =0$ (black solid line), $\alpha =\pi /2$ (blue dashed line), and $\alpha =\pi$ (red dash-dotted line). $P$ is calculated at optimum bias; $W=7\times {10}^{-3}$, $l_1=\xi$, and $l_2=6\xi$. $\tilde{P}$ is defined in Fig. 2.