Reciprocal Space Tomography of 3D Skyrmion Lattice Order in a Chiral Magnet

Reciprocal space tomography of 3D skyrmion lattice order in a chiral magnet

Shilei Zhanga, Gerrit van der Laanb, Jan Muller¨ c, Lukas Heinenc, Markus Garstd, Andreas Bauere, Helmuth Bergerf, Christian Pﬂeiderere, and Thorsten Hesjedala,1

aClarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom; bMagnetic Spectroscopy Group, Diamond Light Source, Didcot, OX11 0DE, United Kingdom; cInstitut fur¨ Theoretische Physik, Universitat¨ zu Koln,¨ 50937 Koln,¨ Germany; dInstitut fur¨ Theoretische Physik, TU Dresden, 01062 Dresden, Germany; ePhysik Department, Technische Universitat¨ Munchen,¨ 85748 Garching, Germany; and fCrystal Growth Facility, Ecole Polytechnique Fed´ erale´ de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

Edited by E. W. Plummer, Louisiana State University, Baton Rouge, LA, and approved May 9, 2018 (received for review February 23, 2018)

It is commonly assumed that surfaces modify the properties of magnetization profile is required, making the interpretation of stable materials within the top few atomic layers of a bulk spec- the experimental data rather demanding. imen only. Exploiting the polarization dependence of resonant To clarify the effects of surfaces on the properties of a bulk elastic X-ray scattering to go beyond conventional diffraction and material deep below a surface, skyrmions in chiral magnets are imaging techniques, we have determined the depth dependence ideally suited from an experimental point of view, as the spins of the full 3D spin structure of skyrmions—that is, topologi- exhibit changes of orientation in all spatial directions. X-rays are cally nontrivial whirls of the magnetization—below the surface perfectly matched to the length scales of the magnetic periodici- of a bulk sample of Cu2OSeO3. We found that the skyrmions ties, suggesting that studies of the spin structure with increasing change exponentially from pure Neel-´ to pure Bloch-twisting depth as controlled by the scattering angle or the photon energy over a distance of several hundred nanometers between the sur- should be possible (12). In addition, representing chiral spin face and the bulk, respectively. Though qualitatively consistent textures, the dichroism of soft X-ray scattering permits recon- with theory, the strength of the Neel-twisting´ at the surface struction of the full 3D spin order, providing information on and the length scale of the variation observed experimentally gradual variations (13, 14). Last but not least, high-quality sin- exceed material-specific modeling substantially. In view of the gle crystals are available as a key precondition for meaningful exceptionally complete quantitative theoretical account of the experimental results. magnetic rigidities and associated static and dynamic properties On a different note, skyrmion materials are also ideally suited of skyrmions in Cu2OSeO3 and related materials, we conclude for the investigation of the effects of surfaces on the bulk from that subtle changes of the materials properties must exist at dis- a theoretical point of view, as a remarkably complete material- tances up to several hundred atomic layers into the bulk, which specific quantitative account has been developed (15–18). At originate in the presence of the surface. This has far-reaching the heart of the formation of skyrmions in bulk chiral mag- implications for the creation of skyrmions in surface-dominated nets is a hierarchy of energy scales comprising, in decreasing systems and identifies, more generally, surface-induced gradual strength, exchange interactions with coupling J , Dzyaloshinskii– variations deep within a bulk material and their impact on tailored Moriya (DM) spin–orbit interaction with coupling D, and higher functionalities as an unchartered scientific territory. order crystal field terms. All of these interactions are controlled by the atomic positions and symmetries of the underlying crystal structure. While the ratio of the exchange coupling and the skyrmions | resonant elastic X-ray scattering | 3D magnetic imaging | magnetic surface effects | spintronics Significance

n extraordinary variety of experimental and theoretical tools Magnetic skyrmion lattices have exceptional properties, mak- Ahas made the exploration and technological exploitation of ing them ideally suited for a new generation of spintronics surfaces of bulk materials one of the most thriving topics in the devices. As devices are surface-dominated, a key question con- natural sciences (1, 2). As the underlying interactions and asso- cerns the influence of surfaces on the spin structure. While a ciated effects of charge screening occur on atomic distances, it is broadly assumed that surfaces do not modify the bulk prop- wide range of methods exist for studies of 2D skyrmions, the erties of inherently stable materials beyond the top few atomic detailed knowledge of 3D magnetic structures has remained layers. Moreover, only a few studies have addressed the evolution elusive. Here we use tomographic resonant elastic X-ray scat- of materials properties with atomic resolution far below sur- tering to reconstruct the depth dependence of a 3D skyrmion faces, since conventional methods do not provide the necessary crystal order in Cu2OSeO3. We find a continuous transforma- information (3–5). tion with increasing depth from complete Neel-type´ winding The local orientation and magnitude of the magnetization in at the surface to the Bloch-type winding in the bulk. Our study spin textures is particularly amenable to track gradual variations suggests the importance of free surfaces has been severely of the interplay of different energy scales over large distances, underestimated and will be of central importance for future reflecting sensitively even the weakest interactions. Such spin skyrmionics applications. textures have been of great fundamental and technological interest for many decades, ranging from field-theoretical questions to Author contributions: S.Z., G.v.d.L., and T.H.designed research; S.Z., G.v.d.L., J.M., L.H., A.B., spintronics applications (6–8). For instance, great efforts have and T.H. performed research; H.B. grew the sample; S.Z. developed the analytical tools; been dedicated to measurements of magnetic domains in bulk S.Z., G.v.d.L., M.G., A.B., C.P.,and T.H. analyzed data; and S.Z., C.P.,and T.H. wrote the paper. materials using polarized neutron imaging (9, 10), but the spatial The authors declare no conflict of interest. resolution of a few micrometers achieved to date exceeds largely This article is a PNAS Direct Submission. the atomic scales of interest. Much higher resolution has recently Published under the PNAS license. been reported using hard X-ray scattering in transmission geom- 1 To whom correspondence should be addressed. Email: [email protected]. etry (11). However, for the micrometer-sized crystals required ac.uk. in these studies, surfaces dominate the scattering volume such This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. that the link with genuine bulk properties cannot be addressed 1073/pnas.1803367115/-/DCSupplemental. unambiguously. Moreover, a sophisticated reconstruction of the Published online June 4, 2018.

6386–6391 | PNAS | June 19, 2018 | vol. 115 | no. 25 www.pnas.org/cgi/doi/10.1073/pnas.1803367115 Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 tdwt eaoa atc opsdo h pncngrtossonin shown configurations spin the (see angle of determined helicity composed the lattice to hexagonal corresponds a with N ated pure in between number shown varying winding patterns configuration with skyrmions spin for planar sphere space a of surface the on 1. Fig. or Bloch- whether determines interactions N DM length, the modulation of character material-specific a at tion interaction, DM hn tal. et Zhang N from magnetization. change the of illustration Schematic MnSi as such magnets chiral cubic Co in Fe stabilize (19), observed former skyrmions the of values types the abundant N featuring and two materials Bloch- the magnetic for in experimentally 1 Fig. in Methods and number als winding angle topological helicity the by characterized aeil ntowy.Frt tm ttesraeeprec a experience surface the at atoms First, ways. two in materials environment. bulk perturbations the in how propagate of state comparisons magnetic In detailed the (30–35). of for reported allows been spin has this collective rigidities, turn, these the materi- reflects of different which for als, account phases magnetic universal different across a excitations prop- particular, dynamic and In static erties. associated under- and the of rigidities terms magnetic in lying exists materials interfacial hosting skyrmion in different as those well to as GaV akin (25–28) interactions (29). as systems systems DM such film and systems anisotropies thin bulk of in interfaces and faces A4 A3 A2 A1 ´ e-wsigi realized. is eel-twisting norsuy eepotdta antcsymosare skyrmions magnetic that exploited we study, our In ufcsrpeetsc etraino krinhosting skyrmion of perturbation a such represent Surfaces of understanding complete exceptionally an together, Taken x Zn lutaino krinodrrnigfo N from ranging order skyrmion of Illustration y Mn 1−x z χ Co = ´ e-yesymos h eiiyangle helicity the skyrmions, eel-type IAppendix SI 2) h atrhv enosre tsur- at observed been have latter the (24), x χ ±90 i(0,FG 2) Cu (21), FeGe (20), Si B1–B4 J o h ahmtcldfiiin.A shown As definition). mathematical the for seFg o nilsrto and illustration an for 1 Fig. (see B3 B2 B1 B4 z /D ◦ y -1 eeae hne ftesi orienta- spin the of changes generates , ihtehdeo ofiuain hw in shown configurations hedgehog the with and , .(C1) Materials ). Supplementary x χ m χ zS 0 = 0 / M eoe ntergthn ieo h aes osqety h hrlt ftesymoscnb straightforwardly be can skyrmions the of chirality the Consequently, panels. the of side right-hand the on denoted 4 S 8 or xiiigsrn uniaxial strong exhibiting , 1 e-t lc-wsigwt nraigdphblwtesrae h oo oigrflcsthe reflects coding color The surface. the below depth increasing with Bloch-twisting to eel- ´ 180 C1 C4 C3 C2 -1.0 q 2 ◦ y OSeO epciey While respectively. , -0.5 eghgsi configuration spin Hedgehog (A1–A4) surface. a below depth increasing with Bloch-twisting to eel- ´ J 0.0 3 /D N .Aseegahcpoeto oncsteplanar the connects projection stereographic A ( B4). Bloch-twisting pure and (B1) eel-twisting ´ q 2,2) and 23), (22, 0.5 x 1 = ≈ χ N χ = λ/2π N 1.0 notably , 180 assumes = n the and 10 Materi- ayn ewe ueN pure between varying 1 10 ◦ the , n (C4) and 180° 127° 155° acltddcri ESdfrcinptenassoci- pattern diffraction REXS dichroic Calculated (C1–C4) respectively. A1–A4, 90° D oee,frteti ael tde,teitrlyo h sur- the abundance an of Moreover, interplay unraveled. be the cannot (42–44). bulk studied, the magnets lamella puta- with face thin chiral electron as the cubic transmission for (37), selected However, Lorentz in bobbers holographic (LTEM) chiral microscopy by (38– so-called observed prediction skyrmions the states, tively of inspired has metastable stability surfaces of the of energetics on same The surfaces 42). of importance distance the a at vanish to expected interactions. be wavelength may with ∼J for perturbation states turn, modulated any In helimagnetically bulk. λ, the of of case rigidities the magnetic associated interac- the and by tions determined be by essentially tilt will a orientation, notably pertur- original possible orientation, largest spin the the even of only, bation layers positions atomic atomic com- few the top the affects the surface to in the according that Hence, view accepted arrangement. monly atomic local the by spins adjacent heterostructures as well as 26) (25, 28). (18, films thin and theoretically in both proven experimentally studied been been surface, not has but latter the theoretically the of experimentally, predicted While been symmetry symmetry. has by inversion former allowed the be of may lack terms 37). interaction the (36, energetically additional bulk to assume the due with to compared Second, surface permit- as orientations the putatively favorable at thus more moments interactions, magnetic local ting of balance different χ h retto fteetnto iemre yawiearrow white a by marked line extinction the of orientation The B1–B4. = eettertcladeprmna tde aeconsidered have studies experimental and theoretical Recent of coupling the in originate scales all on interactions the As /D 90 z ◦ ≈ orsodt ueN pure to correspond y λ/2π x Real- (B1–B4) (A4). Bloch-twisting pure and (A1) eel-twisting ´ hrceitco h aac fteunderlying the of balance the of characteristic , s i and PNAS s j hspoaainms edetermined be must propagation this , | e-adpr lc-wsig epciey (D) respectively. Bloch-twisting, pure and eel- ´ ue1,2018 19, June | o.115 vol. 90 ◦ z | wyfo the from away ieto fthe of direction o 25 no. m -0.5 -1.0 zS / 0.5 0.0 1.0 | M 6387

PHYSICS of subtle structural and compositional defects introduced during A 150 the preparation of the samples cannot be ruled out. L To the best of our knowledge, the propagation of surface- 100 3 induced changes of the spin orientation deep into the bulk at L (nm) 2

distances up to several hundred nanometers has not been stud- a 50 ied experimentally before. In our study, we have addressed this d topic in terms of a quantitative comparison between the rigidi- 0 ties of the magnetic state, as inferred from the helimagnetic modulation length λ, with the propagation length of surface- B induced Néel-twisting, as determined by the helicity angle χ, 160 Experiment both recorded in the same measurement. As our main result, 150 Model fit we find that the Néel-twisting propagates an order of magnitude deeper into the bulk than expected theoretically; that is, the 140 heuristically expected decay length λ/2π and Néel decay length inferred from the helicity χ differ by an order of magnitude. This 130 (deg)

has far-reaching implications for the stabilization of skyrmions m

χ 120 in surface-dominated system as well as surface-induced material properties even deep inside bulk materials. 110

Results 100 Bloch skyrmion To overcome the limitations of conventional imaging and diffrac- 90 tion techniques in studies of the depth dependence of bulk materials at distances well below the surface, we performed resonant 925 930 935 940 945 950 955 elastic X-ray scattering (REXS). By changing the energy of the Energy (eV) incident X-rays, we varied systematically the attenuation depth χ = 155° χ = 127° 10 C m m x10 D to perform a quasi-tomographic mapping of the magnetic state. 0.02 5.0 This goes well beyond previous depth-dependent REXS studies, which pursued chemical modiﬁcations and the identiﬁcation of 2.5 the presence of magnetic phases only (12, 14, 45).

For our study, we chose single-crystal Cu2OSeO3, as high- .l.u.) 0 0.0 quality single crystals are readily available and the material is (r y

known to be chemically and structurally stable and well-behaved. q Typical experimental and calculated diffraction patterns are -2.5 931.25 eV 932.45 eV shown in Figs. 1 and 2. The hexagonally long-range ordered -0.02 skyrmion lattice phase gives rise to a six-fold diffraction pat- -5.0 10 tern, providing direct information on λ and thus the hierarchy E 0.02 x10 F of magnetic interactions (13, 46). Using the circular dichroism 1.0 (CD) from all six magnetic skyrmion lattice peaks—that is, the difference between the intensities obtained using left- and right- 0.5 circularly polarized incident X-rays—an extinction direction of 0

(r.l.u.) 0.0 this diffraction pattern may be observed at which the dichroism y vanishes (white arrows in Figs. 1 and 2). The orientation of the q extinction direction corresponds directly to the helicity angle χ. -0.5 As the magnetic properties in Cu2OSeO3 originate in the mag- -0.02 netic moment of the Cu2+ ions, we tuned the photon energy -0.02 0 0.02 -0.02 0 0.02 -1.0

across the Cu L2,3 edge under the (001) diffraction condition qx (r.l.u.) qx (r.l.u.) (47). Details of the experimental geometry have been reported elsewhere (47) and may also be found in SI Appendix, Sup- Fig. 2. Key aspects of polarized REXS for the determination of the depth plementary Materials. Further, the intensity patterns recorded dependence of skyrmion lattice order in Cu2OSeO3.(A) Energy dependence experimentally for a given incident photon energy represent a of the penetration depth of photons tuned to the Cu L2,3 edge under the weighted average over photons scattered at different depths. (001) diffraction condition in Cu2OSeO3, where da corresponds to the depth The attenuation profile of the intensity, taking into account the at which the scattering intensity has decreased to 1/e. (B) Helicity angle χ actual path length of the photons in the material under the given m as a function of photon energy taken directly from the measured REXS diffraction pattern. The red line corresponds to the depth dependence as scattering angle, was determined by means of radiative transi- inferred self-consistently from the experimental data (see text for details tion probability calculations at the Cu L2,3 edge and found to and Fig. 3). (C and D) Typical experimental REXS intensity pattern observed be essentially exponential (cf., ref. 48 and SI Appendix, Supple- for photon energies of 931.25 eV and 932.45 eV, respectively. All patterns mentary Materials). We carefully confirmed that parasitic signal recorded experimentally were resolution limited with a well-defined extinc- contributions due the effects of natural dichroism and birefrin- tion direction (white arrow). (E and F) Calculated REXS intensity pattern gence do not affect our results as discussed in SI Appendix, based on the intensity attenuation profile taking into account the depth Supplementary Materials. For what follows, we define a sampling dependence inferred from the experiment (cf. Fig. 3). depth normal to the surface, da , representing the point where the scattering intensity has dropped to 1/e, where da as a function of photon energy is shown in Fig. 2A. χm, as a function of energy determined directly from the mea- Data were recorded in the skyrmion phase under an applied sured diffraction patterns. The red curve in Fig. 2B reflects the magnetic field of 32 mT at a temperature of 57 K as a function of depth dependence as inferred self-consistently from the exper- photon energy. All REXS diffraction patterns displayed the same imental data when taking into account the depth-dependent basic orientation of the six-fold intensity pattern akin to the cal- intensity attenuation (see Fig. 3 for details). culated patterns shown in Fig. 1, C1–C4. In addition, all patterns Typical intensity patterns for 931.25 eV and 932.45 eV as asso- displayed a well-defined extinction direction, which was found to ciated with values of da of ∼31 nm and ∼77 nm featuring helicity change as a function of photon energy, characteristic of a helicity angles of 155◦ and 127◦ are shown in Fig. 2 C and D, respec- angle changing with depth. Shown in Fig. 2B is the helicity angle, tively. This suggests that the skyrmions are right-handed (see

6388 | www.pnas.org/cgi/doi/10.1073/pnas.1803367115 Zhang et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 hn tal. et Zhang neatos i o hnea ucino depth. of DM function and a exchange as the change of of not ratio function length did the interactions, modulation a by determined magnetic as purely the is broadening which that radial implied of energy lack photon The broadening, (22). azimuthal literature or radial to of corresponding indications without limited Appendix, SI 1. model for as same where the is faster, much N pure reached assumes is Bloch-twisting and N of strength value interaction determined the experimentally is For surface. length unconstrained decay an helicity observed tally (C (χ Bloch-twisting pure into of dependence depth the represents line pure red into depth The turns Attenuation and (A) bulk nanometers. the d hundred into several deep below penetrates Bloch-twisting surface the at twisting 3. Fig. eemndeprmnal ti orsod oterdcrein curve red the Cu in to angle N corresponds helicity (this the with experimentally of attenuation dependence determined Depth intensity (B) the depth. of increasing dependence energy the self-consistently C A B a e-wsiga h ufc smc ekr(χ weaker much is surface the at eel-twisting e-wsig(χ eel-twisting ´ ´ oprsnof Comparison )

essmaue eiiyangle helicity measured versus z z d (nm) (nm) a (nm) 140 120 100 100 500 400 300 200 100 0.1 80 60 40 20 10 0 0 0 et eedneo krinhlct nCu in helicity skyrmion of dependence Depth 1 90 e-wsiga h ufc ( surface the at eel-twisting .Aldt eeresolution- were data All Materials). Supplementary ´ exp 0 χ = exp λ 179.8 57.58 = ihmtra-pcfi acltos h experimen- The calculations. material-specific with ◦ exp ttesrae(z surface the at ) χ th,1 0 model 1 = 120 m necletareetwt the with agreement excellent in nm, =135° 90 χ m ◦ w ao ifrne r observed: are differences major two λ, ssoni i.2 Fig. in shown as eo eea ude nanometers. hundred several below ) χ (deg) L exp χ th,2 0 = L 0 L th,1 m oe considers 1 Model nm. 62.5 exp = th = 150 180 = )cagsexponentially changes 0) χ experimental exp χ 135 experimental χ L χ ◦ exp 0 th exp th,2 0 0 model 2 aigit account into taking , ,where ), A 2 J ◦ =179.8° = /D OSeO and =179.8° =180° m oe 2 Model nm. 5.4 ae rmthe from taken χ χ respectively. B, χ exp 0 3 ueN Pure . L exp m th = 180 = .Pure A). 179.8 2 nm 5.4 OSeO eel- ´ ◦ λ ), 3 , xetdt oto h ea egho etraino the of are perturbation properties a bulk of the length above. as explained decay as surprising, state the not magnetic control is 2 to and expected 1 models in length decay pure exhibit to surface than N the smaller forcing much when Further, were experiment. 1) the (model surface unconstrained the length for decay helicity a with at excellent lized in follow- dependence bulk experiment. depth the the with exponential in agreement qualitative Bloch-twisting an into essentially changed ing N surface the that the found we at models, both In calculations. Details in our 3C found interaction. Fig. be DM in may of Shown interfacial implementation symmetries numerical as the the to to of due referred terms surface, DM the additional a of assumed presence 2 the model whereas N surface, pure unconstrained an sidered unchanged depth. quantitatively increasing are with ratio interactions the the that both also that that suggested unlikely such extremely change seemed it interactions Since of depth. broadening of ratio function radial the a of that implies absence spots the diffraction the above, emphasized As 50). ture, aeilseicnmrclcluain ftehlct decay helicity the J of calculations length, numerical additionally material-specific field (40). magnetic expected be a recently to have length of decay surfaces presence the modifies of the effects that Moreover, the established observed. of calculations experimentally numerical than shorter scale length significantly a on occur to 9.1 expected heuristically experimentally, is ation observed length length decay 62.5 helicity The Discussion n h ifato atrsi i.2 Fig. in χ patterns diffraction the and energy- an assumed the We account intensity. variation photon exponential into the of taking attenuation self-consistently, dependent Fig. under- evolu- fitted the in the oversample were illustrated data dependence emphasize experimental depth qualitatively better the lying that as to note depth, rotated We 1D. increasing been with has tion plot the of depth sampling of 2 tion Fig. in with illustrated pattern as intensity found, an is and line experiment, pro- extinction the attenuation well-defined from exponential a of each an inferred by to as change contributed according file intensity monotonic weighted the is a where layer depth, Assuming increasing N depth. with between increasing lattice skyrmion with the of tion oooial noapr lc state, length Bloch decay pure a into monotonically (χ Bloch-twisting pure with uksml—hti,apr N pure a is, sample—that bulk parameters. surface, the at exp s ´ e-wsig(oe ) eas on h aesothelicity short same the found also we 2), (model eel-twisting oee,i oe ,tehlct nl ttesraestabi- surface the at angle helicity the 1, model in However, con- 1 Model scenarios. two on focused we calculations, our In ooti eprisgt,w aeteeoecridout carried therefore have we insights, deeper obtain To hw nFg 3A Fig. in Shown of change The efind we 3B, Fig. in shown As i m hti,i sepce oocro itneta is that distance a on occur to expected is it is, that nm; · (z F m em eetvl ls otemgei modulation magnetic the to close deceptively seems nm, s D . ~ j ) h stoi Mitrcini ui rsa struc- crystal cubic a in interaction DM isotropic the ; λ ´ e tt ttesrae h atrmyb utfidwith justified be may latter The surface. the at state eel ij ssoni i.3B. Fig. in shown as χ L 57.58 = th,1 0 · th (s aigit con h xhneinteractions, exchange the account into taking , i ≈ L L × χ exp th 135 s exp 0 ≈ j m oee,a mhszdaoe h relax- the above, emphasized as However, nm. 62.5 = n h fet fdplritrcin (49, interactions dipolar of effects the and ); sacmaio of comparison a is n h eiiydcylength, decay helicity the and , ◦ χ 5.4 PNAS m n h N the and stemaue eiiyangle helicity measured the is χ m bevto ftesm au of value same the of Observation nm. safnto feeg eel varia- a reveals energy of function a as exp d m ent htterdln nFg 2 Fig. in line red the that note We nm. a χ (z | exp c.Fg 2 Fig. (cf. L 90 = ue1,2018 19, June ( = ) th (z ´ e-wsigetne notebulk the into extended eel-twisting ´ e tt.Ti N This state. eel χ ≈ h esrdvalues measured The ). exp 0 ◦ χ 5.4 epi h uk h helicity The bulk. the in deep ) exp 0 179.8 = − m Thus, nm. A E χ 90 χ(z exp and and | ´ e-adBloch-twisting and eel- J ◦ J (z o.115 vol. /D exp(−z ) ◦ ) /D ) → F .Teorientation The B). ttesraeo the of surface the at ihkyrslsof results key with aeit account into take ´ suatrd this unaltered, is e tt changed state eel i o hneas change not did 90 χ L th 0 ◦ | exp nahelicity a on , χ swl as well as IAppendix . SI /L o 25 no. m r free-fit are , ´ eel-twisting safunc- a as exp λ/2π χ 90 + ) L | m exp (d 6389 L L a = ≈ E B χ th th ◦ )

PHYSICS It is important to note that the theoretical and experimental magnetization along the boundary of a specimen that potentially curves in Fig. 3C are roughly parallel; that is, the difference is stabilizes individual skyrmions, skyrmion clusters, and skyrmion over an order of magnitude, regardless of the precise helicity chains (39, 60–62). Further, it has been argued that surface angle reached. Thus, the results of our calculations underscore confinement in combination with finite system thickness affects the heuristic argument that the propagation of perturbations into the equilibrium regime of the skyrmion lattice phase uniformly, the bulk is determined by the balance of exchange and DM inter- causing twisted helicity angles and extra modulations along the actions and are experimentally expected to vanish at distances of thickness (36, 41, 63). Even nonaxisymmetric 3D skyrmion mod- order λ/2π. In fact, the difference between theory and experi- ulations (64–67), as well as metastable surface configurations ment is even over an order of magnitude taking into account the referred to as chiral bobbers, have been proposed (37, 42, 63). effect of the applied field (40). The long-range character of dipo- It is important to emphasize that none of these mechanisms lar interactions, having been included in our calculations, does explains our findings. On the contrary, our observations of a not change these conclusions. This raises the questions, first, of much stronger Néel-twisting induced by the surface of a bulk whether these effects are specific to the skyrmion lattice order crystal than anticipated theoretically imply that all of these and, second, in which way the rigidity of the magnetic order in effects may be much stronger quantitatively than anticipated. the bulk may be affected by the surface. In view that microscopic methods such as LTEM, magnetic Unfortunately, for the scattering and magnetic field geome- force microscopy, and scanning tunneling microscopy used so tries accessible with our experimental set-up, it was not possible far in studies of nanoscale systems hosting skyrmions provide to obtain comparable information on the depth dependence information on selected magnetization components only, quan- of any of the other magnetic phases. However, it is interest- titative tomography of the full 3D structure will be essential. ing to note that LTEM holography has recently been reported Here, REXS offers a new avenue, providing direct access to for FeGe and Fe0.95Co0.05Ge (41, 51). In these studies, Néel- depth-resolved information of the chirality and helicity even for twisting is observed on the surfaces (top and bottom) of a thin small scattering volumes. This represents an important method- lamella. Interestingly, the data reported in these papers entail ological step forward, as it permits us to study, for example, the same discrepancies we report here for bulk Cu2OSeO3 using a possible attractive nature of skyrmion tubes in thin platelets REXS. However, as pointed out above, the sample geometry where soft X-rays are able to probe the entire thickness of the required for LTEM is different, and changes of the crystal struc- specimen, permitting the full reconstruction of the skyrmion ture as well as defects due to the thinning process cannot be ruled characteristics. out. On a final note, our results may also have far-reaching impli- Further, the resolution-limited evidence that the modulation cations for the general understanding of structure versus func- length as a function of photon energy and associated probing tionality relationships beyond the magnetic textures investigated depth does not change renders local changes of the microscopic here. Namely, they indicate that surfaces at least for this broad interactions due to changes of the atomic positions highly unlikely. class of materials play a much more important role than hith- This includes surface-induced magnetic anisotropy, which may erto assumed. This motivates the experimental and theoretical not cause a change from Néel- to Bloch-twisting. These consid- investigation of the precise atomic structure and associated erations point at a dynamic origin of the softening of the mag- dynamical properties in bulk materials deep below their surface, netic rigidity as the most likely explanation—for example, due to representing unchartered territory. thermal fluctuations or magneto-elastic coupling. Concerning the effects of thermal fluctuations, we note that a Materials and Methods remarkably complete theoretical and experimental understand- Sample. Single crystalline Cu2OSeO3 was grown by chemical vapor trans- ing of spin waves in cubic chiral magnets has been developed, port and subsequently characterized by single crystal diffraction (using including Cu2OSeO3 (30, 31, 34, 35, 52–54). This includes the Cu Kα radiation) and electron backscattering diffraction to confirm the effects of critical fluctuations within a Brazovskii scenario of a crystal quality and single-chirality. A carefully polished (001) surface was fluctuation-induced first order transition and the stabilization used for the REXS experiments, which were carried out at the RASOR of the skyrmion phase by virtue of thermal Gaussian fluctua- diffractometer on beamline I10 at the Diamond Light Source, United tions (55, 56). However, when we explored the effects of finite Kingdom. temperatures in our calculations, we did not find any indications of substantial changes of the helicity angle at the surface Skyrmion Lattice Phase. In the skyrmion lattice phase [i.e., at 57 K and in an applied field of 32 mT along the (001)-direction], the six magnetic peaks as well as the helicity decay length. A different mechanism appear as satellites surrounding the (0,0,1) Bragg peak, with the modula- may be connected to the effects of magneto-elastic coupling. tion wavevector q = 0.0158 r.l.u. In the (hk1)-plane, the azimuthal angles Here Raman studies in the related compound MnSi (57) sug- Ψi describe the orientation of the wavevectors qi, where i = 1, 2, ... , 6 and gest unusual magnon–phonon coupling effects. Unfortunately, ◦ ◦ ◦ Ψi+1 = Ψi + 60 , with Ψ ∈ [−180 , 180 ). The coordinate system is illus- ◦ quantitative experimental and theoretical exploration of this pos- trated in Fig. 2, in which we define Ψ = 0 along the qx direction (see also sibility is well beyond the scope of our study and motivates SI Appendix, Fig. S2). further investigation planned for the future. In summary, determining the full 3D spin order in the CD-REXS Technique. For the purpose of our study, we define the CD sig- skyrmion lattice of Cu2OSeO3 by means of polarized REXS nal (the CD-REXS signal) as the difference in diffraction intensities for as a depth-dependent probe, we find pure Néel-twisting at the the same skyrmion peak at the same geometrical condition, as obtained surface that changes gradually into pure Bloch-twisting deep using left- and right-circularly polarized soft X-rays. In our experiment, within the bulk. Combining our results with material-specific the CD-REXS was recorded by integrating a series of reciprocal space- modeling, we are forced to conclude that surfaces in this class map scans for each polarization separately, followed by normalization of crystal structures affect the properties even at distances up and subsequent subtraction of the left- and right-circularly polarized to several hundred atomic layers. These effects are of great patterns (for details, see refs. 13, 47, and 68). The direct CD-REXS pat- importance when tailoring skyrmions in nano-scaled systems, as tern shows six magnetic peaks with varying CD amplitudes ICD(Ψ) and surfaces appear to favor skyrmions energetically much stronger a distinct boundary that separates the positive and negative halves. In than anticipated theoretically (38–41). the (hk1) plane, the extinction vector that characterizes the boundary Whereas early work considered the formation of hexagonal always passes through the center of the plane [i.e., the (001) recip- skyrmion lattices purely as the result of a minimization of the rocal space point], and its direction varies with photon energy. Its total energy (19, 58, 59), recent studies address the additional physical meaning is further discussed in SI Appendix, Supplementary effects of symmetry breaking of surfaces or strain-imposed mod- Materials. ifications of the interactions. For instance, an edge twist effect ACKNOWLEDGMENTS. We thank A. Rosch for stimulating discussions and was proposed and putatively observed, causing a canting of the S. Maier and the staff at the Diamond Light Source for support. The resonant

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