Despite more than 50 years of investigation, it is still unclear how the underlying structure of relaxor ferroelectrics gives rise to their defining properties, such as ultrahigh piezoelectric coefficients, high permittivity over a broad temperature range, diffuse phase transitions, strong frequency dependence in dielectric response, and phonon anomalies1,2,3,4,5,6,7,8,9,10. The model of polar nanoregions inside a non-polar matrix has been widely used to describe the structure of relaxor ferroelectrics11. However, the lack of precise knowledge about the shapes, growth and dipole patterns of polar nanoregions has led to the characterization of relaxors as “hopeless messes”12, and no predictive model for relaxor behaviour is currently available. Here we use molecular dynamics simulations of the prototypical Pb(Mg1/3,Nb2/3)O3–PbTiO3 relaxor material to examine its structure and the spatial and temporal polarization correlations. Our simulations show that the unusual properties of relaxors stem from the presence of a multi-domain state with extremely small domain sizes (2–10 nanometres), and no non-polar matrix, owing to the local dynamics. We find that polar structures in the multi-domain state in relaxors are analogous to those of the slush state of water. The multi-domain structure of relaxors that is revealed by our molecular dynamics simulations is consistent with recent experimental diffuse scattering results and indicates that relaxors have a high density of low-angle domain walls. This insight explains the recently discovered classes of relaxors13 that cannot be described by the polar nanoregion model, and provides guidance for the design and synthesis of new relaxor materials.
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This work was supported by the ONR under grant N00014-12-1-1033. Computational support was provided by the US DOD through a Challenge Grant from the HPCMO. We thank P. M. Gehring for discussions on his experimental diffuse scattering data and the diffuse scattering method.
The authors declare no competing financial interests.
Reviewer Information Nature thanks P. Gehring, G. Guzmán-Verri and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
In the paraelectric phase, dipoles are randomly oriented (upper panel). At Tb, PNRs appear (middle panel). Dipoles align and are still randomly oriented inside and outside of the PNR. The PNRs grow and interact with each other on cooling. At T*, roughly halfway between Tb and Tf, static local distortions were detected using several experimental techniques. As T is lowered further, PNRs coalesce and touch each other and the system undergoes a transition into the frozen phase at Tf (lower panel), in which the PNRs cannot reorient.
We compare our computational DS at 300 K using MD trajectories with 100-ps simulation times to the experimental DS for PMN and PMN–10PT at 300 K (ref. 18). a–e, DS around the (100) Bragg spot for 36 × 36 × 36 (a), 64 × 64 × 64 (b) and 72 × 72 × 72 (c) supercells, the experimental PMN (d) and the experimental PMN–10PT (e). f–j, As a–e, but around the (110) Bragg spot. Experimentally reported butterfly and rod shapes can be seen with increasing clarity for larger MD supercells. Orange arrows in c–e highlight the changes with PT content. Panels d, e, i and j adapted from ref. 18, American Physical Society.
a–g, Around (100) at 100 K, 200 K, 300 K, 400 K, 500 K, 600 K and 700 K, respectively. h–n, As a–g, but around (110).
Extended Data Figure 4 Analysis of the DS intensity of the (110) Bragg spot using scans through q values along the (110) direction, showing the comparison between experimental and theoretical results.
a, DS intensity for PMN, PMN–10PT, PMN–20PT (experimental) and PMN–25PT (from MD simulations) at 300 K. b, Temperature dependence of I0 extracted by fitting a Lorentzian to the DS intensity. c, d, Temperature dependence of the DS intensity for PMN–25PT (theoretical; c) and PMN (experimental; d)33. Image in d reproduced with permission from ref. 33, copyright (2004) by the American Physical Society.
a–e, We multiply the lengths of the displacements by 0.9, 0.8, 0.7, 0.3 and 0.2, respectively. f, DS intensities obtained using unscaled, time-averaged Pb-atom displacements.
a, Paraelectric random displacements. b, Ferroelectric collinear displacements. c, Ferroelectric displacements with 90° domain walls. DS intensities are essentially zero for all three models (a–c). d, A structure with a 20° angle between dipoles in nearest-neighbour cells obtained by Monte Carlo calculation. In all cases (a–d), DS does not assume the butterfly shape.
Extended Data Figure 7 Domain-size dependence of DS intensities for 71° domain-wall model structures.
a–c, Domain sizes are 4 nm, 5.6 nm and 8 nm, respectively. As domain size increases, the model butterfly DS pattern shrinks, implying that growth of the correlated domain with lower temperature would lead to a weaker, less extensive DS pattern, in disagreement with experimental observations of greater DS extent at lower temperatures. However, this observation supports our model of fixed-size domains separated by domain walls.
Extended Data Figure 8 Time-delay-averaged angle correlation functions for Pb atomic pairs between nth neighbour cells.
a–m, Correlations at 600 K (a–c), 500 K (d–f), 400 K (g–i), 300 K (j–l) and 100 K (m–o). Left, middle and right panels correspond to the 〈100〉, 〈110〉 and 〈111〉 directions, respectively.
a, The five subgroups are illustrated for a 6 × 6 × 6 supercell. b, The number of each subgroup in a 72 × 72 × 72 supercell in our MD simulations. c, Overlap of time-delay-averaged angle autocorrelation distributions for Pb-atom displacements in different local environments as a function of temperature. Changes in the temperature dependence of the overlap correspond to the relaxor transition temperatures Tb = 550 K, T* = 480 K and Tf = 380 K.
a, Real part of the dielectric constant for different frequencies f in the gigahertz range. The frequency dispersion and the diffuseness of the temperature dependence of the dielectric constant are similar to the relaxor dielectric response. b, Arrhenius plot of the frequency f as a function of value of the inverse temperature 1/T at which the real part of the dielectric constant is maximum, for the frequencies shown in a. A clear non-Arrhenius dependence of frequency on temperature is observed for f ≤ 0.333 GHz, whereas the plot for the higher f follows the Arrhenius law.
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Takenaka, H., Grinberg, I., Liu, S. et al. Slush-like polar structures in single-crystal relaxors. Nature 546, 391–395 (2017). https://doi.org/10.1038/nature22068
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