Degenerate Soft Modes and Selective Condensation in BaAl$_2$O$_4$ via Inelastic X-ray Scattering

TL;DR

This paper addresses how the ferroelectric transition in BaAl2O4 arises near a structural quantum critical point. The authors combine meV-resolution inelastic x-ray scattering with first-principles phonon calculations to identify two soft modes at the M and K points that soften toward $T_{ m C}$ and are nearly degenerate above it; the M-point mode condenses to form a new acoustic branch below $T_{ m C}$, while the K-point mode does not condense and instead hardens. The near-degeneracy and selective condensation reveal a delicate competition between two instabilities and demonstrate that structural fluctuations from both modes underpin the Sr-substitution–induced structural quantum critical point. This work provides direct experimental evidence for soft-mode-driven structural transitions at zone boundaries and links to rigid-unit modes in tetrahedral tilting, offering insight into lattice-driven quantum criticality.

Abstract

BaAl$_2$O$_4$ is a ferroelectric material that exhibits structural quantum criticality through chemical composition tuning. Although theoretical calculations and several diffraction experiments have suggested the involvement of a soft mode in its ferroelectric structural phase transition, direct experimental verification is still lacking. In this study, we successfully observed two soft modes of BaAl$_2$O$_4$ using x-ray inelastic scattering, providing direct experimental evidence for their role in the structural phase transition. Furthermore, we reveal that the soft modes at the M and K points are nearly degenerate in energy, indicating a delicate balance in which either mode could potentially freeze. The K-point mode simultaneously softens toward the transition temperature ($T_{\rm C}$) in a manner nearly identical to the M-point mode. However, the phase transition condenses only at the M point, with the M-point mode stabilizing as an acoustic mode in the low-temperature structure and the K-point mode hardening as temperature decreases.

Figures (5)

• Figure 1: (a) ($hk$8) cross section of the reciprocal space of the hexagonal lattice. Inelastic x-ray scattering experiments were performed along the $\Gamma$--M and $\Gamma$--K directions, with the scattering vectors $\bm{Q} = (h\; 0\; 8)$ and $\bm{Q} = (h\; h\; 8)$, respectively, as shown by green arrows. (b) Calculated phonon dispersion of the high-temperature phase of BaAl$_2$O$_4$ obtained along the $\Gamma$--M--K--$\Gamma$ path. The three acoustic branches are labeled A, B, and C, and the optical branch just above them is labeled D. Branch E is considered to be an optical branch corresponding to the weak scattering observed in the spectrum at small $h$ around $\pm$13 meV in Fig. 2. Panels (c), (d), and (e) visualize the magnitudes of the phonon eigenvectors of Ba, Al, and O atoms in the $z$ direction, respectively (see also the main text). (f) Dynamical structure factor $S(\bm{Q}, E)$ calculated along the $\Gamma$ (008) -- M (1/2 0 8) -- K (1/3 1/3 8) -- $\Gamma$ (008) path. Larger marker sizes and darker colors indicate higher values of $S(\bm{Q}, E)$.
• Figure 2: (a) IXS spectra measured at 650 K with various scattering vectors $\bm{Q} = (h\; 0\; 8)$ along the $\Gamma$-M direction. The inelastic peaks indicated by triangles are attributed to branch B shown in Fig. 1(f). Additionally, small inelastic peaks are observed around $\pm$13 meV, as indicated by circles, which is attributed to branch E. The inelastic peaks marked by arrows in the spectrum at $h$ = 0.402 correspond to the M-point soft mode. (b) IXS spectra measured at 650 K with various scattering vectors $\bm{Q} = (h\; h\; 8)$ along the $\Gamma$-K direction. Similar to the spectra observed in Panel (a), inelastic peaks are observed as indicated by triangles and circles, which are attributed to branch B and E, respectively. The inelastic peaks indicated by arrows in the spectrum at $h = 0.223$ correspond to the K-point soft mode.
• Figure 3: Temperature dependence of inelastic scattering spectra measured near the M and K points. Panels (a) and (c) show the spectra measured above $T_{\rm C}$, while Panels (b) and (d) show those measured below $T_{\rm C}$. The scattering vectors are: (a) $\bm{Q}$ = (0.45 0 8), (b) $\bm{Q}$ = (0.398 0 8), (c) and (d) $\bm{Q}$ = (0.279 0.279 8). The solid black curves represent the fitting results using the DHO model. Resolution functions corresponding to each spectrum are shown at the bottom of each panel. The parameters of peak center and peak width obtained from the fit are included in Supplemental Material Supple.
• Figure 4: Experimentally obtained phonon dispersions along the (a, c) $\Gamma$-M and (b, d) $\Gamma$-K directions. Panels (a) and (b) show the results measured above $T_{\rm C}$, while Panels (c) and (d) show those below $T_{\rm C}$. The open circles plot the peak positions of the phonon indicated by triangles in Fig. 2 and Fig. 3. The upward and downward triangles represent the peak positions of the low-energy phonons indicated by arrows in Fig. 2 and Fig. 3. The broken curves represent the calculated phonon dispersion. Solid curves are guides to the eye, representing the dispersion relation of branch C at representative temperatures estimated as discussion in the main text.
• Figure 5: Temperature dependence of the squared soft-mode energies measured at the M point (1/2 0 8) and the K point (1/3 1/3 8). The straight lines represent linear fits to the squared energies of each mode as a function of temperature. The temperatures at which the two approximate linear fits cross $E=0$ agree within the experimental uncertainty. This temperature is slightly higher than the reported $T_{\rm C}$ = 450 K. Our previous single-crystal x-ray diffraction experiments show that the superlattice reflections begin to develop a precursor feature at temperatures above $T_{\rm C}$Ishii_PRB93, which may also be reflected in the present result.