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Magnetization process of the vortex state in soft magnetic thin square platelets
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Journal of Magnetism and Magnetic Materials 314 (2007) 105–115 www.elsevier.com/locate/jmmm
Magnetization process of the vortex state in soft magnetic thin square platelets M. Wolf, U.K. Ro¨Xler, R. Scha¨fer IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany Received 29 September 2006 Available online 4 March 2007
Abstract An analysis of the magnetization process in magnetically ultra-soft thin film elements is presented for the case of square platelets, which show a vortex ground state and have a very low thickness. The analysis is based on micromagnetic calculations for systems with 1 mm edge length, 8–20 nm thickness and for material parameters similar to permalloy, but with negligible magnetic anisotropy. For the case of a field applied along the diagonal of the square platelet the evolution of magnetization, critical fields for the expulsion of the vortex and the leading energy terms have been determined from the numerical simulations. We show that a phase theory approximation, by using the position of the vortex as the only variable for the evolution of the vortex pattern, semi-quantitatively describes the main features of the reversible magnetization process and the stability limit. An effective demagnetization factor for the vortex pattern of the squares is determined from the numerical results. This effective parameter enables a quantitative description of the main properties of the vortex state and its magnetization process. Deviations of this phase-theory approximation from the numerical calculations are traced back to the evolution of inhomogeneous internal magnetization states, in particular wide walls that change their profiles under the applied field. r 2007 Elsevier B.V. All rights reserved. PACS: 75.70.Ak; 75.60.Ch Keywords: Micromagnetic; Soft magnetic film; Magnetic domain; Magnetic vortex; Magnetization process; Phase theory; Scaling
1. Introduction The behaviour of magnetically soft nanostructured samples is of large technical interest due to their potential and actual applications [1]. The magnetic patterns in such film elements are mainly determined by demagnetization effects that are governed by the geometry of the magnetized finite volume. For vanishing magnetic anisotropy, the magnetic microstructure in small thin film elements displays regular patterns with typical scales of the order of the sample dimension. The Ne´el-phase approximation [2] developed for bulk samples fails in this case because it presumes the existence of (i) domains with different magnetization directions due to magnetic anisotropy, (ii) domains that are separated by walls with negligible wall Corresponding author. Tel.: +49 351 4659 529.
E-mail address: [email protected] (M. Wolf). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.02.184
energy and (iii) ellipsoidal-shaped samples for which the demagnetization effect may be described by a demagnetization tensor. This description requires a well-defined hierarchy of lengths scales to hold between the macroscopic dimension of the sample, the typical domain size and the width of the domain walls. Generally, it is applicable for bulk anisotropic magnetic materials, but it is not expected to describe well micromagnetic patterns in thin films. However, mathematical theories based on scaling ideas for flux-closure structures [3–7] indicate that the regularity of these demagnetization patterns should allow the development of a deeper understanding and of relatively simple rules for their magnetization process in certain cases. Formulating such rules would enable to analyze the patterns and the magnetization process in finite soft-magnetic film elements, in particular for structures that are too large and complex for brute-force micromagnetic simulations.
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For zero applied field and vanishing anisotropy van den Berg [3,4] and Bryant and Suhl [5,6] analyzed the magnetization patterns in thin film elements, i.e. samples with a finite lateral dimension l and a vanishing thickness t-0. In this limit the demagnetization energy is dominating all the other energy terms and the magnetization distribution J(r) can be described by a two-dimensional field m0 (x, y) ¼ (mx, my) ¼ J/Js with the saturation magnetization Js. This normalized magnetization m0 obeys the equation qx m0x ðx; yÞ þ qy m0y ðx; yÞ ¼ 0, is restricted by the condition |m0 | ¼ 1 and has a vanishing normal component on the boundary. These strong restrictions for a two-dimensional field cannot be fulfilled by a magnetization m0 (x, y) continuous everywhere in the area. The loci of discontinuities, i.e. singularities of m0 (x, y) form the walls. The whole reasoning does not depend on the existence of any crystalline magnetic anisotropy. Thus, for t-0 walls of this type must occur also for vanishing anisotropy. Hence, these walls are very different from classic Bloch/Ne´el walls in usual anisotropic magnets, although the internal structure of the walls may be akin to these wall types. The methods as developed by these authors to construct domain patterns and the position of such walls allow for a variety of patterns in soft thin film elements. As the demagnetization energy enters here only to derive the constraints on m0 (x, y), one cannot decide which of the permitted domain patterns has the lowest energy. In principle, the quasistatic magnetic behaviour of thin film elements follows by minimising the usual micromagnetic energy Z Z m E tot ðmÞ ¼ A d3 rjrmj2 þ 0 d3 r H2dem 2 Z J s d3 r mðrÞ Hext ðrÞ, ð1Þ with appropriate boundary conditions [8,9]. The first term of Eq. (1) describes the exchange interaction, the second is the magnetostatic interaction or demagnetization energy and the last one is the energy of the magnetization distribution Jsm(r) in an applied field Hext. Here Js is the saturation magnetization and m(r) is subject to the constraint |m| ¼ 1. In Eq. (1) magnetic anisotropy energy has been omitted, as we are interested in ultra-soft magnetic materials. Obviously, the first term favours a homogeneous m(r), whereas the second one prefers patterns with flux closure. Minimization of the energy (1) allows for a variety of patterns as well as for different types of magnetization behaviour and hysteresis, because (i) the solution depends on the magnetic field history and/or the specific boundary conditions, (ii) besides the geometrical scales thickness t and lateral dimension l of the sample, the energy Etot itself sets a characteristic scale, the so-called exchange length d, given by d 2 ¼ 2Am0 =J 2s . Thus, generally, various different local minima of problem (1) are visited when the external field Hext is changed. DeSimone et al. [7,10] presented in a number of publications a detailed
scaling analysis of the different energy contributions embodied in the micromagnetic functional, Eq. (1), in the limit of vanishing thickness t-0. These considerations allow important conclusions on the relevance of the different energy contributions for the magnetization behaviour. Moreover, the scaling results suggest that external fields are expelled over wide areas of the film elements. This analysis shows that the internal field and the magnetization structures can follow rather simple rules below certain thresholds for the applied magnetic field, in particular, the internal field can remain homogeneous within wide fractions of the film area. Experimental studies with this background have been presented in Refs. [11–13]. The scaling analysis by DeSimone et al. generalizes the previous constructions of flux-closure structures for ultrasoft thin film elements in the limit of two-dimensional magnetization distributions as discussed by van den Berg [3,4] and Bryant and Suhl [5,6] for the case of zero and weak external fields. DeSimone et al. [11,12] discuss also the case of sizeable external fields, when the flux-closure breaks down and the external field penetrates into certain regions of the magnetized film. It is important to note here that this field penetration takes place above a certain threshold field. Below this threshold, for t-0, the internal field remains zero and homogeneous. According to this analysis, smaller energy contributions are related to the inhomogenously distorted magnetization near walls, etc. These terms, however, are significant for the choice between different competing realizations of fluxclosure structures and for the hysteretic magnetization processes. The formation and details of the wall structures depend on the film thickness. It has to be emphasized that the powerful mathematical scaling ideas give rigorously valid results and bounds only in the limit of vanishing thickness, t-0. The magnetic behaviour of films with finite thickness may display certain deviations from this idealized behaviour and the relevance of the mathematically rigorous limit for samples of finite thicknesses is not obvious. Therefore, it is interesting to analyze the behaviour of magnetic microstructures from numerically ‘‘exact’’ solutions for film elements with finite thickness and larger lateral dimensions for cases in which such calculations are still viable. As the simplest pattern, we have chosen here the vortex state in a square platelet with an applied field along the diagonal of the square. This pattern is already highly complex and displays main features like walls and a vortex that are also found in more extended magnetization patterns (see, e.g., Refs. [14,15]). Nevertheless, it turns out that the pattern and the magnetization process remain rather simple for our model system. Based on an analysis of the exact numerical solutions, we find that the evolution of magnetization pattern can be described by a simple effective theory, which describes the evolution of the vortex state by the vortex position instead of the full magnetization distribution. This theory requires only the vortex position as a single variable and an effective
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demagnetization factor in a rough approximation. The simplified approach, as exemplified here for the vortex pattern, may be rather useful to analyze experimental observations. In fact, similar ideas have been considered already in Ref. [14] for more complex domain patterns in submicron magnetic Fe dots. To be specific, we have studied the magnetization behaviour of square-shaped samples with different thicknesses of 8, 12, 16 and 20 nm, and an edge length of 1 mm by numerically solving the micromagnetic problem. The magnetic material parameters were those typical for permalloy. For these material parameters and the chosen size of the square platelets, a vortex state forms as the global ground state in zero external field, as expected from the phase diagram in Ref. [16]. The ‘‘domains’’ in soft film elements with thicknesses in the considered range are separated by walls, which are expected to show the internal structure of Ne´el walls [17]. In order to compare our results with the scaling analysis, the different contributions to the specific energy density, i.e. energies per unit volume of the film elements, were estimated following DeSimone et al. [7,10]. These estimates from scaling are summarised in Table 1. Here, the magnetization distribution m(r) is decomposed as m(r) ¼ (m0 , mz), where mz is the component perpendicular to the film plane and m0 the projection of m(r) onto the (x, y)plane. Both contributions to the magnetization will generally depend on the coordinates x, y and z. The film boundary in the (x, y) plane (or some inner boundary in the plane) is characterized by its normal n0 and m0 n0 denotes the dot product of the in-plane magnetization m0 with n0 at such boundaries. The third column of Table 1 is equivalent to that presented by DeSimone [10], apart of an additional factor l2t which was introduced because specific energies have to be compared. Note that Table 1 gives the weights of the different contributions. In order to get energy densities any value has to be multiplied by J 2s =2m0 [10]. The first row (a) shows that a non-zero component mz of order one would result in an extremely large energy. Row (b) in Table 1 gives the scale for the exchange energy related to a variation of the magnetization along the z-axis. It reveals that a significant variation of m along the z-axis is suppressed. The terms of the next rows (c) and (d) are
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purely due to the magnetostatic term in the energy functional equation (1). Comparing all these terms it follows, as is well known, that for thin films the magnetization m(r) almost perfectly lies the (x, y)-plane. The main contributions are then expected from terms corresponding to rows (c) and (d). This means that in the cases to be investigated the ‘‘magnetostatic regime’’ [10] is realised. Note that these terms lead to the approximation of van den Berg [3,4] and Bryant and Suhl [5,6]. Quite unexpectedly, only the Bloch line and variations of m0 over the lateral scale l give contributions proportional to the exchange interaction Apd2. The leading contribution of the Ne´el wall is of magnetostatic nature, i.e. proportional to J 2s =2m0 , as are the terms in rows (c) and (d), and depends only weakly (logarithmically) on the exchange interaction d2 [10]. Hence, in the ideal case mz is zero and m0 depends on x and y only. In zero field the most important contributions result from the terms in rows (c) and (d) which, however, can be strongly reduced by forming magnetization patterns with closed flux, i.e. for vanishing m0 n0 . But, for moderate fields m0Hext of few tons mT the Zeeman energy (e) becomes larger than the term (c) due to m0 n0 . This results in a critical field for the stability of the initial magnetization structure, e.g. the vortex state becomes unstable. As m0 n0 gives the magnetic surface charge on the sample boundary, in row (c) (t/l) ln(l/t) is proportional to the in-plane demagnetization factor Nx, in agreement with the asymptotic behaviour of the expression for the demagnetization factor of rectangular prisms given by Aharony [18]. Thus, it is the demagnetization factor Nx of the square that determines the critical field, for which flux closure states become unstable. The energy of Ne´el walls and Bloch lines connected with flux-closure patterns are one or two orders of magnitude smaller (see last three rows) and should be of minor importance for the magnetization process. In a more precise discussion one has to bear in mind that all the weights involve still certain factors, such as m0 n0 , etc. given by the expressions from the first column for a specific magnetization structures. These factors depend on the details of the magnetization pattern and on the applied field and are generally unknown. In summary, the welldefined hierarchy of energy contributions in this scaling
Table 1 Weight of the different energy contributions (per volume). The numerical estimates are for the investigated samples with l ¼ 1 mm and thickness t ¼ 8, 12, 16, and 20 nm, respectively, with an exchange length d ¼ 5.11 nm and a saturation magnetization Js ¼ 1.005 T as for permalloy
(a) (b) (c) (d) (e) (f) (g) (h)
Source of the term
Specific energy
t ¼ 8 nm
Out of plane mz Exchange, qm=qz Non-tangential component m0 n0 In-plane divergence r0 m0 External field Hext Ne´el wall Bloch line Exchange, m0 varying smoothly on scale l
1 d2/t2 (t/l) ln(l/t) t/l 2Hextm0/Js (t/l)/ln(lt/d2) (d/l)2 ln(l/d) (d/l)2
1 0.408 0.03863 0.008
t ¼ 12 nm
t ¼ 16 nm
1 1 0.181 0.102 0.05307 0.06616 0.012 0.016 0.002 m0Hext/mT 0.00139 0.00194 0.00248 about 1 104: order of magnitude smaller 2.5 105: two orders of magnitude smaller
t ¼ 20 nm 1 0.065 0.07824 0.020 0.00299
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limit restricts the magnetization distribution in specific ways and introduces certain bounds: the magnetization distribution cannot have a sizeable out-of-plane component mz, it must avoid a leaking flux at the borders of the film element and internal magnetic charges as much as possible. The competition of these demagnetization effects with the external field determines essentially the demagnetization pattern. The exchange energy is largely irrelevant, as it enters only logarithmically the energetics of domain walls or Bloch lines via the exchange length. Otherwise, bending of the magnetization can arise only over lengths of the film element edges. As shown by DeSimone et al. [10,11] the magnetostatic terms determine the distribution of magnetic poles r0 m0 and the boundary charges m0 n0 . These quantities related to the field (mx, my) are not sufficient to specify uniquely the magnetization distribution m(r). For this the energies of distorted magnetization configurations, i.e. walls, vortices and the exchange terms have to be taken into account. These terms also provide the stability of patterns corresponding to various possible shallow minima in the micromagnetic energy. Hence, these secondary terms in the hierarchy of magnetic energies cause the configurational hysteresis that is found in the magnetization processes of thin film elements. As discussed, the scaling ideas cannot determine the magnetization patterns in all details. For this one has to understand the role of the inhomogeneous parts of these patterns, in particular the structure of the walls. Moreover for finite thickness, the small energy terms related to walls and the like will be influenced by the corrections to scaling from the leading large magnetostatic energy terms. It is therefore interesting to study the behaviour of simple domain patterns determined by ‘‘exact’’ numerical solutions in order to contribute to a better understanding of these smaller energy terms. Clearly, this is also desirable to develop a deeper understanding of the hysteretic magnetization processes in such thin films with a finite thickness. 2. Methods 2.1. Details of the calculation procedure For thin platelets with a square base (edge l ¼ 1 mm) and different thicknesses, t ¼ 8, 12, 16, 20 nm, micromagnetic simulations have been performed. The saturation magnetization and exchange interaction were adapted to typical properties of permalloy with Js ¼ 1.005 T and A ¼ 1.05 1011 VAs/m. Therefore, the exchange length d equals 5.11 nm. In order to avoid possible stability problems a non-vanishing magnetic uniaxial anisotropy was assumed. However, the anisotropy constant Ku was chosen as low as 0.1 VAs/m3 in order to emulate an ultrasoft magnetic material and to simulate a situation close to the scaling considerations. The x-axis within the film plane was presumed as the magnetically easy axis of the magnetic uniaxial anisotropy. The calculations were done using the program ‘‘The LLG Micromagnetics
Simulator’’TM developed by Scheinfein [19]. By solving the time dependent Landau–Lifshitz–Gilbert equation a starting magnetization distribution was allowed to relax into equilibrium, defined by the condition that the change of magnetization in the last step was less than 1 106. A cell size of 5 nm 5 nm 4 nm has been used in all cases. The external field was alwayspparallel to the diagonal of the ffiffiffi square, Hext ¼ H ext ð1; 1; 0Þ= 2. It was increased in steps of typically 0.5 or 1 mT. In the vicinity of the stability limit these steps were reduced to 0.1 or 0.2 mT. The magnetization distribution m(r) obtained from the preceding (lower) field value provided the starting configuration (input) for determining the new structure and its properties. The vortex pattern at Hext ¼ 0 was realized by appropriate boundary conditions. The volume averages mp(1ffiffiffi1 0) of the magnetization along the field direction ð1 1 0Þ= 2 has been determined from the calculated distributions m(r). 2.2. Phase theory approximation The results are discussed in close relation to the phasetheory approximation (see Ref. [2, Section 3.4.2]). Here, we briefly motivate this approach and explain how it can be adapted to our problem. As discussed above, the exchange energy disappears from consideration in the thin film limit. This also means that in wide areas the magnetization pattern will show bending only over the large lateral length scale of the film element. On the other hand, the internal field disappears at least for small external fields. The high penalty from magnetostatic energy also means that any inhomogeneity of the internal field generally will not become large. This suggests that an approach based on a homogeneous (zero) internal field should still work for these flux-closure domain patterns. This description can be reduced to a geometric description of the magnetization distribution, which only considers certain variable areas (domains) with almost homogeneous magnetization. Within this simplification the demagnetization effect is considered only on average, disregarding all inhomogeneities. An applied field below a certain critical field HextpHcr distorts the flux-closure structure (Fig. 1). The vortex still is preserved and only its position is shifted. In our case with an external fieldpffiffiffi applied along the diagonal, Hext ¼ H ext ð1; 1; 0Þ= 2, the vortex moves along the other diagonal, perpendicular to the applied field Hext (cf. Figs. 1 and 3a). For such states, we can estimate the average magnetization m from the areas (domains) with different direction of magnetization. Neglecting all inhomogeneities of the internal field and the contributions of the distorted magnetization near the walls and the vortex, we may write the magnetic energy per volume of this configuration as E ¼ J s Hext m þ
J 2s ^ m, mN 2m0
(2)
^ as the external field and the demagnetization with Hext and N tensor, respectively. In Eq. (2) the average magnetization m
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Fig. 1. Sketch of the magnetization pattern of thin square-shaped platelets in an applied field along the diagonal (1 1 0) of the square. At sufficiently small fields the vortex is preserved and there are four domains with different directions of the magnetization, as sketched by the arrows. This simplified pattern is characterised by the position of the vortex d as shown.
is given (cf. Fig. 1) by m ¼ ½ðv1 v3 Þex þ ðv2 v4 Þey ,
(3)
where v1 and v4 (v2 and v4) are the areas of the pattern, for which the magnetization is mainly parallel to 7ex (7ey), respectively. This is a very rough approximation because the possible rotation of the magnetization in the domains is neglected. In particular, this geometric construction of the domain pattern yields a description where walls are not free of magnetic charges. The volume fractions in Eq. (3) are subject to the conditions v1 þ v2 þ v3 þ v4 ¼ 1,
(4)
and v1 ¼ v2 and v3 ¼ v4 as the of the applied field pffiffidirection ffi Hext is parallel to ð1; 1; 0Þ= 2. This description contains only one parameter, e.g. v1. Obviously, it is equivalent to the choice of the vortex position d as the single parameter for characterising the configuration of Fig. 1. The minimization of the energy density (2) with respect to v1 gives the relation 1 H ext m (5) ¼ pffiffiffi 0 , 2v1 2 2N x J s valid for v1p1/2, as the outermost position of the vortex is the corner of the square. Therefore in this approximation the critical field Hcr at which the vortex is expelled is given by pffiffiffi H cr ¼ N x J s = 2m0 . (6) For HextpHcr the average magnetization m(1 1 0) along the direction of applied field is given by mð1 1 0Þ ¼
H ext m0 . N xJ s
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expressions for the demagnetizing factors of a prism have been derived by Aharoni [18], applicable for homogeneously magnetized samples. When comparing the results of micromagnetic calculations with such a simplified description inspired by the classical phase-theory approximation [2] one has to bear in mind the following crucial differences: (i) The micromagnetic calculations take into account the exchange energy whereas a corresponding term is completely lacking in the phase theory, cf. Eq. (2). As it has been argued in the introduction, these terms determine the choice between different coexisting states of nearly the same energy. They are responsible for hysteretic processes. (ii) The description of the demagnetizing energy by a demagnetizing tensor is correct only for homogeneously magnetized bodies. This precondition is hardly fulfilled for samples with an ideal or distorted vortex. (iii) The phase theory gives only average values, whereas in micromagnetic calculations the corresponding integrals are evaluated from the inhomogeneous magnetization distribution m(r). Conventional phase theory works for bulk samples, in which different coexisting regions or domains with homogeneous magnetization in a system are well defined. These different domains are due to the intrinsic magnetic anisotropy of a system, which results in degenerate magnetic ground-state configurations. The possible magnetization directions are governed by the symmetry of this system, i.e. the existence of different directions of equivalent magnetic easy axes. Furthermore, the size of the magnetized bulk sample should be much larger than the typical domain sizes. This justifies neglecting the inhomogeneities in the internal field and the assumption of negligibly thin domain walls. The approach works ideally for large ellipsoidal samples, which develop nearly homogeneous internal fields on average. In contrast, in a thin film the different regions of homogeneous magnetization are not related to thermodynamically stable magnetic states (phases) as required by the classical phase theory. Rather, it is the requirement of flux closure and the geometrical shape of the film element that govern these patterns (cf. Introduction and Refs. [3,4]). These walls are related to singularities of the two-dimensional magnetization distribution and the boundaries of the thin film element. In our case, however, the domain size is not small compared to the sample dimension, any crystalline magnetic anisotropy is lacking, and the domains are not defined by the anisotropy. Hence, our approach lies beyond the usual legitimation of phase theory. Its applicability has to be tested by a careful examination of the results from exact numerical calculations.
(7)
Because the applied field is in the (x, y)-plane and the sample base is a square, only the in-plane demagnetization factor Nx of the square enters into Eqs. (5)–(7). According to Eqs. (5)–(7) any influence of thickness t should be comprised in the demagnetization factor Nx. Analytic
3. Results and discussion 3.1. Magnetizing the vortex state Applying a field along the square’s diagonal (1 1 0), the magnetization component m(1 1 0) increases (Fig. 2). Up to a
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M. Wolf et al. / Journal of Magnetism and Magnetic Materials 314 (2007) 105–115 Table 2 Magnetometric in-plane demagnetization factor Nx, critical fields Hcr estimated by phase theory and obtained from micromagnetic calculations (mm. calc.)
Fig. 2. Calculated component m(1 1 0) versus applied field Hext of squareshaped platelets (side length l ¼ 1 mm and thickness t ¼ 8, 12, 16, 20 nm). At a critical field Hcr a jump of m(1 1 0) is observed, the vortex is expelled and the leaf-like magnetization pattern is formed above Hcr (cf. Fig. 3b).
Fig. 3. Calculated magnetization patterns of the square with t ¼ 16 nm, l ¼ 1 mm for fields Hext below (a) and above (b) the critical field m0HcrE15 mT at which the vortex is expelled. (a) mx–my domain plot with local magnetization direction (short arrows) for Hext E8.5 mT below Hcr, clearly showing the vortex shifted along the 1¯ 1 0 direction. Local magnetization directions are marked by short arrows and given in different grey tones. The long arrows give the main magnetization direction. (b) leaf-like state for m0HextE25.5 mT4m0Hcr. Over a wide area in the centre m is mainly parallel to the direction of external field. (c) Orientation of the (1 1 0) and 1¯ 1 0 direction with respect to the square edge and to Hext.
critical field Hcr the vortex is preserved (Fig. 3a). At this critical field Hcr a jump-like increase of m(1 1 0) is observed (Fig. 2) and the vortex is expelled. Above Hcr a leaf-like pattern of m(x, y, z) is formed (Fig. 3b). For HextXHcr the magnetization m(1 1 0) is nearly independent of the field and close to saturation (Fig. 2). In accordance with Eq. (7) m(1 1 0) increases with decreasing thickness t at any field value Hext (cf. Fig. 2) as long as the vortex exists, i.e. for HextpHcr, as Nx decreases with decreasing t. This explains
t (nm)
Nx
m0Hcr (mT) phase theory
m0Hcr (mT) (mm. calc.)
8 12 16 20
0.01415 0.01968 0.02478 0.02957
10.0 13.9 17.5 20.9
11.3 13.4 15.0 16.5
also (cf. Eq. (6)) that the field range for which the vortex is preserved shrinks with decreasing thickness (see Table 2, cf. Fig. 2). The critical field from micromagnetic calculations and that estimated by the phase theory are summarized in Table 2, which presents also the in-plane (magnetometric) demagnetizing factors Nx, calculated from the expression given by Aharoni [18]. The figures of Table 2 show that the estimated values for Hcr according to the phase theory are larger than those obtained from micromagnetic calculations (with the exception t ¼ 8 nm), but of the right order of magnitude. There exists no common factor between the values of Hcr estimated from phase theory and those obtained from micomagnetic calculations. The values for Hcr for square permalloy film elements observed experimentally by Kerr microscopy in Refs. [12,13] are also explained fairly well by the demagnetizing factor of the corresponding samples. For instance, the reported experimental critical field in Ref. [13] is about 4.7 mT for a square-shaped permalloy film element l ¼ 57 mm and thicknesss t ¼ 207 nm, which is again smaller than the value of about 5.2 mT estimated from the phase theory. The somewhat reduced critical fields observed in experiments/calculations are reasonable as the theoretical stability limit of the vortex pattern should not be reached in real samples due to the finite size of the vortex. As Hcr is mainly determined by the balance of the Zeeman term and the demagnetization energy, for circular elements a similar dependence of the critical field on the thickness is observed [20]. 3.2. Discussion of the energy contributions In Figs. 4–6, we display the different leading energy contributions from the numerical solutions, i.e. the exchange energy Eexch, demagnetizing energy Edem and field energy Efield. As the film elements have different thickness, the energy densities (i.e. energies per unit volume) have to be compared. The average exchange energy density Eexch decreases with decreasing thickness t, for HextoHcr as well as for Hext4Hcr , see Fig. 4. This decrease of the exchange energy is due to the progressive suppression of the out-of-plane magnetization component mz with decreasing film thickness t, as obtained from the numerical calculations: In zero field the average mz decreases from about 1.86 104 for t ¼ 20 nm to 1.73 104 for t ¼ 8 nm. Thus the
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Fig. 4. The specific exchange energy Eexch obtained by micromagnetic calculations in dependence on the applied field Hext. Note the sharp decrease of Eexch at Hcr, just where the vortex is expelled.
Fig. 5. All values Efield/Nx plotted versus m(1 1 0) form a common curve, i.e. Efield scales with a common factor proportional to Nx. A fit to a parabola is possible for m(1 1 0) up to about 0.59 (solid line).
corresponding small contribution to Eexch due to the variation of mz is reduced. Moreover, these values for the average of mz show that the corresponding energy (second row in Table 1) may be neglected as claimed in the Introduction. The above-mentioned variation of Eexch with t is small compared to the total value of Eexch for Hext ¼ 0. The overall exchange energy Eexch is largely reduced by a large jump at the field Hcr when the vortex is expelled. Thus it is concluded that Eexch is dominated by the contribution from the (Ne´el) walls and/or the vortex. Their disappearance at Hcr results in the drastic reduction of Eexch. The values DEexch ¼ Eexch(Hcr+0)–Eexch(Hcr–0), together with the analogously defined quantity for the demagnetization energy DEdem, as derived from the micromagnetic calcula-
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Fig. 6. In the vortex state all values Edem–Edem,0/Nx plotted versus m(1 1 0) form a parabola-like curve. The m(1 1 0) values belong to successive field values and, as shown for t ¼ 16 nm, at Hcr a jump in Edem and m(1, 1, 0) occurs.
tions, are given in Table 3. This change of DEdem at Hcr (visible in Fig. 6 for t ¼ 16 nm) involves contributions from the large-scale magnetization pattern (cf. Fig. 3) as well as from the walls and vortex. To find out an estimate for the demagnetization energy associated with the inhomogeneous parts of the magnetization distribution at Hcr, we need an approximation for the effect due to the global change of the domain pattern. Within our phase theory approximation we expect a change of the demagnetization energy D0 E dem ¼ J 2s N eff ðmð1 1 0Þ ðH cr þ 0Þ2 mð1 1 0Þ ðH cr 0Þ2 Þ=ð2m0 Þ from the jump of the average magnetization m(1 1 0)(Hext) at the transition from the vortex to the leaf state. For the effective demagnetization factor, we use a reduced value Neff, determined from the global behaviour of the demagnetization energy in the vortex state (see below). These values D0 Edem, given in Table 3, should not depend on the contribution of the walls to DEdem. Then, the difference DEdem–D0 Edem between the numerically derived demagnetization energy DEdem and the contribution D0 Edem associated with the global domain pattern can be attributed to the demagnetization energy that is related to inhomogeneities of the walls and/or the vortex. In Table 3, these differences are listed as DE inh dem . The negative sum of this part from demagnetization DE inh dem and the jump in exchange DEexch gives an estimate for the wall energy in the vortex state DEwall. The comparison with the expected wall energies for thin films from Refs. [2,10] shows that the order of magnitude and the trend is correct for our estimate. When comparing these values one has to be aware that, firstly, in zero field the vortex state gives rise to two 901 Ne´el walls with total length 2 O2 l. But, a 901 Ne´el wall has an energy vastly reduced by a factor of the order 0.12 as compared to a 1801 wall (cf. Ref. [2, Figs. 3.73 and 3.79]). Hence, the wall energies in the last two columns
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Table 3 The jumps of the exchange energy DEexch and demagnetization energy DEdem at the critical field Hcr from micromagnetic calculations with DEexch ¼ Eexch(Hcr+0)–Eexch(Hcr–0) etc t
DEexch
DEdem
D0 Edem
DE inh dem
DEwall
Hubert 1801 wall, l
DeSimone 1801 wall, l
nm
104 W s/cm3
104 W s/cm3
104 W s/cm3
104 W s/cm3
104 W s/cm3
104 Ws/cm3
104 W s/cm3
8 12 16 20
6.17 6.32 6.33 6.35
1.28 6.70 13.99 21.68
11.39 18.41 26.47 34.50
10.11 11.71 12.44 12.82
16.28 18.03 18.81 19.17
9.2 12.1 14.8 16.9
5.52 7.74 9.85 11.90
Within the phase theory approximation D0 E dem ¼ J 2s N eff ðm2 ðH cr þ 0Þ2 m2 ðH cr 0ÞÞ=ð2m0 Þ is an estimate for the change of the demagnetization energy 0 due to the change of the magnetization only, where we used an effective demagnetization factor Neff ¼ 0.66 Nx. Then, DE inh dem ¼ DE dem D E dem is the contribution to Edem from the inhomogeneous magnetization and DE wall ¼ ðDE inh þ DE Þ is an estimate for the wall energies in the vortex state. The exch dem values of DEwall are compared with the energies of straight 1801 walls of length l according to Hubert (Ref. [2], chap. 3.6.4, Fig. 3.79) and DeSimone [10].
should be multiplied by a factor of about 0.3 to derive an estimate of the contribution of the Ne´el walls in the vortex state. Due to the presence of the vortex this is a rough approximate. In an applied external field, the angle between adjacent domains changes and depends on the details of the domain pattern. Moreover, for the squares in an external field, the detailed form of the magnetization distribution in the wall is field dependent and the change of the wall profiles has to be taken into account. All this prevents a reliable quantitative estimation of wall energies in the vortex state. However, it is reasonable that our estimates DEwall are considerably larger than the energies estimated from idealized 901 straight walls, because DEwall sums all contributions due to inhomogeneities in the magnetization distribution and their interactions. It is remarked that the domain wall energies estimated following DeSimone et al. include undetermined factors and, therefore, allow only order-of-magnitude estimates. Within this scaling limit l is a characteristic lateral length, which is of course of the order of the square edge, but can numerically differ from it. This explains that the values of the last two rows in Table 3 differ from the estimates for wall energies following [2] by a nearly constant factor. From Eqs. (2) and (7) within the phase theory the specific field energy for HextpHcr should be given by E field ¼
N x J 2s 2 mð1 1 0Þ , m0
(8)
i.e. all values Efield/Nx plotted versus the calculated m(1 1 0) should form a common parabola with the curvature J 2s =m0 . Fig. 5 shows that the values Efield/Nx indeed form a common curve, i.e. Efield scales with a factor proportional to Nx. However, a fit of a parabola is possible only in the range m(1 1 0)o0.59 (cf. the deviations of the parabola from the solid line in Fig. 5). Moreover, the curvature of this parabola is significantly smaller than predicted and can be ascribed to an effective demagnetization factor NeffE0.71Nx. Within the same approximation, Eq. (2), the demagnetization energy Edem should be given by E dem ¼ N x J 2s m2ð1 1 0Þ =2m0 . In contrast, the micromagnetic calcula-
tions yield also for the demagnetized state m(1 1 0) ¼ 0 nonzero demagnetization energies Edem, 0 ¼ Edem(Hext ¼ 0) with values Edem, 0 in units of W s* cm3: 4.30 104 for t ¼ 8 nm, 5.47 104 (t ¼ 12 nm), 6.44 104 (t ¼ 16 nm) and 7.26 104 (t ¼ 20 nm). Therefore, it is suggested that the differences Edem–Edem, 0 follow a quadratic law ðE dem E dem; 0 Þ=N x ¼ J 2s m2ð1 1 0Þ =2m0 . As shown in Fig. 6 these differences form indeed a common curve, when plotted versus m(1 1 0). With regard to the behaviour of Efield, a quadratic law was fitted up to m(1 1 0) ¼ 0.59. As in the case of the field energy the curvature is again too small, corresponding to NeffE0.64 Nx, which agrees satisfactorily with that obtained from Efield(m(1 1 0))/Nx. The reduction of the effective demagnetization factor can be understood from the large-scale inhomogeneous magnetization pattern in the square platelet that results in a suppressed demagnetization energy as compared to that of homogeneously magnetized body, i.e. the external field has a stronger influence or the effective demagnetization factor becomes effectively smaller (cf. Eq. (7)). In summary, the dependence of Edem, Efield on m(1 1 0) can be semiquantitatively understood by a simplified phase theory. The field energy Efield and demagnetization energy Edem behave in a slightly different way. In micromagnetic calculations they are determined by the whole magnetization distribution m(x, y, z), which enters into the corresponding terms (see Eq. (1)). However, in the approximate expressions for averages of the phase theory using an effective demagnetization factor, these results for Edem, Efield are described by the average m(1 1 0) as a single variable characterizing the magnetization configuration. Furthermore, as revealed by Fig. 4, for small external fields Eexch is not small compared to Edem and Efield, respectively. This is a further reason that the evolution of the field energy and the demagnetization energy cannot fully be understood within the frame of phase theory. The failure to describe the field energy for larger external fields with m(1 1 0)40.59 can be understood as consequence of the penetration of the external field into certain sample regions and the concomitant inhomogeneity of the resulting magnetization state.
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3.3. The scaling behaviour of the magnetization and vortex position From Eq. (7) it is suggested that all scaled magnetization values m(1 1 0)NxJs lie on a common straight line and should be equal m0Hext. Fig. 7 reveals that m(1 1 0) scaled in the described way form a common straight line only for small applied fields. However, the slope in this region is larger than expected from the phase theory (dash-dot line in Fig. 7), which is consistent with an effective demagnetization factor NeffE0.77 Nx. This decreased Neff corresponds to an increased initial susceptibility as reported in Ref. [14]. For larger Hext deviations from the scaling are found and even no common curve is established. Again, the relation between the effective demagnetization factor and Nx is in fair agreement with the corresponding demagnetization factors from the analysis of field and demagnetization energy. The observed behaviour of m(1 1 0) in Fig. 7 is related to deviations of the real magnetization distribution from the assumed constant saturation magnetization in the four parts of the square, as sketched in Fig. 1. On the other hand, the vortex position has been suggested to be a main parameter characterizing the magnetization pattern (cf. Section 2.2). Therefore, it is worthwhile to study the relation between m(1 1 0) and the location of the vortex, characterized by its relative position r ¼ d*O2/l. The distance d of the vortex from the centre of the square with the edge length l (cf. Fig. 1) has been determined for each field from figures like Fig. 3a. The parameter r varies between 0 and 1. The phase theory predicts for the m(1 1 0)plot versus r a common straight line from zero to 1/O2. As displayed by Fig. 8 such a plot results indeed in a common straight line. But the slope of this line is greater than the slope corresponding to the simple magnetization configuration of Fig. 1. That means, the magnetization m(1 1 0) increases more strongly than predicted by this model and
Fig. 8. The magnetization components m(1 1 0) and mð1; 1; 0Þ in dependence on the relative vortex r ¼ d*O2/l. The values of m(1 1 0) of all squares lie on a common straight line, but above the curve expected from phase theory (solid straight line). The average of mð1; 1; 0Þ , restricted over the upper right part of the square (grey shaded triangle), mð1; 1; 0Þ , decreases with increasing r.
described by Eqs. (3) and (4), which suggests an additional redistribution of the magnetization. Averaging the component of m(x, y, z) perpendicular to the direction of the applied field, m(1, 1, 0), only over a triangle, the upper right half of the square (grey shaded part of the inset in Fig. 8), gives a non-zero value mð1; 1; 0Þ . As revealed by Fig. 8 these partial averages mð1; 1; 0Þ lie on a common curve when plotted versus r. However, the decrease of mð1; 1; 0Þ with increasing r cannot be explained by simple assumptions on mðx; y; zÞ corresponding Fig. 1 and Eq. (3). These results show that the vortex position captures the main features of the magnetization process for square platelets with different thickness, but is not sufficient to explain finer details. 3.4. Magnetization profiles
Fig. 7. For small Hext the scaled m(1 1 0) values, m(1 1 0)NxJs, form a common straight line with a slope corresponding to NeffE0.77 Nx. For larger Hext the scaling fails.
The deviations of the simple phase theory approximation for the magnetization process in the vortex state points to the importance of two effects. First, within the ‘‘domains’’ a rotation of the magnetization may take place in the external field. This process is not independent of the magnetization distribution on a whole. The other important contribution stems from the evolution of domain walls and inhomogeneities of the magnetization distribution. For the vortex state in zero applied field in Fig. 9 the profiles of the in-plane magnetization component mx is plotted for different cuts y ¼ constant, i.e. along the x-edge of the square platelet. The profiles demonstrate that homogeneous regions mx ¼ constant, i.e. ‘‘domains’’ exist in a well-defined region. These regions strongly shrink when approaching to the vortex. The behaviour of my (not shown here) is quite similar. However, the ‘‘domain walls’’ with rapidly changing magnetization direction occupy
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In accordance with Fig. 3, in an applied field the magnetization distributions show strong distortions, as displayed by the profiles in Fig. 10, lower panel. In particular, the domain walls that are shifted away from the 1¯ 1 0 -diagonal acquire asymmetric shapes, and the ‘‘domains’’ display a continuously distorted magnetization. 4. Conclusions
Fig. 9. Variation of the magnetization component mx for zero field in the square with t ¼ 16 nm along cuts with different y ¼ constant. For small y certain ‘‘walls’’ may be defined. The cut along y ¼ 500 nm clearly displays the vortex.
Fig. 10. Variation of mx and my across the walls for t ¼ 8 and 20 nm, respectively, in dependence on the coordinate x, dashed line in the square of the inset, for x+y ¼ 0.25 mm in zero (upper panel) and nonzero (lower panel) external fields. The external field Hext||(1 1 0) has been 6.65 mT (t ¼ 8 nm) and 14.12 mT (t ¼ 20 nm), for which the relative vortex position is nearly the same. The vertical straight lines mark the wall positions.
sizeable regions of the pattern and there is a smooth transition into the vortex region. The profiles for the ground-state patterns from the calculations for other thicknesses t are rather similar as shown by the profiles of Fig. 10, upper panel.
We have performed micromagnetic calculations for permalloy squares of different thicknesses of 8, 12, 16 and 20 nm with an edge length of 1 mm under an increasing external field Hext parallel to the square’s diagonal and presented results concerning (i) the field dependence of the average magnetization along the field, m(1 1 0)(Hext), (ii) the evolution of the different energy contributions and (iii) the behaviour of the magnetic microstructure m(r). The initial demagnetized state of the squares, a vortex state, turned out to be preserved up to a critical field Hcr. With decreasing t the critical field Hcr decreases. This reduction can be well explained by the decrease of the in-plane demagnetization factor Nx. For low enough applied fields in the vortex state the scaled magnetization values m(1 1 0) Nx of all four squares lie on a common straight line. For higher fields this scaling is lost. Taking into account the small off-set term Edem(Hext ¼ 0), the differences of the demagnetization energy (Edem–Edem(Hext ¼ 0))/Nx as well as the scaled field energy Efield/Nx of all squares plotted versus m(1 1 0) form a common curve. For m(1 1 0)p0.6 this curve is a parabola, for larger m(1 1 0) deviations from the parabola occur. These results may be understood semiquantitatively within a kind of phase theory, whereby the different magnetic phases are the four regions differing in the main magnetization orientations in the vortex pattern. This is supported by the fact that the vortex position captures important features of the magnetization behaviour. The phase approximation appears very crude. However, the calculated spatial magnetization profiles reveal relatively wide homogeneously magnetized areas (‘‘domains’’) separated by narrow regions, in which the magnetization rapidly changes (Ne´el walls). An inspection of the wall profiles reveals strong distortions by the external field. The dependence of the exchange energy on the applied external field as obtained from micromagnetic calculation clearly shows that this term is related to the inhomogeneous magnetization in the vortex state. This term is of the order of magnitude of energy contributions from proper Ne´el walls that are the relevant walls separating domains with well-defined different magnetization-direction in a continuous thin magnetic film. The change in the demagnetization energy at the instability field of the vortex state Hcr involves contributions from the globally inhomogeneous magnetization distribution, as well as from the walls and related inhomogeneities, which change or disappear when the vortex is driven out. An estimation of the latter part has been given. From the scaling properties of the different
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energy contributions, the dominating role of the demagnetization energy in flux-closure patterns and the behaviour of the regions with a nearly homogeneous magnetization, it is understandable that the phase theory approximation with an effective reduced demagnetization factor works rather well. It is expected that similar simple geometrical schemes be developed also for thin film elements of larger dimensions showing more complex patterns. References [1] B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures, vol. 1–3, Topics in Appl. Phys., vol. 83,87,101, Springer, Berlin, 2001–2006. [2] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, 1998. [3] H.A.M. van den Berg, J. Appl. Phys. 60 (1986) 1104. [4] H.A.M. van den Berg, J. Appl. Phys. 61 (1987) 4194. [5] P. Bryant, H. Suhl, Appl. Phys. Lett. 54 (1989) 78. [6] P. Bryant, H. Suhl, Appl. Phys. Lett. 54 (1989) 2224. [7] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, Comm. Pure Appl. Math. 55 (2002) 1408.
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[8] A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford University, Oxford, 2000. [9] W.F. Brown Jr., Micromagnetics, Interscience Publisher, New York, 1963. [10] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, Recent analytical developments in micromagnetics, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, vol. 2, Academic Press, 2005. [11] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, R. Scha¨fer, Proc. Roy. Soc. A 457 (2001) 2983. [12] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, R. Scha¨fer, J. Magn. Magn. Mater. 242–245 (2002) 1047. [13] R. Scha¨fer, A. DeSimone, IEEE Trans. MAG 38 (2002) 2391. [14] P.-O. Jubert, J.-C. Toussaint, O. Fruchart, C. Meyer, Y. Samson, Europhys. Lett. 63 (2003) 132. [15] O. Fruchart, A. Thiaville, Comp. Rendus Phys. 6 (2005) 92. [16] R.W. Cowburn, M.E. Welland, Appl. Phys. Lett. 72 (1998) 2041. [17] K. Ramsto¨ck, W. Hartung, A. Hubert, Phys. Stat. Sol. (a) 155 (1996) 505. [18] A. Aharoni, J. Appl. Phys. 83 (1998) 3432. [19] M.R. Scheinfein, ‘‘The LLG Micromagnetics Simulator’’(TM) /[email protected], [email protected] 1997–2006. [20] T. Pokhil, D. Song, J. Nowak, J. Appl. Phys. 87 (2000) 6319.
Journal of Magnetism and Magnetic Materials 314 (2007) 105–115 www.elsevier.com/locate/jmmm
Magnetization process of the vortex state in soft magnetic thin square platelets M. Wolf, U.K. Ro¨Xler, R. Scha¨fer IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany Received 29 September 2006 Available online 4 March 2007
Abstract An analysis of the magnetization process in magnetically ultra-soft thin film elements is presented for the case of square platelets, which show a vortex ground state and have a very low thickness. The analysis is based on micromagnetic calculations for systems with 1 mm edge length, 8–20 nm thickness and for material parameters similar to permalloy, but with negligible magnetic anisotropy. For the case of a field applied along the diagonal of the square platelet the evolution of magnetization, critical fields for the expulsion of the vortex and the leading energy terms have been determined from the numerical simulations. We show that a phase theory approximation, by using the position of the vortex as the only variable for the evolution of the vortex pattern, semi-quantitatively describes the main features of the reversible magnetization process and the stability limit. An effective demagnetization factor for the vortex pattern of the squares is determined from the numerical results. This effective parameter enables a quantitative description of the main properties of the vortex state and its magnetization process. Deviations of this phase-theory approximation from the numerical calculations are traced back to the evolution of inhomogeneous internal magnetization states, in particular wide walls that change their profiles under the applied field. r 2007 Elsevier B.V. All rights reserved. PACS: 75.70.Ak; 75.60.Ch Keywords: Micromagnetic; Soft magnetic film; Magnetic domain; Magnetic vortex; Magnetization process; Phase theory; Scaling
1. Introduction The behaviour of magnetically soft nanostructured samples is of large technical interest due to their potential and actual applications [1]. The magnetic patterns in such film elements are mainly determined by demagnetization effects that are governed by the geometry of the magnetized finite volume. For vanishing magnetic anisotropy, the magnetic microstructure in small thin film elements displays regular patterns with typical scales of the order of the sample dimension. The Ne´el-phase approximation [2] developed for bulk samples fails in this case because it presumes the existence of (i) domains with different magnetization directions due to magnetic anisotropy, (ii) domains that are separated by walls with negligible wall Corresponding author. Tel.: +49 351 4659 529.
E-mail address: [email protected] (M. Wolf). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.02.184
energy and (iii) ellipsoidal-shaped samples for which the demagnetization effect may be described by a demagnetization tensor. This description requires a well-defined hierarchy of lengths scales to hold between the macroscopic dimension of the sample, the typical domain size and the width of the domain walls. Generally, it is applicable for bulk anisotropic magnetic materials, but it is not expected to describe well micromagnetic patterns in thin films. However, mathematical theories based on scaling ideas for flux-closure structures [3–7] indicate that the regularity of these demagnetization patterns should allow the development of a deeper understanding and of relatively simple rules for their magnetization process in certain cases. Formulating such rules would enable to analyze the patterns and the magnetization process in finite soft-magnetic film elements, in particular for structures that are too large and complex for brute-force micromagnetic simulations.
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For zero applied field and vanishing anisotropy van den Berg [3,4] and Bryant and Suhl [5,6] analyzed the magnetization patterns in thin film elements, i.e. samples with a finite lateral dimension l and a vanishing thickness t-0. In this limit the demagnetization energy is dominating all the other energy terms and the magnetization distribution J(r) can be described by a two-dimensional field m0 (x, y) ¼ (mx, my) ¼ J/Js with the saturation magnetization Js. This normalized magnetization m0 obeys the equation qx m0x ðx; yÞ þ qy m0y ðx; yÞ ¼ 0, is restricted by the condition |m0 | ¼ 1 and has a vanishing normal component on the boundary. These strong restrictions for a two-dimensional field cannot be fulfilled by a magnetization m0 (x, y) continuous everywhere in the area. The loci of discontinuities, i.e. singularities of m0 (x, y) form the walls. The whole reasoning does not depend on the existence of any crystalline magnetic anisotropy. Thus, for t-0 walls of this type must occur also for vanishing anisotropy. Hence, these walls are very different from classic Bloch/Ne´el walls in usual anisotropic magnets, although the internal structure of the walls may be akin to these wall types. The methods as developed by these authors to construct domain patterns and the position of such walls allow for a variety of patterns in soft thin film elements. As the demagnetization energy enters here only to derive the constraints on m0 (x, y), one cannot decide which of the permitted domain patterns has the lowest energy. In principle, the quasistatic magnetic behaviour of thin film elements follows by minimising the usual micromagnetic energy Z Z m E tot ðmÞ ¼ A d3 rjrmj2 þ 0 d3 r H2dem 2 Z J s d3 r mðrÞ Hext ðrÞ, ð1Þ with appropriate boundary conditions [8,9]. The first term of Eq. (1) describes the exchange interaction, the second is the magnetostatic interaction or demagnetization energy and the last one is the energy of the magnetization distribution Jsm(r) in an applied field Hext. Here Js is the saturation magnetization and m(r) is subject to the constraint |m| ¼ 1. In Eq. (1) magnetic anisotropy energy has been omitted, as we are interested in ultra-soft magnetic materials. Obviously, the first term favours a homogeneous m(r), whereas the second one prefers patterns with flux closure. Minimization of the energy (1) allows for a variety of patterns as well as for different types of magnetization behaviour and hysteresis, because (i) the solution depends on the magnetic field history and/or the specific boundary conditions, (ii) besides the geometrical scales thickness t and lateral dimension l of the sample, the energy Etot itself sets a characteristic scale, the so-called exchange length d, given by d 2 ¼ 2Am0 =J 2s . Thus, generally, various different local minima of problem (1) are visited when the external field Hext is changed. DeSimone et al. [7,10] presented in a number of publications a detailed
scaling analysis of the different energy contributions embodied in the micromagnetic functional, Eq. (1), in the limit of vanishing thickness t-0. These considerations allow important conclusions on the relevance of the different energy contributions for the magnetization behaviour. Moreover, the scaling results suggest that external fields are expelled over wide areas of the film elements. This analysis shows that the internal field and the magnetization structures can follow rather simple rules below certain thresholds for the applied magnetic field, in particular, the internal field can remain homogeneous within wide fractions of the film area. Experimental studies with this background have been presented in Refs. [11–13]. The scaling analysis by DeSimone et al. generalizes the previous constructions of flux-closure structures for ultrasoft thin film elements in the limit of two-dimensional magnetization distributions as discussed by van den Berg [3,4] and Bryant and Suhl [5,6] for the case of zero and weak external fields. DeSimone et al. [11,12] discuss also the case of sizeable external fields, when the flux-closure breaks down and the external field penetrates into certain regions of the magnetized film. It is important to note here that this field penetration takes place above a certain threshold field. Below this threshold, for t-0, the internal field remains zero and homogeneous. According to this analysis, smaller energy contributions are related to the inhomogenously distorted magnetization near walls, etc. These terms, however, are significant for the choice between different competing realizations of fluxclosure structures and for the hysteretic magnetization processes. The formation and details of the wall structures depend on the film thickness. It has to be emphasized that the powerful mathematical scaling ideas give rigorously valid results and bounds only in the limit of vanishing thickness, t-0. The magnetic behaviour of films with finite thickness may display certain deviations from this idealized behaviour and the relevance of the mathematically rigorous limit for samples of finite thicknesses is not obvious. Therefore, it is interesting to analyze the behaviour of magnetic microstructures from numerically ‘‘exact’’ solutions for film elements with finite thickness and larger lateral dimensions for cases in which such calculations are still viable. As the simplest pattern, we have chosen here the vortex state in a square platelet with an applied field along the diagonal of the square. This pattern is already highly complex and displays main features like walls and a vortex that are also found in more extended magnetization patterns (see, e.g., Refs. [14,15]). Nevertheless, it turns out that the pattern and the magnetization process remain rather simple for our model system. Based on an analysis of the exact numerical solutions, we find that the evolution of magnetization pattern can be described by a simple effective theory, which describes the evolution of the vortex state by the vortex position instead of the full magnetization distribution. This theory requires only the vortex position as a single variable and an effective
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demagnetization factor in a rough approximation. The simplified approach, as exemplified here for the vortex pattern, may be rather useful to analyze experimental observations. In fact, similar ideas have been considered already in Ref. [14] for more complex domain patterns in submicron magnetic Fe dots. To be specific, we have studied the magnetization behaviour of square-shaped samples with different thicknesses of 8, 12, 16 and 20 nm, and an edge length of 1 mm by numerically solving the micromagnetic problem. The magnetic material parameters were those typical for permalloy. For these material parameters and the chosen size of the square platelets, a vortex state forms as the global ground state in zero external field, as expected from the phase diagram in Ref. [16]. The ‘‘domains’’ in soft film elements with thicknesses in the considered range are separated by walls, which are expected to show the internal structure of Ne´el walls [17]. In order to compare our results with the scaling analysis, the different contributions to the specific energy density, i.e. energies per unit volume of the film elements, were estimated following DeSimone et al. [7,10]. These estimates from scaling are summarised in Table 1. Here, the magnetization distribution m(r) is decomposed as m(r) ¼ (m0 , mz), where mz is the component perpendicular to the film plane and m0 the projection of m(r) onto the (x, y)plane. Both contributions to the magnetization will generally depend on the coordinates x, y and z. The film boundary in the (x, y) plane (or some inner boundary in the plane) is characterized by its normal n0 and m0 n0 denotes the dot product of the in-plane magnetization m0 with n0 at such boundaries. The third column of Table 1 is equivalent to that presented by DeSimone [10], apart of an additional factor l2t which was introduced because specific energies have to be compared. Note that Table 1 gives the weights of the different contributions. In order to get energy densities any value has to be multiplied by J 2s =2m0 [10]. The first row (a) shows that a non-zero component mz of order one would result in an extremely large energy. Row (b) in Table 1 gives the scale for the exchange energy related to a variation of the magnetization along the z-axis. It reveals that a significant variation of m along the z-axis is suppressed. The terms of the next rows (c) and (d) are
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purely due to the magnetostatic term in the energy functional equation (1). Comparing all these terms it follows, as is well known, that for thin films the magnetization m(r) almost perfectly lies the (x, y)-plane. The main contributions are then expected from terms corresponding to rows (c) and (d). This means that in the cases to be investigated the ‘‘magnetostatic regime’’ [10] is realised. Note that these terms lead to the approximation of van den Berg [3,4] and Bryant and Suhl [5,6]. Quite unexpectedly, only the Bloch line and variations of m0 over the lateral scale l give contributions proportional to the exchange interaction Apd2. The leading contribution of the Ne´el wall is of magnetostatic nature, i.e. proportional to J 2s =2m0 , as are the terms in rows (c) and (d), and depends only weakly (logarithmically) on the exchange interaction d2 [10]. Hence, in the ideal case mz is zero and m0 depends on x and y only. In zero field the most important contributions result from the terms in rows (c) and (d) which, however, can be strongly reduced by forming magnetization patterns with closed flux, i.e. for vanishing m0 n0 . But, for moderate fields m0Hext of few tons mT the Zeeman energy (e) becomes larger than the term (c) due to m0 n0 . This results in a critical field for the stability of the initial magnetization structure, e.g. the vortex state becomes unstable. As m0 n0 gives the magnetic surface charge on the sample boundary, in row (c) (t/l) ln(l/t) is proportional to the in-plane demagnetization factor Nx, in agreement with the asymptotic behaviour of the expression for the demagnetization factor of rectangular prisms given by Aharony [18]. Thus, it is the demagnetization factor Nx of the square that determines the critical field, for which flux closure states become unstable. The energy of Ne´el walls and Bloch lines connected with flux-closure patterns are one or two orders of magnitude smaller (see last three rows) and should be of minor importance for the magnetization process. In a more precise discussion one has to bear in mind that all the weights involve still certain factors, such as m0 n0 , etc. given by the expressions from the first column for a specific magnetization structures. These factors depend on the details of the magnetization pattern and on the applied field and are generally unknown. In summary, the welldefined hierarchy of energy contributions in this scaling
Table 1 Weight of the different energy contributions (per volume). The numerical estimates are for the investigated samples with l ¼ 1 mm and thickness t ¼ 8, 12, 16, and 20 nm, respectively, with an exchange length d ¼ 5.11 nm and a saturation magnetization Js ¼ 1.005 T as for permalloy
(a) (b) (c) (d) (e) (f) (g) (h)
Source of the term
Specific energy
t ¼ 8 nm
Out of plane mz Exchange, qm=qz Non-tangential component m0 n0 In-plane divergence r0 m0 External field Hext Ne´el wall Bloch line Exchange, m0 varying smoothly on scale l
1 d2/t2 (t/l) ln(l/t) t/l 2Hextm0/Js (t/l)/ln(lt/d2) (d/l)2 ln(l/d) (d/l)2
1 0.408 0.03863 0.008
t ¼ 12 nm
t ¼ 16 nm
1 1 0.181 0.102 0.05307 0.06616 0.012 0.016 0.002 m0Hext/mT 0.00139 0.00194 0.00248 about 1 104: order of magnitude smaller 2.5 105: two orders of magnitude smaller
t ¼ 20 nm 1 0.065 0.07824 0.020 0.00299
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limit restricts the magnetization distribution in specific ways and introduces certain bounds: the magnetization distribution cannot have a sizeable out-of-plane component mz, it must avoid a leaking flux at the borders of the film element and internal magnetic charges as much as possible. The competition of these demagnetization effects with the external field determines essentially the demagnetization pattern. The exchange energy is largely irrelevant, as it enters only logarithmically the energetics of domain walls or Bloch lines via the exchange length. Otherwise, bending of the magnetization can arise only over lengths of the film element edges. As shown by DeSimone et al. [10,11] the magnetostatic terms determine the distribution of magnetic poles r0 m0 and the boundary charges m0 n0 . These quantities related to the field (mx, my) are not sufficient to specify uniquely the magnetization distribution m(r). For this the energies of distorted magnetization configurations, i.e. walls, vortices and the exchange terms have to be taken into account. These terms also provide the stability of patterns corresponding to various possible shallow minima in the micromagnetic energy. Hence, these secondary terms in the hierarchy of magnetic energies cause the configurational hysteresis that is found in the magnetization processes of thin film elements. As discussed, the scaling ideas cannot determine the magnetization patterns in all details. For this one has to understand the role of the inhomogeneous parts of these patterns, in particular the structure of the walls. Moreover for finite thickness, the small energy terms related to walls and the like will be influenced by the corrections to scaling from the leading large magnetostatic energy terms. It is therefore interesting to study the behaviour of simple domain patterns determined by ‘‘exact’’ numerical solutions in order to contribute to a better understanding of these smaller energy terms. Clearly, this is also desirable to develop a deeper understanding of the hysteretic magnetization processes in such thin films with a finite thickness. 2. Methods 2.1. Details of the calculation procedure For thin platelets with a square base (edge l ¼ 1 mm) and different thicknesses, t ¼ 8, 12, 16, 20 nm, micromagnetic simulations have been performed. The saturation magnetization and exchange interaction were adapted to typical properties of permalloy with Js ¼ 1.005 T and A ¼ 1.05 1011 VAs/m. Therefore, the exchange length d equals 5.11 nm. In order to avoid possible stability problems a non-vanishing magnetic uniaxial anisotropy was assumed. However, the anisotropy constant Ku was chosen as low as 0.1 VAs/m3 in order to emulate an ultrasoft magnetic material and to simulate a situation close to the scaling considerations. The x-axis within the film plane was presumed as the magnetically easy axis of the magnetic uniaxial anisotropy. The calculations were done using the program ‘‘The LLG Micromagnetics
Simulator’’TM developed by Scheinfein [19]. By solving the time dependent Landau–Lifshitz–Gilbert equation a starting magnetization distribution was allowed to relax into equilibrium, defined by the condition that the change of magnetization in the last step was less than 1 106. A cell size of 5 nm 5 nm 4 nm has been used in all cases. The external field was alwayspparallel to the diagonal of the ffiffiffi square, Hext ¼ H ext ð1; 1; 0Þ= 2. It was increased in steps of typically 0.5 or 1 mT. In the vicinity of the stability limit these steps were reduced to 0.1 or 0.2 mT. The magnetization distribution m(r) obtained from the preceding (lower) field value provided the starting configuration (input) for determining the new structure and its properties. The vortex pattern at Hext ¼ 0 was realized by appropriate boundary conditions. The volume averages mp(1ffiffiffi1 0) of the magnetization along the field direction ð1 1 0Þ= 2 has been determined from the calculated distributions m(r). 2.2. Phase theory approximation The results are discussed in close relation to the phasetheory approximation (see Ref. [2, Section 3.4.2]). Here, we briefly motivate this approach and explain how it can be adapted to our problem. As discussed above, the exchange energy disappears from consideration in the thin film limit. This also means that in wide areas the magnetization pattern will show bending only over the large lateral length scale of the film element. On the other hand, the internal field disappears at least for small external fields. The high penalty from magnetostatic energy also means that any inhomogeneity of the internal field generally will not become large. This suggests that an approach based on a homogeneous (zero) internal field should still work for these flux-closure domain patterns. This description can be reduced to a geometric description of the magnetization distribution, which only considers certain variable areas (domains) with almost homogeneous magnetization. Within this simplification the demagnetization effect is considered only on average, disregarding all inhomogeneities. An applied field below a certain critical field HextpHcr distorts the flux-closure structure (Fig. 1). The vortex still is preserved and only its position is shifted. In our case with an external fieldpffiffiffi applied along the diagonal, Hext ¼ H ext ð1; 1; 0Þ= 2, the vortex moves along the other diagonal, perpendicular to the applied field Hext (cf. Figs. 1 and 3a). For such states, we can estimate the average magnetization m from the areas (domains) with different direction of magnetization. Neglecting all inhomogeneities of the internal field and the contributions of the distorted magnetization near the walls and the vortex, we may write the magnetic energy per volume of this configuration as E ¼ J s Hext m þ
J 2s ^ m, mN 2m0
(2)
^ as the external field and the demagnetization with Hext and N tensor, respectively. In Eq. (2) the average magnetization m
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Fig. 1. Sketch of the magnetization pattern of thin square-shaped platelets in an applied field along the diagonal (1 1 0) of the square. At sufficiently small fields the vortex is preserved and there are four domains with different directions of the magnetization, as sketched by the arrows. This simplified pattern is characterised by the position of the vortex d as shown.
is given (cf. Fig. 1) by m ¼ ½ðv1 v3 Þex þ ðv2 v4 Þey ,
(3)
where v1 and v4 (v2 and v4) are the areas of the pattern, for which the magnetization is mainly parallel to 7ex (7ey), respectively. This is a very rough approximation because the possible rotation of the magnetization in the domains is neglected. In particular, this geometric construction of the domain pattern yields a description where walls are not free of magnetic charges. The volume fractions in Eq. (3) are subject to the conditions v1 þ v2 þ v3 þ v4 ¼ 1,
(4)
and v1 ¼ v2 and v3 ¼ v4 as the of the applied field pffiffidirection ffi Hext is parallel to ð1; 1; 0Þ= 2. This description contains only one parameter, e.g. v1. Obviously, it is equivalent to the choice of the vortex position d as the single parameter for characterising the configuration of Fig. 1. The minimization of the energy density (2) with respect to v1 gives the relation 1 H ext m (5) ¼ pffiffiffi 0 , 2v1 2 2N x J s valid for v1p1/2, as the outermost position of the vortex is the corner of the square. Therefore in this approximation the critical field Hcr at which the vortex is expelled is given by pffiffiffi H cr ¼ N x J s = 2m0 . (6) For HextpHcr the average magnetization m(1 1 0) along the direction of applied field is given by mð1 1 0Þ ¼
H ext m0 . N xJ s
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expressions for the demagnetizing factors of a prism have been derived by Aharoni [18], applicable for homogeneously magnetized samples. When comparing the results of micromagnetic calculations with such a simplified description inspired by the classical phase-theory approximation [2] one has to bear in mind the following crucial differences: (i) The micromagnetic calculations take into account the exchange energy whereas a corresponding term is completely lacking in the phase theory, cf. Eq. (2). As it has been argued in the introduction, these terms determine the choice between different coexisting states of nearly the same energy. They are responsible for hysteretic processes. (ii) The description of the demagnetizing energy by a demagnetizing tensor is correct only for homogeneously magnetized bodies. This precondition is hardly fulfilled for samples with an ideal or distorted vortex. (iii) The phase theory gives only average values, whereas in micromagnetic calculations the corresponding integrals are evaluated from the inhomogeneous magnetization distribution m(r). Conventional phase theory works for bulk samples, in which different coexisting regions or domains with homogeneous magnetization in a system are well defined. These different domains are due to the intrinsic magnetic anisotropy of a system, which results in degenerate magnetic ground-state configurations. The possible magnetization directions are governed by the symmetry of this system, i.e. the existence of different directions of equivalent magnetic easy axes. Furthermore, the size of the magnetized bulk sample should be much larger than the typical domain sizes. This justifies neglecting the inhomogeneities in the internal field and the assumption of negligibly thin domain walls. The approach works ideally for large ellipsoidal samples, which develop nearly homogeneous internal fields on average. In contrast, in a thin film the different regions of homogeneous magnetization are not related to thermodynamically stable magnetic states (phases) as required by the classical phase theory. Rather, it is the requirement of flux closure and the geometrical shape of the film element that govern these patterns (cf. Introduction and Refs. [3,4]). These walls are related to singularities of the two-dimensional magnetization distribution and the boundaries of the thin film element. In our case, however, the domain size is not small compared to the sample dimension, any crystalline magnetic anisotropy is lacking, and the domains are not defined by the anisotropy. Hence, our approach lies beyond the usual legitimation of phase theory. Its applicability has to be tested by a careful examination of the results from exact numerical calculations.
(7)
Because the applied field is in the (x, y)-plane and the sample base is a square, only the in-plane demagnetization factor Nx of the square enters into Eqs. (5)–(7). According to Eqs. (5)–(7) any influence of thickness t should be comprised in the demagnetization factor Nx. Analytic
3. Results and discussion 3.1. Magnetizing the vortex state Applying a field along the square’s diagonal (1 1 0), the magnetization component m(1 1 0) increases (Fig. 2). Up to a
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M. Wolf et al. / Journal of Magnetism and Magnetic Materials 314 (2007) 105–115 Table 2 Magnetometric in-plane demagnetization factor Nx, critical fields Hcr estimated by phase theory and obtained from micromagnetic calculations (mm. calc.)
Fig. 2. Calculated component m(1 1 0) versus applied field Hext of squareshaped platelets (side length l ¼ 1 mm and thickness t ¼ 8, 12, 16, 20 nm). At a critical field Hcr a jump of m(1 1 0) is observed, the vortex is expelled and the leaf-like magnetization pattern is formed above Hcr (cf. Fig. 3b).
Fig. 3. Calculated magnetization patterns of the square with t ¼ 16 nm, l ¼ 1 mm for fields Hext below (a) and above (b) the critical field m0HcrE15 mT at which the vortex is expelled. (a) mx–my domain plot with local magnetization direction (short arrows) for Hext E8.5 mT below Hcr, clearly showing the vortex shifted along the 1¯ 1 0 direction. Local magnetization directions are marked by short arrows and given in different grey tones. The long arrows give the main magnetization direction. (b) leaf-like state for m0HextE25.5 mT4m0Hcr. Over a wide area in the centre m is mainly parallel to the direction of external field. (c) Orientation of the (1 1 0) and 1¯ 1 0 direction with respect to the square edge and to Hext.
critical field Hcr the vortex is preserved (Fig. 3a). At this critical field Hcr a jump-like increase of m(1 1 0) is observed (Fig. 2) and the vortex is expelled. Above Hcr a leaf-like pattern of m(x, y, z) is formed (Fig. 3b). For HextXHcr the magnetization m(1 1 0) is nearly independent of the field and close to saturation (Fig. 2). In accordance with Eq. (7) m(1 1 0) increases with decreasing thickness t at any field value Hext (cf. Fig. 2) as long as the vortex exists, i.e. for HextpHcr, as Nx decreases with decreasing t. This explains
t (nm)
Nx
m0Hcr (mT) phase theory
m0Hcr (mT) (mm. calc.)
8 12 16 20
0.01415 0.01968 0.02478 0.02957
10.0 13.9 17.5 20.9
11.3 13.4 15.0 16.5
also (cf. Eq. (6)) that the field range for which the vortex is preserved shrinks with decreasing thickness (see Table 2, cf. Fig. 2). The critical field from micromagnetic calculations and that estimated by the phase theory are summarized in Table 2, which presents also the in-plane (magnetometric) demagnetizing factors Nx, calculated from the expression given by Aharoni [18]. The figures of Table 2 show that the estimated values for Hcr according to the phase theory are larger than those obtained from micromagnetic calculations (with the exception t ¼ 8 nm), but of the right order of magnitude. There exists no common factor between the values of Hcr estimated from phase theory and those obtained from micomagnetic calculations. The values for Hcr for square permalloy film elements observed experimentally by Kerr microscopy in Refs. [12,13] are also explained fairly well by the demagnetizing factor of the corresponding samples. For instance, the reported experimental critical field in Ref. [13] is about 4.7 mT for a square-shaped permalloy film element l ¼ 57 mm and thicknesss t ¼ 207 nm, which is again smaller than the value of about 5.2 mT estimated from the phase theory. The somewhat reduced critical fields observed in experiments/calculations are reasonable as the theoretical stability limit of the vortex pattern should not be reached in real samples due to the finite size of the vortex. As Hcr is mainly determined by the balance of the Zeeman term and the demagnetization energy, for circular elements a similar dependence of the critical field on the thickness is observed [20]. 3.2. Discussion of the energy contributions In Figs. 4–6, we display the different leading energy contributions from the numerical solutions, i.e. the exchange energy Eexch, demagnetizing energy Edem and field energy Efield. As the film elements have different thickness, the energy densities (i.e. energies per unit volume) have to be compared. The average exchange energy density Eexch decreases with decreasing thickness t, for HextoHcr as well as for Hext4Hcr , see Fig. 4. This decrease of the exchange energy is due to the progressive suppression of the out-of-plane magnetization component mz with decreasing film thickness t, as obtained from the numerical calculations: In zero field the average mz decreases from about 1.86 104 for t ¼ 20 nm to 1.73 104 for t ¼ 8 nm. Thus the
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Fig. 4. The specific exchange energy Eexch obtained by micromagnetic calculations in dependence on the applied field Hext. Note the sharp decrease of Eexch at Hcr, just where the vortex is expelled.
Fig. 5. All values Efield/Nx plotted versus m(1 1 0) form a common curve, i.e. Efield scales with a common factor proportional to Nx. A fit to a parabola is possible for m(1 1 0) up to about 0.59 (solid line).
corresponding small contribution to Eexch due to the variation of mz is reduced. Moreover, these values for the average of mz show that the corresponding energy (second row in Table 1) may be neglected as claimed in the Introduction. The above-mentioned variation of Eexch with t is small compared to the total value of Eexch for Hext ¼ 0. The overall exchange energy Eexch is largely reduced by a large jump at the field Hcr when the vortex is expelled. Thus it is concluded that Eexch is dominated by the contribution from the (Ne´el) walls and/or the vortex. Their disappearance at Hcr results in the drastic reduction of Eexch. The values DEexch ¼ Eexch(Hcr+0)–Eexch(Hcr–0), together with the analogously defined quantity for the demagnetization energy DEdem, as derived from the micromagnetic calcula-
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Fig. 6. In the vortex state all values Edem–Edem,0/Nx plotted versus m(1 1 0) form a parabola-like curve. The m(1 1 0) values belong to successive field values and, as shown for t ¼ 16 nm, at Hcr a jump in Edem and m(1, 1, 0) occurs.
tions, are given in Table 3. This change of DEdem at Hcr (visible in Fig. 6 for t ¼ 16 nm) involves contributions from the large-scale magnetization pattern (cf. Fig. 3) as well as from the walls and vortex. To find out an estimate for the demagnetization energy associated with the inhomogeneous parts of the magnetization distribution at Hcr, we need an approximation for the effect due to the global change of the domain pattern. Within our phase theory approximation we expect a change of the demagnetization energy D0 E dem ¼ J 2s N eff ðmð1 1 0Þ ðH cr þ 0Þ2 mð1 1 0Þ ðH cr 0Þ2 Þ=ð2m0 Þ from the jump of the average magnetization m(1 1 0)(Hext) at the transition from the vortex to the leaf state. For the effective demagnetization factor, we use a reduced value Neff, determined from the global behaviour of the demagnetization energy in the vortex state (see below). These values D0 Edem, given in Table 3, should not depend on the contribution of the walls to DEdem. Then, the difference DEdem–D0 Edem between the numerically derived demagnetization energy DEdem and the contribution D0 Edem associated with the global domain pattern can be attributed to the demagnetization energy that is related to inhomogeneities of the walls and/or the vortex. In Table 3, these differences are listed as DE inh dem . The negative sum of this part from demagnetization DE inh dem and the jump in exchange DEexch gives an estimate for the wall energy in the vortex state DEwall. The comparison with the expected wall energies for thin films from Refs. [2,10] shows that the order of magnitude and the trend is correct for our estimate. When comparing these values one has to be aware that, firstly, in zero field the vortex state gives rise to two 901 Ne´el walls with total length 2 O2 l. But, a 901 Ne´el wall has an energy vastly reduced by a factor of the order 0.12 as compared to a 1801 wall (cf. Ref. [2, Figs. 3.73 and 3.79]). Hence, the wall energies in the last two columns
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Table 3 The jumps of the exchange energy DEexch and demagnetization energy DEdem at the critical field Hcr from micromagnetic calculations with DEexch ¼ Eexch(Hcr+0)–Eexch(Hcr–0) etc t
DEexch
DEdem
D0 Edem
DE inh dem
DEwall
Hubert 1801 wall, l
DeSimone 1801 wall, l
nm
104 W s/cm3
104 W s/cm3
104 W s/cm3
104 W s/cm3
104 W s/cm3
104 Ws/cm3
104 W s/cm3
8 12 16 20
6.17 6.32 6.33 6.35
1.28 6.70 13.99 21.68
11.39 18.41 26.47 34.50
10.11 11.71 12.44 12.82
16.28 18.03 18.81 19.17
9.2 12.1 14.8 16.9
5.52 7.74 9.85 11.90
Within the phase theory approximation D0 E dem ¼ J 2s N eff ðm2 ðH cr þ 0Þ2 m2 ðH cr 0ÞÞ=ð2m0 Þ is an estimate for the change of the demagnetization energy 0 due to the change of the magnetization only, where we used an effective demagnetization factor Neff ¼ 0.66 Nx. Then, DE inh dem ¼ DE dem D E dem is the contribution to Edem from the inhomogeneous magnetization and DE wall ¼ ðDE inh þ DE Þ is an estimate for the wall energies in the vortex state. The exch dem values of DEwall are compared with the energies of straight 1801 walls of length l according to Hubert (Ref. [2], chap. 3.6.4, Fig. 3.79) and DeSimone [10].
should be multiplied by a factor of about 0.3 to derive an estimate of the contribution of the Ne´el walls in the vortex state. Due to the presence of the vortex this is a rough approximate. In an applied external field, the angle between adjacent domains changes and depends on the details of the domain pattern. Moreover, for the squares in an external field, the detailed form of the magnetization distribution in the wall is field dependent and the change of the wall profiles has to be taken into account. All this prevents a reliable quantitative estimation of wall energies in the vortex state. However, it is reasonable that our estimates DEwall are considerably larger than the energies estimated from idealized 901 straight walls, because DEwall sums all contributions due to inhomogeneities in the magnetization distribution and their interactions. It is remarked that the domain wall energies estimated following DeSimone et al. include undetermined factors and, therefore, allow only order-of-magnitude estimates. Within this scaling limit l is a characteristic lateral length, which is of course of the order of the square edge, but can numerically differ from it. This explains that the values of the last two rows in Table 3 differ from the estimates for wall energies following [2] by a nearly constant factor. From Eqs. (2) and (7) within the phase theory the specific field energy for HextpHcr should be given by E field ¼
N x J 2s 2 mð1 1 0Þ , m0
(8)
i.e. all values Efield/Nx plotted versus the calculated m(1 1 0) should form a common parabola with the curvature J 2s =m0 . Fig. 5 shows that the values Efield/Nx indeed form a common curve, i.e. Efield scales with a factor proportional to Nx. However, a fit of a parabola is possible only in the range m(1 1 0)o0.59 (cf. the deviations of the parabola from the solid line in Fig. 5). Moreover, the curvature of this parabola is significantly smaller than predicted and can be ascribed to an effective demagnetization factor NeffE0.71Nx. Within the same approximation, Eq. (2), the demagnetization energy Edem should be given by E dem ¼ N x J 2s m2ð1 1 0Þ =2m0 . In contrast, the micromagnetic calcula-
tions yield also for the demagnetized state m(1 1 0) ¼ 0 nonzero demagnetization energies Edem, 0 ¼ Edem(Hext ¼ 0) with values Edem, 0 in units of W s* cm3: 4.30 104 for t ¼ 8 nm, 5.47 104 (t ¼ 12 nm), 6.44 104 (t ¼ 16 nm) and 7.26 104 (t ¼ 20 nm). Therefore, it is suggested that the differences Edem–Edem, 0 follow a quadratic law ðE dem E dem; 0 Þ=N x ¼ J 2s m2ð1 1 0Þ =2m0 . As shown in Fig. 6 these differences form indeed a common curve, when plotted versus m(1 1 0). With regard to the behaviour of Efield, a quadratic law was fitted up to m(1 1 0) ¼ 0.59. As in the case of the field energy the curvature is again too small, corresponding to NeffE0.64 Nx, which agrees satisfactorily with that obtained from Efield(m(1 1 0))/Nx. The reduction of the effective demagnetization factor can be understood from the large-scale inhomogeneous magnetization pattern in the square platelet that results in a suppressed demagnetization energy as compared to that of homogeneously magnetized body, i.e. the external field has a stronger influence or the effective demagnetization factor becomes effectively smaller (cf. Eq. (7)). In summary, the dependence of Edem, Efield on m(1 1 0) can be semiquantitatively understood by a simplified phase theory. The field energy Efield and demagnetization energy Edem behave in a slightly different way. In micromagnetic calculations they are determined by the whole magnetization distribution m(x, y, z), which enters into the corresponding terms (see Eq. (1)). However, in the approximate expressions for averages of the phase theory using an effective demagnetization factor, these results for Edem, Efield are described by the average m(1 1 0) as a single variable characterizing the magnetization configuration. Furthermore, as revealed by Fig. 4, for small external fields Eexch is not small compared to Edem and Efield, respectively. This is a further reason that the evolution of the field energy and the demagnetization energy cannot fully be understood within the frame of phase theory. The failure to describe the field energy for larger external fields with m(1 1 0)40.59 can be understood as consequence of the penetration of the external field into certain sample regions and the concomitant inhomogeneity of the resulting magnetization state.
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3.3. The scaling behaviour of the magnetization and vortex position From Eq. (7) it is suggested that all scaled magnetization values m(1 1 0)NxJs lie on a common straight line and should be equal m0Hext. Fig. 7 reveals that m(1 1 0) scaled in the described way form a common straight line only for small applied fields. However, the slope in this region is larger than expected from the phase theory (dash-dot line in Fig. 7), which is consistent with an effective demagnetization factor NeffE0.77 Nx. This decreased Neff corresponds to an increased initial susceptibility as reported in Ref. [14]. For larger Hext deviations from the scaling are found and even no common curve is established. Again, the relation between the effective demagnetization factor and Nx is in fair agreement with the corresponding demagnetization factors from the analysis of field and demagnetization energy. The observed behaviour of m(1 1 0) in Fig. 7 is related to deviations of the real magnetization distribution from the assumed constant saturation magnetization in the four parts of the square, as sketched in Fig. 1. On the other hand, the vortex position has been suggested to be a main parameter characterizing the magnetization pattern (cf. Section 2.2). Therefore, it is worthwhile to study the relation between m(1 1 0) and the location of the vortex, characterized by its relative position r ¼ d*O2/l. The distance d of the vortex from the centre of the square with the edge length l (cf. Fig. 1) has been determined for each field from figures like Fig. 3a. The parameter r varies between 0 and 1. The phase theory predicts for the m(1 1 0)plot versus r a common straight line from zero to 1/O2. As displayed by Fig. 8 such a plot results indeed in a common straight line. But the slope of this line is greater than the slope corresponding to the simple magnetization configuration of Fig. 1. That means, the magnetization m(1 1 0) increases more strongly than predicted by this model and
Fig. 8. The magnetization components m(1 1 0) and mð1; 1; 0Þ in dependence on the relative vortex r ¼ d*O2/l. The values of m(1 1 0) of all squares lie on a common straight line, but above the curve expected from phase theory (solid straight line). The average of mð1; 1; 0Þ , restricted over the upper right part of the square (grey shaded triangle), mð1; 1; 0Þ , decreases with increasing r.
described by Eqs. (3) and (4), which suggests an additional redistribution of the magnetization. Averaging the component of m(x, y, z) perpendicular to the direction of the applied field, m(1, 1, 0), only over a triangle, the upper right half of the square (grey shaded part of the inset in Fig. 8), gives a non-zero value mð1; 1; 0Þ . As revealed by Fig. 8 these partial averages mð1; 1; 0Þ lie on a common curve when plotted versus r. However, the decrease of mð1; 1; 0Þ with increasing r cannot be explained by simple assumptions on mðx; y; zÞ corresponding Fig. 1 and Eq. (3). These results show that the vortex position captures the main features of the magnetization process for square platelets with different thickness, but is not sufficient to explain finer details. 3.4. Magnetization profiles
Fig. 7. For small Hext the scaled m(1 1 0) values, m(1 1 0)NxJs, form a common straight line with a slope corresponding to NeffE0.77 Nx. For larger Hext the scaling fails.
The deviations of the simple phase theory approximation for the magnetization process in the vortex state points to the importance of two effects. First, within the ‘‘domains’’ a rotation of the magnetization may take place in the external field. This process is not independent of the magnetization distribution on a whole. The other important contribution stems from the evolution of domain walls and inhomogeneities of the magnetization distribution. For the vortex state in zero applied field in Fig. 9 the profiles of the in-plane magnetization component mx is plotted for different cuts y ¼ constant, i.e. along the x-edge of the square platelet. The profiles demonstrate that homogeneous regions mx ¼ constant, i.e. ‘‘domains’’ exist in a well-defined region. These regions strongly shrink when approaching to the vortex. The behaviour of my (not shown here) is quite similar. However, the ‘‘domain walls’’ with rapidly changing magnetization direction occupy
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In accordance with Fig. 3, in an applied field the magnetization distributions show strong distortions, as displayed by the profiles in Fig. 10, lower panel. In particular, the domain walls that are shifted away from the 1¯ 1 0 -diagonal acquire asymmetric shapes, and the ‘‘domains’’ display a continuously distorted magnetization. 4. Conclusions
Fig. 9. Variation of the magnetization component mx for zero field in the square with t ¼ 16 nm along cuts with different y ¼ constant. For small y certain ‘‘walls’’ may be defined. The cut along y ¼ 500 nm clearly displays the vortex.
Fig. 10. Variation of mx and my across the walls for t ¼ 8 and 20 nm, respectively, in dependence on the coordinate x, dashed line in the square of the inset, for x+y ¼ 0.25 mm in zero (upper panel) and nonzero (lower panel) external fields. The external field Hext||(1 1 0) has been 6.65 mT (t ¼ 8 nm) and 14.12 mT (t ¼ 20 nm), for which the relative vortex position is nearly the same. The vertical straight lines mark the wall positions.
sizeable regions of the pattern and there is a smooth transition into the vortex region. The profiles for the ground-state patterns from the calculations for other thicknesses t are rather similar as shown by the profiles of Fig. 10, upper panel.
We have performed micromagnetic calculations for permalloy squares of different thicknesses of 8, 12, 16 and 20 nm with an edge length of 1 mm under an increasing external field Hext parallel to the square’s diagonal and presented results concerning (i) the field dependence of the average magnetization along the field, m(1 1 0)(Hext), (ii) the evolution of the different energy contributions and (iii) the behaviour of the magnetic microstructure m(r). The initial demagnetized state of the squares, a vortex state, turned out to be preserved up to a critical field Hcr. With decreasing t the critical field Hcr decreases. This reduction can be well explained by the decrease of the in-plane demagnetization factor Nx. For low enough applied fields in the vortex state the scaled magnetization values m(1 1 0) Nx of all four squares lie on a common straight line. For higher fields this scaling is lost. Taking into account the small off-set term Edem(Hext ¼ 0), the differences of the demagnetization energy (Edem–Edem(Hext ¼ 0))/Nx as well as the scaled field energy Efield/Nx of all squares plotted versus m(1 1 0) form a common curve. For m(1 1 0)p0.6 this curve is a parabola, for larger m(1 1 0) deviations from the parabola occur. These results may be understood semiquantitatively within a kind of phase theory, whereby the different magnetic phases are the four regions differing in the main magnetization orientations in the vortex pattern. This is supported by the fact that the vortex position captures important features of the magnetization behaviour. The phase approximation appears very crude. However, the calculated spatial magnetization profiles reveal relatively wide homogeneously magnetized areas (‘‘domains’’) separated by narrow regions, in which the magnetization rapidly changes (Ne´el walls). An inspection of the wall profiles reveals strong distortions by the external field. The dependence of the exchange energy on the applied external field as obtained from micromagnetic calculation clearly shows that this term is related to the inhomogeneous magnetization in the vortex state. This term is of the order of magnitude of energy contributions from proper Ne´el walls that are the relevant walls separating domains with well-defined different magnetization-direction in a continuous thin magnetic film. The change in the demagnetization energy at the instability field of the vortex state Hcr involves contributions from the globally inhomogeneous magnetization distribution, as well as from the walls and related inhomogeneities, which change or disappear when the vortex is driven out. An estimation of the latter part has been given. From the scaling properties of the different
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energy contributions, the dominating role of the demagnetization energy in flux-closure patterns and the behaviour of the regions with a nearly homogeneous magnetization, it is understandable that the phase theory approximation with an effective reduced demagnetization factor works rather well. It is expected that similar simple geometrical schemes be developed also for thin film elements of larger dimensions showing more complex patterns. References [1] B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures, vol. 1–3, Topics in Appl. Phys., vol. 83,87,101, Springer, Berlin, 2001–2006. [2] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, 1998. [3] H.A.M. van den Berg, J. Appl. Phys. 60 (1986) 1104. [4] H.A.M. van den Berg, J. Appl. Phys. 61 (1987) 4194. [5] P. Bryant, H. Suhl, Appl. Phys. Lett. 54 (1989) 78. [6] P. Bryant, H. Suhl, Appl. Phys. Lett. 54 (1989) 2224. [7] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, Comm. Pure Appl. Math. 55 (2002) 1408.
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[8] A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford University, Oxford, 2000. [9] W.F. Brown Jr., Micromagnetics, Interscience Publisher, New York, 1963. [10] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, Recent analytical developments in micromagnetics, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, vol. 2, Academic Press, 2005. [11] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, R. Scha¨fer, Proc. Roy. Soc. A 457 (2001) 2983. [12] A. DeSimone, R.V. Kohn, St. Mu¨ller, F. Otto, R. Scha¨fer, J. Magn. Magn. Mater. 242–245 (2002) 1047. [13] R. Scha¨fer, A. DeSimone, IEEE Trans. MAG 38 (2002) 2391. [14] P.-O. Jubert, J.-C. Toussaint, O. Fruchart, C. Meyer, Y. Samson, Europhys. Lett. 63 (2003) 132. [15] O. Fruchart, A. Thiaville, Comp. Rendus Phys. 6 (2005) 92. [16] R.W. Cowburn, M.E. Welland, Appl. Phys. Lett. 72 (1998) 2041. [17] K. Ramsto¨ck, W. Hartung, A. Hubert, Phys. Stat. Sol. (a) 155 (1996) 505. [18] A. Aharoni, J. Appl. Phys. 83 (1998) 3432. [19] M.R. Scheinfein, ‘‘The LLG Micromagnetics Simulator’’(TM) /[email protected], [email protected] 1997–2006. [20] T. Pokhil, D. Song, J. Nowak, J. Appl. Phys. 87 (2000) 6319.