Figure 1

The cross section of a rectangular superconductor with sides $a$ and $b$. The external magnetic field $\mathbf{H}$ is applied along the $z$ axis, and its value is assumed to be constant outside the sample.Reuse & Permissions

Figure 2

(Color online) The profiles of the confinement energy ${ϵ}_c={ϵ}_i^{\text{shield}}+{ϵ}_{ii}$ (measured in units of $g_0={\Phi }_0^2/8\pi A\cdot 1/{\lambda }^2$, where $A$ is the area of the sample) for mesoscopic superconducting squares with size (a) $a=3\lambda$ and (b) $15\lambda$. The Gibbs free-energy distributions for squares with (c) $a=3\lambda$ and (d) $15\lambda$ for the vortex state with $L=5$.Reuse & Permissions

Figure 3

(Color online) The evolution of vortex configurations for the states with vorticity increasing from $L=1$ to 12, in a superconducting square with $a=3\lambda$ (the same results found for larger squares, e.g., with $a=15\lambda$). The vortices in the outer shell are shown by the blue (black) circles while the inner-shell vortices are shown by the yellow (gray) circles. The formation of the second shell starts when $L=5$.Reuse & Permissions

Figure 4

(Color online) The evolution of vortex configurations for $L=15–18$ (a)–(d), and for (e) $L=25$ to (f) 29, in a superconducting square with $a=3\lambda$. For vorticities [(a)–(d)] $L=15–18$, the outermost shell formed by 12 vortices is complete (commensurate with the square boundary), and with increasing magnetic-field vortices fill inner shells. Note that when the inner shell also becomes complete [(b) $L=16$, state (4,12)], the third shell starts to form for (c) $L=17$. For states with larger vorticities, e.g., (e) $L=25$ and (f) $L=29$, the vortex patterns are very close to a triangular lattice which is distorted near the boundary.Reuse & Permissions

Figure 5

(Color online) The change of the free energy $\left(G\text{−}{G}_0\right)$ versus the displacement $R$ of one of the vortices in the initial pentagon-shaped configuration from its initial position toward the center [two different lines for each configuration correspond to increasing and decreasing $\Delta R$ as shown by the arrows in (a)]. $G_0$ is the free energy associated with external magnetic field and the vortex cores [term “4” in Eq. (15)], which is independent of the positions of the vortices. The two stable states, the pentagonlike state (5) and the square-symmetric state (1,4), are shown in the insets. The vortices are labeled by A, B, C, D, and E. Two different symmetry axes of the configuration (5) are shown by the dash-dotted line in the insets of (a) and (b), respectively. The side of the square is $a=3\lambda$. In both cases, the configuration with one vortex in the center (1,4) has a lower energy than the pentagonlike pattern (5). Note that the curves for B and D (and for A and E) are slightly different due to the fact that the configuration (5) is not perfectly aligned with respect to the symmetry axes.Reuse & Permissions

Figure 6

(Color online) The change of the free energy $\left(G\text{−}{G}_0\right)$ versus the displacement $R$ of one of the vortex in the inner shell of the state (5,12) from its initial position toward the center; $G_0$ is defined in the caption of Fig. 5. The change in the free energy due to the movement of the vortices in the inner shells [i.e., (5,12) $\to$ (1,4,12)] is damped by the movement of the vortices in the outermost shell which act as a “softer” wall than the boundary [in the case of the transition (5) $\to$ (1,4), see Fig. 5]. The movement of the vortices in the outmost shell causes more saddle points. The two states, (5,12) and (1,4,12), have very close free energies.Reuse & Permissions

Figure 7

Scanning electron microscope (SEM) images of vortex configurations observed experimentally for vorticities $L=2–13$. Vortex positions are indicated by clusters of small white (Fe) particles. (a) $L=2$; sample size (side of the square) $a\approx 2.5\mu \text{m}$, $H=20\text{Oe}$; (b) $L=3$; $a\approx 2\mu \text{m}$, $H=35\text{Oe}$; (c) $L=4$; $a\approx 2.4\mu \text{m}$, $H=40\text{Oe}$; (d) $L=5$; $a\approx 2.4\mu \text{m}$, $H=40\text{Oe}$; (e) $L=6$; $a\approx 2.5\mu \text{m}$, $H=40\text{Oe}$; (f) $L=7$; $a\approx 2\mu \text{m}$, $H=60\text{Oe}$; (g) $L=9$; $a\approx 3.5\mu \text{m}$, $H=35\text{Oe}$; (h) $L=10$; $a\approx 3.5\mu \text{m}$, $H=35\text{Oe}$; (i) $L=10$; $a\approx 3.5\mu \text{m}$, $H=35\text{Oe}$; (j) $L=11$; $a\approx 2.5\mu \text{m}$, $H=60\text{Oe}$; (k) $L=12$; $a\approx 2.6\mu \text{m}$, $H=60\text{Oe}$; (l) $L=13$; $a\approx 5\mu \text{m}$, $H=20\text{Oe}$.Reuse & Permissions

Figure 8

Histograms of different vortex states observed for vortices $L=2$, 4 (for squares with $a=2\mu \text{m}$) (a) and $L=5$ (for squares with $a=2\mu \text{m}$) and 6 (b) ($a=2\mu \text{m}$ and $a=2.5\mu \text{m}$). SEM images of the corresponding vortex configurations are shown as insets.Reuse & Permissions

Figure 9

(Color online) Comparison of the experimentally observed positions of vortices within the square dots with those found numerically. Superimposed on the experimental images are vortex configurations shown in Figs. 3c, 3d, 3e, 3f, 3g. Two experimental images for $L=6$ [(d) and (e)] are superimposed on the same theoretical image [Fig. 3f], to demonstrate that the observed configurations did not depend on the sample size or the applied field [for image (d) $H=40\text{Oe}$, $a\approx 2.5\mu \text{m}$, for image (e) $H=60\text{Oe}$, $a\approx 2\mu \text{m}$].Reuse & Permissions