Introduction

Superconductors are known to exclude magnetic fields from their interior, except for type II superconductors, which allow the entry of magnetic fields in the form of vortices. These vortices, referred to as fluxoids, consist of both magnetic fluxes induced by external magnetic fields and circulating supercurrents. Owing to the single-valued nature of the superconducting order parameter, the fluxoid must follow quantization. The quantized value of the fluxoid (ΦF) is determined by the sum of the magnetic flux (ΦB) and the supercurrent flux (ΦJ), expressed as ΦF = ΦB + ΦJ = Φ0, where Φ0 = h/2e, with h and e being Plank’s constant and the electron charge, respectively. When the contribution of the supercurrent is negligible (ΦJ = 0), the quantization of the fluxoid is reduced to the quantization of the magnetic field flux (ΦF = ΦB = Φ0). Conversely, in the presence of significant supercurrents, the quantized entity becomes the fluxoid rather than the magnetic field flux. When the magnetic flux is not quantized, the spatially varying phase factor of the superconducting order parameter gives rise to various intriguing phenomena, including the Little‒Park experiment in thin samples1, the transport current2,3,4, the Josephson effect5,6,7, and p-wave and d-wave pairing symmetry in unconventional superconductors8,9,10.

Recently, there has been significant interest in studying the vortex confinement effect in low-dimensional superconductors to investigate the inhomogeneous nature of superconducting order parameters11,12,13,14,15. In confined two-dimensional (2D) disks, for instance, magnetic fluxes can form, but their flux values are smaller than Φ0 because of incomplete screening of the supercurrent against field penetration. Consequently, the quantization of magnetic flux is not satisfactory in 2D confinement systems14,15,16,17,18,19. Although unquantized magnetic fluxes give rise to a plethora of interesting superconducting phenomena, their investigation is challenging owing to geometric limitations, making it difficult to create, visualize, and manipulate them. Specifically, the exploration of 1D confined fluxoids in conjunction with a purported 1D linear potential is particularly intriguing.

In this work, we present the observation of 1D fluxoid confinement via an unquantized magnetic flux in ultrathin superconducting Nb films via vector-field cryogenic magnetic force microscopy (MFM). By utilizing the local stray magnetic field of an MFM tip, we generate a half-vortex ring, resulting in a vortex‒antivortex dipole pair connected by a 1D confined fluxoid. Our study of the manipulation and thermal behavior of these confined vortex pairs through an unquantized magnetic flux in ultrathin films suggests that there are no dimensional restrictions on the formation of 1D superconducting fluxoids.

Results

Generation of a half superconducting vortex ring via an unquantized magnetic flux

In a thick slab of a superconductor under the influence of magnetic fields, the current and field distributions are depicted in Fig. 1a. The green arrows represent the Meissner current (JM) opposing the external field (Bext), whereas the red arrows illustrate the shielding current (Js) induced by the presence of superconducting vortices carrying a magnetic flux quantum (Φ0). On the other hand, in thinner geometries where the thickness is less than the magnetic penetration depth (t < λ), vortices are absent. However, in such systems, the magnetic flux remains confined within the film but is unquantized, characterized by a flux value of ΦB that is less than the magnetic flux quantum Φ0, as depicted in Fig. 1b. In a thick slab, it is possible to create a half vortex ring by employing a local magnetic dipole field, as shown in Fig. 1c. Notably, this half vortex ring corresponds to a superconducting vortex‒antivortex pair (SVAP) connected through a quantized magnetic flux (Φ0), with the associated shielding current (Js). The line tension of the half vortex ring can be conceptualized as a one-dimensional spring exhibiting a linear potential. However, the situation changes in thin slabs. Consequently, the question arises as to whether SVAPs can still form with an unquantized magnetic flux (ΦB) and exhibit long-range interactions, as depicted in Fig. 1d. Here, we demonstrate the formation of SVAPs with unquantized magnetic flux in ultrathin films, suggesting that superconducting vortices with arbitrary flux values can form regardless of the system’s dimensions.

Fig. 1: Creation of a half vortex ring via a local dipole field.
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a, b Fluxoid quantization of thick and thin slabs. The thick slab accommodates the magnetic flux quantum (Φ0). The thin slab experiences field penetration, but no vortex exists. c, d Creation of a half vortex ring by a local dipole field of an MFM tip in thick slabs and thin slabs. Unquantized magnetic flux can be trapped in the thin slab geometry.

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Creation of SVAPs via the local stray magnetic field of an MFM tip

To investigate the properties of SVAPs, we employed a custom-built magnetic force microscope equipped with a 2–2–9-T vector magnet, as previously described20. All the experiments used a commercial MFM tip (PPP-MFMR from NANOSENSORS). The magnetic contrast in MFM images is expressed by the frequency shift ∆f of the MFM cantilever, which is directly linked to the force gradient ∂F/∂z with the relation ∂F/∂z = −2k∆f/f0, where k and f0 are the cantilever’s spring constant and the bare resonance frequency of the MFM cantilever, respectively. The Nb film under study was deposited on a Si substrate via electron beam deposition and possesses the following characteristics: a thickness (t) of 300 nm, a superconducting transition temperature (Tc) of 8.8 K, a coherence length (ξ) of approximately 15.9 nm and a magnetic penetration depth (λ) of approximately 110 nm at 4.2 K. The values of ξ and λ were determined from the upper critical field (Hc2) (see Supplementary Note 1 for detailed methodology) and single-parameter simultaneous fits of the MFM signal21, respectively. Thinner Nb films with thicknesses ranging from 30 to 100 nm were deposited on α-Al2O3 substrates via a DC magnetron sputtering system. These thinner films exhibited superconducting transition temperatures (Tc) ranging from 8.1 to 8.7 K. The magnetic penetration depths (λ) of these thinner films were estimated via the Meissner curve comparative method with MFM22 and varied from approximately 145 nm to 90 nm (see Supplementary Note 1 for further details).

The presence of a local magnetic field induced by an MFM probe tip in close proximity to a superconducting film can cause a half-fluxoid ring to form, as illustrated schematically in Fig. 2a. This half-fluxoid ring corresponds to a superconducting vortex‒antivortex pair (SVAP) connected by a magnetic flux. Notably, in the absence of pinning, the SVAPs can annihilate because of the attractive line tension between the two ends. However, in real samples, the presence of a pinning landscape prevents their annihilation. The procedure for generating an SVAP via an MFM tip is as follows: (1) Above the critical temperature (Tc), the in-plane magnetized tip is positioned above the Nb film, generating a dipole magnetic field. (2) The sample is then cooled below Tc to enter the superconducting state (see Supplementary Movie 1). Upon cooling, some of the U-shaped magnetic flux lines (red lines) originating from the tip become trapped within the superconducting medium, leading to the formation of SVAPs. The remaining flux lines (green lines) that penetrate through the back of the sample give rise to tip-induced isolated vortices (TIVs) with a cylindrical shape. In addition to these tip-induced vortices, there is an uncompensated weak out-of-plane magnetic field (blue lines) arising from the stray field of our MFM system, which leads to the presence of isolated antivortices. Note that the principal origin of the system’s stray field stems from the magnetization of the MFM tip. Before we commence the MFM experiments at low temperatures, the tip is magnetized using a superconducting magnet, subjecting it to a magnetic field of approximately 1 Tesla to increase the signal-to-noise ratio. As a consequence of magnetizing the MFM tip, a stray magnetic field permeates the entire MFM system, typically manifesting as a few Gauss. This stray field influences the number of superconducting vortices under investigation.

Fig. 2: Creation of a superconducting vortex‒antivortex pair via MFM.
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a Schematics illustrating the creation of superconducting vortex‒antivortex pairs (SVAPs) via an in-plane magnetized MFM tip. SVAPs (red dots) and trapped interstitial vortices (TIVs, green dots) are generated by the tip, whereas regular vortices (blue dots) are formed by the stray field of the system. b Illustration of the SVAP configuration created by the MFM tip. Antivortices (white dots) are induced by both the external stray field and the tip field, whereas SVAPs (red dots) and TIVs (dark green dots) represent confined vortices. The yellow arrow indicates the position of the MFM tip during the SVAP creation process. c, d MFM images obtained from two separate experiments with the same conditions and a shifted scan frame. The yellow arrow indicates the position of the MFM tip, which remains at a fixed tip‒sample distance of 600 nm during SVAP formation. The bright dots in the MFM image correspond to antivortices, whereas the dark dots represent vortices.

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An in-plane magnetized tip, polarized in the +Hx direction by the superconducting vector magnet, was utilized to create SVAPs and TIVs, as depicted schematically in Fig. 2b. Owing to the limited scan size of our microscope scanner, we combined two scans to cover the region of interest, as shown in Fig. 2b. The corresponding experimental data for the blue box and the red box in Fig. 2b are presented in Fig. 2c, d, respectively. Both images were obtained at 4.2 K via an in-plane magnetized tip20. Initially, our focus is on the vortices (dark dots) since distinguishing between the antivortices (bright dots) induced by the tip field (red and green lines) and the stray field (blue lines) is challenging at this stage owing to their identical magnetic polarities. However, subsequently, the antivortices of SVAPs were identified and distinguished on the basis of their thermal behavior. As the SVAP vortex and antivortex are confined, the interaction force between them is expected to differ from that between the TIVs.

The first experimental evidence of a qualitative distinction between SVAPs and TIVs is the observation of a significant spatial gap (indicated by the black dotted box) between the SVAPs (enclosed by the red circle) and the TIVs (enclosed by the green box), as illustrated in Fig. 2d. In the normal state, prior to the medium transitioning into the superconducting phase, the spatial density of magnetic flux lines decreases with increasing distance from the MFM tip location. Note that the inhomogeneous spatial density of superconducting vortices is attributed to the superposition of the inhomogeneous magnetic field from the MFM tip and the system’s homogeneous stray field due to the differential interaction between the SVAP and the stray-field-induced vortices. Upon the sample’s superconducting transition, the ends of the SVAPs start moving toward their corresponding counterparts until they encounter strong pinning centers that counterbalance the attractive force from the other end of the SVAP. In contrast, the TIVs remain in their initial positions. Consequently, a spatial gap emerges, indicating the mobility of SVAPs and the existence of long-range mutual attraction between the constituent ends.

Presence of long-range interactions in the SVAP

To examine the nature of the interaction between the ends of the SVAPs, we applied heat pulses to the sample, allowing the SVAPs to overcome local pinning potentials23. As the thermal energy increased, one end of the SVAP approached the other end and settled at a stronger pinning potential, reducing the distance between them. This process was observed through sequential images in Fig. 3a–d (corresponding to the scan region in Fig. 2c). The gradual shrinking and eventual annihilation of the SVAPs indicated the presence of an attractive interaction between the ends. Remarkably, this interaction occurred over a scale of several micrometers, which was significantly greater than the magnetic penetration depth (λ ≈ 110 nm) of the Nb film. The conventional isolated vortex‒vortex interaction range of λ alone cannot account for such long-range interactions24. Therefore, the confinement of the SVAPs must be responsible for this observed long-range interaction.

Fig. 3: Thermally assisted depinning of SVAPs.
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The thermal evolution of the vortices in Fig. 1c and d are shown in a–d and e–h, respectively. After a thermal pulse with an amplitude of Tset, as denoted in the image, is applied, MFM images are obtained at 4.2 K. The inset curve in d provides a detailed time scale of the applied thermal pulse. The annihilation of each vortex‒antivortex pair is traced by the box‒circle link with a dotted colored line. i Schematic view of the creation of isolated vortices via out-of-plane magnetization. j‒l Thermal evolution of isolated vortices created according to the method described in i. The images demonstrate the isotropic behavior of the vortices during the thermal process. All MFM images were taken at 4.2 K with a tip-to-sample distance of 600 nm.

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Additional evidence for the confinement of the SVAPs was obtained through a comparative experiment between the TIVs and the SVAPs in Fig. 3e–h (corresponding to the scanned region in Fig. 2d). Unlike the SVAPs, the TIVs (green box) remained static even after multiple consecutive heat pulses close to the critical temperature (Tc), indicating the absence of long-range interactions in the TIVs. The distinct thermal behavior of the SVAPs and TIVs further supports the presence of long-range attractive interaction, specifically between the ends of an SVAP.

To gain further insights into the interaction potential of an SVAP, we investigated the thermal behavior of isolated vortices created via an out-of-plane magnetized tip generated by a vector magnet, as shown in Fig. 3i–l. Initially densely packed, these isolated vortices gradually spread in a spatially isotropic manner with increasing heat pulses, as depicted in Fig. 3j–l. This behavior strongly contrasts with the confined vortices of the SVAPs, highlighting the distinct nature of the interaction between the ends of the SVAPs (see Supplementary Movie 2 for further details).

Direct evidence of long-range interaction in the manipulation of a single SVAP

To directly investigate the presence of a long-range potential, we present the manipulation of a single SVAP by an MFM tip and show its pair annihilation via thermal annealing (heat pulses). The starting point is prepared by leaving behind only one end of an SVAP after moving the rest of the SVAPs by the tip stray field out of the scan frame, as shown in Fig. 4a. It is then continuously manipulated up to 17 µm, as shown in Fig. 4b, c. The manipulation process was carried out by scanning a single vortex with a tip‒sample distance of 150 nm to enhance the interaction between the vortex and tip25,26,27,28. To manipulate a single vortex effectively, manipulation was performed at 7 K to decrease the local pinning force, and then, the manipulation results were verified by imaging the single vortex at 4 K. After single vortex manipulation, we applied the same thermal annealing treatment with heat pulses as those described previously. The same gradual shrinking of the SVAP and its eventual annihilation are observed, as shown in Fig. 4d–f, clearly demonstrating that the SVAP remains intact throughout the entire MFM manipulation and thermal treatment, all the way until its annihilation. The fact that the counterpart (marked as the white circle) moves toward the manipulated vortex (marked as the black circle) along the manipulated direction, as shown in Fig. 4d, e, is evidence of a mutually attractive force.

Fig. 4: Pair annihilation of an elongated single SVAP.
figure 4

a–c Manipulation process of one end of an SVAP by an MFM tip. After moving the scan frame step by step via piezo walkers, a final vortex‒antivortex distance of 17 µm is achieved. d–f Thermally assisted depinning of both the vortex and the antivortex and pair annihilation as a result of successive heat pulses. Note that a confining interaction still exists between the antivortex and its previously manipulated counterpart because of the movement of the antivortex, as indicated by the white arrow, which is clearly visible in d and e. All MFM images were taken at 4.2 K with a tip‒sample distance of 600 nm.

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The large distance over which the interaction persists, reaching up to 17 microns in our experiments, is particularly notable considering that it is more than fifty times greater than the thickness of the Nb sample. This remarkable resiliency underscores the unique and intriguing nature of the SVAP and its confinement potential, which enables long-range interactions that cannot be explained by conventional vortex‒antivortex interactions within superconducting systems. These findings contribute to our understanding of the fundamental properties of SVAPs and shed light on the underlying physics of confined superconducting vortex structures. The ability to manipulate and control individual SVAPs opens new possibilities for exploring their behavior and harnessing their unique properties in future applications, such as in the design of novel superconducting quantum technologies.

Confinement of the SVAP through an unquantized magnetic flux in thin films

To investigate the confinement effect of the SVAP connected by an unquantized flux, we examined Nb films with thicknesses smaller than the magnetic penetration depth (λ). Specifically, we prepared films with thicknesses of 100, 50, and 30 nm, with corresponding λ values of 90, 130, and 145 nm, respectively. Among these samples, our focus was on the 30-nm film, which had a t − λ ratio of 0.26. This ratio suggests that the 30 nm film does not exhibit a flux quantum along the in-plane direction. Using the same experimental conditions used before, we created SVAPs in these thin films and performed temperature dependence experiments. Surprisingly, we observed pair annihilation not only in the 100 nm and 50 nm films but also in the 30 nm film, as shown in Fig. 5. The presence of long-range interactions even in the 30 nm film, where t ≪ λ, is intriguing. This finding indicates that the unquantized magnetic flux14,15,16,17,18,19 associated with the SVAP can mediate long-range forces between superconducting vortices. To substantiate our findings of nonquantized vortices and to closely examine the current distribution within the 30 nm Nb film, we carried out comprehensive time-dependent Ginzburg–Landau simulations. These simulations revealed a clear pattern: with decreasing film thickness, there was a proportional reduction in magnetic flux, accompanied by a commensurate increase in current flux. We also examined the contrast differences in the MFM images by analyzing the line profiles of each vortex in the Nb films with thicknesses of 300 nm (d > λ) and 30 nm (d < λ). Unlike the 300 nm film, the 30 nm film shows an intensity difference between isolated and paired vortices, suggesting the formation of unquantized vortices. We refer to the supplementary material for an in-depth analysis. These trends demonstrate the direct relationship between the film thickness and the manifestation of the noninteger quantized flux. This result has significant implications for various applications, including superconducting nanowires, low-dimensional Josephson junctions, and monolayer superconductors.

Fig. 5: Thermal evolution of the SVAPs in thinner Nb films.
figure 5

Thermally assisted depinning and pair annihilation of SVAPs in Nb films of a–d 100 nm, e–h 50 nm, and i–l 30 nm. Note that the superconducting transition temperatures (Tc) and magnetic penetration depths (λ) are different in various Nb films, ranging from ≈ 8.7 K to 8.3 K and from ≈ 90 nm to 145 nm, respectively. The annihilation of each vortex‒antivortex pair is traced by the box‒circle link with a dotted colored line.

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Moreover, the 1D nature of the SVAP provides a robust platform for exploring the possibilities of braiding operations via unquantized magnetic flux in topological superconducting heterostructure devices. These devices are at the forefront of current research in condensed matter physics and topological quantum computing. The ability to confine and manipulate the unquantized flux within SVAPs opens new avenues for studying topological properties and realizing exotic quantum phenomena. Our findings highlight the unique confinement effect of SVAPs connected by unquantized magnetic flux in thin films.

Conclusion

Our study demonstrated the creation and manipulation of a vortex‒antivortex pair connected by an unquantized magnetic flux in ultrathin superconducting Nb films via vector-field MFM. The observed long-range interaction through the unquantized flux provides evidence for the universal formation of such flux regardless of the system size. This finding contributes to our understanding of superconducting magnetic properties at the nanoscale, where the nonquantization of magnetic flux plays a role in the superconducting properties.

Our results offer an experimental platform for directly investigating a linear potential and its phase transition in a simple 1D fluxoid model system. This has implications for studying the fundamental properties of superconductivity and exploring novel quantum phenomena in low-dimensional systems. Additionally, the long-range interaction mediated by the unquantized magnetic flux opens up possibilities for realizing non-Abelian statistics through the manipulation and annihilation of SVAPs. This has potential implications for the development of topological quantum devices and the field of topological quantum computing.