Collective Modes of a Two-Dimensional Wigner Crystal

Calculations of the frequency spectrum of lattice vibrations of a finite two-dimensional electron crystal show a soundlike shear mode and a longitudinal mode that has an anomalous dispersion relation. At long wavelengths the frequency of the longitudinal mode varies as the square root of the wave vector. This arises because of the long-range nature of the Coulomb force between the electrons and is analogous to the longitudinal plasma vibration in a three-dimensional Wigner lattice.