Geometrical matching and its influence on the melting transition of confined vortices in a mesoscopic triangle of Bi2Sr2CaCu2O8+y superconductor

To study the melting transition of a vortex crystal (or cluster) consisting of a small number of vortices confined in a mesoscopic-scale superconductor, $c$-axis magnetoresistance in the highly anisotropic high-${T}_{\mathrm{c}}$ superconductor ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+y}$ (Bi2212) has been measured using a triangle-shaped stack of intrinsic Josephson junctions, which was trimmed out from a single-crystal flake of Bi2212 via a double-sided etching process using a focused ion beam. We observed oscillations of the melting transition line, in which the oscillating part of the melting temperature has sharp peaks when the number of vortices $N$ exactly coincides with the triangular number $n(n+1)/2$ ($n$ is an integer number), at least for $Nl50$. This is in contrast to the case of square shapes where the enhancements appeared as broad peaks around $N={n}^{2}$. Furthermore, the field ranges of the $N$-vortex state expand at the triangular numbers of $N$, suggesting that the triangular number of vortices has a more stable configuration than the others. Numerical studies on the configuration of vortices in a triangle shape show that geometrical matching with no defect structures is realized at the triangular numbers, whereas an edge dislocation often appears in the other nonmatching cases. The degree of suppression of the melting temperatures at nonmatching numbers is reasonably consistent with the value estimated from the increase in free energy with the introduction of an edge dislocation.