Signature of long-ranged spin triplets across a two-dimensional superconductor/helimagnet van der Waals interface

The combination of a superconductor with a magnetically inhomogeneous material has been established as an efficient mechanism for the generation of long-ranged spin-polarized (spin-triplet) Cooper pairs. Evidence for this mechanism, however, has been established based on studies done on three-dimensional systems, where the strong bonds existing at the interface between the superconductor and the magnetic material should in principle enhance proximity effects and strengthen any electronic correlations. Here, we fabricate devices based on van der Waals stacks of flakes of the two-dimensional superconductor $NbS_2$ combined with flakes of $Cr_{1/3}NbS_2$, which has a built-in magnetic inhomogeneity due to its helimagnetic spin texture at low temperatures. We find that the critical temperature of these vdW bilayers is strongly dependent on the magnetic state of $Cr_{1/3}NbS_2$, whose degree of magnetic inhomogeneity can be controlled via an applied magnetic field. Our results demonstrate evidence for the generation of long-ranged spin-triplet pairs across the $Cr_{1/3}NbS_2$/$NbS_2$ vdW interface.


Introduction
The interplay between ferromagnetism and superconductivity has been widely investigated in superconductor/ferromagnet (S/F) thin film multilayers with a conventional S, where the Cooper pairs of electrons are in an antiparallel-aligned (spin-singlet) state.Although early studies on such S/F hybrids showed that magnetic exchange field (hex) of a F quickly suppresses the spin-singlet superconductivity from the S/F interface [1][2][3], over the past 20 years several groups have demonstrated, confirming a theoretical prediction [4], that longranged parallel-aligned (spin-triplet) Cooper pairs of electrons can be generated in S/F devices with non-collinear Fs [5][6][7][8][9] or with an intrinsically-inhomogeneous magnetic material like the helimagnet Ho [10][11][12].Evidence for long-ranged spin triplets in these S/F hybrids has been found via measurements of the dependence of their superconducting critical temperature (Tc) on the magnetic state (i.e., collinear/non-collinear) [7][8][9] or via measurements of a long-ranged supercurrents (compared to the F's coherence length, ξF, for spin-singlet pairs) in S/F/S Josephson junctions (JJs) [4][5][6]10,13].Spectroscopic evidence for spin-triplet states has also been reported via measurements of sub-gap states in the superconducting density of states [12,[14][15] or via measurements of an inverse Meissner effect [11].All these studies have established the research field of superconducting spintronics [16] that aims at exploiting long-ranged spin triplets, which carry a net spin, to do spintronics in the superconducting state with low-energy dissipation.
Van der Waals (vdW) materials can be exfoliated down to the two-dimensional limit and exhibit physical properties that often do not have a counterpart in three-dimensional systems.Over the past years, a variety of vdW Ss and Fs has been explored both as single flakes and stacked to form vdW heterostructures [17][18][19][20][21]. S/F vdW heterostructures can be used to realize superconducting spintronic devices with novel functionalities compared to those based on thin film multilayers.Before this occurs, however, it is essential to determine at which S/F vdW interfaces long-ranged spin triplets can be generated Several groups have recently studied the proximity effect in S/F vdW hybrids, and they have observed signatures of phenomena like a 0-to-π transition in S/F/S vdW JJs [22], which had only been known for JJs made from thin film multilayers [2][3].Spin-triplet generation at the interface between electrodes of NbN (S) and the vdW material Fe0.29TaS2 (F) has also been reported [23], although this S/F interface may not be truly vdW in nature.A longranged coupling has also been measured in lateral NbSe2/Fe3GeTe2/NbSe2 (S/F/S) vdW JJs [24], but the spin-triplet nature of the supercurrent and the mechanism behind its generation remains unclear because Fe3GeTe2 does not have an intrinsic magnetic inhomogeneity.
In this Letter, we explore the superconducting proximity effect across the vdW interface forming between a flake of a vdW S (NbS2) and a magnetic flake of Cr1/3NbS2.Cr1/3NbS2 is helimagnetic at low temperatures [26][27] and therefore has a built-in magnetic inhomogeneity ideal for spin-triplet generation.Studying the evolution of the Tc of Cr1/3NbS2/NbS2 (F/S) stacks as a function of the magnetic state of Cr1/3NbS2, we find that the Tc of these vdW devices strongly depends on the Cr1/3NbS2 magnetization, in a way that cannot be reconciled with a conventional short-ranged S/F proximity effect or with stray fields.Supported also by a theoretical model, we show that our results are consistent with the generation of long-ranged spin-triplet pairs across the Cr1/3NbS2/NbS2 vdW interface.
We use Cr1/3NbS2 flakes with thickness ranging between 200 nm and 500 nm, which are obtained via subsequent mechanical cleaving and exfoliation of Cr1/3NbS2 single crystals synthesized as in ref. [26].
We choose Cr1/3NbS2 as our magnetic flake because it is the closest equivalent to the helimagnet Ho, which previous studies on S/F thin film multilayers have shown to be efficient for spin-triplet generation [10][11][12].Like in Ho, the helimagnetic spin texture of Cr1/3NbS2 can be unzipped by an in-plane magnetic field H which, above a certain value depending on thickness [26][27], makes Cr1/3NbS2 fully ferromagnetic.This property provides a tool to control whether any observed effects is due to long-ranged spin triplets because such triplets should be only generated in the magnetically-inhomogeneous (helimagnetic) state of Cr1/3NbS2 and be suppressed when Cr1/3NbS2 is driven into its magnetically-homogeneous (ferromagnetic) state.
Cr1/3NbS2 is an ionic compound, which makes it difficult to obtain flakes with thickness smaller than 50 nm [26][27].Nonetheless, this is not a limitation for our experiment, since we deliberately choose Cr1/3NbS2 flakes with thickness larger than 200 nm.Thick flakes of Cr1/3NbS2 are necessary because, if long-ranged spin-triplet pairs are generated at the Cr1/3NbS2/NbS2 vdW interface, these pairs must have enough room to propagate into Cr1/3NbS2 for them to affect the Tc of the proximitized NbS2.Theory and experiments on S/F thin film multilayers have in fact shown that, to maximize the effect of long-ranged spin triplets on Tc, a F with thickness dF larger than its ξF is needed [9,28].To increase the effect on Tc, the S should also have thickness dS comparable to or smaller than its superconducting coherence length (ξS) [8][9].
The reason why long-ranged spin triplets can reduce the Tc of a S/F heterostructure with dS ~ ξS and dF > ξF can be understood by thinking of a S as a reservoir of Cooper pairs: once long-ranged spin-triplet pairs are generated, they can propagate deeply into the F (much deeper than ξF), which drains pairs out of the S and reduces Tc.On the other hand, if only short-ranged spin-triplet pairs are generated, the proximity effect remains confined at the S/F interface within ξF (typically a few nanometers [29]) and Tc is higher.
To determine whether long-ranged spin triplets are generated in our Cr1/3NbS2/NbS2 devices, we study the evolution of their resistance versus temperature, R(T), measured across Tc, as a function of an in-plane H (i.e., perpendicular to the Cr1/3NbS2 c-axis).As H is increased, the helimagnetic texture of Cr1/3NbS2 progressively unzips until H reaches a saturation field (Hsat), at which Cr1/3NbS2 becomes ferromagnetic [26][27].
For the Cr1/3NbS2/NbS2 device in Fig. 1(a), the R(T) curve measured in H = 0 shows a peak-like feature, Rpeak, at the onset of the superconducting transition.Rpeak persists in the helimagnetic state of Cr1/3NbS2 as H is increased and it vanishes as H approaches μ 0 Hsat ~ 0.5 Tesla (μ 0 being the vacuum permeability).At H > Hsat, Rpeak coincides with the normalstate resistance of the device (RN) at T ~ 6.0 K (Figs. 1(b) and 1(c)).These observations and the fact that Rpeak is only measured in this device with a specific arrangement of contacts rule out magnetic impurities as explanation for Rpeak.The data reported in Fig. 1(c) also show that, as the H polarity is reversed and H is decreased below Hsat, Rpeak reappears in the helimagnetic state of Cr1/3NbS2 before vanishing again at H ~ -Hsat.These results suggest that Rpeak must be correlated to the magnetically-inhomogeneous state of Cr1/3NbS2.
To understand the Rpeak origin, we note that, for the device in Fig. 1(a), we inject the current with two electrodes (I -and I + ) that are contacting only the bare NbS2, and we collect the voltage signal with two other electrodes (V -and V + ) placed under the Cr1/3NbS2/NbS2 stack.These two regions (i.e., bare NbS2 and Cr1/3NbS2/NbS2 stack) may have different Tc, which should be lower for the S/F stack due to the proximity effect [29].As done in ref. [30], where a similar Rpeak is observed, in the Supplementary Material we show that Rpeak can be reproduced considering a network of resistors equivalent to our sample, where each resistor corresponds to a sample region with different Tc.Our analysis also confirms that the regions in the F/S stack have lower Tc than those in the bare S.
For Rpeak to vanish at || > Hsat, the Tc of the F/S stack (Tc,bi), which is lower than the Tc of the bare S (Tc,bare) for H < Hsat, must become closer to Tc,bare when H > Hsat (Fig. 1(d)).This behavior of Tc,bi is opposite to that expected for a F/S proximity with a homogeneous F or for stray fields, since both effects should get stronger and decrease Tc,bi as Cr1/3NbS2 is driven ferromagnetic (see discussion below).The positive shift of Tc,bi towards Tc,bare for H > Hsat (Fig. 1(d)), which leads to the Rpeak disappearance can be due to an unconventional proximity effect at the Cr1/3NbS2/NbS2 (F/S) vdW interface involving long-ranged spin triplets.This is because, when Cr1/3NbS2 is in its helimagnetic state, long-ranged spin triplets can be generated and can propagate deeply into Cr1/3NbS2 reducing Tc,bi compared to Tc,bare (Fig. 1(d); bottom).As Cr1/3NbS2 is driven ferromagnetic for H > Hsat, only short-ranged Cooper pairs can be generated, which remain confined close to the Cr1/3NbS2/NbS2 interface making Tc,bi approach Tc,bare (Fig. 1

(d); top).
The F/S vdW device in Fig. 1, however, makes it difficult to argue generation of longranged spin triplets across the Cr1/3NbS2/NbS2 vdW interface due the presence of two contributions (from bare NbS2 and Cr1/3NbS2/NbS2 stack) to the total R(T).
To validate our interpretation of the data in Fig. 1, we make another Cr1/3NbS2/NbS2 vdW device (Fig. 2  To determine if long-ranged spin triplets affect the Tc,bi of the device in Fig. 2(a), from each of the R(T) curves in Fig. 2(b), we extract Tc,bi defined as the T where R reaches the 50% of its normal-state value at T = 6.0K (Tc ~ 5.75 K).Based on these Tc,bi values, we study how Tc,bi varies with H, Tc,bi (H), for both increasing and decreasing H (Fig. 3(a)).We always apply H in the normal state of our devices at T = 7.0 K and, at every H, we record R(T) whilst both cooling (from 7.0 K down to 3.0 K) and warming (from 3.0 K to 7.0 K) our samples, to extract the corresponding Tc,bi values.Unless differently specified, the Tc,bi(H) curves shown are based on sample-cooling Tc,bi data.
Tc,bi (H) for the device in Fig. 2(a) is shown in Fig. 3(a), for both H upsweep (blue curve) and downsweep (red curve).As H is increased, Tc,bi (H) shows a first feature (at μ0H < 60 mT) followed by a decrease with different slope at 60 mT < μ0H < μ0Hsat and then by a positive shift at H = Hsat.During the H downsweep, for H > Hsat, Tc,bi (H) coincides with Tc,bi(H) measured during the H upsweep, but shows a difference of ~ 20 mK from the latter for H < Hsat.
The positive shift in Tc,bi (H) at H ~ Hsat in Fig. 3(a) is consistent with our explanation for the data in Fig. 1 and the disappearance of Rpeak at H ~ Hsat due to a positive Tc,bi shift.Also for this device in Fig. 3(a), if long-ranged spin triplets are generated, then Tc,bi(H) in the Cr1/3NbS2 helimagnetic state (H < Hsat) must be lower than Tc,bi(H) in the Cr1/3NbS2 ferromagnetic state (H > Hsat), meaning that Tc,bi should exhibit a positive shift at H ~ Hsat as we observe.Consistently with the disappearance of Rpeak in Fig. 1(b), the jump in Tc,bi(H) at H ~ Hsat in Fig. 3(a) cannot be due to a conventional F/S proximity involving short-ranged Cooper pairs or due to stray fields, since both effects should decrease rather than increase Tc,bi at H ~ Hsat.Vortices also cannot account for our results, as discussed in the Supplementary Material.
To rule out stray fields, like in previous studies on thin film multilayers [15,31], we fabricate a vdW device where a 10-nm-thick insulator (I) of hexagonal boron nitride (hBN) is placed between Cr1/3NbS2 and NbS2 to suppress any superconducting proximity effects.
The Tc profile of this F/I/S vdW heterostructure (Supplementary Fig. S4), Tc,tri(H), shows some features at μ0H < 100 mT like those in Fig. 3(a) for μ0H < 60 mT, which we associate to stray fields, but then follows a parabolic trend different from Tc,bi(H) in Fig. 3(a).This parabolic trend resembles that measured for bare NbS2 (S) (Supplementary Fig. S5).Also, Tc,tri(H) has a negative rather than positive jump at H ~ Hsat, unlike Tc,bi(H) in Fig. 3(a), confirming that stray fields should get stronger and reduce Tc,bi at H ~ Hsat.Based on these results for the F/I/S device, we conclude that only the features in Tc,bi(H) at μ0H < 60 mT in Fig. 3(a) are due to stray fields, whilst all the other features including the positive jump at H ~ Hsat are due to a proximity effect involving long-ranged spin triplets.Spectroscopic studies on Nb/Ho thin films have shown that spin-triplet generation is sensitive to the misalignment of the magnetic moments near the Nb/Ho interface, compared to the bulk Ho helix [12].Like for Nb/Ho [12], interface magnetic moments misaligned to the Cr1/3NbS2 helix can have a non-null out-of-plane component and generate dipolar fields into NbS2, which can account for the Tc,bi(H) variation at low H in our S/F vdW devices.
Previous transport experiments on epitaxial Ho/Nb thin film heterostructures [32] also show, consistently with our results, that Tc,bi decreases as Ho is driven from its helimagnetic to ferromagnetic state by an in-plane H.However, when Ho becomes ferromagnetic, a negative shift in Tc,bi is observed which, as argued by the authors, is opposite to that expected for spin-triplet generation.By contrast, in our F/S vdW devices, we observe a positive shift in Tc,bi(H) at H ~ Hsat.In addition to the F/S vdW device in Fig. 3(a), we have made another F/S vdW device with thinner NbS2 (~ 10 nm), to determine whether the shift in Tc,bi increases as dS gets reduced and closer to ξSwhich would further indicate that the Tc,bi(H) shift is due to a superconducting proximity effect.As shown in Supplementary Fig. 5(a), this device has two regions, one made of a Cr1/3NbS2/NbS2 stack and the other made of bare NbS2 (S) but of the same NbS2 flake used for the F/S stack.Each of these regions has its own set of electrodes to measure Tc,bare(H) independently on Tc,bi(H).For this device, we find that Tc,bare(H) has a parabolic trend consistent with H-induced orbital depairing (Supplementary Fig. 5(b)).In contrast, Tc,bi(H) deviates from Tc,bare(H) for H < Hsat and shows a first drop at μ0H < 80 mT, which we ascribe to stray fields, followed by a slower decrease that we relate to spin-triplet generation (see above).For H > Hsat, where long-ranged spin triplets cannot be generated, Tc,bi(H) follows Tc,bare(H), but with a constant (negative) offset due to hex.For 80 mT < μ 0 H < μ 0 Hsat, the proximity effect must therefore involve long-ranged spin triplets, since a F/S proximity with short-ranged triplets, which occurs for H > Hsat, gives a Tc,bi(H) profile like Tc,bare(H) (see theory model below).Also, the shift in Tc,bi from Tc,bare for this device is ~ 40 mK at H < Hsat, which is larger than that in Fig. 3(a), consistently with the decrease in NbS2 thickness from 20 nm to 10 nm.
To support our claims that long-ranged spin-triplets are generated at the Cr1/3NbS2/NbS2 vdW interface, we develop a model describing the proximity effect in our F/S system.Our theoretical description of the effect is based on the quasiclassical Green's function method in the diffusive limit (Usadel approach).A sketch of our model is reported in the inset of Fig. 3(b).We consider a S of length dS coupled to an infinitely-long helimagnet F with dF >> ξF.The in-plane magnetization (in the y-z plane) of the helimagnet is modeled using the exchange field   = ℎ ex ( y sin  +  z cos ), where  relates to the pitch of the helix (2π/q) defined along the x-direction perpendicular to the S/F interface.The quality of the interface in our model is related to the parameter γ  (see Supplementary Material), which defines the interface transparency, and to γ =  F / S depending on the mismatch between the normal-state conductivities of F ( F ) and S ( S ) .We assume that an in-plane H is applied (along the z-direction), which modulates the F's magnetic textures and penetrates the S affecting Tc,bi, and we only consider H-induced orbital depairing.Also, we simply assume that the transition from a helimagnetic to a ferromagnetic state in F occurs abruptly at Hsat like for the device in Fig. 3(a) and we model it as a step function.
Our hypothesis is that an inhomogeneous (helical) magnetization in a F can generate long-range spin-triplet correlations at the S/F interface thus suppressing the superconducting gap in S. Effectively, this gap suppression is seen in our theoretical curves as a reduction of the superconducting critical temperature Tc,bi.We observe that the theoretical Tc,bi(H) in Fig. 3(b) (blue curve) calculated with our model for a helimagnet becoming ferromagnetic at μ 0 Hsat ~ 0.5 T is in very good agreement with the experimental data in Fig. 3(a).In the same Fig.3(b), we also report the theoretical Tc,bi(H) profile (red curve) calculated for a S/F system with a homogeneous hex confirming a parabolic trend.
The inset of Fig. 3(b) shows the anomalous Green's functions (pairing amplitudes) as a function of the position x inside F. We find that the helical ordering of the F induces a nonvanishing long-ranged pairing amplitude, which suppresses the S gap and reduces Tc,bi.The curves in Fig. 3(b) are calculated using the following parameters: dS = 2 ξS, γ = 0.055, γ  = 0, qξS = 3, h/Δ0 = 100, ξS = 10 nm (Δ0 being the NbS2 superconducting gap).The bulk Tc and upper critical field of S are assumed to be Tc,S = Δ0/(1.764kB)= 6.3K (kB being the Boltzmann constant) and μ 0 Hc,S = 5 T, respectively.The only fitting parameters are γ and

FIG. 1 .
FIG. 1.(a) Optical microscope image of a Cr1/3NbS2 (200 nm)/NbS2 (20 nm) vdW heterostructure on Au (23 nm)/Ti (7 nm) electrodes, with inset showing the materials stack, and corresponding R(T) as a function of an in-plane H in (b).The Rpeak feature vanishes as H is increased above Hsat.(c) Rpeak versus H for H ranges specified in the legend.(d) Model showing the origin of Rpeak due to the presence of regions with different Tc for H < Hsat (bottom panel) assuming equal Tc for H > Hsat (top panel).Red and blue curves refer to bare NbS2 and Cr1/3NbS2/NbS2, respectively.
(a)), where all electrodes are in contact with the Cr1/3NbS2/NbS2 stack.As shown in Fig. 2(b), the R(T) of this device does not exhibit any Rpeak, independently on H, which supports our claim that Rpeak in Fig. 1 originates from two regions with different Tc.

FIG. 2 .
FIG. 2. (a) Optical microscope image of a Cr1/3NbS2 (350 nm)/NbS2 (25 nm) vdW heterostructure made on Au (33 nm)/Ti (7 nm) electrodes with inset showing the materials stack and corresponding R(T) in (b) as a function of an in-plane H (values in the color bar) used to switch Cr1/3NbS2 from a helimagnetic to a ferromagnetic state.

FIG. 3 .
FIG. 3. (a) Tc versus in-plane H of a Cr1/3NbS2/NbS2 device, Tc,bi(H), determined from R(T) curves in Fig. 2(b) for H upsweep (blue curve) and downsweep (red curve).The lower and upper sketches show the magnetic configuration of Cr1/3NbS2 for H < Hsat and H > Hsat, respectively.(b) Theoretical Tc,bi(H) for a S/F with a helimagnet becoming ferromagnetic at Hsat (blue curve) and for S/F with homogenous ferromagnet (red curve).The theoretical model used is shown in the lower panel, whilst the upper inset shows the decay of pairing amplitudes from the S/F interface in the F helimagnetic state: spin-singlets (F0), long-ranged triplets (F2) and short-ranged triplets (F3).