Magnetic moment of a slab of type-I superconductor: Theoretical model and experiment

The magnetic moment $M$ of superconducting samples with a nonzero demagnetizing factor is one of the most important but not always completely understood sample characteristics. Here we calculate the magnetic moment of a slab of type-I superconductor in an arbitrarily oriented magnetic field, applying a recently developed model of the intermediate state (IS) [V. Kozhevnikov et al., Phys. Rev. B 89, 100503(R) (2014)]. We introduce and employ a novel form of the thermodynamic potential, which allows us to derive a general formula for the total free energy of the slab. The magnetic moment, calculated from the total free energy, is consistent with experimental data reported in this paper as well as with data available in the literature. Also, the calculated moment solves a long-standing puzzle of experimental $M\left(H\right)$ curves $(H$ is the applied field) and provides insight in the field driven evolution of the magnetic flux density in the domain structure of the IS. This paper completes our presentation of a theoretical model which consistently addresses all properties of the IS, thus solving the problem of the IS in a slab that was put forward by Landau almost eight decades ago.

Figure 1

Average magnetic induction in (a) and the moment of (b) an infinite slab $\left(\eta =1\right)$ and a cylinder $\left(\eta =1/2\right)$ in a perpendicular field $H$, as expected from the standard theory.

Figure 2

Cross sectional view of the modeled superconducting slab in the intermediate state. $H_{\perp }$ and $H_{\parallel }$ are the perpendicular and parallel components of the applied field, respectively. Domains are rectangular parallelepipeds extended along $H_{\parallel }$. The healing length $L_h$ is the characteristic distance over which the field passing through the sample relaxes to its uniform state. The parallel component of the flux density $B_{\parallel }$ in the N domains is equal to $H_{\parallel }$, while the perpendicular component $B_{\perp }=H_{\perp }{\rho }_n$, where ${\rho }_n=D_n/D$ is the fraction of the N phase.

Figure 3

Magnetic moment of the In-B sample measured in a perpendicular field at 3 K. Relevant data are above 10 Oe; at lower fields the moment is due to the current protecting the flux trapped at pinning centers. $H_c$ is the thermodynamic critical field and $H_{ci}$ is the critical field of the transition from the intermediate to the normal state.

Figure 4

Magnetic moment of the $3.86\mu \mathrm{m}$ thick indium film sample (In-A) in a perpendicular field at the different temperatures indicated on the graphs. The dashed curve in the graph for 3 K represents the moment inferred from our model with $\delta$ calculated using the Pippard coherence length ${\xi }_0=380$ nm, as described in Ref. [12].

Figure 5

Magnetic moment of the $2.79\mu \mathrm{m}$ thick indium film sample (In-B) in a perpendicular field at the different indicated temperatures. The dashed curves are extrapolations of the experimental data consistent with the rule of 1/2.

Figure 6

Magnetic moment of the In-A and In-B samples in a parallel field.