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Article

Understanding Complex Interplay among Different Instabilities in Multiferroic BiMn7O12 Using 57Fe Probe Mössbauer Spectroscopy

1
Department of Chemistry, Lomonosov Moscow State University, Moscow 119991, Russia
2
Department of Chemistry, MSU-BIT University, Shenzhen 517182, China
3
Kurnakov Institute of General and Inorganic Chemistry of Russian Academy of Sciences (RAS), Moscow 119991, Russia
4
Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Ibaraki, Japan
*
Author to whom correspondence should be addressed.
Int. J. Mol. Sci. 2024, 25(3), 1437; https://doi.org/10.3390/ijms25031437
Submission received: 4 January 2024 / Revised: 20 January 2024 / Accepted: 21 January 2024 / Published: 24 January 2024
(This article belongs to the Special Issue Physical Inorganic Chemistry in 2024)

Abstract

:
Here, we report the results of a Mössbauer study on hyperfine electrical and magnetic interactions in quadruple perovskite BiMn7O12 doped with 57Fe probes. Measurements were performed in the temperature range of 10 K < T < 670 K, wherein BiMn6.9657Fe0.04O12 undergoes a cascade of structural (T1 ≈ 590 K, T2 ≈ 442 K, and T3 ≈ 240 K) and magnetic (TN1 ≈ 57 K, TN2 ≈ 50 K, and TN3 ≈ 24 K) phase transitions. The analysis of the electric field gradient (EFG) parameters, including the dipole contribution from Bi3+ ions, confirmed the presence of the local dipole moments pBi, which are randomly oriented in the paraelectric cubic phase (T > T1). The unusual behavior of the parameters of hyperfine interactions between T1 and T2 was attributed to the dynamic Jahn–Teller effect that leads to the softening of the orbital mode of Mn3+ ions. The parameters of the hyperfine interactions of 57Fe in the phases with non-zero spontaneous electrical polarization (Ps), including the P1 ↔ Im transition at T3, were analyzed. On the basis of the structural data and the quadrupole splitting Δ(T) derived from the 57Fe Mössbauer spectra, the algorithm, based on the Born effective charge model, is proposed to describe Ps(T) dependence. The Ps(T) dependence around the ImI2/m phase transition at T2 is analyzed using the effective field approach. Possible reasons for the complex relaxation behavior of the spectra in the magnetically ordered states (T < TN1) are also discussed.

1. Introduction

The variety of structural and magnetic phase transitions in the so-called quadruple perovskite BiMn7O12 [1,2,3,4] and its solid solutions, such as BiMn7−xCuxO12 (0 < x ≤ 1.1) [5,6], has attracted strong interest from researchers. Numerous transitions of different origins are associated with the presence of Mn3+ and Bi3+ cations in the crystal lattice of these oxides, which promotes structural instability [7,8,9,10]. High-spin Jahn–Teller (JT) cations Mn3+(d4) in the non-distorted octahedral oxygen surrounding possess an energetically degenerate configuration, eg1, which provokes, along with a local distortion of polyhedra (MnO6), a cooperative interaction in which the JT centers Mn3+ proper, often called orbital ordering [7,8,11,12]. The distortion driven by easily polarized Bi3+ cations, containing the 6s2 lone electron pair, results in off-centric cation displacements and the formation of local electric dipoles, which are responsible for the ferroelectric properties of many Bi-containing perovskites [7,10].
Non-zero magnetization (M) co-existing with electrical polarization (Ps) is characteristic of multiferroics, which can be grouped into two types [13]. In the first group (type-I multiferroics), M and Ps are independent of each other, i.e., magnetism and ferroelectricity have different origins. In the second group (type-II multiferroics), M and Ps demonstrate a strong mutual influence, i.e., magnetism induces ferroelectricity [13]. Such a magneto-electric coupling strongly correlates with local distortions, and, thus, the roles of local polar and magnetic clusters must be studied. Therefore, these compounds are increasingly studied not only by using diffraction and magnetic methods but also using local nuclear resonance techniques, such as NMR [14,15,16,17,18,19,20,21], NQR [22,23], muon spectroscopy [24,25,26], the spectroscopy of perturbed angular γ–γ correlations [27,28], and Mössbauer spectroscopy [9,29,30]. The temperature dependences of hyperfine magnetic fields (Bhf) and principal components {Vii}X,Y,Z of the tensor of the electrical field gradient (EFG) gained using these methods reproduce the corresponding dependences M(T) and Ps(T). Meanwhile, the relationship Bhf = αM is usually linear for 57Fe, 55Mn, and 53Cr nuclei, and the dependencies of the EFG tensor parameters Vii = f(Ps) and η = f′(Ps) (where η = (VXXVYY)/VZZ is the asymmetry parameter) demonstrate nontrivial behavior. In some works, quadratic dependences Vii = a + bPs2 and ηPs2 are used for approximation [31,32,33]. However, such an approach is rather formal and does not allow for associating the hyperfine parameters of different resonant nuclei with the structural data and physical characteristics of the compounds under study.
In the present work, we present Mössbauer-based research on the hyperfine interactions of 57Fe probes in perovskite BiMn7O12. This manganite demonstrates spontaneous electrical polarization in the temperature range of T < TC ≈ 440 K, whereas, at lower temperatures (T < TN1 ≈ 59 K), it acquires a magnetically ordered state and multiferroic properties [2]. In contrast to perovskites ABO3, in their “quadruple” analogs (AA3)B4O12, the sublattice A is divided into two sublattices formed by cations A′ with a high coordination number (CN = 8–12) and by using JT cations A″ = Cu2+ and Mn3+, which are located in the square planar oxygen coordination [34]. In the case of (Bi3+Mn3+3)[Mn3+4]O12, these sublattices consist of the cations A′ = Bi3+ and A″ = Mn3+, whereas the sublattice with a distorted octahedral oxygen coordination (B) consists of the JT cations Mn3+ that directly initiate orbital ordering (cooperative JT effect) [8]. A combined effect of two cations (Mn3+)B and (Bi3+)A, of which the latter contains the stereochemically active lone-pair electrons [7,9,10], results in a whole cascade of structural and magnetic phase transitions in BiMn7O12 (Figure 1) [3]. However, the mechanisms and driving forces of these phase transitions are still widely discussed despite abundant experimental data and theoretical studies [35].
According to the results of the earlier Mössbauer studies of perovskites AMn6.9657Fe0.04O12 (A = Ca, Sr, Cd, Pb) [35,36,37], the 57Fe probes are localized in the structure solely in the formal oxidation state “3+”, substituting manganese cations only in octahedral sublattice. Moreover, experimental and theoretical studies show that the hyperfine parameters of the 57Fe spectra reflect the features of the local crystal structure of this class of manganites. It is essential to highlight that the studies involving macroscopic diagnostic methods found the influence of the 57Fe probes on physical parameters to be insignificant, as well as the patterns of the structural and magnetic transitions in these oxides. Thus, utilizing Mössbauer spectroscopy to probe a more complicated BiMn7O12 system is justified by the current experimental data, and the successful application of this technique is used to study other isostructural compounds of the AMn7O12 family.
Our work aims to qualitatively obtain new information about the local structure of BiMn7O12 and outline the features of the structural, electrical, and magnetic phase transitions. We describe a close interplay between the orbital and spin degrees of freedom, which is characteristic of systems with a strong electron correlation.
The Results and Discussion section of the manuscript is divided into several parts. The first part is devoted to analyzing the effect of 57Fe probes on structural (T1, T2, and T3) and magnetic (TN1 and TN2) transitions in BiMn7O12 (Figure 1). Based on theoretical calculations of the EFG parameters in the paraelectric range T > T1, the second part discusses the crystal chemistry of Bi3+ cations and their influence on the transition of the bismuth sublattice into the ordered ferroelectric state. In the third part, we consider the dynamic Jahn–Teller effect of Mn3+ cations in octahedral coordination at intermediate temperatures T2 < T < T1. The fourth part presents an analysis of the temperature dependence of the spontaneous polarization in BiMn6.94Fe0.04O12 at temperatures TN1 < T < T3 and T3 < T < T2. The final part describes the hyperfine magnetic interactions of 57Fe probes in the magnetically ordered temperature range T < TN1.

2. Results and Discussion

2.1. Crystallographic, Magnetic, and Thermodynamic Data

X-ray diffraction patterns show no additional reflections corresponding to impurity phases (Figure S1). Having been measured at different temperatures, they suggest that the studied sample retains all the crystal modifications characteristic of an undoped (without Fe) quadruple manganite BiMn7O12 [3]. The observed reflections at 615 K are associated with the cubic ( Im 3 ¯ ) BiMn7O12 phase that is stable at T > T1 (Figure 1a). A part of the reflections split upon transition below T1, which corresponds to the monoclinic symmetry I2/m (Figure 1b). The monoclinic phase reflection (242) splits upon the further cooling of the sample (Figure 1c). As noted in [3], the temperature T3 of the phase transition monoclinic (Im) ↔ triclinic (P1) (Figure 1d) was not detected in the thermodynamic measurements; however, it can be evaluated from the deviation of the α and γ monoclinic unit cell angles from 90° (Figure S2). The estimated point T3 ≈ 240 K for BiMn6.96Fe0.04O12 is noticeably lower than ~290 K for the undoped BiMn7O12 sample [3].
The BiMn6.96Fe0.04O12 powder is characterized by a high degree of crystallinity according to the SEM data (Figure S3). It is inferred from particle agglomeration and the wide distribution of particle sizes ranging from 0.5 to 20 μm. Almost all crystallites have an irregular shape.
The peaks observed in the differential scanning calorimetry (DSC) curves of BiMn6.96Fe0.04O12 (Figure 2a) correspond to the phase transitions at the temperatures T1 ≈ 580–590 K and T2 ≈ 420–440 K, i.e., structural transitions I2/m Im 3 ¯ and ImI2/m, respectively, according to the literature data [3]. It is worth noting that the undoped sample BiMn7O12 demonstrated the same transitions at ~ 608 K and ~ 460 K, respectively, as was reported in the earlier experiments [3]. Further lowering the lattice symmetry to P1 does not affect DSC curves. Measurements upon cooling and heating reveal a difference in the transition points ΔT1 ~ 7 K and ΔT2 ~ 20 K, both of which slightly exceed the corresponding values for the BiMn7O12 sample [3].
By measuring the heat capacity CP/T(T) in BiMn6.96Fe0.04O12 (Figure 2b), we obtained the values of the temperatures of transition to the magnetically ordered states TN2 and TN3. TN2 reached ~ 50 K, and the temperature of the third magnetic transition TN3 ≈ 24 K was 3–5 K lower than that of the undoped manganite [4]. The third phase transition at TN3 is also clearly seen in the temperature profile of the magnetic susceptibility χ(T) and is typical of antiferromagnets (Figure 2c). The parameters of the Curie–Weiss fit (Figure 2c) are in good agreement with the data obtained for the undoped manganite BiMn7O12 [3]. The temperature shift of structural and magnetic transitions upon iron-doping can be caused by the stabilization of a small quantity of iron in the manganite structure, rather than the precipitation of an impurity iron phase or its localization on the crystallite surface.

2.2. Mössbauer Data for T > T1

Figure 3a represents the typical Mössbauer spectra of 57Fe in BiMn6.96Fe0.04O12 measured at high temperatures T > T1. At these temperatures, the spectra consist of a quadrupole doublet with small and virtually temperature-independent splitting Δ ≈ 0.26 mm/s (Figure 4). The value of the isomer shift δ633K ≈ 0.16 mm/s corresponds to Fe3+ cations [38], isovalently substituting Mn3+ in the octahedral positions of Mn2 (Figure 1a). Despite BiMn6.96Fe0.04O12 having a cubic structure ( Im 3 ¯ ) at T > T1 and the local octahedral anion environment of the Mn2 sites being formally considered to be undistorted, the local symmetry of the oxygen sites explains the non-zero quadrupole splitting of the spectrum (Table 1). Although all the Mn2 sites substituted by Fe3+ probes are equivalent, the experimental spectra cannot be satisfactorily fitted using one doublet with unbroadened resonant lines. This suggests the presence of a distribution p(Δ) of Δ values (Figure 3a), i.e., that the crystalline environment of the 57Fe probes is not homogenous.
The EFG parameters were calculated within the “ionic model” to support this assumption. It takes into account monopole ( V Z Z mon ) and dipole ( V Z Z dip ) contributions from ions that are located in the non-centrosymmetric sites in BiMn7O12. Having stereochemically active 6s2 lone-pair electrons, Bi3+ cations are considered to mainly contribute to V Z Z dip . The lone pair induces the displacement of Bi3+ cations from their centrosymmetric positions, which is equivalent to inducing the electric dipole moment pBi. Therefore, only dipole contributions ( V Z Z , B i dip ) from Bi3+ were taken into account when calculating V Z Z dip using variable pBi values in further calculations. Additionally, the dipole moments pBi were considered to be randomly oriented in a crystal lattice since BiMn7O12 is paraelectric at T > T1 [3]. See Appendix A for details.
The inclusion of the dipole contribution from Bi3+ allows us to achieve a good agreement between the theoretical and experimental values of the quadrupole splitting. The calculated dipole moment pBi ≈ 1.2 × 10−29 C·m lays within the range of the corresponding values pBi for other Bi3+ oxide compounds [10]. Most importantly, even with a random orientation of the pBi moments, the Mn2 sites become non-equivalent in terms of the induced lattice contribution V Z Z , Bi dip . This is, in essence, the main cause of the observed broadening of the spectra, i.e., the appearance of the p(Δ) distribution (Figure 3a). Using the calculated values of Δtheor for each Mn2 site within the P1 pseudocell (see Appendix A, Appendix B and Appendix C), which is characterized by the peculiar relative orientation of the surrounding dipole moments pBi, we calculated the mean value of the quadrupole splitting as well as the dispersion D p t h e o r = 0.020 mm2/s2, which was found to be close to D p exp ≈ 0.017 mm2/s2 of the experimental (Table 1) distribution p(Δ).
Thus, the Mössbauer data indicate that, in the paraelectric cubic phase of BiMn6.96Fe0.04O12 at T > T1, Bi3+ cations exist in a locally distorted environment and retain their electric dipole moments pBi that are randomly oriented in the cubic lattice. In this case, transitions to the anti- or ferroelectric states at lower temperatures T < T2 should be accompanied by the ordering of the pBi dipoles, i.e., they should represent the “order-disorder” phase transitions [39], as an alternative to the “displacive” phase transitions [40]. Previously, in refs. [41,42], static dipole moments pBi are assigned to the lone spx-hybrid electron pairs of Bi3+ cations, which are oriented parallel to the direction of the displacement of a bismuth cation from its conventional centrosymmetric site (Figure 5a). Such an approach can qualitatively explain the unusually large thermal ellipsoids of Bi3+ cations reported in [1,3] for BiMn7O12 at T > T1. These ellipsoids may form as a result of the superposition of multidirectional spx-hybrid pairs, whose randomly oriented lobes create a sphere that manifests itself in the diffraction patterns as unusually large bismuth thermal ellipsoids (Figure 5b). However, it should be noted that this approach is a simplified, albeit illustrative, model that has not been experimentally confirmed for the majority of known Bi(III) phases [43,44,45].

2.3. Mössbauer Data for T2 < T < T1

BiMn6.9657Fe0.04O12 undergoes the structural transition at T1 ≈ 590 K, transforming from cubic ( I m 3 ¯ ) to monoclinic (I2/m) lattice symmetry (Figure 1b) with decreasing temperature. Figure 3b illustrates a typical Mössbauer spectrum of 57Fe probes in the monoclinic BiMn7O12, which has the shape of a symmetrically broadened quadrupole doublet. Despite the lowering of the manganite lattice symmetry, the obtained distributions p(Δ) show a single maximum that corresponds to the average <Δ> value which increases drastically upon decreasing temperature (Figure 3 and Figure 4). Considering the fact that the main contribution to the EFG imposed on the spherical Fe3+ cations is accounted for by the distortion of their crystalline surrounding (lattice contribution), it is difficult to explain the observed sharp change in the quadrupole splitting with temperature.
The results of the calculation of the EFG parameters for the different sites of Mn3+ with the monopole contributions from all ions (Bi3+, Mn3+, and O2−), as well as additional dipole contributions from Bi3+ and O2−, show that the values VZZ,Mn4 = 3.76 × 1020 V/m2 and VZZ,Mn5 = 4.21 × 1020 V/m2 are close to each other, which is probably responsible for the presence of only one maximum in p(Δ) (Figure 3b). As expected, the EFG parameters for both Mn sites are almost independent of temperature. Moreover, it was established that the calculated values ΔtheorMnieQ V Z Z , Mni T e o p for sites Mn4 and Mn5 (where Q is the quadrupole moment of 57Fe nuclei) remarkably exceed the corresponding experimental values Δexp (Figure 4; Tables S2–S5).
We suppose that the abovementioned discrepancy between the calculated and experimental values of the quadrupole splitting (Δtheor > Δexp) and their unusually strong temperature dependences can be attributed to the dynamic JT effect of Mn3+ cations occurring in this temperature range [46]. The JT interactions of the Mn3+ cations in BiMn7O12 can result in the so-called orbital ordering, or cooperative JT effect, which is also observed in other perovskite-like Mn(III) oxide systems, namely, RMnO3 [47,48], R1−xAxMnO3 [11], and AMn7O12 [37,49] (R = REE, A = Ca, Sr, Pb). All these systems can exhibit a structural transition to a crystal lattice with enhanced symmetry in the temperature range T > TJT, which is ascribed to the dynamic JT effect, or the “melting” of the cooperative JT distortion [46]. Similar phase transitions can occur through two mechanisms: one involves increasing the symmetry of distorted (Mn3+O6) polyhedra until the uniform population of Mn3+ eg-orbitals is achieved, and the other entails the orientational disordering of distorted (Mn3+O6) polyhedra while preserving the polarization of eg-orbitals even at high temperatures (T >> TJT) [50]. As was noted in several refs. [51,52,53], the local disordering of (Mn3+O6) polyhedra can start at a temperature (T*) that is significantly lower than the temperature of the structural phase transition TJT (>>T*). However, there is still no reliable experimental data on the changes in the structure and electronic state of manganite, which take place in this “intermediate” temperature range.
Local minima are known to appear on the adiabatic potential surface of possible nuclear configurations of O2− anions in the (MnO6) polyhedra if the anharmonicity of vibronic interactions is taken into account. These minima reflect the specific orthorhombic distortions of the corresponding polyhedra. With increasing temperature, the crystalline environment of the JT Mn3+cations stochastically relaxes between these minima due to thermally activated excitations or the tunnel effect [54]. Although the Fe3+ cations per se do not participate in the vibronic interactions, their local crystal environment also fluctuates dynamically due to the cooperative JT effect. Therefore, we suggest that the observed significant reduction in the Δexp values compared to the theoretical calculations can be associated with the relaxation behavior of the Mössbauer spectra in the temperature range T2 < T < T1. In [54,55], it was shown that such spectra can be described with the “two-level model” in the limit of “fast relaxation”, i.e., when ΩR >> Ω0, where ΩR and Ω0 are the frequencies of the relaxation of the oxygen environment and the precession of the 57Fe quadrupole moment around the VZZ direction, respectively. The model adopts the frequencies of forward (Ω12) and reverse (Ω21) transitions between states “1” and “2” as variables connected by the detailed equilibrium principle n1Ω12 = n2Ω21, where n1 and n2 are the populations of states (Figure 6a) [55].
In the monoclinic structure (I2/m) of BiMn7O12, the distortion of (MnO6) polyhedra corresponding to the energy minimum E1 of the adiabatic potential, is described as a “bonding” Q(−)—the linear combination of the orthorhombic (Q2) and tetragonal (Q3) vibrational modes [7,56]. In this case, the distortion with a higher energy E2 is attributed to the “antibonding” vibration mode Q(+). In a local approximation, when only the closest anion environment is considered, the two vibrational modes—Q(−) and Q(+)—correspond to the distortions that exhibit equal magnitudes but opposite signs in their EFG components (VZZ) imposed on the 57Fe nuclei occupying the Mn4 and Mn5 sites [54]. Consequently, when the population of the E1 and E2 levels equalizes with increasing temperature, quadrupole splitting sharply decreases, i.e., Δ(T) ∝ <VZZ>, where <VZZ> is averaged over the energy states E1 and E2 [54]. On the other hand, the monotonous decrease in Δ(T) up to T1 can suggest a gradual enhancement of the (FeO6) symmetry upon approaching the temperature of the structural phase transition I2/m Im 3 ¯ . This conclusion is consistent with the synchrotron X-ray diffraction studies of BiMn7O12, which also demonstrate the gradual decrease in the distortion parameter Δd of (MnO6) polyhedra when TT1 [3]. Thus, these parameters behave similarly when assessed by inherently different characterization techniques. These data suggest the second-order JT phase transition that can be referred to as the displacive structural transition, in contrast to the “order-disorder” transition mechanism.
The fitting of the whole series of spectra within the framework of a two-level model allowed us to estimate the average relaxation frequency ΩR = Ω12Ω21/(Ω12 + Ω21) ≈ (2–7) × 107 Hz, which significantly exceeds the frequency of the quadrupole precession Ω0 ≈ 8.5 × 106 Hz. Increasing temperature leads to a gradual equalization of the populations n1 and n2. This should result in a sharp decrease in the quadrupole splitting and a slight broadening of the doublet components in the limit of fast relaxation ΩR >> Ω0. Indeed, this pattern describes the temperature-related changes of all the spectra at T2 < T < T1 (Figure S4). Using the linear approximation of ln(n1/n2) = f(1/T) (obtained from the Arrhenius equation):
Ω 12 ( 21 ) ( T ) = Ω 1 ( 2 ) 0 exp E 12 ( 21 ) k B T ,
where Ω 1 ( 2 ) 0 are temperature-independent parameters; E12(21) are the activation energies of “1” and “2” states, respectively; kB is the Boltzmann constant) we evaluated the energy difference between the relaxed states ΔE = 69(2) meV and the mean value of the activation energy Eact = 220(9) meV (Figure 6a and Figure S5) that closely corresponds to <Eact> for other perovskite-like Mn(III) manganites [57]. However, a deviation from linearity is observed at higher temperatures (T > 550 K) (Figure 6b). This is likely to result from the changes in the relative position of levels E1 and E2 between which the relaxation occurs. This explanation is indirectly supported by a similar temperature profile of the distortion parameters Δd of polyhedra (MnO6) (Figure 6b), which govern the splitting of the 3d levels of Mn cations under the influence of the ligand field.
It is worth noting that the observed structural changes in BiMn7O12 in the temperature range of the JT transition are similar to those in the isostructural phase of LaMn7O12 [58] but differ crucially from the so-called “traditional” perovskites RMnO3 (R = REE), in which the cooperative JT effect follows the “order-disorder” mechanism [39]. In these oxides, polyhedra (Mn3+O6) remain distorted even at temperatures significantly exceeding TJT. However, these distortions are randomly oriented in the crystal lattice, thus making the structure “macroscopically” more symmetrical compared to the low-temperature phase with orbital ordering.

2.4. Mössbauer Data for Temperature Ranges T3 < T < T2 and TN1 < T < T3

In the temperature range T3 < T < T2, the Mössbauer spectra of BiMn6.96Fe0.04O12 consist of a broadened quadrupole doublet (Figure 3c). The observed bimodal profile of p(Δ) (Figure 3c) suggests the stabilization of 57Fe nuclei at two different crystallographic sites of the manganite. This conclusion agrees with the earlier structural data for BiMn7O12 [3]: in the crystal lattice with the Im symmetry, the JT Mn3+ cations occupy two sites, Mn1 and Mn2, in the very distorted octahedral oxygen environment (Figure 1c). Thus, the high value <Δ> ≈ 0.62 mm/s observed in the spectra of Fe3+ probe cations is conditioned by a low symmetry of the oxygen environment of the JT Mn3+ cations. Based on the p(Δ) distribution, we fitted the whole series of spectra measured in the range T3 < T < T2 as a superposition of two quadrupole doublets, Fe(1) and Fe(2), having close values of isomer shifts, δ1δ2 (Figure 3c).
The analysis above yielded anomalously sharp temperature dependences Δi(T) for both Fe(1) and Fe(2) doublets (Figure 4). Such behavior can stem from the induction of spontaneous BiMn7O12 polarization at T < T2 [3]. To support this assumption quantitatively, we obtained an expression relating the lattice contributions V Z Z lat with the values and mutual orientation of electric moments (pk) in the lattice of BiMn7O12 (for details, see SI). The values pk and their projections pik were calculated using the Born model [59]: pk = ZkΔrk or pik = ZkΔxik, where Zk is the Born charge of the kth ion, which is an isotropic scalar value in our calculation; Δrkxik) is the vector of the displacement of the kth ion (and ith projection) from the centrosymmetric position. The values Δrkk and Δxik were calculated using the crystallographic data for BiMn7O12 obtained at 300 K [3]. To estimate Born charges, we sequentially varied the charges {ZBi, ZMn(i), ZO(i)} and their corresponding dipole moments pik with the given displacement Δxik until the best agreement with the experimental splitting Δi was achieved (Table S1). The values approximated in such a manner that ZBi = +3.30, ZMn1ZMn2 = +3.30, and ZO = −2.20, and all lie within the range of the Born charges obtained earlier for the corresponding ions in other perovskite oxides [60].
Using the above approximations, we derived an equation that describes Δ(T) as a function of the ith projections P i i = x , y , z = k p i k of the spontaneous polarization Ps = (Px2 + Py2 + Pz2)1/2:
Δ ( T ) = ( 1 γ ) eQ 2 k 2 x i 2 q k r k + r i k P i ( T ) r k 3 p k ( T 0 ) P s ( T 0 )
where pk(T0)/Ps(T0) is the ratio of the dipole moment of the kth ion (pk) to the spontaneous polarization in the crystal, which is calculated based on the crystallographic data for BiMn7O12 (Im). The first and the second terms in Equation (1) are the monopole and dipole contributions to the EFG. Using Equation (1), we plotted theoretical dependencies Δ1(Ps) and Δ2(Ps) (Figure 7). Using the numerical solution of the equations Δ1(Ps) = Δ1(T) and Δ2(Ps) = Δ2(T) and accounting for statistical errors allowed us to simulate the temperature dependence Ps(T) (Figure 8).
The same algorithm for constructing dependencies Ps(T) using experimental data Δi(T) was applied to the triclinic phase (P1) of BiMn7O12. The distribution p(Δ) has a trimodal profile for the given structure modification (Figure 3d), which indicates that Fe3+ cations occupy at least three nonequivalent sites. According to the structural data [4], the Mn3+ cations form four equally populated sites (Mn4, Mn5, Mn6, and Mn7) in an octahedral oxygen environment (Figure 1d). The calculated EFG parameters of Mn4 and Mn6 suggest that these atoms are located symmetrically near similar crystalline environments, making them indistinguishable in the 57Fe Mossbauer spectra. Therefore, the spectra at TN1TT3 were fitted with three quadrupole doublets, namely, Fe(1), Fe(2), and Fe(3), with the Fe(3) component being two times more intense than Fe(1) and Fe(2) (Figure 3d). Using the structural data for the triclinic BiMn7O12 phase at 10 K [4] and the described algorithm combining the theoretical dependencies Δi(Ps) with the experimental Δi(T) values, we were able to model Ps(T) across the temperature range under investigation (Figure 8).
The dependences Δ ∝ VZZ = f(Ps) (Figure 7) agree with the results of [24,25,26,27], where, in a general case, the dependence of VZZ(Ps) at low Ps values was represented using a Taylor series expansion:
V zz ( P s ) = V zz ( 0 ) + i V zz x i P s + 1 2 i , j 2 V zz x i x j P s 2 + = V zz ( 0 ) + α P s + β P s 2 + γ P s 4
It was shown in [12] that the linear term vanishes and the quadratic term in the expansion becomes significant (β >> γ ≠ 0) for the centrosymmetrical crystal sites. At the same time, if the center of symmetry is absent, the linear term should be predominant (α >> β >> γ ≠ 0). Since for both polymorphic modifications of the BiMn7O12 octahedral Mn3+ sites are not centrosymmetric, the above dependences VZZ(Ps) can be described using an expansion in a series with nonzero parameters α and β, whose values (Table 2), in order of magnitude, are consistent with similar data from other perovskite-like ferroelectrics [24,25,26,27].
When describing the dependences, Ps(T) was obtained for two temperature ranges, and we used the model of the average effective field [61]. Within the framework of this approach, it is assumed that every ion in the ferroelectric crystal is affected by an effective electric field (Eeff), which can be expressed as
E eff = E 0 + β P s + γ P s 3 + δ P s 5 +
where E0 is the external electrical field, and the following terms correspond to the dipolar, quadrupolar, and octupolar interactions, respectively. In our calculations, E0 = 0. Moreover, only dipolar and quadrupolar interactions were taken into consideration, the latter (γ) serving to describe the phase transitions of both the first and second orders within the united approach. For statistical consideration, the Ps(T) dependence of polarization can have a general form [61]:
P s ( T ) = P 0 tanh E eff P 0 N 1 k B T = P 0 tanh ( β P s + γ P s 3 ) P 0 N 1 k B T
where P0 is the spontaneous saturation polarization, and N is the number of elementary dipoles per unit cell. Using the relationship kBTC = β·P02/N (where TC is the Curie ferroelectric point) and designations of the normalized values σsPs/P0, τT/TC, gγP02/β, one can derive an expression that is convenient for analyzing the experimental data:
σ s ( T ) = tanh σ s ( T )   +   g σ s 3 ( T ) τ
where the parameter g is a quantitative criterion of the order of the transition to the ferroelectric state [61]. The analysis of the experimental dependences Ps(T) using Expression (5) is shown in Figure 8.
The value g ≈ 0.48(3) in the temperature range T3 < T < T2 indicates a significant contribution of the quadrupolar interactions (γ ≠ 0) in Expression (3), which, in turn, is a feature of the first-order transitions [61,62]. A similar behavior was observed for many oxide systems, in particular, those demonstrating multiferroic properties [63]. The transition point TC ≈ 437 K of the ferroelectric state, having been evaluated within the framework of this approach based on the description of Ps(T), insignificantly differs from the temperature of the structural transition ImI2/m (T2 ≈ 442 K) determined for the sample BiMn6.96Fe0.04O12 from the thermodynamic data (Figure 2a). We suggest that the observed first-order transition at T2 stems from an inevitable coupling between the electric polarization and the crystal lattice. There is a simultaneous structural change accompanying this transition as evidenced by the above finding. By using a simple phenomenological approach (see Appendix D), it can be shown that the coupling between spontaneous polarization (Ps) and strain (ε) can switch an otherwise second-order transition to a first-order transition. Moreover, there is a relationship between the strength of the ferroelastic coupling and the size of the hysteresis ~20 K (Figure 2a) of the resultant first-order transition. It is known that the size of hysteresis is determined by the energy barrier at TC (Figure S4), which is largely dependent on the magnitude of the ferroelastic coupling coefficient λ in the fourth-order term of the Landau free energy (see Equation A8). On the other hand, a larger λ also leads to a larger spontaneous lattice distortion upon the first-order phase transition. Therefore, the magnitude of thermal hysteresis increases with an increase in lattice distortion.
In the range TN1TT3, the dependence Ps(T) demonstrates a kink at the point T3, and its course follows Expression (4) with the parameter g ≈ 0, which, upon extrapolation to ~294 K (see Figure 8), should correspond to a “gradual” second-order phase transition [61]. A discussion of the nontrivial course of the dependencies Ps(T) is beyond the scope of our work; however, it may motivate someone to study this unusual system using new independent local and macroscopic methods and theoretical approaches.

2.5. Mössbauer Study in the Temperature Range T < TN1

The quadrupole doublets at temperatures slightly below the Néel point (T < TN1 ≈ 50 K) contain the broadened components that reflect hyperfine magnetic fields Bhf induced on the 57Fe nuclei (Figure 9a). The spectra were fitted via a reconstruction of distributions p(Bhf) characterized by a certain dispersion DP(δ) at a given temperature. The kink on the temperature dependence DP(δ) = f(T) (Figure 9b; Table S6 for data) corresponds to the temperature 57(3) K, which, within the measurement error, coincides with TN1 ≈ 59 K for undoped manganite BiMn7O12 (Figure 2b). This result gives independent confirmation of the stabilization of 57Fe probes in the lattice of the BiMn7O12 manganite under study.
At low temperatures, T << TN1, the Mössbauer spectrum of BiMn6.9657Fe0.04O12 has an asymmetric and slightly broadened Zeeman structure (Figure 10), which can be described by a superposition of four unequally broadened sextets in accordance with the structural data for the triclinic phase BiMn7O12 [4]. With increasing temperature, the profiles of these sextets change noticeably, which is characteristic of the system exhibiting relaxation processes. Earlier studies showed [64] that this behavior could be attributed to the magnetic excitation of paramagnetic impurity ions within magnetically ordered matrices, where competing exchange interactions play a significant role. When embedded within the manganite matrix, impurity cations Fe3+ with half-filled orbitals are surrounded by the JT Mn3+ cations with anisotropic orbital occupation. This can lead to a noticeable weakening of the exchange interactions between the impurity cations and their magnetic environment. Essentially, this suggests that iron cations can undergo lower-energy magnetic excitations, influencing also the neighboring Mn3+ cations rather than only Fe3+ cations. An increase in temperature leads to the progressive occupation of magnetic Fe3+ states characterized by different projections (SZ) of the total spin S = 5/2. As the magnetic interaction of iron with its surroundings is weakened, the relaxation of spin between the states |5/2, SZ> decelerates. If the relaxation period τR closely corresponds to the period of Larmor precession (τL) of the 57Fe nuclear spin around the hyperfine field Bhf, the Mössbauer spectra typically have complex relaxation profiles [65].
Additionally, it should be noted that the SEM data indicates that an average particle size exceeded ~2 µm for the BiMn6.96Fe0.04O12 sample (Figure S3). Consequently, the observed relaxation behavior of the Mössbauer spectra cannot be attributed to the superparamagnetic or superferromagnetic states of small particles [65,66].
Taking into account the considerations presented above, the spectra were fitted using a multilevel relaxation model [67]. This model is based on the assumption that in the effective magnetic Weiss field, spin S = 5/2 of the Fe3+ cation, stochastically relaxes between Zeeman states |5/2, SZ> [67]. Along with the static parameters (δ, eQVZZ and Bhf), the model includes variable relaxation parameters, namely, the relaxation frequency ΩR (=1/τR) and the relative population (s) of the Zeeman sublevels between which the relaxation occurs. A detailed description of this model can be found in [67]. This allowed us to process the whole series of spectra measured in the temperature range 10 K < T < TN1. It should be noted that the static and relaxation parameters of the Mössbauer spectra remain virtually unchanged, which indicates the stability of a complex model of spectrum processing. A smooth and continuous change in the hyperfine magnetic field near TN1 indicates the occurrence of a second-order magnetic phase transition. This conclusion is consistent with the results of a theoretical study of undoped manganite BiMn7O12, according to which the transition at TN1 = 59 K corresponds to the formation of a single E-type AFM structure [68]. At the same time, the magnetic phase transitions at TN2 ≈ 50 K and TN3 ≈ 24 K cannot be seen in the Mössbauer spectra because the former does not change the local magnetic environment of manganese (probe iron) cations in B sites, and the second transition (TN3) leads to the magnetic ordering of Mn3+ in the A″ sublattice.
Characteristic of spin–spin relaxation, there are no obvious changes in the frequency ΩR as a function of temperature. However, the values of ΩR ~ (0.2–0.5) × 109 s−1 are found to be significantly lower than the characteristic frequencies of spin waves ΩSkBT/ħ = 1011–1012 s−1 (T = 1–100 K) in conventional magnetic systems [65]. This indicates that the spin fluctuations involving iron probes are predominantly local. Furthermore, the ΩR value is comparable to the Larmor frequency ΩR ≈ 108 s−1 of the 57Fe nuclear spin in the hyperfine magnetic field <Bhf> ≈ 50 T, thus supporting our assumption about the relaxed nature of the observed spectra.
Figure 11 shows the temperature dependence of the hyperfine field <Bhf(T)> averaged over all partial spectra Fe(i), which was approximated for spin S = 5/2 using the parametric Brillouin function:
B h f ( T ) = B h f ( 0 ) B 5 / 2 ξ 5 2 T N T σ Mn ( T )
where ξ = JFeMn/JMnMn is the ratio of exchange integrals that characterize the magnetic interactions of Fe3+ probes with the surrounding manganese cations (JFeMn) and the averaged interactions between the Mn3+ cations themselves (JMnMn). The value ξ = 0.67(3) obtained from the best fit of the experimental spectra evidences the weakening of the exchanged magnetic interactions of the iron cations with the manganese sublattice, which is equivalent to decreasing the effective Weiss field [69]. This can result from a so-called “orbital dilution”, a phenomenon characteristic of the impurity cations of transition metals with non-degenerate orbital electron states (Fe3+, Cr3+…: <L> = 0, where L is the total orbital momentum) if they are stabilized within the matrix of transition metals with degenerate orbital states (Rh4+, Mn3+…: <L> ≠ 0) [70,71]. It was shown previously that such impurity centers behave as peculiar “orbital defects” due to the absence of orbital degeneration, i.e., orbital degrees of freedom. Even at very low concentrations, they can significantly affect the magnetic structure of the compound. Experimental methods employed to study such systems are currently in their early stages of development.

3. Materials and Methods

The manganite BiMn6.96Fe0.04O12 was synthesized in a high-pressure, “belt”-type chamber. A stoichiometric mixture of Mn2O3 (99.9%, Rare Metallic Co., Tokyo, Japan) Bi2O3 (99.9999%, Rare Metallic Co., Tokyo, Japan), and 57Fe2O3 (95.5% enriched with 57Fe, Trace Sciences International, Richmond Hill, ON, Canada) was filled into a gold capsule in which it was subjected to a pressure of ~6 GPa followed by heating to 1323 K for 10 min. The sample was quenched to room temperature after holding for 120 min. The synthesis of undoped manganite BiMn7O12 is described in more detail in [3].
The X-ray diffraction data were acquired using a synchrotron source of X-rays (SXRPD) in a large Debye–Sherrer chamber with the line BL15XU (SPring-8, Sayo, Japan) and the 2θ value ranging from 3° to 60° with a step of 0.003°. The monochromatic radiation with the wavelength of λ = 0.65298 Å was used. Experiments were performed in a temperature range of between 100 and 670 K. Prior to measuring, the powder samples were tightly packed in a Lindemann glass capillary (for the 100–400 K range) and a quartz capillary (for the 350–670 K range) with an internal diameter of 0.1 mm. The capillaries were cooled using an N2 flow when the low-temperature measurements were performed. The processing of the diffraction patterns and refinement of the crystal lattice parameters were performed using the Rietveld method, using the RIETAN-2000 software similar to the procedure described in [3].
Scanning electron microscopy (SEM) images were taken on a NVision 40 electron microscope (Carl Zeiss; Oberkochen, Germany) equipped with an Oxford Instruments X-Max analyzer. The accelerating voltage varied in the range from 3 to 20 kV.
For measuring differential scanning calorimetry (DSC) curves on a Mettler Toledo DSC1 STARe calorimeter in the temperature range 125–673 K, samples were placed in Al crucibles, the rate of heating/cooling in the nitrogen flow being 10 K/min.
The heat capacity measurements were carried out on a PPMS calorimeter (Quantum Design, San Diego, CA, USA) in the temperature range of 2–300 K in the modes of heating and cooling in external magnetic fields ranging from 0 to 90 kOe.
Magnetic susceptibility was measured on a SQUID MPMS 1T magnetometer (Quantum Design, San Diego, CA, USA) in the temperature range of 2–350 K in the ZFC (cooling without external magnetic field) and FC (cooling in the external magnetic field 10 kOe in strength) modes.
Mössbauer spectra were measured with a conventional electrodynamic-type spectrometer in the constant acceleration mode with a 1450 MBq 57Co(Rh) γ-ray source. The values of the isomer shift are given relative to α-Fe (298 K). Processing of the experimental spectra was performed with the use of the program package “SpectrRelax” [72]. Computations of the EFG parameters were carried out using the “GradientNCMS” software (ver. 8.3) designed by the authors and are represented in more detail in [73].

4. Conclusions

We explored the interplay between the local crystal structure of the multiferroic BiMn7O12 manganite and the processes of its spontaneous polarization and magnetic ordering using 57Fe-probe Mössbauer spectroscopy. It was shown that Fe3+ probes statistically substitute isovalent Mn3+ cations in the octahedral oxygen local environment. The parameters of the electric hyperfine interactions of 57Fe nuclei reflect the symmetry of the crystalline environment of Mn3+ cations in these sites.
The calculations of the EFG parameters, considering both monopole and dipole contributions, are in accordance with our experimental results, demonstrating that, in the paraelectric phase (at T > T2), cations Bi3+, even while existing in locally distorted crystalline environments, maintain their electrical dipole moments pBi, which are randomly oriented within the cubic lattice. As a result, the phase transitions into the ferroelectric state involve the ordering of pBi dipoles, i.e., they may be considered the transitions of the “order-disorder” type.
It was determined that the monotonous decrease in Δ(T) from T2 up to T1 can indicate a gradual increase in the symmetry of (Fe3+O6) polyhedra while approaching the temperature of the structural transition I m 3 ¯ I2/m, which is corroborated by the synchrotron diffraction studies of undoped BiMn7O12. This characteristic behavior has been independently registered through methods that are entirely different in their physical nature. Thus, it strongly indicates the occurrence of the second-order JT phase transition. Its mechanism can be classified as a structural transition of the “displacive” type, in contrast to “order-disorder” transitions.
Using the Born model, we calculated dynamic ion charges that indicate only a minor ion polarization and the predominance of ionic contributions in the spontaneous electrical polarization of the crystal. The observed robust temperature dependence of the quadrupole splitting Δi(T) of the partial spectra Fe(i) is governed by the temperature dependence Ps(T). The dependence Ps(T) on the opposite sides of the phase transition P1 ↔ Im (T3 ≈ 240 K) significantly differs in its behavior. In the range T3 < T < T2, Ps(T) indicates that the ferroelectric–paraelectric phase transition is of the first order. The Curie point TC ≈ 437 K determined from the Mössbauer data closely coincides with the temperature of the structural transition ImI2/m. The proposed algorithm for finding the correlation between the experimental dependencies Δi(T) for the probe 57Fe nuclei and the polarization Ps(T) of the crystal can be applied to other systems with ferroelectric and multiferroic properties.
At low temperatures, T < TN1, and the 57Fe Mössbauer spectra demonstrate relaxation behavior. This can result from a so-called “orbital dilution” characteristic of the impurity cations of transition metals with non-degenerate orbital electron states within the matrix of transition metals with degenerate orbital states.

Supplementary Materials

The supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/ijms25031437/s1.

Author Contributions

Conceptualization, A.V.S., A.A.B. and I.A.P.; methodology, I.S.S., V.I.N. and A.V.S.; software, A.V.S.; validation, I.S.S. and A.A.B.; formal analysis, I.S.S. and V.I.N.; investigation, I.S.S., V.I.N., M.N.S. and A.A.B.; data curation, I.S.S., V.I.N., A.V.S. and A.A.B.; writing—original draft preparation, A.V.S. and I.A.P.; writing—review and editing, A.V.S., A.A.B. and I.A.P.; visualization, I.S.S. and I.A.P.; supervision, A.V.S. and I.A.P.; project administration, A.V.S. and I.A.P.; funding acquisition, I.S.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Russian Science Foundation (RSF, grant No. 19-73-10034). The synchrotron radiation experiments were conducted at the former NIMS beamline (BL15XU) of SPring-8 with the approval of the former NIMS Synchrotron X-ray Station (proposal number: 2017B4502).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

We thank Y. Katsuya and M. Tanaka for their help at SPring-8.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A

Electric Field Gradient Calculation Details

In terms of an ionic approximation, i.e., without regard for the covalence of the chemical bonds between the anionic and cationic sublattices, the EFG lattice tensor components on 57Fe nuclei represent the sum of the monopole (Vmon) and dipole (Vdip) contributions [74]
V i j l a t = V i j m o n + V i j d i p = ( 1 γ ) k 2 x i x j q k r k + r i k p i k r k 3
where γ is the Sternheimer antishielding factor, qk is the effective charge of the kth ion in the lattice, rk is the distance between the kth ion and the 57Fe nucleus, and rik is the projection of the radius vector rk on the direction of the ith projection of the electric dipole moment of the kth ion pik (where i = x, y, z). The dipole moments of the ions (pk) and their projections (pik) were assumed to be calculated in terms of the Born model [59]: pk = ZkΔpk or pik = ZkΔxik, where Zk is the Born charge of the kth ion, which was taken to be an isotropic scalar quantity, and Δpkxik) is the displacement vector of the kth ion (and its ith projection) from its centrosymmetrical position. Δpk and Δxik were calculated using the crystallographic data for the BiMn7O12 manganite obtained at 300 K [3]. To estimate the Born charges, we used the following procedure: charges ZBi, ZMni, and ZOi, and the corresponding dipole moments pik were sequentially varied at displacement Δxik until the best agreement with the experimental quadrupole splittings Δi. At the first stage of this procedure, the desired Born charges of the ions Zk were taken to be equal to their formal oxidation levels in the BiMn7O12 compound. As a result, we obtained the values ZBi = +3.30, ZMn1ZMn2 = +3.30, and ZO = −2.20, which fall within the Born charge range obtained for the corresponding ions in other perovskite-like oxides [60]. Using the projections of the electric dipole moments pik, we can calculate the spontaneous polarization of the manganite,
P s = 1 V i k p i k 2 1 2
The value obtained at T = 300 K (Ps(T0) ~ 9 μC/cm2) agrees with the value (7 μC/cm2 at 300 K) obtained earlier for 57Fe-free manganite BiMn7O12 (space group Im) [3]. All these results demonstrate that, despite the initial assumptions, the proposed calculation scheme is quite self-consistent and can be extend to the temperature range for which structural data are absent.
The Born charges obtained for Bi3+, Mn3+, and O2− ions (Zi) can be compared to the effective charges (S) calculated using the Brown model by using the data on the crystal structure of the plain BiMn7O12 manganite [3].
S = k s k = k exp r 0 r k B
where rk is the metal–oxygen distance for the kth pair M3+-O2− (where M = Bi, Mn), r0 is a constant for the given metal grade (r0 = 2.09 for Bi3+-O2− and r0 = 1.760 for Mn3+-O2−), and B = 0.37 [75]. Using this equation, we calculated the effective charges S of the Bi3+, Mn3+, and O2− ions occupying nonequivalent crystallographic positions for each of the two structural modifications of BiMn7O12.
Charges <SBi>, <SMn>, and <SO> that were averaged for each kind of ions are presented in the Table S1. As would expected, the obtained values of <S> are very close to the formal oxidation levels of the corresponding atoms but are lower than their Born charges. The high Born charges, which are several times higher than the formal oxidation levels for most perovskite-like metal oxides, are associated with the deformation (polarizability) of the electron shell of an ion when it is displaced from the centrosymmetrical position in a crystal [59]. However, in our case, the discrepancy between the Born (Z) and effective (<S>) charges is not so significant (Table S1), which indirectly confirms the validity of the ion model used in this work.
The temperature dependences of the dipole moment pk(T) and the related spontaneous polarization Ps(T), which are likely to be the main cause of the sharp temperature-induced change in the splitting Δ1(T) and Δ2(T), were taken into account using the approximate expressions pk(T) = ξk(T)pk(T0) and Ps(T) = ξk(T)Pk(T0), in which pk(T0) and Pk(T0) are the known values of the corresponding parameters at a given temperature T0 = 300 K. Our preliminary calculations showed that the same type of functions ξk(T) ≡ ξk(T0),
ξ ( T ) = p Bi ( T ) p Bi ( T 0 ) = p Mn ( i ) ( T ) p Mn ( i ) ( T 0 ) = p O ( i ) ( T ) p O ( i ) ( T 0 ) = p S ( T ) p S ( T 0 )
can be taken for all ions in order to achieve agreement with the experiment data and to decrease the number of variable parameters.

Appendix B

Distortion Parameters Calculation Details

To calculate the distortion (Δd) parameters of MnO6 polyhedra, we used this equation:
Δ d = 1 6 n = 1 6 l n 1 6 ( n = 1 6 l n ) 1 6 ( n = 1 6 l n ) 2 = 1 6 n = 1 6 l n l m e d l m e d 2
where ln is the length of the nth Mn-O(n) bond and lmed is the average value of the corresponding bonds in MnO6.

Appendix C

Electric Field Gradient Calculation within P1 pseudocell (T > T1) Details

In the general case the sublattice A′ cations in quadrupole perovskites (which are Bi3+ cations in the perovskite BiMn7O12) are located in the partial position 2a with coordinates (0, 0, 0). However, as noted in [3], the presence of a stereochemically active lone-pair 6s2 electrons directed towards the center of the oxygen dodecahedron Bi3+O12 leads to a displacement of the Bi3+ cation itself along the diagonal form it’s partial position 2a to the new partial position 16f with coordinates (±0.0223(9), ±0.0223(9), ±0.0223(9)).
To account for the influence of the Bi3+ cations displacement to the new partial position 16f, the calculations were carried out within the framework of a pseudocell. The initial crystal lattice was increased by a factor of 8 to the size of a cube with a decrease in the symmetry of the lattice to the space group P1 with parameters (aP1 = bP1 = cP1 = 2aini = 14.96716 Å) and initial cubic angles, while maintaining the coordinates of other types of atoms, however, the coordinates of the Bi3+ cations were chosen randomly from 8 available triple of coordinates for each position with consideration that BiMn7O12 is paraelectric at T > T1. The calculations were carried out on each of the 64 manganese atoms of the pseudocell due to the environment nonequivalence of each of them. Thus, a set of 64 quadruple distribution splittings Δ was used, from which, using methods of mathematical statistics, the average values of the quadruple splitting Δav and the quadruple splitting dispersion D were calculated.

Appendix D

Extension of the Landau Theory to First-Order Phase Transitions

For a ferroelectric system with two order parameters of electrical polarization Pz (primary) and strain ε (secondary), generic Landau free energy (F) can be expressed as [76]:
F ( P z , ε ) = 1 2 a ( T ) P z 2 + 1 4 b P z 4 + 1 6 c P z 6 + 1 2 k ε 2 + λ ε P z 2
This equation consists of three contributions: (i) the electric energy due to the primary order parameter Pz: 1 2 a ( T ) P z 2 + 1 4 b P z 4 + 1 6 c P z 6 , where the coefficient a(T) is assumed to be temperature dependent a(T) = a0(TTC). Usually, it is assumed that the system intrinsically tends to undergo a second-order transition and, thus, the coefficient b (>0) of the fourth-order term is positive; c is the coefficient of the sixth-order term, and c > 0. (ii) The elastic energy due to the second-order parameter ε: 1 2 k ε 2 , where k is the elastic modulus and, thus, k > 0. (iii) The ferroelastic coupling energy is as follows: λ ε P z 2 , where λ is the coupling coefficient.
Minimizing the total energy with respect to the strain, ∂F/∂ε = 0, yields a relation between primary and secondary parameters Pz and ε, respectively,
ε = λ k P z 2
Substituting Equation (A7) into Equation (1), we obtain a renormalized Landau free energy,
F ( P z ) = 1 2 a ( T ) P z 2 + 1 4 b 2 λ k 2 P z 4 + 1 6 c P z 6 = 1 2 a ( T ) P z 2 + 1 4 b P z 4 + 1 6 c P z 6
The most important consequence of the ferroelastic coupling is that the coefficient (b > 0) of the fourth-order term is renormalized and now becomes b′ ≡ (b − 2λ2/k). As the coefficient b is usually a small positive constant and elastic modulus is always positive (k > 0), a coupling coefficient λ of certain magnitude can make b′ < 0. Because a negative fourth-order term in Landau free energy (Equation (A8)) creates an energy barrier in the free energy landscape, this leads to a first-order transition. It is interesting that a renormalization group approach also yields a similar conclusion: a second-order transition is changed into a first-order transition if a primary order parameter (Pz) is coupled to the strain in the fluctuation region near TC [77].
Minimizing F′(Pz) with respect to the polarization Pz, ∂F′/∂Pz = 0, provides the following description, summarized in Figure S6 for the paraelectric-ferroelectric transition. For T > T** = TC + (b′)2/4a0c, the paraelectric phase (Pz = 0) is the only stable phase. At TT** ferroelectric phase (Pz ≠ 0), corresponding to
P z ( T ) = ± b + ( 1 4 ( b ) 2 a 0 c ( T T C ) ) 1 / 2 c 1 / 2
arises as a metastable state, associated with a secondary minimum in the non-equilibrium curve F′(T, Pz). On cooling, the stability increases and paraelectric and ferroelectric phases become equally stable at T* = TC + 3(b′)2/16a0c. Below T* paraelectric phase, metastable decreases to TC, where it becomes unstable. A region of coexistence between paraelectric and ferroelectric phases in the temperature interval ΔT = 1/16(b′)2/a0c, in which the two phases are alternately metastable. The phase transition may occur at any temperature within the ΔT interval. However, the most probable transition temperature is TC, corresponding to an equal stability for the two phases. The existence of a region of coexistence for the two phases implies a thermal hysteresis on cycling across the paraelectric-ferroelectric transition.
The obtained Equation (A8) allows for an important prediction about the relationship between the strength of the ferroelastic coupling and the size of the hysteresis of the resulting first-order phase transition. The size of the hysteresis is determined by the energy barrier at TC (Figure S6), which is largely dependent on the magnitude of the negative fourth-order term 1 4 b P z 4 [6,8]. The more negative this term is, the larger the transition barrier and the thermal hysteresis. A large ferroelastic coupling coefficient λ contributes to a large negative fourth-order term 1 4 b P z 4 and, thus, contributes to a larger transition hysteresis. On the other hand, from Equation (A7), one can see that a larger ferroelastic coupling coefficient λ also leads to a larger spontaneous lattice distortion upon the first-order ferroelectric transition. Therefore, the strength of the coupling coefficient λ can be represented by the magnitude of the spontaneous lattice distortion. As a result, Equation (A8) predicts that the magnitude of thermal hysteresis increases with an increase in lattice distortion.

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Figure 1. Crystal structures of BiMn7O12: (a) At T > T1 in the cubic I m 3 ¯ structure (without Bi splitting); (b) At T2 < T < T1 in the monoclinic I2/m structure; (c) At T3 < T < T2 in the monoclinic Im structure; (d) At T < T3 in the triclinic P1 structure (all viewed along the monoclinic b axis; Elongated Mn–O bonds due to the Jahn–Teller distortions in MnO6 octahedra are marked by red lines; The crystal cells are marked with black lined rectangles. The inset in the center depicts the accordance of the crystal structures and physical properties.
Figure 1. Crystal structures of BiMn7O12: (a) At T > T1 in the cubic I m 3 ¯ structure (without Bi splitting); (b) At T2 < T < T1 in the monoclinic I2/m structure; (c) At T3 < T < T2 in the monoclinic Im structure; (d) At T < T3 in the triclinic P1 structure (all viewed along the monoclinic b axis; Elongated Mn–O bonds due to the Jahn–Teller distortions in MnO6 octahedra are marked by red lines; The crystal cells are marked with black lined rectangles. The inset in the center depicts the accordance of the crystal structures and physical properties.
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Figure 2. (a) Differential scanning calorimetry (DSC) curves of BiMn6.9657Fe0.04O12 upon heating and cooling (three runs were performed to check the reproducibility; since there were no peaks observed, data in the 125–300 K range are not shown); (b) Specific heat, plotted as CP/T versus T, of BiMn7O12 and BiMn6.9657Fe0.04O12 at H = 0 (measurements were performed on cooling); (c) ZFC dc (left scale) and reversed (right scale) magnetic susceptibility curves of BiMn7O12 and BiMn6.9657Fe0.04O12 (dashed vertical lines emphasize magnetic anomalies).
Figure 2. (a) Differential scanning calorimetry (DSC) curves of BiMn6.9657Fe0.04O12 upon heating and cooling (three runs were performed to check the reproducibility; since there were no peaks observed, data in the 125–300 K range are not shown); (b) Specific heat, plotted as CP/T versus T, of BiMn7O12 and BiMn6.9657Fe0.04O12 at H = 0 (measurements were performed on cooling); (c) ZFC dc (left scale) and reversed (right scale) magnetic susceptibility curves of BiMn7O12 and BiMn6.9657Fe0.04O12 (dashed vertical lines emphasize magnetic anomalies).
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Figure 3. Left panel: typical Mössbauer spectra of the 57Fe nuclei in BiMn6.96Fe0.04O12 manganite measured at different temperatures (each at a specific range according to different crystal structures). Right panel: the p(Δ) distributions and their representation as the superposition of normal distributions corresponding to the crystal sites of 57Fe probe nuclei within a manganite structure. (ad) Correspond to particular temperature range (see text).
Figure 3. Left panel: typical Mössbauer spectra of the 57Fe nuclei in BiMn6.96Fe0.04O12 manganite measured at different temperatures (each at a specific range according to different crystal structures). Right panel: the p(Δ) distributions and their representation as the superposition of normal distributions corresponding to the crystal sites of 57Fe probe nuclei within a manganite structure. (ad) Correspond to particular temperature range (see text).
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Figure 4. The temperature dependencies of the experimental values of the quadrupole splittings Δiexp(T) of the partial Fe(i) spectra at specific ranges according to different crystal structures of the BiMn6.96Fe0.04O12 manganite (asterisks show the theoretical values).
Figure 4. The temperature dependencies of the experimental values of the quadrupole splittings Δiexp(T) of the partial Fe(i) spectra at specific ranges according to different crystal structures of the BiMn6.96Fe0.04O12 manganite (asterisks show the theoretical values).
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Figure 5. Schematic representations: (a) the formation of the pBi dipole moment as a result of the displacement of Bi3+ cations (brown balls) from their centrosymmetric positions (balls with a dotted line). The Bi3+ center is shifted toward the lone pair; (b) the random orientation of the lone electron pairs or displacements of Bi3+ cations leads to the zero value of the total crystal polarization (<PBi>) averaged over all directions (the large brown ball represents the ellipse of the thermal vibrations of bismuth).
Figure 5. Schematic representations: (a) the formation of the pBi dipole moment as a result of the displacement of Bi3+ cations (brown balls) from their centrosymmetric positions (balls with a dotted line). The Bi3+ center is shifted toward the lone pair; (b) the random orientation of the lone electron pairs or displacements of Bi3+ cations leads to the zero value of the total crystal polarization (<PBi>) averaged over all directions (the large brown ball represents the ellipse of the thermal vibrations of bismuth).
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Figure 6. (a) The explicative scheme of the two-level relaxation model: Ei—the energies of states “1” and “2”, ni—the probabilities of states, Ωij—the frequencies of transitions between states. (b) The reciprocal temperature dependencies of the logarithm ln(n1/n2) of probabilities n1 and n2 ratio, and the distortion parameters Δd for Mn4O6 and Mn5O6 octahedra, calculated from structural data [3]. Blue lines represent the linear approximation in the selected temperature range and are shown for visual convenience. The shaded part corresponds to the temperature range (T > 500 K) for which a change in the degree of the distortion (Δd) of the MniO6 polyhedra is expected and, as a consequence, so too is a change in the relative position of the energy levels E1 and E2 (see text) The black dots correspond to the left scale, and the circles correspond to the right scale.
Figure 6. (a) The explicative scheme of the two-level relaxation model: Ei—the energies of states “1” and “2”, ni—the probabilities of states, Ωij—the frequencies of transitions between states. (b) The reciprocal temperature dependencies of the logarithm ln(n1/n2) of probabilities n1 and n2 ratio, and the distortion parameters Δd for Mn4O6 and Mn5O6 octahedra, calculated from structural data [3]. Blue lines represent the linear approximation in the selected temperature range and are shown for visual convenience. The shaded part corresponds to the temperature range (T > 500 K) for which a change in the degree of the distortion (Δd) of the MniO6 polyhedra is expected and, as a consequence, so too is a change in the relative position of the energy levels E1 and E2 (see text) The black dots correspond to the left scale, and the circles correspond to the right scale.
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Figure 7. Left panel: The dependencies of the theoretical Δi values versus spontaneous crystal polarization Ps at (a) T = 300 K and (b) T = 10 K. The curves refer to the experimental values Δexpi from the Mössbauer spectroscopy data at (a) 300 K and (b) 101 K temperatures. Shaded areas correspond to evaluated Ps values when the theoretical Δi(Ps) values conform to the experimental Δexpi ones in the best way. The verticle lines in the shaded areas showed the approximate mean position for ease of perception. Right panel: the dependencies of the theoretical VZZi values versus spontaneous crystal polarization Ps at corresponding temperatures fitted with quadratic functions (see text).
Figure 7. Left panel: The dependencies of the theoretical Δi values versus spontaneous crystal polarization Ps at (a) T = 300 K and (b) T = 10 K. The curves refer to the experimental values Δexpi from the Mössbauer spectroscopy data at (a) 300 K and (b) 101 K temperatures. Shaded areas correspond to evaluated Ps values when the theoretical Δi(Ps) values conform to the experimental Δexpi ones in the best way. The verticle lines in the shaded areas showed the approximate mean position for ease of perception. Right panel: the dependencies of the theoretical VZZi values versus spontaneous crystal polarization Ps at corresponding temperatures fitted with quadratic functions (see text).
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Figure 8. The temperature dependencies of the spontaneous polarization Ps(T) for the two crystal structures of the BiMn6.96Fe0.04O12 manganite. The solid and dashed curves represent the fitting in order with theory explained in the text. The dotted line shows the temperature of the phase transition.
Figure 8. The temperature dependencies of the spontaneous polarization Ps(T) for the two crystal structures of the BiMn6.96Fe0.04O12 manganite. The solid and dashed curves represent the fitting in order with theory explained in the text. The dotted line shows the temperature of the phase transition.
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Figure 9. (a) 57Fe Mössbauer spectra of BiMn6.96Fe0.04O12 near TN1 fitted as the distributions of the single Lorentz line; (b) the temperature dependence of the dispersion DP(δ) of the isomer shift δ. The kink was used to evaluate the magnetic phase transition point (see text).
Figure 9. (a) 57Fe Mössbauer spectra of BiMn6.96Fe0.04O12 near TN1 fitted as the distributions of the single Lorentz line; (b) the temperature dependence of the dispersion DP(δ) of the isomer shift δ. The kink was used to evaluate the magnetic phase transition point (see text).
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Figure 10. 57Fe Mössbauer spectra of BiMn6.96Fe0.04O12 at T < TN1, fitted with the multilevel magnetic spin relaxation (see text). Black curves are subspectra obtained during fitting procedure, red curves are summarized fitted spectra. The differecnces between experimental and fitted data are also shown in the bottom of each spectrum.
Figure 10. 57Fe Mössbauer spectra of BiMn6.96Fe0.04O12 at T < TN1, fitted with the multilevel magnetic spin relaxation (see text). Black curves are subspectra obtained during fitting procedure, red curves are summarized fitted spectra. The differecnces between experimental and fitted data are also shown in the bottom of each spectrum.
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Figure 11. The temperature dependence Bhf(T) of the hyperfine magnetic field Bhf at 57Fe nuclei approximated using a modified Brillouin function (see text). The dashed red line shows the “pure” Brillouin law for spin S = 5/2. Black dots are experimental mean values of the hyperfine fields.
Figure 11. The temperature dependence Bhf(T) of the hyperfine magnetic field Bhf at 57Fe nuclei approximated using a modified Brillouin function (see text). The dashed red line shows the “pure” Brillouin law for spin S = 5/2. Black dots are experimental mean values of the hyperfine fields.
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Table 1. 57Fe hyperfine parameters at T > T1 of BiMn6.9657Fe0.04O12.
Table 1. 57Fe hyperfine parameters at T > T1 of BiMn6.9657Fe0.04O12.
T, K<δ>, mm/s<Δ>, mm/s D p exp , mm2/s2Γ, mm/s
6020.18(1)0.28(1)0.016(1)0.24 *
6220.17(1)0.26(1)0.016(1)0.24 *
6330.16(1)0.26(1)0.017(1)0.24 *
6530.15(1)0.25(1)0.016(1)0.24 *
* When processing the spectra, the linewidth Γ was fixed in accordance with the “thin” absorber and properties of the source. <δ> is the mean isomer shift, <Δ> is the mean quadrupole splitting, D p exp is the dispersion of the quadrupole splitting taken from the distribution reconstruction procedure.
Table 2. Taylor expansion parameters VZZ(0), α, and β of the dependences VZZ(Ps) calculated for both polymorphic modifications for all octahedral Mn3+ sites.
Table 2. Taylor expansion parameters VZZ(0), α, and β of the dependences VZZ(Ps) calculated for both polymorphic modifications for all octahedral Mn3+ sites.
T, KSiteVZZ(0) (V/m2 × 1021)α (V/C × 1021)β (V·m2/C2 × 1022)
300Mn10.38(1)−0.6(1)0.78(3)
Mn20.27(1)1.9(1)0.48(2)
101Mn40.385(1)0.71(1)0.12(3)
Mn50.519(3)0.39(2)0.24(6)
Mn60.565(1)−0.85(1)0.195(1)
Mn70.526(1)−0.72(1)0.100(1)
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Soboleva, I.S.; Nitsenko, V.I.; Sobolev, A.V.; Smirnova, M.N.; Belik, A.A.; Presniakov, I.A. Understanding Complex Interplay among Different Instabilities in Multiferroic BiMn7O12 Using 57Fe Probe Mössbauer Spectroscopy. Int. J. Mol. Sci. 2024, 25, 1437. https://doi.org/10.3390/ijms25031437

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Soboleva IS, Nitsenko VI, Sobolev AV, Smirnova MN, Belik AA, Presniakov IA. Understanding Complex Interplay among Different Instabilities in Multiferroic BiMn7O12 Using 57Fe Probe Mössbauer Spectroscopy. International Journal of Molecular Sciences. 2024; 25(3):1437. https://doi.org/10.3390/ijms25031437

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Soboleva, Iana S., Vladimir I. Nitsenko, Alexey V. Sobolev, Maria N. Smirnova, Alexei A. Belik, and Igor A. Presniakov. 2024. "Understanding Complex Interplay among Different Instabilities in Multiferroic BiMn7O12 Using 57Fe Probe Mössbauer Spectroscopy" International Journal of Molecular Sciences 25, no. 3: 1437. https://doi.org/10.3390/ijms25031437

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