Mode decomposition based on crystallographic symmetry in the band-unfolding method

The band-unfolding method is now widely used to calculate the effective band structures of a disordered system from its supercell model. The unfolded band structures are obtained as if the disordered system really has the crystallographic symmetry of its underlying structure. However, it has still been difficult to decompose the unfolded band structures into the modes based on the crystallographic symmetry of the underlying structure. This has prohibited detailed analyses of, e.g., selection rules and avoided band crossings for the unfolded band structures. The authors here develop a procedure to decompose the unfolded band structures according to the small representations (SRs) of the little groups. The decomposition is performed using the projection operators for SRs derived from the group representation theory. Using this procedure, the authors investigate the phonon band structure of disordered face-centered-cubic Cu${}_0.75$Au${}_0.25$, which has large variations of atomic masses and force constants among the atomic sites due to the chemical disorder. In the unfolded phonon band structure, several peculiar behaviors such as discontinuous and split branches are found in the decomposed modes corresponding to specific SRs.