Nonlocal Thermoelectric Effects and Nonlocal Onsager relations in a Three-Terminal Proximity-Coupled Superconductor-Ferromagnet Device

We study thermal and charge transport in a three-terminal setup consisting of one superconducting and two ferromagnetic contacts. We predict that the simultaneous presence of spin filtering and of spin-dependent scattering phase shifts at each of the two interfaces will lead to very large nonlocal thermoelectric effects both in clean and in disordered systems. The symmetries of thermal and electric transport coefficients are related to fundamental thermodynamic principles by the Onsager reciprocity. Our results show that a nonlocal version of the Onsager relations for thermoelectric currents holds in a three-terminal quantum coherent ferromagnet-superconductor heterostructure including a spin-dependent crossed Andreev reflection and coherent electron transfer processes.

Figure 1

(a) The device consisting of two ferromagnets (regions to the left and right in blue) and a superconductor (the green region in the center). Trajectories for electrons (black) and holes (red) illustrate possible transport processes in the ballistic case, as discussed in the text (white arrows denote the spin). (b) Equivalent circuit diagram of the setup shown in (a) for the diffusive limit including the coherence leakage [41]. The interface parameters are discussed in detail beneath Eq. (3).

Figure 2

Density of states $D$ in the contact region for $G_1=G_2=0.1G_S$, $G_1^{\mathrm{MR}}=G_2^{\mathrm{MR}}=0.005G_S$ (10% polarization), and ${\eta }_{\mathrm{Th}}\equiv {\epsilon }_{\mathrm{Th}}{G}_S/G_q=0.5{\Delta }_0$ (with the Thouless energy ${\epsilon }_{\mathrm{Th}}$ of the contact region). (a) Total DOS depending on the spin-mixing term $G^{\varphi }$ for equal ferromagnets. The $G^{\varphi }$ term splits the pseudogap into the different spin directions. (b) shows the asymmetry in the SDOS for spin-down (the spin-up SDOS looks equal but mirrored at the $\epsilon =0$ axis).

Figure 3

Nonlocal thermopower $\mathcal{S}=L_{12}^{qT}/T_S{L}_{11}^{qV}$ for a symmetric setup as a function of polarization $\mathcal{P}$ and a spin-mixing parameter in the (a) clean and (b) dirty limits for $T=T_S=0.1T_c$. We assume equally polarized channels, ${\mathcal{P}}_n\equiv \mathcal{P}$. In (a), ${\mathcal{T}}_{n1}\equiv {\mathcal{T}}_1=0.1={\mathcal{T}}_2\equiv {\mathcal{T}}_{n2}$, $L=0.5{\xi }_0$, $\delta {\Omega }_1=\delta {\Omega }_2=\pi /20$; in (b), $G_1=G_2=0.1G_S$ and ${\eta }_{\mathrm{Th}}=0.5{\Delta }_0$. $\mathcal{S}$ is plotted in units of $g{k}_{\mathrm{B}}/|q|$, where $g=-{\mathcal{T}}_2(1+{\mathcal{P}}^2)\delta {\Omega }_2/2\pi$ in the clean limit and $g=-G_2/(G_2+G_S)$ in the dirty limit.

Figure 4

Temperature dependence of local and nonlocal thermoelectric coefficients for a symmetric setup in the clean and dirty limits for various spin-mixing parameters $\delta \phi$ and $G^{\varphi }$. Both coefficients are normalized to the normal state value of the nonlocal conductance $(L_{12}^{qV}{)}_{T>T_c}$ and are plotted in units of $k_{\mathrm{B}}{T}_c/|q|$. Here, ${\mathcal{P}}_n\equiv \mathcal{P}=0.1$, ${\eta }_{\mathrm{Th}}={\Delta }_0$, and all other parameters are the same as in Fig. 3.