Symmetry-based atomic scale description of lattice dynamics and its applications for phonon mode analysis

Details Symmetry-based atomic scale description of lattice dynamics and its applications for phonon mode analysis Symmetry-based atomic scale description of lattice dynamics and its applications for phonon mode analysis

2007-07-13

arxiv.org/abs/0707.1921v3

Physics

Condensed Matter

Materials Science

13 figures with a new section and new figures on phonon mode analysis

Scientific paper

We present classical and quantum mechanical multiscale descriptions of lattice dynamics, from the atomic to the continuum scale, using atomic scale symmetry modes and their constraint equations. This approach is demonstrated for a one-dimensional chain and a two-dimensional square lattice with a monatomic basis. For the classical description, we find that rigid modes, in addition to the distortional modes found before, are necessary to describe the kinetic energy, and obtain constraint equations among these modes. Lagrangian equations, modified with the Lagrange multiplier terms, are solved for phonon dispersion relations without using displacement variables explicitly. The long wavelength limit of the kinetic energy terms expressed in terms of atomic scale modes is shown to be consistent with the continuum theory, and the leading order corrections are obtained. We also analyze the phonon in terms of symmetry modes, and find how the contribution of different symmetry modes varies depending on the phonon branch and wavevector. For the quantum mechanical description, we find conjugate momenta for the atomic scale symmetry modes. In direct space, graphical rules for their commutation relations are obtained. Commutation relations in the reciprocal space are also calculated. We emphasize that the approach based on atomic scale symmetry modes could be useful for the description of multiscale lattice dynamics, materials with electron-phonon coupling, and the dynamics of structural phase transition, which can be probed by time-resolved x-ray diffraction.

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