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Observation of a gel of quantum vortices in a superconductor at very low magnetic fields

José Benito Llorens, Lior Embon, Alexandre Correa, Jesús David González, Edwin Herrera, Isabel Guillamón, Roberto F. Luccas, Jon Azpeitia, Federico J. Mompeán, Mar García-Hernández, Carmen Munuera, Jazmín Aragón Sánchez, Yanina Fasano, Milorad V. Milošević, Hermann Suderow, and Yonathan Anahory
Phys. Rev. Research 2, 013329 – Published 17 March 2020
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  • INTRODUCTION
  • EXPERIMENTAL
  • RESULTS
  • DISCUSSION
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    A gel consists of a network of particles or molecules formed for example using the sol-gel process, by which a solution transforms into a porous solid. Particles or molecules in a gel are mainly organized on a scaffold that makes up a porous system. Quantized vortices in type-II superconductors mostly form spatially homogeneous ordered or amorphous solids. Here we present high-resolution imaging of the vortex lattice displaying dense vortex clusters separated by sparse or entirely vortex-free regions in β−Bi2Pd superconductor. We find that the intervortex distance diverges upon decreasing the magnetic field and that vortex lattice images follow a multifractal behavior. These properties, characteristic of gels, establish the presence of a novel vortex distribution, distinctly different from the well-studied disordered and glassy phases observed in high-temperature and conventional superconductors. The observed behavior is caused by a scaffold of one-dimensional structural defects with enhanced stress close to the defects. The vortex gel might often occur in type-II superconductors at low magnetic fields. Such vortex distributions should allow to considerably simplify control over vortex positions and manipulation of quantum vortex states.

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    • Received 8 October 2019
    • Revised 9 January 2020
    • Accepted 22 January 2020
    • Corrected 2 September 2020

    DOI:https://doi.org/10.1103/PhysRevResearch.2.013329

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Phase transitions by orderSuperconductivityVortex latticesVortices in superconductors
    1. Physical Systems
    SQUIDType-II superconductors
    1. Techniques
    Landau-Ginzburg theoryMagnetic force microscopy
    Condensed Matter, Materials & Applied Physics

    Corrections

    2 September 2020

    Correction: The omission of a statement in the Acknowledgment section has been fixed.

    Authors & Affiliations

    José Benito Llorens1, Lior Embon2, Alexandre Correa3, Jesús David González4,5, Edwin Herrera1,6, Isabel Guillamón1,7, Roberto F. Luccas3,8, Jon Azpeitia3, Federico J. Mompeán3,7, Mar García-Hernández3,7, Carmen Munuera3,7, Jazmín Aragón Sánchez9, Yanina Fasano9, Milorad V. Milošević5, Hermann Suderow1,7,*, and Yonathan Anahory10,†

    • 1Laboratorio de Bajas Temperaturas y Altos Campos Magnéticos, Departamento de Física de la Materia Condensada, Instituto Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
    • 2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
    • 3Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Científicas (ICMM-CSIC), Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain
    • 4Facultad de ingeniería, Universidad del Magdalena, Santa Marta, Colombia
    • 5Theory of Functional Materials, Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
    • 6Facultad de Ingeniería y Ciencias Básicas, Universidad Central, Bogotá 110311, Colombia
    • 7Unidad Asociada de Bajas Temperaturas y Altos Campos Magnéticos, UAM, CSIC, Cantoblanco, E-28049 Madrid, Spain
    • 8Instituto de Física Rosario, CONICET-UNR, Bv. 27 de Febrero 210bis, S2000EZP Rosario, Santa Fé, Argentina
    • 9Centro Atómico Bariloche and Instituto Balseiro, CNEA and Universidad de Cuyo, Av. E. Bustillo 9500, R8402AGP, S. C. Bariloche, RN, Argentina
    • 10The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

    • *Corresponding author: hermann.suderow@uam.es
    • †Corresponding author: yonathan.anahory@mail.huji.ac.il

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    Vol. 2, Iss. 1 — March - May 2020

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    Images

    • Figure 1
      Figure 1

      (a) shows a typical topographic AFM image, with a line profile that show a 15 nm step in the inset (green line). (b) MFM image showing the vortex lattice at 2 K and 300 G and its Fourier transform (inset). The vortex lattice is clearly hexagonal over the whole area. The color code represents the shift of the resonance frequency of the cantilever where white denotes the normal phase and black the superconducting phase. Dark lines and other darker regions in the MFM image are the result of the nonmagnetic tip-sample interaction. The scale bar in both images is of 1.4μm. (c) Optical image of the SOT at a few tens of μm from the β−Bi2Pd surface. The SOT reflection on the surface is visible on the bottom part. (d)–(i) 20×20μm2 SOT images that represent the out-of-plane field B(x,y) obtained after field cooling the sample in magnetic fields of 2 (d), 3 (e), 6 (f), 12.5 (g), 25 (h), and 50 (i) G. The vortex gel is formed below about 20 Oe. The color scale spans 13 [(d)–(f)], 32 (g), and 27 G [(h)–(i)]. The scale bar in (d) is for all SOT images and is 4μm long. The vortex profiles along the white line in (d) are shown in Fig. 7.

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    • Figure 2
      Figure 2

      Simulation of vortex configurations close to a linear defect. We show vortices as bright spots. To obtain vortex positions, we introduce a critical temperature variation as a function of the position from the defect Δx, taking Tc(Δx)=Tc,defect+(Tc,bulk−Tc,defect)(Δx)2LD2, where LD is the lateral size of the defect. Note that we allow for a slow decay when leaving the linear defect. The position of the defect is marked by a red dashed line. (a)–(c) show a field of view of 10×20μm2, for applied magnetic fields of 2, 5, and 15 G, respectively. We plot the magnetic induction at a height of 300 nm above the sample surface. This gives a span of ≈15 (a), 16 (b), and 18 G (c) in each image. Inset in (a) shows comparative magnetic profiles of two selected vortices, marked in the main panel by green and magenta lines.

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    • Figure 3
      Figure 3

      (a) Histograms of the intervortex nearest neighbor distances for different magnetic fields normalized by the integrated area at 6, 25, 50, and 600 G. In the insets we show SOT (left inset, image taken at 5 G) and MFM (right inset, image taken at 600 G) images, together with their Delaunay triangulation (white lines). (b) Standard deviation (SD) from histograms in (a), normalized to the intervortex distance a0. Data from SOT and MFM are shown as blue dots. The simulations are shown as orange dots. Black points correspond to the data in the quasicrystalline Bragg glass phase nucleated in pristine cuprate samples and violet points to a disordered lattice in presence of columnar defects. The blue line is 1/μ0H. In the inset we show intervortex distance vs the magnetic field for the same images.

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    • Figure 4
      Figure 4

      We show in the main panel the distribution of fractal dimensions f(α) in a cuprate superconductor (black line) for 5 G and in β−Bi2Pd (rest of colored lines) for different magnetic fields from 5 to 600 G. For the cuprate superconductor, larger images or images at other magnetic fields lead to curves with much smaller dispersion of α, centered at α=2. In the inset we show the generalized dimension Dq as a function of the set of exponents q. Note that the curves strongly vary for low magnetic fields, but remain centered around 2 for high magnetic fields and for the results in the cuprate superconductor. Details on the calculation of multifractal parameters are provided in Appendix pp6.

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    • Figure 5
      Figure 5

      Optical and SEM analysis of fractured surfaces in β−Bi2Pd. (a) Forces exerted during the cleavage process of a single crystal. In the opening mode (mode I), there is tensile stress normal to the plane of the crack. The sliding mode (mode II) describes shear stress parallel to the plane of the crack and perpendicular to the crack front. The tearing mode (mode III) a shear stress parallel to the plane of the crack and parallel to the crack front. (b) Optical picture of a sample after cleaving. We identify twist hackles (yellow dashed lines), linear features that seem step edges along crystalline directions (red arrow) and the debonding path, or the direction where the crack front propagated during fracture (blue arrows) . [(c) and (e)] Magnified areas marked by red rectangles in (b) taken using an optical camera. [(f)–(h)] SEM images. We mark places where the sample forms fully detached layers by green arrows. Images are at the red rectangles shown in (b) marked by the corresponding letters. Scale bars are of 0.2 mm (b), 40 μm in (c)–(e), 100 μm in (f), 50 μm (g), and 5 μm in (h).

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    • Figure 6
      Figure 6

      Multifractal properties of vortex lattice and of images of fractured surfaces in β−Bi2Pd. We show the distribution of fractal dimensions f(α) in the main panel for the vortex lattice image obtained at 50 G (red line), an optical image of the surface (blue line), a scanning electron microscope image of the surface (green line) and an image with random noise (black line). The inset shows the generalized dimension Dq as a function of the set of exponents q, also in red (vortex lattice 50 G), blue (optical image), green (scanning electron microscope image) and black (white random noise). Lateral sizes of the images, shown in the panels at the right (with borders of the same colors as in the main panel), are similar, of about 20μm.

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    • Figure 7
      Figure 7

      Vortex field profiles Bz(r) along vortices far from a defect (blue points) and close to a defect along the defect (red points) and along the direction perpendicular to the defect (black points). Profiles are taken from Fig. 1 along the white lines. The fit to the vortex profile leaving the magnetic flux ϕ as a free parameter is shown in blue. Vortices far from defects provide an excellent fit with 1.06Φ0 where Φ0 is the flux quantum. However, close to the defects the fit does not provide a good account of the experimental data. By contrast, a fit of the same vortex profiles using the penetration depth λ as a free parameter leads to orange lines, which much better account for the experimental data in all situations. We find λ=172 nm (which is of order of ∼λ measured using Hall probes, as discussed in the main article) for vortices far from a defect and λ=360 nm close to the average value found for vortices at a defect.

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    • Figure 8
      Figure 8

      Behavior of the hexagonal vortex lattice as a function of temperature measured with MFM. In (a)–(c), the images are taken at 2.75, 3.75, and 4.5 K, respectively at 300 G. The color scale represents the observed frequency shift. Scale bar is 1μm. Blue lines are the Delaunay triangulation of vortex positions. Blue and red points in (a) highlight vortices with seven and five nearest neighbors respectively. The dark arrow at the bottom highlights the position of the vertical line discussed in the text.

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    • Figure 9
      Figure 9

      In (a), an example with vortex positions indicating how boxes decrease its size. In (b), we show ln(Aεq) versus ln(ε) curves in a set from q=−5 to 5. The curves have a clear slope for large ln(ε) and become flat for low ln(ε). The change of slope is due to the density of points. We only take in account the points before the change of slope. This gives us αq. In (c), we show ln(τεq) vs ln(ε) curves at the same set of q as in (b). The same behavior with the size arises and we treat it similarly. The slope of this curve gives us τq.

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    • Figure 10
      Figure 10

      Voronoi tessellation of the vortex lattice. [(a) and (b)] Vortex lattice images and the Voronoi tessellation (white lines). The position of each vortex is shown by a point. (c) percentage of cells with the number of sides n for different magnetic fields (shown in the legend). Dashed lines are a guide to the eye. (d) Average over all areas occupied by cells having sides n for images at different magnetic fields (shown in the legend).

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