Bloch Point Structure in a Magnetic Nanosphere
BBloch Point Structure in a Magnetic Nanosphere
Oleksandr V. Pylypovskyi, Denis D. Sheka,
1, 2, ∗ and Yuri Gaididei Taras Shevchenko National University of Kiev, 01601 Kiev, Ukraine Institute for Theoretical Physics, 03143 Kiev, Ukraine (Dated: September 25, 2018)A Bloch Point singularity can form a metastable state in a magnetic nanosphere. We classifypossible types of Bloch points and derive analytically the shape of magnetization distribution ofdifferent Bloch points. We show that external gradient field can stabilize the Bloch point: the shapeof the Bloch point becomes radial–dependent one. We compute the magnetization structure of thenanosphere, which is in a good agrement with performed spin–lattice simulations.
PACS numbers: 75.10.Hk, 75.40.Mg, 05.45.-a
INTRODUCTION
Topological singularities are widely recognized as keyto understanding the behavior of wide variety condensedmatter systems. Linear topological singularities such asdislocations, disclinations, and vortices, play a crucialrole in low–dimensional phase transitions, crystallineordering on curved surfaces , rotating trapped Bose–Einstein condesates etc. Recent advances in micro–structuring technology have made it possible to fabri-cate various nanoparticles with well–prescribed geome-try. Much recent research in this field has focused onthe statics and dynamics of topological singularities innanoscale confined systems: essentially inhomogeneousstates can be realized in magnetic nanoparticles andferroelectric nanoparticles . As a result of the compe-tition between exchange and magnetic dipole–dipole in-teractions the ground state of magnetic disks with sizeslarger than some tens of nanometers is a flux–closure vor-tex state.Besides linear singularities there exist also so–calledpoint singularities such as monopoles, Bloch points, boo-jums. For example, hedgehog (monopole) singularitiesplay a crucial role in the behavior of matter near quantumphase transitions that are seen in a variety of experimen-tally relevant two–dimensional antiferromagnets, boo-jums are relevant in superfluid He–3, Bloch points alongwith Bloch lines are principle in understanding of mag-netic bubble dynamics.
The concept of point singularities was introduced inmagnetism by Feldtkeller , who considered differentmagnetization distributions inside the singularity andproposed first estimations of the Bloch point shape.Later D¨oring studies how magnetostatic energy gov-erns the Bloch point structure by selecting the rotationangle inside the Bloch point. Bloch point singularitieswere directly observed in yttrium iron garnet crystals. During the last decade Bloch points were also studiedby micromagnetic simulations in nanowires, in bub-ble materials, in disks–shaped and astroid–shapednanodots . The ultrafast switching of the vortex coremagnetization open doors to consider the vortex statenanoparticles as promising candidates for magnetic ele- ments of storage devices. There are different scenariosof the switching process: (i) The symmetric or so–calledpunch–through core reversal takes place under the ac-tion of DC magnetic field applied perpendicularly to themagnet plane. This reversal process as a rule ismediated by creation of two Bloch points. However sin-gle Bloch point scenario was also mentioned in Thiaville et al. . (ii) The switching under the action of differ-ent in–plane AC magnetic fields or by a spin polarizedcurrents, is accompanied by the temporary creationand annihilation of the vortex–antivortex pair. The lat-ter is accompanied by Bloch point creation .The purpose of the current work is to study the mag-netization structure of the Bloch point of the sphericalnanosized particle. As opposed to bubble films, wherethe static Bloch point results from the transition be-tween Bloch lines, and vortex nanodots, where theBloch point dynamically appears during the vortex coreswitching process, the Bloch point in the nanosphereis an example of “pure” singularity without surrounding.Such a singularity is in some respect the only stable sin-gularity in ferromagnet. We consider different types ofBloch point and classify them in terms of vortex parame-ters. The conventional magnetization distribution in theBloch point is generalized for the radial–dependent one.Such radial distribution becomes important for the Blochpoint nanosphere under the action of nonhomogeneousmagnetic field. We show that radial gradient field canstabilize the Bloch point and compute the magnetizationstructure, which is in a good agrement with performedspin–lattice simulations.The paper is organized as follows. In Sec. I we de-scribe the model and present the classification of differ-ent Bloch point types (Sec. I A). The energetic analysisand the Bloch structure is analyzed in Sec. I B. In orderto stabilize the Bloch point inside the nanosphere, weconsider the the influence of external gradient field onthe magnetization structure. The Bloch point solutionbecomes radially dependent: we calculate the magnetiza-tion structure analytically in Sec. II. In Sec. III we studythe Bloch point structure numerically, in particular, theproblem of stability. We discuss our results in Sec. IV. InAppendix A we analyze the Bloch point structure under a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r the influence of weak fields using the linearized equations. I. THE MODEL AND THE BLOCH POINTSOLUTIONS
Let us consider the classical isotropic ferromagneticsphere of the radius R . The continuum dynam-ics of the magnetization can be described in termsof the magnetization unit vector m = M /M S =(sin Θ cos Φ , sin Θ sin Φ , cos Θ), where Θ and Φ are, ingeneral, functions of the coordinates and the time, and M S is the saturation magnetization. The total energy E of such a sphere, normalized by 4 πM S V with V = πR reads: E = E ex + E f + E ms . (1a)The first term in (1a) is dimensionless exchange energy: E ex = 38 π ε (cid:90) d r (cid:104) ( ∇ Θ) + sin Θ ( ∇ Φ) (cid:105) (1b)with ε = (cid:96) /R being reduced exchange length, (cid:96) = (cid:112) A/ πM S being the exchange length, A being the ex-change constant and r = ( x, y, z ) /R being the reducedradius–vector. The second term determines the interac-tion with external magnetic field H : E f = − π (cid:90) d r ( m · h ) , (1c)where h = H / πM S is a reduced external field. We willdiscuss the influence of external field later, see Sec. II.The last term determines the reduced magnetostatic en-ergy: E ms = − π (cid:90) d r ( m · h ms ) , (1d)where h ms = H ms / πM S is a reduced magnetostaticfield H ms . Magnetostatic field h ms satisfies the Maxwellmagnetostatic equations (cid:40) ∇ × h ms = 0 , ∇ · h ms = 4 πλ, (2)which can be solved using magnetostatic potential, h ms = − ∇ ψ . The source of the field h ms are magne-tostatic charges: volume charges λ ≡ − ( ∇ · m ) / π andsurface ones σ ≡ ( m · n ) / π with n being the externalnormal. The magnetostatic potential inside the samplereads: ψ ( r ) = (cid:90) V d r (cid:48) λ ( r (cid:48) ) | r − r (cid:48) | + (cid:90) S d S (cid:48) σ ( r (cid:48) ) | r − r (cid:48) | (3a) ≡ π (cid:90) V d r (cid:48) (cid:16) m ( r (cid:48) ) · ∇ r (cid:48) (cid:17) | r − r (cid:48) | . (3b) The equilibrium magnetization configuration is deter-mined by minimization of the energy functional (1),which leads to the following set of equations: ε ∇ m = ∇ ψ, ∇ ψ = ∇ · m . (4) A. Classification of singularities
Let us start the Bloch point as a particular solution of(4). In the exchange approach the simplest hedgehog–type Bloch point is characterized by the magnetizationdistribution of the form m = r /r with a singularity atthe origin. Using spherical frame of reference for theradius–vector r with the polar angle ϑ and azimuthalone ϕ , one can describe the magnetization angles of sucha Bloch point as follows: Θ = ϑ and Φ = ϕ . The energyof the Bloch point in the exchange approach reads E ex0 = 3 ε, E ex0 = 4 πAR. (5)This interaction is invariant with respect to the joint ro-tation of all magnetization vectors, which gives a possibil-ity to consider family of solutions with different rotationangles. We consider the following singular magnetization dis-tribution:Θ( ϑ ) = pϑ + π (1 − p ) / , Φ( ϕ ) = qϕ + γ, p, q = ± , (6)which describes a three–parameter Bloch point. We re-fer to the parameter q = ± p = ± γ describes the azimuthal rotational angle of the Blochpoint. We refer to the micromagnetic singularity (6) as toBP pq . For example, the hedgehog–type Bloch point is avortex Bloch point with positive polarity ( p = 1, q = 1, γ = 0). The schematic of magnetization distribution indifferent types of Bloch points is presented on Fig. 1. Theanalogy between Bloch point and vortices comes from thesymmetric or punch–through vortex polarity switchingprocess under the action of DC perpendicular magneticfield. This reversal process as a rule is mediated by cre-ation of two Bloch points. For example, two singulari-ties, BP and BP − describe intermediate state betweentwo vortices with opposite polarities, see Fig. 1c. It isalso instructive to mention that a single Bloch point canbe imagined as a composite of two vortices with oppo-site polarities: such a singularity can appear in 3D Eu-clidean space during the vortex polarity switching processin antiferromagnets . All four distributions for differ-ent signs of p and q can be observed during symmetricalBloch points injection in polarity switching process ofvortices and antivortices .Topological properties of the Bloch point can be de-scribed by the topological (Pontryagin) index Q = 14 π (cid:90) sin Θ( r )dΘ( r )dΦ( r ) = pq. (7) (a) p = q = 1 (b) p = − q = 1 (c) Vortexswitching (d) p = 1 q = − p = q = − FIG. 1: Schematic of different types of Bloch points. Magnetization distribution in azimuthal vortex Bloch points insphere, see Figs. 1a, 1b, and both Bloch points in axial part of cylinder-shaped sample during the vortex polarityswitching process, see Fig. 1c. The same is for azimuthal antivortex Bloch points, see Figs. 1d, Fig. 1e, and bothsingularities in axial part of astroid–shaped sample during the switching, see Fig. 1f.Different Bloch point distributions with equal Q are topo-logically equivalent: e.g., BP − − can be obtained fromBP by simultaneous rotation of all magnetization vec-tors by π in vertical plane, and BP − transforms to BP − by rotation by π/ for magnetic bubbles. B. Magnetization structure of Bloch points
The most strong exchange interaction is invariant withrespect to the rotation angle γ . Such degeneracy is re-moved under account of magnetostatic interaction. It isworth noting that the problem of stray field influence onthe Bloch point energetics has a long story. Feldtkeller inhis pioneer work used a so–called pole avoidance princi-ple, see e.g. Ref. 29: the magnetostatic tries to avoid anysort of volume or surface charge. In this way he calcu-lated the angle γ from the condition that the total volumemagnetostatic charge (cid:82) λ ( r )d r = 0, where λ ( r ) is thecharge density. For the Bloch point given by Ansatz (6)it has a form λ ( r ) = − (cid:2) p sin ϑ + cos γ (cos ϑ + 1) (cid:3) / πr and leads to the rotation angle γ F = arccos (cid:16) − p (cid:17) = (cid:40) ◦ , p = +1 , ◦ , p = − . (8)In it interesting to note that the same value γ F alsocorresponds to absence of the total surface charge, (cid:82) σ ( r )d S = 0, where the surface charge density σ ( r ) = (cid:0) p cos ϑ + cos γ sin ϑ (cid:1) / π .Another approach was put forward by D¨oring , whodetermined the equilibrium angle of γ by minimizing theenergy E msD = 38 π (cid:90) V d r ( h ms ) (9) and obtained γ D = arccos (cid:18) − (cid:19) ≈ . ◦ . (10)However one has to emphasize that the equilibrium angle(10) minimizes only the inner part of the magnetostaticenergy because the integration in (9) is carried over thesample volume V while the outer part of stray field isignored. Note the similar approach was used in quiterecent paper, where a magnetization contraction wastaken into account.The aim of this section is to find the equilibrium rota-tion angle which minimizes the total magnetostatic en-ergy. In order to derive the magnetostatic energy ofBloch points (6), we calculate first magnetostatic poten-tial (3b) using an expansion of 1 / | r − r (cid:48) | over the sphericalharmonics,1 | r − r (cid:48) | = 1 r > ∞ (cid:88) l =0 l (cid:88) m = − l π l + 1 (cid:18) r < r > (cid:19) l Y lm ( ϑ, ϕ ) Y (cid:63)lm ( ϑ (cid:48) , ϕ (cid:48) )with r < = min( r, r (cid:48) ) and r > = max( r, r (cid:48) ) which resultsin ψ pq =1 ( r ) = pπr + π r −
8) cos γ + πr ( p − cos γ ) cos ϑ,ψ pq = − ( r ) = pπr (1 + cos ϑ ) + πr cos(2 ϕ + γ ) sin ϑ. Simple calculations show that the magnetostatic energyof the antivortex Bloch point does not depend on γ and E ms q = − = 7 / ≈ .
23. In contrast to this, the vortexBloch point energy depends on the rotation angle γ andhas the form E ms pq =1 ( γ ) = 130 (7 + 4 p cos γ + 4 cos 2 γ ) . (12)The equilibrium value of rotation angle γ corresponds E m s ( γ ) Angle γ , ◦ γ γ D γ F FIG. 2: (Color online.) The Bloch point energy vs rotation angle for BP : analytical result (12) (solidcurve) and simulations (symbols). Simulationsparameters: sphere diameter 2 R = 35 a , exchangelength (cid:96) = 3 . a , damping parameter η = 0 . γ = arccos (cid:16) − p (cid:17) ≈ (cid:40) ◦ , p = +1 , ◦ , p = − . (13)Let us compare Bloch point energies (12) for abovementioned approaches: the energy of Feldtkeller Blochpoint E ms pq =1 ( γ F ) = 0 .
1, for D¨oring Bloch point onehas E ms ( γ D ) ≈ . is E ms1 ( γ EV ) ≈ . γ , see (13): E ms pq =1 ( γ ) = 112 ≈ . . (14)In order to verify our results we performed numericalspin–lattice simulations, see details in Sec. III. We com-pare analytical dependence E ms p =1 q =1 ( γ ), see Eq. (12), withthe discrete energy (24), extracted from simulations, seeFig. 2. Both dependencies are matched in maximum at γ = 0. Comparison can be provided by calculating en-ergy gain ∆ E ( γ ) = E msmax − E ms ( γ ) for different rotationangles γ . According to simulation results the energy gainfor mentioned above angles read:∆ E ( γ F ) ≈ . , ∆ E ( γ D ) ≈ . , ∆ E ( γ ) ≈ . . The maximum energy gain takes place for γ , which cor-responds to the energy minimum in a good agrement withour analytical result (13). II. THE BLOCH POINTS IN EXTERNAL FIELD
The Bloch point does not form a ground state of amagnetic sphere. It corresponds to the saddle point(sphaleron) of the energy functional . This brings up the question: How to stabilize the Bloch point? In thissection we show that one way to achieve this goal is toapply a magnetic field which has the same symmetry asthe hedgehog Bloch point with m = r /r , i. e. a radialsymmetric magnetic gradient magnetic field in the form h = b r . (15)Under the action of the space dependent magnetic field(15) the magnetization distribution also becomes spacedependent. We take into account possible dependence bythe following radial Bloch point AnsatzΘ( ϑ ) = pϑ + π (1 − p ) / , Φ( r, ϕ ) = qϕ + γ ( r ) (16)with a radially dependent parameter γ ( r ) in comparisonwith Eq. (6). The form of this Ansatz will justified bynumerical simulations in Sec. III.Inserting Eq. (16) into Eq. (1b) for the exchange en-ergy of such magnetization distribution we get E ex = 3 ε + ε (cid:90) (cid:18) d γ d r (cid:19) r d r. (17a)The magnetostatical potential of the Bloch point (16)reads ψ p =1 q =1 ( r ) = − π (cid:90) r [1 + 2 cos γ ( r (cid:48) )] d r (cid:48) −− π ϑ − r r (cid:90) r (cid:48) [cos γ ( r (cid:48) ) −
1] d r (cid:48) . Here and below we consider the case of BP only. Themagnetostatic energy of such a Bloch point has the form E ms = 110 (cid:90) r [7 + 4 cos γ ( r ) + 4 cos 2 γ ( r )] d r. (17b)From Eq. (1c) we obtain that the Bloch point interactionwith magnetic field can be expressed as follows E f = − b (cid:90) r cos γ ( r )d r. (17c)By minimizing the total energy, δ E /δγ = 0, we obtainthat the equilibrium distribution γ ( r ) is a solution of thefollowing nonlinear differential equation ε d γ d r + 2 εr d γ d r + 15 sin γ + 25 sin 2 γ − br sin γ = 0 (18)augmented by boundary conditions of the formd γ d r (cid:12)(cid:12)(cid:12)(cid:12) r =0 = d γ d r (cid:12)(cid:12)(cid:12)(cid:12) r =1 = 0 . (19) -1-0.9-0.8-0.7-0.6 0 0.25 0.5 0.75 1 g ε ( r ) Distance r Analytics b = 0 . b = 0 . FIG. 3: (Color online.) Reduced rotation angle g ε ( r ),see (20) for different field intensities and ε = 0 . γ ( r ) ≈ γ + bg ε ( r ) , | b | (cid:28) . (20)An explicit form of the function g ε ( r ) is calculated inAppendix A. The comparison with numerical solution ofEq. (18) shows a quite good agreement up to relativelystrong fields ( b (cid:46) γ = 0 (mod π ).To describe the behavior of the Bloch point in a criti-cal region b ≈ b c where the spatially non-uniform distri-bution transforms to the spatially uniform one, we use avariational approach with a two–harmonics trial function γ ( r ) ≈ α + α cos πr . Near the critical point α , α (cid:28) α and to the second orderwith respect to α . By excluding α and keeping termsnot higher than α , we get E ( γ ) ≈ E + p ( b, ε ) α + p ( b, ε ) α , (21)The energy (21) as a function of α has a double–wellshape ( p ( b, ε ) <
0) for b < b c with the critical magneticfield b c given by b c ( ε ) ≈ . − . ε + (cid:112) . − . ε + 467 ε (22)In the critical region when 0 < b c ( ε ) − b (cid:28) b c ( ε ) ,α ( b ) ≈ a ( ε ) (cid:112) b c ( ε ) − b (23)For b > b c , p > α = 0. It corresponds to γ = 0. Numer-ical integration of Eq. (18) for ε = 0 .
05 shows that the phase transition occurs when b c ≈ . b c (0 . ≈ .
465 obtained fromEq. (22). The critical behavior predicted by (23) is alsoconfirmed by our numerical simulations (see Fig. 4a).
III. NUMERICAL STUDY OF THE BLOCHPOINT STRUCTURE
In order to check analytical results about Blochpoint structure, we performed simulations using in–housedeveloped spin–lattice simulator
SLaSi that solvesLandau–Lifshitz–Gilbert equation in terms of spinsd S n d t = − (cid:126) (cid:20) S n × ∂ H ∂ S n (cid:21) − ηS (cid:20) S n × d S n d t (cid:21) , where H is a lattice Hamiltonian of the classical ferro-magnet: H = − J (cid:88) ( n,δ ) S n · S n + δ + 2 µ B H (cid:88) n S n + 2 µ (cid:88) n (cid:54) = k (cid:20) ( S n · S k ) r nk − S n · r nk )( S k · r nk ) r nk (cid:21) . (24)Here S n is a classical spin vector with fixed length S in units of action on the site n of a three–dimensionalcubic lattice with lattice constant a , J is the exchangeintegral, µ B is Bohr magneton, r nk is the radius–vectorbetween n -th and k -th nodes, η is a damping param-eter, H is external magnetic field and δ runs over sixnearest neighbors. Integration is performed by modified4–5 order Runge–Kutta–Fehlberg method (RKF45) andfree spins on the surface of the sample. Numerically we checked the Bloch point structure,given by the radial–dependent Ansatz (16). by model-ing spherically–shaped sample with diameter 2 R = 35 a (such a sample consists of 24 464 nodes with nonzerospin), and exchange length (cid:96) = 3 . a ( ε = 0 . b = 1 .
0. By modeling theoverdamped dynamics we observed that the Bloch pointstructure quickly relaxes to the state similar to one,given by (16): The polar Bloch point angle Θ( r ) doesnot deviate from ϑ within the accuracy 0 . γ ( r ), see Fig. 5. Simula-tions were performed for crystallographic directions [111]( ϑ = π/
4) and [110] ( ϑ ≈ π/
2, the plane is shifted by z = − . a from the origin). One can see from Fig. 5 thatnumerical data are well confirmed by analytical curve γ ( r ), calculated as numerical solution of (18).To validate our theory we performed also direct stabil-ity check. Numerically we check the stability of the Blochpoint against the shift of its position. We start simula-tions with the Bloch point state using Ansatz–function(16), which is shifted along ˆ z –axis by ∆ z = − a . We R o t a t i o n a n g l e γ Field b γ (0) γ (1) γ th (0) γ th (1) b c (a) Critical behaviour R e du ce d e x c h a n g e l e n g h t ε Field b Nonhomogeneous00.020.040.060.080.1 1.4 1.6 1.8 2 2.2 2.4Hedgehog (b) Phase diagram
FIG. 4: (Color online.) Bloch point under the action of the gradient field. (a): Rotation angle vs field intensity b near the critical field b c ≈ .
473 from numerical solution of Eq. (18) (blue curves) and theoretical estimation byEq. (23) (red curves) with ε = 0 .
05. Solid lines correspond to the rotation angle γ (0) and dashed line to γ (1). (b):Phase diagram for solutions of Eq. (18). The upper (hedgehog) phase correspond to the solution γ = 0, the lower(nonhomogeneous) one to the radial–dependent Bloch point with γ ( r ). Dashed lines correspond to analytical resultfor the critical field b c = 1 .
465 for ε = 0 .
05, see text. R o t a t i o n a n g l e γ ( r ) Distance r TheoryData ( ϑ = π/ )Data ( ϑ = π/ ) FIG. 5: (Color online.) Radial dependence of rotationangle γ in spherical particle. Line: numericalintegration of (18). Symbols: SLaSi simulations forcrystallographic directions [110] and [111]. Parametersare the same as in Fig. 2.also apply γ ( r, t = 0) = 3 ◦ in order to break the sym-metry. For rapid relaxation we used in most of simu-lations the overdamped regime (the damping parame-ter η = 0 . S tot x = S tot y = S tot z = 0.The temporal evolution of initially shifted Bloch pointis presented in Fig. 6 for the Bloch point sample with2 R = 35 a (24 456 nodes) in applied field with b = 1, see also the supplementary video . Originally the Blochpoint was shifted down from the origin which correspondsto S tot z >
0, see inset (a). During the evolution a numberof magnons are generated, inset (b). After quick damp-ing of oscillations, the micromagnetic singularity goes tothe sample origin, see inset (c). The relaxation processconsists of two parts: (i) The rotation angle γ ( r ) changesits value from initial uniform one to the final nonhomo-geneous state during a time τ γ ≈ ω − . (ii) The re-laxation of S tot z component of total spin of the sampletooks approximately the same time. During all simula-tions time | S tot x | ≈ (cid:12)(cid:12) S tot y (cid:12)(cid:12) (cid:46) − . IV. CONCLUSION
To summarize, we study the magnetization structureof the Bloch point. In spite of the fact that the Blochpoint as a simplest 3D topological singularity was stud-ied during a long time, from the pioneer papers byFeldtkeller and D¨oring , see also for review Refs. 4and 10, the problem of the Bloch point structure stillcauses discussions. The point is that the moststrong exchange interaction determines only the relativemagnetization distribution accurate within the rotationangle γ . This rotation angle, which is determined by themagnetostatic interaction, is most questionable: its valueis equal to 120 ◦ according to Feldtkeller , to 112 . ◦ fol-lowing D¨oring and 113 ◦ following El´ıas and Verga .We analyze the origin of all these results and calculatedthe equilibrium value, about 105 ◦ , see (12), which mini-mizes the total magnetostatic energy, not only the part -0.0500.050.10.150.20.250.3 0 100 200 300 400 500 600 700 800 900 T o t a l s p i n S t o t z / S u , t o t z Time t , ω − Dynamics of S tot z and plane z = − . a (a) (b) (c) (d) Plane y = − . a (a) t = ω − (b) t = 200 ω − (c) t = 400 ω − (d) t = 900 ω − -1-0.500.51 FIG. 6: (Color online.) Dynamics of total spin along z –axis of the sample. The Bloch point is initially shifted by∆ z = − a from center of the sample. Insets show magnetization distribution in z = − . a and y = − . a planesin different times. Color bar indicates S z, n for different lattice nodes. Applied field amplitude b = 1, otherparameters are the same as in Fig. 2.of it.The next problem appears in modeling of the Blochpoint. As is was discussed by Thiaville et al. , the mod-eling of singularity is mesh–dependent. In particular, amesh–friction effect and a strong mesh dependence of theswitching field during the Bloch–point–mediated vortexswitching process was detected using OOMMF micromag-netic simulations. The reason is that micromagneticsimulators consider the numerically discretized Landau–Lifstitz equation, which are valid in continuum theory.Since the Bloch point appears as a singularity of con-tinuum theory, it is always located between mesh points,and causes the mesh–dependent effects and therefore maybe insufficient for describing near–field Bloch point dis-tribution. In contrast to this, spin–lattice simulations arefree from these shortage. From the beginning we considerdiscrete spins, located on the cubic lattice, and their dy-namics is governed by the discrete versions of Landau–Lifshitz equations. The lattice Hamiltonian allows us tocalculate the discrete energy of the Bloch point similarto the atomiclike calculations by Reinhardt .Using in–house developed spin–lattice SLaSi simula-tor we modeled the Bloch point state nanosphere andchecked our analytical predictions about Bloch pointstructure. We stabilized the singularity inside the spher-ical particle by applied gradient magnetic field. The fieldcauses the new type of Bloch point with radial–dependentrotation angle γ ( r ). ACKNOWLEDGMENTS
Authors acknowledge computing time on the high–performance computing cluster of National TarasShevchenko University of Kyiv and SKIT-3 Comput-ing Cluster of Glushkov Institute of Cybernetic of NAS of Ukraine . This work was supported by the Grant ofthe President of Ukraine No. F35/538-2011. We thankV. Kravchuk for helpful discussions. Appendix A: Bloch point structure in a weak field
We consider here the magnetization structure of aBloch point under the action of weak magnetic field. Onehas to linearize Eq. (18) on the background of the unper-turbed rotation angle γ , see (20), which can be presentedas follows: γ ( r ) ≈ γ + bg ε ( r ) , g ε ( r ) = 2 √ ε f ( λr ) , λ = 12 (cid:114) ε . Here the function f ( ξ ) satisfies the linearized version ofEq. (18): d f d ξ + 2 ξ d f d ξ − f = ξ, which can be easily integrated: f ( ξ ) = C λ sinh ξξ + 2 cosh ξ − ξ − ξ,C λ = λ − λ sinh λ + 2 cosh λ − λ cosh λ − sinh λ . (A1)The graphics of the g ε ( r ) for ε = 0 .
05 is presented inFig. 3 together with numerical solution of Eq. (18) byshooting method. In spite of limitation of our analysisby the case of weak field, | b | (cid:28)
1, the function g ε ( r ) pro-vides a good approximation for the solution of nonlinearEq. (18) up to to very strong fields b ≤ (cid:12)(cid:12) [ γ ( r ) num − γ ( r ) theor ] /γ ( r ) num (cid:12)(cid:12) ≤ . -1.2-1-0.8-0.6-0.4-0.20 0 0.25 0.5 0.75 1 g ε ( ) a nd g ε ( ) ε = ℓ /R g ε (0) g ε (1) FIG. 7: (Color online.) Reduced rotation angle g ε vs reduced exchange length ε : at r = 0 (solid curve) and r = 1 (dashed curve). The rotation angle in the Bloch point is essentiallyinfluenced by the exchange parameter ε , see Fig. 7. In thelimit case of small particle ( ε (cid:29)
1) the role of exchangeis dominant, which results in the constant angle g ∞ = −√ / ≈ − .
97. In the opposite case ε (cid:28)
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