Scanning SQUID microscope study of vortex polygons and shells in weak-pinning disks of an amorphous superconducting film

Direct observation of vortices by the scanning superconducting quantum interference device microscopy was made on large mesoscopic disks of an amorphous MoGe thin film. Owing to the weak pinning nature of the amorphous film, vortices are able to form geometry induced, (quasi-)symmetric configurations of polygons, and concentric shells in the large disks. Systematic measurements made on selected disks allow us to trace not only how the vortex pattern evolves with magnetic field, but also how the vortex polygons change in size and rotate with respect to the disk center. The results are in good agreement with theoretical considerations for mesoscopic disks with sufficiently large diameter. A series of vortex images obtained in a disk with a pinning site reveals a unique line symmetry in vortex configurations, resulting in modifications of the shell filling rule and the magic number.

Figure 1

(Color online) Scanning SQUID microscope images of vortices in an amorphous MoGe disk with $34\mu \text{m}$ diameter for vorticities $L=1–5$ obtained after the disk was cooled to 3.2 K in different magnetic fields. (a) $L=1$ at $8.0\mu \text{T}$; (b) $L=2$ at $10\mu \text{T}$; (c) $L=3$ at $16\mu \text{T}$; (d) $L=4$ at $18\mu \text{T}$; (e) $L=5$ at $21\mu \text{T}$. (f) An additional image of the vortex pair at $14\mu \text{T}$. All images are the same in size of $66\times 66\mu {\text{m}}^2$. Color bars indicate the magnitude of the magnetic flux detected in the pick-up loop.

Figure 2

(Color online) (a) Vorticity vs normalized magnetic flux in the $34\mu \text{m}$ disk in 3.2 K. A solid line represents a condition of ${\Phi }_d/{\Phi }_0=L$. The inset to (a) shows the results around the transition between $L=5$ and 6 obtained by repeated field-cool measurements. (b) The normalized diameter of the polygon ring vs. the normalized magnetic flux. The dotted curves represent the theoretical results for polygon sizes with different $N=2–5$ as denoted. The solid curve indicates the upper limit of the polygon size. The inset to (b) illustrates how we determine the diameter $D_{\text{ring}}$ of the polygon ring.

Figure 3

(Color online) Images of vortices in disks with different diameters for large vorticities. Images of (a), (b), and (c) are obtained in a $56\mu \text{m}$ disk in 3.9 K at fields of 8.0, 9.0, and $10\mu \text{T}$, respectively. Other images of (d), (e), and (f) are observed in a $106\mu \text{m}$ disk in 5.0 K at fields of 3.8, 4.6, and $6.0\mu \text{T}$, respectively. For clarity, the corresponding vortex configurations for these images of (d), (e), and (f) are sketched in (g), (h), and (i), respectively.

Figure 4

Vorticity vs. normalized magnetic flux in the $106\mu \text{m}$ disk in 5.0 K. The configurations of vortices are denoted. A solid line represents a condition of ${\Phi }_d/{\Phi }_0=L$.

Figure 5

(Color online) Selected images of vortices observed in a $56\mu \text{m}$ disk with a pinning center taken in 3.3 K for different vorticities. (a) $L=1$ at $4.0\mu \text{T}$; (b) $L=2$ at $6.0\mu \text{T}$; (c) $L=3$ at $7.5\mu \text{T}$; (d) $L=4$ at $9.0\mu \text{T}$; (e) $L=5$ at $10\mu \text{T}$; (f) $L=6$ at $11\mu \text{T}$; (g) $L=7$ at $13\mu \text{T}$; (h) $L=8$ at $13.5\mu \text{T}$; (i) $L=13$ at $20\mu \text{T}$. A red point in each image marks the position of the pinning site in the disk. For clarity, the vortex configurations for the images of (a), (h), and (i) are sketched in (j), (k), and (l), respectively. The dotted line for each sketch represents a symmetric line defined by the pinning site and the disk center $\mathbf{O}$.