On-demand thermoelectric generation of equal-spin Cooper pairs

TL;DR

This paper proposes a quantum heat engine using a topological insulator junction to generate and control equal-spin Cooper pairs and supercurrents on-demand without magnetic manipulation, based on nonlocal thermoelectric effects.

Contribution

It introduces a novel thermoelectric mechanism in a S-F-S junction on a topological insulator to produce spin-triplet pairs without magnetic control, enabling on-demand supercurrents.

Findings

Purely nonlocal Andreev thermoelectric effect observed

Supercurrent can be switched on/off by phase tuning

Low fluctuations of thermoelectric current for small temperature gradients

Abstract

Superconducting spintronics is based on the creation of spin-triplet Cooper pairs in ferromagnet-superconductor (F-S) hybrid junctions. Previous proposals to manipulate spin-polarized supercurrents on-demand typically require the ability to carefully control magnetic materials. We, instead, propose a quantum heat engine that generates equal-spin Cooper pairs and drives supercurrents on-demand without manipulating magnetic components. We consider a S-F-S junction, connecting two leads at different temperatures, on top of the helical edge of a two-dimensional topological insulator. Heat and charge currents generated by the thermal bias are caused by different transport processes, where electron cotunneling is responsible for the heat flow to the cold lead and, strikingly, only crossed Andreev reflections contribute to the charge current. Such a purely nonlocal Andreev thermoelectric effect injects spin-polarized Cooper pairs at the superconductors, generating a supercurrent that can be switched on/off by tuning their relative phase. We further demonstrate that signatures of spin-triplet pairing are facilitated by rather low fluctuations of the thermoelectric current for temperature gradients smaller than the superconducting gap.

Equations

$$H_{\mathrm{BdG}}=H_0+H_{\mathrm{S}}+H_{\mathrm{F}},$$

$$I_{\mathrm{R}}^{he}$$

$$I_{\mathrm{R}}^{ee}$$

$$T_{\mathrm{RL}}^{he}(E,\varphi )=\gamma (E){\cos }^2\left[\varphi /2+\left(d_{\mathrm{NR}}-d_{\mathrm{NL}}\right)E\right],$$

$$T_{\mathrm{RL}}^{he}(-E)-T_{\mathrm{RL}}^{he}(E)=\gamma (E)\sin [2E(d_{\mathrm{NR}}-d_{\mathrm{NL}})]\sin \varphi .$$

$$\begin{gathered} S_{\mathrm{RR}}=e{I}_0{\int }_{-\infty }^{\infty }\frac{\mathrm{d}E}{{\Delta }_0}\sum _{\alpha ,\beta ,\gamma ,\delta ,k,l}\mathrm{sgn}(\alpha )\mathrm{sgn}(\beta ) \\ \times A_{k\gamma ,l\delta }(\mathrm{R}\alpha ,E)A_{l\delta ,k\gamma }(\mathrm{R}\beta ,E)f_{k\gamma }(E)\left[1-f_{l\delta }(E)\right], \end{gathered}$$

$$A_{k\gamma ,l\delta }(i\alpha ,E)={\delta }_{i,k}{\delta }_{i,l}{\delta }_{\alpha ,\gamma }{\delta }_{\alpha ,\delta }-{\left[s_{ik}^{\alpha \gamma }(E)\right]}^{\ast }{s}_{il}^{\alpha \delta }(E),$$

$$I_{\mathrm{R}}=\frac{e}{2h}{\int }_{-\infty }^{\infty }\mathrm{d}E\sum _{\alpha ,\beta ,j}\mathrm{sgn}(\alpha )\left[{\delta }_{\mathrm{R},j}{\delta }_{\alpha ,\beta }-T_{\mathrm{R}j}^{\alpha \beta }(E)\right]f_{j\beta }(E),$$

$$\sum _{j,\beta }{T}_{ij}^{\alpha \beta }=\sum _{i,\alpha }{T}_{ij}^{\alpha \beta }=1,$$

$$\begin{array}{c}{I}_{\mathrm{R}}=\frac{e}{2h}{\int }_{-\infty }^{\infty }\mathrm{d}E\{\left[1-T_{\mathrm{RR}}^{ee}(E)+T_{\mathrm{RR}}^{he}(E)\right]\left[f_{\mathrm{R}e}(E)-f_0(E)\right]-\left[1-T_{\mathrm{RR}}^{hh}(E)+T_{\mathrm{RR}}^{eh}(E)\right]\left[f_{\mathrm{R}h}(E)-f_0(E)\right]\\ +\left[T_{\mathrm{RL}}^{he}(E)-T_{\mathrm{RL}}^{ee}(E)\right]\left[f_{\mathrm{L}e}(E)-f_0(E)\right]-\left[T_{\mathrm{RL}}^{eh}(E)-T_{\mathrm{RL}}^{hh}(E)\right]\left[f_{\mathrm{L}h}(E)-f_0(E)\right]\}.\end{array}$$

$$\begin{array}{c}{I}_{\mathrm{R}}=\frac{e}{h}{\int }_{-\infty }^{\infty }\mathrm{d}E\left\{\left[1-T_{\mathrm{RR}}^{ee}(E)+T_{\mathrm{RR}}^{he}(E)\right]\left[f_{\mathrm{R}e}(E)-f_0(E)\right]+\left[T_{\mathrm{RL}}^{he}(E)-T_{\mathrm{RL}}^{ee}(E)\right]\left[f_{\mathrm{L}e}(E)-f_0(E)\right]\right\}.\end{array}$$

$$I_{\mathrm{R}}^{he}=I_0{\int }_{-\infty }^{\infty }\frac{\mathrm{dE}}{{\Delta }_0}{T}_{\mathrm{RL}}^{he}(E)\left[f_{\mathrm{L}e}(E)-f_0(E)\right],$$

$$I_{\mathrm{R}}^{ee}=-I_0{\int }_{-\infty }^{\infty }\frac{\mathrm{dE}}{{\Delta }_0}{T}_{\mathrm{RL}}^{ee}(E)\left[f_{\mathrm{L}e}(E)-f_0(E)\right],$$

$$\hat{I}(x)=\frac{1}{2}e{v}_{\mathrm{F}}{\Psi }^{†}\mathrm{diag}\left(1,-1,1,-1\right)\Psi ,$$

$$\Psi (x)=\sum _{i\in \{1,2,3,4\}}{\int }_{-\infty }^{\infty }\frac{\mathrm{d}E}{\sqrt{h{v}_F}}{\varphi }_i(x){\hat{a}}_i,$$

$$⟨\hat{I}(x)⟩=\frac{e}{2h}{\int }_{-\infty }^{\infty }\mathrm{d}E\sum _i\left\{|u_{\uparrow ,i}(x){|}^2{f}_i-|u_{\downarrow ,i}(x){|}^2{f}_i+|v_{\downarrow ,i}(x){|}^2{f}_i-|v_{\uparrow ,i}(x){|}^2{f}_i\right\},$$

$$\langle S\rangle=-2e\Delta_{0}\sum_{i\in\{1,2,3,4\}}\int\mathrm{d}E\hskip 2.15277pt\Big{[}\operatorname{Im}(\mathrm{e}^{-\mathrm{i}\phi}u_{\uparrow,i}^{*}v_{\downarrow,i})+\operatorname{Im}(\mathrm{e}^{-\mathrm{i}\phi}u_{\downarrow,i}^{*}v_{\uparrow,i})\Big{]}f_{i}.$$

$$\int_{x_{0}}^{x}\mathrm{d}x^{\prime}\hskip 2.15277pt\langle S\rangle=-2e\Delta_{0}\int_{x_{0}}^{x}\mathrm{d}x^{\prime}\hskip 2.15277pt\sum_{i\in\{1,2,3,4\}}\int\mathrm{d}E\hskip 2.15277pt\Big{[}\operatorname{Im}(\mathrm{e}^{-\mathrm{i}\phi}u_{\uparrow,i}^{*}(x^{\prime})v_{\downarrow,i}(x^{\prime}))+\operatorname{Im}(\mathrm{e}^{-\mathrm{i}\phi}u_{\downarrow,i}^{*}(x^{\prime})v_{\uparrow,i}(x^{\prime}))\Big{]}f_{i}.$$

$$\left(\begin{array}{c}{b}_e^{(1)}\\ b_h^{(1)}\\ b_e^{(6)}\\ b_h^{(6)}\end{array}\right)=S\left(\begin{array}{c}{a}_e^{(1)}\\ a_h^{(1)}\\ a_e^{(6)}\\ a_h^{(6)}\end{array}\right),$$

$$S=\left(\begin{array}{ccc}R& & T^{\prime }\\ T& & R^{\prime }\end{array}\right)\text{and}R=\left(\begin{array}{ccc}{r}_{\mathrm{LL}}^{ee}& & r_{\mathrm{LL}}^{eh}\\ r_{\mathrm{LL}}^{he}& & r_{\mathrm{LL}}^{hh}\end{array}\right),R^{\prime }=\left(\begin{array}{ccc}{r}_{\mathrm{RR}}^{ee}& & r_{\mathrm{RR}}^{eh}\\ r_{\mathrm{RR}}^{he}& & r_{\mathrm{RR}}^{hh}\end{array}\right),T=\left(\begin{array}{ccc}{t}_{\mathrm{RL}}^{ee}& & t_{\mathrm{RL}}^{eh}\\ t_{\mathrm{RL}}^{he}& & t_{\mathrm{RL}}^{hh}\end{array}\right),T^{\prime }=\left(\begin{array}{ccc}{t}_{\mathrm{LR}}^{ee}& & t_{\mathrm{LR}}^{eh}\\ t_{\mathrm{LR}}^{he}& & t_{\mathrm{LR}}^{hh}\end{array}\right).$$

$$\left(\begin{array}{c}{b}_e^{(1)}\\ b_h^{(1)}\\ p_e^{(2)}\\ p_h^{(2)}\end{array}\right)$$

$$\left(\begin{array}{c}{m}_e^{(2)}\\ m_h^{(2)}\\ p_e^{(3)}\\ p_h^{(3)}\end{array}\right)$$

$$\left(\begin{array}{c}{m}_e^{(3)}\\ m_h^{(3)}\\ p_e^{(4)}\\ p_h^{(4)}\end{array}\right)$$

$$\left(\begin{array}{c}{m}_e^{(4)}\\ m_h^{(4)}\\ p_e^{(5)}\\ p_h^{(5)}\end{array}\right)$$

$$\left(\begin{array}{c}{m}_e^{(5)}\\ m_h^{(5)}\\ b_e^{(6)}\\ b_h^{(6)}\end{array}\right)$$

$${\varphi }_{+}^e(x)={\mathrm{e}}^{\mathrm{i}Ex}\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),{\varphi }_{-}^e(x)={\mathrm{e}}^{-\mathrm{i}Ex}\left(\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right),{\varphi }_{-}^h(x)={\mathrm{e}}^{-\mathrm{i}Ex}\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),{\varphi }_{+}^h(x)={\mathrm{e}}^{\mathrm{i}Ex}\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right),$$

$$t_{\mathrm{RL}}^{he}=\frac{t_1+t_2}{1-\Sigma }=\left(t_1+t_2\right)\sum _{n=0}^{\infty }{\Sigma }^n,$$

$$\Sigma ={\Sigma }_1^{\mathrm{L}}+{\Sigma }_1^{\mathrm{R}}-{\Sigma }_1^{\mathrm{L}}{\Sigma }_1^{\mathrm{R}}+{\Sigma }_2^{\uparrow }+{\Sigma }_2^{\downarrow }-{\Sigma }_2^{\uparrow }{\Sigma }_2^{\downarrow }+{\Sigma }_3^e+{\Sigma }_3^h.$$

$$t_1$$

$$t_2$$

$${\Sigma }_1^{\mathrm{L}}$$

$${\Sigma }_1^{\mathrm{R}}$$

$${\Sigma }_2^{\uparrow }$$

$${\Sigma }_2^{\downarrow }$$

$${\Sigma }_3^e$$

Full text

Introduction. — The new field of superconducting spintronics has emerged since the creation of spin-triplet Cooper pairs in experiments Eschrig (2010); Linder and Robinson (2015); Eschrig (2015). The development of spintronics had already benefited from the use of superconducting materials, resulting in longer spin lifetimes and energy-efficient components Yang et al. (2010); Hübler et al. (2012). Now, triplet supercurrents formed by spin-polarized Cooper pairs add the possibility of transporting a net spin component at zero resistance and thus pave the way for spintronic devices that are less liable to overheat Sosnin et al. (2006); Keizer et al. (2006); Robinson et al. (2010a, b); Khaire et al. (2010); Klose et al. (2012); Robinson et al. (2012, 2014); Srivastava et al. (2017); Diesch et al. (2018); Jeon et al. (2018). The key challenge in the field is the nonequilibrium and on-demand generation of equal-spin Cooper pairs in a viable fashion Banerjee et al. (2014); Bathen and Linder (2017); Breunig et al. (2018); Ouassou et al. (2018); He et al. (2019), desirably avoiding the complicated manipulation of magnetic components.

In this Letter, we propose a thermoelectric engine that creates spin-polarized Cooper pairs driving a supercurrent on demand from a temperature gradient. We consider a superconductor–ferromagnetic-insulator–superconductor (S-F-S) junction on top of the helical edge state of a quantum spin Hall insulator (QSHI) König et al. (2007); Brüne et al. (2010); Knez et al. (2011); Brüne et al. (2012); Reis et al. (2017); Kammhuber et al. (2017); Wu et al. (2018) connecting a hot and a cold bath, cf. Fig. 1(a). Only two microscopic transport processes couple the baths: quantum tunneling of electrons, known as electron cotunneling (EC), and crossed Andreev reflection (CAR). The QSHI edge states comprise one-dimensional Dirac fermions characterized by spin-momentum locking Wu et al. (2006); Xu and Moore (2006). Therefore, while EC amounts to a spin-polarized normal current, the peculiar transport properties of the helical edge states guarantee that CAR always converts electrons into holes with the same spin, creating equal-spin Cooper pairs at the superconductors Adroguer et al. (2010); Crépin et al. (2015); Keidel et al. (2018). Our key finding is that the nonlocal current becomes dominated by CAR processes, with almost complete suppression of the EC contribution. This is only possible due to a unique interference effect for CAR processes in our setup. As sketched in Fig. 1(b), CAR requires a spin-flip process at the central ferromagnet and an Andreev reflection at either the left or the right superconductor. In an asymmetric junction, the different phases acquired in each path constitute interference, making CAR transmission strongly asymmetric in energy and thus creating an Andreev-dominated thermoelectric current in the right lead, cf. Fig. 1(c,d).

Harvesting waste heat by quantum thermoelectric effects has become essential in modern nanoscale devices Roche et al. (2015). While tackling this problem in S-F hybrid junctions can lead to potentially strong thermoelectric effects Machon et al. (2013); Kalenkov and Zaikin (2014, 2015); Ozaeta et al. (2014); Hwang et al. (2016); Kolenda et al. (2016); Beiranvand and Hamzehpour (2017); Cao et al. (2015); Hwang et al. (2018); Bergeret et al. (2018), it requires a careful control of magnetic elements and usually features a low heat-to-supercurrent conversion. In this proposal, the unique interference of CAR processes, together with the spin-polarization induced by the helical edge state, creates a strong spin-polarized thermoelectric current that can be controlled by tuning the phase difference between the superconducting leads and does not rely on manipulating the ferromagnetic domain. We further demonstrate that the thermoelectric current is enhanced over its fluctuations for temperature gradients comparable to the superconducting gap, facilitating the experimental realization of our proposal by thermovoltage Kolenda et al. (2016); Shelly et al. (2016) or thermophase measurements across the junction Giazotto et al. (2015a, b).

Setup. — The spin polarization of nonlocal transport and the absence of backscattering at the helical edge of a QSHI is of great interest for traditional spintronics. Moreover, proximity-induced superconductivity and ferromagnetism can confine the helical edge states, opening new scattering channels Adroguer et al. (2010); Crépin et al. (2015); Keidel et al. (2018) that can lead to the emergence of Majorana bound states Fu and Kane (2009); Crépin et al. (2014); Crépin and Trauzettel (2014); Fleckenstein et al. (2018a) or exotic odd-frequency superconducting pairing Crépin et al. (2015); Cayao and Black-Schaffer (2017); Keidel et al. (2018); Fleckenstein et al. (2018b). Given recent advances in the experimental realization of helical edge states Reis et al. (2017); Kammhuber et al. (2017); Wu et al. (2018), hybrid structures like the one sketched in Fig. 1(a) are within reach: superconductors Knez et al. (2012); Hart et al. (2014); Sajadi et al. (2018); Fatemi et al. (2018) have been successfully coupled to QSHIs Wiedenmann et al. (2016); Bocquillon et al. (2017), and monolayer QSHIs provide a new promising platform to induce ferromagnetic order Wu et al. (2018); Fatemi et al. (2018). The observation of Majorana modes in helical hinge states of Bi(111) films under the influence of superconductivity and magnetic iron clusters has recently been reported in Ref. Jäck et al. (2019).

We theoretically describe the one-dimensional helical edge states of a QSHI in proximity to superconducting and ferromagnetic order by a Bogoliubov–de Gennes Hamiltonian in the Nambu basis $\Psi(x)=(\psi_{\uparrow},\psi_{\downarrow},\psi^{\dagger}_{\downarrow},-\psi^{\dagger}_{\uparrow})$ of the form ($\hbar\!=v_{\mathrm{F}}\!=\!1$)

[TABLE]

with $H_{0}\!=\!\hat{p}_{x}\hat{\tau}_{3}\hat{\sigma}_{3}-\mu\hat{\tau}_{3}\hat{\sigma}_{0}$ the Hamiltonian of the free helical edge, $H_{\mathrm{S}}\!=\![\Delta(x)\cos\phi(x)\hat{\tau}_{1}+\Delta(x)\sin\phi(x)\hat{\tau}_{2}]\hat{\sigma}_{0}$ the proximity-induced superconductivity, and $H_{\mathrm{F}}\!=\!\hat{\tau}_{0}\mathbf{m}(x)\!\cdot\!\bm{\sigma}\!\equiv\!\hat{\tau}_{0}(m_{\parallel}\cos\lambda\,\hat{\sigma}_{1}+m_{\parallel}\sin\lambda\,\hat{\sigma}_{2}+m_{z}\hat{\sigma}_{3})$ describing the effect of the ferromagnetic barrier. Here, $\hat{p}_{x}\!=\!-\mathrm{i}\partial_{x}$ and $\hat{\sigma}_{i}$ ($\hat{\tau}_{i}$) are Pauli matrices acting in spin (Nambu) space. We consider a system with two S regions (named SL and SR) separated by two normal regions (NL and NR) surrounding one ferromagnetic insulator (F); their respective widths are $d_{\mathrm{X}}$ for $\mathrm{X}\!\in\!\{\mathrm{SL,NL,F,NR,SR}\}$. The pair potential is assumed equal for both superconductors and constant, $\Delta(x)\!=\!\Delta_{0}$, a valid approximation as long as the Fermi wavelength in each superconductor is much smaller than the proximity-induced coherence length $\xi_{0}\!=\!1/\Delta_{0}$. For simplicity, we take the phase of the pair potential $\phi(x)\!=\!\phi$ in SR and zero otherwise. The F region is modeled by constant $m_{\parallel}(x)\!=\!m_{0}$ within F, and we choose $m_{z}\!=\!0$ since its effect can be absorbed in the phase difference $\phi$ between the superconductors Nilsson et al. (2008); Crépin et al. (2014); Keidel et al. (2018). Without loss of generality the angle $\lambda$ is set to zero. Finally, we assume that all regions reside at the same chemical potential, i.e., $\mu(x)\!=\!0$ everywhere.

In the following, we consider that all leads except $\mathrm{L}$ are at the same temperature 111To account for finite temperatures in the superconductors, the gap is approximated as $\Delta(T)\!\simeq\!\Delta_{0}\tanh(1.74\sqrt{T_{c}/T-1})$, with $T_{c}$ the critical temperature and $T$ the superconductor’s temperature. ($T_{\mathrm{SL}}\!=\!T_{\mathrm{SR}}\!=\!T_{\mathrm{R}}\!\equiv\!T_{0}$) and set $T_{\mathrm{L}}\!=\!T_{0}+\theta$, introducing the temperature difference $\theta$. The electric current in the right lead after a temperature bias is applied to the left lead is given by $I_{\mathrm{R}}\!=\!I_{\mathrm{R}}^{he}+I_{\mathrm{R}}^{ee}$, where 222See Supplemental Material, which includes References Anantram and Datta (1996); Sivan and Imry (1986); Bardas and Averin (1995); Linder and Bathen (2016); Butcher (1990); Lumbroso et al. (2018); Sigrist and Ueda (1991); Leggett (1975), at [URL will be inserted by publisher] for details of the derivation of the nonlocal current and the supercurrent, an analysis of the CAR interference, the calculation of the anomalous Green function, and more information on the current fluctuations in the right lead.

[TABLE]

with $T^{he}_{\mathrm{R}\mathrm{L}}$ the CAR probability, $T^{ee}_{\mathrm{R}\mathrm{L}}$ the EC probability, $I_{0}\!=\!e\Delta_{0}/h$, $\delta\!f(E)\!=\!f[E,0,k_{\mathrm{B}}(T_{0}+\theta)]-f(E,0,k_{\mathrm{B}}T_{0})$, and $f(E,\mu,\tau)\!=\!\{1+\exp[(E-\mu)/\tau]\}^{-1}$ the Fermi distribution function. The probabilities are obtained by solving the scattering problem defined by the solutions of Eq. 1 in every region Blonder et al. (1982); Büttiker (1992); Takane and Ebisawa (1992); Anantram and Datta (1996); Nazarov and Blanter (2009); Mello and Kumar (2004); Crépin et al. (2014, 2015); Keidel et al. (2018). Similarly, the current at each superconductor $\text{S}\!=\!\mathrm{SL,SR}$ is given by $I_{\text{S}}\!=\!\langle J_{\text{S}}\rangle\!+\!\int_{0}^{d_{\text{S}}}\mathrm{d}x\langle S_{\text{S}}\rangle$, where $J_{\text{S}}\!=\!e(\psi^{\dagger}_{\text{S}\uparrow}\psi_{\text{S}\uparrow}-\psi^{\dagger}_{\text{S}\downarrow}\psi_{\text{S}\downarrow})$ is the quasiparticle current operator and $S_{\text{S}}\!=\!\mathrm{i}e\Delta_{0}(\mathrm{e}^{-\mathrm{i}\phi_{\text{S}}}\psi^{\dagger}_{\text{S}\uparrow}\psi^{\dagger}_{\text{S}\downarrow}+\text{h.c.})$ the source term operator Note (2).

As we describe in detail below, an interference of CAR processes depicted in Fig. 1(b) leads to a particular thermoelectric effect, where the current can be completely dominated by equal-spin Andreev processes, see Fig. 2. At the same time, the energy current is only given by the symmetric part of the transmissions; therefore, it can be dominated by EC processes. Such a decoupling of transport processes for the heat and charge currents is a special feature of this setup.

Generation of equal-spin Cooper pairs. — Helicity determines that particles arriving to the right lead have the same spin polarization as the injected particles on the left lead. While this does not restrict the quantum tunneling of electrons through the junction (EC), CAR processes are only possible if injected electrons and transmitted holes have the same spin Adroguer et al. (2010); Crépin et al. (2015); Keidel et al. (2018). By breaking time-reversal symmetry, the $\mathrm{F}$ region facilitates equal-spin CAR processes. As sketched in Fig. 1(b), incident electrons can be transmitted as holes through the junction if at least one spin-flip process takes place at the $\mathrm{F}$ region and one Andreev reflection occurs at either superconductor. Crucially, scattering events involving an Andreev reflection at the right superconductor will acquire an extra phase $\phi$ and a phase shift $d_{\mathrm{NR}}E$ compared to the ones where the reflection takes place at $\mathrm{SL}$, which are only shifted by $d_{\mathrm{NL}}E$ (we measure $d_{\mathrm{NL,NR}}$ in units of $\xi_{0}$). The interference between these two processes is a unique property of CAR, not present in EC, resulting in an unusually strong asymmetry of the transmission probability with the energy, cf. Fig. 1(c).

CAR processes thus require proximity-induced equal-spin pairing Crépin et al. (2015); Keidel et al. (2018), which we analyze by computing the retarded Green function associated to Eq. 1 Crépin et al. (2015); Breunig et al. (2018); Keidel et al. (2018). In the basis defined above, the anomalous part of the retarded Green function is written as $\hat{G}^{\mathrm{R}}_{\mathrm{eh}}(x,x^{\prime},E)\!=\!f_{0}^{\mathrm{R}}(x,x^{\prime},E)\hat{\sigma}_{0}+\mathbf{f}^{\mathrm{R}}(x,x^{\prime},E)\cdot\mathbf{\hat{\sigma}}$, where $f_{0}^{\mathrm{R}}$ is the singlet and $\mathbf{f}^{\mathrm{R}}$ the vector of triplet amplitudes Note (2). To quantify the net spin carried by a Cooper pair, we define the polarization vector $\mathbf{p}(x,x^{\prime},E)\!=\!\mathrm{i}\mathbf{f}^{\mathrm{R}}\times\mathbf{f}^{\mathrm{R}*}$. Note that a finite triplet amplitude is not enough to obtain a polarization. That is the case without magnetic impurity $\mathrm{F}$, where we have $\mathbf{f}^{\mathrm{R}}\!=\!(0,0,f^{\mathrm{R}}_{3})$, but $\mathbf{p}\!=\!0$. By contrast, when time-reversal symmetry is broken by the $\mathrm{F}$ region ($m_{0}\!\neq\!0$), we find a finite axial polarization $\mathbf{p}(x,x,E)\!=\!(0,p_{2},0)$ evidencing that the Cooper pairs develop a net spin.

We show in Fig. 3(a) the polarization of Cooper pairs computed at the interface between each superconductor and the inner normal regions, see Fig. 1. First, polarizations at each superconductor have different signs, indicating that Cooper pairs with opposite net spin have been transferred to each superconductor. Second, the polarization is maximum for the resonant energies of the S-F-S junction. Resonant scattering at each S-F region always gives rise to zero-energy Majorana (quasi-)bound states, with additional finite energy Andreev states depending on the cavity’s width Fu and Kane (2009); Crépin et al. (2014, 2015); Keidel et al. (2018); Fleckenstein et al. (2018a). The hybridization between the bound states at each S-F cavity is controlled by the phase difference between the superconductors Nilsson et al. (2008); Keidel et al. (2018). This, in turn, allows for the control of the electric current through the junction.

Andreev-dominated thermoelectric effect. — Given a positive temperature gradient, we find that a finite thermoelectric current, $I_{\mathrm{R}}$, is completely dominated by Andreev processes when three requirements are fulfilled (see Fig. 2):

(i) the base temperature $T_{0}$ is sufficiently large, i.e., $T_{0}\gtrsim T_{c}/2$;

(ii) the junction is asymmetric, which we realize by setting $d_{\mathrm{NL}}\neq d_{\mathrm{NR}}$; and

(iii) the phase difference $\phi$ is not an integer multiple of $\pi$.

Under these conditions, the energy asymmetry of the CAR transmission is comparable to the energy-antisymmetric bias $\delta\!f$ as illustrated in Fig. 1(c), whereas the asymmetry in the EC probability occurs on a much smaller energy scale 333The physical origin of the asymmetry of EC is the spin-splitting of bound states, and it is thus of the order of the hybridization energy. By contrast, the asymmetry in CAR is the result of an interference effect. Note that the location of the extremes of $\delta\!f$, and thus the width of the integration window in Eq. 2, are mainly determined by the base temperature $T_{0}$.. As a result, the Andreev current $I_{\mathrm{R}}^{he}$ becomes much larger than the EC current $I_{\mathrm{R}}^{ee}$ as the temperature grows. The CAR contribution is suppressed as the base temperature approaches $T_{c}$, where the induced gap vanishes. It is a good consistency check that simultaneously $I_{\mathrm{R}}^{ee}\!\rightarrow\!0$, since without superconductivity the resonant tunneling at the S-F regions disappears and so does the thermoelectric effect [Here; wediscussthecaseofaferromagneticinsulator; asopposedtoanotherrecentproposaltousetheedgestatesofaquantumspinHallinsulatorforefficientthermoelectricity; ]Gresta2019.

The interference effect of CAR processes is caused by an asymmetric S-F-S junction. When $d_{\mathrm{NL}}\!=\!d_{\mathrm{NR}}$, $I_{\mathrm{R}}^{he}$ vanishes since the two paths in Fig. 1(b) destructively interfere. In general, these two contributions acquire a different energy-dependent phase coming from the fact that the Andreev reflection for each path takes place at different superconductors. Consequently, a CAR-dominated thermoelectric current requires that the S-F-S junction is asymmetric and there is a finite phase difference between superconductors, as shown in Fig. 1(d). The interference effect on the CAR probability is written as Note (2)

[TABLE]

where $\gamma(E)$ is an even function of the energy and $\phi$ is the phase acquired by Andreev reflections at $\mathrm{SR}$. Importantly, all higher order contributions are equal for both paths and even in energy Note (2), so they are included into the parameter $\gamma(E)$. Since only the odd part of $T_{\mathrm{RL}}^{he}$ contributes to the integration, it can be more conveniently expressed as

[TABLE]

The sinusoidal behaviour of the current with $\phi$ is shown in the inset of Fig. 2(a), revealing the phase difference as an ideal knob to tune the thermoelectric effect. Eq. 4 clearly displays two of the three conditions for the Andreev-dominated thermoelectric effect. A finite electric current, $I_{\mathrm{R}}$, is obtained when the phase difference and the asymmetry result in a finite contribution to Eq. 4 that is comparable to the integration window determined by the temperature bias $\delta\!f(T_{0},\theta)$. We also note that the finite thermoelectric effect indicates the simultaneous presence of both even- and odd-frequency pairing amplitudes in our setup Hwang et al. (2018).

Signatures of equal-spin Cooper pairs. — An Andreev-dominated thermoelectric effect stems from the injection of Cooper pairs with opposite net spin into each superconductor. The resulting current, $I_{\mathrm{R}}$, could be measured as a voltage drop on the right lead after heating up the left one Kolenda et al. (2016); Karimi and Pekola (2018). Our proposal is also phase-tuneable so the thermoelectric power generation can be controlled by a small magnetic field Shelly et al. (2016). Increasing the temperature gradient drives larger thermoelectric currents [see Fig. 2 (a)], but also potentially larger fluctuations Kheradsoud et al. (2019). It is thus essential for the characterization of the proposed heat engine to identify a parameter regime where the fluctuations are the smallest with respect to the average current. That is, where the Fano factor $F=S_{\mathrm{RR}}/\lvert 2eI_{\mathrm{R}}\rvert$, with $S_{\mathrm{RR}}$ the current fluctuations in the right lead, is minimal. The zero-frequency fluctuation of $I_{\mathrm{R}}$ is given by Anantram and Datta (1996)

[TABLE]

with

[TABLE]

where Greek letters label Nambu indices, with $\operatorname{sgn}(\alpha)\!=\!\pm 1$ for $\alpha\!=\!e,h$, Latin symbols represent reservoirs $\{\mathrm{L},\mathrm{R},\mathrm{SL},\mathrm{SR}\}$, $s_{ik}^{\alpha\gamma}$ denotes the amplitude for a particle of type $\gamma$ in reservoir $k$ to be scattered into reservoir $i$ as a particle of type $\alpha$, and $f_{j\beta}(E)\!=\!f(E,\operatorname{sgn}(\beta)\mu_{j},k_{\mathrm{B}}T_{j})$ is the Fermi distribution for particles $\beta$ in reservoir $j$ 444Since we neglect quasiparticle injection in the superconductors, the sum over the reservoirs in Eq. 5 is effectively restricted to L and R..

For an asymmetric junction, the phase difference $\phi$ controls the thermoelectric current, see Eq. 4. The fluctuations are, however, almost independent of $\phi$, indicating that they are mostly caused by thermal noise Note (2). Due to the carrier-selective heat and charge transfer in this setup, thermal noise is caused by normal scattering processes that do not experience interference and $S_{\mathrm{RR}}$ increases steadily with the temperature bias $\theta$. By contrast, the Andreev-dominated current increases rapidly for $\theta\!<\!T_{c}$ and appears to saturate for higher bias. Importantly, for experimentally relevant values Kolenda et al. (2016); Karimi and Pekola (2018), when the current is maximum, the Fano factor becomes minimum, see Fig. 3(b), thus demonstrating that the current is enhanced over its fluctuations. Note that an Andreev-dominated current requires $T_{0}\!\gtrsim\!T_{c}/2$, resulting in rather large Fano factors Note (2). Recently, the electronic noise due to temperature differences in mesoscopic conductors, different than thermal or shot noise, was measured and proposed as an accurate temperature probe Lumbroso et al. (2018).

Finally, the injection of Cooper pairs with opposite net spin into each superconductor produces a non-equilibrium supercurrent $I_{\mathrm{SL}}(\phi)\!=\!I_{\mathrm{SR}}(-\phi)$, see Fig. 3(c). At equilibrium ($\theta\!=\!0$), even though the current at the normal leads vanishes, $I_{\mathrm{L,R}}\!=\!0$, there is a finite, Josephson-like supercurrent at each superconductor. Interestingly, without the F region (*i.e., *with $m_{0}\!=\!0$ and no CAR), the equilibrium current is barely affected by the temperature bias. By contrast, in the presence of F, CAR processes pump equal-spin Cooper pairs into the superconductors resulting in large non-equilibrium supercurrents. The temperature bias thus creates supercurrents with opposite sign at each superconductor, that could be measured separately or after connecting them through a loop, as depicted in Fig. 1. Measuring the flux inside the loop with and without temperature gradient, one could determine a thermophase Giazotto et al. (2015a). Within our estimations, for biases $\theta\!<\!T_{c}$, close to the minimum of fluctuations, the magnitude of the temperature-induced supercurrent is comparable to $I_{0}$, the zero-temperature maximum Josephson current, with a typical value of $\sim\!1\mu$A.

Summary. — We propose a quantum heat engine that can be electrically controlled to generate spin-polarized Cooper pairs and drive supercurrents from a temperature bias on demand. Our proposal is based on a unique transport mechanism taking place at a S-F-S junction on the helical edge of a QSHI. Nonlocal Andreev processes through the junction experience an interference effect between the contributions from each superconductor. This interference is not present for normal processes, resulting in carrier-selective heat and charge currents where normal processes transfer heat and Andreev processes transfer charge. Due to the strong spin-orbit coupling at the helical edge state, the thermoelectric current is completely dominated by equal-spin Andreev processes. We discussed how the proposed spin-triplet thermoelectric effect could be measured as a thermophase appearing between the superconductors. The measurement is further facilitated by the low fluctuations of the spin-polarized nonlocal current.

The authors are grateful to M. Moskalets for valuable discussions. We acknowledge support from the DFG (SPP 1666 and SFB 1170, Project-ID 258499086), the Cluster of Excellence EXC 2147 (Project-ID 39085490), the Ministry of Innovation NRW via the “Programm zur Förderung der Rückkehr des hochqualifizierten Forschungsnachwuchses aus dem Ausland”, the Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant No. 743884 and the Academy of Finland (project 312299).

I Supplemental material to “On-demand thermoelectric generation of equal-spin Cooper pairs”

In this supplemental material, we provide more details regarding the derivation of the expression for the current in the right lead and the crossed Andreev reflection amplitude. We also describe the Green function techniques used to compute the Cooper pair polarization. Finally, we show the behavior of the current fluctuations in the right lead as a function of the base temperature.

II Current in the right lead

In order to derive Eq. (2) of the main text, we follow the formalism developed in Refs. Sivan and Imry (1986); Bardas and Averin (1995); Linder and Bathen (2016); Butcher (1990). As a starting point, by virtue of the standard approach in mesoscopic physics Anantram and Datta (1996) and to recapitulate the main text, the charge current on the right side of the setup $I_{\mathrm{R}}$ is given by

[TABLE]

where Greek summation indices $\alpha,\beta\in\{e,h\}$ run over the electron/hole degree of freedom with $\operatorname{sgn}(\alpha)=\pm 1$ for $\alpha=e/h$, the Latin index $j\in\{\mathrm{L},\mathrm{R},\mathrm{SL},\mathrm{SR}\}$ runs over all reservoirs, $T_{Rj}^{\alpha\beta}\!=\!|s_{Rj}^{\alpha\beta}|^{2}$, with $s_{Rj}^{\alpha\beta}$ the scattering amplitude for a particle of type $\beta$ in reservoir $j$ to be scattered into reservoir $\mathrm{R}$ as a particle of type $\alpha$, and $f_{j\beta}(E)=f(E,\operatorname{sgn}(\beta)\mu_{j},k_{\mathrm{B}}T_{j})$, with $f(E,\mu,\tau)=\{1+\exp[(E-\mu)/\tau]\}^{-1}$, is the Fermi distribution for particles $\beta$ in reservoir $j$.

The lengthy expression arising from performing the sum in Eq. S 1 can be simplified substantially. Using our assumption of grounded superconductors, i.e., $\mu_{\mathrm{SL}}=\mu_{\mathrm{SR}}=0$, and equal superconductor temperatures, i.e., $T_{\mathrm{SL}}=T_{\mathrm{SR}}\equiv T_{0}$, the Fermi functions in the superconductors coincide and $f_{\mathrm{SL}e}(E)=f_{\mathrm{SL}h}(E)=f_{\mathrm{SR}e}(E)=f_{\mathrm{SR}h}(E)\equiv f_{0}(E)=f(E,0,k_{\mathrm{B}}T_{0})$ follows. Furthermore, by employing unitarity of the scattering matrix and conservation of quasiparticle current, which implies

[TABLE]

we can eliminate the coefficients involving $\mathrm{SL},\mathrm{SR}$ and arrive at

[TABLE]

Next, by recognizing that $f_{ih}(E)-f_{0}(E)=f_{0}(-E)-f_{ie}(-E)$ and that particle-hole symmetry enforces $T_{ij}^{\alpha\beta}(E)=T_{ij}^{\bar{\alpha}\bar{\beta}}(-E)$ where $\bar{\alpha}=h,e\text{ if }\alpha=e,h$, the terms in Eq. S 3 corresponding to the injection of holes can be folded back onto their charge conjugated counterparts, yielding

[TABLE]

The first term in Eq. S 4 corresponds to local current contributions in the right lead resulting from injection from the right reservoir, while the second term in Eq. S 4 describes the charge current in the right lead rooting in CAR and EC processes of particles injected from the left reservoir.

We stress that for our choice of chemical potentials and temperatures Eq. S 4 is general for a four terminal setup with two superconducting leads, irrespective of the specific scattering problem at hand. Eq. S 4 holds even when including quasiparticle injection from and into the superconductors.

Importantly, since we assume equilibrium between the right reservoir and the superconductors, i.e., $T_{\mathrm{R}}=T_{0}$, the local current contribution vanishes identically. Therefore, the current in the right lead is a purely nonlocal effect and solely given by $I_{\mathrm{R}}=I^{he}_{\mathrm{R}}+I^{ee}_{\mathrm{R}}$, with

[TABLE]

where $I_{0}=e\Delta_{0}/h$. Upon identifying $f_{\mathrm{L}e}(E)-f_{0}(E)\equiv\delta\!f(E)$, Eqs. S 5a and S 5b correspond to Eqs. (2a) and (2b) of the main text.

The helicity of the QSHI edge states profoundly affects the resulting current. By convention, incoming and right-moving particles from the left and outgoing right-moving particles and holes on the right of the system must have spin $\uparrow$. Consequently, the edge states act as a spin filter for nonlocally driven current.

III Current in the superconductors

The current operator in Nambu and spin space is defined as

[TABLE]

with $\Psi=(\psi_{\uparrow},\psi_{\downarrow},\psi_{\downarrow}^{\dagger},-\psi_{\uparrow}^{\dagger})^{T}$. The field operators can be rewritten using the scattering states, according to

[TABLE]

where the $\phi_{i}$ are the scattering states and thus solutions to the BdG Hamiltonian, and the sum over the index $i$ runs over the four scattering states ($i=1$: e from left, $i=2$: h from left, $i=3$: e from right, $h$ from right). Here, $\hat{a}_{i}$ is the electron or hole annihilation operator at the corresponding lead, L or R. This means that the index $i$ contains both lead and particle type index from the sum in Eq. (23) of Ref. Anantram and Datta (1996). The components of the scattering states are given by $\phi_{i}(x)=(u_{\uparrow,i},u_{\downarrow,i},v_{\downarrow,i},v_{\uparrow,i})^{T}$. With that, the average current can be written as

[TABLE]

with $f_{1,2}=f_{\mathrm{L}e,\mathrm{L}h}$ and $f_{3,4}=f_{\mathrm{R}e,\mathrm{R}h}$. One can check that this gives back the same result as Eq. (29) of Ref. Anantram and Datta (1996), and therefore the expression we used for the current in the normal leads, for $I_{\mathrm{L,R}}$ when applied to a two-lead setup (for $I_{\mathrm{R}}$ one needs to insert a minus sign in order to follow the convention of positive currents always flowing out of the reservoirs). Eq. S 8 now allows us to calculate the (quasiparticle) current at any position using the known scattering states.

We now follow Eq. S 8 and apply it to the definition of the source term in the main text, resulting in

[TABLE]

After integration, the supercurrent is given by

[TABLE]

Note that, to take into account the temperature of the superconductor, $\Delta_{0}$ should be replaced by $\Delta(T_{\mathrm{S}})$. Furthermore, note that within the superconducting regions, there is also a quasiparticle contribution of the same form as the current in the normal regions in Eq. S 8.

IV Analysis of the crossed Andreev reflection amplitude

In this section of the supplementary material, we discuss the CAR amplitude in more detail and provide a derivation of Eqs. (3) and (4) of the main text. To that end, we decompose the full scattering problem into simpler pieces, namely the superconducting barriers SL and SR, the ferromagnetic region F, and the intermediate normal domains NL and NR. We follow the same approximations as stated in the main text. For the sake of readability we slightly change notation compared to the main text and denote transmissions and reflections as $t_{ij}^{\alpha\beta}$ with $i\neq j$ and $r_{ii}^{\alpha\beta}$, respectively.

In order to write the S-matrix elements of the full system in terms of the scattering coefficients of the single constituents, we proceed as follows. Globally, the amplitudes of incoming and outgoing modes at the outmost interfaces $(1)$ and $(6)$ (see Fig. S 1) are related by the full scattering matrix $S$ according to

[TABLE]

with

[TABLE]

Here, $a^{(i)}_{\alpha}$ ($b^{(i)}_{\alpha}$) with $i=1,6$ denotes the amplitude of an incoming (outgoing) mode at interface $(i)$ of particle/hole type $\alpha$. For the scattering coefficients of the total system we choose the convention that the right (left) sub- and superscript refers to the incoming (outgoing) particle, e.g. $t_{\mathrm{RL}}^{he}$ is the amplitude for an electron to be crossed Andreev reflected from left to right.

Within the system the amplitudes between positions $(i)$ and $(i+1)$ are related by scattering matrices associated with single S and F barriers or the intermediate NL and NR regions. We denote the amplitude of rightmovers (leftmovers) of type $\alpha$ at interface $(i)$ with $p_{\alpha}^{(i)}$ ($m_{\alpha}^{(i)}$). Specifically, they are related by

[TABLE]

In Eqs. S 13a, S 13b, S 13c, S 13d and S 13e, $r^{\alpha\beta}_{X(ii)}$ corresponds to a reflection process of a particle of type $\beta$ into type $\alpha$ at interface $(ii)$ of region X, whereas $t^{\alpha\alpha}_{X(ij)}$ represents a transmission of particle $\alpha$ from interface $j$ to $i$ through region X with $X\in\{\mathrm{SL},\mathrm{NL},\mathrm{F},\mathrm{NR},\mathrm{SR}\}$.

The scattering problems need to be set up such that the scattering coefficients alone capture the phase shifts picked up due to propagation. Specifically, there are four solutions of the Bogoliubov–de Gennes Hamiltonian at each interface $(i)$ given by (note that we set $\hbar=v_{\mathrm{F}}=1$)

[TABLE]

corresponding to right- ($+$) and leftmoving ($-$) electrons and holes ($e$ and $h$, respectively). The scattering matrix relating interfaces $(i)$ and $(i+1)$ is then obtained by constructing scattering states out of the solutions $\phi^{\alpha}_{\pm}(x-x_{(i)})$ and $\phi^{\alpha}_{\pm}(x-x_{(i+1)})$, respectively, where $x_{(}j)$ denotes the location of interface $(j)$.

One can now use all 16 subequations of Eqs. S 13a, S 13b, S 13c, S 13d and S 13e not involving the outgoing amplitudes $b^{(i)}_{\alpha}$ in order to write the 16 coefficients $p^{(i)}_{\alpha}$, $m^{(i)}_{\alpha}$ in terms of the ingoing amplitudes $a^{(i)}_{\alpha}$. Subsequently, the remaining 4 linear equations can be used to relate the outgoing and ingoing amplitudes $b^{(i)}_{\alpha}$ and $a^{(i)}_{\alpha}$ in the form of Eq. S 11 and we can read off $t^{he}_{\mathrm{RL}}$.

We obtain

[TABLE]

where we define self-energies as

[TABLE]

Before we provide the lengthy expressions for $t_{1}$, $t_{2}$ and $\bm{\varSigma}$ in terms of the elements of the scattering matrices defined in Eq. S 13, we first give a graphical explanation of Eq. S 15 (see Fig. S 2). The CAR coefficient $t_{\mathrm{RL}}^{he}$ is given by the sum of the two lowest order processes $t_{1}$ and $t_{2}$ necessary to convert an electron incoming from the left into a hole leaving the heterostructure to the right as also shown in the main text, augmented by the insertion of all possible closed loops, denoted $\Sigma^{\mathrm{L,R}}_{1}$, $\Sigma^{\uparrow,\downarrow}_{2}$ and $\Sigma^{e,h}_{3}$.

The schematic paths in Fig. S 2 correspond to the expressions

[TABLE]

for the two lowest order paths for CAR, *i.e.,

(i) transmission through SL, reflection at F, local Andreev reflection at SL under electron-hole conversion, and then transmission all the way through to the right lead ($t_{1}$);

(ii) transmission through SL and F, local Andreev reflection at SR under electron-hole conversion, reflection at F followed by tunneling to the right lead ($t_{2}$),

[TABLE]

for the two closed loops in the left (L) and right (R) S-F cavities,

[TABLE]

for the two closed loops between the SL and SR barriers - *i.e., *one involving spin-up electrons and spin-down holes (denoted with superscript $\uparrow$), and the other one built from spin-down electrons and spin-up holes ($\downarrow$) - and finally

[TABLE]

for the loops between the S regions with an additional detour in each of the cavities (see Fig. S 2). The superscript refers to the particle type along the long paths connecting the S regions.

In order to derive Eq. (3) of the main text, one can now analyze the energy dependence of $\bm{\varSigma}$ by making use of particle hole symmetry. The terms $\Sigma_{1}^{\mathrm{L}}$ and $\Sigma_{1}^{\mathrm{R}}$ involve all four modes and all possible reflections and thus are invariant under charge conjugation, i.e.,

[TABLE]

which implies that the real and imaginary parts fulfill

[TABLE]

By contrast, $\Sigma_{2}^{\uparrow/\downarrow}$ and $\Sigma_{3}^{e/h}$ are charge conjugated partners of one another, such that

[TABLE]

and hence

[TABLE]

Next, we consider the complex valued functions $u(E),w(E),z(E)$ defined as

[TABLE]

such that $u+w+z=\bm{\varSigma}$. Importantly,

[TABLE]

because of Eqs. S 21 and S 23. Eq. S 26 immediately leads to the crucial result

[TABLE]

demonstrating that $\lvert\bm{\varSigma}\rvert$ is indeed even in $E$. Going back to the full crossed Andreev reflection coefficient, we can rewrite Eq. S 15 as

[TABLE]

where the denominator $d(E)=1+\lvert\bm{\varSigma}(E)\rvert^{2}-\bm{\varSigma}(E)-\left(\bm{\varSigma}(E)\right)^{*}$ is real and obeys

[TABLE]

while finally the numerator of the second term in Eq. S 28 is $n(E)=1-\left(\bm{\varSigma}(E)\right)^{*}$ with the property

[TABLE]

In conclusion, the modulus of the contribution of all higher order corrections given by $1/(1-\bm{\varSigma})=d/n$ is indeed even in energy.

As the final step, we turn to the first term in Eq. S 28 responsible for the interference effect. From Eq. S 17, we first write

[TABLE]

Now, using the explicit form of the coefficients we provide in Eq. S 37 below, we have

[TABLE]

Thus, Eq. S 31 simplifies to

[TABLE]

which is of the form

[TABLE]

Here, we have introduced the phase difference between the two paths $\Delta\varphi=\phi+2(d_{\mathrm{NR}}-d_{\mathrm{NL}})E$, an unimportant global phase $\tau$, and the modulus of a product of scattering coefficients fulfilling $\lvert t(E)\rvert=\lvert t(-E)\rvert$ (see below). Combining Eqs. S 28, S 29, S 30 and S 34, we thus show that the absolute squared of the CAR coefficient can be written in the form

[TABLE]

where $\gamma(E,\phi)$ is even in $E$ and given by (we have restored the dependence on the phase difference $\phi$)

[TABLE]

as stated in Eq. (3) in the main text.

In order to show that $\lvert t(E)\rvert$ is indeed even in $E$, we define $A_{\mathrm{S}}(E)=\operatorname{arccosh}(E/\Delta)$, $A_{\mathrm{F}}(E)=\operatorname{arccosh}(E/m_{0})$ and

[TABLE]

The coefficients are then found to be [together with Eq. S 32]

[TABLE]

From Eq. S 37 one can check that the modulus of all coefficients is an even function of $E$, and thus $\lvert t(E)\rvert$ must be even as well.

V Green function techniques and superconducting pairing

The retarded Green function is a $4\times 4$-matrix in Nambu and spin space related to the Hamiltonian in Eq. (1) of the main text. In the basis $\Psi(x)=(\psi_{\uparrow},\psi_{\downarrow},\psi^{\dagger}_{\downarrow},-\psi^{\dagger}_{\uparrow})$, it takes the form

[TABLE]

Note that due to the choice of basis the functions $[G^{R}_{eh}]^{\uparrow\uparrow/\downarrow\uparrow}$ and $[G^{R}_{he}]^{\uparrow\uparrow/\uparrow\downarrow}$ are the negative of the corresponding correlator. Following Ref. Keidel et al., 2018, we can use particle-hole symmetry to relate the electron-hole and hole-electron blocks. Explicitly, particle-hole symmetry relates the blocks according to

[TABLE]

with $\mathbf{P}=\hat{\tau}_{2}\hat{\sigma}_{2}\mathbf{K}$ the charge conjugation operator and $\mathbf{K}$ complex conjugation. This translates into

[TABLE]

which, anticipating what we need in the following, can be recast as

[TABLE]

The spin components of the anomalous part of the Green function are extracted by writing

[TABLE]

where $\sigma_{k}$ are the Pauli matrices in spin space. The singlet is given by the $k=0$ component. Inverting results in

[TABLE]

Note that our choice of basis leads to positive sign for $f_{0}$ and the relative minus sign for $f_{3}$. We also define the shorthand notation $f_{+}=[G^{R}_{eh}]^{\uparrow\uparrow}=f_{1}-\mathrm{i}f_{2}$ and $f_{-}=[G^{R}_{eh}]^{\downarrow\downarrow}=f_{1}+\mathrm{i}f_{2}$ for the equal-spin pairings.

We can now analyze the equal-spin components of the Green function $[G^{R}_{eh}]^{\uparrow\uparrow/\downarrow\downarrow}$ and $f_{0}$ and $f_{3}$ (see Fig. S 3), or simply all components $f_{k}$.

Another option is to consider the polarization vector

[TABLE]

where $\vec{f}=(f_{1},f_{2},f_{3})$ Sigrist and Ueda (1991); Leggett (1975). Similarly, we can decompose the hole-electron block of the anomalous Green function through

[TABLE]

which implies

[TABLE]

Put together with the relation between the blocks in Eq. S 38, the vectors $\vec{f}$ and $\vec{f^{\prime}}$ are related via

[TABLE]

such that the polarization vector for the hole-electron block fulfills

[TABLE]

The local equal-spin pairings are exactly equal, i.e. $[G^{R}_{eh}]^{\uparrow\uparrow}=[G^{R}_{eh}]^{\downarrow\downarrow}$, meaning that $\langle\psi^{\dagger}_{\uparrow}\psi^{\dagger}_{\uparrow}\rangle=-\langle\psi^{\dagger}_{\downarrow}\psi^{\dagger}_{\downarrow}\rangle$, which directly implies that $f_{2}=0$. From Eq. S 44 we immediately see that $p_{1}=p_{3}=0$, i.e. $\vec{p}$ points in the $y$-direction. We show in Fig. S 3 the squared moduli of $f_{\pm},f_{3},f_{0}$ for phase differences $0,0.5\pi,\pi$. The resulting polarization vector $\vec{p}$, more precisely the only nonzero component $p_{2}$, is shown in Fig. S 4.

VI Temperature dependence of the noise in the right lead

As we describe in the main text, the noise in the right lead, $S_{\mathrm{RR}}$, defined in Eq. (5) of the main text, is almost independent of the phase difference, see Fig. S 5(a). This is a result of the noise being dominated by the thermal contribution, which is caused by normal processes that do not experience interference. On the other hand, the thermoelectric current in the right lead, $I_{\mathrm{R}}$, depends on the phase difference $\phi$ between the superconductors, due to the interference effect of CAR processes, see Fig. S 5(b). Consequently, the Fano factor strongly depends on the phase difference, see Fig. S 5(c), and is minimum when the current is maximum. Moreover, as the inset of Fig. S 5(c) indicates, the minimum of the Fano factor is rather stable for temperature biases $\theta\sim T_{c}$.

In the main text, we focus on the behavior of the current fluctuations and the Fano factor as a function of the temperature difference $\theta$ at fixed base temperature $T_{0}$. Here, we provide some information about the dependence on $T_{0}$. We fix the phase difference between the superconductors to be at the optimal operating value $\phi=\pi/2$. The temperature difference is set to be $\theta=2T_{0}$ (plotted in cyan) and $\theta=T_{c}$ (orange). Note that at $T_{0}=T_{c}/2$, both cases coincide and we also recover the results of the main text.

Fig. S 6 (a) shows the noise in the right lead $S_{\mathrm{RR}}$. For all choices for the temperature gradient $\theta$ the fluctuations do not grow monotonically up to $T_{0}=T_{c}$, but reach a maximum below that value. This is connected to the fact that the scattering problem and thus the conductance change dramatically when the two superconducting barriers are removed. Note that there are several contributions to the noise in setups like this, namely an equilibrium contribution due to finite temperatures, a non-equilibrium contribution due to a temperature gradient and, additionally, a contribution due to the thermoelectrically created nonlocal charge current. The interplay of these sources of fluctuations has been carefully studied in Lumbroso et al. (2018). In Fig. S 6 (b), we plot the total current in the right lead produced by the temperature gradient. For smaller values of $T_{0}$, the current for $\theta=T_{c}$ is of course much larger than for $\theta=2T_{0}$, but for both cases it is not dominated by Andreev contributions below $T_{0}\approx T_{c}/2$. The quite sharp drop of the $\theta=T_{c}$ case for $T_{0}$ smaller than $T_{c}/2$ is mostly due to the vanishing of the normal contribution. Finally, Fig. S 6 (c) displays the Fano factor $F=S_{\mathrm{RR}}/\lvert 2eI_{\mathrm{R}}\rvert$ resulting from noise and average current in Fig. S 6 (a) and (b). For small base temperatures, Fano factors below 10 are possible. However, note that a dominating Andreev contribution to the current requires $T_{0}$ to be of the order of $T_{c}/2$, which increases the Fano factor at the optimal working point of our device.