Exploring metamagnetism of single crystalline \eun\ by neutron scattering
X. Fabreges, A. Gukasov, P. Bonville, A. Maurya, A. Thamizhavel, S. K. Dhar
EExploring metamagnetism of single crystalline EuNiGe by neutron scattering X. Fabr`eges , A. Gukasov , P. Bonville , A. Maurya , A. Thamizhavel and S. K. Dhar Laboratoire L´eon Brillouin, CEA, CNRS, Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France SPEC, CEA, CNRS, Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-Sur-Yvette, France and Department of Condensed Matter Physics and Materials Science,Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India
We present here a neutron di ff raction study, both in zero field and as a function of magnetic field, of themagnetic structure of the tetragonal intermetallic EuNiGe on a single crystalline sample. This material is knownto undergo a cascade of transitions, first at 13.2 K towards an incommensurate modulated magnetic structure,then at 10.5 K to an equal moment, yet undetermined, antiferromagnetic structure. We show here that the lowtemperature phase presents a spiral moment arrangement with wave-vector k = ( , δ, c -axis, the square root of the scattering intensity of a chosen reflection matches verywell the complex metamagnetic behavior of the magnetization along c measured previously. For the magneticfield applied along the b -axis, two magnetic transitions are observed below the transition to a fully polarizedstate. PACS numbers:
I. INTRODUCTION
Neutron di ff raction on Eu materials is inherently di ffi cultbecause of the very strong absorption cross section of naturaleuropium. Nevertheless, magnetic structure determinationswere carried out a few decades ago in single crystallineEuAs and in EuCo P . Interestingly, antiferromagneticEuAs presents a feature which was to be found in many Euintermetallics studied later: a first transition to an incom-mensurate phase, extending only over a few K, followed bya transition to an equal moment phase . But most of theinformation about the magnetic structure of Eu compoundshas been quite often inferred only through single crystalmagnetization measurements or M¨ossbauer spectroscopy onthe isotope Eu, like in EuPdSb . In the last few years,however, neutron di ff raction with thermal neutrons was suc-cessfully employed to unravel the magnetic structure of someintermetallic divalent Eu materials . Of the two valencesEu + and Eu + , only the divalent, with a half-filled 4 f shellwith L = = /
2, has an intrinsic magnetic moment of7 µ B . Despite the quite weak anisotropy of the Eu + iondue to its vanishing orbital moment, a variety of structureswas found, ranging from ferromagnetic in EuFe P andEu PdMg , collinear antiferromagnetic (AF) in EuCu Sb to incommensurate spiral in EuCo P and EuCu Ge . Thisindicates that the interionic interactions are quite complexin Eu intermetallics, most probably due to the oscillatingcharacter of the RKKY exchange and also to the relativeimportance of the dipole-dipole interactions between ratherlarge Eu + moments of 7 µ B . As a result, the deduction oftheir magnetic structure from solely macroscopic measure-ments is often impossible.In this work, we present a neutron di ff raction study of sin-gle crystalline EuNiGe . EuNiGe was the subject of two pre-vious studies, on a polycrystalline sample and on a singlecrystal . It crystallizes in a body-centered tetragonal struc-ture (space group I mm ) and presents two magnetic transi-tions, at T N = FIG. 1: Magnetization curves at 1.8 K in EuNiGe with the fieldalong b and c taken from Ref.15. The insert shows the low fieldpart of the curve for H // b , where a dip is clearly seen. commensurate moment modulated phase, then at T N = c axis shows a particularly complex behavior at1.8 K , with two spin-flop like magnetization jumps at 2 and3 T followed by a saturation in the field induced ferromagneticphase at 4 T (see Fig.1). When the field is applied along the a ( b ) axis, the magnetization curve shows no such anomaly andreaches saturation at 6 T. However, a small deviation from lin-earity is observed for this direction at low field, as shown inthe insert of Fig.1, and the linear behaviour is recovered above1.3 T.Assuming simple AF structures with propagation vectors k = (001) or ( / / c using a molecular field modelinvolving two nearest neighbor exchange constants (along a and along c ), the dipolar field and a weak crystal fieldinteraction . Clearly, an experimental determination of thezero field magnetic structure is needed in order to go furtherin the understanding of EuNiGe . This was the original aim a r X i v : . [ c ond - m a t . s t r- e l ] M a y of the present work, but while exploring the in-field meta-magnetic behavior of EuNiGe , we have found a number ofmagnetic phase transitions which were not detected by macro-scopic measurements. Here we give the detailed descriptionof these transitions with the field oriented along the b ( a ) and c directions and present a molecular field model with 4 ex-change integrals which partially succeeds in reproducing boththe zero field structure and the magnetization curves. II. EXPERIMENTAL DETAILS
Details of the preparation method of the EuNiGe singlecrystals, grown with In flux, can be found in Ref.15. Forthe neutron di ff raction study, a 3x3.7x1 mm single crystalwas mounted with the c axis or the b axis vertical in thevariable temperature insert of a 7.5 T split-coil cryomagnet.Experiments were performed on the neutron di ff ractometerSuper-6T2 (Orph´ee-LLB) . Scattering intensity maps weremeasured at λ = ◦ stepsand recording the di ff raction patterns with a position-sensitivedetector (PSD). This allowed to detect all transformationsof the magnetic structure under magnetic field by directinspection of the 3D crystal reciprocal space obtained bytransformation of the measured sets of PSD images. Forquantitative refinements and studies of the magnetic fielddependence, single (lifting) counter mode was used. Theresults were analyzed using the Cambridge CrystallographySubroutine Library(CCSL) .Prior to magnetic structure studies, the nuclear structurewas verified in zero field at 15 K. A total of 213 reflectionswere measured and 94 unique ones (74 > σ ) were obtainedby merging equivalents, using space group I mm . Since Eu isa strongly absorbing neutron material, the absorption correc-tions are of major importance in the merging procedure.Theywere made using the ABSMSF program of the CCSL whichproperly accounts for the crystal shape. The absorption cor-rection was found very important, yielding an absorption co-e ffi cient µ = − , which resulted in up to 80% reduc-tion in the intensity of some measured reflections. Absorptioncorrection yielded an improvement of the internal factors ofnuclear reflections from R int = .
32 (without corrections) to R int = .
07 and it was applied to all measured magnetic datasets. The nuclear structure parameters obtained in the refine-ment were found in agreement with those published earlier ,with lattice parameters a = b = . c = . FIG. 2: Nuclear reflections and magnetic satellites at 1.6 K inEuNiGe in the (h k 0) plane. Satellites observed around (1 1 0)can be indexed with a k = ( / δ
0) propagation vector.
III. THE MAGNETIC STRUCTURE IN ZERO FIELD
The zero field magnetic structure of EuNiGe was first stud-ied using a PSD. Figure 2 shows a bidimensional (h k 0) in-tensity cut in the reciprocal space at 1.6 K. Apart from thenuclear reflections being located at integer positions, thereare additional satellites which can be assigned to an antifer-romagnetic contribution. Eight magnetic satellites can be dis-tinguished around (1 1 0) and indexed using a k = ( ± / ± δ δ = k domains are la-belled k = ± ( / δ k = ± ( / - δ k = ± ( δ /
0) and k = ± (- δ / I mm space groupand correspond to a rather complex antiferromagnetic struc-ture with a very large unit cell, whose details are discussedbelow. For instance, k = ( / δ
0) corresponds to a magneticcell four times larger than the crystallographic one along a and20 times along b . Actually, it is not possible to decide whether k is incommensurate with the lattice spacing or not although,generally, such a small δ value points to an incommensuratestructure.The temperature evolution of k = ( δ /
0) magnetic reflec-tions were followed to monitor the transitions from the anti-ferromagnetic to the paramagnetic state. The value δ (cid:39) = / + ions, with a molecular field constant | λ | (cid:39) / µ B . Such a fit gives an excellent agreement be-tween calculated and experimental data, with a transition tem- FIG. 3: Position of the magnetic satellite at 5 and 12 K. At 12 K aclear shift of the δ value is observed. perature T t (cid:39) + δ /
1) satellite in-tensity deviates from the mean field function, and vanishesabove 13.5 K. In this temperature range, the observed smallshift of the δ value corresponds to the intermediate phase re-ported in the M¨ossbauer investigation , which shows an in-commensurate modulation of Eu moments. Therefore, thevalue δ (cid:48) = IV. THE FIELD VARIATION OF MAGNETIC STRUCTURE
With the magnetic field applied along c , the behaviour ofthe magnetization is quite peculiar (see Fig.1). We monitoredthe scattering intensity of the (1 0 1) reflection for H // c asa function of field. This reflection contains both nuclear andmagnetic contributions, the magnetic one being proportionalto the square of the induced magnetization. Figure 4 bottomshows the field evolution of the square root of the (1 0 1) mag-netic scattered intensity (after substraction of the nuclear com-ponent) compared with the magnetization data. A very goodagreement between the two probes is observed, with two welldefined jumps at respectively 2 and 3 T, followed by the spin-flip transition at H (cid:39) + momentof 7 µ B . The top panel of figure 5 shows the scattering in-tensity along the ( / δ
0) direction at 1.6 K for H =
0, 2 and2.5 T. In zero field, two well defined peaks are observed at δ = ± b ∗ . At 2 T, the fieldof the first magnetization jump, a first order transition occurswith the appearance of two new satellites with δ ∗ = δ = b ∗ direction. At 2.5 T,the zero field δ = δ ∗ = = / δ δ = δ ∗ = δ = δ ∗ = FIG. 4:
Top : (1 + δ /
1) scattering intensity vs temperature (red cir-cles) and fit to a (squared) S = / Bot-tom : magnetization at 1.8 K (black line) and square root of the (1 0 1)scattered intensity at 1.6 K (red circles) vs field applied along c . one in turn vanishes at the second critical field of 3 T, abovewhich only ferromagnetic (FM) contributions remain. Finally,the intensity of (FM) reflections reach saturation at 4 T corre-sponding to the spin-flip field.With the magnetic field applied along b , we monitored thescattering intensities corresponding to the k = ( / δ
0) and k = ( δ /
0) domains between 2 and 14 K in fields up to6 T. Figure 6 top presents the evolution of k and k inten-sities at 8 K. Below µ H = k vanishes at the benefitof k . This first transition corresponds to a spin-flop transi-tion selecting the (a,c) magnetic domains in which momentsare orthogonal to the applied magnetic field. Above 2.5 Tthe reverse process occurs with the sudden extinction of the k signal at the benefit of k . Finally, no antiferromagneticcontribution is observed above 4.3 T, the sample being fullypolarized. The corresponding phase diagram extracted fromneutron di ff raction data is presented in Fig.6 bottom. Com-paring with the phase diagram for H // [100] in Ref.15, ex-tracted from macroscopic measurements, one sees that the lat-ter could not catch the first transition at low field. Besides this,the overall agreement is good, except for the behaviour nearthe transition at T N (cid:39) . FIG. 5: At 1.6 K for H // c : Top : ( / δ
0) scans at 0, 2 and 2.5 T;
Bot-tom : ( / δ
0) scattering intensities vs field for δ = δ = H // b : Top : Evolution of k and k satellites versusmagnetic field at 8 K. Intensities have been normalized to one in bothcases for clarity. Three transitions are observed at 0.4, 2.5 and 4.3T; Bottom : corresponding (H,T) phase diagram.
V. MAGNETIC STRUCTURES REFINEMENT
For the zero field and in-field magnetic structure determina-tion, integrated intensity measurements were performed usinga single counter.In zero field, and for each k domain, 46 satellites werecollected at 1.6 K, of which about 25 (0 T) were statisticallyrelevant ( I > σ ) and used in the refinement. The mag-netic structure was analyzed by using the propagation vectorformalism. Tetragonal symmetry and the highly symmetri-cal (0 0 z) Wyko ff position occupied by the Eu + ion limitspossible magnetic structures to amplitude modulated and he-licoidal ones. First, models of a circular helix with momentsconstrained in the planes perpendicular to the highest symme-try axes ( a , b , c ) were tested. For all four propagation vectorsthe best fit was obtained for the helix envelope with the ma-jor axes of 7.6(3) µ B lying in the plane perpendicular to thelargest component of the propagation vector, namely the (b,c)plane for k and k , and the (a,c) plane for k and k . Inthis case, the moment rotates by φ = ◦ along the main com-ponent of the propagation vector. The corresponding mag-netic structure is presented in Figure 7 left. The refinementyielded the following populations of domains in zero field: k = k = k = k = < R w < k is alternatively φ and π − φ . Thus, theincremental angle φ cannot be determined from the neutrondata and both the circular and elliptic solutions are valid can-didates.With a field of 2.5 T along the c -axis (i.e. between thetwo metamagnetic transitions), for each k vector, 46 satelliteswere collected at 1.6 K, of which about twelve were statis-tically relevant and used in the refinement. In this case theantiferromagnetic contribution is well described by a similarcircular structure with reduced ordered magnetic moment of m = . µ B . We note that the associated error bars are big-ger, with R w = . b , integrated intensities of k relatedreflections were collected at 5 K and 4.5 T. A total of 111 re-flections were collected, of which 12 were statistically rele-vant ( I > σ ) and used in the refinement. The antiferromag-netic contribution could not be refined with such a small setof reflections. However, neutrons are only sensitive to mag-netic contributions orthogonal to the probed Q-vector. In ourdataset, the ( / δ L) reflections are not observed indicating thelack of an ordered magnetic moment in the orthogonal (a,b)plane. Therefore, one can describe the ordered moments asbeing antiferromagnetically coupled and collinear along the c -axis.
4 1 !!!!
3 2 bccbc ! Best refinement Calculated structurec
FIG. 7: a) k refined magnetic structure in zero-field, with φ = ◦ .Note that other structures allowing for an “elliptical” envelope, i.e.with φ < ◦ , are also possible. b) Calculated magnetic structurefrom the model with 4 exchange integrals, dipolar interactions andanisotropy described in section VI. The labels 1-4 indices indicatethe spin positions along the a -axis, with φ = ◦ . VI. MODELLING THE MAGNETIC PROPERTIESA. The mean field self-consistent calculation
We have searched for a set of exchange integrals that wouldreproduce the zero field magnetic structure and the behaviorof the magnetization, using a self-consistent calculation of themoment arrangement in the presence of exchange and dipole-dipole interactions among Eu + ions. Since this calculationcannot involve a large number of magnetic sites, it cannot in-tegrate the δ components of the propagation vector evidencedby neutron di ff raction in both zero-field and in-field structures.Based on the neutron di ff raction results, we chose a propa-gation vector k = ( / z = z = / , i.e. 8 ions. The calculated structure con-sists of ferromagnetic (b,c) planes. The calculation does noteither consider the twinning when computing the magnetiza-tion. Consequently, it cannot be expected to reproduce all ofthe experimental features and must be considered as approxi-mate.We consider 4 isotropic exchange integrals, 3 of them intra-sublattice and one inter-sublattice, the sublattices in questionbeing the simple tetragonal lattice and that obtained by thebody centering translation with vector ( / / / ) (see Fig.8).The hamiltonian of the problem contains an exchange part(a negative integral means an antiferromagnetic coupling), adipolar interaction part and also an anisotropy, or crystal field,part. The two latter terms are needed for a realistic descriptionof the system since they are of the same order of magnitudeand their balance determines the direction of the moments.Each ion is linked by exchange to its neighbors according tothe paths described in Fig.8, and a molecular field is calcu- J a J J J c b ac FIG. 8: Definitions of the 4 exchange integrals involved in the meanfield calculations of the magnetic structure of EuNiGe . The twosimple tetragonal sub-lattices are sketched by blue and red hexagons. lated for each of the 8 ions in the cell. The infinite range dipo-lar field acting on each ion is calculated using an Ewald typesummation method . The dipolar field has no free parameter,it depends only on the way the magnetic cell is chosen. Anaxial anisotropy (crystal field) term is added with the form: H an = DS z , where D is a coe ffi cient with magnitude a few0.1 K and S z is the component along c of the Eu + spin. For D <
0, this term favors a moment alignment along c and for D > T N (cid:39)
13 K and ofthe paramagnetic Curie temperature θ p (cid:39) .We have tried to obtain a zero-field structure like that shownin Fig.7 a, which is the closest to the actual structure, neglect-ing the small δ component of the propagation vector. First,one finds that the D coe ffi cient must be taken negative, oth-erwise the moments have a strong a ffi nity to lie in the (a,b)plane. Then, one must take J > J a < a . The other integrals J a and J c have no obviously requiredsign. Exploring the { J α } space of exchange integrals with rea-sonable values, we found that the magnetization jumps at 2and 3 T for H // c cannot be reproduced together. The parame-ter set we propose reproduces the spin-flop at an intermediatevalue 2.5 T and the spin-flip at 4 T for H // c , the spin-flip at6 T for H // a or b , and the correct T N and θ p values. It yieldsa zero-field structure of “elliptic” type, i.e. the moments liein the (b,c) plane with an incremental angle φ = ◦ both forthe ions with z = z = ( / , / , / ) (see Fig.7 b). Neitherthe phase shift between the two spirals nor their absolute po-sition can be ascertained in the calculation. The parameter setreads: J a = J c = J = J a = − D = − H // a (orange curve), the magnetization islinear with the field, which is to be expected since H is per-pendicular to the (b,c) plane of the spiral, inducing a conicalmoment arrangement. For H // b (magenta curve), a dip is ob-served, which is due to the fact that the conical structure is notrealized, when the field lies in the plane of the spiral, until athreshold spin-flop field is reached, here 1.3 T. In the insert of Magnetic field (T) M µ B / E u ) H // aH // bH // c
H (T) M ( µ B / E u ) (a,b) average a gn e ti za ti on ( FIG. 9: Calculated magnetization curves along the 3 symmetry di-rections for the spiral structure of EuNiGe with k = ( / a and b simulating the experimental data with H // (100)or (010). Fig.9, the red curve is an average of the magnetizations along a and b , simulating the presence of domains. It is readilycomparable with the data shown in the insert of Fig.1, and thecalculated spin-flop field of 1.3 T is in very good agreementwith the measured value. B. Discussion
In the above parameter set, the absolute value of J a , 0.6 K,is six times larger than that of J a , 0.1 K. This may seem puz-zling, since the next nearest neighbour distance along a istwice the nearest neighbour distance. The dominant exchangein EuNiGe is probably the RKKY interaction, which varieswith distance as 1 / r , but which is an oscillating function of r . Then, one may speculate that a large | J a | / J a ratio can hap-pen if RKKY exchange is close to a node for r = a and ismaximum for r = a .The Dzyaloshinski-Moriya (DM) exchange was not in-cluded in the calculation, although the nearest neighbour ionpairs allow for a non-zero DM vector, their midpoint not be-ing an inversion center. Introduction of DM exchange couldinduce the observed incommensurability, but it is likely thatit cannot account for another puzzling feature of the magneticstructure of EuNiGe : the symmetry breaking between the a and b axes. Indeed, the zero-field propagation vector, for in-stance k = ( / δ a and b .At present, we have no explanation as to the source of thisasymmetry in a tetragonal compound.Among Eu intermetallics of the type EuMX , where M is a d metal and X Ge or Si, EuNiGe is the only one where themagnetic structure, of spiral type, has been determined. Wethink that the germanides EuRhGe , EuIrGe , EuPtGe andthe silicide EuPtSi , which show a low field dip in theirmagnetization curves, should also present a spiral magneticstructure. It is of interest to gather the information about the number of magnetic transitions and the magnetic structure ofthe low temperature phase in the EuMX intermetallics (puta-tive, except for EuNiGe ), as shown in Table I. It comes out TABLE I: Magnetic characteristics in the EuMX series. A * denotesthat the spiral plane is deduced from single crystal magnetizationdata, not from neutron di ff raction measurements.material spiral plane nb. of transitionsEuNiGe (this work) (b,c),(a,c) 2EuPtGe (a,b) ∗ (a,b) ∗ (b,c),(a,c) ∗ (b,c),(a,c) ∗ no spiral 2EuIrSi ? (no single crystal) 2 that all the studied EuMX materials present a spiral structure,except EuRhSi , the situation in EuIrSi being unknown sinceno single crystal could be grown. There seems to be a corre-lation between the number of transitions and the plane of thespiral structure: one observes one transition if the spiral liesin the (a,b) plane, and two transitions if the spiral lies in the(b,c) or (a,c) plane, or if there is no spiral. In all the com-pounds, the intermediate phase between the two transitions isan incommensurate modulated phase, probably collinear, asdetermined by M¨ossbauer spectroscopy. One can conjecturethat a spiral lying in the (a,b) plane is more stable than a spiralin the (b,c) or (a,c) planes, which breaks the tetragonal sym-metry, as mentioned above. In the latter case, the transitionfrom paramagnetism would therefore occur first towards theintermediate phase, then to the spiral phase. VII. CONCLUSION
We have studied the magnetic order in EuNiGe versus tem-perature and magnetic field by single crystal neutron di ff rac-tion. Despite the strong Eu absorption and a limited dataset,the complete (B,T) phase diagram in the low temperaturephase was extracted. The zero-field magnetic structure wasfound to be an equal moment helicoidal phase, with an in-commensurate wave-vector k = ( / δ δ (cid:39) .
05. Apply-ing the field along the tetragonal axis, we found the peculiarbehaviour that δ changes from 0.05 to 0.072 at 2 T, where afirst magnetization jump occurs, and vanishes at 3 T, wherethe second magnetization jump takes place. All the structureswere refined with good accuracy.These results are in perfect agreement with previousmacroscopic measurements (magnetization and magneto-resistivity). The local information extracted from neutrondi ff raction allowed us to identify an additional transition un-der magnetic field. Most of these features (except the smallincommensurate component of the propagation vector) werewell reproduced by a self-consistent mean field calculation. T. Chattopadhyay, H. G. von Schnering and P. J. Brown, J. Magn.Magn. Mat. , 247 (1982) T. Chattopadhyay, P. J. Brown, P. Thalmeier and H. G. vonSchnering, Phys. Rev. Lett. , 372 (1986) A. Reehuis, W. Jeitschko, M. H. M¨oller and P. J. Brown, J. Phys.Chem. Solids , 687 (1992) P. Bonville, J. A. Hodges, M. Shirakawa, M. Kasaya and D.Schmitt, Eur. Phys. J. B , 349 (2001) N. Kumar, S. K. Dhar, A. Thamizhavel and P. Bonville, Phys. Rev.B , 144414 (2010) N. Kumar, P. K. Das, R. Kulkarni, A. Thamizhavel, S. K. Dharand P. Bonville, J. Phys.: Condens. Matter , 036005 (2012) A. Maurya, A. Thamizhavel, S. K. Dhar and P. Bonville, Sci. Rep. , 12021, doi: 10.1038 / srep12021 (2015) A. Maurya, P. Bonville, A. Thamizhavel and S. K. Dhar, J. Phys.:Condens. Matter , 366001 (2015) A. Maurya, P. Bonville, R. Kulkarni, A. Thamizhavel and S. K.Dhar, J. Magn. Magn. Mat. , 823 (2016) D. H. Ryan, J. M. Cadogan, S. Xu, Z. Xu and G. Cao, Phys. Rev. B , 132403 (2011) W. N. Rowan-Weetaluktuk, D. H. Ryan, P. Lemoine and J. M.Cadogan, J. Appl. Phys , 17E101 (2014) D. H. Ryan, A. Legros, O. Niehaus, R. P¨ottgen, J. M. Cadoganand R. Flacau, J. Appl. Phys. , 17D108 (2015) D. H. Ryan, J. M. Cadogan, V. K. Anand, D. C. Johnston and R.Flacau, J. Phys.: Condens. Matter , 206002 (2015) R. J. Goetsch, V. K. Anand and D. C. Johnston, Phys. Rev. B ,064406 (2013) A. Maurya, P. Bonville, A. Thamizhavel and S. K. Dhar, J. Phys.:Condens. Matter , 216001 (2014) A. Gukasov, A. Goujon, J.-L. Meuriot, C. Person, G. Exil and G.Koskas, Physica B , 131 (2007) P. J. Brown and J. C. Matthewman, 1993 CCSL-RAL-93-009 andhttp: // / dif / ccsl / html / ccsldoc.html Z. Wang and C. Holm, J. Chem. Phys. , 6351 (2001) I. Dzialoshinski, J. Phys. Chem. Solids , 241 (1958); T. Moriya,Phys. Rev.120