Charging spectrum and configurations of a Wigner crystal island

Charging of a clean two-dimensional island is studied in the regime of small concentration of electrons when they form the Wigner crystal. Two forms of electron-electron interaction potential are studied: the pure Coulomb interaction and the exponential interaction corresponding to the screening by a pair of close metallic gates. The electrons are assumed to reside in a parabolic external confining potential. Due to the crystalline symmetry the center of the confinement can be situated at distinct positions with respect to the crystal. With the increasing number of electrons $N$ the center periodically hops from one such a location to another providing the lowest total energy. These events occur with the period $\sim N^{1/2}$. At these moments in the case of the pure Coulomb interaction the charging energy of the island has a negative correction $\approx 15\%$. For the case of the exponential interaction at the moments of switching the capacitance becomes negative and $\sim N^{1/4}$ new electrons enter the island simultaneously. The configurations of disclinations and dislocations in the island are also studied.