Computed epitaxial monolayer structures: I. One-dimensional model: Comparison of computed and analytical results

An exact computational procedure is developed which, when applied to Frank and van der Merwe's one-dimensional dislocation model, yields the equilibrium structures of a one-dimensional atom chain with elastic interatomic forces and sinusoidal substrate potential. This forms the first part of the development of a general procedure for calculating the equilibrium structures of two-dimensional monolayers with elastic interatomic forces and substrate potentials of more complex character. The minimum energy principle is applied to distinguish the stable structures from the computed equilibrium ones. The results aie in close agreement with the analytical approximate solutions of Frank and van der Merwe. The atomic displacements and limiting misfits agree almost perfectly. The curves of lowest energy against misfit are likewise in excellent agreement for long chains. For relatively short chains, containing only few dislocations, the curves are composed of segments, one segment for each additional dislocation. The discreteness of dislocations in finite chains is also borne out in other properties. The calculations further show (i) that, for a finite chain with odd numbers of dislocations and atoms, the stable configuration is one with the central atom on a potential crest and (ii) that the cusped minima present in the interfacial energy curve of thick bicrystals disappear when complete or partial accommodation of lattice parameters is energetically possible as in thin films.