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Nofar Fridman, Tomer Daniel Feld, Avia Noah, Ayelet Zalic, Maya Markman, T. R. Devidas, Yishay Zur, Einav Grynszpan, Alon Gutfreund, Itai Keren, Atzmon Vakahi, Sergei Remennik, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Martin Emile Huber, Igor Aleiner, Hadar Steinberg, Oded Agam, Yonathan Anahory

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[Anomalous thickness dependence of the vortex pearl length in few-layer NbSe2](https://mdr.nims.go.jp/datasets/0d3c041c-5cc2-490a-ab2f-13c17e2b2987)

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Anomalous thickness dependence of the vortex pearl length in few-layer NbSe2Article https://doi.org/10.1038/s41467-025-57817-3Anomalous thickness dependence of thevortex pearl length in few-layer NbSe2Nofar Fridman 1,2 , Tomer Daniel Feld1,2, Avia Noah 1,2,3, Ayelet Zalic 1,2,Maya Markman1,2, T. R. Devidas 1,2, Yishay Zur1,2, Einav Grynszpan1,2,Alon Gutfreund 1,2, Itai Keren 1,2, Atzmon Vakahi2, Sergei Remennik 2,Kenji Watanabe 4, Takashi Taniguchi 5, Martin Emile Huber 6, Igor Aleiner7,Hadar Steinberg1,2, Oded Agam 1 & Yonathan Anahory 1,2The coexistence of multiple types of orders is a common thread in condensedmatter physics and unconventional superconductors. The nature of super-conducting orders may be unveiled by analyzing local perturbations such asvortices. For thin films, the vortex magnetic profile is characterized by thePearl-length Λ, which is inversely proportional to the 2D superfluid density;hence, normally, also inversely proportional to the film thickness, d. Here weemploy the scanning SQUID-on-tip microscopy to measure Λ in NbSe2 flakeswith thicknesses ranging from N = 3 to 53 layers. For N > 10, we find theexpected dependence Λ / 1=d. However, six-layer films show a sharp increaseof Λ deviating by a factor of three from the expected value. This value remainsfixed for N = 3 to 6. This unexpected behavior suggests the competitionbetween two orders; one residing only on the first and last layers of the filmwhile the other prevails in all layers.Superconductivity in the presence of competing or intertwined orderparameters has generated much interest over the last decades.Detecting the presence of an additional order parameter and deter-mining its influence on superconductivity is crucial in unveiling thepairing mechanism. In particular, order-parameter competition couldinvolve two superconducting order parameters with different pairingsymmetries1 or a single superconducting channel and an order para-meter related to another degree of freedom, such as charge densitywave2,3 or spin density wave4.Although usually discussed in the context of high-temperaturesuperconductors5,6, competing orders are also relevant to many otherlayered materials, such as twisted bilayer graphene7 and NbSe21.Moreover, in the case of thin films with few atomic layers, the com-petition betweendistinct order parameters residing on the surface andin the bulk also becomes a possibility. This is due to the disparity inproperties exhibited by the surface and bulk regions, notably exem-plified by the presence of Rashba spin-orbit coupling on the surface8.However, as far as we are aware, the examination of competitionbetween surface and bulk order parameters, both experimentally andtheoretically, has not been documented in prior studies. In this work,such a competition is uncovered through measurements of the mag-netic field profile of superconducting vortices in NbSe2. NbSe2 is alayered superconductor which sustains superconductivity withTc >4:2 K for any thickness above 3 layers, and thus is uniquely suitablefor such an experiment.Superconducting vortices are often exploited as local perturba-tions used to probe the properties of the order parameter9–12. Vorticesconsist of a core where the superconductivity is locally suppressed onthe scale of the coherence length ξ where the magnetic field canpenetrate. This field is screened by the surrounding superconductingReceived: 2 June 2024Accepted: 3 March 2025Check for updates1The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel. 2Center for Nanoscience and Nanotechnology, Hebrew University of Jerusalem,Jerusalem, Israel. 3Facultyof Engineering,RuppinAcademicCenter, Emek-Hefer 40250Monash, Israel. 4ResearchCenter for ElectronicandOpticalMaterials,National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan. 5Research Center for Materials Nanoarchitectonics, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba, Japan. 6Departments of Physics and Electrical Engineering, University of Colorado Denver, Denver, CO, USA. 7Google QuantumAI, Santa Barbara, CA, USA. e-mail: nofarfri.friedman@mail.huji.ac.il; agam.oded@gmail.com; yonathan.anahory@mail.huji.ac.ilNature Communications |         (2025) 16:2696 11234567890():,;1234567890():,;http://orcid.org/0000-0002-2037-569Xhttp://orcid.org/0000-0002-2037-569Xhttp://orcid.org/0000-0002-2037-569Xhttp://orcid.org/0000-0002-2037-569Xhttp://orcid.org/0000-0002-2037-569Xhttp://orcid.org/0000-0001-6498-2910http://orcid.org/0000-0001-6498-2910http://orcid.org/0000-0001-6498-2910http://orcid.org/0000-0001-6498-2910http://orcid.org/0000-0001-6498-2910http://orcid.org/0000-0002-2958-4228http://orcid.org/0000-0002-2958-4228http://orcid.org/0000-0002-2958-4228http://orcid.org/0000-0002-2958-4228http://orcid.org/0000-0002-2958-4228http://orcid.org/0000-0002-6855-7505http://orcid.org/0000-0002-6855-7505http://orcid.org/0000-0002-6855-7505http://orcid.org/0000-0002-6855-7505http://orcid.org/0000-0002-6855-7505http://orcid.org/0000-0002-3891-2908http://orcid.org/0000-0002-3891-2908http://orcid.org/0000-0002-3891-2908http://orcid.org/0000-0002-3891-2908http://orcid.org/0000-0002-3891-2908http://orcid.org/0000-0002-1677-4049http://orcid.org/0000-0002-1677-4049http://orcid.org/0000-0002-1677-4049http://orcid.org/0000-0002-1677-4049http://orcid.org/0000-0002-1677-4049http://orcid.org/0000-0002-6408-5424http://orcid.org/0000-0002-6408-5424http://orcid.org/0000-0002-6408-5424http://orcid.org/0000-0002-6408-5424http://orcid.org/0000-0002-6408-5424http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-7638-902Xhttp://orcid.org/0000-0002-7638-902Xhttp://orcid.org/0000-0002-7638-902Xhttp://orcid.org/0000-0002-7638-902Xhttp://orcid.org/0000-0002-7638-902Xhttp://orcid.org/0000-0002-4118-8586http://orcid.org/0000-0002-4118-8586http://orcid.org/0000-0002-4118-8586http://orcid.org/0000-0002-4118-8586http://orcid.org/0000-0002-4118-8586http://orcid.org/0000-0002-9368-1129http://orcid.org/0000-0002-9368-1129http://orcid.org/0000-0002-9368-1129http://orcid.org/0000-0002-9368-1129http://orcid.org/0000-0002-9368-1129http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57817-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57817-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57817-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-57817-3&domain=pdfmailto:nofarfri.friedman@mail.huji.ac.ilmailto:agam.oded@gmail.commailto:yonathan.anahory@mail.huji.ac.ilwww.nature.com/naturecommunicationscurrents and results in magnetic flux quantization. In bulk super-conductors, the magnetic field is screened exponentially with a char-acteristic scale knownas the Londonpenetration depth λL. By contrast,for a film of thickness d<λL, screening is less effective and is governedby the Pearl lengthΛ=2λ2L=d13–15. In this limit, themagneticfielddecaysas 1=Λr near the vortex core and as Λ=r3 for distances greater than Λ,where r is the distance from the vortex’s center. Therefore, near thevortex core, there is no characteristic length scale for the magneticfield screening. In this work, we measure the Pearl length, which is afundamental characterizer of superconductors that is highly sensitiveto variation of the two-dimensional superfluid density. This sensitivityis particularly important when studying phase transitions that involvechanges in the superconducting order, as such transitions naturallyalter the superfluid density.We have measured the thickness dependence of the Pearl lengthin NbSe2 flakes with thicknesses ranging from N =3 to 53 layers. Forthis purpose we employ a highly sensitive microscopy technique ofSQUID-on-tip16,17 (SOT, see Fig. 1) coupled to a tuning fork designed tomeasure gradients in minute magnetic signals emitted by a vortex,particularly in cases where the Pearl length significantly exceeds thesize of micron-scale flakes. Our data presents the anticipated 1=ddependence for flakes of thicknesses N≳10 layers. However, strikingly,Λ largely deviates from the expected 1=d dependence for thinner films.Such deviation has not been previously reported in NbSe2 neither intransport18 nor in tunneling3,9,19–21 studies. We suggest that the sharpjump in Λ can be attributed to the competition between bulk andsurface superconductivity.ResultsNbSe2 was mechanically exfoliated to obtain flakes with thicknessesranging from N =3 to 53 layers (d = 1:9 to 33 nm). To prevent sampledegradation, thinflakeswithN ≤ 14 layerswere encapsulatedwith h-BNfrom both sides (see Methods). Figure 2a, b depict representativeoptical images of thin films, showcasing the presence of atomically flatterraces. The number of layers, which corresponds to the thickness, isindicated for eachobserved area. For thin flakes (N ≤ 14 layers), samplethickness was measured using cross-sectional scanning transmissionelectron microscopy (STEM), as described in the Methods and shownin Supplementary Fig. 1. For thicker samples with N>14 layers, thethickness was measured using an atomic force microscope (AFM).Energy-dispersiveX-ray spectroscopy (EDS) analysiswas performedonthe flakes (N ≤ 14 layers) and showed no surface contamination (Sup-plementary Fig. 2).The samples are mounted in a scanning SOT microscope to con-duct magnetic imaging. In particular, we image the out-of-planecomponent of the magnetic field Bz ðh, rÞ at the surface of super-conductingNbSe2 in the presence of vortices at 4.2 K (Fig. 1), whereh isthe distance of the tip from the surface and r = ðx, yÞ is the in-planecoordinate. Vortices are visually identified as bright regions in theacquired images (see Figs. 1 and2c). To ensure non-overlapping signalsfrom individual vortices, we adjusted the external magnetic fieldμ0Hz � 1 mT, that controls the vortex density. The positions of thevorticesmeasured at different magnetic fields aremarked with orangedots in Fig. 2a, b (see also Supplementary Fig. 3).We commence by clarifying our capacity to extract the Pearllength even in cases where it considerably surpasses the flake size. Themagnetic field profile of the Pearl vortex exhibits a characteristic gra-dual transition from the short-distance asymptotic behavior, descri-bedby 1=Λr, when h≪ r≪Λ to the long-distance behavior ofΛ=r3 whenr≫Λ. Examples of such profiles are presented in Fig. 2d for h=260 nmwith Λ= 1:5 μm and Λ = 100 μm and plotted for a typical SOT field ofview ( rj j ≤ 1:1 μm). For this region of interest, h≪ r≪Λ, both profilesare governed by the samepower-lawdecay. Therefore, the twoprofilesdiffer solely by their magnitude, which is inversely proportional to Λ.The slow 1=r decay implies that magnetic fields resulting from neigh-boring vortices and the Meissner current associated with the sample’sedges add up to an approximately constant background. This back-ground can be effectively eliminated by measuring the gradient of thesignal yielding a localized signal from which the Pearl length can beextracted with an error of order ðh=LÞ2, where L is the typical size ofthe flake or the distance to a neighboring vortex (see SupplementaryNote 1). As we typically measure at h≲0:4 µm and L is typically largerthan 1 µm, this error is small. To substantiate this result, we fit simu-lated isolated vortices and compare them with fits for vortices sur-rounded by randomly distributed vortices at a typical distance weencounter in experimental conditions (∼2 µm). Our results, presentedin Supplementary Fig. 4, show that the influence of these additionalvortices on Λ is up to 10%. Thus, when themagnetic field is sufficientlyweak, it is feasible to measure Pearl lengths in the range of a fewhundred microns, even with a limited field of view of just a fewmicrons. The capability to measure screening lengths of such magni-tude is essential for the conclusions drawn in this work.To measure the gradient of the out-of-plane component of themagnetic field, Bz ðh, rÞ, we mechanically couple the SOT to a tuningfork, as shown in Fig. 2f. The tuning fork is set to oscillate, resulting in aperiodic lateral motion of the SQUID loop at a frequency of approxi-mately 32 kHz, as indicated by the blue double-headed arrow in theinset of the Figure.When the amplitude of the tuning fork oscillation issmall, the in-phase ac signal,Bacz h, rð Þ, is proportional to the gradient ofthe correspondingdc signal along a chosendirection. Setting the x axisalong this direction, Bacz h, rð Þ ffi xacdBz h, rð Þ=dx, where xac is theoscillation amplitude of the SQUID loop. Using an independent mea-surement of the oscillation amplitude (see Methods and Supplemen-tary Note 2), we are able to deduce the spatial derivative of the staticsignal Bz ðh, rÞ, shown in Fig. 2e. Typically we set xac ≲ 100 nm, whichyields a large signal while keeping the following approximationBacz h, rð Þ ffi xacdBz h, rð Þ=dx within our experimental uncertainty.Moreover, by conducting measurements at a frequency of 32 kHz, weare able to effectively eliminate the 1=f noise, which is typically at leasttwo orders of magnitude higher below 1 kHz16,17.Figure 3a–d display typical images of the spatial derivative alongthe x axis of the out-of-plane component of the magnetic field(Bacz ðh, rÞ=xac) measured for various thicknesses (N = 3, 6, 7, and 14layers). Figure 3e–h show the best fits achieved for each image to theFig. 1 | Schematic diagram of the experimental setup. Two NbSe2 flakes withN =6 and 7 layers. SQUID-on-tip (SOT) magnetic images of vortices representingthe magnitude of the out-of-plane component of the magnetic field Bz ðh, rÞ areoverlayed on the top surface, whereh is the distanceof the tip from the surface andr = ðx, yÞ is the in-plane coordinate. The contour of each layer is colored accordingto the superconducting order parameters:ψ (blue) residing in all layers, andφ (red)is a different superconducting order parameter confined to the surface. The colorintensity encodes the amplitude of each order parameter, where gray represent thenormal state ψ���� =0. For N = 7, the two order parameters are finite, while for N =6,ψ���� =0, while φ���� remains finite and confined to the first and last layer.Article https://doi.org/10.1038/s41467-025-57817-3Nature Communications |         (2025) 16:2696 2www.nature.com/naturecommunicationsPearl model. This model requires two parameters, the height h and thePearl length Λ. We determine h by sensing the sample surface with thetuning fork, as in an AFM, then retract a known amount, leaving Λ asour sole fitting parameter. The uncertainty on themeasurement of h is15 nm, which propagates to an uncertainty in the Pearl length of 8–10%(Supplementary Note 3). In addition, we take into account our finiteSOT resolution by convoluting the modeled image with a circle cor-responding to the SQUID’s diameter, which we determine by mea-suring the field period of the critical current oscillations16,17. Todemonstrate the good agreement between fits andmeasurements, weshow the cross-sections of derivatives of the model and theBacz h, rð Þ=xac signal along the x axis of the image (Fig. 3i–l). From thesefits we obtain Λ= 111, 101, 30, and 12 μm for N = 3, 6, 7, and 14,respectively.Notably, themeasured signalmagnitude is approximatelytwice as large for N =6 compared to N =3, despite obtaining similar Λvalues for both cases. The difference of the signals is attributed to thedifference in the tip height at which the measurement was conducted:h=360± 15 and 260± 15 nm for N = 3 and 6, respectively. The nearlyconstant value of Λ for two different thicknesses manifests a puzzlingdeviation from the expected 1=d dependence.Figure 4 summarizes the measured values of the Pearl length Λ asa function of the thickness d ranging fromN = 3 to 53 layers plotted ona logarithmic scale. According to the Pearlmodel, this plot is expectedto exhibit a slope of −1 (representing the d�1 dependence of the Pearllength) along with an offset determined by the London penetrationdepth λL. Indeed, for N ≳ 10 layers, such dependence is observed,enabling us to estimate λL = 230 nm in good agreementwith bulk valueλL = 200 nm measured at 4.2 K22–25. Notice that, for thicker films, thethickness was measured with AFM, and the flakes were not encapsu-lated. Consequently, the uncertainty on the thickness is larger, whichexplains partially the scatter of the data points around 1=d.For N =6, we observe a drastic increase in Λ, to 101 μm, which isthree times larger than the expected value of 28 μmaccording to Pearlmodel. Moreover,Λ remains surprisingly constant, within 100± 15 μm,for N =3, 4, 5, and 6 layers. However, take note that in a differentsample with N =6, we obtained a significantly different value ofΛ=39 µm. This value is somewhat closer to the expected value fromPearl’s model of 28 µm. These findings suggest that near a criticalthickness of N =6, where the system undergoes a phase transition,finite size effects are significant.Two key features characterize the measured Pearl length depen-dence on the film thickness. The constant value of Λ below the criticalthickness, and the sharp jump near N =6. In what follows, we discusspotential experimental issues, and explain why their impact on ourfindings is negligible. Consider first the issue of sensitivity limitations.In Fig. 3i, we compare the profile of the imagemeasured in Fig. 3a withthree simulated profiles. The best fit (in yellow) and two other profilesobtained by changing Λ by ± 15% (blue and red). The figureab500 nmec60 T/m346765Fig. 2 | Pearl model and SQUID-on-tip images of NbSe2 at 4.2K. a, b Opticalimages of the ultra-thin flakes and the locations of the imaged vortices. The num-bers indicate the number of layers N, and orange lines outline the terraces’ edges.a The blue dashed line shows the area covered with top and bottom hBN. b Theentire field of view is encapsulated with top and bottom hBN (c) SQUID-on-tip(SOT) image of the out-of-plane component of the magnetic field Bz ðh, rÞ of anisolated vortex located in theN = 7 layer region shown in panel a, where h is the tip-to-surface distance and r = ðx, yÞ is the in-plane coordinate. d Calculated magneticprofile Bz ðh, xÞ of vortices using the Pearl model with Pearl length Λ= 1:5 (blue) and100 μm (yellow). Both profiles decay with the same power law ð1=ΛrÞ for r≪Λ.(inset) Illustration of a SOT scanning the surface monotonically as opposed to (f).e The same vortex as in c, measured while the tip oscillates along the x axis. Theimage shows the field component oscillating in phasewith the SQUID loop Bacz ðh, rÞdivided by the motion amplitude xac resulting in spatial derivative of the imageshown in (c) along the x axis. f Same as (d) but showing the spatial derivative alongthe x axis Bacz ðh, xÞ=xac. (inset) Illustration of the SOT coupled to a tuning forkoscillating along the x axis (blue double-headed arrow).Article https://doi.org/10.1038/s41467-025-57817-3Nature Communications |         (2025) 16:2696 3www.nature.com/naturecommunicationsdemonstrates that it is possible to distinguish variations in Λ of theorder of 15%, given our measurement sensitivity (SupplementaryNote 3). This resolution is largely sufficient when compared to ourclaims. Another potential issue is the disorder and surface roughness,which might introduce possible limitation on the measured filmthickness. To exclude this mechanism, cross-sectional STEM imageswere taken (Supplementary Fig. 1) to ensure the crystal quality andassess contamination for all thin films (N ≤ 14). The flakes were foundto be atomically flat without any trace of contamination over micron-size terraces where vortices were imaged.DiscussionThe sharp jump of Λ near N =6 indicates a sharp reduction in the totalsuperfluid density. NbSe2 is known to be a two-band superconductor26,with the larger energy band associated with the Nb orbitals, and thelower energy band with the Se orbitals. Hence a possible explanationfor this reduction is the suppression of one of the bands with thethickness of the film18,20,21,27,28. Although that could explain the jump inΛ, it does not account for the saturation of Λ in thinner films as the 1=ddependence should persist albeit with a different offset. Moreover, inNbSe2, the spectral signature of the Se-derived band disappearsgradually at films in the d � 10 to 20 nm thickness range21, well abovethe thickness of 3:5 nm (N =6) where we observe the transition.The distinctive behavior of Λ presented in Fig. 4 suggests thepresence of a phase transition wherein a partial disappearance ofsuperconductivity occurs as the film thickness diminishes, thus leadingto a notable jump in the Pearl length. At the same time, the saturation inthe Pearl length implies the persistence of the superfluid density inde-pendent of the film thickness. This feature indicates that the remainingsuperconducting order parameter is confined in specific layers, and istherefore unaffected by the film thickness. Thus, the order parameter,which is suppressed for thinner films, prevails elsewhere for N >6.The observed jump in the Pearl length dependence as a functionof the number of layers in the film, ΛðNÞ, can be naturally explainedwithin the framework of the two-component Ginzburg-Landaudescription, in which φ and ψ, represent superconducting orderparameters belonging to different superconducting classes. In thisdescription, the Pearl length is given by the sum of two contributions,1ΛðNÞ =Njψj2Λb+jφj2Λsð1ÞFig. 3 | Images of the magnetic field gradient near vortices in samples of dis-tinct thicknesses. a–d Spatial derivative along the x axis of the out-of-planecomponent of magnetic field Bacz ðh, rÞ=xac of a vortex located in a region of N = 3a, 6 b, 7 c, and 14 d layers, where h is the tip-to-surface distance, r = ðx, yÞ is the in-plane coordinate, and xac is the motion amplitude of the SQUID loop. Imageswere acquired from h= 360 a, 260 b 360 c, and 360 d nm (e–h) Calculatedtheoretical magnetic image to obtain the best fit of the images shown in (a–d). ThePearl lengths obtained are Λ= 111 e, 101 f, 30 g, and 12 h µm (i–l) Profile of theexperimental data Bacz ðh, xÞ=xac (brown), and the calculated vortex (yellow).i Calculated vortex profile with Λ=94 (blue), 128 (red) μm, which is a ± 15%deviation from the best fit, all curves are multiplied by a factor 6 for clarity.Article https://doi.org/10.1038/s41467-025-57817-3Nature Communications |         (2025) 16:2696 4www.nature.com/naturecommunicationswhereΛb is a length scale characterizing the bulk, whileΛs is associatedwith the surface. The order parameters φ and ψ, are determined byminimization of the dimensionless energy density of an effectivelytwo-dimensional system:FN =n*jφj2ð�2 + jφj2 +μjψj2Þ+Njψj2ð�2 + jψj2Þ ð2Þwhere n* and μ, are fitting parameters that control the competitionbetween the two order parameters. The length scales, Λb and Λs, areuniquely determined from the asymptotic limits of the Pearl length atlarge and small film thicknesses. They are found to be: Λb =2λ2L=a andΛs =0:6Λb, where λL =230 nm is the bulk London penetration depth,while a=0:63 nm is the spacing between neighboring layers. The twoother fitting parameters, n* and μ, are chosen such that ΛðNÞ followsthe data points and the systemundergoes a first-order phase transitionat thickness of approximately N � 6 layers. This transition transformsthe system from a phase where both ψ and φ are non-zero (N ≥ 7) to aphase where ψ vanishes while φ remains non-zero (N ≤6). Theexperimental data is perfectly described by n* = 6:5 and μ= 1:9, asshown in Fig. 4. In this model, the energy associated with theφ-component is independent of the number of layers,N, therefore it isassociatedwith a contribution coming from the surface of the film (i.e.,the first and last layers). Furthermore, the experimental data does notcorroborate the presence of a term proportional to φ*ψ+φψ* in thefree energy. Hence, ψ and φ must belong to distinct irreduciblerepresentations of the system’s superconducting symmetry group,linked to the point groups D3h and D3d for films with an odd or evennumber of layers, respectively. These irreducible representations areboth one-dimensional; otherwise, the system should exhibit anadditional type of symmetry breaking. The sensitivity of our measure-ments is insufficient for unveiling such a symmetry-breaking.The phenomenological model proposed herein assumes thepresence of a superconducting order parameter localized near thesample surface, distinct from that of the bulk. This order parametermay originate from various factors, such as a strong Rashba spin-orbitcoupling at the surface or the presence of strain induced by latticemismatch between hexagonal h-BN and NbSe2. Both, Rashba-spin-orbit and strain gradient, induce an effective pseudomagnetic fieldacting with opposite signs on electrons associated with distinctvalleys29. The strain engendered by the lattice mismatch permeatesinto the bulk to a depth of approximately the effective lattice constantof the resulting moiré patterns. This depth is estimated to be on theorder of 1 nm, significantly smaller than the sample thickness at whichthe transition manifests and thus consistent with our data. Addition-ally, it is noted that the coexistence of superconductivity with a dif-ferent order parameter, such as charge density waves, does not alterthe fundamental framework of our phenomenological model. How-ever, it may influence its parameters and serve as a mediator for theinteraction between bulk and surface superconducting orders.The current study identifies a superfluid-density transition thathas not been previously observed in transport or tunnelinganalyses3,9,19–21. Notably, previous tunneling investigations were con-ducted at temperatures of 1.2K3, 300 mK20 and below 100mK9,21,whereas our experimentwas conducted atT =4:2 K. This suggests thatthe observed transition may be a distinct feature of intermediatetemperature regime. While tunneling experiments have also beenconducted at T =4:2 K, these were limited to bulk crystals, wheresuppression of surface states is significant. It is also important tohighlight that tunneling experiments using STM cannot fully encap-sulate the sample, as the top surface must remain exposed. Ourexperimental study suggests that encapsulating the sample with hBNon both surfacesmight be crucial to prevent degradation and enablingmeasurement of the intrinsic properties of the sample or to strain theNbSe2 as discussed above. This limitation could be addressed throughdevice-based tunneling experiments, where the top tunneling elec-trode is deposited on a thin hBN layer. Future SOT measurements atmillikelvin temperatures will be necessary to directly compare theseresults with tunneling experiments.One might expect that an abrupt change in the superfluid densityas a function of film thickness would also manifest in the criticaltemperature. However, previous transport studies of the Tc thicknessdependence revealed only a small change between the bulk value andthat observed in 5–6 layer devices18. Notably, the ultrathin NbSe2 flakesin ref. 18were exfoliatedonSiO2 and solely top-encapsulated,while thesamples in our study are encapsulated with hBN on both sides.Moreover, assessing Tc in two-dimensional superconductors throughtransport measurements faces several inherent limitations that aredifficult to circumvent. Firstly, these measurements are highly sensi-tive to the nature of the vortex dynamics and pinning effects30, whichcanpotentiallymask the observed transition, which ismeasured underequilibrium conditions. Secondly, transport measurements requiremetallic leads, which influence the superconducting state through thereverse proximity effect. Thirdly, Tc measurements are sensitive to theorder parameter only near the critical temperature, making it impos-sible to probe transitions occurring at intermediate temperatures as inour experiment.Measurements of the in-plane critical magnetic field as a functionof temperature could provide valuable evidence supporting our find-ings, provided orbital depairing effects play a significant role. Thisassumption is plausible, as the zero-temperature out-of-plane coher-ence length (approximately 2.3 nm31) is considerably larger than theinterlayer spacing (0.63 nm). Under these conditions, two distincttransitions as a function of the in-plane magnetic field are anticipated:one corresponding to the bulk superconducting order parameter andanother, at a higher field, associated with the surface superconductingorder parameter.Such experimental observations have indeed been reported inref. 32, where a kink in the critical field versus temperature curve wasidentified. This kink is interpreted as a first-order transition into aFulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state. Here,based on our data, we offer an alternative interpretation—namely thatthe observed kink is in fact the hallmark of the surface super-conducting order parameter. Furthermore, consistent with ourL2Fig. 4 | Thickness dependence of the Pearl length.Measured Pearl length Λ (reddots) obtained from SQUID-on-tip (SOT) images and the best fit to our phenom-enological model (blue dots). The black dashed line represents Pearl length thick-ness dependence according to the Pearl model considering λL = 230 nm. The errorbars represent the total uncertainty resulting from the systematical and statistical(one standard deviation) uncertainties as described in Supplementary Note 3 andSupplementary Figs. 4, 5. The sudden suppression of bulk superconductivity ψdepicted in Fig. 1 is reflected by the sharp increase of Λ at N =6 layers.Article https://doi.org/10.1038/s41467-025-57817-3Nature Communications |         (2025) 16:2696 5www.nature.com/naturecommunicationstheoretical explanation, the kink became progressively weaker as thefilm thickness increased. For films thinner than 5 nm—near the thick-ness at which we observed the transition—the critical field no longerexhibited the kink associated with the additional transition. Thesediffering interpretations highlight the need for further investigationsto distinguish between these scenarios or to reconcile them within aunified framework.ConclusionsIn summary, the present study has unveiled a remarkable anomaly inthe Pearl length dependence in the few-layer limit. This anomaly isinterpreted as a phase transition in thin films of NbSe2, wherein areduction in film thickness triggers the suppression of bulk super-conductivity, giving rise exclusively to surface superconductivity. Ourapproach provides invaluable insights into the underlying super-conductive characteristics of the system. Specifically, irrespective ofthe theoretical framework, our experimental methodology serves as apowerful tool for detecting surface superconductivity and the con-current suppression of bulk superconductivity.The utility of Pearl-length measurements extends beyond thisinvestigation, as they serve as a sensitive tool for probing the super-conducting nature of thin films under various experimental condi-tions, such as elastic strain, electric field perturbations, or alterationsin temperature. This versatility underscores the broader implicationsof our findings, advancing our understanding of superconductivity indiverse contexts and paving the way for further studies.MethodsSample fabricationhBN was exfoliated onto 285 nm SiO2/Si substrates, yielding flakes withthicknesses ranging from 5 to 15 nm for the upper layer and 10–20nmfor the lower hBN layer. To prepare NbSe2 flakes, an initial exfoliationwas carried out onto PDMS, followed by a transfer onto 285 nm SiO2/Sisubstrates. This process resulted in the production of large, thin flakesdisplaying uniform steps. Flakes with consistent and sizable steps wereidentified utilizing optical microscopy, encompassing various thick-nesses. Subsequently, a polycarbonate (PC) pickup technique wasemployed, as detailed in ref. 33. The pickup started with top hBN, fol-lowed by the NbSe2 flake, and finally the lower hBN layer. The resultantstructure was positioned onto a SiO2 chip, near a predeposited goldheater used for SQUID-on-tip localization. Both the pickup procedureand the exfoliation of NbSe2 were executed within an argon environ-ment. This approach aimed to prevent degradation of the samples andmaintain their integrity throughout the experimental process.SQUID-on-tip fabricationThe SOT was prepared using a self-aligned three-step process invol-ving thermal deposition of Pb at cryogenic temperatures, as outlinedin earlier works16,17. The SQUID loop employed in this work have adiameter of approximately 250nm. This choice of a relatively largerdiameter is strategic, as it provides the enhanced magnetic sensitivityrequired formeasuring weak extended signals obtained by Λ= 100 μmvortices. Additionally, the larger diameter causes lower period of thequantum interference pattern of the SQUID critical current versusapplied field. This enhances the zero-field magnetic sensitivity16,17,which is essential for measurements at low vortex densities.SQUID-on-tip measurementsAll measurements were carried out at a temperature of 4.2K, which isbelow the critical temperature of 5.5 K20 for NbSe2 flakes with threelayers or more. To initiate the experimental process, we applied small0:3 mT magnetic fields to induce the creation of multiple vorticeswithin the sample. Subsequently, we gradually reduced the magneticfield to a range of approximately �0:05 mT to 0:05 mT. Our primaryobjective during this phase was to identify a relatively isolated vortexpositioned at a distance of 1 μmormore from the edges of the uniformsteps. Once such a vortex was located, we employed the Tuning fork(TF) to establish a distance akin to that of an AFM. Subsequently,images of these vortices were captured for further analysis.Sample characterizationThe thicknesses of the NbSe2 flakes were assessed utilizing ScanningTransmission Electron Microscopy (STEM). Specific areas of interestwithin the sample were carefully selected to encompass the approx-imate regions where the vortices were observed. These selected areaswere subsequently fabricated into thin lamellas using a Focused IonBeam (FIB) technique. These lamellas were then subjected to STEMimaging, with the outcomes presented in Supplementary Fig. 1. Toensure the integrity of the samples, thorough examinations for oxi-dation and degradation were conducted through Energy DispersiveX-ray Spectroscopy (EDS). The analysis confirmed an organized crystalstructure, with no discernible signs of oxidation or degradation.Tip-to-sample distance measurementThe SOT is attached to a tuning fork excited electrically near its reso-nance frequency at � 32 kHz. The quality factor of the tuning forkcoupled with the SOT varies between 10000 and 30000 at 4 K with a10mbar helium gas pressure. The electrical signal is amplified at roomtemperature using a homemade operational amplifier. The amplifiedsignal is fed to a nanonis lock-in amplifier in phase-locked loop mode.As the tip approaches within a few nm of the surface, a change in thephase is detected and the tip is retracted by a known safe distance. Itwas previously calibrated that once the phase variation is detected, theedge of the tip is within 1–2 nm from the sample surface. Thus, thesample-tip distance is set by the distance at which we retract the tip(typically around 300 nm).Data availabilityThe data that supports the findings of this study have been depositedin the GitHub database: https://github.com/QIL123/NbSe2_Thin_vortex, https://doi.org/10.5281/zenodo.14852386.Code availabilityThe MATLAB scripts that analyze the raw data and reproduce the fig-ures appearing in this paper have been deposited in the GitHub data-base: https://github.com/QIL123/NbSe2_Thin_vortex, https://doi.org/10.5281/zenodo.14852386.References1. Cho, C. W. et al. Nodal and nematic superconducting phases inNbSe2 monolayers from competing superconducting channels.Phys. Rev. Lett. 129, 087002 (2022).2. Chang, J. et al. Direct observation of competition between super-conductivity and charge density wave order in YBa2Cu3O6.67. Nat.Phys. 8, 871–876 (2012).3. Pásztor, Á. et al. Multiband charge density wave exposed in atransition metal dichalcogenide. Nat. Commun. 12, 6037 (2021).4. 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Fast pick up technique for high quality heterostructuresof bilayer graphene and hexagonal boron nitride. Appl. Phys. Lett.105, 013101 (2014).AcknowledgementsWe would like to thank Hermann Suderow, Avraham Klein, Isabel Guil-lamón,MaximKhodas, LeonidGlazman, Boris Shapiro, CharisQuay HueiLi, Marco Aprili, Eli Zeldov, and Oded Millo for fruitful discussions. Wethank Snir Gazit for the support in the data analysis. Y.A. acknowledgesthe support from the European ResearchCouncil (ERC) startupgrantNo.802952 (STRONG) and consolidator grant No. 101124770 (MAJOR). H.S.acknowledges funding by Israel Science Foundation grant 164/23 andDFG Priority program grant 443404566. K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grant Numbers 21H05233 and23H02052) and World Premier International Research Center Initiative(WPI), MEXT, Japan.Author contributionsY.A.,H.S.,O.A., T.D.F., andN.F. conceived theexperiment. A.Z., N.F., T.R.D,E.G, and I.K fabricated the NbSe2 devices. N.F. and T.D.F. conducted thescanning SOT measurements. O.A. and I.A. conceived the theoreticalmodel. N.F., T.D.F, A.N, Y.Z, and A.G fabricated the SOT sensor. A.G. andT.D.F fabricated the Tuning forks. N.F., T.D.F., and Y.A. generated thenumerical simulations. N.F., T.R.D., A.V., S.R., and T.D.F. characterized theNbSe2 samples. N.F, T.D.F, and M.M. analyzed the data. Y.A. and A.N.constructed the scanning SOT microscope. M.E.H. Conceived the SOTreadout electronics. K.W. and T.T. synthesized the hB.N. N.F., H.S., O.A.,and Y.A. wrote the article with contributions from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-57817-3.Correspondence and requests for materials should be addressed toNofar Fridman, Oded Agam or Yonathan Anahory.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-57817-3Nature Communications |         (2025) 16:2696 7https://doi.org/10.1038/s41467-025-57817-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Anomalous thickness dependence of the vortex pearl length in few-layer NbSe2 Results Discussion Conclusions Methods Sample fabrication SQUID-on-tip fabrication SQUID-on-tip measurements Sample characterization Tip-to-sample distance measurement Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information