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

Journal Article · Wed Jun 17 00:00:00 EDT 2015 · Physical Review. B, Condensed Matter and Materials Physics

DOI:https://doi.org/10.1103/PhysRevB.91.224305· OSTI ID:1193218

Meng, Qingping ^[1]; Wu, Lijun ^[1]; Welch, David O. ^[1]; Zhu, Yimei ^[1]

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1. Brookhaven National Lab. (BNL), Upton, NY (United States)

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. In addition, 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 non-zero 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 one-dimensional system at a finite temperature may become indefinitely close to the energy of an incommensurate state. This finding implies that the higher-order incommensurate-commensurate transitions are continuous ones and that the phase transition may exhibit a “devil's staircase” behavior at a finite temperature.

• Research Organization: Brookhaven National Laboratory (BNL), Upton, NY (United States)
• Sponsoring Organization: USDOE; USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)
• Grant/Contract Number: AC02-98CH10886
• OSTI ID: 1193218
• Alternate ID(s): OSTI ID: 1185135
• Report Number(s): BNL--108038-2015-JA; KC0201010
• Journal Information: Physical Review. B, Condensed Matter and Materials Physics, Journal Name: Physical Review. B, Condensed Matter and Materials Physics Journal Issue: 22 Vol. 91; ISSN 1098-0121; ISSN PRBMDO
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

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