4. Other versions of near-surface properties

The rounded corners and the field inhomogeneity near the surface yield an excess energy of the system favoring to a fine laminar structure (directly proportional to a period D of the onedimensional laminar lattice). On the other hand, there is an excess energy associated with the surface tension at the S/N interface in the bulk, which favors to a coarse structure (reversely proportional to D). Optimizing sum of these two energy contributions in the specimen free

Figure 5. Cross-sectional views of the S and N laminae and of the field distribution (in A, C, and D) near the surface(s) of a type-I plane-parallel slab in perpendicular magnetic field. (A) Landau [18], (B) Landau [19], (C) Tinkham [3], (D) Abrikosov [7], and (E) Marchenko [20]. Letters s and n designate superconducting and normal phases, respectively; v designates the free space. In (C) v also designates a void in the static field outside the sample. See text for other notations.

<sup>D</sup><sup>2</sup> <sup>¼</sup> <sup>δ</sup><sup>d</sup>

where δ is a wall-energy parameter characterizing the S/N surface tension and associated with the coherence length [2, 3] and f <sup>L</sup>ð Þh is the Landau spacing function determined by the shape of the corners and the near-surface field inhomogeneity and h ¼ H=Hc. f <sup>L</sup>ð Þh was calculated numerically in [21], and an analytical form of this function was obtained in [22] (see also [2]). Soon thereafter Landau abandoned this model, admitting that the proposed flux structure does not correspond to a minimum of the free energy [23]. So, he suggested another so-called

<sup>f</sup> <sup>L</sup>ð Þ<sup>h</sup> , (12)

energy, Landau calculated the period:

96 Superfluids and Superconductors

One of the important consequences of the Landau models is demonstration of significance of the near-surface field distribution and domain shape (FDDS) for forming and stabilizing the flux structure of the IS. On that reason it is worth to briefly overview other available scenarios for FDDS.

There are two simplified modifications of the original (non-branching) Landau's version of FDDS.

Tinkham [3] proposed that the dominant contribution in the surface-related properties comes from field inhomogeneities outside the sample extending over a "healing length" Lh as shown in Figure 5C. Lh <sup>¼</sup> <sup>D</sup>�<sup>1</sup> <sup>n</sup> <sup>þ</sup> <sup>D</sup>�<sup>1</sup> s �<sup>1</sup> , where Dn and Ds are the widths of the normal and superconducting laminae, respectively. Correspondingly, Tinkham neglects the roundness of the laminae corners (b and c in Figure 5A). This version meets the limiting cases—D ! 0 when either Ds ! 0 or Dn ! 0—and is consistent with images of the IS flux structure (see, e.g., [13, 24, 25]). Tinkham's FDDS works surprisingly well for the IS [25, 26]; it was also successfully validated for the mixed state in type-II superconductors [27]. Note that all of these are in spite of apparent contradiction of the Tinkham's scenario with basics of magnetostatics, since it allows for existence of voids in the static magnetic field near the sample (e.g., in a region designated by v in Figure 5C).

Abrikosov [7] proposed another simplified version of Landau's FDDS. He assumed that major role is played by the round corners and therefore neglected the field inhomogeneity outside the specimen. However, the latter means that the field near the surface is uniform, and therefore this scenario is inconsistent with images of the IS flux structure. Abrikosov's version of FDDS is shown in Figure 5D, where size of the corners c is the same as Lh in the Tinkham's scenario.

An interesting result for a possible domain shapes was obtained by Marchenko [20]. Like Landau [18], Marchenko used conformal mapping to calculate the domain shape in infinite slab but in a tilted field. He found that in a strongly tilted field width of the S-domains can increase as shown in Figure 5E. We note that in such case, the field lines should leave the N domains converging instead of diverging as in Figure 5A–D, because bending of the lines over sharp corners (marked a in Figure 5E) would take enormous energy [2]. Therefore this scenario also allows for existence of the voids in the field outside the specimen; and moreover, it may lead to appearance of a maximum in the field magnitude in the free space above the N laminae.

To conclude this section on theoretically predicted scenarios for the near-surface properties of the IS, we note that neither of them is consistent simultaneously with the classical magnetostatics and with experimental images of the flux structure. So far no experimental results on FDDS in the IS have been reported. Hence measurements of these properties are open and important (see, e.g., Landau's papers [18, 19, 23]) problem of fundamental superconductivity<sup>3</sup> .

Landau branching model, according to which the field near the surface is uniform and the flux structure can be observed only in a narrow gap inside the specimen provided the gap width is less than some critical value estimated by Landau [19]. It turned out that there is no critical gap and the field is inhomogeneous both inside (in the gap) and outside the specimen. These results unambiguously turned down the branching model. Typical images and diagrams for

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 99

Further progress in imaging the IS structure was reached using Bitter or powder technique and magneto-optics [13]. It was established that the flux pattern in flat plates in perpendicular field consists of irregular corrugated laminae transforming into N (S) fractional laminae and tubes near the low (high) end of the IS field range. A numerous variety of different flux patterns were

A detailed study of the IS flux pattern was conducted by Faber with tin and high-purity aluminum parallel-plane plate specimens [31]. It was found that at high reduced temperature ( ≈ 0:9Tc) in a broad field range, the structure is pass-independent (i.e., reproducible at increasing at decreasing fields) and consists of corrugated laminae. Therefore Faber concluded that the laminar flux structure is equilibrium structure of the IS. Typical images of the passindependent flux pattern in perpendicular field from the Faber's work are shown in Figure 7. A breakthrough in forming regular and controllable IS flux structure was achieved by Sharvin [32]. Applying the field tilted with respect to a single-crystal Sn specimen, Sharvin obtained a regular linear laminar structure as shown in Figure 8. Measuring period of the structure and using Landau's formula, Eq. (12), corrected to account the field inclination, Sharvin calculated the wall-energy parameter δ. Similar experiments and calculations Sharvin performed for In [32].

The aforementioned difference between the critical field HI observed in resistive measurements and theoretically expected value for this field 1ð Þ � η Hc was investigated by Desirant and Shoenberg in a detailed study of magnetization of long cylindrical specimens of different radii in transverse field [33]. Apart from confirmation of the resistive results, Desirant and

Figure 7. Typical images of pass-independent flux structures of the IS obtained with aluminum parallel-plane plate specimen in perpendicular field at temperature 0.92Tc and the field 0.38Hc (a) and 0.53Hc (b). Dark areas are

superconducting. After Faber [31].

the field distribution obtained by Meshkovsky and Shalnikov are available in [1].

reported when samples are in nonequilibrium state [12].
