Incommensurately modulated structures in Pb(Zr_1−xSn_x)O₃ single crystals by x-ray diffraction

Antiferroelectric materials have been attracting the attention of scientists for many years because of their potential applications in micromechanical systems, energy storage devices,^1,2 and electrical refrigeration devices.³ Solid solutions of antiferroelectrics with ferroelectrics also possess several useful properties, such as strong piezoelectric and pyroelectric effects.^4–6 There are many reports that the most useful characteristics of these compounds reach the highest values in the region of phase transition, which are usually associated with the structural instabilities. In this context, the study of the phase transition mechanisms in such materials is of high interest.

While the ferroelectric phase transition mechanism via the softening of the TO optic phonon mode at Γ –point [q_Γ = (0, 0, 0)] has already been quite well described,^7–9 the mechanism of the antiferroelectric phase transition is still not fully understood. Lead zirconate, PbZrO₃, the most popular antiferroelectric material, has the ground state with the Pbam space-group and the unit cell being an eightfold multiplication of the primitive cubic unit cell (Pm-3m, Z = 1). This ground state can be rationalized as a combination of lattice distortions characterized by the wave vectors q_Σ = (0.25, 0.25, 0) and q_R = (0.5, 0.5, 0.5), the first one associated with the antiparallel displacements of Pb ions and the second with the octahedral tilt mode. As the phase transition at ∼500 K is marked by a substantial dielectric anomaly, Γ-point instability must play a role in this puzzle as well.^10–12 Several recent works proposed various scenarios of an interplay of these polarization and tilting modes.^11–16 The central feature in this context is a Σ₃ phonon branch populated with transverse vibrations of Pb ions. This branch (connecting Γ and M points) was shown to be soft as a whole, making no discrimination between commensurate and incommensurate modulation of lead displacements.^11–13 

In lead zirconate, the Σ modulation below T_C is normally commensurate; however, on applying elevated hydrostatic pressure, an incommensurate modulation becomes possible in a narrow temperature range.¹⁷ Our experimental investigations revealed that the hydrostatic pressure and a partial substitution of Zr⁴⁺ with smaller Sn⁴⁺ ions on the perovskite B-site position have similar effects on the phase transitions’ sequence.¹⁸ In particular, the temperature–composition phase diagram for PbZr_1−xSn_xO₃ solid solution (in abbreviated form PZS-x) developed based on our Raman, dielectric, and thermodynamic measurements^18–21 indicated the existence of the intermediate antiferroelectric phase (AFE2) similar to that observed under hydrostatic pressure.^17,22 Many experimental investigations performed previously—such as Raman and Brillouin light scattering and also preliminary powder x-ray scattering²³—suggested that this phase may be orthorhombic with incommensurate modulations, however, without a definitive proof (such conjecture could also be done based on earlier electron diffraction measurements on ceramics of heavily substituted PbZrO₃²⁴). Another interesting feature of PZS-x compounds with x > 0.25 is the existence of the next intermediate phase (IM-phase), which seems to be of highly ferroelastic character,²⁵ but the structure of the IM phase is not yet determined. In particular, neither of these new phases has been crystallographically characterized, yet.

The purpose of this work is to investigate the structural properties of Pb(Zr_1−xSn_x)O₃—PZS-x solid solution single crystals through the synchrotron x-ray diffraction. The main motivation is to better understand the phase diagram established for PZS solid solutions¹⁸ and also to confirm the existence of incommensurate modulations in these compounds. An analysis of the diffraction images of PZS-x single crystals with three different concentrations of Sn ions (x-values) yields essential information concerning the different stable phases existing in PZS solid solution including modulated structures and pretransitional effects. We pay special attention to the temperature dependence of the diffraction signal in the parts of the reciprocal space associated with the discussed lattice distortions. This information should be instrumental in understanding the mechanisms governing the phase sequence and associated properties.

PbZr_1−xSn_xO₃ single crystals with a nominal composition of x between 0.05 and 0.3 were grown by the flux method from the high-temperature solution in a Pb₃O₄–B₂O₃ solvent. The composition of the melt used in our experiments was as follows: 2.4 mol. % of PbZr_1−xSn_xO₃, 77 mol. % of Pb₃O₄, and 20.6 mol. % of B₂O₃. The Pb₃O₄ was used instead of PbO to avoid oxygen deficiency and vacancies in the grown crystals. The crystallization was carried out in a platinum crucible covered with a platinum lid heated up to 1350 K under the conditions of a temperature gradient. The temperature gradient in the melt was kept around 5 K/cm. The temperature, measured under the bottom of the crucible, was kept for at least 3 h to enable complete the soaking of the melt, and then, it was lowered at the rate of 2 K/h down to 1120 K. The as-grown PbZr_1−xSn_xO₃ single crystals were etched in diluted acetic acid to remove residues of the solidified flux. The chemical compositions of the obtained crystals were verified using energy-dispersive x-ray spectroscopy (EDS). For use in x-ray diffraction and diffuse scattering experiments, samples were prepared in the form of needles with 80 × 80 × 500 µm³ dimensions. Three samples with different compositions from different places of the phase diagram¹⁸ were chosen for the experiments. The chosen samples had the real composition as follows: x = 0.04 (±0.005), 0.1 (±0.005), 0.28 (±0.006).

The diffraction experiments were carried out at the XRD1 beamline of the Synchrotron Electra Trieste. The data were collected using x rays with an energy of 13 keV using a Pilatus 2M detector. The energy of x rays was set to be below the absorption L3-edge for Pb,²⁶ which in combination with a thin sample ensures an optimal diffraction signal and probing of the bulk of the crystal. To collect a full three-dimensional volume of reciprocal space, the sample was rotated through 180° in ω with a step size of 0.2°. For each crystal orientation, an exposure of 2 s was recorded. The temperature was controlled and monitored with a gas heat blower. The reciprocal space maps of scattering intensity were reconstructed using the CRYSALIS PRO software package,²⁷ and the SNBL Toolbox was used for image preprocessing.²⁸ The structure distortions were analyzed on the basis of superstructure reflections corresponding to certain points of the pseudo-cubic Brillouin zone. Hereafter, we use the following standard notation for such points: R (h + 1/2, k + 1/2, l + 1/2), M (h + 1/2, k + 1/2, l), X (h + 1/2, k, l), Σ (h + 1/4, k + 1/4, l), and S (h + 1/4, k + 1/4, l + 1/2), where h, k, and l are the Miller indices of the (pseudo-)cubic setting. Permutations of indices lead to cubic symmetry-equivalent positions in the case of M, X, Σ, and S points.

The information about the phase diagram was obtained from our previous dielectric and differential scanning calorimetry (DSC) measurement; nevertheless, for each crystal used in the x-ray scattering experiments, the dielectric measurements have been repeated before the synchrotron experiment to have the exact temperatures of the subsequent phase transitions, i.e., AFE1–AFE2, AFE2–IM, and IM–PE. The measurements of dielectric permittivity were performed for the frequency of 100 kHz with the use of an Agilent 4363 LCR meter and a programmable temperature controller Lake Shore (model 331). Measured crystals were covered by silver electrodes and placed in the furnace in which the temperature was controlled by a thermocouple with the accuracy of 0.1 K and the temperature rate was 1 K/min.

The stability and the temperature range of subsequent phases existing in PZS solid solutions, namely, PE (cubic)–IM (yet unknown symmetry)–AFE2 (orthorhombic)–AFE1 (orthorhombic), depend strictly on the composition.¹⁸ It was found that the mechanism of the AFE1–AFE2 phase transition is purely displacive for all compositions of PZS solid solution.²¹ Numerous experiments revealed distinct differences in the physical properties of single crystals with compositions of x below and above 0.25. It is believed that all of this is due to the tricritical point, the existence of which was postulated in early studies.¹⁸ It means that around this concentration, the change from the first- to second-order phase transition at T_C takes place. In our earlier studies of specific heat,¹⁹ we found that above the value of x = 0.25, the latent heat at T_C at IM–PE phase transition is very small and of lambda shape, suggesting the change of the character of the phase transition to second order. In order to have an overview of the phase transition temperatures, the temperature dependences of the dielectric permittivity for PZS-0.04 and PZS-0.1 (a) and also PZS-0.28 (b) are presented in Fig. 1.

There is no significant difference in DS shape anisotropy among different compositions of PZS. The hk0 images show that intensities around the Bragg reflections stretch into crossing streaks enclosing a checkerboard-like background. The intensity of the streaks shows maxima at Γ points with lowering intensity toward M points. In general, the intensity distribution is very similar to that observed for pure PbZrO₃^11,29 and not far from that observed for some chemically disordered Pb-based perovskites.³⁰ In Ref. 29, it was concluded that Pb atoms produce most of the scattering along Γ–M lines, which is in accordance with the measurements presented in Ref. 12, suggesting that the Γ–M line is connected with the lowest frequency Pb-related modes. This is the direction of the mentioned soft Σ phonon branch that, as will be clear from the data for other phases, will host the characteristic modulations. In this sense, one can view changes of the intensity along the Γ–M line as signatures of ordering within the system of atomic (predominantly Pb) displacements already anisotropically correlated in the cubic phase.

Additionally, characteristic narrow type DS lines running along M–R directions were observed on the (hk1/2) plane [Figs. 2(d)–2(f)]. Those lines are present in the paraelectric phase up to high temperatures with gradually decreasing intensity. The observed diffuse scattering seems to be somewhat dependent on composition, i.e., concentration of Sn ions in PZS. R-point (h + 1/2, k + 1/2, l + 1/2) intensities are significantly sharper for PZS-0.1 and PZS-0.28, which suggests a correlation that leads to the doubling of the cubic cell in all directions. In general, the existence of R-point intensity can be associated with anti-phase octahedral tilts, while the in-phase tilts produce M-point reflections.³¹ The M–R diffuse line points to the system of tilts with a phase of rotation disordered from layer to layer.³² In our case, rather broad intensity maxima appear in the vicinity of M-points. As shown earlier for PZO,²⁹ also displacements of Pb atoms (responsible for the whole Γ–M line signal) can contribute to this intensity. Taking this into account, the relative intensity of M and R points suggests that, of the two tilting systems in the paraelectric phase, the in-phase one, if present, is much less significant. It should be made clear that the intensities of observed features related to tilts are far from the level of Bragg diffraction; therefore, the correlation of tilts has a short-range character in the cubic phase.³² From the current experiment, we cannot decisively say if it is static or dynamic.

We now move to the other end of the phase diagrams and show the results for the low-temperature antiferroelectric phase AFE1. The same ground-state structure existing in all samples of PZS is orthorhombic with the Pbam space group,¹⁸ just as in pure PbZrO₃.³³ Figure 3 shows the section of the reciprocal space maps again on the pseudo-cubic hk0 and hk1/2 planes at room temperature for all three samples. All observed spots are Bragg reflections of the AFE1 structure; Σ, M, R, S, and X points of the cubic Brillouin zone become all zone centers in the Pbam space group. As shown in Figs. 3(a) and 3(b), Σ spots related to the fourfold modulation involving Pb antiparallel displacements appear in both diagonal directions, which is due to the extensive twinning observed for PZS-0.04 and PZS-0.1 (for the latter we have noted that all possible orientational domain states are populated in the sample). The only significant DS signal present in all structures has the form of sharp streaks passing through Σ spots.

This type of DS is related to planar defects or stacking faults and, in the context of well-established domains, can be assigned to anti-phase boundaries separating phase-shifted regions of the same modulation wave.

With samples heated to the AFE2 phase, a clear change in the diffraction pattern can be observed (Fig. 4). A complex diffraction pattern appears in all samples generally similar to each other although some significant differences can also be noted. This phase has been predicted to be incommensurate,¹⁸ and indeed, the pseudo-cubic Γ–M direction is populated with incommensurate reflections up to the third-order (marked as Σ′, Σ″, and Σ‴) in the case of all three concentrations [faint spots of the fourth-order can also be discerned, especially in Fig. 4(b)]. In the case of PZS-0.04 and PZS-0.1, Σ reflection originating from the commensurate orthorhombic AFE1 phase is also clearly visible. The intensity of the incommensurate reflections is proportionally larger in Sn-richer phases; in addition, the shape of the Σ′ is characteristically elongated, and a thin DS line runs along the whole Γ–M direction. Half layers [Figs. 4(d)–4(f)] confirm the slightly different status of PZS-0.04 with signatures of commensurate modulation only, while for PZS-0.1 and PZS-0.28, the presence of S′ and S″ reflections confirms the domination of the incommensurate structure.

To better understand the nature of the coexistence of modulations, we plot intensity profiles along Γ–M for PZS-0.04 and PZS-0.1 in Fig. 5. First, we note that the incommensurate character of Σ′ reflections is clear from the reading of the modulation vector, which is of about q = 0.154 of pseudocubic reciprocal lattice units. It means that the periodicity of modulation in this phase should be between 6 and 7 of pseudocubic {110}c planes. For two different directional variants of modulation in PZS-0.04, we observe a signal of similar strength for Σ′ and Σ with a slight prevalence of incommensurate modulation in one direction and the commensurate in the orthogonal one. This implies that in the narrow transitional region in PZS-0.04, both types of modulations coexist. This narrow transitional region corresponds to a change in the slope of the ε(T) dependence just before the T_C [i.e., in the AFE2 region as indicated in Fig. 1(a)]. In the case of PZS-0.1 crystal, as indicated in the same figure, the AFE2 phase can be divided into two regions: the first one [region I or AFE1 + (AFE2)] in which the incommensurate phase starts to develop, i.e., a precursor satellite reflection starts to appear. With increasing temperature [region II or AFE2 + (AFE1)], satellite reflections become dominant and the commensurate phase gradually disappears.

As we already noted, the PZS-0.28 sample is special in the sense that it features a pure incommensurately modulated phase as well as an additional intermediate state (Fig. 1) before reaching the paraelectric phase upon heating. The relatively large amount of Sn ions may result in a more homogeneous distribution of B-site atoms. It is therefore worth paying special attention to the evolution of polar and tilting order with temperature in this composition. In Fig. 6, we present intensity profiles for the Γ–M direction at several temperatures. Indeed the AFE1 and AFE2 phases are well separated with a full change of pattern between 167 °C and 172 °C. Several broad Σ′ reflections are accompanied by the second- and third-order satellites in the AFE2 phase. The broadness of Σ′ indicates that large deviations of the modulation vector exist in the real structure of this phase. With increasing the temperature above 197 °C, i.e., in the IM phase, the intensity of the second- and third-order reflections (Σ″ and Σ”’) goes down. Σ′ reflections remain but with orders of magnitude lower intensity, and they gradually spread into streaks merging with the DS centered at Γ reflections. In Figs. 7(a) and 7(b), we evidence the state at 204.5 °C in the IM phase, where one reciprocal plane has still a signature of incomensuration, while the distribution of DS on the orthogonal plane has side shoulders akin to those observed for disordered PbCd_1/3Nb_2/3O₃.³⁴ The observed distribution is more and more diffuse with increasing temperature through the IM phase forming finally the structure characteristic for diffuse scattering in the PE phase. The changes in the scattering pattern along the Γ–M line between IM and PE phase are gradual, which correlates with the behavior of permittivity (Fig. 1) with a rounded peak and smooth change of slope. The most prominent structural characteristic of the IM phase is the presence of sharp and intense spots at R points [Figs. 7(c) and 7(d)]. While there is some reminiscence of the underlying DS M–R line in Fig. 7(d), the sharpness of spots indicates that the associated tilt system has a long-range order. This agrees with the expectation that the IM phase is ferroelastic. While it is clear from the absence of both M-point reflections and R-point ones with h = k = l that we are dealing with the antiphase (or “minus”) tilts,³¹ a full structure refinement is needed for assigning a corresponding tilt system.

In the context of the highly discussed interplay of polar and tilting distortions in PbZrO₃ and related compounds, it is important to underline that in PZS-0.28, it is the latter distortion that settles first upon cooling. It can also be related to the situation in a similar compound PbHfO₃ that has the incommensurate intermediate phase of similar character,³⁵ which has been argued to be triggered by the soft tilting mode.³⁶ This mode is, just like in the current case of PZS-0.28, considered to drive the antiferrodistortive phase transition from the paraelectric phase to the analogous IM phase in Sn doped PbHfO₃ before the appearance of the incommensurate phase.³⁷ 

We examined the phase transitions’ sequence in PZS single crystals as a function of the composition and temperature by x-ray scattering for three chosen samples with different sequences of the phase transitions. The summary of the reflections existing in different phases of the three measured samples is presented in Table I. Special attention was paid to the intermediate phases occurring in PZS compositions (see the phase diagram in Ref. 18). We observed the appearance of the incommensurate modulations in all compositions that were detected in the AFE2-orthorhombic phase. While for PZS-0.04 and PZS-0.1, the coexistence of commensurate and incommensurate Σ-type peaks is observed, the AFE2 phase in PZS-0.28 is characterized by strong incommensurate reflections only. As the amount of Sn increases in the PZS compounds, the incommensurate modulations become more stable, with the temperature range of existence of the dominating AFE2 phase being directly proportional to the Sn content.

The correlated disorder of octahedral tilts (see also Refs. 38 and 39) and displacements of lead ions characterize the paraelectric phase of all studied compositions, as evidenced by highly structured diffuse scattering. In PZS-0.28, the first lattice distortion to decrease the symmetry from cubic upon cooling is related to the appearance of R-type reflections. This implies that the resulting IM phase has a long-range order of oxygen octahedral rotations, and Pb atomic displacements are still correlated only on a short range with some signatures of local modulation. The gradual increase in the R-point intensities upon cooling suggests that the transition at T_C is close to the second order.

This work was financially supported by the Strategy AV21 framework of the Czech Academy of Sciences (program Efficient Energy Conversion and Storage). We are grateful to Dr. J. Fábry for his help with the sample preparation and the synchrotron measurement. For Synchrotron research, support was received from the CALIPSOplus project under Grant Agreement No. 730872 from the EU’s HORIZON 2020 Research and Innovation Framework Program.

The data that support the findings of this study are available from the corresponding author upon reasonable request.