Oscillatory Behavior of Vortex-Lattice Melting Transition Line in Mesoscopic ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+y}$ Superconductors

The vortex-lattice melting transition of a limited number of vortices confined in mesoscopic square superconductors was studied by $c$-axis resistance measurements using stacks of intrinsic Josephson junctions in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+y}$. In contrast to the melting transition in bulk crystals, we have first found a clear oscillatory behavior in the field dependence of the melting temperature in small samples of $5–10\mu m$ square. The periods of the oscillations roughly obey the regularity of the matching conditions of square vortex lattices surrounded by a square boundary and the melting temperatures are enhanced around the vortex number of $i^2$ (where $i$ is an integer). The results suggest that a confinement effect by the square boundary stabilizes square lattice structures which are realized around $i^2$ vortex number even in competition with the favorable Abrikosov triangular lattice in the bulk.

Figure 1

Magnetic field dependence of $c$-axis resistance with different applied currents in $5\times 5\mu {\mathrm{m}}^2$ IJJs. Although there is a large current dependence, the melting transitions are observed as steplike enhancements of the resistance in both cases indicated by arrows. The inset shows a top view of $5\times 5\mu {\mathrm{m}}^2$ $z$-shaped IJJs by a SIM.

Figure 2

$d\log (R_c+\delta R)/dT$ map on the $H$-$T$ plane of $5\times 5\mu {\mathrm{m}}^2$ IJJs. $\delta R$ is set to $1\mathrm{m}\mathrm{\Omega }$ to improve the appearance of the image. The melting transition line and the individual vortex penetrations can be recognized as a white oscillating line and as fine white slash lines, respectively. The red dotted lines are used as a guide to count the number of vortices. The $i^2$th lines of the individual penetrations are emphasized by the red lines.

Figure 3

Melting transition field (red open circles) and first penetration field (blue open diamonds) as a function of temperature, extracted from the map of Fig. 2 with a fitting curve by the decoupling theory. The inset shows the difference from the fitting curve of the melting temperature ($\mathrm{\Delta }{T}_m$) as a function of the magnetic field. The period of oscillatory behavior for the melting transition is extracted from this plot, as indicated by the arrows.

Figure 4

Change of vortex numbers in one period of oscillations as a function of the total vortex number for four different sized samples. The vortex numbers were converted from the magnetic field dependence of the oscillation periods shown in the inset by dividing both axes by the period for the individual vortex penetrations ($\mathrm{\Delta }{H}_p$). With respect to the $10\mu \mathrm{m}$ IJJs, measurements were performed twice (solid and open circles) to check the reproducibility. In the inset, the magnetic field of the horizontal axis is taken from the smaller dip field of both dips that define one period in the $\mathrm{\Delta }{T}_m$ oscillations.