Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

Abstract

We studied the lattice vibrations of two inter-penetrating atomic sublattices via the Frenkel-Kontorova (FK) model of a linear chain of harmonically interacting atoms subjected to an on-site potential, using the technique of thermodynamic Green's functions based on quantum field-theoretical methods. General expressions were deduced for the phonon frequency-wave-vector dispersion relations, number density, and energy of the FK model system. As the application of the theory, we investigated in detail cases of linear chains with various periods of the on-site potential of the FK model. Some unusual but interesting features for different amplitudes of the on-site potential of the FK model are discussed. In the commensurate structure, the phonon spectrum always starts at a finite frequency, and the gaps of the spectrum are true ones with a zero density of modes. In the incommensurate structure, the phonon spectrum starts from zero frequency but at a nonzero wave vector; there are some modes inside these gap regions, but their density is very low. In our approximation, the energy of a higher-order commensurate state of the onedimensional system at finite temperature may become indefinitely close to the energy of an incommensurate state. This finding implies that the higher-order incommensuratecommensurate transitions are continuous ones, and that the phase transition may exhibit a "devil's staircase" behavior at a finite temperature.

Lattice vibrations in the Frenkel-Kontorova Model. II. Thermal conductivity

“…In the first paper of this series [1], termed number Ⅰ, we derived general expressions for vibrational properties of the lattice of the Frenkel-Kontorova (FK) model, number density, and the energy of the system. These derivations utilized the technique of thermodynamic Green's functions, based on quantum field-theoretic methods.…”

Comment on “Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy”

“…The article by Meng et al [1] (paper I) explores a quantum treatment for the modes of the Frenkel-Kontorova (FK) problem, and it compares those modes to the classical modes found by Novaco [2] (paper II) using a mass-density-wave (MDW) analysis. There are many similarities between these results, but some differences were found that affect the general conclusions of these papers.…”

Nonlinear acoustics in periodic media: from fundamental effects to applications

“…This model is one of the most important models for discrete systems. It has received much attention because of its relevance to a wide variety of physical problems [40]. The first purpose of Frenkel and Kontorova was to model dislocations in epitaxial monolayers [41], but its surprising ability to describe many physically important phenomena, such as the dynamics of absorbed layers of atoms on crystal surfaces [42], or charge-density wave in two specific condensed matter systems [43,44], opened the range of applications further than originally expected.…”