Incipient modulated phase in Sr$_{1-x}$Ca$_{x}$TiO$_3$

Abstract

Nanometer-scale modulations can spontaneously emerge in complex materials when multiple degrees of freedom interact. Here we demonstrate that ferroelectric Sr$_{1-x}$Ca$_x$TiO$_3$ lies in close proximity to an incipient structurally modulated phase. Using inelastic neutron and X-ray scattering, we show that upon cooling, dipolar fluctuations strongly couple to and soften the $c_{44}$ transverse acoustic mode. We identify the wavevector at which this softening is maximal, thereby defining the characteristic length scale of the modulation. Calcium substitution enhances both the amplitude and the wavevector of the softening by strengthening the ferroelectric and antiferrodistortive instabilities. Our results demonstrate that nonlinear flexoelectric phonon coupling tends to stabilize a modulated state that cooperates with, rather than competes against, the other lattice instabilities in SrTiO$_3$.

Figures (4)

• Figure 1: Lattice instabilities in Sr$_{1-x}$Ca$_{x}$TiO$_3$ (a) Antiferrodistortive (AFD) transition at $T_{\mathrm{AFD}}$. (b) Polar distortion: Ca$^{2+}$ substitution stabilizes ferroelectricity along the [110] direction. (c) Schematic of TO–TA coupling in Sr$_{1-x}$Ca$_x$TiO$_3$: softening of the transverse-optical (TO) mode induces a concomitant softening of the transverse-acoustic (TA, $c_{44}$) branch, which is maximal at $Q_{\mathrm{min}}$, signaling a tendency toward a modulated phase. (d) Phase diagram of Sr$_{1-x}$Ca$_x$TiO$_3$: $T_{\mathrm{AFD}}$ and Curie temperature $T_C$ as a function of Ca concentration, compared with previous studies Lima2015Bednorz1984. (e) Dielectric constant as a function of temperature for three Ca concentrations; the inset highlights the low-temperature region below 35 K. (f) Real-space illustration of TO–TA coupling: dipolar fluctuations are accompanied by a transverse strain modulation with wavelength $\sim 1/Q{\mathrm{min}} \simeq 15$ nm in pristine SrTiO$_3$.
• Figure 2: TA softening in Sr$_{1-x}$Ca$_x$TiO$_3$ : (a–b) Energy scans at $\mathbf{Q} = (H,H,2)$ for $H = K = 0.014$–0.032 at (a) $T = 150$ K and (b) $T = 20$ K for $x$ = 1.5 $\%$. Fits including a convolution with the experimental resolution are shown in dotted lines (see SM)(c) Dispersion of the TO (squares) and TA (circles) modes for $x = 0.8\%$ at $T = 150$ K (dark blue: INS; light blue: IXS) and $T = 20$ K (orange: INS; red: IXS). (d) Same as (c) for $x = 1.5\%$. Both dispersions are fitted using the mean-field Landau–Ginzburg–Devonshire solution Morozovska2017 (see text). At the highest doping, the model fails to capture the abrupt change of the TA dispersion around $H = K = 0.04$.
• Figure 3: Temperature dependence of the TA softening in Sr$_{1-x}$Ca$_x$TiO$_3$ : Energy scans at $\mathbf{Q} = (0.027,0.027,2)$ in the Ca-doped sample ($x = 1.5\%$) from $T = 10$–200 K (high-resolution mode). Fits including a convolution with the experimental resolution are shown in dotted lines (see SM) (b–c) Temperature dependence of the TA mode: (b) energy and (c) intensity for $x$=0 (dark blue) at $H$=$K$=0.017 and 1.5$\%$ (dark red) at $H$=$K$=0.027. The dashed vertical line indicates the structural transition $T_{\text{AFD}}$. The TA mode softening begins below $T_{\text{AFD}}$.
• Figure 4: Doping evolution of the TA softening in Sr$_{1-x}$Ca$_x$TiO$_3$: Q-dependence of the TA softening between 20 K and 150 K, $\frac{\Delta \omega_{TA}}{\omega_{TA}} = [\omega_{TA}(20~\mathrm{K}) - \omega_{TA}(150~\mathrm{K})]/\omega_{TA}(150~\mathrm{K})$, for (a) $x = 0$, (b) $x = 0.8\%$, and (c) $x = 1.5\%$ in Sr$_{1-x}$Ca$_x$TiO$_3$. Dashed lines are a guide to the eye. Closed circles are results from this work; open circles are taken from Ref. Fauque2022. (d) Doping evolution of $Q_{\rm min}$. The gray band indicates the wavevector where the TO–TA coupling was found to be maximal in Ref. orenstein2025. (e) Dispersion of the TO (dotted lines) and TA (solid lines) modes in the presence of a flexoelectric coupling $f$Morozovska2017. As the TO mode softens, $f$ approaches the critical value $f_c$, above which an incommensurate phase is stabilized. Closed squares and circles denote the TO and TA modes, respectively, for $x = 0.8\%$. (f) Real-space schematic illustrating the two characteristic length scales in Sr$_{1-x}$Ca$_x$TiO$_3$: around a Ca dopant, a ferroelectric moment forms over a length $\ell_0$, which clusters with other dipoles over a longer scale $\ell_{\rm min} = 2\sqrt{2}\pi/Q_{\rm min}$.