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Experimental Study of the Magnetic Field Distribution and Shape of Domains Near the Surface of a Type-I Superconductor in the Intermediate State

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Abstract

The importance of accounting for the inhomogeneity of the magnetic field distribution and roundness of domain walls near the surface of type-I superconductors in the intermediate state for forming the equilibrium flux structure was predicted by Landau eight decades ago. Further studies confirmed this prediction and extended it to all equilibrium properties of this state. Here we report on direct depth-resolved measurements of the field distribution and shape of domains near the surface of high-purity type-I (indium) films in a perpendicular field using Low-Energy Muon Spin Rotation spectroscopy. We find that at low applied fields (in about half of the field range of the intermediate state) the field distribution and domains’ shape agrees with that proposed by Tinkham. However, for high fields our data suggest that reality differs from theoretical expectations. In particular, the width of the superconducting laminae can expand near the surface leading to formation of a maximum in the static magnetic field in the current-free space outside the sample. A possible interpretation of these experimental results is discussed.

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Notes

  1. The healing length Lh is an effective width of a near-surface spatial layer over which the strongly non-uniform induction inside the sample relaxes to its uniform state away from it. The term “effective” implies, that within this layer the field distribution in both in- and out-of-plain projections remains the same as the induction distribution inside the sample; and at distances larger than Lh the field is undisturbed and equal to the applied field H0.

  2. In spite of the large value of the mean free path, an ultimate test of purity of a superconducting sample is reproducibility of its magnetization curve measured with the sample cooled in zero and in non-zero field (see [12] for more details).

  3. This can be understood as follows. Thermodynamics of the IS is based on condition of minimization of the sample free energy [6, 8]. Bending of the field lines makes positive contribution in the free energy and therefore it cannot be strong due to requirement of thermodynamics (see [12] for more details).

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Acknowledgments

The muon measurements were performed at the Swiss Muon Source Sμ S, Paul Scherrer Institute, Villigen, Switzerland.

Funding

This work was supported by the National Science Foundation (Grant No. DMR 0904157), by the Research Foundation – Flanders (FWO, Belgium) and by the Flemish Concerted Research Action (BOF KU Leuven, GOA/14/007) research program. V.K. received support from the sabbatical fund of the Tulsa Community College.

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Authors and Affiliations

  1. Tulsa Community College, Tulsa, OK, 74119, USA

    V. Kozhevnikov

  2. Paul Scherrer Institut, 5232, Villigen, Switzerland

    A. Suter & T. Prokscha

  3. Solid State Physics and Magnetism Section, KU Leuven, 3001, Leuven, Belgium

    C. Van Haesendonck

Authors
  1. V. Kozhevnikov
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  2. A. Suter
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  3. T. Prokscha
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  4. C. Van Haesendonck
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Correspondence to V. Kozhevnikov.

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Appendix

Appendix

The muon stopping distribution for a given material is calculated with the Monte Carlo code TRIM.SP [27, 28]. Stopping profiles for various muon implantation energies for indium are shown in Fig.13. Figure 14 depicts the muon stopping distribution in solid N2 for a muon implantation energy of E = 14.3 keV.

Fig. 13
figure 13

Depth profile of the stopping distances of muons of different energies in indium

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Fig. 14
figure 14

Muon stopping distribution in solid nitrogen for an implantation energy E = 14.3 keV

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The average stopping distance (implantation depth) \(\overline {x}(E)\) at given muon energy E is calculated as

$$ \overline{x}(E)={\int}_{0}^{\infty} xf(x,E)dx, $$

where x is the stopping distance and f(x,E) is the stopping distances distribution shown in Figs. 13 and 14.

The out-of-plane coordinate z (shown in Fig. 1b) for the data obtain with muons implanted into the sample equals \(-\overline {x}(E)\); and z for the data measured outside is \(z={\Delta } - \overline {x}(E)\), where Δ is the thickness of the nitrogen overlayer. Details about the depth profiles of the low-energy muons are available in [28, 42].

Figures 15 and 16 show representative data for depolarization rates σTB and σLR, respectively, recorded with IN-A sample without (panels (a)) and with (panels (b)) N2 overlayes.

Fig. 15
figure 15

Depolarization rates σTB recorded by the Top and Bottom counters inside (a) and outside (b) of IN-A sample at distances from the surface as indicated. Vertical dashed line marks the critical field Hci

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Fig. 16
figure 16

Depolarization rates σLR recorded by the Left and Right counters inside (a) and outside (b) of IN-A sample at distances from the surface as indicated. Vertical dashed line marks the critical field Hci

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Kozhevnikov, V., Suter, A., Prokscha, T. et al. Experimental Study of the Magnetic Field Distribution and Shape of Domains Near the Surface of a Type-I Superconductor in the Intermediate State. J Supercond Nov Magn 33, 3361–3376 (2020). https://doi.org/10.1007/s10948-020-05576-1

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