Straight motion of half-integer topological defects in thin Fe-N magnetic films with stripe domains

In thin magnetic films with perpendicular magnetic anisotropy, a periodic “up-down” stripe-domain structure can be originated at remanence, on a mesoscopic scale (~100 nm) comparable with film thickness, by the competition between short-range exchange coupling and long-range dipolar interaction. However, translational order is perturbed because magnetic edge dislocations are spontaneously nucleated. Such topological defects play an important role in magnetic films since they promote the in-plane magnetization reversal of stripes and, in superconductor/ferromagnet hybrids, the creation of superconducting vortex clusters. Combining magnetic force microscopy experiments and micromagnetic simulations, we investigated the motion of two classes of magnetic edge dislocations, randomly distributed in an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{N}}}_{2}^{+}$$\end{document}N2+-implanted Fe film. They were found to move in opposite directions along straight trajectories parallel to the stripes axis, when driven by a moderate dc magnetic field. Using the approximate Thiele equation, analytical expressions for the forces acting on such magnetic defects and a microscopic explanation for the direction of their motion could be obtained. Straight trajectories are related to the presence of a periodic stripe domain pattern, which imposes the gyrotropic force to vanish even if a nonzero, half-integer topological charge is carried by the defects in some layers across the film thickness.

FIG. 1. Schematic description of the method used to measure the displacement of magnetic edge dislocations. a) AFM image at a reversal field near remanence (H = −35 Oe): two main particles are clearly visible; b) we mark them with green circles on the figure and note down their exact surface coordinates; c) we create an empty mask, where we report only the green circles in the positions determined in b); d) we superimpose the mask in c) on the MFM image; now we can choose a particular dislocation and mark it with a red cross; e) we apply a magnetic field H = −126 Oe and measure the same area, where the two particles are still present in the AFM image; we repeat the procedure described from a) to d) and obtain figure f); f) the red cross is maintained in the same position of figure d), as deduced from the coordinates of the green circles, showing that the dislocation has now moved with respect to the red cross. 22 Topographic imperfections are generally considered a bother in the Magnetic Force Microscopy (MFM) technique 23 but, in our experiment, they were a help to create a map of the analyzed surface. It is well known that the adhesion of 24 fine particles of contaminants on the surface of a sample measured in air is unavoidable. For this reason we protected 25 our Fe-N film with a gold capping layer. In this way, the particles (maximum size below 10 nm) can be detected 26 during the Atomic Force Microsopy (AFM) scan, but they do not affect the MFM results, since the lift height of 27 the magnetic analysis was well above 30 nm. In our measurements we marked the particles present on the gold 28 surface, detected by AFM, and, superimposing the relative MFM image, we transferred the markers position on the 29 stripes. This procedure was repeated for all MFM images taken at different fields, referring to the same archipelago 30 of particles, that did not move from one measurement to the other. In this way, one can assure that the analyzed 31 area is the same at every field and can deduce the precise position of the magnetic edge dislocations with respect to 32 the topography, investigating their evolution and motion in the presence of a magnetic field. A schematic example of 33 the technique is reported in the figure 1. This is just an example, demonstrating how with this procedure one can 34 get exact coordinates of all the points included in the examined area, starting from the position of surface particles, 35 and transfer the deduced map on the MFM image. This allowed us to follow the motion of the magnetic dislocations, 36 occurring under the protecting gold layer, in the Fe-N film.

38
DATA TAKEN AT REMANENCE. 39 Here we present some supplementary MFM data, taken on the same Fe-N sample as the one used to obtain the 40 MFM data in Fig. 1 of the Main Article. It is important to note that all the supplementary MFM images in Fig. 41 2 have been taken at remanence, using a different experimental apparatus located in Paris. In the supplementary 42 MFM images, blue rectangles and squares denote corresponding areas of the sample surface, while blue crosses mark 43 topographical defects taken as landmarks (using the method described in the previous Section). Green and red circles 44 denote the positions of two different types of magnetic edge dislocations, while green and red arrows denote the 45 direction of their motion after the application of an in-plane magnetic field H app . We remind that the field was 46 removed before taking each MFM image. Note that, in Fig. 2, a positive in-plane magnetic field was applied in the 47 North-East (NE) direction, while a negative field in the South-West (SW) direction.

FIG. 2.
Effect of reversing an in-plane applied magnetic field on MFM data taken at remanence. MFM images of the Fe-N film taken at remanence after a magnetic field Happ, of variable intensity and sign in the various panels, was applied in plane. Blue rectangles and squares denote corresponding areas of the sample surface, while blue crosses mark topographical defects taken as landmarks using the method illustrated in Fig. 1. Green and red circles denote the positions of two different types of magnetic edge dislocations, while green and red arrows denote the direction of their motion after the application of the in-plane magnetic field Happ and its subsequent removal. A positive field was applied in plane in the North-East (NE) direction (panels (a) and (d)), while a negative field in the South-West (SW) direction (panels (b) and (c)), as indicated by the black thick arrows. One can infer that some dislocations exist, for which the inversion of the in-plane magnetic field Happ induces an inversion in their displacement; moreover, opposite dislocations in a pair appear to move into opposite directions. field was removed. A stripe pattern with two types of magnetic edge dislocations, marked by red and green crosses, 50 developed: see Fig. 2(a). Next, a negative magnetic field (H app =-150 Oe) was applied and subsequently removed: 51 see Fig. 2(b). One can observe that the "green" dislocations moved to NE and the "red" dislocations moved to SW 52 with respect to the blue crosses (i.e., the still landmarks). 53 Afterwards, different regions of the sample surface were analyzed after application, and subsequent removal, of a 54 moderate negative field: see Fig. 2(c) (H app =-150 Oe). Next, the same analysis was performed after application, and 55 subsequent removal, of a positive field: see Fig. 2(d) (H app =1800 Oe). One can observe that, in this case, the "green" 56 dislocations moved to SW and the "red" dislocations moved to NE with respect to the blue crosses.

57
From these MFM data taken at remanence, one can therefore infer that some dislocations exist, for which the inver-58 sion of the in-plane applied magnetic field induces an inversion in their displacement; moreover, opposite dislocations 59 in a pair appear to move into opposite directions.

61
The evolution in time of a given magnetic texture, n = M/M s , where M s is the saturation magnetization, is 62 phenomenologically described by the Landau-Lifshitz-Gilbert (LLG) equations of motion 1,2 , possibly modified 3,4 to 63 include the contribution of spin transfer torques 5,6 where γ e = |e| 2me g e = µB g e is the electron gyromagnetic ratio; e, m e and g e are the electron charge, mass and g-factor, 65 respectively; µ B the Bohr magneton and the Planck constant; H eff the local effective field; α > 0 the Gilbert 66 damping and β the non-adiabaticity parameter 7 . The last two terms in (1) respectively represent the adiabatic 4 67 and non-adiabatic 5,6 contribution to the torque, acting on n due to an electrical current, with density j and spin 68 polarization P s ; u = geµB 2|e|Ms P s j is the velocity associated with the spin-polarized electrical current.

69
Starting with the LLG equations of motion (1), Thiele 8,9 derived an approximate equation 10,11 in order to describe 70 the steady-state dynamics of a domain wall. In the presence of both an external magnetic field and a spin-polarized 71 electrical current, the Thiele equation reads 12 where v is the drift velocity induced by an external magnetic field, and u is the velocity associated with a spin-73 polarized electrical current. According to the Thiele equation (2), the total force, F tot , acting on the domain wall 74 in the magnetic texture, is the sum of three terms: (i) an external force, F ext = −∇U ; (ii) a gyrotropic force, 75 F gyro = G × (v − u); (iii) a dissipative force, F diss = D · (αv − βu). Note that, in this work, we set u = 0 in Eq. 2 76 because no spin-polarized electrical current is applied to the system.

77
It is important to note a fundamental difference between the micromagnetic simulations and the approximate 78 Thiele dynamics of the dislocations. The micromagnetic simulations are performed by numerically solving the LLG 79 equations of motion for an inhomogeneous magnetization distribution (in our case, a magnetic stripe pattern with edge 80 dislocations, generated by the procedure described in Methods: i.e., tilting the magnetic field with respect to the film 81 surface by an angle typically ranging from 1 to 3 degrees). In Eq. 1, all the forces due to reversible effects are included 82 in the effective field H eff : both the internal forces (due to exchange energy, demagnetizating energy, and anisotropy 83 energy) and the external forces (due to Zeeman energy, in our case). Dissipative effects are also taken into account 84 through the Gilbert damping α. The Thiele equation of motion (2) describes, instead, the steady-state motion of an 85 idealized inhomogeneous distribution of the magnetization, under the effect of an external force, F ext (in our case, 86 the external force is due to the applied magnetic field). The gyrotropic force acting on the moving magnetization 87 distribution accounts for its topological properties. Also in the Thiele equation dissipative effects are taken into 88 account through the Gilbert damping, but the magnetization distribution is assumed to perform a displacement, at 89 constant velocity v, without any dynamical deformation. Therefore, internal forces must not be included explicitly 90 in Eq. 2.

91
Let us consider the driving effect of a dc external field on the magnetic edge dislocations of a regular stripe domain 92 pattern with period P . To fix ideas, let us assume that a strong magnetic field is first applied in plane along the 93 versor of the stripes axis, e x , in order to saturate the sample. Subsequently the field is decreased towards remanence, 94 where the stripe domain pattern develops, with e x as the stripes axis. Next, the field is reversed: i.e., a dc magnetic 95 field, H rev , is applied antiparallel to e x . In the following, we will also consider the case of a dc magnetic field, H, 96 parallel to e x . In any case, the intensity of the driving magnetic field, externally applied to the stripe domain pattern, 97 is supposed to be moderate: i.e., smaller than H c , the coercive field.

98
The unit magnetization vector is expressed in polar coordinates as where r = (r cos φ, r sin φ) is the position vector of n in the film plane, xy; Θ(r) is the canting angle formed by 100 n with the normal to the film plane, e z ; and Φ(φ) is the azimuthal angle, formed by the in-plane magnetization, 101 n IP = (m x , m y ), with the stripes axis, e x .

102
The Zeeman potential energy, U , the gyrovector, G, and the dissipation dyadic, D, are respectively given by 8,9,11 where the integrals are extended to the film volume.

106
For sufficiently small intensity of a dc magnetic field applied along the stripes axis, x, either parallel (H x > 0) or 107 antiparallel (H x < 0) to the unit vector, e x , we put forward the hypothesis that the complex magnetic structure of a 108 given magnetic edge dislocation can be approximated by the idealized spin configuration shown in Fig. 3 for the top, 109 central and bottom film layers, respectively.

110
For the sake of simplicity, we assume the topological defect to have a semicircular shape with radius R = P/4. Note 111 that, owing to presence of closure "cap" domains 13-16 near the top and bottom surfaces of the film, the magnetic 112 configuration in the semicircular region of the dislocation varies along the film thickness.

113
For the canting angle, Θ(r), we assumed the following dependence where Θ(r) is the angle formed by n with the normal to the film plane, e z . Note that, in our idealized spin configuration 115 (Fig. 3), the angle Θ(r) was supposed to be independent of the layer. The parameter p is the z-polarization. For 116 p = +1, as r increases from 0 to ∞, the angle Θ(r) increases monotonically from Θ = 0 ("up" magnetization) to 117 Θ = π ("down" magnetization). For p = −1, as r increases from 0 to ∞, the angle Θ(r) decreases monotonically 118 from Θ = π ("down" magnetization) to Θ = 0 ("up" magnetization). We suppose that ∆ ≪ R is the width of the 119 transition region between two domains with opposite z-component of the magnetization.

120
For the azimuthal angle, Φ(φ), we assumed the following dependence where Φ(φ) is the angle formed by the in-plane magnetization, n IP , with the stripes axis, e x . The parameters m and 122 γ are, respectively, the vorticity and helicity 17 of the in-plane magnetization, n IP , in the transition region between 123 domains with opposite values of m z . One has m = +1 for a vortex arrangement of the in-plane magnetization, and 124 m = −1 for an antivortex arrangement; the helicity angle, γ, in principle can take any value. Note that in our 125 idealized spin configuration (Fig. 3) the same vorticity m = +1 was taken for all layers, while the helicity angle γ 126 was supposed to depend on the layer.

127
At the top film surface, the presence of magnetic-flux-closure "cap" domains makes the in-plane magnetization 128 vector, n IP , point outward along the radius of the semicircle, namely the helicity angle is γ = 0.

129
At the bottom film surface, the magnetization in the magnetic-flux-closure "cap" domains is oppositely directed 130 with respect to the top surface. Then, the in-plane magnetization vector, n IP , points inward along the radius of the 131 semicircle, namely the helicity angle is γ = π. 132 In the central film layer, closure domains are absent. Therefore, the in-plane magnetization, n IP , is tangentially 133 disposed with respect to the semicircumference: i.e., the helicity angle is |γ| = π 2 . Moreover, at remanence (or for 134 sufficiently small intensity of the dc driving magnetic field), we heuristically assume that a symmetric configuration 135 is realized, with a constant vorticity (m = +1) and with a change in the sign of the helicity angle occurring just at 136 the endpoint of the stripe domain (i.e. for φ = 0). Namely, in the central film layer, at remanence (or for sufficiently 137 small intensity of the dc driving magnetic field), we suppose either a head-to-head configuration or a tail-to-tail one 138 to be realized We remind that γ = + π 2 is associated with a counter-clockwise circulation of the in-plane magnetization, while γ = − π 2 140 is associated with a clockwise circulation. Clearly, it depends on the history of the sample whether a head-to head 141 configuration or a tail-to-tail one is realized, for a given topology of the defect. E.g. the head-to-head configuration 142 shown in the central panel of Fig. 3 is expected to be realized, at remanence (or for sufficiently small intensity of the 143 driving dc magnetic field), in a film which was previously saturated in plane along e x .

144
In the following, we are going to establish a link between the dynamical behavior and the topological properties of 145 the in-plane magnetization distribution of the defect. Namely, within a fixed film layer, the in-plane magnetization 146 distribution in the magnetic edge dislocation can be associated with a topological number 17 (or, equivalently, a 147 topological charge 18 ) The φ-integration simply provides πm. Note that this result holds not only for the central film layer and for the head-149 to-head configuration explicitly considered on the r.h.s. of Eq. 10. Rather, it holds for any configuration of a given 150 film layer, provided only that a constant vorticity (m = dΦ dφ ) is assumed for the in-plane magnetization configuration 151 along the semicircumference. As regards the r-integration, one has to consider that the defect is associated with a 152 localized magnetization texture with a finite radius, R. Therefore, the integration range can be limited between 0 and 153 2R. Depending on the z-polarization parameter (p = ±1), one therefore obtains that the topological number, N sk , 154 associated with the in-plane magnetization distribution of the defect in each film layer is a half integer The sign of N sk is solely determined by the vorticity (m) and the z-polarization (p = ±1), but not by the helicity 156 (γ) of the in-plane magnetization configuration. E.g., for the idealized in-plane magnetization distribution of Fig. 3, 157 one has, in each film layer, m = 1 and p = −1, so that N sk = + 1 2 does not change from layer to layer: i.e., N sk is 158 independent of the helicity angle, γ.

160
In this Section, we perform an explicit calculation of the external force, F ext , the gyrovector, G and the dissipation 161 dyadic, D, for a magnetic edge dislocation embedded in a stripe domain structure. The magnetization configuration 162 in the central film plane, and the forces resulting from such a calculation, are schematically depicted in Fig. 4 in the 163 case of a reversal magnetic field applied in the film plane along the stripes axis, and in Fig. 5 in the case of a positive 164 magnetic field. The dc magnetic field is supposed to have such a low intensity that, in the framework of the idealized 165 model presented in the previous section, either a head-to-head (a) or a tail-to-tail (b) magnetization configuration is 166 assumed to be realized in the central film plane, depending on the topology of the dislocation and on the history of 167 the film. 168 1. External force 169 Malozemoff and Slonczewski 11 provided an argument to calculate the static force, F ext , tending to displace a non-170 equilibrium distribution of magnetization, M(x − X), in the presence of some externally applied field distribution, 171 H ext (x). X is a vector representing the position of the distribution (in our case, of the center of the semicircular 172 region) and x is the position vector of the magnetization in the local frame of reference. A distortion will be described 173 by a change in the functional form M(x); a displacement only by a change in X. 174 The external force F ext = −∂U /∂X is obtained differentiating the potential energy, U , of the magnetization 175 distribution with respect to its position, X. Any variation δU can be expressed as a volume integral of local density 176 variations δu(x − X); therefore the external force can be written in two alternative forms 11 In the following, we use the second expression (on the right of Eq. 12). Moreover, we take into account that where e r = cos φe x + sin φe y and e φ = − sin φe x + cos φe y respectively denote the radial and tangential unit vector. 179 The external force, F ext , associated with a magnetic field applied along the stripes axis, H = H x e x , can be expressed 180 in Cartesian coordinates as where the φ-integration is extended to the semicircumference. 182 We now take into account, in an approximate way, the layer dependence of the azimuthal angle, Φ(φ) = mφ + γ, 183 by assuming the following z-dependence for the helicity, γ: In the Appendix A, we explicitated the calculation of F ext , by separating the contributions from the top, bottom 185 Whereas, the contribution to the external force from the central film layers in the dislocation is always parallel to the 187 applied magnetic field for a head-to-head configuration, and antiparallel to the field for a tail-to-tail configuration where δ denotes the average thickness of the surface film layers where closure "cap" domains are present, and we have 189 made the approximation of a domain wall width much smaller than the stripe width, ∆ ≪ 2R. The gyrovector G is defined in Eq. 5. In the case of a magnetic edge dislocation with a semicircular shape as in 192 Fig. 3, G Note that, in the third line of Eq. 18, the integrand takes the same form as in Eq. 10. However, there is a 194 fundamental difference: now, the integration range spans the whole film volume. Therefore, in the expression (18) for 195 G, the r-integration is extended to ∞, and there is a z-integration over the film thickness, t. 196 The other fundamental feature, to be taken into account for the calculation of G, is that the defect is embedded 197 in a periodic stripe domain pattern. Consequently, when z and φ (i.e., the layer index and the in-plane direction) 198 are fixed, the integrand, F(r) = [− sin Θ(r) dΘ(r) dr ], is a periodic function of r oscillating between equal and opposite 199 values. Therefore, albeit the period of the integrand is not constant, if the defect is embedded in a stripe domain 200 pattern, the r-integration yields a vanishing gyrovector and a vanishing gyrotropic force As mentioned above, the period of F(r) is not constant. Rather, when |φ| decreases from π 2 to 0 (i.e., when the 202 in-plane direction changes from e y to e x ), the period monotonically increases from 4R (i.e., the same period as the 203 stripe domain pattern) to ∞. 204 Finally, it is worth observing that the result in Eq. 19, namely the vanishing of the gyrotropic force for a magnetic 205 edge dislocation embedded in a stripe domain pattern, is quite general: i.e., it holds irrespective of the value, m, for 206 the vorticity and γ, for the helicity of the film layer configuration. In fact, for fixed φ and fixed z, the result (19) 207 follows just as a consequence of the vanishing of the r-integration in Eq. 18, provided that the upper limit is properly 208 extended to ∞. 209

Dissipative force 210
The dissipation dyadic tensor, D, defined in Eq. 6, is diagonal in the polar coordinates representation of n(r) where, taking into account that ( dΦ dφ ) 2 = m 2 = 1, the nonzero tensor elements take the form The dissipative force F diss = D · αv will now be calculated in Cartesian coordinates. In the general case, v = 213 v x e x + v y e y , one has where the tensor elements are given by Now, we observe that the limits in the φ-integration are either − π 2 < φ < + π 2 or + π 2 < φ < 3 2 π. Since the primitives 218 are ∫ dφ sin 2 φ = φ 2 − 1 4 sin(2φ), ∫ dφ sin φ cos φ = 1 2 sin 2 φ, by parity considerations 219 it follows immediately that, whatever the in-plane magnetization configuration, one has D xx < 0, D yy < 0 and 220 D xy = D yx = 0. Moreover, we observe that the r-integration can be limited between 0 and 2R, because for ∆ ≪ 2R 221 the integrand takes the form of a narrow peak centered at r = R, and the results 21 are independent of the z-222 polarization parameter (p = ±1). Note that, in deriving D φφ , it was taken into account that dΦ dφ = m and m 2 = 1. 223 Therefore the results for the tensor elements are also independent of the vorticity, m, and the helicity, γ. 224 In the particular case of a translational motion of the magnetic edge dislocation, with velocity v = v x e x parallel to 225 the stripes axis, the dissipative force takes the form Since α > 0, it results that the dissipative force F diss has always the effect of hindering the motion of a dislocation. Let us explicitate the calculation of the external force F ext , Eq. 14, in a few special cases, γ = 0, | π 2 |, π,, which 230 respectively correspond to the contributions from the top, central, and bottom layers of the film as represented in 231 Fig. 3. We always consider the case of a configuration with vorticity m = +1, so that one has dΦ(φ) dφ = 1 in Eq. 14.

233
This is the simplest case, where Φ = ϕ. The external force takes the form where δ is the average thickness of the surface film layers where closure "cap" domains are present.

237
Now, taking into account that ∫ dφ sin φ cos φ = 1 2 sin 2 φ, by parity considerations it follows immediately that 238 the contribution of the top film layers to the y-component of the external force is zero.

239
Using less trivial considerations, it turns out that also the contribution to the x-component vanishes.

240
This is proved as follows.

242
After φ-integration, one then has It is now useful to define the two primitives I s (r) and I c (r) where the relationship between I c (r) and I s (r) was obtained integrating by parts.

246
Taking for Θ(r) the explicit form (7), and performing the r-integration, the primitive I s (r) can be expressed as 247 where Si(x) = ∫ x 0 sin t t dt denotes the Sine Integral function. Note that the result in Eq. 31 holds irrespective of 248 p = ±1. Namely, I s (r) does not depend on the z-polarization in the domains. 249 Moreover, in the approximation of a domain wall width, ∆, much smaller than the stripe width, 2R, the function 250 sin Θ(r) takes the form of a well localized and symmetric peak, centered at r = R. Therefore, the integrand 251 in Eq. 28 is unaffected by the stripe domain structure: i.e., the r-integration in Eq. 28 can safely be limited 252 between 0 and 2R. 253 Now, we observe that in the approximation ∆ ≪ 2R, of a domain wall width much smaller than the stripe 254 width, one has This is another simple case, where Φ = ϕ + π. Therefore one has Using the same symmetry considerations as for the case of the top surface, one therefore obtains that the 266 contribution from the bottom film layers to the external force is vanishing 269 -(a) Head-to-head configuration. In this case, the φ-integration limits are − π 2 < φ < π 2 and, taking into 270 account the change of helicity at the vertex of the dislocation, (i.e., γ = + π 2 for − π 2 < φ < 0 and γ = − π 2 271 for 0 < φ < π 2 ), F ext can be expressed as -(b) Tail-to-tail configuration. In this case, the φ-integration limits are π 2 < φ < 3 2 π and, taking into 273 account the change of helicity at the vertex of the dislocation, (i.e., γ = + π 2 for π 2 < φ < π and γ = − π 2 274 for π < φ < 3 2 π), F ext can be expressed as On the basis of parity considerations about the φ-integration, one immediately finds that the Cartesian 276 y-component of the external force identically vanishes, for both cases of a head-to-head and a tail-to-tail 277 magnetization configuration.

278
Summarizing, the contribution from the central film layers to the external force can be written Now, taking into account that, in the approximation of a domain wall width much smaller than the stripe width 280 (∆ ≪ 2R), the function Θ(r) takes the form of a well localized and symmetric peak, centered at r = R, the 281 integration range in Eq. 41 can safely be limited between 0 and 2R. Taking Eq. 35 into account, one finally 282 obtains Namely, the contribution from a central film region, with thickness t ′ = (t − 2δ), to the external force is nonzero. 284 Finally, we note that the sign of the external force depends on the in-plane magnetization configuration, whether 285 head-to-head or tail-to-tail, in the transition region between two opposite domains. In general, the force does 286 not depend on the sign of the z-polarization parameter (p = ±1), see Eq. 42.
head-to-head configuration in Fig. 6a is expected to be realized at remanence, or for extremely small values of an 292 in-plane dc magnetic field applied along the stripes axis. Whereas, the other configurations are likely to occur when 293 the application of the external magnetic field produces deviations from the above idealized symmetric configuration. 294 We start from the general expression of the external force in Cartesian coordinates (14) 295 where the φ-integration is extended to the semicircumference. Now we observe that, for all configurations in Fig. 6, 296 one can write = + cos(mφ) sin where γ = c π 2 and the parameter c is associated with a counterclockwise (c = +1) or clockwise (c = −1) rotation of 298 the in-plane magnetization vector in the semicircular transition region between two opposite domains. 299 Taking into account that dΦ(φ) dφ = m, after some simple algebra one therefore obtains • (a) Central layer: constant vorticity (m = +1) and change of helicity (c = ±1) for φ = 0.

301
The first configuration in Fig. 6a is characterized by constant vorticity (m = +1) and by a change of sign in the 302 helicity (γ = ± π 2 , or equivalently c = ±1) occurring just at the endpoint of the stripe domain (i.e., for φ = 0). 303 The associated topological number is N sk = 1 2 .

304
For this head-to-head configuration, one has mc = +1 for − π 2 < φ < 0 and mc = −1 for 0 < φ < + π 2 . The 305 external force has already been explicitly calculated in the previous Appendix A and turns out to be directed 306 along the stripes axis, e x , while the y component vanishes (see Eq. 39) (Fig. 6a) (46) where we remind (see previous Appendix A) that 2δ is the total thickness of the two surface film layers where 308 closure "cap" domains are present, and the factor (2 × 1.852∆) is the result of the r-integration (∆ << 2R is 309 the width of the transition region between two domains with opposite z component of the magnetization). The configuration in Fig. 6b is characterized by a constant vorticity (m = +1) and a constant helicity γ = π 2 , 312 or equivalently c = +1) in the whole semicircumference, − π 2 ≤ φ ≤ π 2 . In fact, a change of sign in the helicity 313 (γ = ± π 2 , or equivalently c = ±1) is supposed to occur just for φ = π 2 .

314
The associated topological number is N sk = 1 2 .

315
Note that, in this case (b), one has (mc) = 1 in the whole range of integration − π 2 < φ < + π 2 . The explicit 316 calculation of the external force leads to a vanishing result, for both the x and y components The proof of the above result (48) proceeds as in the previous Appendix A, see Eqs. 28-36.

321
Finally it can be shown that, in case the change of helicity occurred for an angle φ 0 different from π 2 (e.g. 322 comprised in the range 0 < φ 0 < π 2 ), both the x and y component of the external force would not vanish any 323 more.  The configuration in Fig. 6c is characterized by a change of vorticity (m = ±1) occurring just at the endpoint 326 of the stripe domain (i.e., for φ = 0) and by a constant helicity (γ = π 2 , i.e. c = +1).

327
Note that, in this case, the associated topological number vanishes, N sk = 0.

328
In this case (c), one has mc = +1 for − π 2 < φ < 0 and mc = −1 for 0 < φ < + π 2 : namely, the product mc 329 takes exactly the same values as in the case (a) of a head-to-head configuration calculated before. Therefore, 330 the calculation of the external force, using Eq. 45, leads exactly to the same result for both configurations in 331 Fig. 6a and in Fig. 6c. Namely, the external force turns out to be directed along the stripes axis, e x , while the 332 y component vanishes 333 F ext central = +(t − 2δ) (2 × 1.852 ∆) M s H x e x (Fig. 6c) (49) 334 In conclusion, the external force, exerted on a dislocation by a dc magnetic field applied along the stripes axis, 335 was found to be the same for two idealized configurations characterized by a different topological number (N sk ), 336 but sharing a common feature: the product of vorticity and helicity changes sign (mγ = ± π 2 ) at the vertex of the 337 dislocation, see Fig. 6a and 6c. Whereas, for the configuration in Fig. 6b, where mγ remains constant in sign and 338 value along the whole semicircumference, the external force vanishes. Therefore, the product of vorticity and helicity, 339 mγ, should be regarded as a parameter more relevant than just the topological number, N sk , as far as a finite external 340 force is concerned.