Mapping the structural transitions controlled by the trilinear coupling in Ca_3-xSr_xTi₂O₇

Proper ferroelectrics (FEs) are characterized by polarization which is the primary order parameter originating from zone-center polar lattice instability. On the other hand, in hybrid improper ferroelectrics (HIF), polarization is part of a more complex lattice distortion arising from a combination of two nonpolar lattice modes with different symmetries, the so-called trilinear coupling.^1,2 In perovskites, these modes correspond to oxygen polyhedral distortions, commonly rotations or tilts, which can, in addition, induce magnetoelectricity and weak ferromagnetism. Such an electric-field-controllable mechanism can be extremely attractive for device applications if operated at room temperature and hereby attracts enormous attention.³ Moreover, it has been shown for Ca_3–xSr_xTi₂O₇ crystals recently that there is a unique domain topology associated with various types of oxygen octahedral distortions having a direct impact on ferroelectric properties.^4,5,6

The family of HIF compounds reveals an amazingly rich set of structure transitions, as was demonstrated on Ca_3–xSr_xTi₂O₇,⁴ Sr₃Zr₂O₇,⁷ and Ca₃Mn₂O₇,⁸ recently. These materials belong to the Ruddlesden-Popper phases described by the general formula A_n+1B_nO_3n+1 (n = 2). The structure consists of perovskite ABO₃ building blocks stacked along the [001]-axis with a rocksalt AO layer interspersing every two perovskite unit cells. In the ferroelectric (FE) state, A₃B₂O₇ forms the orthorhombic polar space group A2₁am, which corresponds to the distorted tetragonal I4/mmm structure where the paraelectric (PE) phase is eventually established upon heating. How the orthorhombic structure transforms into the distortion-free aristotype phase upon heating has been the subject of many theoretical works.^1,2,9 Experimentally, it has been demonstrated that each of these compounds exhibits its own unique route (summarized in Table I).

Improper hybrid ferroelectricity has been experimentally demonstrated for the first time in the orthorhombic Sr-doped Ca₃Ti₂O₇.³ This distortion includes the rotation of the BO₆ octahedra around the [001]-axis described by the in-phase rotation mode $X2+$ and the diagonal (with respect to the tetragonal-like basal plane) tilting mode $X3−$ around the [110]-axis, which are simultaneously coupled to the $Γ5−$ polar zone-center mode, thereby creating the trilinear coupling.¹ By introducing the Sr-substitution, one can tune the robustness of distortions as the paraelectric Sr₃Ti₂O₇ phase crystallizes in the I4/mmm space group. Theoretical works predicted the existence of an intermediate phase, either Amam or Acam;^1,9 however, the combined study of Ca_3–xSr_xTi₂O₇ provided a different picture⁴ by discovering the intermediate paraelectric P4₂/mnm phase in Ca₂SrTi₂O₇ characterized by the $X3−$ [100] tilting mode oriented along the tetragonal-like directions. The impact of the discovered structure on physical properties has been studied experimentally¹⁰ and also theoretically by a first-principles study recently.¹¹ The A2₁am-P4₂/mnm transition corresponds to the relaxation of the in-phase rotation $X2+$⁠.⁴ Crossing the P4₂/mnm-I4/mmm border is related to the relaxation of the $X3−$ tilting mode.

Sr₃Zr₂O₇ undergoes a first-order FE-PE transition, where the paraelectric polymorph competes with the polar phase and emerges from a trilinear coupling of rotation and tilt modes interacting with an antipolar mode.⁷ The structure transforms from A2₁am to I4/mmm via two intermediate paraelectric phases upon heating: first, the $X1−$ rotation mode is lost in the second-order Pnab → Amam transition and then the tilting mode $X3−$ is lost in the second-order Amam → I4/mmm transition. The A2₁am → Pnab structure transition reveals hysteresis corresponding to a first-order transition linked to the competition of the $X1−$ out-of-phase and the $X2+$ in-phase rotations.

In Ca₃Mn₂O₇, the high-temperature I4/mmm structure undergoes a transition to an intermediate PE orthorhombic Acaa structure upon cooling and then changes to A2₁am.¹² As the temperature is further lowered, an antiferromagnetic order sets in at 115 K. The Acaa symmetry is generated by the out-of-phase rotation $X1−;$ thus, the transition pathway from Acaa to A2₁am results in the competition of the $X1−$ out-of-phase and $X2+$ in-phase rotations, thereby related to peculiar physical phenomena.⁸ It is worth noting that the Mn substitution on the Ti-site of Ca₃Ti₂O₇ has also been studied both experimentally^13,14 and theoretically;¹⁵ for the Ti-rich side, I4/mmm goes directly to polar A21am symmetry upon cooling while the structural transition processes via the Acaa symmetry for the Mn-rich side.¹³ 

In this work, we present the temperature evolution of structure phase transitions revealed by varying the strontium concentration in Ca_3–xSr_xTi₂O₇ (0.5 ≤ x ≤ 0.9) using in situ high-temperature x-ray diffraction. We show that on the Ca-rich side the structure transition processes directly from I4/mmm to polar A21am (x < 0.8), while the intermediate phase P4₂/mnm is preferred in the transition pathway for the Sr-rich compositions. The structure-Sr concentration phase diagram is constructed and discussed within the framework of related HIF compounds.

The polycrystalline Ca_3–xSr_xTi₂O₇ samples (x = 0.5, 0.8, 0.9) were prepared using a solid-state reaction method as described in Ref. 4. The Ca_2.15Sr_0.85Ti₂O₇ single crystals were grown by an optical floating zone method.³ The samples with x = 0.5, 0.8, and 0.9 were characterized using the high-resolution x-ray diffractometer (Bruker XRD D8 Discover with Cu Kα source) equipped with the high-temperature chamber HTK 1200 N. Additionally, the sample with x = 0.5 was measured on the high-resolution synchrotron powder diffraction beamline I11 at Diamond Light Source using the multianalyzer crystal detector stages and 15 keV x-rays calibrated against NIST SRM 640c silicon powder. Samples were loaded in quartz capillaries and variable temperature heating provided by a cyberstar hot air blower.¹⁶ The neutron diffraction of pulverized single crystals with x = 0.85 was measured on the D2B high-resolution powder diffractometer (ILL). The single-crystal diffraction on the x = 0.85 sample from a different batch was measured using the hot neutron four-circle D9 diffractometer (ILL) equipped with a four-cycle furnace. The sets of D2B and D9 data can be found in Ref. 17. The diffraction data were analyzed using FullProf¹⁸ software. The differential scanning calorimetry (DSC) was measured using SETSYS Evolution 24 instrument (SETARAM) in the He atmosphere. The heating/cooling rate was 10 K/min and the transition temperatures were determined by the onset of the observed peaks.

Powder neutron diffraction study of a sample with x = 0.85 revealed the decay of the (237) and (033) reflections at ∼550 °C upon increasing temperature as shown in Figs. 1(a) and 1(b). The single-crystal neutron diffraction of a sample from a different batch observed the decay of the (033) Bragg peak at slightly higher temperature ∼580 °C, which might be caused by a negligible variation of the strontium concentration.

As the (237) reflection has a lower intensity and higher 2θ in the x-ray diffraction data compared to the (033) reflection, the latter was chosen to identify transitions across various structure phases. The integrated intensities I₍₀₃₃₎ of the superlattice peaks are expected to scale as the square of the order parameter Q for these phase transitions, where Q is given by the rotation and tilting angles ω_R and ω_T of the oxygen octahedra, respectively. The angles are given by the strontium concentration as was shown by the synchrotron x-ray diffraction experiment.⁴ Simulation of the (033) reflection integrated intensity for different octahedral rotations (tilts) which corresponds to the increasing strontium concentration is shown in Fig. 1(c). It clearly shows the decrease of intensity by ∼15% for the sample with x = 0.8 and ∼18% for sample with x = 0.9 from the undoped Ca₃Ti₂O₇. The values of I₍₀₃₃₎ were obtained by fitting of the peak profile using the Lorentz function. The (033) reflection of the x = 0.8, 0.85, and 0.9 samples was scanned in the 2θ region from 51.1° to 53.1° upon cooling using the Bruker diffractometer. In the case of x = 0.5 sample, the reflection was tracked in the 2θ region from 16.0° to 16.5° upon warming using the I11 instrument. The resulting scans upon decreasing temperature are summarized in Fig. 2. In all four compounds, a gradual decrease of the (033) intensity with increasing temperature is revealed. The free energy of the HIF superlattice system in terms of Ginzburg-Landau (GL) theory^19,20 is given by

where $ω=(1√2ωR,1√2ωR,ωT)$⁠, E is the external electric field, and P is the electrical polarization of Ca₃Ti₂O₇. The quadratic term and the biquadratic terms coupling the rotations and polarization stated in Eq. (1)²⁰ are omitted with respect to the trilinear term for temperatures close to T_C. Assuming the first-order nature of the phase transition and the collaborative trilinear interaction, the coefficients A₁, A₃, and C₄ are expected to be positive while A₂ and C₃ negative, respectively. The equilibrium solution providing non-negative ω² obtained by minimizing the free energy is

where the term $C3′$ is defined analogically to Ref. 4, providing a first-order transition for A₂ < 0, tricritical transition for A₂ = 0, and a continuous transition for A₂ > 0. With respect to the I₍₀₃₃₎(T) error bars, qualitative results, i.e., the signs are given for the fitting coefficients except the transition temperatures.

In the case of Ca_2.5Sr_0.5Ti₂O₇, the intensity decreases monotonically down to background values at T_C using data combined from the I11 and Bruker measurements as shown in Fig. 2(a). The structure of the room-temperature phase corresponds to the A2₁am space group, while the phase above T_C can be described by the I4/mmm space group. The fit of the data for the Ca_2.5Sr_0.5Ti₂O₇ sample offer T_C = 605(15) °C, positive A₁ and A₃ parameters, negative $C3′$ coefficient in line with the trilinear coupling, and the A₂ coefficient approaching zero values within the large error bars proposing the composition with x_Sr = 0.5 is very close to the tricritical point.

In Figs. 2(b) and 2(c), we observe a different situation. Upon cooling, the temperature dependence of the (033) reflection intensity is characterized by a steplike structure emerging at T_C1 = 360(25) °C in Ca_2.2Sr_0.8Ti₂O₇ and T_C1 = 280(30) °C in Ca_2.15Sr_0.85Ti₂O₇. This remarkable change of slope signifies a transition from the room-temperature A2₁am space group to the tetragonal phase described by the P4₂/mnm symmetry. The corresponding coefficients A₁ and A₃ are positive, while $C3′$ and A₂ are negative for both systems, pointing to the first-order nature of the A2₁am-P4₂/mnm structural transition as expected from the group-subgroup relations. The (033) intensity decays completely at T_C2 = 480(15) °C in Ca_2.2Sr_0.8Ti₂O₇ (T_C2= 550(18) °C in Ca_2.15Sr_0.85Ti₂O₇) when the high-temperature I4/mmm structure is established. The fitting parameter A₂ is clearly positive for Ca_2.2Sr_0.8Ti₂O₇ and Ca_2.15Sr_0.85Ti₂O₇, respectively, revealing a continuous P4₂/mnm-I4/mmm transition in agreement with the Landau theory. To verify the order of the transitions obtained by the fits, we have measured the Ca_2.15Sr_0.85Ti₂O₇ sample both upon cooling and warming. A clear declination from the warming curve signifying hysteretic behavior and hence a discontinuous transition is observed below ∼350 °C, although lower temperatures could not be measured due to the gradual degradation of the sample in the furnace.

The intensity evolution in Ca_2.1Sr_0.9Ti₂O₇ is different from the previous two samples, as shown in Fig. 2(d). Here, the smooth decay of the (033) intensity suggests a second-order, single structure transition between the two tetragonal phases P4₂/mnm and I4/mmm at T_C2= 510(10) °C with the positive A₂ parameter.

We further note that the DSC experiment on Ca_2.15Sr_0.85Ti₂O₇ does not show any anomaly up to 1300 °C [see Fig. 3(b)], which might be caused by the structure disorder increasing for higher strontium concentration, smearing out the discontinuous character of the transition at T_C1∼ 280 °C. On the other hand, the heating and cooling curves obtained on Ca₃Ti₂O₇ [inset of Fig. 3(b)] still show a transition at T_C∼ 760 °C with hysteresis ΔT ∼ 10 °C, which agrees well with the DSC data presented in Ref. 21.

The data are summarized in the temperature-concentration phase diagram shown in Fig. 3(a). It can be separated into three regions O, T′, and T defined by the A2₁am, P4₂/mnm, and I4/mmm space groups. Clearly, the Ca-rich side reveals a direct transition from the orthorhombic to the undistorted tetragonal structure, while a narrow region of the intermediate phase opens in the Sr concentration range 0.6 < x < 0.8 and broadens with increasing x. According to Ref. 4, the undistorted I4/mmm phase is finally stabilized for x > 1. The phase diagram includes data obtained from the in situ x-ray diffraction, ex situ observation of twin-change under polarized optical microscopy (POM) after various heat treatments, in situ transmission electron microscopy (TEM) of Ca_2.1Sr_0.9Ti₂O₇,⁴ and DSC (this work and Ref. 10).

Turning our attention to the character of the structure transitions in HIF, we observe an agreement between our and previously published data: a first-order phase transition has been observed on the DSC curves in Ca₃Ti₂O₇²¹ and Ca_3–xSr_xTi₂O₇ for x < 0.5,¹⁰ corresponding to the O → T crossover in line with the group-subgroup relations. Similarly, a discontinuous change of the tilting order parameter when crossing the O → T′ boundary was reported in the Sr-rich composition with x = 0.9.^4,5 The character of the transitions can be revealed also by the domain topology which has been heavily discussed recently.^4,5,6 The formation of Z₂xZ₂ domains with Z₄ vortices is associated with a second-order transition in the Sr-rich compounds. On the other hand, the phases with x < 0.95 form the Z₄xZ₂ domains with Z₃ vortices which are related to discontinuous transitions.⁵ Our data show a discontinuous transition in the case of the A2₁am-P4₂/mnm paths when the trilinear term in the GL fit is considered, while a second-order transition for the P4₂/mnm-I4/mmm crossover is observed in all measured compositions. Interestingly, the GL fit of the A2₁am-I4/mmm transition in Ca_2.5Sr_0.5Ti₂O₇ suggests that this concentration might be on the verge of the tricritical point as indicated in the phase diagram.

In a loose analogy with the mean-field calculations using a simple microscopic Hamiltonian applied on the Aurivillius compounds, we can find a close resemblance with the Ca_3–xSr_xTi₂O₇ phase diagram shown in Fig. 2^,22 and the order of the structure transitions. The temperature evolution of the order parameters ϕ₁ and ϕ₃ [Fig. 1(b)^,22] corresponds very well to the temperature evolution of the (033) intensity in Ca_2.15Sr_0.85Ti₂O₇ and Ca_2.2Sr_0.8Ti₂O₇. Depending on the coupling strength γ (strontium concentration in our case), the character of the O → T′ and T′ → T transitions can be either second- or first-order. Increasing γ leads to a single first-order transition [Fig. 1(c) ²²] which, in this simplified view, corresponds to the Ca_2.5Sr_0.5Ti₂O₇ data. In further analogy with these schemes, we should expect the T′ → T transition to reveal a second-order character in Ca_2.1Sr_0.9Ti₂O₇, and both types of transitions can emerge in the intermediate compositions. Such a scenario implies the existence of a tricritical point, where a first-order transition changes into a second-order one. It has been suggested that the density of domain walls is expected to be smaller for a tricritical phase transition, compared to a second-order one.²³ The tricritical behavior near the x_Sr = 0.5 concentration demands experiments focused on the region of the O, T′, and T phase coexistence investigating a variety of strontium compositions.

Using in situ high-temperature x-ray diffraction and other techniques, we have studied the structural transitions in the Ca_3–xSr_xTi₂O₇ HIF compounds tuning the trilinear coupling strength by varying the Sr-doping level. Consistent with previous studies, the Ca-rich compositions reveal a single direct transition from the orthorhombic ferroelectric A2₁am phase to the tetragonal paraelectric I4/mmm phase. For x ≥ 0.8, an intermediate paraelectric P4₂/mnm phase emerges wedged in between the room-temperature and the high-temperature phase as revealed by two successive transitions. Above x ∼ 0.9, we observe a single phase transition from P4₂/mnm to I4/mmm at high temperatures. The temperature dependence of the I₍₀₃₃₎ intensities and the phase diagram corresponds qualitatively very well to the calculated diagrams of the Aurivillius phases. Using the GL fit with the trilinear term included, we show that the A2₁am-P4₂/mnm structural transition across the series is first-order-like, while the P4₂/mnm-I4/mmm crossover reveals a continuous character, consistently with previous experiments. Moreover, the data suggest the presence of the tricritical behavior for the x_Sr = 0.5 concentration. We have also shown that the simple x-ray diffraction method can be used to identify the order of the transitions using the modified GL fit for the HIF systems. To clarify the related domain configuration and the properties of the domain walls in the vicinity of the critical point where the three phases merge, the next step is to prepare high-quality samples from the concentration range 0.6 < x < 0.8 and explore this part of the Ca_3–xSr_xTi₂O₇ phase diagram by further x-ray and neutron diffraction experiments.

This work was supported by the Institute for Basic Science (IBS) in Korea (No. IBS-R009-G1). We acknowledge the Institute Laue-Langevin (ILL), Grenoble, France, for the allocation of time on D2B and D9 diffractometers. The work was supported within the program of Large Infrastructures for Research, Experimental Development and Innovation (Project Nos. LM2015050 and LTT17019) financed by the Ministry of Education, Youth and Sports, Czech Republic. We acknowledge the support and beam time under Award No. “EE16074” at Diamond Light Source in providing synchrotron research facilities used in this work. The DSC experiments were performed in MGML (https://mgml.eu/) through the program of Czech Research Infrastructures (No. LM2011025). The work at Rutgers University was supported by the DOE under Grant No. DOE: DE-FG02-07ER46382.