Evolution of the 2D antiferromagnetism with temperature and magnetic field in multiferroic Ba CoGe O 2 2 7 V. Hutanu,1,2,∗ A.P. Sazonov,1,2 M. Meven,1,2 G. Roth,1 A. Gukasov,3 H. Murakawa,4 Y. Tokura,4,5 D. Szaller,6 S. Bord´acs,7 I. K´ezsm´arki,6 V.K. Guduru,8 L.C.J.M. Peters,8 U. Zeitler,8 J. Romhanyi,9 and B. N´afr´adi10 1RWTH Aachen University, Institut fu¨r Kristallographie, D-52056 Aachen, Germany 2Forschungszentrum Ju¨lich GmbH, Ju¨lich Centre for Neutron Science at MLZ, D-85747 Garching, Germany 3CEA, Centre de Saclay, DSM/IRAMIS/Laboratoire L´eon Brillouin, F-91191 Gif-sur-Yvette, France 4RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan 5Department Applied Physics, University of Tokyo, Tokyo 113-8656, Japan 6Department of Physics, Budapest University of Technology and Economics and Condensed Matter Research Group of the Hungarian Academy of Sciences, H-1111 Budapest, Hungary 7UniversityofTokyo,DepartmentofAppliedPhysicsandQuantum-PhaseElectronicsCenter(QPEC),Tokyo113-8656,Japan 4 8High Field Magnet Laboratory, Institute of Molecules and Materials, 1 Radboud University Nijmegen, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands 0 9Leibniz Institute for Solid State and Materials Research, IFW Dresden, D-01069, Germany 2 10E´cole Polytechnique F´ed´erale de Lausanne, Laboratory of Nanostructures n and Novel Electronic Materials, CH-1015 Lausanne, Switzerland a J Wereportonsphericalneutronpolarimetryandunpolarizedneutrondiffractioninzeromagnetic 8 field as well as flipping ratio and static magnetization measurements in high magnetic fields on the 1 multiferroic square lattice antiferromagnet Ba2CoGe2O7. We found that in zero magnetic field the magnetic space group is Cm(cid:48)m2(cid:48) with sublattice magnetization parallel to the [100] axis of this ] orthorhombicsetting. Thespincantinghasbeenfoundtobesmallerthan0.2◦ inthegroundstate. l e This assignment is in agreement with the field-induced changes of the magnetic domain structure - below 40mT as resolved by spherical neutron polarimetry. The magnitude of the ordered moment r has been precisely determined. Above the magnetic ordering temperature short-range magnetic t s fluctuationsareobserved. Basedonthehigh-fieldmagnetizationdata,werefinedtheparametersof . t the recently proposed microscopic spin model describing the multiferroic phase of Ba2CoGe2O7. a m - I. INTRODUCTION romagnetism. Recently, using both conventional unpo- d larized neutron diffraction data13 and magnetic symme- n try analysis3,4 we have studied the magnetic structure o c Emergence of ferroelectricity in several members of Ba2CoGe2O7 at 2.2K, below TN ≈ 6.7K. The re- of the melilite family, including Ba CoGe O , below [ 2 2 7 sultsshowedanantiferromagnetic(AFM)orderoftheCo their magnetic ordering temperature has been recently magnetic moments within the (a,b) plane, while neigh- 1 discovered.1,2 The remarkable and complex response of boring planes stacked along the c axis are ordered ferro- v these materials to magnetic and electric fields can be 7 magetically (FM). Throughout the paper we index the 1 predicted by considering the magnetic point group sym- momentum-space coordinates q = (h,k,l) in the cor- 5 metries of both the paramagnetic and magnetically or- responding reciprocal lattice units (r.l.u.) of the or- 4 dered phases.3,4 The field dependence of the ferroelec- thorhombic Cmm2 crystallographic unit cell proposed . tric polarization in Ba CoGe O was reproduced by ab- 1 2 2 7 previously in Ref. 13, where the two mirror planes are initio calculations,5 however, the magnitude of the pre- 0 the (100) and (010) planes and the two-fold axis points 4 dicted polarization was considerably smaller than the along the [001] direction. The relation between the unit 1 experimental value. The spin-wave excitation spec- cells based on the space groups P¯42 m and Cmm2 is il- 1 : trum of this material together with the strong opti- v lustrated in Ref. 13. The direction of the Co magnetic i cal magnetoelectric effect exhibited by these magnon moments was assumed to be parallel to the [100] direc- X modes are captured by a microscopic spin Hamilto- tion of the Cm(cid:48)m2(cid:48) cell, based on bulk magnetization r nian where single-ion anisotropy dominates over mag- measurements in our former work,13 while it was tenta- a netic exchange interaction.6–10 Nevertheless, some of the tively assigned to be parallel to the [110] axis in early magnonmodesappearinginintermediatemagneticfields neutron diffraction studies.14 Nevertheless, the moment (5T< B <14T) remained unexplained by the theory. direction within the (a,b) plane cannot be determined In Ba CoGe O , weak ferromagnetism was observed be- 2 2 7 unambiguously by unpolarized neutron diffraction due low the antiferromagnetic ordering temperature of T ≈ N to the presence of energetically equivalent magnetic do- 6.7K (Refs. 11 and 12) as a result of a small about mains with equal population. Moreover, the magnitude 0.1◦ canting of the spins within the (a,b) plane induced of the small canting (Fig. 1) cannot be measured with by the Dzyaloshinskii-Moriya interaction (ϕ(cid:48) in Fig. 1). high precision by unpolarized neutron diffraction. Po- While the canting predicted based on density functional larized neutron diffraction techniques are fast develop- theory calculations5 is small it is much less than 0.1◦ in ing experimental methods well suited for precise deter- zero field in contradiction with the proposed weak fer- 2 mination of magnetic structures, spin canting, magnetic II. EXPERIMENTAL domain structures and fluctuations.15–17 Therefore, we revisit the magnetic symmetry of the ground state and High quality single crystals of Ba CoGe O were 2 2 7 refine the parameters previously obtained for magnetic grown by floating-zone technique and characterized in interactions and anisotropies using a combination of po- previous studies.9,11,13,18 larized and unpolarized neutron diffraction methods and SNP measurements were performed at 4K with a high-field magnetization experiments. Cryopad on the polarized single-crystal diffractometer Inthiswork,wepresentpolarizedandunpolarizedneu- POLI@HEiDi at the hot source of the FRMII reactor trondiffractionresultsofBa2CoGe2O7 singlecrystalsto- in Garching, Germany.19,20 A Ge (311) monochromator gether with magnetization measurements. We refine its was used to generate a monochromatic neutron beam magnetic structure in the zero-field ground state (mag- with 1.17˚A wavelength. The polarization of both the netic space group, MSG, Cm(cid:48)m2(cid:48)) and study the influ- incoming and scattered beam was controlled by polariz- enceoftheappliedfieldonthemagneticdomainpopula- ing3Heneutronspinfilters. Inordertocontrolthedecay tion. By unpolarized neutron diffraction experiments we of the filter polarization the incoming beam polarization investigated the temperature dependence of the sublat- was measured by a transmission monitor. The scattered ticemagnetization. Basedontheresultsofbulkmagneti- beam polarization was also systematically monitored on zationmeasurementsathighmagneticfieldupto32Twe the (440) structural reflection. Polarization corrections havedeterminedthemagneticinteractionandanisotropy described in detail in Ref. 20 were applied. With this parameters. method1%precisiononpolarizationmatrixelementscan The paper is organized as follows. The experimental be reliably reached.20 The sample was mounted with the proceduresaredescribedinSec.II.InSec.IIIAthedirec- [110] direction perpendicular to the scattering plane in a tionoftheprimaryAFMorderisdeterminedbymeansof specialFRMIIclosedcyclecryostatsuitabletobehosted sphericalneutronpolarimetry(SNP).Thezerofieldmag- inside the Cryopad. Stable temperatures down to 3.9K neticdomainpopulationsandtheeffectofmagneticfield have been reached at the sample position in this setup. on the magnetic domain structure is also analyzed. In Forzerofieldcooledmeasurementsthesamplewascooled Sec.IIIB,weestimatethecantinganglebyanothertype insidetheCryopad(strayfield<5mG).Tostudythein- ofpolarizedneutrondiffractiontechnique,namelybythe fluenceofexternalfieldonthemagneticdomaindistribu- flipping-ratio method. Sec. IIIC compares the temper- tionthesamplewaswarmedto15KoutsidetheCryopad. ature evolution of the magnetic moment to predictions Anexternalfieldofmaximum20mTparalleltothe[110] by molecular field models. The critical exponent of the direction has been applied using resistive coils outside antiferromagneticphasetransitionisalsodetermined. In the cryostat. The sample has been cooled over T down N Sec.IIID,themagneticexchangeandanisotropyparam- to 4K in the applied field. Finally the magnetic field etersaredeterminedusinghigh-fieldmagnetizationdata. was switched off and the cryostat was placed back into Theorderedmagneticmomentobtainedbyneutronscat- the Cryopad for the SNP measurements without warm- teringiscomparedtothevaluedeterminedfromthemag- ingitoverthetransitiontemperature. Fortherefinement netic susceptibility data in the paramagnetic phase. The of the SNP data the program SNPSQ of the Cambridge paper is concluded in Sec. IV. Crystallography Subroutine Library was used.21 Polarized neutron flipping-ratios were measured on the Super-6T2 diffractometer at the Orph´ee reactor of LLB.22Theexperimentsweredoneinanappliedexternal magnetic field of 6.2T both above and below the mag- netic transition temperature at T =10K and T =1.6K, respectively. Additional flipping ratio measurements at 1.6K in 0.5T, 1T and 4T magnetic fields were also per- formed. TheprogramCHILSQ(Ref.21)wasusedforthe [010] leastsquaresrefinementsoftheflippingratiosinthelocal [100] susceptibility approach with the atomic site susceptibil- ity tensor χ (Ref. 23). ij [001] Unpolarized single-crystal neutron diffraction stud- ϕ′ ies were done on the four-circle diffractometer HEiDi (Refs. 19 and 24) at the hot source of FRM II. The tem- perature dependence of selected magnetic Bragg reflec- tions were measured with wavelength λ = 0.87˚A in the temperature range 2.2–15K. Magnetization measurements at T =4K in a 32T bit- ter magnet were performed in the High Field Magnet FIG.1. (Coloronline)MagneticstructureofBa2CoGe2O7 at Laboratory,Nijmegen. Themagnetizationwasmeasured 2.2K: View from the [001] direction. in fields parallel to [110], [100] and [001] axes. The abso- 3 lute magnetic moment was confirmed by magnetization In case of Ba CoGe O two sets of 180◦ domains ro- 2 2 7 measurements performed in 0-14T field range by ACMS tated by 90◦ with respect to each other are allowed by inPhysicalPropertyMeasurementSystem(PPMS)from symmetry(Fig.2leftpanel). Ifoneofthemisdominant, Quantum Design. significantnon-zerotermsoccurinallsixoff-diagonalel- ements of the polarization matrix. If only domains type I and II are present, only P and P terms occur with xz zx III. RESULTS AND DISCUSSION opposite signs, other off-diagonal elements are zeroes. In the case of domains type I and IV are present, only P yz A. Polarized neutron diffraction: Spherical and P are non-vanishing. If only 180◦ domains e.g. zy neutron polarimetry type I and III is present, then elements P and P are xy yx non-zero and they change sign when domains II and IV In a neutron scattering experiment the relationship are present. between the polarization of the incident and scattered In order to determine the equilibrium domain struc- beams P and P(cid:48) can be conveniently expressed by the ture and the MSG of Ba CoGe O SNP measurements 2 2 7 tensor equation:25 have been performed on a single crystal with vertically P(cid:48) =PP +P(cid:48)(cid:48) or in components P(cid:48) =P P +P(cid:48)(cid:48), oriented[110]axis. Thisgeometrygaveaccessto(h,h,l) i ij j i type reflections. Usually using SNP measurement even where tensor P describes the rotation of the polarization few magnetic reflections is sufficient to precisely deter- and P(cid:48)(cid:48) is the polarization created in the scattering pro- minethedirectionofthemagneticinteractionvector.25,26 cess. The experimental quantities which are obtained in Thefullpolarizationmatrixofthe(440),(111)and(112) an SNP experiment, for each Bragg reflection, are the reflections and some of their equivalents were measured. components Pij of the 3×3 polarization matrix P The sample was prepared in three different magnetic do- mainstatesZFC,FC110andFC¯1¯10afterzero-fieldcool- I++−I+− P = ij ij , (1) ing, cooled in 20mT parallel to the [110] axis and cooled ij I+++I+− antiparalleltothe[110]axis,respectively. Asanexample, ij ij the polarization matrices measured for the (112) Bragg where the indices i and j refer to one of the three right- handed Cartesian coordinates x(cid:48), y(cid:48) or z(cid:48) defined by the reflectionat4KafterZFC,FC110andFC¯1¯10procedures experiment. Direction x(cid:48) is parallel to the the scatter- are presented in Table I. ing vector Q and z(cid:48) is vertical (normal to the scattering Measured polarization matrices were treated within plane). Thefirstsubscriptcorrespondstothedirectionof two magnetic structure models: Calc110 and Calc100. the initial polarization, while the second is the direction ForthemodelCalc110theAFMcomponentisfixedalong of the analysis. I is the measured intensity with spins [110] (MSG P2(cid:48)1212(cid:48)1) while for the Calc100 model the parallel (++) and antiparallel (+−) to j. AFM component is along [100] (MSG Cm(cid:48)m2(cid:48)). For the Thepolarizationmatrixiscloselyrelatedtothepolar- calculations of the expected P the lattice constants and ization tensor as structural parameters from previous measurements were (cid:28)P P +P(cid:48)(cid:48)(cid:29) used.13,18 The magnitude of the magnetic moments for P = i ij j , (2) the Co ions were initially set to values obtained from ij P i domains Ref. 13 and afterwards refined together with the do- where the angle brackets indicate an average over all the main ratios. Both models fail to explain the observed different magnetic domains which contribute to the re- polarization matrices assuming a single-domain state. flection. Considering for magnetic domains allowed by symmetry It was indicated in former studies that energetically with equal populations gave much better agreement for equivalentmagneticdomainsarepresentinBa CoGe O the ZFC case for both models. The calculated P with 2 2 7 in zero magnetic field.13 As a result, it is impossi- refined magnetic domain populations are given in Ta- ble to distinguish with conventional unpolarized neu- ble I for both Calc110 and Calc100 models. As demon- tron diffraction between three possible MSG P2(cid:48)2 2(cid:48), strated in Table I the agreement between the measured 1 1 1 Cm(cid:48)m2(cid:48), and P112(cid:48).3 On the other hand, SNP can be and calculated components of Pij for all three ZFC, used to determine th1e magnetic domain populations and FC110 and FC¯1¯10 domain states is much better for the thus the MSG of the system. The magnetic interaction model Calc100 (χ2 = 7%) than for the model Calc110 vectorscorrespondingto180◦domainspresentinanequi- (χ2 =25%). Hence, the model Calc110 can be excluded domain antiferromagnetic structure rotate the neutron and the model Calc100 with sublattice magnetization beam polarization in opposite directions. Thus an equi- parallel to [100] is found to be the magnetic structure domain crystal would be characterized by a polarization with Cm(cid:48)m2(cid:48) MSG. matrix with non-vanishing elements only in the diagonal Now we focus on the magnetic domain population (P ) for mixed nuclear and magnetic Bragg reflections. refined within Calc100 model. Figure 2 schematically ii A crystal containing unequal volumes of magnetic do- demonstrates the influence of fields parallel to the [110] mains, however, has also non-zero off-diagonal elements and [¯1¯10] axes on the domain imbalance. Following P in the polarization matrix. ZFCprotocolthealloweddomainsareequallypopulated ij 4 TABLEI. Polarizationmatriceson(112)mixednuclearandmagneticBraggreflectionofBa2CoGe2O7 measuredat4Kafter zero-field cooling (ZFC), field cooling with B (cid:107) [110] and field cooling with field in opposite direction B (cid:107) [¯1¯10]. Calculated matrices from two magnetic models Calc110 and Calc100 (described in text) are also shown. ZFC FC,B(cid:107)[110] FC,B(cid:107)[¯1¯10] Pij x(cid:48) y(cid:48) z(cid:48) x(cid:48) y(cid:48) z(cid:48) x(cid:48) y(cid:48) z(cid:48) Observed x(cid:48) 0.73(1) 0.01(2) −0.06(4) 0.74(1) 0.03(2) 0.30(2) 0.72(2) −0.03(2) −0.25(1) y(cid:48) 0.07(6) 0.76(2) 0.04(4) 0.00(3) 0.82(2) −0.02(6) 0.06(3) 0.81(1) 0.01(6) z(cid:48) 0.04(2) 0.04(1) 0.76(4) −0.29(1) 0.00(3) 0.79(2) 0.29(1) 0.00(1) 0.79(1) Calc100 x(cid:48) 0.78 −0.01 −0.05 0.78 0.01 0.22 0.78 −0.02 −0.24 y(cid:48) 0.01 0.88 0.00 −0.01 0.88 0.00 0.02 0.88 0.00 z(cid:48) 0.05 0.00 0.89 −0.22 0.00 0.89 0.24 0.00 0.89 Calc110 x(cid:48) 0.89 0.00 −0.03 0.89 0.00 0.17 0.89 −0.01 −0.20 y(cid:48) 0.00 0.94 0.00 0.00 0.94 0.00 0.01 0.94 0.00 z(cid:48) 0.03 0.00 0.95 −0.17 0.00 0.95 0.20 0.00 0.95 ZFC FC, B (cid:107)[110] FC, B (cid:107)[¯1¯10] H H I II I II I II IV III IV III IV III I II III IV I II III IV I II III IV 28(3)% 24(3)% 21(3)% 27(3)% 37(4)% 38(4)% 12(3)% 13(3)% 11(3)% 13(3)% 36(4)% 40(4)% FIG. 2. (Color online) Influence of field cooling on domain imbalance. Left panel: Zero-field cooling. Middle panel: Field coolinginB (cid:107)[110]. Rightpanel: FieldcoolinginB (cid:107)[¯1¯10]. Schematicviewofthespinstructureformthe[001]direction. Black solid arrows represent the Co magnetic moments. Red empty arrows show the direction of the field-induced FM component. Refined domain population is presented as a table below each panel. withintheexperimentalprecision(Fig.2left). Noprefer- The change in volume ratio of the magnetic domain ential domain orientation in ZFC experiment was found. population is linear with field strength between B = 0, Memory effect in the AFM domain population was ab- 10 and 20mT fields. This extrapolates to about 40mT sent in subsequent thermal cycles between 4-15K. Field- applied along [110], which is required to fully suppress cooling even in a small 10mT field applied parallel to the energetically unfavored domains in agreement with [110] axis induces observable unbalance in the domain static magnetization measurements (Sec. IIID, Ref. 27). population. In B =20mT domains I and II are energet- This field value is much smaller than the critical field ically favorable compared to domains III and IV (Fig. 2 of abut 1T where the field induced electric polariza- center). Theirvolumecoverabout3/4ofthecrystalvol- tion disappears,1,11 supporting the presence of an anti- ume. As the field is applied along [110], the population ferromagnetic polarization-polarization coupling present within the I-II and III-IV pairs is expected to be equal in the spin Hamiltonian.28 taking into account their symmetry. Indeed, the refined valuesofdomainpopulationisinagreementwiththisex- Whenthemagneticandnuclearunitcellsareidentical pectation. Cooling with the same field strength applied and magnetic and nuclear intensity occurs at the same alongtheoppositedirection,i.e. along[¯1¯10],reversesthe positioninreciprocalspace,likeinBa2CoGe2O7,SNPal- situation;domainsIIIandIVbecomedominantandtake lowstodeterminethemagneticstructurefactorandthus about 3/4 of crystal volume (see Fig. 2 right). Experi- the magnitude of the ordered magnetic moment. This mentswithotherfielddirectionsshowedthesamedomain calculationyields2.7µB/Coingoodagreementwiththe formation supporting that domains are equienergetic. results of unpolarized neutron diffraction discussed be- low (Sec. IIIC). SNP is sensitive not only to the magni- 5 tude but also to the direction of the magnetic moment. 1.2 Thus we tried to use it to determine the magnitude of spin canting. Our calculations showed, however, that for Ba2CoGe2O7 canting angle less than ∼ 2.5◦ introduces 0.9 differences in the polarization matrices smaller than the experimental error, at least for the accessible Bragg re- ) B flections, and so is not measurable reliably. Therefore, µ ( 0.6 to estimate the canting angle more precisely we rely on M F polarized neutron flipping-ratio measurements as well as µ magnetization data (see Sec. IIIB). 0.3 B. Polarized neutron diffraction: Flipping-ratio measurements 0 0 2 4 6 Classical polarized neutron flipping-ratio method,29 H (T) is used to study the magnetization distribution around magnetic atoms in ferromagnetic and paramagnetic ma- FIG. 3. (Color online) Field dependence of the induced FM terials. In antiferromagnets the scattering cross-section componentµFMperpendiculartothedirectionoftheprimary is usually polarization independent and the classical AFM ordering (dark rectangles). Error bars are within the method is not applicable.30 Polarized neutron flipping- symbols. The line is a linear fit to the neutron flipping-ratio data. Circles are magnetization data taken from Ref. 27. ratio measurements in antiferromagnetic compounds are therefore performed in special conditions: above T in N the paramagnetic state and in external magnetic fields. as22,34 For each Bragg reflection, the flipping ratio, R, mea- (cid:12) (cid:18) (cid:19)(cid:12) sured by polarized neutron diffraction is ϕ(cid:48) =(cid:12)(cid:12)1tan−1 √−2Dab (cid:12)(cid:12) (4) (cid:12)2 3J −D (cid:12) I+ (F +F⊥)2 c R= = N M , (3) with D /J = (g −g )/g , where g = 2.0023 is the I− (F −F⊥)2 i i e i e N M free electrons g-value. Index i denotes the crystallo- where I is the intensity of neutrons diffracted with spins graphic orientations. Calculations based on g-value and parallel(+)andantiparallel(−)totheappliedmagnetic J parameters obtained in Sec. IIID yield Dc = 0.12 K, field, FN is the nuclear structure factor and FM⊥ is the Dab = 0.32 K and ϕ(cid:48) = 0.085◦ in good agreement with projection of the magnetic structure factor F to the our experimental upper limit of ϕ(cid:48) <0.2◦. M scattering plane. In a real experiment one has to take into account the degree of polarization of the neutrons, the efficiency of the flipping and the angle between F C. Unpolarized neutron diffraction M and the scattering vector. Theexperimentaldataatboth1.6Kand10Ktemper- In order to follow the temperature evolution of the atures measured in B =6.2T is well fitted to the model magneticstructureofBa CoGe O ,severalintensemag- 2 2 7 ofsphericaldistributionofthe magneticmomentaround netic and structural Bragg reflections were collected in Co atoms. No significant local anisotropy was foundand the temperature range of 2.2–15K. Their integrated in- themagneticsusceptibilitytensorisdescribedbyasingle tensitieswereusedtorefinethemagnitudeoftheComag- non-zero parameter χ =χ =χ =0.166(3)µ /T. netic moment. All other parameters such as the atomic 11 22 33 B Additional low temperature flipping ratio measure- positional parameters, the isotropic temperature factors, ments were performed at different magnetic fields to ex- the scale and the extinction parameters were fixed ac- tract the field induced ferromagnetic (FM) component, cordingtoourpreviousstudyofthenuclearandmagnetic µ , of Ba CoGe O . Figure 3 shows µ perpendicu- structuresatfixedT =2.2Kand10.4Ktemperatures.13 FM 2 2 7 FM lar to the direction of the primary AFM ordering. The The upper panel of Figure 4 shows the temperature extrapolation of µ (H) to zero field gives 0.01(1)µ , dependence of the integrated intensity of the magnetic FM B which is in a good agreement with bulk magnetization (110) Bragg reflection as an example. The intensity measurements.27 Taking into account the magnitude of of this reflection decreases continuously with increas- Co magnetic moment from unpolarized neutron diffrac- ing temperature and becomes constant above T . The N tionwecanestimatethevalueofcanting,ϕ(cid:48),inzerofield temperature-independentintensityaboveT isduetothe N to be less than 0.2(2)◦. structuralcontributiontotheBraggreflection. Itshould Similar values for the canting angle were re- be noted that nuclear intensity is forbidden for corre- ported for other Dzyaloshinskii-Moriya (DM) sponding (010) reflection in the tetragonal space group antiferromagnets.31–33 The magnitude of ϕ(cid:48), is de- P¯42 m. However,theleastsquaresfitgivestemperature- 1 termined by, D, the strength of the DM interaction independent nuclear contribution of about 3% of the 6 100 effect. The integrated intensity, I, of magnetic Bragg reflec- tionsfollowsthesquareofthemagneticorderparameter. 75 The data were fitted close to T assuming a power law N s) dependence to the equation26,35 t ni u 50 b. (cid:18)T −T(cid:19)2β r I =I +I N , (5) a n 0 ( TN I 25 where I is the nuclear (structural) contribution to the n intensity, I is the magnetic intensity at T = 0 and β is 0 0 the critical exponent. The fit yields β = 0.21 ± 0.04 as the critical exponent, however it should be noted 0 0.5 1 1.5 that only a limited number of data point is available in the close vicinity of T . Nevertheless, this value is N unusual, it is inconsistent with two-dimensional Ising (β ≈ 0.13), three-dimensional Ising (β ≈ 0.33) or three- dimensional Heisenberg model (β ≈ 0.37).36 It is close to the value found for layered antiferromagnets with XY anisotropy.37,38 It was suggested theoretically that β = 0.23 is an universal property of the finite-size XY model.39 This universal value expected to hold over an extended, but not universal temperature regime is in good agreement with our observations. Our experimen- tal β = 0.21±0.04 value is also close to that expected for a tricritical transition (β = 0.25).40 In this scenario the other fluctuating order is likely the ferroelectric one. However, in order to deffinitively settle the value of β further experimental would be useful. The lower panel of Fig. 4 shows the temperature de- pendenceoftherefinedComagneticmoment. Forasim- ple antiferromagnetic structure, the temperature depen- FIG. 4. (Color online) Upper panel: Temperature depen- dence of the magnetic moment, µ, in the conventional denceoftheintegratedintensityofthemagnetic(110)Bragg molecular-field model can be expressed as reflection. The experimental data (shown by circles) is taken (cid:18) (cid:19) by unpolarised single-crystal neutron diffraction. Solid line µ 3S T µ N =B , (6) shows a fit to Eq. 5. The dashed line represents the nuclear µ S S+1 T µ 0 0 (structural) contribution. Lower panel: Temperature depen- dence of the Co magnetic moment for Ba2CoGe2O7. The whereS isthemagneticmomentofthesystem, µ0 isthe experimental data from the single-crystal neutron diffraction magnetic moment at T = 0K, and BS is the Brillouin measurements are shown by circles. The solid line is a result function. of a modified molecular field model (Eq. 7). The dotted line This simple model fails to reproduce the experimental is shown to illustrate the deviation of µ(T) from the conven- dataasshownbydashedlineinthelowerpanelofFig.4 tional molecular field model (Eq. 6). withS =3/2[high-spin(HS)stateofCo2+, t5 e2]. Note 2g g that the ordered moment at T = 0 is µ = 2.81 µ /Co B whichissomewhatlessthanthefullmomentcorrespond- magnetic intensity at T = 0. This contribution is small ingtoS =3/2. Whilethemolecularfieldtheorypredicts but experimentally clearly observable at all equivalent asharponsetoftheorderparameterbelowT ,theexper- N positions up to room temperature according to the neu- imental magnetic moment values start to grow at higher tron diffraction measurements both at HEiDi and 6T2. temperatures above T . Moreover, the experimental µ N This forbidden intensity could be attributed both to valueisalwayshigherthanthecurvedescribedbyEq.6. small orthorhombic distortion (Refs. 13 and 18) and to We analyzed the data in a modified molecular field Renninger scattering (Ref. 14). Observations of a large model41 number of forbidden peaks at different wavelengths, at µ (cid:18)h 3S T [1+a(µ/µ )2] µ (cid:19) N 0 different instruments and in different samples as well as =BS + , (7) µ T S+1 T µ performed ψ scans suggest that observed intensities are 0 0 due to distortion. However, the intensities are only par- where h is a fictive magnetic field modeling the effect of tially described within the orthorhombic Cmm2 model, short-range magnetic order above T , and a is a magne- N suggestingthatatleastpartofitisduetotheRenninger toelastic parameter describing the magnetostrictive shift 7 The magnetic and the structural unit cells coincide, TABLEII. Parametersobtainedfromthefitusingthemod- thus, we search for the ground state in a site factorized ified molecular field model (Eq. 7) with S =3/2. (cid:81) (cid:81) form, |Ψ(cid:105) = |ψ (cid:105)|ψ (cid:105). The variational wave i∈A j∈B i j h a µ0 (µB) R44 functions |ψi(cid:105) are states of the four dimensional local 0.09±0.05 0.43±0.07 2.81±0.05 0.996 Hilbert space, spanned by an S = 3/2 spin. The varia- tionalparametersareobtainedbyminimizingtheground state energy E = (cid:104)Ψ|H|Ψ(cid:105). (cid:104)Ψ|Ψ(cid:105) ofT (Refs.41and42). ThefitusingEq.7forHSCo2+ Calculations based on Eq. 8 with parameters J = N is shown by solid line in Fig. 4 lower panel. This later 2.3K, J = 1.8K, Λ = 14K, g = 2.24, g = 2.18 and z x y approach yields a remarkably good account to the data. g = 2.1 closely reproduce the observed data. Due to z Table II summarizes the fitted parameters. A small but large single-ion anisotropy, Λ, the S =1/2 ground state z finitehisresponsiblefortheincreaseofµaboveTN. We ofCo2+ inBa2CoGe2O7 isseparatedbyagapofapprox- suggestthathisduetothefluctuatingshortrangeorder imately 4meV from the S = ±3/2 spin states. This z persisting above TN which was also observed for other shows up as an increase of the field dependent magne- layered antiferromagnets.43 tization when the Zeeman energy becomes equal to the anisotropygap. Indeedataround10Tthemagnetization in [110] and [100] directions deviate from a linear behav- D. Magnetization measurements ior and show an upward curvature. It is clearly seen in the field derivatives of the Ba CoGe O magnetization, 2 2 7 The field dependence of the magnetization measured dM/dH, (Fig. 6). The sudden drop in the derivative upto32TatT =4KisplottedinFigure5. Themagne- around 15T indicates that at this field the spin config- tization increases continuously with increasing field and uration becomes a fully collinear ferromagnet. At zero starts to saturate at approximately 15T for [100] and temperature, this would correspond to a metamagnetic [110] directions, while it continues to increase signifi- transition(spin-floptransition),whichhasbeenobserved cantly up to 32T for [001] direction. The reduced slope in the softening of a magnon mode in previous THz ab- of the magnetization for fields parallel to the [001] axis sorption spectroscopy studies.7 Further increase of the clearly shows the easy-plane character of the magnetic magnetic field changes the magnitude of the magnetic structure. The saturation magnetization is about 5% moment by mixing the Sz = ±3/2 spin states into the higher in the [100] direction compared to the [110] direc- ground state. Therefore, the high-field saturation mo- tion,indicatingfiniteg-factoranisotropywithinthe(a,b) ment is considerably larger than the ordered moment plane. The highest magnetization of about 3.3µ /Co is observed by neutron scattering in zero field (or in the B measured in B = 32T parallel to the [100] axis. The low-field range). The inset of Fig. 6 focuses on the field value µ ≈ 3.3µ /Co is significantly higher than the or- rangeof0–1T.Theweakcurvaturein0–0.1Tfieldrange B dered moment obtained from zero-field neutron diffrac- istheconsequenceofthein-planedomainrearrangement tion experiments indicating the presences of single ion in agreement with the spherical polarimetry data. anisotropy. We also investigate the anisotropy of the spin sys- To reproduce the field dependence of the magnetiza- tem by analyzing the susceptibility data in the high- tion we follow Refs. 7 and 8, and take the anisotropic temperature phase reproduced from Ref. 12 in Fig. 7. Hamiltonian: The inverse magnetic susceptibilities are almost linear inthetemperaturerangefromabout30Kupto300Kin- (cid:88)(cid:16) (cid:17) (cid:88) H=J SˆxSˆx+SˆySˆy +J SˆzSˆz dicating a paramagnetic behavior at high temperatures. i j i j z i j (i,j) (i,j) Overthistemperatureregion,thedatawerefittedbythe (cid:88) (cid:88) Curie-Weiss model +Λ (Sˆz)2−hg Sˆ . (8) i i C i i χ= , (9) T −θ where the (i,j) pairs denote nearest-neighbor sites. The CW axes x, y and z are parallel to the [100], [010] and [001] whereC istheCurieconstantandθ istheCurie-Weiss CW crystallographic directions, respectively. The Hamilto- temperature. The best fits were obtained with θ = CW nianin Eq. 8 includes a strong single-ion anisotropy Λ, −33.4±0.3K,µ =4.35±0.01µ forB (cid:107)[001]direction eff B as well as an exchange anisotropy J (cid:54)=J . Suggested by whileθ =−20.8±1.1Kandµ =4.88±0.03µ was z CW eff B theorthorhombicCm(cid:48)m2(cid:48)MSGandthedirectiondepen- foundforB ⊥[001]direction. Thecorrespondingfitsare dence of the saturation magnetization in the (a,b) plane shown in Fig. 7. θ is negative in agreement with the CW different g and g values were allowed in the g-factor antiferromagnetic nature of the dominant Co-Co near- x y tensor describing the Zeeman interaction. Although the est neighbor exchange. By comparing the Curie-Weiss lattice symmetries allow for the Dzyaloshinskii-Moriya temperature, |θ |, with the 3D ordering temperature, CW (DM)interactionD(S ×S )–itseffectonthemagne- T , a ratio |θ |/T = 5, can be obtained. This in- A B N CW N tizationintheintermediate-andhigh-fieldregioncanbe dicates a significant suppression of the 3D ordering, as neglected. a result of quasi-2D anisotropy. Indeed we found that 8 3.5 3 2.5 )B 2 µ (o C 1.5 M 1 B||[100] 0.5 B||[110] B||[001] 0 0 5 10 15 20 25 30 35 40 B (T) FIG. 5. (Color online) Magnetization of Ba2CoGe2O7 with FIG.7. (Coloronline)Temperaturedependenceoftheinverse fields applied along [100], [110] and [001] directions (symbols magnetic susceptibilities with magnetic field applied perpen- from top to bottom respectively). Solid lines are results of dicular (green/lower symbols) and parallel (red/upper sym- calculations described in the text with parameters indicated bols) to the [001] axis of Ba2CoGe2O7 as reproduced from in the text. Ref. 12. The lines show Curie-Weiss fit to the data in the 30–300K range (see text). 0.4 B||[100] 0.35 B B||[110] d 0.2 0.3 B||[001] M/ 0.1 that the measured values for the effective magnetic mo- B 0.25 d ment of Co2+ in Ba2CoGe2O7 are close to those mea- M/d 0.2 0 0 0.3 0.6 0.9 sured in other Co oxides, e.g., CoO and Co2SiO4 with d 0.15 B (T) µeff ≈4.4−4.9µB (Refs. 46 and 47). 0.1 0.05 0 IV. CONCLUSION 0 5 10 15 20 25 30 35 B (T) By a combination of bulk magnetization measure- ments, polarized and unpolarized neutron diffraction experiments we determined, with high-precision, the FIG. 6. (Color online) Field derivatives of magnetization, dM/dH,ofBa2CoGe2O7 withfieldsappliedalong[100],[110] ground state magnetic structure of Ba2CoGe2O7 and and [001] directions (symbols from top to bottom respec- its evolution with magnetic field and temperature. The tively). Inset shows the low-field region below 1T. magnetic space group is Cm(cid:48)m2(cid:48) with the AFM sublat- tice magnetization laying parallel to the [100] direction. Magnetic field dependent SNP identified a change in the Ba CoGe O is a two dimensional antiferromagnet. The AFM domain structure below 0.04T in-plane fields in 2 2 7 in-plane nearest neighbor antiferromagntic exchange in- goodagreementwithfielddependentmagnetizationmea- teraction is J =2.3K. The inter-plane ferromagnetic ex- surements. The resultsare compatiblewith small <0.2◦ change interaction is about an order of magnitude lower. cantingofthespinswithinthe(a,b)plane. Themagnetic It is estimated to be J(cid:48) = −0.2K based on a mean field orderingtemperatureT =6.7Kissignificantlyreduced N approximation.45 relative to the mean field value estimated from the high- Based on the parameters obtained from the Curie- temperaturesusceptibilitydata. Thisweattributetothe Weiss fits, we also calculated the g-factor according to strongquasi-2DrealspaceanisotropiesinthespinHamil- tonian. The temperature dependence of the order pa- (cid:115) 3k C rameter exhibits an unusual β ∼ 0.21 critical exponent, B g = , (10) N S(S+1)µ2 which is indeed compatible with the predictions for the A B 2D XY spin model. The value of β might also signal the where k is the Boltzmann constant and N is the Avo- vicinity of a tricritical point where the other fluctuating B A gadro’s number. We obtain g = 2.2 and g = 2.6 phase is ferroelectric. At low fields below 6T polarized (cid:107) ⊥ for the directions parallel and perpendicular to the [001] neutron diffraction data shows no significant local mag- axis, respectively. The easy-plane anisotropy is in agree- netic anisotropy within the (a,b) plane. The magnetic ment with diffraction measurements and the high-field susceptibility tensor can be well described by a single magnetization data (Sec. IIID). It should be noted, non-zero parameter χ = χ = χ = 0.166(3)µ /T 11 22 33 B 9 in agreement with magnetization measurements. How- ported by BMBF contract 05K10PA2 and by the Euro- ever, using high-field magnetization (up to 32T) a slight pean Commission under the 7th Framework Programme in-plane g-factor anisotropy was observed pointing to through the ’Research Infrastructures’ action of the ’Ca- the orthorhombic character of the magnetic symmetry. pacities’ Programme, NMI3-II Grant number 283883, by At zero magnetic field the ordered magnetic moment is Hungarian Research Funds OTKA K108918, TA´MOP- µ = 2.81 µ /Co while the high-field saturation value is 4.2.1.B-09/1/KMR-2010-0001, TA´MOP 4.2.4.A/2-11-1- B significantly higher it exceeds µ = 3.3 µ Co. This is a 2012-0001 and the Funding Program for World-Leading B/ consequence of the spin gap of 4meV induced by single Innovative R&D on Science and Technology (FIRST ion anisotropy. Program), Japan. J.R. acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG) and the Emmy-Noether program. The work in Lausanne ACKNOWLEDGMENTS was supported by the Swiss National Science Founda- tion (SNSF). We acknowledge support of the HFML- The first author dedicates this paper to the mem- RU/FOM,memberoftheEuropeanMagneticFieldLab- ory of his mother who suddenly passed away during oratory (EMFL). Part of this work has been supported the preparation of the manuscript. We thank K.Penc by EuroMagNET II under EU contract number 228043. for discussions. This research project has been sup- ∗ [email protected] 18 V. Hutanu, A. Sazonov, H. Murakawa, Y. Tokura, 1 H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and B.Na´fra´di, andD.Chernyshov,Phys.Rev.B84,212101 Y. Tokura, Phys. Rev. B 85, 174106 (2012). (2011). 2 M. Akaki, H. Iwamoto, T. Kihara, M. Tokunaga, and 19 V. Hutanu, M. Meven, Leli`evre-Berna, and G. Heger, H. Kuwahara, Phys. Rev. B 86, 060413 (2012). Physica B 404, 2633 (2009). 3 J.M.Perez-MatoandJ.L.Ribeiro,ActaCryst.A67,264 20 V. Hutanu, M. Meven, S. Masalovich, G. Heger, and (2011). G. Roth, J. Phys.: Conf. Ser. 294, 012012 (2011). 4 P. Toledano, D. D. Khalyavin, and L. C. Chapon, Phys. 21 P. J. Brown and J. C. Matthewman, Rev. B 84, 094421 (2011). http://www.ill.fr/dif/ccsl/html/ccsldoc.html (2000). 5 K.Yamauchi,P.Barone, andS.Picozzi,Phys.Rev.B84, 22 A.Gukasov,A.Goujon,J.-L.Meuriot,C.Person,G.Exil, 165137 (2011). and G. Koskas, Physica B: Condensed Matter 397, 131 6 S.MiyaharaandN.Furukawa,JournalofthePhysicalSo- (2007). ciety of Japan 80, 073708 (2011). 23 A. Gukasov and P. J. Brown, J. Phys.: Condens. Matter 7 K. Penc, J. Romha´nyi, T. Ro˜o˜m, U. Nagel, A. An- 14, 8831 (2002). tal, T. Feh´er, A. Ja´nossy, H. Engelkamp, H. Murakawa, 24 M. Meven, V. Hutanu, and G. Heger, Neutron News 18, Y.Tokura,D.Szaller,S.Borda´cs, andI.K´ezsm´arki,Phys. 19 (2007). Rev. Lett. 108, 257203 (2012). 25 P. Brown, Physica B 297, 198 (2001). 8 J.Romha´nyiandK.Penc,Phys.Rev.B86,174428(2012). 26 P. Brown, Neutron Scattering from Magnetic Materials, 9 I. K´ezsma´rki, N. Kida, H. Murakawa, S. Borda´cs, editedbyT.Chatterji(Elsevier,Amsterdam,2006)p.215. Y. Onose, and Y. Tokura, Phys. Rev. Lett. 106, 057403 27 H. T. Yi, Y. J. Choi, S. Lee, and S.-W. Cheong, Appl. (2011). Phys. Lett. 92, 212904 (2008). 10 S. Bordacs, I. Kezsmarki, D. Szaller, L. Demko, N. Kida, 28 J. Romha´nyi, M. Lajko´, and K. Penc, Phys. Rev. B 84, H.Murakawa,Y.Onose,R.Shimano,T.Room,U.Nagel, 224419 (2011). S. Miyahara, N. Furukawa, and Y. Tokura, NATURE 29 R.Nathans,C.G.Shull,G.Shirane, andA.Andresen,J. PHYSICS 8, 734 (2012). Phys. Chem. Solids 10, 138 (1959). 11 H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and 30 P.J.Brown,J.B.Forsyth, andF.Tasset,PhysicaB267- Y. Tokura, Phys. Rev. Lett. 105, 137202 (2010). 268, 215 (1999). 12 T.Sato, T.Masuda, andK.Uchinokura,PhysicaB329- 31 T.Thio,T.R.Thurston,N.W.Preyer,P.J.Picone,M.A. 333, 880 (2003). Kastner, H. P. Jenssen, D. R. Gabbe, C. Y. Chen, R. J. 13 V. Hutanu, A. Sazonov, M. Meven, H. Murakawa, Birgeneau, andA.Aharony,Phys.Rev.B38,905(1988). Y. Tokura, S. Borda´cs, I. K´ezsma´rki, and B. Na´fra´di, 32 G. M. De Luca, G. Ghiringhelli, M. Moretti Sala, Phys. Rev. B 86, 104401 (2012). S. Di Matteo, M. W. Haverkort, H. Berger, V. Bisogni, 14 A. Zheludev, T. Sato, T. Masuda, K. Uchinokura, G. Shi- J. C. Cezar, N. B. Brookes, and M. Salluzzo, Phys. Rev. rane, and B. Roessli, Phys. Rev. B 68, 024428 (2003). B 82, 214504 (2010). 15 in Neutron Scattering from Magnetic Materials, edited by 33 B. Bohnenbuck, I. Zegkinoglou, J. Strempfer, C. S. Nel- T. Chatterji (Elsevier Science, Amsterdam, 2006). son, H.-H. Wu, C. Schu¨ßler-Langeheine, M. Reehuis, 16 B. Na´fra´di, T. Keller, H. Manaka, A. Zheludev, and E.Schierle,P.Leininger,T.Herrmannsdo¨rfer,J.C.Lang, B. Keimer, Phys. Rev. Lett. 106, 177202 (2011). G. Srajer, C. T. Lin, and B. Keimer, Phys. Rev. Lett. 17 B. Na´fra´di, T. Keller, H. Manaka, U. Stuhr, A. Zheludev, 102, 037205 (2009). and B. Keimer, Phys. Rev. B 87, 020408 (2013). 34 M. Elhajal, B. Canals, and C. Lacroix, Phys. Rev. B 66, 014422 (2002). 10 35 E. E. Rodriguez, C. Stock, K. L. Krycka, C. F. Majkrzak, 41 A. B. Beznosov, B. I. Belevtsev, E. L. Fertman, V. A. P. Zajdel, K. Kirshenbaum, N. P. Butch, S. R. Saha, Desnenko, D. G. Naugle, K. D. D. Rathnayaka, and J. Paglione, and M. A. Green, Phys. Rev. B 83, 134438 A. Parasiris, Low Temp. Phys. 28, 556 (2002). (2011). 42 A. B. Beznosov, E. L. Fertman, V. V. Eremenko, P. P. 36 M. F. Collins, Magnetic Critical Scattering (Oxford Uni- Pal-Val, V. P. Popov, and N.N. Chebotayev, LowTemp. versity Press, New York, 1989). Phys. 27, 320 (2001). 37 M. Greven, R. J. Birgeneau, Y. Endoh, M. A. Kastner, 43 A. Antal, T. Feh´er, B. Na´fra´di, L. Forro´, andA. Ja´nossy, B.Keimer,M.Matsuda,G.Shirane, andT.R.Thurston, arXiv 1210.5381. Phys. Rev. Lett. 72, 1096 (1994). 44 Foraroughestimateofhowwellthemodelfitsthedatawe 38 M. Greven, R. J. Birgeneau, Y. Endoh, M. A. Kastner, use R=1−((cid:80)(µfit−µobs)2)/((cid:80)(µobs−(cid:104)µobs(cid:105))2) where M. Matsuda, and G. Shirane, Zeitschrift Fur Physik B- µobs arethemeasuredvalues,(cid:104)µobs(cid:105)istheaverageofµobs, condensed Matter 96, 465 (1995). and µfit are the calculated values from the fit. 39 S. T. Bramwell and P. C. W. Holdsworth, Phys. Rev. B 45 N.AshcroftandN.Mermin,SolidStatePhysics (Saunders 49, 8811 (1994). College, Philadelphia, 1976). 40 T.ThioandA.Aharony,Phys.Rev.Lett.73,894(1994). 46 R. A. Verhelst, R. W. Kline, A. M. de Graat, and H. O. Hooper, Phys. Rev. B 11, 4427 (1975). 47 A. Sazonov, M. Meven, V. Hutanu, G. Heger, T. Hansen, and A. Gukasov, Acta Cryst. B 65, 664 (2009).