Predicted Very Large Thermoelectric Effect in Ferromagnet-Superconductor Junctions in the Presence of a Spin-Splitting Magnetic Field

We show that a huge thermoelectric effect can be observed by contacting a superconductor whose density of states is spin split by a Zeeman field with a ferromagnet with a nonzero polarization. The resulting thermopower exceeds $k_B/e$ by a large factor, and the thermoelectric figure of merit $ZT$ can far exceed unity, leading to heat engine efficiencies close to the Carnot limit. We also show that spin-polarized currents can be generated in the superconductor by applying a temperature bias.

Figure 1

Top: schematic systems studied in this work. In both of them a ferromagnet (F) is coupled via a tunneling contact to a thin-film superconductor (S), whose tunneling density of states is modified by an exchange field. In (a) the exchange field is induced by the proximity of a ferromagnetic insulator (FI), whereas in (b) it is induced by the Zeeman energy due to an applied magnetic field $\mathrm{B⃗}$ parallel with the easy axis of the ferromagnet. Bottom: tunneling densities of states for spin $\uparrow /\downarrow$, averaged over spin ($N_0$), and the difference of them ($N_z$) obtained for an exchange field $h=\mathrm{\Delta }/2$.

Figure 2

Thermoelectric coefficients vs exchange field $h$ at $k_{B}T/{\mathrm{\Delta }}_0=0.1$ (black lines), 0.2 (blue lines) and 0.3 (red lines). From top left to bottom right: conductance, heat conductance, thermoelectric coefficient, and thermopower. The solid lines are numerical integrals of Eqs. (5), the dashed lines are the approximations in Eqs. (6)),(8). The curves have been calculated for $\mathrm{\Gamma }={10}^{-6}{\mathrm{\Delta }}_0$. ${\mathrm{\Delta }}_0$ is the superconducting order parameter at $T=0$ and $h=0$.

Figure 3

Figure of merit $ZT$ as a function of exchange field at $k_{B}T/{\mathrm{\Delta }}_0=0.1$ (black lines), 0.2 (blue lines), and 0.3 (red lines), and $P=0.9$. The solid lines are the exact results and the dashed lines the results obtained from Eq. (9). The dotted line indicates the zero-temperature limit $ZT=P^2/(1-P^2)$.

Figure 4

Maximum power $\stackrel{.}{W}={\max }_V[-IV]$ generated by the FIS junction from a temperature difference $T_F-T_S$ (solid line), and the corresponding heat engine efficiency $\eta$ (dashed line). We fix $P=1.0$, $k_B{T}_S=0.2{\mathrm{\Delta }}_0$, $h=0.6{\mathrm{\Delta }}_0$, and $\mathrm{\Gamma }={10}^{-5}{\mathrm{\Delta }}_0$. The linear-response result $\eta ={\eta }_{\mathrm{CA}}ZT/(ZT+2)$ for $ZT=4.04$ and the Carnot efficiencies ${\eta }_C=1-T_{\text{cold}}/T_{\text{hot}}$ are also shown (dotted lines).