Models for monolayers adsorbed on a square substrate

Motivated by recent experimental studies of Hg and Pb monolayers on Cu(001) we introduce a zero temperature model of a monolayer adsorbed on a square substrate. Lennard-Jones potentials are used to describe the interaction between pairs of adlayer-adlayer and adlayer-substrate atoms. We study a special case in which the monolayer atoms form a perfect square structure and the lattice constant, position and orientation with respect to the substrate can vary to minimize the energy. We introduce a rule based on the Farey tree construction to generate systematically the most energetically favored phases and use it to calculate the phase diagram in this model.