Spin pumping in NbRe/Co superconductor-ferromagnet heterostructures

Ferromagnetic resonance (FMR) spectroscopy measurements were performed on NbRe/Co/NbRe trilayers in order to probe spin pumping across the superconductor/ferromagnet interface and to detect the possible presence of spin-triplet pairing in the superconducting NbRe layer. FMR spectra were acquired as a function of frequency, magnetic field, and temperature, and reveal that the Gilbert damping parameter associated with spin pumping remains almost constant as temperature goes down through the superconducting transition. Additionally, the dependence of the Gilbert damping parameter on the thickness of the NbRe layer in trilayers is used to determine the values of the spin mixing conductance at the interface ( 18−21  nm−2) and the spin diffusion length ( 7.1−12.5 nm) in the NbRe layer. These findings may suggest that spin pumping would still be effective even though NbRe becomes superconducting, which would indicate that the spin-triplet would be the dominant pairing mechanism. Future experiments are proposed in light of these results.


Introduction
Superconducting spintronics is a promising emerging field, which may represent a paradigm shift for many applications working with a logic based on spin currents, with * Authors to whom any correspondence should be addressed.
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unprecedented gain in terms of energy dissipation [1]. The building block of a superconducting-based spintronic device is a superconductor/ferromagnet (SC/FM) heterostructure in which it is possible to exploit the coexistence of these two antagonistic orders, coupled by the proximity effect (PE), to manipulate spin states [2,3]. In this sense, a fundamental role is played by spin-polarized Cooper pairs, the so-called triplets, that, under appropriate conditions, can be generated at the interface of SC/FM hybrids [4]. In the last years, several combinations of materials and device architectures have been explored to create and manipulate triplets, for example, spin valves based on half-metallic ferromagnets [5], Josephson junctions fabricated from common metals [6] or conical magnets [7] and domain walls in notched geometry [8].
Recently, spin pumping by ferromagnetic resonance (FMR) has been recognized as a powerful method to investigate spin transport in superconducting hybrids and to disclose new mechanisms useful for the design of spintronic devices [9][10][11][12][13][14][15][16]. In this technique, an angular momentum transfers from an FM to an SC layer, due to the precession of the magnetization in the FM. The transfer of angular momentum (either spin or orbital current) across the interface between both layers and its relaxation has an effect on the Gilbert damping parameter of the FM layer, which tells about the overall dissipation process [10,11,16]. An enhancement of the Gilbert damping below the superconducting critical temperature, T c , is a signature of the presence of angular momentum in the SC layer, that is, the existence of spin supercurrents. These currents are supported by the presence of spin triplet correlations which may originate, for instance, from the coexistence of spin-orbit coupling (SOC) and ferromagnetic ordering [11], non-homogeneous magnetization at the interface [16,17] or by the varying-in-time magnetization itself [18].
NbRe is a non-centrosymmetric superconductor (NCS) recently obtained in thin film form [19]. Due to its peculiar crystal structure, its superconducting order parameter may in principle consist of a mixture of spin-singlet and spin-triplet components [20]. However, this occurrence is still debated in the literature [21][22][23][24]. In this respect, both the realization of NbRe-based SC/FM heterostructures and spin pumping experiments can be useful to shed light on this question. First, SC/FM hybrids can lead to a better understanding of the nature of the NbRe superconducting order parameter. In these systems, it is possible to discriminate between different pairings since, for instance, unlike singlets, a triplet component can propagate over longer distances in the ferromagnet with respect to conventional Cooper pairs [25] or can survive the ferromagnetic filtering [26]. Second, FMR experiments can successfully reveal the presence of equal spin pairing, since spin transfer into an SC is possible only in the presence of a triplet channel [16].
In this work, we report spin pumping experiments in NbRe/Co-based heterostructures as a means to explore the pairing symmetry in NbRe, as well as to estimate its spin transport parameters. Cobalt was chosen since it is a valuable material for the realization of high-quality clean limit SC/FM devices [27]. We measure FMR in bilayers and trilayers and study the temperature dependence of the resonance linewidth for different thicknesses of the NbRe layer. The analysis of such dependence allows estimating values for the spin mixing conductance at the interface and the spin diffusion length in the NbRe layer. The possibility of the occurrence of the spin triplet state in the NbRe layer is discussed.
This paper is organized as follows. Section 2 describes the sample preparation and the experimental details behind every measuring technique that was used. Section 3 presents the results obtained in trilayers and bilayers, gives an interpretation in the framework of spin pumping through an interface, and discusses the pairing mechanism that may possibly be found in the NbRe layer that integrates these samples in light of the results achieved. Finally, section 4 summarizes this work and makes some considerations on the possible development of this study.

Experimental
NbRe/Co/NbRe trilayers were deposited on Si(100) substrates by UHV dc diode magnetron sputtering operating at a base pressure of about 2 × 10 −8 mbar. The depositions were performed at room temperature, at a sputtering power of 100 W and Ar pressure of 10 −2 mbar. The typical deposition rates were about 0.12 nm s −1 for both materials. More details about the sputtering procedure can be found elsewhere [28]. The trilayers have constant Co thickness, d Co = 20 nm, which assures a good signal-to-noise ratio and NbRe thickness in the range d NbRe = 8 − 50 nm, to evaluate the effect of such thickness on the spin relaxation process. All trilayers were deposited in the same sputtering run, thanks to a movable shutter, which selectively covers the substrates. This ensures the deposition of a series of hybrids under identical conditions. In order to compare the results obtained in different structures [18], a NbRe/Co bilayer with d Co = 20 nm and d NbRe = 15 nm was also deposited. To prevent Co oxidation, the bilayer was covered in situ by a non-superconducting Nb cap layer of 2 nm, which in air forms a protective insulating barrier of Nb 2 O 5 for the underlying Co.
The interface properties of the hybrids were investigated by low-angle x-ray reflectometry (XRR) measurements using a Philips X-Pert MRD high resolution diffractometer with a Cu-K α radiation (λ =1.5406 Å). The superconducting electric transport properties of the films were measured in a 4 He cryostat using a standard dc four-probe technique on unpatterned samples.
The setup for performing FMR spectroscopy is illustrated in figure 1(a). A vector network analyzer (VNA) was used in conjunction with a coplanar waveguide (CPW) to convert the rf-electrical signal into an oscillatory magnetic field acting on the SC/FM heterostructure. The samples were mounted in a flip-chip geometry on the CPW and an external static magnetic field (µ 0 H) was applied in the sample plane, perpendicular to the oscillatory magnetic fields generated by the CPW (h RF ). The CPW was mounted in a probe and introduced in a cryostat capable of cooling down to 1.8 K and applying magnetic fields up to 6 T.
To obtain a map of FMR, the rf-electrical signal transmission through the CPW, S 12 , was recorded as a function of the external field and microwave frequency (f ) using the VNA, and the overall procedure was repeated as a function of temperature (T). The external magnetic field varied from −0.4 T to 0.4 T, and the microwave frequencies ranged from 10 MHz to 20 GHz. Figure 1(b) shows illustrative transmission curves as a function of the applied field measured at T = 10.0 K for different frequencies in the SC/FM trilayer with d NbRe = 25 nm.
In addition to FMR spectroscopy, we also performed inverse spin Hall effect (ISHE) measurements [29,30] using a similar experimental setup. ISHE is the reciprocal effect of the spin Hall effect (SHE) [31] and it originates from spin-orbit interaction-a spin current generates a transverse charge current in a conductor, NbRe in our case. We made electric contacts in our sample and measured the voltage induced while sweeping the applied magnetic field at a fixed frequency and temperature, in a way equivalent to the transmission curves shown in figure 1

X-ray reflectivity
The specular reflectivity profile of a deliberately fabricated NbRe/Co bilayer with nominally d NbRe = d Co = 6 nm is shown in figure 2 (black circles). The red line is the simulation curve obtained by using the Parratt and Nevot-Croce recursion relation [32,33] in order to determine the exact values of the layer thickness as well as the roughness at the SC/FM interface. The modelling gives d NbRe = 6.3 nm, d Co = 4.5 nm, with an interfacial roughness of about 0.3 nm. Moreover, the simulation reveals the presence of a thermal oxide layer of CoO of thickness d CoO = 0.9 nm with a roughness of about 0.8 nm. This result confirms the good layering and interface quality of the NbRe/Co heterostructures investigated in this work.

Electric transport
The NbRe/Co/NbRe trilayers were preliminarily characterized by electric transport measurements to determine their superconducting critical temperature. A constant current, I b = 0.1 mA, was applied to bias the films. T c was defined at the midpoint of the resistive transition, R(T). In figure 3 the dependence of T c on d NbRe is shown with solid squares. A representative R(T) curve for the trilayer with d NbRe = 50 nm, showing a superconducting transition temperature of T c ≃ 5.9 K, is reported in the inset of this figure. As expected, T c is strongly suppressed for thinner samples with d NbRe comparable to the NbRe superconducting coherence length, which is  about ξ ≈ 5 nm [19]. The T c of the NbRe/Co bilayer is also shown with a crossed open square in the same figure.

Ferromagnetic resonance
The shape of the FMR absorption line, which represents the absorption of microwave energy in the sample at a fixed frequency as a function of the external field, is typically described by a symmetrical Lorentzian form, although it may be asymmetrical if the detection method involves modulating the external field and utilizing lock-in techniques. However, the presence of eddy currents in conductive samples may impact the symmetry of the line shape [34]. Our study accounts for this by fitting the FMR absorption to a combination of symmetrical and asymmetrical contributions, as shown respectively by the yellow and purple lines in figure 4(a) [35].
The position of the resonance frequency, f, as a function of the applied field, µ 0 H, when such a field is applied in the film plane is described by the Kittel equation [36] where γ is the gyromagnetic ratio (γ ≈ 28 GHz T −1 ) and M eff = M S − H k is the effective field given by the saturation magnetization, M S , and the out-of-plane anisotropy field, H k . Figure 4(b), which plots the f (H) data obtained at T = 2.0 K for the trilayer with d NbRe = 25 nm, shows that our fit of the measured resonance position agrees well with the Kittel dispersion given by equation (1) [12,18], where the thickness of the superconducting material exceeded the London penetration depth (⩾100 nm).
The full width at half maximum of the FMR absorption lineshape serves as a measure of the underlying magnetic relaxation mechanisms. We aim at measuring the magnetodynamic damping and the variations caused by NbRe as a function of temperature. Thus, we studied the FMR linewidth, µ 0 ∆H, as a function of microwave frequency, f, and extracted the Gilbert damping parameter, α, following the relation [38] where µ 0 ∆H in is the inhomogeneous linewidth broadening, which does not depend on frequency and usually accounts for magnetic inhomogeneities in the sample. Figure 5(a) shows the frequency dependence of the linewidth of the FMR peaks obtained for the trilayer with d NbRe = 25 nm at three temperatures, above (T = 6.0 K, green squares) and below (blue circles at T = 2.0 K, and yellow triangles at T = 4.0 K) the superconducting transition temperature determined from resistance measurements (T c ≃ 5.6 K). At each temperature, the linewidth grows linearly as the frequency increases. The lines shown in this panel are the fits of the data to equation (2). As the temperature decreases, the whole set of data shifts downwards, although the slope remains almost the same. This suggests that the damping parameter is almost temperature independent, while the ordinate of the origin, that is the inhomogeneous contribution to damping, progressively reduces as the temperature lowers.
Similar measurements were made for all trilayers at several temperatures in the same range (2.0 − 12.0 K). Figure 5(b) plots the temperature dependence of the linewidth at 15 GHz for the sample with d NbRe = 10 nm (lowermost, blue data) and the sample with d NbRe = 25 nm (topmost, red data). In this panel, the linewidth of the sample with d NbRe = 25 nm first increases slightly in the normal regime and then shrinks from 24 to 22 mT as the sample is cooled from 7.0 K down to 2.0 K, mostly due to the reduction of the inhomogeneous damping. The sample with d NbRe = 10 nm also presents a decreasing behavior, although in this case the variation seems to be less steep and the linewidth remains almost invariant in the superconducting regime. These results suggest that a change could be taking place at the superconducting transition temperature of these trilayers (T c ≃ 5.6 K for d NbRe = 25 nm, T c ≃ 4.7 K for d NbRe = 10 nm). In contrast to these sharp changes, figure 5(c) shows a much unclear evolution with temperature for the Gilbert damping parameter extracted by fitting the data in panel (a) using equation (2), with a slight decrease below T c for the sample with d NbRe = 25 nm (topmost, red dots) and We note here that although the Gilbert damping in the Co layer hardly varies across the superconducting transition of the NbRe layers, it does vary with the thickness of such layers, indicating a possible diffusion channel for spin precession in Co. Figure 6 presents the results obtained at 2.0 K for the trilayers with d NbRe = 8 nm (lowermost, blue circles), 10 nm (yellow triangles), and 25 nm (topmost, green squares). Similar results obtained for d NbRe = 50 nm are not shown for the sake of clarity. Fitting the data for each sample to equation (2) produced the lines shown in the figure. We can see that the ordinate of the origin, that is the inhomogeneous damping, remains virtually identical for the three samples, indicating that the variation of the thickness of the NbRe layer does not affect the magnetodynamic properties of the Co layer, neither does the possible superconducting screening of dipolar fields from Co [12,18]. However, the data indicate that the slope, that is the Gilbert damping parameter, increases with d NbRe , suggesting that a thicker SC layer causes a larger dissipation in the spin precession in the FM layer, as was previously observed both in the normal [13] and superconducting [11] state for NbN-and Nb-based SC/FM hybrids, respectively. We obtained α values of 0.0169, 0.0184, 0.0198 and 0.0194 for NbRe thicknesses of 8, 10, 25 and 50 nm, respectively, indicating that the Gilbert damping saturates at about 25 nm.
Different mechanisms may contribute to modifying the damping coefficient of Co in the presence of NbRe adjacent layers. Interfacial effects are expected to be mostly independent of the thickness of the NbRe layers, so we will assume that the additional damping on the precessing Co spin moments may be a measure of the spin pumping efficiency across the NbRe/Co interface. The quantity measuring the efficiency of spin transport across an interface is the spin mixing conductance, g ↑↓ , which is given by [39] where g is the Landé factor, µ B is the Bohr magneton, and ∆α = α NbRe/Co − α Co , with α NbRe/Co and α Co the damping parameters at the interface NbRe/Co and the Co layer, respectively. Equation (3) neglects the role of the spin diffusion length λ sd within the absorbing NbRe layer, which is taken into account in the following equation [40]: The inset of figure 6 presents the Gilbert damping parameter as a function of the thickness of the NbRe layer. The red line was obtained by fitting the data to equation (4) with g ↑↓ and λ sd as fitting parameters. We measured a Gilbert damping parameter of 0.007 for a 20-nm-thick bare film of Co, in agreement with values previously reported [41], and considered the experimental error from measurements to obtain estimates for the spin mixing conductance, g ↑↓ = 18 − 21 nm −2 , and the spin diffusion length, λ sd = 7.1 − 12.5 nm. These values are in the ballpark of those previously found in Nb/Py bilayers [42], and allow understanding the decrease of α for NbRe thicknesses below 15 nm, that is, comparable to λ sd , as reported in the inset of figure 6.
FMR measurements were also carried out in the NbRe/Co bilayer with d NbRe = 15 nm. Following the analysis made for trilayers, we first plot in figure 7(a) the linewidth of the resonance peaks as a function of frequency at temperatures above (T = 6.0 K, green squares) and below (blue circles at T = 2.0 K, yellow triangles at T = 4.0 K) the superconducting transition temperature determined from resistance measurements (T c ≃ 5.0 K). Other results obtained at several temperatures in the 2.0 − 12.0 K range were omitted for the sake of readability of the figure. Similarly to what happened for trilayers, the linewidth shows a linear dependence with frequency (following equation (2)), although in this case all points fall upon a single curve and there is no sign of shift between curves, i.e. both the Gilbert damping parameter and the inhomogeneous damping (which happens to be almost negligible) remain constant with temperature across T c . This is illustrated by figure 7(b), in which the linewidth obtained at 15 GHz does not seem to change significantly with temperature but shows an almost constant value around 18.5 mT. The Gilbert damping parameter is plotted in figure 7(c), showing a very small variation, if any, with temperature. Both the linewidth at 15 GHz and the Gilbert damping are smaller than those obtained for trilayers (see figures 5(b) and (c)), but still larger than those for bare Co layers. This could be explained considering that NbRe/Co/NbRe samples have two interfaces that may contribute to spin dissipation from Co into NbRe, and this would enhance spin pumping in trilayers with respect to bilayers.

Inverse spin Hall effect
In order to gain further confirmation of the spin current injection by FMR into NbRe, inverse spin Hall measurements were performed on the trilayer with d NbRe = 25 nm. As shown in figure 8, a voltage (V ISHE ) is induced at the resonance fields at which FMR peaks occurred at 15 GHz (around 130 mT), as spins are pumped from Co into NbRe and this results in charge accumulation. Lorentzian peaks appear in the dc voltage and invert polarity by reversing the magnetic field, consistently with the symmetry of ISHE, which accounts only for the symmetric part of the voltage peaks [29,30,43]. As the temperature decreases, V ISHE manifests from 8.0 K down to 3.0 K, and then drops rapidly below the sensitivity of our measurement setup, similarly to what was previously found in experiments on Nb/Py bilayers [42]. This voltage drop is probably due to the reduction in NbRe resistivity, while the spin mixing conductance appears to be constant, as suggested by our FMR measurements. We note here that the MW power used for the detection of ISHE is 10 dBm, which is much larger than that used in the FMR experiments (−5 dBm), so we expect that a local heating of the sample takes place. This would explain the voltage drop at around 2.5 K instead of dropping at the T c value of 5.6 K measured for this sample.

Discussion
The results presented here might provide some insight into the nature of the pairing state in NbRe. When an SC and a FM are placed in contact, a PE occurs at the SC/FM interface, that is superconducting pair correlations stretch into the FM layer, while normal electrons leak into the SC layer [4]. The range of penetration of the pair correlations in the FM layer depends on the nature of the pairing state: spin singlet and antiparallel-spin triplet pairing correlations are characterized by a short-range decay (ξ S ), while parallel-spin triplet pairing shows a longrange decay into the FM layer (ξ T ) which is mostly limited by the spin diffusion length in the FM [17]. Due to their particular crystal structure, NCSs lack a center of inversion symmetry. As a consequence, a Rashba-type SOC develops [44] and a mixture of both singlet and triplet states is possible. Thus, when considering PE at the interface, both correlations can manifest in the FM layer, the larger the triplet component, the larger the range of decay [45,46]. In these circumstances, spin-polarized supercurrents are expected to occur in NCS materials [47][48][49]. For this reason, the observation, when NbRe is in the superconducting state, of a damping parameter value comparable to the one exhibited when such a layer is normal points out that spin pumping, instead of being forbidden in the SC layer, is actually maintained, suggesting that parallel triplet pairing correlations could be dominant [45]. Nonetheless, this should be deemed as indecisive, because Co can assure a long penetration of both the singlet correlations, ξ S = 3.0 nm [27], and the triplet ones, due to its long spin diffusion length of about 60 nm [50]. This complicates the interpretation of the results, since both kinds of correlations contribute to the damping differently. In this respect, it will be interesting to perform the same experiment by using a ferromagnet in the dirty-limit, for example Py as in previous experiments [10-12, 16, 42], because singlet correlations in such system should be almost negligible.
Finally, we would like to mention that the almost constant Gilbert damping parameter below the critical temperature cannot be due to a poor inverse PE (IPE). Indeed, as it was recently demonstrated [51], spin injection processes are suppressed in the case of low IPE, which takes place for large values of the ratio σ SC /σ FM , where σ is the normal state electrical conductivity. In our case, we have ρ NbRe = σ -1 NbRe ≈ 100 µΩ · cm [19] and ρ Co = σ -1 Co ≈ 30 µΩ · cm as estimated from 4-contact Van der Pauw resistivity (ρ) measurements on a deliberately deposited 20-nm-thick Co film. Therefore, we have σ NbRe /σ Co ≈ 0.3 and, as a consequence, an efficient polarized current transfer should in principle be possible at the NbRe/Co interface.

Conclusions
To summarize, we measured spin pumping in NbRe/Co-based heterostructures by FMR spectroscopy and ISHE voltage detection. We have an indication that the spin accumulation that occurs in NbRe when this material is normal remains almost the same in the superconducting state, as we do not see a clear change in the Gilbert damping parameter obtained from the experimental dispersion curves. We analyze the dependence of this parameter on the thickness of NbRe in the framework of a theoretical model for spin transport across an interface between a magnetic and a non-magnetic material, and we determine the spin mixing conductance through the interface and the spin diffusion length in NbRe. The fact that spin transport appears to be almost temperature independent when NbRe is driven into the superconducting state suggests that parallelspin triplet pairing could be the dominant mechanism in this system, even though this is to be taken as inconclusive. Future work should consider improving the quality of samples and focus on experiments in epitaxial thin films to try and find the ultimate proof for the nature of the pairing state. Additionally, the superconducting state could be further explored by preparing samples in the dirty limit, using for instance Py, to compare with the results found in Co, which happens to be in the clean limit.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.