Coulomb interactions in two-dimensional lattice structures.

Abstract

Using the Van der Hoff--Benson method, general relations are derived for electric potentials and dipole-dipole interaction tensors of two-dimensional lattice systems which consist of charges or dipole moments changing periodically from site to site. Numerical values of the potentials and dipole-dipole interaction tensors as well as their asymptotics near the symmetric points of the first Brillouin zone are presented for square, triangular, and hexagonal (honeycomb) lattices. In particular, an asymptotic responsible for the formation of an incommensurate phase in triangular antiferromagnets is given. Dispersion laws for collective excitations of these lattices are calculated and compared to those derived in the nearest-neighbor approximation. The symmetry properties and lattice-sublattice relations are determined, which enables us to minimize the number of independently calculable lattice sums. For a two-dimensional Bravais lattice constituted by similar charges, at a constant unit-cell area, a dependence of the Coulomb energy on geometrical parameters of a lattice is derived. It is shown that with parameters of a triangular lattice, this dependence reaches the minimum which specifies all coefficients of long-wavelength quadratic asymptotics of the dipole-dipole interaction tensor. The low-temperature states are considered for a triangular lattice of dipoles with their orientations degenerate in the lattice plane. The temperature of the orientational phase transition is estimated by various methods and its significant characteristics are discussed. \textcopyright{} 1996 The American Physical Society.