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[Andreas Dönni](https://orcid.org/0000-0002-7300-9175), Vladimir Y. Pomjakushin, [Alexei A. Belik](https://orcid.org/0000-0001-9031-2355)

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[Er-driven incommensurate to commensurate magnetic phase transition of Fe in the spin-chain compound <math>  <msub>    <mi>BaErFeO</mi>    <mn>4</mn>  </msub></math>](https://mdr.nims.go.jp/datasets/fc6d9d5e-c271-484e-9390-14b824f63b0a)

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Er-driven incommensurate to commensurate magnetic phase transition of Fe in the spin-chain compound ${\rm BaErFeO}_{4}$PHYSICAL REVIEW B 109, 064403 (2024)Er-driven incommensurate to commensurate magnetic phase transitionof Fe in the spin-chain compound BaErFeO4Andreas Dönni ,1 Vladimir Y. Pomjakushin ,2 and Alexei A. Belik 1,*1Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan2Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland(Received 18 September 2023; revised 30 November 2023; accepted 8 January 2024; published 2 February 2024)Magnetic phase transitions and structures of the spin-chain compound BaErFeO4 were investigated by mea-surements of magnetic properties (specific heat, magnetic susceptibility) and neutron diffraction. The lattice ge-ometry of the orthorhombic crystal structure (space group Pnma) of the BaRFeO4 compounds (R = Dy–Yb, Y)supports frustrations which lead to multiple magnetic phase transitions with complex magnetic structures.BaErFeO4 undergoes three successive magnetic phase transitions at TN1 = 49 K, TN2 = 33.4 K, and TN3 = 3.4 K.In contrast with the previously investigated BaRFeO4 (R = Y, Tm, Yb) compounds, all with incommensuratemagnetic propagation vectors k1 = (0, 0, kz ), BaErFeO4 is the first member in this series that shows a phasetransition from an incommensurate (k1 below TN1) to a commensurate magnetic structure with k2 = ( 12 , 0, 12 )below TN2. In the crystal structure, all magnetic ions (Fe1, Fe2, Er1, and Er2) are part of chains propagating alongthe b axis. Below TN1, strong antiferromagnetic (AFM) Fe-Fe spin-exchange coupling between square pyramidal(Fe1) and octahedral (Fe2) centers generates a collinear AFM structure with a constant size of the ordered Femoments and a constant magnetic phase inside each chain of Fe3+ cations. Exchange coupling between theFe chains is much weaker. At TN2, 3d-4 f exchange interactions induce an ordered moment at the Er3+ ions,which results in a change of the direction of the ordered Fe moments from the b direction (above TN2) to insidethe ac plane (below TN2) and a change from an incommensurate (k1 above TN2) to a commensurate (k2 belowTN2) AFM structure. Toward lower temperature, 4 f -4 f exchange interactions become stronger and create atTN3 a constant magnetic phase inside each chain of Er3+ cations. At TN2 and TN3, the magnetic susceptibilityshows sharp decreases that coincide with large increases of the correlation length of the magnetic structure. Theunique magnetic structures of BaErFeO4 are compared with those of other BaRFeO4 compounds by consideringexperimental and theoretical aspects.DOI: 10.1103/PhysRevB.109.064403I. INTRODUCTIONFirst-principles calculations are powerful enough to predictcorrect crystal ground-state structures of many materials [1],but they are not yet capable of predicting real magnetic groundstates; they can only give the lowest-energy structure amongdifferent calculated models. Neutron diffraction is a uniquedirect method to determine the magnetic structure of a crystal.Magnetic structure solutions from neutron diffraction are stillmainly a trial-and-error approach to fit the experimental datato all possible symmetry-adapted magnetic structures (repre-sentation analysis, calculated by a computer program) for agiven crystal structure and a propagation vector (a point onthe Brillouin zone) [2–4].Magnetic exchange interactions between 3d and 4 f elec-trons may account for complex magnetic properties ofintermetallic compounds, as illustrated by recent examplesbased on neutron diffraction. In the centrosymmetric greenphase compound Gd2BaCuO5 [5], the coupling of Cu2+ andGd3+ spins is important in inducing ferroelectricity. Theinterplay of 3d-4 f interactions demonstrates an alternative*alexei.belik@nims.go.jproute to find magnetoelectric materials. In the Mn self-dopedperovskite (Yb0.667Mn0.333)MnO3 [6], Yb–Yb interactionscreate an antiferromagnetic (AFM) structure at TN ≈ 40 K,whereas Mn–Yb interactions induce a small but nonzero fer-romagnetic (FM) Yb3+ moment at much higher temperature(Tc = 106 K).We have recently started to investigate the rare-earth seriesof isostructural quarternary ferrites, BaRFeO4 with R = Dy toYb [7,8], and found that each compound shows unique mag-netic and dielectric properties. Detailed magnetic, dielectric,and ferroelectric properties of BaHoFeO4 were investigatedby other authors [9,10], and the appearance of a hidden mul-tiferroic state under a magnetic field has been discovered[9]. Members of the BaRFeO4 series are isostructural withthe parent compound BaYFeO4 [11–16], which undergoestwo successive magnetic phase transitions at TN1 = 48 K andTN2 = 36 K. Below TN2, a cycloidal incommensurate mag-netic structure and a spin-driven multiferroic state have beenobserved [12,13]. Due to the peculiar orthorhombic crystalstructure, where magnetic ions form spin chains, BaRFeO4ferrite compounds exhibit multiple phase transitions andadopt complex magnetic structures. Recently, we have deter-mined the magnetic structures of the compounds BaYbFeO4and BaTmFeO4 [17] with three successive magnetic phase2469-9950/2024/109(6)/064403(12) 064403-1 ©2024 American Physical Societyhttps://orcid.org/0000-0002-7300-9175https://orcid.org/0000-0003-2180-8730https://orcid.org/0000-0001-9031-2355https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.109.064403&domain=pdf&date_stamp=2024-02-02https://doi.org/10.1103/PhysRevB.109.064403DÖNNI, POMJAKUSHIN, AND BELIK PHYSICAL REVIEW B 109, 064403 (2024)transitions and found that 3d-4 f magnetic exchange inter-actions play an important role. Based on neutron diffractionexperiments, all magnetic structures of the parent compoundBaYFeO4 [12] and BaRFeO4 (R = Yb, Tm) [17] were foundto be incommensurate with a magnetic propagation vectork1 = (0, 0, kz ).In this paper, we present measurements of specific heat,magnetic susceptibility, and neutron diffraction to extend thedetermination of the magnetic structures to BaErFeO4 withthree successive magnetic phase transitions at TN1 = 49 K,TN2 = 33.4 K, and TN3 = 3.4 K. BaErFeO4 is found to be thefirst member in the series of BaRFeO4 ferrites that shows aphase transition from an incommensurate magnetic structurewith k1 = (0, 0, kz ) below TN1 to a commensurate magneticstructure with k2 = ( 12 , 0, 12 ) below TN2. Magnetic exchangeinteractions among and between 3d and 4 f electron spinshave different strengths and shape the magnetic structures atdifferent temperatures. Here, 3d-3d coupling at TN1, TN2, andTN3, 3d-4 f interactions at TN2, and 4 f -4 f coupling at TN3.Magnetic properties of BaErFeO4 with sharp decreases of themagnetic susceptibility at TN2 and TN3 are unique in the seriesof BaRFeO4 ferrites.II. EXPERIMENTALA large amount (∼11 g) of polycrystalline BaErFeO4 fer-rite sample was synthesized from stoichiometric mixtures ofBaCO3 (99.9%), Er2O3 (99.9%), and Fe2O3 (99.999%) usinga conventional solid-state annealing method. The obtainedmixture was pressed into pellets and annealed in air on Pt foilsuccessively at (1) 1430 K for 40 h, (2) 1520 K for 30 h, (3)1520 K for 36 h, and (4) 1520 K for 40 h, with grinding be-tween each of the four steps. The resulting BaErFeO4 samplecontained no detectable impurity phase based on x-ray andneutron diffraction data. All measurements reported in thispaper were performed on this BaErFeO4 ferrite sample.To investigate the magnetic phase transitions, the specificheat of BaErFeO4 was measured in several magnetic fieldsfrom 0 to 90 kOe at temperatures between 2 K and room tem-perature using a commercial calorimeter (Quantum DesignPPMS) by the pulse relaxation method. Magnetic susceptibil-ity measurements were performed on a SQUID magnetometer(Quantum Design MPMS-XL-7 T) at temperatures between 2and 400 K on cooling and warming in applied magnetic fieldsfrom 0.1 to 10 kOe.To determine the magnetic structures of BaErFeO4, powderneutron diffraction experiments were performed at the PaulScherrer Institute, Switzerland, on the high-resolution pow-der diffractometer for thermal neutrons (HRPT) [18] usingan incident neutron wavelength of λ = 1.886 Å. Data werecollected in the magnetically ordered and paramagnetic statesat temperatures between 1.8 and 70 K for a 2θ range of3.55◦–164.50◦ and a step width of 0.05◦. We did not measurethe experimental absorption correction coefficient µR of ourBaErFeO4 sample. Er has a moderate macroscopic absorptioncross-section per unit volume � of 5.1861 cm−1 [19], relativeto huge values of some other rare-earth elements, such asDy (29.731 cm−1), Sm (171.23 cm−1), or Gd (1494.1 cm−1).A rough estimation gives µR ∼ 0.8(=5.1861 × 0.394 × 0.4)based on an atomic mass ratio of 39.4% of Er in BaErFeO4and R = 0.4 cm of the cylindrical sample holder. The diffrac-tion patterns were analyzed with µR = 0.8 by the Rietveldmethod using the FULLPROF suite [20]. The peak shape func-tion used in the refinements was Thompson-Cox-Hastingspseudo-Voigt convoluted with axial divergence asymmetryfunction [21]. Possible models for the magnetic structureswere deducted based on a group theory analysis using theprograms ISODISTORT [22] and BASIREPS in the FULLPROF suiteprogram package [20].III. RESULTS AND DISCUSSIONA. Crystal structure of BaErFeO4The crystal structure of BaErFeO4 at room tempera-ture has been determined by synchrotron x-ray diffraction[8]. BaErFeO4 crystallizes in the orthorhombic space groupPnma (No. 62) and is isostructural with BaYFeO4 [11]. TheBaYFeO4 structure type is formed for BaRFeO4 compoundswith R = Dy − Yb [7,8]. The structure parameters of param-agnetic BaErFeO4 at T = 70 K refined from powder neutrondiffraction data are summarized in Table I. They are in goodagreement with the reported room-temperature synchrotronx-ray results [8]. Refinements performed for different val-ues of the absorption correction coefficient (0.0 � µR � 2.0)showed a strong correlation with the Debye Waller factorsand no correlation for the other structural parameters (valueswithin the errors given in Table I) and the agreement valuesof the fit. As a conclusion, except for the Debye-Waller fac-tors, Er absorption can be neglected in the crystal structurerefinement.The arrangement of the magnetic ions (Fe1, Fe2, Er1,and Er2) in the crystal structure of BaErFeO4 is shown inFig. 1 as a projection onto the bc and ac planes. All magneticions are all located on 4c sites (x, 14 , z) with the structureparameters given in Table I. There are rings consisting of 4magnetic ions [for example, Fe21, Fe12, Fe23, and Fe14, seeFig. 1(b)], and each ring forms a chain propagating along theb axis. All magnetic ions are part of rings and chains. Theorthorhombic crystal structure of BaErFeO4 is illustrated inFig. 2. The chains of Fe3+ ions are built by alternate cornershared units of [FeO5]7− square pyramids (Fe1, blue) and[FeO6]9− octahedra (Fe2, red) [11].Figure 3 shows the temperature dependence of the latticeconstants a, b, and c and the volume V of BaErFeO4 be-low T = 70 K based on our neutron diffraction data. For allparameters, a monotonic decrease is observed in the com-mensurate magnetic phase below TN2. When decreasing thetemperature from the paramagnetic state (T = 70 K) to theincommensurate magnetic phase (between TN1 and TN2), spin-lattice coupling creates a small increase for the lattice constantb but not for a, c, and V .B. Magnetic phase transitions of BaErFeO4Figures 4 and 5 show the temperature dependence of thespecific heat Cp/T at different magnetic fields. No specificheat anomaly was observed at TN1 during the transition to theincommensurate magnetic phase in agreement with results forother members of the BaRFeO4 family (R = Y [7], Dy [7],and Ho [9]). A very weak specific heat anomaly was detected064403-2ER-DRIVEN INCOMMENSURATE TO COMMENSURATE … PHYSICAL REVIEW B 109, 064403 (2024)TABLE I. Structure parameters of paramagnetic BaErFeO4 at T = 70 K refined from powder neutron diffraction data (HRPT, λ = 1.886 Å,µR = 0.8). Space group Pnma (No. 62); Z = 8. WP: Wyckoff position. The occupation factor g = 1 for all the sites.BaErFeO4, T = 70 K:a = 13.0844(1) Å; b = 5.6736(1) Å; c = 10.2055(1) Å; V = 757.62(1) Å3.χ 2 = 3.73; Rwp = 2.57%; Rexp = 1.33%; RBragg = 2.35%.Site WP x y z B(Å2)Ba1 4c 0.2110(2) 0.25 0.6738(3) =B(Er1)Ba2 4c 0.4146(3) 0.25 0.3956(3) =B(Er1)Er1 4c 0.4147(1) 0.25 0.0140(2) 0.26(2)Er2 4c 0.1435(1) 0.25 0.3102(2) =B(Er1)Fe1 4c 0.4681(1) 0.25 0.7156(1) 0.24(2)Fe2 4c 0.1902(1) 0.25 0.0227(1) =B(Fe1)O1 4c 0.5867(3) 0.25 0.6156(2) 0.49(2)O2 4c 0.2907(2) 0.25 0.1820(3) =B(O1)O3 8d 0.0056(1) 0.5093(3) 0.3597(1) =B(O1)O4 8d 0.2175(1) 0.5086(3) 0.4408(1) =B(O1)O5 8d 0.1117(1) 0.0000(4) 0.1312(1) =B(O1)FIG. 1. Arrangement of magnetic Fe3+ and Er3+ ions in thecrystal structure of BaErFeO4 shown as a projection onto (a) the bcplane and (b) the ac plane. Drawings were made using VESTA [23].at TN2, in agreement with results for BaDyFeO4 [7]; on theother hand, no specific heat anomaly was detected at TN2in BaYFeO4 [7] and BaHoFeO4 [9]. A very strong specificheat anomaly appeared at TN3, in agreement with results forBaRFeO4 with magnetic rare-earth elements [7,9]. However,small magnetic fields, as small as 2 kOe, completely sup-pressed the specific heat anomaly at TN2 in BaErFeO4 (insetof Fig. 4) and also slightly suppressed the anomaly at TN3(Fig. 5). With the increase of an applied magnetic field, theanomaly at TN3 continued to be suppressed, and an effect ofa large magnetic field (90 kOe) on specific heat was observedfar above TN1 up to ∼120 K. It means that a large portion oflow-temperature magnetic entropy (observed below ∼14 K)moved to a high-temperature region.FIG. 2. Orthorhombic crystal structure of BaErFeO4 consistingof corner-shared units of [FeO5]7− square pyramids (Fe1, blue)and [FeO6]9− octahedra (Fe2, red). The drawing was made usingVESTA [23].064403-3DÖNNI, POMJAKUSHIN, AND BELIK PHYSICAL REVIEW B 109, 064403 (2024)FIG. 3. Temperature dependence of the lattice constants a, b, andc and the volume V of BaErFeO4 below T = 70 K based on neutrondiffraction data.Magnetic susceptibility data showed that the behavior ofBaErFeO4 strongly depended on the measurement magneticfield especially below ∼10 K (Fig. 6), where a noticeable FM-like contribution appeared above about H = 1.5 kOe, and thetransition moved to higher temperatures. Here, TN1 remainedalmost unaffected by magnetic fields, while TN2 slightly de-creased with the increase of a magnetic field. The magneticsusceptibility curves exhibited strong decreases near TN2 andTN3 below about H = 1 kOe, which are unique in the seriesof BaRFeO4 (R = Dy − Yb, Y) compounds [7,12,11,15] andFIG. 4. Specific heat of BaErFeO4 as Cp/T vs T at 0 Oe (blackfilled circles with line) and 90 kOe (red empty circles without line),measured on cooling. The inset shows zoomed parts of the Cp/T vsT curves at 0 Oe and 2 kOe on cooling and heating.FIG. 5. Specific heat of BaErFeO4 as Cp/T vs T below 80 K atdifferent magnetic fields, measured on cooling. (a) H = 0, 2, 5, and10 kOe and (b) H = 10, 30, 50, 70, and 90 kOe.indicated that the (zero field) magnetic structures of R = Erare partly different from the previously investigated mate-rials with R = Yb, Tm, Y [12,17]. The complex magneticbehavior of BaErFeO4 is illustrated in Fig. S1 in the Supple-mental Material [24], where magnetic susceptibility data atH = 1.5 kOe are plotted in different presentations as an ex-ample. The χ−1 vs T curve (field-cooled on cooling, 10 kOe)of BaErFeO4 shown in Fig. 7 could be well fitted by theCurie-Weiss law at high temperatures between 250 and 395 K.However, a noticeable deviation from the Curie-Weiss lawis observed below ∼150 K, that is, far above TN1. The ex-perimental effective magnetic moment μeff = 10.267(12) µBFIG. 6. Magnetic susceptibility curves, χ vs T , of BaErFeO4 atdifferent magnetic fields from 1.0 to 2.8 kOe with a step of 0.2kOe (left-hand axis) and at 100 Oe (right-hand axis), measured oncooling.064403-4ER-DRIVEN INCOMMENSURATE TO COMMENSURATE … PHYSICAL REVIEW B 109, 064403 (2024)FIG. 7. χ−1 vs T curves [zero-field-cooling (ZFC) and field-cooled on cooling (FCC), 10 kOe] of BaErFeO4 with a Curie-Weissfit (black line) in the temperature range between 250 and 395 K.Obtained fitting parameters are given in the figure.is found to be ∼8% smaller than the calculated one μcalc =11.192 µB. A similar tendency was observed in BaDyFeO4(μeff = 11.28 µB vs μcalc = 12.14 µB) [7]. The Weiss tem-perature θ = −36.6(8) K in BaErFeO4 is close to that ofBaDyFeO4 (θ = −34 K) [7] and BaHoFeO4 (θ = −30 K)[10]. We note that the Curie-Weiss parameters in BaRFeO4could be affected by the presence of impurities with hightransition temperatures.Therefore, both specific heat (at zero magnetic field)and magnetic susceptibility data revealed the presence ofthree magnetic phase transitions in BaErFeO4 at TN1 = 49 K,TN2 = 33.4 K, and TN3 = 3.4 K (the temperatures are given atzero or very small magnetic fields). The results of our (zerofield) neutron diffraction measurements, described below,confirmed the presence of three magnetic phase transitionsat TN1, TN2, and TN3 in BaErFeO4 with an incommensuratemagnetic structure below TN1 and commensurate magneticstructures below TN2.C. Magnetic structures of BaErFeO4Figure 8 shows the refinement of neutron diffraction pat-terns of BaErFeO4 measured in the magnetically orderedstates at (a) T = 40 K (between TN1 and TN2), (b) T = 22 K(between TN2 and TN3), and (c) T = 1.8 K (below TN3). Simul-taneous refinements of crystal and magnetic structures wereperformed in the full range of scattering angles 2θ up to 162°.The refinements in the full 2θ range are shown in Fig. S2 inthe Supplemental Material [24]. The inset of Fig. 8 comparesobserved neutron intensities measured in the paramagneticstate at T = 70 K, the incommensurate phase at T = 40 K,and the commensurate phase at T = 22 K.For space group is Pnma (No. 62), the list of all possiblemagnetic propagation vectors (k) with different symmetryand the corresponding complex irreducible representations(irreps) are tabulated in Ref. [22] and summarized in TableS1 in the Supplemental Material [24]. To solve the mag-netic structures of BaErFeO4, we have systematically checkedFIG. 8. Experimental (black dots), calculated (red line), and dif-ference (blue line) neutron diffraction patterns of BaErFeO4 in themagnetically ordered state at (a) T = 40 K, (b) T = 22 K, and (c)T = 1.8 K. Tick marks indicate Bragg peak positions. The first rowis for the nuclear peaks, and the second row is for the magneticpeaks. The inset compares observed neutron intensities measured atT = 70, 40, and 22 K.which of the commensurate and incommensurate magneticpropagation vectors from this list can explain peak positionsof the experimentally observed magnetic Bragg peaks.At T = 40 K, all magnetic Bragg peaks of BaErFeO4 canbe indexed with an incommensurate propagation vector k1 =(0, 0, kz ). Magnetic 3d-3d exchange interactions inducedmagnetic order at the Fe3+ ions, and Er3+ ions remain disor-dered. For the space group Pnma, site (4c) and the propagationvector k1, representation analysis for the possible magneticstructures gives the result summarized in Table II. There arefour irreps, mLD1, mLD2, mLD3, and mLD4, with differentsymmetry, and the basis vectors are complex. Among the fourmagnetic ions on site 4c (e.g., Fe14, Fe11, Fe12, and Fe13),only the pairs separated by �z = 12 are coupled by symmetry(e.g., Fe14 with Fe11 and Fe12 with Fe13). The result of therefinement of the magnetic structure at T = 40 K is given inTable III. The incommensurate magnetic structure ofBaErFeO4 at T = 40 K corresponds to the irrep mLD4 likethat in BaYbFeO4 at T = 42 K. Details of the refinementfor BaYbFeO4 are described in Ref. [17]. For BaErFeO4 atT = 40 K, the propagation vector is k1 = (0, 0, kz ), withkz = 0.396(3) and ordered Fe moments of my = 2.50(9) µB064403-5DÖNNI, POMJAKUSHIN, AND BELIK PHYSICAL REVIEW B 109, 064403 (2024)TABLE II. Group theory analysis for the magnetic structure of BaErFeO4 between TN1 and TN2 calculated using ISODISTORT [22] andBASIREPS [20]. The crystallographic space group is Pnma (No. 62). The magnetic propagation vector is k1 = (0, 0, kz ). a = exp(iπkz ).a∗ = exp(−iπkz ). All magnetic ions (Fe) are located on 4c sites, and the positions are shown in Fig. 1(b). Components of the magneticmoments are expressed using (u1, v1, w1) and (u2, v2, w2) in orbits 1 and 2, respectively. Irrep denotes irreducible representation. The charactersets correspond to the four symmetry elements [20].airrep irrep Orbit 1 Orbit 1 Orbit 2 Orbit 2(ISODISTORT) (BASIREPS) (x, y, z) (−x + 12 , −y, z + 12 ) (−x, y + 12 , −z) (x + 12 , −y + 12 , −z + 12 )mLD1 IRrep(1) (0, v1, 0) (0, −v1, 0) · a∗ (0, v2, 0) (0,−v2 0) · a∗mLD2 IRrep(2) (u1, 0,w1) (−u1, 0, w1) · a∗ (u2, 0, w2) (−u2, 0, w2) · a∗mLD3 IRrep(4) (u1, 0,w1) (u1, 0, −w1) · a∗ (u2, 0, w2) (u2, 0, −w2) · a∗mLD4 IRrep(3) (0, v1, 0) (0, v1, 0) · a∗ (0, v2, 0) (0, v20) · a∗Fe14 Fe11 Fe12 Fe13Fe21 Fe24 Fe23 Fe22aPositions of magnetic ions: Fe14 (0.032, 0.75, 0.216); Fe11 (0.468, 0.25, 0.716); Fe12 (−0.032, 0.25, −0.216); Fe13 (0.532, 0.75, 0.284);Fe21 (0.190, 0.25, 0.023); Fe24 (0.310, 0.75, 0.523); Fe23 (−0.190, 0.75, −0.023); Fe22 (0.690, 0.25, 0.477). Symmetry elements: Symm(1):1; Symm(2): 2 (0, 0, 12 ) 14 , 0, z; Symm(3): m x, 14 , z; Symm(4): n (0, 12 , 12 ) 14 , y, z. Character sets: mLD1 (1, a, 1, a); mLD2 (1, a, −1, −a);mLD3 (1, −a, −1, a); mLD4 (1, −a, 1, −a).are oriented along the b direction, perpendicular to k1. Thecollinear incommensurate magnetic structure is illustrated inFig. 9(a). It is a spin density wave (SDW). The size ofthe ordered moments and the magnetic phase is constantwithin each ring and chain of Fe3+ ions. Nearest-neighborand second-nearest-neighbor Fe3+ ions [see J1 and J2 inFig. 1(b)] are coupled AFM. Along the c direction, the sizeof ordered Fe moments changes according to the kz com-ponent of the propagation vector k1. Between TN1 and TN2,we performed neutron diffraction measurements at two tem-peratures T = 40 and 36 K. The refinement of the magneticstructure at T = 36 K, closer to TN2 = 33.4 K, yielded kz =0.396(3), my = 2.70(9) µB. The temperature dependence ofthe magnetic propagation vector is very weak (zero within ex-perimental accuracy) with the value kz = 0.396(3) remainingfar away from, for example, kz = 0.5. The ordered Fe momentmy slightly increased toward lower temperature. Between TN1and TN2, our results for BaErFeO4 agree with a more de-tailed investigation [12] of the temperature dependence of theincommensurate magnetic structure of BaYFeO4. Magneticordering of Mn3+ ions in perovskite TmMnO3 [25] is anexample, where below TN1 = 42 K, the magnetic propagationvector k1 = (kx, 0, 0) exhibits a strong temperature depen-dence (0.45 < kx � 0.5) toward a commensurate structurebelow TN2 = 32 K with k2 = ( 12 , 0, 0).At T = 22 and 1.8 K (below TN2), all magnetic Braggpeaks of BaErFeO4 can be indexed with a commensuratepropagation vector k2 = ( 12 , 0, 12 ). As illustrated in Fig. S3in the Supplemental Material [24], experimental data disagreewith the positions of magnetic Bragg peaks for the propaga-tion vectors k = (0, 0, 12 ) and k = (0, 0, kz ). Below TN2,all magnetic Fe3+ and Er3+ ions are ordered. For the spacegroup Pnma, site (4c) and the propagation vector k2, represen-TABLE III. Result of the refinement of the magnetic structures of BaErFeO4 at T = 40, 22, and 1.8 K based on powder neutron diffractiondata (HRPT, λ = 1.886 Å, µR = 0.8). Positions of the magnetic ions (Fe3+, Er3+) are labeled as shown in Fig. 1(b). Magnetic phases δ0, δFe,δEr21, and δEr14 are in units of 2π .aT = 40 K (TN2 < T < TN1): k1 = (0, 0, kz ); irrep: mLD4kz = 0.396(3); δ0 = 0Fe14: (0, v1, 0)δ0; v1 = 2.50(9) µB; Fe12: (0, v2, 0)δ0; v2 = 2.50(9) µB;Fe21 : (0, v1, 0)δ0; v1 = −2.50(9) µB; Fe23 : (0, v2, 0)δ0; v2 = −2.50(9) µB;χ 2 = 2.13; Rwp = 3.10%; Rexp = 2.12%; RBragg = 2.29%; Rmag = 24.7%T = 22 K (T < TN2): k2 = ( 12 , 0, 12 ); irrep: mU2+δFe = 0.125; δEr21 = −0.235(7); δEr14 = 0.053(6)Fe14: (u, 0, w)δFe; u = −2.20(5) µB; w = 2.22(7) µB; Er14: (u, 0, w)δEr14; u = 2.63(6) µB; w = −0.42(10) µB;Fe21: (u, 0, w)δFe; u = 2.20(5) µB; w = −2.22(7) µB; Er21: (u, 0, w)δEr21; u = −0.45(7) µB; w = 2.32(8) µB;χ 2 = 1.87; Rwp = 2.85%; Rexp = 2.09%; RBragg = 1.71%; Rmag = 3.97%T = 1.8 K (T < TN3): k2 = ( 12 , 0, 12 ); irrep: mU2+δFe = 0.125; δEr21 = −0.107(2); δEr14 = −0.106(2)Fe14: (u, 0, w)δFe; u = −2.13(4) µB; w = 3.16(5) µB; Er14: (u, 0, w)δEr14; u = 8.14(5) µB; w = −2.38(6) µB;Fe21: (u, 0, w)δFe; u = 2.13(4) µB; w = −3.16(5) µB; Er21: (u, 0, w)δEr21; u = −1.08(4) µB; w = 7.62(5) µB;χ 2 = 4.31; Rwp = 4.40%; Rexp = 2.12%; RBragg = 2.45%; Rmag = 3.46%aPositions of magnetic ions: Fe14 (0.032, 0.75, 0.216); Fe12 (−0.032, 0.25, −0.216); Fe21 (0.190, 0.25, 0.023); Fe23 (−0.190, 0.75, −0.023);Er21 (0.143, 0.25, 0.310); Er14 (0.085, 0.75, 0.514).064403-6ER-DRIVEN INCOMMENSURATE TO COMMENSURATE … PHYSICAL REVIEW B 109, 064403 (2024)FIG. 9. Illustration of the magnetic structures of BaErFeO4 at (a)T = 40 K, (b) T = 22 K, and (c) T = 1.8 K. Drawings were madeusing VESTA [23].tation analysis for the possible magnetic structures gives theresult summarized in Table IV. There are eight irreps, mU1+,mU1−, mU2+, mU2−, mU3+, mU3−, mU4+, and mU4−,with different symmetry and the basis vectors are complex.All four magnetic ions on site 4c (e.g., Fe14, Fe12, Fe11, andFe13) are coupled by symmetry. The result of the refinementof the magnetic structures at T = 22 and 1.8 K is given inTable III. The commensurate magnetic structures ofBaErFeO4 at T = 22 and 1.8 K correspond to the irrep mU2+and all magnetic ions Fe1, Fe2, Er1, and Er2 are orderedwithin the ac plane. The commensurate magnetic structuresat T = 22 and 1.8 K are illustrated in Figs. 9(b) and 9(c),respectively.For the propagation vector k2 = ( 12 , 0, 12 ), the magneticFe and Er ions with the positions indicated in Fig. 1(b) areconnected by the symmetry operations of site (4c) given inTable IV. For mU2+, there is a FM coupling between theordered moments at (x, y, z) and (−x, y + 12 , −z) as well asbetween (−x + 12 , −y, z + 12 ) and (x + 12 , −y + 12 , −z + 12 ).The first two symmetry operations transform Fe14 into Fe12inside the same ring shown in Fig. 1(b). The symmetry opera-tions connect all Fe1 and all Fe2 ions with positions indicatedin Fig. 1(b) without translations by a lattice constant a or c.(Translations along b can be ignored because of the FM cou-pling of k2 along the b direction.) As a result, for mU2+, thereis a FM coupling inside each ring for all Fe1 ions as well as forall Fe2 ions. However, this is not true for the positions of Er1and Er2 ions. The first two symmetry operations transformEr14 into Er12-c. For k2, a translation by the lattice constant creverses the direction of the ordered Er moment. As a result,for mU2+, there is an AFM coupling inside each ring for allEr1 ions as well as for all Er2 ions.At TN2, a first-order phase transition from an incom-mensurate magnetic propagation vector k1 = (0, 0, kz ) toa commensurate propagation vector k2 = ( 12 , 0, 12 ) occursin BaErFeO4. All ordered Fe moments change the direc-tion from along the b direction (above TN2) to inside theac plane (below TN2). Above TN2, the size of orderedFe moments changes between different chains along thec direction. Below TN2, the size of all ordered Fe1 and Fe2moments is constant with a noncollinear coupling betweendifferent chains along the [1, 0, 1] direction. All Fe momentsare approximately parallel or antiparallel to [1, 0, 1] or [1,0, −1] directions [see Figs. 9(b) and 9(c)]. The tempera-ture dependence of the magnetic phase and the ordered Femoments below TN2 is shown in Fig. 10 in blue color. Themagnetic phase δFe = 0.125 is equal for all Fe1 and Fe2 ionsand temperature independent [Fig. 10(a)]. The temperaturedependence of the ordered Fe moment is weak [Fig. 10(c)] andslightly increases from 2.21(6) µB at T = 22 K to 2.70(4) µBat T = 1.8 K (Table V) due to a small increase of the mzcomponent below T ≈ 12 K [Fig. 11(a)].Magnetic Er3+ ions order at the first-order phase transitionat TN2 = 33.4 K. At T = 22 K, 3d-4 f exchange interactionsinduced a strongly noncollinear structure with an orderedmoment of 2.52(11) µB predominantly along the a direction atEr14 and Er12 and an ordered moment of 2.36(11) µB predom-inantly along the c direction at Er22 and Er24 [see Figs. 9(b)and 1(b) and Table V]. In contrast, the ordered moments atEr11, Er13, Er21, and Er23 are much smaller and close to zero.For k2, all Er1 and Er2 ions are coupled by symmetry. Withineach ring of Er ions [Fig. 1(b)], the ordered Er momentsand the magnetic phases are not constant (Fig. 10, Table III),resulting in a large magnetic phase difference between Er14and Er21 of 0.29(1).064403-7DÖNNI, POMJAKUSHIN, AND BELIK PHYSICAL REVIEW B 109, 064403 (2024)TABLE IV. Group theory analysis for the magnetic structure of BaErFeO4 below TN2 calculated using ISODISTORT [22] and BASIREPS [20].The crystallographic space group is Pnma (No. 62). The magnetic propagation vector is k2 = ( 12 , 0, 12 ). All magnetic ions (Fe, Er) are locatedon 4c sites, and the location is shown in Fig. 1(b). The components of the magnetic moments are expressed using (u, v, w). irrep denotesirreducible representation. The character sets correspond to the eight symmetry elements [20].airrep irrep(ISODISTORT) (BASIREPS) (x, y, z) (−x, y + 12 , −z) (−x + 12 , −y, z + 12 ) (x + 12 , −y + 12 , −z + 12 )mU1+ IRrep(1) (0, v, 0) (0, v, 0) (0, v, 0) · i (0, v, 0) · imU1− IRrep(2) (u, 0, w) (−u, 0, −w) (u, 0, −w) ·i (−u, 0, w) ·imU2+ IRrep(3) (u, 0, w) (u, 0, w) (u, 0, −w) ·i (u, 0, −w) ·imU2− IRrep(4) (0, v, 0) (0, −v, 0) (0, v, 0) ·i (0, −v, 0) ·imU3+ IRrep(7) (u, 0, w) (u, 0, w) (−u, 0, w) ·i (−u, 0, w) ·imU3− IRrep(8) (0, v, 0) (0, −v, 0) (0, −v, 0) ·i (0, v, 0) ·imU4+ IRrep(5) (0, v, 0) (0, v, 0) (0, −v, 0) ·i (0, −v, 0) ·imU4− IRrep(6) (u, 0, w) (−u, 0, −w) (−u, 0, w) ·i (u, 0, −w) ·iFe14 Fe12 Fe11 Fe13Fe21 Fe23 Fe24 Fe22Er14 Er12 −c Er11 + c Er13Er21 Er23 −c Er24 + c Er22aPositions of magnetic ions: Fe14 (0.032, 0.75, 0.216); Fe12 (−0.032, 0.25, −0.216); Fe11 (0.468, 0.25, 0.716); Fe13 (0.532, 0.75, 0.284);Fe21 (0.190, 0.25, 0.023); Fe23 (−0.190, 0.75, −0.023); Fe24 (0.310, 0.75, 0.523); Fe22 (0.690, 0.25, 0.477); Er14 (0.085, 0.75, 0.514); Er12(−0.085, 0.25, 0.486); Er11 (0.415, 0.25, 0.014); Er13 (0.585, 0.75, −0.014); Er21 (0.143, 0.25, 0.310); Er23 (−0.143, 0.75, 0.690); Er24(0.357, 0.75, −0.190); Er22 (0.643, 0.25, 0.190). Symmetry elements: Symm(1): 1; Symm(2): 2 (0, 0, 12 ) 14 , 0, z; Symm(3): 2 (0, 12 , 0) 0, y,0; Symm(4): 2 ( 12 , 0, 0) x, 14 , 14 ; Symm(5): −1 0, 0, 0; Symm(6): a x, y, 14 ; Symm(7): m x, 14 , z; Symm(8): n (0, 12 , 12 ) 14 , y, z. Character sets:mU1+ (1, i, 1, i, 1, i, 1, i); mU1− (1, i, 1, i, −1, −i, −1, −i); mU2+ (1, i, −1, −i, 1, i, −1, −i); mU2− (1, i, −1, −i, −1, −i, 1, i); mU3+ (1,−i, −1, i, 1, −i, −1, i); mU3 − (1,−i, −1, i, −1, i, 1, −i); mU4 + (1,−i, 1, −i, 1, −i, 1, −i); mU4 − (1,−i, 1, −i, −1, i, −1, i).With decreasing temperature, the specific heat Cp/T startsto increase below ∼12 K toward a strong peak at TN3 = 3.4 K(Fig. 4). The magnetic entropy of this Cp/T peak correspondsto a change of the magnetic structure induced by 4 f -4 f ex-change interactions. Inside each ring of Er ions, the largemagnetic phase difference becomes smaller and reaches zeroat TN3 [Figs. 10(a) and 10(b)]. At the same time, the smallordered Er moments start to increase [Fig. 10(c)]. At T =1.8 K, all ordered Er moments have large values between 4.8and 6.7 µB (Table V). Below TN3, Er ions in the same ringhave a constant magnetic phase (like the Fe ions), but themagnetic structure is noncollinear (different from the Fe ions).The temperature dependence of the ordered Er moments andthe components (mx, mz ) is shown in Figs. 10 and 11, re-spectively. As shown in Fig. 10(a), at T = 22 K, the magneticphases of δ(Er14) and δ(Er21) have different signs of positiveand negative, respectively. Toward the zero-phase differenceat TN3, the sign of δ(Er14) changes from positive to negativenear 7 K, where the ordered moments of Er11 and Er13 exhibita minimum of zero [Fig. 10(c)]. This minimum is caused bya reversal of the direction of both components mx and mz, asshown in Fig. 11(c) for Er11. Such a minimum is not observedfor Er2 because the magnetic phase δ(Er21) remains negative[Fig. 10(a)] between 1.8 and 30 K.We checked the effect of absorption (for 0.0 � µR � 2.0)on the refined magnetic structures given in Table III. Correla-tions with the ordered moments and phase factors were foundto be very weak (values within the errors given in Table III).As a conclusion, the Er absorption can be neglected in ourmagnetic structure refinements.The Thompson-Cox-Hastings pseudo-Voigt function usedin the refinement of the neutron data consists of a Gaus-sian and a Lorentzian component. The correlation lengths (L)of the magnetic and the crystal structure can be comparedfrom the Lorentzian peak broadening of the resolution pa-rameter Y (Ym and Yn for the magnetic and crystal structure,respectively). From the refinement of the crystal structurein the paramagnetic state at T = 70 K, we obtained Yn =0.0487(18). This value of Yn was kept fixed for the refinementsof the magnetic structures. We assume that the Lorentzian sizeeffect contribution to the width of nuclear peaks is resolutionlimited because the Yn parameter of the Cagliotti function[20,26] is close to the instrumental value. The microstraincontribution given by the resolution parameter U of the Gaus-sian component was fixed by nuclear peaks for the magneticpeaks. In the incommensurate phase at T = 40 K, a largeLorentzian peak broadening with Ym = 0.36(6) or Ym/Yn =7.4(1.3) indicates a magnetic structure with a much shortercorrelation length. Using the well-known Debye-Scherrer for-mula σ1 = Ym − Yn = λ/L [27], where λ = 1.886 Å is theneutron wavelength, the correlation length of the magneticstructure at T = 40 K can be estimated as L = 350(70) Å. Inthe commensurate phase, the Lorentzian peak broadening ismuch smaller with Ym = 0.070(9) or Ym/Yn = 1.4(2) at T =22 K and Ym = 0.055(3) or Ym/Yn = 1.1(1) at T = 1.8 K.Compared with the crystal structure, the correlation lengthof the magnetic structure is only slightly shorter at T = 22 Kand almost equal at T = 1.8 K. At TN2, the magnetic suscep-tibility at zero field exhibits a large steep decrease (Fig. 6)and the correlation length of the magnetic structure shows alarge increase between the incommensurate (T = 40 K) andthe commensurate structure (T = 22 K). At TN3, the magneticsusceptibility exhibits another large steep decrease (Fig. 6),and the correlation length of the magnetic structure increasesfrom a slightly shorter (T = 22 K) to an almost equal value(T = 1.8 K) compared with that of the crystal structure.064403-8ER-DRIVEN INCOMMENSURATE TO COMMENSURATE … PHYSICAL REVIEW B 109, 064403 (2024)FIG. 10. Temperature dependence of (a) the magnetic phase δ,(b) the phase difference �δ, and (c) the ordered magnetic Fe3+ andEr3+ moments of BaErFeO4 below TN2. δ and �δ are given in unitsof 2π .D. Comparison of magnetic structures of BaRFeO4 compoundsUp to date, the determination of magnetic structures ofisostructural BaRFeO4 ferrite compounds by neutron diffrac-tion has been reported for R = Yb [17], Tm [17], Er (thispaper), and Y [12]. These compounds are compared inTable VI in order of increasing R3+ ionic radii. Three succes-sive magnetic phase transitions are observed for the materialswith magnetic R3+ cations, and two phase transitions forBaYFeO4 with the nonmagnetic Y3+ cation. The BaRFeO4compounds with R = Yb, Tm, Y, adopt incommensurate mag-netic structures below TN1 with the propagation vector k1 =(0, 0, kz ) being stable down to the lowest measured tempera-ture. A phase transition to a commensurate magnetic structurehas not been observed. The BaErFeO4 compound reportedin this paper is the first example in this series where theincommensurate magnetic structure with k1 changes at TN2to a commensurate AFM structure with k2 = ( 12 , 0, 12 ).The relative values of the various spin exchange constantsof the parent compound BaYFeO4 were calculated in a the-oretical study using extended Hückel spin dimer analysis[11]. The result suggested the presence of strong AFM Fe-Fespin-exchange coupling between square pyramidal (Fe1) andoctahedral (Fe2) centers inside each Fe chain compared withmuch weaker inter-chain magnetic Fe-Fe exchange interac-tions. Such a behavior has been experimentally observed forthe magnetic structures of BaRFeO4 ferrites with R = Yb,Tm, and Y [12,17]. A collinear AFM structure with a constantordered Fe moment and a constant magnetic phase appearsinside each Fe chain for all magnetic structures of thesecompounds. According to our neutron diffraction results, alsoBaErFeO4 shows such a behavior for all magnetic structures(see Fig. 9). Interchain magnetic Fe-Fe exchange interactionsalong various possible pathways are much weaker and de-termine the magnetic propagation vector and the choice ofthe irrep. The different behavior between the BaRFeO4 com-pounds reflects the different magnetic anisotropy and size ofthe magnetic R3+ ions. Compared with the ferrite compoundswith smaller R3+ ions (R = Yb, Tm), BaErFeO4 belongs to adifferent group of magnetic structures. It would be interestingTABLE V. Ordered magnetic moments M(mx, my, mz ) in units of µB at T = 40, 22, and 1.8 K calculated for the rings of BaErFeO4.Positions of the magnetic ions (Fe3+, Er3+) are labeled as shown in Fig. 1(b).aIon T = 40 K T = 22 K T = 1.8 KFe14; Fe12 2.50 (0, 2.50, 0) 2.21 (−1.55, 0, 1.57) 2.70 (−1.51, 0, 2.24)Fe21; Fe23 2.50 (0, −2.50, 0) 2.21 (1.55, 0, −1.57) 2.70 (1.51, 0, −2.24)Fe11; Fe13 2.50 (0, 2.50, 0) 2.21 (−1.55, 0, −1.57) 2.70 (−1.51, 0, −2.24)Fe24; Fe22 2.50 (0, −2.50, 0) 2.21 (1.55, 0, 1.57) 2.70 (1.51, 0, 2.24)Er14; −Er12 – 2.52 (2.49, 0, −0.40) 6.68 (6.41, 0, −1.87)Er21; −Er23 – 0.23 (−0.04, 0, 0.22) 6.02 (−0.85, 0, 5.96)Er11; −Er13 – 0.87 (−0.85, 0, −0.14) 5.23 (5.02, 0, 1.47)Er22; −Er24 – 2.36 (0.45, 0, 2.31) 4.80 (0.67, 0, 4.75)aPositions of magnetic ions: Fe14 (0.032, 0.75, 0.216); Fe12 (−0.032, 0.25, −0.216); Fe11 (0.468, 0.25, 0.716); Fe13 (0.532, 0.75, 0.284);Fe21 (0.190, 0.25, 0.023); Fe23 (−0.190, 0.75, −0.023); Fe24 (0.310, 0.75, 0.523); Fe22 (0.690, 0.25, 0.477); Er14 (0.085, 0.75, 0.514); Er12(−0.085, 0.25, 0.486); Er11 (0.415, 0.25, 0.014); Er13 (0.585, 0.75, −0.014); Er21 (0.143, 0.25, 0.310); Er23 (−0.143, 0.75, 0.690); Er22(0.643, 0.25, 0.190); Er24 (0.357, 0.75, −0.190). Ordered magnetic moments with errors: (Fe, T = 40 K) 2.50(9); (Fe, T = 22 K) 2.21(6);1.55(4); 1.57(5); (Fe, T = 1.8 K) 2.70(4); 1.51(3); 2,24(4); (Er, T = 22 K) 2.52(11); 2.49(6); 0.40(9); 0.23(1); 0.04(1); 0.22(1); 0.87(4);0.85(2); 0.14(3); 2.36(11); 0.45(7); 2.31(8); (Er, T = 1.8 K) 6.68(6); 6.41(4); 1.87(5); 6.02(5); 0.85(4); 5.96(4); 5.23(5); 5.02(3); 1.47(4);4.80(4); 0.67(3); 4.75(3).064403-9DÖNNI, POMJAKUSHIN, AND BELIK PHYSICAL REVIEW B 109, 064403 (2024)FIG. 11. Temperature dependence of the components of orderedmagnetic Fe3+ and Er3+ moments (mx, mz ) along the a and c direc-tions of BaErFeO4 below TN2.to verify by neutron diffraction whether the commensuratemagnetic structure with k2 = ( 12 , 0, 12 ) is present in thecompounds with larger magnetic R3+ ions (R = Ho, Dy).The ionic radius of Y3+ is larger than that of Er3+, but thephase transition has not been observed in BaYFeO4, possiblybecause Y3+ is nonmagnetic. We note that the spin dimercalculations [11] predicted for BaYFeO4 a magnetic structurewith k = (0, 0, 0). Experimentally, such a propagation vectorhas not been observed in any of the investigated BaRFeO4ferrites (R = Er, Tm, Yb, Y).For all compounds of Table VI, magnetic ordering of Fe3+ions occurs in two successive magnetic phase transitions atTN1 and TN2. Below TN1, all compounds adopt a SDW mag-netic structure with the same incommensurate propagationvector k1 = (0, 0, kz ) and the same irrep mLD4. OrderedFe moments are oriented along the b direction and rare-earthions are disordered. For the compounds with R = Yb, Tm,and Y, the magnetic propagation vector k1 is stable at TN2and rare-earth ions remain disordered. For T < TN2, orderedFe moments appear inside the ac plane and coexist with theb component of the SDW structure (TN2 < T < TN1). Theresulting incommensurate magnetic structures belong to twodifferent irreps and have two order parameters. They corre-spond to a cycloidal spiral (R = Yb, Tm) or a cycloid (R =Y). In contrast, for BaErFeO4, the change of the magneticpropagation vector to k2 = ( 12 , 0, 12 ) at TN2 coincides to theonset of magnetic ordering of rare-earth ions. Ordered Femoments (as well as ordered Er moments) appear inside the acplane and the b component (of the SDW structure) disappears.The resulting AFM structure belongs to a single irrep with oneorder parameter. In this sense, the phase transition from k1 tok2 at TN2 in BaErFeO4 is driven by the appearance of magneticordering of Er3+ ions through 3d-4 f exchange interactions.The symmetry of the magnetic structures with k1 and k2is different. Therefore, below TN2 the magnetic structuresof BaRFeO4 compounds with different propagation vectorscannot be compared directly. For the nonmagnetic R = Y3+ion, the third magnetic phase transition at TN3 is absent. ForR = Yb and Tm, with k1, the onset of magnetic order ofthe rare-earth ions appears at TN3, at much lower temperaturethan TN2. For R = Yb, 4 f -4 f exchange interactions producea collinear AFM structure, where the size and the magneticphase of ordered Yb moments is constant inside each Ybchain. For R = Tm, 3d-4 f coupling induces magnetic order atpart of the Tm ions. Ordered Tm2 ions coexist with disorderedTm1 ions due to frustration of magnetic exchange interac-tions. Inside each Tm chain, both size and the magnetic phaseof the ordered Tm moments are not constant. Down to thelowest measured temperature (T = 1.6 K), 4 f -4 f exchangeinteractions are too weak to produce magnetic order of Tm1ions in BaTmFeO4. For R = Er with k2, 3d-4 f exchangeinteractions induce magnetic order at TN2. Because of thesymmetry of k2, all Er ions order at TN2. Inside each Er chain,both size and magnetic phase of ordered Er moments are notconstant close to TN2. With decreasing temperature, the 4 f -4 fexchange interactions gradually get stronger and modify themagnetic structure of the Er ions. At TN3, each Er chainreached a constant magnetic phase [Fig. 10(a)] and a compa-rable large size of the Er1 and Er2 ions with a noncollineararrangement [Figs. 9(c) and 10(c)]. Interestingly, magneticstructures of rare-earth ions stabilized by 3d-4 f exchangeinteractions (observed for BaTmFeO4 at TN3) and by 4 f -4 fexchange interactions (observed for BaYbFeO4 at TN3) bothappear in BaErFeO4 at TN2 and TN3, respectively.IV. SUMMARYIn the orthorhombic crystal structure (space group Pnma)of the ferrite compound BaErFeO4, magnetic ions (Fe1 andFe2, as well as Er1 and Er2) form spin chains propagatingalong the b direction. The lattice geometry supports frustrationof magnetic exchange interactions that can lead to complexmagnetic structures with multiple phase transitions. Basedon macroscopic measurements (specific heat and magneticsusceptibility), we showed that BaErFeO4 undergoes threesuccessive magnetic phase transitions at TN1 = 49 K, TN2 =33.4 K, and TN3 = 3.4 K. We employed neutron diffraction todetermine the magnetic structures. In this paper, we show thatBaErFeO4 is the first member in the series of the BaRFeO4 (R:rare earth) ferrite compounds that shows a rare-earth driven064403-10ER-DRIVEN INCOMMENSURATE TO COMMENSURATE … PHYSICAL REVIEW B 109, 064403 (2024)TABLE VI. Comparison of the magnetic structures of BaRFeO4, (R = Yb, Tm, Er, Y) compounds in order of increasing ionic R3+ionic radii. Magnetic phase transitions temperatures (TN1, TN2, TN3). Magnetic propagation vectors k1 = (0, 0, kz ), k2 = ( 12 , 0, 12 ). Irrep:irreducible representation. Maximum (Mmax) and minimum (Mmin ) ordered moment. Ordered components: mx , my, mz. *: no neutron dataavailable.BaYbFeO4 [17] BaTmFeO4 [17] BaErFeO4 (this paper) BaYFeO4 [12]TN1, TN2, TN3 (K) 57, 36, ≈ 18 47.5, 46, ≈ 6 49, 33.4, 3.4 48, 36, –Ordering of Fe3+ Fe3+; TN2 < T < TN1 Fe3+; TN2 < T < TN1 Fe3+; TN2 < T < TN1 Fe3+; TN2 < T < TN1Propagation vector k1 = (0, 0, kz ) k1 = (0, 0, kz ) k1 = (0, 0, kz ) k1 = (0, 0, kz )irrep mLD4 mLD4 mLD4 mLD4Structure type SDW SDW SDW SDWkz; T 0.315(1); 42 K * 0.396(3); 40 K 0.333; 38 KMmax (µB); T 2.70(7); 42 K * 2.50(9); 40 K 2.2(2); 3.0(2); 38 KOrdered components my * my myOrdering of Fe3+ Fe3+; T < TN2 Fe3+; T < TN2 Fe3+; T < TN2 Fe3+; T < TN2Propagation vector k1 = (0, 0, kz ) k1 = (0, 0, kz ) k2 = ( 12 , 0, 12 ) k1 = (0, 0, kz )irrep mLD4 + mLD3 mLD4 + mLD2 mU2+ mLD4 + mLD2Structure type Cycloidal spiral Cycloidal spiral Commensurate, AFM Cycloidkz; T 0.314(1); 25 K 0.374(1); 28 K – 0.358(2); 6 K0.293(2); 1.6 K 0.375(1); 1.6 KMmax; Mmin(μB); T 3.96; 2.87; 1.6 K 3.42; 3.06; 1.6 K 2.70; 2.70; 1.8 K 3.0(1); 2.8(1); 6 KOrdered components mx , my, mz mx , my, mz mx , mz my, mzOrdering of R3+ Yb3+; T < TN3 Tm2; T < TN3 Er3+; T < TN2 –Propagation vector k1 = (0, 0, kz ) k1 = (0, 0, kz ) k2 = ( 12 , 0, 12 ) –irrep mLD3 mLD2 mU2+ –Structure type SDW SDW Commensurate, AFM –Stabilized by 4 f -4 f (T < TN3) 3d-4 f (T < TN3) 3d-4 f (T < TN2) –4 f -4 f (T < TN3) –kz; T 0.293(2); 1.6 K 0.375(1); 1.6 K – –Mmax (μB); T 2.00(7); 1.6 K 3.20(7); 1.6 K 4.80–6.67; 1.8 K –Ordered components mx mx , mz mx , mz –phase transition from an incommensurate SDW structure withk1 = (0, 0, kz ) below TN1 to a commensurate AFM structurewith k2 = ( 12 , 0, 12 ) below TN2. At TN1, 3d-3d exchange in-teractions produce collinear AFM ordering of Fe3+ cationswith a constant magnetic phase in each Fe spin chain. AtTN3, 4 f -4 f exchange interactions stabilize noncollinear AFMordering of Er3+ cations with a constant magnetic phase ineach Er spin chain. At TN2, 3d-4 f exchange coupling inducesstrongly noncollinear magnetic order at all Er3+ cations witha large magnetic phase difference in each Er spin chain. Theappearance of magnetic Er3+ ordering at TN2 coincides witha change of the direction of the Fe3+ moments from the bdirection (above TN2) to inside the ac plane (below TN2) and aphase transition from k1 to k2. For the BaRFeO4 ferrites withsmaller R3+ cations (R = Tm, Yb) and nonmagnetic R = Y,the rare-earth ions remain disordered at TN2 and the magneticstructure remains incommensurate with a propagation vectork1. The magnetic susceptibility of BaErFeO4 shows sharpdecreases at TN2 and at TN3 that coincide with large increasesof the correlation length of the magnetic structure. Magneticproperties of BaErFeO4 are unique in the series of BaRFeO4ferrites.ACKNOWLEDGMENTSThis paper is partially based on experiments performed onHRPT diffractometer (Proposal No. 20202060) at the SwissSpallation Neutron Source SINQ, Paul Scherrer Institute,Switzerland.[1] A. R. Oganov, C. J. Pickard, Q. Zhu, and R. J. Needs, Structureprediction drives materials discovery, Nat. Rev. Mater 4, 331(2019).[2] Y. A. Izyumov, V. E. Naish, and R. P. Ozerov, Neutron Diffrac-tion of Magnetic Materials (Springer, New York, 1991).[3] J. M. Perez-Mato, J. L. Ribeiro, V. Petricek, and M. Aroyo,Magnetic superspace groups and symmetry constraints in in-commensurate magnetic phases, J. Phys.: Condens. Matter 24,163201 (2012).[4] H. T. Stokes, B. J. Campbell, and R. 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