The stability of vortex-like structures in uniaxial ferromagnets

*Corresponding author. Tel.: (49)-9131-852-7635; fax: (49)- 9131-852-8495; e-mail: [email protected]. 1Permanent address: Physicotechnical Institute, 340114 Donetsk, Ukraine. Journal of Magnetism and Magnetic Materials 195 (1999) 182—192 The stability of vortex-like structures in uniaxial ferromagnets A. Bogdanov1, A. Hubert* Institut fu( r Werksto⁄wissenschaften, Lehrstuhl Werksto⁄e der Elektrotechnik, Universita( t Erlangen-Nu( rnberg, Martensstr. 7, D 91058, Erlangen, Germany Received 15 September 1998; received in revised form 2 December 1998 Abstract Two-dimensional localized states in the form of isolated vortices are studied systematically in uniaxial ferromagnets with an antisymmetric ÔDzyaloshinskyÕ exchange interaction. In addition to previously investigated n-vortices, new types of localized solutions were found. Their structure and equilibrium parameters were calculated by numerically solving the di⁄erential equations. We studied the stability of all solutions with respect to small radial distortions by solving the eigenvalue problem for the perturbation energy. It turned out that single vortices as well as multiple vortices with a magnetization rotation kn (k"2, 3, 2) are stable in certain parameter regions, while other solutions of the di⁄erential equations such as vortices with nodes and large or blown-up vortices are always radially unstable. The stability analysis also answered the question of the decay modes of the stable solutions at their stability limits. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic vortices; Stability; Helimagnets 1. Introduction The formation and evolution of inhomogeneous localized patterns (topological defects or localized states) are a subject of intensive research in many nonlinear physical systems [1—3]. Up to now, non- singular, time-independent localized states were studied mostly for cases of structures which vary along one deÞnite spatial direction only (one-di- mensional localized structures). Planar magnetic domain walls are the best-known example for such structures, for which the most extensive experi- mental and theoretical investigations were carried out [4—6]. The situation changes radically for structures which depend on more than one spatial coordinate (multidimensional localized states). Such solutions were proved to be unstable for interaction func- tionals of rather general form according to the Hobard—Derrick theorem [7,8]. However, it was shown later that this theorem does not apply to functionals with terms that are linear in the Þrst spatial derivatives [9,10]. Among these are the so- called Lifshitz invariants, and it was proved by direct calculations that such terms do stabilize two-dimensional localized structures in the form of non-singular vortices [9,11—13]. 0304-8853/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 1 0 3 8 - 5 Lifshitz invariants occur in systems without in- version symmetry in certain groups of magnetic materials [14], ferroelectrics and liquid crystals [15]. In magnetic crystals these energy terms have the form [14]: J i ›J j ›g !J j ›J i ›g , (1) where J(r) is the magnetisation Þeld and g is a spa- tial coordinate. They describe an antisymmetric spin coupling, which can be considered as the in- homogeneous part of the Dzyaloshinsky—Moriya interaction [16,17] and which was Þrst introduced in Ref. [14]. Following our earlier contributions [9,13] we will call it a Dzyaloshinsky interaction. Such interactions also occur in frustrated systems [18—20], and can be induced by nonmagnetic impu- rities with strong spin—orbit coupling [21]. The theoretical study of isolated vortices in mag- netic materials without inversion center started in Refs. [11—13]. Systematic theoretical investigations of isolated vortices were carried out in Ref. [8]. For a suƒciently strong Dzyaloshinsky interaction, ex- tended modulated magnetic structures have a lower energy than the homogeneous state [14]. In uniaxial ferromagnets (in particular in the two sym- metry classes D n and C n7 ) one-dimensional, peri- odic helix structures are stable at low magnetic Þelds, while two-dimensional vortex lattices proved to be thermodynamically stable in a range of larger magnetic Þelds along the easy axis [13]. In a much wider parameter range isolated vortices are pos- sible as metastable solutions. The equilibrium vor- tex size is comparable to the domain wall width. Unlike the periodic, modulated states, isolated vortices also remain stable for very weak Dzyaloshinsky interactions as long as this energy term does not disappear altogether. The generation of vortices out of ßuctuations near the Curie temperature was demonstrated by numerical simulations in Ref. [22]. Up to now no systematic experimental search for isolated vortices has been reported. However, recently magnetic vortices have been identiÞed in NiMn [20]. In this frustrated system the Dzyaloshinsky inter- action is due to a special chiral spin mechanism [18,19]. The present contribution deals with a further development of the theory of vortex states. While the previous papers were mainly dedicated to the calculations of the equilibrium structures of vor- tices, in this paper all possible axial structures are studied particularly with respect to their stability. The di⁄erential equations for vortex states and the boundary conditions are introduced in Section 2.1. A systematical search for possible solutions leans on an auxiliary Cauchy (initial value) ap- proach as explained in Section 2.2. Special solu- tions of the Cauchy problem are used as starting solutions for a Þnite di⁄erence calculation of the boundary problem deÞned in Section 2.1. The radial stability of the solutions found in this way is explored by solving the eigenvalue problem for corresponding di⁄erential operators as explained in Section 2.3. Section 3 is dedicated to a discussion of stable and unstable solutions and their decay modes. It turns out that only conventional n-vor- tices as well as multiple, ÔwindingÕ kn-vortices can be stable, all other possible solutions are unstable. 2. Micromagnetic equations and methods 2.1. The boundary value problem for vortex states To investigate localized states we use the same model of a uniaxial ferromagnet as in Ref. [9]. The functional of the energy density was there: …"A+ i A Lm Lx i B 2!Km2 z !J 4 m z H(%) z !1 2 J 4 m )H $ #Du D , (2) where A is the exchange sti⁄ness constant, m(r)"J(r)/J 4 is the reduced magnetization (m2"1), J 4 is the saturation magnetization, K is the anisotropy which deÞnes the z-axis as the easy axis if it is positive, H(%) is the external magnetic Þeld parallel to the easy axis, H $ is the stray Þeld, and D is the coeƒcient of the Dzyaloshinsky interaction energy. Depending on crystal symmetry the Dzyaloshinsky energy functional u D consists of some of the terms in Eq. (1) as elaborated in [13] and as given explicitly in cylindrical coordinates in Eqs. (5) and (6). 183A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192 The equilibrium conÞgurations of m(r) and the stray Þeld H $ can be calculated by solving the equations minimizing the energy (2) together with the equations of magnetostatics. We consider an isolated vortex as an axisymmetric magnetization distribution that is independent of the z-coordinate. For non-singular structures the magnetization in the center should be parallel to the vortex axis (z-axis). On the other hand, when the distance from the center goes to inÞnity, the vector m(r) again approaches the easy axis. Unlike conventional magnetic inhomogeneities, vortices are stabilized mainly by interior interactions. Certainly surface stray Þelds would modify the vortex structure if we assume the vortex to end at a surface, but for initial investigations it is reasonable to ignore surface e⁄ects by considering inÞnitely extended samples only and concentrate on the basic internal proper- ties. Following Ref. [9] we introduce new variables based on the anisotropy Þeld H ! "2K/J 4 and the domain wall width ‚ B- "JA/K: h"H/H ! , r8"r/‚ B- , u"…/(H ! J 4 ). (3) We express the magnetization vector m(r) in terms of spherical coordinates, and the spatial variables in cylindrical coordinates m"(sin h cos t, sin h sin t, cos h), r8"(o cos u, o sin u, z). (4) The Dzyaloshinsky energy u D for systems with D n symmetry is then expressed in these variables as (where ho,dh/do, etc.) u(1) D "sin(u!t)ho!sin h cos h cos(u!t)to #o~1cos(u!t)hr #o~1sin h cos h sin(u!t)tr. (5) In this case charge-free solutions with t"u!n/2 are preferred. For crystals of C n7 symmetry u D is given by u(2) D "cos(u!t)ho#sin h cos h sin(u!t)to #o~1sin(u!t)hr #o~1sin h cos h cos(u!t)tr. (6) which assumes the lowest energy for t"u. The magnetostatic problem for this structure can be solved rigorously [13] and the stray Þeld energy can be expressed as an e⁄ective anisotropy. The total energies for both structures can be reduced to a common functional form. After integ- ration with respect to u the di⁄erence between the vortex energy per unit length and that of the uni- form state with h"0 is E"nP = 0 u8 (h, o) do (7) with u8 "Ch2o#o~2sin2 h#sin2 h#2h(1!cos h) #4i8 n (ho#o~1sin h cos h)D o. (8) Here the e⁄ects of anisotropy, of the Dzyaloshinsky interaction, and of the stray Þeld energy have been collected in the e⁄ective reduced material constant i8 . This is only possible if all these energy contribu- tions are local in nature. Due to the internal stray Þelds the problem would in general have a nonlocal character. However, for uniaxial crystals of D n and C n7 symmetries, and for cylindrical symmetry of the investigated structures, the stray Þeld energy can be expressed by a local energy density [9,13]. If this is true, the material constant i8 can be expressed as follows: KI "K (D n symmetry), KI "K#K $ "K(1#Q~1) (C n7 symmetry), (9) i8 " nD 4JAKI , with Q"K/K $ , K $ "J2 4 /2k 0 . (10) The parameter i8 plays a similar role as the Ginz- burg—Landau parameter i in the theory of super- conductivity. It describes the relative contribution of the Dzyaloshinsky energy term. Modulated structures can be demonstrated to be possible as thermodynamically stable states only in i8 exceeds the value of 1. This means that a critical value of the Dzyaloshinsky interaction coeƒcient D is neces- sary for modulated states to be stable solutions. Metastable isolated vortex solutions are, however, A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192184 Fig. 1. Magnetization distribution of isolated vortices in a crys- tal of C n7 symmetry. (a) n-vortex; (b) 2n-vortex. possible for all materials with i8 ’0 as will be shown. Every rotationally symmetric structure must then obey the Euler equation which follows from varying (7): d2h do2 #1 o dh do ! 1 o2 sin h cos h #4i8 n sin2h o !h sin h! sin h cos h "0. (11) For all solutions the magnetization m should be parallel or antiparallel to the easy axis both in the centre and at inÞnity. This means h(0)"k 1 n and h(R)"k 2 n if k 1 and k 2 are integers. The chiral character of the Dzyaloshinsky interaction imposes further restrictions on the boundary conditions. For interactions of type (1) only the proper sense of rotation leads to a decrease of the system energy and to a stabilization of the vortex structure. In particular, for positive values of i8 the angle h should decrease with increasing o. DeÞning as in Ref. [9] h at inÞnity as zero we may write the boundary conditions as h(0)"kn, h(R)"0, with k integer’0. (12) Such solutions represent localized vortices in a homogeneously magnetized matrix. The angular amplitude *h"Dh(0)!h(R)D"kn describes the total variation of h in the vortices when o goes from zero to inÞnity. We call these solutions n-vortices, 2n-vortices and so on. The structures of the Þrst two of these localized states in crystals with C n7 sym- metry are shown in Fig. 1. 2.2. Numerical procedures In previous papers [11—13] we investigated regu- lar vortices (k"1) as solutions of the di⁄erential equation (11) with the boundary conditions (12) by analytical and numerical methods. In numerical investigations we used a Þnite-di⁄erence method, which, however, converges only if the starting func- tions are rather close to the Þnal solutions. Our present intention is to Þnd and study all possible localized solutions of the equations. In search of possible solutions of the boundary value problem we study an initial value problem as an auxiliary approach [23,24] by solving the di⁄er- ential equation (11) with the initial conditions h(0)"n, dh/do(0)"!a (13) for di⁄erent values of a (0(a(R). It is clear that any localized solution h k (o) of the boundary value problem (11), (12) is among the set of solutions h (a) (o) (0(a(R) of the Cauchy problem (11), (13) and corresponds to a certain Þxed value of a denoted as a k . A qualitative analysis of possible trajectories in phase space (h, ho) [25] makes it possible to reveal localized structures among other trajectories. Typical phase trajectories ho(h) for the Cauchy problem (11), (13) are presented in Fig. 2. For arbit- rary values of a the lines ho(h) normally end by spiraling around one of the attractors (h i , 0) where cos(h i )"!h. (14) Such trajectories do not fulÞll the boundary condi- tions (13). Only for certain discrete values of a k the lines ho(h) end in the points (kn, 0). They correspond to solutions of the boundary value problem (11), (12) for kn-vortices, examples of which are shown in Fig. 3 for k"1—4. These special curves ho(h, ak) 185A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192 Fig. 2. Typical phase trajectories for the solutions of the Cauchy problem (11), (13). For most values of a the phase trajectories spiral around one of the poles. Only for certain discrete values a k (a 1 "0.126, a 2 "0.345, a 3 "0.753) the curves end in the points (kn, 0) and represent localized solutions of the boundary value problem (11), (12). Fig. 3. The magnetization proÞles for kn-vortices (k"1,2, 4). separate sets of phase trajectories h (a) (o) ending at di⁄erent attractors (Fig. 2). Thus in the points a"a k the integral trajectories ho(h) change their topology. One should keep in mind the di⁄erence between the initial value problem introduced above, and a regular Cauchy problem for dynamic systems. In the latter case the spiraling curves describe stable oscillatory solutions, while the separatrix solutions of the type ho(h, ak) are unstable with respect to small perturbations of the initial conditions (13) (see in this connection the discussion in Ref. [22]). In our case the auxiliary Cauchy problem is solved only as a selection procedure for the solutions of the corresponding boundary value problem (12). Despite the fact that the proÞles for localized solu- tions formally coincide with the separatrix curves ho(h, ak), they cannot be considered equivalent. For the boundary value problem (11), (12) the perturba- tions m(o) of the solutions should not violate the conditions (12), which means m(0)"m(R)"0. (15) Therefore perturbations with m(R)O0, which cause the separatrix solutions in the Cauchy prob- lem to become unstable, are excluded from consid- eration in the boundary value problem. Nevertheless, we can utilize the Cauchy problem as an approach to Þnd solutions of the boundary value problem by the following procedure: 1. For given values of i8 and h and di⁄erent a the Cauchy problem (11), (13) is solved by the Runge—Kutta method, and values of a where phase trajectories ho(h) changes the attractors from one side of a k to the other are determined roughly. 2. By repeating calculation with di⁄erent a, Ôshoot- ingÕ at the boundary value, the proÞle h a (o) for the localized structure is interpolated more ac- curately. 3. The proÞle obtained in (ii) is improved by using a Þnite-di⁄erence method for the boundary value problem as in Eqs. (11) and (12), with the solutions of (ii) as the starting function. A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192186 Fig. 4. The region of existence (white area) for n-vortices. Example proÞles of two metastable vortex solutions — valid for the positions in the phase diagram marked by black dots — are shown as insets. 2.3. Radial stability of vortex solutions After localized proÞles have thus been calculated, we have to check the stability of these solutions. Since vortex states have a higher energy than the homogeneous matrix, they can only represent rela- tive energy minima or metastable states. Some important results about vortex stability can be derived by model calculations. If a n-vortex is represented by a linear ansatz h"n(1!o/o 0 ), its energy has a local minimum for a Þnite radius o 0 , and this value is proportional to the Dzyaloshinsky coeƒcient D [11]. If D would be zero, this min- imum would disappear. Also an analysis of the vortex energy under scaling transformations dem- onstrates the crucial role of the Lifshitz invariant in the stabilization of localized vortex states [9,10]. The region of the metastable existence of simple n-vortices was calculated in Ref. [9]. For i8 ’1, periodic modulated states are thermodynamically stable in a certain region of the applied Þeld, while isolated n-vortices can exist at higher magnetic Þelds. At lower Þelds they either condense into the vortex lattice on the line h* (see Fig. 4), or they strip out into the spiral structure when the Þeld is lowered below the elliptic instability Þeld h %-- . For small i8 n-vortices can exist even at negative Þelds. The magnetization in the vortex core is then oriented along the Þeld, while the surrounding matrix is magnetized in the opposite direction. Thus the vortex size increases with increasing Þeld. Finally, when h reaches a certain critical value h " (i8 ), the vortex ÔburstsÕ into the homogeneous state with the magnetization parallel to the applied Þeld. Such considerations do not solve the problem of vortex stability completely, however, although they o⁄er considerable insight. Continuous radial defor- mations other than the mentioned scaling trans- formations of the vortex proÞle might exist, which might lead to a decrease of vortex energy and thus to instability. To check stability systematically we explore the energy change under small perturba- tions of the vortex structures. We can exclude from the consideration such distortions as bending, pinching or twisting in the basic plane because they increase the vortex energy. The only contribution decreasing the total energy of the vortex and stabilizing it is due to the Dzyaloshinsky interac- tion and connected to the radial rotation of the magnetization. Hence we conclude that outside the region of periodic, modulated states radial distor- tions of type m(o) are the most dangerous in the sense that they most probably lead to instabilities. Let us therefore consider a small arbitrary radial distortion m(o) of a localized solution of the bound- ary value problem (11), (12). We insert hI (o)" h(o)#m(o) into the energy functional (7) and keep only terms up to second order in m(o). Because h(o) is a solution, the Þrst-order terms must vanish, yielding EI "E#E(2), (16) where E(2)"P = 0 CA dm doB 2 #G(o)m2Do do (17) is the perturbation energy with G(o)"(1#o~2)cos 2h#h cos h !(4i/n)o~1sin 2h. (18) Radial stability of the function h(o) means that the functional E(2) is positive for all functions m(o) which obey constraint (15). Correspondingly, the solutions will be unstable if there is a function m(o) that leads to a negative energy (17). Thus, the prob- lem of radial stability is reduced to the solution of 187A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192 Fig. 5. A typical magnetization proÞle (a) for the n-vortex. The Þrst three excitation modes are shown in (b). The structure is radially stable because the smallest eigenvalue j 1 is positive. the spectral problem for functional (17). We solve it by expanding m(o) in a Fourier series: m(o)" =+ k/1 b k sin[kh(o)]. (19) Inserting this reduces the perturbation energy (17) to the following quadratic form: E(2)" =+ l, k/1 A kl b k b l , (20) where A kl "P = 0 CklA dh doB 2 cos(kh) cos(lh) #G(o) sin(kh) sin(lh)Do do. (21) To establish radial stability of a solution, one has to determine the smallest eigenvalue j 1 of the sym- metric matrix A (21). If j 1 is positive, the solution h(o) is stable with respect to small radial perturba- tions. Otherwise it is unstable. A standard inverse power method [26] was used to calculate numer- ically the smallest eigenvalue j 1 and the corre- sponding eigenvector b(1). 3. Results 3.1. Regular and wide single vortices We start the discussion of the solutions and their stability with ordinary n-vortices. A typical vortex proÞle with the three Þrst perturbation modes and the corresponding eigenvalues is shown in Fig. 5. For our vortex solutions the eigenmode m n (o) corre- sponding to the nth eigenvalue (j n ) consists mainly of the function sin[nh(o)], with small admixtures of other harmonics. In particular, the eigenmode cor- responding to the smallest eigenvalue j 1 can be written as m 1 (o)"sin[h(o)]# =+ k/2 e k sin[kh(o)], (22) where e k @1 in most cases. The function m 1 (o) de- scribes a displacement of the vortex front. Thus the lowest perturbation of the structure is connected with an expansion or compression of the proÞle. The calculations showed that in the region of existence of n-vortex solutions matrix (21) has only positive eigenvalues, and thus these solutions are radially stable. For 0(i8 (1 further solutions for n-vortices can be constructed in a certain interval of negative Þelds, using the procedure elaborated in Section 2.2. Compared to normal n-vortices they have a lar- ger core size (Fig. 6). The smallest eigenvalues of these large vortices are always negative, however, and hence these structures are unstable, either with respect to an inÞnite expansion of the core, or to a contraction into a normal vortex. Following Ref. [9] we can characterize the vor- tex radius by the distance between the origin and the point where the tangent at the inßection point A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192188 Fig. 6. The proÞle of a large n-vortex (a) with the Þrst three excitation modes (b). The structure is unstable with respect to the Þrst excitation mode because j 1 is negative. 2The solution for large vortices in regular ferromagnets was Þrst obtained in Ref. [27] (see also Ref. [28]). Fig. 7. The equilibrium radius R 0 of the n- and 2n-vortices as a function of an applied magnetic Þeld. Solid lines indicate basic radial stable vortices and dashed lines correspond to large structures. The vortex radii for normal and large vortices be- come the same in the critical points of the bursting instability. (o 0 , h 0 ) intersects the o axis: R 0 "o 0 #h 0 (dh/do)~1 0 . (23) The radius of ordinary vortices goes to zero as i8 approaches zero, and these structures thus disap- pear by collapsing to a singular line in regular ferromagnets [9]. In contrast, large vortices have a Þnite size2 even for i8 "0, but they are still unsta- ble with respect to the perturbations of type (22). The instability of time-independent two-dimen- sional localized states in regular ferromagnets (i8 "0) may be also demonstrated by a scaling transformation analysis of the solutions as shown in Ref. [9]. In small negative Þelds the radius and the energy of large vortices go to inÞnity. In a stronger nega- tive Þeld the proÞle of a normal vortex approaches the proÞle of its large twin, and the structures become the same at a certain critical negative Þeld h " (i8 ) (Fig. 7). No solutions for n-vortices exist at stronger negative Þelds beyond h " (i8 ). The vortex radius increases in the vicinity of this critical Þeld and becomes equal to that of the unstable vortex in the critical point. The Þeld dependence of the smallest eigenvalues in the vicinity of the bursting Þeld (Fig. 8) also demonstrates the merging of the solutions in the critical point and yields a precise method for the calculation of the critical line h " (i8 ), which was used in Fig. 4. At the critical point both eigenvalues are zero. This means that perturbation (22) does not change the energy of the critical vortex structure. To inves- tigate the character of the transition we consider the energy change of vortices under a radial expan- sion or contraction. Instead of modes (22) which are valid only for small initial distortions of the solutions we consider the following transformation of the discretized proÞle: o@"ko for i(N 0 , o@"o for N 0 )i)N, (24) 189A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192 Fig. 8. The smallest eigenvalues for normal and large n-vortices as a functions of an applied magnetic Þeld in the vicinity of the critical point of the bursting instability. Fig. 9. Nonequilibrium energy of localized structures as a func- tions of the vortex size in the vicinity of the bursting Þeld. As the Þeld approaches the critical value h " "!0.005894, the local minimum (corresponding to the normal n-vortex) joins the local maximum (corresponding to the wide n-vortex), forming an unstable inßection point. A solution at the critical point decays by an unlimited expansion. Fig. 10. The proÞles of a radially stable 2n-vortex and an unstable large twin structure at negative magnetic Þeld. At the critical Þeld both proÞles coincide and the vortex becomes unstable with respect to the transition into the n-structure. where N is a number of points used in the discretiz- ation, and N 0 (N. Depending on the value of the k transformation (24) describes either can expansion (k’1) or a con- traction (k(1) of the structure. In Fig. 9 the energy of the distorted vortex as a function of its radius (23) is plotted for stronger magnetic Þelds close to the transition point. When the Þeld approaches the critical value, the local minimum corresponding to the vortex state transforms to an inßection point, demonstrating the tendency of the structure to an inÞnite expansion. 3.2. Multiple vortices In addition to p-vortices also vortices with a ro- tation by an angle kn (k*2) exist in certain regions of the phase diagram (Fig. 4). Example proÞles of regular multiple solutions can be found in Fig. 3. For directions of the magnetization close to the easy-axis the proÞles have a smaller slope, while in intermediate regions the function h(o) practically coincide with the solution for a planar domain wall [4]. Numerical analysis of the eigenvalue problem revealed that also these structures are radially stable, and the lowest excitation mode has again the form (22). As for the simple vortices, there exist also wide multiple vortices. They are shown for the case of 2n-vortices in Fig. 10. As earlier for the n-vortices, the solutions with the larger core size are radially unstable. At a critical value of the magnetic Þeld both proÞles merge, and by the unlimited expan- sion of the region with h’n the structure trans- forms to an ordinary n-structure. The critical Þeld for 2n-vortices is included in Fig. 7. 3.3. Vortices with nodes In addition to the localized structures considered above, a further class of solutions, proÞles with nodes, can be constructed in certain (usually nar- row) Þeld intervals. A simple vortex with a node is shown in Fig. 11. Unlike all other localized states A. Bogdanov, A. Hubert / Journal of Magnetism and Magnetic Materials 195 (1999) 182—192190 Fig. 11. The proÞle (a) and the Þrst excitation mode (b) for a vortex with a node. The structure is unstable with respect to the transition into a normal n-vortex. these structures have sectors with a reverse rotation of the magnetization vector. The solution of the spectral problem reveals their instability with re- spect to perturbations that remove the energetically disadvantageous humps (Fig. 11). This instability develops by merging the regions with the reverse rotation sense, a process which may be compared to the annihilation of unwinding domain walls [4]. In contrast, no annihilation of domain walls take place in regular, multiple vortices because their Ôdomain wallsÕ have a winding character. 4. Conclusions Isolated vortices and their stability regimes were investigated systematically for uniaxial ferromag- nets with Dzyaloshinsky interaction terms. The phase space of the solutions of the micromagnetic equations was studied by a geometrical approach, which allowed us to explore all possible localized solutions of cylindrical symmetry. Equilibrium structures were then calculated by a Þnite-di⁄er- ence method, and their stability with respect to small radial perturbations was investigated by an eigenvalue analysis. In addition to previously investigated regular n-vortices, new types of localized solutions have been constructed. They include vortices with rota- tion of the magnetization by angles kn (k’2), vor- tices with nodes and so called large vortices. By solving the eigenvalue problem for the perturbation functionals we proved the radial stability of all kn-vortices (k"1,22). In contrast, large vortices and vortices with nodes were found to be unstable. The kn-vortices are therefore the only stable rota- tionally symmetric vortex solutions in uniaxial fer- romagnets with a Dzyaloshinsky interaction. Acknowledgements Support by the Alexander von Humboldt Foun- dation and Deutsche Forschungsgemeinschaft is gratefully acknowledged. We would like to express our gratitude to A. Shestakov for useful dis- cussions. One of the authors (A.B) is very grateful to G. Cuntze, J. McCord, K. Ramsto‹ ck, K. Reber, and L. Wenzel for a friendly attitude and practical help during his stay in Erlangen. References [1] A. Buka, L. Kramer, Pattern Formation in Liquid Crys- tals, Springer, New York, 1995. [2] C. Rebbi, G. Soliani, Solitons and Particles, World Scien- tiÞc, Singapore, 1984. [3] S.E. Trullinger, V.E. Zakharov, V.L. Pokrovsky, Solitons, Vol. 17, North-Holland, Amsterdam, 1986. [4] A. Hubert, Theorie der Doma‹ nenwa‹ nde in Geordneten Medien, Springer, Berlin, 1974. [5] A.P. Malozemo⁄, J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials, Academic Press, New York, 1979. [6] A. Hubert, R. Scha‹ fer, Magnetic Domains. The Analysis of Magnetic Microstructures, Springer, Berlin, 1998. 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