Large Noncollinearity and Spin Reorientation in the Novel Mn2RhSn Heusler Magnet
O. Meshcheriakova, S. Chadov, A. K. Nayak, U. K. Rößler, J. Kübler, G. André, A. A. Tsirlin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle, M. Nicklas, C. Felser
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Large non-collinearity and spin-reorientation in the novel Mn RhSn Heusler magnet
O. Meshcheriakova , , S. Chadov , A. K. Nayak , U. K. R¨oßler , J. K¨ubler , G. Andr´e ,A. A. Tsirlin , J. Kiss , S. Hausdorf , A. Kalache , W. Schnelle , M. Nicklas , C. Felser Graduate School of Excellence ”Materials Science in Mainz” Johannes Gutenberg - Universtit¨at, 55099 Mainz, Germany Max-Planck-Institut f¨ur Chemische Physik fester Stoffe, N¨othnitzer Str. 40, 01187 Dresden, Germany Leibniz-Institut f¨ur Festk¨orper- und Werkstoffforschung IFW, Helmholtz Str. 20, 01171 Dresden, Germany Institut f¨ur Festk¨orperphysik, Technische Universit¨at Darmstadt, Germany and Laboratoire L´eon Brillouin, CEA-CNRS Saclay, Gif-sur-Yvette Cedex, France
Non-collinear magnets provide essential ingredients for the next generation memory technology.It is a new prospect for the Heusler materials, already well-known due to the diverse range ofother fundamental characteristics. Here we present combined experimental/theoretical study ofnovel non-collinear tetragonal Mn RhSn Heusler material exhibiting unusually strong canting ofits magnetic sublattices. It undergoes a spin-reorientation transition, induced by a temperaturechange and suppressed by an external magnetic field. Due to the presence of Dzyaloshinskii-Moriyaexchange and magnetic anisotropy, Mn RhSn is suggested to be a promising candidate for realizingthe skyrmion state in the Heusler family.
The art of controlling magnetic degrees of freedom hasled to a broad range of applications that make up therapidly developing field of spintronics. Up to now, mostof the exploited compounds have been so-called collinearmagnets, i.e. materials in which the magnetization isformed by local magnetic moments aligned parallel orantiparallel to one another. Yet, the possibility of in-fluencing their mutual orientation opens new horizonsfor the field of spintronics. Non-collinear magnets canbe widely applied in current-induced spin-dynamics [1],magnetic tunnel junctions [2], molecular spintronics [3],spin-torque transfer by small switching currents [4] andanomalous exchange bias [5]. Impressive improvementof the critical current density by five orders of magni-tude [6–8] is offered by non-collinear magnets driven intothe skyrmion phase [6–14]. While such exotic magneticarrangements are sensitive to external conditions (mag-netic field and temperature), an expansion of the relatedmaterial base is important for their stabilization.Flexible tuning of the magnetic properties can ulti-mately be realized in multicomponent systems of sev-eral magnetic sublattices with competing types of inter-actions such as magneto-crystalline anisotropy, dipole-dipole and Dzyaloshinsky-Moriya (DM) interactions [15,16]. Heusler compounds, of which there are over 1000members, provide a rich variety of parameters for al-most any material engineering task (e.g. half-metallicferromagnetism [17, 18], shape memory [19], exchangebias [20], topological insulators [21], spin-gapless semi-conductivity [22], spin-resolved electron localization [23]and superconductivity [24]). Furthermore, the major-ity of Mn YZ (Y – transition metal, Z – main-groupatom) systems are non-centrosymmetric; this togetherwith the magneto-crystalline anisotropy induced by in-trinsic tetragonal distortion makes such systems attrac-tive for skyrmion research.First, we will discuss here the unusual ground-statemagnetic canting observed in Mn RhSn together with the subsequent temperature-induced spin-reorientationinto the collinear ferrimagnetic mode. Further, we willgive a detailed micromagnetic analysis which suggeststhis collinear regime to provide perfect conditions for theskyrmion formation, in agreement with the earlier theo-retical studies [9, 25, 26].In a non-relativistic case, the magnetic non-collinearityis a result of the competition between antiparallel andparallel exchange interactions (or between several typesof antiparallel interactions). Such a situation is oftenencountered in Mn YZ compounds, but not all of themexhibit non-collinearity. In general, these materials crys-tallize in the non-centrosymmetric I m I at 2 b (0 , / ,
0) and Mn II at 2 d (0 , / , / ). Zand Y elements occupy the 2 a (0 , ,
0) and 2 c (0 , / , / )positions, respectively (Fig. 1 a). The most significantexchange coupling between the nearest Mn I and Mn II atoms is characterized by a large exchange constant( J Mn I − Mn II ∼ −
20 meV) (e.g. [27]) that leads to a typi-cal collinear FiM (ferrimagnetic) state. Despite the factthat the in-plane interaction of Mn atoms can be rathercomplicated (e.g. the nearest in-plane neighbours coupleparallel, the next-nearest couple antiparallel or parallel,and so on), these interactions are rather weak comparedto J Mn I − Mn II , which always aligns the Mn spin momentsof the same plane parallel to one another (Fig. 1 b). Forthis reason, we initially do not consider the in-plane in-teractions but will expand the description in terms of theeffective inter-plane exchange coupling J , which indicatesthe interaction of a certain Mn atom ( i ) with all otherMn atoms ( i ′ ) in a different plane, i.e. J = P i ′ J ii ′ .Since the collinear order being substantially deter-mined by the nearest-plane J interaction becomes evenmore stable if the Y atom is magnetic (as e. g. in caseof Mn CoZ systems [27]), our further consideration con-cerns Mn YZ Heusler materials with the non-magneticheavy Y elements (such as Rh or Ir, since in case of
FIG. 1. (a) Crystal and magnetic structures of Mn YZHeusler compounds. Due to the magneto-crystallineanisotropy induced by the tetragonal distortion, the Mn I magnetic moments are oriented along the c axis; the momentson Mn II are canted in an alternating manner with respect tothe c axis. (b) Schematic picture of the leading magnetic ex-change interactions between different atomic layers in Mn YZ(atomic planes containing Z and Y elements are shown inblue and red, respectively). The arrows show the orienta-tion of the spin moments on Mn and the springs show theexchange interactions between different planes. Consideringonly the nearest antiparallel interactions J (between Mn I -Zand Mn II -Y planes) leaves the magnetic structure collinear;introducing the next-nearest antiparallel coupling j (betweenMn II -Y planes) leads to the alternating canting of Mn II mo-ments by θ and 2 π − θ . light elements, such as Ti or V, Mn atoms occupy equiv-alent 2 c and 2 d positions). In this case, the collinearitycan be perturbed by the next important interaction j between the next-nearest planes, e.g. between pairs ofMn II -Y planes as shown in Fig. 1 b. This interactionis antiparallel due to its indirect origin realized throughthe main-group element Z (super-exchange) [28]. Since j tends to rotate the moments of the nearest Mn II -Yplanes antiparallel to each other, it competes with thestrong antiparallel exchange J , and may then result ina non-trivial canting angle ( θ = 0 ◦ , ◦ , Fig. 1 b). Therelevant θ -dependent part of the Heisenberg Hamiltonianwill contain only antiparallel interactions: H θ = − J cos θ − / · j cos 2( π − θ ) , (1)where the first term is the coupling of the nearest planes(Mn I -Z with Mn II -Y) and the second is that of the next-nearest (Mn II -Y) planes. The factor / accounts forthe twice sparser entrance of the next-nearest plane cou-plings. The extrema of H θ are found from:sin θ (cid:18) / + jJ cos θ (cid:19) = 0 , (2)and θ , = 180 ◦ ± arccos (cid:16) J j (cid:17) non-collinear solutions aregiven subject to the condition j/J > / , which meansthat the canting occurs only if the next-nearest antipar-allel exchange j is sufficiently strong.To justify the proposed magnetic order we performed ab initio calculations (Supporting Information [31], Mn YZ m Mn I m Mn II θ , [ ◦ ] m Y M M exp Mn RhSn 3.51 3.08 180 ±
55 0.14 1.9 1.97Mn PtIn 3.38 3.30 180 ±
50 0.12 1.4 1.6Mn IrSn 3.52 3.08 180 ±
44 0.09 1.4 1.5TABLE I. Computed atomic magnetic moments m , cant-ing angles θ , and total magnetization per formula unit M = m Mn I + m Mn II · cos θ + m Y , compared to the experi-mentally measured magnetization M exp . Values of magneticmoments/magnetization are given in µ B . (b) ( )( ) ( ) (a) FIG. 2. (a) Temperature-dependent neutron diffraction spec-tra. The (002)-peak decays over 1.8-80 K. (b) Weakeningof the in-plane magnetism (produced by Mn II x -component)releases the z -component of Mn I , while the z -component ofMn II evolves rather insignificantly. Sec. V) for Mn RhSn and another two similar Heuslersystems, Mn PtIn and Mn IrSn. For Mn RhSn, the plotof the total energy as a function of θ indeed exhibits twoenergy minima corresponding to the non-trivial cantingangles θ , = 180 ◦ ± ◦ . Similar plots were obtained foranother two compounds (Supporting Information [31],Fig. S8). Calculated local moments, their orientations,total magnetization, and experimentally measured one,are summarized in Tab. I. These magnetic properties maybe significantly affected by those kinds of disorder whichare typical for Heusler systems. The details of this aspectare discussed in Supporting Information [31] (Sec. IV).Powder neutron scattering data convincingly demon-strate the predicted ground-state non-collinearity(Fig. 2). At 1.8 K the magnetic moments are 3.59 and3.47 µ B on Mn I and Mn II . The value obtained for themore localized Mn I correlates with the calculated result(Tab. I), while the Mn II moment is larger: it is definedless precisely as the scattering events on itinerant mo-ments are more dispersed. The magnetic structure wasfound to be canted by about θ , = (180 ± . ◦ withinalternating Mn II -Rh planes. It is important to note,that such strong magnetic canting was never reportedfor the Heusler materials, in which it is typically of anorder few degrees at most.Being non-collinear in the ground state, the magneticconfiguration evolves with changes to the temperatureand external field. Observation of the (002)-peak in-tensity for T ≤
80 K indicates the presence of in-planemagnetism (Fig. 2 b). As the temperature increases, thepeak gradually decreases and subsequently vanishes for
T >
80 K, suggesting that the in-plane magnetic com-ponent is suppressed (Fig. 2 b). This is attributed tothe gradual spin-reorientation of the Mn II sublattice; thecanting angle decreases until a collinear FiM order setsin at 80 K. Such behavior is strongly pronounced in the M ( T ) curves measured in weak fields (0.1-0.5 T, Fig. 3 a)and suppressed in stronger fields (5 T). This is evidentlyan intrinsic effect as the applied fields are larger than thecoercive field ( H c = 0 .
065 T). It is not only the mutualorientation of the site-specific moments that changes buttheir absolute values also change (Fig. 3 c). In the cantedlowest-temperature state, the Mn I moment is somewhatcompensated by the equally strong Mn II . As the temper-ature increases, the moments of Mn II delocalize furtherand release the Mn I to reach 4.5 µ B . This occurs grad-ually, and the slope of the zero-field heat capacity curvechanges (Fig. 3 e); the spin-wave term is sufficiently weakin comparison to the electronic and phonon contributionsthat no sharp anomaly is visible. However, the onset ofthe FiM phase is characterized by the explicit step-likeincrease in the ac-susceptibility signal (Fig. 3 b). Mea-sured values of χ ′ and χ ′′ were found to be indepen-dent of the frequency, suggesting a high magnetic ho-mogeneity. The evolution of the magnetism with tem-perature is echoed by the crystal structure (Fig. 3 f).Although the a -parameter increases monotonically, thechange in c -parameter is non-linear and corresponds tothe ac-susceptibility behavior. The sudden rise in thevicinity of 280 K is an anomaly corresponding to T C .The c -parameter eventually decreases until a transitionto the cubic phase occurs at about 570 ◦ C (SupportingInformation [31], Fig. S5).By systematic coarse-graining of the spin-lattice modela micromagnetic continuum theory has been developed(Supporting Information [31], Sec. VI). Considering onlythe leading Heisenberg-like exchange, the analysis showsthat in tetragonal inverse Mn YZ Heusler alloys, themagnetic ordering displays coexisting magnetic modeswith ferrimagnetism (FiM) of the two sublattices and anantiferromagnetic mode (AFM) on the Mn II -sublattice.These systems, thus, are close to a bicritical (or tetra-critical) point in their magnetic phase diagram. In FIG. 3. Evolution of the magnetic structure with the tem-perature. Canted (red), collinear ferrimagnetic (yellow) anddisordered (blue) magnetic states of Mn RhSn. (a) Zero-field-cooled (ZFC), field-cooled (FC) and field-heated (FH) mag-netization as a function of the temperature measured at in-duction fields of 0.1, 0.5, and 5 T. (b) Real ( χ ′ ) and imaginary( χ ′′ ) ac-susceptibility components are frequency independentand show a pronounced step at the onset of the FiM phase.(c) The change in the canting angle occurs because of the si-multaneous re-alignment of the Mn II moment and a decreasein its absolute value. This, in turn, releases the previouslysuppressed Mn I moment from 3.5 to 4.5 µ B . (d) The sum ofthe total and z -components of the Mn I(II) moments follows theac-susceptibility behavior. No in-plane component is presentafter 80 K. (e) A change in the slope of the heat capacity curveis observed in the vicinity of the spin-reorientation. (f) Evo-lution of the lattice parameters with temperature: the changein the magnetism is echoed mainly by the c -parameter, while a evolves monotonically. Mn RhSn, the thermodynamic potential favours a dom-inating collinear FiM order for
T >
80 K. Below thistemperature the AFM sets in. By the crystal sym-metry of Mn YZ, chiral inhomogeneous DM couplings exist in spatial directions perpendicular to the crystalaxis [25, 26] that cause a spiral twist of these magneticmodes with long pitch. The micromagnetic model forthe FiM state is exactly the Dzyaloshinskii-model fora magnetic order in acentric tetragonal crystals from42 m ( D d ) class [25, 29]. Therefore, in the collinearFiM state, chiral skyrmions and skyrmion lattices ex-ist in these magnets, as predicted in Ref. [9, 25]. Themicromagnetic model predicts a chiral twisting lengthΛ ∼
130 nm, which corresponds to the diameter of theFiM-state skyrmions. These chiral skyrmion states ex-ist in the inverse Heusler alloys without the need of anyadditional effects not accounted for by the basic mag-netic couplings, i.e. Heisenberg-like and DM-exchangeand leading anisotropies, and at arbitrary temperature.This is in contrast to chiral cubic helimagnets, whichrequire fine-tuned additional effects for the existence ofskyrmionic states. Because the tetragonal lattice also in-duces a sizeable easy-axis magneto-crystalline anisotropyin Mn RhSn, as calculated by relativistic DFT, the mag-netic phase diagram is not expected to display a field-driven condensed skyrmion phase in this FiM-state. Theratio of easy-axis anisotropy to DM coupling is large.Using the universal phase diagram of chiral magnets [9],skyrmions do exist as nonlinear solitonic excitations ofthe collinear state in Mn RhSn. Therefore, this inverseHeusler alloy is an ideal system to realize reconfigurablenanomagnetic patterns composed of its two-dimensionalfree skyrmions at elevated temperatures.Coexistence of FiM and AFM orders in the cantedstate will be the subject to different DM-couplings. Thus,in the ground-state, novel types of chirally twisted tex-tures can exist in Mn YZ alloys. These chiral DM-couplings favour different twisting lengths for the twomodes and their interaction. Then, the magnetic ordermay become quasiperiodic and exceedingly complex, asrecently described for the similar case of textures in bi-axial nematic liquid crystals [30]. The presence of sev-eral DM-terms and anisotropies affecting the coexistingmagnetic modes promise a rich behavior of chiral tex-tures in tetragonal inverse Heusler alloys Mn YZ. E.g.,closely below the onset of spin-reorientation temperature,the chiral skyrmion of the FiM-state is superimposed bya vortex-like AFM-configuration on the Mn II -sublatticewith a defect in the core of the soliton configuration. Inthe ground-state, such configurations may become insta-ble, depending on the stiffness of the AFM order. Upto now, such complex configurations have been analysedonly for the simpler case of chiral AFMs with a coexistingweak-FM mode [26].As we have demonstrated theoretically and experi-mentally, the design of non-collinear magnets within theHeusler family of materials can be based on Mn YZcompositions, with Y and Z being a non- or weakly-magnetic transition-metal and a main-group element, re-spectively. The choice of the Mn YZ Heusler group al-lows to control the canting angles by, e.g. combining theY and Z elements or varying the Mn content. The useof heavy transition metals (e.g. as in the present case,Y=Pt, Rh, Ir and Z=Sn, In) amplifies the magnetically-relevant relativistic effects that are already present inthese systems, such as the DM interaction and magneto-crystalline anisotropy. Such multiple magnetic degreesof freedom together with the possibility of their manip-ulation provided by the family of Mn YZ Heusler ma-terials is vital for efficient engineering and stabilizationof various magnetic orders. In particular, Mn RhSn issuggested to be a promising candidate for realizing theskyrmion state in the Heusler family.Authors thank A. Bogdanov, R. Stinshoff and A. Be-leanu for helpful discussions. The support of the Euro- pean Commission under the 7th Framework Programme:Integrated Infrastructure Initiative for Neutron Scat-tering and Muon Spectroscopy: NMI3-II/FP7 - Con-tract No. 283883, the Graduate School of Excellence ’Ma-terials Science in Mainz’, the DFG project FOR 1464’ASPIMATT’ and the European Research Council (ERC)for ’Idea Heusler!’ are gratefully acknowledged. [1] P. Bal´a˘z, M. Gmitra, and J. Barna´s, Phys. Rev. B ,174404 (2009)[2] Y. Miura, K. Abe, and M. Shirai, Phys. Rev. B ,214411 (2011)[3] A. Soncini and L. F. Chibotaru, Phys. Rev. B , 132403(2010)[4] N. L. Chung, M. B. A. Jalil, S. G. Tan, J. Guo, and S. B.Kumar, J. Appl. Physics , 084502 (2008)[5] Y. F. Tian, J. F. Ding, W. N. Lin, Z. H. Chen, A. David,M. He, W. J. Hu, L. Chen, and T. Wu, Sci. Reports ,1094 (2013)[6] F. Jonietz, S. M¨uhlbauer, C. Pfleiderer, A. Neubauer,W. M¨unzer, A. Bauer, T. Adams, R. Georgii, P. B¨oni,R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Sci-ence , 1648 (2010)[7] X. Z. 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O. Meshcheriakova, , S. Chadov, A. K. Nayak, U. K. R¨oßler, J. K¨ubler, G. Andr´e, A. A. Tsirlin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle, M. Nicklas, and C. Felser Graduate School of Excellence ”Materials Science in Mainz” Johannes Gutenberg - Universtit¨at, 55099 Mainz, Germany Max-Planck-Institut f¨ur Chemische Physik fester Stoffe,N¨othnitzer Strasse 40, 01187 Dresden, Germany Leibniz-Institut f¨ur Festk¨orper- und Werkstoffforschung IFW, Helmholtz Strasse 20, 01171 Dresden, Germany Institut f¨ur Festk¨orperphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany and Laboratoire Lon Brillouin, CEA-CNRS Saclay, 91191 Gif-sur-Yvette Cedex, France
I. SYNTHESIS OF Mn RhSn AND Mn IrSn
Polycrystalline samples were repeatedly arcmeltedfrom stoichiometric amounts of high-purity commerciallyavailable elements in an Ar atmosphere with an overallmass loss of less than 0.5 wt.%. In each melting run, apiece of titanium was used to purify the residual atmo-sphere. A two-step process was employed for Mn RhSn:a premelt of Rh-Sn was prepared, and this was thenplaced on the Mn. According to the respective phase di-agrams, Rh and Sn react well with each other and forma stable phase. In the second step, when the premelt isheated, it absorbs the Mn pieces, and the evaporation ofMn is minimized. To ensure homogeneity, the sampleswere melted 3 times on each side. As a result, after theMn is absorbed by the phase, evaporation of the completephase, not the single elements, takes place.The Mn IrSn sample was prepared by induction heat-ing. The procedure was repeated several times to ensurehomogeneity: six repetitions of arcmelting and two of in-duction heat. In the latter process, the sample was main-tained in the liquid state for 5 min. After 1 week of an-nealing, the arcmelted samples were fast cooled, whereasthe induction-heated samples were cooled slowly. Theingots were then wrapped in Ta foil and annealed inevacuated silica tubes at 800 ◦ C for 1 week. To reducethe amount of surface oxidation, Mn pieces were prelim-inary sealed in evacuated silica tubes and left overnightat 900 ◦ C for purification. These pieces were processedrepeatedly until a shiny silver-coloured surface was ob-tained.
II. PRIOR CHARACTERIZATION
Metallographic analysis by scanning electron (Fig. S1)and optical (Fig. S2) microscopy revealed that the sam-ples are single-phase materials with a homogeneous com-position distribution. The composition was characterizedby energy-dispersive X-ray (EDX) spectroscopy (valuesare summarized in Tab. S1). Since the electron penetra-tion depth is on the order of nanometres, a well-polishedsample surface is essential for eliminating morphology ef-fects. For this reason, the samples were embedded inepoxy resin blocks, and a smooth surface was prepared.
FIG. S1. SEM image of Mn RhSn; composition analysis wasperformed at the areas marked. The Mn RhSn stoichiome-try is constant across the whole sample; a minor impurity ofMn . RhSn is present at around 1 at.% and does not bias thepresented results.FIG. S2. An optical microscope image of the homogeneousMn RhSn phase.
The measured composition deviates from the target val-ues by 0.5 at.%, which is within the range of experimentalerror.
A. X-ray diffraction
Powder X-ray diffraction patterns (Fig. S3) were ob-
Spot Mn (at.%) Rh (at.%) Sn (at.%)1) 49.73 25.55 24.712) 48.79 25.73 25.483) 47.84 26.23 25.93TABLE S1. EDX analysis of the Mn RhSn sample takenfrom the areas indicated in Fig. S1. The composition is wellreproduced across the whole observed area.
30 40 50 60 70 80 909 12 15 18 21 24
I-4m2
T = 300 K I [ a . u . ] T = 100 K T = 50 K
I-4m2 I [ a . u . ]
2 [ o ] FIG. S3. Powder X-ray patterns obtained at room temper-ature (top) and 50 and 100 K (bottom). The coloured andblack lines correspond to the observed and calculated intensi-ties, respectively. Incident light wavelengths of λ = 1 . I -4 m tained using a Guinier camera (Cu K α radiation) withLaB acting as the internal standard. The samples werefirst sieved through a 40 µ m mesh. Rietveld refinementwas performed using the FullProf software [1] for thestructure analysis.In the case of Mn RhSn [2], low-temperature X-raydiffraction patterns were measured at ESRF, Grenoble,with an incident beam wavelength of 0.43046 ˚A for ahigh-resolution data analysis. The Mn RhSn compoundcrystallizes in an inverse tetragonal 119 Heusler structure(with Wyckoff sites of Mn I : (0, 0, ); Mn II : (0, , ); Rh:(0, , ); and Sn: (0, 0, 0)). Mn IrSn and Mn PtIn [3]crystallize in the same structure with lattice parameters -6 -4 -2 0 2 4 6external magnetic field [T]-2-1012 m agne t i z a t i on [ µ B / f . u . ] FIG. S4. Magnetic hysteresis loops measured at 1.8 K ofpolycrystalline Mn RhSn [2] (red), Mn PtIn [3] (green), andMn IrSn (blue; present work) samples. of a = 4 .
29 ˚A, c = 6 .
59 ˚A and a = 4 .
32 ˚A, c = 6 .
77 ˚A,respectively.
B. Magnetic and thermal measurements
Magnetization measurements were performed in con-stant field sweeps at different temperatures using theQuantum Design MPMS XL superconducting quantuminterference device (SQUID) magnetometer. The totalmagnetization was obtained from the hysteresis loop at1.8 K (Fig. S4) and was 1.97 µ B (per formula unit). Inthe zero-field-cooled (ZFC) mode, the sample was ini-tially cooled in the absence of a field down to 2 K, anddata were collected as the temperature was increased inthe applied field. In the field-cooled (FC) mode, datawere collected while the sample was cooled in the field,and subsequently, data were also collected while the sam-ple was heated in the field during the field-heated (FH)mode. The real ( χ ′ ) and imaginary ( χ ′′ ) parts of the ac-susceptibility were obtained simultaneously at the low-est possible dc-field of 50 Oe. Various field frequenciesfrom 33 to 9997 GHz were applied over the temperaturerange of 2 to 300 K. The heat capacity measurementswere performed in zero field over the same temperaturerange. A transition to the high-temperature cubic phase(Fig. S5) was observed with the help of differential scan-ning calorimetry (DSC) at a moderate rate of 10 K/min;the powder sample was encapsulated in an Al O crucibleand measured in an Ar atmosphere. III. NEUTRON SCATTERING
A two-axis diffractometer equipped with a vertical fo-cusing pyrolytic graphite monochromator and a cold neu-tron guide was used for the neutron scattering mea-surements. The sample was encapsulated in a vana-
480 510 540 570 600 630 6600.20.30.40.50.6 D S C [ V ] T [ o C] Mn RhSn 10 K/min
FIG. S5. The transition from the high-temperature cubicto low-temperature tetragonal phase occurs between 537 and594 ◦ C. Red and blue curves correspond to the heating andcooling regimes, respectively.Mn I [ µ B ] Mn II [ µ B ] θ [ ◦ ] a [˚A] c [˚A] M z = M tot M x M z M tot − RhSn sampleobtained from neutron scattering measurements. dium crucible, and the sample environment was con-stant throughout the measurement. The magnetic andnuclear structures were refined by the Rietveld methodusing the FullProf software [1]. The nuclear phase wasfirst optimized above the magnetic ordering temperature,and the obtained parameters were later used for the re-finement of the magnetic phase. This significantly im-proved the sigma deviations and R factors. The back-ground was modelled by interpolation between manuallyselected points. The peak shape profile was described bya pseudo-Voigt function with a refined ratio between theGaussian and Lorentzian contributions. Different oxida-tion states, Mn and Mn , were assumed to calculatethe magnetic scattering form factors. The results of theneutron scattering measurements are listed in Tab. S2.Several refinement approaches were used by fixing orreleasing certain, all, or some parameters. In total, wevaried 10 parameters: the scale, zero-shift, a -constant, c -constant, projections of the spin [ M z (Mn I ), M x (Mn II ),and M z (Mn II )], Lorenztian-to-Gaussian ratio in the peakshape (varied separately for the Bragg and magneticphase), and overall isotropic displacement (temperature)factor. The room-temperature pattern was refined firstto eliminate any contribution of the magnetic signal, andthe obtained zero-shift value was then fixed to avoid er-rors in the lattice parameters.The lattice constants are robust and independent ofthe specific refinement procedure. We also found that theMn I moment was indeed strongly localized and alignedalong the crystallographic z -axis. If a slight deviation
15 30 45 60 750 150 0 150 0 150 300 (d)(c)(b)
I-4m2 (119) 1.8 K 70 K 200 K 299 K I [ a . u . ] ( )( ) ( ) (a) (101) I r e l [ a . u . ] (002) T [K] (110)
FIG. S6. Neutron data of Mn RhSn. (a) Evolution of the neu-tron spectra with temperature (1.8 K, 70 K, 200 K and 299 K);peak positions from the Bragg and magnetic phases are dis-tinguished by vertical markers. Inset: evolution of (002) peakwithin the whole temperature regime. Normalized intensityof the (101), (002) and (110) peaks ((b), (c), (d) panels, re-spectively) together with the corresponding crystal planes. from this direction is introduced, then the refinementdoes not converge. In contrast, the Mn II moment prefersa canted orientation, and the obtained absolute value ofthe Mn II moment was slightly higher than that predictedtheoretically: 3.47 µ B as opposed to 3.08 µ B . The valueof the Mn I moment, however, was in good agreementwith the predicted value: 3.59 µ B to 3.51 µ B .The magnetic state appears to be well-analyzable dueto its noticeable contribution to the overall intensity:e.g., (101)-peak increases by nearly 65 % with temper-ature decrease from T = 299 K (paramagnetic) to 1.8 K.The additional intensity contains only that magneticcomponent, which is perpendicular to the scattering vec-tor; therefore the neutrons reflected from the (101), (002)and (110) crystal planes give us an estimate how the mag-netic moment evolves within these planes (see Fig. S6).For T >
80 K the magnetic contribution to the (002)peak vanishes and only the Bragg intensity is observed.The in-plane magnetism is realized by canting of the Mn II moment. IV. DISORDER EFFECTS AND THE PHASESTABILITY OF Mn RhSn
Chemical disorder, which often occurs in multicompo-nent systems, such as Heusler alloys, may severely in-fluence the magnetic properties. Random exchange be-tween Rh and Mn I would increase the amount of Mn II type (magnetically antiparallel to Mn I ). In turn, the ex-change between Sn and Mn II will increase the amountof Mn I type. The exchange between Rh and Sn will lo-cally convert Mn I into Mn II within Mn I –Sn planes, andMn II into Mn I within Mn II –Rh planes. In which wayparticular type of disorder will affect the system exactly,is rather complicated question, as it implies not just astraightforward redistribution of different Mn types, butthe change of the whole magnetic coupling picture. Itis easy to show (e.g., by first-principle calculations) thatin all cases the total energy drastically increases, whichindicates that such events are of small probability (oncethe system holds the correct stoichiometry and is prop-erly annealed). In any case, if such situation would occur,the interpretation of the neutron spectra using the pro-posed magnetic picture would be unreliable.For the present samples such straightforward reasonfor chemical disorder as deviation between the actual andthe target compositions can be excluded due to the highquality evidenced by EDX and also XRD (see Fig. S2,Tab. S1). Within this restriction, certain random in-termixing of different atomic types would be still pos-sible. However, any mixtures involving Sn are unlikely,as it affects the zinc-blende sub-structure, which is the“skeleton” of any Heusler alloy (as one can represent theHeusler system as zinc-blende plus extra transition ele-ment). Certain intermixing within Rh–Mn II plane wouldstill be possible (indeed, there are Heusler systems sta-bilized by such mechanisms, e.g. Fe CuGa which ex-hibits Fe-Cu intermixing [4]). However such disorderat high rates would lead to the statistical emergence ofthe inversion symmetry, which contradicts to the presentXRD data (Fig. S7), convincingly deducing the noncen-trosymmetric group I -4 m I /mmm (No. 139).Experimentally the compositional stability of the com-pound was investigated by considering Mn- and Rh-excess regimes. The Mn-excess series (Mn − x Rh x Sn,with the composition step of 0.1) always exhibit the phaseseparation in a form of a growing amount of Mn-richMn Sn phase (group No. 194, hexagonal) and the “host”phase of Mn RhSn. There is no direct transition be-tween these crystal structures for the symmetry reasons:right at Mn . Rh . Sn composition the phase separation
30 40 50 60 70 80 90 I exp I calc I-4m2 I4/mmm I no r m [ a . u . ]
2 [ o ] FIG. S7. XRD powder pattern (red) of Mn RhSn. The mea-sured pattern has been refined using the noncentrosymmetrictetragonal Heusler structure I -4 m I -4 m I /mmm (139) structuresare showed below (blue and gray, respectively). sets in, as observed with the help of XRD, EDX, opticaland electron microscopy. Introduction of additional Rhin Mn RhSn system reduces the c/a ratio and graduallybrings the structure to the cubic phase. The smallestRh content, which is enough to form a cubic structure isMn Rh . Sn. Thus, the Mn RhSn phase is rather sensi-tive to a slight stoichiometric deviation of Rh or Mn. Forthe working composition (i.e., Mn RhSn) the Rietveldrefinement of the 2:1:1 sample shows the R -Bragg factorof 3.998. If present, the disorder between Mn and Rhatoms would contribute additional intensities to (002),(110), (202) and (310) peaks whereas the (101), (103),(211), (301), (321) and (215) would be suppressed (seeFig. S7) V. COMPUTATIONAL DETAILS
To justify the proposed magnetic order, first-principles calculations using the spin-polarized relativis-tic Korringa–Kohn–Rostoker (SPR-KKR) Green’s func-tion method [5] within a local density approximation [6]were performed for several Mn Y Z
Heusler materials:the recently synthesized Mn PtIn [3], Mn RhSn [2], andMn IrSn, which is reported for the first time in thepresent work. To determine the magnetic ground states,we started with the experimental lattice parameters andallowed for the self-consistent determination of local mo-ments including their amplitudes, directions, and period-icity.From the total energies obtained as a function of θ (Fig. S8), not only both the Sn-containing compoundsbut also the In-containing compound exhibit noncollinearmagnetic order characterized by canting of the Mn II lo-cal moment direction: θ , = 180 ◦ ± ◦ , 180 ◦ ± ◦ , and θ [ o ] E t o t a l [ e V / f . u . ] Mn RhSn: 180°±55° Mn PtIn: 180°±50° Mn IrSn: 180°±44°
FIG. S8. Total energy per formula unit computed as a func-tion of the orientation of the Mn II magnetic moment charac-terized by angle θ . The energy minima (indicated by arrows)occur at θ , = 180 ◦ ± ◦ , 180 ◦ ± ◦ , and 180 ◦ ± ◦ in thecase of Mn RhSn, Mn PtIn, and Mn IrSn, indicated respec-tively by red, green, and blue. ◦ ± ◦ for Mn RhSn, Mn PtIn and Mn IrSn, respec-tively. Upon closer examination of Fig. S8, the largestenergy scale ( ∼ RhSn,Mn PtIn, and Mn IrSn, respectively) is revealed to bethe difference between two collinear configurations: theFM (ferromagnetic, θ = 0 ◦ ) and FiM (ferrimagnetic, θ = 180 ◦ ) configuration. Thus, the small energy (0.06,0.04, and 0.09 eV for Mn RhSn, Mn PtIn, and Mn IrSn,respectively) gained by canting can be considered as aperturbation of the collinear ferrimagnetic state, whichis typical for most Mn -based Heusler systems.To ensure that the canted magnetic state that is ob-tained is due to the proposed mechanism, we computedthe exchange coupling constants (using the approach inRef. [7]) for the model Eq. (1, 3) and calculated the cant-ing directly from minimizing the Heisenberg Hamilto-nian (1). For example, the values obtained for Mn RhSn( J = − .
46 and j = − .
05 meV) satisfy criterion (2): j/J = 0 . > / . This then leads to θ , = 180 ◦ ± . ◦ ,which is in reasonable agreement with the ab initio calcu-lated value in Tab. I (in the main part). The same holdsalso for the In-based compound Mn PtIn: J = − . j = − .
67 meV gives j/J = 0 . > / , leading to θ , = 180 ◦ ± . ◦ .In order to analyze the possible long-range magneticorders, and in particular, the possibility of skyrmionsin Mn RhSn, we computed the absolute values of theDzyaloshinskii-Moriya (DM) vectors for Mn RhSn, byfollowing the scheme intoduced in Ref. [8]. In additionwe computed the magnetocrystalline anisotropy as theenergy difference between orientations of the total mag-netization along the c -axis and within ab -plane. Due to
90 120 150 180 210 240 θ [ o ] E t o t a l [ m e V / f . u . ]
120 125 1301234
FIG. S9. Total energy per Mn RhSn formula unit computedas a function of the orientation of the Mn II magnetic mo-ment characterized by angle θ . We compare three magneticorientations: black - total magnetization is along the c -axis;blue and red - total magnetization within ab -plane, but withMn II -moments staggering within ab and within ac -planes, re-spectively. Inset shows the detailed energy trends near to thecanting minimum at about 125 ◦ . canting, for the second case we distinguish two orienta-tions - first, when Mn II moments stagger within ab -planeand second - within ac -plane (see Fig. S9). As it followsfrom the inset, the energy minima for the ab -orientationsare shifted by few degrees. Their absolute values areabout 2 . ab -plane) and 1 . ac -plane). VI. CONTINUUM MODEL OF MAGNETICORDER IN Mn RhSn
The phenomenological continuum theory for the mag-netism of Mn RhSn can be written in terms of the foursublattices ( l = 1 , , ,
4) consisting of the two sublat-tices Mn I on Wyckoff site 2 a and Mn II on site 2 c . Weuse standard methods to derive a quantitative model inthe shape of this phenomenological theory by a system-atic coarse graining of a microscopic model, where di-rect and antisymmetric DM exchange are calculated withDFT methods (discussed in previous section). Eventu-ally, by adding magnetic anisotropies, also from DFTcalculations and Zeeman energy, a micromagnetic low-temperature continuum model can be constructed. Forthe thermal phase diagram, empirical input is needed towrite a Landau-Ginzburg functional for coupled magneticmodes. As the low acentric symmetry of Mn RhSn allowsfor the presence of chiral inhomogeneous DM couplings,the resulting model has the form of a Dzyaloshinskii-model [9] marked by the presence of Lifshitz-type invari-ants that couple different magnetic order parameters.The magnetic moments S l in each unit cell of the lat-tice R n , n = ( i, j, k ) of each sublattice are expressed bya continuous functions m l ( r ) with the property m l ( R n + b l ) = S l ( R n ) , (1)where b l are the base vectors of the sublattice sites.The magnetic free energy is expressed by a standardgradient expansion up to square terms, w = X l,m X αβ A αβlm ∂ α m l · ∂ β m m + X γ X l,m X αβ D ( γ ) αβlm m αl ¯ ∂ γ m βm + w ( { m l } ) . (2)The first two lines describe the inhomogeneous exchangeby a set of (anisotropic) constants A αβlm , where αβ arelabels of Cartesian coordinates, and the second line givesthe inhomogeneous DM couplings D ( γ ) αβlm that arise inlow symmetry crystals. The Lifshitz-type invariants arewritten in short form, a ¯ ∂b ≡ a∂b − b∂a . Finally w col-lects the terms which are homogeneous in the set of func-tions { m l } . Within our ansatz this contribution canbe written as w = w + w a + w h . . . , where w collectscontributions deriving from isotropic exchange only, and w a , w h contains anisotropic and Zeeman terms, includingthe demagnetization energy.This coarse grained continuum theory for the ground-state can be derived from the microscopic classical Hamil-tonian of the lattice and symmetry constraints. Usingthe results of the ab initio calculations the magneticenergy can be expressed by a model including direct(isotropic) Heisenberg-like exchange couplings and theDM-couplings: H = − X n p X lm J lm ( R p − R n ) S l ( R n ) · S m ( R p )+ X n p X lm D lm ( R p − R n ) · ( S l ( R n ) × S m ( R p )) . (3)Expanding the continuous functions for the sublatticesinto a Taylor series, m l ( r − r ) = m l ( r ) + X ν ν ! [( r − r ) · ∇ ] ν m l ( r ) , (4)and using Eq.(1) in this Heisenberg-DM-Hamiltonian,the continuum theory can be derived. For Mn RhSn withthe tetragonal lattice described by space group I ¯4 m m D d ), the effective model simplifiesto w = X lm X x = a,b,c A xlm X α ( ∂ x m αl ∂ x m αm )+ X lm [ D alm m zl ¯ ∂ a m bm + D blm m zl ¯ ∂ b m am ]+ X lm J lm m l · m m , (5)where the surface terms and constants have been omitted.The gradients ∂ a , ∂ b , ∂ d are written along and in units ofthe tetragonal lattice cell. The coefficients J lm (Tab. S3), l \ m J lm [meV/ µ ]. A a A b A c l \ m A a,b,clm [meV/( µ a )]. D a D b l \ m D a,blm [meV/( µ a )]. A xlm (Tab. S4), D xlm (Tab. S5), now describe effectivecoarse grained exchange and DM-couplings.It must be noted that there is no weak ferromag-netism or weak antiferromagnetism in the tetragonal in-verse Heusler structure, i.e., there are no bilinear cou-pling terms between components of the staggered andferromagnetic vectors derived from the DM-exchange, be-cause the four sublattices are related by non-primitivetranslations. After quantification of the exchange cou-plings terms, it is convenient to analyse the 4-sublatticesystem by using staggered and ferromagnetic vectors oftwo sublattices, m Mn I · L = ( m − m )2 ; m Mn I · F = ( m + m )2 ; m Mn II · l = ( m − m )2 ; m Mn II · f = ( m + m )2 , (6)where the spin vectors L and F are related to Mn I , and l and f – to Mn II sublattice. In the ground-state config-urations, these vectors fulfill L + F = 1; L · F = 0 l + f = 1; l · f = 0 . (7)Dropping irrelevant terms, the homogeneous part of thecontinuum theory now is expressed as˜ w = J F F · F + J L L · L + J f f · f + J l l · l + J c F · f + J ′ F · l − m Mn I · F + m Mn II · f ) · H , (8)with coefficients in [meV]: J F J L J f J l J c J F f J F l J Lf J ′ J Ll -285.4 -83.1 111.3 -920.0 898.2 898.2 0 0 4.8 4.8 .Here we use an obvious notation for the effective ex-change, internal to the ordering modes and between themodes, and using the spin moments from the DFT calcu-lations m Mn I(II) = 3 .
51 (3 . µ B (see Sec. V). The field H in (8) is the internal magnetic field. It is seen that theexchange couplings have a clear hierarchy, showing thatmagnetic ordering is either dominated by the FM orderon Mn I sublattice or by staggered AFM order on Mn II .This AFM order is only very weakly coupled via the stag-gered vectors L , which however is not the dominatingmagnetic mode on Mn I sublattice. Hence, Mn RhSn isclose to a tetracritical (or bicritical) point, where thesetwo magnetic modes would coexist with the paramag-netic state. The FM mode f is a secondary magneticorder for sublattice Mn II , which is antiparallel to F viathe very strong coupling J c . The superposition of F , l and f determines a canted state for the magnetic orderwith the magnetic cell equivalent to the crystallographicunit cell, i.e. a Γ-point mode. Ground state.
The homogeneous ground state can befound by neglecting in (8) the small coupling J ′ , and writ-ing a coplanar canted state arbitrarily in the ac -plane,using F ≡ F = (0 , , l ≡ l = (sin[ π − ϑ ] , , f ≡ f = (0 , , cos[ π − ϑ ]). Its energy is given by w = J F + J l + ( J f − J l ) ξ + J c ξ ; ξ = cos[ π − ϑ ] . (9)The solution for the canting angle is ϑ = π − arccos J c J f − J l ) . (10)Using the parameters above, the canting angle is ϑ = 64 . ◦ , which is in reasonable agreement with the ab initio calculations finding θ , = 55 ◦ , considering thatthe exchange approximation in Eq. (8) neglects theanisotropy which has an easy-axis character for thecanted sublattice Mn II (see Fig. S9). The ground state,thus, is composed of the two modes F and l , which are al-most decoupled for J ′ ≃
0, and by the induced FM mode f on Mn II . However, the conditions (7) provide a non-linear coupling between these different modes in a propermicromagnetic model. Inhomogeneous states.
The micromagnetic model re-quires now to include the exchange terms and the in-homogeneous DM-couplings, i.e., the gradient energy isgiven by w = w J + w D , (11) where we have the squared gradient terms derived fromthe isotropic exchange in the form w J = A ⊥ F [ ∂ a F · ∂ a F + ∂ b F · ∂ b F ] + A || F ∂ c F · ∂ c F + A ⊥ l [ ∂ a l · ∂ a l + ∂ b l · ∂ b l ] + A || l ∂ c l · ∂ c l + A ⊥ f [ ∂ a f · ∂ a f + ∂ b f · ∂ b f ] + A || f ∂ c f · ∂ c f + A ⊥ F f [ ∂ a F · ∂ a f + ∂ b F · ∂ b f ] + A || F f ∂ c F · ∂ c f (12)with coefficients in [meV/ a ] units: A ⊥ F A || F A ⊥ f A || f A ⊥ l A || l A ⊥ F f A || F f A ⊥ F l A || F l ≃ ≃ w D = D F [ F c ¯ ∂ b F a + F c ¯ ∂ a F b ]+ D l [ l c ¯ ∂ b l a + l c ¯ ∂ a l b ]+ D f [ f c ¯ ∂ b f a + l c ¯ ∂ a f b ]+ D c [ F c ¯ ∂ b f a + F c ¯ ∂ a f b + f c ¯ ∂ b F a + f c ¯ ∂ a F b ] , (13)with coefficients in [meV/ a ] units: D F D l D f D c -9.8 5.1 7.0 -9.8 .An inhomogeneous modification of the ground-statetakes place on long lengths, owing to the weakness of theDM-exchange compared to the direct exchange. Consid-ering that the DM-couplings and also applied magneticfields and anisotropies are small in comparison to thestrong local exchange forces, the basic canted structureis preserved in each unit cell. But it can be slowly rotatedover length of many unit cells. Landau-Ginzburg functional.
In order to complete thephenomenological theory, we briefly discuss the form ofan appropriate Landau-Ginzburg functional which couldbe used to model the thermal phase diagram and thephase transitions. The two primary order parameters(OPs) are the FiM and the AFM modes, F and l . Theyand the coupling between them have to be consideredwith respect to the secondary OP, which is the FMmode f . The complete Landau-Ginzburg (LG) func-tional contains Lifshitz invariants w D from Eqs. (13).Hence, this LG-functional is not a simple extension of aproper Landau-theory with the set of applicable squared-gradient terms w J from (12), as the magnetic free en-ergy violates the Lifshitz condition. Consequently, this Dzyaloshinskii model should be understood as a pseudo-microscopic continuum theory. Still, the LG functionalfor Mn RhSn can be written by using standard Landauexpansion for the homogeneous coupling terms, insteadof ˜ w : w = a F F · F + b F ( F · F ) + a l l · l + b f ( l · l ) + a f f · f + c f F · f + c ′ F · l + b F f | F | | f | + b F l | F | | l | + b fl | f | | l | + b c ( F · f ) + b ′ ( F · l ) + h.o.t. − F + f ) · H , (14)where all magnetizations F , f , and l are now consideredas variable length 3-component vectors. The completeLG functional for the magnetic free energy then is givenby w LG = w + w J + w D + w a , (15)where the w a collects anisotropic contributions, not con-tained in the first three terms, i.e., the magnetocrys-talline and exchange anisotropies.A complete microscopic derivation of the terms in theLG-functional would require a detailed finite tempera-ture statistical theory beyond the input from the DFT-calculations for ground-states. And, for the behavior atlower temperatures, the higher-order-terms (h.o.t.’s) arerequired in the thermodynamic potential. However, asemi-quantitative model could be written in the usualmanner by restricting the temperature dependence of themodel to the coefficients of the square terms in the pri-mary OPs: a F ( T ) = α F ( T − T C ) a l ( T ) = α l ( T − T N ) (16)In Mn RhSn the bare FiM Curie-temperature T C ∼
280 K < T C , and the T N ∼
80 K, i.e. closeto the onset of the AFM mode on Mn II -sublattice.These bare or ideal transition temperatures should notdeviate strongly from the observed magnetic transitiontemperatures, because the corrections due to DM-exchange [10] and anisotropy are expected to remainsmall. Magnitude of the remaining coefficients a f andof the quartic terms with coefficients b ν could roughlybe fixed to the empirical ordered moments. Here, weonly note that the suppression of the FM mode F onMn I -sublattice below the onset of the AFM signals,a bicritical behavior with a repulsive interaction, i.e. b F l >
0, while b fl may be small. This suggests that thethermal magnetic phase diagram of Mn RhSn is closeto a bicritical behavior, where the FiM and AFM modesrather compete and inhomogenous textures can occur.
Skyrmions in the ferrimagnetic state . The qualitativediscussion of the LG-functional is sufficient to understandthe basic features of inhomogeneous state in the interme-diate temperature range T N < T < T C where only theFiM collinear state exists. In that case, the FM modeon the Mn II -sublattice is antiparallel to FM mode on the Mn I -sublattice, f = − F . Inserting this into w J + w D inEqs. (12)-(13) and adding Zeeman term an magnetocrys-talline uniaxial anisotropy yields a functional describinginhomogeneous FiM states in the ab -plane :˜ w FiM = ˜ A [ ∂ a F · ∂ a F + ∂ b F · ∂ b F ]+ ˜ D [ F c ¯ ∂ b F a + F c ¯ ∂ a F b ] − m Mn I − m Mn II ) F · H − K ( F c ) . (17)This free-energy functional for the FiM state is exactlyequivalent to the FM Dzyaloshinskii-model for chiralmagnets from crystal class 42 m studied earlier [11, 12]. Inparticular, the solutions for isolated and condensed chiral“vortices” and the magnetic phase diagram presented inthis pioneering work desribe what is now known as chiralskyrmions in the FiM state of acentric Mn Y Z inversetetragonal Heusler alloys.The parameters of this model for Mn RhSncan be calculated from the microscopic inputas ˜ A = A F + A f − A F f = 16 . /a ] and˜ D = D F + D f − D c = 7 . /a ]. The DFT cal-culations of anisotropy (see Sec. V) suggest an effectiveeasy-axis anisotropy of the order K ≃ RhSn at elevated temperatures and evaluatethe sizes or stability of skyrmions and the phase diagramin Mn RhSn in the FiM state. The chiral magnetictwisting lengths is given byΛ = 4 π ˜ A/ ˜ D, (18)which means Λ ∼ a ∼
130 nm for Mn RhSn. Thestrengths of the easy-axis anisotropy determines whethera modulated spiral ground-state exists and whether afield-induced skyrmion lattice in an effective field point-ing along the c -axis occurs (see Figs. 9 and 10 in Ref. 12).The different cases can be distinguished by the parameter κ = π ˜ D/ p ˜ A K . (19)For κ <
1, the ground state is collinear as the stronganisotropy suppresses the spiral state. For κ > . c .For the estimated coefficients, we find κ ≃ .
95 as areasonable value for Mn RhSn but close to the criti-cal κ . The spiral magnetic states in crystals with 42 m crystals can have demagnetizing fields, as they are ofcycloidal (N´eel)-like character when propagating along(110)-directions, while they are of helical (Bloch)-like FIG. S10. (a) Shape of the double-twisted skyrmion configu-ration in the tetragonal inverse Heusler alloys of 42 m symme-try. The FM magnetization F on sublattice Mn I parametrizesthe FiM collinear state at higher temperatures in Mn RhSn.The corresponding magnetization f on sublattice Mn II isstrictly antiparallel to F . (b) Projection of the skyrmion inthe ab -plane. (c) Close to the reorientation transition, theAFM mode l on Mn II -sublattice sets in: | l | ≪ | F | and also | l | ≪ | f | . l is perpendicular to F and rotates with it in thesame plane in each radial direction. In the center | l | = 0.(d) Projection of l onto ab -plane. character for propagation along (100)-directions. Thedemagnetizing field further reduces the effective κ andoblique/skew spirals for propagation directions in be-tween, as discussed in Ref. 12. Hence, the quantitativeestimates for the micromagnetic model ˜ w FiM suggest acollinear FiM state in Mn RhSn.The solutions for isolated chiral skyrmions with a singleFM ordering mode have been presented in Ref. 12 for thebasic model Eqs. (17). The shape of an isolated skyrmionin Mn RhSn is sketched in Fig. S10. Close to the reori-entation transition, where the AFM ordering sets in, themagnitude of the l -mode is small and can be modulated( | l | 6 =const). As long this mode is subjugated to the FiMorder it remains perpendicular to the F -mode. Owing toits softness, it will not only rotate in a manner, so as tominimize the energy of its Lifshitz-invariants Eq. (13), italso will be modulated with a zero, l = 0, in the centerin form of a vortex-like defect. In Mn RhSn, the coef-ficients ˜ D and D l have similar magnitude and the samesign, so that the screw-sense of the rotation of F and l are not in conflict. Hence, close to T N the l -mode followsand co-rotates the FiM mode F while being modulatdin lengths, Fig. S10 (c, d). At lower temperatures, thenecessity to have a defect of the l -mode in the skyrmioncenter and the associated large defect-energy most likelywill destabilize the skyrmions. However, there may existother localized solitonic textures in this acentric coupledmagnetic system, which may cause inhomogeneous mag-netic states to exist in the acentric Mn Y Z alloys. [1] J. Rodr´ıguez-Carvajal,
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