Ho and Fe magnetic ordering in multiferroic HoFe3(BO3)O4
D. K. Shukla, S. Francoual, A. Skaugen, M. v. Zimmermann, H. C. Walker, L. N. Bezmaternykh, I. A. Gudim, V. L. Temerov, J. Strempfer
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Ho and Fe magnetic ordering in multiferroic HoFe (BO ) D. K. Shukla, ∗ S. Francoual, A. Skaugen, M. v. Zimmermann, H. C.Walker, L. N. Bezmaternykh, I. A. Gudim, V. L. Temerov, and J. Strempfer Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany L.V. Kirensky Institute of Physics, Siberian Branch of Russian Academy of Sciences, Krasnoyarsk 660036, Russia (Dated: September 27, 2018)Resonant and non-resonant X-ray scattering studies on HoFe (BO ) reveal competing magneticordering of Ho and Fe moments. Temperature and X-ray polarization dependent measurementsemployed at the Ho L edge directly reveal a spiral spin order of the induced Ho moments in the ab -plane propagating along the c -axis, a screw-type magnetic structure. At about 22.5 K the Fespins are observed to rotate within the basal plane inducing spontaneous electric polarization, P .Components of P in the basal plane and along the c -axis can be scaled with the separated magneticX-ray scattering intensities of the Fe and Ho magnetic sublattices, respectively. PACS numbers: 61.05.cp, 75.25+z, 75.80.+qKeywords: Multiferroics, X-ray diffraction, spin arrangements in magnetically ordered materials, structure-property relationship
I. INTRODUCTION
Spontaneous electric polarization in a magneticallyordered phase and a coupling between the two orderparameters defining a magnetoelectric behavior havebeen observed in many materials , and understoodphenomenologically . However, recent observations ofimproper ferroelectricity , anomalous even within therealm of magnetoelectricity, require an even better un-derstanding of the subject. Rare earth iron borates R Fe (BO ) , where R = rare earth , are multifer-roics, characterized by long range magnetic wave vectors q k c * , and show a large magnetoelectric effect ( P ∼ µCm − ). The R Fe (BO ) crystal structure al-lows a dominant Fe-Fe exchange interaction, which is re-flected by a T N in a narrow temperature range (30-40K) for compounds with different R . The rare earth ex-change interaction takes place via O and B i.e. R-O-B-O-R. However, there is a more direct superexchange pathbetween Ho and Fe through the R-O-Fe chains, causing f − d exchange interaction. Evidence of the f − d ex-change interaction has been reported for several of thesecompounds, based on the observation of splitting of theground-state doublet crystal-field level of Kramer’s R ions using optical spectroscopy . A spin reorientationat low temperature is shown by the R = Gd and Ho compounds due to the large magnetic moment at the Rsite with a different magnetic anisotropy (usually easy-axis) than at the Fe site (easy-plane). However in thecase of R = Tb , the rare earth anisotropy plays such astrong role that it already achieves an easy-axis magneticstructure at T N .The key question regarding the spin arrangement inthe basal plane of the aforementioned easy-plane ferrob-orates, which is essential for understanding the inducedelectric polarization, has remained unanswered. To un-derstand the magnetic structure of these compounds neu-tron powder diffraction on R = Y, Pr, Nd, Tb, Ho andEr , resonant magnetic X-ray scattering on R = Gd and non-resonant magnetic X-ray scattering on R = Nd,Gd, Tb and Y have been performed. From neutronpowder diffraction it has not been possible to determinethe spin direction in the basal plane, since these com-pounds have a trigonal crystal structure. Concerning theX-ray scattering measurements on these compounds, un-til now, it has not been possible to directly determinethe magnetic anisotropy of the rare earth, as the mea-sured intensity variation as a function of the azimuthalangle about the scattering vector is flat . Also reso-nant soft X-ray scattering measurements at the Fe L , absorption edges, which in principle could directly probethe Fe magnetic anisotropy, are not possible, as alreadythe lowest indexed magnetic reflection (0 0 1.5) in thesecompounds is not accessible at these energies.HoFe (BO ) undergoes a structural transition fromR32 to P3
21 ( T S ∼
427 K) and shows spontaneous po-larization and magnetoelectricity below ∼
23 K, while thelong range magnetic order sets in at T N = 38 K .A spin re-orientation occurs at low temperature ( T SR ∼ . Below T SR , the polarization drops to zero . Wehere combine resonant X-ray scattering measurements atthe Ho L -edge and high-energy (non-resonant) X-raymagnetic scattering (HEXMS) measurements at 100 keV,which allow the Ho and the Fe magnetic anisotropies tobe directly determined. Full X-ray polarization analy-sis has been performed at the Ho L resonance, fromwhich an investigation of the Ho moments within thebasal plane becomes possible. HEXMS measurementsare used to investigate the combined Ho and Fe momentsoriented in the ab plane. The Fe moments are foundto rearrange in the ab plane at around 22.5 K inducingmagneto-electric behavior. The purpose of the presentX-ray scattering study is to disentangle the Ho and Femagnetic ordering and attempt to understand the asso-ciated spontaneous electric polarization. II. EXPERIMENTAL DETAILS
Single crystals of HoFe (BO ) were grown using theflux method . The resonant X-ray scattering (RXS)measurements at the Ho L -edge were carried out in ver-tical scattering geometry at beamline P09 at the PETRAIII storage ring at DESY Hamburg, Germany. The sam-ple was mounted on a Psi-diffractometer so that c k U (see Fig. 1). For full polarization analysis, variable lin-early polarized incident X-rays were generated using two400 µ m diamond quarter-wave plates in series . Thepolarization of the scattered signal was analyzed using aPG(006) analyzer crystal. Fig. 1 represents the scatteringconfiguration at P09. The HEXMS investigations werecarried out in horizontal scattering geometry, in trans-mission, at beamline BW5 at the DORIS III storage ringat DESY at a photon energy of 100 keV . The absorp-tion length in HoFe (BO ) is 2.12 mm at 100 keV, whichallows one to probe directly the bulk of the sample in asimilar fashion to neutron measurements. The beam sizewas 1 x 1 mm . At high photon energies the magneticscattering cross-section does not depend on the polariza-tion of the X-rays . III. RESULTS AND DISCUSSION
FIG. 1. (Color online) Vertical scattering geometry at beam-line P09. θ is the Bragg angle, α is the angle between themagnetic moment m and the scattering vector. η and η ′ de-fine the angle of the incident polarization and the rotation ofthe analyzer crystal, respectively. k and k ′ are the incidentand scattered wave vectors. σ ( σ ′ ) and π ( π ′ ) denote the po-larization perpendicular and parallel to the diffraction planeof the incident (scattered) beam, respectively. U , U and U define a basis for the magnetic structure . In Fig. 2, the energy dependence of the intensity ofthe (0 0 4.5) reflection at 6 K in the σ - π ′ channel isshown. The resonant enhancement at the Ho L -edgein the σ - π ′ channel confirms the magnetic origin of thisreflection. Non-resonant (at energies away from the Ho L -edge) intensity was observed in neither the σ - σ ′ northe σ - π ′ channel at this position, as seen from the inset of Fig. 2. ’ ’ I n t e n s it y ( i n ( c t s / s ec a t m A ) Energy (eV) (0 0 4.5) -0.2 -0.1 0.0 0.1 0.20700140021002800 resonance off-resonance I n t e n s it y ( a r b . un it s ) scans (degrees) FIG. 2. (Color online) Energy dependence at the magnetic(0 0 4.5) reflection in the σ - π ′ channel at T ∼ σ - π ′ channel. Full X-ray polarization analysis of the magnetic (0 04.5) reflection has been performed at the Ho L reso-nance at 6 K. The incident linear polarization is variedaround the incident wave vector k using phase plates andat the same time the polarization of the scattered beamis analyzed using a polarization analyzer. For each in-cident polarization state, rocking scans of the analyzercrystal at different analyzer positions η ′ (with steps of25 ◦ between 0 ◦ to 150 ◦ ) are performed upon rotatingthe analyzer and detector assembly around the scatteredwave vector k ′ (see Fig. 1). The measured intensities arefitted by the following relation : I ( η, η ′ ) = I P ′ ( η ) cos η ′ + P ′ ( η ) sin η ′ ) (1) I and the Poincar´ e -Stokes parameters, P ′ and P ′ ,are the fitting parameters. P ′ and P ′ define the linearpolarization state of the scattered X rays . Figure 3shows P ′ (solid circles) and P ′ (hollow circles) measuredfor the magnetic (0 0 4.5) reflection as a function of in-cident polarization angle η . It can be seen that P ′ is aminimum for σ incident light and a maximum for π in-cident light, opposite to the behaviour of a charge peak.In order to fit the parameters P ′ and P ′ we use thedensity matrix formalism and the matrix for the res-onant magnetic cross-section as introduced by Hill andMcMorrow . Only electric dipole processes are consid-ered.To fit the experimental data, instead of directly as-suming an ab plane spin spiral, a formalism derived fora more general c -axis conical spiral structure was used .The best fit of P ′ and P ′ for the magnetic (0 0 4.5)reflection is obtained when Ho moments are forming a -50 0 50 100 150-1.0-0.8-0.6-0.4-0.20.00.20.40.60.8 P ’ , P ’ Incident Polarization, (degree)
FIG. 3. (Color online) P ′ (solid circles) and P ′ (hollow cir-cles) measured at the magnetic (0 0 4.5) refection at reso-nance as a function of the incident polarization at T = 6 K.Solid lines are fits corresponding to the Ho moments form-ing an ab -plane spiral along the c -axis. The dashed line isthe simulated P ′ variation when moments are assumed tobe oriented along the c -axis. The dashed-dotted and dottedlines are simulated P ′ and P ′ variations, respectively, whenthere are equally populated magnetic domains with momentsin the ab -planes. Our data therefore conclusively excludesthese alternative hypotheses for the absence of any azimuthalvariation in the intensity. basal plane spin spiral around the c axis. In this crystalstructure (s.g. P3
21) the wave vector (0 0 3/2) repre-senting a doubling of the unit cell along the c axis allowsthe realization of a basal-plane spiral with a rotation ofthe Ho moment of 60 degrees from one crystallographicplane to another crystallographic plane. This is in agree-ment with the absence of an intensity variation with thevariation of azimuth as observed in previous X-ray scat-tering studies , which is observed when either spinsare aligned along the scattering vector ( k to the c axis)or form a spin spiral structure around it. For these twoextreme cases a significant difference is expected for P ′ . P ′ should have a maximum at η = 45 ◦ for spins orientedalong the c axis (dashed line in Fig. 3). Sometimes natu-rally occurring defects i.e. magnetic domains might alsocause an insensitivity of the resonant magnetic scatteringto azimuthal rotation. To rule out such a possibility wehave also simulated the full polarization analysis resultsfor magnetic domains. For the simulation, moments areconsidered to have an ab-plane collinear magnetic struc-ture i.e. there are three (six) equally populated domainswith the moments lying at 120 ◦ (60 ◦ ) from each other.The calculated variation of P ′ (dashed-dotted line) and P ′ (dotted line) from the magnetic domains cannot re-produce the experimental results. Fitting curves are sim-ulated assuming 67 % linearly polarized incident light,determined from direct beam measurement. Similar re-sults, corresponding to the ab -plane spiral, are obtained when measured at a sample temperature of 14 K. ( Feab)2 50% Fe + 50% Ho (0 0 1.5) at 100 keV I / I( ) , P a / P a ( ) , P c / P c ( ) T (K) (0 0 1.5) I n t e g r a t e d i n t e n s it y ( a r b . un it s ) T (K) (0 0 4.5) at Ho L3 edge Pa Pc FIG. 4. (Color online) Temperature dependences of the nor-malized intensities measured at the magnetic (0 0 1.5) reflec-tion using 100 keV photon energy (solid triangles) and at themagnetic (0 0 4.5) reflection at the Ho L resonance (solidcircles) in σ - π ′ . The inset shows the region around T SR , mea-sured with attenuated flux to reduce beam heating. Hollowsymbols are the electric polarization data, in the basal plane(triangles) and along the crystal axis (circles), from Ref. .Dashed and dotted lines show the squared Fe ab plane mag-netic moments from Ref. and the expected signal for 50 %Fe + 50 % Ho, respectively. The y axis of the data are nor-malized to the (extrapolated) saturation values. The integrated HEXMS intensity of the (0 0 1.5) re-flection measured at 100 keV is shown as a function oftemperature in Fig. 4 together with the integrated inten-sities of the (0 0 4.5) reflection measured as a functionof temperature in the σ - π ′ channel at the Ho resonance.The completely different behavior of the intensities withtemperature is obvious. This can be explained by theelement specifity of RXS on one hand and by the simpli-fied magnetic scattering cross-section at high energies on the other hand. The intensity of the (0 0 4.5) reflec-tion at the Ho L -edge results from the ordering of theHo moments. The intensity evolution of the Ho RXS sig-nal with decreasing temperature is characteristic of thepolarization of the Ho moments by the ordering of theFe moments. This indicates that the magnetic scatter-ing in the resonant condition is due to an induced Homoment. As described in Ref. , for small scatteringangles HEXMS probes the component of the spin pro-jected onto the normal of the scattering plane i.e. along U in Fig. 1. The sample was mounted so that the ab plane, specifically the [h -h 0] direction, is oriented per-pendicular to the diffraction plane and hence momentsin the ab plane along [h -h 0] are probed. In order tohave large enough intensities, measurements were per-formed at the lowest possible position in Q , (0 0 1.5),since the non-resonant X-ray scattering cross-section in-cludes the magnetic form factor, which decreases rapidlywith increasing Q . At 100 keV photon energies, thescattering angles are small, i.e. 1.41 and 4.23 degreesfor the (0 0 1.5) and (0 0 4.5) reflections, respectively.Therefore, an increase in the HEXMS intensity e.g. withtemperature of any reflection along [0, 0, l] directly in-dicates an increasing magnetic moment perpendicular tothe scattering plane. For the ab -plane Ho spin spirals,the intensities measured at resonance and off-resonanceare proportional to the square of the magnetization, sothat the normalized intensities can be directly compared.Therefore, the HEXMS intensity of the (0 0 1.5) reflec-tion consists of a combination of Ho and Fe moments,which have large components in the ab plane. The com-plete vanishing of the HEXMS signal at T SR =4.5 K, asshown in the inset of Fig. 4, means that all Fe as wellas Ho moments undergo a spin-reorientation, easy planeto easy axis, below T SR . We would like to mention thatin neutron diffraction , below T SR , Ho moments are ob-served as oriented by 60 ◦ with respect to each other alongthe c axis, instead. The small ab -component of the Homoments below T SR in neutron diffraction measurementshas been determined from an additional magnetic peakappearing below T SR . We were unable to measure thisreflection at the Ho L resonance due to beam heating,and it was too weak to be detected by HEXMS. At thispoint we can conclude that the ordering of the Fe mo-ments in the ab plane is probed through non-resonantmagnetic X-ray scattering, and the observation of zeropolarization below T SR is due to nearly coinciding mo-ment directions of Fe and Ho.In the HEXMS measurement we see a combination ofHo and Fe moments, as both moments lie in the ab plane.To model the two contributions, we sum 50% of the sig-nal from Ho taken from RXS measurement and 50% ofthe signal from the ab -plane component of Fe taken fromRef. , normalized to their saturation values at 5 K. The50%-50% ratio agrees with saturation magnetization val-ues of about 5 µ B for both magnetic sublattice . Thesimulated intensity is in agreement with the experimen-tal data down to 22.5 K (see Fig. 4). However, to fit theHEXMS data further below 22.5 K, the Fe contributionmust be increased significantly. This clearly indicatesthat at about 22.5 K the Fe moments rotate within the ab plane out of the scattering plane. The total moment inthe basal plane remains unchanged for a rotation of themoments within the basal plane. Neutron powder diffrac-tion can not observe a rotation of the Fe spins in the basalplane in this compound, since neutron diffraction is sen-sitive to the total magnetic moment perpendicular to Q i.e. moments in the basal plane in the present case. Thisis different for HEXMS, where the moment componentperpendicular to the diffraction plane contributes to themagnetic signal. An Fe spin rotation within the ab -planeseems to be directly connected with the second step ofthe two step variation in the Ho RXS signal, (i) a slowincrease down to ∼
23 K and then (ii) a fast up-rise be-low this temperature. Ritter et al. also observe a rapidincrease of the Ho ab plane magnetic moment and a slow increase of the c component of the Fe moments below ∼
23 K.The magnetic scattering intensity is proportional tothe magnetization squared, I = AM , and in the caseof spiral magnets spontaneous electric polarization P ∝ M . These can thus directly be compared ( I and P ).The temperature dependences of HEXMS and RXS in-tensities in the basal plane are very similar to the spon-taneous electric polarization measured by Chaudhury etal. . Zero field electric polarization data in warming cy-cles (normalized to the saturation values) along the a axis and the c axis from Ref. are plotted together withour measurements (see Fig. 4). The close relationshipstrongly suggests that the spontaneous electric polariza-tions arising along the a and c axes, are due to distinctcontributions coming from the Fe and Ho magnetic or-dering, respectively.It should be noted that the inverse Dzyaloshinskii-Moriya (DM) interaction which has been used toexplain the direction of P in the spiral magnets would notbe valid in this case. According to the inverse DM inter-action, P ∝ e ij × (S i × S j ), where S i and S j are the spinmoments on two neighboring magnetic sites and e ij is theunit vector connecting the two sites. For the screw-typemagnetic structure e ij being parallel to S i × S j shouldresult into P = 0. Numerous other examples of systemsexist in which the inverse DM interaction does not drivethe ferroelectric polarization. For example in HoMnO ,the origin of P was explained by exchange striction of Ho-Mn (for P along the c axis) and Mn-Mn (for P along the a axis) . Meanwhile, very recently it was found that inthe helical spin spiral system CaMn O , P develops asa result of an axial lattice distortion constituting an axialvector which remains invariant under inversion. Return-ing to HoFe (BO ) , one possible mechanism to explainthe observed P , could be a symmetric exchange inter-action, similar to that used to explain the anomalouslylarge P in DyMnO . According to this mechanism, P in HoFe (BO ) would arise through a symmetric ex-change interaction between the Ho and Fe moments or-dering with the same periodicities, τ Ho = τ F e = . Thiswould cause a polar lattice modulation with q = 0, induc-ing spontaneous electric polarization. Also, in the presentstudy, due to a subtle observation of the Fe spin rotationwithin the ab -plane at ∼
23 K, which has been identifiedas coinciding with the onset of P , HoFe (BO ) indicatesimilarities to the R MnO compounds . In R MnO ( R =Tb and Dy) , there exist three distinct phase transitionswhich are characterized as 1) T N ∼
41 K, where Mn spinsorder with a propagation vector along the b -direction;2) T S ∼
28 K for R = Tb and ∼
18 K for R = Dy, identifiedas an onset of P , where the Mn spins develop a compo-nent along the c -axis; and 3) a final transition T N ∼ b direction. Similar such transitions in HoFe (BO ) could be identified as 1) T N ∼
39 K, where Ho and Feboth order antiferromagnetically; 2) T S ∼
23 K, the on-set of P , where Fe spins are found to rotate within the ab -plane; and 3) a final spin reorientation transition at T SR ∼ P to the interactionsamong specific ions (Ho-Ho, Fe-Fe and Ho-Fe). Never-theless, we would like to emphasize our observation of ascaling of the Ho squared magnetization with P c and theFe squared magnetization with P a . In addition to simi-larities with the extensively studied R MnO compounds,the direct observation of the screw-type magnetic struc-ture of the Ho sublattice and scaling of the ferroelectricpolarization with the separated contributions of the Hoand Fe magnetizations in HoFe (BO ) open up a newavenue for understanding the multiferroicity mechanismin similar compounds. IV. SUMMARY
In conclusion, Ho moments form an ab -plane spin spiralpropagating along the c axis i.e. a screw-type magneticstructure. An accelerated increase of the magnetizationof the Ho magnetic sublattice below 22.5 K seems to be a driving force behind the observed rotation of the Fespins within the basal plane. This temperature is identi-fied as the onset of the spontaneous electric polarizationin the system. The Ho RXS signal directly scales withthe polarization developed along the c axis below 18 K.However, polarization developed along the a axis , be-low 15 K, scales with the HEXMS intensity which hasbeen identified as a contribution mostly from Fe mag-netic moments. These observations demonstrate that themacroscopic electric polarization in these systems devel-ops in close connection with spin rearrangement withinthe magnetic ordered phase, due to competition betweenthe different magnetic order parameters and magneticanisotropies. This study is useful to understand the spon-taneous electric polarization in a broad range of multifer-roic materials, where, below the onset of T N , magnetiza-tion from more than one magnetic sublattice and strongstructural and magnetic anisotropies are involved. ACKNOWLEDGMENTS
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