Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Intrinsic structure of relaxor ferroelectrics from first principles

Xinyu Xu1,2,*, Kehan Cai3,*, Yubai Shi4, Peichen Zhong4, and Pinchen Xie1,†

  • *These authors contributed equally to this work.
  • †Contact author: pinchenxie@lbl.gov

Phys. Rev. B 114, 024204 – Published 6 July, 2026Erratum Phys. Rev. B 114, 139901 (2026)

DOI: https://doi.org/10.1103/fz5h-1t5b

Abstract

We introduce FIRE-Swap, a first-principles-based framework for sampling representative compositional structures in complex perovskites using machine-learning interatomic potentials. By combining Monte Carlo chemical swaps with geometric relaxation, the method treats chemical and geometric degrees of freedom concurrently. Applied to lead magnesium niobate (PMN), lead zirconate titanate (PZT), and lead strontium titanate (PST), FIRE-Swap robustly predicts rock-salt-like chemical order in PMN, but not in the isovalent solid solutions PZT and PST with the same mixing ratio. For PMN, it further reveals anchored Nb clusters that host interconnected, noncoarsened polar nanoregions. These results provide a mesoscale structural basis for understanding relaxor ferroelectricity and establish FIRE-Swap as a practical route toward systematic, first-principles modeling of complex perovskites and related substitutionally disordered materials.

View figure in article

Physics Subject Headings (PhySH)

Erratum

Erratum: Intrinsic structure of relaxor ferroelectrics from first principles [Phys. Rev. B 114, 024204 (2026)]

Xinyu Xu, Kehan Cai, Yubai Shi, Peichen Zhong, and Pinchen Xie
Phys. Rev. B 114, 139901 (2026)

Article Text

Supplemental Material

References (93)

  1. G. A. Smolensky, V. A. Isupov, A. A. Agranovskaya, and S. N. Popov, Ferroelectrics with a diffuse phase transition, Sov. Phys. Solid State 2, 2584 (1961) [Fiz. Tverd. Tela (Leningrad) 2, 2906 (1961)].
  2. G. Burns and F. H. Dacol, Glassy polarization behavior in ferroelectric compounds Pb(Mg1/3Nb2/3)O3 and Pb(Zn1/3Nb2/3)O3, Solid State Commun. 48, 853 (1983).
  3. G. Burns and F. H. Dacol, Crystalline ferroelectrics with glassy polarization behavior, Phys. Rev. B 28, 2527 (1983).
  4. D. Viehland, S. Jang, L. E. Cross, and M. Wuttig, Freezing of the polarization fluctuations in lead magnesium niobate relaxors, J. Appl. Phys. 68, 2916 (1990).
  5. D. Viehland, M. Wuttig, and L. Cross, The glassy behavior of relaxor ferroelectrics, Ferroelectrics 120, 71 (1991).
  6. A. Levstik, Z. Kutnjak, C. Filipič, and R. Pirc, Glassy freezing in relaxor ferroelectric lead magnesium niobate, Phys. Rev. B 57, 11204 (1998).
  7. R. A. Cowley, S. N. Gvasaliya, S. G. Lushnikov, B. Roessli, and G. M. Rotaru, Relaxing with relaxors: A review of relaxor ferroelectrics, Adv. Phys. 60, 229 (2011).
  8. A. A. Bokov and Z.-G. Ye, Recent progress in relaxor ferroelectrics with perovskite structure, Prog. Adv. Dielectr. 1, 105 (2020).
  9. K. M. Rabe, C. H. Ahn, and J.-M. Triscone, Physics of Ferroelectrics: A Modern Perspective, Topics in Applied Physics, Vol. 105 (Springer, Berlin, 2007).
  10. D. Sherrington, BZT: A soft pseudospin glass, Phys. Rev. Lett. 111, 227601 (2013).
  11. G. V. Lecomte, H. v. Löhneysen, and E. F. Wassermann, Frequency dependent magnetic susceptibility and spin glass freezing inPtMn alloys, Z. Phys. B 50, 239 (1983).
  12. D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys (Revised Reprint) (CRC Press, Boca Raton, FL, 2009).
  13. See Appendix pp3 for further discussion.
  14. W. Zhong, D. Vanderbilt, and K. M. Rabe, First-principles theory of ferroelectric phase transitions for perovskites: The case of BaTiO3, Phys. Rev. B 52, 6301 (1995).
  15. Y.-H. Shin, V. R. Cooper, I. Grinberg, and A. M. Rappe, Development of a bond-valence molecular-dynamics model for complex oxides, Phys. Rev. B 71, 054104 (2005).
  16. H. Takenaka, I. Grinberg, S. Liu, and A. M. Rappe, Slush-like polar structures in single-crystal relaxors, Nature (London) 546, 391 (2017).
  17. A. R. Akbarzadeh, S. Prosandeev, E. J. Walter, A. Al-Barakaty, and L. Bellaiche, Finite-temperature properties of Ba(Zr,Ti)O3 relaxors from first principles, Phys. Rev. Lett. 108, 257601 (2012).
  18. J. Zhang, L. Liu, A. A. Bokov, N. Zhang, D. Wang, Z.-G. Ye, and C.-L. Jia, Compositional ordering in relaxor ferroelectric Pb(BB′)O3: Nearest-neighbor approach, Phys. Rev. B 103, 054201 (2021).
  19. M. Kopecký, J. Kub, J. Fábry, and J. Hlinka, Nanometer-range atomic order directly recovered from resonant diffuse scattering, Phys. Rev. B 93, 054202 (2016).
  20. M. J. Cabral, S. Zhang, E. C. Dickey, and J. M. LeBeau, Gradient chemical order in the relaxor Pb(Mg1/3Nb2/3)O3, Appl. Phys. Lett. 112, 082901 (2018).
  21. M. Eremenko, V. Krayzman, A. Bosak, H. Y. Playford, K. W. Chapman, J. C. Woicik, B. Ravel, and I. Levin, Local atomic order and hierarchical polar nanoregions in a classical relaxor ferroelectric, Nat. Commun. 10, 2728 (2019).
  22. S. Pennycook and L. Boatner, Chemically sensitive structure-imaging with a scanning transmission electron microscope, Nature (London) 336, 565 (1988).
  23. See Supplemental Material at http://link.aps.org/supplemental/10.1103/fz5h-1t5b for details on (1) fire-swap simulations of pmn close to the melting temperature, (2) evolution of cluster size in fire-swap simulations of PMN, PZT, and PST, (3) finite size analysis (3) pair and radial distribution functions associated to different compositional structures, (4) distribution of Pb-based local order parameter, and (5) error distribution of deep potential model.
  24. E. C. Neyts and A. Bogaerts, Combining molecular dynamics with Monte Carlo simulations: Implementations and applications, Theor. Chem. Acc. 132, 1320 (2013).
  25. M. Widom, W. P. Huhn, S. Maiti, and W. Steurer, Hybrid Monte Carlo/molecular dynamics simulation of a refractory metal high entropy alloy, Metall. Mater. Trans. A 45, 196 (2014).
  26. E. Antillon, C. Woodward, S. Rao, B. Akdim, and T. Parthasarathy, Chemical short range order strengthening in a model fcc high entropy alloy, Acta Mater. 190, 29 (2020).
  27. E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and P. Gumbsch, Structural relaxation made simple, Phys. Rev. Lett. 97, 170201 (2006).
  28. F. H. Stillinger and T. A. Weber, Inherent structure in water, J. Phys. Chem. 87, 2833 (1983).
  29. D. S. Corti, P. G. Debenedetti, S. Sastry, and F. H. Stillinger, Constraints, metastability, and inherent structures in liquids, Phys. Rev. E 55, 5522 (1997).
  30. N. Nakagawa and M. Peyrard, The inherent structure landscape of a protein, Proc. Natl. Acad. Sci. USA 103, 5279 (2006).
  31. The mixing ratios of PZT and PST are chosen to match PMN's Mg:Nb ratio, allowing PZT and PST to serve as isovalent control systems. This isolates whether FIRE-Swap predicts chemical order specifically due to PMN's heterovalent chemistry, rather than the mixing ratio itself.
  32. J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Phys. Rev. Lett. 100, 136406 (2008).
  33. J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly constrained and appropriately normed semilocal density functional, Phys. Rev. Lett. 115, 036402 (2015).
  34. Y. Zhang, J. Sun, J. P. Perdew, and X. Wu, Comparative first-principles studies of prototypical ferroelectric materials by LDA, GGA, and SCAN meta-GGA, Phys. Rev. B 96, 035143 (2017).
  35. J. Han, L. Zhang, R. Car, and W. E, Deep potential: A general representation of a many-body potential energy surface, Commun. Comput. Phys. 23, 629 (2025).
  36. L. Zhang, J. Han, H. Wang, R. Car, and W. E, Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics, Phys. Rev. Lett. 120, 143001 (2018).
  37. L. Zhang, J. Han, H. Wang, W. A. Saidi, R. Car, et al., End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems, in Advances in Neural Information Processing Systems (Curran Associates, Red Hook, NY, 2018), Vol. 31.
  38. B. Cheng, Cartesian atomic cluster expansion for machine learning interatomic potentials, npj Comput. Mater. 10, 157 (2024).
  39. D. S. King, D. Kim, P. Zhong, and B. Cheng, Machine learning of charges and long-range interactions from energies and forces, Nat. Commun. 16, 8763 (2025).
  40. P. Zhong, D. Kim, D. S. King, and B. Cheng, Machine learning interatomic potential can infer electrical response, npj Comput. Mater. 11, 384 (2025).
  41. J. Wu, J. Yang, Y.-J. Liu, D. Zhang, Y. Yang, Y. Zhang, L. Zhang, and S. Liu, Universal interatomic potential for perovskite oxides, Phys. Rev. B 108, L180104 (2023).
  42. B. Cheng, Latent Ewald summation for machine learning of long-range interactions, npj Comput. Mater. 11, 80 (2025).
  43. J. Chen, H. M. Chan, and M. P. Harmer, Glassy polarization behavior in ferroelectric compounds Pb(Mg1/3Nb2/3)O3 and Pb(Zn1/3Nb2/3)O3, J. Am. Ceram. Soc. 72, 593 (1989).
  44. C. A. Randall, A. S. Bhalla, T. R. Shrout, and L. E. Cross, Classification and consequences of complex lead perovskite ferroelectrics with regard to B-site cation order, J. Mater. Res. 5, 829 (1990).
  45. C. Boulesteix, F. Varnier, A. Llebaria, and E. Husson, Numerical determination of the local ordering of Pb(Mg)1/3Nb2/3O3 (PMN) from high-resolution electron microscopy images, J. Solid State Chem. 108, 141 (1994).
  46. Z. Kutnjak, C. Filipič, R. Pirc, A. Levstik, R. Farhi, and M. El Marssi, Slow dynamics and ergodicity breaking in a lanthanum-modified lead zirconate titanate relaxor system, Phys. Rev. B 59, 294 (1999).
  47. P. Davies and M. Akbas, Chemical order in PMN-related relaxors: Structure, stability, modification, and impact on properties, J. Phys. Chem. Solids 61, 159 (2000).
  48. M. A. Akbas and P. K. Davies, Thermally induced coarsening of the chemically ordered domains in Pb(Mg1/3Nb2/3)O3 (PMN)-based relaxor ferroelectrics, J. Am. Ceram. Soc. 83, 119 (2000).
  49. Z. Xu, S. M. Gupta, D. Viehland, Y. Yan, and S. J. Pennycook, Direct imaging of atomic ordering in undoped and La-doped Pb(Mg1/3Nb2/3)O3, J. Am. Ceram. Soc. 83, 181 (2000).
  50. L. Farber and P. K. Davies, Influence of cation order on the dielectric properties of Pb(Mg1/3Nb2/3)O3–Pb(Sc1/2Nb1/2)O3 (PMN–PSN) relaxor ferroelectrics, J. Am. Ceram. Soc. 86, 1861 (2003).
  51. D. Fu, H. Taniguchi, M. Itoh, S. y. Koshihara, N. Yamamoto, and S. Mori, Relaxor Pb(Mg1/3Nb2/3)O3: A ferroelectric with multiple inhomogeneities, Phys. Rev. Lett. 103, 207601 (2009).
  52. The formation of Nb-rich sublattice prevents the adjacency of the Mg sites that presumably contribute to greater electrostatic repulsion than the adjacent Mg-Nb and Nb-Nb sites.
  53. M. Cantoni, S. Bharadwaja, S. Gentil, and N. Setter, Direct observation of the B-site cationic order in the ferroelectric relaxor Pb(Mg1/3Ta2/3)O3, J. Appl. Phys. 96, 3870 (2004).
  54. A. Kumar, J. N. Baker, P. C. Bowes, M. J. Cabral, S. Zhang, E. C. Dickey, D. L. Irving, and J. M. LeBeau, Atomic-resolution electron microscopy of nanoscale local structure in lead-based relaxor ferroelectrics, Nat. Mater. 20, 62 (2021).
  55. Antiphase boundaries are disordered regions where the βI/II classification flips.
  56. E. F. Moore, The shortest path through a maze, in Proceedings of the International Symposium on the Theory of Switching (Harvard University Press, Cambridge, MA, 1959), pp. 285–292.
  57. C. Y. Lee, An algorithm for path connections and its applications, IRE Trans. Electron. Comput. EC-10, 346 (1961).
  58. The upper bound corresponds to the maximal separation between Nb clusters and rock-salt clusters. The upper bound cannot be reached because our definition of a Nb cluster omits sites at the interface.
  59. We verify this by tracking the number of Nb sites in the rock-salt cluster, where an Nb site belongs to the rock-salt cluster if it has at most one Nb neighbor. Details are omitted here but are available in the Supplemental Data and Code [77].
  60. Visualization is provided later in Fig. 4.
  61. Pb: 9.405 fm; Mg: -3.73 fm; Nb: 7.054 fm; O: 5.803 fm.
  62. See Supplemental Fig. 1 of Ref. [21].
  63. See Supplemental Fig. 17(a) of Ref. [21].
  64. This highlights potential nonuniqueness in CS inferred from reverse Monte Carlo methods.
  65. S. Wakimoto, C. Stock, R. J. Birgeneau, Z.-G. Ye, W. Chen, W. J. L. Buyers, P. M. Gehring, and G. Shirane, Ferroelectric ordering in the relaxor Pb(Mg1/3Nb2/3)O3 as evidenced by low-temperature phonon anomalies, Phys. Rev. B 65, 172105 (2002).
  66. R. Blinc, V. V. Laguta, and B. Zalar, Field-cooled and zero-field-cooled Pb207 NMR and the local structure of relaxor Pb(Mg1/3Nb2/3)O3, Phys. Rev. Lett. 91, 247601 (2003).
  67. I.-K. Jeong, T. W. Darling, J. K. Lee, T. Proffen, R. H. Heffner, J. S. Park, K. S. Hong, W. Dmowski, and T. Egami, Direct observation of the formation of polar nanoregions in Pb(Mg1/3Nb2/3)O3 using neutron pair distribution function analysis, Phys. Rev. Lett. 94, 147602 (2005).
  68. P. M. Gehring, H. Hiraka, C. Stock, S.-H. Lee, W. Chen, Z.-G. Ye, S. B. Vakhrushev, and Z. Chowdhuri, Reassessment of the Burns temperature and its relationship to the diffuse scattering, lattice dynamics, and thermal expansion in relaxor Pb(Mg1/3Nb2/3)O3, Phys. Rev. B 79, 224109 (2009).
  69. The local dipole moment can be rigorously defined from the Wannier-center representation of polarization prescribed by Berry-phase theory [92, 93], as in Ref. [79].
  70. D. Fu, H. Taniguchi, M. Itoh, and S. Mori, Pb(Mg1/3Nb2/3)O3 (PMN) relaxor: Dipole glass or nano-domain ferroelectric? in Advances in Ferroelectrics (IntechOpen, London, 2012).
  71. K. Hino, D. Morikawa, D. Fu, M. Itoh, and K. Tsuda, Direct imaging of temperature evolution of polar nanoregions and chemically ordered regions in PMN relaxor: Evidence for polar phase percolation, Appl. Phys. Lett. 128, 142901 (2026).
  72. B. Deng, P. Zhong, K. Jun, J. Riebesell, K. Han, C. J. Bartel, and G. Ceder, Chgnet as a pretrained universal neural network potential for charge-informed atomistic modelling, Nat. Mach. Intell. 5, 1031 (2023).
  73. D. Zhang, X. Liu, X. Zhang, C. Zhang, C. Cai, H. Bi, Y. Du, X. Qin, A. Peng, J. Huang, et al., DPA-2: A large atomic model as a multi-task learner, npj Comput. Mater. 10, 293 (2024).
  74. I. Batatia, P. Benner, Y. Chiang, A. M. Elena, D. P. Kovács, J. Riebesell, X. R. Advincula, M. Asta, M. Avaylon, W. J. Baldwin, et al., A foundation model for atomistic materials chemistry, J. Chem. Phys. 163, 184110 (2025).
  75. PeroStruc, https://github.com/Kehan-Cai-nanako/PeroStruc.
  76. A. Hjorth Larsen, J. Jørgen Mortensen, J. Blomqvist, I. E. Castelli, R. Christensen, M. Dułak, J. Friis, M. N. Groves, B. Hammer, C. Hargus, et al., The atomic simulation environment—A python library for working with atoms, J. Phys.: Condens. Matter 29, 273002 (2017).
  77. Supplemental data and code, https://github.com/xuxinyu2820/PMN_COR.
  78. B. Yang, P. Xie, and R. Car, Deuteration removes quantum dipolar defects from KDP crystals, npj Comput. Mater. 10, 241 (2024).
  79. P. Xie, Y. Chen, W. E, and R. Car, Thermal disorder and phonon softening in the ferroelectric phase transition of lead titanate, Phys. Rev. B 111, 094113 (2025).
  80. P. Xie, Y. Chen, X. Xu, Z. Yao, W. E, and R. Car, Ab initio bulk free energy surface of proper ferroelectrics, Phys. Rev. Lett. 136, 056801 (2026).
  81. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  82. G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  83. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  84. L. Zhang, D.-Y. Lin, H. Wang, R. Car, and W. E, Active learning of uniformly accurate interatomic potentials for materials simulation, Phys. Rev. Mater. 3, 023804 (2019).
  85. Y. Zhang, H. Wang, W. Chen, J. Zeng, L. Zhang, H. Wang, and W. E, DP-GEN: A concurrent learning platform for the generation of reliable deep learning based potential energy models, Comput. Phys. Commun. 253, 107206 (2020).
  86. H. Wang, L. Zhang, J. Han, and W. E, DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics, Comput. Phys. Commun. 228, 178 (2018).
  87. D. Lu, H. Wang, M. Chen, L. Lin, R. Car, W. E, W. Jia, and L. Zhang, 86 PFLOPS deep potential molecular dynamics simulation of 100 million atoms with ab initio accuracy, Comput. Phys. Commun. 259, 107624 (2021).
  88. J. Zeng, D. Zhang, D. Lu, P. Mo, L. Zhang, H. Wang, W. E, R. Car, et al., DeePMD-kit v2: A software package for deep potential models, J. Chem. Phys. 159, 054801 (2023).
  89. R. Wang, Y. Gao, H. Wu, and Z. Zhong, Pre-training, fine-tuning, and distillation (PFD): Automatically generating machine learning force fields from universal models, Phys. Rev. Mater. 9, 113802 (2025).
  90. D. Zhang, A. Peng, C. Cai, W. Li, Y. Zhou, J. Zeng, M. Guo, C. Zhang, B. Li, H. Jiang, et al., A graph neural network for the era of large atomistic models, arXiv:2506.01686.
  91. D. Zhang, H. Bi, F.-Z. Dai, W. Jiang, X. Liu, L. Zhang, and H. Wang, Pretraining of attention-based deep learning potential model for molecular simulation, npj Comput. Mater. 10, 94 (2024).
  92. R. Resta and D. Vanderbilt, Theory of polarization: A modern approach, in Physics of Ferroelectrics: A Modern Perspective (Springer, Berlin, Heidelberg, 2007), pp. 31–68.
  93. N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation