On-demand thermoelectric generation of equal-spin Cooper pairs

Felix Keidel, Sun-Yong Hwang, Björn Trauzettel, 3 Björn Sothmann, and Pablo Burset Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany Theoretische Physik, Universität Duisburg-Essen and CENIDE, D-47048 Duisburg, Germany Würzburg-Dresden Cluster of Excellence ct.qmat, Germany Department of Applied Physics, Aalto University, 00076 Aalto, Finland (Dated: July 9, 2019)

Introduction. -The new field of superconducting spintronics has emerged since the creation of spin-triplet Cooper pairs in experiments [1][2][3]. The development of spintronics had already benefited from the use of superconducting materials, resulting in longer spin lifetimes and energy-efficient components [4,5]. 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 [6][7][8][9][10][11][12][13][14][15][16]. The key challenge in the field is the nonequilibrium and on-demand generation of equal-spin Cooper pairs in a viable fashion [17][18][19][20][21], desirably avoiding the complicated manipulation of magnetic components.
In this Letter, we propose a thermoelectric engine that produces a spin-polarized supercurrent on demand from a temperature gradient. We consider a superconductorferromagnetic-insulator-superconductor (S-F-S) junction on top of the helical edge state of a quantum spin Hall insulator (QSHI) [22][23][24][25][26][27][28] 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 [29,30]. 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 [31][32][33]. Our key finding is that the heatto-supercurrent conversion of this engine is near-perfect, with almost complete suppression of the normal particle current. 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, cf. Fig. 1(c,d).
Harvesting waste heat by quantum thermoelectric effects has become essential in modern nanoscale de- vices [34]. While tackling this problem in S-F hybrid junctions can lead to potentially strong thermoelectric effects [35][36][37][38][39][40][41][42][43][44], 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, create a strong spinpolarized thermal supercurrent between the superconductors that can be simply 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 or thermophase measurements across the junction [45,46].
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 [31][32][33] that can lead to the emergence of Majorana bound states [47][48][49][50] or exotic odd-frequency superconducting pairing [32,33,51,52]. Given recent advances in the experimental realization of helical edge states [26][27][28], hybrid structures like the one sketched in Fig. 1(a) are within reach: superconductors [53][54][55][56] have been successfully coupled to QSHIs [57,58], and monolayer QSHIs provide a new promising platform to induce ferromagnetic order [28,56]. 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. [59].
We theoretically describe the one-dimensional helical edge states of a QSHI in proximity to superconducting and ferromagnetic order by a Bogoliubovde Gennes Hamiltonian in the Nambu basis Ψ( with H 0 =p xτ3σ3 − µτ 3σ0 the Hamiltonian of the free helical edge, H S = [∆(x) cos φ(x)τ 1 + ∆(x) sin φ(x)τ 2 ]σ 0 the proximity-induced superconductivity, and H F =τ 0 m(x)· σ ≡τ 0 (m cos λσ 1 + m sin λσ 2 + m zσ3 ) describing the effect of the ferromagnetic barrier. Here,p x = −i∂ x and σ i (τ 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 X for X ∈ {SL, NL, F, NR, SR}. The pair potential is assumed equal for both superconductors and constant, ∆(x) = ∆ 0 , a valid approximation as long as the Fermi wavelength in each superconductor is much smaller than the proximity-induced coherence length. For simplicity, we take the phase of the pair potential φ(x) = φ in SR and zero otherwise. The F region is modeled by constant m (x) = m 0 within F, and we choose m z = 0 since its effect can be absorbed in the phase difference φ between the superconductors [33,48,60]. Without loss of generality the angle λ is set to zero. Finally, we assume that all regions reside at the same chemical potential, i.e., µ(x) = 0 everywhere. In the following, we consider that all leads except L are at the same temperature [61] (T SL = T SR = T R ≡ T 0 ) and set T L = T 0 + θ, introducing the temperature difference θ. The electric current in the right lead after a temperature bias is applied to the left lead is given by with T he RL the CAR probability, T ee RL the EC probability, The probabilities are obtained by solving the scattering problem defined by the solutions of Eq. (1) in every region [32,33,48,[63][64][65][66][67][68].
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, i.e., EC processes, CAR processes are only possible if injected electrons and transmitted holes have the same spin [31][32][33]. By breaking time-reversal symmetry, the 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 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 φ and a phase shift d NR E compared to the ones where the reflection takes place at SL, which are only shifted by d NL E (note that we set = v F = 1). 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). Furthermore, since δf (E) is odd in E, only the antisymmetric part of the transmissions will contribute to the charge current.
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 [32,33,47,48,50]. The hybridization between the bound states at each S-F cavity is controlled by the phase difference between the superconductors [33,60]. This, in turn, allows for the control of the electric current through the junction. Total current as well as normal and Andreev contributions as a function of (a) the temperature difference (phase difference φ in the inset) and (b) the base temperature with θ = 2T0 fixed. Unless specified otherwise, we use the parameters dSL = dSR = ξ0, dFM = 0.6ξ0, dNL = 0.4ξ0, dNR = 0.9ξ0, m0 = 1.5∆0, T0 = 0.5Tc, θ = Tc, and φ = π/2, with ξ0 = 1/∆0 and Tc = ∆0.
As we describe in detail below, the interference of CAR processes leads to a particular thermoelectric effect, where the current can be completely dominated by equal-spin Andreev processes. 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.
Andreev-dominated thermoelectric effect. -Given a positive temperature gradient, we find that a finite thermoelectric current is completely dominated by Andreev processes when three requirements are fulfilled [cf. Fig. 2 (a,b)]: (i) the base temperature T 0 is sufficiently large, i.e., T 0 T c /2; (ii) the junction is asymmetric, which we realize by setting d NL = d NR ; and (iii) the phase difference φ is not an integer multiple of π. Under these conditions, the energy asymmetry of the CAR transmission is comparable to the energy-antisymmetric bias δf as illustrated in Fig. 1 (c), whereas the asymmetry in the EC probability occurs on a much smaller energy scale [69]. As a result, the Andreev current I he R becomes much larger than the EC current I ee R as the temperature grows. The 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 that we explain below. 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 ee R → 0, since without superconductivity the resonant tunneling at the S-F regions disappears and so does the thermoelectric effect [70].
The second condition for a CAR-dominated thermoelectric current is that the S-F-S junction is asymmetric, as shown in Fig. 1(d). This is a direct result of the interference between different contributions to the CAR current. Indeed, when d NL = d NR , I he R vanishes since the two paths in Fig. 1(b) destructively interfere: their contributions to the total scattering coefficient are equal and given by an even function of the energy. In general, these two contributions acquire an energy-dependent phase coming from the fact that the Andreev reflection takes place at different superconductors. For each path, the electron or hole propagation at the S-F cavity where the Andreev reflection takes place results in a different accumulated phase. The interference effect on the CAR probability can thus be written as [62] where γ(E) is an even function of the energy and φ is the phase acquired by Andreev reflections at SR. Importantly, all higher order contributions are equal for both paths and even in energy [62], so they are included into the parameter γ(E).
Since only the odd part of T he RL contributes to the integration, it can be more conveniently expressed as The sinusoidal behaviour of the current with φ 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 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 δf (T 0 , θ). We also note that the finite thermoelectric effect indicates the simultaneous presence of both even-and odd-frequency pairing amplitudes in our setup [43].
Detection of the spin-polarized supercurrent. -Having determined how the Andreev-dominated thermoelectric effect takes place, we can now analyze the best conditions for its observation.
Increasing the temperature gradient drives larger thermoelectric currents [see Fig. 2 (a)], but also potentially larger fluctuations [71]. 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 RR /|2eI R |, with S RR the current fluctuations in the right lead, is minimal. The zero-frequency fluctuation of I R is given by [66]  where Greek letters label Nambu indices, with sgn(α) = ±1 for α = e, h, Latin symbols represent reservoirs {L, R, SL, SR}, s αγ ik denotes the amplitude for a particle of type γ in reservoir k to be scattered into reservoir i as a particle of type α, and f jβ (E) = f (E, sgn(β)µ j , k B T j ) is the Fermi distribution for particles β in reservoir j [72].
After identifying the phase difference φ as the tuning parameter to control the thermoelectric current, it is striking to see that the fluctuations are almost independent of it, cf. Fig. 3(a). This indicates that they are mostly caused by thermal noise. Because of the carrierselective heat and charge transfer in this setup, thermal noise is due to normal processes that do not experience interference and S RR increases steadily with the temperature bias θ. By contrast, the Andreev-dominated current appears to saturate for higher bias, see Fig. 3(b). Importantly, when the current is maximum, the Fano factor becomes minimum, see Fig. 3(c), thus demonstrating that the current is enhanced over its fluctuations. Moreover, as the inset of Fig. 3(c) indicates, the minimum of the Fano factor is rather stable for temperature biases θ T c . Note that in order to find an Andreev-dominated current we need T 0 to be sufficiently large (T 0 ≈ T c /2), which always leads to rather large Fano factors. We provide more details on the T 0 -dependence of the noise in the supplemental material [62]. 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 [73].
Finally, an Andreev-dominated thermoelectric current in R fulfills I L = −I R , with I L caused by local Andreev processes. Importantly, even though current conservation demands that the currents on the superconducting leads must be balanced, both I SL and I SR are nonzero for finite θ and fulfill I SL (φ) = I SR (−φ), cf. inset in Fig. 3(b). In our setup, CAR processes due to a temperature gradient are only possible by the simultaneous creation of equal-spin Cooper pairs on one superconductor and their annihilation on the other. The temperature bias thus creates opposite sign supercurrents in the superconductors that could be measured as a finite thermophase in a setup similar to the one depicted in Fig. 1(a) [45]. Moreover, for a bias θ ∼ 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 ∼ 1µA. The sizeable spin-polarized thermoelectric current proposed here is thus within experimental reach and its detection and control should be accessible for temperature biases comparable to the superconducting gap.
Summary. -We propose a quantum heat engine that can be electrically controlled to drive spin-polarized 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 equalspin 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 Sk lodowska-Curie Grant No. 743884 and the Academy of Finland (project 312299).

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. Furthermore, we show the behavior of the current fluctuations in the right lead as a function of the base temperature.

Current in right lead
In order to derive Eq. (2) of the main text, we follow the formalism developed in Refs. [74][75][76][77]. As a starting point, by virtue of the standard approach in mesoscopic physics [66] and to recapitulate the main text, the charge current on the right side of the setup I R is given by where Greek summation indices α, β ∈ {e, h} run over the electron/hole degree of freedom with sgn(α) = ±1 for α = e/h, the Latin index j ∈ {L, R, SL, SR} runs over all reservoirs, T αβ Rj = |s αβ Rj | 2 , with s αβ Rj the scattering amplitude for a particle of type β in reservoir j to be scattered into reservoir R as a particle of type α, and The lengthy expression arising from performing the sum in Eq. (S 1) can be simplified substantially. Using our assumption of grounded superconductors, i.e., µ SL = µ SR = 0, and equal superconductor temperatures, i.e., T SL = T SR ≡ T 0 , the Fermi functions in the superconductors coincide and f SLe ( Furthermore, by employing unitarity of the scattering matrix and conservation of quasiparticle current, which implies we can eliminate the coefficients involving SL, SR and arrive at

(S 3)
Next, by recognizing that f ih (E) − f 0 (E) = f 0 (−E) − f ie (−E) and that particle-hole symmetry enforces T αβ ij (E) = Tᾱβ ij (−E) whereᾱ = h, e if α = e, h, the terms in Eq. (S 3) corresponding to the injection of holes can be folded back onto their charge conjugated counterparts, yielding 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 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 R = I he R + I ee R , with and The helicity of the QSHI edge states profoundly affects the resulting current. By convention, incoming and rightmoving particles from the left and outgoing rightmoving particles and holes on the right of the system must have spin ↑. Consequently, the edge states act as a spin filter for nonlocally driven current.

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 αβ ij with i = j and r αβ ii , 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 Here, a (i) α ) with i = 1, 6 denotes the amplitude of an incoming (outgoing) mode at interface (i) of particle/hole type α. 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 he RL 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 α at interface (i) with p . Specifically, they are related by In Eqs. (S 8a) to (S 8e), r αβ X(ii) corresponds to a reflection process of a particle of type β into type α at interface (ii) of region X, whereas t αα X(ij) represents a transmission of particle α from interface j to i through region X with X ∈ {SL, NL, F, NR, 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 = v F = 1) 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 φ α ± (x − x (i) ) and φ α ± (x − x (i+1) ), respectively, where x ( j) denotes the location of interface (j). One can now use all 16 subequations of Eqs. (S 8a) to (S 8e) not involving the outgoing amplitudes b   We obtain where we define self-energies as Before we provide the lengthy expressions for t 1 , t 2 and Σ in terms of the elements of the scattering matrices defined in Eq. (S 8), we first give a graphical explanation of Eq. (S 10) (see Fig. S 2). The CAR coefficient t he RL 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 Σ L,R 1 , Σ ↑,↓ 2 and Σ e,h 3 . The schematic paths in Fig. S 2 (22) , (S 15) 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 Σ by making use of particle hole symmetry. The terms Σ L 1 and Σ R 1 involve all four modes and all possible reflections and thus are invariant under charge conjugation, i.e., Σ L/R 1 while finally the numerator of the second term in Eq. (S 23) is n(E) = 1 − (Σ(E)) * with the property In conclusion, the modulus of the contribution of all higher order corrections given by 1/(1 − Σ) = d/n is indeed even in energy.
As the final step, we turn to the first term in Eq. (S 23) responsible for the interference effect. From Eq. (S 12), we first write  (a) Noise, (b) average current, and (c) Fano factor in the right lead as a function of the base temperature T0 with temperature differences θ = 2T0 (cyan) and θ = Tc (orange) fixed. The dashed black line in (a) shows the noise in the equilibrium case for θ = 0. The phase difference is φ = π/2 for all panels, and the other parameters are chosen as in the main text.
Dependence of the noise in the right lead on the base temperature In the main text, we focus on the behavior of the current fluctuations and the Fano factor as a function of the temperature difference θ 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 φ = π/2. The temperature difference is set to be θ = 2T 0 (plotted in cyan) and θ = 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 3 (a) shows the noise in the right lead S RR . For all choices for the temperature gradient θ 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 [73]. In Fig. S 3 (b), we plot the total current in the right lead produced by the temperature gradient. For smaller values of T 0 , the current for θ = T c is of course much larger than for θ = 2T 0 , but for both cases it is not dominated by Andreev contributions below T 0 ≈ T c /2. The quite sharp drop of the θ = T c case for T 0 smaller than T c /2 is mostly due to the vanishing of the normal contribution. Finally, Fig. S 3 (c) displays the Fano factor F = S RR /|2eI R | resulting from noise and average current in Fig. S 3 (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.