Free sliding in lattices with two incommensurate periodicities

Abstract

The equilibrium configurations of the Frenkel-Kontorova model have been studied numerically. This model, which consists of a chain of ions connected by springs in the presence of a static sinusoidal potential, can be used to describe sliding charge-density waves in solids, incommensurate chain compounds such as ${\mathrm{Hg}}_{3-\delta }\mathrm{As}{\mathrm{F}}_6$, superionic conductors, and epitaxial crystal growth. If the natural periodicity of the chain is incommensurate with the periodicity of the sinusoidal potential, there exists, in the thermodynamic limit at some critical strength of the sinusoidal potential, a transition from a state in which the chain is pinned in place to a state in which it can accelerate freely when an arbitrarily small force is applied to each ion. The lattice vibration spectrum in the sliding regime exhibits a zero-frequency mode involving ionic vibrations that are more complicated than those found in the continuum limit of this model. Chains with free ends appear to require an activation energy to slide. Comparisons between the exact numerical results and those generated through approximation schemes are made along with an assessment of the validity of these schemes for various problems. The results of this work have direct application to the study of the conductivity, lattice dynamics, and elastic neutron and x-ray scattering of the systems previously mentioned.