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

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 BiMn_6.96⁵⁷Fe_0.04O₁₂ undergoes a cascade of structural (T₁ ≈ 590 K, T₂ ≈ 442 K, and T₃ ≈ 240 K) and magnetic (T_N1 ≈ 57 K, T_N2 ≈ 50 K, and T_N3 ≈ 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 p_Bi, 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 (P_s), 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 P_s(T) dependence. The P_s(T) dependence around the Im ↔ I2/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 < T_N1) 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 BiMn_7−xCu_xO₁₂ (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, e_g¹, 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 (P_s) is characteristic of multiferroics, which can be grouped into two types [13]. In the first group (type-I multiferroics), M and P_s are independent of each other, i.e., magnetism and ferroelectricity have different origins. In the second group (type-II multiferroics), M and P_s 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 (B_hf) and principal components {V_ii}_X,Y,Z of the tensor of the electrical field gradient (EFG) gained using these methods reproduce the corresponding dependences M(T) and P_s(T). Meanwhile, the relationship B_hf = αM is usually linear for ⁵⁷Fe, ⁵⁵Mn, and ⁵³Cr nuclei, and the dependencies of the EFG tensor parameters V_ii = f(P_s) and η = f′(P_s) (where η = (V_XX − V_YY)/V_ZZ is the asymmetry parameter) demonstrate nontrivial behavior. In some works, quadratic dependences V_ii = a + bP_s² and η ∝ P_s² 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 < T_C ≈ 440 K, whereas, at lower temperatures (T < T_N1 ≈ 59 K), it acquires a magnetically ordered state and multiferroic properties [2]. In contrast to perovskites ABO₃, in their “quadruple” analogs (A′A″₃)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 AMn_6.96⁵⁷Fe_0.04O₁₂ (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 (T_N1 and T_N2) 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 BiMn_6.94Fe_0.04O₁₂ at temperatures T_N1 < T < T₃ and T₃ < T < T₂. The final part describes the hyperfine magnetic interactions of ⁵⁷Fe probes in the magnetically ordered temperature range T < T_N1.

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 ($\mathit{Im}\overline{3}$) 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 BiMn_6.96Fe_0.04O₁₂ is noticeably lower than ~290 K for the undoped BiMn₇O₁₂ sample [3].

The BiMn_6.96Fe_0.04O₁₂ 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 BiMn_6.96Fe_0.04O₁₂ (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 ↔ $\mathit{Im}\overline{3}$ and Im ↔ I2/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 C_P/T(T) in BiMn_6.96Fe_0.04O₁₂ (Figure 2b), we obtained the values of the temperatures of transition to the magnetically ordered states T_N2 and T_N3. T_N2 reached ~ 50 K, and the temperature of the third magnetic transition T_N3 ≈ 24 K was 3–5 K lower than that of the undoped manganite [4]. The third phase transition at T_N3 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 BiMn_6.96Fe_0.04O₁₂ 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 δ_633K ≈ 0.16 mm/s corresponds to Fe³⁺ cations [38], isovalently substituting Mn³⁺ in the octahedral positions of Mn2 (Figure 1a). Despite BiMn_6.96Fe_0.04O₁₂ having a cubic structure ($\mathit{Im}\overline{3}$) 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. ⁵⁷Fe hyperfine parameters at T > T₁ of BiMn_6.96⁵⁷Fe_0.04O₁₂.

T, K	<δ>, mm/s	<Δ>, mm/s	${\mathit{D}}_{\mathit{p}}^{\mathbf{exp}}$, mm2/s2	Γ, mm/s
602	0.18(1)	0.28(1)	0.016(1)	0.24 *
622	0.17(1)	0.26(1)	0.016(1)	0.24 *
633	0.16(1)	0.26(1)	0.017(1)	0.24 *
653	0.15(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.

). 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 ($V_{ZZ}^{\mathrm{mon}}$) and dipole ($V_{ZZ}^{\mathrm{dip}}$) 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 $V_{ZZ}^{\mathrm{dip}}$. The lone pair induces the displacement of Bi³⁺ cations from their centrosymmetric positions, which is equivalent to inducing the electric dipole moment p_Bi. Therefore, only dipole contributions ($V_{ZZ,Bi}^{\mathrm{dip}}$) from Bi³⁺ were taken into account when calculating $V_{ZZ}^{\mathrm{dip}}$ using variable p_Bi values in further calculations. Additionally, the dipole moments p_Bi 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 p_Bi ≈ 1.2 × 10⁻²⁹ C·m lays within the range of the corresponding values p_Bi for other Bi³⁺ oxide compounds [10]. Most importantly, even with a random orientation of the p_Bi moments, the Mn2 sites become non-equivalent in terms of the induced lattice contribution $V_{ZZ,\mathrm{Bi}}^{\mathrm{dip}}$. 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 Δ^theor 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 p_Bi, we calculated the mean value of the quadrupole splitting as well as the dispersion $D_p^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}$= 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 BiMn_6.96Fe_0.04O₁₂ at T > T₁, Bi³⁺ cations exist in a locally distorted environment and retain their electric dipole moments p_Bi 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 p_Bi 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 p_Bi are assigned to the lone sp^x-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 sp^x-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

BiMn_6.96⁵⁷Fe_0.04O₁₂ undergoes the structural transition at T₁ ≈ 590 K, transforming from cubic ($Im\overline{3}$) 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 V_ZZ,Mn4 = 3.76 × 10²⁰ V/m² and V_ZZ,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 Δ^theor_Mni ≈ eQ$V_{ZZ,\mathrm{Mni}}^{\mathrm{T}\mathrm{e}\mathrm{o}\mathrm{p}}$ for sites Mn4 and Mn5 (where Q is the quadrupole moment of ⁵⁷Fe nuclei) remarkably exceed the corresponding experimental values Δ^exp (Figure 4; Tables S2–S5).

We suppose that the abovementioned discrepancy between the calculated and experimental values of the quadrupole splitting (Δ^theor > Δ^exp) 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], R_1−xA_xMnO₃ [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 > T_JT, 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⁺ e_g-orbitals is achieved, and the other entails the orientational disordering of distorted (Mn³⁺O₆) polyhedra while preserving the polarization of e_g-orbitals even at high temperatures (T >> T_JT) [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 T_JT (>>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 Δ^exp 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 Ω_R >> Ω₀, where Ω_R and Ω₀ are the frequencies of the relaxation of the oxygen environment and the precession of the ⁵⁷Fe quadrupole moment around the V_ZZ 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 (V_ZZ) 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) ∝ <V_ZZ>, where <V_ZZ> 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 → $\mathit{Im}\overline{3}$. This conclusion is consistent with the synchrotron X-ray diffraction studies of BiMn₇O₁₂, which also demonstrate the gradual decrease in the distortion parameter Δ_d of (MnO₆) polyhedra when T → T₁ [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 Ω_R = 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 Ω_R >> 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):

$${\Omega }_{12(21)}(T)={\Omega }_{1(2)}^0\exp \left(-\frac{E_{12(21)}}{k_{B}T}\right),$$

where ${\Omega }_{1(2)}^0$ are temperature-independent parameters; E₁₂₍₂₁₎ are the activation energies of “1” and “2” states, respectively; k_B 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 E_act = 220(9) meV (Figure 6a and Figure S5) that closely corresponds to <E_act> 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 Δ_d 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 T_JT. 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 T_N1 < T < T₃

In the temperature range T₃ < T < T₂, the Mössbauer spectra of BiMn_6.96Fe_0.04O₁₂ 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, δ₁ ≈ δ₂ (Figure 3c).

The analysis above yielded anomalously sharp temperature dependences Δ_i(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 $V_{ZZ}^{\mathrm{lat}}$ with the values and mutual orientation of electric moments (p_k) in the lattice of BiMn₇O₁₂ (for details, see SI). The values p_k and their projections p_ik were calculated using the Born model [59]: p_k = Z_kΔr_k or p_ik = Z_kΔx_ik, where Z_k is the Born charge of the kth ion, which is an isotropic scalar value in our calculation; Δr_k (Δx_ik) is the vector of the displacement of the kth ion (and ith projection) from the centrosymmetric position. The values Δr_kk and Δx_ik were calculated using the crystallographic data for BiMn₇O₁₂ obtained at 300 K [3]. To estimate Born charges, we sequentially varied the charges {Z_Bi, Z_Mn(i), Z_O(i)} and their corresponding dipole moments p_ik with the given displacement Δx_ik until the best agreement with the experimental splitting Δ_i was achieved (Table S1). The values approximated in such a manner that Z_Bi = +3.30, Z_Mn1 ≈ Z_Mn2 = +3.30, and Z_O = −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 ${\left\{P_i\right\}}_{i=x,y,z}={\displaystyle \sum _k{p}_{ik}}$ of the spontaneous polarization P_s = (P_x² + P_y² + P_z²)^1/2:

$$\Delta (T)=(1-{\mathsf{\gamma }}_{\infty })\frac{\mathit{eQ}}{2}{\displaystyle \sum _k\frac{{\partial }^2}{\partial x_i^2}}\left[\frac{q_k}{r_k}+\frac{r_{ik}{P}_i\left(T\right)}{r_k^3}\left(\frac{p_k\left(T_0\right)}{P_s\left(T_0\right)}\right)\right]$$

(1)

where p_k(T₀)/P_s(T₀) is the ratio of the dipole moment of the kth ion (p_k) 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(P_s) and Δ₂(P_s) (Figure 7). Using the numerical solution of the equations Δ₁(P_s) = Δ₁(T) and Δ₂(P_s) = Δ₂(T) and accounting for statistical errors allowed us to simulate the temperature dependence P_s(T) (Figure 8).

The same algorithm for constructing dependencies P_s(T) using experimental data Δ_i(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 T_N1 ≤ T ≤ T₃ 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 Δ_i(P_s) with the experimental Δ_i(T) values, we were able to model P_s(T) across the temperature range under investigation (Figure 8).

The dependences Δ ∝ V_ZZ = f(P_s) (Figure 7) agree with the results of [24,25,26,27], where, in a general case, the dependence of V_ZZ(P_s) at low P_s values was represented using a Taylor series expansion:

$$V_{\mathrm{zz}}(P_{\mathrm{s}})=V_{\mathrm{zz}}^{\left(0\right)}+{\displaystyle \sum _i\left(\frac{\partial V_{\mathrm{zz}}}{\partial x_{\mathrm{i}}}\right)P_{\mathrm{s}}}+\frac{1}{2}{\displaystyle \sum _{i,j}\left(\frac{{\partial }^2{V}_{\mathrm{zz}}}{\partial x_i\partial x_j}\right)P_{\mathrm{s}}^2}+\dots =V_{\mathrm{zz}}^{\left(0\right)}+\alpha P_{\mathrm{s}}+\hspace{0.17em}\hspace{0.17em}\beta P_{\mathrm{s}}^2+\gamma P_{\mathrm{s}}^4\hspace{0.17em}\dots$$

(2)

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 V_ZZ(P_s) can be described using an expansion in a series with nonzero parameters α and β, whose values (

Table 2. Taylor expansion parameters V_ZZ⁽⁰⁾, α, and β of the dependences V_ZZ(P_s) calculated for both polymorphic modifications for all octahedral Mn³⁺ sites.

T, K	Site	VZZ(0) (V/m2 × 1021)	α (V/C × 1021)	β (V·m2/C2 × 1022)
300	Mn1	0.38(1)	−0.6(1)	0.78(3)
	Mn2	0.27(1)	1.9(1)	0.48(2)
101	Mn4	0.385(1)	0.71(1)	0.12(3)
	Mn5	0.519(3)	0.39(2)	0.24(6)
	Mn6	0.565(1)	−0.85(1)	0.195(1)
	Mn7	0.526(1)	−0.72(1)	0.100(1)

), in order of magnitude, are consistent with similar data from other perovskite-like ferroelectrics [24,25,26,27].

When describing the dependences, P_s(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 (E_eff), which can be expressed as

$$E_{\mathrm{eff}}=E_0\hspace{0.17em}+\beta P_{\mathrm{s}}+\hspace{0.17em}\gamma P_{\mathrm{s}}^3+\hspace{0.17em}\delta P_{\mathrm{s}}^5+\dots$$

(3)

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 P_s(T) dependence of polarization can have a general form [61]:

$$P_s(T)=P_0\tanh \left(\frac{E_{\mathrm{eff}}{P}_0{N}^{-1}}{k_{B}T}\right)=P_0\tanh \left(\frac{(\beta P_{\mathrm{s}}+\gamma P_{\mathrm{s}}^3)P_0{N}^{-1}}{k_{B}T}\right)$$

(4)

where P₀ is the spontaneous saturation polarization, and N is the number of elementary dipoles per unit cell. Using the relationship k_BT_C = β·P₀²/N (where T_C is the Curie ferroelectric point) and designations of the normalized values σ_s ≡ P_s/P₀, τ ≡ T/T_C, g ≡ γP₀²/β, one can derive an expression that is convenient for analyzing the experimental data:

$${\sigma }_s(T)=\tanh \left(\frac{{\sigma }_s\left(T) + g{\sigma }_s^3\right(T)}{\tau }\right)$$

(5)

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 P_s(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 T_C ≈ 437 K of the ferroelectric state, having been evaluated within the framework of this approach based on the description of P_s(T), insignificantly differs from the temperature of the structural transition Im ↔ I2/m (T₂ ≈ 442 K) determined for the sample BiMn_6.96Fe_0.04O₁₂ 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 (P_s) 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 T_C (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 T_N1 ≤ T ≤ T₃, the dependence P_s(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 P_s(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 < T_N1

The quadrupole doublets at temperatures slightly below the Néel point (T < T_N1 ≈ 50 K) contain the broadened components that reflect hyperfine magnetic fields B_hf induced on the ⁵⁷Fe nuclei (Figure 9a). The spectra were fitted via a reconstruction of distributions p(B_hf) characterized by a certain dispersion D_P(δ) at a given temperature. The kink on the temperature dependence D_P(δ) = f(T) (Figure 9b; Table S6 for data) corresponds to the temperature 57(3) K, which, within the measurement error, coincides with T_N1 ≈ 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 << T_N1, the Mössbauer spectrum of BiMn_6.96⁵⁷Fe_0.04O₁₂ 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 (S_Z) 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, S_Z> decelerates. If the relaxation period τ_R closely corresponds to the period of Larmor precession (τ_L) of the ⁵⁷Fe nuclear spin around the hyperfine field B_hf, 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 BiMn_6.96Fe_0.04O₁₂ 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, S_Z> [67]. Along with the static parameters (δ, eQV_ZZ and B_hf), the model includes variable relaxation parameters, namely, the relaxation frequency Ω_R (=1/τ_R) 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 < T_N1. 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 T_N1 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 T_N1 = 59 K corresponds to the formation of a single E-type AFM structure [68]. At the same time, the magnetic phase transitions at T_N2 ≈ 50 K and T_N3 ≈ 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 (T_N3) leads to the magnetic ordering of Mn³⁺ in the A″ sublattice.

Characteristic of spin–spin relaxation, there are no obvious changes in the frequency Ω_R as a function of temperature. However, the values of Ω_R ~ (0.2–0.5) × 10⁹ s⁻¹ are found to be significantly lower than the characteristic frequencies of spin waves Ω_S ≈ k_BT/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 Ω_R value is comparable to the Larmor frequency Ω_R ≈ 10⁸ s⁻¹ of the ⁵⁷Fe nuclear spin in the hyperfine magnetic field <B_hf> ≈ 50 T, thus supporting our assumption about the relaxed nature of the observed spectra.

Figure 11 shows the temperature dependence of the hyperfine field <B_hf(T)> averaged over all partial spectra Fe(i), which was approximated for spin S = 5/2 using the parametric Brillouin function:

$$〈B_{hf}(T)〉=〈B_{hf}(0)〉B_{5/2}\left[\xi \frac{5}{2}\left(\frac{T_N}{T}\right){\sigma }_{\mathrm{Mn}}(T)\right]$$

(6)

where ξ = J_FeMn/J_MnMn is the ratio of exchange integrals that characterize the magnetic interactions of Fe³⁺ probes with the surrounding manganese cations (J_FeMn) and the averaged interactions between the Mn³⁺ cations themselves (J_MnMn). 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 BiMn_6.96Fe_0.04O₁₂ 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 STAR^e 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 p_Bi, which are randomly oriented within the cubic lattice. As a result, the phase transitions into the ferroelectric state involve the ordering of p_Bi 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 $Im\overline{3}$ → 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 Δ_i(T) of the partial spectra Fe(i) is governed by the temperature dependence P_s(T). The dependence P_s(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, P_s(T) indicates that the ferroelectric–paraelectric phase transition is of the first order. The Curie point T_C ≈ 437 K determined from the Mössbauer data closely coincides with the temperature of the structural transition Im ↔ I2/m. The proposed algorithm for finding the correlation between the experimental dependencies Δ_i(T) for the probe ⁵⁷Fe nuclei and the polarization P_s(T) of the crystal can be applied to other systems with ferroelectric and multiferroic properties.

At low temperatures, T < T_N1, 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.