Large Tunable Thermoelectric Effects in
Superconducting Spin Valves with Commercially
Available Materials

Pablo Tuero, Johanne Bratland Tjernshaugen, Carlos Sanchez, César

Gonzalez-Ruano, Yuan Lu, Jacob Linder, Farkhad G Aliev

► To cite this version:

Pablo Tuero, Johanne Bratland Tjernshaugen, Carlos Sanchez, César Gonzalez-Ruano, Yuan Lu, et al. Large Tunable Thermoelectric Effects in Superconducting Spin Valves with Commercially Available Materials. PRX Energy, 2025, 4 (3), pp.033003. 10.1103/xbgg-gttc. hal-05367417

HAL Id: hal-05367417

https://hal.science/hal-05367417v1

Submitted on 15 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Large tunable thermoelectric effects in superconducting spin valves
with commercially available materials

Pablo Tuero†,¹ Johanne Bratland Tjernshaugen †,² Carlos Sanchez,¹ César Gonzalez-Ruano,¹ Yuan Lu,³ Jacob Linder,²,\* and Farkhad G. Aliev⁴,†

¹Departamento Física de la Materia Condensada C-III,
Universidad Autónoma de Madrid, Madrid 28049, Spain

²Center for Quantum Spintronics, Department of Physics,
Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

³Université de Lorraine, CNRS, IJL, F-54000 Nancy, France

⁴Departamento Física de la Materia Condensada C-III,
Instituto Nicolás Cabrera (INC) and Condensed Matter Physics Institute (IFIMAC),
Universidad Autónoma de Madrid, Madrid 28049, Spain

Recent studies have revealed magnetically controllable thermoelectric effects in superconductor/ferromagnet (S/F) structures. A tunable cryogenic thermoelectric generator needs not only a high conversion factor between electricity and heat, but also a large change in the thermoelectric output when switching the magnetic state of the device. However, the reported modifications in thermoelectric power are either minimal, involve superconductors with relatively low critical temperatures (below 1 K), or do not utilize commercially available spintronic materials. Here, we experimentally measure and numerically model thermoelectric effects in fully epitaxial F/S/F junctions based on commercially available, easily grown materials, as well as their dependence on the magnetic configuration of the F electrodes. We observe sizeable Seebeck coefficients for the parallel alignment of the ferromagnetic electrodes, reaching values of about 100 µV/K. Importantly, we find a decrease of the thermoelectric signal of more than an order of magnitude when switching from a parallel to an antiparallel configuration, constituting a large thermoelectric spin-valve effect. Theoretical modeling based on a self-consistent non-equilibrium Keldysh-Usadel Green's function theory, combined with micromagnetic simulations, qualitatively reproduce the experimental findings. The thermoelectric effect is optimized when there is a large spin-dependent electron-hole asymmetry in the superconductor combined with spin-dependent transmission through the interfaces. These findings pave the way for the development of efficient and versatile cryogenic thermoelectric heat engines.

INTRODUCTION

The conversion of heat into electricity and vice versa is known as thermoelectricity [1, 2]. This field of research has received increasing attention over the last decades, not only for the purpose of gaining insight into the fundamental physics of new materials [3, 4], but also due to the large number of industrial applications thermoelectricity enables. These include vastly different areas such as temperature control (cooling) of electronic devices, air conditioning, food refrigeration, and ultrasensitive detection of electromagnetic radiation [5–9].

A prominent challenge with thermoelectricity occurs in the low-temperature regime. Whereas conventional thermoelectric materials typically operate in the regime 100 K – 400 K, their performance, in terms of for instance the Seebeck coefficient, drastically drops at lower temperatures [10]. Therefore, there exists a need to design better material platforms that can yield efficient thermoelectricity even under cryogenic conditions. One promising application of thermoelectricity under cryogenic conditions is localized cryogenic cooling. This can be achieved using either thermoelectric materials highly efficient at low temperatures [11] or superconductor based microscale refrigerators [12] which would complement the existing liquid nitrogen, helium or magnetic salt cooling procedures.

Superconductors are excellent thermoelectric materials when combined with ferromagnets [2, 13]. This emergent phenomenon has been predicted more than a decade ago [14, 15] and subsequently experimentally confirmed [16]. The key

mechanism behind this emergent thermoelectricity is a spin-dependent particle-hole asymmetry in a superconductor that coexists with a spin-splitting field, which in turn is utilized by using the spin-selective transport properties of ferromagnets. Combining superconductors with antiferromagnets also produces enhanced thermoelectric effects [17]. Moreover, large thermoelectric effects have been predicted [18] and experimentally confirmed [19] in tunnel junctions with two different superconductors in the non-linear regime due to spontaneous breaking of electron-hole symmetry.

The investigation of heat dissipation and energy harvesting in superconductor/ferromagnet (S/F) hybrids and in quantum devices in general [20] is an important area of research. For example, quantum error correction operations produce heating [21], potentially harvestable by low-temperature Peltier elements. In that context, obtaining the maximum possible Seebeck coefficients at the lowest possible temperatures (where actual quantum computers operate) and with minimum possible temperature gradients, could be important for the potential applications of magnetic state controlled thermoelectricity. Magnetic control over thermoelectric effects could improve localized cooling for quantum devices, such as superconducting qubits and ultra-sensitive detectors [22] by dynamically tuning heat flow. Magnetically controlled thermoelectric transport could also be used to manipulate heat flux (thermal diodes) [23], in a way analogous to how electronics manipulate electric energy. Last but not least, the above-mentioned waste heat from superconducting or other low-temperature electronic
systems could be converted into usable electrical energy via magnetically tunable thermoelectric generators [13, 24].

Very recently, it has been predicted [25] and experimentally demonstrated [24, 26] that interfacing a superconductor with two ferromagnets allows the Seebeck coefficient of the system to be tuned via the magnetic alignment of the ferromagnetic regions. However, in order for such an effect to be viable for cryogenic thermoelectric applications, a number of conditions have to be met. Firstly, one would need to use commercially available materials with high spin polarization which are easily grown. Secondly, the resulting magnitude of the thermoelectric effect would need to be sizeable at temperatures well below $T_c$, and there should be a large difference in the Seebeck coefficient for the parallel (P) and antiparallel (AP) magnetic states of the spin-valve.

In this work, we report the experimental observation of a large superconducting thermoelectric spin-valve effect that meets all of the key criteria stated above. This represents a major advance compared to previous works on related superconductor/ferromagnet structures [24, 26]. Specifically, we study the spin-dependent transport and thermoelectric response in Fe/MgO/V/MgO/Fe/Co structures as a function of the relative alignment between the soft (Fe) and hard (Fe/Co) F electrodes. An important reason to use vanadium in our Fe/MgO/V based junctions, is its nearly perfect crystalline structure matching with Fe and MgO, allowing the superconductor to be grown on top of an almost fully spin polarized (Fe/MgO) ferromagnet with effective spin polarization exceeding 80%. Our findings demonstrate a large thermoelectric effect, with a Seebeck coefficient exceeding 100 µV/K at base temperatures around $0.1T_c$, where $T_c$ is the critical temperature of the V film. Importantly, we report a large change in the thermoelectric response – greater than a factor of 10 – when switching the device from the antiparallel (AP) to the parallel (P) alignment of the electrodes. Analyzing the results, we find a high sensitivity of the thermoelectric response to magnetic domain rotation and motion, as highlighted by the comparison of experimental thermoelectric and magnetoresistance data with micromagnetic simulations and theoretical modeling based on the non-equilibrium Keldysh-Usadel Green's function formalism.

## EXPERIMENTAL RESULTS

Figure 1(a) illustrates the experimental setup and the specific junctions under study. We measure magnetoresistance (MR) and thermoelectric (TE) effects in epitaxial Fe(45)/MgO(2.3)/V(40)/MgO(2.3)/Fe(10)/Co(20) junctions grown on MgO(001) substrates, with layer thicknesses provided in nanometers. Further details on junction growth, characterization, experimental setup, and simulation procedures are available in the Supplemental Material [27]. In these F1/S/F2 junctions, V is the BCS superconductor (S), Fe serves as the soft magnetic electrode (F1) and Fe/Co functions as the magnetically hard electrode (F2), while the MgO acts as a symmetry-filtering

tunnel barrier, facilitating high spin polarization (p) in Fe, exceeding 0.7-0.8 [28, 29]. Due to the weak antiferromagnetic coupling of the junctions [30–33], two distinct magnetic states are experimentally observed: at low fields, the junctions tend toward an antiparallel alignment, while a moderate external magnetic field (H) of 0.1–0.2 T will drive them into a parallel magnetic state. Figure 1(b) provides a low-bias electron transport characterization of the junctions in the superconducting state for both magnetic alignments. The oscillations in the conductance curves are likely due to McMillan resonances [33, 34] that may occur for clean samples. Figure 1(c) provides temperature-dependent subgap conductance measurements that show a superconducting critical temperature of $T_c = 4.7$ K marked by an abrupt change of slope. This value is close to the one expected from the dependence of $T_c$ on the thickness for vanadium thin films grown in ultra high vacuum conditions [35]. The steeper decrease in conductance at biases less than the superconducting gap and below $T_c$ is due to the reduced density of states of vanadium at the Fermi level in the superconducting state. An alternative method to determine the superconducting critical temperature, which provides similar results, is related to the detection of the emergence of superconducting quasiparticle interference effects in conductance as a function of temperature [33].

In order to perform thermoelectric response measurements, we induce temperature gradients across the junctions by controlling the power dissipated by a light emitting diode (LED) placed above them. This creates an inward heat flux on the top surface of the samples that can be tuned by biasing the LED at different voltages $V_{LED}$. We estimate the temperature profile and the total temperature difference $\Delta T$ across the heterostructure by

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Sketch of the F/S/F junctions when heated by a light emitting diode (LED). The black arrows show the crystalline axes and the direction of the applied magnetic field. The layer thicknesses are given in nm. (b) Conductance–bias curves under different magnetic configurations measured at $T = 0.07T_c$. Conductance is normalized so that $G(V = 7$ mV) = 1$ when $\mu_0H = 0$. (c) Temperature dependence of the zero bias conductance measured with a current bias $I_{bias} = \pm 0.5$ nA with $\mu_0H = 0$, showing a superconducting critical temperature of 4.7 K indicated by the dashed line. The decrease in conductance below $T_c$ corresponds to the opening of a pseudogap in the conductance due to superconductivity.</figcaption></figure>
solving the heat diffusion equation in a one-dimensional model that approximates our samples [27]. We note that the estimated values of ΔT are in the order of 0.1 – 0.2 K, which indicates that the device could operate in slightly nonlinear regime in measurements with a base temperature of 0.3 K.In the absence of applied current (I = 0), the TE voltage ΔV generated under a given ΔT is obtained by subtracting the background voltage signal from the voltage measured under heating (see [27] for the discussion of the origin of the background TE signal).

Figure 2(a) presents our main experimental observation: a substantial (over 10×) change in ΔV between the AP and P alignments of the ferromagnetic electrodes. The Seebeck coefficient for the parallel alignment, calculated based on the estimated temperature gradients, reaches a value of approximately 100 µV/K [27]. The strong magnetic field dependence of the TE signal disappears above $T_c$. The data shown in Figure 2 corresponds to an upward sweep of the external in-plane magnetic field. Starting from the most negative H, as its magnitude decreases and the magnetic configuration transitions from the P to AP state, the TE signal changes relatively smoothly. In contrast, the transition from AP to P in the positive H branch leads to an abrupt increase in the TE response, accompanied by a subtle but reproducible “overshoot” effect manifested as a peak followed by an immediate drop, shown in detail in Figure 2(b).

Magnetoresistance (MR) measurements performed during the TE experiments are shown in Figure 2(c). Resistance R is measured at a bias of V ~ 5 mV under each value of applied in-plane magnetic field H in thermal equilibrium before heating the junction with the LED. We define the magnetic field dependent MR as:

$$MR(H) = \frac{R(H) - R_{min}}{R_{min}} \times 100\%. \quad (1)$$

Here, $R_{min}$ is the minimum measured value of resistance for a given base temperature, and therefore corresponding to the magnetic state closest to a fully AP state.

Interestingly, we find that resistance is higher in the P state than in the AP state, contrary to the seminal Fe/MgO/Fe-based heterojunctions [26, 36, 37]. We believe that the main reason for the negative MR in our F/S/F (F/N/F above $T_c$) junctions is that two Fe layers are separated by a relatively thick 40 nm vanadium film. Therefore, the simple picture explaining the conventional tunneling MR in Fe/MgO/Fe junctions may not be applicable to our structure. Importantly, early works on tunneling MR demonstrated that the presence of different non-magnetic spacers in between two ferromagnetic electrodes can alter not only its magnitude but also its sign [38]. More recently, the introduction of a vanadium film between Fe and MgO/Fe has been predicted to invert the MR sign [39].

We observe that the MR curve remains qualitatively similar above and below $T_c$, i.e., it decreases more smoothly in the P to AP transition in the negative H branch and increases more abruptly in the AP to P transition in the positive H branch. As we discuss below, based on micromagnetic simulations, the difference between P-AP and AP-P transitions could be due to

<figure><img src="image_3.png" /><figcaption>FIG. 2. Thermoelectric response of a F/S/F junction as a function of the upward swept applied in-plane (IP) magnetic field at T = 0.3 K and two different evaluated ΔT values. For comparison, a much smaller thermoelectric response observed above the critical temperature (at T = 11 K) is also shown. Part (b) zooms the low field region in (a), where an abrupt transition from the AP to the P state takes place. (c) Typical magnetoresistance (MR) curves vs IP magnetic field at a bias of V = 5 mV at T = 0.3 K and 11 K. The blue arrows show the relative magnetic alignment of the ferromagnetic electrodes while the black arrow shows the direction of variation of the magnetic field. (d) Dependence of the TE response on the evaluated temperature difference between the ferromagnetic electrodes, measured at different IP magnetic states as indicated by the colored points in (c).</figcaption></figure>

a rapid change in the relative magnetization configuration of the F electrodes during AP-P transition due to the combined effects of macrospin rotation and domain wall formation and displacement. The overall magnitude of the MR increases by approximately a factor of three in the superconducting state. We believe that the enhanced negative MR observed well below $T_c$ could be attributed to a more effective spin signal transfer between the two Fe electrodes through the 40 nm thick V layer due to the generation of superconducting spin-triplet states [29, 40].

Figure 2(d) illustrates the TE response as a function of the temperature gradient at various fixed magnetic fields, color-coded to correspond with the MR plot in Figure 2(c), confirming a strong dependence of the TE response with the magnetic state of the junctions.

## MICROMAGNETIC SIMULATIONS

Numerical simulations of the TE response require knowledge about the average angle between the magnetizations of the two
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Angle difference between the macrospin of the two F regions for an AF coupling of RKKY = -0.5 mJ/m² (selected points with numbers). (b) Normalized magnetoresistance for two different couplings and calculated using two methods explained in the text, where R₀ is the resistance at μ₀H = 0. (c) Magnetization map for the interface of both F layers on the selected points in (a).</figcaption></figure>

ferromagnets enclosing the superconductor. This information was obtained through micromagnetic simulations using the open-source software MuMax³ [41]. At the same time, in order to mimic the experimental setup of Figure 1(a), the system was divided into three different regions. Region 1 models the soft 45 nm thick Fe layer, while region 2 is the 10 nm thick Fe layer and Region 3 the 20 nm thick Co layer, that together form the hard Fe/Co electrode (see [27] for simulation details). To simulate a more realistic system, the presence of defects in the crystalline structure of the layers was taken into account. A concentration of 25% for superficial and 10% for bulk defects was considered. [27].

In our epitaxial Fe/MgO/V/MgO/Fe/Co vertically patterned junctions grown on MgO(001) substrate, the Fe(001) surface follows the cubic symmetry of the underlying MgO(001) lattice, leading to a dominant fourfold anisotropy with two easy axes along the two diagonals of the rectangular patterned junctions (along the MgO sides). This stabilizes two equivalent Fe (100) and Fe (010) magnetization directions marked (x) and (y) in Figure 3(c) indicating both equivalent directions. As sketched in Figure 1(a), Fe/Co are two separate layers, i.e. not intermixed Fe-Co. The first Fe(10nm) layer interfacing MgO has a fourfold crystalline anisotropy, similarly to the soft Fe layer, and is hardened by the Co(20nm) layer. The

micromagnetic simulations did not assign any in-plane or out-of-plane anisotropy to the Co layer to simulate magnetization reorientation and magnetoresistance. The possible out of plane anisotropy is irrelevant in our case as the Co layer thickness substantially exceeds 1-2 nm, where perpendicular magnetic anisotropy has been observed [42].

Moreover, owing to the presence of the low field antiferromagnetic (AF) coupling between the two F layers, a negative exchange interaction was introduced between regions 1 and 2 by means of a negative Ruderman–Kittel–Kasuya–Yosida (RKKY) coupling constant. Different RKKY constants from -0.5 mJ/m² to -2 mJ/m² were considered. The simulations were performed by varying the magnetic field starting from -0.5 T and up to 0.5 T. The results for the angle difference Δθ(H) in Figure 3(a) and Figure 4(c) were computed based on the macrospin of each region. In other words, the average magnetization vector for AF coupled regions 1 and 2 (see Fig.S5 in [27]) was computed separately, and then the angle between them was calculated.

To calculate the MR we followed a simplified model [43] to approximate the magnetic state dependent conductance G for our junctions (see [27] for details). We simulated the MR using two approaches. The first one (AvAngle) calculates conductance as a function of the angle of the averaged mag-
netization in the two layers. The second approach divides the magnetic layers into a number of interfacial cells and then averages the conductance between each cell (discretization size of $2.34 \times 2.34 \times 1.67 \text{ nm}^3$ was used, see [27] for the details) from the one electrode and a randomly chosen cell of the opposite electrode. Then, the electrodes were interchanged and same procedure was repeated. The second method (RndDiff.) thus models elastic impurity scattering of the electrons within the 40nm thick V layer separating the ferromagnetic electrodes. As one observes in Figure 3(b), using the lowest AF coupling value of RKKY= -0.5 mJ/m², the above mentioned concentration of defects and the RndDiff. method, we obtain an MR curve that is qualitatively similar to the experimental measurements in Figure 2(c). The magnetoresistance saturates above 0.3 T and the “overshoot” effect is reduced as in MR experiments.

THERMOELECTRICITY VIA NON-EQUILIBRIUM QUASICLASSICAL THEORY

We use non-equilibrium Keldysh-Usadel Green's function theory [2, 44] to numerically explore the setup in Figure 1(a). We first provide an overview of the model used. A more technical description of the modeling is provided in [27]. Second, we combine the results from the quasiclassical simulations with the micromagnetic simulations to model how the thermoelectricity depends on the external magnetic field, and we compare the result with the experimental data in Figure 2(a).

The ferromagnets in Figure 1(a) are treated as nonsuperconducting metallic reservoirs at temperatures $T$ and $T + \Delta T$ for the soft (MgO-Fe) and hard (Fe-Co) ferromagnets, respectively. The voltages in the reservoirs are $\pm\Delta V/2$, respectively. The voltages are chosen such that the electrical current $I$ through the system is zero. The interfaces to the superconductor are treated using spin-active tunneling interfaces with spin polarization $p$, tunneling conductance $G_0$, spin-mixing $G_\phi$ and the average interface magnetization direction $m$. The spin polarization filters incoming spins, while the spin-mixing generates a spin splitting in the superconductor. The average interface magnetizations lie in the plane of the interfaces, and the angle difference between the two interfaces is $\Delta\theta = \theta_2 - \theta_1$. This angle difference depends on the external magnetic field, as explained in the previous section. The superconductor is treated as a Bardeen-Cooper-Schrieffer (BCS) [45] superconductor of length $l$. The model is illustrated in Figure 4(a). We assume that the system is diffusive, and we solve the Usadel equation [46] in the superconductor. This is done self-consistently for the superconducting order parameter. When the Usadel equation is solved, the current $I(\Delta V)$ is calculated. The thermovoltage is determined by solving $I(\Delta V) = 0$.

In the simulations, we use the values $G_0/G = 1/3.5$, $l = 1.5\xi$ and $p = 0.8$. Here, $G$ is the normal state bulk conductance and $\xi$ is the diffusive coherence length of the superconductor. The spin-mixing angle is typically taken to be a fitting parameter and is set to $G_\phi/G_0 = 2.75$. The temperature is set to $T = 0.057T_c^B$.

<figure><img src="image_5.png" /><figcaption>FIG. 4. (a) The model used in the numerical calculations. The superconductor S is sandwiched between two ferromagnets, modeled as non-superconducting reservoirs N with spin-active interfaces FI. The angle difference between the average in-plane magnetizations $m_1$ and $m_2$ of the interfaces is given by $\Delta\theta = \theta_2 - \theta_1$. The external magnetic field $\mu_0H$ affects $m_1$ and $m_2$ differently. (b) Induced thermovoltage $\Delta V$ as a function of the angle difference. $\Delta_0$ is the bulk superconducting gap at zero temperature. (c) Simulated angle difference for an external magnetic field $\mu_0H$ with four different RKKY couplings in units mJ/m². (d) Thermovoltage as a function of the external magnetic field. For comparison, the inset shows the experimental results in Fig. 2(a) for $\Delta T = 0.2$ K.</figcaption></figure>

and the temperature difference is $\Delta T = 0.038T_c^B$, where $T_c^B$ is the bulk critical temperature of the superconductor. After solving the Usadel equation, we insert the numerical values $T_c^B = 5.3$ K and $\xi = 26$ nm [29, 47], which yields $T = 0.3$ K, $\Delta T = 0.2$ K, and $l = 39$ nm. These are the values used to convert from dimensionless to numerical values in Figures 4 and 5. We also use the BCS relation $\Delta_0/k_B T_c^B = 1.76$ between the bulk gap $\Delta_0$ at zero temperature and the bulk critical temperature. The dependence of $\Delta V$ on $\Delta\theta$ is shown in Figure 4(b). We note
that with the chosen interface parameters, the quasiclassical
simulations qualitatively reproduce the conductance curve in
Figure 1(c), although with a lower critical temperature of the
superconducting film (not shown here).

A point requiring clarification is that while the temperature
gradient and resulting thermovoltage are measured across the
entire stack of materials in the experiment, the temperature
gradient and thermovoltage are measured across the super-
conducting region in the theoretical modeling. However, this
procedure is warranted for two reasons. As shown in [27], the
simulations for the local temperature profile reveal that the vast
majority of the temperature drop occurs precisely across the
Fe/MgO/V interfaces and the superconductor, and not in the
ferromagnets themselves. Moreover, the voltage drop across
the junction has its primary contribution coming from the in-
terfaces between the ferromagnets and the superconductor due
to the MgO-barrier providing a high resistance. This matches
the assumptions made in the theoretical model.

We use micromagnetic simulations for four different RKKY
couplings, as shown in Figure 4(c), to model the thermoelectric
response to an applied magnetic field. The thermovoltage as a
function of the applied magnetic field is presented in Figure
4(d), and does qualitatively match the experimental data in
the inset. The difference in magnitude of the thermovoltage
between the simulations and measurements may be attributed
to the restriction in quasiclassical theory regarding how the
reservoirs are treated. Normal and ferromagnetic reservoirs are
described by the same Green's function due to the requirement
of a weak spin polarization in the quasiclassical approximation,
unlike the strongly polarized ferromagnet Fe used in the
experiment, and this likely reduces the numerically simulated
value for ΔV. No such methodology restriction applies to the
interfacial polarization in our treatment, however, which can
model strongly polarized ferromagnetic interfaces [48].

DISCUSSION

Our experimental results demonstrate that the magnetic
configuration of the superconducting spin-valve changes the
thermoelectric signal by more than an order of magnitude
when switching from a parallel to an antiparallel state. After
achieving a qualitative agreement between the experimental
results and simulations of both the TE and MR effects, we
conclude that the thermoelectric effects are highly sensitive to
the collective domain rotation within the ferromagnetic layers,
as shown in Figure 3(c). At H = 0, due to the negative exchange
interaction, the layers tend to align in an antiparallel configu-
ration, resulting in the observed minima in resistance and TE
signals. When an in-plane magnetic field is applied along one
of the easy axes, the reorientation of the magnetization in the
two ferromagnetic layers involves complex magnetic dynamics,
including the formation of inner and edge domain walls, as
illustrated in Figure 3(c). Due to magnetocrystalline anisotropy,
inner magnetic moments preferentially align towards the crys-

<figure><img src="image_6.png" /><figcaption>FIG. 5. The thermovoltage dependence on (a) the ratio of the bulk conductance to the interface conductance $G/G_0$, (b) the spin mixing $G_\phi$, and (c) the interface polarization $p$ in the parallel (P) and antiparallel (AP) configuration.</figcaption></figure>

tal's easy axes, while the edge located spins tend to minimize
their normal component to reduce the stray field related energy
contribution.

In Figure 3, we also observe how the correlated rotation
between the two ferromagnetic layers can induce an “overshoot”
in the TE response (Figure 2(a)). The ”overshoot” is seen as
a sharp dip that occurs in the angle misalignment between
the average magnetizations in Figure 3(a). This correlated
rotation of two antiferromagnetically coupled Fe layers from
points 2 to 4 as marked in Figure 3(a) is explained by the
corresponding magnetization maps in 3(c). At point 3, one
layer is mainly aligned along the x-axis, while the other aligns
along the y-axis. At point 4, the alignment is interchanged.
Following this transition, the layers continue to rotate towards
a parallel alignment at high fields. This effect is observed in
simulations and in the experiment only for AP → P transition
demonstrating the asymmetry between the P → AP and AP →
P transitions.

The mechanism behind the thermoelectric effect was originally proposed in Refs. [14, 15], and is described in detail in Ref. [25]. The effect relies on two properties of the structure. First, the superconducting density of states must be spin split. Second, the interfaces must filter the spins such that the density of states available for the thermally excited electrons and holes
is asymmetric around the Fermi energy.

The thermoelectric effect is optimized when there is a large gap and spin splitting in the superconductor, and the interface polarization is high. For the parameter sets in Figure 4, the gap value is approximately 75% of the bulk value $\Delta_0$ for all angle configurations of the interfaces. If the interface conductance is good, corresponding to low $G/G_0$, the gap is suppressed due to the inverse proximity effect. This is seen in the P configuration in Figure 5(a). As the interface conductance decreases, the gap value and thus the thermovoltage increases. By reducing the interface conductance further, the superconductor becomes more isolated from the reservoirs, and the transport across the superconductor is reduced. The spin mixing $G_\varphi$ induces the spin splitting in the superconductor, and the thermovoltage dependence on the spin mixing is shown in Figure 5(b). When $G_\varphi$ is zero, there is no spin splitting, while when $G_\varphi$ is high, the spin splitting is large and superconductivity is destroyed. The optimal $G/G_0$ and $G_\varphi$ values also depend on the length of the superconductor. A longer superconductor can withstand a higher interface conductance and spin mixing, thus increasing the thermovoltage. Moreover, sandwiching the superconductor in between the ferromagnets in the F/S/F structure increases the Seebeck coefficient compared to the S/F/F in Ref. [26] due to the increased spin splitting in the superconductor. Figure 5(c) shows that the thermovoltage increases with increasing interface polarization. This provides a way to optimize the thermoelectric effect. As seen in Figure 5, the thermovoltage in the AP configuration is significantly smaller in magnitude than in the P configuration. This is because the average spin splitting in the AP configuration is smaller than in the P configuration.

Finally, we note that the large Seebeck coefficient in the P configuration remains large and more than an order of magnitude greater than in the AP configuration as the base temperature is increased, meaning that the effect is robust against an increase in temperature [27]. These theoretical predictions will be tested and verified in forthcoming experiments currently in preparation.

## CONCLUSIONS

We report the experimental observation of a large superconducting thermoelectric spin-valve effect which meets a number of key criteria required for cryogenic thermoelectric applications: (i) we use commercially available materials with high spin polarization which are easily grown; (ii) a considerable thermoelectric effect, with a Seebeck coefficient exceeding 100 µV/K, (iii) a large change in the thermoelectric response – greater than a factor of 10 – when switching the device from AP to P alignment. Micromagnetic simulations and theoretical modeling based on the non-equilibrium Keldysh-Usadel Green's function formalism are consistent with both the thermoelectric and magnetoresistance results. As real cryogenic quantum systems are expected to operate under thermal gradients, integrating low-temperature Peltier elements into quantum computers could significantly

enhance their performance. One promising future application of ferromagnet/superconductor/ferromagnet-based devices is the recovery of heat inevitably generated during quantum error correction [21]. This is particularly relevant since most current quantum computers use qubits based on superconducting hybrid structures.

## METHODS

A more detailed explanation of the matters treated in this section is provided in the Supplemental Material [27].

**Sample growth and experimental procedure:** The F/S/F junctions studied in this work were grown by molecular beam epitaxy (MBE) and litographed into $20 \times 20 \ \mu m^2$ lateral size samples. Both MR and TE response measurements highlighted in Figure 2 were performed at temperatures of 0.3 K and 11 K since these temperatures can be sustained in the cryogenic system (Janis He³ cryostat) without the use of external heaters, minimizing extrinsic thermal gradients. The TE voltage $\Delta V$ results from subtracting the background voltage signal from the voltage measured under heating at zero current bias. Different temperature gradients are induced by controlling the power dissipated by the LED by tuning its voltage bias $V_{LED}$. In MR measurements performed at thermal equilibrium, voltage is measured at bias currents of $I = \pm 10$ nA and the resistance is calculated as $R = V/(2|I|)$. This current bias yields a voltage of ~ 5 mV.

**Modeling of the temperature profile:** In order to estimate the total temperature difference $\Delta T$ across the junction when heated under different $V_{LED}$, we solve the heat diffusion equation in a 1D model of our junctions. We impose an inward heat flux $q$ on one of the edges to account for the LED heating. The value of $q$ is estimated by taking into account the power dissipated by the LED ($I_{LED} \times V_{LED}$) and its directionality through the radiation pattern provided by the manufacturer, the distance between the LED and the sample, and the contact area of the junctions that can absorb heat. The other edge of the system is fixed at a given temperature. We use the calculated values of $\Delta T$ to estimate the Seebeck coefficient under different magnetic fields, base temperatures, and temperature gradients.

**Micromagnetic simulations:** The relative angle between the ferromagnetic layers is calculated using micromagnetic simulations performed in MuMax³ [41]. We simulate a 3D model of the magnetic layers our junctions with matching vertical dimensions. A phenomenological antiferromagnetic coupling is set between the two Fe layers of the system to reproduce the behavior under applied in-plane magnetic fields inferred from the MR measurements in Figure 2(c). Superficial and bulk defects are implemented by lowering the saturation magnetization and adding disorder to the local magnetocrystalline anisotropy, respectively. The calculated MR displayed in Figure 3(b) is obtained by using the
Slonczewski formula [43] with a sign change to account for the negative MR induced by the presence of the V layers in between the two ferromagnetic electrodes.

Thermoelectricity via non-equilibirum quasiclassical theory: We use non-equilibrium Keldysh-Usadel Green's function theory [2, 44, 46] to model the thermoelectric effect in the F/S/F structure numerically. The ferromagnets are treated as non-superconducting metallic reservoirs with spin-active interfaces. We calculate the quasiclassical Green's function in the superconducting layer numerically by solving the Usadel equation self-consistently for the superconducting order parameter, and find the voltage ΔV such that the charge current through the system disappears. The thermovoltage ΔV depends on the external magnetic field through the relative angle of the average interface magnetizations.

Acknowledgments. The work in Madrid was supported by Spanish Ministry of Science and Innovation (PID2021-124585NB-C32, TED2021-130196B-C22 and PID2024-155399NB-I00). F.G.A. also acknowledges financial support from the Spanish Ministry of Science and Innovation through the María de Maeztu Programme for Units of Excellence in R&D (CEX2023-001316-M) and TEC-2024/TEC-380 (Mag4TIC-CM) project. The work in Trondheim was supported by the Research Council of Norway through Grant No. 323766 and its Centres of Excellence funding scheme Grant No. 262633 "QuSpin." Support from Sigma2 - the National Infrastructure for High Performance Computing and Data Storage in Norway, project NN9577K, is acknowledged.

†P.T. and J.B.T. contributed equally to the manuscript.

<footnote>* e-mail: jacob.linder@ntnu.no</footnote>

<footnote>† e-mail: farkhad.aliev@uam.es</footnote>

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