Role of Two-Dimensional Ising Superconductivity in the Nonequilibrium Quasiparticle Spin-to-Charge Conversion Efficiency

Abstract

Nonequilibrium studies of two-dimensional (2D) superconductors (SCs) with Ising spin–orbit coupling are prerequisite for their successful application to equilibrium spin-triplet Cooper pairs and, potentially, Majorana Fermions. By taking advantage of the recent discoveries of 2D SCs and their compatibility with any other materials, we fabricate here nonlocal magnon devices to examine how such 2D Ising superconductivity affects the conversion efficiency of magnon spin to quasiparticle charge in superconducting flakes of 2H-NbSe₂ transferred onto ferrimagnetic insulating Y₃Fe₅O₁₂. Comparison with a reference device based on a conventionally paired superconductor shows that the Y₃Fe₅O₁₂-induced in-plane (IP) exchange spin-splitting in the NbSe₂ flake is hindered by its inherent out-of-plane (OOP) spin–orbit field, which, in turn, limits the transition-state enhancement of the spin-to-charge conversion efficiency. Our out-of-equilibrium study highlights the significance of symmetry matching between underlying Cooper pairs and exchange-induced spin-splitting for the giant transition-state spin-to-charge conversion and may have implications toward proximity-engineered spin-polarized triplet pairing via tuning the relative strength of IP exchange and OOP spin–orbit fields in ferromagnetic insulator/2D Ising SC bilayers.

Injection and excitation of electrons, typically called Bogoliubov quasiparticles (QPs), in a superconductor (SC) with either an external (Zeeman) or internal (exchange) spin-splitting field¹⁻³ under nonequilibrium conditions (i.e., voltage bias or temperature gradient) have been one of the central research topics in superconducting spintronics.¹⁻⁷ This is because their exotic transport properties, derived from the superconductivity-facilitated coupling between different nonequilibrium imbalances (e.g., spin, charge, heat, and spin-heat), can considerably improve the functionality and performance of spintronic devices. Various nonequilibrium phenomena mediated by QPs have been observed in SC-based devices with either Zeeman or exchange spin-splitting: long-range spin signals,⁸⁻¹⁰ pure thermal spin currents,¹¹ large (spin-dependent) thermoelectric currents,¹² and spectroscopic evidence of spin-heat transport.¹³

Recently, a magnon spin-transport experiment¹⁴ has reported that the conversion efficiency of thermal-magnon spin to QP charge via an inverse spin-Hall effect (iSHE)¹⁵ in an exchange-spin-split Nb layer can be significantly enhanced by up to 3 orders of magnitude in the normal-to-superconducting transition regime. This giant transition-state QP iSHE has been semi-quantitatively explained in terms of two competing mechanisms of the superconducting coherence versus the exchange-field-frozen QP relaxation. A very recent theory¹⁶ has pointed out that the electron–hole symmetry breaking present in SC/FMI (FMI = ferromagnetic insulator) bilayers mixes the spin and heat imbalances and can cause the enhancement of QP spin accumulation by several orders of magnitude relative to the normal state. Both these studies^14,15 emphasize the crucial role of the spin-splitting of QP density-of-states (DOS) and the resulting electron–hole asymmetry in enhancing the spin sensitivity of the SC detector.^5,15

The advent of two-dimensional (2D) SCs¹⁷⁻²¹ and their compatibility with any other materials via circumventing the need for lattice matching between adjacent material systems provide platforms to explore intriguing physical phenomena in various geometries,²² including van der Waals (vdW) heterostructures with a twist, and in proximity combination with magnetic vdW flakes and/or thin films.^23,24 Because excited QPs and Cooper pairs in the superconducting condensate state are intimately correlated,¹⁻⁶ studies of nonequilibrium QP spin properties in such 2D SCs are of fundamental importance for understanding equilibrium spin-polarized triplet Cooper pairing¹⁻⁶ and the possible stabilization of Majorana Fermions.²⁵⁻²⁷

2D superconductivity has been recently discovered in monolayer transition metal dicalcogenides (TMDs)¹⁷ such as gated 2H-MoS₂^18,19 and 2H-NbSe₂.²⁰ Interestingly, the in-plane (IP) upper critical field is found to far exceed the Pauli paramagnetic limit of isotropic Bardeen–Cooper–Schrieffer (BCS) SCs ≈ 1.84T_c,²⁸ where Zeeman spin-splitting fields are the predominant mechanism for Cooper pair breaking in the 2D limit and T_c is the superconducting transition temperature. Such an enhancement of is explained by Ising spin–orbit coupling (SOC),¹⁷⁻²¹ rooted in the broken IP crystal inversion symmetry plus the large SOC due to heavy transition metal atoms in TMDs. The Ising SO field μ₀H_SO (as large as several hundred Tesla in the monolayer limit)¹⁷⁻²¹ strongly pins Cooper pair spins at K and K’ points of the hexagonal Brillouin zone to opposite out-of-plane (OOP) directions over IP applied magnetic fields. This stabilizes OOP Cooper pairing and forms so-called Ising superconductivity.¹⁷⁻²¹

We here investigate how the 2D Ising superconductivity influences the transition-state enhancement of magnon spin to QP charge conversion in a superconducting flake of 2H-NbSe₂^20,29−31 (Figure 1a) and compare its efficiency with a conventional superconducting thin film of Nb¹⁴ (BCS SC). We first demonstrate that the normal-state spin-to-charge conversion functionality of the 2H-NbSe₂ flake can be 4 times more efficient than that of the Nb film. We then find distinctively different transition-state conversion behaviors (e.g., modest transition-state enhancement, rather weak thickness dependence) in the 2H-NbSe₂ and attribute these to OOP Cooper pairing that hampers proximity penetration of IP exchange spin-splitting from the adjacent ferrimagnetic insulating Y₃Fe₅O₁₂. Notably, the maximum enhancement of spin-to-charge conversion appears at a critical thickness over which the IP crystal symmetry is recovered (equivalently, OOP Ising pairing is no longer protected), allowing the IP exchange field to penetrate. This provides a guideline as to how to tune the relative strength of these two phenomena for a desired proximity effect.^32,33 We believe that, along with recent advances in 2D SCs of various intriguing properties (e.g., type-I/-II Ising, Rashba, topological SCs),^22,34 our approach helps find right material combinations for developing superconducting spintronic devices over conventional BCS SCs.

Figure 1.

Nonlocal magnon spin-transport device with Ising superconductor. (a) Device layout and measurement scheme. 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 Y₃Fe₅O₁₂ (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 via the inverse spin-Hall effect (iSHE). Such a conversion process also occurs for the central 2H-NbSe₂ flake and thereby . Note that, unlike spin-singlet (S = 0) Cooper pairs in a coherent ground state, the excited quasiparticles (QPs) can carry spin angular momentum in the superconducting state. How out-of-plane (OOP) Cooper pairing of the 2H-NbSe₂ affects the transition-state enhancement of QP iSHE will be discussed in this study. (b) Crystal structure of the 2H-NbSe₂, where in-plane inversion symmetry breaking by Se plus spin–orbit coupling of Nb lead to OOP spin-singlet (S = 0) Cooper pairs, constituting Ising superconductivity. (c,e,g,i) Optical micrographs of the fabricated devices. Atomic force microscopy (AFM) scans of the transferred 2H-NbSe₂ flakes (d,f,h) and the deposited Nb thin film (j).

Results and Discussion

Our nonlocal magnon spin-transport devices (Figure 1a) are composed of two identical Pt electrodes and a central 2-H NbSe₂ flake transferred onto 200 nm thick single-crystalline Y₃Fe₅O₁₂ (YIG) films (see Methods and Supplementary section 1 for details), which are grown by liquid phase epitaxy on a (111)-oriented single-crystalline Gd₃Ga₅O₁₂, (GGG) wafer. Bulk 2H-NbSe₂ is a layered type-II SC, having anisotropy²⁹ in both the IP (OOP) coherence length () ≈ 10 (3) nm and the IP (OOP) London penetration depth () ≈ 70 (230) nm at zero temperature T = 0. As shown in Figure 1b, it has a hexagonal crystal structure with lattice constants, a = b ≈ 0.3 nm and c ≈ 1.3 nm and each unit cell consists of two AB stacked NbSe₂ layers.^30,31 On a single-piece YIG film, we prepared several independent devices with different 2H-NbSe₂ flake thicknesses (Figure 1c–h) as well as reference devices in which Nb thin film is directly deposited¹⁴ (Figure 1i,j). The Nb thickness t_Nb is fixed at 15 nm, which is comparable to its dirty-limit coherence length ξ_Nb, so that the YIG-induced exchange spin-splitting-field can penetrate the Nb layer while retaining the superconducting coherence, thereby maximizing the transition-state QP iSHE.¹⁴

In this device structure (Figure 1c,e,g,i), we pass a dc current I_dc through one Pt electrode (using leads 1 and 2) while measuring the IP magnetic-field-angle α dependence of the nonlocal open-circuit voltages [, ] using the other Pt electrode (leads 7 and 8) and the central NbSe₂ (or Nb) (leads 3 and 4). Since we apply an external IP magnetic field μ₀H_ext = 5 mT that is larger than the coercive field of YIG, α is simply defined as the relative angle of μ₀H_ext (//M_YIG) to the long axis of the two Pt electrodes which are collinear.¹⁴ The total voltage measured across the detector is then given by . Here, and developed via iSHE (spin-to-charge conversion)¹⁵ in the detector are proportional to the magnon spin current and accumulation created electrically [SHE (charge-to-spin conversion)¹⁵ ∝ I_dc] and thermally [spin-Seebeck effect (SSE, heat-to-spin conversion)³⁵ ∝(I_dc)²], respectively.^14,35 By inverting the polarity of I_dc, one can determine the magnitude of each component based on their characteristic angular dependences:^14,36

and

where V₀ is a spin-independent offset voltage. Below, our discussion will focus on , since it remains detectably large at low T for reasonable |I_dc| such that Joule heating does not destroy the superconducting phase of the 2H-NbSe flake (or Nb thin film).

Let us first discuss the electrical transport properties of the transferred 2H-NbSe₂ flake. In the plot of its resistance versus temperature T (Figure 2a) for = 9 nm, a resistance anomaly appears around 26 K, which is indicative of its phase transition from a normal metal to an incommensurate charge density wave (CDW) phase.³⁷ Note that the strongly suppressed CDW phase transition temperature, T_CDW = 26 K for our = 9 nm flake, is presumably due to the proximity coupling of the CDW with the magnetic order of YIG. In analogy with the Pauli effect²⁸ in conventional SCs, the Zeeman (or exchange) energy competes with the CDW condensation energy and hence T_CDW is predicted to decrease in the presence of external (and/or internal) spin-splitting fields.³⁸ As T is reduced further, 2H-NbSe₂ becomes superconducting below ∼6.75 K. From the T-dependent upper critical field (Figure 2d), that is obtained by applying an external magnetic field either parallel μ₀H^∥ (Figure 2b) or perpendicular μ₀H^⊥ (Figure 2c) to the interface plane, we find ≈ 8 nm and ≈ 3 nm using Ginzburg–Landau (GL) theory³⁹ (see Methods for a detailed discussion), so confirming the anisotropic superconducting state of 2H-NbSe₂.²⁹ The extrapolated value of at lower T goes beyond = 12.4 T. Because the = 9 nm flake corresponds to 7× the unit cell and is much smaller than ≈ 230 nm, neither the IP crystal inversion symmetry nor orbital effect (i.e., interlayer Meissner screening current) is fully recovered.¹⁷ So Ising Cooper pairing¹⁷⁻²¹ would account for the increase of over . Note that a rather linear behavior for the intermediate = 9 nm suggests that not only Ising SOC²⁰ but also Abrikosov vortex occupation³⁹ causes Cooper pair breaking (see Methods for details). These multiple characteristics are a measure of the high quality of our transferred 2H-NbSe₂ flake. In contrast, the deposited Nb thin film of t_Nb = 15 nm has isotropic coherence lengths ≈ ≈ 12–13 nm (Figure 2h) and its low-T value is below = 8.3 T (Figure 2f–h), as would be expected from an isotropic BCS SC.

Figure 2.

Electrical characterization of the transferred 2H-NbSe₂ flake. (a) 2H-NbSe₂ resistance as a function of temperature, T, for the transferred 2H-NbSe₂ flake ( 9 nm) measured using a 4-terminal current–voltage method (using leads 3–6 in Figure 1e). Typical –T curves measured by applying an external magnetic field either parallel μ₀H^∥ (b) or perpendicular μ₀H^⊥ (c) to the interface plane. The T-dependent IP (OOP) upper critical field () is determined from the point where R = 0.5R_T=8K. (d) Summary of the and data. The blue dashed line represents the Pauli paramagnetic limit ≈ 1.84T_c.²⁸ The red and violet solid lines in (b) are theoretical fits using Ginzburg–Landau (GL)³⁹ and pair breaking (PB)²⁰ theories, respectively. (e–h) Data equivalent to (a)–(d) but for the t_Nb = 15 nm reference device (Figure 1j).

We now focus on how the conversion efficiency of magnon-carried spin to QP charge varies when the 2H-NbSe₂ becomes superconducting. Figure 3a,d,g shows the thermally driven nonlocal signal for the = 4, 9, and 46 nm devices at various base temperatures T_base around the superconducting transition T_c. In the normal state (T_base/T_c > 1), a negative (<0) of a few tens of nanovolts is observed for I_dc = |0.5| mA (J_dc = |3.0| MA/cm²). Given > 0 (Supplementary section 2) and < 0 (Figure 3j), this indicates that the 4d heavy element Nb, having a negative spin-Hall angle θ_SH (<0), governs spin-to-charge conversion characteristics in the normal-state 2H-NbSe₂. Upon entering the superconducting state (T_base/T_c < 1), a clear enhancement of up to around 100 nV appears immediately below T_c (T_base/T_c ≈ 0.99) and then it decays toward zero, deep into the superconducting state. It is noteworthy that, for the normal state (T_base > T_c), of the transferred 2H-NbSe₂ flakes go beyond of the deposited Nb film, in particular, the = 2.5 nm device reveals 4 times greater signals (Supplementary section 3), indicating high spin mixing conductance and spin transparency at the interface between our transferred 2H-NbSe₂ flakes and YIG film.

Figure 3.

Enhancement of nonlocal signals in the transition state of the 2H-NbSe₂ detector. (a,d,g) Thermally driven nonlocal voltages as a function of IP field angle α for the 4, 9, and 46 nm devices, taken at I_dc = |0.5| mA around the superconducting transition, T_c, of the 2H-NbSe₂. The black solid lines are sin(α) fits. Note that dips in at α ≈ 90° and 270° near T_c which are pronounced for a thicker flake arise from Abrikosov-vortex-flow-driven spin-independent Hall effect¹⁴ under a transverse magnetic field that is close to the upper critical field μ₀H_c2 of type-II SC (i.e., vortex melting field). (b,e,h) Normalized 2H-NbSe₂ resistance versus T_base plots for the 4, 9, and 46 nm devices, measured using a four-terminal current–voltage method (using leads 3–6 in Figure 1c,e,g) with varying I_dc in the Pt injector. The critical temperature T_c is defined as the point where . The inset summarizes the measured T_c as a function of I_dc (or J_dc). (c,f,i) Estimated magnitude of as a function of T_base for the 4, 9, and 46 nm devices. (j–l) Data equivalent to (a)–(c) but for the t_Nb = 15 nm reference device.

We systematically measure the T_base dependence of the normalized (Figure 3b,e,h) and (Figure 3c,f,i) with varying I_dc in the Pt injector. The results are qualitatively similar to the magnon devices with Nb detectors¹⁴ and also to the t_Nb = 15 nm reference device studied here (Figure 3j–l). As I_dc increases, T_c of the 2H-NbSe₂ detector is progressively reduced (inset of Figure 3c,f,i) and the transition width broadens. As a result of this depressed superconductivity, caused by the combined effect of more populated spin-polarized QPs⁵ and increased heat dissipation in the 2H-NbSe₂ at a high I_dc, a peak of the enhancement occurring in the vicinity of T_c (Figure 3c,f,i) shifts to a low T_base and the enhancement regime widens. These demonstrate that the spin-to-charge conversion efficiency indeed rises when mediated by QPs in the transition state of 2H-NbSe₂/YIG bilayer, that is the enhanced spin-detection functionality of a 2D Ising SC in the normal-to-superconducting transition regime.

We next plot the normalized voltages (Figure 4a–c) and (Figure 4d) as a function of the normalized temperature T_base/T_c for a quantitative analysis. With increasing I_dc, the peak amplitude strongly diminishes, the full-width-at-half-maximum (fwhm) broadens, and the peak position is away from T_c (inset of Figure 4a–d). In addition to these generic features, one can find important quantitative differences between the 2H-NbSe₂ and Nb detectors¹⁴ from the thickness dependence of the amplitude, fwhm and position (Figure 4f).

Figure 4.

2H-NbSe₂ thickness dependence of the transition-state enhancement and comparison with the Nb detector. (a–c) versus T_base/T_c plot for the 4, 9, and 46 nm devices. Each inset displays the |I_dc| (or |J_dc|) dependence of the peak amplitude, width, and position. (d) Data equivalent to (a) but for the t_Nb = 15 nm device. Note that unlike the amplitude, the width and position can be approximately estimated based on data below T_c (Figure 3c,f,i,l) where the transition-state enhancement of QP iSHE provides a detectable amplitude of . (e) -dependent T_c. (f) -dependent peak amplitude, width, and position. Abrupt changes of T_c, peak width and position below = 3 nm, coinciding with the OOP coherence length (black vertical line in e and f), are likely due to thermal-fluctuation-enhanced T_c suppression at the 2D limit.^20,39 Detailed results of the nm device can be found in Supplementary section 3. In (e) and (f), data from the t_Nb = 15 nm reference device are also included for quantitative comparison.

First, the enhancement amplitude attained in the 2H-NbSe₂ detectors is relatively small compared with the t_Nb = 15 nm reference device with a similar lateral dimension, even though the 2H-NbSe₂ flakes (e.g., = 4, 9 nm) possess a higher T_c in thinner layers (Figure 4e). Second, the peak width and position abruptly change across 3 nm, coinciding with (black vertical line in Figure 4e,f) below which thermal-fluctuation-enhanced T_c suppression at the 2D limit is expected,^20,39 and they become almost -independent for thicker flakes. Note that the Nb dectectors¹⁴ reveal a monotonic narrowing of fwhm and a peak shift closer to T_c with increasing t_Nb. Third, unlike the Nb detectors,¹⁴ the maximum enhancement in the spin-to-charge conversion does not appear at ≈ and the -dependent enhancement is rather weak.

To account for these distinctively different conversion phenomena, we consider the layer thickness-dependent Ising superconductivity.^20,40 For a few monolayer 2H-NbSe₂, the IP crystal inversion symmetry is strongly broken by Se atoms (Figure 1b) and thus OOP Cooper pairing is protected and stabilized by the resulting Ising SO-field (76 meV in the monolayer limit).^20,41 In this regime, the YIG-induced IP exchange field (<1 meV)^14,41 hardly spin-splits the QP DOS of the 2H-NbSe₂ and the transition-state enhancement of QP iSHE thus relies mostly on the superconducting-coherence-relevant resonant absorption,^14,16,42 leading to a modest enhancement. As the flake becomes thicker, the IP bulk crystal inversion symmetry is restored, which weakens the OOP Ising pairing and, in turn, enables the YIG-induced IP exchange field to propagate through. This explains why we obtain the maximum enhancement of the transition-state QP iSHE at = 9 nm . Note that, as a critical thickness value that is necessary to fully restore the IP bulk inversion symmetry (equivalently, to diminish Ising pairing) is larger than the coherence length, beyond this critical value, proximity extension of the YIG-induced IP exchange spin-splitting over the entire 2H-NbSe₂ layers is not very effective, limiting the enhancement amplitude. Furthermore, a Γ-centered Se-electron Fermi pocket, constituting a second band with a smaller superconducting gap, emerges in the 2H-NbSe₂ thicker than a few monolayers.⁴³ This second band whose gap energy seems weakly dependent on (43) can provide another path for spin-polarized QPs to enter the 2H-NbSe detector, effectively weakening the -dependent transition-state enhancement.

Our out-of-equilibrium study highlights the importance of symmetry matching between underlying Cooper pairs and exchange-induced spin-splitting for the giant transition-state enhancement of QP iSHE.^14,16 Based on this, we would predict a greater transition-state QP iSHE, for instance, in MnPS₃/NbSe₂ bilayers, where exchange spin-splitting⁴⁴ and SO fields are both OOP and thus match in the symmetry each other. Similarly, Rashba SC/YIG bilayers, where the Rashba SC has IP SO-fields,³⁴ would be another symmetry-matching combination. Our results may also provide a guideline for the proximity engineering of hybrid quantum materials that allow for exotic quantum phases (e.g., topological superconductivity with spin-polarized triplet pairs and/or Majorana zero modes)²⁵⁻²⁷ at zero field in equilibrium.

Conclusions

Our magnon spin-transport experiments with 2H-NbSe₂ detectors have shown that OOP Cooper pairing of Ising SC, derived by IP inversion symmetry breaking and strong SOC, hinders the proximity propagation of IP exchange spin-splitting, in turn limiting the transition-state enhancement of QP iSHE. Contrary to the magnon devices with Nb (BCS SC) detectors,¹⁴ the maximum enhancement does not appear at ≈ but at a different critical thickness over which the IP crystal symmetry is recovered and so the OOP Ising pairing is no longer protected, allowing the IP exchange field to penetrate. This result should be taken into account for better proximity engineering of Ising SC triplet Josephson junctions with IP ferromagnets.⁴⁵ We believe that, with the layer thickness-tunable OOP Cooper pairing^20,40 and IP exchange spin-splitting, 2D Ising SC/FMI bilayers have desirable material properties for the topological protection of spin-polarized triplet Cooper pairs²⁵ and Majorana Fermions.^26,27 Our findings, together with recent progress in 2D SCs and magnetic vdW crystals,^22,24 also raise the possibility of developing highly efficient atomically thin spin-to-charge converters via symmetry engineering.

Methods

Device Fabrication

We fabricated the magnon spin-transport devices (Figure 1c,e,g,i) based on 200 nm thick single-crystalline YIG films (from Matesy GmbH, https://www.matesy.de/en/products/materials/yig-single-crystal) as follows. We first defined a pair of Pt electrodes with an area of 1.5 × 50 μm², 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 magnon spin-diffusion length estimated from our previous study.¹⁴ For the reference device (Figure 1i), we defined the central 15 nm thick Nb detector with a lateral dimension of 9 × 12 μm², which was grown by Ar-ion beam sputtering at a working pressure of 1.5 × 10^–4 mbar. Subsequently, we defined the outer Au(80 nm)/Ru(2 nm) leads and bonding pads, which were deposited by Ar-ion beam sputtering.

We next selected NbSe₂ flakes of suitable geometry and thickness, which were mechanically exfoliated from a high-quality single crystal (from HQ Graphene, http://www.hqgraphene.com/NbSe2.php) and first transferred onto SiO₂(300 nm)/Si substrates, via optical microscopy inspection. We then picked up the selected NbSe₂ flake and transferred it onto the central region of each magnon device (Figure 1c,e,g) using a polydimethylsiloxane-based dry transfer method (see Supplementary section 1 for full details). All these processes have been conducted in an inert atmosphere glovebox to prevent oxidation and degradation of the 2H-NbSe₂. Note that the 2H-NbSe₂ flakes and Nb thin film were prepared on the same-piece YIG film, confirming almost identical SHE/iSHE properties of the Pt injectors/detectors.

To prevent the unintentional contribution of iSHE from inner Au/Ru leads themselves to total voltage signals, we electrically isolate them from the active regime of magnon spin-transport by depositing a 10 nm thick Al₂O₃ oxide layer in-between apart from the electrical contact parts on top of the central 2H-NbSe₂ (or Nb). Finally, we defined the inner Au(10 nm)/Ru(2 nm) leads, which were deposited by Ar-ion beam sputtering. Before depositing the inner Au/Ru leads, the NbSe₂ (or Nb) and Pt surface were gently Ar-ion beam etched for transparent electrical contacts between them.

Superconducting Transition Measurement

To characterize superconducting properties, dc electrical transport measurements were conducted on either transferred NbSe₂ flakes or deposited Nb thin films of the fabricated magnon devices attached on either IP (Figure 2b) or OOP (Figure 2c) rotatable holder in a Quantum Design Physical Property Measurement System (PPMS). Using electrical leads 3–6 (Figure 1c,e,g,i) with a four-probe configuration, we measured the resistance Rversus temperature T curves at the applied current I ≤ 10 μA while decreasing T. The T-dependent IP (OOP) upper critical field () of Figure 2d (Figure 2g) was obtained by applying an external magnetic field μ₀H^∥ (μ₀H^⊥) parallel (perpendicular) to the interface plan. The and values are determined from the point where R = 0.5R_T=8K.

We estimated the and values of the transferred 2H-NbSe₂ flake ( = 9 nm) from the and data (Figure 2d), respectively, using an anisotropic GL theory³⁹ for :

1a

1b

where is the magnetic flux quantum. It is noteworthy that as is reduced and reaches the atomically thin limit (), the dominant Cooper-pair breaking mechanism under application of μ₀H^∥ changes from Abrikosov vortex occupation to Ising SOC as recently discussed.¹⁷⁻²¹ For , eq 1a can thus be rewritten as

1c

where is the strength of Ising SO field. For completeness, we also fitted the data (violet solid line, Figure 2d) with this formula.

On the other hand, for the deposited Nb thin film of t_Nb = 15 nm ≤ , the T-dependent upper critical fields (Figure 2h) were fitted with³⁹

2a

2b

Note that, unlike bulk Nb, the occupation energy of Abrikosov vortices in a superconducting Nb thin film (t_Nb ≤ ) under μ₀H^∥ is higher than that under μ₀H^⊥, differentiating formulas (eq 2a and 2b) for the T-dependent IP/OOP upper critical fields.³⁹ This is because the density of Cooper pairs cannot change much on a length scale shorter than the coherence length and hence IP Abrikosov vortices cannot efficiently accommodate magnetic flux.³⁹ When the Nb (BCS SC) film becomes sufficiently thin (t_Nb ≪ ), Abrikosov vortex occupation under μ₀H^∥ is strongly suppressed and a μ₀H^∥-driven dominant Cooper-pair breaker is now the Pauli paramagnetic effect (i.e., Zeeman spin-splitting).²⁸ Accordingly, eq 2a can be rewritten by

2c

Nonlocal Measurements

We measured the nonlocal magnon spin-transport (Figure 1a) on the magnon devices attached on an IP rotatable sample holder in the Quantum Design PPMS at various T between 2 and 300 K. A dc current I_dc in the range of 0.1–1 mA was applied to the first Pt using a Keithley 6221 current source, and the nonlocal voltages [, ] across the second Pt and the central 2H-NbSe₂ (or Nb) are simultaneously recorded as a function of IP magnetic-field-angle α with rotating the IP sample holder by a Keithley 2182A nanovoltmeter. Note that α is defined as the relative angle of μ₀H_ext (//M_YIG) to the long axis of two Pt electrodes which are collinear.

The Oersted field μ₀H_Oe induced from I_dc applied to the Pt electrode is estimated using Ampere’s law

Here μ₀ = 4π × 10^–7 Tm/A is the permeability of free space, w_Pt is the width (1.5 μm) of the Pt electrode, and d is the distance from the Pt/YIG interface. For the maximum I_dc = 1.0 mA used, we get μ₀H_Oe = 0.3–0.4 mT at d = 100 nm and it decreases to 0.02–0.03 mT at d = 7.5 μm. These estimated values are too weak to perturb the magnetization direction of ferrimagnetic insulating YIG¹⁴ under application of μ₀H^∥ = 5 mT (Figure 1c,e,g,i) and to suppress the superconducting properties of 2H-NbSe₂ flakes and a Nb thin film whose upper critical fields in the transition state are larger than 0.5 T (Figure 2b,c,f,g).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.1c07192.

• Dry transfer of 2H-NbSe₂ flakes onto magnon spin-transport devices, Nonlocal spin signals detected by the Pt detector across T_c of the 2H-NbSe₂ flake, Transition-state enhancement of QP iSHE for the t_NbSe2 = 2.5 nm device (PDF)

Author Contributions

^† K.-R.J. and K.C. contributed equally to this work. K.-R.J. and S.P.P.P. conceived and designed the experiments. The magnon spin-transport devices were fabricated by K.-R.J. with help from J.Y., J.-C.J., A.C., H.H., J.-K.K., and K.C. K.-R.J. performed exfoliation/pick-up/transfer of 2H-NbSe₂ under the guidance of K.C. The nonlocal transport measurements were carried out by K.-R.J. with the help of J.Y. and J.-C.J. K.-R.J. performed the data analysis. S.P.P.P. supervised the project. All authors discussed the results and commented on the manuscript, which was written by K.-R.J., K.C., and S.S.P.P.

Open access funded by Max Planck Society.

The authors declare no competing financial interest.