{'fraction_non_alphanumeric': 0.03762062374714466, 'fraction_numerical': 0.01804111696424409, 'mean_word_length': 5.097640704945992, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Connecting the Multiscale Structure with Macroscopic Response of Relaxor Ferroelectrics

Lead‐based relaxor ferroelectrics are characterized by outstanding piezoelectric and dielectric properties, making them useful in a wide range of applications. Despite the numerous models proposed to describe the relation between their nanoscale polar structure and the large properties, the multiple contributions to these properties are not yet revealed. Here, by combining atomistic and mesoscopic‐scale structural analyses with macroscopic piezoelectric and dielectric measurements across the (100–x)Pb(Mg1/3Nb2/3)O3–xPbTiO3 (PMN–xPT) phase diagram, a direct link is established between the multiscale structure and the large nonlinear macroscopic response observed in the monoclinic PMN‐xPT compositions. The approach reveals a previously unrecognized softening effect, which is common to Pb‐based relaxor ferroelectrics and arises from the displacements of low‐angle nanodomain walls, facilitated by the nanoscale polar character and lattice strain disorder. This comprehensive comparative study points to the multiple, distinct mechanisms that are responsible for the large piezoelectric response in relaxor ferroelectrics.

polarization. On the other hand, the model by Takenaka et al. [7] predicts a high density of low-angle domain walls in a PND or slush-like polar structure, arising from anisotropically coupled dipoles, ultimately contributing to a greater flexibility for polarization rotation and thus large properties. Coupling between PNRs and ferroelectric polarization was experimentally observed by X-ray and neutron scattering experiments, [20][21][22] and the role of PNRs on the large piezoelectricity of relaxor ferroelectrics has also been discussed. [23,24] Polarization rotation, which was identified as the main mechanism responsible for the property enhancement in relaxor ferroelectrics, [25] was directly evidenced during field application by in situ pair distribution function analysis on PMN-PT. [26] While the described models explain the effects of PNR dynamics or the presence of high density low-angle domain walls on the lattice response under external fields in terms of the polarization rotation mechanism, it is not clear how this nanoscale structure in relaxor-based materials affects the motion of domain walls and thus the macroscopic piezoelectric response, and whether it can relate to high properties of compositions at different parts of the phase diagram. This is particularly important for the high performance polycrystalline relaxor ferroelectrics where, unlike in domain-engineered single crystals, the nonlinear and hysteretic domain-wall contributions may dominate the total piezoelectric response. [27] In domain engineered single crystals, on the other hand, the best properties are observed in AC-poled crystals where the domainwall motion does occur during poling even if its contribution is not as obvious as in ceramics. [28] In this work we present a broad picture of the multiscale structure of relaxor ferroelectrics, and relate it to the piezoelectric and dielectric nonlinear response by studying polycrystalline PMN-xPT over a wide compositional range, spanning from the relaxor PMN end member (x = 0) to the ferroelectric tetragonal phase region (x = 40). By nonlinear macroscopic measurements we identify three distinct regions in the dynamic electromechanical and dielectric response, which we associate with specific atomic-and mesoscopic-scale structures of the PMN-xPT solid solutions. In contrast to the nonlinear response of the morphotropic phase boundary (MPB) and tetragonal PMN-xPT compositions (i.e., x ≥ 35) where polarization rotation dominates, the results reveal additional large hysteretic and anhysteretic nonlinear contributions to the response in a wide range of compositions with monoclinic symmetry (20 ≤ x < 33.5), which are characterized by a complex nanodomain structure and relaxor behavior. Based on the detailed compositional comparison of the atomic and nanoscale structures, we show that this additional nonlinear response in the monoclinic compositions, ascribed to a hitherto unreported softening effect, is related to the nanoscale polar entities and to the A-sublattice strain disorder, which makes the nanodomain walls exceedingly mobile. We thus present here a comprehensive structure-property investigation that enables us to experimentally demonstrate a new softening effect, which originates in the displacement of nanodomain walls, boosting the piezoelectric response of relaxor ferroelectrics. This effect contrasts the widely considered polarization rotation mechanism that dominates the piezoelectric response of PMN-xPT compositions close to the MPB with the tetragonal phase.

Relaxor Character, Domain Structure, and Atomic-Scale Structure of PMN-xPT
We begin by presenting in Figure 1 the phase composition, the relaxor character, and the domain structure of the analyzed PMN-xPT compositions. Figure 1a shows the phase diagram of the PMN-xPT solid solution system derived from X-ray powder diffraction (XRD) analysis and temperature-dependent permittivity measurements of the PMN-xPT samples (Section S1, Supporting Information) possessing comparable (3-5 µm) average grain sizes (Section S2, Supporting Information). The phase diagram agrees well with the one proposed by Singh et al. [29] The distortion of the perovskite lattice of PMN progressively increases with the increasing PT content, evolving from pseudocubic (Pc) to monoclinic (M) M B symmetry (rhombohedrallike M distortion), then to M C (tetragonal (T)-like M distortion) and finally to the T phase (P4mm space group) ( Figure 1a; Section S1, Supporting Information). M B and M C notations correspond to the Cm and Pm space groups, respectively (after Vanderbilt and Cohen [30] ). Figure 1b illustrates the relaxor character of the samples by plotting the frequency dispersion of the permittivity maximum, defined as @100 kHz @ 1 kHz max m ax max , as a function of composition, where @100 kHz max T ε and @1 kHz max T ε correspond to the temperature of the permittivity maximum measured at 100 and 1 kHz, respectively. The compositional region in the PMN-xPT phase diagram exhibiting a dispersive permittivity maximum (indicated by orange color in Figure 1a,b) is commonly linked to the presence of nanosized polar entities and short-range ordering, giving rise to the relaxor behavior, [11,31] as supported by previous diffuse scattering analyses. [21,22,32,33] From the highest frequency dispersion measured in the canonical PMN relaxor ( max ∆ ε T = 10 °C), the dispersion tends to approach zero as the PT content is increased to x = 33.5 and beyond. The trend of the decreasing max T ∆ ε with increasing PT content reflects the well-known gradual weakening of the relaxor character and the emergence of the long-range ferroelectric order when approaching the T phase. [22] Next, we investigate the compositional relaxor-to-ferroelectric evolution of pristine samples from the domain structure perspective. Figure 1c-e presents the micro-and nanoscale domain structures of compositions representative of the M (30PT), MPB (35PT), and T (40PT) phase regions. The monoclinic 30PT composition is characterized by a hierarchical arrangement of differently oriented sets of ≈5-10 nm wide striation-like nanodomains, generally extending along ⟨111⟩ (see Section S3, Supporting Information), separated by irregular wedge-shaped microscale domains (Figure 1c). In the morphotropic 35PT composition, the hierarchical domain arrangement still persists, however, the nanodomains are embedded in straighter and more regular microscale lamellar domains of up to ≈300 nm in width ( Figure 1d). Finally, in tetragonal 40PT the domain hierarchy is no longer present and the domain structure consists of straight lamellar domains of up to ≈400 nm in width (Figure 1e). Full transmission electron microscopy (TEM) analysis of the PMN-xPT compositional series, reported in Section S3 of the Supporting Information, further confirms the domain-structure development from hierarchical nanodomain arrangement in 20 ≤ x ≤ 32, to nanodomains merging at x = 33.5 and formation of straight lamellar domains with embedded nanodomains at x = 35 (i.e., at the onset of T symmetry), up to the disappearance of the hierarchical domains at x > 37. The observed domainstructure evolution with the increasing PT content is consistent with recent theoretical predictions, [7] experimental diffuse scattering measurements in PMN-xPT [32] and previously reported domain evolution in PMN-PT ceramics [34] and single crystals. [35] The presented domain-structure evolution already suggests that the high piezoelectric response present in PMN-xPT over a wide range of compositions cannot have a unique origin (such as polarization rotation alone).
We further note that despite the domain-structure analysis performed on unpoled samples, in the case of the monoclinic 30PT composition we observe qualitatively the same nanoscale features before and after poling, i.e., the nano domains are also present after poling, but their orientation is changed depending on the direction of the applied external field (see ref. [36]). In the case of the MPB 35PT composition, on the other hand, the microscale lamellar domains become thinner with an applied field and settle in their preferred orientation, partially eliminating the hierarchically arranged nanodomains, [36] and likely making the accumulated domain walls of high density harder to move after initial poling, as compared to the monoclinic compositions. As shown later, these observations are consistent with the nonlinear piezoelectric analysis.
From a comparison of the presented results, it becomes clear that the drastic drop of max T ∆ ε at x = 33.5 (see Figure 1b) coincides with the striation-like nanodomains starting to merge and pattern into more regular lamellar-like domains (Section S3, Supporting Information). Following this observation, it is also evident that the absence of the relaxor character (∆ εmax T ≈ 0) at x > 35 is related to the emergence of the T phase and the disappearance of the hierarchical domain structure. The provided analyses thus confirm an intimate relationship between the average symmetry, relaxor behavior, and the domain structure in PMN-xPT.
We next present in Figure 2 the analysis of the three representative compositions at the atomic scale by high-angle annular dark field imaging with scanning transmission electron microscopy (HAADF-STEM; original HAADF images are shown in Section S4, Supporting Information). Motivated by theoretical studies, which predict a strong effect of localized Pb displacements from their average position, [13,14] as well as the influence of Ti-concentration in PMN-xPT on both the correlated B-site and localized Pb displacements, [7,37] we base our analysis on the B-atom off-center displacements within its Pbcage (Figure 2a As shown in Figure 2a, the 30PT M composition is characterized by small B-atom displacements (represented by the arrows) with island-like patches of a few-unit-cell semiuniform displacements of up to ≈20 pm in magnitude (darker yellow regions). In between these patches the displacements are smaller than ≈14 pm and appear as uncorrelated, randomly pointing in all directions (see the polar plot underneath Figure 2a), which illustrates an average picture of the displacements). By contrast, the morphotropic 35PT shows much larger and more correlated B-atom displacements of ≈30 pm on average, displaced roughly along [011] direction, but slightly tilted toward [001] (which in the (100) projection corresponds to the distortion of an M cell away from [111] and toward [001]-see arrow in the polar plot underneath Figure 2b; all indexing is given in pseudocubic perovskite setting). Homogeneous areas of stronger displacements are connected into patches that extend along [01-1], i.e., perpendicular to the direction of the displacements (dark red areas in Figure 2b). These regions likely correspond to the striped nanodomains within microscale lamellar domains, as observed in Figure 1d. Finally, Figure 2c shows the tetragonal 40PT composition with the strongest, up to ≈60 pm B-atom displacements and the largest regions of uniform displacements. The average B-atom displacements along [001] direction coincide with the expected T distortion (see polar plot underneath Figure 2c), and the regions of larger displacements , is suggested to correspond to a 90° T domain (with the T distortion oriented normal to the imaging plane; see polar plot in Figure 2c showing strongly dispersed points from a combination of in-plane and out-of-plane polarization due to the scanned area crossing adjacent domains). We also note that the strongest displacements in the 40PT composition (up to 60 pm) suggest largest c/a ratio of the unit cell (see also the largest 002/200 peak splitting in the XRD pattern of the Section S1, Supporting Information), and thus strongest intrinsic polarization in this PT-rich composition.
Figure 2d-f shows the positional disorder of the Pb atoms, derived from the same HAADF images as those used for analysis in Figure 2a-c, and determined by the distortion angle Φ Pb , which is defined as the angle of displacement of each next Pb atom from the equatorial plane of the previous Pb in the horizontal direction (see schematics on the left-hand side and detailed description in the Experimental Section). This lattice distortion reflects the localized nature of Pb off-center displacements, believed to play the key role in the nanoscale polar structure and the relaxor nature of PMN. [3,4,8,13,33,38] We note that imaging was performed at room temperature, thus below the corresponding maximum in the permittivity and freezing temperature T f of the samples (see T f values in Section S1, Supporting Information), meaning that the offcenter atom displacements are temporally and spatially frozen, [13,38] making the visualization of the associated average Pb-sublattice positional disorder on the atomic scale possible. This disorder is observed in all the representative compositions (see colored contour maps in Figure 2d-f); however, the average distortion angle and its lateral fluctuation are clearly larger in the M composition ( Pb Φ = 0.96° ± 0.73°), as compared to the MPB ( Pb Φ = 0.67° ± 0.50°) and the T composition ( ΦPb = 0.68° ± 0.54°). Furthermore, the largest local Pb distortions with highest Φ Pb angles are observed in the 30PT M composition (red-circled spots in Figure 2d); see also the broader distortion-angle distribution curve below the map, and its longer tail, as indicated by the arrow). These strong local Pb distortions are believed to play a key role in the short-range ordering and thus the relaxor behavior of the relaxor-ferroelectric compositions.
The presented analysis is consistent with the atomistic model of the relaxor nature in PMN-xPT, which explains the atomic origin of the relaxor-type nanoscale structure in this perovskite. [37] The model predicts that at low Ti concentrations, the B-site chemical disorder strongly affects the BO bonds, causing the oxygen with more surrounding Nb 5+ to be overbonded while oxygen in the Mg 2+ -rich environment to be underbonded. This further affects the position of Pb atoms that tend to move closer to the Mg-rich faces, which leads to localized Pb off-center displacements and structural disorder, causing relaxor behavior. In contrast, high Ti concentrations (i.e., x ≥ 35PT) cause the formation of strongly hybridized TiO bonds, contributing to more correlated (less disordered) Pb displacements and larger average B-atom displacements (along the [001] direction), leading to suppression of the dielectric frequency dispersion and to the long-range ferroelectric ordering. The atomic analysis provided here experimentally confirms these theoretical predictions and illustrates the compositionally induced relaxor-to-ferroelectric crossover in PMN-xPT at the atomic level.
The observed stronger Pb-positional disorder (Figure 2d), which is believed to be the source of the relaxor nature of the M phases of PMN-xPT (Figure 1b), further influences the diffraction contrast in TEM, enabling visualization of the nanodomains (Figure 1c). These nanodomains were previously defined as an assembly of low-angle and low-energy domain walls, [7,39] and were predicted to arise from nanoscale variations in polar displacements with weak correlations across the nanodomains. Therefore, we suggest that the observed local Pb distortions, characteristic for the relaxor ferroelectric PMN-xPT compositions, strongly influence the domain-wall mobility and thus the piezoelectric and dielectric response of these compositions.

Field-Dependent Piezoelectric and Dielectric Response
To confirm our hypothesis of the influence of local Pb distortions on the domain-wall mobility, we analyzed the converse piezoelectric response of the PMN-xPT compositional series (20 ≤ x ≤ 40) as a function of electric-field amplitude. It is assumed that the weak-field dynamic nonlinearity and the phase lag in the response are sensitive to the interaction of the domain walls with the identified atomic and nanoscale structural characteristics of the relaxor-ferroelectric (M) compositions. A common approach to represent the response is to use the complex longitudinal piezoelectric coefficients, i.e., the real d 33 ′ coefficient along with the tangent of the piezoelectric phase angle tanδ p , defined as the ratio between the imaginary d 33 ″ and real d 33 ′ coefficient (tanδ p = d 33 ″/d 33 ′). [40] Figure 3a presents the field-dependent d 33 ′ data, from which we assess the nonlinearity (i.e., field-dependence of d 33 ′), while Figure 3b shows the field-dependent tanδ p , correlated to the piezoelectric hysteresis. All the compositions exhibit increasing d 33 ′ and tanδ p values as a function of the field amplitude. In addition, tanδ p of all compositions approaches zero in the proximity of zero field amplitude, confirming that the hysteresis is indeed field-induced. In ferroelectrics, this behavior is commonly attributed to the contribution from irreversible displacements of non-180° domain walls [27,[40][41][42] or other types of dynamic interfaces [43] (see Section S5, Supporting Information). Close inspection of the data reveals a transition in the nonlinear behavior between x = 33.5 and x = 32 (arrow in Figure 3a). For compositions x ≥ 33.5, the d 33 ′ versus field curves show a slightly descending (sublinear) trend, while for x < 33.5 these curves change to a strongly ascending (superlinear) trend. This transition is accompanied by an anomalous increase in the field dependence of tanδ p , occurring at the same compositional point (see arrow in Figure 3b). The same two nonlinear regimes, observed in the piezoelectric coefficient (Figure 3a), along with the anomaly in the phase angle (Figure 3b), were also identified in the field-dependent permittivity response (see Figure S7 in Section S6, Supporting Information), and are additionally highlighted in the schematic in Figure 3c. We note that the crossover between the two regimes coincides with the In the next step we quantify the data shown in Figure 3a by determining the reversible ( 33 init d ) and irreversible (α*) coefficients following the Rayleigh law approach [41] (all the details on the Rayleigh formalism are provided in Section S5, Supporting Information) where d E ′( ) 33 0 is the total, field-dependent longitudinal piezoelectric coefficient (data shown in Figure 3a), and E 0 is the field amplitude. The two coefficients extracted from the analysis, 33 init d and α*, represent the zero-field intercept and the fielddependent slope of the d 33 ′-E 0 curves, respectively. In order to compare the PMN-xPT compositions exhibiting different coercive fields (Section S7, Supporting Information), the irreversible α* coefficient is calculated for the electric field corresponding to 30% of the coercive field (E/E C = 0.3; see Section S5 of the Supporting Information for details).
As observed from the black curve in Figure 3d Figure 3d). 33 init d peaking close to the two MPBs was also observed in PMN-xPT single crystals [44] and is commonly attributed to the MPB-related enhancement of the lattice response arising from the polarization rotation mechanism. [45] In strong contrast to 33 init d , α* (which represents the irreversible displacements of dynamic interfaces) does not peak at or close to the MPBs (red curve in Figure 3d Figure 3d). We note that the value of α* in the M B region (up to ≈20 × 10 −16 m 2 V −2 ) is almost an order of magnitude higher than that reported for morphotropic "soft" Pb(Zr,Ti)O 3 (PZT) under similar driving conditions (in the range of 2 × 10 −16 -4 × 10 −16 m 2 V −2 ; [27,40] ). Even though the 33 init d is low is low in the monoclinic PMN-xPT compositions, the large, the large α* leads to piezoelectric coefficients which at the high-field end may even exceed that of MPB compositions (compare the green and red curve in Figure 3a).
Consistent with the compositional evolution in the nonlinearity (α*), a similar trend with PT content is revealed in tanδ p (see red arrow in Figure 3e), with large phase angle values (up to tanδ p = 0.31) confined to the M B phase region. The analysis therefore suggests that the piezoelectric response in the M B phases is characterized by an enhancement of both nonlinearity and hysteresis (also illustrated by the different piezoelectric hysteresis evolution of representative M B and MPB samples shown in the insets of Figure 3e). Note that the compositional trends shown in Figure 3 for the converse piezoelectric response are consistently observed also in the direct piezoelectric and dielectric response (Sections S6 and S8, Supporting Information). This supports the hypothesis that the enhanced nonlinearity and hysteresis in poled M B compositions are likely dominated by the displacements of ferroelectric/ferroelastic non-180° domain walls.
To complement the analysis of the first-harmonic response, we next show the results of the weak-field third-harmonic polarization measurements, which are highly sensitive to nonlinear domain-wall dynamics. [46] Here, we analyze the dependence of P 3 ″ (Figure 4a) and P 3 ′ (Figure 4b) on the electric-field amplitude, where P 3 ″ and P 3 ′ are the amplitudes of the out-of-phase and in-phase polarization of the third harmonic, respectively, along with the third-harmonic phase angle δ 3 (Figure 4c), which is defined as tan 3  ). The inset above shows the frequency dispersion of the maximum permittivity ( T ∆ ε max ), reproduced from Figure 1b; middle red dashed line links the drop in T ∆ ε max at 32PT and the transition between the RE-FE and FE regions. In the middle are schematics of the hysteresis loops characteristic for these three types of responses, derived from the corresponding signs of P 3 ′ and P 3 ″, and the δ 3 value or range (see color bands in (a-c)). Loop deformations due to the third-harmonic contribution are indicated by arrows. Right-most inset of (d) shows the P 3 ″-P 3 ′ phasor diagram with the hysteresis-loop deformations, dependent on a particular combination of ±P 3 ″ and ±P 3 ′. The diagram presents the phasors (colored arrows) related to the three identified third-harmonic responses (curved gray arrow indicates the δ 3 transition observed in RE response; curved red arrow indicates the transition from FE to RE-FE-type nonlinear response with decreasing PT content). The red and blue shaded triangles surrounding the colored arrows indicate the span of the δ 3 values identified at the high-field end of the RE-FE and FE response (see colored bands in (c)).  Figure 4d) with the adjacent schematic loop deformations by the third-harmonic response for these three types of responses).
The FE-type response, approximately confined to the region 33.5 ≤ x ≤ 40, is characterized by three features (follow blue-shaded regions in Figure 4a-c): i) a negative P 3 ″ evolving with field with a power exponent close to 2, ii) P 3 ′ settling around zero in the whole field range, and iii) δ 3 varying with field in between extremes of ≈−105° and ≈−75°, averaging around −90°. These three features of the third-harmonic response are close to those predicted by the Rayleigh relations (Section S5, Supporting Information), thus suggesting a Rayleigh-like response (see also blue loop schematic and the blue arrow in the phasor diagram in Figure 4d). This kind of response was previously reported for "soft" PZT compositions [27,40,41,46,47] and is illustrated with the characteristic third-harmonic polarization response for a Nbdoped morphotropic PZT in Section S9 of the Supporting Information.
The nonlinear response in the RE-FE region (20 ≤ x ≤ 32) is distinctly different from that in the FE region, and it is identified by (follow red-shaded regions in Figure 4a-c): i) a stronger negative P 3 ″ evolving with field with a power exponent of ≈2.5, ii) a field-induced negative P 3 ′, and iii) δ 3 that still varies with field, however, around a mean value of ≈−125° (see red arrow in the phasor diagram in Figure 4d). The RE-FE response clearly deviates from the Rayleigh predictions. We note that the strong field-induced third-harmonic contribution −P 3 ″, which is hysteretic in nature, is consistent with the large first-harmonic hysteresis measured in the M B phases (see Figure 3e and inset with red borders). In addition, the RE-FE response is characterized by the emergence of −P 3 ′, which represents an anhysteretic (reversible) contribution, responsible for the enhancement of the total polarization amplitude, as indicated with arrows in the red loop schematics in Figure 4d.
Finally, for a thorough comparison, we show a third type of response confined to close-to-PMN region, denoted as RE-type (see phase diagram in Figure 4d). This response is characterized by (follow gray-shaded regions in Figure 4a-c): i) a nearly zero P 3 ″, ii) a field induced positive P 3 ′ and iii) δ 3 showing a clear transition with field from ≈−180° to ≈0° (see gray curved arrow in the phasor diagram in Figure 4d). This transition between two anhysteretic responses with polarization divergence at weak and saturation at high fields (see gray loop schematics in Figure 4d) was previously suggested to arise from evolving dynamics of PNRs with field. [48,49] Considering that the bridging RE-FE-type response qualitatively deviates from both the RE-type and FE-type, as illustrated in the phasor diagram in Figure 4d, the presented data suggest that this response cannot be simply assigned to the PNR dynamics, nor to the classical Rayleigh-like dynamics. While the underlying mechanisms may be complex, our data provide the key link between this particular nonlinear behavior and the relaxor nature of the M B compositions, which is evidenced by two observations. First, the transition from the RE-FE to the FE-type response (see overlapping red/ blue area in the phase diagram in Figure 4d) and the corresponding anomaly in the first-harmonic piezoelectric response (arrows in Figure 3a,b) occur close to x = 32. This is the compositional point above which the dispersion of permittivity maximum T max ∆ ε drops to nearly zero (see inset linked to the phase diagram in Figure 4d), and where the nano domains begin to merge into regular and larger T-like lamellar domains (Section S3, Supporting Information). Therefore, the RE-FEtype response is confined to the monoclinic compositions that demonstrate relaxor characteristics. Second, a qualitatively similar response to that of the RE-FE-type was measured in the relaxor-ferroelectric phase of PMN below the freezing point (see Section S10, Supporting Information), supporting the idea that the observed RE-FE response is likely dominated by PMN, i.e., the relaxor end-member of the PMN-xPT solid solution. In support to this claim is the large nonlinearity and hysteresis measured in PMN-7PT under DC bias. [50] Additionally, the RE-FE-type characteristics are also measured in two other lead-based relaxor-ferroelectric materials, i.e., Pb(Fe 0.5 Nb 0.5 )O 3 (PFN) and Pb(Sc 0.5 Nb 0.5 )O 3 (PSN) (Section S9, Supporting Information). We observe up to ≈200% increase of the piezoelectric coefficient in the relaxor ferroelectrics, while the "classical" ferroelectric compositions achieve merely an ≈80% increase. This clearly points to the quantitative importance of the observed nonlinear response in relaxor ferroelectrics. The comparison also demonstrates that the relaxor-induced softening effect is, in relative terms, much stronger than that induced in "soft" PZT by the donor dopant (in this case Nb), which is widely exploited in current applications, such as in multilayer piezoelectric actuators. [51,52] All these results indicate that the identified nonlinear (RE-FE-type) behavior is not related to MPB compositions, as it is also observed in PFN and PSN, but is intimately associated with the relaxor nature of the analyzed compositions and thus common to Pb-based relaxor ferroelectrics. Furthermore, there are two key features which lead to the softening of the response positional disorder on the A site, reflecting lone-pair electron hybridization with oxygen, and B-site charge disorder, both crucial for the RE-FE response. Considering the similar electronic structure of Pb and Bi, [53] it would be interesting to further explore lead-free Bi-based relaxor ferroelectrics with B-site charge disorder.

Link between Nanoscale Domains and Atomic Structure
The macroscopic piezoelectric data strongly suggest that the observed large increase in the d 33 ′ with field amplitude in monoclinic PMN-xPT and similar relaxor-ferroelectric compositions (see Figure 5) has its origin in the domain-wall dynamics. To further explore the characteristic nano-and atomic-scale structural environment of these domain walls, we next show a combined TEM and STEM analysis on the representative monoclinic 30PT composition. Figure 6a shows a TEM image of the striation-like nanodomains in the 30PT sample. This image was processed for its pixel intensities, which reflect differences in diffraction contrast due to lattice distortion, to obtain the 3D intensity map given in Figure 6b. To compare these nanoscale features with those on the atomic scale, we captured an integrated differential phase contrast (iDPC) STEM image, shown in Figure 6c, which allows detection of the positions of all atoms including those of the oxygen (see Figure 6c-e). The analysis of the iDPC image enabled us to measure the displacements of Pb atoms from the center of their oxygen cage, obtaining information about the local polar structure. We observe that stronger Pb displacements, creating clusters of strong uniform polar distortion, are coherent only within a few nm (Figure 6d), which corresponds to the width of nanodomains (see the indicated scale relation between Figures 6b,e). Similar clusters are observed for B-site displacements from their oxygen cage, with slightly smaller magnitudes but of the same direction as those of Pb (this analysis is presented in Section S11, Supporting Information), confirming the unique nanoscale polar order. The angles between the dipoles in subsequent unit cells vary rather smoothly (see neighboring arrows in Figure 6d), which is in line with what was previously defined as low-angle domain walls. [7,8] From the atomic-to-nanoscale structural comparison it is now clear that 1) there are no straight domain walls, nor can their thickness be unambiguously defined, 2) clustering of larger polar displacements into nanodomains obeys crystallographic orientation (i.e., extending along ⟨111⟩ directions; see Figure S5 in Section S3, Supporting Information), and 3) there are no sharp boundaries at the edges of polar clusters, indicating a continuously varying polarization across different polar-cluster regions. This atomic-to nanoscale comparison indicates that the nanoscale clusters of stronger polarity are directly linked to the observed nanodomains and, as suggested by the macroscopic data (Figures 3 and 4), are greatly affected by external fields, causing easy movement of the nanodomain walls.
By combining the compositional evolution of the subswitching electrical and electromechanical response of PMN-xPT (Figures 3-5) with the meso-to-atomic-scale structural evolution (Figures 1, 2, and 6), we can now propose a link across these different scales, identifying the characteristic response types for each group of materials (i.e., relaxor, relaxor ferroelectric, and ferroelectric). Furthermore, the large nonlinear piezoelectric and dielectric response of the monoclinic PMN-xPT compositions can now be explained. We suggest that the large RE-FE-type response arises from the displacements of the low-angle nanodomain walls, which are mirroring the atomic-scale disorder, i.e., the short-range Pb-sublattice positional disorder (Figure 2d) and the nanoscale regions of stronger polar distortion (Figure 6d,e). This relaxor-specific nanoscale structure is responsible for the increased mobility of the nanodomain walls, [36,54] representing thus a new type of electromechanical softening, distinctly different from those reported earlier in the frame of the adaptive phase theory, [39] or MPB-related polarization rotation mechanism assisted by rotation of PNRs. [12] The key role of the low-angle domain walls has been discussed by Takenaka et al., [7] however, theoretical modeling in that study describes the mechanism on the lattice scale, pointing to an easy rotation of polar vectors, and is hindered from detecting the domain-wall-motion contribution and the associated softening effect. Despite previous studies identifying the nanodomain structure in PMN-PT, our work provides insights into the relationship between these nanoscale features and the ultrahigh response, revealing multiple contribution to the functional properties. Furthermore, our analysis made it possible to clearly separate the low-angle domain wall dynamics from the polarization rotation mechanism that underpins the response at the MPB with the T phase. Thus, there are more mechanisms contributing to a large piezoelectric response in a given relaxor ferroelectric system, and the dominant mechanism evolves with the composition.

Conclusion
In summary, we show here three distinct regions in the dynamic electromechanical response of the polycrystalline PMN-xPT, i.e., the RE, RE-FE, and FE states. We identified a large nonlinear dielectric and piezoelectric response that is confined to a wide monoclinic compositional range of the PMN-xPT (20 ≤ x ≤32), representing a new type of softening of the electromechanical response, also observed in other Pb-based relaxor ferroelectrics beyond PMN-PT. We further suggest a link between this macroscopic response and the nano-and atomic-scale structure of relaxor ferroelectrics, revealing a softening effect that is different from the usually assumed enhanced polarization rotation at MPBs. Furthermore, the herein observed Pbpositional disorder and the nanoscale polar clusters forming highly mobile low-angle nanodomain walls are suggested to be the key features leading to the exceedingly large nonlinear piezoelectric and dielectric response of the monoclinic PMN-xPT and similar Pb-based relaxor ferroelectrics.  5, 10, 20, 27, 30, 32, 33.5, 35, 37, and 40, were synthesized via a mechanochemical activation route. Details on the synthesis procedure can be found elsewhere. [55] The activated powders were uniaxially pressed into pellets at 50 MPa, isostatically pressed at 300 MPa, and then sintered in closed alumina crucibles buried into a packing powder consisting of the same composition as the pellet. The same sintering procedure, i.e., 1200 °C for 16 h with a heating/cooling rate of 2 °C min −1 was employed for all compositions to ensure comparable microstructures (see Section S2, Supporting Information).
The phase composition of the sintered samples was determined by XRD analysis, using PANalytical X'Pert Pro diffractometer with Cu Kα1 radiation (λ = 1.54056 Å) and X'Celerator detector. Roomtemperature XRD patterns of the powders, obtained by crushing the pellets, were collected over the range of 20°-120° with a step size of 0.008°.
Microstructural features were observed using a field-emission scanning electron microscope (SEM; JSM-7600, Jeol Ltd., Tokyo, Japan) on samples that were ground, polished, and thermally etched at 900 °C for 15 min. Average grain size of the samples was estimated from the SEM images. The density was measured in accordance with the Archimedes' principle.
TEM (JEM-2100, Jeol Ltd., Tokyo, Japan) operated at 200 kV with a beryllium double-tilt specimen holder was used to determine the domain structure of the samples. TEM samples were prepared by cutting disks, mechanical thinning, and dimpling, then coldstage Ar-ion milling until perforation (RES 010, Bal-Tec AG, Balzers, Liechtenstein).
All structural, microstructural, and atomic-scale investigations were performed on pristine samples, thus in their unpoled state.
STEM and Microanalysis: Samples for STEM analysis were produced by wedge polishing using a Multiprep polishing system (Allied High Tech Product Inc., Compton, USA) and cold-stage low-energy Ar-ionmilling (model 1050 TEM Mill, Fishione Instruments, Corporate Circle, USA), and final carbon-coating (PECS system, Gatan Inc., Pleasanton, USA) to prevent charging.
For atomic-resolution imaging a probe-corrected STEM (Titan G2, FEI, Hilsboro, USA) was used, equipped with a Schottky field emission gun operated at 200 kV, with the beam current of 30 pA and a convergence semiangle of 19.6 mrad. For HAADF imaging the annular dark-field inner collection semiangle was 28-150 mrad. Integrated differential phase contrast (iDPC) STEM imaging was performed simultaneously with annular dark field imaging at the collection semiangle of 7-28 mrad, using a 4-quadrant segmented detector. [56] Analysis of the HAADF (Section S4, Supporting Information) and iDPC images (Section S11, Supporting Information) for determining positions of atom columns was performed using a 2D Gaussian fit (by Gaussian fit-on-spot plugin in ImageJ software), after which the relative displacements were calculated.
The Pb-sublattice positional disorder (presented in Figure 2d-f) was measured in the following way. First, the HAADF image was rotated so that Pb atomic columns along the [010] direction were on average aligned with the horizontal line. Then, an angle was calculated between a line connecting two neighboring Pb columns and the reference horizontal line (as represented in the scheme on the left-hand side of Figure 2d). This calculation was repeated for each next unit cell, meaning that the angle was determined for each unit cell independently. All angles for each horizontal Pb-Pb distance were then plotted in the form of a contour plot (Figure 2d-f).
Piezoelectric and Dielectric Measurements: Cylindrical ceramic samples for electrical measurements were first thinned to ≈0.5 mm, surface polished, and then annealed to 600 °C for 1 h with heating and cooling rate of 5 °C min −1 and 1 °C min −1 , respectively, to relax mechanical stresses possibly induced during the polishing procedure. The samples were then electroded with Au by radio frequency magnetron sputtering and poled at room temperature using a DC field of 60 kV cm −1 for 30 min. The subsequent electrical and electromechanical measurements were performed at least 24 h after the poling procedure.
Temperature-dependent permittivity measurements were performed on poled samples during heating by an LCR bridge (Agilent E4980A Precision LCR meter) in the temperature range from 25 to 300 °C, using 1 V of AC applied voltage and 1, 10, and 100 kHz driving frequencies (for PMN and PMN-10PT, the measurements were performed down to −50 °C; Section S1, Supporting Information). The heating rate was 2 °C min −1 .
Converse piezoelectric measurements were performed using a fiberoptic displacement sensor (MTI 2100 Fotonic Sensor), a low-distortion voltage generator (SRS DS360), a voltage amplifier (Trek 609E-6), a high-voltage probe (Textronix P6015A), an oscilloscope (LeCroy 9310C), and two lock-in amplifiers (SR830 DSP). A cantilever was used as the upper mechanical contact. The strain signal was monitored both by the oscilloscope and a lock-in amplifier. The second lock-in amplifier was used to monitor the voltage on the sample using the high-voltage probe. The measurements were carried out at room temperature using a continuous bipolar 10 Hz sinusoidal electric field with increasing amplitude. For more details on the measurement system and determination of the piezoelectric d 33 coefficient and phase angle tanδ p see ref. [57].
Dielectric permittivity was determined using the driving-voltage setup as described above for the converse piezoelectric measurements, but equipped with a custom-made resistor box, consisting of selectioncontrolled precision resistors connected in series with the sample. The capacitive current was determined by measuring the voltage drop on a selected resistor by the lock-in technique, which made possible to extract the first and third-harmonic polarization responses; further details regarding the measurement technique and procedures are reported by Damjanovic and co-workers. [49,58] The dielectric measurements were performed under the identical driving field conditions as used for the piezoelectric measurements. The dielectric hysteresis loops (Section S8, Supporting Information) were measured using a charge amplifier (Kistler 5018A) and oscilloscope (LeCroy 9310C) using the same driving-voltage instruments as described above.
Direct piezoelectric measurements (Section S8, Supporting Information) were performed using a dynamic Berlincourt-type press, equipped with charge amplifiers, as reported by Barzegar et al. [59] The samples were driven with an AC stress of 1 Hz frequency and amplitudes in the range of ≈0.3-3.5 MPa. The samples were clamped with a constant prestress of ≈3 MPa, applied with a step motor.
Large-signal P−E hysteresis loops (Section S7, Supporting Information) were measured using a commercial aixACCT TF 2000 analyzer at 1 Hz of sinusoidal driving frequency and 60 kV cm −1 of field amplitude. The samples were immersed in silicone oil to prevent arching. The large-signal P−E hysteresis loops and third-harmonic polarization response below room temperature (Section S10, Supporting Information) were measured as described above using a commercial aixACCT low-temperature sample holder.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.