Equilibrium Topology of the Intermediate State in Type-I Superconductors of Different Shapes

A high-resolution magneto-optical technique was used to analyze flux patterns in the intermediate state of bulk Pb samples of various shapes—cones, hemispheres, and discs. Combined with the measurements of macroscopic magnetization, these results allowed studying the effect of bulk pinning and geometric barrier on the equilibrium structure of the intermediate state. Zero-bulk pinning discs and slabs show hysteretic behavior due to topological hysteresis—flux tubes on penetration and lamellae on flux exit. (Hemi)spheres and cones do not have a geometric barrier and show no hysteresis with flux tubes dominating the intermediate field region in both regimes. It is concluded that flux tubes represent the equilibrium topology of the intermediate state. Real-time video is available in the EPAPS Document No. ..

Figure 1

$M(H)$ loop measured in a Pb single crystal at $T=4.5\mathrm{K}$. Larger open symbols show measurements each after field cooling at indicated field to 4.5 K. Small open symbols show minor hysteresis loops.

Figure 2

Structure of the intermediate state in a disc-shaped Pb single crystal at 5 K. Left column—increasing magnetic field after ZFC. Right column—decreasing field.

Figure 3

Magnetization loop in a “perfect superconducting” sphere (solid symbols) and a hemisphere (open symbols) at $T=4.5\mathrm{K}$.

Figure 4

Flux penetration (left column) and exit (right column) in a Pb hemisphere ($d=4\mathrm{mm}$) at $T=4\mathrm{K}$ and indicated magnetic fields. Inset shows tubular pattern in a different hemisphere sample.

Figure 5

Magnetization loop in a cone at $T=4.5\mathrm{K}$.

Figure 6

Flux penetration (left column) and exit (right column) into a cone-shaped sample at 4 K and at indicated fields. Inset shows a zoom of the tubular pattern.