Confinement of Skyrmion states in noncentrosymmetric magnets

A. A. Leonov, A. N. Bogdanov, U. K. RoßlerIFW Dresden, Postfach 270116, D-01171 Dresden, Germany

(Dated: November 3, 2018)

Skyrmionic states in noncentrosymmetric magnets with Lifshitz invariants are investigated withinthe phenomenological Dzyaloshinskii model. The interaction between the chiral Skyrmions, beingrepulsive in a broad temperature range, changes into attraction at high temperatures. This leadsto a remarkable confinement effect: near the ordering temperature Skyrmions exist only as boundstates, and Skyrmion lattices are formed by an unusual instability-type nucleation transition. Nu-merical investigations on two-dimensional models demonstrate the confinement and the occurrenceof different Skyrmion lattice precursor states near the ordering transition that can become thermo-dynamically stable by anisotropy or longitudinal softness in cubic helimagnets. We introduce a newfundamental parameter, confinement temperature Tp separating the peculiar region with ”confined”chiral modulations from the main part of the phase diagram with regular helical and Skyrmionstates.

PACS numbers: 75.10.-b 75.30.Kz 61.30.Mp 03.75.Lm

In certain noncentrosymmetric magnetic systems, theasymmetric Dzyaloshinskii-Moriya (DM) exchange re-sults in chiral couplings that can stabilize long-period,non-collinear modulations of the magnetization with afixed sense of rotation [1, 2]. These chiral couplingsare phenomenologically described by Lifshitz invariantsthat destroy the homogeneity of ordered phases [1]. Inhighly symmetric systems with Lifshitz invariants multi-ple modulations occur as textures with localized twists intwo or more spatial directions [3, 4]. In noncentrosym-metric magnetic systems with Lifshitz invariants thesedouble-twist motifs exist as smooth (baby)-Skyrmions[5–7], static solitonic textures localized in two spatialdirections, which can be extended into the third direc-tion as Skyrmion strings or Hopfions. These magneticSkyrmions are stabilized solely by the chiral DM cou-plings [6, 7], which prevent a spontaneous collapse intotopological singularities. Skyrmionic matter is createdby the condensation of these solitonic units, similar tovortex matter in type-II superconductors [5]. Just suchchiral Skyrmions have been recently observed in thinlayers of noncentrosymmetric ferromagnet (Fe,Co)Si [8].Skyrmionic states stabilized by Lifshitz-type invariantsmay exist and form extended mesophases in various con-densed matter systems, as chiral liquid crystals, ferro-electrics, multiferroics, and in confined achiral systems(e.g., thin magnetic layers) [4, 9–11].

Skyrmionic textures usually form through nucleation,following a classification introduced by DeGennes [12]for (continuous) transitions into incommensurate mod-ulated phases. As demonstrated in Refs. [5, 6], for thelow-temperature micromagnetic model of chiral magnets,at the transition from the field-driven Skyrmion latticesinto the polarized homogeneous state the lattice perioddiverges and the Skyrmions are set free as localized exci-tations. As the Skyrmions retain their size and axisym-metric shape, there is a full spectrum of lattice modes upto the transition, in contrary to an instability type tran-

sition where the amplitude of one fundamental mode actsas small parameter of the transition [12].

Here, we show for the standard model of chiralisotropic ferromagnets [1, 7, 13] that Skyrmions are con-fined very close to the ordering temperature. In thatpart of the phase diagram, the creation of Skyrmions asstable units and their condensation into extended tex-tures occurs simultaneously through a rare case of aninstability-type nucleation transition [14], but the con-fined Skyrmions as discernible units may arrange in dif-ferent mesophases. This is a consequence of the couplingbetween the magnitude and the angular part of the orderparameter. Thus, near the ordering transitions, the localmagnetization is not only multiply twisted but also lon-gitudinally modulated. From numerical investigations on2D models of isotropic chiral ferromagnets, a staggeredhalf-Skyrmion square lattice at zero and low fields and ahexagonal Skyrmion lattice at larger fields are found inoverlapping regions of the phase diagram near the tran-sition temperature. Furthermore, the thermodynamicstability of Skyrmionic states can be favoured with re-spect to one-dimensional modulations by supplementingthe model with cubic exchange anisotropy or in a modi-fied model for metallic chiral magnets [7].

Within the standard phenomenological (Dzyaloshin-skii) theory [1] the magnetic energy density of a non-centrosymmetric ferromagnet can be written in the di-mensionless form [7, 13]

Φ = (gradm)2 − wD(m)− h(n ·m) + am2 +m4. (1)

Here, reduced values of the spatial variable x = r/LD,the magnetization m = M/M0, and the applied magneticfield hn (h = |H|/H0, n is a unity vector along H), areexpressed via the parameters of the energy density [13]w(M) = A(gradM)2 +DwD(M)−H ·M+f(M) (wheref(M) = a1M

2 + a2M4, and M = |M|): LD = A/D,

H0 = κM0, M0 = (κ/a2)1/2, a = a1/κ, κ = D2/(A).

arX

iv:1

001.

1292

v3 [

cond

-mat

.sta

t-m

ech]

17

Dec

201

0

2

m/m

0

1.0

0.5

0.05 10 15

00

(a) (b) (c)

�

�

�

FIG. 1: (Color online) (a) Equilibrium solutions of local-ized Skyrmions, shown as cross section in the inset, displaystrong localization of θ(ρ) (blue) and weak variation of themagnetization modulus m(ρ) (red) in a broad range of tem-perature and field (a, h). Exemplary profiles for a = −0.25,h = 0.25. (b) At lower fields, isolated Skyrmions condenseinto a dense-packed hexagonal lattice. (c) Near the orderingtemperature ac = 0.25 square lattice solutions with staggeredhalf-Skyrmion cells arise.

DM energy wD consists of Lifshitz invariants [1]

L(k)ij = mi(∂mj/∂xk)−mj(∂mi/∂xk). (2)

For noncentrosymmetric uniaxial ferromagnets DM func-tionals wD are listed in [5]. In particular, for isotropic

and cubic helimagnets wD(m) = L(y)xz + L(x)

zy + L(z)xy

= m·rotm [13] (for details see appendix). Functional (1)includes only the basic (isotropic) interactions essentialto stabilize chiral modulations. This depends on threeinternal variables (components of the magnetization vec-tor m) and two control parameters, the reduced magneticfield amplutude h and the ”effective” temperature a(T ).In chiral magnetism models (1) play a fundamental roleand are similar to the Frank energy in liquid crystals andLandau-Ginzburg functional in superconductivity. Weconsider 2D chiral modulations homogeneous along theapplied field h||z and modulated in the plane perpendic-ular h.

Isolated and embedded Skyrmions. The equations min-imizing functional (1) include solutions for axisymmet-ric localized states (isolated Skyrmions), ψ = ψ(φ), andθ(ρ), m(ρ) where we use spherical coordinates for themagnetization (m, θ, ψ) and cylindrical coordinates forthe spatial variables (ρ, φ, z). The solutions ψ(φ) for allnoncentrosymmetric classes have been derived in [5]. Forcubic helimagnets and uniaxial systems with n22 sym-metry, ψ = φ− π/2 (Fig. 1 a). Profiles θ(ρ) and m(ρ) ofisolated Skyrmions are derived from numerical solutionsof the Euler equations, as given in Ref. 7 and common forall noncentrosymmetric classes with Lifshitz invariants,with the boundary conditions θ(0) = π, dm/dρ(0) = 0,θ(∞) = 0, m(∞) = m0 (m0 is the magnetization inthe saturated state). For extended textures of two-dimensional models, the functional has been investigatedby numerical energy minimization using finite-differencediscretization on rectangular grids with adjustable gridspacings [7].

Isolated Skyrmions (Fig. 1 (a)) exist only below a crit-ical line h0 and condense into a hexagonal lattice (Fig. 2)

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

(a)

( )c

I

III

pA

B

u

hc

h0

hn

hn

hn

h

q

h

Skyrmion Lattice

Hexagonal

0.4

0.2

0.0-1.0 -0.5 0.0 0.25 0.23 0.25a

5

0

h1

02

�

� Helicoid

Hexagonal SLSquare

SL�

0

0.1

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

(b)Bhc

h0

0.1 0.2 a

hcII

IsolatedSkyrmions

FIG. 2: (Color online) Field h vs. temperature a phasediagram. (a) Areas with repulsive (I) (shaded), attractive(II) Skyrmions (hatched), and strictly confined pocket (III)(cross-hatched) are separated by h? (dashed line), Eq. (3).Isolated Skyrmions collapse at critical line h0. Above thisline no static Skyrmions exist. Below line hc Skyrmions con-dense into a hexagonal lattice In region (II), the hexagonalSkyrmion lattice (SL) exists as metastable state up to thenucleation field hn. For temperatures between points A andB, hn < h0, for larger temperatures aB < a < ac = 0.25 theisolated Skyrmions disappear at lower fields than the denseSL, h0 > hn. For clarity, line hn is only schematically given inpanel (a), numerically exact data are shown in panel (b). De-tail of the phase diagram (c) near the ordering temperatureshows the existence regions for different modulated states.Lines for first order transitions: α square half-SL ↔ hexago-nal SL, β helicoid↔ hexagonal SL. Line γ marks the continu-ous transition from the conical equilibrium phase in isotropicsystems to the paramagnetic phase.

below a field hc, which marks the transition between theSkyrmion lattice (SL) and the homogeneous paramag-netic state (see Fig. 2 (a), (b)). Near the ordering tem-perature a square half-Skyrmion lattice (Fig. 1 (c)) haslower energy than the hexagonal lattice. Half-SLs con-sist of cells with up and down magnetization in the cen-ter and in-plane magnetization along the cell boundaries.Such cells have a topological charge 1/2. In the hexag-onal SLs the magnetization at the boundaries (center)is (anti)parallel to the applied field. The cells bear unittopological charges.

Confinement. By solving the linearized Euler equa-tions for the asymptotics of isolated Skyrmions ∆m =(m−m0), θ ∝ exp(−κρ) (ρ� 1) one finds three distinctregions in the magnetic phase diagram with differentcharacter of Skyrmion-Skyrmion interactions (Fig. 2 (a)):repulsive interactions in a broad temperature range (area(I)) are changed to attractive interaction at higher tem-peratures (area (II)). Finally in area (III) near the order-ing temperature ac = 0.25 strictly confined Skyrmionsexist. These regions are separated by the line

h? =√

2± P (a)(a+ 1± P (a)/2), P (a) =√

3 + 4a (3)

with turning points p (−0.75,√

2/4), q (0.06, 0.032√

5),and u (-0.5, 0) (dashed line in Fig. 2 (a)). In the majorpart of the phase diagram, the evolution of SLs under a

3

FIG. 3: (Color online) Numerically exact solutions for hexag-onal (a)–(c) and square (d)–(f) Skyrmion lattices within theconfinement pocket. Contour plots of Mz(x, y) and M(x, y)(a),(d) and their diagonal cross-sections (b),( e), (c),(f) arederived by minimization of Φ (1) with a = 0.23 and differentvalues of the applied field. Solid (dashed) lines in panels (e),(f) indicate profiles for cells with (anti)parallel magnetizationin the center.

magnetic field closely agrees with the behavior studiedearlier for the low-temperature limit [5, 6] and the tran-sition mechanism at the high-field limit is of the nucle-ation type with isolated Skyrmion excitations appearingbelow the instability line h0. The temperature of theturning point p (ap = −0.75), a confinement tempera-ture Tc determines the frontier between low- (T < Tp)and high-temperature (Tp < a < Tc) chiral modulations.Usually ∆T = Tc− Tp � Tc and high-temperature mod-ulations are restricted by a narrow vicinity of the Neeltemperature. According to calculations in [7] for MnSihigh-temperature Skyrmions exist in temperature inter-val ∆T = Tc − Tp = 2 K below the Curie temperatureTc.

Evolution of confined Skyrmions. In the vicinity of theordering temperature the Skyrmion lattices exist withinthe confinement pocket (III). In this region Skyrmionstates drastically differ from those in the main part ofthe phase diagram. Due to the ”softness” of the magne-

tization modulus the field-driven transformation of theSkyrmion lattices evolves by distortions of the modulusprofiles m(ρ) both in the hexagonal and square Skyrmionlattices while the equilibrium periods of the lattices donot change strongly with increasing applied field (Fig.3). Despite the strong transformation of their internalstructures the Skyrmion lattices preserve axisymmetricdistribution of the magnetization near the centers of theSkyrmion lattice cells (Fig. 3). This remarkable propertyreflects the basic physical mechanism underlying the for-mation of Skyrmion lattices. The local energetic advan-tage of Skyrmion lattices over helicoids is due to a largerenergy reduction in the “double-twisted” Skyrmion cellcore compared to “single-twisted” helical states [4, 7].This explains the unusual axial symmetry of the cell coresand their stability. An increasing magnetic field gradu-ally suppresses the antiparallel magnetization in the cellcore reducing the energetic advantage of the “double-twist” and increases the overall energy of the condensedSkyrmion lattice. At critical field hc a first-order tran-sition occurs into the polarized paramagnetic state. Inthe interval hc < h < hn the hexagonal Skyrmion latticeexists as a metastable state. At the lability field hn themagnetization modulus in the cell center becomes zero(see magnetization profile for h = 0.042 in Fig. 3 (c)).At this field, the ”double-twist” region is suppressed, andthe lattice loses its stability.

At zero field with increasing temperature a the mag-netization modulus m in hexagonal and square latticesgradually decreases to zero at the ordering temperature.This is the instability-type nucleation transition into theparamagnetic phase : the order parameter m becomeszero at the transition point (as in the instability mode),however, the lattices retain their symmetry and the ar-rangement of axisymmetric Skyrmions up to the criti-cal point. In finite fields in the region (II) the attrac-tive Skyrmion-Skyrmion interaction means that multi-Skyrmions can always form bound states. Near ac suchclusters of Skyrmions are more stable than the isolatedSkyrmions. This is seen in Fig. 2 (a),(b). For tempera-tures above point B, aB < a < ac, the metastable SL asthe densest infinite cluster is more stable than isolatedSkyrmions. Thus, in region (II) of the phase diagram,there is clustering and, for higher temperatures, confine-ment of isolated Skyrmion excitations.

The predicted Skyrmionic textures and the pre-cursor phenomena associated with the confinement ofSkyrmions near the ordering transition are observablein magnetic materials with appropriate symmetry. In alarge group of uniaxial noncentrosymmetric magnets theDM energy is described by gradients only along directionsperpendicular to the axis (e.g. multiferroic BiFeO3, spacegroup R3c and antiferromagnetic Ba2CuGe2O7, spacegroup P 421m [2, 15]). In such magnets one-dimensionalmodulations with the propagation vector in the basalplane (helicoids) exist in broad ranges of the magnetic

4

0.21 0.23 0.250

2

4

a

Half-SkyrmionLattice

Cone

(a)

HexagonalSkyrmion

Lattice

h10

2

a0.18 0.22 0.26

Cone

SaturatedState

SaturatedState

Helicoid

(b)

0

4

8

Half-Skyrmion Lattice

h10

2

hx

�

�

FIG. 4: (Color online) Magnetic phase diagrams of cubic heli-magnets with exchange anisotropy b = −0.05 and the appliedfield along (111) (solid) and (001) (dashed) axes (a) containsregions with thermodynamically stable hexagonal SLs andhalf-SLs. Magnetic phase diagram of the modified isotropicmodel with η = 0.8 (b) has extended domains with thermo-dynamically stable helicoid, half-SLs and hexagonal SLs withthe magnetization along the cell axes parallel to the magneticfield (hatched area).

fields [1, 16]. In the confinement pocket of the phase di-agram (III) the helicoids also exist only as bound states[16]. They have the lowest energy at lower fields, and thehexagonal SL corresponds to the global energy minimumat higher fields (Fig. 2 (c)).

Cubic helimagnets. In other groups of chiral mag-nets the DM energy includes contributions with gradientsalong all three spatial directions. This stabilizes chiralmodulations with propagation along the direction of anapplied field as cone phases [13]. For the isotropic modelΦ(m) (1) the cone phase solution with the fixed magne-tization modulus and rotation of m around the appliedmagnetic field: ψ = z, cos(θ) = h/m, m = |a − 1/2|/2,is the global energy minimum in the whole region wheremodulated states exist. For cubic helimagnets, the en-ergy density (1) has to be supplemented by anisotropiccontributions, Φa = b

∑i(∂mi/∂xi)

2 + k∑im

4i , where

b and k are reduced values of exchange and cubicanisotropies [13]. These anisotropic interactions impairthe ideal harmonic twisting of the cone phase and leadto the thermodynamic stability of Skyrmion states, asshown in the equilibrium phase diagram Fig. 4 (a). Thedifference between the energy of the hexagonal Skyrmionlattice Wsk and of the cone phase Wcone calculated forthe isotropic model, ∆Wmin = Wsk −Wcone, has min-ima along a curve hx(a) which reaches the critical pointhx(ac) = 0 as ∆Wmin = 0.0784(0.25 − a)). Weak ex-change anisotropy of a cubic helimagnet, therefore, cre-ates a pocket around ac, where the hexagonal Skyrmionlattice becomes the global energy minimum in a field(Fig. 4 (a)). This case is realized in cubic helimag-nets with negative exchange anisotropy (b < 0) as inMnSi [13]. This anisotropy effect provides a basic mech-anism, by which a Skyrmionic texture is stabilized inapplied fields, as observed experimentally as so-called”A-phase” in MnSi [17–19]. The exchange anisotropy

b < 0 also leads to the thermodynamic stability of half-SLs (Fig. 4 (a)). The stabilization of such textures maybe responsible for anomalous precursor effects in cubichelimagnets in zero field [7, 20, 21].

Chiral modulations in non-Heisenberg models. A gen-eralization of isotropic chiral magnets proposed in [7]replaces the usual Heisenberg-like exchange model bya non-linear sigma-model coupled to a modulus fieldwith different stiffnesses. This yields a generalized gra-dient energy for a chiral isotropic system with a vec-tor order parameter, which is equivalent to the phe-nomenological theory in the director formalism [4, 7]∑i,j(∂imj)

2 →∑i,j(∂imj)

2 + (1 − η)∑i,j(∂im)2 =

m2∑i,j(∂inj)

2 + η∑i(∂im)2. Parameter η equals unity

for a “Heisenberg” model, in chiral nematics η = 1/3 [4].Confined chiral modulations are very sensitive to valuesof η < 1. The magnetic phase diagram calculated forη = 0.8 includes pockets with square half-SLs, hexagonalSLs with the magnetization in the center of the cells par-allel to the applied field, and helicoids with propagationtransverse to the applied field (Fig. 4 b).

The confinement effects on chiral Skyrmions stronglychanges the picture of the formation and evolution of chi-ral modulated textures and shed new light on the prob-lem of precursor states observed as blue phases in chiralnematics [4] and in chiral magnets [7, 17–21]. The re-sults show that confinement / deconfinement transitionsof localized string-like solitons can be realized in thesecondensed matter systems. They provide counterpartsof formation mechanisms for extended microscopic mat-ter from topological solitons, as devised originally in theSkyrme model [22].

Acknowledgment. We thank K. v. Bergmann, G.Bihlmayer, S. Blugel, H. Eschrig, S.V. Grigoriev, J.-H.Han, S. Heinze, R. Mossner, C. Pappas, W. Selke, H.Wilhelm, and M. Zhitomirsky for discussions. Supportby DFG project RO 2238/9-1 is gratefully acknowledged.

APPENDIX:DZYALOSHINSKII THEORY OFCHIRAL HELIMAGNETISM

The appendix includes technical comments on themathematical structure of the interaction functional (1),on the mathematical relation between this model andthose for different classes of noncentrosymmetric mag-netic systems and chiral liquid crystals. The equationsfor isolated and bound Skyrmions are provided alongwith a short information on the numerical methods tosolve these problems.

1. Basic model.

In noncentrosymmetric ferromagnets chiral asymmetryof exchange coupling related to the quantum-mechanical

5

Dzyaloshinskii-Moriya interactions [23] induces long-period modulations of the magnetization vector M witha fixed rotation sense [1]. The equilibrium chiral modula-tions are derived by minimization of the magnetic energywhich can be written in the following general form [1]

W = A∑i,j

(∂Mj/∂xi)2−M ·H+W0(M)+WD(M) (4)

where A is the exchange stiffness constant, H is an ap-plied magnetic field, W0 collects short-range magnetic in-teractions independent on spatial derivatives of the mag-netization, and Dzyaloshinsky-Moriya energy WD is com-posed of antisymmetric terms linear with respect to firstspatial derivatives of the magnetization vector M(r)

L(k)ij = Mi (∂Mj/∂xk)−Mi (∂Mj/∂xk) .

Functional forms of WD energy contributions are deter-mined by crystallographic symmetry of a noncentrosym-metric magnetic crystal [1, 5, 13]. Particularly, for im-portant cases of noncentrosymmetric ferromagnets be-longing to cubic (23, 432) and uniaxial crystallographicclasses (nmm, 42m) the DM energy contributions havethe following form [1, 5, 13]

WD = D (L(z)yx + L(y)

xz + L(x)zy ) = DM · rotM

for (23), (432);

WD = D (L(x)xz + L(y)

yz ) for (nmm);

WD = D (L(y)xz + L(x)

yz ) for (42m) (5)

where n = 3, 4, 6 and D is a Dzyaloshinskii constant.For other noncentrosymmetric classes this WD energy in-cludes two or more Dzyaloshinskii constants [5].

Near the ordering temperatures the magnetization am-plitude varies under the influence of the applied field andtemperature. Commonly this process is described by in-cluding into the magnetic energy an additional term [13]

W →W + a1M2 + a2M

4, a1 = J(T − Tc), a2 > 0 (6)

This model is fundamental to the systematic phe-nomenological description of magnetic states in noncen-trosymmetric magnetic systems [1]. Interaction func-tional (4) plays in chiral magnetism a similar role asthe Frank energy in liquid crystals [24] or Ginzburg-Landau functional in physics of superconductivity [25].Its form is dictated by the natural extension of Landau’sapproach to ordering transtions for modulated systemswhich break the third Lifshitz condition, as pioneered byDzyaloshinskii in particular in magnetism [26]. As a re-sult, this type of theory constitutes the canonical formof statistical field theories for condensed matter systems,where particular couplings and symmetry enable Lifshitzinvariants to induce modulated phases. Extensions in-cluding higher-order secondary effects in magnetism (likeanisotropies or dipolar interactions) are widely used to

describe evolution of modulated states in many chiralmagnets [2, 6, 7, 27], including metamagnetic transitionsin cubic helimagnets induced by high pressure [13, 28, 29].

By rescaling the spatial variable in (1),(6) x = r/LD,the magnetic field h = H/H0, the magnetization m =M/M0

LD = A/D, H0 = κM0, M0 = (κ/a2)1/2,

κ = D2/(2A), a = a1/κ = J(T − Tc)/κ. (7)

energy W (6) can be written in the following reducedform (Eq. (1))

Φ = (gradm)2 − wD(m)− h(n ·m) + am2 +m4, (8)

where h = |h| and n is a unity vector along the ap-plied magnetic field. Functional (8) includes three inter-nal variables (components of the magnetization vectorm) and two control parameters, the reduced magneticfield amplutude h and the ”effective” temperature a(T )(7). By direct minimization of Eq. (8) we have derivedone-dimensional (kinks, helicoids, conical helices or spi-rals) and two-dimensional solutions (isolated and boundSkyrmions) for arbitrary values of the control parame-ters. These results are collected in the phase diagram ofsolutions (Fig. 2).

Energy (8) includes only basic interactions essensialto stabilize Skyrmionic and helical phases. Solutionsfor chiral modulated phases and their most general fea-tures attributed to all chiral ferromagnets are deter-mined by interactions functional (8). Generically, thereare only small energy differences between these differ-ent modulated states. On the other hand, weaker en-ergy contributions (as magnetic anisotropy, stray-fields,magneto-elastic couplings) impose distortions on solu-tions of model (8) which reflect crystallographic symme-try and values of magnetic interactions in individual chi-ral magnets. These weaker interactions determine thestability limits of the different modulated states (Fig.4). The fact that thermodynamical stability of individualphases and conditions of phase transfomations betweenthem are determined by magnetocrystalline anisotropyand other relativistic or weaker interactions means that(i) the basic theory only determines a set of differentand unusual modulated phases, while (ii) the transitionsbetween these modulated states, and their evolution inmagnetization processes depends on symmetry and de-tails of magnetic secondary effects in chiral magnets, inparticular the strengths of relativistic magnetic interac-tions. Thus functional (8) is the generic model for a man-ifold of interaction functionals describing different groupsof noncentrosymmetric magnetic crystals, because it al-lows to identify the basic modulated structures that maybe found in them.

6

2. Equations for isolated and bound Skyrmions

The Euler equations for functional (8) have solutionsfor axisymmetric isolated structures of type ψ = ψ(φ),θ(ρ), m(ρ) (we introduce here spherical coordinates forthe magnetization (m, θ, ψ) and cylindrical coordinatesfor the spatial variables (ρ, φ, z)) [7]. Solutions ψ = ψ(φ)are known for all uniaxial and cubic noncentrosymmet-ric ferromagnets [5]. Particularly, ψ = φ − π/2 for cu-bic helimagnets, ψ = φ and ψ = −φ − π/2 for uniax-ial ferromagnets with (nmm) and 42m symmetry, corre-spondingly. After substitution of solutions ψ = ψ(φ)into (8) and integration with respect to φ the totalenergy E of a Skyrmion (per unit length along z) isE = 2π

∫∞0

Φ(m, θ)ρdρ with energy density

Φ = m2ρ +m2

[θ2ρ +

sin2 θ

ρ2− θρ −

sin θ cos θ

ρ

]+ am2 +

+m4 − hm cos θ (9)

where a common convention ∂f/∂x ≡ fx is applied. TheEuler equations for the functional (9)

m

[θρρ +

θρρ− sin θ cos θ

ρ2− 2 sin2 θ

ρ− h sin(θ)

]+

+2 (θρ − 1)mρ = 0,

mρρ +mρ

ρ+m

[θ2ρ +

sin2 θ

ρ2− θρ −

sin θ cos θ

ρ

]+

+2am+ 4m3 − h cos(θ) = 0 (10)

with boundary conditions θ(0) = π, θ(∞) = 0, m(∞) =m0 describe the structure of isolated Skyrmions (themagnetization of the homogeneous phase m0 is derivedfrom equation 2am + 4m3 − h = 0). Typical solutionsθ(ρ), m(ρ) of Eqs. (10) are plotted in Fig. 1.

For present work, we have evaluated numerically thesolutions for the energy functionals of type Eq. (1) withina standard approach using finite differences for gradi-ent terms and adjustable grids to accommmodate mod-ulated states with periodic boundary conditions. Searchfor modulated states and energy minimization was doneby Monte Carlo simulated annealing. For 2-dimensionalmodels, this approach can give converged results, aschecked by comparison with analytical results. Includingsecondary effects, the approach is able and has been usedby us to calculate thermodynamic stability of phases,and to study the complete temperature-field phase di-agrams in terms of equilbrium (mean-field) states of theenergy functionals Eq. (1). As we are investigating here

long-period modulated phases, the phase diagrams areexpected to give qualitatively the correct results for thesephenomenological models.

[1] I.E. Dzyaloshinskii, Sov. Phys. JETP 19, 960.[2] A. N. Bogdanov et al., Phys. Rev. B 66, 214410 (2002);[3] R.M. Hornreich et al., Phys. Rev. Lett. 48, 1404 (1982).[4] D. C Wright, N. D. Mermin, Rev. Mod. Phys. 61, 385

(1989).[5] A.N. Bogdanov and D.A. Yablonsky, Sov. Phys. JETP

68, 101 (1989).[6] A. Bogdanov, A. Hubert, J. Magn. Magn. Mater. 138,

255 (1994); ibid. 195, 182 (1999); A. Bogdanov, JETPLett. 62, 247 (1995); A.N. Bogdanov et al., Physica B359, 1162 (2005).

[7] U.K. Roßler et al., Nature 442, 797 (2006).[8] X. Z. Yu et al., Nature 465, 901 (2010).[9] A. N. Bogdanov, U. K. Roßler, Phys. Rev. Lett. 87,

037203 (2001).[10] M. Bode et al., Nature 447 190 (2007).[11] U. Tkalec et al., Phys. Rev. Lett. 103, 127801 (2009).[12] P.G. de Gennes, in Fluctuations, Instabilities, and Phase

transitions, ed. T. Riste, NATO ASI Ser. B, vol. 2(Plenum, New York, 1975).

[13] P. Bak and M. H. Jensen, J. Phys.C: Solid State Phys.13, L881 (1980); O. Nakanishi et al., Solid State Comm.35, 995 (1980).

[14] J. W. Felix et al., Phys. Rev. Lett. 57, 2180 (1986).[15] I. Sosnowska et al., J. Phys. C 15, 4835 (1982); A. Zhe-

ludev et al. Phys. Rev. Lett. 78, 4857 (1997).[16] B. Schaub, D. Mukamel, Phys. Rev. B 32, 6385 (1985).[17] C. I. Gregory et al., J. Magn. Magn. Mater. 104-107,

689 (1992); D. Lamago et al., Physica B 385-386, 385(2006).

[18] B. Lebech et al., J. Magn. Magn. Mater. 140, 119 (1995).[19] S. Muhlbauer et al., Science, 323, 915 (2009).[20] C. Pfleiderer et al., Nature 427, 227 (2004).[21] C. Pappas et al., Phys. Rev. Lett. 102, 197202 (2009).[22] T. H. R. Skyrme, Proc. Roy. Soc. Lon. 260, 127 (1961).[23] I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957), T.

Moriya, Phys. Rev. 120, 91 (1960).[24] P. G. De Gennes and J. Prost, The Physics of Liquid

Crystals (Oxford University Press, Oxford, 1993), 2nded., D. C Wright, N. D. Mermin, Rev. Mod. Phys. 61,385 (1989).

[25] E. H. Brandt, Rep. Prog. Phys. 58, 1465 (1995), Phys.Rev. B, 68, 054506 (2003).

[26] See, e.g., J.-C. Toledano, P. Toledano, The Landau The-ory of Phase Transitions, World Scientific, Singapore1987, chap. V and chap. VI,6.

[27] Yu. A. Izyumov, Sov. Phys. Usp. 27, 845 (1984).[28] G. G. Lonzarich, L. Taillefer, J. Phys. C. 18, 4339 (1985),

M. Yamada et al. J. Alloys Comp. 364, 37, (2004).[29] M. L. Plumer, M. B. Walker, J. Phys. C. 14, 4689 (1981),

M. L. Plumer, J. Phys.:Cond Matt. 2, 7503 (1990).