Three-mode coupling interference patterns in the dynamic structure factor of a relaxor ferroelectric

A longstanding controversy for relaxor ferroelectrics has been the origin of the “waterfall” effect in the phonon dispersion curves, in which low-energy transverse phonons cascade into vertical columns. Originally interpreted as phonons interacting with polar nanoregions (PNRs), it was later explained as an interference effect of coupling damped optic and acoustic phonons. In light of a recently discovered PNR vibrational mode near the “waterfall” wave vector [M. E. Manley, J. W. Lynn, D. L. Abernathy, E. D. Specht, O. Delaire, A. R. Bishop, R. Sahul, and J. D. Budai, Nat. Commun. 5, 3683 (2014)], we have reexamined this feature using neutron scattering on [100]-poled PMN-30%PT $[0.6\mathrm{Pb}\left(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3}\right){\mathrm{O}}_3-0.3\mathrm{PbTi}{\mathrm{O}}_3]$. We find that the PNR mode couples to both optic and acoustic phonons and that this results in complex patterns in the dynamic structure factor, including intensity pockets and peaks localized in momentum-energy space. These features are fully explained by extending the mode-coupling model to include three coupled damped harmonic oscillators representing the transverse optic, acoustic, and PNR modes.

Figure 1

Inelastic neutron-scattering measurements of the phonon dispersion curves for [100]-poled PMN-30%PT. (a) In-plane dispersion along $\mathit{Q}=[2,K,0]$, which selects for transverse modes with atom displacements parallel to the [100] poling direction. The TO, PNR, and TA mode dispersion curves form arcs that are symmetric with respect to positive and negative values of $K$, and all three peak at the zone boundaries at $K=+0.5$ and $-0.5$. The red lines indicate the trajectories of data collected at fixed crystal angles corresponding to Bragg conditions. These trajectories do not follow the crystal symmetry (see text). (b) Out-of-plane dispersion along $\mathit{Q}=[2,-0.4,L]$. The TO, PNR, and TA modes are symmetric with respect to positive and negative values of $L$. The multiple scattering with flange features is a spurious scattering effect and is not symmetric with respect to positive and negative values of $L$. These slices of the data, and other similar ones, are used to separate real features from spurious effects.

Figure 2

Inelastic neutron scattering of the “waterfall” feature sliced into out-of-plane sections in energy-momentum space. The in-plane image plots shown for reference at the top are after Ref. [26]. (a)–(c) Perpendicular to the poling direction, near $(0,-2,0)$, no PNR mode [12] is detected between the TO and TA modes and a single “waterfall” feature appears. (d)–(f) Along the poling direction, near (200), a separate PNR mode appears between the TO and TA modes. (e) A “mini-waterfall” feature appears between the TO and PNR mode. (f) TO-TA coupling dominates the main “waterfall” feature. (g) “Mini-waterfall” appearance changes in (300) zone.

Figure 3

Mode-coupling model calculations describing the “waterfall” feature sliced into out-of-plane sections in energy-momentum space in Fig. 2 (see text for details). The panels (a)–(g) correspond to the experimental results with the same labels shown in Fig. 2.