Role of random electric fields in relaxors

Static Signature: Short-Range Polar Order.

To answer these questions we compared the nanoscale structures of PMN and PZT using neutron-based measurements of the elastic diffuse scattering, which is arguably the single most interesting and intensely examined property of lead-oxide relaxors to date. Strong, temperature-dependent and highly anisotropic diffuse scattering has been documented in many different lead-oxide perovskite relaxors using X-ray and neutron-scattering methods (20⇓⇓⇓⇓⇓⇓–27). Fig. 1C, Inset illustrates the anisotropic nature of this diffuse scattering in PMN via a log-scale intensity map of the well-known, butterfly-shaped diffuse-scattering contours that decorate the (001) Bragg peak in the scattering plane defined by the pseudocubic [110] and [001] crystallographic axes. We measured the diffuse-scattering intensity at very low temperature (1.7 K), where it is strongest, by scanning the neutron momentum transfer along the trajectory defined by the dashed blue arrow in Fig. 1C. These data are given by the solid blue dots, and two strong peaks are evident where this trajectory clips the two lower wings of the butterfly-shaped contours. The origin of this diffuse scattering is extremely controversial. Numerous models of the underlying short-range structural order have been proposed to explain the experimental data (25, 28⇓⇓⇓–32). Neutron backscattering measurements (33, 34), which provide very sharp energy resolution , show that truly elastic (i.e., static) diffuse scattering in PMN first appears at a temperature K that coincides with that at which the lowest-frequency transverse optic (TO) phonon is softest and overdamped; it also grows monotonically on cooling (35). A monotonic increase of the diffuse scattering on cooling is observed in all pseudorhombohedral compositions of PMN-xPT and PZN-xPT (25, 36). Other neutron and X-ray studies have demonstrated that relaxor diffuse scattering is strongly affected by external electric fields (37, 38) and vanishes at high pressures (27, 39). For these reasons many researchers identify the diffuse scattering with static, nanometer-scale regions of short-range polar order known as polar nanoregions that condense from the soft TO mode on cooling from high temperatures well above (23, 30). We emphasize that all lead-oxide perovskite relaxors for which single crystals have been examined with neutron or X-ray scattering techniques exhibit similar anisotropic, temperature-dependent diffuse-scattering contours including PMN, PZN, Pb(Sc_1/2Nb_1/2)O₃, Pb(Mg_1/3Ta_2/3)O₃, Pb(In_1/2Nb_1/2)O₃, and Pb(Sc_1/2Ta_1/2)O₃ (20, 22, 24⇓⇓–27, 30, 31). All of these compounds are composed of heterovalent B-site cations; hence . In addition, a recent neutron-scattering study discovered that the lead-free relaxor (Na_1/2Bi_1/2)TiO₃ (NBT), an A-site disordered perovskite composed of heterovalent Na⁺ and Bi³⁺ cations, displays static diffuse-scattering contours characterized by a temperature dependence and anisotropy like that found in PMN (40). This suggests that the physics that governs this diffuse scattering is not necessarily limited to lead-based perovskite relaxors.

By contrasting the elastic diffuse scattering from PMN to that from PZT, we can isolate the influence of quenched REFs on the nanometer-scale structure of lead-oxide perovskite relaxors. To this end we examined the neutron diffuse scattering from a single crystal of PZT with a Ti concentration under exactly the same conditions as those used to measure the diffuse scattering from the PMN single crystal. To account precisely for differences in crystal size and beam illumination, the diffuse-scattering data from both crystals were normalized by the integrated intensity of a transverse acoustic (TA) phonon also measured under identical conditions. The resulting data, given by the solid blue dots in Fig. 1D, shows that the diffuse scattering for the nonrelaxor PZT is at least a factor of 20 weaker than that in PMN. Fig. 1D, Inset shows a log-scale intensity map of the elastic scattering from the same PZT crystal at very low temperature (10 K) in the scattering plane near (001), and no evidence of diffuse scattering is present. Additional data measured on PZT in the scattering plane at both low (16 K) and high (620 K) temperatures confirm this finding (Fig. S1). Static diffuse scattering was recently reported in a single crystal of PZT with using X-ray inelastic scattering techniques (41). However, these data were obtained with an energy resolution nearly 10 times coarser than ours; hence there is a possibility of phonon contamination. This static diffuse scattering was modeled using a Huang formalism, which is inconsistent with that seen in relaxors, and was attributed to elastic deformations induced by defect centers of tetragonal symmetry. This is not unexpected for a composition so close to the MPB. Indeed such Huang scattering would have impaired our search for relaxor-like diffuse scattering except for the fact that our PZT composition lies in the purely pseudorhombohedral regime.

As the only significant difference between PMN and PZT lies in the huge disparity in REF strength, our neutron data establish a direct causal link between strong REFs and the temperature-dependent, anisotropic diffuse scattering in lead-oxide perovskite relaxors. In the same vein, our dielectric permittivity measurements demonstrate that strong REFs are responsible for the relaxor-like behavior. Together these results imply the existence of a fundamental connection between the relaxor phase and the anisotropic, temperature-dependent diffuse scattering. We motivate this picture through a heuristic model, developed in the SI Materials and Methods, that allows us to relate the strength of the REFs to the diffuse-scattering intensity. We thus identify the relaxor-order parameter with the static, anisotropic diffuse scattering associated with short-range polar order, but only in the sense that it marks the transition temperature from the high-temperature paraelectric phase (where it is zero) to the low-temperature relaxor state (where it is nonzero). Indeed, as noted by Cowley et al. (13) and Prosandeev et al. (42) the strictly elastic diffuse scattering in PMN increases on cooling in a manner similar to that of an order parameter at a conventional phase transition, and this also holds true for all pseudorhombohedral compositions of PMN-xPT and PZN-xPT. As first discussed by Westphal et al. (10) and later Stock et al. (11), this identification is consistent with the ideas of Imry and Ma who showed that magnetic systems with continuous symmetry will break up into short-range ordered domains in the presence of random magnetic fields (43). In relaxors like PMN, long-range ferroelectric order is stifled by the formation of analogous polar domains on cooling. By contrast, our data show that no such domains form in ferroelectric PZT because the REFs are too weak. This picture naturally explains why the diffuse scattering in relaxors vanishes for compositions on the tetragonal side of the MPB; the uniaxial anisotropy there stabilizes the ferroelectric state.

Dynamic Signature: The Waterfall Effect Versus Zone-Boundary Soft Modes.

We next compare the lattice dynamical properties of PZT and PMN to look for a dynamic signature of the relaxor phase. PMN exhibits several anomalous dynamical features that are also seen in other relaxor compounds such as PZN-xPT and Pb(Mg_1/3Ta_2/3)O₃; thus the availability of a high-quality PZT single crystal provides us with a rare opportunity to determine if any of these are specific to lead-based perovskite relaxors. In particular, the diffuse scattering in PMN, which is purely relaxational above K (44), has profound effects on the lifetimes of low-frequency TA modes (45, 46). The best-known dynamic anomaly is the “waterfall effect,” a term coined to characterize the false impression that the soft TO phonon branch in PZN-8%PT drops steeply into the TA branch at a nonzero wave vector (47). The waterfall effect results from a strong damping of the TO mode that occurs at significantly larger wave vectors than happens in cubic PbTiO₃ (48). The cause of this damping is also controversial. Model simulations based on standard TO–TA mode coupling theory provide qualitative agreement with experimental data measured on PZN-8%PT (49). However, TO–TA mode coupling has been shown to be weak in PMN and therefore unlikely to be the origin of the waterfall effect (45, 46). An alternative explanation is that the waterfall effect arises from disorder introduced through the heterovalent nature of the B-site cations, i.e., random dipolar fields (46). If this idea is correct, then the waterfall effect should be absent in PZT or qualitatively different from that observed in PMN and other relaxors. Another unusual dynamical feature is the simultaneous appearance of highly damped zone-boundary soft modes at the R- and M-point Brillouin zone boundaries in PMN (50), which exhibit a temperature dependence that tracks that of the soft, zone-center TO mode, and which give rise to broad superlattice peaks at the same locations. This behavior indicates that competing antiferroelectric and ferroelectric fluctuations are present in PMN, a situation that has been applied theoretically to SrTiO₃ (51). It was recently shown that the waterfall effect is present in PMN-60%PT, a composition that lies well beyond the MPB (52). Because PMN-60%PT exhibits no diffuse scattering and no discernable frequency dependence to the dielectric permittivity, the waterfall effect cannot be associated with the relaxor phase. The observation of a waterfall effect in PZT would provide additional evidence for this conclusion. By contrast, the soft and highly damped R- and M-point zone boundary modes seen in PMN are absent in PMN-60%PT (52) but have been observed in the relaxor PZN-5.5%PT; thus these broad modes may constitute a genuine dynamical signature of relaxors.

To determine if these dynamical anomalies are intrinsic to the relaxor state, we present neutron inelastic scattering data on PZT, which were obtained in the scattering plane near the (220) Brillouin zone using a series of constant- and constant- scans. These scans probe TA and TO phonons polarized along [110] and propagating along [001], and they are directly comparable to those published on PbTiO₃ (48) and PZN-8%PT (47), all of which were measured in the same scattering plane and in the same Brillouin zone. The TA and TO phonon dispersion curves for PZT measured at 650 K (60 K above T_C1) are shown in Fig. 2A. We find that the TA phonon branch in PZT is similar to, but slightly softer than, the corresponding TA phonon branch in PbTiO₃, which is shown in Fig. 2B (48). The TO phonon branch in PZT drops to very low energies at small reduced-wave vectors and is similar in this regard to the soft TO phonon branch in PbTiO₃ measured at 783 K (20 K above ). However, at large wave vectors the TO phonon branch in PZT is significantly softer than that in PbTiO₃ and closely resembles the TO phonon branch in the relaxor PZN-8%PT, which is also shown in Fig. 2B. In particular, the TO phonon branch in PZT appears to intercept the TA phonon branch at a nonzero wave vector near 0.05 reciprocal lattice units (rlu), albeit this is substantially smaller than the corresponding wave vector (0.14 rlu) in PZN-8%PT. This behavior suggests that the waterfall effect is present in PZT.

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Fig. 2.

Dispersions of the TA and soft TO modes measured in the scattering plane near (220) in the cubic phases of (A) PZT and (B) PbTiO₃ (48) and PZN-8%PT (47). The green triangles/black circles in A denote the locations of maximum scattering intensity in constant- scans, respectively; Inset shows constant- scans measured at 650 K at 6, 8, and 10 meV (data have been displaced vertically for clarity). All lines are guides to the eye.

The nature of the waterfall effect in PZT is more fully revealed by an analysis of the constant- scans shown in Fig. 3 A–D, which were measured above and below the cubic-to-rhombohedral phase transition temperature K. The TO mode at is well-defined and softens on cooling from 900 to 650 K, but it is nearly overdamped at the same temperatures at . In fact for wave vectors rlu, the TO mode is so damped that it is not possible to locate the peak in a constant- scan. This -dependent broadening of the soft TO mode is the hallmark of the waterfall effect and has been reported in various compositions of PMN-xPT and PZN-xPT. The solid lines in these panels represent fits of the data to the sum of two uncoupled damped harmonic oscillators convolved with the instrumental resolution function. The quality of the fits indicates that TA–TO mode coupling is weak in PZT for phonons propagating along [001]. A series of constant- scans were measured at between 300 and 900 K to characterize the temperature dependence of the TA phonon linewidth and energy. Fig. 3E shows that the TA phonon linewidth in PZT broadens on cooling from above and narrows slightly below. This behavior is markedly different from that in PMN (dashed lines) for which the TA phonon linewidth broadens enormously on cooling. The broadening in PMN has been shown to result from a coupling to a low-energy relaxational mode associated with the strong diffuse scattering (44, 45). Assuming that a coupling of similar strength is present in PZT, then the comparatively weak temperature-dependent TA phonon broadening would further support our finding of little or no diffuse scattering in this compound. In Fig. 3F the TA phonon energy in PZT is shown to vary monotonically and changes little with temperature. Indeed, the net shift in the PZT TA phonon energy over this broad temperature range is almost identical to that in PMN (dashed lines) (45, 46). We therefore exclude TA–TO mode coupling as the origin of the waterfall effect in PZT. We also exclude REFs as the mechanism responsible for the waterfall effect. At lower temperature (295 K), only the TA mode is visible because the TO mode has hardened to energies that lie outside the range of these scans. The softening of the TO mode on cooling from 900 to 650 K and the subsequent hardening is consistent with the cubic-to-rhombohedral phase transition at being driven by a soft, zone-center mode instability (53).

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Fig. 3.

Constant- scans measured at (A) (2,2,0), (B) (2,2,−0.1), (C) (2,2,−0.15), and (D) (2,2,−0.2) at temperatures above and below K. Temperature dependencies of the TA phonon (E) linewidth and (F) energy for PZT (open circles) and PMN (dashed lines). The solid line is a guide to the eye.

The lower-temperature phase transition for this composition of PZT at K corresponds to a doubling of the unit cell that is manifested by the appearance of new Bragg reflections at the R points , where h and l are odd and . This suggests that there may be a soft mode at the R point. In PMN, highly damped (i.e., short-lived), soft zone-boundary modes are observed at both R and M points (Fig. S3), and structure-factor considerations preclude these from being octahedral tilt modes. To learn if PZT exhibits similar dynamics, we measured constant- scans at the R-point at 290, 650, and 900 K, which are shown in Fig. 4A, Upper. A broad distribution of scattering is observed in PZT below 10 meV at 900 and 650 K but is absent at 290 K. The shift in spectral weight from high to low energies on cooling from 900 K is particularly apparent near meV where an increase in scattering is seen at 290 K. To clarify this behavior we measured the scattering intensity at and meV as a function of temperature. The data (Fig. 4A, Lower) show that the scattering at 2 meV increases on cooling from 900 K to a maximum at near 450 K and then decreases. This behavior is precisely that expected for a soft mode: for the soft mode is located above 2 meV, which results in little or no scattering; but on cooling through the soft mode drops to lower energies, thus moving into and then out of the instrumental window and causing the scattering intensity to first increase and then decrease. This procedure mimics that used by Swainson et al. to demonstrate the softening of the R-point zone-boundary mode in PMN (50).

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Fig. 4.

(A, Upper) Constant- scans measured at the R point at 900, 650, and 290 K. (Lower) Background subtracted intensity, adjusted by the Bose factor , measured at the same and at fixed meV from 900 to 300 K. (B, Upper) Neutron-scattering intensity measured at the M points (solid red circles) and (solid blue triangles) from 900 to 300 K. The background has been subtracted from the total scattering intensity and the result adjusted by the Bose factor . (Lower) Analogous data for the relaxor PMN measured by Swainson et al. (50) at .

We performed identical measurements on PZT at the M points and at the same energy transfer meV. These data, shown in Fig. 4B, Upper, reveal completely different behavior as comparatively little temperature dependence is observed. This is consistent with the fact that no superlattice reflections are seen at M-point locations in this composition of PZT (53). It also points to a fundamental difference in the dynamics between PZT and PMN for which concurrent, broad, soft zone-boundary modes were observed at both R and M points as indicated in Fig. 4B, Lower. At higher Ti concentrations PZT exhibits a low-temperature monoclinic phase, and X-ray inelastic measurements have observed soft M-point modes (54). However, no soft R-point mode was reported for this monoclinic composition. We thus propose that a competition between ferroelectric and antiferroelectric fluctuations, as revealed by the simultaneous presence of heavily damped (short-lived) soft modes at the R, M, and Γ points, may represent a true dynamical signature of relaxor behavior.

Ultrahigh Piezoelectricity.

It is well known that pseudorhombohedral compositions of PMN-xPT and PZN-xPT located near the MPB exhibit enormous piezoelectric coefficients; the values of the longitudinal piezoelectric coefficient for single-crystal MPB compositions of PMN-xPT and PZN-xPT poled along [001] can exceed 2,800 pC/N and are the largest known (6, 55, 56). The concepts of polarization rotation (2) and adaptive phases (3) were proposed to explain the ultrahigh electromechanical response in MPB systems like PZT and PMN-xPT. More recently, Kutnjak et al. reported that the origin of the giant electromechanical response at the MPB is related to the proximity of the system to a critical end point (4). We note here that the q-integrated neutron elastic diffuse-scattering intensity in PMN-xPT, which from the fluctuation-dissipation theorem is a measure of the static susceptibility, and share a similar composition dependence (36, 57). Both properties (in this case measured along [110]) peak near the MPB and then decrease precipitously in the tetragonal (Ti rich) regime in which the relaxor phase is replaced by a conventional ferroelectric phase (52). Given our finding that PZT exhibits essentially no such diffuse scattering one might expect that the corresponding value of in PZT should be significantly smaller than that in PMN-xPT. We find that this is indeed true: the value of that we measured on a [001]-poled single crystal of PZT with a composition close to the MPB is 1,200 pC/N. Thus, the optimal values of in single-crystal PMN-xPT and PZN-xPT are ∼100% greater than that in single-crystal PZT. Insofar as PZT and PMN-xPT are identical systems save for the strength of the REFs, our results indicate that the underlying nanometer-scale, static, polar order in relaxors that is associated with the temperature-dependent diffuse scattering plays a seminal role in greatly amplifying the piezoelectric response beyond that in the nonrelaxor PZT, which does not display an equivalent nanometer-scale polar order. This conclusion is consistent with the theoretical results of Pirc et al. that suggest polar nanoregions play an important role in the electrostriction in relaxors (58). Kutnjak et al. also showed that the phase transitions in relaxor-based ferroelectrics are first order as an external electric field must be applied to reach the critical end point. By contrast, PZT compositions near the MPB exhibit second-order phase transitions (critical points) at T_C in zero field (59). This suggests that the presence of REFs is at least as important to the large piezoelectric response as is the proximity of the system to a critical end point. We note that this behavior is not without precedent. In single crystals of highly magnetostrictive galfenol, an alloy of magnetic Fe and nonmagnetic Ga that is often likened to a magnetic relaxor and which should possess strong random magnetic fields, both the neutron elastic diffuse scattering intensity and the longitudinal magnetostrictive coefficient increase dramatically as the system is doped toward a phase boundary and then decrease sharply as the boundary is crossed (60).