Figure 4: Resonance modes coherently trapping ferroelectric modes.

(a) Single resonance mode (RM) radiating energy through the TO phonon with which it is in resonance. (b) Anderson localization scheme for ferroelectric TO phonons eminating from and then scattered by resonance modes. Because they remain in phase, radiating ferroelectric phonons following two equal but opposite scattering paths (I, II) constructively interfere on returning to the source, which tends to coherently trap waves in disordered systems³⁵. (c) Modification to the dispersion curves from an Anderson-localized standing TO phonon at the resonance wave vector. (d) Phonons trapped via Anderson localization at the resonance decay in real space exponentially with a localization length L, which transforms in reciprocal space to a q-space convolution of the phonon peak at q₀ with a Lorenztian width 1/πL (full-width at half-maximum). (e) Local mode intensities versus Q=[2, K, 0]. At 688 K the local mode intensity is flat, which is consistent with a fully localized resonance mode. At 488 K (red points), a Lorenztian function fit (red line) to the data results in L=2.0±0.2 nm and K₀=0.196±0.005 (which is at the TO resonance crossing), a form consistent with Anderson-localized TO phonons. Local mode intensities (peak areas) were determined from peak fits to the data in Figs 1 and 2 (see Supplementary Fig. 6 for details). (f) Effect of anisotropy in the TO phonon (red surfaces) on the trapping wave vectors, q₀, determined by the intersection of the TO phonon surface with the resonance mode (black surface).