Lattice match between coexisting cubic and tetragonal phases in PMN-PT at the phase transition

(1−x)Pb(Mg_1/3Nb_2/3)O₃-xPbTiO₃ (PMN-PT) solid solutions attract interest because of their exceptional electromechanical properties. Single crystals of PMN-PT, with the composition near the morphotropic phase boundary (MPB), exhibit effective piezoelectric coefficients of ≈2500 pC/N,^1,2 surpassing those of PbZr_1−xTi_xO₃ and α-SiO₂ (quartz) by one and three orders of magnitude, respectively.^3–6 In addition to its technological relevance, PMN-PT has been the subject of extensive academic research^7–9 trying to understand the complex origins of the electromechanical response in this system.

The phase diagram of PMN-PT is featured by a transition between the high-temperature cubic paraelectric and the low-temperature tetragonal (x > 0.35) or rhombohedral (x < 0.28) phases (Fig. 1).^10–14 These low-temperature phases are separated by a monoclinic phase (approximately, 0.28 < x < 0.35)¹⁵ that encompasses compositions displaying the enhanced properties and thus is of most practical and fundamental interest.¹⁶ All phase transitions in PMN-PT are associated with cooperative polar cation shifts,¹⁷ which remain partly disordered in the low-temperature phases with intricate local correlations.^18,19 The nature of this correlated disorder resulting in complex assemblages of ferroelastic and ferroelectric domains remains debatable.

Here, we employ variable-temperature high-resolution reciprocal space (RS) mapping to probe the transition between the cubic (C) and tetragonal (T) polymorphs in a single crystal of 0.65PMN-0.35PT. We identify a narrow temperature range of ≈4 K, where these two phases coexist. This two-phase state exhibits a 1.8 K hysteresis between heating and cooling, supporting the first-order nature of this transition. We establish that the resulting assemblage of the cubic and tetragonal phases is characterized by a remarkable matching of the cubic and orientationally averaged tetragonal pseudocubic lattice parameters. The implications of this matching and its possible mechanism are discussed.

A single crystal of 0.65PMN-0.35PT was produced using a flux-growth technique. This method exploits spontaneous nucleation through the gradual cooling of a supersaturated solution from high temperature.²⁰ Initial high-resolution RS mapping was performed on an “as-grown” unpoled sample using the four-circle x-ray diffractometer in Tel Aviv University.²¹ These experiments identified an “intermediate state” along the transition path between the C and T phases: Fig. 1 highlights the relevant phase-diagram region. Detailed studies were then carried out at the ID28 beamline²² of the European Synchrotron Radiation Facility. For these measurements, the same sample was etched to a cylinder-like shape with a diameter of ≈50  $μm$ and a length of ≈200  $μm$⁠. This sample preparation procedure was followed by two heating–cooling cycles, erasing any occasional poling history. The wavelength of the incident x-ray beam was $0.98 Å$⁠, which yields an average x-ray penetration depth of $16.4 μm$⁠. X-ray diffraction intensities were acquired with the PILATUS 1M detector by continuously rotating the crystal around the cylinder axis and saving the integrated detector images every $0.1°$⁠. The measurements were performed at high-scattering angles, $2θ$⁠, ranging between 110 $°$ and 130 $°$ to enable highest possible RS resolution.

The data were collected at 25 temperatures between 403 and 458 K, with the sample temperature being controlled using the ID28-based heat blower. The anticipated phase-transition range (424–442 K) was scanned with the temperature step of 1 K. The heating/cooling rate was 0.3 K/min, but the RS measurements were taken only after achieving each target temperature. At each temperature, the data were used both to determine the crystal-orientation matrix using CrysAlis Pro software²³ and to reconstruct high-resolution diffraction intensity distributions around 32 distinct Bragg peaks. The reconstruction of RS volumes (RSV) was accomplished using the custom MATLAB scripts, following procedures detailed elsewhere.^21,24,25

Figure 2 shows three-dimensional diffraction intensity distribution $I B x , B y , B z$ around one of the collected Bragg peaks (⁠ $7 1 ¯ 2$ in this case), in the Cartesian coordinate system $XYZ$⁠, with $X$ set nearly parallel to the scattering vector $B$⁠. The individual panels contain three projections of the intensity distribution along $Z, Y$⁠, and $X$ axes, defined as, e.g., $I z B x , B y= ∫ I B x , B y , B z d B z$⁠. The individual sub-peaks in the RS maps are diffracted from a specific set of twin domains. The quantitative analysis of the separation between the sub-peaks can be done as described in Ref. 24. Specifically, the vectors between the individual components can be used as an input data for the assignment of the sub-peaks to the individual domain variants. The simplest version of this analysis relies on the RS separation along the Cartesian $X$-axis because the component $B x$ is nearly equal to the length of the scattering vector $B$⁠. The presence of one-to-one correspondence between $B$ and the scattering angle (⁠ $2θ)$ means that this separation can be observed using high-resolution powder diffraction experiment^26–28 and allows determination of the free lattice parameters (⁠ $a T$ and $c T$ for the case of T-domains).

The supplementary material contains an animated version of the same figure, showing the temperature dependence of the RSV around all the collected Bragg peaks. The transition to the C-phase is evidenced by the merging of all the sub-peaks. Figure 3(a) illustrates this temperature dependence using a one-dimensional intensity distribution $I=I(B)$⁠. Panels (b)–(d) in Fig. 3 add three $I z( B x B y)$ intensity projections at 440, 437, and 426 K, which correspond to the nominally cubic, “intermediate,” and tetragonal states, respectively. Figure 3(c) reveals a different topology of the sub-peaks, compared to that in the nominally tetragonal region [Fig. 3(d)]. The “intermediate state,” sandwiched between the C- and T-phases, exists over a narrow temperature interval of 4 K and is therefore difficult to detect. The C↔T transition temperature determined by our measurements is ≈10 K higher than that reported previously.^10–14 Possible reasons for this discrepancy include differences between the ceramic samples used to determine the published phase diagrams and the single crystal studied here, and the coarser temperature steps employed in previous works.

Figure 3(c) confirms that one of the subpeaks (labeled “C”) ascribed to the intermediate state can be treated as a continuation of the corresponding cubic peak—an observation common to all the collected Bragg-peak maps (see the supplementary material). These subpeaks can be used to estimate the corresponding volume fraction of the C-phase as $η C(T)= ∑ C I ( T ) ∑ ALL I ( T )$⁠. Here, the sums in the numerator and the denominator run over the sub-peaks attributed to the C-phase and all the sub-peaks, respectively. The peak intensities, $I(T)$⁠, were determined by integrating the intensity distributions $I( B x, B y, B z)$ over the radius $R= 10 − 3 Å − 1$ around selected sub-peaks. Figure 4 displays the resulting $η C(T)$ dependence, highlighting the 4 K wide phase-coexistence region and the 1.8 K-wide hysteresis observed during the heating-cooling cycle. The presence of this hysteresis suggests that the observed phase coexistence is intrinsic rather than caused by a compositional segregation and consistent with the first-order type of the C↔T transition in PMN-PT.

Figure 5 reveals a remarkable matching of the “cubic” and “averaged tetragonal” lengths of the lattice basis vector in the phase-coexistence range. The difference $Δ C T= a C − a C T a C$ is less than $10 − 4$⁠, which is below the relative standard uncertainty of $a C$⁠, $a T$⁠, and $c T$⁠.

Thus, HP between the C phase and ATP can reside on the (011) plane (Fig. 6). The basis vectors, forming the corresponding 2D unit cell in this plane, would be $[100]$ and $[0 1 ¯1]$ (these directions are explicitly displayed in Fig. 6). For the C-phase, the lengths of these vectors are $a C$ and $a C 2$⁠, respectively. For the ATP, these lengths are $a C$ and $a C 2+O(Δ a T 2)$⁠. Neglecting the second power of $Δ a T 2$⁠, we can state that the (101)-micro-twinned tetragonal phase can grow mismatch-free on the (011) plane of the C-phase.

The original “adaptive phase” concept^30–33 involves the miniaturization of tetragonal domains along the domain-wall normal, which, on average, results in the effective monoclinic symmetry. Our experiments show that the tetragonal domains in the two-phase state are large enough to produce Bragg peaks, which can be clearly assigned to the tetragonal, rather than a monoclinic phase. Assuming that the domain size is the only contribution to the width of the observed Bragg peaks (0.002–0.005 Å⁻¹), we estimate this size as several hundred nanometers. Similar domain sizes, for 0.65PMN-0.35PT and other compositions near the MPB, were observed in Refs. 34–36. We propose that the lattice matching of the C- and T-phases in the intermediate, two-phase state observed here is enabled by the assembly of the T-domains in a fashion similar to ATP but without the domain miniaturization. We used the splitting of reflections in our 3D RS reconstructions to confirm the corresponding {110}-type HP orientations; the details of this analysis will be reported separately. Real space characterization of domain arrangements can be continued using, e.g., dark-field x-ray microscopy.^37–39 

In conclusion, we performed in situ high-resolution x-ray diffraction on a single crystal of 0.65PMN-0.35PT using a fine-step sampling of temperatures across the cubic-tetragonal phase transition. We discovered a two-phase state, only 4 K wide, characterized by the lattice parameter (⁠ $a C$⁠) of the C-phase matching the orientationally averaged $( 2 3 a T+ 1 3 c T$⁠) lattice metrics of the T-phase. This matching effectively corresponds to the unit-cell volume conservation. In principle, such conditions enable {110}-type habit planes connecting the C-phase and a micro-twinned assembly of T-domains.