Incommensurability and recursivity: lattice dynamics of modulated crystals

Abstract

The lattice dynamics of incommensurate crystals have been studied on three different one-dimensional models: a linear chain with modulated spring constants, the Frenkel-Kontorova model and a model that is equivalent to the model studied by Hofstadter (1979) for a Bloch electron in a uniform magnetic field. For the modulated spring model it turns out that the detailed structure of the spectrum of eigenvibrations as a function of modulation wavevector is related to the continued-fraction expansion of this wavevector, which is very similar to what Hofstadter found for his model. For a limiting value of the modulation amplitude the spectrum has a recursive structure, but for weaker modulation the recursivity is washed out and the structure may be calculated by perturbative methods. The Frenkel-Kontorova model is studied in the regime where the contribution of the background to the total energy is relatively small compared with the contribution due to the interaction between the atoms. The results are similar to those for the modulated spring model, although the mechanism that leads to modulation is quite different for the two models. This also holds for the optical activity of both systems. In general, the number of optically active vibration modes is small. The oscillator strengths corresponding to these modes depend in a smooth way on the modulation wavevector on the period of the background potential; therefore in this respect it is valid to make a superstructure approximation.