# Fileset [PhysRevB.110.134426-PrMn7O12-neutron.pdf](https://mdr.nims.go.jp/filesets/3afb41e6-0690-412c-b648-965df9891ae1/download) ## Creator Dylan Behr, [LIU Ran](https://orcid.org/0000-0002-1659-2325), [YAMAURA Kazunari](https://orcid.org/0000-0003-0390-8244), [BELIK Alexei](https://orcid.org/0000-0001-9031-2355), Dmitry D. Khalyavin, Roger D. Johnson ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Magnetic structures of

{PrMn}_{7} O_{12}

: Intersublattice magnetoelastic coupling and incommensurate spin canting](https://mdr.nims.go.jp/datasets/29145e9e-b3d9-4cf4-a59c-a8bf1a2dc9d3) ## Fulltext Magnetic structures of ${{\rm PrMn}_7{\rm O}_{12}}$: Intersublattice magnetoelastic coupling and incommensurate spin cantingPHYSICAL REVIEW B 110, 134426 (2024)Magnetic structures of PrMn7O12: Intersublattice magnetoelastic couplingand incommensurate spin cantingDylan Behr,1,* Ran Liu,2,3,† Kazunari Yamaura ,2,3 Alexei A. Belik ,2 Dmitry D. Khalyavin,4 and Roger D. Johnson 1,51Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom2Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan3Graduate School of Chemical Sciences and Engineering, Hokkaido University, North 10 West 8, Kita-ku,Sapporo, Hokkaido 060-0810, Japan4ISIS facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot OX11 0QX, United Kingdom5London Centre for Nanotechnology, University College London, London WC1E 6BT, United Kingdom(Received 6 August 2024; accepted 1 October 2024; published 16 October 2024)We report on the magnetic structures of PrMn7O12, resolving both a high-temperature structure in commonwith the other R3+Mn7O12 systems and an additional low-temperature phase characterized by a superposed in-commensurate (ICM) spin canting. A phenomenological model is developed to account for large magnetoelasticcoupling immediately below the Curie temperature, demonstrating that A′B exchange is primarily responsible forthe ferrimagnetic structures observed across the R3+Mn7O12 family. Finally, an analytical mean-field minimalmodel is presented that successfully accounts for all magnetic structures observed in this family of materials,including the canted commensurate and ICM ground states.DOI: 10.1103/PhysRevB.110.134426I. INTRODUCTIONThe ABO3 (B = Mn) perovskite manganites have under-gone intensive study since the late 1940s, a period duringwhich research was spurred on by successive milestones ofsynthesis and theoretical and experimental discovery [1,2].These include the observation of metal-insulator transi-tions [1], colossal magnetoresistance [3], multiferroic andmagnetodielectric behavior [4], as well as demonstration offunctionality as catalysts [5], electrolytes [6], and thermoelec-tric ceramics and sensors [7]. Their structural and chemicalflexibility underpins their diverse physical properties, makingthem canonical systems for the study of structure-propertyrelationships in strongly correlated electron systems.Owing to their high-pressure and high-temperature syn-thesis requirements, the AA′3B4O12 (A′ = Mn, B = Mn)quadruple perovskite manganites (QPMs) are relatively lesswell studied, and significant ambiguities still exist in thecharacterization of their structural, electronic, and magneticfeatures [8,9]. Derived from their simple perovskite ABO3analogs, QPMs are distinguished by a chemical orderingwhere three out of four A-site cations (here labeled A′) are*Contact author: dylan.behr.20@ucl.ac.uk†Present address: Institute of Scientific and Industrial Research,Osaka University, Mihogaoka 8-1, Ibaraki, Osaka 567-0047, Japan.Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.replaced by Mn, stabilized by a pattern of large octahedraltilts (a+a+a+ in Glazer notation) [10,11]. The resulting aris-totypical structure has cubic Im3̄ symmetry and hosts twodistinct sublattices of magnetic Mn cations (A′ and B). Thevarious structures exhibited in QPMs are strongly influencedby the choice of A-site cation: The A-site valence determinesthe proportion of Mn3+ : Mn4+ on B sites, which in turn de-termines the ratio of Jahn-Teller (JT) active octahedral MnO6units to non-JT-active units [9,12–16]. Here, A3+ cations per-mit homovalent Mn3+ ions at all B sites, which typicallystabilize monoclinic I2/m crystal structures at room temper-ature [17,18] due to orbital order analogous to that observedin LaMnO3, where MnO6 octahedra are alternately elongatedalong approximately a and c in the ac plane, indicating acheckerboard arrangement of half-occupied d3x2-r2 and d3z2-r2orbitals of the Mn3+ 3d4 B-site cations [19–21].Based upon the Goodenough-Kanamori-Anderson(GKA) [2,22,23] rules taken in the limit of 180◦ Mn-O-Mnbond angles, the orbital order is expected to stabilize anA-type antiferromagnetic (AFM) structure on the B sites, withferromagnetic (FM) exchange between half-filled and emptyeg orbitals oriented within the ac plane and AFM exchangebetween empty eg B-site orbitals separated along b. However,complicating this network of B-site magnetic exchange isthe presence of (i) the additional magnetic A′-site sublatticeof Mn3+ ions and (ii) the pattern of large octahedral tilts.The former extends the total set of exchange interactionspresent, introducing A′B and AA′ nearest-neighbor exchange,which may compete with and frustrate BB exchanges and oneanother. The latter can modulate the strength of competingexchange pathways between magnetic ions, which dependstrongly on M-O-M bond angles.2469-9950/2024/110(13)/134426(9) 134426-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-0390-8244https://orcid.org/0000-0001-9031-2355https://orcid.org/0000-0001-7241-792Xhttps://ror.org/02jx3x895https://ror.org/026v1ze26https://ror.org/02e16g702https://ror.org/01t8fg661https://ror.org/03gq8fr08https://ror.org/04ptp8872https://ror.org/02jx3x895https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.110.134426&domain=pdf&date_stamp=2024-10-16https://doi.org/10.1103/PhysRevB.110.134426https://creativecommons.org/licenses/by/4.0/DYLAN BEHR et al. PHYSICAL REVIEW B 110, 134426 (2024)Authors of previous studies on the R3+Mn7O12 systemshave identified a �-point ferrimagnetic (FIM) phase com-mon to all trivalent-R quadruple perovskites, comprising aC-type AFM structure on B sites (AFM planes staked FM)and a C-type-derived FIM structure on A′ sites which or-der concomitantly on cooling through the magnetic transitiontemperature T1 [24]. Indeed, these magnetic structures areobserved across R = La, Ce, Nd, Sm, and Eu QPMs, withmagnetic transition temperatures of T1 = 79.5, 80, 83, 85, 87,and 87 K, respectively, increasing with decreasing ionic ra-dius. Furthermore, trivalent-A QPMs BiMn7O12, DyMn7O12,and YMn7O12 exhibit the same or similar FIM modes astheir aforementioned family members, albeit with additionalexotic magnetic and structural phenomena: BiMn7O12 hostsadditional magnetic modes, including a polar AFM E-typeorder that couples to a polar distortion of the crystal struc-ture [25], while YMn7O12 and DyMn7O12 exhibit emergentdipolar glass transitions that, in the case of DyMn7O12, coupleto the magnetic order parameter [26,27].In R3+Mn7O12 compounds with a larger rare-earth ion,namely, R = La and Nd, a further magnetic transition isobserved on cooling below T2 = 22.5 and 12 K associ-ated with the onset of superposed orthogonal modes on Bsites, with propagation vectors (0,1,0) and (kx, 1, kz ), respec-tively. In LaMn7O12, the additional magnetic mode resemblescommensurate (CM) A-type AFM order, with AFM spin cor-relations between nearest-neighbor B sites along b and FMcorrelations along a and c. The additional magnetic modein NdMn7O12 is similar, augmented by an incommensurate(ICM) modulation within the ac plane. In BiMn7O12, whichalso hosts a large A-site cation, a further magnetic mode withpropagation vector (0,1,0) is also observed below T3 = 27 K,although details of the associated magnetic structure have notyet been determined. Whether CM or ICM with the nuclearlattice, these superposed magnetic modes manifest as spincanting of the higher-temperature FIM magnetic structures,resulting in highly noncollinear ground states.The apparently universal behavior below T1, contrastedwith the exotic noncollinear CM or ICM ground states foundonly in certain compositions below T2, demands a generalmodel of magnetism in the QPMs that outlines the respon-sible set of interactions for the observed structures and trends.Herein, we make progress toward this aim through the studyof PrMn7O12, an as yet relatively understudied member of theR3+Mn7O12 family. PrMn7O12 has been reported to crystallizein both trigonal (R3̄) and monoclinic (I2/m) polymorphs [28].The monoclinic polymorph, pertinent to this paper, has beenobserved from DC-susceptibility measurements to undergo amagnetic transition at T1 concomitant with the development ofa net macroscopic magnetization. However, to our knowledge,no microscopic study of magnetism has yet been performedon the system. In this paper, we report on the magneticstructures of monoclinic PrMn7O12 based on neutron pow-der diffraction (NPD) experiments, and we evidence strongmagnetoelastic coupling in the system. A phenomenologicaltheory of the magnetoelastic coupling is formulated whichis then mapped onto the microscopic degrees of freedompresent. Our observations and analyses are used to inform ageneral model of magnetism in the QPM family, accountingfor the presence of pure �-point FIM, CM, and ICM cantedground states, outlining the competing instabilities responsi-ble. The paper is organized as follows: In Sec. II, we brieflydescribe the experimental methods; in Sec. III, we presentour experimental results from which the magnetic phases ofPrMn7O12 are characterized; and in Sec. IV, we present aphenomenological model for intersublattice magnetoelasticcoupling and comment on the wider magnetic phase dia-gram for the RMn7O12 family derived from an empiricallyconstrained minimal mean-field Heisenberg model presentedin Appendix B. Finally, we summarize our conclusions inSec. V.II. EXPERIMENTPolycrystalline PrMn7O12 samples were prepared fromstoichiometric mixtures of Mn2O3, Mn3O4, and Pr6O11. Themixture was placed in four Pt capsules and treated at 6 GPaand ∼1600 K for 2 h (heating time to the synthesis tem-perature was 10 min) in a belt-type high-pressure apparatus.After the heat treatments, the samples were quenched to roomtemperature, and the pressure was slowly released. All ob-tained samples were black pellets. Magnetic susceptibilitymeasurements were performed on a Quantum Design mag-netic property measurement system (MPMS-3) between 2 and100 K in a 100 Oe field under both zero-field-cooled (ZFC)and field-cooled-on-cooling (FCC) conditions. A magnet-reset procedure was applied before the ZFC measurement.Isothermal magnetization measurements were performed be-tween −70 and 70 kOe at 5, 25, and 60 K. Heat capacitymeasurements were made in 0 and 90 kOe fields using a Quan-tum Design physical property measurement system (PPMS).NPD measurements were performed on the WISH time-of-flight diffractometer [29] at ISIS, the UK neutron and muonspallation source. The sample (∼0.65 g) was lightly packedinto a thin 3-mm-diameter cylindrical vanadium can andmounted within a 4He cryostat. Data were collected with highcounting statistics at a fixed temperature within each magneticphase, including the paramagnetic phase for reference. Datawere also collected with lower counting statistics on warmingin the temperature range 1.5–300 K to ascertain thermal trendsin crystal structure and magnetic order. All diffraction datawere refined using FULLPROF [30]. Symmetry analyses wereperformed using the ISOTROPY software suite, particularly theISODISTORT tool for modal decomposition of nuclear struc-tures and identification of magnetic normal modes, and theISOTROPY interactive command line tool for the systematicderivation of invariants [31,32].III. RESULTSSpecific heat capacity [Fig. 1(a)] and magnetic suscepti-bility [Fig. 1(b)] measurements indicate the presence of twomagnetic transitions at T1 = 82 K and T2 = 12 K, the first ofwhich coincides with the development of a bulk magnetic mo-ment [Fig. 1(b)]. Indeed, a significant remnant magnetizationwas measured in isothermal field-dependent magnetizationmeasurements at 60, 25, and 5 K, consistent with long-range (FM) FIM order (see Fig. 1 inset). Both specific heatanomalies shift to higher temperature in an applied field,with the lower-temperature feature becoming sharper and the134426-2MAGNETIC STRUCTURES OF PrMn7O12: … PHYSICAL REVIEW B 110, 134426 (2024)FIG. 1. (a) Specific heat capacity over temperature of PrMn7O12in zero applied field and 9 T. Magnetic transitions inferredfrom anomalous features in the heat capacity are indicated byvertical dashed lines. (b) Zero-field-cooled (ZFC) and field-cooled-on-cooling (FCC) magnetization measurements of polycrystallinePrMn7O12 under an applied DC field of 100 Oe. The inset depictsisothermal moment vs field measurements at temperatures 5, 25, and60 K.higher-temperature feature becoming broader. The origin ofthis behavior is not clear, but the observed changes at T1 areconsistent with the field coupling to the net magnetization.NPD data collected with high counting statistics at 100 Kwithin the paramagnetic phase of PrMn7O12 were usedto refine a structural model based on the published I2/mphase [28], with excellent agreement achieved [see Fig. 2(a)].The structural parameters are detailed in Table I. No signif-icant impurities were found in the sample, with all nuclearTABLE I. Crystal structure parameters of PrMn7O12(space group I2/m) refined at 100 K. The lattice parameterswere determined to be a = 7.49813(7) Å, b = 7.34352(6) Å,c = 7.49291(7) Å, and β = 91.2485(8)◦. Atomic Wyckoff positionsare Pr: 2a [0,0,0]; Mn1: 2d [ 12 , 12 , 0]; Mn2: 2c [ 12 , 0, 0]; Mn3: 2b[0, 12 , 0]; Mn4: 4f [ 34 , 34 , 14 ]; Mn5: 4e [ 34 , 34 , 34 ]; O1,O4: 8j [x, y, z];O2,O3: 4i [x, 0, z]. Uiso values were constrained to be the same forall Mn and all O, respectively. They were, Pr: 0.2(1); Mn: 0.6(4); O:0.7(2) in units ×10−2 Å2. The Pr site was occupied by 91% Pr, 9%Mn.Atom x y zO1 0.9851(3) 0.6906(3) 0.8272(3)O2 0.8210(4) – 0.3125(4)O3 0.1673(3) – 0.3073(3)O4 0.6890(2) 0.8252(3) 0.0130(2)FIG. 2. Neutron powder diffraction patterns with Rietveld re-fined models in the (a) paramagnetic, (b) FIM I, and (c) FIM IIphases. The observed intensity pattern is depicted with red pointswhile the calculated intensity is overlaid in black. Magnetic inten-sities appearing below T1 and T2 are filled in lime green. Greenticks below indicate the calculated positions of reflections originatingfrom the nuclear structure and �-point magnetic order (upper row),and incommensurate (ICM) order with propagation vector (kx, 1, kz )(lower row). All inset plots of (a) and the left-hand insets of (b) and(c) depict the fitting of selected peaks from a high-resolution detectorbank, while those depicted to the right in plots (b) and (c) highlight aselected region of the main plots ∼7.1 Å.peaks being accounted for by the I2/m model. It is worthstressing that the R3̄ is absent from our sample. Refinementsindicate A sites are partially occupied by Mn, with Mn ac-counting for ∼10% and Pr the remaining 90%.Below T1, additional magnetic intensities appeared at posi-tions of allowed nuclear intensity, unambiguously indexing atkχ = (0, 0, 0) [see Fig. 2(b)]. Informed by the results of prioranalyses on isostructural RMn7O12 systems [24], a magneticstructure transforming as the m�+2 irreducible representation(irrep) of the paramagnetic I2/m parent was found to faith-fully reproduce the observed magnetic intensities (see Fig. 3).Here, B sites order with a Cb-type AFM structure wherenearest-neighbor spins separated along b are coaligned, whilethose along a and c are antialigned. Here, A′ sites are found toorder simultaneously with a FIM mode derived from a Ca-typestructure, whereby nearest-neighbor spins separated along aare coaligned, while those along b and c are antialigned.The moments on A′ and B sublattices are collinear with one134426-3DYLAN BEHR et al. PHYSICAL REVIEW B 110, 134426 (2024)FIG. 3. The refined magnetic structure of PrMn7O12 in the FIMI phase at 40 K. The A′ sublattice is presented in (a) with sites Mn1,Mn2, and Mn3 in red, green, and blue, respectively, while the Bsublattice is presented in (b) with sites Mn4 and Mn5 in silver andgold, respectively.another and lie within the ac plane. No ordered moments werefound on the Pr A site. That the modes on both Mn sublat-tices transform as the same irrep of the I2/m parent permitstheir cross-coupling and simultaneous ordering. We hereafterrefer to the A′- and B-site modes as χa and χb, respectively(see Tables II and III for the definition of magnetic modes).The corresponding phase has one uncompensated Mn A′-sitemoment per formula unit and is labeled FIM I, which wellaccounts for the observed net magnetization (see Table IV fordetails).On cooling below T2 the �-point magnetic intensities con-tinue to grow monotonically, while a new broad magneticintensity appears in the vicinity of ∼7.1 Å that indexeswith the propagation vector kα = (0.207(6), 1,−0.03(2))[see Fig. 2(c)]. Systematic testing of candidate structures atthe indexed propagation vector demonstrates that the diffrac-tion pattern is well fit by an additional ICM AFM mode onthe B sites that resembles an Ab-type structure in the k =(0, 1, 0) CM limit (nearest neighbors separated along b areantialigned, while those along a and c are aligned), augmentedTABLE II. k = (0, 0, 0) magnetic modes of PrMn7O12 defined interms of relative mode amplitudes m of magnetic sites in the y = 0(A′ sites) and y = 14 (B sites) planes of the unit cell. Sites relatedby I-centering have the same phase as per the propagation vector.Subscripts differentiate between the unique directions of χ AFMmodes, while superscript i denotes any associated moment directionsin a Cartesian basis coincident with the parent cubic axes.Frac. coords. Mag. modesA′ site x y z F i(A′) χ ia(A′) χ ib(A′) χ ic(A′)Mn1 0 0 12 +m +m +m −mMn2 12 0 0 +m −m +m +mMn3 12 0 12 +m +m −m +mB site F i(B) χ ia(B) χ ib(B) χ ic(B)Mn41141434 +m +m +m +mMn51141414 +m −m −m +mMn42341414 +m −m +m −mMn52341434 +m +m −m −mTABLE III. k = (kx, 1, kz ) B-site magnetic modes of PrMn7O12defined in terms of relative mode amplitudes m of magneticsites in the y = 14 plane of a unit cell at lattice site R. The sitephases are modulated by the propagation vector as follows:m41 = m[exp(ik · R)]; m51 = m[exp(−ikz/2) exp(ik · R)]; m42 =m[exp(−i(kx + kz )/2) exp(ik · R)]; m52 = m[exp(−ikx/2) exp(ik ·R)]. Partner sites in the adjacent plane at y = 34 here are πout-of-phase with corresponding sites below as per the propagationvector. Notation and fractional coordinates are as given in Table II,with subscripts differentiating between the unique directions of αAFM modes.Mag. modesB sites γ i(B) αia(B) αib(B) αic(B)Mn41 +m41 +m41 +m41 +m41Mn51 −m51 +m51 +m51 −m51Mn42 +m41 −m42 +m42 −m42Mn52 −m52 −m52 +m52 +m52by a long wavelength modulation. Following the formalism ofprior works, we label this mode αb [24]. In refinements, a bestminimal model is achieved when the moments associated withthis mode are constrained to the ac plane in a direction perpen-dicular to the preexisting χb mode. In this case, the additionalαb magnetic component corresponds to a planar rocking of B-site moments about the average FIM I structure. However, ourdata analysis could not exclude a conical rotation of the spinsabout the FIM I structure (the same ambiguity was reportedfor the Nd system [24]). Indeed, it is typical for NPD thatspin density waves or helicoidal modulations cannot readilybe distinguished. The resulting canted magnetic structure islabeled FIM II, as shown in Fig. 4.The crystal and magnetic structures were refined againstvariable-temperature NPD data to establish the evolution ofstructural and magnetic properties. In Fig. 5(a), the mon-oclinic lattice parameter β is plotted against temperature,TABLE IV. Magnetic structure parameters refined in the FIMI and FIM II phases of PrMn7O12. Moment directions are givenin spherical coordinates defined such that mz = m cos(θ ) ‖ c, mx =m cos(φ) sin(θ ) ‖ a∗, and my = m sin(φ) sin(θ ) ‖ b, with φ fixed to0 to constrain moments to the ac plane. Here, m = [mx, my, mz] isthe polarization of the respective magnetic modes χa, χb, and αb,which are defined in terms of relative mode amplitudes m in Tables IIand III.T (K) 40 K 1.5 KMn1 χa χam (μB) 2.80(1) 3.33(1)θ (◦) 145.9(2) 144.6(1)Mn4 χb χb αbm (μB) 2.71(1) 3.22(1) 1.46(3)θ (◦) 145.9(2) 144.6(1) 54.6(1)R (%) 3.74 3.92wR (%) 3.72 4.17RMag (%) 0.97 2.23 7.82134426-4MAGNETIC STRUCTURES OF PrMn7O12: … PHYSICAL REVIEW B 110, 134426 (2024)FIG. 4. The refined magnetic structure of PrMn7O12 in the FIMII phase at 1.5 K projected along b, presented in planes of constanty as indicated and spanning 4 unit cells along a. The A′ sublatticeis presented with sites Mn1, Mn2, and Mn3 in red, green, and blue,respectively, while the B sublattice is presented with sites Mn4 andMn5 in silver and gold, respectively. The incommensurate αb mod-ulation is depicted here with thin black arrows superposed on theB-site moments, which rock about their average Cb structure.with magnetic transitions indicated. Remarkably, a significantanomalous increase in the monoclinic angle is exhibited oncooling below T1, which had otherwise increased with a shal-low linear gradient on cooling through the paramagnetic phase(depicted in Fig. 5(a) with the red dashed line). We define themagnetostrictive component of β as the deviation of β fromits linear trend in the paramagnetic state. It should be notedthat PrMn7O12 is not unique in exhibiting such anomalousbehavior of β at T1, with LaMn7O12 and BiMn7O12 tentativelyevidencing similar behavior [24,28]. Figure 5(b) depicts thethermal evolution of magnetic moments on A′ and B sublat-tices with the B-site moment decomposed into its χb and αbcomponents. In Fig. 5(c), the magnetostrictive β componentis superimposed on the squared magnetic moments on eachsublattice. The two quantities follow the same trend with tem-perature and indeed are shown to be proportionate in the insetof the figure. This result implies a linear-quadratic coupling ofthe lattice to the magnetic order parameter, which is discussedin detail in the following section.FIG. 5. (a) Distortion of monoclinic β angle (left axis, black)against temperature with the cube-rooted unit cell volume (rightaxis, green) overlaid. A linear fitting of β against temperature in theparamagnetic phase is depicted in red. The deviation of β from this fitis assumed to be magnetostrictive. (b) Thermal evolution of magneticmoment components on Mn3+ sites. Orthogonal B-site componentsassociated with �-point (χ ) and ICM (α) magnetic order are indi-cated with triangles and crosses, respectively. (c) Magnetostrictive(MS) change in β (right axis) with the χ -mode moments squared(left axis) overlaid. The inset depicts the average χ -mode momentacross A′ and B sites squared plotted against the change in β.IV. DISCUSSIONThe results outlined above invite a group-theoretical anal-ysis of the coupling of structural strain or displacive modes tothe �-point χi magnetic modes. Taking the Im3̄ QPM aristo-type as the parent, monoclinic distortions lowering the crystalsymmetry to I2/m transform as the (0, δb, 0) order parameterdirection (OPD) of the �+4 irrep. Among the symmetry-adapted normal modes of this irrep are those associated withcooperative JT distortions akin to those observed in LaMnO3.Indeed, it is the orbital instability of the octahedrally coordi-nated Mn3+ B sites that establish the monoclinic structure athigh temperature. Also, transforming by this irrep and OPDis a shear lattice strain described by the infinitesimal straintensor: ⎡⎣ 0 0 exz0 0 0exz 0 0⎤⎦.This lattice strain is precisely that associated with the devia-tion of β from its cubic value of 90◦. Hence, the anomalousincrease in β observed on cooling below T1 indicates a cou-pling of �+4 (0, δb, 0) structural distortions to the magneticmodes. It should be noted that, for all symmetry-adaptedmodes of the �+4 irrep, the special OPD simply determines134426-5DYLAN BEHR et al. PHYSICAL REVIEW B 110, 134426 (2024)TABLE V. �-point symmetry-adapted magnetic modes ofPrMn7O12 with respect to the Im3̄ parent. The modes are labeled Fand χ , as defined in Tables II and III. Subscripts differentiate betweenthe unique directions of χ and α AFM modes, while superscripts de-note the associated moment directions in a Cartesian basis coincidentwith the parent cubic axes.Irrep OPD A-site modes B-site modesm�+4 (ηa, 0, 0) F x, χ xa , χ xb , χ xc F x, χ yc , χ zb(0, ηb, 0) F y, χ ya , χyb , χ yc F y, χ za, χxc(0, 0, ηc ) F z, χ za, χzb, χzc F z, χ xb , χ yawhich of the parent basis vectors are established as the uniqueaxis of the I2/m isotropy subgroup, with (δa, 0, 0) establish-ing a, (0, δb, 0) establishing b, and (0, 0, δc) establishing c.We take b to be the unique axis, maintaining convention forthe monoclinic cell and ensuring that lattice basis vectors ofthe parent cell correspond to those of the subgroup.Of the magnetic modes observed, the FIM I structure trans-forms as the (ηa, 0, ηc) OPD of the m�+4 magnetic irrep. Asummary of the magnetic modes associated with each of theobserved irreps is presented in Table V. A systematic search offree-energy invariants coupling the �+4 structural distortionsto the m�+4 magnetic modes of the FIM I phase reveals thetrilinear invariant:F (�+4 : m�+4 ) = δaηbηc + δbηcηa + δcηaηb. (1)Notably, with reference to the magnetic modes in Table V,one can show that the above invariant cannot be mapped ontointrasublattice Heisenberg-like interactions consistent withthe empirically determined magnetic structure. Rather, the in-variant is mapped onto magnetostrictive Heisenberg-like A′Bexchange interactions. For example, the second term relevantto our b unique convention maps onto a mean-field exchangeenergy of the form:E (�+4 : m�+4 ) ∝ (β − β0)[χ xa (A′)χ xb (B) + χ za (A′)χ zb (B)],(2)where the bracketed letters indicate the sublattice hosting themagnetic modes and β0 = 90◦. While other intrasublatticeinteractions surely exist, the large magnetostrictive distortionnonetheless evidences that strong intersublattice interactionsbetween nearest-neighbor A′ and B sites are important in sta-bilizing the magnetic structure of the FIM I phase.The wider phase diagram is explored in Appendix B,where a minimal Heisenberg model is developed, heav-ily constrained to the empirical magnetic structures foundacross the RMn7O12 family. The model, which includes av-erage nearest-neighbor A′B and BB exchange (Fig. 6) andnext-nearest-neighbor BB exchange, captures all magneticstructures observed, including canted CM and ICM groundstates. It is apparent that no single tuning parameter explainsthe different phases observed across the rare-earth QPMs.However, our minimal model indicates that the larger rare-earth ions lead to a larger JBB/JA′B ratio, giving rise to spincanting, which are then separated in propagation vectors byJ ′BB/JBB.FIG. 6. Exchange interaction pathways viewed down the b axis,for slices through the unit cell 0 � y � 14 (left) and 14 � y � 12(right). The region 12 � y � 1 is equivalent by mirror symmetry. Allsymmetry-inequivalent A′B interactions are colored, with solid anddashed lines set to be ferromagnetic and antiferromagnetic, respec-tively, in the mean-field calculations. BB interactions are indicatedby thick black lines (next-nearest-neighbor BB exchange interactionsand BB exchange ‖ b are omitted for clarity), and the unit cellis shown in thin black lines. Transition metal atoms are numberedaccording to Table II, and Pr atoms are shown as yellow spheres.V. CONCLUSIONSIn summary, we have solved the magnetic structures ofPrMn7O12, resolving two distinct magnetic phases. The high-temperature magnetic structure of the FIM I phase resemblesthose observed in related trivalent-R QPM systems, consist-ing of FIM order on A′ sites and AFM order on B sites.At low temperature, an additional magnetic phase transitionoccurs to a highly canted FIM II phase, characterized bya superposed ICM modulation of the high-temperature or-der on B sites, resembling the ground state of NdMn7O12.We demonstrate strong magnetoelastic coupling between thesymmetry-adapted magnetic modes of the FIM I phase and asheer lattice strain, which can be understood in terms of a phe-nomenological model that demonstrates that A′B exchange isprimarily responsible for the FIM structures observed acrossthe R3+ QPM family. Additionally, an empirically constrainedanalytical minimal model is formulated and shown to captureall magnetic structures observed, including canted CM andICM phases.ACKNOWLEDGMENTSThis paper was partially supported by a Grant-in-Aid forScientific Research (Grant No. JP22H04601) from the JapanSociety for the Promotion of Science and the KazuchikaOkura Memorial Foundation (Grant No. 2022-11). MANAwas supported by the World Premier International ResearchCenter Initiative, MEXT, Japan.APPENDIX A: LATTICE PARAMETERSFigure 7 shows the monoclinic I2/m lattice parameters a,b, and c refined against NPD data measured as a function oftemperature. Note that the monoclinic angle β and unit cellvolume are shown in Fig. 5.134426-6MAGNETIC STRUCTURES OF PrMn7O12: … PHYSICAL REVIEW B 110, 134426 (2024)FIG. 7. Temperature dependence of the monoclinic I2/m latticeparameters a, b, and c.APPENDIX B: MEAN-FIELD HEISENBERG MODELIt was previously suggested that the second magnetic phasetransition found in RMn7O12 (R = La and Nd) at T2 likelyoriginates in the competition between A′B and BB exchange,where A′B interactions were assumed to be large (now con-firmed above), while BB exchange is compromised by thelarge degree of octahedral tilting [24]. It was further assumedthat A′A′ exchange is negligible. We now test this proposalby developing a minimal mean-field Heisenberg model. Westress that this model is greatly simplified by heavy constraintsbased upon the observed RMn7O12 magnetic structures. Assuch, it serves only as a proof of principle regarding the sta-bilization of CM and ICM canted ground states and indicatesthe hierarchy of exchange interactions, their competition, andtuning between different compositions.We begin by partitioning the mean-field spin Hamilto-nian into a term that stabilizes the �-point χ -type magneticstructure on the A′ and B sites and a term that stabilizes ak = (k, 1,−k) α-type magnetic structure on the B sites (notewe have set |kx| = |kz| for simplicity):H = Hχ + Hα, (B1)whereHχ = −48JA′BSχA′SχB + 24JBB(SχB)2 + 8J ′BB(SχB)2, (B2)andHα = −16JBB(SαB)2cos(πk) − 8JBB(SαB)2+ 8J ′BB(SαB)2cos(2πk), (B3)where SiA′ and SiB are the magnetic moment components onthe A′ and B sublattices, associated with the ith mode (i =χ, α). To make the problem tractable yet informative, wehave imposed the following simplifications. The A′B inter-actions parameterized as JA′B were set to all be equivalentin magnitude and with signs supporting the empirical FIM IFIG. 8. Phase diagrams in JBB and J ′BB depicting (a) the cantingangle θ , defined as tan(θ ) = SHBS�Band (b) component k in propagationvector (k, 1, −k). On both, the positions of R = Nd and Pr systemshave been marked using knowledge of their empirically determinedcanting angles and propagation vectors, while the canting angle ofthe La system has been used to mark the locus of points it mayoccupy on the phase diagram. The position of R = Eu, Sm, andCe systems could not be determined since they adopt pure FIM Iground-state magnetic structure, but given the positions of the otherrare-earth ions with similar atomic radii, one can assume they lie inthe vicinity of the triple point at JBB = 12 , J ′BB = 14 .structure (Fig. 6). Similarly, the nearest-neighbor BB interac-tions were set to be AFM along b and FM along a and c,all equal in magnitude parameterized by JBB. Next-nearest-neighbor BB interactions were implemented along the [0.5,0, −0.5] directions, parameterized as J ′BB, which introduce aninstability toward ICM α modes. Similar super-superexchangeinteractions were found to be significant in the orthorhombicperovskites due to the octahedral tilting distortions bringingtwo apical oxygen ions close together along this cubic per-ovskite face diagonal [33]. We impose the constant moment134426-7DYLAN BEHR et al. PHYSICAL REVIEW B 110, 134426 (2024)condition (Sχ )2 + (Sα )2 = S2, givingH = 8(SχB)2[4JBB + J ′BB + J (k)] − 48JA′BSχB SχA′− 8JBBS2 − 8J (k)S2, (B4)where the propagation vector-dependent part is defined asJ (k) = 2JBB cos(πk) − J ′BB cos(2πk). (B5)We have assumed that the α mode results in spin canting,which in the ICM case is manifest as an additional helicoidalorder with spin components in the plane perpendicular to theχ modes. Trivially, Sα = 0 corresponds to the collinear FIM Istructure. Differentiating with respect to k, one finds∂H∂k= 16π sin(πk)(S2 − SχB2)[JBB − 2J ′BB cos(πk)], (B6)which gives stationary points at(1) SχB = S, corresponding to a fully saturated momentand hence zero canting;(2) k = 0, which for SχB < S corresponds to the CM cantedstate of LaMn7O12;(3) cos(πk) = JBB2JBB′ , which for SχB < S corresponds to theICM canted states of NdMn7O12 and PrMn7O12.By evaluating the stability of these stationary points withSχB < S, one finds that the phase boundary between the CMand ICM canted phases is defined byJ ′BBJBB= 12. (B7)Differentiating with respect to SχB , one finds∂H∂SχB= 8{2SχB [4JBB + J ′BB + J (k)] − 6JA′BSχA′}, (B8)which yields a stationary point atSχBSχA′= 3JA′B[4JBB + J ′BB + J (k)]. (B9)The observed absence of canting on the A′ sites implies SχA′ =S, and hence, SχB /SχA′ � 1. Equation (B9) then gives4JBB + J ′BB + J (k) � 3JA′B. (B10)The boundary between the FIM I phase and the canted CMphase can then be obtained by setting k = 0 in Eq. (B10):JBBJA′B= 12. (B11)Substituting cos(πk) = JBB2JBB′ into Eq. (B10) yields the bound-ary between FIM I and the ICM canted phase, which isdescribed by the equation:J ′BBJA′B= 14[(3 − 4JBBJA′B)+√(3 − 2JBBJA′B)(3 − 6JBBJA′B)]. (B12)The above analytical results are summarized as a phasediagram in Fig. 8. The canting angle generated by orthogonalχ and α components is shaded in the top pane, while thepropagation vector component k is shaded in the lower pane.The above phase boundaries are indicated by solid white lines.Using the values of the canting angle and propagation vector,the approximate locations of QPM systems with differentrare-earth A-site cations are shown on the diagram. The resultssuggest that the large rare-earth systems R = Nd, Pr, andLa lie within the vicinity of the triple point JBB/JA′B = 12 ,J ′BB/JA′B = 14 . The ratio JBB/JA′B must be sufficiently large toproduce finite canting, while the CM La system lies belowthe phase boundary J ′BB/JBB = 12 , separating it from the ICMground states of the Nd and Pr systems. The rare-earth cationsystems exhibiting only the pure FIM I phase in the groundstate, i.e., R = Eu, Sm, and Ce, lie within the leftmost phasefor small JBB/JA′B. It is apparent that no single tuning parame-ter explains the different phases observed across the rare-earthQPMs. However, our minimal model indicates that the largerrare-earth ions lead to a larger JBB/JA′B ratio, which are thenseparated by J ′BB/JBB.[1] J. M. Coey, M. Viret, and S. Von Molnár, Mixed-valence man-ganites, Adv. Phys. 48, 167 (1999).[2] J. B. Goodenough, Theory of the role of covalence in theperovskite-type manganites [La,M(II)]MnO3, Phys. Rev. 100,564 (1955).[3] S. Jin, T. H. Tiefel, M. McCormack, R. A. Fastnacht, R.Ramesh, and L. H. Chen, Thousandfold change in resistiv-ity in magnetoresistive La-Ca-Mn-O films, Science 264, 413(1994).[4] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y.Tokura, Magnetic control of ferroelectric polarization, Nature(London) 426, 55 (2003).[5] I. Yamada, H. Fujii, A. Takamatsu, H. Ikeno, K. Wada, H.Tsukasaki, S. Kawaguchi, S. Mori, and S. Yagi, Bifunctionaloxygen reaction catalysis of quadruple manganese perovskites,Adv. Mater. 29, 1603004 (2017).[6] M. B. Hanif, S. Rauf, M. Motola, Z. U. D. Babar, C. J. Li,and C. X. Li, Recent progress of perovskite-based electrolytematerials for solid oxide fuel cells and performance optimizingstrategies for energy storage applications, Mater. Res. Bull. 146,111612 (2022).[7] Z. Shi, J. Zhang, J. Wei, X. Hou, S. Cao, S. Tong, S. Liu, X.Li, and Y. Zhang, A-site deficiency improved the thermoelec-tric performance of high-entropy perovskite manganite-basedceramics, J. Mater. Chem. C 10, 15582 (2022).[8] A. N. Vasil’ev and O. S. Volkova, New functional materialsAC3B4O12 (Review), Low Temp. Phys. 33, 895 (2007).[9] A. A. Belik, R. D. Johnson, and D. D. Khalyavin, The richphysics of A-site-ordered quadruple perovskite manganitesAMn7O12, Dalton Trans. 50, 15458 (2021).[10] A. M. Glazer, The classification of tilted octahedra in per-ovskites, Acta Cryst. B 28, 3384 (1972).134426-8https://doi.org/10.1080/000187399243455https://doi.org/10.1103/PhysRev.100.564https://doi.org/10.1126/science.264.5157.413https://doi.org/10.1038/nature02018https://doi.org/10.1002/adma.201603004https://doi.org/10.1016/j.materresbull.2021.111612https://doi.org/10.1039/D2TC02952Ahttps://doi.org/10.1063/1.2747047https://doi.org/10.1039/D1DT02992Dhttps://doi.org/10.1107/S0567740872007976MAGNETIC STRUCTURES OF PrMn7O12: … PHYSICAL REVIEW B 110, 134426 (2024)[11] M. V. Talanov, Group-theoretical analysis of 1:3 A-site-orderedperovskite formation, Acta Cryst. Sect. A 75, 379 (2019).[12] J. Chenavas, F. Sayetat, A. Collomb, J. C. Joubert, and M.Marezio, A perovskite-like compound, Solid State Commun.16, 1129 (1975).[13] S. V. Ovsyannikov, A. M. Abakumov, A. A. Tsirlin, W.Schnelle, R. Egoavil, J. Verbeeck, G. Van Tendeloo, K. V.Glazyrin, M. Hanfland, and L. Dubrovinsky, Perovskite-likeMn2O3: A path to new manganites, Angew. Chem. Int. Ed. 52,1494 (2013).[14] R. D. Johnson, L. C. Chapon, D. D. Khalyavin, P. Manuel,P. G. Radaelli, and C. Martin, Giant improper ferroelectricity inthe ferroaxial magnet CaMn7O12, Phys. Rev. Lett. 108, 067201(2012).[15] S. V. Streltsov and D. I. Khomskii, Jahn-Teller distortion andcharge, orbital, and magnetic order in NaMn7O12, Phys. Rev. B89, 201115(R) (2014).[16] A. A. Belik, Y. Matsushita, M. Tanaka, R. D. Johnson, andD. D. Khalyavin, A plethora of structural transitions, distortionsand modulations in Cu-doped BiMn7O12 quadruple perovskites,J. Mater. Chem. C 9, 10232 (2021).[17] A. A. Belik, Y. Matsushita, Y. Kumagai, Y. Katsuya, M.Tanaka, S. Y. Stefanovich, B. I. Lazoryak, F. Oba, and K.Yamaura, Complex structural behavior of BiMn7O12 quadrupleperovskite, Inorg. Chem. 56, 12272 (2017).[18] A. Prodi, E. Gilioli, R. Cabassi, F. Bolzoni, F. Licci, Q. Huang,J. W. Lynn, M. Affronte, A. Gauzzi, and M. Marezio, Magneticstructure of the high-density single-valent eg Jahn-Teller systemLaMn7O12, Phys. Rev. B 79, 085105 (2009).[19] Z. Popovic and S. Satpathy, Cooperative Jahn-Teller couplingin the manganites, Phys. Rev. Lett. 84, 1603 (2000).[20] Y. Murakami, J. P. Hill, D. Gibbs, M. Blume, I. Koyama, M.Tanaka, H. Kawata, T. Arima, Y. Tokura, K. Hirota et al., Res-onant x-ray scattering from orbital ordering in LaMnO3, Phys.Rev. Lett. 81, 582 (1998).[21] X. Qiu, T. Proffen, J. F. Mitchell, and S. J. L. Billinge, Orbitalcorrelations in the pseudocubic O and rhombohedral R phasesof LaMnO3, Phys. Rev. Lett. 94, 177203 (2005).[22] J. Kanamori, Superexchange interaction and symmetry proper-ties of electron orbitals, J. Phys. Chem. Solids 10, 87 (1959).[23] P. W. Anderson, Antiferromagnetism. Theory of superexchangeinteraction, Phys. Rev. 79, 350 (1950).[24] R. D. Johnson, D. D. Khalyavin, P. Manuel, L. Zhang, K.Yamaura, and A. A. Belik, Magnetic structures of the rare-earthquadruple perovskite manganites RMn7O12, Phys. Rev. B 98,104423 (2018).[25] D. Behr, A. A. Belik, D. D. Khalyavin, and R. D. Johnson,BiMn7O12: Polar antiferromagnetism by inverse exchange stric-tion, Phys. Rev. B 107, L140402 (2023).[26] R. D. Johnson, D. D. Khalyavin, P. Manuel, Y. Katsuya, M.Tanaka, Y. Matsushita, L. Zhang, K. Yamaura, and A. A.Belik, Displacive structural phase transitions and the magneticground state of quadruple perovskite YMn7O12, Phys. Rev. B99, 024107 (2019).[27] R. D. Johnson, D. D. Khalyavin, P. Manuel, L. Zhang, K.Yamaura, and A. A. Belik, Emergence of a magnetostructuraldipolar glass in the quadruple perovskite Dy1-δMn7+δO12, Phys.Rev. Lett. 125, 097601 (2020).[28] F. Mezzadri, M. Calicchio, E. Gilioli, R. Cabassi, F. Bolzoni,G. Calestani, and F. Bissoli, High-pressure synthesis and char-acterization of PrMn7O12 polymorphs, Phys. Rev. B 79, 014420(2009).[29] L. C. Chapon, P. Manuel, P. G. Radaelli, C. Benson, L. Perrott,S. Ansell, N. J. Rhodes, D. Raspino, D. Duxbury, E. Spillet al., WISH: The new powder and single crystal magneticdiffractometer on the second target station, Neutron News 22,22 (2011).[30] J. Rodríguez-Carvajal, Recent advances in magnetic structuredetermination by neutron powder diffraction, Phys. B: Condens.Matter 192, 55 (1993).[31] B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch,ISODISPLACE: A web-based tool for exploring structural distor-tions, J. Appl. Cryst. 39, 607 (2006).[32] H. T. Stokes, D. M. Hatch, and B. J. Campbell, ISOTROPYsoftware suite, https://iso.byu.edu.[33] N. S. Fedorova, Y. W. Windsor, C. Findler, M. Ramakrishnan,A. Bortis, L. Rettig, K. Shimamoto, E. M. Bothschafter, M.Porer, V. Esposito et al., Relationship between crystal structureand multiferroic orders in orthorhombic perovskite manganites,Phys. Rev. Mater. 2, 104414 (2018).134426-9https://doi.org/10.1107/S2053273318018338https://doi.org/10.1016/0038-1098(75)90130-1https://doi.org/10.1002/anie.201208553https://doi.org/10.1103/PhysRevLett.108.067201https://doi.org/10.1103/PhysRevB.89.201115https://doi.org/10.1039/D1TC02344Fhttps://doi.org/10.1021/acs.inorgchem.7b01723https://doi.org/10.1103/PhysRevB.79.085105https://doi.org/10.1103/PhysRevLett.84.1603https://doi.org/10.1103/PhysRevLett.81.582https://doi.org/10.1103/PhysRevLett.94.177203https://doi.org/10.1016/0022-3697(59)90061-7https://doi.org/10.1103/PhysRev.79.350https://doi.org/10.1103/PhysRevB.98.104423https://doi.org/10.1103/PhysRevB.107.L140402https://doi.org/10.1103/PhysRevB.99.024107https://doi.org/10.1103/PhysRevLett.125.097601https://doi.org/10.1103/PhysRevB.79.014420https://doi.org/10.1080/10448632.2011.569650https://doi.org/10.1016/0921-4526(93)90108-Ihttps://doi.org/10.1107/S0021889806014075https://iso.byu.eduhttps://doi.org/10.1103/PhysRevMaterials.2.104414