Chiral skyrmions in cubic helimagnet films: the role of uniaxial anisotropy
M. N. Wilson, A. B. Butenko, A. N. Bogdanov, T. L. Monchesky
CChiral skyrmions in cubic helimagnet films: the role of uniaxial anisotropy
M. N. Wilson, A. B. Butenko,
2, 3
A. N. Bogdanov, and T. L. Monchesky Department of Physics and Atmospheric Science,Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 IFW Dresden, Postfach 270016, D-01171 Dresden, Germany Institute of Applied Physics, University of Hamburg, D-20355 Hamburg, Germany (Dated: April 8, 2018)This paper reports on magnetometry and magnetoresistance measurements of MnSi epilayersperformed in out-of-plane magnetic fields. We present a theoretical analysis of the chiral modulationsthat arise in confined cubic helimagnets where the uniaxial anisotropy axis and magnetic field areboth out-of-plane. In contrast to in-plane field measurements (Wilson et al. , Phys. Rev. B ,144420 (2012)), the hard-axis uniaxial anisotropy in MnSi/Si(111) increases the energy of (111)-oriented skyrmions and in-plane helicoids relative to the cone phase, and makes the cone phasethe only stable magnetic texture below the saturation field. While induced uniaxial anisotropy isimportant in stabilizing skyrmion lattices and helicoids in other confined cubic helimagnets, theparticular anisotropy in MnSi/Si(111) entirely suppresses these states in an out-of-plane magneticfield. However, it is predicted that isolated skyrmions with enlarged sizes exist in MnSi/Si(111)epilayers in a broad range of out-of-plane magnetic fields. These results reveal the importance ofthe symmetry of the anisotropies in bulk and confined cubic helimagnets in the formation of chiralmodulations and they provide additional evidence of the physical nature of the A -phase states inother B20-compounds. PACS numbers: 75.25.-j, 75.30.-m, 75.70.Ak
I. INTRODUCTION
Broken inversion symmetry in magnetic crystals cre-ates both of one-dimensional (1D) helical modulations, and two-dimensional (2D) localized structures ( chiralskyrmions ). These textures are due to
Dzyaloshinskii-Moriya (DM) interactions imposed by the chirality of theunderlying crystal structure.
Similar interactions inother condensed matter systems that lack inversion sym-metry (such as multiferroics, ferroelectrics, chiral liquidcrystals ) can also stabilize skyrmionic states. Impor-tantly, multi-dimensional solitons are unstable in mostachiral nonlinear systems and collapse spontaneously intopoint or linear singularities. This fact attaches specialimportance to chiral condensed matter systems as a par-ticular class of materials where skyrmion states can exist.Among noncentrosymmetric magnetic compounds,easy-axis ferromagnets with n mm ( C nv ) and ¯42 m ( D d ) symmetries can be considered as the most suit-able crystals to observe chiral skyrmions. In thesecompounds, condensed 2D chiral skyrmion textures( skyrmion lattices ) can exist as thermodynamically sta-ble states in a broad range of applied magnetic fieldsand temperatures . In other classes of noncentrosym-metric magnets, skyrmion lattices compete with one-dimensional modulations ( helicies ) and arise only forcertain ranges of the material parameters. A num-ber of recent investigations indicate the possible exis-tence of chiral skyrmions in noncentrosymmetric uni-axial ferromagnets. In cubic helimagnets, the situa-tion is even more difficult for skyrmion formation: one-dimensional single-harmonic modulations ( cone phases)correspond to the global energy minimum in practi- cally the entire region where chiral modulations exist,while skyrmion lattices and helicoids can exist only asmetastable states.
Skyrmionic states and other mul-tidimensional modulated textures are reported to existonly in close vicinity to the Curie temperatures ( T C ) ofbulk cubic helimagnets as so called precursor states . Beyond the precursor region, condensed skyrmionphases and other thermodynamically stable nontrivialmodulations are expected to exist only in cubic heli-magnets where additional stabilizing effects are present.Theoretical analysis and experimental observations showthat surface/interface induced uniaxial distortions and finite size effects effectively suppress unwantedcone states and stabilize helicoids and skyrmion lat-tices in confined cubic helimagnets. Recently, the chal-lenge of creating and observing such textures was over-come by the fabrication of free-standing nano-layers ofcubic helimagnets and the synthesis of epitaxial thinfilms of these materials on Si(111) substrates. De-spite numerous indirect indications of skyrmionic statesin different nonlinear systems confined cubic heli-magnets still remain the only class of materials whereskyrmionic states can be induced, observed, and ma-nipulated in a broad range of the thermodynamicalparameters.
Investigations of chiral skyrmionsin cubic helimagnet nano-layers have gained importanceafter the discovery of similar skyrmionic states stabilizedby surface/interface DM interactions in nano-layers ofcommon magnetic metals and perspectives of theirapplications in data storage technologies.
Anisotropy plays a decisive role in the structure ofskyrmions in epitaxial films of cubic helimagnets. Dueto the lattice mismatch between the B20 crystal and a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r the Si(111) substrate, strain induces a uniaxial mag-netic anisotropy through magnetoelastic coupling. Thisuniaxial anisotropy can lead to two kinds of regularskyrmions. The (111)-easy-plane uniaxial anisotropy inMnSi/Si(111) stabilizes skyrmions with their cores lyingalong the in-plane direction, whereas the (111)-easy-axis anisotropy in FeGe/Si(111) produces skyrmions withtheir cores aligned along the [111] direction. In addition, specific effects imposed by a confined ge-ometry of nano-layers also contribute to the stability ofcomplex magnetic textures over a broad range of ther-modynamic parameters.
We address these finite-sizeeffects in a separate paper. In this work we concentrateon effects imposed by induced uniaxial distortions.In our previous polarized neutron reflectometry (PNR)and magnetometry study, we showed that the groundstate of MnSi thin films in a thickness range 7 nm ≤ d ≤
40 nm is helimagnetic with a propagation vector ori-ented along the out-of-plane [111] direction. Measure-ments with both techniques yield a helical wavelength of L D = 13 . Following the introduction of a newclass of magnetic materials in the form of epilayers of cu-bic helimagnets in Refs. 26 and 27, we conducted detailedinvestigations of the magnetic states in MnSi/Si(111)films for in-plane magnetic fields.
This paper investigates the magnetic properties ofMnSi/Si(111) nano-layers in out-of-plane magnetic fields.We extend earlier calculations to include solutionsfor basic chiral modulations in cubic helimagnets withhard-axis uniaxial distortions and construct the magneticphase diagrams of the solutions (Section II).The measurements in an out-of-plane magnetic fieldconfirm the absence of thermodynamically stable (111)-skyrmions lattices and in-plane helicoidal phases (Sec-tion III). This conclusion is supported by theoreticalcalculations, which demonstrate that the cone phase isthe only thermodynamically stable phase below the sat-uration field over the entire magnetic phase diagram. Weexplain the difference in the behavior of epilayers andbulk crystals in Section IV. In Section IV D we comparethe magnetization processes observed in bulk to those inconfined cubic helimagnets and update the T − H phasediagram.Finally, we overview the existing observations in con-fined chiral systems within the framework of our results(Section V). II. CHIRAL MODULATIONS IN CUBICHELIMAGNETS WITH UNIAXIALDISTORTIONS
Modulated states that arise in cubic helimagnets havebeen described within the Dzyaloshinskii theory of chiralhelimagnets in Refs. 41 and 42. The energy functionalintroduced by Bak and Jensen became the basic modeland formed a conceptual framework for magnetism ofcubic helimagnets. It is well-established that a strong uniaxial magneticanisotropy arises in epilayers of cubic helimagnets as a re-sult of surface/interface interactions and epitaxially in-duced strain. The magnetic states in thesessystems can be derived by minimization of a Bak-Jensenfunctional (Eq. (1) in Ref. 41) that includes an additionaluniaxial anisotropy with constant K . In this paperwe write the energy density w ( M ) for a cubic helimagnetnano-layer in a magnetic field perpendicular to the filmsurface ( H || z ) as a sum of three energy contributions, w = w ( M ) + w c ( M ) + f ( M ): w ( M ) = A (grad M ) − D M · rot M − HM z − KM z , (1) skyrmion lattice isolated skyrmions cone (b) (d) (c) helicoid (a) x y z FIG. 1. . (a,b) one-dimensional and (c,d) two-dimensionalchiral modulations that can exist as either stable ormetastable states. w c ( M ) = (cid:88) i =1 (cid:2) B ( ∂M i /∂x i ) + B c M i (cid:3) , (2) f ( M ) = J ( T − T ) M + bM . (3)The functional w ( M ) describes the main magnetic in-teractions in terms of the exchange interaction with ex-change stiffness constant A , the Dzyaloshinskii-Moriya(DM) coupling with constant D , the Zeeman energy,and the induced uniaxial anisotropy. The energy con-tribution w c includes exchange anisotropy ( B ) and cubicmagnetocrystalline anisotropy ( B c ) (the x i are the com-ponents of the spatial variable). In cubic helimagnets,exchange and magnetocrystalline anisotropies are muchsmaller than the interactions included in w . The energydensity f ( M ) comprises magnetic interactions imposedby the variation of the magnetization modulus M ≡ | M | and is written in the spirit of the Landau theory as anexpansion of the free energy with respect to the orderparameter M , with coefficients J and b . The character-istic temperature T is related to the Curie temperatureof a cubic helimagnet, T C = T + D / (4 JA ). In a broadtemperature range, the magnetization vector practicallydoes not change its length, and the non-uniform magneticstates only include a rotation of M . Spatial modulationsof the magnetization modulus become a sizeable effect inthe precursor region and lead to the specific effects ob-served in this region in close vicinity to T C (see Ref. 12and the bibliography therein).In this paper we investigate a model that has a fixedmagnetization modulus M = const, Eq. (1). We discusspossible distortions of the basic magnetic phases imposedby cubic anisotropy, stray-fields, and spatial variations of M at the end of the paper.For w with H || z and easy-axis anisotropy ( K > cone phase, the heli-coid , isolated skyrmions , and skyrmion lattices (Fig.1).
In this section we investigate Eq. (1) with H || z and K <
0, which describes chiral modulations inMnSi/Si(111) films with a hard-axis anisotropy, and con-struct the phase diagram of the solutions for functional(1) (Fig. 3).(1) Conical helices. By introducing spherical coordi-nates for the magnetization vector, M = M (sin θ cos ψ, sin θ sin ψ, cos θ ) , (4)one can readily derive analytical solutions for the conephase, cos θ = HH C , ψ = 2 πzL D , H C = H D (cid:18) − KK (cid:19) , (5)where L D = 4 πA/ | D | , H D = D M/ (2 A ) . (6) FIG. 2. (color on-line) The equilibrium periods of the mod-ulated phases shown in Fig. 1 and isolated skyrmion sizes asa function of the applied field for K = 0. This is representa-tive of the main features of chiral modulations in films with K >
K <
0, and shows the transition of the helicoidsinto a set of isolated domain walls ( kinks ) at H h = 0 . H D and the skyrmion lattice into a “gas” of isolated skyrmionsat H s = 0 . H D . Isolated skyrmions exist above the el-liptical instability field H el = 0 . H D indicated by the greyshaded region. Inset shows the equilibrium skyrmion sizesat high magnetic fields. The lower panel shows the differencesbetween the energy densities for the helicoids (solid lines) andthe skyrmion lattice (dashed lines) relative to the cone phaseas functions of the applied field.
The helical wavelength at zero field and zero anisotropy( L D ) and the saturation field of the cone phase ( H D )for K = 0 represent two of the characteristic materialparameters of cubic helimagnets (see Table 1 in Ref. 12).As the field increases along the propagation direction,the spins cant towards the field and produce the single-harmonic modulation described by Eqs. (5) and shownin Fig. 1(b). The magnetic field competes with the DM-interaction, which is represented by the effective easy-plane anisotropy K = D / (4 A ), and transforms the conecontinuously into the saturated state ( θ = 0) at the crit-ical field H C ( K ).(2) Helicoids. Distorted helical modulations, knownas helicoids, are shown in Fig. 1 (a) for the case ofan in-plane propagation vector. The transverse distor-tions imposed by applied magnetic fields and/or uniaxialanisotropy are described by solutions to the well-knowndifferential equations for the non-linear pendulum. In bulk helimagnets, a helicoid evolves continuouslyfrom a single-harmonic helix with period L D (Eq. (6)) FIG. 3. (color on-line) The phase diagram of the equilibriumstates for model (1) with the two control parameters of model(1), in reduced values of applied magnetic field h = H/H D ( H (cid:107) ˆz ) and uniaxial anisotropy k = K/K , as independentvariables (the details are given in Fig. 12). Filled areas in-dicate the regions of global stability for the cone (green), he-licoid (red), and skyrmion lattice (blue). In the saturatedstate (grey area), skyrmions exist as isolated (noninteract-ing) objects. Thin dotted lines designate critical lines for themetastable helicoid ( h h ) and skyrmion lattice ( h s ); h ( k ) isthe first-order transition line between the stable helicoid andskyrmion lattice. Inset shows the induced uniaxial anisotropyas a function of the film thickness in hard-axis MnSi/Si(111) and easy-axis FeGe/Si(111) epitaxial films. The inducedanisotropy ranges for these compounds are indicated alongthe
K/K axis. into a one-dimensional soliton lattice at high fields. Thelattice transforms into a set of isolated domain walls(kinks) at a critical field H h ( K ) (Figs. 2 and 3). Thisresult is achieved by ignoring the weak demagnetizingfield contribution. Contrary to the case H ⊥ z describedin Ref. 24, these helicoids would have a continuous fielddependence similar to bulk crystals.(3) Isolated and embedded skyrmions. For a mag-netization given in spherical coordinates (Eq. 4), andthe spatial variables in cylindrical coordinates, r =( ρ cos ϕ, ρ sin ϕ, z ), axisymmetric localized solutions (iso-lated skyrmions) for Eq. (1) are described by ψ = ϕ + π/ θ = θ ( ρ ), which are derived from the Euler equation, d θdρ + 1 ρ dθdρ − ρ sin θ cos θ + 2 ρ sin θ − ( K/K ) sin θ cos θ − ( H/H D ) sin θ = 0 , (7)with boundary conditions, θ (0) = π , θ ( ∞ ) = 0. Typi-cal solutions θ ( ρ ) for negative K are plotted in Fig. 4 to-gether with magnetization profiles for isotropic ( K = 0)and easy-axis ( K = 0 . K ) helimagnets.Analysis shows that in a broad range of control param-eters, chiral modulations are qualitatively similar in he-limagnets with different signs of K . For the case K = 0, the shaded region in Fig. 2 shows the fields where iso-lated skyrmions form. At the highest fields, the field in-duced saturated state is the lowest energy state, althoughisolated skyrmions can form inside this phase. Whenthe field is lowered to H s (0) = 0 . H D , metastableskyrmion lattices are able to condense. Then, as the fieldreaches H h (0) = 0 . H D , isolated domain walls con-dense into metastable helicoids, while skyrmion latticesand isolated skyrmions remain as metastable solutions.Below the strip-out field H el (0) = 0 . H D , the isolatedskyrmions become unstable and collapse into the stablehelicoid phase. FIG. 4. Solutions of Eq. (7) for isolated skyrmions with thecontrol parameters (
K/K , H/H D ): (1) - ( 0.5, 1.0); (2) -(0,1.0); (3) (-0.5, 1.0); (4) - (-0.5, 0.8). Inset shows the skyrmionsizes in films with different types of uniaxial anisotropy anddifferent values of the applied magnetic field. Ensembles of isolated skyrmions have been observedin Fe . Co . Si mechanically thinned films and FePdnano-layers as a result of a skyrmion lattice expansionin a high magnetic field ( H > H s ). The equilibriumenergy densities of the skyrmion lattice (∆ E s = E s − E c )and the helicoid (∆ E h = E h − E c ) relative to that of thecone phase ( E c ) are calculated from the model given byEq. (1) and are plotted in the lower panel of Fig. 2 asa function of the reduced magnetic field, h = H/H D ,for the isotropic case ( K = 0), as well as the easy-axis( K = 0 . K ) and hard-axis ( K = − . K ) anisotropies.The K − H phase diagram in Fig. 3 (see also Fig. 12in the Appendix) overviews magnetic properties of con-fined cubic helimagnets with different signs of uniaxialanisotropy in perpendicular magnetic fields. A corre-sponding phase diagram for hard-axis systems in in-planemagnetic fields has been constructed in Ref. 22 and hasbeen applied to analyze magnetic states in MnSi/Si films. III. EXPERIMENTAL RESULTSA. Sample Preparation
The 25.4-nm thick MnSi thin film was grown on aSi(111) high resistivity wafer ( ρ ≥ ≤ d ≤
40 as our previous work has shownthat the magnetic behaviour is qualitatively similar inthis range. The 25.4-nm sample was annealed ex-situ under an Ar atmosphere for one hour at 400 ◦ C to trans-form the residual manganese rich phase that was presentin the film into MnSi. X-ray diffraction (XRD) measure-ments in the region 2 θ = 30 ◦ − ◦ presented in Fig. 5(b)show no detectable impurity phase in this sample afterthe annealing and the Kiessig fringes in the inset demon-strate the high interfacial quality and uniformity of thefilm. This annealing did not affect the magnetic statesof the film other than to increase the saturation magne-tization due to the increased MnSi volume. FIG. 5. XRD curves of the 25.4 nm sample before (a), andafter (b) annealing. In both figures the intensity is normalizedto the maximum height of the Si(222) substrate peak whichremains unchanged through the annealing. The inset showsa fit to the Kiessig fringes.
As a second check of the sample quality, we determinedthe residual resistivity ratio (RRR) between T = 299 Kand T = 2 K from magnetoresistance (MR) measure-ments. For these measurements, we photolithographi-cally patterned a portion of the MnSi film into a Hall-barusing SPR220-3.0 photoresist and Ar-ion etching. Wethen attached Au-wire leads onto the surface using Insolder for four point resistivity measurements. The highRRR = 26.8 is further evidence of the high sample qual-ity. B. Magnetometry
We explored the phase diagram by measuring the mag-netization M as a function of applied magnetic field H M agne t i z a t i on M ( k A / m ) d = 25.4 nm (a)(b)55K 5K - d M / d H ( k A / m / T ) Applied Field H (T)
FIG. 6. (color on-line) (a) Magnetization curves from a d = 25 . H (cid:107) [111]. Temperaturesshown are 5, 10, 15, 20, 25, 30, 35, 40, 41, 42, 42.5, 43, 44,45, 46, 47, 48, 49, 50 and 55 K. (b) The static susceptibilityis obtained by calculating dM/dH from the the data in (a). and as a function of temperature T with a QuantumDesign MPMS-XL SQUID magnetometer with the ap-plied magnetic field pointing out of plane along the MnSi[111] direction. The magnetic susceptibility is a com-mon tool for mapping the phase diagram in magneticsystems. In bulk cubic helimagnets, peaks in themagnetic susceptibility ( dM/dH ) are signatures of thefirst-order magnetic phase transitions that separate thecone phase from two adjacent areas: the low-field regionwith multi-domain helical states and a small closed re-gion near the ordering temperature, the A -phase pocket(Fig. 10(b)). The first-order character of the tran-sition in and out of the A -phase is a reflection of thedifferences in the topology between these phases. In con-trast, a second-order transition, identified by a minimumin d M/dH , exists between the conical phase and thefield induced ferromagnetic state.From M − H scans, we calculated the static susceptibil-ity, dM/dH , as a function of both temperature and fieldin order to search for any indication of a magnetic phasetransition below the saturation field H C . In Fig. 6, wepresent the measured M − H curves obtained between5 K and 50 K, which are qualitatively similar to what (a) d = 25.4 nm M agne t i z a t i on M ( k A / m ) (b) S u sc ep t i b ili t y - d M / d H ( k A / m / T ) Temperature T (K)
FIG. 7. Field cooled magnetization measurements for a d =25 . H (cid:107) [111]. (a) Data sets shownin blue are for field values in steps of 0.05 T from 0.05 to 0.6 Tand steps of 0.1 T from 0.7 T to 1.0 T, and the data sets in redare each measured at a field 10 mT higher than the blue. (b)Field cooled static susceptibility dM/dH calculated from thepairs of red and blue M ( T )-curves. Field values shown are insteps of 0.05 T from 0.055 to 0.605 T and steps of 0.1 T from0.705 T to 1.005 T from top to bottom, and are separated by20 kA/m/T for clarity. is found in bulk. We present each of these curves ononly a single branch, alternating increasing and decreas-ing field, as we saw no hysteresis in the full hysteresisloops taken over several temperatures between 5 K and T C .Unlike the case for bulk MnSi samples and MnSi epi-layers with in-plane magnetic fields, there are no peaksin the dM/dH of Fig. 6(b) that would signal the exis-tence of chiral modulations other than the cone phase.The only magnetic phase transition that is visible is thesecond order transition delineated by the inflection pointin the dM/dH -curves at a field H ⊥ sat that we attributeto the onset of the saturated state at H C . We presentthe temperature dependence of H ⊥ C in Fig. 10. For fieldsnear 0.1 T, dM/dH drops for all scans measured below T C . We attribute this to a small sample misalignment in the straw used to hold samples for SQUID measurementswhich mixes in a small amount of the uncompensatedin-plane magnetic moment into the out-of-plane M − H measurements. We confirm the absence of hysteresis inthe out-of-plane M − H loops by MR measurements pre-sented in the next section.To screen for first-order transitions that may havephase boundaries along a vertical line on a T − H plot, wecalculated the static susceptibility from field-cooled mag-netization measurements. For an in-plane magnetic field,such measurements produced clear peaks in dM/dH at the skyrmion phase boundary of MnSi thin films, and measurements of bulk samples have produced peakscorresponding to the transition in and out of the A -phase. Samples were cooled in a fixed magnetic fieldfrom T = 100 K to 5 K, and the magnetization wasmeasured on warming. The curves from these measure-ments are shown in Fig. 7. In Fig. 7(b) we constructed dM/dH -curves from the pairs of data sets separated by H = 10 mT in Fig. 7(a). These figures show no peaksthat would indicate the presence of a first-order magneticphase transition. C. Magnetoresistance
The existence of the A -phase pocket is also observ-able in magnetoresistance (MR) measurements, as shownby Kadowaki et al. who observe hysteretic peaks inthe MR near the A -phase boundary. We use MR mea-surements as further evidence of the absence of first-order magnetic phase transitions in MnSi/Si(111) inout-of-plane magnetic fields, and to probe the magneticphase diagram with a higher density of field-temperaturepoints. While such features are present at the skyrmion-helicoid boundaries in MnSi thin films for in-plane mag-netic fields, the out-of-plane MR do not show such trade-marks.Fig. 8 shows representative resistivity curves for bothincreasing and decreasing field scans measured at T =10 K. The sample mount used for MR measurements al-lowed very accurate sample alignment perpendicular tothe applied magnetic field, in contrast to the SQUID mea-surements where the alignment is only accurate to withina few degrees. The magnetoresistance data is thus a morereliable indicator of the true out-of-plane hysteresis of thesample, and the lack of any hysteresis in this configura-tion is supporting evidence that the drop in dM/dH ob-served at 0.1 T in Fig. 6 is not intrinsic, but is rather dueto a small sample misalignment. Furthermore, the lack ofhysteresis or peaks in the MR is additional evidence forthe absence of the A -phase pocket in this sample. Thetemperature dependence of the MR =( ρ ( H ) − ρ (0)) /ρ (0)in Fig. 9 demonstrates that it varies smoothly over allfields and temperatures and supports the conclusion thatno skyrmions exist in out-of-plane magnetic fields. -4 -2 0 2 49.009.259.509.7510.00 -0.4 -0.2 0.0 0.2 0.49.889.929.9610.00 T = 10K
Long i t ud i na l R e s i s t i v i t y (- c m ) Applied Field H (T)
FIG. 8. Resistivity of a 25.4-nm thick MnSi/Si(111) film at T = 10 K, H (cid:107) [111], for increasing (red filled circles) and de-creasing fields (blue open circles). No hysteresis is observedin this data or at any other temperature. M agne t o r e s i s t an c e Applied Field H (T)
10 K
FIG. 9. Magnetoresistance ( ρ ( H ) − ρ (0)) /ρ (0) of a 25.4-nmMnSi/Si(111) film with H (cid:107) [111]. Temperatures shown are 10,15, 20, 25, 30, 35, 40, 41, 42, 43, 44, 45, and 46K. Curves areoffset by 0.015 for clarity. D. Phase diagram of MnSi/Si(111) for H (cid:107) [111]
We summarize the experimental results of Figs. 6, 7,8 and 9 with a construction of the magnetic phase di-agram for MnSi/Si(111) films in terms of temperatureand the perpendicular applied field (Fig. 10(a)). Thecone phase and the field-induced ferromagnetic state arethe only thermodynamically stable states below the or- dering temperature. The critical fields H C ( T ) obtainedfrom the minima in d M/dH separate these two regionsand are consistent with the features in the MR data inFig. 9. Above the Curie temperature, the minimum in d M/dH persists and is shown by the open circles inFig. 10(a). In addition, there is a weak feature visiblein the dM/dH data of Fig. 7(b) at higher fields. Thesetwo features are representative of the broad cross-overregion between the field induced saturated state and theparamagnetic state, as observed in bulk MnSi, andFeGe. We obtain additional confirmation of H C ( T ) from theinflection point in the dM/dH data of Fig. 7(b). Whilethe critical fields obtained from this method are consis-tent with the H C ( T ) values from Fig. 6(b), the M ( T )data were less noisy. We therefore used the minimain d M/dT as a measure of the temperatures of thephase transition at a given value of H . For higher fields,a second minimum is present in d M/dT due to thecross-over region and is shown by the open squares inFig. 10(a).The magnetic phase diagram for MnSi epilayers (Fig.10 (a)) differs from the corresponding phase diagramfor bulk cubic helimagnets (Fig. 10(b)). The regionwith multi-domain helicoid states bounded by the crit-ical line H C ( T ) and a tiny closed area near the Curietemperature, namely the A -phase pocket that exists bulkMnSi, both disappear from the phase diagram ofMnSi/Si(111) epilayers. In Section IV, we discuss thephysical mechanisms underlying the formation of theseareas in bulk cubic helimagnets and their modification inconfined samples. IV. DISCUSSIONA. Absence of H C1 in MnSi/Si(111) Multi-domain helicoid states arise as a result of the de-generate ground state in bulk cubic helimagnets. At zerofield, the helices propagate along the directions imposedby cubic anisotropy, which are the (cid:104) (cid:105) directions in thecase of MnSi. The applied magnetic field lifts the de-generacy of these propagation directions, selects the onealong the direction of the applied field, and transformsthe helix into the cone phase. In bulk cubic helimagnets amagnetic-field induced reorientation of the helices devel-ops a complex process including a displacement of the do-main boundaries and a rotation of the propagation direc-tions within the domains.
These processes are sim-ilar to magnetic-field induced transformations of multi-domain states observed in many classes of magneticallyordered materials and are decribed by common micro-magnetic equations.
In cubic helimagnets, the reori-entation of the helicoids ends at the critical field H C ( T )with the formation of a single-domain cone phase.In epitaxial MnSi nano-layers, a strong hard-axisanisotropy favors helices with the propagation direction FIG. 10. (a) The phase diagram of a d = 25 . H (cid:107) [111]. The filled circles (squares) show transitionfields obtained from the minima in d M/dH ( d M/dT ) calculated from the data in Fig. 6 (Fig. 7). Only a single phaseboundary is seen, the boundary between the conical and ferromagnetic phases. The area with multi-domain helicoid statesand the A -pocket, characteristic of bulk MnSi and other cubic helimagnets (b), are suppressed by the strong hard-axis uniaxialanisotropy. In addition, transition fields in between the saturated state and paramagnetic states are shown by the open circlesand squares. (b) The T − H phase diagrams of bulk MnSi near T C is compared to a Ge-doped MnSi crystal in the inset(b). Along the dotted line H ∗ ( T ), the difference between the energy densities of the skyrmion lattice and the cone phase isminimal (see inset in Fig. 3). perpendicular to the film surfaces and suppresses heliceswith other propagation directions. As a result, the prop-agation direction is homogeneous in the ground state ofsuch films. Due to the existence of inversion domains inthe crystal structure, there are variations in the mag-netic chirality on the length scale of the order of 1 µ m. The magnetic frustration between these regions createsmagnetic domains that display a glassy-magnetic behav-ior for fields applied in-plane. Nevertheless, the unifor-mity of the propagation direction explains the absenceof the multi-domain helicoids in out-of-plane magneticfields and the lack of a critical line H C ( T ) in the mag-netic phase diagram of MnSi films. Results that are con-sistent with these facts have been reported for 9 nm and19 nm thick epitaxial MnSi films by others. B. Nature of skyrmions in MnSi nano-layers
The strong uniaxial anisotropy that arises in epitaxialfilms of cubic helimagnets drastically changes the energybalance between the various modulated states, as shownin Fig. 3. The strong
K > lies within the[ K B , K C ] interval of the calculated K − H phase dia-gram in Fig. 3. Huang et al. report the existence ofskyrmion lattices in a range of magnetic fields that arein agreement with the calculated critical fields H and H s for the skyrmions lattice in Fig. 3. The calculations show that in easy-axis FeGe films, the uniaxial anisotropyeffectively suppresses the cone phase and stabilizes theskyrmion lattice in a broad range of the applied fieldsand temperatures. In contrast, MnSi/Si(111) epilayersexhibit a strong hard-axis uniaxial anisotropy. Figure 3shows the range of anisotropies spanned by the MnSifilms in Ref. 22. In such nano-layers, elliptically distortedskyrmions have been found to exist in a broad range ofin-plane magnetic fields. For a perpendicular magneticfield on the other hand, the hard-axis uniaxial anisotropy(
K <
0) in MnSi/Si(111) shifts the energy balance in fa-vor of the cone phase (lower panel of Fig. 2). As aresult, the
K < exclude the existence ofin-plane helicoids and (111)-oriented skyrmions (Fig. 1)in hard-axis MnSi/Si(111) epilayers and establish that ahelix with a propagation direction along (111) is the onlymagnetic ground state.These findings, however, have been recently disputedin Ref. 57. Based on Lorenz microscopy measurementsof a 10-nm thick MnSi/Si(111) epilayer, the authors ofRef. 57 claim that in-plane helicoids and skyrmions lat-tices exist in a broad range of out-of-plane magneticfields in contrast to the theoretical results summarizedin Fig. 3, which show that these states would only bepresent for
K >
0. Loudon recently demonstrated thatthe contrast in the Lorentz images published by Li et al. are due to structural artifacts and are not of magneticorigin: the same features observed in Ref. 57 are alsoobserved at room temperature, far above T C . It is im-portant to point out that the interpretation put forwardin Ref. 57 is not only at odds with theoretical calcula-tions, but it also contradicts the experimentally estab-lished facts. The striped pattern in Fig. 1(a) in Ref. 57 isincorrectly interpreted as an in-plane helicoidal groundstate. In-plane helicoids would have zero remanent mag-netization, contrary to what is reported by others.
MnSi/Si(111) films clearly show oscillations in the re-manent magnetization with thickness with a wavelengthgiven by the pitch of the helix that rules out the ex-istence of in-plane helicoids. Furthermore, PNR con-clusively shows that the propagation vector of the helixpoints out-of-plane.
The striped pattern is, how-ever, explained by moire fringes and is perfect agreementwith the strain reported in Ref. 22.
C. Why the A -phase exists The suppression of an A -phase pocket in MnSi/Si(111)near the ordering temperature in Fig. 10 due to K < A -phase in bulk MnSi crystals. Analy-sis of the magnetic-field-driven evolution of the skyrmionlattice period and the energy ∆ E h ( H/H D ) (Fig. 2) allowsone to understand the physical mechanism that leads tothe formation of the A -phase. For K = 0 (bulk helimag-nets), the cone phase is the global minimum of the func-tional w over the whole magnetic field range where mod-ulated states exist ( 0 < H < H D ), as indicated by thefact that both ∆ E h and ∆ E s are always positive. How-ever, two-dimensional chiral modulations provide a largerreduction of the DM interaction energy in skyrmion lat-tices compared to one-dimensional helical modulations.Calculations within the model of Eq. (1) show that thisreduction increases with increasing field up to a field H ∗ as the equilibrium sizes of the skyrmion cell decreases. At H ∗ , the skyrmion lattice reaches its highest densityand lowest energy difference ∆ E s ( H ∗ ) ≡ min[∆ E s ], asseen by a comparison between the K = 0 line in thelower panel of Fig. 2 to the point (a) in the upper panel.At higher fields H ∗ < H < H s the skyrmion lattice grad-ually expands into a set of isolated skyrmions at a criticalfield H s = 0 . H D . Nevertheless, skyrmion lattices are only metastable for K = 0 and additional interactions are required to stabi-lize them. The size of the anisotropy given by Eq. 2gradually decreases as min[∆ E s ( T )] decreases with in-creasing T and becomes zero at T C . This means thateven small perturbations can suppress the cone phase andlead to the formation of a skyrmion lattice in a pocketabout the H ∗ ( T ) line (see Fig. 7 in Ref. 12). The smallsize of the energy imbalance and low potential barriersthat characterize this region make the A -phase pocket extremely sensitive to small interactions, such as the soft-ening of the magnetization modulus, dipolar interactions,fluctuations, and anisotropy. In particular, calculationsshow that an exchange anisotropy as small as B = 0 . K (Eq. (2)) is sufficient to create a thermodynamically sta-ble skyrmion lattice in a certain field range near H ∗ . The importance of this anisotropy is evidenced by a num-ber of experimental results, including the variation in thesize of the A -phase pocket with the orientation of themagnetic field, and the increase in the size of the A -phase region in MnSi by doping with a larger spin-orbitinteraction element. The behavior in bulk crystals contrasts the behaviorin epilayers discussed in Section IV B, where a uniax-ial anisotropy dominates the small interactions discussedabove, and either suppresses the A -phase entirely, or re-sults in the stabilization of a skyrmion lattice over largeregions of the phase diagram. To explore the evolutionof the skyrmion phase between the behaviors observed inbulk and those in films, it would be interesting to inves-tigate the influence of a uniaxial pressure on bulk cubichelimagnets.The sensitivity of chiral modulations in the A -phase to weak interactions leads to several complexmagnetic states as observed in many cubic chiralhelimagnets. This complex behavior is demon-strated theoretically when a soft magnetization modu-lus is included in the calculation. However, the exactstructure of the complex magnetic textures reported inRefs. 12, 16–20, 48, 50, 60–63 and the particular physicalmechanism underlying the stabilization of these statesare still unresolved and remains the subject of contro-versy between different research groups. (For details see review papers ). Our results on theprecursor states evolution in confined cubic helimagnets(and, particularly, the conclusion about the suppressionof the A-phase pocket in easy-plane MnSi/Si (111) epi-layers) are based on the analysis of the general energybalance between the competing cone phase and the mul-tidimensional magnetic modulations in the A-phase, butdo not depend on the specific details of the textures inthis problematic region.
D. T − H phase diagrams of bulk crystals revisited
The T − H phase diagram presented in Fig. 10(b) wasconstructed from the first papers dedicated to magneticproperties of MnSi and other cubic helimagnets. This diagram was later explained through several theo-retical and experimental efforts (see, e.g., Refs. 53 and69). However, recent progress in our understanding ofthe magnetization processes now enables us to updatethe ‘canonical’ magnetic phase diagram of cubic helimag-nets to include regions of metastability and the precur-sor region, both of which are necessary to understandthe collection of measurements of these materials. Withthis purpose, we address here a problem of metastable0
FIG. 11. Calculated T − H phase diagram for a bulk MnSibased on the solutions for K = 0 in Fig. 3. Thin lines boundthe existing areas for the helicoids ( H h ( T )), the skyrmion lat-tice ( H sk ( T )), isolated skyrmions ( H el ( T )); H is the line ofthe phase equilibrium between the metastable helicoid andskyrmion lattice. Two characteristic temperatures, confine-ment temperature T cf and nucleation temperature T D boundthe precursor region ( T cf < T < T D ) .Typical T − H phasediagrams for cubic helimagnets with easy-axis (b) and easy-plane (c) uniaxial distortions. states and discuss a ‘hierarchy’ of magnetic states aris-ing in cubic helimagnets. Nearly all representations of thephase diagram consider only the equilibrium phases andignore the metastable states. In Section II, however, wepresent regions of stability and metastability. Metastablestates are important in magnetization processes in gen-eral, as described in Ref. 55. This is true for the first-order transitions between states with different topology.Metastable states are seen in the regions of mixed phasein (Fe,Co)Si nano-layers, and MnSi thin films. Morerecently, field cooling experiments in bulk Fe . Co . Simanaged to form a metastable skyrmion lattice. Theisolated skyrmions reported in Refs. 25 and 37 are an-other example of metastability. To facilitate a compari-son between theory and experiments, we produce a the-oretical T − H phase diagram in Fig. 11(a) that includesregions of metastability based on the results of Section II.In a broad temperature range, the magnetization mod-ulus is practically uniform over the material but hasa temperature dependence M ( T ) = M (0) σ ( T ). Thetemperature dependence in Fig. 11 is obtained from so-lutions to Eq. (1) by using the reduced magnetization σ ( T ) and H C ( T ) for bulk MnSi. In Fig. 11(a) we usethe calculated critical fields of the modulated states for K = 0 (Fig. 3) to obtain the theoretical equilibriumphase boundaries for the cone phase ( H < H C (0) σ ( T )),the metastable helicoid ( H < H h σ ( T )), skyrmion lat-tice ( H < H s σ ( T )), and for isolated skyrmions ( H >H el σ ( T )). This provides a good description of the MnSi phase diagram over most of the phase diagram.The updated T − H phase diagram for MnSi in Fig. 11provides a framework in which to understand the fieldcooling experiments in bulk Fe . Co . Si. By coolingthrough the precursor region at an appropriate field be-low H sk = 0 . H C , stable skyrmion lattices are nucle-ated in this region characterized by low energy barriers.As the temperature drops below the precursor region,the skyrmion lattices become metastable. However, thebarrier heights increase with decreasing temperature andprovide the robustness of these metastable states. Whenthe field is then reduced at fixed T , H el is eventuallyreached where the skyrmions strip-out into helicoids, asobserved in Ref. 70.The temperature dependence for K (cid:54) = 0 is shown inFig. 11(b) and (c) by again using the results of Section IIwith σ ( T ) for bulk MnSi. These figures facilitate a com-parison with the thin film experiments. Figure 11(b) cap-tures well the qualitative behavior of MnSi in Fig. 10,while Fig. 11(c) is able to reproduce the stable skrymionand helicoid regions of FeGe/Si(111) in Ref. 29. Spatialmodulations of the magnetization modulus, while negli-gible in a broad temperature range, become a sizeableeffect in the vicinity of the ordering temperature. Inthis precursor region, the magnetic textures also displayspatial variations of the magnetization modulus, whichstrongly modifies their properties compared to regular modulations that arise at lower temperatures. A the-oretical treatment within the Dzyaloshinskii-Bak-Jensenmodel that accounts for this additional degree of freedomreveals two characteristic temperatures that define theprecursor region, namely the confinement temperature, T cf = T − D / (4 JA ) and the nucleation temperature, T D = T + D / (2 JA ) shown in Fig. 11(a). These sep-arate the precursor region from the paramagnetic phaseon one side from the region with regular chiral modula-tions on the other. The peculiarities of the T − H phasediagram in this region are discussed in Ref. 12.Finally, we note that basic chiral modulations shownin Fig. 1 arise in cubic helimagnets as a result of com-petition between the main magnetic interactions and aredescribed by regular solutions of the Dzyaloshinskii-Bak-Jensen model. These should not be confused with “weak”magnetic states in the A -pocket where a clear hierarchyof interactions disappears and the energy barriers thatprotect the states are small. E. Surface effects in chiral ferromagnets
Recent experimental and theoretical findings demon-strate that surface effects may stabilize specific chiralmodulations in confined cubic helimagnets as skyrmionsmodulated along three spatial directions, or twistedstates at high in-plane fields. Theoretical analysisshows that the DM interactions near the surfaces of cu-bic helimagnets induce specific chiral modulations withthe propagation direction perpendicular to the sample1surfaces ( chiral twists ) . In chiral helimagnets filmswhere d ≤ L D , such surface twists become a size-able effect and strongly modify the skyrmion energeticsand provide a thermodynamic stability to the skyrmionlattice in a broad range of applied magnetic fields. These results elucidate recent observations of skyrmionlattices in free-standing cubic helimagnets nano-layers(see e.g Ref. 34), whereas skyrmions are suppressed byone-dimensional (conical) modulations in bulk crystalsof the same material. It was also established by nu-merical calculations that similar surface modulation in-stabilities strongly influence the structure of isolatedskyrmions in magnetic nanodots.
Furthermore, wenote that in MnSi/Si (111) epilayers, surface effects com-pete with the in-plane uniaxial anisotropy. However,for the d = 25 . . L D film investigated in thispaper, a strong in-plane anisotropy sufficiently weakensthe finite size effects in a broad range of magnetic fieldsand temperatures. V. MATERIALS AND PERSPECTIVES
In this section we briefly overview the existing groupsof confined noncentrosymmetric magnets and discusshow the induced and intrisic uniaxial anisotropy influ-ences chiral modulations in these compounds.
A. Free-standing films of cubic helimagnets.
The first images of chiral skyrmions have been observedin 20-nm thick mechanically thinned B20 Fe . Co . Sisamples. Subsequently, the formation and evolutionof skyrmions and helicoids were investigated in simi-lar free-standing layers of other B20 compound (see e.g.Refs. 34, 35, and 74). Unfortunately no M ( H ) measure-ments have been carried out in these films and valuesof the induced uniaxial anisotropy are unknown. How-ever, the observed magnetization processes in these com-pounds demonstrate features characteristic of easy-axistype of anisotropy.The different B20 material systems display a range ofbehaviours in external magnetic fields that are explainedwith the aid of the phase diagram in Fig. 3 by differencesin the size of K . The magnetic-field-induced evolution ofmagnetic states observed in (Fe,Co)Si free-standing filmstransforms from the helicoid ⇒ the skyrmion lattice ⇒ the saturated state with isolated skyrmions (this corre-sponds to interval (5), K B < K < K C = 1 .
90 in theAppendix). In FeGe nano-layers by contrast, the ob-served sequence of magnetic configurations follows from the helicoid ⇒ the skyrmion lattice ⇒ the cone phase ⇒ the saturated state (which is characteristic for interval(4), K A < K < K B = 0 .
363 in the Appendix).
B. Chiral helimagnetic epilayers
The fabrication of MnSi nano-layers on Si(111) opened the possibility of exploring the magnetic prop-erties of chiral thin films. This work has introduced anew class of nanomagnetic systems, epitaxial chiral he-limagnet thin films which are more amenable than me-chanically thinned layers to the investigation of skyrmionstates and other nontrivial chiral modulations with mul-tiple techniques. . Investigations in other B20epilayers, FeGe/Si, (Fe,Co)Si, and MnGe present awide range of material parameters to explore. So far, de-tailed measurements of the induced uniaxial anisotropyhave been carried out in epitaxial MnSi andFeGe , which span complementary ranges of K (Fig.3). The first measurements of the magnetic anisotropyin (Fe,Co)Si/Si(111) appeared following the preparationof this manuscript. Like the case of MnSi/Si(111), thesefilms have an out-of-plane hard-axis. PNR measurementsshows that the helical ground state of the films also havea propagation vector pointing out-of-plane. However,the presence of hysteresis in both the in-plane and out-of-plane M ( H ) curves indicates that the magnetic be-haviour differs from that of MnSi/Si(111). C. Fe and FePd nano-layers.
The induced interfacial DM interactions in ultra-thinlayers of common magnetic metals are capable of stabi-lizing skyrmion lattices, as well as isolated skyrmionsin large out-of-plane magnetic fields. In the K − H phase diagram of Fig. 3, such isolated skyrmions exist asmetasable objects within the saturated phase. Furthermeasurements to determine the parameters of skyrmionsand to map out the phase diagram for this system willprovide important comparisons with the theoretical pre-dictions and observations in nanolayers of cubic helimag-nets. D. Relations to bulk uniaxial helimagnets.
In uniaxial noncentrosymmetric ferromagnets, an in-trinsic uniaxial magnetic anisotropy stabilizes similar chi-ral modulations as those found in cubic helimagnets withan induced uniaxial anisotropy . For example, the chiralmagnet Cr / NbS (space group P6
22) develops long-range helimagnetic order below T C with a period L D = 48.0 nm. The propagation direction of the helixalong the hexagonal axis indicates a hard-axis type ofuniaxial anisotropy in this helimagnet. This implies thatCr / NbS should exhibit magnetic properties similar tothose observed in easy-plane epitaxial films MnSi/Si(111)in Refs. 22 and 28 and in the present paper. Contrary tohigh symmetry cubic helimagnets where the DM inter-actions provide three equivalent propagation directions2( w D = D M · rot M in Eq. (1)), uniaxial noncentrosym-metric magnets have a DM energy that is more complexand may include several material parameters. Particu-larly, for Cr / NbS the DM energy contribution can bewritten as w D = − D M · rot M − D (cid:18) M x ∂M y ∂z − M y ∂M x ∂z (cid:19) . (8)The last term in Eq. (8) imposes a difference betweenin-plane modulations and those along the hexagonal axis( z ). The K − H phase diagram for Cr / NbS dependson the additional material parameter D /D , and can beobtained from Fig. 3 by extending ( D >
0) or shrink-ing ( D <
0) the cone phase region. Peculiarities of themagnetic properties observed for in-plane fields imply theexistence of skyrmionic states in this helimagnet. Thesefindings correlate with theoretical predictions and ex-perimental observations of elliptically distorted in-planeskyrmions in easy-plane epitaxial MnSi/Si(111) films. In tetragonal magnets Cr Ge , and Mn RhSn, which belong to the ¯42 m ( D d ) chiral point group, theDM interactions only stabilize modulations propagatingin the plane perpendicular to the tetragonal axis. Chiralferromagnets of this class are suitable objects for inves-tigations of skyrmions and helicoid states. VI. CONCLUSIONS
We present experimental investigations of the magneticstates in epitaxial MnSi/Si films in perpendicular mag-netic fields and theoretical analysis of chiral modulationsunder the influence of an induced uniaxial anisotropy.The K − H phase diagram of the solution in cubic heli-magnets with induced uniaxial anisotropy (Fig. 3) pro-vides an effective tool to calculate the magnetizationcurves and the magnetic phase diagrams in bulk and con-fined helimagnets (Fig. 11).Our findings show that a subtle balance between thecone and the skyrmion lattice energies (Fig. 2) is vio-lated near the ordering temperature and results in theformation of a small closed area where the skyrmion lat-tice becomes thermodynamically stable. We argue thatthe ‘canonical’ T − H phase diagram of a cubic heli-magnet (Fig. 10(b)) includes (i) stable regions consist-ing of the cone phase and the saturated state that re-sult from the strongest interactions, and (ii) regions withmulti-domain helicoids and the complex modulations inthe A -phase that are induced by much weaker forces.The area of the phase diagram occupied by these weakstates can be easily modified by external and internaldistortions and even be totally suppressed, as observedin MnSi/Si(111) epilayers (Fig. 10(a)). We constructthe updated T − H phase diagram that includes bothstable and metastable solutions derived within the basicmodel of Eq. (1). We show that a strong induced uniax-ial anisotropy in hard-axis MnSi/Si epilayers completely suppresses the A -phase area and argue that uniaxial pres-sure applied to a bulk cubic helimagnets would providean effective method to investigate this phenomenon. To-gether with earlier findings, our results create aconsistent picture of uniaxial anisotropy effects arising inconfined cubic helimagnets and uniaxial bulk ferromag-nets.More detailed experimental investigations are requiredin both induced and intrinsic chiral magnets to providea comprehensive picture of how anisotropy affects themagnetic properties of skyrmions and other chiral mod-ulations and to observe the complete range of behaviorpredicted by the phase diagram in Fig. 3. From the theo-retical side, finite-size effects indicated in Ref. 23 and 24should be thoroughly investigated to complete the theo-retical description of confined chiral modulations withinbasic model of Eq. (1). ACKNOWLEDGMENTS
We would like to thank Filipp Rybakov, JamesLoudon and Mike Robertson for helpful discussions.ABB acknowledges financial support from the DeutscheForschungsgemeinschaft (SFB 668) and the HamburgCluster of Excellence NANOSPINTRONICS. TLM andMNW acknowledge support from NSERC, and the sup-port of the Canada Foundation for Innovation, the At-lantic Innovation Fund, and other partners which fundthe Facilities for Materials Characterization, managedby the Institute for Research In Materials. We wouldalso like to thank Eric Karhu, Mike Johnson and SimonMeynell for technical assistance.
Appendix: K-H phase diagram details
We present in Fig. 12 the K − H phase diagram overa larger range of K than presented in Fig. 3 and collectthe coordinates of the critical and characteristic points inTable 1. Depending on values of K , the H − K phase dia-gram indicates seven different types of the magnetizationcurves and T − H phase diagrams:1. K ≤
0. In this region the cone phase correspondsto the global minimum of functional w in Eq. (1)in the whole region where the modulated states ex-ist. This describes magnetization processes in bulk( K = 0) and confined cubic helimagnets with hard-axis uniaxial anisotropy ( K < < K < K β = 0 . K . The helicoid remains theenergy minimum at low fields and transforms intothe cone phase along line ( α − B ).3. K β < K < K A = 0 . K . In this narrow intervalthe cone phase is separated from the alternative3modulated states by three first order transitionslines, ( α − A ), ( A − β ), and ( β − B ).4. K A < K < K B = 0 . K . Here the skyrmionlattice area is bounded by the first order lines ( A − C ) and ( A − β − B ) correspondingly into the helicoidand the cone.5. K B < K < K C = 1 . K . In this extended inter-val the evolution of the modulated states followsthe scenario characteristic for noncentrosymmetricuniaxial ferromagnets : at low fields (line ( A − C ))the helicoid transforms by the first-order processinto the skyrmion lattice which gradually trans-forms into a set of isolated skyrmions at criticalline B − C (see Fig. 2).6. K C < K < K E = 2 . K . In this interval thehelicoid directly transforms into the saturated stateat line ( C − E ).7. For K > K E modulated states are totally sup-pressed. In this case isolated skyrmions can existeven at zero field in systems with arbitrary largeanisotropy. FIG. 12. K − H phase diagram in a broad range of inducedanisotropy K and perpendicular field H includes the completeexistence areas of the (meta)stable modulated states. Insetshows the detailed phase diagram within the stability area ofthe skyrmion lattice. TABLE I. Critical and characteristic points in the K − H phase diagram A B C E β γ δ (cid:15)K/K H/H D Archetypical T − H phase diagrams for easy-axis sys-tems with K B < K < K C = 1 .
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