Helmut Kr onm ul ler on the asion of his 70th birthday Riccardo Hertel MaxPlanckInstitut ur Mikr ostrukturphysik Weinb er 2 06120 Hal le Germany Zero57356eld magnetization states in ermallo thin 57356lm elemen ts of rectangular shap are calculated m ID: 26360 Download Pdf

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Helmut Kr onm ul ler on the asion of his 70th birthday Riccardo Hertel MaxPlanckInstitut ur Mikr ostrukturphysik Weinb er 2 06120 Hal le Germany Zero57356eld magnetization states in ermallo thin 57356lm elemen ts of rectangular shap are calculated m

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Thic kness dep endence of magnetization structures in thin ermallo rectangles De dic ate to Pr ofessor Dr. Helmut Kr onm ul ler on the asion of his 70th birthday Riccardo Hertel Max-Planck{Institut ur Mikr ostrukturphysik, Weinb er 2, 06120 Hal le, Germany Zero-eld magnetization states in ermallo thin lm elemen ts of rectangular shap are calculated means of nite elemen micromagnetic mo deling. The energies of sev eral ossible magnetization patterns are determined for dieren sizes and thic knesses of the sp ecimen. Based on these data, phase

diagram of the lo est-energy conguration is set up for rectangles of edge lengths et een 250 nm and 1000 nm. It is sho wn that thic kness- and size-dep enden transition from quasi- homogeneous single-domain states to demagnetized ux-closure patterns ccurs in that range. The regime of sizes and thic kness of the phase diagram in whic the lo est-energy conguration of the thin lm elemen is single-domain state is particularly imp ortan t, since in this case the sample is exp ected to sho go stabilit with resp ect to thermal demagnetization, whic is required

criterion to mak it suitable for tec hnological applications in magneto-electronic devices. Keyw ords: Micromagnetic Mo delling; Finite Elemen Metho d; ermallo Thin-Film Elemen ts; Single- Domain Limit I. INTR ODUCTION In Magnetic Random Access Memory (MRAM) de- vices [1 ], soft magnetic thin lm elemen ts are used as unit cells of information. Ideally eac thin-lm elemen of the device is homogeneously magnetized, with ossible directions of the magnetization, et een whic the par- ticle can switc hed. According to its magnetization di- rection, the information stored in an elemen

is ev aluated as either logical "one" or logical "zero". In real orld, ho ev er, suc soft magnetic platelets are generally not magnetically bistable systems. Instead, umerous dier- en magnetic structures can observ ed at zero eld, de- ending on the particle’s shap e, size, material, thic kness, its microstructure and its magnetic history Most of these magnetization states are undesirable from tec hnological oin of view, ecause once the magnetic structure splits up in to ux-closure magnetic domain pattern, the cell is no longer in state whic can unam biguously assigned

to logical alue. In ermallo thin-lm elemen ts of rect- angular shap e, large set of ossible magnetic structures has een found oth exp erimen tally[2 {4 and umerically [5, 6]. The agreemen et een calculated and observ ed structures is remark ably go d. The domain structures can categorized in to remanen states with high net magnetization, and ux closure patterns with anishing erage momen t. The domain pattern of the latter ones are in erfect agreemen with the construction sc heme set up an Den Berg [7], an ingen uous geometrical al- gorithm to construct ux-closure

patterns in ideally soft thin-lm elemen ts of arbitrary shap e. The formation of these magnetic structures can usually understo considering Bro wn’s ole oidance principle[8 on one Electronic address: hertel@mpi- halle.mpg. de phone: +49-(0)345-5582-592 ax: +49-(0)345-5511223 FIG. 1: Some basic magnetization structures in ermallo rectangles. Demagnetized domain structures: (a) Landau, (b) Diamond; States with high remanence: (c) S-state, (d): C-state. The terms C-state and S-state are coined the ux lines through the rectangle, ha ving similar shap es as the resp ectiv letters.

side and the tendency to oid inhomogeneities of the magnetic structure on the other side. Remanen mag- netic structures in soft magnetic elemen ts are the out- come of comp etition et een stra eld energy and exc hange energy When ysteresis lo op is erformed with an in-plane eld, it is generally not clear what ex- actly determines the resulting zero eld conguration, esp ecially if the sample drops in demagnetized state. Whic path, e.g. the magnetization tak es from satu- rated state to demagnetized state with sev en domains (diamond state[9]) as compared to the

path from satura- tion to Landau-t yp structure, either with or without cross-tie alls, and wh the sp ecimen ho oses to drop in to one state rather than the other are dicult questions whic need to examined with accurate time-resolv ed micromagnetic sim ulations. Ob viously the formation of end domains along those edges of the sp ecimen whic are aligned erp endicular to the external in-plane eld is decisiv initial breaking of symmetry whic leads to dieren remanen states, kno wn as C-state and S- state [10 ]. The C-state seems to connected with the Landau-structure, while

the S-state app ears to related with the diamond structure[11 ]. In order to suppress closed-ux domain structures

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whic ma originate from suc end domain patterns, elongated thin-lms elemen ts with tap ered ends ha een considered [2 ]. Indeed, these particles resulted to ha one ell-dened remanen state with almost uni- form magnetization. Am biguities connected with the ossibilit to obtain either C-state or S-state as remanen states and the resulting reduced stabilit with resp ect to the formation of ux-closure patterns ere oided. Ho ev er other

problems arose. While the remanen state as ell-dened, the rev ersal mec hanism as not, sim- ply ecause the sources of ossible instabilities had een successfully suppressed. As consequence of this, the lo cation at whic the rev ersal ucleates is almost unpre- dictable in suc particles. Ev en computations lead to dieren results, some claiming the rev ersal starts in the middle of the sample[12 ], others stating that ortex enetrates the sample from the side of the sp ecimen[13 ], oth of whic are lik ely to ossible. No ada ys, the formation of end domains is seen as helpful

feature, ecause they allo the magnetization to rev erse con tin- uously means of edge soliton propagation. In this case, the magnetization of the particle switc hes the ucleation of head-on domain alls mo ving along the edges, whic is preceded an expansion of the end do- mains. Ho ev er, it is imp ortan to ha con trol of the end domains, in order to allo for repro ducible switc h- ing pro cess. It has een sho wn earlier, that the S-state switc hes in con tin uous fashion [5, 14 ], with edge solitons whic propagate along the edge without in ter- secting. Therefore, switc hing pro cess et een S-

states is generally not problematic, whereas C-states are dicult to switc h[6 ]. Recen tly Arrott [29 has prop osed the use of thin-lm elemen ts with sp ecially rounded cor- ners similar to the shap of ean in order to fa or the C-state as remanen state, to suppress acciden- tal switc hing. The rev ersal of this particle should ccur means of an additional bias eld, whic con erts the structure in to the S-t yp e, th us allo wing to switc easily One of the ma jor hallenges in the tec hnical applica- tion of magnetic thin lm elemen ts for MRAM devices is the

required reliabilit and repro ducibilit of the switc h- ing pro cess. Ov er more than 10 13 cycles of magnetization rev ersal, the particle is required to remain magnetized in ell-dened high-remanence state. Therefore, the um- er of ossible demagnetized states needs to ept as small as ossible, in order to reduce the probabilit of an acciden tal demagnetization, driv en, e.g. thermal ef- fects. Con trary to the aforemen tioned approac hes, whic fo cus on dieren shap es of thin lm elemen ts, this pap er rep orts on the inuence of the thickness of thin lm

el- emen on the total energy of domain structures. If thin lm elemen is single-domain particle in the sense that ux-closure magnetization pattern has higher energy than structure with high remanence, it is unlik ely to drop acciden tally in demagnetized states, and it is im- ortan to kno whether this prop ert can obtained simply hanging the particle’s thic kness. Sev eral magnetic structures in ermallo thin lm ele- men ts of rectangular shap (asp ect ratio 2:1) are in esti- gated, and their energy is compared as function of the sp ecimen’s thic kness. The in estigations

are carried out using micromagnetic nite elemen algorithm. I. MICR OMA GNETIC BA CK GR OUND The most signican energy terms to considered for the calculation of remanen magnetic structures in the framew ork of micromagnetism are exc hange, anisotrop and stra eld energy: tot x;y ;z (1) In the case of ermallo surface anisotrop is neg- ligible, and since are in terested in zero eld con- gurations, the Zeeman term is omitted. mimic the prop erties of ermallo an exc hange constan =1.3 10 11 J/m, uniaxial anisotrop constan 500 J/m and saturation magnetization

96 10 A/m corresp onding to saturation olariza- tion of 1.00 T, are assumed. In Eq. (1), denotes the reduced magnetization and is unit ector parallel to the easy axis, whic is is along the long edge of the rectangle. The stra eld is the magnetic eld arising from the magnetic momen ts of the sample. detailed description of the calculation of this con tribution is giv en in the next section. Equilibrium congurations of the magnetization can obtained minimizing the total energy tot with re- sp ect to the directional eld of the magnetization, i.e. solving the

ariational problem tot 0, where tot is uniquely determined the magnetic structure ). Stable magnetic structures corresp ond to lo cal energy minima of the magnetic congurational space. Gener- ally analytic solutions of this ariational problem are only ailable for samples of sp ecial, simple geometry and making use of simplifying assumptions concern- ing the magnetic structure and the demagnetizing eld Ho ev er, means of umerical calculations, mag- netization structures can calculated from rst prin- ciples in the framew ork of micromagnetism. Suc sim- ulations generally

represen dicult task, and they are usually restricted to particles in the sub-micrometer range. This is mostly so ecause of the high system requiremen ts (memory pro cessing time) in olv ed with the umerical calculation of the long-range in teraction giv en the stra eld, whic gro rapidly with in- creasing um er of discretization cells. The restrictions in size are also due to the disparate length scales in- olv ed in micromagnetic problems. Widely extended re- gions of homogeneous regions (the magnetic domains) are connected narro regions of strongly inhomoge- neous

magnetization (the domain alls). The imp ortan length scales describing these features, rst in tro duced Kronm uller[15 ], are the so-called exc hange lengths. They

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describ the ypical extension of magnetic microstruc- tures. In soft magnetic materials the exc hange length of the stra eld A= is decisiv e, whereas in hard magnetic materials the exc hange length due to the anisotrop A=K needs to considered. These exc hange lengths are connected with the extension of eel alls and Blo alls[16 ], resp ectiv ely er- ful metho to accoun for the dieren length

scales in micromagnetic sim ulations has een presen ted Hertel and Kronm uller, who dev elop ed dieren adaptiv mesh renemen sc hemes for three-dimensional micro- magnetic calculations, th us allo wing to accurately sim u- late the magnetic structure in samples of unpreceden ted size [17 ]. Increased exp erimen tal abilit of nano-patterning of thin lm elemen ts, on one side, and the steadily gro wing computational er com bined with impro ving umeri- cal tec hniques, on the other side, ha shifted the ypical size of samples whic can calculated reliably in the range of

particles as they are exp erimen tally pro duced and tec hnologically relev an t. In fact, it is no ada ys os- sible to erform micromagnetic sim ulations on elemen ts of larger size than those planned to used in future magneto-electronic devices. oten tial memory elemen as it could used in an MRAM should ha lateral extension in the sub-micron range. I. FINITE ELEMENT MICR OMA GNETIC MODELING calculate umerically the eld of the magnetiza- tion, discretized form of it is rst required. or this purp ose, the sample is sub divided in to simplex elemen ts, i.e. tetrahedral elemen ts

in the three-dimensional case. The orien tation and the shap of these elemen ts is gen- erally irregular, whic allo ws to appro ximate the shap of samples of virtually arbitrary shap e. This is in strong con trast to the commonly used nite-dierence sc hemes, whic require regular discretization mesh of cub e- shap ed cells of equal size. In the latter case, inclined or curv ed surfaces need to appro ximated with "stair- case" appro ximation. Only in o-dimensional sim ula- tions, and with considerable eort, it has recen tly ecome ossible to obtain some remedy of the

errors arising from this appro ximation [18 ]. The geometrical exibilit of the nite elemen metho has een exploited to erform sim ulations of materials with complicated, realistic grain structures [19 20 and to adaptiv ely rene sp ecic parts of the computational region, wherev er higher discretiza- tion densit is required [21 ]. Numerically the minimization of the total energy (1) is erformed with conjugate gradien metho [22 ]. The constrain of constan magnitude of the magnetization const needs to observ ed during the minimiza- tion pro cess, whic can ac hiev ed

easily represen ting the magnetization ector with spherical co ordinates It is notew orth that within eac elemen linear in ter- FIG. 2: Finite elemen mesh used for the sim ulations. The edges of the elemen ts are represen ted as wireframe. The mesh consists of 15 000 no des placed on regular grid, whic are the corner oin ts of 58 212 tetrahedral elemen ts. Inside the elemen ts the magnetization is in terp olated linearly olation of the magnetization is used. The in terp olation is erformed using so-called shap functions, whic al- lo to determine alue of discretized function inside eac

nite elemen means of in terp olation of the al- ues on the discretization oin ts, whic are just the corner oin ts of the elemen ts in this case. Compared with the zero-order appro ximations used in algorithms based on nite dierences and the ev aluation of the stra eld means of ast ourier ransformation (FFT) [23 ], where the magnetization is assumed to piecewise homo ge- ne ous this rst-order in terp olation sc heme with piece- wise line ar appro ximation is more precise concerning the calculation of the stra eld term. By using linear in- terp

olation sc heme, inhomogeneous magnetization pat- tern do not erform sudden, unph ysical discon tin uities at the oundaries of the discretization cells, con trary to zero-order sc hemes, where this eect can lead to ctitious uncomp ensated harges, whic ma sources of errors in the calculation of the stra eld term. The dra wbac of the nite elemen metho is that the implemen tation is quite complicated and the memory requiremen ts are higher than those needed in nite-dierence sc hemes. Ha ving the discretization sc heme with the corresp ond- ing in

terp olation functions, the ev aluation of the micro- magnetic energy terms is generally straigh tforw ard. The only term whic requires closer lo ok is the long-range term resulting from the stra eld. calculate the stra eld, scalar oten tial is in tro duced, from whic is deriv ed as gradien eld The oten tial satises oisson’s equation (2) with the oundary conditions (3)

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(4) and lim !1 (5) where and are the in ard and the out ard limit of the oten tial at the surface and is the normal ector directed out of the sample. solv oisson’s equation with

prop er considera- tion of the oundary conditions, the oundary lemen etho (BEM) is com bined with the inite lemen etho (FEM). Highly accurate calculations can ob- tained means of this ybrid FEM/BEM sc heme, whic as rst applied to micromagnetic problems redkin and Ko ehler[24 ]. The ansatz used to solv Eq. (2) is to split the oten tial in to parts One part of the oten tial, is zero outside the magnetic particle and satises oisson’s equation inside the sam- ple. The Neumann oundary conditions (4) are natural conditions in the solution of The other part, is solution of Laplace’s

equation. The Diric hlet oundary conditions for whic uniquely determine the solution, are obtained from the alues of at the particle’s sur- face means of oundary in tegral[24 ( (6) where the in tegral is extended er the surface and ( is the solid angle subtended at the surface oin Numerically the elliptical dieren tial equations are solv ed with the Galerkin metho d, means of whic the discretized problem is transformed in to (large) set of linear equations that can solv ed with the biconju- gate gradien metho d. After the stra eld of magnetic structure is

calculated, the minim um energy arrangemen of the magnetization is determined. The pro cess of calcu- lating the magnetic structure and its stra eld is itera- tiv ely rep eated un til self-consisten solution is obtained. IV. MA GNETIC STR UCTURES IN PERMALLO RECT ANGLES Soft magnetic thin lm elemen ts are magnetized in- plane ecause of the predominan inuence of the stra eld energy ("shap anisotrop y"). The tendency to arrange the in-plane magnetization in to form magnetic domains is due to the nite lateral extension. Surface harges at the oundary of the

sample can oided aligning the magnetization parallel to the edges, as is the case in ux-closure patterns. Ob viously the driving force to form magnetic domains diminishes 0.2 0.4 0.6 0.8 c/a 0.05 0.1 0.15 0.2 0.25 stray [K V] 0.01 0.02 0.03 0.04 0.01 0.02 0.03 0.04 0.05 FIG. 3: Av erage stra eld energy densit of rectangular par- ticles with asp ect ratio =2:1, homogeneously magnetized along the long edge as function of the relativ thic kness c=a The alues are calculated using the equations due to Aharoni. The energy densit stra =V is giv en in units of the stra eld

constan 2. with the particle’s thic kness, ecause the amoun of sur- face harges arising from homogeneous in-plane mag- netization dep ends on the area of the surface to whic the magnetization is erp endicularly orien ted. Analytic expressions on the stra eld energy of homogeneously magnetized prisms ha een pro vided Aharoni[25 ]. Quan titativ ely the thic kness dep endence of the erage energy densit of homogeneously magnetized slab is sho wn in the diagram of Fig. 3. As can seen in the inset, esp ecially in the thin-lm regime the increase in energy with thic kness is

particularly steep, and almost linear. The energies resulting from the analytical calculations represen upp er ounds for real rectangles with single- domain in-plane magnetization. If rectangular thin lm elemen is magnetized along the long edge, surface harges (and stra eld energy) are reduced the for- mation of end domains, whic lead to the aforemen tioned C- and S-states. Still, these structures ha consider- able stra eld energy whic can reduced means of ux-closure arrangemen ts, ho ev er at the exp ense of exc hange energy due to the domain alls. Con trary to

the stra eld energy densit of homogeneously magne- tized thin lm elemen t, the energy densit of an idealized ulti-domain structure with ux closure do es not dep end on the thic kness, pro vided that the magnetization do es not ary along the thic kness. This thic kness dep endence of the stra eld energy ev en tually leads to transition from single-domain state to ulti-domain state. giv an erview of the dep endence on size and thic kness of magnetic structures in thin ermallo rect- angles, the energy of arious magnetization states is calculated in dieren

platelets. The particles ha edge lengths ranging et een 250 nm 125 nm and 1000 nm 500 nm, whic is in the order of the tec hnologi-

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FIG. 4: The ma jor domains of Landau-t yp structure can separated cross-tie all. This all yp is often found in soft magnetic thin lms of in termediate thic kness (10-50 nm) cally relev an size. The thic kness of the rectangles is ar- ied et een nm and 32 nm. Since the asp ect ratio of the rectangles is equal to 2:1 in all cases, the long edge and the thic kness are sucien to sp ecify the sample’s dimensions. In addition to the

magnetic pat- terns sho wn in Fig. cross-tie all structure as sho wn in Fig. is tak en in to consideration. The resulting energies are sho wn in Fig. 5, as function of the thic kness for four rectangles of dieren size. Gen- erally the erage energy densit of the high-remanence and -states increases with the thic kness, in accor- dance with the eha vior exp ected from analytic calcula- tions, cf. Fig. 3. Although not recognizable from Fig. 5, in all cases the -state has lo er energy than the state. The reason for this is probably that the surface harges of opp osite sign, whic attract

eac other, are closer together in the -state than in the -state. In the range of size and thic kness considered here, the cross-tie structure is nev er lo est-energy arrangemen t. or ery thin and small platelets, the cross-tie structure is umerically unstable and con erts in to the Landau- structure. With increasing size, ob viously the tendency to form ux-closure domain patterns increases with the size (b oth thic kness and edge length), so that the v- erage energy densit of Landau, Diamond and cross-tie structures decreases with the edge length. Although the energy of the

dieren patterns can dif- fer signican tly with the size, the magnetic structures are largely indep enden on size and thic kness. As sho wn in Fig. 6, the patterns are mostly in arian with resp ect to scaling. In the -state, ho ev er, the domain alls ecome somewhat sharp er and the sub division in to do- mains clearer as the particle’s size increases. It is dicult task to iden tify structure as mag- netic ground state of particle, ecause it can not ruled out strictly that some structure whic has not een tak en in to consideration has lo er total energy than the lo

est-energy arrangemen of the calculated structures. This situation ccurred in Standard Prob- lem No. [26 27 in whic small ferromagnetic cub as considered, and the transition et een magneti- zation states (Flo er State and ortex State) as calcu- lated. This as supp osed to size-dep enden transi- tion et een magnetic ground states. Later, Hertel and Kronm uller ha sho wn that third state, whic as named Twisted Flo er State, has lo er energy at the relev an edge lengths [28 ]. In small thin lm ele- 10 15 20 25 30 thickness [nm] 0.00 0.02 0.04 0.06 0.08 energy density [K V] C-state S-state

cross-tie Landau Diamond 250 nm x 125 nm C-state Landau 10 20 30 thickness [nm] 0.00 0.01 0.02 0.03 0.04 0.05 energy density [K V] C-state S-state cross-tie Landau Diamond 500 nm x 250 nm Landau Diamond C-state 10 15 20 25 30 thickness [nm] 0.000 0.005 0.010 0.015 0.020 0.025 0.030 energy density [K V] C-state S-state cross-tie Landau Diamond 750 nm x 375 nm Landau Diamond C-state 10 15 20 25 30 thickness [nm] 0.000 0.005 0.010 0.015 0.020 0.025 energy density [K V] C-state S-state cross-tie Landau Diamond 1000 nm x 500 nm Landau Diamond C-state FIG. 5: Av erage energy densities of

dieren magnetization states in rectangular platelets as function of the thic kness. The lo est-energy arrangemen is mark ed in the corresp ond- ing regions.

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FIG. 6: ersp ectiv view on platelets of dieren size, ho ev er with same asp ect ratio. On the left side, the platelets are in the diamond state, on the righ the magnetization is in the C-state. The ma jor edge lengths are 1000 nm, 500 nm, and 250 nm, the thic knesses are 30, 15, and 7.5 nm, resp ectiv ely Note that the particles are only graphically placed next to eac other. The micromagnetic calculations

are erformed for isolated, non-in teracting platelets. 400 600 800 1000 major edge length [nm] 10 15 20 25 thickness [nm] Landau Diamond C-state FIG. 7: Phase diagram of the lo est-energy arrangemen in ermallo rectangles of asp ect ratio 2:1. men ts, where the magnetization structure is essen tially o-dimensional, the manifoldness of ossible magnetic structures is probably smaller, so that the set of magneti- zation states presen tly considered is lik ely to sucien t. The symmetric in-plane Flo er state has not een con- sidered (in the nomenclature of and -state, this one could

referred to as -state"). Though this struc- ture migh ccur in ery small platelets, it is unstable in larger rectangles and there it has higher energy than or -state due to the inecien reduction of surface harges. Also not considered is double cross-tie all structure, whic could ecome stable in thic and large platelets but is unstable at smaller sizes. Moreo er, not ev en the single cross-tie pattern is an energetically con- enien arrangemen in the range of size considered here, and o-fold cross-tie structure requires larger lateral extensions than single one to sustain the cross-tie

alls. Based on the data as presen ted in Fig. phase di- agram according to Fig. can set up. According to this, single-domain regime, where the high-remanence structures C-state and S-state ha the lo est energy can clearly iden tied dep ending on size and thic kness. or rectangular platelet with dimensions corresp onding to this regime, ux-closure patterns ha higher energy than the quasi-homogeneous states. Suc particles can therefore exp ected to more unlik ely to drop ac- ciden tally in to tec hnologically undesirable ux-closure domain state. [1] L. Zh u, Y. Zheng, and

G. Prinz, J. Appl. Ph ys. 87 6668 (2000). [2] K. J. Kirk, J. N. Chapman, and C. D. W. Wilkinson, J. Appl. Ph ys. 85 5237 (1999). [3] R. D. Gomez, T. Luu, A. ak, I. Ma ergo yz, K. Kirk, and J. Chapman, J. Appl. Ph ys. 85 4598 (1999). [4] A. Hub ert and R. Sc h afer, Magnetic Domains The nalysis of Magnetic Micr ostructur es (Berlin, New ork, Heidelb erg: Springer, 1998). [5] H. Kronm uller and R. Hertel, J. Magn. Magn. Mater. 215-216 11 (2000). [6] W. Ra and A. Hub ert, IEEE rans. Magn. 36 3886 (2000). [7] H. A. M. an Den Berg and A. H. J. an Den Brandt, J. Appl. Ph ys. 62 1952 (1987). [8] W.

F. Bro wn, Jr., Magnetostatic Principles in err omag- netism (North-Holland publishing compan Amsterdam, 1962). [9] R. Hertel and H. Kronm uller, J. Appl. Ph ys. 85 6190 (1999). [10] Y. Zheng and J. Zh u, J. Appl. Ph ys. 81 4336 (1997). [11] R. Hertel and H. Kronm uller, Ph ysica 275 (2000). [12] J. Gadb ois, J. G. Zh u, W. vra, and A. Hurst, IEEE rans. Magn. 34 1066 (1998). [13] T. Sc hre, J. Fidler, K. J. Kirk, and J. N. Chapman, J. Magn. Magn. Mater. 175 193 (1997). [14] R. H. Ko h, D. J. G., W. Abraham, rouilloud, R. Alt- man, Y. Lu, W. J. Gallagher, S. R.E., and S. arkin, Ph ys.

Rev. Lett. 81 4512 (1998). [15] H. Kronm uller, Zeitsc hrift ur Ph ysik 168 478 (1962). [16] H. Kronm uller and M. Lam ec k, in ehrbuch der Ex- erimentalphysik edited W. de Gruyter (Bergmann, Sc haefer, Berlin, New ork, 1992), ol. 6, pp. 715{791. [17] R. Hertel and H. Kronm uller, Ph ys. Rev. 60 7366 (1999). [18] Z. Gim butas, C. Garcia-Cerv era, and E. einan, submit- ted to Journal of. Comp. Ph ys. (2001). [19] T. Sc hre, H. Kronm uller, and J. Fidler, J. Magn. Magn. Mater. 127 L273 (1993). [20] R. Fisc her and H. Kronm uller, Ph ys.Rev.B 54 7284 (1996). [21] R. Hertel and H. Kronm

uller, IEEE rans. Magn. 34 3922 (1998). [22] W. H. Press, B. Flannery S. A. euk olsky and W. T. etterling, Numeric al cipies: The art of scientic om- puting (Cam brige Univ erist Press, Cam brige, New ork,

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1986). [23] D. Berk v, K. Ramst oc k, and A. Hub ert, Ph ys. Stat. Sol. (a) 137 207 (1993). [24] D. R. redkin and T. R. Ko ehler, IEEE rans. Magn. 26 415 (1990). [25] A. Aharoni, J. Appl. Ph ys. 83 3432 (1998). [26] W. Ra e, K. abian, and A. Hub ert, J. Magn. Magn. Mater. 190 332 (1998). [27] R. D. McMic hael, Standar pr oblem numb er 3, Pr oblem sp cic ation

and ep orte solutions (Micromagnetic Mo d- eling Activit Group, http://www.ctcms.nist.gov/ rd m/mum ag.h tml 1998). [28] R. Hertel and H. Kronm uller, J. Magn. Magn. Mater. 238 185 (2002). [29] A.S. Arrott, priv ate comm unication (2001)

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