Topological defects in helical magnets
JJETP, November 2018
Topological defects in helical magnets
T. Nattermann and V.L. Pokrovsky
2, 3 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany Department of Physics, Texas A&M University, College Station, TX 77843-4242 Landau Institute for Theoretical Physics, Chernogolovka, Moscow District 142432, Russia (Dated: October 2, 2018)Helical magnets which violated space inversion symmetry have rather peculiar topological defects.In isotropic helical magnets with exchange and Dzyaloshinskii-Moriya interactions there are onlythree types of linear defects: ± π and 2 π -disclinations. Weak crystal anysotropy suppresses lineardefects on large scale. Instead planar defects appear: domain walls that separate domains withdifferent preferential directions of helical wave vectors. The appearance of such domain walls inthe bulk helical magnets and some of their properties were predicted in the work . In a recentwork by an international team of experimenters and theorists the existence of new types of domainwalls on crystal faces of helical magnet FeGe was discovered. They have many features predictedby theory , but display also unexpected properties, one of them is the possibility of arbitrary anglebetween helical wave vectors. Depending on this angle the domain walls observed in can be dividedin two classes: smooth and zig-zag. This article contains a mini-review of the existing theory andexperiment. It also contains new results that explain why in a system with continuos orientation ofhelical wave vectors domain walls are possible. We discuss why and at what conditions smooth andzig-zag domain walls appear, analyze spin textures associated with helical domain walls and findthe dependence of their width on angle between helical wave vectors. PACS numbers:
I. INTRODUCTION
We are happy to congratulate Lev Pitaevskii on oc-casion of his Jubilee. We are amazed by the depth andbrilliance of his scientific achievements that include suchpearls as the
Gross-Pitaevskii equation and the
Theoryof electromagnetic fluctuations in dispersive media . Butnot less surprising is his universalism, so rare in our time,and his irreproachable scientific integrity. These quali-ties together made him a unique person, able to continuethe magnificient Landau-Lifshitz compendium of moderntheoretical physics. The fact that it is still not simplyalive after the death of both inital authors, but is anindispensable belonging of any private and institutionalphysical library, is the result of his tireless work. One ofus (VP) has the privilege and pleasure to be in friendshipwith him starting from the end of 1950-th. He learned alot from scientific discussions with Lev but not only that.Lev’s natural kindness and sincerity is an important partof life for all his friends. We wish him good health andhigh spirit.In this article, written in his honour, we review theoryand experiment on topological defects in helical magnets.Topological defects are among the most facscinating ob-jects of condensed matter physics and quantum fieldtheory . They almost unavoidably appear in orderedphases, either as a result of initial conditions like cosmicstrings , or as equilibrium configurations like vortex lat-tices in type-II superconductors and superfluids . De-fects counteract the emergent rigidity of the condensateand hence are fundamental both from the point of viewof basic science as well as of practical applications Topological defects are different in systems with dif-ferent symmetries and types of its violation in the or-dered states. Some of them are well known, for examplequantized vortices in quantum liquids or domain wallsin magnets. Skyrmions are more sophisticated butobtained a broad advertisement recently due to experi-ments in 2-dimensional magnets and nuclear matter .Defects in superfluid helium 3 with its complex orderingare more exotic. Their description and literature can befound in the book by Volovik .Topological defects in helical magnets are interestingnot only as a subject of pure science, but also becausethey interact with electric current and thus may servefor transformation of magnetic signals into electric onesand back, that is a basic element of sensing, transfer andstoring of information and energy.In the year 2012 Fuxiang Li and the authors publishedan article on domain walls in helical magnets, in whichwe predicted rather unusual properties of these defects .Most of these predictions were confirmed in a fundamen-tal experiments performed by the international team ofexperimenters and theorists in 2018 , but not all andsome unexpected features appeared. In the following textwe analyze these experiments and explain some of dis-crepancies between our theory and the experimental ob-servations. Explanations of other facts require a furtherdevelopment of theory. The main reason of discrepanciesis that Magnetic Force Microscope (MFM), used in theseexperiments, gives information on magnetic textures oncrystal faces of samples, whereas our theory had in mindthe bulk.The content of the following article is as follows. In a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t he second section we present a general analysis of thetopological defects in isotropic helical magnets. In thethird section we consider in more details domain walls inthe bulk helical magnets. The fourth section is dedicatedto helical domain walls at crystal faces. It contains abrief description of the experimental observations madein the work and modifications of the domain walls theorynecessary in this situation. In this section we comparetheory given in our article and in the cited publicationby international group and experiment. In Conclusion wesummarize new results of this article and discuss unsolvedproblems.We conclude the introduction by a brief descriptionof the interactions essential for helical magnets and thestructure of helical ordering without defects. In this ar-ticle we consider systems without inversion symmetry.The violation of this symmetry in helical magnets is as-sociated with Dzyaloshinskii-Moriya interaction. Typicalhelical magnets with these properties are alloys MnSi ,FeGe , Fe x Co − x Si . The Hamiltonian of spin systemin these helical magnets in continuous approximation canbe represented as follows:H¸ = (cid:90) d x (cid:34) J ∇ m ) + g m ( ∇ × m ) + v (cid:88) α =1 m α (cid:35) . (1)Here we measured all lengths in units of the lattice pa-rameter a . The first term in (1) represents the ex-change interaction, the second is the Dzyaloshinskii-Moriya (DM) interaction and the third term is the crystalfield energy corresponding to cubic anisotropy of the lat-tice. The dimensionless magnetization vector m in thisapproach has unit length. The hierarchy of interactionsis J (cid:29) g (cid:29) v .In the absence of crystal field the energy of systemdescribed by Hamiltonian (1) has minimum at magneti-zation field equal to m ( r ) = ˆ e cos q · r + ˆ e sin q · r , (2)where q is a vector with fixed modulus q = q ≡ ( g/J )ˆ q . The three mutually perpendicular unit vectors { ˆ e , ˆ e , ˆ q } ≡ T form a right tripod if g > g <
0. Its orientation in space is arbitrary. Anadditional phase in the argument of sine and cosine canbe absorbed in a rotation of T around ˆ q . The only lengthscale which appears in (2) is (cid:96) = 2 πJ/g which is the pitchof the helix. In FeGe, (cid:96) = 70 a . In this structure magne-tization is constant in planes perpendicular to the wavevector of helix q and rotates at motion along the helixvector. It means that the vectors q and − q correspondto the same helical structure.Weak cubic anisotropy lifts the degeneracy of the helixenergy with respect to direction of q . It pins the helixaxes to the direction of one of the cube diagonals (three-fold axis) if v > v <
0. The anisotropy also produces lit-tle distortions of helical structure which will be neglected.Neutron magnetic diffraction experiments have shown that in FeGe the axis of helix coincides with the direction(100) or equivalent. It means that the constant of cubicanisotropy v in this material is negative. II. TOPOLOGICAL DEFECTS IN ISOTROPICHELICAL MAGNETS
The central object of the topological classification ofdefects is the degeneracy space R (or order parameterspace), i.e. the manifold of internal states possessing thesame free energy . The value of order parameter ineach point of the d -dimensional (sub)space V d surround-ing the defect determines its mapping onto the degener-acy space R . This mapping can be classified into ensem-bles of equivalent maps, which form the d − th homotopygroup π d . If R is disconnected, as in systems with adiscrete symmetry, then one type of defects are domainwalls. Inside the wall the order parameter changes be-tween its values in the domains, its width is commonlyrelated to the strength of the anisotropy responsible forthe discrete symmetry.To classify defects in helical magnets we consider firstthe degeneracy space. Homogeneous rotations of the tri-pod T induce transitions to other states with the samefree energy. Any rotation in 3-dimensional vector spacecan be parametrized by a vector ω , whose direction de-fines the axis of rotation and absolute value ω definesrotation angle. Acting onto a vector a it transforms itinto a vector a (cid:48) defined by equation: a (cid:48) = a cos ω +( ˆ ω × a ) sin ω + ˆ ω ( ˆ ω · a )(1 − cos ω ) ≡ e ω ˆ Ω a = e ω × a . (3)In the second line we introduced the linear operator ˆ Ω defined by its action onto a vector of 3-dimensional space:ˆ Ω a ≡ ˆ ω × a . The rotation angle ω is confined to the in-terval 0 ≤ ω ≤ π . Negative angles will be described asrotations about − ˆ ω . The set of all rotation vectors ω fills a 3-dimensional sphere of the radius π . Since rota-tions around ˆ ω and − ˆ ω by π lead to the same result,diametrically opposite points t ± = ± π ˆ ω on the sphere’ssurface are equivalent. For equivalence of two operationswe will use symbol ∼ . Thus, t + ∼ t − . This considera-tion shows that the order parameter space for isotropichelical magnets is isomorphic to the group of rotation in3-dimensional space SO .Helix configuration (2) can be considered as a rotationof vector e about ˆ q by an angle q · r , i.e. m ( r ) = e ( q · r ) ˆ q × ˆ e . (4)Eq. (4) is invariant under the replacement of q by − q .The latter can be treated as a rotation of q around e by π . Thus, the invariance with respect to change of sign q can be formulated as e π ˆ e × m = m . In other words,rotations around e by ω and about − e by ( π − ω ) areequivalent: e ω ˆ e × m = e − ( π − ω )ˆ e × m . (5)2herefore, also the inner points of the sphere u + = ω ˆ e and u − = ( ω − π )ˆ e are equivalent, that is u + ∼ u − .Correspondingly, the order parameter space R is reducedto R = SO / Z . This is a sphere of radius π with points t + and t − , and u + and u − identified. Its first homotopygroup is π ( SO / Z ) = Z , (6)which is the group of integers modulo 4 . This resultimplies that there are three types of topologically stableline defects in helical magnets called disclinations.Analogous defects appear in cholesteric liquid crystalsand superfluid He .In disclinations orientation of the tripod T variesin space. Different disclination are characterised bychange of the tripod orientation in course of circulationof the coordinate vector r along a closed curves sur-rounding the central line of defect. These we parametrizeas r ( s ), where s denotes a contour variable chaging inthe interval (0 , r (0) = r (1). Eachclosed curve in the real space maps into a closed curvein the order parameter space SO / Z which we denoteas ω ( s ). Closed curves in the order parameter space canbe classified according to their total rotation angle. t +
0. (7) θ has a branch cut at x = 0 , y ≤
0. The rotated vectorsare ˆ e (cid:48) = ˆ e ≡ ˆ z , q (cid:48) = q cos θ + ˆ e × q sin θ and hence m (cid:48) ( r ) = ˆ e cos q (cid:48) · r + ˆ q (cid:48) × ˆ e sin q (cid:48) · r . (8)The function θ ( x, y ) has been used to create the magne-tization profile in Fig. 1 III. DOMAIN WALLS IN THE BULK
Since the anisotropy allows several discrete orienta-tions of the helix, wave vector domains with differenthelix orientation separated by domain walls (hDW) canappear. It was shown in that DWs in helical magnetsare fundamentally different from common Bloch and Neelwalls . These DWs are one-dimensional textures inwhich magnetization rotates around a fixed axis. ThehDWs are generically two-dimensional textures with ro-tating axis of rotation. For a range of orientations thehDWs contain a regular lattice of disclinations. ThehDWs that are bisector planes between two helix wavevectors in different domains are free of disclinations andhave minimal surface energy.Let the helix wave vectors in two domains be q and q .They are not collinear. In the bulk the angles betweenthem is either π/ v < if v >
0. Since theasymptotic dependence of magnetization in two limitingcases cannot be described by one variable, the texturethat connects the two asymptotical helixes must dependat least on two variables.The second important fact is that the wave vector in-side the domain wall necessarily changes its length. In-deed, any continuous distribution of magnetization canbe described by function φ ( r ) whose gradient is the localvalue of wave vector q ( r ) = ∇ φ ( r ). It is reasonable toassume that φ depends only on coordinates in the planeof the two wave vectors. In other words, the domain wallmust be a plane perpendicular to the plane ( q , q ). Letus assume that the lines of wave vectors are continuouscurves asymptotically approaching straight lines in di-rection q and q in different domains. Magnetization3s described by eq. (2) in which the argument of sineand cosine is replaced by φ ( r ). It is possible to take unitvector ˆ e perpendicular to the plane ( q , q ). The localvector ˆ e ( r ) is uniquely determined by the vectors q ( r )and ˆ e . The requirement of constant modulus for wavevector leads to equation ( ∇ φ ) = const. This equationcoincides with the stationary Hamilton-Jacobi equationfor free particle. The vector ∇ φ is the momentum ofthis particle. But free particle can not change its mo-mentum. Therefore, there is no solution of such equationwith asymptotics of ∇ φ equal to one constant vector inone domain and another constant vector in another do-main.Another consequence of this consideration is that thewidth of the hDW has the order of magnitude of the pitchof helix (cid:96) . Indeed, the change of wave vector modulus vi-olates the balance of exchange and DM forces. Namelythey define the variation of magnetization inside the do-main wall and restore theis balance outside. The con-tribution of anisotropy can be neglected. The only scaleof the isotropic Hamiltonian is (cid:96) . This peculiarity of thehDW is also unusual. Commonly the width of domainwall is determined by competition of exchange force andanisotropy. Such a domain wall would be much wider.Consider the domain wall that is a bisector plane be-tween vectors q and q . Its normal is directed along theunit vector ˆ q − where q ± = 12 ( q ± q ) (9)Both vectors q + and q − lay in the plane perpendicularto the domain wall. Let assume that vector field q ( r )asymptotically approaches q in the domain ˆ q − · r > q in the domain ˆ q − · r < φ ( r )in the presence of domain wall has a form: φ ( r ) = q + r + q − w ln (cid:20) (cid:18) ˆq − · r w (cid:19)(cid:21) (10)The vector field q ( r ) for this trial function reads: q ( r ) = q + + q − tanh (cid:18) ˆq − · r w (cid:19) (11)The magnetization in such texture can be represented asfollows: m ( x , y ) =ˆ z cos φ ( x , y ) + ˆ q × ˆ z sin φ ( x , y ) (12)Here ˆ z is unit vector in the direction of vector q × q , ˆ q isunit vector in direction of q ( r ); x − axis is parallel to thevector q + . In Fig. 2 the regions of positive and negativeprojections of m ( r ) to the z − axis and local direction of q ( r ) are schematically shown for w = 0 . (cid:96) . Minimiza-tion of energy (see section 4) for mutually perpendicular q and q gives w ≈ . (cid:96) . In order to make magnetictexture inside the hDW clearly visible we display in Fig.2 a thicker domain wall. FIG. 2: Distribution of the magnetization projection m z inthe bisector domain wall according to trial function (10) with w = 0 . (cid:96) . The value of m z is shown by color Note that there are two bisectors for any pair of wavevectors in domains. Their trial functions differ by per-mutation of vectors q + and q − . They both realize localangular minima of the surface energy, but their energiesper unit area are different. Namely, the bisector of theangle α between wave vectors less than π has smaller sur-face energy than bisector of complementary angle π − α (see Fig.3). Any attempt to extend such Ansatz to the
5s the distance to the center. After integration it givesln
L(cid:96) by order of magnitude. Thus, the total energy ofdisclination hDW per unit area is roughly equal to E dw ∼ Jq L + J ln L(cid:96) L (18)Its minimization gives L ∼ q − ∼ (cid:96) . More accurate co-efficients in this relation can be found numerically. Itrequires sufficiently accurate solution of static Landau-Lifshitz equation with singularities or a proper trial func-tion for φ ( x, y ) that we did not find so far. IV. DOMAIN WALLS AT CRYSTAL FACES
In the cited work the authors studied about 90 sam-ples of FeGe single crystals with typical sizes 0.5 × × ◦ C. The crystal structure was checkedby Laue diffraction. The samples then were cut and pol-ished to yield faces (100) and (110) with roughness 1nm. MFM pictures and measurements were performedin Trondheim by P. Schoenherr under supervision of D.Meier. The magnetic tip in the MFM had radius about50 nm. It was scanned with the distance of the tip to face30 nm. Standard dual-pass MFM measurements allowedthe resolution 10-15 nm. Measurements were performedat temperature 260-273K maintained by permanent wa-ter flux. M. Garst and A. Rosch supervised theoreticalpart of work.The first experimental fact discovered in the MFMstudies of helical magnet is that on the crystal face,the wave vectors of helix lay in the plane of face. Thatwas checked for the faces (1,0,0) and (1,1,0). The secondsurprising fact is that unlike in the bulk, the orienta-tion of the surface helix wave vector is not confined to adefinite crystallographic directions within the face plane.These two facts can be explained if spin-orbit interactionnear the face creates uniaxial easy-axis anisotropy. Letus denote n the normal vector to the face. Then surfaceanisotropy energy is H sa = − λ (cid:90) [ n · m ( r )] d xa (19)The unit magnetization vector field is given by eq. (12).Let choose e = n × ˆq | n × ˆq | and e = ˆq × e = n − ˆ q ( nˆq ) | n × ˆq | .Then nm ( r ) = ne sin φ ( r ) = | n × ˆq | sin φ ( r ). Average (cid:68) [ n · m ( r )] (cid:69) over phase φ is equal to ( n × ˆq ) = sin θ where θ is the angle between helix wave vector q andthe normal to the face n . Thus, the surface anisotropyenergy per unit area is σ = − λ sin θ . It has minimumat θ = π , i.e. for the helix wave vector in the planeof face. The value of surface anisotropy λ appears asrelativistic correction of the second order, whereas the constant of cubic anisotropy v appears only in the fourthorder relativistic correction. Therefore it is reasonableto assume that λ (cid:29) v . This fact explains why the he-lix wave vector tends to turn parallel to the face plane.The in-face anisotropy is too weak to confine these vec-tors to crystal directions with small indices in the face.In infinite perfect samples the helix vector on surface isdetermined by its bulk value and the normal to the facewith some exceptions. For the face (001) and the samehelix wave vector in the bulk, any direction of surfacehelix wave vector has the same energy. The authors ofarticle concluded that there is no dependence betweenbulk and surface wave vectors. When the bulk vectoris not perpendicular to the crystal face as happens forthe face (110), this result looks surprising. It may hap-pen if the parameters J and g rapidly change in closevicinity of the boundary. Then surface layer becomes tosome extent magnetically independent of the bulk. Analternative explanation proposed in is the closeness oftemperature 260-273 K at which measurements were per-formed to the Neel point T N = 278 K . In this range oftemperature the magnetization is still small and energyof bulk anisotropy proportional to the fourth power ofmagnetization becomes negligible in comparison to othercontribution to energy quadratic in m .Due to insensitivity of energy to the direction of surfacehelix wave vectors, the angle α between wave vectors q and q in different domains changes from sample to sam-ple. Symmetry of wave vectors with respect to changeof sign implies that α varies in the limits between 0 and π . Let us denote β the angle formed by one of the helixwave vectors and domain wall. Experimental graph ofdependence of β on α is shown in Fig. 5d.It shows that at 0 < α < ◦ (DW of the type I in termi-nology accepted in ) and at 140 ◦ < α < ◦ (DW of typeIII), β follows well defined dependence β = α/ . Theseare domain walls whose plane are bisectors between vec-tors q and q that are well described by variational eqs.(10,11). Experimental data imply that domain walls inthis ranges of α relax to the closest bisector direction.Note that the angle between any possible initial direc-tion of the hDW and bisector in this range of angles isless than 40 ◦ . In the interval 140 ◦ < α < ◦ (DWof the type II), β is not a function of α. Experimentalvalues of β at fixed α are scattered in this range of α more or less uniformly between α/ α . Such domainwalls must be supplied with zig-zag chain of disclinations-antidisclinations as discussed in Section 3. However, at α = 90 ◦ and a fixed value β (cid:54) = 0 , π , the two lines withina primitive period have different lengths. It follows fromgeometrical constraints, i.e. fixed values α and β andangles (120 ◦ or 90 ◦ ) at vertices of zig-zag line. MFMpictures of the domain walls in this range of α con-firm the existence of disclination-antidisclination zig-zagstructure, though in real crystals it is not so ideally pe-riodic and domain wall median is not ideal straight lineIn theoretical part of the article , the authors per-formed micromagnetic calculations of domain wall con-6 IG. 5: MFM pictures of Helimagnetic domain walls in FeGe.Reproduced from article by courtesy of Profs. D. Meier andY. Tokura. a. Bisector domain wall at α < ◦ (type I); b.Zig-zag disclination wall (type III);c. Bisector domain wall at α > ◦ (dislocation wall, type II); d. Graph of the angle β between domain wall and one of the helix wave vectors (ournotations differ from those accepted in ). See comments inthe text. figuration in two dimensions. They demonstrated thatat small and large angles α , minimum energy is real-ized by smooth non-singular bisector domain walls. Atangles α in the range near 90 ◦ , the domain walls has zig-zag shape with regularly intermitting disclinations andanti-disclinations. Theory even describes the irregularityof this chain considering them as random fluctuations.This is without doubt a success. However some principalquestions remain unanswered.The main such question is what keeps orientation ofthe helix wave vectors? We understand that in the bulk it is anisotropy that reduces initial SO(3) symmetry ofthe exchange and DM interactions to discrete group ofcube without inversion. But experimenters tell us thatanisotropy plays no role at the crystal faces. Then thereis no topological reason for appearance of domain walls.Another idea is that the helix on the surface is fixed by itscoupling with the helix in the bulk. In this case orienta-tion of the wave vector at the surface could be arbitraryin plane of the face (001), but not in the face (110). Italso may be the edge or random pinning that fixes theorientation of wave vector near the boundaries or defectsand then these fixed pieces serve as nuclei for domaingrowth.But if the direction of helix vector can vary continu-ously another principal question appears. If symmetryof a system does not require domain walls as topologicaldefects, can they nevertheless appear if different possi-ble values of order parameter are fixed ’by hands” nearboundaries of the sample? For simple systems such asHeisenberg or XY (planar) magnets the answer to thisquestion is no. In both cases, if spins on two sides ofa big stripe or slab are fixed artificially, the transitionfrom one to another orientation in the sample proceedssmoothly. Instead of domain wall we see spins slowly ro-tating in space. The principal question is whether thesame is correct for an isotropic helical magnets. The dif-ference with simpler systems is that there is no smoothtexture that changes its direction conserving the mod-ulus of wave vector (see Section 3). In this article weissue a proof of the statement that in isotropic helicalmagnet, the transition between two different helix wavevectors fixed near boundaries proceeds by formation ofthe hDW whose width depends on the angle α betweenfixed wave vectors. We calculate explicitly the angulardependence of the width for bisector domain walls givenby variational ansatz (10).The derivation of this result employs general equation(17) with q ( r ) given by Ansatz (11). It represents thedomain wall energy as a sum of two positive contributions E (1) dw and E (2) dw originating from deviation of q ( r ) from itsenergy preferable value q and from rotation of the wavevector ˆ q ( r ), respectively. The integrals involved in theseexpressions can be calculated explicitly. The result is asfollows: E (1) dw = 12 Jq wI , (20)where I ( α ) = sin α (cid:18) − a − a + A ln a + B ln 2 aa + 1 (cid:19) (21)and a = cot π − α , A = 2 (cid:0) a + 1 (cid:1) a , B = 2 (cid:0) a + 1 (cid:1) a ( a − . (22)The rotational part of energy reads E (2) dw = Jw I where I ( α ) = 12 ( α − cot α ) (23)7inimization of the total DW energy E dw = E (1) dw + E (2) dw over w leads to the result: w ( α ) = q − (cid:114) I I (24)This equation shows that the width of bisector domainwall becomes infinite at α = 0, has a minimum and againgoes to infinity at α = π . The curve w ( α ) (see Fig. 5)is not symmetric with respect to the point α = π/
2, i.e. w ( α ) (cid:54) = w ( π − α ). This asymmetry seemingly contradictsto the established invariance of the helix to the flip ofone of wave vectors. The reason of this asymmetry isthe accepted assumption that domain wall is the bisectorof a smaller angle between wave vectors entire interval0 < α < π . Certainly, the second half of this intervalcan be reduced to the first by the flip, for definiteness, of q , i.e. by transformation α ↔ π − α . However, at thisflip the domain wall that forms an angle ± ( π − α ) / α ↔ π − α and simultaneously q + ↔ q − . As we statedearlier, the surface energy of the bisector of larger an-gle hDW is larger than the analogous energy for smallerangle. Note that both are local minima of the surfaceenergy σ as function of the angle β between plane of do-main wall and one of the wave vectors at fixed value of α . In the bulk the angle between different wave vectorsis α = π . Then a = √ ≈ .
414 and the domain wallwidth of bisector domain wall is w ( π/ ≈ . (cid:96) . ↵
2. It is asymmetric with respect to reflectionin the point α = π/
2. At this value of α , both anglesbetween axis of rotations are equal. Therefore, in thiscase the width and the energy of two branches of thecurves w ( a ) and E ( α ) arrive at the common limit.We start the proof of these statements with the caseof small α . In general case the wave vector q ( x, y ) must go to the limit ˆ xq at α →
0. Corrections to this limitat small α must start at least with α since at permuta-tion of two wave vectors the distribution of magnetizationdoes not change. It means that modulus of wave vector q ( x, y ) differs from q by a small value of the relativeorder α . As a consequence | q ( x, y ) − q | has order ofmagnitude α q and I ∼ α . The unit vector ˆ q ( x, y )rotates in the domain wall to the angle α . Therefore, ∇ ˆ q is ∼ α/w by the order of magnitude. Thus, I ∼ α . Weconclude that w ( α ) ∝ /α at α → α approaching π , the x − component of wave vectortends to zero. Neglecting it, we arrive at wave vector thathas only y − component. Due to symmetry, it changessign at crossing the median of hDW y = 0. Therefore,the unit vector of its direction is equall to sign of y andits derivative ∇ ˆ q = 2 δ ( y ). Thus, I = w (cid:82) ∞ ( ∇ ˆ q ) dη diverges at α approaching π . On the other hand, integral I remains finite in this limit since q is even function of y that becomes zero at y = 0 and tends to the value q at y → ±∞ . Thus | q − q | has maximum equal to q at y = 0 and rapidly decreases outside domain wall.Let consider briefly the energetics of domain walls. Itfollows from exact equation valid for bisector hDW’s andis not associated with any variational ansatz: E dw = Jq (cid:112) I I . (25)From previous analysis we find that at α →
0, the sur-face energy of the bisector hDW goes to zero as | α | . At α = → π this energy becomes infinite. This result seemsto be inconsistent with equivalence of α = π and α = 0.However, as we have shown earlier , just at the value of α = π , x − component of wave vector is equal to zero atany x and y , whereas its y -component turns into zero atthe median of the hDW. Thus, the limit of the bisectorhDW at α → π leads to the specific minimization prob-lem for a helical configuration with wave vector directedeverywhere along y − axis, changing from − q to q andtaking value 0 at y = 0. Energy of such configuration isinfinite as we have shown earlier.The micromagnetic calculations in qualitatively agreewith our consideration in the range 0 < α < ◦ , thoughquantitatively this dependence is closer to α instead ofour result α . But in the second range of bisector hDW140 ◦ < α < ◦ it results in monotonic decrease of sur-face energy to zero value at α = π in contrast to infi-nite surface energy predicted by our theory. The reasonof these discrepancies presumably is the finite size of a”sample” used in micromagnetic calculations. It does notexceed 20 (cid:96) . It becomes smaller than the width of the do-main wall at α sufficiently close to 0 or π invalidating thecalculation of surface energy.As for zig-zag domain walls we have seen already thatat α = 90 ◦ and β = 0, zig-zag line is symmetric, its sideshave equal length. Therefore it could be expected thatit realizes minimum of energy in comparison with either β (cid:54) = 0 or α (cid:54) = 90 ◦ . Indeed such a minimum was foundin the same micromagnetic calculations. However, ener-getically unfavorable configuration do not relax to this8nergy minimum. The metastability of these configura-tions may be associated with their complicated topolog-ical structure and small distances between disclinationsand antidisclinations that increase their rigidity.In conclusion of this section we propose an approach tothe problem of zig-zag domain walls at arbitrary angle α between the wave vectors in the two domains based on itssimilarity with the devils staircase in the commenurate-incommensurate transition . Let consider a domain walltilted with the angle γ = β − α/ l i = (cid:96)/ cos( α/ ∓ γ ); i = 1 ,
2. The mismatch sums upto a full period if ( n + 1) l = nl (we considered thesimplest commensurate situation). This condition is sat-isfied if γ = γ n = arctan[cot( α n +1 )]. The mismatch canbe avoided by introduction of a pair ± π -disclinations.The distance between two such dipoles must be L = n(cid:96) .Let the distance between two disclination in the dipole is d . The mismatch though small ( q − q ∝ αγq at small α and γ ) persists at least at the distance L generating thedeviation energy of the order J ( γq αd ) /L per unit area.The rotation energy per unit area is ∼ Jα [ln( L/d )] /L .Minimization over d gives d ∼ (cid:96)/γ , but the energy de-creases with L . We conclude that in the infinite system L must be infinite at small angles α and β . Thus, wehave proved that the domain wall tilted to bisector atsmall angle γ are unstable. This stament agrees withexperiment that did not observe such domain walls atsmall α . The situation is different at large angles α and γ since in this case the only scale of length for the hDWis L .Sch¨onherr et al. argued theoretically and have shownnumerically that surface energy of the disclination hDWhas minimum at angle between wave vectors α = π/ β = γ = 0. Our consideration shows that it shouldhave more shallow minima at the same α and an infi-nite discrete set of γ = γ n corresponding to simple ratio-nal mismatch. Another rational mismatches of two stepsof zig-zag and also zig-zags consisting of more than twosteps would give a Devil’s staircase of local minima, butonly few of them with minimal denominators will be seenat finite temperature. V. CONCLUSIONS
Here we discuss open questions. Some of them arerelated to experiment. It would be very instructive to perform measurements on the same or other samples,but at temperature lower than 260K in FeGe to ensurethat the volume anisotropy is more significant. Will arbi-trary directions of wave vectors and angles between thempersist? It would be also interesting to apply polarizedelectron or neutron scattering to get information on thesame objects not only at the surface, but at least partlyin the bulk. Though the samples used in experiment were sufficiently large, there were no long regular hDWon presented pictures. This fact interferes quantitativecomparison of theory and experiment.Among theoretical problems we would like to mention,first is the problem of the bulk-surface coupling or de-coupling. So far there is no satisfactory theory of thisphenomenon. It is very important to develop variationalmethods for structure and energy of the hDWs with thegoal to get a desirable precision. So far we even couldnot compare our variational calculations of bisector do-main walls made for infinite sample with micromagneticcalculations in since the latter were performed for finiteand not too large samples. Their authors have foundsignificant finite size effects. Thus, it would be usefulto develop variational approach to finite samples. Vari-ational approach to theory of disclination domain wallsso far did not achieve quantiative level. We are lookingnow for a satisfactory trial function for magnetization.Important statements that were proved in this articleinclude stability of the smooth (bisector) domain walls atall angles between wave vectors and instability of zig-zagdomain walls at small angles. VI. ACKNOWLEDGENENTS
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