Spin and dipole order in geometrically frustrated mixed-valence manganite Pb₃Mn₇O₁₅

Manganites are materials with properties relevant for both fundamental research and practical applications, owing to the cross-correlation of their charge, spin, orbital and lattice degrees of freedom. Certain manganites have for example been found to display colossal magnetoresistance [1‐4], multiferroicity and magneto(di)electric effects [5‐7]. Lone-pair Pb²⁺ cations may induce polar distortions that couple charge and spin orders. In this context, oxide compounds from the phase diagram of the PbO-MnO-O system, combining Pb and Mn cations, are very interesting as they may display concomitant and coupled dielectric and magnetic orders. At atmospheric pressure, the following phases of the PbO-MnO-O system have been established (see Fig. 1): Pb_1+xMn₈O₁₆ hollandite-type structure [8, 9], Pb₃Mn₇O₁₅ [8, 10‐13, 15‐18], PbMn₂O₄ [11, 12], and Pb₄Mn₉O₂₀ [19] with close packed structures; Pb_0.25MnO₂, Pb_0.25MnO_2−x (x = 0,01) [13] with unknown structures, as well as Pb₂MnO₄ [20, 21]. Utilizing high pressure (>7 GPa) and high temperature (T > 800 °C) techniques the phases PbMnO_2.75 [22], PbMnO₃ [23], Pb₁₃Mn₉O₂₅ [24], and PbMn₇O₁₂ [25] with perovskite-based structure have been obtained.

Pb₃Mn₇O₁₅ (PMO) is a mixed valence manganite (Mn³⁺/Mn⁴⁺), whose structural [14‐17], magnetic and dielectric [10‐13, 26‐30] properties have been studied in detail. The material is a dielectric relaxor, and becomes antiferromagnetic below T_N ~ 67 K. However, there is a long-standing controversy about the structure of this material, and its microscopic magnetic structure is yet unknown. The first structure determination of PMO was reported by Darriet et al. [14] who concluded that the symmetry at 295 K is orthorhombic. The orthorhombic description was later supported by Marsh et al. [31]. Le Page et al. [15] subsequently argued that this structure is really hexagonal. More recently, Volkov et al. [27] reported as well an hexagonal structure, s.g. P6 ₃ /mcm, similar to that the mineral Zenzenite Pb₃(Fe,Mn)₇O₁₅ [32]. However a high resolution study by Rasch et al. [16] using synchrotron data demonstrated instead an orthorhombic symmetry. In another study, Volkov et al. [17] and Kimber [28] also reported an orthorhombic Pnma structure for PMO at 298 K, albeit noting that it was close to a structural phase transition, with a pseudo-hexagonal cell at 298 K, evolving into an hexagonal structure at higher temperatures [17, 28]. Stabilization of hexagonal structure at room-temperature after thermal treatment is not possible [17]. Interestingly, the hexagonal structure may also be stabilized by doping, e.g. by replacing only 5 % of Mn by Fe³⁺ [29]—like Zenzenite mineral. Doping by 5 % Ga³⁺ or Ge⁴⁺ keeps the structure orthorhombic. Relatively small changes of the magnetic properties were observed for low doping levels [33].

There remain unsolved issues as to both the nuclear and the magnetic structure as well as the dielectric and macroscopic magnetic properties of PMO. In this article, the structural, thermal, magnetic and dielectric properties of stoichiometric single crystals of Pb₃Mn₇O₁₅ are investigated. Powder X-ray and neutron diffraction data recorded on crushed single-crystals confirm a pseudo-hexagonal orthorhombic structure, and allow us to determine the magnetic structure of the low-temperature antiferromagnetic state of Pb₃Mn₇O₁₅.

High-quality single crystals of Pb₃Mn₇O₁₅ were grown in Pt crucibles in a PbO flux in an intimate mixture of 9PbO·Mn₂O₃, as described in Bush et al. [12]. Excess flux was dissolved in HNO₃. Shiny, opaque, black and plate-like crystals of dimensions up to 2 × 10 × 10 mm³ were obtained. The crystals occurred with hexagonal habits, showing the main forms as {0001}, {10–12} including also poorly developed {11–21} forms. For Mössbauer studies 2 % Fe-doped Pb₃(Mn_0.98Fe_0.02)₇O₁₅ single crystals were grown under similar conditions adding 2 mol % ⁵⁷Fe₂O₃ (instead of the same amount of Mn₂O₃). Phase purity was checked by powder X-ray diffraction.

The phase stability of PMO was investigated by thermogravimetry analysis, and the cations stoichiometry by energy-dispersive spectroscopy (EDS). Several iodometric titrations were made on Pb₃Mn₇O_15+δ single crystals to determine the oxygen content. Second harmonic generation (SHG) measurements at room temperature evidenced the centrosymmetric character of the crystals.

Crystal and magnetic structures were investigated by single crystal and powder X-ray diffraction (XRPD), and neutron powder diffraction on crushed single-crystals (NPD; data collected at the LLB, Saclay, France). Refinements of the XRPD and NPD data were performed by the Rietveld method using the FULLPROF software [34]. An analysis of coordination polyhedra of cations was performed using the IVTON software [35].

The Mössbauer spectroscopy measurements were performed using a flow type helium cryostat and a conventional constant acceleration type spectrometer in transmission geometry with a ⁵⁷Co(Cr) source, which had an activity of 5 mCi and was kept at 300 K. The Mössbauer spectra were analyzed using the SPECTR program from the MSTools package [36] in the Lorentz line approximation. Isomer shifts were determined in relation to the spectrum centroid of α-Fe.

Magnetic, thermal, and dielectric properties of PMO were investigated on single crystals. Magnetization data was recorded as a function of temperature and magnetic field on an MPMS squid magnetometer and a PPMS system with VSM option from Quantum Design Inc. In the same setup, heat capacity data was collected using a relaxation method, and magneto(di)electric and pyroelectric measurements were performed under magnetic fields using external LCR meter and electrometer.

The Supplemental Material gives further experimental details.

The stoichiometry of the single crystals was investigated using EDS and iodine titration. A Pb:Mn cation ratio of 1:2.335(2) was found. For the crystal selected for X-ray single crystal diffraction and study of physical properties, the average oxygen index δ was positive and of the order of 0.02(1). In the case of the powder sample which was used for neutron powder diffraction studies, an additional batch of single crystals was used and the resulting δ was of the order of −0.03(1).

Single-crystal XRD data at 295 K indicates a hexagonal structure s.g. P6 ₃ /mcm, with lattice parameters: a = 9.979(1) Å, c = 13.598(3) Å. The main structural results are presented in Table 1 and Table SM1 in the Supplemental Materials. On the other hand, the best refinements of powder XRD data on crushed single crystals are obtained for an orthorhombic Pnma structure (see Fig. 2). Comparison of hexagonal and orthorhombic models derived from XRPD data gave worse results for first one (R_p, R_wp, R_B = 2.39, 3.46 and 4.79 respectively for hexagonal model, and 1.96, 2.73, and 4.03 for orthorhombic one) and the peak shape of several reflections clearly showed a splitting (see inset of Fig. 2); thus confirming that the space group of PMO is Pnma or Pn2 ₁ a. Centrosymmetric model Pnma was selected based on SHG results. Conversely, high-resolution diffraction studies using X-ray single crystal data showed no detectable metric deviation from a hexagonal lattice. Refinement of the same experimental set of intensities based on orthorhombic model (with increased number of refined parameters) gave us a worse agreement between the measured data sets and calculations (R_B = 4.6 vs. 3.96). Crystallographic results for PMO and associated polyhedral analysis are summarized in Tables 1 and 2. As illustrated in Fig. 3, the orthorhombic and hexagonal structures are closely related, and may be compared, considering that the hexagonal basal plane (a,b) corresponds to the (b,c) plane in the orthorhombic structure (a_ortho ≈ c_hexa, b_ortho ≈ 2 a_{hexa +} b_hexa, and c_ortho ≈ b_hexa).

The existence of pseudo-symmetry in the PMO crystal structure is indicative of a distorted structure of higher symmetry. If the distortion is small enough, it can be expected that the crystal acquires this more symmetric configuration at a higher temperature. At high temperatures, hexagonal symmetry with space group P6 ₃ /mcm is indeed observed [17, 28], while as the temperature decreases, a distortion occurs and the structure becomes orthorhombic, with space group Pnma, i.e. a subgroup of P6 ₃ /mcm.

The displacive phase transition from P6 ₃ /mcm to Pnma in PMO can be quantified in terms of the amplitudes of the condensing modes, which generate the observed static distortion. The character and number of these modes can be derived by the group-subgroup relation between the hexagonal and the orthorhombic phases. The amplitudes, character, and number of condensing modes were derived using the AMPLIMODES software [37, 38]. Given the high- and the low-symmetry structures, AMPLIMODES determines the atomic displacements that relate them, defines a basis of symmetry-adapted modes, and calculates the amplitudes and polarization vectors of the distortion modes. The results of this decomposition are reported in Table SM2 in the Supplemental Materials. The structural transition is associated with three main modes. These primary GM1+, GM5+ and M2− modes have the largest amplitudes with values of 0.44, 0.76 and 1.36 Å, respectively. We find a total distortion amplitude (the amplitude of the sum of all condensing mode vectors) of 2.29 Å with atomic displacements ranging from 0.05 to 0.36 Å.

Figure 4 shows the temperature dependence of the dielectric parameters ε and tanδ recorded under magnetic fields up to 9 T (E along the hexagonal c-axis, i.e. [001], H along [1–10] in the (a, b)-plane). ε increases sharply at around 100–200 K depending on the frequency, qualitatively in a similar way as earlier reported [30] (note that the crystals dimensions—area and thickness after polishing—were only roughly estimated in our experiments, and that other experiments suggest ε ~ 30 at low temperatures, akin to that estimated in [30]). The sharp steps in ε correspond to well-defined maxima in the tanδ curves. Fitting the observed relaxation to the Arrhenius behavior τ = τ_oexp(E_a/k_BT) [40, 41], (k_B is the Boltzmann constant, τ = 1/2πf) yields an activation energy E_a ~ 0.18 eV and a relaxation time τ_o ~ 10⁻¹¹ s. The derived relaxation parameters (Ea, τ_o) which were found for PMO are characteristic of polaron relaxation [30]. As seen in Fig. 4, some magnetic field dependence, albeit weak, is observed in ε and tanδ curves. No magneto-(di)electric effects could be observed, as in the earlier dielectric study of PMO reported in [30].

The Mössbauer spectra of the Fe-doped PMO sample registered just above T_N (70 K) and at low temperatures (10 K) are shown in Fig. 5. The latter spectrum demonstrates a typical broadened sextet of lines showing the presence of a magnetic order while the latter indicates a typical paramagnetic doublet and hence shows the absence of both long and short range magnetic order. The spectrum at 10 K was fitted using a superposition of two sextets and a singlet (Fig. 5, lower panel). The spectrum at T = 70 K was fitted using a superposition of two paramagnetic doublets and a singlet (Fig. 5, upper panel). The main parameters extracted from the Mössbauer spectra, isomer shift—IS with respect to metallic iron, quadrupole splitting—QS, the effective magnetic field H and the relative area A of partial spectra, are collected in Table 3. From the obtained parameters of the sextets and doublets it is possible to conclude that most of the Fe ions (average on two spectra A = 91(1)  %) are in the six-fold oxygen coordination in Fe³⁺ high-spin state. As for the singlet partial spectrum (A = 9(1)  %) it is likely due to an impurity phase containing Fe³⁺ ions in the four-fold oxygen coordination with cubic symmetry. These data are consistent with earlier reported Mössbauer data obtained at room temperature [28], according to which for low concentrations of Fe doping into the 4 Mn sites (Mn1 (12i), Mn2 (8 h), Mn3 (6f), and Mn4 (2d); using the hexagonal space group P6₃/mcm), the Fe atoms were mainly localized in Mn2 (8 h) and Mn3 (6f) sites.

Magnetic properties of PMO single crystals have been reported earlier [27]. A paramagnetic behavior was observed above 250 K followed at lower temperatures by several magnetic transitions. A first anomaly near 160 K was interpreted as short-range antiferromagnetic ordering within the basal planes of the structure. Then at 67 K long-range order occurs where, although the overall structure is antiferromagnetic, weak ferromagnetism appears due to incomplete cancellation of the spins. Finally, at 20 K another transition interpreted as a reorientation of the spins was reported [27].

The magnetization of our single crystals was measured as a function of temperature and magnetic field (Figs. 6 and 7). Low-field magnetization M and heat capacity data (inset; plotted as C/T) suggest an antiferromagnetic transition near 67 K, with an excess moment in the hexagonal basal plane, similar to that reported in Ref. [27]. The remanent magnetization at 2 K in hysteresis loops amounts to 0.38 emu/g (c.f. about 0.5 emu/g in Ref. [27]), which corresponds to a moment of 0.09 μ_B/f.u.

As seen in the M-T data recorded in a larger field (H = 1 kOe, Fig. 6) in the plane along the [1–10] direction, an additional anomaly is observed (broad maximum near T* ~ 125 K). This is reminiscent of a similar anomaly in PMO compounds near 160 K interpreted to be related to short-range antiferromagnetic order and/or charge-orbital order in Refs. [26, 27]. The magnetization does not follow a Curie–Weiss behavior in the range of temperature investigated in Fig. 6. However, a Curie–Weiss temperature θ_CW ~ 520 K was reported in a study of magnetic properties up to higher temperatures [27], thus yielding a frustration parameter f = θ_CW/T_N ~ 8.

Furthermore, low-temperature behavior of the ZFC magnetization curve recorded in 1 kOe (Fig. 6) and M-H hysteresis curves recorded with the magnetic field applied along two different directions in the hexagonal basal plane (Fig. 7) suggest some additional in-plane anisotropy of the excess moment; M-H curves recorded with H//[1–10] showing a two-step switching behavior [42].

The neutron powder diffraction patterns collected above (70 K) and below (1.8 K) T_N are presented in Fig. 8. The neutron patterns show additional contributions of intensity for several nuclear peaks below T_N. All observed superstructure magnetic reflections could be indexed with a magnetic propagation vector k = (0, 0, 0). This means that no loss in translational symmetry arises from the spin arrangement and identity of the magnetic and chemical cells. The chemical space group of PMO remains Pnma down to 1.8 K according to our neutron diffraction data. The nine independent magnetic Mn ions in PMO are distributed on four different sites of the space group Pnma, namely the: 8d (5 Mn), 4a (1 Mn), 4b (1 Mn) and 4c (2 Mn) sites. This combination of magnetic cations produces 9 independent magnetic sublattices associated with these sites (the coordinates are given in Table SM1 in the Supplemental Materials and a graphical representation is presented in Fig. 9). In order to generate all possible magnetic structures that are compatible with the crystal structure, we have used the program for calculating irreducible representations BASIREPS from the FULLPROF suite [33]. In the framework of the representation analysis, which has been developed by Bertaut [43], it was shown that there are only eight possible one-dimensional irreducible representations Gi (i = 1, 2,…, 8) of mmm associated with the propagation vector k = (0, 0, 0). They are summarized in Table 4 for the case of the crystallographic space group Pnma. The corresponding magnetic configurations for the 8d, 4a, 4b and 4c sites (Table 4) are also given. The available positions allow one ferromagnetic (F) and three antiferromagnetic (G, C and A) configurations or modes: F = S1 + S2 + S3 + S4, and G = S1 − S2 + S3 − S4, A = S1 − S2 − S3 + S4, C = S1 + S2 − S3 − S4 where Sn indicates the spin for the n-fold position. The available set of magnetic reflections allows a preliminary statement to be made concerning the magnetic structure model. Firstly, it was considered that all 9 Mn cations carry magnetic moments and that the basis functions for 8d, 4a, 4b and 4c belong to the same irreducible representation (this is usually the case). The representations G2, G4, G6 and G8 exclude any magnetic moment on Mn cations at 4a and 4b and cannot take part in the solution. The remaining models were hence G1 (AxGyCz), G3 (GxAyFz), G5 (CxFyAz) and G7 (FxCyGz). As an additional condition the susceptibility measurements were used, which suggest a canted antiferromagnetic structure and non-compensated magnetic moment in (b,c) plane (i.e. the (a,b) plane in the hexagonal setting considered in magnetic measurements). The lower panel of Fig. 8 shows a typical neutron powder diffraction pattern of PMO measured at 1.8 K as well as the results of the Rietveld refinement. The nuclear and magnetic structures were refined simultaneously in the antiferromagnetic region. Detailed symmetry analysis of the magnetic structure shows that our experimental NPD pattern corresponds to the magnetic configuration G3 (GxAyFz).

Good agreement between the observed and calculated intensities of the PMO neutron diffraction patterns (see Fig. 8 as an illustration) indicates the correctness of the model used. The resulting magnetic structure at 1.8 K is depicted in different views in Fig. 10. If the spin components of 9 different Mn cations belong to various one-dimensional irreducible representations, different magnetic groups would be involved and the real magnetic symmetry could be the intersection of these different magnetic groups. However the case where spin components belong to different representations was outside the scope of our analysis.

The magnetic structure of PMO is characterized by commensurate magnetic ordering of the magnetic moments on nine magnetic sites with mainly antiferromagnetic interactions between edge-sharing Mn–octahedra within partially filled Kagomé planes. The magnetic moments of the different sites are not identical and vary in the frame of 1.2(5)–4.4(4)μ_B (see also Table 4). However we recognize that this value can also be found to be significantly higher, e.g. ∼5.0 μ_B. An explanation for the deviation from a higher value could be given by the fact that the magnetic moment is lowered from the covalency of bonds and such covalent effects might be different for existing Mn sites. For PMO, we have found that the magnetic moment of the Mn6 site is located mainly along the a-axis (see Table 4). The Mn1 and Mn4 cations (8d) were found to show basically the same magnetic moment of ~3.2 μ_B but oppositely oriented along the c-axis. The Mn2 and Mn3 cations (8d) were found to have similar magnetic moments of ~3.6 μ_B and oppositely oriented along the a-axis. Further analysis shows that the magnetic moments of several of the cations refined along the b-axis are relatively small and there is a clear tendency that the smallest magnetic moment component is found along this direction (see Table 4). The y-component of Mn5, both the y and z components of Mn6 and the z-component of Mn2, Mn3 and Mn9 spins were slightly above or within standard deviation values (0.4–0.5). These components were set to zero in the final stage of the refinements. The proposed model of the magnetic structure G3 (GxAyFz) with resulting orientation of overall ferrimagnetic moments along the c-axis agrees well with our magnetic measurements and an earlier study from Volkov et al., who reported that the magnetic easy axis must lie within the (011)-plane (referred to as the hexagonal (001)-plane) [27].

Interplay of ferromagnetic and antiferromagnetic interactions in PMO is plausible due to the numerous inequivalent exchange pathways (see lower panel of Fig. 9) and the competition between antiferromagnetic (superexchange) and ferromagnetic (super exchange and double exchange) interaction [44‐46]. Considering these possible exchange interactions, AFM alignment seems to be the most favourable interaction for Mn spins within the Kagomé planes. However, the triangular arrangements of magnetic moments with each pair coupled antiferromagnetically tend to hinder the system to find a unique ground state, i.e., the geometrical frustration [47, 48]. As seen in the lower panel of Fig. 9, the average Mn–Mn distances vary between 2.44 and 3.65 Å; Mn1, Mn3 and Mn9 being the closest, as indicated by the shaded Mn-triangles in Fig. 9. The chemical formula Pb₃Mn₇O₁₅ indicates a mixed-valence system with coexisting of Mn³⁺ and Mn⁴⁺ (3 Mn⁴⁺ and 4 Mn³⁺ per formula unit). The distribution of Mn³⁺/Mn⁴⁺ between possible sites is influenced by the lone electron pairs of Pb²⁺ cations. Bond valence calculations (see Table 2) indicate some charge ordering of Mn³⁺ and Mn⁴⁺ cations for PMO but a complete description of this charge ordering is still under investigation [26, 28]. Considering the d⁴ electronic configuration of Mn³⁺ and associated Jahn–Teller distortions, it was suggested that the Mn³⁺ ions would occupy the Mn4 and Mn5 positions while the Mn⁴⁺ ions would occupy the less distorted Mn6 and Mn7 positions [16]. Such a distribution gives a Mn³⁺/Mn⁴⁺ ratio of 4:3 which is equal to the value obtained when assuming that the unit cell has a zero electric charge. Another argument supporting this model was that the oxygen octahedra around Mn⁴⁺ ions are almost undistorted. From Table 2 it is clear that all of the Mn-octahedra are differently distorted. However, it is not straightforward to determine the Mn³⁺ and Mn⁴⁺ sites from crystal data alone. One possible way is to place Mn⁴⁺ at sites with shortest average Mn–O distance around them.

The structural, magnetic, dielectric properties of Pb₃Mn₇O₁₅ have been investigated in detail. Neutron powder diffraction on crushed single crystals is used to determine the low-temperature antiferromagnetic magnetic structure. Mn has average valence +3.4 in the material, and is distributed onto 9 independent octahedral sites in a specific orthorhombic structure with partly filled Kagomé layers connected by ribbons of edge-sharing MnO₆ octahedra and intercalated Pb²⁺ cations. Structural distortion originates from lone-electron pair of Pb²⁺ cations and from the presence of particular charge ordering of Mn³⁺ and Mn⁴⁺ cations in the octahedral sites, 57 % of which are occupied by the Jahn–Teller active Mn³⁺ cations. We discuss in detail the associated geometrical frustration, and complex interplay between structural distortions and magnetic interaction paths in Pb₃Mn₇O₁₅. It is unclear whether the oxygen content may influence the phase stability in PMO, as the oxygen index δ was slightly larger than zero for single-crystals with hexagonal space group, and slightly lower than zero for the powders with orthorhombic one.