3. Landau laminar models

After Peierls [11] this inhomogeneous state in type-I superconductors is named the intermediate state. Properties of the IS were (and in some extent still are) one of the longest-standing challenges of physics of superconductivity. Below we will expose the main theoretical ideas and key experimental achievements addressing these properties. Comprehensive reviews of the experimental and theoretical works on the IS published before 1970 are available in [1, 12, 13]; for references to more recent publications, we recommend papers by Brandt and Das [14] and Clem

Figure 3. Cross section of the cylindrical sample in case if superconducting phase (S, colored in gray) is gradually

The first successful theoretical model of the IS magnetic properties was developed in 1936 independently by Peierls [11] and London [16]. In this model properties of ellipsoidal samples are considered in an averaged limit, in which the nonuniform induction B is replaced by average B. This allowed to use Eq. (6) with demagnetizing factor η calculated for uniform ellipsoid. However Eq. (6) has two unknowns, B and Hi, both of which are needed to calculate the specimen magnetic moment. Basing on a paradigm that the N phase is unstable at Hi < Hc, Peierls and London postulated that inside the specimen in the IS (i.e., at 1ð Þ � η Hc < H < Hc),

Eqs. (2), (6), and (9) constitute a complete system of equations. Solving it one finds B, Hi, and M:

8 ><

>:

8 ><

>:

B ¼ 0

Hi ¼ Hc

Hi ¼ H=ð Þ 1 � η M ¼ HV=4πð Þ 1 � η

B ¼ ð Þ H � Hcð Þ 1 � η =η

M ¼ V Hð Þ � Hc =4πη

H ≤ Hcð Þ 1 � η

Hcð Þ 1 � η ≤ H ≤ Hc

Hi ¼ Hc: (9)

:

(10)

(11)

et al. [15].

94 Superfluids and Superconductors

2. Model of Peierls and London

replaced by the normal (N, colored in blue) phase filled by the field.

Magnetic flux structure of the IS was for the first time considered by Landau [18] for an infinite parallel-plane plate (slab) in perpendicular field, i.e., for the sample-field configuration shown in Figure 1c. In such a specimen the surface current (and hence the Meissner state) is absent because <sup>B</sup> <sup>¼</sup> <sup>H</sup>, and therefore <sup>g</sup> <sup>¼</sup> <sup>H</sup> � <sup>B</sup> <sup>c</sup>=4<sup>π</sup> <sup>¼</sup> 0 at any <sup>H</sup> from zero to Hcr. Due to that the IS starts at H right above zero, no matter how small is this field. Magnetic moment of this specimen (Landau considered thick plate) is M Hð Þ¼ �ð Þ Hc þ H V=4π; graphs for B and M are shown by the green lines in Figure 4a and b.

Assuming that (i) the plate is split for regularly structured S and N laminae and (ii) the boundary of a cross section of the S laminae is the line of induction B with magnitude Hc at the S/N interface, Landau calculated shape of rounded corners of the S laminae near the sample surface. Landau's scenario for cross section of the S-lamina near the surface is shown in Figure 5a. To meet the second assumption, Landau splits a central field line for two branches (oba and ocd in Figure 5a) making a sharp (90�) turn at the splitting point (o). Hence, in this scenario the field fills all space outside the specimen, as it is supposed to be the case in magnetostatics. On the other hand, splitting the field line challenges the magnetostatics rules [4], and the sharp turn of the line may cost the system too much energy [2, 19].

Figure 4. Peierls and London model. Average magnetic induction (a) and magnetic moment (b) for specimens with demagnetizing factor η ¼ 1 (infinite slab in perpendicular field, green line), η ¼ 1=2 (long cylinder in perpendicular field, blue line) and η ¼ 0 (long cylinder in parallel field, red line). NS designates the normal state (black line).

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.

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 energy, Landau calculated the period:

$$D^2 = \frac{\delta d}{f\_L(h)'} \tag{12}$$

branching model [19, 23] (see also [1, 8]), in which N laminae near the surface split for many thin branches as shown in Figure 5B, so that the flux emerges from the sample uniformly over the whole surface. However, this branching model was disproved by Meshkovskii and

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

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

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

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

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

, where Dn and Ds are the widths of the normal and

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 97

Shalnikov when they for the first time directly measured flux structure of the IS [24].

4. Other versions of near-surface properties

<sup>n</sup> <sup>þ</sup> <sup>D</sup>�<sup>1</sup> s �<sup>1</sup>

for FDDS.

in Figure 5C. Lh <sup>¼</sup> <sup>D</sup>�<sup>1</sup>

designated by v in Figure 5C).

FDDS.

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 branching model [19, 23] (see also [1, 8]), in which N laminae near the surface split for many thin branches as shown in Figure 5B, so that the flux emerges from the sample uniformly over the whole surface. However, this branching model was disproved by Meshkovskii and Shalnikov when they for the first time directly measured flux structure of the IS [24].
