Giant Transition-State Quasiparticle Spin-Hall Effect in an Exchange-Spin-Split Superconductor Detected by Nonlocal Magnon Spin Transport

Although recent experiments and theories have shown a variety of exotic transport properties of nonequilibrium quasiparticles (QPs) in superconductor (SC)-based devices with either Zeeman or exchange spin-splitting, how a QP interplays with magnon spin currents remains elusive. Here, using nonlocal magnon spin-transport devices where a singlet SC (Nb) on top of a ferrimagnetic insulator (Y3Fe5O12) serves as a magnon spin detector, we demonstrate that the conversion efficiency of magnon spin to QP charge via inverse spin-Hall effect (iSHE) in such an exchange-spin-split SC can be greatly enhanced by up to 3 orders of magnitude compared with that in the normal state, particularly when its interface superconducting gap matches the magnon spin accumulation. Through systematic measurements by varying the current density and SC thickness, we identify that superconducting coherence peaks and exchange spin-splitting of the QP density-of-states, yielding a larger spin excitation while retaining a modest QP charge-imbalance relaxation, are responsible for the giant QP iSHE. The latter exchange-field-modified QP relaxation is experimentally proved by spatially resolved measurements with varying the separation of electrical contacts on the spin-split Nb.

O ver the past decade, it has been shown that the combination of superconductivity with spintronics leads to a variety of phenomena that do not exist separately. 1−8 In particular, recent discovery and progress in the proximity generation and control of spin-polarized triplet Cooper pairs 1−3 at carefully engineered superconductor (SC)/ ferromagnet (FM) interfaces in equilibrium allow for the development of nondissipative spin-based logic and memory technologies.
Besides triplet Cooper pairs, nonequilibrium quasiparticles (QPs) in a spin-split SC 4−6 have also raised considerable interest. This is because their exotic properties resulting from the mutual coupling between different nonequilibrium imbalances of spin, charge, heat, and spin-heat can greatly enhance spintronics functionality. 5 For example, the coupling of spin and heat imbalances gives rise to long-range QP spin signals as observed in Al-based nonlocal spin valves 9−11 with a Zeeman spin-splitting field. In addition, a temperature gradient between a normal metal (NM) and a spin-split SC separated by a tunnel barrier induces a pure QP spin current 12 without an accompanying net charge current, analogous to the spindependent Seebeck tunneling. 13,14 Substituting the NM by a FM, one can achieve large (spin-dependent) thermoelectric currents 15,16 far beyond those commonly found in all-metallic structures.
Magnon spintronics 17−19 has been an emerging approach toward computing devices in which magnons, the quanta of spin waves, are used to carry, transport, and process spin information instead of conduction electrons. Especially in the low-damping ferrimagnetic insulator yttrium−iron−garnet (Y 3 Fe 5 O 12 = YIG), 20 a magnon-carried spin current can propagate over extremely long distances (centimeters at best), and it is free from ohmic dissipation due to the absence of electrons in motion. 17−19 Despite many recent advances 17−19 in this research field, how magnon spin current interacts with and is converted to QP spin and charge currents in a spin-split SC is yet to be investigated.
In this paper, we report three key aspects of the conversion behavior of magnon spin to QP charge via the inverse spin-Hall effect (iSHE) in an exchange-spin-split SC (Nb), directly probed by nonlocal magnon spin-transport 18 (Figure 1a). First, the iSHE in the superconducting state of Nb becomes up to 3 orders of magnitude greater than in the normal state. Second, this enhancement appears only in the vicinity of the superconducting transition temperature T c when the magnon spin current has an energy comparable to the (singlet) superconducting gap 2Δ SC of Nb ( Figure 1a). Lastly, its characteristic dependence on a dc current density J dc and the Nb thickness t Nb indicates that a singularity near the gap edge and a spin-splitting field are both essential for the giant transition-state QP iSHE, the latter of which is experimentally confirmed by performing spatial profiling of the transition-state enhancement by varying the separation distance of electrical contacts on the spin-split Nb layer. When a dc charge current I dc is applied to the right Pt injector, either electrically or thermally driven magnons accumulate in the ferrimagnetic insulator Y 3 Fe 5 O 12 (YIG) underneath and diffuse toward the left Pt detector. These magnon (s = +1) currents are then absorbed by the left Pt detector, resulting in the electron spin accumulation that is, in turn, converted to a nonlocal charge voltage V nl Pt via the inverse spin-Hall effect (iSHE). Such a conversion process also occurs for the central Nb and thereby V nl Nb . However, the conversion efficiency changes dramatically when turning superconducting due to the development of quasiparticle (QP) density-of-states with exchange spin-splitting ΔE ex . Note that in contrast to spin-singlet (S = 0) Cooper pairs in a coherent ground state, the excited QPs can carry spin angular momentum in the superconducting state. (b) Optical micrographs of the fabricated devices with and without a 10-nmthick Al 2 O 3 spin-blocking layer. (c) In-plane (IP) magnetization hysteresis m(H) curves of a bare YIG film, measured at a temperature T of 2−300 K. The inset summarizes the T dependence of the saturation magnetic moment.

RESULTS AND DISCUSSION
The nonlocal magnon spin-transport devices ( Figure 1b) we study consist of two identical Pt electrodes and a central Nb layer on top of 200-nm-thick YIG films, which are liquid-phase epitaxially grown on a (111)-oriented single-crystalline gadolinium gallium garnet (Gd 3 Ga 5 O 12 , GGG) wafer (see Methods). Control devices, in which a 10-nm-thick Al 2 O 3 spinblocking layer is inserted between Nb and YIG in an otherwise identical structure, are also prepared for comparison ( Figure  1b). Here, we send a dc current I dc through one Pt (using leads 1 and 2 in Figure 1b) while measuring the in-plane (IP) magnetic-field-angle α dependence of the nonlocal open-circuit voltages [V nl Pt (α), V nl Nb (α)] using both the other Pt (leads 7 and 8) and the central Nb (leads 3 and 4). Note that we apply an external in-plane magnetic field μ 0 H ext of 5 mT, larger than the coercive field of YIG (Figure 1c), to fully align its magnetization M YIG along the field direction. α is defined as the relative angle of μ 0 H ext (//M YIG ) to the long axis of two Pt electrodes, which are collinear.
As schematically illustrated in Figure 1a, the right Pt acts as a NM injector of magnon spin current across the Pt/YIG interface via either electron-mediated SHE (charge-to-spin conversion) 21 or spin Seebeck effect (SSE) (heat-to-spin conversion) 22 due to the accompanying Joule heating [ΔT ∝ (I dc ) 2 ]. The left Pt serves as an NM detector of the magnon spin current, diffusing through a YIG channel, via electronmediated iSHE (spin-to-charge conversion), whereas in the same device, the middle Nb functions as an exchange-spin-split SC detector of the diffusive magnon current via QP-mediated iSHE below T c . 8 The total voltage measured across the detector is given by V nl tot = ΔV nl el + ΔV nl th + V 0 , where ΔV nl el and ΔV nl th are proportional to the magnon spin current and accumulation created electrically (SHE ∝ I dc ) 21 and thermally [SSE ∝ (I dc ) 2 ], 22 respectively. These electrical and thermal magnon currents can be separated straightforwardly by reversing the polarity of I dc , allowing us to determine the magnitude of each component based on their distinctive angular dependences; 18 dc nl tot dc a n d ACS Nano www.acsnano.org Article . V 0 is an offset voltage that is independent of the magnon spin-transport.
The typical result of such a measurement using the Pt detector at 300 K is displayed in Figure 1d−i, for the t Nb = 15 nm devices with and without the Al 2 O 3 spin-blocking layer. This evidences that both electrically (Figure 1e and h) and thermally (Figure 1f and i) excited magnons transport spin angular momentum over a long distance of 15 μm at room temperature, which is consistent with the original work. 18 We note that from reference devices having the Pt injector/ detector only, the room-temperature magnon spin-diffusion length l sd m of the YIG is estimated to be around 11(9) μm for the electrically (thermally) driven magnons (Supplementary Section 1). The transporting spin current is absorbed by the middle Nb to a certain extent, given by the difference between the signals with versus without the Al 2 O 3 insertion (see Supplementary Section 2 for the quantitative analysis). Figure 2a,b,d,e show the temperature T evolution of ΔV nl el (α) and ΔV nl th (α) for the t Nb = 15 nm devices measured by the Pt detector at a fixed I dc = |0.5| mA (J dc = |3.3| MA/ cm 2 ). As summarized in Figure 2f and g, ⌈ΔV nl el ⌉ Pt diminishes with decreasing the base temperature T base , and it almost vanishes for T base ≤ 10 K, whereas ⌈ΔV nl th ⌉ Pt significantly increases at low T base . Such distinct T base -dependences are in line with previous experiments 23, 24 and theoretical considerations 25,26 that the injection mechanisms for electrical and thermal magnons across the Pt/YIG interface (parametrized by the effective spin conductance and the interface spin Seebeck coefficient, respectively) differ fundamentally. Furthermore, the energy-dependent magnon diffusion and relaxation of the YIG channel may play a role in the transport process. 27,28 We below focus on the nonlocal signal from the thermally generated magnons (ΔV nl th ) since it remains sufficiently large at low T base for allowing a reliable analysis across T c . In Figure 2f Figure 2f) is also explained by the development of a (singlet) superconducting gap and the freeze out of the QP population at a lower T base . 29, 34−38 Next, using the Nb detector in the same device, we confirm the above interpretation and demonstrate that the conversion efficiency of magnon spin to QP charge can be dramatically enhanced in the vicinity of T c . Figure 3a shows the thermally driven nonlocal signal ⌈ΔV nl th ⌉ Nb for the t Nb = 15 nm devices with and without the Al 2 O 3 (spin-blocking) layer at various T base around the superconducting transition of the Nb. In the normal state (T base /T c > 1), a negative ⌈ΔV nl th ⌉ Nb (<0) with several tens of nanovolts is clearly observed. Given ⌈ΔV nl th ⌉ Pt > 0 (see Figure 2b), this evidences that Nb and Pt have opposite signs in the spin-Hall angle θ SH , which is in agreement with recent theoretical and experimental studies. 38−40 Intriguingly, upon entering the superconducting state (T base /T c < 1), a significant enhancement of ⌈ΔV nl th ⌉ Nb up to a few microvolts appears immediately below T c (T base /T c ≈ 0.96), and then it decays toward zero deep in the superconducting state. We note that there exist visible dips in ⌈ΔV nl th ⌉ Nb at α ≈ 90°and 270°n ear T c (Figure 3a), which are also present for the Al 2 O 3inserted control device (Figure 3b) and thus have nothing to do with the magnon spin-transport and QP iSHE. Similar spinindependent signals have been observed in local measurements on NbN/YIG 32 and MoGe/YIG 41 bilayers as well and are explained in terms of an Abrikosov-vortex-flow-driven Hall effect under a transverse magnetic field that is close to the upper critical field μ 0 H c2 of (type-II) SC.
To examine the effect of heating power, we measure the T base dependence of ⌈ΔV nl th ⌉ Nb (Figure 3b and c) and the normalized R Nb /R T=7K Nb (Figure 3d) at various I dc . As I dc increases, T c of the Nb detector is systematically reduced and the transition width becomes broad (Figure 3d). We note that the stronger decay of T c for I dc > 0.5 mA is likely caused by the greater injection/excitation of spin-polarized QPs into the Nb detector (see Supplementary Section 3 for a comparison analysis of T c data between the Al 2 O 3 -absent and Al 2 O 3 -present devices). Accordingly, not only a peak of the ⌈ΔV nl th ⌉ Nb enhancement shifts to a low T base , but the enhancement regime widens (Figure 3e). For I dc ≥ |0.7| mA (J dc ≥ |4.2| MA/cm 2 ), the Nb does not turn fully superconducting down to the lowest T base = 2 K studied (inset of Figure 3d). The corresponding ⌈ΔV nl th ⌉ Nb then remains nonzero at 2 K and is larger than the normal state value (Figure 3e). For a quantitative analysis, we plot the normalized voltage ⌈ΔV nl th ⌉ Nb /⌈ΔV nl th ⌉ T=7K Nb as a function of the normalized temperature T base /T c in Figure 3f. We then find that the transition-state enhancement of ⌈ΔV nl th ⌉ Nb /⌈ΔV nl th ⌉ T=7K Nb can ACS Nano www.acsnano.org Article reach up to 3 orders of magnitude at the smallest I dc = |0.1| mA (J dc = |0.6| MA/cm 2 ). With increasing I dc , its peak amplitude decays rapidly, the full-width-at-half-maximum (fwhm) broadens, and the peak is positioned farther away from T c (inset of Figure 3f). These results ensure that the depressed superconductivity with increasing the heating power has a negative effect on the transition-state enhancement of the QP iSHE. We perform similar measurements on an additional set of devices with different t Nb (Figure 4a−f), comparable to or smaller than the superconducting coherence length ξ SC , and thereby strong t Nb -dependent superconducting properties (e.g., QP band structure and DOS). Since thin Nb films usually contain a larger amount of grain boundaries, defects, and disorders from the structural inhomogeneity near the growth interface than thick bulk Nb, 42,43 the associated scattering effectively weakens electron−electron and electron−phonon interactions and therefore the smearing-out effect of the QP DOS around the gap edge. 44 One would predict a greater enhancement of the QP iSHE if the Nb detector is thicker.
However, experiments give a very different result (Figure 4g−i). As t Nb increases, the peak amplitude of ⌈ΔV nl th ⌉ Nb / ⌈ΔV nl th ⌉ T=7K Nb rises until reaching 15 nm and then drops strongly for thicker Nb detectors, leading to a maximum at t Nb = 15 nm ( Figure 4j). The width and position of the transition-state enhancement, on the other hand, behave as expected for highly and quickly developed coherence peaks in the QP DOS of thick Nb when T c is crossed: a progressive narrowing of fwhm and a peak shift closer to T c , respectively, with the increase of t Nb (inset of Figure 4j). The nontrivial t Nb -dependent enhancement (Figure 4j) indicates that there is another key ingredient that controls the enhancement amplitude, that is to say, the exchange spin-splitting field, 4−6 which has turned out to considerably modify the QP spin relaxation mechanism via a freezing out of elastic/intravalley spin-flip scattering. 4−6 Below, we discuss how this exchange-field-frozen spin-flip scattering 4−6 is linked to and modifies the QP charge relaxation.
To theoretically describe our results, we first calculate the excited QP spin current density J s0 qp at the YIG/superconducting Nb interface as a function of the normalized temperature T/T c for different values of the magnon spin accumulation Δμ m relative to the zero-T energy gap 2Δ 0 SC (Figure 5a and b). For this calculation, we employ the recent models 29,30 that explicitly take the superconducting coherence factor into account (see Supplementary Section 4 for full details). Note that the characteristic energy of incoherent magnons which excite spin-polarized QPs in the Nb detector is ACS Nano www.acsnano.org Article set by Δμ m , and the T c (or 2Δ 0 SC ) suppression at a larger Δμ m is inferred from our data set (Figures 3 and 4). For a quantitative comparison, J s0 qp is normalized to its normal state value J s0 .
The calculated J s0 qp /J s0 increases largely near T c (0.8T c − 0.9T c ), and it decreases exponentially when T < 0.8T c , reflecting the singularity behavior in a nonequilibrium population of spin-polarized QPs. 29,30,42 In addition, the peak amplitude of J s0 qp /J s0 is inversely proportional to Δμ m / 2Δ 0 SC (inset of Figure 5a and b), explaining qualitatively the heating power dependence of the transition-state enhancement (Figure 3f). Nonetheless, this analysis based on the superconducting coherence factor does not capture the mechanism behind the nontrivial t Nb dependence (Figure 4j). We next consider the QP resistivity ρ SC qp (Figure 5c and d) and the volume fraction of QP charge imbalance v Q (Figure 5e and f), which together determine the effective resistivity ρ SC * (=ρ SC qp v Q , inset of Figure 5e and f) of the superconducting Nb. 38,45 Here 38,45 where l y is the spinactive length of the Nb detector, given approximately by the sum of the length of the Pt injector l y Pt and l sd m in our device geometry, and λ Q is the QP charge-imbalance relaxation length. ρ SC qp and ρ SC * are normalized by their normal-state ones ρ 0 and ρ 0 * , respectively. We note that if the SC thickness is comparable to or smaller than the QP spin transport length, as relevant to our system, 38,46 the QP-mediated iSHE voltage V i S H E q p i n t h e S C c a n b e a p p r o x i m a t e d a s ⌉ Nb as a function of the base temperature T base for the devices with different d s . In these measurements, I dc is fixed at |0.5| mA and the magnetic field μ 0 H ext at 5 mT. The inset shows the normalized Nb resistance R Nb /R T=7K Nb versus T base plot, which confirms nearly identical superconducting transition T c of the Nb layer. Note that a relatively higher T c of the 15-nm-thick Nb layer in these devices than that of the prior device (Figure 3d) is due to the better initial base pressure (<1 × 10 −9 mbar) before film deposition. (c) ⌈ΔV nl th ⌉ Nb / ⌈ΔV nl th ⌉ T=7K Nb as a function of T base /T c . The right inset displays the d s dependence of the peak amplitude, width, and position: the black solid line is an exponential fit, whereas the black dashed lines are given as a guide to the eye. A magnified plot of the peaks is also shown in the left inset. d s -dependent ⌈ΔV nl th ⌉ Nb above (d) and immediately below (e) T c of the Nb layer. In e, the solid lines are fitting curves to estimate the QP charge-imbalance relaxation length λ Q (see Supplementary Section 5 for explicit formulas). which is predicted to slightly increase near T c 45,47 (see Supplementary Section 4 for details), e is the electron charge, and ℏ is the reduced Planck constant. Consequently, J s0 qp and ρ SC * appear to be governing parameters in V iSHE qp . The most salient aspect of the calculations is that in the vicinity of T c , v Q dominates the T-dependent ρ SC * over ρ SC qp , resulting in V iSHE qp ∝ λ Q for given J s0 qp and l y values. This signifies that the QP charge imbalance relaxation is likely responsible for the nontrivial t Nb -dependent transition-state enhancement (Figure 4j) observed in our system.

ACS
We thus propose the following mechanism. If QP charge relaxes through the spin-flip scattering 1/τ sf qp and the inelastic scattering 1/τ in , and 1/τ sf qp > 1/τ in , the effective relaxation time τ Q * for the QP charge imbalance 48 is given by τ * ≈ . Qualitatively, we can understand the t Nb -dependent transition-state enhancement (Figure 4j) in the following manner. When t Nb ≪ ξ Nb , the superconducting coherence is too weak to inject/ excite large QP spin currents across the YIG/Nb interface. In contrast, for t Nb > ξ Nb , the exchange spin-splitting-field cannot propagate over the entire depth of such thick Nb and hence the converted QP charge relaxes faster primarily via the spinflip scattering process. Overall, these two competing effects control the amplitude of the transition-state enhancement by which one would expect a maximum at the intermediate t Nb ≈ ξ Nb (around 15 nm for Nb thin films). Note also that the enhancement width and position are determined by J s0 , the latter of which decays rapidly to zero below T c for a strong superconducting Nb.
To check the validity of this proposal, we experimentally investigate how the transition-state enhancement scales with the separation distance d s between Au/Ru electrical contracts on the exchange-spin-split Nb layer (Figure 6a, see Supplementary Section 5). Importantly, while the peak position and width of the transition-state enhancement are almost independent of d s (Figure 6b and c), the peak amplitude increases quasi-exponentially with the increase of d s (inset of Figure 6c), reflecting the characteristics of the QP charge-imbalance relaxation effect (see Supplementary Section 5). From the d s -dependent ⌈ΔV nl th ⌉ Nb (Figure 6e), we are able to estimate λ Q in the vicinity of T c (T base /T c = 0.94−0.98) for the spin-split Nb to be around 90 μm. This is surprisingly a few orders of magnitude larger than either commonly assumed 48 or hitherto reported in Nb films without the presence of spinsplitting fields 49 and thereby should indicate the significantly exchange-field-modified QP relaxation in our system.
Finally, we briefly mention other relevant experiments. It has been previously shown that in all-metallic nonlocal spin-Hall devices, 8 the giant iSHE (∼2000 times at most) is created by electrical spin injection from Ni 8 Fe 2 through Cu into superconducting NbN far below T c (T base /T c = 0.3) and attributed to the exponentially increasing QP resistivity at a lower T. By contrast, a recent experiment has reported that for a YIG/NbN vertical junction 33 the 2−3 times enhanced iSHE voltage by local SSE is measurable only in a limited T range right below T c (T base /T c = 0.96). In this work, the superconducting coherence factor is pointed out as a main source for such an enhancement, and a quantitative description of the data is also provided. In metallic/conducting Nb/Ni 8 Fe 2 bilayers, 38 a monotonic decay of spin-pumping-induced iSHE appears across T c , indicating no superconducting coherence effect detectable.

CONCLUSIONS
The key findings of our study that help understand these puzzling results are as follows. The spin-to-charge conversion mediated by QPs is substantially enhanced in the normal-tosuperconducting transition regime, where the interface superconducting gap matches the magnon spin accumulation. The conversion efficiency and characteristics depend crucially on the driving/heating power and the SC thickness, which is understood based on the two competing effects: the superconducting coherence 29,30,42 and the exchange-field-modified QP relaxation. [4][5][6]48 The validity of these competing mechanisms is experimentally confirmed by spatially resolved measurements with varying the separation of electrical contacts on the spin-split Nb layer. A quantitative reproduction of the result remains an open question for a theory. The coupling between different nonequilibrium imbalances (magnon, spin, charge, heat, magnon-heat, and spin-heat) 4,12 with exchange spin-splitting and the nonlinear kinetic equations 4 in the superconducting state should be taken into account rigorously. Moreover, how the magnetic-field-induced screening supercurrents in a spin-split SC contribute to the QP spin-to-charge conversion when coupled with these nonequilibrium modes 50 remains to be addressed. We speculate that the giant transition-state QP SHE is generic in any FMI/SC system, and its efficiency gets even larger especially with twodimensional (2D) SCs 51 where the exchange spin-splitting can readily proximity-penetrate the entire depth of the 2D SCs. We also anticipate that such a giant spin-to-charge conversion phenomenon (involving nonequilibrium QPs) can be used as an extremely sensitive probe of spin currents in emergent quantum materials. 52

METHODS
Device Fabrication. We fabricated the magnon spin-transport devices (Figure 1b) based on 200-nm-thick single-crystalline YIG films (from Matesy GmbH) by repeating a sequence of optical lithography, deposition, and lift-off steps. Note that these YIG films exhibited a very low Gilbert damping of 0.6 × 10 −4 at room temperature, determined via ferromagnetic resonance line width measurements (by Matesy GmbH, https://www.matesy.de/en/ products/materials/yig-single-crystal). We first defined the central Nb detector with a lateral dimension of 9 × 90 μm 2 , which was grown by accelerated Ar-ion beam sputtering at a working pressure of 1.5 × 10 −4 mbar. For the control device, a 10-nm-thick Al 2 O 3 spin-blocking layer was in situ deposited prior to the Nb deposition. We then defined a pair of Pt electrodes of 1.5 × 50 μm 2 , which were deposited by dc magnetron plasma sputtering at an Ar pressure of 4 × 10 −3 mbar. These Pt electrodes are separated by a center-to-center distance d Pt−Pt of 15 μm, which is comparable to the typical l sd m of singlecrystalline YIG films 18 and also to the estimated values from our Ptonly reference devices with different d Pt−Pt (Supplementary Section 1). The Nb thickness ranges from 10 to 35 nm, whereas the Pt thickness is fixed at 10 nm. Finally, we defined Au(80 nm)/Ru(2 nm) electrical leads and bonding pads, which were deposited by the Ar-ion beam sputtering. Before depositing the Au/Ru layers, the Nb and Pt surfaces were gently Ar-ion beam etched for transparent electrical contacts between them. ACS Nano www.acsnano.org Article Nonlocal Measurement. We measured the nonlocal magnon spin-transport (Figure 1a and b) in a quantum design physical property measurement system at a temperature varying between 2 and 300 K. A dc current I dc in the range of 0.1 to 1 mA was applied to the first Pt using a Keithley 6221 current source, and the nonlocal voltages [V nl Pt (α), V nl Nb (α)] across the second Pt and the central Nb are simultaneously recorded as a function of in-plane magnetic-field-angle α by a Keithley 2182A nanovoltmeter. α is defined as the relative angle of μ 0 H ext (//M YIG ) to the long axis of two Pt electrodes that are collinear.
Estimation of the magnon spin-diffusion length of YIG, quantification of spin currents leaking into the central Nb at room temperature, first-order estimate of the YIGinduced internal field at the Nb/YIG interface, theoretical description of the conversion efficiency of magnon spin to QP charge in the superconducting Nb, spatially resolved measurements by varying the separation of electrical contacts on the spin-split Nb layer (PDF)