Metamagnetic texture in a polar antiferromagnet
D. A. Sokolov, N. Kikugawa, T. Helm, H. Borrmann, U. Burkhardt, R. Cubitt, E. Ressouche, M. Bleuel, K. Kummer, A. P. Mackenzie, U. K. Rößler
MMetamagnetic texture in a polar antiferromagnet
D. A. Sokolov ∗ , N. Kikugawa, T. Helm, H. Borrmann, U. Burkhardt, R. Cubitt, J. S.White, E. Ressouche, M. Bleuel,
6, 7
K. Kummer, A. P. Mackenzie,
1, 9 and U. K. R¨oßler Max-Planck-Institut f¨ur Chemische Physik fester Stoffe, D-01187 Dresden, Germany National Institute for Materials Science, Tsukuba 305-0003, Japan Institut Laue-Langevin, 6 Rue Jules Horowitz, F-38042 Grenoble, France Laboratory for Neutron Scattering and Imaging (LNS),Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland Univ. Grenoble Alpes, CEA, INAC-MEM, 38000 Grenoble, France NIST Center for Neutron Research National Institute of Standards and Technology Gaithersburg, MD 20988-8562, USA Department of Materials Science and Engineering University of Maryland, College Park, MD 20742-2115, USA ESRF, 71 avenue des Martyrs, 38000 Grenoble, France Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy,University of St Andrews, St Andrews KY16 9SS, United Kingdom IFW Dresden, PO Box 270116, D-01171 Dresden, Germany
The notion of a simple ordered state implies homogeneity. If the order is established by a broken symmetry,the elementary Landau theory of phase transitions shows that only one symmetry mode describes this state.Precisely at points of phase coexistence domain states formed of large regions of different phases can be sta-bilized by long range interactions. In uniaxial antiferromagnets the so-called metamagnetism is an exampleof such a behavior, when antiferromagnetic and field-induced spin-polarized paramagnetic/ferromagnetic statesco-exist at a jump-like transition in the magnetic phase diagram. Here, combining experiment with theoreticalanalysis, we show that a different type of mixed state between antiferromagnetism and ferromagnetism can becreated in certain non-centrosymmetric materials. In the small-angle neutron scattering experiments we observea field-driven spin-state in the layered antiferromagnet Ca Ru O , which is modulated on a scale between 8and 20 nm and has both antiferromagnetic and ferromagnetic parts. We call this state a metamagnetic texture and explain its appearance by the chiral twisting effects of the asymmetric Dzyaloshinskii-Moriya (DM) ex-change. The observation can be understood as an extraordinary coexistence, in one thermodynamic state, ofspin-orders belonging to different symmetries. Experimentally, the complex nature of this metamagnetic stateis demonstrated by measurements of anomalies in electronic transport which reflect the spin-polarization in themetamagnetic texture, determination of the magnetic orbital moments, which supports the existence of strongspin-orbit effects, a pre-requisite for the mechanism of twisted magnetic states in this material. Our findingsprovide an example of a rich and largely unexplored class of textured states. Such textures mediate betweendifferent ordering modes near phase co-existence, and engender extremely rich phase diagrams. I. INTRODUCTION
The term metamagnetism , possibly coined by Kramers asa joke, was used to describe the bizarre properties of certainmagnetic materials that have been investigated for more than100 years. They appeared to be paramagnetic or antiferro-magnetic in the ground state, but ferromagnetic in appliedfields[1–3].
Metamagnetism now labels a sudden rise or cross-over of the magnetization under applied field and is observedin various classes of materials. Once N´eel’s notion of antifer-romagnetism had been accepted, one type of metamagneticbehavior could easily be explained as the jump-like transi-tion between a collinear antiferromagnetic up-down state anda spin-polarized up-up state when a field overcomes the ex-change between sublattice spins constrained to collinear con-figurations by a strong easy-axis magnetic anisotropy [3, 4].In the generic magnetic phase diagrams of such materials,a first-order phase transition occurs between the two spin-orders. Long-range classical dipolar interactions or magne-tostrictive interactions can stabilize domain states in which thetwo spin-orders co-exist. These classical domain structures atphase-coexistence points are well understood [5].Ordered states with rotatable order parameters may alsodisplay intrinsically inhomogeneous phases. Here twisting short-range forces cause a continuous modification of theorder-parameter direction. In non-centrosymmetric magnet-ically ordered materials, spin-orbit effects on the magneticexchange interactions cause chiral spiral ordering[6]. Phe-nomenological theory is able to predict and describe suchmodulated states in a wide range of condensed matter sys-tems, such as incommensurable states in certain crystals un-dergoing lattice instabilities [7, 8], or chiral liquid crystals[9–11]. In such modulated textures the direction of a mul-ticomponent order parameter spatially rotates from one ori-entation to another. Chiral helimagnetic order is a paragonof such textures in which spin-orbit coupling twists an ele-mentary spin-ordered pattern over long periods [6, 12, 13].For this type of directional order in systems with a twistingshort-range force, static multidimensional solitons are theoret-ically predicted[14], which now are called chiral skyrmions inthe case of chiral ferromagnets or N´eel antiferromagnets [15].The condensation of such particle-like states can yield richphase diagrams [16, 17], which have become a major topicin condensed matter magnetism over the last decade [18, 19].Although a chiral helimagnet macroscopically behaves as anantiferromagnet, the primary magnetic order is a simple fer-romagnetic spin order, twisted into a helix over long distances[12, 13]. a r X i v : . [ c ond - m a t . s t r- e l ] N ov Here we show how a spiral magnetic order emerges in amaterial with antiferromagnetic order parameter. The spi-ral propagates in a direction perpendicular to the wavevectorof antiferromagnetic order. Materials displaying such com-plex textures may also host new types of antiferromagneticskyrmions, which are a subject of intense theoretical andexperimental research[20]. We present the first experimen-tal realization of a magnetic texture composed of an antifer-romagnetic ground state and a ferromagnetic spin-polarizedstate. We identified the layered orthorhombic antiferromag-netic oxide Ca Ru O , as suitable for a focused search for ametamagnetic texture. This material crystallizes in the non-centrosymmetric polar structure described by space-group Bb2 m , which belongs to polar point-group C v . The crystalstructure consists of RuO bilayers with corner sharing RuO octahedra, which are rotated around the crystallographic c -axis and tilted with respect to the ab -plane[21]. The basic anti-ferromagnetic order-parameter in Ca Ru O was identified indetailed neutron diffraction studies[22]. This magnetic-orderparameter is described by a simple collinear ordering-mode,which does not allow for a canting of moments into a weakferromagnetic state. Thus, Ca Ru O meets the elementarysymmetry conditions for modulated magnetism. A further re-quirement is relevant spin-orbit couplings, that affect the pri-mary magnetic order. We have used X-ray magnetic circulardichroism (XMCD) spectroscopy on the Ru ions to measureits orbital magnetic moment. We find relatively large mo-ments with a ratio of orbital to spin moment of about 0.15, seeSFIG.1 in Supplementary materials in agreement with earlier[23] and our own theoretical investigations. This indicates thatpossibly strong antisymmetric DM exchange interactions doaffect the magnetic order in Ca Ru O . The material ordersantiferromagnetically below the N´eel temperature, T N = 56 Kwith the ordered moments along the a -axis and the magneticpropagation vector along the [001] direction. Within the bi-layer the Ru moments are coupled ferromagnetically, whereasthe coupling between the adjacent bilayers is antiferromag-netic [21, 22]. This state is normally referred to as AFM-a. Oncooling below 48 K, the ordered moments within the bilayerspontaneously re-orient to point along the b -axis, and this stateis known as AFM-b. The coupling between the adjacent bi-layers remains antiferromagnetic. The moment re-orientationis accompanied by the first order structural transition at 48K[24].The isothermal magnetization at low temperatures displaysa single metamagnetic transition and reaches ∼ µ B perRu ion, a slightly reduced value compared to 2 µ B for the fullmoment expected for Ru , FIG 1a. At T ≥
43 K the magne-tization shows two metamagnetic transitions, which becomeincreasingly separated in field as the temperature increased to48 K, resulting in the “funnel”-type structure in the magneticsusceptibility, dm/dH plotted as a function of temperature andmagnetic field, FIG 1b. The higher field transition at H > R xy for current along the a -axis and with magneticfield along the b -axis displays two features that are markedby local maxima in the derivative dR xy /dH. These maximasplit towards lower and higher fields upon increasing tem-perature, a behavior similar to dm/dH, see further discussionand SFIG.2,3 in Supplementary materials. Our analysis, fol-lowing the general approach for the Anomalous Hall effect(AHE)[25], indicates that in addition to a strong AHE com-ponent there is an intrinsic additional contribution in the re-gion between the two metamagnetic transitions, see Supple-mentary materials. This suggests the presence of a magnetictexture with either topological features or non-collinear com-plex modulations in this magnetic state that can contributean extraordinary off-diagonal components of the resistivitytensor. A quantitative extraction of this extraordinary Hall-resistivity may only become possible by taking into accountfield-induced changes in the band structure and the exactstructure of the new magnetic order, and lies beyond the scopeof this work.The results of our bulk measurements refine the publishedphase diagram of Ca Ru O in the region between the linesseparating AFM-b and CAFM states[26–28]. Until now mostof the neutron scattering measurements on Ca Ru O wereperformed at commensurate wavevectors. In this Article wefocus on the nature of the magnetic state near AFM-a to AFM-b transition at magnetic fields between 2 T and 5 T and reportthe magnetic modulation in a previously unexplored region ofthe reciprocal space near the wavevector Q=0.SANS experiments were performed to search directly for abulk long-wavelength magnetic modulation in the “funnel”-type region of the µ H-T phase diagram near Q=0. Typi-cal SANS patterns obtained at 4 different fields are shownin FIG 2. Each pattern consists of the images of the maintwo-dimensional low-Q detector and 4 additional high-Q de-tectors as detailed in Ref.[29]. The magnetic field was ap-plied parallel to the b -axis, which points out of the plane ofthe detector. Data were collected after zero field cooling thesample to 2 K, applying the field at 2 K and measuring thepattern at several increasing temperatures. For each field anon-magnetic background collected at 65 K was subtracted.The most striking feature found in our experiments is a pair ofsatellites at Q MMT =( ± ∆ ,0,0), which correspond to a mag-netic modulation propagating along the a-axis with a repeat !" &’’ !" ( ) * + , - . / ’ ’" )72 )82)92 ):2 &6"&6 &6’’6;’6$’6"’6 ’6’ : < => + : $" ’ 4 ’ ’ ’ ’ ) &6’’6 :@+:5)4 A +32 6’&6 @ ) B A + < C %$ ’ DE@ DE@ (DE@ @@3 FIG. 1: Bulk properties of Ca Ru O measured with the magnetic field along the b-axis on the same single crystal: (a) Magnetic fielddependent magnetization at various temperatures spanning the region of interest in the phase diagram. Up(down) arrow corresponds to increase(decrease) of the field. (b) Differential susceptibility dm/dH obtained by differentiating magnetization in (a) with respect to the field. The colorscale represents a magnitude of dm/dH. Dotted green line encircles the region in which the metamagnetic texture (MMT) was observed inSANS measurements, see text. Solid black line corresponds to transitions inferred from low field m(T) measurements. Black filled trianglesin (b) correspond to the maximum in dR xy /dH in the Hall effect measurements. Blue filled dots in (b) are inferred from the field dependentmaxima in the specific heat. AFM-a, AFM-b, and CAFM mark two antiferromagnetic and the canted antiferromagnetic regions of the phasediagram, see text. (c) Temperature dependence of the specific heat. The sharp maximum at zero field marks the moment re-orientationtransition. (d) Derivative of the Hall resistivity with respect to the magnetic field. distance of 2 π / ∆ , FIG 2. We observed the satellites at fieldsfrom 2 T up to 5 T in the temperature range, which showsa hysteretic behaviour of the magnetisation shown in FIG 1a.No satellites were observed at fields above 5 T, suggesting thatthe “funnel”- type region of the phase diagram is not a uni-form magnetically ordered state. The satellites develop fromthe strong intensity near Q=0 at the temperature of AFM-ato AFM-b transition. The scattering is broad with respect towavevector near the onset temperature, FIG 2a. The apparentdiffuse nature of the scattering is most likely due to a quasi-long-range ordering. The wavevector of satellites initially in-creases on heating, although in FIG 3 we show that the tem-perature dependence of the wavevector is not monotonic at allfields. This pattern was observed at all fields except for 2 T,where satellites exist only in a very small temperature range ina proximity of the metamagnetic transition. We also observeda second harmonic of the primary satellites at 2Q MMT at 2 T, 2.5 T and 3 T, which could correspond to higher order peaksor represent a double scattering.The magnetic field and the wavevector dependence ofSANS intensity is summarized in FIG 3. Increasing the mag-netic field suppresses intensity of satellites at the correspond-ing temperatures, FIG 3a. The intensity of the satellites is thestrongest at the lowest temperature at which we can resolvethe satellites from the strong scattering near the direct beam.The wavevector of the modulation is strongly temperature de-pendent. For µ H ≥ MMT < − at the onset temperatureup to Q MMT =0.08 ˚A − at T ≥
50 K. In contrast, for µ H ≤
3T Q
MMT displays a sharp maximum just above the onsettemperature, FIG 3b. The repeat distance of the modulation, ∆ =2 π /Q MMT reaches ∼
200 ˚A at the lowest temperature ofobservation of the satellites at 5 T. Near 50 K, at 5 T the repeatdistance decreases to ∼
80 ˚A. The competing character of sev- c a µ Η (a) (b)(d)(c) Q L ( Å - ) Q L ( Å - ) Q L ( Å - ) Q H ( Å - ) Q L ( Å - ) Q L ( Å - ) Q L ( Å - ) Q H ( Å - ) Q L ( Å - ) Q L ( Å - ) Q L ( Å - ) Q H ( Å - ) Q L ( Å - ) Q L ( Å - ) Q L ( Å - ) Q H ( Å - ) log I(a.u.) FIG. 2: Typical SANS patterns measured at 48 K in magnetic fields from 2 T (a) to 4.5 T (d) applied parallel to the b-axis. 4 smaller panelsare SANS detectors positioned at 1.2 m from the sample and thus able to detect diffraction from 001 magnetic reflection. The panel in thecentre is the main SANS detector at 2 m from the sample. The metamagnetic texture (MMT) propagates along the a -axis with Q MMT reaching(0.08,0,0) ˚A − . Higher order reflections were observed at 2 T (a) and 2.5 T (b). A split along the a -axis reflection at Q AF =(0,0,-0.32) ˚A − wasobserved at 4.5 T(d) and also at 5 T, see Supplementary materials. No such splitting was observed below the onset of the metamagnetic texture,T MMT . Note the same logarithmic scale of the intensity for all fields. The non-magnetic background at 65 K measured at the correspondingfield was subtracted from all the patterns. The filled blue circle at Q = 0 is a mask applied to cover the direct neutron beam. Q L and Q H arewavevectors along (00L) and (H00) directions. eral coupling terms, which have different temperature depen-dencies, is most likely the origin of non-monotonic tempera-ture dependence of the wavevector of the metamagnetic tex-ture, see Supplemental materials for details. Using the rock-ing curve measurements we estimated the correlation lengthof the modulation along the b -axis, FIG 3c. For details of esti-mate, see SFIG.4 in supplementary materials. The correlationlength ξ b is not resolution limited and reaches 2280 ˚A at 2.5 T,comparable to the correlation length of 5500 ˚A in the A-phaseof MnSi[18]. Magnetic field suppresses correlations along b axis at µ H > a -axis is not measured in our experiment.The observation of the satellites along the a -axis in the ac -plane indicates that the ordered moment of the modulation canhave components parallel to the c and b -axes. In AFM-b andAFM-a regions of the phase diagram, which border the regionof the metamagnetic texture the ordered moment has no com-ponent along the c -axis. We also note that in none of the Feand Mn-doped Ca Ru O does the ordered moment acquirea component along the c -axis[30, 31]. It is therefore likelythat the ordered moment of the modulation is along the b -axis, but the component along the a -axis cannot be ruled out.Our modulation is then either a helix or a cycloid if it has acomponent of the ordered moment along the propagation vec-tor. Further experiments with polarized neutrons are requiredto identify the type of the modulation. The strong intensitynear Q AF =(0,0, ± − or simply (001) corresponds tothe bulk antiferromagnetism, which propagates along the c -axis in agreement with Ref.[22]. The scattering is very broadnear Q AF at the temperatures at which the satellites at Q MMT are observed, but turns into a well-defined sharp reflection atthe lowest temperatures, where no satellites are observed. Thebroad, ring-like shaped features near Q AF possibly originatefrom a short-range or fluctuating antiferromagnetic order. Weobserved reflection at Q AF =(00-1) only, no reflection was ob-served at Q AF =(001) due to a small tilt of the crystal with re-spect to the vertical direction. We also note that the antiferro-magnetic reflection acquires a modulation, which propagatesalong the a -axis, which becomes resolvable above 4 T. Thewavevector of the modulation, Q mAF =( δ δ reaches0.059 ˚A − at 5 T. This behaviour is reminiscent of a magneticfield-induced commensurate-to-incommensurate transition inthe DM antiferromagnet Ba CuGe O , in which the magneticfield was applied in the plane of the rotation of the spins[32].A commensurate to incommensurate antiferromagnetic tran-sition was also reported for Fe and Mn-doped Ca Ru O inRef.[30, 31]. A cycloidal modulation propagating along the a -axis was identified for both types of doping. These obser-vations suggest that the antiferromagnetism in Ca Ru O canbe easily destabilized by application of the magnetic field ordoping and is prone to host magnetic solitons.The onset of the magnetic texture with the ordered momentalong b -axis requires the ordered moment in the AFM a stateto rotate from the a -axis to the b -axis locally, on the length-scale of the magnetic texture. We propose that such defects inthe magnetic structure break the long-range three-dimensionalantiferromagnetism in Ca Ru O . The intensity near the an-tiferromagnetic wavevectors Q AF is maximised near the tem-perature at which the magnetic satellites at Q MMT disappear.We note that in a previous neutron scattering work the inten-sity of the antiferromagnetic reflections measured between 3and 4 T showed a reduced intensity on cooling [22]. We ar-gue that the emergence of the magnetic texture is the originof the reduced integrated intensity of the antiferromagneticBragg peaks at Q AF . It is unlikely therefore, that the mag-netic texture and the bulk antiferromagnetism co-exist in anon-equilibrium state. Instead, the magnetic texture devel-ops from the antiferromagnetism as an equilibrium state inthe presence of the magnetic field, which enhances the effectof the DM interactions. Further neutron diffraction experi-ments are needed to describe the splitting of (001) magneticreflection.Summarizing the experiments, Ca Ru O displays aspirally-modulated magnetic order in broad temperature-fieldregion, previously regarded as a crossover[22, 28]. The propa-gation vector of the spiral is aligned perpendicular to the mag- netic field and the staggered magnetization is likely parallel toit. The magnetic field applied along the polar b -axis desta-bilizes the antiferromagnetic ground state by flipping spinsfrom the a -axis to b -axis locally on a scale between 8 and20 nm depending on the magnetic field and temperature asillustrated schematically in FIG 4b. Then, a mixed state mod-ulating between antiferromagnetic and ferromagnetic spin-configurations with long periods, is observed. Our observa-tion of metamagnetic textures in Ca Ru O invites a com-parison with spiral spin states driven by a competition be-tween direct exchange interactions such as in Ca Co O [33]and MnSc S [34] or rare-earth elements such as Tb, Dy, Ho,which demonstrate a helically modulated magnetic structuredue to nesting of the Fermi surface or a Kohn anomaly[35].Although the theoretical description of the helical modula-tion in Ca Co O is still lacking, it is considered to resultfrom a competition of antiferromagnetic and ferromagneticdirect exchange interactions. A magnetic vortex state re-ported in MnSc S results most likely due to a competitionbetween nearest and the next nearest neighbour exchangeinteractions[34]. Elemental rare-earths such as Tb, Dy, andHo order via long-range exchange interactions carried by con-duction electrons (RKKY) at a finite wavevector dictated byRKKY. As the temperature is lowered the effects of crys-talline electric field and magnetic anisotropy lead to a reduc-tion of the ordering wavevector and transition into ferromag-netic state. The theories explaining the magnetic structures ofTb, Dy, Ho consider a high density of states near the Fermilevel, which drives the nesting. The phenomenon of metam-agnetic textures in Ca Ru O is distinctly different from bothabove mentioned examples as the textured state results froma coupling of ferromagnetic and antiferromagnetic order pa-rameters via so-called Lifshitz invariants, see Supplementarymaterials for details. A rather generic character of such a termin the free-energy expansion suggests that more spin texturedstates driven by DM interaction are awaiting discovery.Elementary considerations are sufficient to explain why thismodulated state exists, and in fact is an expected behavior in apolar antiferromagnet such as Ca Ru O when, under a mag-netic field, its magnetic states is transformed into a ferromag-netic configuration. Its non-centrosymmetric crystal structureand layered antiferromagnetic ordering (FIG 4a) enable spe-cific couplings between the two co-existing order parameters,the antiferromagnetic staggered magnetization and the ferro-magnetic spin polarization. Ultimately, these couplings de-rive from the DM interactions (DMI) in this material. Thehierarchy between (i) the strong spin-exchange, stabilizing acertain antiferromagnetic spin-pattern as the ground state, (ii)the twisting effects of the DMIs on this ground-state, and (iii)the possibility to tune the system into the spin-polarized stateby external magnetic fields, while temperature is used to tunethe weaker magnetic anisotropies, makes this layered chiralmagnet with a polar structure ideally suited for a modulationof the desired type.In contrast to a “proper” Dzyaloshinskii texture where onedirectional ordering mode is twisted, this is a more complextexture generated near the co-existence points of two phases.Here, the magnetic order is spatially wavering between the Å Å Å - FIG. 3: Metamagnetic textures in Ca Ru O : (a) Magnetic field dependence of the wavevector-dependent azimuthally averaged SANS in-tensity on the detector plane capturing magnetic satellite at Q MMT measured at 48 K. (b) Temperature dependence of the wavevector of themodulation measured at fields from 2 T to 5 T. (c) Magnetic field dependence of the correlation length along b -axis at 48 K measured insamples 1 and 2 on D33, ILL and on SANS-II, PSI, see Methods. Where not shown explicitly, the errorbars (one sigma standard deviation)are smaller than markers. Lines are a guide to the eye. co-existing ferromagnetic and antiferrromagnetic configura-tions, as sketched in FIG 4b. Therefore, the term “metamag-netic texture” appears appropriate for this modulated phase inCa Ru O .Qualitatively, the mechanism enabling such complex mixedstates can be stated by using symmetries to construct the phe-nomenological continuums theory for these ordering modes,i.e. by constructing the Landau-Ginzburg free energy forthe coexisting and coupled ordering modes as reported forCa Ru O in the Supplementary materials. The specificmechanism then is described by free energy terms knownas “Lifshitz-type invariants”. Such terms are linear in spa-tial gradients of one mode and couple it to the other mode.These Lifshitz-type terms describe a frustrated coupling be-tween different pure modes. The expectation that such termscause modulations of thermodynamic mixed phases has beenput forth theoretically for a long time [36–41]. However, theseeffects can become relevant only if a system can be tunedtowards special multicritical regions of the phase diagram,where the two primary modes co-exist. This may be the rea-son why concrete examples for the effects of such terms havebeen scarce. Typically such effects have been discussed forfrustrated (magnetic) systems of low symmetry where differ-ent order parameters already co-exist in the ground state[42].Recently, the importance of such couplings has been raisedin the context of “phase co-existence” in materials with mul-tiple electronic instabilities such as the manganite or cuprateperovskites[41]. The phase diagrams of these materials mayinclude multicritical points and also co-existence of ferro- andantiferromagnetic phases, but the role of Lifshitz-type cou-plings for mixed states is difficult to establish for the elec- tronic or structural order parameters. Hence, simple experi-mental systems displaying mixed textures composed of dif-ferent ordering modes have remained elusive. The discoveryof field-driven modulated magnetic state in Ca Ru O nowprovides an example of a modulated state with mixed symme-tries.The phenomenological theory describing its metamagneticbehavior is detailed in the section VI of the Supplemental ma-terials. The Landau-Ginzburg free energy displays Lifshitz-type invariants that are anisotropic in spin directions andfavour modulated coupled states between FM and AFM spin-structure. The specific form of these terms reveals that theyare caused by spin-orbit interactions and encode the twistinginfluence of the antisymmetric DM-exchange on the magneticorder.The observations demonstrate that metamagnetic crystalswith appropriate non-centrosymmetric structure are an idealplayground for creating such textures. As can be justified fromthe phenomenological Landau theory, the spin-twisting DMIsin these magnets preclude homogeneous phases, unless stabi-lized by additional strong anisotropies, and generically favourmixed AFM-FM states near the metamagnetic transition.FIG 4c,d shows schematically possible magnetic phase di-agrams that can be realized in antiferromagnets with non-centrosymmetric, in particular polar symmetry.The tricritical region with first-order phase transitions in thecase of magnets with strong easy-axis anisotropy is replacedby transitional regions covered by modulated mesophases.Similarly, for systems with weak or absent anisotropies, mod-ulated states still can occur near the transition towards thespin-polarized paramagnetic phase at elevated temperatures, T N T N TH afmfm pm afmfm pm polar centrosymmetric cd T N T H T N T afmfm pm afm pmfm sf Ca Ru IIRu I O a b fm fm afm FIG. 4: Crystal structure of Ca Ru O , panel a shows the The Ru atoms occupy a single crystallographic site. The antiferromagnetic primaryorder splits these positions into two double layers, Ru (I) and Ru (II), which have internally ferromagnetic spin-configurations and are antipar-allel. Above 48 K transition the collinear spin-structure has moments directed along the a -axis of the orthorhombic cell. The polar axis is b .Cartesian coordinates xyz are used for spin and spatial gradients, with z along b as indicated. Panel b shows a metamagnetic one-dimensionaltexture propagating along the a axis. In an applied field along the polar axis b , the spin configuration oscillates from fm to afm and back to fmover a repeat period Λ . c , d , schematic phase diagrams of bipartite antiferromagnets with easy-axis anisotropy. Panel c the temperature-fieldphase diagram for the case of large anisotropy. A first-order transition between an antiferromagnetic (afm) collinear and a spin polarized (fm)state occurs along the double line. For temperatures above a tricritical point (triangle) the transition is continuous. The afm-fm co-existencecan be replaced in a polar magnet by a region of modulated phases µ . Towards the paramagnetic state at elevated temperatures, anomaloustransitions into precursor states ( π , π ) then are expected. The dotted yellow line indicates the transition between the improper, metamagnetictexture and the precursor state µ → π . The precursor of type π only implies modulations of primary afm modes, while π states can bemetamagnetic, being composed of modulations between afm and fm modes. Panel d displays the phase diagram of a system with a weakanisotropy. The double line signifies a first order transition of the spin-flop type, which is only a re-orientation of the ordered moment. Themarked point is a bi-critical point (almond mark). In an antiferromagnet with a polar structure, metamagnetic textures µ can still occur atelevated temperatures and for fields higher than the spin-flop field, when a sizeable net moment and antiferromagnetic order compete. As inthe case of large anisotropy, different types of precursors, i.e. a proper antiferromagnetic texture π or metamagnetic textures with coupledmodulations between afm and fm mode π can occur. where antiferromagnetic and ferromagnetic order parame-ters have similar magnitude and can become intertwined.The transition from the paramagnetic state to the modulatedstates is expected to be unconventional, implying inhomoge-neous pre-cursor states. The stabilization of this mixed mag-netic texture relies on the unavoidable asymmetric exchangethrough spin-orbit couplings. It can be predicted for manysystems to occur. So far, in Ca Ru O we have observed onlya mixed state with a one-dimensional modulation. The basicmechanism that enables the generation of this mixed state canact also in different spatial directions. Then, it may becomepossible to create mixed textures similar to the chiral magneticskyrmions in non-centrosymmetric ferromagnets. Therefore,appropriate non-centrosymmetric metamagnets may also bearlocalized or multidimensional lumps of one type order im-mersed in another one. Phase transition involving such tex-tures may allow to create condensates of such lumps to formtextured states that are simultaneously modulated in differentdirections, akin to skyrmion lattices, but composed of dif-ferent co-existing ordering modes. We propose to call such states improper Dzyaloshinskii textures , as they are composedof two ordering modes with different symmetry. II. METHODSA. Crystal growth and bulk characterization
High quality single crystals of Ca Ru O were grown usinga floating zone method in a mirror furnace. The single crys-tals were oriented using a white beam backscattering LaueX-ray diffraction method. SFIG.5 in supplementary materi-als shows the corresponding Laue diffraction image indexedwith the Bb2 m-structure and room temperature lattice pa-rameters. The Laue diffraction image shows sharp reflections,which indicate the excellent quality of the sample. The crys-talline quality was further confirmed by measuring the rockingcurve at (10,0,0) strong nuclear reflection in a neutron beam,SFIG.6. Measurements of the magnetization were performedusing the vibrating sample magnetometer; specific heat wasmeasured using the physical property measurements systemby Quantum Design. B. X-ray diffraction and structure refinement
As crystals of Ca Ru O are easily cleaved, great care hadto be taken to isolate a single crystal of adequate quality. Fi-nally an irregular chip (162 × × µm ) was selectedand used in single crystal X-ray measurements on RigakuAFC7 diffractometer with a Saturn 724+ CCD detector. Afterpreliminary unit cell determination oscillation images aroundthe unit cell axes proved good crystal quality without indica-tions of partial cleavage or twinning, see SFIG.7. All diffrac-tion experiments were performed at 295 K applying graphite-monochromated Mo-K α radiation ( λ = 0.71073 ˚A) collimatedwith a mono-capillary. A total of three full ϕ -scans resultedin 2250 images from which after integration and scaling 7089Bragg intensities were obtained. After averaging 1328 uniquereflections were used in structure refinement. Derived latticeparameters, a = 5.3824(6) ˚A, b = 5.5254(4) ˚A, c = 19.5946(15)˚A (non-standard Bb2 m ) are in a very good agreement withliterature data. Refinement of the established model in spacegroup Cmc2 (standard setting of no. 36) converged in anexcellent fit of 60 parameters vs. all 1328 independent reflec-tions. Agreement based on F including isotropic extinctioncorrection is indicated by wR = 0.05 and goodness-of-fit =1.12. However, a clear assignment of the absolute structurewas not possible as refinement of a twin by inversion resultedin a volume ratio of 0.45(10) : 0.55. Based on careful analysesa centrosymmetric model as well as pseudo-tetragonal twin-ning had to be excluded. In an ongoing investigation we tryto clarify if these observations point to a structural phase tran-sition at high temperatures and are in accordance with anti-phase domain type features in optical micrographs. Details ofdiffraction experiment and structure determination are avail-able from Cambridge Structural Database, CCDC 1901958. C. Small angle neutron scattering
Two samples of a similar size from different growths werestudied with SANS. We have carried out our SANS measure-ments on sample 1 using D33 SANS instrument at ILL in hor-izontal magnetic fields at temperatures between 10 K and 61K. The measurements were performed using unpolarized neu-trons with the wavelength λ =4.8 ˚A. The neutron beam wascollimated over 2.8 m before the sample. The sample to detec-tor distance equaled 2 m. We have used the ILL Blue Charly
8T horizontal magnet with the field oriented parallel to the neu-tron incident momentum. Typically, each scan was collectedover 30 minutes to obtain a good statistics. The sample wascooled in a zero field to 2 K, at which the field was appliedand the data was collected on heating. The temperature wasthen raised to 65 K (above T N = 56 K), at which the field wasreduced to zero, the sample was cooled to 2 K in zero field,then the next field was applied. SANS measurements on sam-ple 2 were performed at the SANS-II instrument at PSI using a similar setup as at D33, with λ =4.93 ˚A. Some of our earlierSANS measurements were performed at NG7 instrument atthe NIST Center for Neutron Research. D. Transport and XMCD measurements
For the transport measurements we prepared a microstruc-tured device using a standard focused ion beam procedure.We fabricated a Hall bar device from an oriented single crys-tal of Ca Ru O by the application of focused ion beam (FIB)as described elsewhere[43]. We cut a thin rectangular slice,with dimensions × × µm from the crystal and trans-ferred it into non-conductive epoxy on a sapphire substrate.Ohmic contacts with approximately 10 Ω contact resistanceswere produced by sputter coating Au and annealing at ◦ C. The magnetoresistance measurements were performed in aLOT Quantum design magnet system. XMCD measurementswere performed on ID32 beamline at ESRF.
E. Theoretical considerations
III. ACKNOWLEDGEMENTS
We thank U. Nitzsche for technical support with FPLO.N. K. acknowledges the support from JSPS KAKNHI (No.JP17H06136). D. A. S. thanks C. Geibel for the critical read-ing of the manuscript and constructive comments. Access toNG7 SANS was provided by the Center for High ResolutionNeutron Scattering, a partnership between the National Insti-tute of Standards and Technology and the National ScienceFoundation under Agreement No. DMR-1508249. We thankJ. Krzywon and Y. Qiang for technical support during SANSexperiment at NIST. This work is partly based on experimentsperformed at the Swiss spallation neutron source SINQ, PaulScherrer Institute, Villigen, Switzerland.
IV. AUTHOR CONTRIBUTIONS
U.K.R. conceived the project. U.K.R., A.P.M. and D.A.S.supervised the project. N.K. and D.A.S. grew single crystals.D.A.S oriented and characterised samples. H.B. and U.B.analysed the crystal structure. T.H. performed the electri-cal transport measurements and analysed the Hall effect data.K.K. performed XMCD measurements. D.A.S., R.C., J.S.W.,and M.B. performed SANS measurements. D.A.S. and E.R.carried neutron diffraction experiments. U.K.R. carried outDFT calculations and developed the Landau-Ginzburg-typefree energy theory. D.A.S. and U.K.R wrote the manuscriptwith contributions from all co-authors.
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(2018), Probing the mag-netic field driven modulated structure and exotic mag-netism in Ca Ru O . (Institut Laue-Langevin, 2018)https://doi.ill.fr/10.5291/ILL-DATA.5-42-462. upplementary Information for Metamagnetic texture in a polar antiferromagnet D. A. Sokolov, N. Kikugawa, T. Helm, H. Borrmann, U. Burkhardt, R. Cubitt, J. S.White, E. Ressouche, M. Bleuel, K. Kummer, A. P. Mackenzie,
1, 8 and U.K. R¨oßler Max-Planck-Institut f¨ur Chemische Physik fester Stoffe, D-01187 Dresden, Germany National Institute for Materials Science, Tsukuba 305-0047, Japan Institut Laue-Langevin, 6 Rue Jules Horowitz, F-38042 Grenoble, France Laboratory for Neutron Scattering and Imaging (LNS),Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland Univ. Grenoble Alpes, CEA, INAC-MEM, 38000 Grenoble, France NIST Center for Neutron Research, NIST, Gaithersburg, MD 20899, USA ESRF, 71 avenue des Martyrs, 38000 Grenoble, France Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy,University of St Andrews, St Andrews KY16 9SS, United Kingdom IFW Dresden, PO Box 270116, D-01171 Dresden, Germany
I. X-RAY MAGNETIC CIRCULAR DICHROISM (XMCD)
The X-ray magnetic circular dichroism (XMCD) experi-ment was performed at the ESRF beamline ID32 using thehigh-field magnet endstation . The samples were cleaved insitu in the high field magnet at low temperature and a pressurebetter than × − mbar leaving behind a clean ab surface.The magnetic field was applied along the beam direction. The b axis of the samples was titled by ◦ towards the c axis withrespect to the field direction, resulting in a small field com-ponent along c but zero field along a . The X-ray absorptionspectra were taken in total electron yield mode with circularleft and circular right polarization for both positive as wellas negative field direction. The XMCD was determined fromall four spectra as the difference between spectra taken withopposite helicity.Fig. 1a shows the XAS spectra for positive and negative he-licity and the XMCD obtained as the difference of the two inthe CAFM phase at low temperature and high field. In addi-tion to the Ru M and M absorption lines there is an addi-tional broad X-ray absorption features around 450 eV whichdisplays no XMCD and is due to Ca L , absorption. Wefollowed the XMCD signal at the Ru M , as a function ofmagnetic field and temperature in the relevant region of phasespace. The obtained spectra are shown in Fig. 1b both withthe relative intensities as measured (left panel) and with theintensities normalized to peak value (right panel). The in-tensity of the XMCD signal (left panel) scales well with themacroscopic magnetization curves shown in Fig. 2a of themanuscript. The lineshape of the XMCD, however, does notchange, neither with field nor temperature, across the meta-magnetic, magnetic and MIT transitions. Using the XMCDsum rules, we can extract the ratio of orbital to spin mo-ment aligned along the field which only depends on the ratioof the integrated XMCD signal at the M and M absorp-tion edge, respectively . As the XMCD lineshape does notchange across the phase diagram we always find the samevalue < m L > / < m S > ≈ . ± . , confirming thepresence of a sizable orbital moment. In principle, spin- andorbital moments can also be extracted individually from theXMCD signals . We refrained from doing so here because of the complex background of the XAS spectra which makes itdifficult to extract reliable numbers.The robustness of the XMCD lineshape and the < m L >/ < m S > ratio suggests that the local crystal field experi-enced by the Ru ions in the RuO octahedra does not changesignificantly in the interesting region of the phase diagram. II. TRANSPORT MEASUREMENTS
We carried out magnetotransport measurements on a sam-ple in the Hall-bar geometry, prepared by focused Ionbeam (FIB) microfabrication. A lamella with dimensions(3 × × µ m was cut from a single-crystal using Gal-lium FIB. The device is shown in the inset of Fig 2a. Weused a standard four-terminal Lock-In method for measure-ments of electrical resistivity. We applied a current of 100 µ Awith frequency f = 177 Hz along the a -axis while the magneticfield was applied along the b axis. The zero-field resistivitycurve shown in Fig 2a resembles data reported previously anddemonstrates the high quality of micromachined devices. Thein-plane resistivity, ρ xx , as well as the Hall resistivity, ρ yx ,exhibit a step-like behavior at low temperatures (see Fig 2band c). The sharp change evolves into a broader transitionas temperature approaches 50 K. Above 50 K, both ρ xx and ρ yx follow an overall positive slope. Most interestingly, weobserve a two-step-like behavior in ρ yx for temperatures be-tween 45 K and 49 K, the range in which the metamagnetictexture was observed in small-angle neutron scattering mea-surements. In general, the Hall resistivity can be a composi-tion of three components: ρ yx = ρ Nyx + ρ Ayx + ρ Tyx , (1)where N, A, T denote the normal, anomalous and topologicalHall effect contributions . Using the expression: ρ yx H = R + S A ρ αxx MH + ρ Tyx
H , (2)where H is the magnetic field and M is the magnetization,we can extract the normal and anomalous Hall coefficients, a r X i v : . [ c ond - m a t . s t r- e l ] N ov
440 450 460 470 480 490 500 510 520Photon energy (eV)1.21.31.41.51.6 X A S ( a r b . un i t s ) H = 7 T, T = 39.8 K×10 (a) XAS, + XAS, XMCD, +
450 470 490 510Photon energy (eV) X M C D
54 K36 K45 K (b)
450 470 490 510Photon energy (eV)TTTTTT normalised38 40 42 44 46 48 50 52Temperature (K)0.000.050.100.150.200.25 m L / m S (c) FIG. 1: (a) X-ray absorption spectra for both experimental helici-ties and the corresponding XMCD in the CAFM phase. (b) XMCDas a function of field and temperature in the relevant region of thephase diagram. The XMCD intensities are shown both as measuredto ease comparison with macroscopic magnetization measurements(left) and normalised to average intensity to better compare the line-shapes between the XMCD spectra (right). (c) Ratio of orbital tospin moment as extracted from the XMCD data. R and S A (see Fig 3a). The respective intercept and slopeof the linear fits to the high-field part of ρ yx /H plotted ver-sus M ρ xx /H are listed in Table 1. Again using these materialsparameters and M ( H ) data, the Hall resistivity ρ yx can be simulated under assumption that only the normal and anoma-lous contributions exist. In Fig 3b we show the simulationresult for T = 47 K compared to the raw data, M, ρ xx and ρ yx . Both case α =0 and 1 are shown, which correspond toan anomalous contribution dominated by intrinsic or extrinsicscattering mechanisms, respectively . As can be seen fromFig 3c deviations between simulated Hall resistivity and theexperimental data appear for temperatures below 49 K. Thisindicates the existence of a so-called topological contributionto ρ yx in the state corresponding to the H − T -region of thephase diagram, where the metamagnetic texture is observed.It may be related to either a topological contribution or an-other non-trivial contribution that is not considered through normal and anomalous Hall resistivity. In particular, it mayderive from effects of a non-collinear spin-structure . T R S A
39 -6.4 2.1e-441 -7.1 2.5e-443 -6.1 2.7e-445 -5.1 2.8e-447 -5.2 3.3e-449 -4.0 3.2e-451 0.7 2.2e-453 0.63 2.1e-4TABLE I: Linear fit parameters extracted from Fig 3a.
III. SMALL-ANGLE NEUTRON SCATTERINGMEASUREMENTS
Our SANS sample 1 measured at ILL was a 238 mg singlecrystal, mounted on Al sample holder with the c-axis verti-cal and the b-axis parallel to the field and the neutron inci-dent momentum. The approximate sample dimensions alongthe major crystallographic axes were a=8.3 mm, b=7.7 mm,c=1.5 mm. The SANS sample 2 measured at PSI was a 215mg single crystal with approximate dimensions of a=6 mm,b=5.3 mm, c=2.8 mm. The sample 2 was measured in thesame orientation as sample 1. ξ b , the correlation length of themagnetic texture along the b -axis, directed along the incidentmomentum of neutrons was calculated from the rocking curvemeasurement (rotation around the c-axis), schematically illus-trated as ω rotation in Fig 4a. The rocking curve at 2 T and48 K, obtained by rotating the sample together with a cry-omagnet was fitted by a Gaussian lineshape, which yieldedFWHM=3.1 and 2.7 degrees for the satellites contained in leftand right boxes, Fig 4b,c. ξ b = 8 ln Q tan ( b ) , (3)where b is the full width at half maximum of the rockingcurve and Q is the wavevector of the observed reflection. Forestimates of ξ a and ξ c , the correlation lengths along the a and c axes we used the tangential and radial widths of the satellites,which were fitted to Gaussian lineshapes. The experimentalresolution in a typical SANS geometry is largely determinedby the wavelength spread, ∆ λ/λ and d-spacing spread of asample, ∆ d/d. For the estimates of ξ b we ignored the effects ofthe instrumental resolution since b >> Θ∆ λ/λ and the widthof the rocking curve is much greater than the angular size ofthe direct beam ( ∼ Θ is the scatteringangle. ξ a = 1 Q (∆ d/d ) , (4)where ∆ d/d is the d-spacing spread of the lattice, which is
10 µm xx ( c m ) H (T)
39K 41K 43K 45K 47K 49K 51K 53K0 100 200 300200400600800 xx ( c m ) T (K) yx ( c m ) H (T) (a) (b) (c) FIG. 2: Transport measurements on a microstructured single crystal of Ca Ru O : (a) Temperature dependence of the in-plane resistivity.Inset: False color SEM image of the FIB-microfabricated Hall-bar device (purple) with gold contacts (yellow). Current runs along the a-axis.(b) In-plane magnetoresistivity and (c) Hall resistivity curves at various temperatures for field applied along the b-axis.
012 0 3 6 9 12 02040 M ( µ B ) u R / ρ yx ( µ Ω c m ) exp. data α = 0 α = 1 T = 47 K ρ xx ( Ω µ ) m c µ H (T) α = 0 ρ yx ( Ω µ ) m c µ H (T)
20 30 40 50 60 700510 39 K 41 K 43 K 45 K 47 K 49 K 51 K 53 K fits α = 0 ρ yx / H ( µ Ω ) T / m c M ρ / H ( µ B m Ω cm /T) (a) (b) (c) FIG. 3: (a) Linear fits (orange lines) to the Hall data from Fig 2 (c) to Eq. 2 as described in the text. The respective fit parameters we show inTable 1. (b) Example curves (black) of magnetization, in-plane resistivity and Hall resistivity, at 47 K used for fitting the Hall data accordingto Eq. 2. Pink and red curves are two simulations according to Eq. 2 with α =0,1, respectively. (c) Hall resistivity data from Fig 2c, offset forbetter visibility. Orange curves are simulations according to Eq. 2 with α =0 using parameters from Table 1, see text for further details. estimated from the radial width, R w using the following, ( R w / = ( a / + (∆ d/d ) + (∆ λ/λ ) , (5)where a is the angular size of the direct beam as detailed inRef. . ξ c = 8 ln Q tan ( t/2 ) , (6)where t is the intrinsic tangential variation of the lattice, whichis estimated from the tangential widths of the reflection, T w and direct beam, az using the following, T w = p t + az , (7)The field dependence of ξ a and ξ c is shown in Fig 5. Whereasno clear field dependence was observed for ξ a , a moderatesuppression of ξ c by the field was evidenced on both instru-ments. In our SANS geometry the resolution in the detectorplane (our a -, and c -axes) is approximately one of order ofmagnitude lower than along the neutron flux direction (our b -axis). Therefore, the values of ξ a and ξ c should be regardedas a lower limit on correlation length in the ac plane. IV. SINGLE CRYSTAL GROWTH AND ORIENTATION OFSAMPLES
Single crystals of Ca Ru O were grown using a floatingzone method in a mirror furnace (Canon Machinery, model SCI-MDH)), as reported elsewhere . The crystal growth wasperformed in an atmosphere of the mixture of Ar and O (Ar:O =85:15).The single crystals were oriented using the X-ray Lauebackscattering method utilising a home-built instrument. Thetypical pattern shown in Fig 6 demonstrates very sharp reflec-tions and allows to distinguish between the a - and b -axes. Thefull width at the half maximum (FWHM)=0.32 degree of therocking curve measured at the strong nuclear reflection (10 00) in a neutron beam with the wavelength λ =1.272 ˚A on D23instrument at ILL indicated an excellent quality of our crystal,Fig 7.Oscillation images around crystallographic axes confirmthat crystal quality was maintained for the small single crystal,Fig 8. V. DFT CALCULATIONS
We have ascertained the presence of a sizeable effect ofspin-orbit couplings (SOC) and orbital magnetic moments bystandard DFT-calculations within the full-potential local or-bital (FPLO) approach . We used the generalized gradientapproximation (GGA) as exchange-correlation functional .Correlations beyond have been included by the GGA+Umethod for the 4d-states of Ru with an effective Hubbard-like U=0 to 3 eV. The fully relativistic FPLO code in-cludes SOC to all orders, being based on solutions of the 4- Θ K i K f B c ab sample ω (c)(a)(b) FIG. 4: (a) Geometry of SANS experiment on D33 instrument at ILL. (b) Raw SANS pattern collected at 48 K in magnetic field of 2 T. (c)Rocking scan of magnetic Bragg peaks shown in (b). Intensities in the left and right boxes in panel (b) are indicated by red and black markers,respectively. Solid lines are best fits to Gaussian lineshapes.
Å Å
FIG. 5: Magnetic field dependence of the correlation length along the a (a) and c -axis (b) at 48 K. Dashed lines are a guide to the eye. spinor Kohn-Sham-Dirac equations. As a relevant example,for ferromagnetic spin configurations we find spin-moments m s = µ B / Ru ion and orbital moments m o = µ B / Ru ion for a representative value of U = 2.25 eV. However,there are large uncertainties regarding exact values of spin andorbital moments, as seen from calculated results in Table IIand the appropriate values for the DFT+U-correction in the metallic state at temperatures above the metal-insulator tran-sition are uncertain. For U ≥ eV a gap opens and the band-structure would correspond to an insulating ground-state, ingood agreement with earlier DFT-results by Liu . However,the results indicate the presence of relatively strong SOC ef-fects in the collinear fully polarized state. This also suggeststhat antisymmetric Dzyaloshinskii-Moriya exchange is rela- FIG. 6: The room temperature X-ray Laue diffraction pattern of an as-grown (010) facet of a Ca Ru O single crystal. The red spots andassigned Miller indices show the calculated diffraction pattern of the Bb2 m orthorhombic-space group. !"! % & ’ ()* + , -+ . / $! " :;3!<&$!=!=!1>?@A;!4" FIG. 7: The rocking curve at (10 0 0) reflection measured in a thermal neutron beam. The peak is fitted by a Gaussian lineshape with the fullwidth at the half maximum FWHM=0.32 degree. tively strong in Ca Ru O . In view of the complex bi-layerstructure of Ca Ru O and its correlated metallic character atthe relevant higher temperatures, a credible microscopic eval-uation of the Dzyaloshinskii-Moriya interactions (DMIs) ishardly feasible. However, as the Ru ions occupy the general8b Wyckoff positions in Ca Ru O , the microscopic DMIsbetween the spins s i on these sites, D ij · ( s i × s j ) are allowedfor all pairs of sites with a general Dzyaloshinksii vector D ij ,which is only determined by the SOC in the spin-split elec-tronic bandstructure. VI. LANDAU-GINZBURG FREE ENERGY
The primary magnetic order in Ca Ru O has been iden-tified as a simple antiferromagnetic two-sublattice structure,where ferromagnetically coupled Ru-bilayers alternate withantiparallel moments stacked in c -direction . The antiferro-magnetic order breaks the B-centering operation with the vec-tor t = (1 / , / , (in the Cartesian coordinate system ofFig. 4 a, which will be used for spatial and spin-coordinates U m s m o eV µ B / Ru0 1.53 0.091.00 1.46 0.121.50 1.53 0.092.00 1.52 0.202.25 1.56 0.193.00 1.74 0.02TABLE II: Magnetic spin moment m s and orbital moment m o on Ruin Ca Ru O from GGA+U density functional theory calculations. in the following). The metamagnetic behavior of this two-sublattice order, and the tricritical behavior Fig. 4 c, can bedescribed by the Landau theory for the coupling of the twoequivalent sublattices M I and M II , but this expansion requireshigher-order terms . In the vicinity of the tricritical point the FIG. 8: (a), (b) and (c) show X-ray diffraction patterns of a small Ca Ru O crystal after rotation about the crystallographic a , b , and c -axisrespectively. The rotation axis in each case is vertical. expansion can be expressed by a free energy density F = B ( | M I | + | M II | )+ A M I · M II + B ( | M I | + | M II | )+ A ( M I · M II ) + B ( | M I | + | M II | ) . (8)(9)Representing this phenomenological theory in terms of thestaggered vector l = (1 / M I − M II ) of antiferromagnetismand the net magnetic moment f = (1 / M I + M II ) leads toa free energy, which should include at least 6 th order terms inthe Landau expansion to describe the tricritical point and thephase coexistence between antiferromagnetism and ferromag-netic field-enforced states, w = a l | l | + a f | f | + b l | l | + b f | f | + c | l | | f | + c l | l | + c f | f | + c | l | | f | + c | l | | f | . (10)The two co-existing symmetry modes l and f and their Carte-sian spin-components belong to odd and even representationsof the Cmc2 space-group (which is a standard setting ofspace group No36 equivalent to Bb2 m) with respect to thepartial t -translation, i.e. they have different symmetry. As thiscrystal lattice of Ca Ru O has a non-centrosymmetric or-thorhombic symmetry belonging to Laue class 2mm, Landau-Ginzburg free energies for these two modes can have Lifshitzinvariants, i.e. terms linear in spatial gradients of Cartesiancomponents of either of these modes. These terms derive fromthe Dzyaloshinskii-Moriya interactions and can be written ascombinations of bilinear antisymmetric forms, Γ ( γ ) ij ( x ) ≡ ( x i ∂ γ x j − x j ∂ γ x i ) . (11)For the 2mm symmetry and the simple antiferromagnetism in Ca Ru O , the corresponding free energy contributions are w D = D x Γ ( x ) zx ( l ) + D y Γ ( y ) yy ( l ) (12) w F = F x Γ ( x ) zx ( f ) + F y Γ ( y ) yz ( f ) , (13)where the coefficients D x,y , F x,y are materials constants.These contributions, in particular w D lead to the spirallingcycloidal modulations of the magnetic order , which we call Dzyaloshinskii textures . In an antiferromagnet where only the w D invariants are acting, an antiferromagnetic spiral wouldbe composed only of the l -symmetry mode. Therefore, wecan refer to such a sprial as a “proper” texture. In theschematic phase diagrams, Fig.4c and Fig.4d, the presenceof term w D can lead to spiralling or other precursor texturesdesignated π in particular at higher temperatures above thetemperature range, where the anisotropies enforce a homo-geneous antiferromagnetic state, which seems to be the casein Ca Ru O . Otherwise, the w D term can affect the anti-ferromagnetic order-parameter and could lead to a spirallingantiferromagnetic ground-state for weak enough anisotropies.The existence of these terms also means that the thermalphase transition from the paramagnetic to the antiferromag-netic state does not obey the Lifshitz criterion of the Landautheory for a continuous phase transition. Therefore, the ther-mal ordering transition in a material with a contribution of theform w D is expected to be anomalous. In particular, a fluctu-ating precursor states can arise above the magnetic ordering.A complete phenomenological theory then requires also theusual squared gradient terms of the order-parameter, w E = A l ( ∇ l ) + A f ( ∇ f ) + . . . , (14)where the ellipses stand for anisotropic exchanges terms. In-deed, in Ca Ru O strong additional anisotropies, suppress amodulation in the antiferromagnetic ground state. A completeLandau theory for the homogeneous states would require ad-ditional terms, the leading magnetocrystalline anisotropy be-ing described by w a = K z l z + k z f z + κ x l x + κ xy l x l y + κ y l y + ν x f x + ν xy l x l y + ν y l y , (15)with anisotropy coefficients K z , κ, ν . We note that fordoped compounds Ca (Ru − x TM x ) O , where TM is Fe orMn, incommensurately modulated antiferromagnetic ground-states have been observed and have been interpreted asDzyaloshinskii spirals . The observation suggests that thesubstitituion on the magnetic site weakens the anisotropy andreveals the presence of the inhomogeneous terms w D , suchthat the w D terms overcomes the anisotropies w a .In the region of the metamagnetic phase co-existence, ad-ditional higher order Lifshitz terms become operative, whichcouple l and f . There are a great number improper of cou-plings between these modes in Ca Ru O . With the aim toillustrate the complexities of possible effects, we give here acomplete list of these terms. The mixed higher order terms areLifshitz-type invariants as follows: w µ = X α = x,y,z X β = x,y (cid:16) a α f α Γ ( β ) β z )( l ) + b α l α Γ ( β ) β z ( m ) (cid:17) (16)and w ∆ = ∆ f x f y Γ ( z ) xy ( l )+ ∆ f x f y Γ ( z ) zx ( l )+ ∆ f x f y Γ ( x ) yz ( l )+ ∆ f x f y Γ ( z ) yz ( l )+ ∆ f x f y Γ ( y ) zx ( l )+ ∆ f z f x Γ ( y ) xy ( l )+ ∆ f z f x Γ ( z ) zx ( l )+ ∆ f y f z Γ ( x ) xy ( l )+ ∆ f y f z Γ ( z ) yz ( l )+ Ξ l x l y Γ ( z ) xy ( f )+ Ξ l x l y Γ ( z ) zx ( f )+ Ξ l x l y Γ ( x ) yz ( f )+ Ξ l x l y Γ ( z ) yz ( f )+ Ξ l x l y Γ ( y ) zx ( f )+ Ξ l z l x Γ ( y ) xy ( f )+ Ξ l z l x Γ ( z ) zx ( f )+ Ξ l y l z Γ ( x ) xy ( f )+ Ξ l y l z Γ ( z ) yz ( l ) , (17)where coefficients ∆ and Ξ are materials constants. When en-forced by the external field or near a multicritical point, the an-tiferromagnetic mode l and f can co-exist, these mixed termsbecome operational and will allow the formation of modu-lated states composed of the two different modes. Therefore,we can call these modulated states “improper textures” as theyare enabled by mixed terms coupling modes of differeent sym-metry.Aditionally, there also exist higher order Lifshitz invariants that are quartic in either l , f w = X α = x,y,z X β = x,y ( η α f α Γ ( β ) β z ( f ) + τ α l α Γ ( β ) β z ( l ))+ σ f x f y Γ ( x ) yz ( f )+ σ f x f y Γ ( y ) zx ( f )+ σ f y f z Γ ( y ) zx ( f )+ σ f z f x Γ ( x ) yz ( f )+ σ l x l y Γ ( x ) yz ( l )+ σ l x l y Γ ( y ) zx ( l )+ σ l y l z Γ ( y ) zx ( l )+ σ l z l x Γ ( x ) yz ( l ) . (18)Depending on many materials parameters a x,yz , b x,y,z , ∆ , Ξ , η, τ , and σ , these terms describe pos-sible coupled modulations of coexisting primary symmetrymodes l , f , which can take place in distinct fashion in allthree spatial directions. In addition the coupling can displaymarkedly anharmonic effects.We observe that the Lifshitz-(type)-invariants couple differ-ent Cartesian components of the order-parameters to differ-ent spatial directions via the gradient term. Thus, these termsbreak isotropy in spin-space. This implies that their micro-scopic origin is the relativistic spin-orbit interaction. Specif-ically, a microscopic mechanism to explain these terms is theantisymmetric pairwise Dzyaloshinskii-Moriya exchange, orappropriate generalization for magnetic systems with a moreitinerant character of spin-ordering,A complete Landau-Ginzburg free energy density for a po-lar metamagnet would collect all these terms w = w E + w + w D + w F + w a + w µ + w ∆ + w (19)Minimizing this free energy functional maps out all possi-bilities of metamagnetic modulations around a tricritical pointfor the specific antiferromagnetic order parameter in spacegroup Cmc2 . The corresponding Euler-Lagrange equationsfor the variational problem will constitute a system of cou-pled partial differential equations for the degrees of freedomdescribed by the fields l ( r ) , and f ( r ) .A dedicated theory for a specific material could be dis-tilled from this most general functional by restricting to afew crucial terms. For a simple case, which may pertain toCa Ru O , it may suffice to consider only one-dimensionalmodulations in y -direction and spin-components in the yz -plane. This is the case sketched in Fig.4b. The most im-portant terms are then proper Lifshitz invariant F y Γ ( y ) yz )( f ) and the mixed Lifshitz-invariants a z f z Γ ( y ) yz )( l ) and ( b y l y + b z l z ) Γ ( y ) y z ( f ) . These inhomogeneous contributions to the freeenergy imply that the presence of a net magnetization f z inan applied field along the polar axis favours an instability to-wards an antiferromagnetic modulation in the yz -plane. But,the local ferromagnetic modulation is also unstable with re-spect to modulations through the proper Lifshitz invariants.Only strong anisotropies can prevent the instability of thespin-system towards mixed states where ferromagnetic andantiferromagnetic configurations are simultaneously presentin a spatially modulated fashion. The presence of these dif-ferent effective couplings then leads to modulations with acompeting character, as different coupling terms co-operateand frustrate each other. In our observations, this competingcharacter of the modulation is noticable, as the characteristicmodulation length displays a pronounced temperature depen-dence. For an ordinary Dzyaloshinskii spiral, this behavioris unusual and unexpected , as in that case there is only onecoupling term that rules the frustration of one simple symme-try mode. Also, the presence of the higher order term couldlead to marked anharmonicities in metamagnetic textures thatare driven by the higher order mixed terms. For the particular antiferromagnetic order in Ca Ru O , themixed Lifshitz-type terms are of higher order and affect themagnetic spin-structure only in the region of a metamagneticco-existence. It is the underlying tricritical point which re-veals the presence of these terms, Fig.4c. However, appro-priate symmetry of an antiferromagnetic mode can also allowmixed Lifshitz type invariants with a bilinear form like, C ( γ ) ij ( f i ∂ γ l j − l j ∂ γ f i ) . (20)In the vicinity of the bi-critical point, in the case of a sys-tem with weak anisotropy, sketched in Fig. 4 d, these termsdrive the existence of metamagnetic textures with modulationbetween antiferromagnetic ground-state in spin-flopped con-figuration and the field-enforced ferromagnetism. Kummer, K. et al.
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