Diffusional dynamic process of irreversible ferroelectric to antiferroelectric phase transition and the rate limit

Abstract

The ferroelectric to antiferroelectric phase transition in perovskite structures and its underlying mechanism form the physical foundation for numerous applications such as high-performance energy storage, ultrafast information storage, and electrocaloric refrigeration. Understanding of ferroelectric to antiferroelectric phase transition, especially the description of the transition dynamic process, is crucial for material design and device reliability enhancement. Most of the previous studies demonstrated that ferroelectric to antiferroelectric phase transition are first-order transitions driven by atomic rearrangement, generally regarded as prototypical fast kinetic processes. Here, we report an irreversible ferroelectric to antiferroelectric phase transition at low temperatures in Pb(Zr_0.97Ti_0.03)O₃ as seen by neutron diffraction patterns. The in-situ variable-temperature transmission electron microscopy reveals that the ferroelectric to antiferroelectric phase transition exhibits relaxation behavior and is a slow diffusional dynamic process, which corrects the previous understanding. Density functional theory simulations reveals that the irreversible transition behavior is due to variation of the transition energy barrier with temperature. Based on Density functional theory and experimental data, a multiscale phase-field model was developed to describe the transition dynamics and provide the theory limitation for the rate of ferroelectric to antiferroelectric phase transition in Pb(Zr_0.97Ti_0.03)O₃ at low temperatures, offering a physical basis for future material applications and device design.

The authors reveal that the ferroelectric to antiferroelectric phase transition in Pb(Zr_0.97Ti_0.03)O₃-1wt%Nb₂O₅ at low temperatures is an irreversible process exhibiting relaxation behavior and diffusional kinetics.

Introduction

The ferroelectric to antiferroelectric (FE-AFE) phase transition involves crystal symmetry changing, dipole reordering, and significant changes in energy and entropy, representing a frontier interdisciplinary field in condensed matter physics and materials science^1–5. The dynamic mechanism of the FE-AFE phase transition in perovskite structures forms the physical foundation for applications such as high-performance energy storage, ultrafast information storage, and electrocaloric refrigeration^6–9. For instance, the giant isothermal entropy change (electrocaloric effect) generated during the FE-AFE transition enables all-solid-state refrigeration^10–12; the instantaneous release of substantial polarization energy accompanying the FE-AFE transition underpins the ultrahigh energy storage density in antiferroelectric capacitors^13–15. Critically, these functionalities are inherently governed not only by the initial and final phases but also by the kinetics and pathways of the transition process itself. Therefore, a deep understanding of the FE-AFE phase transition — particularly a detailed depiction of its dynamic evolution is essential for guiding rational material design, optimizing functional performance, and ensuring long-term device reliability.

To date, there are two main approaches that reveal the structural evolution of the FE-AFE phase transition: static structural observation and analysis of dynamic evolution. Static structural observation focuses on characterizing the stable phases before and after the transition, predominantly utilizing techniques such as Transmission Electron Microscopy (TEM) and various diffraction methods^16–23. For example, the AFE-FE transformation of (Bi_0.5Na_0.5)TiO₃-0.07BaTiO₃ was revealed by using Neutron Powder Diffraction (NPD) technology¹⁶. In Sm-doped BiFeO₃ materials, High-Angle Annular Dark-Field Scanning Transmission Electron Microscopy (HAADF-STEM) was applied to statically observe the FE-AFE structure transformation under an electric field¹⁷. Although TEM techniques are effective for observing atomic-scale phase structures, dynamically capturing the phase transformation process is challenging, necessitating the waiting for opportune moments and requiring rapid response capabilities.

Dynamic evolution studies of FE-AFE phase transitions rely heavily on electrical characterization techniques and theoretical simulations to infer the kinetics, such as analyzing current transient response curves or performing computational modeling^{10,16,24–32}. For instance, in PbZrO₃-based ceramics, high-time-resolution strain measurements revealed that the field-induced AFE-FE transition is an almost instantaneously lattice adjustment process¹⁶. The FE-AFE phase transitions in BiFeO₃ ceramics could be completed instantly by measuring the strain¹⁶. The field-induced AFE-FE phase transition of AgNbO₃ was an instantaneous process shown by the electrical response²⁴. Time-Dependent Density Functional Theory - Molecular Dynamics (TDDFT-MD) simulations indicated that a laser pulse can trigger polarization switching in CuInP₂S₆ within approximately 400 femtoseconds²⁵. Using voltage pulse excitation and broadband current response measurements, it was observed that the AFE-FE phase transition in Hf_0.25Zr_0.75O₂ (HZO) thin films can be completed in approximately 2-4.5 nanoseconds²⁶. Most dynamic studies have demonstrated that FE-AFE phase transitions are first-order phase transitions driven by atomic rearrangements, generally regarded as prototypical fast kinetic processes. However, due to the lack of direct in-situ observational evidence, significant controversies remain in understanding the dynamic process of the FE-AFE transition, such as: What are the characteristics of phase evolution? How fast is the phase transition?

In this work, the neutron diffraction technology was applied to reveal that Pb(Zr_0.97Ti_0.03)O₃-1%Nb₂O₅ (PZT97/3) would undergo an irreversible FE-AFE phase transition at about −20 °C, which updated the phase diagram of Pb(Zr_1-xTi_x)O₃ compositions (PZT). PZT ferroelectrics were chosen because they are one of the most famous perovskite oxides for their versatile applications and complex phase structures. Variable-temperature transmission electron microscopy (in-situ TEM) was employed to capture this FE-AFE transition process. The in-situ videos, for the first time, revealed that the FE-AFE transition exhibits relaxation behavior and is a diffusive dynamic process. Based on Density functional theory (DFT) simulations and experimental data, a multiscale phase-field model was developed to describe the transition dynamics and provide the theoretical limitation for the rate of FE-AFE transition in PZT97/3 at low temperatures, offering a physical basis for future material applications and device design.

Results and discussion

Through variable-temperature neutron diffraction measurements, we systematically investigated the crystal structural evolution of PZT97/3 during a cooling cycle from 25 ^oC to −50 ^oC and subsequent reheating back to 25 ^oC, as illustrated in Fig. 1a–c. At the initial state (25 ^oC), PZT97/3 exhibits the typical ferroelectric rhombohedral structure with space group R3c. Upon cooled to −50 ^oC, the ferroelectric structure transforms into an antiferroelectric phase with the space group Pbam. During the structural phase transition, new diffraction peaks emerge at approximately 2θ = 48°, 56°, and 83°. These peaks are assigned to the 1/4(161)_P, 1/4(-160)_P, and 1/4(152)_P planes, characteristic of the Pbam phase, providing clear evidence for the R3c-Pbam phase transition. Remarkably, this Pbam structure remains stable upon reheating back to 25 ^oC. Corresponding structural parameter changes are detailed in Table S1 and Fig. S1. Supporting X-ray diffraction patterns (XRD, Fig. S2) confirm the sequence of phase transitions R3c-Pbam-Pbam occurring during 25 ^oC → −50 ^oC → 25 ^oC. Furthermore, the dielectric temperature spectrum (Fig. S3) exhibits a sharp peak at approximately −20 ^oC, signifying the occurrence of the FE-AFE phase transition.

Fig. 1. FE-AFE phase transition of PZT97/3 ceramics driven by temperature.

The neutron diffraction patterns of the PZT97/3 ceramics at 25 ^oC, −50 ^oC, and 25 ^oC, respectively (a). b–d are the enlarged images. The updated phase diagram is shown in (e). f–h show the crystal structure transforms with temperature changing. The structure transits from Rhombohedral (R3c) into Orthorhombic (Pbam) from 25 ^oC to −50 ^oC. This phase transformation is irreversible and the structure is still Orthorhombic (Pbam) when the sample was heated back to 25 ^oC.

This trend is closely related to lattice reconstruction and enhanced structural ordering. For piezoelectric ceramic materials, the conventional phase diagram indicates that during the cooling process from 300 K to 200 K, they will transform from the antiferroelectric phase to the ferroelectric phase (Fig. 1e). However, theoretical calculations and optical analyses by Li et al.³³ suggest an additional FE-AFE phase transition occurring in the low-temperature region. In this work, we demonstrate that such a phase transition indeed exists by undoubted experimental evidence. This discovery has been experimentally verified to confirm that the update of the PZT phase diagram by Li et al. is correct. As shown in Fig. 1(e), we have redrawn the updated phase diagram. Figure 1f–h depict the unit cells change during the FE-AFE transformation, which is manifested as the displacement of Pb atoms along the a-axis direction, giving rise to the antiferroelectric phase. During the subsequent reheating process back to 25 ^oC, the atomic positions remain essentially unchanged, exhibiting a slight volumetric expansion of the unit cell (Table S1).

Building on the evidence of phase transition in PZT97/3, the in-situ TEM analysis provides unprecedented nanoscale insight into the structural evolution. At 25 ^oC, the selected area electron diffraction (SAED) pattern along the [110]_p zone axis exhibits prominent 1/2{111} superlattice reflections³⁴, confirmed the FE R3c phase (Fig. 2a, b). Upon immediately cooling to −50 ^oC, the R3c phase remains stable — evidenced by the SAED pattern showing exclusively sharp 1/2{111} reflections with no new features (Fig. 2c-d). Crucially, only after a subsequent 2-hour relaxation at −50 ^oC, the system could transit to an AFE Pbam phase, confirmed by the appearance of well-defined 1/2{001}, 1/2{110} and 1/2{111} superlattice reflections in the SAED pattern (Fig. 2e, f)^35–39. The AFE state persists upon reheating to 100 ^oC, confirming that thermal cycling alone cannot reverse the FE-AFE transformation as shown in Fig. S4.

Fig. 2. In-situ TEM images of the dynamic process of FE-AFE phase transition.

The Bright-field (BF) TEM images and SAED patterns taken along the [110]_p zone axis of PZT97/3 at 25 °C (a, b), −50 °C (c, d) and sample maintained at −50 °C for 2 hours (e, f). The scale bars in (a) and (b) are 1μm and 2 nm^-1, respectively. The images show the sample is ferroelectric phase with R3c at 25 °C and the structure was not change when it was just cooled at −50 °C. When the sample was hold at −50 °C for 2 h, the structure changed into antiferroelectric phase with Pbam. The in-situ BF-TEM images at −50 °C presented this phase transition is a typical diffusional dynamic process (g–l).

To probe the dynamics of the FE-AFE transition in real time, we collected time-resolved TEM images (Fig. 2g–l and Video S1). At t = 0 s, a sharp interface demarcates coexisting FE and AFE regions. As the transformation proceeds, AFE domains systematically advance into the FE matrix, with the propagation front locally obstructed at FE domain boundaries by t = 20 s (Fig. 2i), indicative of microstructural pinning. Despite this hindrance, this diffusion-controlled boundary migration fully consumes the FE phase within 50 s - a second-timescale transition revealing unexpectedly sluggish dynamics - and culminating in a homogeneous AFE state (Fig. 2l).

Atomically resolved high-angle annular darkfield (HAADF)-STEM imaging (Fig. 3a, b) directly captures a phase boundary separating the rhombohedral FE (R3c) and orthorhombic AFE (Pbam) phases of PZT97/3, with their coexistence confirmed by wide-field atomic-resolution images and corresponding fast Fourier transform (FFT) patterns (Fig. S5). Remarkably, an atomically thin interfacial region exhibits structural signatures distinct from both bulk phases, suggesting a boundary phase (BP) that mediates the transition. Color-coded Pb–Pb distance maps (Fig. 3a) show a gradual modulation of lattice parameters across the interface: the BP exhibits only moderate, spatially heterogeneous distortions, in contrast to the severe distortions of the AFE Pbam region and the nearly undistorted FE R3c region. Simultaneously, atomic displacement vectors (Fig. 3b) trace the polarization evolution from parallel alignment in R3c to antiparallel alignment in Pbam. Within the BP region, periodic modulations resembling AFE order appear with uncompensated dipoles, indicating a coherent rotation of polarization with reduced amplitude. The scatter plot (Fig. S6), compiled from all vectors in Fig. 3b, reveals a continuous 180^o rotation of vector orientations within the BP region, from ~45^o to 225^o, accompanied by a gradual decrease in magnitude, collectively signifying a suppressed yet coherent polarization transition.

Fig. 3. Atomic-scale structural analysis of phase transitions.

a, b Atomic-scale HAADF STEM images overlaid the spatial variation of interatomic distances within the Pb sublattice (a) and mapping Pb displacement vectors (b) across the FE, boundary, and AFE phases. The boundary phase is highlighted by a yellow box. Scale bar, 2 nm. c HAADF image of the same region depicted in (a) and (b). Individual atomic columns (with a focus on the Pb columns) are color-coded to distinguish the different structural phases (purple for FE R phase, orange for boundary phase and blue for AFE phase). This assignment is based on local structural parameters (lattice spacing and polarization direction) extracted from panels (a) and (b). Yellow arrows mark the direction of local Pb displacements, illustrating the smooth rotation of dipole vectors across the phase boundary.d Spatial profile of the lattice parameter extracted from the pale yellow box in (a), revealing a distortion zone in the boundary phase and oscillations in the AFE phase. e Corresponding Pb displacement magnitude (solid dots) and angle (hollow squares) across the FE–BP–AFE region extracted from the pale blue box in (b), showing a gradual transition from the parallel state, through quasi-antiparallel ordering, to antiparallel dipole configurations.

As illustrated in Fig. 3c, the region undergoing the phase transition can be partitioned into three structurally distinct subregions, based on variations in lattice parameters and Pb displacements: R3c region (R), the boundary phase region (BP) and Pbam region (O). Structural reconfiguration occurs specifically within the BP region, which acts as a strained coherent interface. The BP creates a strain-buffering zone that hinders antiferroelectric domain growth across the BP, effectively acting as a dynamic barrier to the FE-AFE transition⁴⁰. Consequently, the interfacial energy barrier fundamentally prohibits transformation pathways, constraining the system to a sustained diffusion-controlled process.

Line-profile analysis of lattice spacing (Fig. 3d, extracted from the pale yellow box in Fig. 3a) quantitatively illustrates the interface evolution. The R3c region shows an almost uniform spacing (Δd <5 pm), whereas the BP exhibits pronounced fluctuations (~40 pm) before the profile transitions to large antipolar oscillations in the Pbam (~80 pm). This progression confirms that the BP functions as a structural-adaptor layer: it imposes intermediate local distortions to buffer the lattice mismatch and mediate symmetry reconstruction during the phase transformation. Pb displacement profiles provide further insight into polarization dynamics (Fig. 3e, extracted from pale blue box in Fig. 3b). In R3c, Pb ions exhibit uniform, large displacements (~30–35 pm) at a constant orientation, whereas in Pbam they alternate in direction by ~180^o, reflecting antiparallel dipoles. Within the BP region, two distinct local displacement patterns emerge with a characteristic fourfold periodicity. In one pattern, pairs of Pb atoms retain large displacements similar to those in R3c, with local lattice distortions accommodating adjacent dipole reorientation. In the other, Pb displacements are strongly suppressed (~10 pm), akin to a paraelectric-like, frustrated polarization state.

Within the transitional region preceding the Pbam phase, polarization vectors gradually rotate, and Pb displacements evolve toward an antiparallel configuration, marking the dynamical onset of antiferroelectric order. In the region adjacent to the R3c phase, dipoles with reduced amplitude remain closely aligned with those in the untransformed R3c matrix, implying that polarization reversal occurs first by reducing displacement magnitude prior to directional rotation. Collectively, the Pb displacement data delineate a sequential transition pathway: the BP phase first suppresses dipole amplitudes, then facilitates their directional reversal. This stepwise coordination of lattice distortion and polarization rotation by the BP phase during the time-dependent evolution across the FE–AFE boundary mediates a diffusion-mediated phase transition, as evidenced by the progressive propagation of structural changes and polarization reorientations across the phase boundary. Critically, the lattice mismatch at the BP interface (quantified by 40 pm lattice fluctuations) establishes a substantial energy barrier. Overcoming this barrier necessitates the observed sequential suppression and rotation of dipoles, rendering the transformation a sustained diffusion-controlled process rather than a transient event.

To better understand this irreversible and relax phase transition process, the thermodynamic analysis has been carried out shown in Fig. 4. The calculated total energies of FE and AFE phase as a function of temperature through the method of phonon are displayed in Fig. 4a. It can be found that the rhombohedral R3c structure exhibits lower energy at high temperatures (> −10 ^oC), whereas the orthorhombic Pbam structure has lower energy at low temperatures (<−10 ^oC). This indicates that the FE phase is more stable at high temperatures, whereas the AFE phase is more stable at low temperatures. It is noteworthy that the energies of the two structures intersect around −10 ^oC, suggesting the potential for phase coexistence in this temperature range. Furthermore, the phase transition energy barriers between FE and AFE phases were calculated at different temperatures based on the BP structure captured in our experiments, as shown in Fig. 4b, c. It can be found that the energy barrier for the transition from FE to AFE is 25 meV at −50 ^oC. Consequently, this 25meV energy barrier requires a relaxation time for the R3c structure to gradually transform into the Pbam structure. It is worth noting that the energy barrier ΔE_R-O decreases with decreasing temperature (Fig. S7). This suggests that the structural transition becomes relatively easier at lower temperatures. Consequently, as temperature decreases, the relaxation time shortens and the transition kinetics accelerate (Video S2). However, the energy barrier for the transition from AFE to FE is as high as 43 meV at 25 ^oC. Therefore, the temperature field effect alone is insufficient to overcome the 43 meV energy barrier and facilitate the reverse phase transition from Pbam to R3c. As show in Fig. S8, after the FE-AFE transition is completed upon cooling, reheating to room temperature does not trigger the reverse transformation under zero-field conditions, suggesting that thermal energy alone is insufficient to overcome the transition barrier. In contrast, when an external electric field is applied at room temperature, the AFE superlattice reflections disappear, indicating that the AFE-FE transition can occur under the applied field. This behavior is consistent with previous observations in PZT-based ceramics^41–44. The intervention of other external field energy, such as an electric field or an irradiation field, is necessary for the AFE-FE transition.

Fig. 4. First principle simulation and Thermodynamic analysis of FE-AFE phase transition.

a The calculated total energies of FE and AFE phase as a function of temperature through the method of phonon. b The calculated pathway for phase transition from AFE to FE at −50 ^oC. c The calculated pathway for phase transition from FE to AFE at 25 ^oC.

To describe the dynamic process of the FE–AFE phase transition with high fidelity, a multiscale theoretical framework combining first-principles energetics and mesoscale kinetics was established. We conducted two-dimensional (2D) phase-field simulations to investigate the FE-AFE phase transition at low temperatures, with the free energy (ΔG) directly derived from density functional theory (DFT) calculations. In this framework, the DFT calculations provide the atomic-scale foundation for constructing the Landau-Ginzburg-Devonshire (LGD) free-energy functional. Specifically, the DFT-calculated total energies of the FE and AFE phases were fitted by a sixth-order Landau polynomial expansion, where the coefficients were optimized to reproduce the DFT-computed energy hierarchy and its temperature evolution. The temperature dependence of the second-order Landau coefficient follows the Curie-Weiss law, capturing the thermodynamic evolution of the free-energy difference between the FE and AFE phases. This DFT-based parameterization ensures that the phase-field simulations inherit a quantitatively accurate free-energy landscape derived from first principles, thereby bridging the atomic and mesoscale descriptions.

To ensure a physically consistent connection between atomic-scale energetics and mesoscale evolution, the parameterized Landau free-energy functions were incorporated into the phase-field simulations as the thermodynamic driving force for phase transformation, while the elastic, electrostrictive, and gradient energy coefficients were taken from experimental or literature values. The thermodynamic driving force arises from the local free-energy difference (Δf) between the FE and AFE states at a given temperature, which represents the functional derivative of the total free energy with respect to the polarization order parameter in the time-dependent Ginzburg-Landau (TDGL) equation. Physically, this quantity determines the direction and rate of domain evolution as the system seeks to minimize its total free energy.

Among these parameters, the gradient energy coefficient controls the domain-wall width and energy, whereas the kinetic coefficient (L) governs the dynamic response of domain-wall motion. In the TDGL framework, the diffusion of the domain wall is jointly determined by the thermodynamic driving force (Δf) and the kinetic coefficient L, modulated by the interfacial and elastic energy terms that affect the wall mobility and shape. Accordingly, the domain-wall velocity V follows a linear relationship with the local free-energy difference and the kinetic coefficient, expressed as $V\propto L$ ΔG ^45,46, where L represents the temperature-dependent interfacial mobility.

To capture microstructural effects, localized perturbations were introduced into the order parameter field to mimic structural defects such as dislocations or compositional inhomogeneities. In the simulations, several grid points were randomly assigned small fluctuations (within±5%) in the local polarization, thereby producing spatial variations in the free-energy landscape. These perturbations act as pinning centers or nucleation sites, leading to curved domain walls, interface bending, and heterogeneous phase evolution.

The temperature dependence of the domain-wall velocity originates from the explicit temperature dependence of both the Landau coefficient (thermodynamic term) and the kinetic coefficient (dynamic term). The Landau coefficient governs the curvature and depth of the double-well potential, thus controlling the free-energy difference between the FE and AFE states, while the kinetic coefficient describes the rate of polarization relaxation. As temperature decreases, the potential wells deepen, increasing the free-energy imbalance (Δf) and accelerating domain-wall motion; however, at lower temperatures, kinetic limitations dominate due to the saturation of L.

Using this multiscale model, we performed simulations of the FE-AFE phase transition under low-temperature conditions. Below the critical transition temperature, a local structural perturbation-such as a microcrack, inclusion, or dislocation- was introduced to initiate nucleation. As illustrated in Fig. 5a, the antiferroelectric phase preferentially nucleates at the perturbation site and subsequently propagates outward, forming a curved domain wall that advances with a diffusion-like velocity V. This process progressively transforms the ferroelectric phase into the antiferroelectric phase over time (Video S2).

Fig. 5. Phase field simulation and Kinetics analysis of FE-AFE phase transition.

a–d Simulated FE–AFE phase distributions at different time steps using the phase-field method at -100 °C (Black bar is 10 nm). The black squares indicate artificially introduced perturbation sites. The AFE phase initially nucleates and grows at these locations, then propagates outward in an arc-shaped manner at a velocity (v), during which the surrounding FE phase gradually transforms into the AFE phase. e Temperature dependence of the diffusion rate for antiferroelectric phase, and the simulated values fit well with the experimental values (purple stars). The velocity increases with decreasing temperature and eventually saturates at theory limit (V_limit = 8.05 nm/s).

Figure 5e presents the computed domain-wall velocity as a function of temperature, directly obtained from the nondimensionalized phase-field simulations and then converted to physical units through the kinetic coefficient described above. As shown in Fig. S7, the simulations reveal that the lattice-transformation energy barrier gradually decreases as temperature decreases, leading to a larger ΔG and a correspondingly stronger thermodynamic driving force. This explains the initial rapid increase of the phase-transition velocity at low temperatures (Fig. 5e). However, once the temperature falls below a critical threshold, the energy barrier becomes nearly eliminated, and the system approaches its minimum-energy configuration. In this regime, the domain-wall motion is no longer limited by the thermodynamic driving force but by intrinsic kinetic factors-such as structural rearrangement and polarization relaxation^45,46-which cause the kinetic coefficient to saturate. Consequently, the overall domain-wall velocity V reaches a limiting value V_max, representing the intrinsic dynamical limit of the lattice-transformation process rather than thermodynamic control.

Overall, these simulations demonstrate that the phase-field model quantitatively links first-principles energetics with mesoscale domain dynamics, clarifying how the mathematical framework governs the physical features observed in Fig. 5, including nucleation, defect-induced domain-wall curvature, and temperature-dependent interface velocity. The excellent agreement between the simulated and experimental data demonstrates that the self-consistent coupling of polarization, elastic, and thermal fields in the model successfully captures the intrinsic temperature-dependent kinetics of the FE-AFE interface migration.

Crucially, our work identifies and quantifies, for the first time, the theoretical kinetic limit of the FE–AFE phase transition driven by temperature. This limit corresponds to the minimum achievable transformation timescale dictated by intrinsic thermodynamic and elastic constraints of the material system. It provides a physically meaningful benchmark for the fastest possible response of ferroelectric materials driven by temperature and establishes a rigorous boundary for the design of ferroelectric devices.

Methods

Sample preparation

Pb(Zr_0.97Ti_0.03)O₃-1%Nb₂O₅ (PZT97/3) ceramics were fabricated using a conventional solid-state reaction method. Stoichiometric amounts of analytical-grade PbO, ZrO₂, TiO₂, and Nb₂O₅ powders were weighed with an additional 1 wt.% PbO to compensate for volatilization loss. The powders were thoroughly mixed by ball milling in ethanol with zirconia media for 12 h, followed by drying and sieving. The resulting mixture was calcined at 850 °C for 3 h in air with a heating rate of 5 °C/min and a cooling rate of 10 °C/min to promote phase formation. The calcined powders were ground, pressed into pellets under a uniaxial pressure of 30 MPa, and sintered in covered alumina crucibles at 1200 °C for 2 h under identical heating and cooling rates to obtain dense ceramics. The sintered samples were polished and subsequently annealed at 400 °C for 1 h in air to relieve residual stress and ensure structural stability before electrical and structural characterization. In this study, 1 wt.% Nb₂O₅ was incorporated into Pb(Zr_0.97Ti_0.03)O₃ to improve electrical performance. Previous reports indicate that Nb doping enhances electromechanical response while causing only minor shifts in phase boundaries without altering lattice symmetry⁴⁷. Thus, Nb₂O₅ primarily stabilizes the perovskite structure while preserving the intrinsic phase-transition pathway of Zr-rich PZT.

Neutron diffraction

To investigate the evolution of the crystal structure of PZT97/3 ceramics with temperature, variable-temperature neutron diffraction experiments were conducted on the High-Resolution Neutron Diffractometer (HRND) at the China Academy of Engineering Physics. The sample was synthesized using the conventional solid-state reaction method. The neutron wavelength was set to 1.8838 Å. The sample was mounted in a high-precision cryogenic temperature control environment. Measurements were sequentially performed at the following temperature points: room temperature (25 ^oC), low temperature (−50 ^oC), and upon reheating back to 25 ^oC, to observe structural changes during both cooling and heating paths. At each temperature point, neutron diffraction patterns were collected over a 2θ range of 20^o to 150^o, with a minimum integration time of 6 hours to ensure sufficient statistical precision in the diffraction peak intensities. For the precise determination of crystal structure parameters, all diffraction data were refined using the Rietveld method via the FullProf program suite. During the refinement process, the background, polynomial peak shape function, zero-point error, lattice parameters, and atomic positional parameters were sequentially fitted. The initial structural models used were the ferroelectric phase (R3c) and the antiferroelectric phase (Pbam).

Transmission electron microscopy

Specimens for TEM were first mechanically thinned and then Ar⁺ ion–milled, with the ion-milling voltage stepped down from 3 keV to 0.5 keV to minimize beam damage. A thin carbon coat was applied to suppress charging under the electron beam. Bright-field (BF) images, dark-field (DF) images and selected-area electron diffraction (SAED) patterns were collected on a JEOL JEM‑F200 microscope. In-situ low-temperature TEM experiments were performed on a JEOL JEM‑F200 equipped with a Gatan model 636 cooling holder. High-angle annular dark-field (HAADF) imaging at atomic resolution was conducted on a Cs-corrected ThermoFisher Spectra 300 microscope with the accelerating voltage of 300 kV.

First-principles calculation

Our first-principles calculations were performed with the CASTEP code based on the density functional theory (DFT) using ultrasoft pseudopotentials and a plane-wave expansion of the wave functions. The lattice parameters of R3c, Pbam, PI are based on the refined structure of Rietveld and the TEM data. The exchange correlation potentials were described with the generalized gradient approximation (GGA). The kinetic energy cutoff of plane waves was set as 400 eV. For relaxation of structures, the Brillouin zone was sampled by Monkhorst-Packgrid grip with k-spacing of 0.2 Å⁻¹. The values of the kinetic energy cutoff and the k-point grid were determined by ensuring the convergence of total energies within an accuracy of 1 meV/atom. The self-consistent field (SCF) tolerance of $5.0\times$10^-5 eV/atom was used during structural relaxation via Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization. In order to investigate the free energy of AFE and FE structures under different temperatures, we apply the quasi-harmonic phonons model in which the nonequilibrium Gibbs function G can be written as follows:

$$\mathrm{G}=\mathrm{U}+\mathrm{PV}-\mathrm{TS}$$

Where T is temperature and $\mathrm{F}=\mathrm{U}-\mathrm{TS}$ is the Helmholtz free energy.

Phase field simulation

To describe the phase transition from ferroelectric (FE) to antiferroelectric (AFE) states which occurs in ferroelectric materials when the temperature falls below a critical threshold, the ferroelectric polarization (P_F) and antiferroelectric polarization (P_A) are commonly chosen as order parameters to characterize the degree of lattice polarization⁴⁸. The total free energy density of the system comprises several contributions, including the standard Landau–Devonshire energy density, the elastic energy density, the polarization gradient energy density, and the electrostatic energy density, as well as the coupling energy density arising from the electrostrictive interactions between stress and the FE/AFE polarizations. Accordingly, the total free energy density is formulated as:

$$F={\int }_{\mathrm{V}}\left(f_{\mathrm{landua}}+f_{\mathrm{elas}}+f_{\mathrm{grad}}+f_{\mathrm{elec}}+f_{\mathrm{coup}}\right)dV$$ 1.1

The Landau energy density is formulated as⁴⁹:

$$f_{\mathrm{landau}}=\sum _{\mathrm{i}}\left({\alpha }_1{P}_{\mathrm{Fi}}^2+{\alpha }_{11}{P}_{\mathrm{Fi}}^4+{\alpha }_{111}{P}_{\mathrm{Fi}}^6\right)+\frac{1}{2}{\alpha }_{12}\sum _{\mathrm{i}\ne \mathrm{j}}{P}_{\mathrm{Fi}}^2{P}_{\mathrm{Fj}}^2+{\alpha }_{112}\sum _{\mathrm{i}\ne \mathrm{j}}{P}_{\mathrm{Fi}}^4{P}_{\mathrm{Fj}}^2+{\alpha }_{123}{\prod }_{\mathrm{i}}{P}_{\mathrm{Fi}}^2+\sum _{\mathrm{i}}\left({\beta }_1{P}_{\mathrm{Ai}}^2+{\beta }_{11}{P}_{\mathrm{Ai}}^4+{\beta }_{111}{P}_{\mathrm{Ai}}^6\right)+\frac{1}{2}{\beta }_{12}\sum _{\mathrm{i}\ne \mathrm{j}}{P}_{\mathrm{Ai}}^2{P}_{\mathrm{Aj}}^2+{\beta }_{112}\sum _{\mathrm{i}\ne \mathrm{j}}{P}_{\mathrm{Ai}}^4{P}_{\mathrm{Aj}}^2+{\beta }_{123}{\prod }_{\mathrm{i}}{P}_{\mathrm{Ai}}^2+{\mu }_{11}\sum _{\mathrm{i}}{P}_{\mathrm{Fi}}^2{P}_{\mathrm{Ai}}^2+{\mu }_{12}\sum _{\mathrm{i}\ne \mathrm{j}}{P}_{\mathrm{Fi}}^2{P}_{\mathrm{Aj}}^2+\frac{1}{2}{\mu }_{44}\sum _{\mathrm{i}\ne \mathrm{j}}{P}_{\mathrm{Fi}}{P}_{\mathrm{Fj}}{P}_{\mathrm{Ai}}{P}_{\mathrm{Aj}}$$ 1.2

where the subscripts i, j denote Cartesian components (i, j = 1, 2, 3); ${\mu }_{ij}$ accounts for the coupling between the FE and AFE polarizations; α_i, ${\alpha }_{\mathrm{ij}}$, ${\alpha }_{\mathrm{ijk}}$, and ${\beta }_i$, ${\beta }_{\mathrm{ij}}$, ${\beta }_{\mathrm{ijk}}$ are the FE and AFE dielectric stiffnesses, respectively, under constant stress. The coefficients α₁, β₁ are related to temperature, which are determined by the Curie–Weiss law:

$${\alpha }_1\left(T\right)=\frac{T-T_0}{2{\epsilon }_{0}C},{\beta }_1\left(T\right)=\frac{T-T_{\mathrm{0AFE}}}{2{\epsilon }_{0}C}$$ 1.3

where $T$ is the temperature, ε₀ is the vacuum permittivity, and $C$ is the Curie constant. T₀ and T_0AFE represent the Curie-Weiss temperatures of the ferroelectric and antiferroelectric phases, respectively.

The elastic energy density is defined as:

$$f_{elas}=\frac{1}{2}{C}_{\mathrm{ijkl}}{e}_{\mathrm{ij}}{e}_{\mathrm{kl}}=\frac{1}{2}{C}_{\mathrm{ijkl}}\left({\epsilon }_{\mathrm{ij}}-{\epsilon }_{\mathrm{ij}}^0\right)\left({\epsilon }_{\mathrm{kl}}-{\epsilon }_{\mathrm{kl}}^0\right)$$ 1.4

where C_ijkl is the elastic stiffness tensor, and e_ij, ${\epsilon }_{\mathrm{ij}}$ and ${\epsilon }_{\mathrm{ij}}^0$ denote the elastic strain, total strain and spontaneous strain, respectively. The spontaneous strain can be expressed by ${\epsilon }_{\mathrm{ij}}^0=Q_{\mathrm{ijkl}}{P}_{\mathrm{Fk}}{P}_{\mathrm{Fl}}+Z_{\mathrm{ijkl}}{P}_{\mathrm{Ak}}{P}_{\mathrm{Al}}$, with $Q_{\mathrm{ijkl}}$ and $Z_{\mathrm{ijkl}}$ being the electrostrictive coefficient of the FE and AFE phases. The total strain ε_ij can be derived from the displacement field u_i, ${\epsilon }_{\mathrm{ij}}=\frac{1}{2}\left(u_{\mathrm{i},\mathrm{j}}+u_{\mathrm{j},\mathrm{i}}\right)$. In Fourier space, the elastic displacement field is solved by refs. ^50,51:

$${\tilde{u}}_{\mathrm{i}}=i{A}_{\mathrm{ik}}^{-1}{\mathrm{\xi }}_{\mathrm{l}}{\tilde{\mathrm{\pi }}}_{\mathrm{kl}}=X_{\mathrm{j}}\frac{N_{\mathrm{ij}}\left(\xi \right)}{D\left(\xi \right)}$$ 1.5

where ${\tilde{\pi }}_{\mathrm{ij}}=-C_{\mathrm{ijkl}}{\tilde{\epsilon }}_{\mathrm{kl}}^0$, $X_{\mathrm{i}}=-i{C}_{\mathrm{ijkl}}{\tilde{\epsilon }}_{\mathrm{kl}}^0{\xi }_{\mathrm{j}}$, $i=\sqrt{-1}$. The wavy line on the symbol indicates its Fourier transform, and $\xi$ is the wave vector. For cubic crystals, the explicit expressions for relevant quantities such as $N_{ij}\left(\xi \right)$ and $D\left(\xi \right)$ take the forms⁵¹:

$$D\left(\xi \right)={\mu }^2\mu \left(\lambda +2\mu +{\mu }^{\prime }\right){\mathrm{\xi }}^6{+\mathrm{\mu }}^{\prime 2}\mathrm{\mu }\left(3\lambda +3\mu +{\mu }^{\prime }\right){\xi }_1^2{\xi }_2^2{\xi }_3^2+{\mu }^{\prime }\left(2\lambda +2\mu +{\mu }^{\prime }\right){\xi }^2\left({\xi }_1^2{\xi }_2^2+{\xi }_2^2{\mathrm{\xi }}_3^2+{\xi }_3^2{\mathrm{\xi }}_1^2\right)$$ 1.6

$$N_{11}\left(\xi \right)={\mu }^2{\xi }^4+\theta {\xi }^2\left({\xi }_2^2+{\xi }_3^2\right)+\gamma {\xi }_2^2{\mathrm{\xi }}_3^2$$ 1.7

$$N_{12}\left(\xi \right)=-\left(\lambda +\mu \right){\xi }_1{\xi }_2\left(\mu {\xi }^2+{\mu }^{\prime }{\xi }_3^2\right)$$ 1.8

where ${\xi }^2={\xi }_{\mathrm{i}}{\xi }_{\mathrm{i}}$, $\theta =\mu \left(\lambda +\mu +{\mu }^{\prime }\right)$, $\gamma ={\mu }^{\prime }\left(2\lambda +2\mu +{\mu }^{\prime }\right)$, $\lambda =c_{12}$, $\mu =c_{44}$, ${\mu }^{\prime }=c_{11}-c_{12}-2c_{44}$. $c_{\mathrm{ij}}$ is the elastic modulus of the material.

According to Ginzburg–Landau theory, the free energy also includes the spatial gradient of the order parameters, which accounts for the domain wall energy in systems. For simplicity, the lowest-order gradient energy density is given by:

$$f_{\mathrm{grad}}=\frac{1}{2}G{F}_{11}\left(\right)P_{\mathrm{F1},1}^2+P_{\mathrm{F2},2}^2+P_{\mathrm{F3},3}^2\left)\right)+G{F}_{12}\left(\right)P_{\mathrm{F1},1}{P}_{\mathrm{F2},2}+P_{\mathrm{F2},2}{P}_{\mathrm{F3},3}+P_{\mathrm{F3},3}{P}_{\mathrm{F1},1}\left)\right)+\frac{1}{2}G{F}_{44}\left[{\left(P_{\mathrm{F1},2}+P_{\mathrm{F2},1}\right)}^2+{\left(P_{\mathrm{F2},3}+P_{\mathrm{F3},2}\right)}^2+{\left(P_{\mathrm{F1},3}+P_{\mathrm{F3},1}\right)}^2\right]+\frac{1}{2}G{F}_{44}^{\prime }\left[{\left(P_{\mathrm{F1},2}-P_{\mathrm{F2},1}\right)}^2+{\left(P_{\mathrm{F2},3}-P_{\mathrm{F3},2}\right)}^2+{\left(P_{F1,3}-P_{F3,1}\right)}^2\right]+\frac{1}{2}G{A}_{11}\left(\right)P_{\mathrm{A1},1}^2+P_{\mathrm{A2},2}^2+P_{\mathrm{A3},3}^2\left)\right)+G{A}_{12}\left(\right)P_{\mathrm{A1},1}{P}_{\mathrm{A2},2}+P_{\mathrm{A2},2}{P}_{\mathrm{A3},3}+P_{\mathrm{A3},3}{P}_{\mathrm{A1},1}\left)\right)+\frac{1}{2}G{A}_{44}\left[{\left(P_{\mathrm{A1},2}+P_{\mathrm{A2},1}\right)}^2+{\left(P_{\mathrm{A2},3}+P_{\mathrm{A3},2}\right)}^2+{\left(P_{\mathrm{A1},3}+P_{\mathrm{A3},1}\right)}^2\right]+\frac{1}{2}G{A}_{44}^{\prime }\left[{\left(P_{\mathrm{A1},2}-P_{\mathrm{A2},1}\right)}^2+{\left(P_{\mathrm{A2},3}-P_{\mathrm{A3},2}\right)}^2+{\left(P_{\mathrm{A1},3}-P_{\mathrm{A3},1}\right)}^2\right]$$ 1.9

where $G{F}_{11}$, $G{\mathrm{F}}_{12}$, $G{F}_{44}$, $G{F}_{44}^{\prime }$ and $G{A}_{11}$, $G{A}_{12}$, $G{A}_{44}$, $G{A}_{44}^{\prime }$ are the gradient energy coefficients for FE and AFE polarizations, respectively. $P_{\mathrm{Fi},\mathrm{j}}$/$P_{\mathrm{Ai},\mathrm{j}}$ denotes the derivative of the ith component of the polarization vector, $P_{\mathrm{Fi}/\mathrm{Ai}}$, with respect to the ith coordinate.

The electrical energy density is expressed as⁵²:

$$f_{\mathrm{elec}}=-\frac{1}{2}{\kappa }_{\mathrm{ij}}{E}_{\mathrm{i}}{E}_{\mathrm{j}}-P_{\mathrm{i}}{E}_{\mathrm{i}}$$ 1.10

where the total electric field, E_i, has two contributions, the depolarization field, $E_{\mathrm{i}}^{\mathrm{d}}$, and the external electric field, $E_{\mathrm{i}}^{\mathrm{ext}}$, such that: $E_{\mathrm{i}}=E_{\mathrm{i}}^{\mathrm{d}}+E_{\mathrm{i}}^{\mathrm{ext}}$, ${\kappa }_{ij}$ is the dielectric tensor. The depolarization field $E_{\mathrm{i}}^{\mathrm{d}}$ in Fourier space can be written as⁵³:

$${\tilde{E}}_{\mathrm{i}}^{\mathrm{d}}=-i{\xi }_{\mathrm{i}}\tilde{\varphi }=\frac{{\xi }_{\mathrm{i}}}{{\epsilon }_0}\frac{\left({\xi }_1{\tilde{P}}_1+{\xi }_2{\tilde{P}}_2+{\xi }_3{\tilde{P}}_3\right)}{\left({\kappa }_{11}{\xi }_1^2+{\kappa }_{22}{\xi }_2^2+{\kappa }_{33}{\xi }_3^2\right)}$$ 1.11

The expression for the coupling energy density between ferroelectric and antiferroelectric polarization and stress ${\sigma }_{\mathrm{ij}}$ is given by:

$$\begin{array}{c}\hfill f_{\mathrm{coup}}=-Q_{11}\sum _{\mathrm{i}}{\sigma }_{\mathrm{ii}}{P}_{\mathrm{Fi}}^2-Q_{12}\sum _{\mathrm{i}\ne \mathrm{j}}{\sigma }_{\mathrm{ii}}{P}_{\mathrm{Fj}}^2-\frac{1}{2}{Q}_{44}\sum _{\mathrm{i}\ne \mathrm{j}}{\sigma }_{\mathrm{ij}}{P}_{\mathrm{Fi}}{P}_{\mathrm{Fj}}\hfill \\ \hfill -Z_{11}\sum _{\mathrm{i}}{\sigma }_{\mathrm{ii}}{P}_{\mathrm{Ai}}^2-Q_{12}\sum _{\mathrm{i}\ne \mathrm{j}}{\sigma }_{\mathrm{ii}}{P}_{\mathrm{Aj}}^2-\frac{1}{2}{Z}_{44}\sum _{\mathrm{i}\ne \mathrm{j}}{\sigma }_{\mathrm{ij}}{P}_{\mathrm{Ai}}{P}_{\mathrm{Aj}}\hfill \end{array}$$ 1.12

As noted above, the elastic strain energy depends on both the polarization state and applied mechanical load. An applied electric field can modify the polarization, altering the strain energy; conversely, external stress can affect the polarization, and hence influence the electrostatic energy. The temporal evolution of the polarization fields under applied electric and/or mechanical loads is governed by the time-dependent Ginzburg–Landau (TDGL) equation^54,55:

$$\frac{\partial P_{Fi}}{\partial t}=-L_{\mathrm{FE}}\frac{\delta F}{\delta P_{\mathrm{Fi}}}$$ 1.13

$$\frac{\partial P_{Ai}}{\partial t}=-L_{\mathrm{AFE}}\frac{\delta F}{\delta P_{\mathrm{Ai}}}$$ 1.14

where $L_{\mathrm{FE}/\mathrm{AFE}}$ is the kinetic coefficient, F is the total free energy, and $\frac{\delta F}{\delta P_{\mathrm{Fi}/\mathrm{Ai}}}$ is the thermodynamic driving force for the polarization evolution, $t$ denotes time. The semi-implicit Fourier-spectral method was employed to solve the partial differential Eqs. (1.13) and (1.14) in the present work.

For computational efficiency and generality, the following set of dimensionless variables is adopted:

$$r^{*}=\sqrt{\frac{\mid {\mathrm{\alpha }}_0\mid }{G_{110}\mathrm{r}}}{,t}^{*}=\mid {\mathrm{\alpha }}_0\mid {Lt,P}^{*}=\frac{P}{P_0},$$

$${{\alpha }_1}^{*}=\frac{{\alpha }_1}{∣{\alpha }_0∣},{{\alpha }_{11}}^{*}=\frac{{\alpha }_{11}}{∣{\alpha }_0∣},{{\alpha }_{12}}^{*}=\frac{{\alpha }_{12}}{∣{\alpha }_0∣},$$

$${{\alpha }_{111}}^{*}=\frac{{\alpha }_{111}{P}_0^4}{∣{\alpha }_0∣},{{\alpha }_{112}}^{*}=\frac{{\alpha }_{112}{P}_0^4}{∣{\alpha }_0∣},{{\alpha }_{123}}^{*}=\frac{{\alpha }_{123}{P}_0^4}{∣{\alpha }_0∣},$$

$${{\beta }_1}^{*}=\frac{{\beta }_1}{∣{\alpha }_0∣},{{\beta }_{11}}^{*}=\frac{{\beta }_{11}}{∣{\alpha }_0∣},{{\beta }_{12}}^{*}=\frac{{\beta }_{12}}{∣{\alpha }_0∣},$$

$${{\beta }_{111}}^{*}=\frac{{\beta }_{111}{P}_0^4}{∣{\alpha }_0∣},{{\beta }_{112}}^{*}=\frac{{\beta }_{112}{P}_0^4}{∣{\alpha }_0∣},{{\beta }_{123}}^{*}=\frac{{\beta }_{123}{P}_0^4}{∣{\alpha }_0∣},$$

$${{\mu }_{11}}^{*}=\frac{{\mu }_{11}{P}_0^2}{∣{\alpha }_0∣},{{\mu }_{12}}^{*}=\frac{{\mu }_{12}{P}_0^2}{∣{\alpha }_0∣},{{\mu }_{44}}^{*}=\frac{{\mu }_{44}{P}_0^2}{∣{\alpha }_0∣}$$

$${{\mathrm{Q}}_{11}}^{*}={\mathrm{Q}}_{11}{P}_0^2,{{\mathrm{Q}}_{12}}^{*}={\mathrm{Q}}_{12}{P}_0^2,{{\mathrm{Q}}_{44}}^{*}={\mathrm{Q}}_{44}{P}_0^2,$$

$${{\mathrm{Z}}_{11}}^{*}={\mathrm{Z}}_{11}{P}_0^2,{{\mathrm{Z}}_{12}}^{*}={\mathrm{Z}}_{12}{P}_0^2,{{\mathrm{Z}}_{44}}^{*}={\mathrm{Z}}_{44}{P}_0^2,$$

$${\mathrm{E}}^{*}=\frac{\mathrm{E}}{\left(∣{\alpha }_0∣P_0\right)},{\sigma }^{*}=\frac{\sigma }{\left(∣{\alpha }_0∣P_0^2\right)},{\kappa }^{*}=\frac{\kappa }{∣{\alpha }_0∣}$$

$${{\mathrm{c}}_{11}}^{*}=\frac{{\mathrm{c}}_{11}}{\left(∣{\alpha }_0∣P_0^2\right)},{{\mathrm{c}}_{12}}^{*}=\frac{{\mathrm{c}}_{12}}{\left(∣{\alpha }_0∣P_0^2\right)},{{\mathrm{c}}_{44}}^{*}=\frac{{\mathrm{c}}_{44}}{\left(∣{\alpha }_0∣P_0^2\right)}$$

$$\mathrm{G}{{\mathrm{F}}_{11}}^{*}=\frac{\mathrm{G}{\mathrm{F}}_{11}}{{\mathrm{G}}_0,}\mathrm{G}{{\mathrm{F}}_{12}}^{*}=\frac{\mathrm{G}{\mathrm{F}}_{12}}{{\mathrm{G}}_0,}\mathrm{G}{{\mathrm{F}}_{44}}^{*}=\frac{\mathrm{G}{\mathrm{F}}_{44}}{{\mathrm{G}}_0,}\mathrm{G}{{\mathrm{A}}_{44}^{\prime }}^{*}=\frac{\mathrm{G}{\mathrm{F}}_{44}^{\prime }}{{\mathrm{G}}_0,}$$

$$\mathrm{G}{{\mathrm{A}}_{11}}^{*}=\frac{\mathrm{G}{\mathrm{A}}_{11}}{{\mathrm{G}}_0,}\mathrm{G}{{\mathrm{A}}_{12}}^{*}=\frac{\mathrm{G}{\mathrm{A}}_{12}}{{\mathrm{G}}_0,}\mathrm{G}{{\mathrm{A}}_{44}}^{*}=\frac{\mathrm{G}{\mathrm{A}}_{44}}{{\mathrm{G}}_0,}\mathrm{G}{{\mathrm{A}}_{44}^{\prime }}^{*}=\frac{\mathrm{G}{\mathrm{A}}_{44}^{\prime }}{{\mathrm{G}}_0,}$$ 1.15

where $P_0=0.4\mathrm{C}{\mathrm{m}}^{-2}$ is the spontaneous polarization at room temperature, ${\alpha }_0=\left(22-479\right)3.8\times 10^5\frac{{\mathrm{m}}^{2}N}{C^2}$, and $G_0=1.73\times 10^{-10}\frac{m^{4}N}{C^2}$ is a reference value for the gradient energy coefficient.

In this work, the simulations began with an initially mono-domain ferroelectric state, characterized by assigning $P_{\mathrm{Fi}}=1$ and $P_{\mathrm{Ai}}=0$. A total of 256×256 discrete grid points were employed with a cell size of $\Delta x^{*}=\Delta y^{*}=1$, and periodic boundary conditions were imposed in both the x and y directions. These conditions emulate an infinite single crystal, eliminating edge effects and isolating intrinsic material behavior.

Author contributions

Z.F. and Z.G. conceived the idea of this work. M.G., B.H., and Z.X. performed the experiments. M.L., A.S., and L.F. analyzed the data and carried out theoretical simulations. H.N., X.C., and G. W. prepared the samples. Z.F. and Z.G. wrote the draft, and all authors contributed to the paper revision. Z.G. guided the project. Z.G., Y. Z., and J.L. provided financial and technical support for this work.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

Original experimental data, including high-resolution transmission electron microscopy image sequences, are protected and not publicly available due to size limitations and ongoing related research. If you need access to these restricted data, results related to neutron irradiation and others can be obtained by contacting the author M.Y. Ge (email: 3104370554@qq.com). For TEM and other scanning data, you can contact the author B. Han (email: hanbing24@mails.ucas.ac.cn). For theoretical calculation data such as first-principles calculations, please contact the author M.Q. Liu (email: 18228166863@163.com).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-67974-0.