Kinks in the Frenkel-Kontorova model with long-range interparticle interactions

Abstract

We study a nonlocal Frenkel-Kontorova model that describes a one-dimensional chain of atoms moving in a periodic external potential and repulsing one another according to a long-range law, e.g., the power law \ensuremath{\sim}${\mathit{x}}^{\mathrm{\ensuremath{-}}\mathit{n}}$. The investigation is carried out both numerically and analytically in approximations of a weak or strong bond between atoms. Static characteristics of kinks (topological solitons) such as the effective mass, shape, and amplitude of the Peierls potential, the interaction energy of kinks, and the creation energy of kink-antikink pairs are calculated for different (exponential and power with n=1 and 3) laws of the interparticle interaction and various concentrations of atoms, i.e., ratio between the external potential period and the average spacing of atoms in the chain. The anharmonicity of the interaction potential between atoms is shown to result in differences between kink and antikink parameters, which are proportional to the value of the anharmonicity that rises with increasing exponent n of the interaction potential as well as at a changeover from a complex to a simpler unit cell. It is noted that at a power law of the interparticle repulsion this law describes also the asymptotics of the kink shape as well as the interaction energy of the kinks. Because of this, the dependence of, e.g., the amplitude of the Peierls potential versus the atom concentration, is similar to the ``devil's staircase.'' The applicability of the extended Frenkel-Kontorova model for describing diffusion characteristics of a quasi-one-dimensional layer adsorbed on a crystal surface is discussed.