After more than seven decades of studies we are back in the era of relaxor ferroelectrics (FEs), a group of materials consisting of the unique combination of a disordered relaxor (RE) component and a long-range-ordered FE component. The group of RE-FEs includes a vast range of organic and inorganic materials, as well as lead-based and lead-free oxide compositions of various structural types1,2,3. The most discussed is the pseudo-binary (1–x)Pb(Mg1/3Nb2/3)O3–xPbTiO3 (PMN–PT) solid solution where PMN and PT are the RE and FE end-members, respectively (note that throughout this paper PMN–PT will be in the focus and the terms “RE” and “RE-FE” will specifically refer to lead-based perovskite compositions). In contrast to “normal” FEs, canonical REs are distinguished by the temperature-dependent dielectric permittivity maximum that is typically broad and frequency dispersed (Fig. 1a). Among other useful properties of RE-FEs, the large permittivity over a broad temperature range is advantageous in diverse application areas, such as energy storage4, electrocaloric cooling5 and electrostrictive actuation6.

Fig. 1: Dielectric signatures and large piezoelectric response of lead-based RE-FE single crystals and ceramics.
figure 1

a Frequency dispersion of the temperature-dependent dielectric permittivity (ε’) maximum of PMN single crystal, representing the characteristic feature of RE-based materials1,3. b Piezoelectric d33 coefficient of RE-FE Pb(Zn1/3Nb2/3)O3–PbTiO3 (PZN-PT) single crystals as a function of composition (PT content) and orientation (with respect to applied electric field), illustrating the large response of <001> -oriented rhombohedral (R) crystals close to the morphotropic phase boundary (MPB) region7 (see red arrow). The source of the d33 data is reported in the original paper (Ref. 7). The upper inset shows schematically the mechanism responsible for the large response, i.e., the easy rotation of the R [111]pc polarization vector (pc denotes pseudo cubic) toward the tetragonal (T) [001]pc polar direction in the case of the electric field (E) applied along [001]pc (see text for detailed explanation). c Piezoelectric d33 coefficient of PMN–PT ceramics as a function of PT content for undoped and Sm-doped ceramic samples, illustrating the ultrahigh piezoelectricity achieved by Sm doping. The data for undoped and Sm-doped PMN–PT were reproduced from Kelly et al.54 and Li et al.12, respectively. Figures shown in (a) and (b) were reprinted from Ref. 3 and Ref. 7 by permission from John Wiley & Sons and AIP Publishing, respectively.

A true interest in RE-FE single crystals began in 1997 when Park et al.7 reported on the unusually large piezoelectric response (piezoelectric coefficient up to ~2500 pC/N) of PMN- and PZN-based rhombohedral crystals when measured along non-polar axis and in the vicinity of the morphotropic phase boundary (MPB; see red arrow in Fig. 1b). The large response was initially explained by first-principles computations using the concept of polarization rotation8. A simpler phenomenological approach later explained that the propensity for the polarization rotation has its origin in the structural instability related to the proximity of rhombohedral (R) and tetragonal (T) phases close to the MPB9. Therefore, by applying an electric field on the R crystal along its off-polar [001]pc direction, the field-driven rotation of the [111]pc R polarization vector toward the [001]pc T direction becomes easier in the proximity of the T phase (see inset of Fig. 1b), resulting in large piezoelectric coefficients measured  in compositions close to the MPB (Fig. 1b, red arrow). Following the discoveries of the bridging monoclinic phases at MPBs in PZT and PMN–PT10, the concept of polarization rotation has been universally used to explain MPB property enhancement11. As one of the points of discussion later in this contribution, it is important to emphasized that the polarization rotation in PMN–PT and similar RE-FEs is not as simple as that in FE PZT because it is complicated by the disordered polar structure inherent to RE-based compositions (see next section for details).

PT-based RE-FE crystals has recently been overshadowed by the discovery of ultrahigh piezoelectricity in rare-earth doped PMN–PT ceramics12. With the strongest effect achieved using Sm doping, the authors reported on piezoelectric coefficients exceeding 1500 pC/N, which is more than doubled with respect to the response of the morphotropic undoped PMN–PT ceramics (see the maximum d33 values at the MPB in the two plots of Fig. 1c, i.e., 1500 pC/N for Sm-doped versus 650 pC/N for undoped PMN–PT ceramics). The piezoelectric enhancement was attributed to the local structural disorder introduced by the donor dopant13, adding up to the already present disorder inherent to RE materials14. The multiple origins of the large response of donor-doped PMN–PT are still under debate with open questions regarding charged point defects and domain-wall pinning mechanisms influenced by the dopant15.

In addition to their usefulness, the highly performant lead-based RE-FEs are also rich from a fundamental perspective and perhaps the most difficult to understand, particularly when it comes to structure-property relations1. Several different models have been proposed to explain the characteristic frequency dispersion of the temperature-dependent permittivity maximum16,17,18,19,20 (Fig. 1a) and the polar structure of PMN and similar REs14,21,22,23,24,25,26. The key feature of RE-based materials is the existence of polar states correlated over a local, nanometer scale, traditionally referred to as the polar nanoregions (PNRs). Strongly supported by diffuse scattering data (see subsequent section for details and references), the dynamic behavior of PNRs under applied fields, alongside their freezing at lower temperatures, has been correlated with the characteristic dielectric dispersion17. The key question that has bothered the community for a long time is how and to which extent is the RE nature of these materials, and thus the nanoscale polar nature, implicated in their ultrahigh piezoelectric properties? Allowed by the fast development of theoretical and experimental methodologies (some of them are mentioned in subsequent sections), great advances have recently been made in deciphering the complex polar structure of RE-FEs and its role in the functional response. In this contribution, I will briefly review these issues focusing on the PMN–PT model system. The analysis of the current results suggests that there are still a number of details to be figured out, especially in relation to the complex link between the multiscale structure and macroscopic functional response. Filling such gaps may possibly allow to transfer the knowledge from lead-based to environmentally friendlier lead-free RE-FE materials.

Polar structure of RE-FE PMN–PT

The traditional view of the polar structure of REs consisting of PNRs dates back to the original study by Burns and Scott27. In their pioneering work the presence of small regions of polarization (size of several unit cells) below the critical temperature (later to be called the Burns temperature) was inferred from macroscopic optical measurements on (Pb1–xLax)(Zr1–y Tiy)O3 ceramics and PMN single crystals. With no intention of criticizing any work, it is important to mention that derivations of microscopic mechanisms purely from macroscopic behavior of complex materials should be done with caution because it is not impossible that different microscopic mechanisms may result in the same or similar macroscopic physical properties. Nevertheless, the accepted picture of PNRs embedded into a non-polar matrix1, depicted in Fig. 2a, has been extensively used for modelling neutron and X-ray diffuse scattering data of different RE and RE-FE materials23 and also extended to involve a polar (FE) matrix instead of the non-polar (paraelectric) matrix14,22,28. Additional information on the local and average structure of PMN and PMN–PT, as provided by Neutron and X-ray scattering experiments, can be found in the review article by Cowley et al.29 (see also references therein).

Fig. 2: Polar structure of PMN–PT.
figure 2

a Representations of two different microscopic polar states of RE-based materials consisting of (a) polar nanoregions (PNRs) embedded into a non-polar or ferroelectric (FE) matrix and (b) polar nanodomains (PNDs) separated by low-angle domain walls. The latter schematic was reproduced based on the study by Takenaka et al.25 on PMN-PT and Eremenko et al.26 on PMN; note that both these studies adopted combined theoretical and experimental approaches. c Correlation between Pb displacements from the center of the PbO12 cage (see inset) and number of surrounding B-site Mg ions in PMN; d projected Pb displacement map for PMN with arrows and colors indicating the direction and magnitude of the displacements, respectively. The data shown in (c, d) were extracted using reverse Monte Carlo simulations as reported by Eremenko et al.26. To be noted is that this method reported in Ref. 26 is based on PMN structural modelling by performing simultaneous refinements using a wide range of experimental dataset, including Neutron and X-ray powder total-scattering, extended X-ray absorption fine structure, 3D X-ray single-crystal diffuse-intensity distributions and Neutron Bragg profiles. e Projected polarization map of a region in PMN single crystal with 30 mol% PT (PMN–30PT) as determined from the difference of the center-of-mass positions of cations and anions using scanning transmission electron microscopy (STEM) analysis (see Kumar et al.37). The direction and magnitude of the off-centric displacements is indicated by the different colors and color brightness, respectively. The inset shows the extracted statistical distribution of domain-wall angles, confirming the presence of a large fraction of low-angle domain walls (peak at ~20°). f Maps of Pb distortion angle ΦA (see inset) in PMN–30PT and PMN–40PT ceramics, illustrating the compositional dependence of the Pb positional disorder (original data are reported in Otonicar et al.36). Figures shown in (e) were reprinted from Ref. 37 by permission from Springer Nature.

The classical view of the polar state of REs (Fig. 2a) has been questioned by several authors21,23,25 (it is encouraged to read the excellent comment-paper on this topic published by Hlinka30). In agreement with earlier diffuse scattering studies by Pasciak et al.21, using molecular dynamics simulations on PMN–PT with 25 mol% PT (PMN-25PT), combined with experimental diffuse scattering data, Takenaka et al.25 proposed an alternative scenario where the matrix of the PNRs is inexistent and the material is instead composed of densely packed polar nanodomains (PNDs) of several nanometers in size (~2–10 nm) separated by domain walls (Fig. 2b; note that the term PND is here used instead of PNR only to distinguish between the two different polar states shown in Fig. 2a and b). Based on the coexistence of multiple polar symmetries in PMN–25PT, it was reported that the change of polarization direction at the walls adopt, in average, low angles (~55°). The wide distribution of the domain-wall angles was proposed to arise due to the small energy differences between tetragonal-, orthorhombic- and rhombohedral-like orientations of local polarization. This is very different from the situation in normal FEs where polarization directions are restricted by the particular (single) polar symmetry, meaning that fewer domain walls and angles are possible (e.g., 180° and 90° walls in tetragonal FEs). The newly proposed polar structure of RE-FEs has opened interesting questions on how the so-called low-angle domain walls move and switch under applied external fields. To be noted, however, is that such dynamics was already discussed at least 25 years earlier within the model based on random fields in PMN18 and in the frame of the PNR-breathing model19.

The simplest crystal-chemical explanation for the disordered polar structure of lead-based REs considers the displacements of Pb cations, which crucially influence the local polarization31. To be recalled is that Pb displacements also play a fundamental role in the long-range polar ordering of conventional FEs, such as PbTiO332. Polar displacements of Pb cations in PMN are experimentally supported by various short-range-structure analyses, including nuclear magnetic resonance33 and pair distribution function34. In lead-based perovskites the Pb cations are more sensitive to the B-site charge and strain disorder than the B-site cations themselves31. Figure 2c illustrates this point for PMN by revealing a nearly linear correlation between the Pb off-centric displacements from the center of PbO12 cage and the B-site chemistry represented here by the number of Mg cations in the surrounding B sites (see inset of Fig. 2c). The tendency of increased magnitude of Pb displacements with increasing Mg/Nb ratio is attributed to the Pb-O-Mg/Nb bonding interactions: the low ionic charge of Mg (2+) leads to underbonded oxygen, which is compensated by shorter Pb-O bonds and thus larger Pb displacements26,31,35. This chemically-driven disorder in the Pb displacements ultimately prevents the formation of a long-range FE order, leading to a disorder relaxor phase in which Pb displacements show quasi-ordered correlations extending on a short (nanoscale) range. Such correlations are evident in the Pb displacement map for PMN extracted using reverse Monte Carlo simulations (see reddish regions in Fig. 2d), which uses structural refinements to different experimental diffraction and spectroscopic datasets. Interestingly, while this polar picture is overall compatible with the view of PNDs (Fig. 2b), the simulations revealed additional details of the hierarchical polar arrangements in PMN (details are out of scope of this Perspective paper and can be found in Ref. 26). The chemical effect illustrated in Fig. 2c is also consistent with theoretical studies, confirming the essential role of the random electric fields (arising due to the random distribution of Mg2+ and Nb5+ cations on B sites) in the development of the nanoscale-correlated polar structure of PMN24.

As the essential element of the PND scenario (Fig. 2b), the low-angle domain walls have been experimentally identified by atomic-scale scanning transmission electron microscopy (STEM) analysis in a wide range of PMN–PT compositions, first in ceramic samples36, followed by single crystals (Fig. 2e; see also the inset showing the wide distribution of the domain-wall angles)37. It was found that these interfaces tend to be located at or close to the central regions of local structural heterogeneities, such as the chemical ordered regions and oxygen distortion and tilt regions37. This spatial correlation confirms that the heterogeneities play a role in disrupting the local polarization, affecting the domain structure and thus low-angle domain walls.

The positional disorder of Pb cations in PMN, led by the bonding interactions with the B-site cations (see Fig. 2c), is strongly affected by the incorporation of Ti cations on the B sites when forming the PMN–PT solid solution. Using atomic-resolution STEM imaging36, a stronger disorder in the Pb-atom positions was consistently observed in PMN–PT ceramics with lower PT content (Fig. 2f). First-principles calculations revealed that the reduced Pb positional disorder by the addition of PT has to do with the effect of the Ti ions as they tend to stabilize collinear (ordered) Pb displacements of (100)-type31. The effect of Ti extends over multiple scales and shows up not only in the atomic structure (Fig. 2f), but also in the nano- and micro-scale domain structure, short- and long-range crystal symmetry and phase composition of PMN–PT as all these features develop with increasing PT content36,38,39,40,41. The implications of this multiscale structural evolution on the piezoelectric response of PMN–PT compositional series is presented in the next section.

Implications of the polar structure in the piezoelectric response of RE-FE PMN–PT

A great deal of studies has been performed to understand the implications of the RE nature on the large piezoelectric response of PMN–PT and similar RE-FEs. Different explanations have been proposed, including those based on polarization rotation with the bridging monoclinic phase (see first section and Refs therein), presence of nanoscale domains around MPB38,42, critical points in the electric-field–temperature phase diagram43, random electric fields44, interaction between PNRs45 or between PNRs and spontaneous polarization14,46, and dynamics of low-angle domain walls25,36. All these complex mechanisms likely play a role and may dominate depending on the particular case. The most recent model by Li et al.14,  which was introduced using phase-field modeling, assumes the existence of PNRs inside a FE matrix (in accordance to Fig. 2a) where the PNR orientation changes with temperature. At cryogenic temperatures the polarization vectors of PNRs are presumably non-collinear with respect to the spontaneous FE polarization, Ps (Fig. 3a), whereby the collinear arrangement is achieved with increasing temperature. In this collinear state, under electric fields applied perpendicularly to the Ps direction, the PNRs can easily rotate helping the rotation of Ps in nearby regions (Fig. 3b). It is this mutual rotation mechanism that the authors used to explain why RE-FE PMN–PT single crystals have superior piezoelectric response than normal FEs, such as PZT.

Fig. 3: Proposed mechanisms for the large piezoelectric response of PMN–PT single crystals and ceramics.
figure 3

a Phase-field simulated polar state at 100 K of a composite consisting of [100]-poled FE matrix (green) and inclusions of PNRs (red, yellow and blue). The x-and y-axes denote [100] and [010] direction, respectively, with the colors represent the direction of the local polarization. The right-hand inset shows an enlarged region highlighting the nano-sized PNRs with the different direction of polarization (different symmetry) with respect to that of the matrix FE phase. b Proposed mechanism of enhanced piezoelectric response of lead-based RE-FEs based on the polar state shown in (a). Green area and blue arrows represent the FE matrix and the associated polarization (Ps), respectively; yellow circles and purple arrows represent PNRs and their local polarization, respectively. The schematic shows the initial state at room temperature under no applied electric field E (left), where PNRs and Ps are nearly collinear, and under an applied E perpendicularly to the spontaneous polarization (Ps) of the FE matrix (middle and right schematic). The large piezoelectric response is explained by the rotation of PNRs along the E direction, which helps rotating the spontaneous polarization of the nearby FE region (details are reported in Ref. 14). c Reversible (d33init) and irreversible (α*) Rayleigh coefficients, and phase angle (tanδp) of PMN–PT ceramics as a function of PT content extracted from the field dependence of the converse piezoelectric response. The different colors mark the individual phase regions (MB and MC stand for monoclinic and T for tetragonal). The insets show piezoelectric hysteresis loops (displayed for increasing field amplitudes) representative of MB (top) and MPB (bottom) composition. d Phase diagram of PMN–PT denoting the different types of nonlinear responses in the RE (gray), RE-FE (red) and FE (blue) regions of the PMN–PT system identified based on distinct third-harmonic responses (for details see Ref. 36). The inset shows the frequency dispersion of the temperature-dependent maximum of the dielectric permittivity (ΔTεmax) as a function of PT content. The connecting line denotes the compositional point (~32 mol% PT) of the transition between the RE-FE-type and FE-type response coinciding with the drop of ΔTεmax to zero.

To be noted is that the PNR-based mechanism presented in Fig. 3b is consistent with previous diffuse scattering studies, explaining the key role of the random electric fields on the high piezoelectricity of PMN–PT (as opposed to PZT in which such fields are absent)44. By invoking PNRs as characteristic polar features of REs, the model also provides a step forward in the understanding of the generalized polarization rotation mechanism as the origin of morphotropic property enhancements in PZT and PMN–PT8,9,10,11. In this aspect, the work by Manley et al.46 has also to be emphasized. Based on a detailed neutron scattering analysis, the authors explain the dynamic role of PNRs in the elastic shear softening of the macroscopic polarization rotation in PMN–PT single crystals. In view of the criticism that this work received47,48, it should be still mentioned that the general conclusion on the essential role of the nanoscale polar features in the piezoelectric response of PMN–PT is well supported by other studies14,25,36,44,45, while the exact underlying mechanism(s) are still subject of debates.

Interestingly, the scenario proposed by Li et al. (Fig. 3a,b) assumes the existence of PNRs residing inside a long-range-ordered FE matrix (see Fig. 2a) and is thus in apparent contradiction with the picture of highly dense PNDs and low-angle domain walls where the matrix is nonexistent (see Fig. 2b). In addition, recent advances in diffuse scattering techniques38 made it possible to identify a correlation between the increased properties at the MPB and the nanodomains, supporting the idea of the adaptive phase model42. It should be emphasized that, on the one hand, a number of microscopy12,49 and diffuse scattering studies22,28,50 report on the existence of different symmetries in PMN–PT and similar RE-FEs when the material is probed on different length scales, indirectly supporting the scenario depicted in Fig. 2a. However, on the other hand, it is equally important to mention the recent criticism to the diffuse scattering data on REs whose interpretation may be ambiguous considering that the data can be well represented by assuming either of the two polar states illustrated in Fig. 2a and b21,30. It is the personal opinion of the author of this paper that all these considerations leave open questions regarding the details of the mechanism(s) responsible for the large piezoelectricity of lead-based RE-FE ceramics and single crystals.

Most of the mechanisms used to describe the effect of the characteristic nanopolar structural features on the large piezoelectric response of PMN–PT are based on polarization rotation and thus lattice response to external fields. In view of the discovery of low-angle domain walls (see Fig. 2e), however, it is interesting to consider how these interfaces will respond to external fields and how their dynamics will be reflected in the macroscopic piezoelectricity across different parts of the PMN–PT phase diagram, not only close to the MPB.

Recent harmonic analysis of the nonlinear piezoelectric and dielectric properties of PMN–PT ceramics revealed a wide range of PMN-richer monoclinic compositions with the RE behavior exhibiting enhanced nonlinear response36. From the quasi-plateau of the irreversible Rayleigh coefficient, α*, and the corresponding piezoelectric phase angle, tanδp, reached in the monoclinic MB region (see the two red curves and red arrows in Fig. 3c), it was suggested that the enhanced irreversible (hysteretic) domain-wall contribution of these MB compositions cannot be associated with the MPB effect. In fact, the MPB enhancements are clearly evident in the peaks of the reversible Rayleigh coefficient, d33init, occurring at the MPBs (see black curve in Fig. 3c). Additional third-harmonic polarization analysis made it possible to identify distinct regions in the phase diagram of PMN–PT, each showing a unique type of piezoelectric and dielectric nonlinear response (denoted in Fig. 3d as RE-, RE-FE- and FE-type). The pronounced hysteretic and nonlinear response of PMN-richer MB composition showing RE behavior (see red-box inset of Fig. 3c) was contrasted with the more common Rayleigh-like response of PMN-poorer compositions where the RE behavior is not observed (see blue-box inset of Fig. 3c). The transition between these two types of responses at ~32 mol% PT was shown to occur at the point when the frequency dispersion of the temperature-dependent permittivity maximum drops to zero (see plot above the phase diagram in Fig. 3d and connecting line). Since this dispersion represents the fingerprint of the RE behavior, the enhanced nonlinear piezoelectricity of MB compositions was attributed to the irreversible and reversible dynamics of low-angle domain walls. By quantifying the dynamic contribution of low-angle domain walls to the macroscopic piezoelectric response of PMN–PT ceramics, the results clearly point toward the extended effect of these RE-specific interfaces going well beyond MPB effects. While a similar nonlinear effect was also identified in lead-based compositions other than PMN–PT36, it remains largely unexplored in lead-free RE-FE compositions.


In this contribution, I reviewed and briefly discussed some of the current issues related to the polar structure of PMN-PT and similar lead-based RE-FEs alongside the implications of this structure in the large piezoelectric response of these materials. Despite intensive studies, it is found that the actual polar structure of lead-based RE-FEs is still an open issue with many details requiring further investigations. Considering that a clear microscopic picture of this structure is the precondition to understand the large macroscopic piezoelectric response of RE-FEs, it is not surprising that different, often incompatible explanations have been reported so far. A consensus on the unified microscopic description of the polar state in REs and RE-FEs has not been reached yet. However, the question is whether this is indeed necessary. As a matter of fact, it is not impossible that different polar states (such as those discussed in Fig. 2a, b) play a dominant role depending on the temperature, exact material composition (e.g., content of the RE phase, i.e., PMN, in PMN–PT) and material form (single crystal, ceramic or thin film), as, for example, indicated by some recent studies51. Complementary experimental and theoretical studies over wide material’s length scale should be able to resolve this issue and provide guidance as to whether multiple models need to be elaborated.

The Pb-atom positional disorder triggered by the chemical bonding interactions with the multivalent B-site cations has been found to lie at the origin of the characteristic polar structure of PMN-PT. In the next step, it would be important to understand how this disorder is affected by the application of an electric field, requiring state-of-the-art in situ STEM techniques. Although complicated and time consuming, in situ structural investigations on the atomic level should make it possible to visualize and comprehend the complex dynamics of the low-angle domain walls. In addition, an important advantage is that the problem of the local nature of the STEM analyses has been solved by statistical sampling37. The results of such in situ studies may have broader consequences considering that the low-angle-domain-wall dynamics was found to be responsible for the large nonlinear piezoelectric and dielectric response of a variety of lead-based RE-FEs with different B-site cations, including PMN–PT, Pb(Sc0.5Nb0.5)O3 and Pb(Fe0.5Nb0.5)O336.

An interesting point of discussion is whether the current knowledge of the mechanisms responsible for the large piezoelectric response of PMN–PT could be used to tailor functional properties of lead-free RE-FE materials. A system to possibly consider is the BiFeO3–BaTiO3 solid solution considering its robust, compositional-dependent RE-FE behavior52 and the presence of Bi3+ cations with similar electronic structure as Pb2+ cations. While a rigorous local structural analysis of BiFeO3–BaTiO3 is still missing, it is interesting that the recently reported high piezoelectric activity of these ceramics achieved by a combination of Ga doping and quenching was found to be correlated with the increased lattice strain and tetragonality53, reminiscent of the effect of Sm doping on the ultrahigh piezoelectricity of PMN-PT12,13. As a final note, it must be mentioned that systematic data on domain-wall contributions to the nonlinear dielectric and piezoelectric response of lead-free RE-FE ceramics, measured over wide compositional, temperature and driving-field ranges, are largely lacking. Such measurements would be useful to identify the piezoelectric contribution arising from the RE character of these materials, as recently demonstrated for PMN–PT36.