Lattice dynamics of the hybrid improper ferroelectrics ${(\mathrm{Ca},\mathrm{Sr})}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$

The structure and lattice dynamics of the hybrid improper ferroelectric compound ${\mathrm{Ca}}_{3-x}{\mathrm{Sr}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ (for $x=0,0.6,$ and 0.9) have been studied with a combination of diffraction, inelastic scattering experiments, Raman spectroscopy, and calorimetry measurements, as well as first-principles simulations. Using inelastic neutron scattering, we have measured the phonon density of states (DOS) for $x=0.9$, which revealed a strong broadening but little change in phonon energies on heating from 10 K to 728 K across the ferroelectric phase transition temperature, $T_{FE}$. Using inelastic x-ray scattering, the momentum-resolved phonon dispersions were measured from 80 K to 950 K on a single crystal (for $x=0.6$), and also revealed a strong phonon broadening but a small energy shift for acoustic modes on heating across $T_{FE}$. Our Raman measurements (for $x=0.6$) showed robust rotational and oxygen breathing modes but soft tilt modes, consistent with previous measurements on similar compounds. Our density functional calculations achieve good agreement with both the phonon DOS and dispersions measured. We did not observe any unusual quadratic dispersion for $c$-polarized transverse acoustic modes, at odds with a recently predicted quasi-2D character, for either undoped ($x=0$) or doped ($x=0.6$) compounds.

Figure 1

(a) ${\mathrm{Ca}}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$ crystal structure in the low-temperature $A{2}_{1}am$ (space group 36) polar phase showing the rotation and tilt of ${\mathrm{TiO}}_6$ polyhedra and in-plane displacement of Ca ions. (b)–(d) Decomposition of the total distortion into the symmetry-adapted modes $X_3^{-}$ (tilt) and $X_2^{+}$ (in-plane rotation), and ${\mathrm{\Gamma }}_5^{-}$ modes of the high-temperature $I4/mmm$ (space group 139) structure.

Figure 2

(a) Heat flow measurements of ${\mathrm{Ca}}_{2.4}{\mathrm{Sr}}_{0.6}{\mathrm{Ti}}_2{\mathrm{O}}_7$ sample across $T_{FE}$ on heating and cooling showing the transition at 751 and 719 K, respectively. (b) Ferroelectric transition temperature $T_{FE}$ as a function of $x$ in ${\mathrm{Ca}}_{3-x}{\mathrm{Sr}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ from Li et al. [16] and from the present study showing nearly linear decrease with increasing $x$.

Figure 3

XRD from a ${\mathrm{Ca}}_{2.4}{\mathrm{Sr}}_{0.6}{\mathrm{Ti}}_2{\mathrm{O}}_7$ single crystal (a) below and (c) above $T_{FE}$ in the $\left[HK0\right]$ scattering plane, and comparison with the respective simulated intensities in panels (b) and (d). In the simulated pattern, the superlattice peaks for $H=2n$ and $K=2n+1$ (i.e., (2,1,0), (2,−1,0)) emerge below $T_{FE}$. Experimental data in panel (a) also show the presence of the superlattice peaks for $H=2n+1$ and $K=2n$ due to twinning of the sample in the orthorhombic phase. Selected superlattice peaks are circled in panels (a) and (b).

Figure 4

Dynamical structure factor $S(\left|Q\right|,E)$ of ${\mathrm{Ca}}_{2.1}{\mathrm{Sr}}_{0.9}{\mathrm{Ti}}_2{\mathrm{O}}_7$ powder sample at multiple temperatures (indicated on each panel) across $T_{FE}\sim 660\mathrm{K}$, measured on ARCS with incident neutron energy $E_i=70\mathrm{meV}$. The intensity is in log scale.

Figure 5

Dynamical structure factor $S(\left|Q\right|,E)$ of ${\mathrm{Ca}}_{2.1}{\mathrm{Sr}}_{0.9}{\mathrm{Ti}}_2{\mathrm{O}}_7$ powder sample at multiple temperatures (indicated on each panel) across $T_{FE}\sim 660\mathrm{K}$, measured on ARCS with incident neutron energy $E_i=140\mathrm{meV}$. The intensity is in log scale.

Figure 6

Dynamical susceptibility ${\chi }^{\prime \prime }\left(E\right)$ of ${\mathrm{Ca}}_{2.1}{\mathrm{Sr}}_{0.9}{\mathrm{Ti}}_2{\mathrm{O}}_7$ calculated from the dynamical structure factor $S\left(E\right)$ using ${\chi }^{\prime \prime }\left(E\right)=[1-exp(-E/k_{\mathrm{B}}T)]S\left(E\right)$ by integrating the data in Figs. 4 and 5 between $3.5<\left|Q\right|<7.5$ and $4.75<\left|Q\right|<10.75{\AA }^{-1}$, respectively, for the incident energy $E_i$ of (a) 70 and (b) 140 meV. Curves in both the panels are offset by 0.15 for clarity.

Figure 7

Phonon density of states (DOS) of ${\mathrm{Ca}}_{2.1}{\mathrm{Sr}}_{0.9}{\mathrm{Ti}}_2{\mathrm{O}}_7$ sample at multiple temperatures (indicated on top of each curve) across $T_{FE}\sim 660\mathrm{K}$ measured at the ARCS instrument with incident neutron energy $E_i=140$ meV. Experimental data are corrected for background, phonon occupation factor, and multiple and multiphonon scattering. On the bottom, DFT-simulated neutron-weighted partial and total phonon DOS of ${\mathrm{Ca}}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$ showing an excellent agreement. The area is normalized to 1 in each case. Curves are offset by 0.01 for clarity.

Figure 8

Phonon dispersions of ${\mathrm{Ca}}_{2.4}{\mathrm{Sr}}_{0.6}{\mathrm{Ti}}_2{\mathrm{O}}_7$ at multiple temperatures measured along different high-symmetry directions below and above $T_{FE}\sim 751\mathrm{K}$ with HERIX. Error bars represent HWHM of the phonon linewidth on either side of the marker. Error bars in phonon energies from DHO fitting are smaller than the size of markers. Thin curves are the simulated phonon dispersions for ${\mathrm{Ca}}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 9

Phonon dispersions of undoped ($x=0$) and doped ($x=0.6$) ${\mathrm{Ca}}_{3-x}{\mathrm{Sr}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ at 300 K along different high-symmetry directions measured with HERIX. Error bars represent HWHM of the phonon linewidth on either side of the marker. Error bars in phonon energies are smaller than the size of markers. Experimental markers are overplotted on DFT-simulated phonon dispersion of ${\mathrm{Ca}}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$ showing an excellent agreement.

Figure 10

(a) Temperature-dependent Raman spectra from ${\mathrm{Ca}}_{2.4}{\mathrm{Sr}}_{0.6}{\mathrm{Ti}}_2{\mathrm{O}}_7$ single-crystal (measurement temperature indicated on each curve) across $T_{FE}\sim 751\mathrm{K}$. The labels from 1 to 12 at the bottom refer to the mode eigenvectors shown in Fig. 11. Vertical dashed lines are guides to the eye showing the relative softening and broadening of the peaks below and above $T_{FE}$. Experimental data are corrected for the background. Curves are offset by 500 units for clarity. The gap from 561 to 617 ${\mathrm{cm}}^{-1}$ is due to the data collections with different spectrometer position in the low and high Raman shift region. (b) Change in phonon frequency of selected peaks as a function of temperature across $T_{FE}$. (c) Temperature dependence of Raman shift of $X_2^{+}$ and $X_3^{-}$ modes below $T_{FE}$.

Figure 11

DFT-simulated phonon energy and corresponding powder-averaged Raman intensity of all 69 Raman-active modes of ${\mathrm{Ca}}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$ below $T_{FE}$. Eigenvectors (normalized by the square root of the mass of the atoms) corresponding to the peaks marked in the top panel are also shown. The peak labeled ${\mathrm{P}}_2$ has an overlap with the high-$T{X}_3^{-}$ distortion shown in Fig. 1. Similarly, peaks ${\mathrm{P}}_3$ and ${\mathrm{P}}_4$ have an overlap with the polar ${\mathrm{\Gamma }}_5^{-}$ distortion and peaks ${\mathrm{P}}_5$ and ${\mathrm{P}}_6$ have an overlap with the in-plane $X_2^{+}$ distortion.

Figure 12

Constant-$\mathit{Q}$ IXS spectra for the transverse acoustic mode at $\mathit{Q}=(0.15,4.0,0)$ and $(0.0,0.2,-10)$ for ${\mathrm{Ca}}_{2.4}{\mathrm{Sr}}_{0.6}{\mathrm{Ti}}_2{\mathrm{O}}_7$. The lines are fits using a damped harmonic oscillator profile convoluted with the instrument resolution (see text). The values of $E$ (in meV) and $\mathrm{\Gamma }$ (in meV) listed in the inset are the fitted phonon energy and phonon linewidth (corrected for resolution), respectively. Negative $E$ corresponds to phonon annihilation and positive $E$ corresponds to phonon creation.