5. Key experiments

Although we began this chapter from theoretical models, a real story of the IS has started from experiment. Measuring electrical resistance R of tin wires, De Haas with collaborators revealed a strong dependence of R Hð Þ on direction of the applied field H: instead of a sharp S/N transition at a threshold field (Hc) in the parallel field, R returns to its full value gradually at the field range from about Hc=2 to Hc when the field is perpendicular [29, 30]. Later it was shown that reproducible R Hð Þ in the perpendicular field is liner [28]; one of the graphs for R Hð Þ from [28] is reproduced in Figure 6. The linear R Hð Þ is consistent with the Peierls-London model; however, it was revealed that transition from the Meissner state to the IS takes place at HI, which is somewhat greater than 1ð Þ � η Hc ¼0.5Hc.

The first observation of the IS magnetic structure was achieved by Meshkovsky and Shalnikov, who mapped the field in a gap between two tin hemispheres with radius 2 cm using a resistive probe made of a tiny bismuth wire [24]. Originally this experiment was designed to verify the

Figure 6. Relative resistance of a high-purity tin cylindrical wire of 0.4 mm in diameter and 5 cm in length at temperature 1.666 K (a) in increasing and in decreasing transverse field and (b) in increasing longitudinal field. After Andrew [28].

<sup>3</sup> First results of direct measurements of FDDS were recently presented in V. Kozhevnikov, A. Suter, T. Prokscha, C. Van Haesendonck arXiv:1802.08299v1 [cond-mat.supr-con] (2018).

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

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>

Although we began this chapter from theoretical models, a real story of the IS has started from experiment. Measuring electrical resistance R of tin wires, De Haas with collaborators revealed a strong dependence of R Hð Þ on direction of the applied field H: instead of a sharp S/N transition at a threshold field (Hc) in the parallel field, R returns to its full value gradually at the field range from about Hc=2 to Hc when the field is perpendicular [29, 30]. Later it was shown that reproducible R Hð Þ in the perpendicular field is liner [28]; one of the graphs for R Hð Þ from [28] is reproduced in Figure 6. The linear R Hð Þ is consistent with the Peierls-London model; however, it was revealed that transition from the Meissner state to the IS takes place at

The first observation of the IS magnetic structure was achieved by Meshkovsky and Shalnikov, who mapped the field in a gap between two tin hemispheres with radius 2 cm using a resistive probe made of a tiny bismuth wire [24]. Originally this experiment was designed to verify the

Figure 6. Relative resistance of a high-purity tin cylindrical wire of 0.4 mm in diameter and 5 cm in length at temperature 1.666 K (a) in increasing and in decreasing transverse field and (b) in increasing longitudinal field. After Andrew [28].

First results of direct measurements of FDDS were recently presented in V. Kozhevnikov, A. Suter, T. Prokscha, C. Van

5. Key experiments

98 Superfluids and Superconductors

3

HI, which is somewhat greater than 1ð Þ � η Hc ¼0.5Hc.

Haesendonck arXiv:1802.08299v1 [cond-mat.supr-con] (2018).

.

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 reported when samples are in nonequilibrium state [12].

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].

Shoenberg revealed that the critical field of the IS/NS transition Hcr is appreciably smaller than the thermodynamic critical field Hc measured in parallel field. It was also found that the differences ΔHI ¼ HI � ð Þ 1 � η Hc and ΔHcr ¼ Hc � Hcr depend on the specimen radius: the smaller the radius, the greater the differences. One of magnetization curves reported in [34] is reproduced in Figure 9.

The differences of ΔHI and ΔHcr are usually interpreted as a price paid by the specimen for the extra energy needed to create the S/N interfaces in assumption that ΔHI and ΔHcr are small [3, 8]. We note that this explanation is not full because significant part of the extra free energy is associated with the field inhomogeneity near the specimen surface. On the other hand, the observed extension of the Meissner state (up to HI > ð Þ 1 � η Hc) means that 4πM=V at HI is greater than Hc, the value following from the PL model. This "excess magnetic moment" is consistent with the rule of 1/2, and it is indeed seen in Figure 9 and in other data reported by Desirant and Shoenberg. However this feature can hardly be attributed to the S/N surface tension. Egorov et al. [35] measured induction B in the bulk of N domains of a high-purity single-

results are shown in Figure 10. Ht in this graph corresponds to Hcr in our notations. The tubular phase mentioned in the caption most probably corresponds to the filament state

Results of Egorov et al. show that B in N domains is Hc at low applied field and decreases with increasing field down to Hcr at the IS/N transition. But induction B in N domains equals to the field strength Hi. Therefore the original postulate used in the PL and Landau models (Hi ¼ Hc)

Recently the IS problem was revisited by Kozhevnikov et al. [25, 26] via magneto-optics and measurements of electrical resistivity and magnetization in high-purity indium films of different thickness in the fields of different orientations. An immediate motivation for this research was discrepancy in values of the coherence length for Sn and In following from Sharvin's results for the IS structure [32, 33] and those obtained from the measured magnetic field profile in the

Figure 10. Induction in N domains of the Sn single-crystal plate at temperature 0.08 K measured at increasing (circles, solid line) and decreasing (triangles, dashed line) applied field. For decreasing field the N state is field supercooled down to Hscl. At increasing field the laminar structure transforms to one with tubular S regions at Ht. After Egorov et al. [35].

is correct for the low reduced fields, but it can be not so at higher fields.

) in perpendicular field using μSR spectroscopy. Reported

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 101

crystal tin slab (18 � <sup>12</sup> � 0.56 mm<sup>3</sup>

discussed in [36].

Figure 8. Photograph of the IS flux structures taken with a single-crystal tin disc-shaped specimen (<sup>∅</sup> <sup>50</sup> � 2 mm2 ) in the field tilted for 15� with respect to the specimen at temperature 0.58Tc and field 0.95Hc . Light areas are normal. After Sharvin [32].

Figure 9. Magnetization curve (m ¼ M=V) of cylindrical mercury specimen with radius 23 μm in transverse field at temperature 2.12 K measured at increasing (⊙) and decreasing (þ) fields. Hc is thermodynamic critical field measured in parallel field. Solid line based on landau branching model [19] with wall-energy parameter adjusted for best fit. After Desirant and Shoenberg [34].

The differences of ΔHI and ΔHcr are usually interpreted as a price paid by the specimen for the extra energy needed to create the S/N interfaces in assumption that ΔHI and ΔHcr are small [3, 8]. We note that this explanation is not full because significant part of the extra free energy is associated with the field inhomogeneity near the specimen surface. On the other hand, the observed extension of the Meissner state (up to HI > ð Þ 1 � η Hc) means that 4πM=V at HI is greater than Hc, the value following from the PL model. This "excess magnetic moment" is consistent with the rule of 1/2, and it is indeed seen in Figure 9 and in other data reported by Desirant and Shoenberg. However this feature can hardly be attributed to the S/N surface tension.

Shoenberg revealed that the critical field of the IS/NS transition Hcr is appreciably smaller than the thermodynamic critical field Hc measured in parallel field. It was also found that the differences ΔHI ¼ HI � ð Þ 1 � η Hc and ΔHcr ¼ Hc � Hcr depend on the specimen radius: the smaller the radius, the greater the differences. One of magnetization curves reported in [34] is

Figure 8. Photograph of the IS flux structures taken with a single-crystal tin disc-shaped specimen (<sup>∅</sup> <sup>50</sup> � 2 mm2

field tilted for 15� with respect to the specimen at temperature 0.58Tc and field 0.95Hc . Light areas are normal. After

Figure 9. Magnetization curve (m ¼ M=V) of cylindrical mercury specimen with radius 23 μm in transverse field at temperature 2.12 K measured at increasing (⊙) and decreasing (þ) fields. Hc is thermodynamic critical field measured in parallel field. Solid line based on landau branching model [19] with wall-energy parameter adjusted for best fit. After

) in the

reproduced in Figure 9.

100 Superfluids and Superconductors

Sharvin [32].

Desirant and Shoenberg [34].

Egorov et al. [35] measured induction B in the bulk of N domains of a high-purity singlecrystal tin slab (18 � <sup>12</sup> � 0.56 mm<sup>3</sup> ) in perpendicular field using μSR spectroscopy. Reported results are shown in Figure 10. Ht in this graph corresponds to Hcr in our notations. The tubular phase mentioned in the caption most probably corresponds to the filament state discussed in [36].

Results of Egorov et al. show that B in N domains is Hc at low applied field and decreases with increasing field down to Hcr at the IS/N transition. But induction B in N domains equals to the field strength Hi. Therefore the original postulate used in the PL and Landau models (Hi ¼ Hc) is correct for the low reduced fields, but it can be not so at higher fields.

Recently the IS problem was revisited by Kozhevnikov et al. [25, 26] via magneto-optics and measurements of electrical resistivity and magnetization in high-purity indium films of different thickness in the fields of different orientations. An immediate motivation for this research was discrepancy in values of the coherence length for Sn and In following from Sharvin's results for the IS structure [32, 33] and those obtained from the measured magnetic field profile in the

Figure 10. Induction in N domains of the Sn single-crystal plate at temperature 0.08 K measured at increasing (circles, solid line) and decreasing (triangles, dashed line) applied field. For decreasing field the N state is field supercooled down to Hscl. At increasing field the laminar structure transforms to one with tubular S regions at Ht. After Egorov et al. [35].

Meissner state [37]. In Figure 11 we reproduce typical magneto-optical images obtained for a 2.5-μm-thick film. The most unexpected result revealed with this specimen is that in perpendicular field the critical field Hcr ≈ 0:4Hc at T ! 0. A typical magnetization curve obtained with another (3.86-μm-thick) film is shown in Figure 12. Hcr for this specimen at 2.5 K is 0.65Hc and

4πM(0)/V = 1.6Hc. All data were well reproducible, and the area under magnetization curves plotted in reduced coordinate is close to 1/2, meaning that the obtained experimental results reflect the equilibrium properties of the IS. However these results conflict with available theoretical models. A new model, consistently addressing outcomes of this work and explaining earlier revealed "anomalies," is presented in [25, 26]. We discuss it in the following section.

The simplest of experientially observed equilibrium domain structures of the IS is onedimensional laminar lattice in slab-like specimens placed in a tilted field. Therefore such a specimen/field configuration is the most convenient for modeling. A laminar model for tilted field (LMTF) was developed in [25, 26]. Schematics of the specimen in the LMFT is shown in Figure 13.

(II) Specimen thickness d ≫ λ. This means that negative surface tension of S/V (V stands for

(III) Longitudinal sizes of the specimen (along x and y axes) are much greater than thickness d, i.e., the slab is considered infinite. This means that flux of the perpendicular component of the applied field H<sup>⊥</sup> is conserved, and therefore H<sup>⊥</sup> ¼ B<sup>⊥</sup> ¼ B⊥rn, where B<sup>⊥</sup> is average perpendicular component of the induction over the specimen, B<sup>⊥</sup> is perpendicular component of the induction in N domains (considered uniform), and r<sup>n</sup> is volume fraction of the N phase: r<sup>n</sup> ¼ Dn=D ¼ Vn=V with Dn and Vn designating the width of the

(IV) B<sup>∥</sup> ¼ ð Þ Hi <sup>∥</sup> ¼ H<sup>∥</sup> due to the absence of the demagnetizing field along y-axis or along the

(V) Tinkham's version of the FDDS (see Figure 5C) is adopted due to its simplicity and

We start from construction of a thermodynamic potential F T <sup>~</sup>ð Þ ; <sup>V</sup>; Hi , which is the Legendre transform of the Helmholtz free energy F Tð Þ ; V; B to the variables ð Þ T; V; Hi . It is often referred

> <sup>V</sup> � <sup>B</sup>⊥Hi<sup>⊥</sup> 4π

where F Tð Þ ; V; B is Helmoltz free energy. The term ð Þ B � Hi=4π V reflects work done by the magnet power supply to keep the set field H when the flux in the system changes [2]. In our case the flux of the perpendicular component is fixed, and therefore the term ð Þ B⊥Hi⊥=4π V

It should be remembered that canonical Gibbs free energy is function of pressure, but not volume, as in this case.

<sup>V</sup> <sup>¼</sup> <sup>F</sup> � <sup>B</sup>∥Hi<sup>∥</sup>

4π

<sup>V</sup> <sup>¼</sup> <sup>F</sup> � <sup>H</sup><sup>2</sup>

∥ 4π

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 103

V, (13)

6. Laminar model for flat slab in tilted field

(I) Specimen is in the free space (vacuum).

vacuum) interfaces due to nonzero H<sup>∥</sup> is neglected.

N laminae and a total volume of the N phase, respectively.

parallel component of the applied field H∥.

consistency with the experimental images.

:

<sup>V</sup> <sup>¼</sup> <sup>F</sup> � <sup>B</sup>∥Hi<sup>∥</sup>

4π

Setting of the model is:

to as the Gibbs free energy<sup>4</sup>

4

<sup>F</sup><sup>~</sup> <sup>¼</sup> <sup>F</sup> � <sup>B</sup> � <sup>H</sup><sup>i</sup> 4π

Figure 11. Magneto-optical images taken with 2.5-μm-thick in film at 2.5 K. [H∥, H<sup>⊥</sup> in Oe]: (a) [0, 1], (b) [60, 8], (c) [100, 6], (d) [110, 3], and (e) [115, 1.3]. Superconducting regions are black. After Kozhevnikov et al. [25].

Figure 12. Magnetization curve of 3.86-μm-thick indium film measured in perpendicular field at 2.5 K. Green (orange) circles represent the data measured at increasing (decreasing) field. Shadowed area represents the specimen condensation energy (1/2 in the reduced coordinates of this graph). Hc was determined from magnetization curve in parallel field, and the specimen volume was determined from the slope of that curve. After Kozhevnikov et al. [25].

4πM(0)/V = 1.6Hc. All data were well reproducible, and the area under magnetization curves plotted in reduced coordinate is close to 1/2, meaning that the obtained experimental results reflect the equilibrium properties of the IS. However these results conflict with available theoretical models. A new model, consistently addressing outcomes of this work and explaining earlier revealed "anomalies," is presented in [25, 26]. We discuss it in the following section.
