Energy barriers, structure, and two-stage melting of microclusters of vortices

The melting of two-dimensional microclusters of “particles” with logarithmic repulsive interaction and confined by an external parabolic potential is considered. The model describes the behavior of vortices in a small island or grain of a type-II superconductor with a thickness smaller than the coherent length, vortices in a rotating vessel with superfluid, or electrons in a semiconductor nanostructure, surrounded by a media with a dielectric constant essentially smaller than that for the nanostructure. Shell configurations corresponding to the local and global minima of the potential energy for microclusters (“Periodic Table” for a two-dimensional Thomson atom) are calculated, image potentials being taken into account. Due to image forces, configurations with larger numbers of vortices in internal shells become more stable. Rearrangements of the structure due to the anisotropy of confinement are studied. By the analysis of the temperature dependence of structure, radial, and angular rms displacements, the melting of clusters is analyzed. Two-stage melting of microclusters of vortices takes place: at lower temperature rotatory reorientation of neighboring “crystalline” shells (“orientational melting”) arises; at much greater temperatures the radial shell order disappears. Two-stage melting is connected with the fact that barrier of shell rotation $U_2$ is less than the barrier of intershell particle jump $U_1$, the ratio $U_2{/U}_1$ drops essentially for small microclusters. For clusters with a larger number of particles, orientational melting takes place only for external pairs of shells. This last fact is connected with approximate equality of barriers $U_1\approx U_2$ for inner shells.