Stability of Néel-type skyrmion lattice against oblique magnetic fields in GaV 4 S 8 and GaV 4 Se 8
B. Gross, S. Philipp, K. Geirhos, A. Mehlin, S. Bordács V. Tsurkan, A. Leonov, I. Kézsmárki, M. Poggio
SStability of N´eel-type skyrmion lattice against oblique magnetic fields in GaV S andGaV Se B. Gross, S. Philipp, K. Geirhos, A. Mehlin, S. Bord´acs,
3, 4
V. Tsurkan,
2, 5
A. Leonov, I. K´ezsm´arki, and M. Poggio
1, 7 Department of Physics, University of Basel, 4056 Basel, Switzerland Experimental Physics V, Center for Electronic Correlations and Magnetism,University of Augsburg, 86159 Augsburg, Germany Department of Physics, Budapest University of Technology and Economics, 1111 Budapest, Hungary Hungarian Academy of Sciences, Premium Postdoctor Program, 1051 Budapest, Hungary Institute of Applied Physics, MD-2028 Chisinau, Republic of Moldova Department of Chemistry, Faculty of Science, Hiroshima UniversityKagamiyama, Higashi Hiroshima, Hiroshima 739-8526, Japan Swiss Nanoscience Institute, University of Basel, 4056 Basel, Switzerland
The orientation of N´eel-type skyrmions in the lacunar spinels GaV S and GaV Se is tied tothe polar axes of their underlying crystal structure through the Dzyaloshinskii-Moriya interaction.In these crystals, the skyrmion lattice phase exists for externally applied magnetic fields parallelto these axes and withstands oblique magnetic fields up to some critical angle. Here, we mapout the stability of the skyrmion lattice phase in both crystals as a function of field angle andmagnitude using dynamic cantilever magnetometry. The measured phase diagrams reproduce themajor features predicted by a recent theoretical model, including a reentrant cycloidal phase inGaV Se . Nonetheless, we observe a greater robustness of the skyrmion phase to oblique fields,suggesting possible refinements to the model. Besides identifying transitions between the cycloidal,skyrmion lattice, and ferromagnetic states in the bulk, we measure additional anomalies in GaV Se and assign them to magnetic states confined to polar structural domain walls. I. INTRODUCTION
The discovery of the nanometer-scale magnetiza-tion configurations known as magnetic skyrmions hasspurred renewed interest in non-centrosymmetric mag-nets. The lack of inversion symmetry in these crystalsgives rise to an asymmetric exchange coupling, knownas the Dzyaloshinskii-Moriya interaction (DMI), whichmanifests itself in the continuum approximation of themagnetic order parameter as Lifshitz invariants (LIs) in-volving first derivatives of the magnetization M with re-spect to the spatial coordinates: L ( k ) i,j = M i ∂M j ∂x k − M j ∂M i ∂x k . (1)Crystal symmetry determines the allowed LIs, i.e. a cer-tain combination of first order derivatives, which – incompetition with the spin stiffness – stabilize modulatedspin-textures such as spirals and skyrmions and deter-mine their internal structure . Both skyrmion lattices(SkLs) and isolated skyrmions have now been ob-served in either bulk or nanostructured noncentrosym-metric crystals. Their topologically protected spin-texture, which is stable even at room temperature , theirnanometer-scale size, and their easy manipulation viaelectric currents and fields make skyrmions a promis-ing platform for information storage and processing ap-plications .Until recently, most investigations in bulk crystals havefocused on Bloch-type skyrmions, in which the local mag-netization rotates perpendicular to the radial directionmoving from the skyrmion core to the far field. This type of skyrmion has been observed in chiral cubic heli-magnets with B20 structure such as MnSi , FeGe , orCu OSeO . Recently, N´eel-type skyrmions, in whichthe local magnetization rotates in a plane containing theradial direction, have been observed in bulk GaV S ,GaV Se , and GaMo S . These materials crys-tallize in the cubic lacunar spinel structure , whichbecomes polar below ∼
45 K and the point symmetry isreduced from T d to C v . Since the magnetic orderdevelops in the polar phase, these compounds are multi-ferroic. Furthermore, the skyrmions posses a non-trivialelectric polarization pattern due to the magnetoelectriceffect , which may enable nearly dissipation free manip-ulation of the magnetic order by electric fields .In addition to obvious differences in the spin texture ofBloch- and N´eel-type skyrmions, the phase diagrams ofcubic helimagnets and polar skyrmion hosts are markedlydifferent. In cubic helimagnets, the LI has an isotropicform w DMI = M · ( ∇ × M ). Therefore, the plane of theSkL aligns itself to be nearly perpendicular to the appliedmagnetic field, irrespective of the field’s direction. Theisotropic LI also results in a narrow stability range of forBloch-type skyrmions in the vicinity of the magnetic or-dering temperature, due to competition with the longitu-dinal conical phase . In contrast, C nv ( n ≥
3) symme-try only allows an axially symmetric LI. Therefore, in po-lar skyrmion hosts, modulated magnetic structures withwave vectors perpendicular to the high symmetry, polaraxis are favoured. In these compounds, the orientation ofN´eel skyrmions is locked to the polar axis rather than theapplied magnetic field. Thus, instead of tilting the planeof the SkL, oblique applied fields distort the configuration a r X i v : . [ c ond - m a t . s t r- e l ] J un of the N´eel skyrmions and displace their cores . Thisproperty has two consequences on the magnetic phasediagram of such materials: 1) the SkL phase is more ro-bust than in cubic helimagnets, because the conical phaseis suppressed, and 2) its stability range depends on thedirection of the applied field. In addition to the polarLI, the second order magnetic anisotropy allowed in thissymmetry can also modify the phase diagram. In thecase of GaV S , strong easy-axis anisotropy suppressesthe modulated phases at low temperature , whereas inGaV Se easy-plane anisotropy helps to stabilize the SkLphase down to the lowest temperatures .Here, we use dynamic cantilever magnetometry(DCM) to map the magnetic phase boundaries inGaV S and GaV Se as a function of the strength andorientation of magnetic field. We determine the cor-responding phase diagrams, which reproduce the ma-jor features predicted by a recent theoretical model .The measurements constitute a direct experimental con-firmation of the robustness of N´eel-type skyrmions tooblique magnetic fields in two materials with uniaxialmagnetic anisotropy of opposite signs. In addition tomagnetic transitions between the cycloidal, SkL, andfield-polarized ferromagnetic states, in GaV Se , we alsoobserve sharp anomalies in the torque, which we assign tofield-driven transformations of magnetic states confinedto polar domain walls (DWs). II. DYNAMIC CANTILEVERMAGNETOMETRY
In DCM, the sample under investigation is attached tothe end of a cantilever, which is driven into self-oscillationat its resonance frequency f . Changes in this resonancefrequency ∆ f = f − f are measured as a function of theuniform applied magnetic field H , where f is the reso-nance frequency at H = 0. ∆ f reveals the curvature ofthe magnetic energy E m with respect to rotations aboutthe cantilever oscillation axis :∆ f = f k l e (cid:32) ∂ E m ∂θ c (cid:12)(cid:12)(cid:12)(cid:12) θ c =0 (cid:33) , (2)where k is the cantilever’s spring constant, l e its effectivelength, and θ c its angle of oscillation. Measurements ofthis magnetic curvature are particularly useful for iden-tifying magnetic phase transitions , since – just as themagnetic susceptibility – it should be discontinuous forboth first and second order phase transitions .DCM measurements are carried out in a vibration-isolated closed-cycle cryostat. The pressure in the sam-ple chamber is less than 10 − mbar and the temperaturecan be stabilized between 4 and 300 K. Using an exter-nal rotatable superconducting magnet, magnetic fieldsup to 4 . ◦ in the plane of cantilever oscillation, as shown inFig. 1. ˆx in our coordinate system is defined by the cantilever’s long axis, while ˆy coincides with its axis ofoscillation. β is the angle between H and ˆx in the xz -plane. The cantilever’s motion is read out using a op-tical fiber interferometer using 100 nW of laser light at1550 nm . A piezoelectric actuator mechanically drivesthe cantilever at f with a constant oscillation amplitudeof a few tens of nanometers (corresponding to oscilla-tion angles of tens of microradians) using a feedbackloop implemented by a field-programmable gate array.This process enables the fast and accurate extraction of f from the cantilever deflection signal as well as pro-viding a measure of the dissipation Γ, which describedthe system’s rate of energy loss: dE/dt = − Γ l e ˙ θ c . Inorder to maintain a constant oscillation amplitude, thecantilever must be driven with a force F = Γ l e ˙ θ c , suchthat any losses due to dissipation are compensated. Thevoltage amplitude used to drive the piezoelectric actu-ator is therefore proportional to Γ = Γ + Γ m whereΓ is the cantilever’s intrinsic mechanical dissipation at H = 0 and Γ m represents magnetic losses. Given that Γ m reflects the sample’s magnetic relaxation, Γ should un-dergo abrupt changes at magnetic phase transitions. Wetherefore use both measurements of the magnetic cur-vature and dissipation, combined with knowledge fromother measurements , to map the low-temperaturemagnetic phase diagrams of GaV S and GaV Se as afunction of H . III. SAMPLES
Single crystals of GaV S and GaV Se are grownby a chemical transport reaction method using iodineas a transport agent . X-ray diffraction measurementsof both sample materials show impurity-free single-crystals . For the DCM measurement, we attach indi-vidual crystals of GaV S and GaV Se , which are a fewtens of micrometers in size, to the ends of commercial Sicantilevers (Nanosensors TM TL-cont) using non-magneticepoxy, as shown in Fig. 1. These cantilevers are 440 µ m-long, 50 µ m-wide, and 2 . µ m-thick. Unloaded, they haveresonance frequencies of about 16 kHz, quality factorsaround 5 × , and spring constants of 300 mN / m. Dueto the additional mass of the samples, the resonance fre-quency of a loaded cantilever shifts to around 3 kHz.Both samples are attached near the free end of thecantilever with the (001) surface pressed flat against theSi surface. The orientation of the GaV S and GaV Se samples differs and can be roughly estimated from opticaland scanning electron microscope (SEM) images. Theresultant direction of each sample’s crystalline axes withrespect to the cantilever is shown in Fig. 1: specificallythe approximate orientation of the four cubic (cid:104) (cid:105) axes ˆc i ( i = 1 , , ,
4) is shown in black, red, green, and blue.Both GaV S and GaV Se undergo a Jahn-Tellerstructural phase transition from a non-centrosymmetriccubic to a rhombohedral structure at 44 K and 42 K, re-spectively . The transition is characterized by Figure 1. Schematics of the measurement setup. (a) shows the coordinate system and the definition of β as the angle between H and ˆx . (b) and (c) show the cantilever, its oscillation angle θ c , and the crystalline axes of the measured sample. Black, red,green, and blue lines correspond to the four ˆc i . (d) shows the orientation of H with respect to an optical image of a sample andcantilever. (e) Composite optical and scanning electron micrographs of the measured GaV S and GaV Se samples mountedon their respective cantilevers. a stretching of the cubic unit cell along one of the fourcubic body diagonals ˆc i , resulting in four different struc-tural domains. The rhombohedral distortion also givesrise to polarization along ˆc i , making these the polar axesof the system. The multi-domain state is composed ofsub-micrometer-thick sheets of these four different rhom-bohedral polar domains, which we label P i . Thepolar axis ˆc i also corresponds to the axis of magneticanisotropy in the respective rhombohedral domain state.In GaV S , the uniaxial anisotropy is of easy-axis type,while in GaV Se it is of easy-plane type . In bothmaterials, measurements indicate the presence of modu-lated magnetic phases including a cycloidal (Cyc) state, aN´eel-type SkL, and a field polarized ferromagnetic (FM)phase . The population of multiple rhombohedral do-mains at low temperature complicates the determinationof the magnetic phase diagram, because for any given ori-entation of the applied field H , there can be up to fourdifferent angles, α i , between H and ˆc i . As a result, foran arbitrary orientation of H , a single phase transition can appear at up to four different values of H , dependingon the projections of H on each ˆc i . Although the appli-cation of a large electric field upon cooling through thestructural phase transition has been shown to polarizeGaV S and GaV Se samples such that only a singledomain is populated , it is practically challenging toapply such fields in a DCM apparatus. IV. MEASUREMENTSA. GaV S Fig. 2 shows DCM measurements of ∆ f ( H ) andΓ m ( H ) in GaV S for different temperatures T . Datashown in Fig. 2 (a) and (b) are collected with H alignedalong the cantilever’s long axis ( β = 0), i.e. approxi-mately H (cid:107) [100]. In this configuration, the angles α i between H and the four ˆc i are the same within theprecision of the sample orientation, i.e. within a few T (K) H ( m T ) (c) SkLFM Cyc PM
0 50 100 15005101520253035 H (mT) f ( H z ) (a)
15 K12 K7 K
0 50 100 1500246810 H (mT) ( a . u . ) (b)0 25 50 75 100 125-10-5051015 H (mT) f ( H z ) (a) 0 100 200 300 400-200-150-100-500 H (mT) f ( H z ) (b)
0 20 40 60 80 -2-10 H (mT) f ( H z ) (c) H Figure 2. Temperature and field dependence of magnetic phase transitions measured by DCM in GaV S . (a) DCM measure-ments of (a) ∆ f ( H ) and (b) Γ( H ) taken at T = 7, 12, and 15 K in cyan, maroon, and brown, respectively. Curves are shiftedfor better visibility. β = 0, i.e. approximately H (cid:107) [100]. Arrows indicate features corresponding to phase transitions. (c)Sketch of the expected magnetic phase diagram as a function of temperature and applied field for H (cid:107) [100] . Color-codeddashed lines and points correspond to temperatures and measured features in (a) and (b). degrees. Consequently, each magnetic phase transitionshould occur at a similar value of H for each domain.In this particularly simple case, we compare ∆ f ( H )and Γ( H ) at different temperatures to the correspondingmagnetic phase diagram measured by K´ezsm´arki et al. and shown schematically in Fig. 2 (c). Where metamag-netic transitions are expected, they manifest themselvesas dips in ∆ f ( H ) and peaks in Γ m ( H ). At T = 12 K,the two features at 45 and 100 mT (indicated by arrows)correspond to the Cyc-to-SkL and the SkL-to-FM phasetransitions, respectively. The double dip (peak) featurein ∆ f ( H ) (Γ( H )) comes from the imperfect alignmentof the sample’s crystalline axes with the coordinate sys-tem of our measurement setup, resulting in a differencein α i for each domain. At T = 7 K only one feature isfound, corresponding to the Cyc-to-FM transition, whileat T = 15 K, which is above the magnetic ordering tem-perature, no features are observed. H is rotated approximately in the (010) plane suchthat, in general, by changing β , we change each α i dif-ferently. As a result, the number of features related tophase transitions and the fields at which they occur canalso change. The dependence that we observe is consis-tent with the orientation of our sample and previous mea-surements by K´ezsm´arki et al. In particular, we note thatbecause of the crystal’s alignment and its cubic symme-try, the measured curves should repeat themselves uponrotating β by 90 ◦ . This periodic behavior can be seenin Fig. 3 (a), where two DCM curves with β = 0 and90 ◦ nearly overlap; differences, including the splitting ofthe dips in ∆ f ( H ) into two dips, are again related to theslight misalignment of the sample’s crystalline axes withrespect to the applied field, resulting in slightly different α i for each domain. In the curve taken with β = 40 ◦ (ap-proximately H (cid:107) [101]) shown in Fig. 3 (b), we observefour features. The features observed at 35 and 60 mT are the Cyc-to-SkL and the SkL-to-FM phase transitions, re-spectively, also observed by K´ezsm´arki et al. These tran-sitions correspond to the P and P domains (blue andblack in Fig. 1) with α = 31 . ◦ and α = 39 . ◦ . Thetwo transitions at 320 mT and 370 mT correspond to theCyc-to-FM transitions in the P and P domains (greenand red in Fig. 1), where α = 84 . ◦ and α = 88 . ◦ .As before, the mismatches α (cid:54) = α and α (cid:54) = α andthe resulting pair of split features are due to the crystal’simperfect alignment with the applied field.Using the measured features in ∆ f ( H ) and Γ m ( H ),we map the magnetic phase transitions of GaV S as afunction of H and β . After initializing the system witha large external field H = 1 T, DCM measurements aremade by stepping H toward zero at a fixed β and T . Theangular dependence over the range − ◦ < β < ◦ isrecorded at T = 11 K by changing β in steps of 2 . ◦ andrepeating the measurement. We plot the features identi-fied in these measurements as open circles in Figs 4 (a)and (b). By comparing our data taken for a few mag-netic field orientations with the phase diagram reportedby K´ezsm´arki et al. , we assign each feature to a certaintype of transition (i.e. Cyc-to-FM, Cyc-to-SkL, SkL-to-FM) occurring in a certain domain state (P , P , P ,P ).Next, we determine the dependence of the phaseboundaries on the orientation of the magnetic field withrespect to the axis of the uniaxial magnetic anisotropy.The measured signatures shown as open circles inFigs. 4 (a) and (b) can be fit by assuming that eachof the four rhombohedral domains of GaV S obeys themagnetic phase diagram shown in (d), plotted as a func-tion of H (cid:107) and H ⊥ , the components of H parallel andperpendicular to the rhombohedral axis ˆc i , respectively.A feature in ∆ f and Γ observed at certain H and β cor-responds to a transition of a particular domain P i for T (K) H ( m T ) (c) SkLFM Cyc PM
0 50 100 15005101520253035 H (mT) f ( H z ) (a)
15 K12 K7 K
0 50 100 1500246810 H (mT) ( a . u . ) (b)0 25 50 75 100 125-10-5051015 H (mT) f ( H z ) (a)
0 100 200 300 400-200-150-100-500 H (mT) f ( H z ) (b) H (mT) f ( H z ) (c) H Figure 3. Angular dependence of magnetic phase transitions measured by DCM in GaV S . (a), (b) ∆ f ( H ) at T = 11 K forat β = 0, 40, and 90 ◦ in maroon, cyan, and brown, respectively. Arrows indicate features corresponding to phase transitions.Inset: zoomed view of the low-field region. (c) Schematic diagram showing the three measured orientations relative to thesample-loaded cantilever. a field of magnitude H and angle α i with respect to ˆc i , as shown in Fig. 4 (c). The magnitude H and theangle α i at which each feature occurs, correspond to apoint on a phase boundary in the diagram of Fig. 4 (d),through H (cid:107) = H cos α i and H ⊥ = H sin α i . This phasediagram reflects the general form suggested by Leonovand K´ezsm´arki . Phase boundaries corresponding tothe diagram are also plotted as a function of β and H inFigs. 4 (a) and (b) to show their agreement with the mea-surements. They appear as solid lines, which are color-coded according to the domain to which they belong.An Euler rotation of the crystal (-5.0, 0.2 and 10 . ◦ )with respect to ideal configuration, shown in Fig. 1 (b),is required such that the phase boundaries correspond-ing to the different domain states collapse onto the singleboundary diagram of Fig. 4 (d).The agreement between the measured features and fitphase boundaries allows us to eliminate complicationsarising from the multi-domain nature of the crystal and,thus, to extract a the general magnetic phase diagramof GaV S as function of field applied parallel and per-pendicular to the anisotropy axis. The position of theintersection between the different phase transitions inFig. 4 (d) shows that the SkL phase in GaV S per-sists in oblique fields up to a threshold angle as large as α max = 77 ◦ . For larger α , the cycloidal state directlytransforms to the ferromagnetic state upon increasing H . The extent of the SkL phase shows stronger stabil-ity against fields applied perpendicular to the anisotropyaxis (up to H ⊥ = 200 mT) than fields applied parallel(up to H (cid:107) = 65 mT). This critical angle α max is largerthan predicted by Leonov and K´ezsm´arki . B. GaV Se We apply the same experimental procedure to explorethe magnetic phase diagram of GaV Se . In this case, H is rotated approximately in the (1¯10) plane. Figs. 5 (a)and (b) show the angular dependence of the features,as extracted from measurements of ∆ f ( H ) and Γ m ( H )at T = 12 K. Using previous measurements made byBord´acs et al. along particular crystalline directions ,as well as neutron diffraction data by Geirhos et al. forguidance, we assign each feature to a transition betweenCyc, SkL, or FM states for a certain domain and color-code it accordingly.Once again, the measured features are shown as opencircles in Figs. 5 (a) and (b) and can be fit by assum-ing that each of the four rhombohedral domains obeys asingle magnetic phase diagram shown in (d). The mag-nitude of the applied field H and its angle α i with re-spect to the assigned domain’s rhombohedral axis ˆc i puteach feature on one of the phase boundaries depicted inFig. 5 (d). Phase boundaries corresponding to the phasediagram are plotted in Figs. 5 (a) and (b) for compari-son with the measured data. They appear as solid lines,which are color-coded according to the domain. Similarlyto GaV S , the overall form of the phase diagram agreeswith that suggested by Leonov and K´ezsm´arki , al-though there are minor quantitative differences betweenour results and the theoretical predictions. Note thatthe rotation plane of H , approximately (1¯10), contains ˆc and ˆc , but not ˆc and ˆc . An Euler rotation of thecrystal (-14, -1 and 7 ◦ ) with respect to ideal configura-tion, shown in Fig. 1 (c), is required such that the phaseboundaries corresponding to the different domain states(P , P , P , P ) collapse onto the single boundary di-agram of Fig. 5 (d). We find additional anomalies inboth ∆ f ( H ) and Γ m ( H ), that cannot be ascribed to the (°) i ( ° ) (c)0 100200300400 H ( m T ) (b)0 100200300400 H ( m T ) (a)0 100 200 300 4000 100200300 H (mT) H || ( m T ) GaV S T = 11 K(d) SkL FMCyc
Figure 4. Magnetic phase transitions measured in GaV S at T = 11 K. Features extracted from DCM measurementof (a) ∆ f ( H ) and (b) Γ( H ) are plotted as open circles as afunction of β . Black, red, green, and blue circles correspond todomains P , P , P , and P , respectively. Color-coded linesindicate phase boundaries for each domain according to thephase diagram in (d). (c) Angle α i between the corresponding ˆc i and the external field H vs. β for all four rhombohedraldomains, using the same color code as in (a) and (b). (d)Best-fit magnetic phase diagram for single-domain GaV S as a function of field applied perpendicular and parallel tothe axis of symmetry. boundaries between the Cyc, SkL, and FM phases. Wesuspect that these anomalies originate from the forma-tion of magnetic textures localized at structural DWs, asdiscussed in section IV C.For the black and red domains, which are the onlytwo experiencing sufficient H (cid:107) to reach the SkL phase,the boundaries of the SkL state appear as prominentrain-drop-like shapes in Figs 5 (a) and (b). From theintersection of the SkL with the Cyc phase boundary in (d), we extract a threshold angle α max = 31 ◦ forthe SkL phase in GaV Se at T = 12 K. Contrary toGaV S , the extent of the SkL phase shows stronger sta-bility against fields applied parallel to the anisotropy axis(up to H (cid:107) = 340 mT) than fields applied perpendicular(up to H ⊥ = 75 mT). Furthermore, we note the presenceof a reentrant Cyc phase for angles 19 ◦ < α i < ◦ , aspredicted by Leonov and K´ezsm´arki . For this rangeof α i , two successive first-order phase transitions fromCyc to SkL and back occur as a function of increasingfield. The signature of this behavior in DCM is shown inFig. 5 (e). C. Magnetic States Confined to Domain Walls inGaV Se Geirhos et al. observed anomalies in various macro-scopic thermodynamic properties of GaV Se , emergingexclusively in crystals with polar multi-domain structure.They suggest a possible scenario for the formation ofmagnetic states at the structural DWs of the lacunarspinel GaV Se . Magnetic interactions change step-wise at the DWs and spin textures with different spi-ral planes, hosted by neighboring domains, need to bematched there. This can, for example, lead to conicalmagnetic states at the DWs with a different closing fieldmagnitude than bulk magnetic states. Here, we adoptand modify this model in order to analyze its applica-bility to anomalies observed in our DCM measurementsof GaV Se , which cannot be assigned to bulk magneticphase transitions.In the rhombohedral phase of the studied lacunarspinels, mechanically compatible and charge neutralDWs are normal to ˆc i + ˆc j , the sum of the two polardirections of the domain states P i and P j separated bythe DW, as shown in Fig. 6 . For example, me-chanically and electrically compatible DWs connecting aP (black) and a P (red) domain are parallel to (001)planes, cf. Fig. 6. The same is true for DWs between P (green) and P (blue) domains.For an arbitrary orientation of the external magneticfield, magnetic states confined to DWs with different ori-entations are expected to undergo field-induced transi-tions, similarly to the bulk (in-domain) magnetic states.However, in this case the situation is more complex: Thestability of the magnetic states confined to DWs is deter-mined by the orientation of the field with respect to themagnetic anisotropy axes of adjacent domains and to theDW itself.It is reasonable to assume, that the angle, γ n , between H and the normal of the DW planes, given by ˆc i + ˆc j ,plays a decisive role in setting the angular range, acrosswhich confined states are stable. This leads to three pairsof DWs, as shown in Fig. 6, each sharing the same γ n fora given H . For DWs in a pair, however, the relativeorientation between the magnetic anisotropy axes of thetwo domains involved and H is not the same. For ex- (°) i ( ° ) (c) 0 100 200 300 H ( m T ) (b) 0 100 200 300 H ( m T ) (a) 0 100 200 300 4000 100200300 H (mT) H || ( m T ) (d) SkL FMCyc i = 26.6°0 100 200 300-20246810 H (mT) f ( H z ) / ( a . u . ) (e) SkL FMCyc Cyc i = 26.6° Figure 5. Magnetic phase transitions measured in GaV Se at T = 12 K. Transitions extracted from DCM measurement of (a)∆ f ( H ) and (b) Γ( H ) are plotted as open circles as a function of β . Black, red, green, and blue circles correspond to transitionsfor domain P , P , P , and P , respectively. Color-coded lines correspond to phase boundaries for the each color-coded domainas indicated by lines in the phase diagram (d). (c) Angle α i between corresponding polar axis and the external field H vs β for all four rhombohedral domains, using the same color code as in (a) and (b). (d) Best-fit magnetic phase diagram forsingle-domain GaV Se as a function of field applied perpendicular and parallel to the axis of symmetry. (e) DCM measurementof ∆ f ( H ) for α i = 26 . ◦ ( β = 7 . ◦ ) showing the reentrant Cyc phase. These measurements corresponds to line-cuts along thedashed vertical lines in (a) and (b) and the dashed diagonal line in (d). ample, consider the P P /P P pair: the rotation planeof H (1¯10) contains the anisotropy axes of P and P ,but not the anisotropy axes of P and P ; they span 54 ◦ with this plane. We therefore introduce another angle, γ p , between H and the difference of the two polar vec-tors ˆc i − ˆc j , which lies in the DW plane. Both theseangles γ n ( β ) and γ p ( β ), plotted in Fig. 7 (a) and (b),respectively, are expected to affect the stability of theDW-confined magnetic states.In the angular dependent torque measurements, shownin Figs. 7 (c)-(f), we observe at most four anomalies(open circles) for a given field orientation. Since thereare six types of DWs, distinguished by γ n and γ p , sometransitions, which occur simultaneously in different typesof DWs appear as a single anomaly, while some transi-tions appear not to be experimentally observable. In thefollowing analysis, we take into account an additionalanomaly (crosses) between β (cid:39)
40 and 130 ◦ at field val-ues around 100 mT, which is not present in our DCMmeasurements, but has been observed in magnetoelectricmeasurements .As a first scenario, we suggest the following assign-ment of the observed anomalies, shown in Fig. 7 (c). The anomalies are labeled A to F with an additionalindex 1 or 2, indicating if they appear for β < ◦ or β > ◦ , respectively. A and B anomalies are assignedto P P DWs; A and B anomalies to P P DWs; C and E anomalies to P P DWs; the C anomaly to P P DWs; D and D anomalies to P P DWs; and F andF anomalies to P P DWs. In this scenario, all observedanomalies are assigned to transitions of magnetic statesconfined to DWs, as shown in Fig. 7 (c). In all cases,both domains adjacent to the DWs host the Cyc stateand the DW-confined state emerges due to the matchingof these two cycloidal patterns. As shown in Fig. 7 (c),no anomaly is observed in angular ranges, where the ad-jacent domains host magnetic states other than the Cyc.This is true for the all the transitions meeting at β ≈ ◦ .For example, the B anomaly, which is assigned to tran-sitions on P P DWs, would progress above 150 mT for β > ◦ , but because in-domain states within the P domain (blue axis) transform from the Cyc to the FMstate for β > ◦ and H >
150 mT, the B anomalydisappears for larger angles. Similarly, the D and D anomalies assigned to transitions in the P P DWs arelimited by the two skyrmion pockets of the P and P Figure 6. Schematic for understanding the orientation of the 6different domain walls types. Top left: Directions of the fourpossible polar axes, P -P , which are the axes of magneticanisotropy within the corresponding domains. The transpar-ent blue plane indicates the approximate plane of rotation ofthe external magnetic field. Top right: Mechanically compat-ible and charge neutral DWs separating P and P domainsare parallel to the (001) plane, just as DWs between P andP domains. The former and latter DWs are referred to asP P and P P , respectively. γ n ( γ p ), the angle between H and ˆc i + ˆc j ( ˆc i − ˆc j ), is shown for both DW pairs. Bottom:The other two pairs of DWs sharing the same orientation.The normal vector of the corresponding planes and their la-bels are indicated for the three cases, as well as the differencevector ˆc i − ˆc j , unique to each DW type. domains. The A and A anomalies assigned to P P and P P , respectively, also do not extend above 150 mTwere P and P domains transform from the Cyc to theFM state.An alternative scenario is an extension of the one sug-gested by Geirhos et al. , shown in Fig. 7 (d). Here,B and B anomalies are assigned to transitions at P P DWs; A , A , C and C anomalies to transitions atP P DWs; D , E , and F anomalies to transitions atP P as well as P P DWs; and D and F anomalies totransitions at P P as well as P P DWs. This scenarioallows some DW transitions to persist even when one ofthe adjacent domains leaves the Cyc phase. Such a situa-tion occurs for the P P DW transition, which penetratesboth the P and the P SkL pockets.In both scenarios, the mirror symmetry expected across β (cid:39) ◦ , as dictated by γ n ( β ) and γ p ( β ) is ful-filled: the transition lines are either symmetric to thispoint or they have a symmetry-related counterpart. Thebasis for both scenarios is the occurrence of a distinctmagnetic state confined to DWs, and its transition tothe FM state at certain critical field, observed as an ad-ditional anomaly in the DCM measurement.The angle of the applied field with the DW-normal, γ n ,and the orientation of its component in the DW-plane, γ p , appear to be a important parameters in determiningthe critical field of the DW states. V. CONCLUSION
We extract the magnetic phase diagrams as a functionapplied field magnitude and direction for both GaV S and GaV Se that are in good qualitative agreement withthe theoretical predictions of Leonov and K´ezsm´arki ,confirming the general validity of their model. Thisagreement, in turn, provides indirect confirmation that,under oblique applied magnetic field, the axes of N´eel-type skyrmions stay locked to the anisotropy axis whiletheir structure distorts and their core displaces. Themeasurements reproduce the overall structure of thephase diagrams, imposing a maximum angle α max ofmagnetic field applied with respect to the anisotropy axis,for which a SkL phase persists. In addition, they showthat easy-axis anisotropy – as found in GaV S – en-hances the robustness of N´eel-skyrmions against mag-netic fields applied perpendicular to the symmetry axis,while easy-plane anisotropy – as found in GaV Se – in-creases their stability for fields parallel to this axis. Ourresults also confirm the existence of a reentrant Cyc phasein GaV Se , which was anticipated to occur for certainvalues of easy-plane anisotropy. Finally, anomalies in∆ f ( H ) and Γ m ( H ), which cannot be explained as bulkdomain transitions, are consistent with distinct magneticstates confined to polar structural DWs and their tran-sition from the Cyc to FM state, as proposed by Geirhoset al. .Nevertheless, the measured magnetic phase diagramsare not in strict quantitative agreement with the pre-dicted ones. For both GaV S and GaV Se , we areunable to tune the uniaxial anisotropy of the model tomatch the measured values of threshold angle of the SkLphase α max = 77 ◦ for GaV S at T = 11 K and 31 ◦ forGaV Se at T = 12 K. This discrepancy suggests thatapproximations made in the model ignore important de-tails, thus preventing it from capturing the full behaviorof the system. Possible improvements to the model in-clude consideration of the anisotropic exchange interac-tion or extension the model from two to three dimensions.Also, further experimental investigation – especially real-space imaging – of anomalies assigned to transitions ofDW-confined magnetic states is required to characterizethe spin pattern associated with these states. n ( ° ) (a)020406080 p ( ° ) (b) x-P P x-P P y-P P y-P P z-P P z-P P H ( m T ) (e) f ( H z ) (°) H ( m T ) (f) - l og ( a . u . )
100 200 300 A A A B B C C D D E E F F H ( m T ) (d) 100 200 300 A A A B B C C D D E E F F H ( m T ) (c) Figure 7. Anomalies in ∆ f ( H ) and Γ( H ) assigned to transitions of DW rather than bulk magnetic states. Arrows at the topindicate from left to right the approximate angle β corresponding to the [111], [001] and [11 −
1] directions, respectively. (a)Angle γ n between the normal vector of a DW and H plotted against β . The color of the dashed lines shows their correspondenceto a DW type in the legend. (b) Angle γ p between the vector formed by the sum of the polar axis vectors of the two adjacentdomains of a DW and H plotted against β . (c) Transitions extracted from both ∆ f and Γ that are not assigned to a domaintransition (dark gray circles). Crosses show transitions extracted from magnetoelectric measurements , scaled by about 0.9 tomatch the DCM data. Colored lines show the suggested assignment of the transitions to DW types as denoted in the legend.(d) Same data as in (c) with a different assignment of transitions. Color map of (e) ∆ f ( H, β ) and (f) − log Γ m ( H, β ). ACKNOWLEDGMENTS
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