Abstract
Phase transition describes a mutational behavior of matter states at a critical transition temperature or external field. Despite the phasetransition orders are well sorted by classic thermodynamic theory, ambiguous situations interposed between the first and secondorder transitions were exposed one after another. Here, we report discovery of phasetransition frustration near a tricritical composition point in ferroelectric Pb(Zr_{1x}Ti_{x})O_{3}. Our multiscale transmission electron microscopy characterization reveals a number of geometrically frustrated microstructure features such as selfassembled hierarchical domain structure, degeneracy of mesoscale domain tetragonality and decoupled polarizationstrain relationship. Associated with deviation from the classic meanfield theory, dielectric critical exponent anomalies and temperature dependent birefringence data unveil that the frustrated transition order stems from intricate competition of shortrange polar orders and their decoupling to longrange lattice deformation. With supports from effective Hamiltonian Monte Carlo simulations, our findings point out a potentially universal mechanism to comprehend the abnormal critical phenomena occurring in phasetransition materials.
Introduction
As a timehonored research topic, phase transition (PhT) covers a broad range of intriguing physical phenomena such as giant electromechanical response^{1,2}, magnetoelectric multiferroicity^{3}, superconductivity^{4}, etc. The classic thermodynamic theory^{5} expounds that the first and secondorder transitions are characteristic of specific signatures such as latent heat, volume change and divergent correlation length. However, an affirmative judgment on order of the transition cannot be made for cases with ambiguous situations^{6,7,8,9}, which therefore leaves an unfilled gap with the existing theory^{10,11,12}. Geometric frustration depicts an intrinsic incompatibility of some fundamental interactions with respect to the underlying lattice geometry. Stimulated by this, a plethora of unusual phenomena and intriguing effects have been reported, e.g., spin liquids and spin ice^{13}, glasstocrystalline transitions^{14}, and exotic spiral ferroelectric states^{15,16}. Given their commonality in phenomenological abnormality about PhT, a potential competition between the first and secondorder transitions at the tricritical point of ferroelectric Pb(Zr_{1−x}Ti_{x})O_{3} (PZT, x ≥ 0.50) is investigated in this work.
Results
Mesoscopicscale structural anomaly
Experimental and theoretical studies have revealed that there exist three tricritical points in the phase diagram of PZT (Fig. 1a). Given the complex structural evolution on the rhombohedral side (0.06 ≤ \({x}_{\mathrm {tcr}}^{{{\mathrm R}}}\) ≤ 0.26)^{17,18} and at the morphotropic phase boundary (MPB, \({x}_{\mathrm {tcr}}^{{{{{{\rm{tri}}}}}}}\) ≈ 0.45)^{12,19,20}, we hereby focus our attention on the tetragonalside tricritical point, \({x}_{\mathrm {tcr}}^{{{\mathrm T}}}\), which was reported to locate in the 0.6 ≤ \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) ≤ 0.7 composition range^{12,17,21}. By probing temperaturedependent spontaneous polarization (P_{S}), dielectric constant (ε), and lattice tetragonality (c/a ratio), a continuoustodiscontinuous transition of the physical quantities at Curie temperature (T_{C}) evidences that the secondorder transition changes to the firstorder one at \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) ≈ 0.65 with increasing x (Fig. 1b and Supplementary Fig. 1). On this basis, the correlation of ferroelectric domain morphology with the PhT order is probed using darkfield transmission electron microscopy (TEM). To distinguish the domain polarization orientation, the failure of Friedel’s law due to dynamical scattering^{22} is considered, i.e., the domains show bright contrast once the P.g > 0 (P, a component of P_{S}; g, scattering vector) criterion is satisfied under twobeam conditions.
We find that the PZT crystals with x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) are characteristic of nesting ferroelectric tetragonal (FE_{T}) and monoclinic (FE_{M}) domains at nanometer scale (Fig. 1c and Supplementary Fig. 2). In sharp contrast, the x = 0.65 crystal is comprised of regularly arranged wide c domains (~1 µm) and narrow a domains (~100 nm), which is analogous to PbTiO_{3} (refs. ^{23,24}) undergoing the firstorder transition (Fig. 1d). However, diffraction contrast analysis unveils that the ferroelastic domains are composed of selfassembled FE_{T} and FE_{M} nanodomains (inset in Fig. 1d). Being consistent with the neutron diffraction result^{25} at x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\), our effective Hamiltonian Monte Carlo simulations further reveal that the structural phase of firstordertransition PZT is dominated by the FE_{T} phase at x > \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) (Supplementary Fig. 3). Given the stable coexistence of different phases at temperature t < T_{C}, the unique hierarchical domain configuration observed in \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) = 0.65 crystal therefore implies a geometric frustration between the PhT orders.
For the x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) crystals, analysis on Xray diffraction data shows that lattice tetragonality of the FE_{T} phase is larger than that of the FE_{M} phase, which agrees well with the data of PZT ceramics^{25,26,27,28} and is corroborated by our TEM data obtained from local regions (Fig. 1e, f). For the x = 0.65 crystal, the c/a ratio of the tetragonal phase is also suggested to be larger than the monoclinic phase, (c/a)_{T} = 1.033 and (c/a)_{M} = 1.030, both of which are smaller than the expected values in the overall trend. However, our selected area electron diffraction (SAED) experiments surprisingly reveal that the two distinct lattice ratios stem from the ferroelastic a and c domains separately, which are measured as (c/a)_{a} = 1.034 and (c/a)_{c} = 1.029. After monthlevel storage, the domain lattice ratio is further increased to (c/a)_{a} = 1.044 ± 0.003 and decreased to (c/a)_{c} = 1.014 ± 0.003, which is confirmed by realspace mapping using the 4D scanning TEM technique (Fig. 1g, h and Supplementary Fig. 4). The degeneracy of phase tetragonality within mesoscale domains further suggests the scenario of PhT frustration occurred at the \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\). It is noteworthy that the structural anomaly is irrelevant to compositional segregation, which is evidenced by energy dispersive Xray spectroscopy maps of the elements (Fig. 1i–l).
Decoupled polarization–strain relationship
Our statistical measurement shows that the average domain size is very small, <d > ≈ 5.3 nm in the PZT crystals (Supplementary Fig. 5). According to the classic domain theory^{5}, d ∝ \(\sqrt{{E}_{\mathrm {DW}}}\) (E_{DW}: domain wall energy), this suggests that the E_{DW} is very low, which monotonically decreases from ca. 250 to 38 mJ m^{−2}, via a concave inflection point at \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\), with decreasing x in terms of our calculation (Supplementary Figs. 6 and 7). Further, atomicscale domain structures were acquired using the negative sphericalaberration imaging technique^{29}. Owing to instability of ordered state to random fields^{30}, the nanodomains form irregular configuration in the x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) crystals^{31} and the lattice tetragonality shows obvious fluctuation in real space. This is well manifested by a topological vortex structure, where the average lattice ratio for the FE_{T} and FE_{M} phase is measured as (c/a)_{T} ≈ 1.036 and (c/a)_{M} ≈ 1.022 (Fig. 2a, b and Supplementary Fig. 8a–d). A unitcellwise correlation of lattice ratio with polar displacement of oxygen (δ_{O2Pb}) is plotted for further data statistics and analysis (Fig. 2c). We find that the c/a ratio and δ_{O2Pb} both follow a Gaussiantype distribution and separately peak at 1.028 (fullwidth at halfmaximum, FWHM = 0.066) and 22.9 pm. Together with individual phase analysis, this evidences a coupled polarization–strain relation, \({P}_{\mathrm S}^{2}=\)σ (σ = c/a − 1)^{32}, in the secondordertransition crystals (Supplementary Fig. 8e, f).
In the x = 0.65 crystal, coexisting tetragonal and monoclinic phases and degeneracy of their lattice tetragonality are also confirmed by analysis on atomicscale TEM images, e.g., the c/a ratio histogram peaks at 1.012 for the c domain and its FWHM is narrowed to 0.054 (Fig. 2d–f). Specifically, we find that the polar displacement of oxygen (δ_{O2Pb}) exhibits a bimodal distribution, whose peaks locate around 16.1 and 32.4 pm, respectively. Clearly, this evidences a decoupled P_{S} ~ σ relation in the tricritical ferroelectric, which is further supported by strongly charged but unstable ferroelastic domain walls if the coupling relation still holds (see “Methods”). On the firstorder transition side (x > \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\)), the coupled P_{S} ~ σ relation is verified again by our quantitative measurement on atomically resolved tetragonal ferroelastic domains (Supplementary Fig. 9). These atomicscale details further indicate a frustrated behavior of PhT at the tricritical point.
Abnormal critical exponents
To verify the frustration scenario at the \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\), critical exponents were analyzed from temperature dependent dielectric constant (ε) of the PZT crystals (Fig. 3a, b and Supplementary Fig. 10). By fitting ε at t > T_{C} using a modified Curie–Weiss law, 1/ε − 1/ε_{m} = (t − T_{0})^{γ}/C (ε_{m}: the maximum of ε, C: Curie constant, T_{0}: Curie–Weiss temperature), we find that the x ≤ \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) PZT crystals are characteristic of a pronounced precursor behavior^{33}. This is manifested by a large gamma exponent (γ > 1) and deviation from the Curie–Weiss law due to formation of polar clusters at t > T_{C}^{6,7,15}. A systematic change of γ is unveiled by our effective Hamiltonian Monte Carlo simulations, which decreases from 2.26 to 1.17 with increasing x at x ≤ \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\), above which the classic meanfield value (γ = 1) is observed (Fig. 3c). In addition to this, Curieconstant ratio (C^{−}/C^{+}) is measured by fitting ε below (C^{−}) and above (C^{+}) T_{C} using CurieWeiss law to testify the PhT orders according to the Landau–Devonshire theory^{5}. Associated with a stepwise increase from 2 (second order) to 8 (first order) with x, the C^{−}/C^{+}ratios of x = 0.54 and 0.60 crystals are found to be 3.45 ± 0.19 and 3.83 ± 0.26, respectively. This ratio is measured as C^{−}/C^{+} ≈ 5.4 for the x = 0.65 crystal (Fig. 3d), which differs from the meanfield value of 4 at the tricritical point^{34}.
The critical exponent anomalies verify the existence of a frustrated PhT order at the \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) (Fig. 3e). Given the nearly coincident loss tangent (tanδ) of the x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) crystals, the discernible thermal hysteresis at T_{C} of x = 0.65 crystal further hints an anomaly of the PhT (Fig. 3a, b). To find out origin of the frustration, the precursor dynamics reflecting opticacoustic modemode coupling at t > T_{C} was probed by analyzing local γ exponent within a restricted temperature interval (Δt = 10 K)^{35}. For the x = 0.65 crystal, two localized polar glassy states featured by gamma (γ) peaks were identified at T_{g1} = t − T_{C} ≈ 40 K and T_{g2} = t − T_{C} ≈ 63 K, respectively (Fig. 3f and Supplementary Fig. 11). Around T_{g1}, the gamma has a peak value of γ ≈ 1.25 in both heating and coolingcycle profiles, while the peak value at T_{g2} decreases from ~1.46 in the heating cycle to ~1.30 in the cooling cycle. Critical behavior study indicates that corresponding to the characteristic γ value, the polar glass states adopt shortrange 3DIsing (γ ≈ 1.25) and longrange 3DrandomIsing (γ ≈ 1.46) universality classes^{6}, respectively. This reveals that accompanied with heating and coolingdependent structural changes, the frustrated PhT order may result from an intricate interplay of competing shortrange dipolar orders with spontaneously developed longrange ones^{36} around t T_{C}. Despite the γ exponent being large (1.22 ≤ γ ≤ 1.44) in the x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) crystals as well, a similar competition is not established due to presence of single γ peak around either T_{g1} or T_{g2} in the heating or cooling cycle. Additionally, we also notice that the tanδ in x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) crystals is one order of magnitude higher than that in the x = 0.65 crystal, which can be attributed to the formation of conductive channels due to random arrangement of nanodomains^{31,37,38}.
Birefringent evidence of PhT frustration
Being consistent with the precursor dynamics identified at t > T_{C}, our variabletemperature polarized light microscopy (PLM) experiments directly reveal the competion of PhT orders at t ≤ T_{C}. From heatingcycle PLM snapshots, we see that the mesoscale bandlike ferroelastic domain array is well preserved until t ≈ T_{C}, at which the domain width and wall position start to evolve dynamically (Fig. 4a). Preservation of the longrange dipolar correlation indicates that the firstorder transition overbears the secondorder one^{39}, which is evidenced by an abrupt drop of birefringence (Δn) as t approaching T_{C} (Fig. 4c). However, the formation of bandlike ferroelastic domains is much delayed in the cooling cycle (Fig. 4b), which can be identified from the domaincontrast change at the identical t points ranging from 697 to 635 K. The enhanced shortrange dipolar correlation suggests that the secondorder transition prevails over the firstorder one^{40}, as proved by a smooth transition of Δn around T_{C} (Fig. 4c). On the physical property aspect, the PhT frustration is also manifested by abnormal ferroelectric property at room temperature (Fig. 4d). With respect to the symmetric polarization–electric field (P–E) loops of PZT undergoing the first^{41} and secondorder transitions, we find that the remnant polarization is reduced by ~20.4% relative to the x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) PZT crystals. Furthermore, a very large builtin field (ΔE_{C} ≈ 20 kV cm^{−1}), an order of magnitude higher than that of x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) crystals, is observed in the tricritical ferroelectric.
Monte Carlo simulations
To further verify the experimental results, we performed effective Hamiltonian Monte Carlo simulations on 80 × 80 × 5 supercells of PZT near \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) at room temperature, in which only tetragonal ferroelastic c and a domains with equivalent volume, identical dipolar magnitude, and orientation along [001]_{T} are configured at the initial state. After relaxation for 40,000 Monte Carlo sweeps, we find that the initial c and a domains in the secondordertransition PZT (x < \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\)) supercells disappear and the entire structure transforms into coexisting tetragonal or monoclinic nanodomains, whose characteristic size is several nanometers (Fig. 5a). In good agreement with the experimental result, the hierarchical domain structure is nicely reproduced in frustratedordertransition PZT supercells with the Ti concentration around \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\). Meanwhile, the nanodomain sizes are found to increase gradually with decreasing x owing to enhanced flexibility in the orientation of local dipoles (Fig. 5b). On the firstordertransition (x > \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\)) side, the ferroelastic 90° domains are preserved due to dominance of the tetragonal phase, albeit with unequal width for the c and a domains (Fig. 5c). In addition, the P_{S} ~ σ coupling and decoupling behaviors across the \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) are also confirmed in our Monte Carlo simulations (Fig. 5d, e and Supplementary Fig. 12).
Discussion
Besides the evidences of the critical exponent and birefringence data, the PhT frustration at \({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) can also be understood from the lattice distortion point of view^{36}. With respect to the PE_{C} phase, freezing of the glassy states at t < T_{C} gives rise to locally favored tetragonal and monoclinic ferroelectric orders, which couple with their individual lattice symmetry^{31}. However, as being subject to the longrange lattice deformation, the shortrange polar orders compete within the mesoscale domains and fail to simultaneously couple with the uniformly distorted lattices. This leads to geometrically frustrated structural features, e.g., the decoupled P_{S} ~ σ relation, which can be attributed to the “slower breathing” of dipolar inhomogeneity according to firstprinciplebased simulations^{42}. Accordingly, the greatly diminished polarization anisotropy (see Supplementary Figs. 6 and 7) becomes responsible for the configurational change of ferroelastic domains under varied boundary conditions^{12}. Therefore, analogous to the spin and charge frustration observed in ferroic materials^{13,43}, the PhT frustration gives rise to an ambiguous PhT order at the tricritical composition point. It is noteworthy that the frustrated transition order is distinct from the coexistence or mixture of first order with secondorder transitions^{44}, which can easily be identified from the correlation of PhT order with the corresponding structural feature.
Regarding the critical exponent γ that is larger than one, it reflects the effective dimensionality of the order parameter and the interactions at play, rather than any disorder or frustration. To verify the existence of geometric frustration, incompatible fundamental interplay of a physical quantity with the lattice geometry and ground state degeneracy should be fulfilled simultaneously^{6,13,16,43}. In our Monte Carlo simulations, we find that the tricritical transition takes place in a composition interval of Δ\({x}_{\mathrm {tcr}}^{{{{{{\rm{T}}}}}}}\) ≈ 0.10, which is denoted by dashed lines in Fig. 1e. For ferroic and superconducting systems with apparent tricritical points^{4,8,34,45}, one may foresee that the PhT frustration occurs as the tricritical point is accessed by tuning composition, electric, magnetic field, and pressure. For systems with hidden tricritical points, e.g., BaTiO_{3}, the frustration scenario probably validates as well given the simultaneous presence of critical exponent anomaly (γ ≈ 1.08)^{7}, structural hierarchy and degenerate lattice ratio for the coexisting tetragonal and monoclinic phases^{46,47}. This even applies to the thermotropic phase boundary^{47}, where the firstorder ferroelectric transition competes with the excited secondorder transition, which is evidenced by presence of lowsymmetric structural phases^{48}.
In summary, we report a frustrated PhT order at the tricritical point of ferroelectric PZT solid solution. Given the ubiquitous tricritical point, our findings suggest a generic mechanism to decipher PhTrelated unusual critical phenomena in ferroic materials, which are featured by abnormal critical exponents, structural hierarchy, degenerate lattice tetragonality for coexisting phases, and may possibly be extended to ferroelectrics with diffuse PhTs near the MPBs^{7,49}. Particularly, the frustrated PhT scenario offers an important degree of freedom to engineer hierarchical domains, which has been reported to play important roles in improving material performances such as piezoelectricity^{50}, magnetoelectric effect^{3}, shape memory^{51}, and electrostatic energy storage^{52}. Therefore, it is believed that this work may inspire extensive research interest on exploring PhTrelated frustrated states and design of material functionality in a more flexible way.
Methods
Materials preparation
Lead zirconate titanate PZT (x = 0.54, 0.60, and 0.65) single crystals were grown by a topseeded solution growth technique^{53}, and the PZT (x = 0.90) thin films were grown by pulsed laser deposition^{54,55}. PLM (Olympus BX60) was used to characterize temperaturedependent domain evolution data. FEI Helios NanoLab 400s focused ion beam (FIB) system was used for preparing the lamella specimens, before which an Au layer (thickness ~25 nm) was coated on the sample surfaces. To protect the samples from being damaged by Ga ions, electroninduced carbon (~180 nm) and ioninduced Pt (~4 μm) layers were deposited on the region of interests (area ~20 µm × 2 µm). After taking out the lamella by making trenches on both sides, the lamella was welded to the TEM grids, milled at 30 kV with 2.8 nA–93 pA currents and followed by a final cleaning at 5 kV and 47 pA. To remove the surface contamination and damage layer, the NanoMill Model 1040 system operated at 500 V was used to further clean and thin down the lamella specimens.
Electron microscopy imaging experiment
The darkfield imaging and SAED experiments were performed on an FEI Tecnai F20 microscope operated at 200 kV. Referring to the SrTiO_{3} standard, the lattice ratios of PZT crystals were measured in a quantitative way from the SAED patterns. The atomicresolution TEM and 4DSTEM experiments were performed on image and probecorrected FEI Titan 80300 microscopes operated at 300 kV, respectively. By fitting atomic column peak intensities using 2DGaussianfunctionbased maximum likelihood estimation^{56,57}, we simulated the atomicresolution images using CrystalKitMacTempas software package. An FEI Titan 80200 ChemiSTEM microscope, equipped with a SuperX energy dispersive Xray spectrometer, was used for compositional analysis.
Estimation of bound charges at ferroelectric domain wall
Supposing the \({P}_{\mathrm S}^{2}=\)σ relation holds in the x = 0.65 crystal, referring to PbTiO_{3} standard^{54} (P_{S} = 96.8 μC cm^{−2}, c/a = 1.0643), the coupling coefficient is determined as к = 145,727 μC^{2} cm^{−4}. For the ferroelastic domains with (c/a)_{a} = 1.044 and (c/a)_{c} = 1.014, our calculation reveals that the density of bound charges is ΔP_{S⊥} = 24.7 μC cm^{−2} normal to the ferroelastic wall plane, which is equal to the P_{S} of BaTiO_{3}^{58}. To lower energy of the strongly charged wall structures, an especially high density of free carriers, beyond the available limit of the material itself^{59}, is needed. To minimize the electrostatic energy, the charged domain walls usually show curved morphology^{60}. This is also different from our experimental observations. These results therefore alternatively refute the P_{S} ~ σ coupling relation in the x = 0.65 crystal.
Dielectric and ferroelectric property measurement
The dielectric properties of PZT single crystals were measured using a Novocontrol Alpha highresolution broadband dielectric spectrometer. The dimensions of the crystals, oriented along (001)_{p}, (011)_{p,} and (001)_{p}, used for the measurement are 1.4 × 0.8 × 0.162, 1.5 × 1.67 × 0.05, and 1.9 × 1.2 × 0.22 mm^{3} for the x = 0.54, 0.60, and 0.65 crystal, respectively. Typically, a small signal ac electric field (1 V_{rms}) was applied for the standard dielectric spectroscopy measurements. The Curieconstant ratio was measured beside T_{C} in a temperature interval of 10–20 K. A standardized ferroelectric analyzer system (TF Analyzer 2000; aixACCT, Germany) was used to measure the ferroelectric property at room temperature.
Monte Carlo simulations
Monte Carlo simulations on PZT bulks were performed using an ab initiobased effective Hamiltonian model^{61,62}, which parametrizes the Born–Oppenheimer energy landscape in terms of local modes about Bsite polar displacement, oxygen octahedral rotation, and takes into account acoustic phonon branches parameterized through homogeneous strain tensor and Asite displacement within each unit cell. The alloying effects are mimicked by introducing local fields through breaking local cubic symmetry in the PE state via compositional disorder and different values of onsite coefficients of the effective Hamiltonian. The supercell lateral sizes were chosen to be of 12 × 12 × 12 or 80 × 80 × 5 unit cells along the pseudocubic (p) [100]_{p}, [010]_{p}, and [001]_{p} axis, respectively. The former and latter geometry was used to obtain temperaturedependent macroscopic property and domain structure, respectively. To compute equilibrium property for each considered composition, we firstly performed temperature annealing simulations to obtain the raw values of T_{C}, where the system was cooled from 1500 K down to 100 K with a step size of 50 K. For x = 54, 60, 65, and 100, the annealing simulations were then repeated under hydrostatic pressure varying from −5 to 5 GPa with an increment of 0.1 GPa. This helps to define the effective pressure and to calibrate the simulated T_{C} with respect to the experimental one. By interpolating the effective pressure values, we obtain the external pressure for all considered compositions in the range of x = 0.50–0.70. This procedure allows to correct for errors induced by LDA approximation while constructing the effective Hamiltonian model, and the errors related to the absence of anharmonic elastic contributions that are responsible for thermal lattice expansion.
Data availability
The authors declare that all data supporting the findings of this study are available within the paper and its Supplementary information files.
Code availability
The codes used in this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
X.K.W. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG; German Research Foundation) under Germany’s Excellence StrategyCluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1390534769. S.P., Y.N. and L.B. thank the DARPA Grant No. HR0011727183D18AP00010 (TEE Program), Vannevar Bush Faculty Fellowship (VBFF) Grant No. N000142012834 from the Department of Defense, and an Arkansas Research Alliance (ARA) Impact Grant. B.X.W., Y.J.X. and Z.G.Y. acknowledge the U.S. Office of Naval Research (ONR Grant No. N000141613106 and No. N000142112085) and the Natural Sciences & Engineering Research Council of Canada (NSERC, Grant No. RGPIN1706915) for support. Z.L. thanks the Natural Science Foundation of China (Grant No. 51902244), the China Postdoctoral Science Foundation (Grant No. 2018M643632) and the Natural Science Foundation of Shaanxi Province of China (Grant No. 2019JQ389) for support. The authors thank N. Setter for initial discussion and useful advice, and M. Kruth and D. Meertens for their help in FIB sample preparation.
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X.K.W. conceived the research idea and coordinated the study together with L.B. and Z.G.Y. X.K.W. carried out the electron microscopy experiments and data analysis with support from C.L.J., R.E.D.B. and J.M. S.P., Y.N. and L.B. performed the effective Hamiltonian Monte Carlo simulations. B.X.W., Y.J.X., Z.L. and Z.G.Y. prepared the PZT crystals, examined the crystal structures using Xray diffraction, and performed dielectric, ferroelectric, and optical property characterizations. X.K.W. wrote the manuscript and all authors discussed the results and revised the manuscript.
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Wei, XK., Prokhorenko, S., Wang, BX. et al. Ferroelectric phasetransition frustration near a tricritical composition point. Nat Commun 12, 5322 (2021). https://doi.org/10.1038/s41467021255431
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DOI: https://doi.org/10.1038/s41467021255431
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