Topological magnetic irreversibility in superconducting Pb samples of various shapes

The magnetic-field dependence of the magnetization of cylinders, disks, and spheres of pure type-I superconducting lead was investigated by means of isothermal measurements of first magnetization curves and hysteresis cycles. Depending on the geometry of the sample and the direction and intensity of the applied magnetic field, the intermediate state exhibits different irreversible features that become particularly highlighted in minor hysteresis cycles. The irreversibility is noticeably observed in cylinders and disks only when the magnetic field is parallel to the axis of revolution and is very subtle in spheres. When the magnetic field decreases from the normal state, the irreversibility appears at a temperature-dependent value whose distance to the thermodynamic critical field depends on the sample geometry. The irreversible features in the disks are altered when they are submitted to an annealing process. These results agree well with very recent high-resolution magneto-optical experiments in similar materials that were interpreted in terms of transitions between different topological structures for the flux configuration in the intermediate state. A discussion of the relative role of geometrical barriers for flux entry and exit and pinning effects as responsible for the magnetic irreversibility is given.

Figure 1

First magnetization curves and hysteresis cycles $M\left(H\right)$ measured at 3 K on sample $A$ (cylinder) when the magnetic field is applied either parallel (a) or perpendicular (b) to the $z$ direction after a zero-field-cooling process. The inset in the upper panel shows the temperature dependence of the magnetic field at the onset of the intermediate state ($H_c^{\prime }$, solid circles), the magnetic field at the onset of the irreversibility ($H_c^{\ast }$, open squares), and the thermodynamic critical field ($H_c$, solid triangles).

Figure 2

(Color online) Minor $M\left(H\right)$ loop (open triangles) measured on sample $A$ (cylinder) at 3 K when the magnetic field is applied parallel to the $z$ direction after a zero-field-cooling process. The magnetic field goes from 0 to 450 Oe (point 1), from 450 to $-500\text{Oe}$ (point 2), from $-500$ to 550 Oe (point 3), and back to zero. A full $M\left(H\right)$ curve (open circles) is shown superimposed. Points $2^{\prime }$ and $3^{\prime }$ indicate where the minor loop merges into the full curve.

Figure 3

(Color online) Minor $M\left(H\right)$ loops measured on sample $B$ (disk) at 3 K with the magnetic field applied parallel to the $z$ axis after a zero-field-cooling process. Each loop was obtained starting at the normal state $\left(H>H_c\right)$ going right down to a certain $H_m$ value and then back to the initial state by slowly measuring the magnetization. The loops are represented with different symbols corresponding to $H_m$ values equally spaced 50 Oe in the range from $-300\text{Oe}$ to 300 Oe with an extra value at $-70\text{Oe}$ (see legend for details). The entire $M\left(H\right)$ cycle measured at the same temperature is also plotted (outermost open circles).

Figure 4

(Color online) First magnetization curves and hysteresis cycles measured at 3 K on all samples ($A$, solid squares; $B$, open circles; $C$, solid triangles; and $D$, open rhombuses) when the magnetic field is applied parallel to the $z$ direction after a zero-field-cooling process. The reduced magnetization $m$ was defined for each sample as the ratio of the magnetization $M$ to the absolute value at the peak of the ascending branch of the corresponding hysteresis cycle $\left|M_{\text{asc}}\left(H_p\right)\right|$.