7. Concluding remarks

The first term in Eq. (22) is

106 Superfluids and Superconductors

magnetic moment is

�M<sup>⊥</sup> <sup>¼</sup> <sup>∂</sup>F~<sup>M</sup> ∂H<sup>⊥</sup>

yields

�M<sup>∥</sup> <sup>¼</sup> <sup>∂</sup>F~<sup>M</sup> ∂H<sup>∥</sup>

> ¼ V Hc

and M ¼ M<sup>∥</sup> (Eq. (24)) converts to Eq. (1):

∂F~<sup>M</sup> ∂H<sup>∥</sup>

¼ V Hc H2 c 8π

And the perpendicular component of the moment is

¼ � <sup>V</sup> Hc

<sup>∂</sup>~<sup>f</sup> <sup>M</sup> ∂h<sup>⊥</sup> ¼ V Hc � <sup>∂</sup>~<sup>f</sup> <sup>M</sup> ∂h<sup>∥</sup>

2ð Þ b<sup>⊥</sup> � h<sup>⊥</sup>

2H<sup>2</sup> c <sup>8</sup><sup>π</sup> ð Þ <sup>b</sup><sup>⊥</sup> � <sup>h</sup><sup>⊥</sup>

measured in [36]. Here we confine our discussion by limiting cases.

<sup>~</sup><sup>f</sup> <sup>M</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup> � <sup>H</sup><sup>2</sup>

4πMð Þ0 <sup>V</sup> <sup>¼</sup> <sup>V</sup> 4π

the excess magnetization at HI as it is seen, e.g., in Figures 9 and 12.

¼ V Hc � H2 c 8π

Since ∂b⊥=∂h<sup>∥</sup> ¼ h∥=b<sup>⊥</sup> (see Eq. (20)), the final form of the parallel component of the specimen

<sup>¼</sup> VHc 4π

> ∂b<sup>⊥</sup> ∂h<sup>⊥</sup> � 1

All obtained formulas are analyzed in detail in [25, 26], where it is shown that the model correctly describes experimental data. In particular, the coherence length calculated from measured D using Eq. (16) agrees well with that obtained from the magnetic field profile

In parallel field (H<sup>⊥</sup> ¼ 0) the model (Eq. (18)) yields r<sup>n</sup> = 0, meaning that the specimen is in the

<sup>M</sup><sup>∥</sup> <sup>¼</sup> <sup>M</sup> ¼ � <sup>V</sup>

In perpendicular field (H<sup>∥</sup> ¼ 0) one can see that hcrð Þ ¼ hcr<sup>⊥</sup> decreases with decreasing thickness d (Eq. (19)) in accord with the experimental data [25, 26, 33], and the induction Bð Þ ¼ b � Hc in N domains equals to Hc at H ¼ HI ¼ 0 and decreases with increasing H (Eq. (20)), as it was found experimentally in [34]. For magnetization 4πM=V at H ! 0, when r<sup>n</sup> ¼ 0, the model (Eq. (25))

Since B decreases with increasing H, ð Þ ∂B⊥=∂H<sup>⊥</sup> < 0, and therefore the expression in parentheses is greater than unity. This makes 4πMð Þ0 =V greater than Hc, thus explaining appearance of

The infinite slab in perpendicular field represents ellipsoid with η = 1. If the slab is thick (i.e., d ≫ δ), the LMTF model converts to the PL model for specimens with unity demagnetization.

<sup>1</sup> � <sup>h</sup><sup>2</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup> � <sup>H</sup><sup>2</sup>

4π

<sup>1</sup> � <sup>∂</sup>B<sup>⊥</sup> ∂H<sup>⊥</sup> 

Meissner state where the N phase is absent. Then <sup>~</sup><sup>f</sup> <sup>M</sup> (Eq. (17)) converts to Eq. (5):

c 8π

h∥ b⊥ 2ð Þ b<sup>⊥</sup> � h<sup>⊥</sup>

<sup>1</sup> � <sup>h</sup><sup>⊥</sup> b⊥ 

> ¼ V 4π

c 8π þ H2

∂b<sup>⊥</sup> ∂h<sup>∥</sup>

<sup>h</sup><sup>∥</sup> <sup>¼</sup> <sup>V</sup> 4π

<sup>1</sup> � <sup>r</sup><sup>n</sup> ð Þ <sup>1</sup> � <sup>∂</sup>B<sup>⊥</sup>

: (23)

1 � r<sup>n</sup> ð ÞH∥: (24)

B⊥: (25)

∂H<sup>⊥</sup> 

<sup>8</sup><sup>π</sup> , (26)

H: (27)

Hc: (28)

More than three decades starting from the 1930s, the problem of the IS was in the main focus of experimental and theoretical researches on superconductivity. This resulted in significant progress reached in understanding properties of the IS as well as properties of superconducting state as a whole. Excellent reviews of these researches are available in [1, 12]. However some puzzles in the IS properties remained open until their possible explanations emerged in studies of recent years. In this chapter we mostly focused at results of these studies.

In particular, we discussed a recently developed phenomenological model of the IS composed for infinite slabs in arbitrary tilted magnetic field. Naturally, this model is not and cannot be free of disadvantages. One of them can be associated with the use of an oversimplified Tinkham approximation for the field distribution and domain shape near the surface through which the flux enters and leaves the specimen. We believe that modern experimental capabilities associated, e.g., with muon spectroscopy and noninvasive scanning magnetic microscopy, can help to resolve this important and very interesting issue, which we discussed in the Section IV. The new model discussed in Section VI is restricted by the slab-like specimens. Its extension to all ellipsoidal shapes covered in the model of Peierls and London is another possible avenue of research on the IS.

Finally, it is important to remind that the IS is one of two inhomogeneous superconducting states. The second state is the mixed state in type-II superconductors, taking place in vast majority of superconducting materials, including those used in practical applications. Therefore understanding of properties of the IS can help to understand properties of the mixed state. As an example, the field distribution and shape of the normal domains (vortices in type-II materials) near the specimen surface should be similar in both these inhomogeneous states.
