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

Ido Biran, Alexei Bosak, Zuo-Guang Ye, Igor Levin, Semën Gorfman

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Ido Biran, Alexei Bosak, Zuo-Guang Ye, Igor Levin, Semën Gorfman. Lattice match between coexisting cubic and tetragonal phases in PMN-PT at the phase transition. Applied Physics Letters, 2024, 124 (24), pp.242901. 10.1063/5.0202576. hal-05095811

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

Ido Biran Ⓔ ; Alexei Bosak Ⓔ ; Zuo-Guang Ye Ⓔ ; Igor Levin Ⓔ ; Semën Gorfman Ⓔ

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Appl. Phys. Lett. 124, 242901 (2024)

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

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Cite as: Appl. Phys. Lett. **124**, 242901 (2024); doi: 10.1063/5.0202576

Submitted: 5 February 2024 · Accepted: 26 May 2024 ·

Published Online: 11 June 2024

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Ido Biran,<fnref n="1" /> ID Alexei Bosak,<fnref n="2" /> ID Zuo-Guang Ye,<fnref n="3" /> ID Igor Levin,<fnref n="4" /> ID and Semën Gorfman<fnref n="1" />,<fnref n="4" />

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## AFFILIATIONS

## ABSTRACT

(1-x)Pb(Mg₁/₃Nb₂/₃)O₃-xPbTiO₃ (PMN-PT) perovskite-like solid solutions are recognized for their outstanding electromechanical properties, which are of technological importance. However, some significant aspects of the crystal structures and domain assemblages in this system and the role of these characteristics in defining the functional performance of PMN-PT remain uncertain. Here, we used synchrotron x-ray diffraction to investigate the phase transition linking the paraelectric (cubic) and ferroelectric (tetragonal) phases in a single crystal of 0.65PMN-0.35PT. We analyzed the evolution of reciprocal-space maps across this transition. These maps were collected using small temperature step (1 K) and a high reciprocal-space resolution to reveal changes in the splitting of Bragg peaks caused by the formation of ferroelastic domains in the low-symmetry phase. Our results uncovered a two-phase state, cubic plus tetragonal phases, which exists over a narrow temperature range of only ≈4 K and exhibits a thermal hysteresis of ≈1.8 K. Remarkably, within this state, the lattice parameter of the cubic phase, a_c, matches the orientational average of the lattice parameters for the tetragonal polymorph. 2/3 a_T + 1/3 c_T. We discuss the implications of this matching, highlighting the possibility of it being realized by the formation of an assemblage of tetragonal twin domains separated from the cubic phase by a strain-free {110} boundary, as in the “adaptive phase” but without domain miniaturization.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0202576

(1-x)Pb(Mg₁/₃Nb₂/₃)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,<fnref n="1" />,<fnref n="2" /> surpassing those of PbZr₁-xTiₓO₃ and α-SiO₂ (quartz) by one and three orders of magnitude, respectively.<fnref n="3" />⁻⁶ In addition to its technological relevance, PMN-PT has been the subject of extensive academic research⁷⁻⁹ 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.

<footnote n="1">¹Department of Materials Science and Engineering, Tel Aviv University, Tel Aviv, Israel</footnote>

<footnote n="2">²European Synchrotron Radiation Facility, Grenoble, France</footnote>

<footnote n="3">³Department of Chemistry and 4D LABS, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada</footnote>

<footnote n="4">⁴Materials Measurement Science Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA</footnote>

<footnote>a) Author to whom correspondence should be addressed: gorfman@tauex.tau.ac.il</footnote>
<figure><img src="image_12.png" /><figcaption>**FIG. 1.** Schematic phase diagram of PMN-PT (redrawn based on Refs. 10–14). Labels C, R, M, and T mark the cubic, rhombohedral, monoclinic, and tetragonal phase areas. A vertical gray line marks the investigated 0.65PMN-0.35PT composition with the newly identified intermediate (C + T) state delimited using horizontal lines and temperature labels. The C↔T transition temperature for this composition is 10 K higher than previously reported (see the text for discussion).</figcaption></figure>

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., $I_z(B_x, B_y) = \int I(B_x, B_y, B_z)dB_z$. 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

<figure><img src="image_13.png" /><figcaption>**FIG. 2.** $I_z(B_x, B_y)$ (a), $I_y(B_x, B_z)$ (b), and $I_x(B_y, B_z)$ (c) projections of the three-dimensional diffraction intensity distribution around the 7 1 2 Bragg peak. The Cartesian **X**-axis is parallel to the reciprocal lattice vector [712]*.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>FIG. 3. (a) Temperature dependence of I(B) obtained from the three-dimensional diffraction intensity distribution around the 712 reflection (heating). (b)–(d) BxB_y projections of the RSVs corresponding to the cubic phase, “intermediate state,” and tetragonal phase, respectively. The labels “C” and “T_1” and “T_2” show the assignment of the sub-peaks to the cubic phase or corresponding tetragonal domains.</figcaption></figure>

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_xB_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 sub-peaks can be used to estimate the corresponding volume fraction of the C-phase as η_C(T) = $\sum_{C} \frac{I(T)}{\sum_{ALL} I(T)}$. 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.

We refined lattice parameters in the cubic, intermediate, and tetragonal states. The refinements were accomplished by identifying the B_xB_yB_z positions of all the sub-peaks (i.e., labeled as “1,” “2,” and “3,” in Fig. 2) and minimizing the sum of squared differences between the lengths of the reciprocal lattice vectors for each subpeak,

$$ \chi^2 = \sum \frac{(B_{\text{obs}} - B_{\text{calc}})^2}{\sigma_{\text{obs}}^2}. \qquad (1) $$

Here, $B_{\text{obs}} = (B_x^2 + B_y^2 + B_z^2)^{1/2}$, $B_{\text{calc}} = (G_{ij}^{*(m)} H_i H_j)^{1/2}$, $H_1$, $H_2$, and $H_3$ are the indices of the reflection, and $[G^{*(m)}]$ is the matrix of the dot products between the reciprocal basis vectors of the domain m. For the C-phase (lattice parameter a_C), $G_{ij}^{*(0)} = a_C^{-2} \delta_{ij}$. For the T-phase, we get

<figure><img src="image_15.png" /><figcaption>FIG. 4. Temperature dependence of the C-phase fraction η_C(T) during heating and cooling across the transition. The dashed lines indicate the eye-guiding fit, which helps to identify the ≈1.8 K hysteresis.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>FIG. 5. Temperature dependence of the T- (a$_T$, c$_T$) and C- (a$_C$) lattice parameters during (a) heating and (b) cooling. The dashed lines indicate the temperature dependence of the orientational average a$_{CT}$ = 2/3 a$_T$ + 1/3 c$_T$.</figcaption></figure>

$$
\begin{align*}
[G^{*(1)}] &= \begin{pmatrix} c_T^{-2} & 0 & 0 \\ 0 & a_T^{-2} & 0 \\ 0 & 0 & a_T^{-2} \end{pmatrix} &
[G^{*(2)}] &= \begin{pmatrix} a_T^{-2} & 0 & 0 \\ 0 & c_T^{-2} & 0 \\ 0 & 0 & a_T^{-2} \end{pmatrix} \\
[G^{*(3)}] &= \begin{pmatrix} a_T^{-2} & 0 & 0 \\ 0 & a_T^{-2} & 0 \\ 0 & 0 & c_T^{-2} \end{pmatrix}.
\end{align*}
$$

The domain index *m* was assigned to each sub-peak to achieve the best match between the corresponding *B*<sub>obs</sub> and *B*<sub>calc.</sub> We minimized χ<sup>2</sup> independently for each temperature using the custom MATLAB-based scripts. Figure 5 presents the temperature dependence of *a*<sub>T</sub>, *c*<sub>T</sub>, and *a*<sub>C</sub> obtained using such minimization on heating [Fig. 5(a)] and cooling [Fig. 5(b)]. The dashed line reflects the temperature dependence of the orientational average of the tetragonal lattice parameters calculated as

$$
a_{CT} = \frac{2}{3} a_{T} + \frac{1}{3} c_{T}. \qquad (2)
$$

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 $\Delta_{CT} = \frac{|a_C - a_{CT}|}{a_C}$ is less than $10^{-4}$, which is below the relative standard uncertainty of $a_C$, $a_T$, and $c_T$.

The observed matching has implications for the mechanism of the C + T phase coexistence. First, the condition $a_C = a_{CT}$ is nearly equivalent to the conservation of the unit-cell volume $a_C^3 = a_T^2 c_T$. Indeed, expressing the matching condition $a_C = a_{CT}$ as

$$
\begin{align}
a_T &= a_C - \Delta a_T, \nonumber \\
c_T &= a_C + 2\Delta a_T, \tag{3}
\end{align}
$$

where $\Delta a_T$ is a small value, we can see that $a_T^2 c_T = a_C^3 + O(\Delta a_T^2)$, with the second term containing the negligibly small second and third power of $\Delta a_T$. Second, we can show that (3) enables a favorable geometrical connection of the coexisting C and T phase volumes via a low-index habit plane (HP). A HP is a lattice plane along which both

phases exhibit matching 2D lattice metrics.²⁹ The existence of a HP is
defined by the following condition:

$$
\det([G'] - [G'']) = 0, \qquad (4)
$$

where [G'] and [G''] are the matrices of the dot products between the
lattice-basis vectors calculated for each phase. Thus, for the T and C
phases, a HP is permissible only if a_T = a_C or c_T = a_C. Alternatively,
condition (4) is satisfied for a HP between C and a micro-twinned T-
phase, also known as “adaptive” tetragonal phase (ATP). According to
Refs. 30–33, ATP represents an assemblage of T-domains with alter-
nating orientations of the longer axis and joined by strain-free domain
walls of the {110} type. Miniaturization of such tetragonal domains
results in their macroscopic averaging into a pseudo-monoclinic (M_C-
type) phase with the lattice parameters a_A, b_A, c_A, and β_A ≈ 90° (with
subscript A denoting “adaptive” phase). Condition (4) is fulfilled when,
e.g., tetragonal c- and a-domains are mixed with their volumetric ratios
of 2/3 and 1/3, respectively (as shown schematically in Fig. 6), so that the
corresponding ATP lattice parameters are

$$
\begin{align}
a_A &= a_C, \nonumber \\
b_A &= a_C - \Delta a_T \nonumber \\
c_A &= a_C + \Delta a_T. \tag{5}
\end{align}
$$

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 [011] (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(Δa_T²). Neglecting the second power of Δa_T², 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<sup>30–33</sup> involves the miniatur-
ization of tetragonal domains along the domain-wall normal, which,
on average, results in the effective monoclinic symmetry. Our experi-
ments 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<sup>−1</sup>), we estimate this size as several hun-
dred 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 inter-
mediate, 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 orienta-
tions; 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.<sup>37–39</sup>

In conclusion, we performed *in situ* high-resolution x-ray diffrac-
tion on a single crystal of 0.65PMN-0.35PT using a fine-step sampling
of temperatures across the cubic-tetragonal phase transition. We dis-
covered a two-phase state, only 4 K wide, characterized by the lattice
parameter (a<sub>C</sub>) of the C-phase matching the orientationally averaged
(2/3 a<sub>T</sub> + 1/3 c<sub>T</sub>) 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.
<figure><img src="image_17.png" /><figcaption>FIG. 6. Schematic illustration of the proposed C- and T-phases coexistence mechanism. The illustration contains (100) plane of the lattices, enabling this coexistence. T-phase is represented by the assemblage of tetragonal c- and a-domains with their polar axes shown by the alternating upward arrow and a cross sign. The domains are joint along mismatch-free (101)-walls. The C-phase (top-right) resides on the micro-twinned T-phase via (011)-habit plane. The mechanism of matching between C and micro-twinned T-phase is demonstrated by joint unit cells with the relevant lattice lengths indicated. The unit cells are enlarged for clarity.</figcaption></figure>

See the **supplementary material** for the figures, similar to Figs. 2 and 3 but for all the measured Bragg peaks and temperatures.

We acknowledge Youli Li (University of California, Santa Barbara) for his help with and support of the Tel Aviv University-based x-ray diffractometer and the temperature controller. We acknowledge the following funding: Israel Science Foundation (Grant Nos. 1561/18, 3455/21, 1365/23 to Semën Gorfman); United States – Israel Binational Science Foundation (Award No. 2018161 to Semën Gorfman and Igor Levin); the U.S. Office of Naval Research (ONR Grant No. N00014-21-1-2085); and the Natural Science and Engineering Research Council of Canada (NSERC, DG, RGPIN-2023-04416).

Certain equipment, instruments, software, or materials, commercial or noncommercial, are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement of any product or service by NIST, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

**AUTHOR DECLARATIONS**

**Conflict of Interest**

The authors have no conflicts to disclose.

**Author Contributions**

**Ido Biran:** Data curation (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Alexei Bosak:** Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Zuo-Guang Ye:** Funding acquisition (equal); Investigation (supporting); Methodology (supporting); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Igor Levin:** Conceptualization (lead); Funding acquisition (supporting); Investigation (supporting); Methodology (supporting); Project administration (lead); Resources (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). **Semën Gorfman:** Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal).

**DATA AVAILABILITY**

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