Phonon dispersion in ${\text{Sr}}_2{\text{RuO}}_4$ studied by a first-principles cumulative force-constant approach

The phonon frequencies along several high symmetry lines for ${\text{Sr}}_2{\text{RuO}}_4$ were calculated using the real-space cumulative force-constant approach. Except for the highest ${\Delta }_1$ $\left(\xi 00\right)$ and ${\Sigma }_3$ $\left(\xi \xi 0\right)$ dispersions, all other 54 branches were quantitatively described very well, including the anomalous softening of the ${\text{RuO}}_6$ octahedra rotational mode which was the lowest ${\Sigma }_3$ dispersion along the $\left(\xi \xi 0\right)$ direction. We did not see any imaginary phonon modes for ${\text{Sr}}_2{\text{RuO}}_4$ and demonstrated that the softening of the rotational mode of the ${\text{RuO}}_6$ octahedra was not due to the anharmonic effects.

Figure 1

(Color online) Calculated phonon dispersions (solid curves) of ${\text{Sr}}_2{\text{RuO}}_4$ using a $2a\times 2a\times 2c$ supercell and the finite difference approach. The symbols are the measured data by Braden et al. (Ref. 22) with, from left to right, ○: ${\Delta }_1$, ●: ${\Delta }_2$, △: ${\Delta }_3$, and ▲: ${\Delta }_4$ representing $\Gamma \text{-Z}$ $\left(\xi 00\right)$ dispersions; ○: ${\Lambda }_1$, ●: ${\Lambda }_2$, and △: ${\Lambda }_3$ for $\text{Z-}\Gamma$ $\left(00\xi \right)$ dispersions; and ○: ${\Sigma }_1$, ●: ${\Sigma }_2$, △: ${\Sigma }_3$, and ▲: ${\Sigma }_4$ for $\Gamma \text{-X}$ $\left(\xi \xi 0\right)$ dispersions.

Figure 2

(Color online) Calculated phonon dispersions (solid curves) of ${\text{Sr}}_2{\text{RuO}}_4$ using a $2\sqrt{2}a\times 2\sqrt{2}a\times c$ supercell and the finite difference approach. The symbols are the measured data by Braden et al. (Ref. 22) with the same meaning as Fig. 1.

Figure 3

(Color online) Calculated phonon dispersions (solid curves) of ${\text{Sr}}_2{\text{RuO}}_4$ using a $2\sqrt{2}a\times 2\sqrt{2}a\times c$ supercell and the linear response approach. The symbols are the measured data by Braden et al. (Ref. 22) with the same meaning as Fig. 1.