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

The band-unfolding method is widely used to calculate the effective band structures of a disordered system from its supercell model. The unfolded band structures show the crystallographic symmetry of the underlying structure, where the difference of chemical components and the local atomic relaxation are ignored. However, it has still been difficult to decompose the unfolded band structures into the modes based on the crystallographic symmetry of the underlying structure, and therefore detailed analyses of the unfolded band structures have been restricted. In this study, a procedure to decompose the unfolded band structures according to the small representations (SRs) of the little groups is developed. The decomposition is performed using the projection operators for SRs derived from the group representation theory. The current method is employed to investigate the phonon band structure of disordered face-centered-cubic ${\mathrm{Cu}}_{0.75}{\mathrm{Au}}_{0.25}$, 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. They are found to occur because different combinations of the chemical elements contribute to different regions of frequency.

Figure 1

Two-dimensional representation of the relation between a supercell model and its underlying structure in reciprocal space. White and black circles represent reciprocal lattice points of the supercell model, and the white circles also correspond to reciprocal lattice points of the underlying structure. Green arrows represent ${\mathbf{G}}_i$ in Eq. (12).

Figure 2

Two-dimensional representation of how the projection operators for wave vectors ${\widehat{P}}^{\mathbf{k}}$ in Eq. (17) and for SRs ${\widehat{P}}^{\mathbf{k}\mu }$ in Eq. (24) work on a phonon-mode eigenvector of a supercell model of a disordered system. Circles represent atoms in the system; red and green ones represent the chemical elements X and ${\mathrm{X}}^{\prime }$, respectively, while white ones indicate that the chemical elements are no longer distinguished. Blue arrows on the circles represent the real parts of a mode eigenvector on atoms or its projections. (a) Hypothetical phonon-mode eigenvector $\tilde{\mathbf{v}}$. (b), (c) Projection of $\tilde{\mathbf{v}}$ by ${\widehat{P}}^{\mathbf{k}}$ and ${\widehat{P}}^{{\mathbf{k}}^{\prime }}$ for different wave vectors $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$, respectively. (d)–(g) Further projection of ${\widehat{P}}^{\mathbf{k}}\tilde{\mathbf{v}}\left({\widehat{P}}^{{\mathbf{k}}^{\prime }}\tilde{\mathbf{v}}\right)$ by ${\widehat{P}}^{\mathbf{k}\mu }$ and ${\widehat{P}}^{\mathbf{k}\nu }$ (${\widehat{P}}^{{\mathbf{k}}^{\prime }{\mu }^{\prime }}$ and ${\widehat{P}}^{{\mathbf{k}}^{\prime }{\nu }^{\prime }}$), where $\mu$ and $\nu$ (${\mu }^{\prime }$ and ${\nu }^{\prime }$) are indices for the SRs of ${\mathcal{G}}^{\mathbf{k}}\left({\mathcal{G}}^{{\mathbf{k}}^{\prime }}\right)$.

Figure 3

Two-dimensional representation of how the projection operators for chemical elements ${\widehat{P}}^{\mathrm{X}}$ in Eq. (27) work on a phonon-mode eigenvector of a supercell model of a disordered system. The symbols are used in the same way as those in Fig. 2. (a) Hypothetical mode eigenvector $\tilde{\mathbf{v}}$. (b), (c) Projection of $\tilde{\mathbf{v}}$ by ${\widehat{P}}^{\mathrm{X}}$ and ${\widehat{P}}^{{\mathrm{X}}^{\prime }}$. (d), (e) Further projection of ${\widehat{P}}^{\mathrm{X}}\tilde{\mathbf{v}}$ and ${\widehat{P}}^{{\mathrm{X}}^{\prime }}\tilde{\mathbf{v}}$ by ${\widehat{P}}^{\mathbf{k}}$. (f)–(i) Further projection of ${\widehat{P}}^{\mathbf{k}}{\widehat{P}}^{\mathrm{X}}\tilde{\mathbf{v}}$ and ${\widehat{P}}^{\mathbf{k}}{\widehat{P}}^{{\mathrm{X}}^{\prime }}\tilde{\mathbf{v}}$ by ${\widehat{P}}^{\mathbf{k}\mu }$ and ${\widehat{P}}^{\mathbf{k}\nu }$.

Figure 4

Phonon band structure of disordered fcc ${\mathrm{Cu}}_{0.75}{\mathrm{Au}}_{0.25}$ calculated using the band-unfolding method. The upper and the lower panels show the results obtained from the 32- and the 108-atom supercell models, respectively. White circles represent experimental data at room temperature [22], and blue dashed curves represent the result calculated using the ICPA method [23].

Figure 5

(a) Phonon band structure of a typical pure fcc metal. Here the result of fcc Cu obtained using first-principles calculations is shown. The label of the SR is also shown for each phonon branch in the Mulliken notation in blue text. (b) Atomic displacements of the $B_2$ mode at the wave vector $\langle 0.5,0.5,0.0\rangle$ at a certain moment. Squares represent the conventional fcc unit cells, white and black circles represent the atoms in different layers along the $\langle 001\rangle$ direction, and blue arrows represent the atomic displacements.

Figure 6

Decomposition of the unfolded phonon band structure of disordered fcc ${\mathrm{Cu}}_{0.75}{\mathrm{Au}}_{0.25}$ according to the SRs along (a) the $\langle 100\rangle$, (b) the $\langle 110\rangle$, and (c) the $\langle 111\rangle$ directions. The result is obtained from the 108-atom supercell model. The leftmost panels show the total spectral function, while the other panels show the partial spectral functions corresponding to different SRs specified at the upper left of the panels.

Figure 7

(a) Decomposition of the partial spectral function for the $B_2$ mode along the $\langle 110\rangle$ direction into the contributions of the combinations of the chemical elements for disordered fcc ${\mathrm{Cu}}_{0.75}{\mathrm{Au}}_{0.25}$. The result is obtained from the 108-atom supercell model. The leftmost panel shows the partial spectral function for the $B_2$ mode in total, while the other panels show the contributions of the combinations of chemical elements specified at the upper left in the panels. (b) The same as (a), but for the doubly degenerate $E$ modes along the $\langle 111\rangle$ direction.

Figure 8

Unfolded phonon band structure of disordered fcc ${\mathrm{Cu}}_{0.75}{\mathrm{Au}}_{0.25}$ calculated using the EAM interatomic potentials. The sizes of the supercell models are shown at the upper left in the panels. White circles represent experimental data at room temperature [22].

Figure 9

Distributions of the second-order force constants between the 1NN atomic pairs with respect to interatomic distance for disordered fcc ${\mathrm{Cu}}_{0.75}{\mathrm{Au}}_{0.25}$ obtained from the 108-atom supercell model. Red circles, blue squares, and green triangles are for Cu-Cu, Cu-Au (Au-Cu), and Au-Au pairs, respectively. The first and the second chemical components are supposed to be on $(0,0,0)$ and $(1/2,1/2,0)$, respectively, in fractional coordinates for the conventional fcc unit cell. Each panel corresponds to the symmetrically inequivalent element of the force constants specified at the upper left in the panel in Cartesian coordinates, where the first and the second symbols are for the first and the second chemical components, respectively.