It has been known for several decades that whenever a system lacks periodicity, either compositional or structural, the normal modes of vibration can still be determined (in the harmonic limit), but the solutions take on different characters and many modes may be non-plane wave modulated. Previous work has classified the types of vibrations into three primary categories, namely propagons, diffusons and locons. Methods exist for distinguishing locons, from propagons and diffusons, which measure the extent to which a mode is localized. However, distinguishing between propagons and diffusons has remained a challenge, since both are spatially delocalized. In this work, I developed a new computational method that quantifies the extent to which a mode’s character corresponds to a propagating mode, e.g., exhibits plane wave modulation. This then allows for clear and quantitative distinctions between propagons and diffusons. By resolving this issue quantitatively, one can now automate the classification of modes for any arbitrary material or structure.

The method uses equilibrium atomic positions and eigenvectors of atoms in each vibrational mode and then calculates the degree of periodicity in the mode’s velocity field – termed eigenvector periodicity (EP). It then compares the EP of a mode to another fictitious mode that has pure sinusoidal modulation. In this way, the method normalizes the EP so that every mode falls between zero and unity. The extremes of zero and unity then correspond 0% and 100% sinusoidal/propagating velocity field periodicity for a given mode. The key here is that calculation of the EP for a mode is well-defined for any normal mode of vibration and can be evaluated in its entirety for a single mode, without any reference or relative scaling to the values of other modes.

I demonstrated the application of this method to several crystalline and amorphous solids (please see the following figures), allows us to clearly quantify what fraction of the modes in a given structure are propagons as a function of the degree of disorder. This new approach will form the basis for the development of a new physical picture for phonon transport as it is expanded to included non-propagating mode contributions. The right vertical axis in these figures shows the participation ratio (PR) which is a measure of the degree of localization of vibrational modes. Therefore, by combining the results of EP and PR, one can identify diffusons in the solid.

FIG. 1. Eigenvector periodicity for crystalline and amorphous silicon and Germanium (c-Si, a-Si, c-Ge, and a-Ge). The system size for this calculation is small (216 atoms)

FIG. 2. Illustration of the velocity field for example, normal modes in a-Si System as identified by their EP in Fig.1
FIG. 3. Eigenvector periodicity and participation ratio for a-Si
FIG. 3. Eigenvector periodicity and participation ratio for a-Ge
FIG. 4. Eigenvector periodicity and participation ratio for a-SiO2
FIG. 5. Mode population for c-Si with different defect concentrations.