Abstract
We construct a Landau–Ginzburg thermodynamic potential, and the corresponding phase diagram for pristine and slightly doped bismuth ferrite, a ferroelectric antiferromagnet at room temperature. The potential is developed based on new Xray and neutron diffraction experiments complementing available data. We demonstrate that a strong biquadratic antiferrodistortivetype coupling is the key to a quantitative description of Bi_{1−x }La_{ x }FeO_{3} multiferroic phase diagram including the temperature stability of the antiferromagnetic, ferroelectric, and antiferrodistortive phases, as well as for the prediction of novel intermediate structural phases. Furthermore, we show that “rotomagnetic” antiferrodistortive–antiferromagnetic coupling is very important to describe the ferroelectric polarization and antiferrodistortive tilt behavior in the R3c phase of BiFeO_{3}. The Landau–Ginzburg thermodynamic potential is able to describe the sequence of serial and triggertype phase transitions, the temperaturedependent behavior of the order parameters, and the corresponding susceptibilities to external stimuli. It can also be employed to predict the corresponding ferroelectric and antiferrodistortive properties of Bi_{1−x }La_{ x }FeO_{3} thin films and nanoparticles by incorporating the gradient and surface energy terms that are strongly dependent on the shape, size, and preparation method.
Introduction
Stateofthe art
Multiferroics, defined as ferroics with more than one longrange order, are ideal systems for fundamental studies of couplings among the ferroelectric (FE) polarization, structural antiferrodistortion, and antiferromagnetic (AFM) order parameters.^{1,2,3,4,5} BiFeO_{3} (BFO) is the one of the rare multiferroics with a strong FE polarization, antiferromagnetism at room temperature as well as conduction and magnetotransport at domain walls.^{6,7,8}
Multiferroic properties have also been extensively demonstrated BiFeO_{3} thin films and heterostructures.^{9,10,11,12} In particular, the studies of thin epitaxial BiFeO_{3} films revealed a universal field induced phase transition, modulated phases and microstructure changes as a function of rareearth (RE) elements (e.g., Dy, Sm, Ho) doping concentration.^{13,14,15,16,17} Further RE doping effect was studied systematically for thin BiFeO_{3} films, and it was shown experimentally and theoretically^{16,17,18} that the doping of BiFeO_{3} by Sm can lead to the enhancement of dielectric properties and tunability, as well as to the stabilization of the polar phase over a wide range of temperatures for the thin (Bi,Sm)FeO_{3} films and the short period superlattices BiFeO_{3}–(Bi,Sm)FeO_{3}.
Bulk BiFeO_{3} exhibits antiferrodistortive (AFD) order at temperatures below 1200 K; it is FE with a large spontaneous polarization below 1100 K and is AFM below Neel temperature T _{N} ≈ 650 K.^{19, 20} The very high AFD transition temperature of multiferroic BiFeO_{3} offers the unique possibility to study the influence of AFD order on the FE and AFM phase transitions. Despite extensive experimental and theoretical studies of BiFeO_{3} physical properties,^{21,22,23,24,25,26,27} many important issues concerning the physical mechanisms responsible for the emergence and manifestation of its multiferroic and electrophysical properties remain unclear.^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26} For example, the possibility of rare “triggertype” phase transitions,^{28} in which different order parameters appear simultaneously at the same transition temperature, is under debate in BiFeO_{3}.^{29} Further, reliable experimental results and analysis^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26} indicate the importance of the AFDtype couplings in BiFeO_{3}, such as biquadratic AFD–AFM and AFD–FE couplings, and different AFD order parameters regarded as “AFD–AFD” coupling. The antiferrodistortion comes from the static rotation of some atomic groups with respect to other parts of the crystal.^{30} In this work, the term AFDsymmetry means only the static rotational symmetry of the oxygen octahedra MO_{6} with respect to the cube A_{8} cell in AFD perovskites with the structural formula AMO_{3}. Oxygen atoms are displaced with respect to the centers of the cube faces A_{8} in the AFD phase, and the angle or the value of the corresponding displacement is a structural order parameter (see for e.g., ref. 31). According to group theory, the aforementioned biquadratic coupling can exist in all AFD multiferroics.^{32}
Research motivation and impact
Our primary goal is to explain and describe quantitatively the experimental data obtained in pure BiFeO_{3} and ab initio calculations performed previously, and in the process, construct a thermodynamic potential of Landau–Ginzburg (LG) type, which describes FE and AFD properties in pure BiFeO_{3}. Such a LGpotential is currently lacking that can be used to reconstruct the FE polarization and AFD tilts in the low temperature AFM phase as well as high temperature phases, which are known in addition to a new phase that is disclosed in this work. Further discussion testifies to the essential impact of the AFD order on the FE and AFM longrange order and provides theoretical background to the appearance of triggertype phase transitions due to the strong biquadratic coupling.
Though the main subject of the investigations is pure bismuth ferrite, we have also included the case of slightly Ladoping^{4} (5%), which does not significantly change the structural parameters of the compound as compared to the pristine BiFeO_{3} (which is necessary for theoretical fitting), but notably lowers the critical transition temperatures and thus increases the reliability of the structural measurements at elevated temperatures. Thus the slightly doped compound with virtually no difference in the multiferroic behavior and a phase diagram as compared with pristine BiFeO_{3} ideally suits our purposes as a reference material for the theoretical fitting. The compounds with larger content of La ions are characterized by the orthorhombic phases without remnant polarization at room temperature and thus beyond the scope of the current study.^{33,34,35}
Results
Experimental results
Structural data obtained for the BiFeO_{3} compounds during the Xray diffraction (XRD) and neutron powder diffraction (NPD) measurements confirmed their singlephase rhombohedral structure (Rphase), which is stable from room temperature up to approximately 810 °C for the pristine compound and ~780 °C for the doped one. The results of the diffraction measurements show a gradual expansion of the unit cell parameters and a volume increase with temperature increase. Chemical substitution of the bismuth ions by small amount of lanthanum ions (5%) only slightly modifies the structural parameters. We did not observe any notable structural anomaly near the magnetic transition temperature, implying a weak correlation between magnetic order and crystal structure in contrast to ref. 35.
Normalized lattice parameters shown in Fig. 1a, b displays tetragonallike distortion of the reduced unit cell associated with the polar displacements of the cations directed along the cparameter (in rhombohedral lattice with hexagonal settings). Evolution of the structural parameters estimated for the doped compound shows faster elongation of the aparameter as compared to the cparameter, resulting in a decrease in tetragonality. Temperature dependences of the unit cell volume and tetragonality parameter are shown in Fig. 2a, c.
It is known that doping with lanthanum as well as with other RE elements diminishes the covalency of the Bi(La)–O chemical bond^{36, 37} and thus reduces the polar displacements of the cations and the tetragonality of the structure. Reduction in the polar displacement of the lightly doped Bi_{1−x } RE _{ x }FeO_{3} compounds has been confirmed experimentally by neutron diffraction data as well as piezoresponse force microscopy technique.^{38, 39} Taking into account the evolution of tetragonality observed for the compounds, one can estimate the transition temperature of the tetragonal structure to the cubiclike structure at about 1100 °C for the pristine BiFeO_{3} and about 1000 °C for the doped compound.
It should be noted that the tetragonality estimated for both compounds gradually increases with temperature up to about 500 °C and then quite rapidly reduces down to the value below the calculated one for the room temperature structure. The rhombohedral angle calculated for the pseudocubic metric shows the opposite trend with a decrease up to T ~ 500 °C followed by gradual increase (see Table S1 in the Supplementary Materials). The critical temperature about 500 °C is also noticed by tracing the evolution of the structural parameters discussed below.
The structural data show that the chemical doping with lanthanum leads to an increase of the rhombohedral angle of the lattice, thus pointing to the reduction of the structural distortion,^{34, 40} wherein the tetragonality and the unit cell volume gradually decreases (e.g., for 16% of La compound, the unit cell volume is about 61.7 Å;^{3} rhombohedral angle alfa _{(R)} ~ 89.68°). The structural parameters, such as the chemical bond angle Fe–O–Fe, gradually increase with an increase in temperature, while the bond lengths of Fe–O exhibit a different behavior for short and long bonds. The short bond length shows a progressive increase with temperature, while the long bond displays a restoring behavior similar to that observed for the tetragonality parameter and the rhombohedral angle (the inflection point is also about 500 °C). Similar evolution of the structural parameters has been observed for the Ladoped compound and the pure compound, thus indicating that the same factors are causing the structural modification observed in both these compounds (Fig. 1b, d). The mentioned structural rearrangement is most probably associated with the oxygen octahedra tilts and rotations, which become more pronounced across the phase transition into the orthorhombic phase as discussed below.
In contrast to the evolution of tetragonality, the magnitude of polar ionic displacement shows a gradual decrease with increase in temperature for both compounds (Figs. 1 and 2). Because of the symmetry restrictions (viz. the presence of the 3rotation axis in the rhombohedral lattice), the ionic shift is directed strictly along the c axis of the lattice. The polar displacement calculated for the bismuth ions in the BiFeO_{3} is about 0.65 Å at room temperature, which leads to a polarization of ~67 µC/cm^{2} (assuming simple point charge model); the dipole moment attributed to the doped compound is about 4% smaller, thus resulting in proportionally lower polarization. The difference in the Fe–O bond lengths (the bond lengths for initial compound are 2.115 and 1.943 Å at room temperature) also contributes to the dipole moment of the compounds, accounting about 30% of overall polarization (Fig. S1 in Supplementary Materials). Temperature increase causes a modification of the ionic coordinates (Table S1 in Supplementary Materials) leading to a gradual decrease of the polar displacements of the Bi ions and nearly negligible changes for the Fe ions (Fig. S1 in Supplementary Materials). The reduction in the displacements of the Bi ions becomes more pronounced at high temperatures showing certain anomaly at ~500 °C for the initial compound.
Thermal evolution of the bond lengths Bi(La)–O observed for the pristine bismuth ferrite shows certain anomaly at ~500 °C, followed by faster decrease in the magnitude as compared to the dependence estimated for the doped compound. The difference in the behavior of the ionic displacement is associated with a modification of the character of the chemical bonds because of different covalent component in the Bi(La)–O lengths as confirmed by change density analysis ref. 36. More pronounced covalent component estimated for the initial compound makes it more stable at high temperatures as confirmed by the given structural data. The displacement estimated for the Fe ions shows only slight decrease with temperature increase for both compounds.
Significant displacement of the bismuth/lanthanum ions from their ideal perovskite positions is often associated with anomalous atomic displacement parameters.^{41} Refinement of the NPD patterns using anisotropic displacement parameters has considerably improved the reliability factors. The atomic displacement parameters calculated for the rhombohedral phase testify to an elongation of the thermal ellipsoids associated with bismuth/lanthanum ions along the polar axis [111]_{p}.
The NPD patterns recorded for the compounds at high temperatures (~800 °C) have revealed drastic changes in the crystal structure associated with the rhombohedralorthorhombic phase transition.^{42, 43} The diffraction data shows a narrow temperature range (about 20–30 °C) of the twophase mixture state, thus confirming the high chemical homogeneity of the compounds. Further temperature increase leads to the formation of a single phase state with nonpolar orthorhombic symmetry described by the space group Pnma (Ophase).
The R–O structural transition for the initial compound is observed at ~840 °C; this transition is accompanied by an abrupt decrease in the reduced unit cell parameters, which becomes much closer in their magnitudes resulting in a pseudocubic symmetry. The unit cell volume calculated for the pristine compound BiFeO_{3} shows a drastic decrease (~1.6%); the doped compound also exhibits a comparable decrease in the unit cell volume (~1.8%) (Fig. 1), while the structural transition is shifted towards lower temperature and completes at about 780 °C. The changes of the unit cell parameters across the transition (Fig. 1) as well as the evolution of the polar ionic shifts imply a firstorder displacive phase transition into the nonpolar orthorhombic state. Lower transition temperature observed for the doped compound has allowed us to perform reliable refinement of the structural parameters as the diffraction patterns are not deteriorated in quality because of decreased signaltonoise and chemical decomposition started at T ~ 850 °С for the initial compound.
The diffraction results cannot provide reliable structural data necessary to reveal the origin of these phase transitions, because of uncertainty in the unit cell parameters calculated near the decomposition temperature. To clarify the structural evolution of the compounds at high temperatures, we have performed theoretical modeling of the phase stability of the different crystal structures as well as the estimated presence of the intermediate phases.
Structural data veracity
It is known^{44,45,46} that a substitution of the bismuth ions by lanthanum ones lowers the temperatures of the structural transitions and increases structural stability of the compounds. The DTA/DSC measurements performed in the research have clarified the phase transition temperatures and an application of the obtained results has permitted us to accomplish a careful estimation of the structural parameters based on the neutron diffraction measurements. The phase purity of the compounds has been verified by the diffraction measurements and the obtained structural data are in accordance with the results declared in previous seminal works.^{13, 34, 44}
It is known that the chemical decomposition of the pristine compound occurs at about 930 °C^{20, 35} while one can assume some release of the oxygen and bismuth ions at lower temperatures; the structural data used for theoretical calculations have been taken at a lower temperature range where any decomposition process is negligible. Moreover, the neutron diffraction process assumes that an interaction of the neutrons with matter occurs in a thick layer of the crystallites (about 100 nm) so that the diffraction data reflect the average structural parameters throughout the grains (not just their surface layers). Taking into account these arguments, one can have high confidence in the reliability of the provided structural results.
Theoretical background
Extraction of the octahedral tilt and polarization
The temperature dependence of the oxygen octahedra tilt has been determined from the full set of oxygen positions, directly extracted from the structural data measured in R3c phase in this work. According to refs. 43, 47, for the R3c space group, the fractional coordinates of oxygen atoms in the one of 18 equivalent Wyckoff position have the form O: \(\left( {\frac{1}{2} + 2e  2d,\quad  4d,\quad 1  s} \right)\), where the temperature dependences e, d and s values will be determined from the experiment. Based on the coordinates, we determine the angle of oxygen octahedra tilt as \(\omega {\rm{ = }}\arctan \left( {\frac{{4\sqrt 3 e}}{{1  12d}}} \right)\). In the same coordinate settings, Bi atom has the following fractional coordinates Bi: (0, 0, \({\frac{1}{4}}\) + s). Components of the oxygen atom displacement Φ are equal to \({{\bf{\Phi }}_1} = {{\bf{\Phi }}_2} = {{\bf{\Phi }}_3} = \frac{a}{{\sqrt 8 }}\tan \frac{\omega }{{\sqrt 3 }} \approx \frac{a}{{\sqrt 2 }}\frac{{2e}}{{1  12d}}\), where a is the lattice constant. Comparison with experiment gives the following values e ~ −0.03; d ~ −0.05; s ~ 0.05 and ω ~ 11°, which vary with temperature as anticipated. The calculated temperature dependence of the oxygen displacement Φ for the R3c phase is shown in Fig. 3a. Empty and filled diamonds correspond to pure BiFeO_{3} and BiFeO_{3} doped with 5% of La respectively.
The temperature dependence of FE polarization was calculated using the following relation \(P{\rm{ = }}\frac{{6{q_e}\left( {{q_{{\rm{Bi}}}}{\rm{\delta }}{z_{{\rm{Bi}}}} + {q_{{\rm{Fe}}}}\delta {z_{{\rm{Fe}}}}} \right)}}{{{V_{{\rm{u}}{\rm{.c}}{\rm{.}}}}}}\), where q _{ e } is the elementary charge of electron, q _{Bi} = 4.38 and q _{Fe} = 3.61 are corresponding Born effective charges taken from,^{48} where δz _{Bi} and δz _{Fe} are displacements of Bi and Fe atoms from their centrosymmetric positions, c is the lattice constant along polar axis, and V _{u.c.} is the unit cell volume, \({V_{{\rm{u}}{\rm{.c}}{\rm{.}}}}{\rm{ = }}\left( {\sqrt 3 {\rm{/2}}} \right){a^2}c\). The factor of “6” takes into account the presence of six formula units inside the unit cell. The temperature dependence of the recalculated spontaneous polarization P in R3c phase is shown in Fig. 3b. Empty and filled diamonds correspond to experimental results for pure BiFeO_{3} and doped with 5% of La, respectively measured in this work.
Thermodynamic potential
Our primary goal is not only to explain and describe quantitatively the experimental data, but also to construct a thermodynamic potential of LG type, that describes AFM, FE and AFD properties for pure BiFeO_{3} and BiFeO_{3} slightly doped with La, including the FEAFD R3c phase, different structural AFD phases (orthorhombic Pbnm, tetragonal I4/mcm) and cubic Pm3m phase. For this purpose, we include the AFD, FE and AFM contributions and the biquadratic coupling among them:
The AFD energy is a sixorder expansion on the two tilt vectors (Φ _{ i } and Ψ _{ i }):
Here Φ _{ i } and Ψ _{ i } are components of pseudovectors, determining outofphase and inphase static rotations of oxygen octahedral groups (eigenvectors of AFD modes of lattice vibrations), and Einstein summation convention is employed. These vectors correspond to Glazer tilt systems a ^{−} b ^{−} c ^{−} and a ^{+} b ^{+} c ^{+}, respectively and have different transformation properties. AFD order parameters Φ _{ i } and Ψ _{ i } could be measured as either tilt angles of oxygen octahedra or oxygen atoms displacement from the symmetric position in an ideal cubic perovskite structure.
FE energy ΔG _{FE} is a sixthorder expansion on the polarization vector P _{ i }.
Sixthorder expansion on the tilts Φ _{ i } (Eq. 2a) is necessary to describe the firstorder transition from the cubic Pm3m to tetragonal I4/mcm phases, and we find that a sixthorder expansions for Ψ_{ i } and P _{ i } are necessary to quantitatively describe the temperature dependence of different order parameters. AFM energy ΔG _{AFM} is a fourthorder expansion in terms of the AFM order parameter vector L _{ i } because this phase transition in BiFeO_{3} is known to be the second order one.
In accordance with the classical LG theory, we assume that the coefficients \(a_i^{(\Phi )}\), \(a_j^{(\Psi )}\) and \(a_k^{({\rm{P}})}\) are temperature dependent according to Barrett law,^{49} \(a_i^{(\Phi )}{\rm{ = }}a_T^{(\Phi )}{T_{q\Phi }}\left( {\coth \left( {{T_{q\Phi }}{\rm{/}}T} \right)  \left( {{T_{q\Phi }}{\rm{/}}{T_\Phi }} \right)} \right)\), \(a_j^{(\Psi )}{\rm{ = }}a_{\rm{T}}^{(\Psi )}{T_{{\rm{q}}\Psi }}\left( {\coth \left( {{T_{{\rm{q\Psi }}}}{\rm{/T}}} \right)  \coth \left( {{T_{q\Psi }}{\rm{/}}{{\rm{T}}_\Psi }} \right)} \right)\) and \(a_k^{(P)}{\rm{ = }}a_T^{(P)}\left( {{T_{qP}}\coth \left( {T_{qP}/T} \right)  {T_{\rm{C}}}} \right)\), where T _{Φ}, T _{Ψ} and T _{C} are corresponding virtual Curie temperatures, T _{ qΦ}, T _{ qΨ} and T _{ qP } are characteristic temperatures.^{50}
As it was shown recently,^{51} similar Barretttype expressions can be used for AFM coefficient \(a_i^{\rm{L}}(T)\) of pure bismuth ferrite \(a_i^{\rm{L}}(T) = {\alpha _{{\rm{LT}}}}{T_{\rm{L}}}\left( {\coth \left( {{T_{\rm{L}}}/T} \right)  \coth \left( {{T_{\rm{L}}}/{T_{\rm{N}}}} \right)} \right)\) with the Neel temperature T _{N} = 645 K and characteristic temperature T _{L} = 550 K. The expression \(L\sim\sqrt {a_1^{\rm{L}}\left( T \right){\rm{/}}{{\rm{a}}_{11}}} \), valid in the isotropic approximation, describes quantitatively both the temperature dependence of the AFM order parameter measured experimentally in BiFeO_{3} by neutron scattering by Fischer et al.^{19} and anomalous AFM contribution to the specific heat behavior near the Neel temperature measured experimentally by Kallaev et al.^{52}. The Neel temperature is a bit higher (T _{N} = 650 K) for Bi_{0.95}La_{0.05}FeO_{3} ^{53} (see the first column in Table 1, where all observed transition temperatures in BiFeO_{3} and Bi_{0.95}La_{0.0}5FeO_{3} are listed).
The AFD–FE–AFM coupling energy ΔG _{BQC} is a biquadratic form of the order parameters L _{ i }, P _{ i }, Φ _{ i } and Ψ _{ i }. The detailed forms of these contributions, which account for the parent phase symmetry Pm3m of BiFeO_{3} and different transformation properties of the outofphase tilt Φ _{ i } and inphase tilt Ψ _{ i }, are listed in the Appendix B, Supplementary Materials. The short form of ΔG _{BQC} is
For a given symmetry, the coupling energy in Eq. 2d includes nine unknown tensorial coefficients in Voight notation for the AFD–AFD (ξ_{11}, ξ_{12}) and AFD–FE (ζ_{44}, ζ_{11}, ζ_{12}, η _{11} η _{12}) biquadratic couplings. Due to the lack of experimental data, FE–AFM and AFD–AFM constants λ _{ ij } and κ _{ ij } ^{32} are assumed to be isotropic, \({\lambda _{ij}}{\rm{ = }}\lambda {\delta _{ij}}\) and \({\kappa _{ij}}{\rm{ = }}\kappa {\delta _{ij}}\).
Note, that DM interaction was not included in the potential (1), because we have focused on the influence of the simplest and omnipresent biquadratic couplings between the tilt, polarization, and (anti)ferromagnetism. Also we see no serious grounds to include more complex DMrelated coupling like L × M × P × Φ in the bulk functional, which can be allowed by the symmetry. Also we have neglected all gradient and flexotype coupling terms, because we restricted ourselves to the consideration of homogeneous bulk BiFeO_{3}.
LG potential coefficients
LG potential coefficients have been determined from experiments using the fitting procedure. The procedure started from the high temperature structural phases (cubic, tetragonal I4/mcm and orthorhombic Pbnm) and then goes down to the low temperature FErhombohedral R3c phase. Expansion coefficients of the LG potential for pure BiFeO_{3} and BiFeO_{3} doped with 5% of La extracted from the fitting of measured the AFD and FE order parameters in rhombohedral R3c, orthorhombic Pbnm and tetragonal I4/mcm phases are listed in the Table 2.
Note, that for the complex sequence of the firstorder phase transitions, the virtual Curie temperatures T _{C}, T _{Φ} and T _{Ψ}, listed in the Table 2, can be different from the corresponding transition temperatures from R3c to Pbnm phase, from Pbnm to I4/mcm phase and from I4/mcm to Pm3m phase measured experimentally and listed in the Table 1. At the first glance, virtual Curie temperatures, T _{C} = (1300–1380) K, and T _{Φ} = (1440–1470) K listed in the Table 2, seem noticeably higher than the corresponding transition temperatures T _{FE} = 1100 K and T _{S2} = (1350–1400) K observed experimentally. The temperature T _{Ψ} = 1200 K is lower than the transition one, T _{S1} = 1250 K. Actually, the Curie temperatures should coincide with the transition temperatures only for the single secondorder transition between e.g. nonstructural cubic paraelectric phase with P _{ i } = 0 and pure FE phase with P _{ j } ≠ 0. For the complex sequence of the firstorder structuralphase transitions andpolar phase transitions, observed for the BiFeO_{3} (R3c with nonzero P _{1} = P _{2} = P _{3} ≠ 0 and Φ _{1} = Φ _{2} = Φ _{3} ≠ 0 ⇒ Pbnm phase with nonzero Φ _{1} = Φ _{2} ≠ 0 and Ψ _{3} ≠ 0 ⇒ I4/mcm phase with the only nonzero Φ _{1} ≠ 0 ⇒ Pm3m phase without any longrange order), the virtual Curie temperatures T _{C}, T _{Φ} and T _{Ψ} can be different from corresponding transition temperatures T _{FE}, T _{S1} and T _{S2} due to the 6thorder nonlinear terms of the order parameter, like \(a_{ijk}^{({\rm{P}})}P_i^2P_j^2P_k^2\) or \(a_{ijk}^{(\Phi )}{\bf{\Phi }}_i^2{\bf{\Phi }}_j^2{\bf{\Phi }}_k^2\), as well as due to the transition temperature shifts by the biquadratic (or even bilinear) coupling terms in Eq. 2d.^{32, 51} Furthermore, at the transition temperatures from R3c and Pnma, and from Pnma to I4/mcm, the two involved phases have the same free energy, and thus usually the transition temperatures are lower than the corresponding Curie temperatures. It appears that the shift caused by the 6thorder nonlinearity is typically small in pure and slightly doped BiFeO_{3}, not more than (10–50) K, but the shift caused by the biquadratic coupling \({\rm{\zeta }}_{ijkl}^{}{\bf{\Phi }}_i^{}{\bf{\Phi }}_j^{}P_k^{}P_l^{}\) decreases the virtual Curie temperature T _{C} by more than 150 K towards experimentally observed value T _{FE}. The difference of about (50–100) K between the observed AFD transition temperatures and corresponding Curie temperatures are mostly related to the coupling between the different structural order parameters described by the term \({\rm{\xi }}_{ij}^{}{\bf{\Phi }}_i^2{\bf{\Psi }}_j^2\) in Eq. 2d. Moreover, the coefficients of the nonlinear terms should exhibit temperature dependence in order to have better agreement between the theory and the experiments.
Let us emphasize that an intermediate metastable monoclinic phase with Φ _{1} = Φ _{2} ≠ Φ _{3} and P _{1} = P _{2} ≠ P _{3} can appear between the rhombohedral and orthorhombic phases (e.g., around (1100–1200 K) in addition to the observed phases. The energy of the monoclinic phase decreases with changing coefficients ζ_{44} and ζ_{12}, (at that ζ_{12} becomes larger with ζ_{44} decrease at the fixed value of ζ_{11} +2 ζ_{12} + ζ_{44}). The phase may become stable in BiFeO_{3} slightly doped with Ca or Mn.^{54, 55}
The fitting results, which illustrate how the considered functional fit the experimentally measured temperature dependence of the AFD order parameter and FE polarization in the FEAFD R3c phase of the pure BiFeO_{3} and BiFeO_{3} with 5% of La, are shown in the Fig. 3a, b. Note that the temperature dependence of the dielectric permittivity measured by Kamba et al.^{56}, as well data obtained Lobo et al.^{57}, was taken into account (Fig. S2c, Supplementary Material), when we chose the optimal fitting parameters for polarization temperature dependence in R3c phase. Dashed curves for BiFeO_{3} are calculated without biquadratic AFD–AFM and FE–AFM couplings terms \(\left( {\kappa {\bf{\Phi }}_{}^2 + \lambda P_{}^2} \right)L_{}^2\). Thus AFD–AFM and FE–AFM couplings are sufficient to cause the small fractures of AFD and FE order parameters at Neel temperature, and to increase their saturation rate in the AFM phase, as shown in Fig. 3a, b. However AFDAFM and FEAFM couplings are relatively weak to shift the lowest AFM transition temperature by 645 K towards R3cPbnm transition at 1100 K. The corresponding temperature dependence of BiFeO_{3} AFM order parameter (theoretical fitting and the neutron scattering by Fischer et al.^{19}) is shown in Fig. S2a, Supplementary Material.
Figure 4 illustrates the temperature dependence of oxygen displacement components (Φ _{ i } and Ψ _{ i }) for different phases of BiFeO_{3}. Experimental data of Arnold et al.^{58} and ab initio calculation results of Kornev and Bellaiche^{59} are compared with our fitting (solid curves). Figure 4 shows a pronounced set of the firstorder phase transitions. By analyzing the tilt absolute value, we can conclude that the outofphase tilt vector Φ first appears at the boundary between cubic Pm3m and tetragonal I4/mcm phases in accordance with the firstorder phase transition scenario, and then it mostly rotates at the borders between different structural phases (Fig. S3b, Supplementary Material). Meanwhile another inphase tilt vector Ψ exists in the Pbnm phase only, and disappears at its boundaries in accordance with the firstorder phase transition scenario. The temperature behavior of the outofphase and inphase tilts is similar to that in CaTiO_{3}.
Role of biquadratic couplings
The temperature dependences of the free energies for different phases (tetragonal I4/mcm, orthorhombic Pbnm and rhombohedral R3c, and nonobservable “intermediate” Imbm) are shown in the Fig. 5a. As one can see from the figure, the predicted “intermediate” Imbm phase with Φ _{1} = Φ _{2} ≠ 0 has the lowest energy in a narrow temperature range between 1305 and 1322 K. However, its energy is almost the same as the energy of the predicted by DFT I4/mcm phase. Hence the phase can be indeed (meta) stable in the narrow temperature interval of 17 K width, and we believe that this prediction could be up for debate, similar to the discussion of the intermediate phases separating R3c and Pnma phases in Cadoped BiFeO_{3} ^{46} and pure CaTiO_{3}.^{60} Note that this phase (without changing the sequence of other phases) can be eliminated by simultaneously changing the coefficients \(a_{12}^{(\Phi )}\) and \(a_{112}^{(\Phi )}\) by about 25%, but the changes lead to the visible disagreement between the theoretical curves and measured experimental data, namely in the shift of the phase transition temperature between R3c and Pbnm phases to 1150 K. In this case, although the Imbm phase is thermodynamically stable in the narrow temperature region, it has not been experimentally observed due to the high kinetic energy barriers of the firstorder transition from I4/mcm to Imbm phases.
Hence, we can achieve a quantitative agreement with experimental results for a definite set of values for the expansion coefficients in Eqs. 2a,b,c,d, which satisfy the definite inequalities for the coefficients listed in the Table 2. All the information we extracted about the biquadratic coupling coefficients are listed in the table. Unexpectedly, the temperature dependence of the coupling coefficients \({\rm{\xi }}_{12}^{}\) appears noticeable for pure and lightly doped BiFeO_{3}.
The biquadratic coupling strength is estimated in different phases in Appendix B, Supplementary Materials.
The ΨΦcoupling between inphase and outof phase tilts can be regarded strong enough if the following inequality is valid \(\left( {2a_{11}^{(\Phi )} + a_{12}^{(\Phi )}} \right)a_{11}^{(\Psi )} \le {\left( {{\rm{\xi }}_{12}^{(\Phi \Psi )}} \right)^2}\). The coupling is weak for the opposite sign >> in the inequality. The positive ΨPcoupling is very strong everywhere to prevent the polarization appearance in Pbnm phase, because \(4\left( {a_{11}^{({\rm{P}})}{\rm{ + }}a_{12}^{({\rm{P}})}} \right)\left( {a_{11}^{(\Psi )}{\rm{ + }}a_{12}^{(\Psi )}} \right){\rm{ < < }}{\left( {\eta _{11}^{(\Psi {\rm{P}})}{\rm{ + }}2\eta _{12}^{(\Psi {\rm{P}})}} \right)^2}\) under the required condition \(\eta _{ij}^{(\Psi {\rm{P}})}\)≥10.^{30} Appeared that the ΨΦcoupling is weak only in the vicinity of the PbnmI4 mcm phase transition and becomes relatively noticeable outside it, and strong in R3c phase (see blue area in Fig. 5b). The coupling between the FE polarization and outofphase tilts can be regarded as strong if the following inequality is valid: \(4\left( {a_{11}^{(P)} + a_{12}^{({\rm{P}})}} \right)\left( {a_{11}^{(\Phi )} + a_{12}^{(\Phi )}} \right) \le {\left( {{\rm{\zeta }}_{11}^{(\Phi P)} + 2{\rm{\zeta }}_{12}^{(\Phi P)} + {\rm{\zeta }}_{44}^{(\Phi P)}} \right)^2}\). Conversely, this coupling can be regarded as weak for the opposite sign “>>” of the inequality. It appears that the PΦcoupling is very strong for the entire R3c phase (see Fig. 5c).
Note that a triggertype phase transition is possible only for the sufficiently large negative value of the coupling coefficient.^{28} Since the best fitting corresponds to an unexpectedly strong ΨΦ and PΦbiquadratic couplings, and some of the coupling coefficients are negative (ζ_{11} < 0, ζ_{44} < 0 and ξ_{12} < 0), this result is a strong indication for the triggertype^{28} phase transition between R3c and Pbnm phases, where three components of polarization (P _{1} = P _{2} = P _{3}) and one outofphase tilt component (Φ _{3}) disappear and inphase tilt (Ψ) appear simultaneously. However, the transition between Pbnm and I4/mcm phases, where Ψ and Φ _{2} disappear, is successive rather than the triggertype due to the small coupling value in the temperature range and possible existence of intermediate Imbm phase within a super narrow temperature range. The transition from I4/mcm to Pm3m phase may be of a triggertype, if the transition were of the second order. However the suggestion contradicts to Kornev et al. results for tetragonal phase.^{59}
Substitution of La doping by other dopants
Chemical doping with RE elements leads to the decrease of the phase transitions temperatures and increases the structural stability of these compounds. The obtained results testify close LandauGinzburgDevonshire (LGD) coefficients for both initial and slightly Ladoped BFO compounds [see Table 2]. The result allows us to expand our theoretical approach to the case of other dopants. It is assumed that addition of the small amount “x” (less than 5%) of RE doping affects the potential expansion coefficients of BFO and LGD in a linear way, \(a_i^{({\rm{P}},\Phi ,\Psi )}\left( {x,T} \right){\rm{ = }}a_i^{({\rm{P}},\Phi ,\Psi )}\left( {0,T} \right){\rm{ + }}x\delta a_i^{({\rm{P}},\Phi ,\Psi )}\), due to the joint action of electrostriction and rotostriction couplings, and Vegard strains (other name is chemical pressure)^{20} produced by the dopants and/or vacancies inclusion.^{16,17,18, 61} The Vegard strains δu _{ ij } can be written as the product of Vegard tensor coefficients W _{ ij } and the instant dopant concentration x(r), δu _{ ij }(r) = W _{ ij } x(r). Allowing for electrostriction and rotostriction couplings, the coefficients \(\delta a_i^{({\rm{P}},\Phi ,\Psi )}\) are proportional to the convolution of the electrostriction (Q _{ ijkl }) and rotostriction (R _{ ijkl }) tensor coefficients with the Vegard strains, namely that the corresponding terms are \(x\delta a_i^{({\rm{P}})}{\rm{ = }}{Q_{iikl}}\delta {u_{kl}}({\bf{r}}) \cong {Q_{iikl}}{W_{kl}}x({\bf{r}})\), \(x\delta a_i^{(\Phi )}{\rm{ = }}R_{iikl}^{(\Phi )}\delta {u_{kl}}({\bf{r}}) \cong R_{iikl}^{(\Phi )}{W_{kl}}x({\bf{r}})\) and \(x\delta a_i^{(\Psi )}{\rm{ = }}R_{iikl}^{(\Psi )}\delta {u_{kl}}({\bf{r}}) \cong R_{iikl}^{(\Psi )}{W_{kl}}x({\bf{r}})\). Our approach reconstructs the coefficients \(a_i^{({\rm{P}},\Phi ,\Psi )}\left( {0,T} \right)\), while the coefficients \(\delta a_i^{({\rm{P}},\Phi ,\Psi )}\) can be calculated (for a definite doping) in a straightforward way if the tensors W _{ ij }, Q _{ ijkl } and R _{ ijkl } are known from microscopic calculations or experiments, where not only the amount of RE dopant, but also the degree of nonstoichiometry (e.g., the vacancies concentration) is controlled with high accuracy. The nonstoichiometry control is especially important because it can affect the free energy coefficients via electrostriction and rotostriction coupling with the Vegard strains produced by the vacancies similarly to the doping effect.
Discussion
To resume the theoretical results, we can state that using the experimental temperature dependences obtained in this work, together with other independently available experimental data and ab initio calculations, we develop a LG type thermodynamic potential of pristine and slightly doped bismuth ferrite that describes quantitatively their multiferroic phase diagrams with the AFM (below 645 K) and FEAFD R3c (below 1100 K) phases, different structural AFD phases (orthorhombic Pbnm at 1100 K < T < 1250 K, tetragonal I4/mcm 1250 K < T < 1350 K) and cubic parent Pm3m phase at T > (1350–1400 K).
By the fitting to experiment and ab initio data for the temperature behavior of the sublattice magnetization L, FE polarization P, outofphase and inphase octahedral tilts Φ and Ψ, we determined the full set of 466LG thermodynamic expansion coefficients, some of which reveal a rather strong temperature dependence. As anticipated, the Barretttype temperature dependences appear to be valid for all generalized stiffness α_{ i }(T) determining the “isotropic” quadratic terms like α_{L}(T)L ^{2}, α_{P}(T)P ^{2}, α_{Φ}(T)Φ ^{2} and α _{Ψ}(T)Ψ ^{2}. Unexpectedly, some of the tensorial expansion coefficients of 4th and 6th order structural terms \(a_{ij}^{(\Phi )}{\bf{\Phi }}_i^2{\bf{\Phi }}_j^2\), \(a_{ijk}^{(\Phi )}{\bf{\Phi }}_i^2{\bf{\Phi }}_j^2{\bf{\Phi }}_k^2\), \(a_{ij}^{(\Psi )}{\bf{\Psi }}_i^2{\bf{\Psi }}_j^2\), and \(a_{ijk}^{(\Psi )}{\bf{\Psi }}_i^2{\bf{\Psi }}_j^2{\bf{\Psi }}_k^2\) the coupling coefficients \({\rm{\xi }}_{ij}^{}{\bf{\Phi }}_i^2{\bf{\Psi }}_j^2\) should have specific (e.g., linear or saturating) temperature dependences. Therefore, a conventional LGtheory first developed for primary ferroics with the secondorder phase transitions, that postulated the temperaturedependent coefficient only for the 2nd order coefficients α_{L}(T), α_{P}(T), and α_{Φ}(T), is insufficient for multiferroics with the strong interactions between different order parameters.
Hence we conclude that the AFD part of BiFeO_{3} thermodynamic potential is much more complex than the one in CaTiO_{3} ^{50} that is a primary ferroic. Moreover, the strong biquadratic AFDtype coupling between the AFD, polar and AFM subsystems is critical to the quantitative description of the available experimental data for both pristine and slightly doped with La bismuth ferrite. Keeping in mind that the triggertype phase transitions originate from the same ion contributions, the strong interaction between different order parameters can be expected. Since the best fitting for BiFeO_{3} corresponds to the unexpectedly strong ΨΦ and PΦbiquadratic couplings and some of the coupling coefficients are negative (ζ_{11 }< 0, ζ_{44} < 0 and ξ_{12} < 0), this result is the independent and rather strong support for a triggertype^{28} phase transition between R3c and Pbnm phases.
XRay and NPD measurements determine a full set of the unit cell parameters, bond angles, and lengths, as well as the ionic displacements in pristine and slightly doped with 5% La bismuth ferrite. The structural parameters, FE polarization and AFD octahedral tilt have been extracted from the measured data in the temperature range 300–1150 K. The experimental data and the results of the theoretical modeling are integrated to determine the evolution of the FE and AFD longrange order parameters, thermodynamic potentials and coefficients across the temperaturedriven phase transition into the nonpolar structural phase. The obtained data have detailed the structural phase diagram, focusing on high temperature phases, which could not be observed experimentally because of the decomposition process. In particular, a new structural phase described by space group Imbm has been predicted.
Using the experimental results obtained in this work, together with other independent available experimental data and ab initio calculations, we have developed a LG type thermodynamic potential of pristine and slightly doped bismuth ferrite that describes quantitatively their multiferroic phase diagrams with the AFM and FEAFD R3c phases, different structural AFD phases and cubic parent phase. We have proved a triggertype transition between R3c and Pbnm phases. We are confident that this transition happens without any “virtual” intermediate phase appearance. The wellpronounced and temperatureseparated triggertype transitions are surprisingly uncommon for ferroic AFD FEs. Complementary to the triggertype transition, which are surprisingly uncommon for ferroic AFD FEs, we have predicted new intermediate phase described by Imbm phase between Pbnm and I4/mcm phases. The impossibility of avoiding this phase proves that the transition between Pbnm and I4/mcm cannot be of a triggertype, as regarded previously.
Based on the obtained experimental data, we have extracted the spontaneous polarization of BiFeO_{3} in R3c phase and estimated AFD parameters including AFDtype biquadratic couplings across the phase transition into the orthorhombic phase for the first time. Furthermore, we have proved that “rotomagnetic” AFDAFM coupling is very important to describe the FE polarization and AFD tilt behavior in the R3c phase of BiFeO_{3}. The coupling was neither considered for BiFeO_{3}, nor probed experimentally for other perovskites, while it was predicted for EuTiO_{3}.^{32}
The benefit of the LGtype thermodynamic potential is not only in its capability to describe the observed sequence of the phase transitions along with the temperature behavior of the order parameters, corresponding susceptibilities to external stimuli in the one of the most promising multiferroic, but also it opens the way to model theoretically polar and antiferrodistoritive properties in thin films and nanoparticles of pristine and slightly doped with RE ions bismuth ferrite by incorporating the effect of the interfacial energy and electrostrictive interaction.^{12}
Methods
Ceramic samples of Ladoped BiFeO_{3} with dopant concentrations x = 0, 0.05 were prepared by the twostage solidstate reaction using high purity oxides (Alfa Aesar) as described in refs 40, 62. The oxides Bi_{2}O_{3}, La_{2}O_{3,} and Fe_{2}O_{3} taken in stoichiometric ratio were thoroughly mixed using a planetary ball mill (Retsch PM 200). Pure BiFeO_{3} was synthesized at 870 °C for 10 min. LaDoped compound was prepared at 950 °C for 15 h. The XRD patterns were collected at room temperature using a Rigaku D/MAXB diffractometer (CuK _{ α } radiation) equipped with an Anton Paar heating stage. NPD measurements were performed with the highresolution neutron powder diffractometer FIREPOD (λ = 1.7982 Å, E9 instrument, HZB). Diffraction data were analyzed by the Rietveld method using the FullProf software package.^{63} Thermal analysis was carried out using differential scanning calorimeter Mettler Toledo 822e in argon atmosphere.
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Acknowledgements
The work of D.V.K., M.V.S., S.A.G. and I.O.T. was supported by the Russian Science Foundation (project # 151920038). Neutron diffraction experiments have been supported by the European Commission under the 7th Framework Programme through the ‘Research Infrastructure’ action of the ‘Capacities’ Programme, NMI3II Grant number 283883’. The work at Penn State is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award FG0207ER46417 (FX and LQC) and by the NSF MRSEC under Grant No. DMR1420620 (F.X. and V.G.). A.N.M. and E.A.E. acknowledge the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility, CNMS2016061.
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D.V.K., I.O.T., A.F. and M.V.S. conducted structural measurements and wrote the experimental part of the paper. E.A.E. and F.X. performed theoretical calculations, supervised by M.D.G., V.G. and L.Q.C., I.O.T. and S.A.G. prepared the samples. A.N.M. proposed theoretical research ideas and formulated the problem, wrote the introductive, theoretical and discussion parts of the paper. All coauthors densely worked on the results analyses and text improvements.
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Karpinsky, D.V., Eliseev, E.A., Xue, F. et al. Thermodynamic potential and phase diagram for multiferroic bismuth ferrite (BiFeO_{ 3 }). npj Comput Mater 3, 20 (2017). https://doi.org/10.1038/s4152401700213
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