1518 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 5, SEPTEMBER 1990

NUMERICAL MICRORt AGNETICS: RECTANGIJLAR PARALLELOPIPEDS D. R. Fredkin*

Department of Physics and

Center for Magnetic Recording Research Ilniversity of California, San Diego

La Jolla, CA 92093 and

T. R. Kochler IRM Rescarch Division, Almaden Research Center

650 IIarry lioad, San Jose, CA 95120

ABSTRACT Results of simulations of magnetization reversal in rectangular

permalloy particles with aspect ratios 10 x 5 x 1 are given. Mag- netization patterns during nucleation of reversal, in the remanent state and after the applied fiild is reversed are shown in detail for the case of an applied field along the long axis of the particle. The nucleation and remanent state bchaviors are also illustrated in lesser detail for other applied ficld directions. The remanent state is the classical four domain pattern, although the seven domain pattern also appears. The longest wall in the four domain pattern is a vortex wall.

INTRODUCTION We have applied our numcrical micromagnetics program' to

the study of magnctization reversal in a variety of permalloy rec- tangular parallelopipeds of various sizes and shapes. Typically, the change of magnetization pattern as size was varied for'a particular aspect ratio was observed over the range of sizes for which the particles support at most a small number of magnetic domains. Thus, the lengths ranged from about 0.5 to 10 prn. Such particles can be made by lithographic techniques and are currently under experimental invcqtigation2.

The reversal process was studied by recording thc magnetiza- tion patterns during the course of numerical experiments which produced hysteresis loops for a magnetic field applied to the parti- cles. We emphasize that no hints about possible domain structure were given to ttic program. In contrast with the quite uniform patterns previously' observed at the start of the loop for spheres and prolate spheroids, there is a xonounccd dispersion of the magnetization at thc corners and edges due to the demagnetizing fields even a t ratker high applied field. As the applied field is low- ered, the dispersion increases while most of the sample remains uniformly magnctized until, depending on the size of the particle, one of two things happens. For small enough particles, there is a sudden reversal of the magneti7ation in the bulk -of the particle. This always occur:: at a reversed applied field, so there is a sub- stantial remanence for thcse particles. For larger particles closure patterns nucleate, suddenly or gradually, near opposite ends of the particle.

Figure l.(a) seven and (b) four domain patterns.

Preliminary results for two permalloy plate-like particles with aspect ratios of IO x 5 x 1 wcre given in Ref. 1. These were of spe- cial interest becaute the remanent states of the two particles wcrc quite different: that for the smaller particle (length = 1.37pm) was the seven domain pattern shown schematically in Fig. l a and that for the larger particle (length = 2.17pm) was the four domain pat- tern shown in Fig. Ib. In addition, the 180" wall in the four domain pattern was found to be a vortcx wd13-s when viewed at the cross section indicated by the dotted line in Fig. Ib. Later detailed sim- ulations for particles of this aspect ratio substantiate that remanent states of each type occur, but indicate that applied ficld direction, applied field step size and meshing details are the mo!;t important factors in determining the remanent state configuration.

The four domain pattern is more prevalent, and its nucleation and the behavior of the vortex wall in the case of an applied field along the long axis of the particle will be shown in detail. Nucleation detail!: and remanent state patterns will also be shown for other applied &:Id dircctions.

* Supportcd in p i t by a grant frorn the National Scicncc Foundation

REStJLTS Results for the simulation of reversal in rectangular permalloy

particles with aslioct ratios 10 x 5 x 1 and varying lengths 8 will be presented in this section. The coordinate system used to indicate the direction of thi: applied field M will have the x-axis along the long axis of the ixrrticle, the y-axis along the short axis and the z-axis perpendicular to the plane of the article.

Although thc remancnt statc and t!e details of the nucleation process depend on the particular case and are most influenced by the direction of 11, there were ccrtain common features of the rc- versa1 processes. The formation of the domain structure can be considered to evolve from the bending of the magnetization at the corners away frorn the applied field direction. The bending is al- ready present at high Gclds and transforms easily into patterns which are precurc:ors to vortices in some of the corners. For parti- cles of this size, the domain wall thickness is a significant fraction of the particle size. Once formcd, the walls are stable and move through the particlc in response to changes in the applicd field. The domain wall nucleation and motion will be discussed in the follow- ing subsections.

(a) 13 = 2000 Oe

( 1 ) ) Ji = I000 Oe

(c) I1 = 700 Oe Figure 2. Gradual pattcrn dcvelopmcnt away from high field for

H 11 ( I , 0,O) (a) Mostly uniform with splaying at corners at high ficld. (h) Prcparation for vortex formation at lowcr coriiers. (c) Vorticcs almost formcd at lower cor- ncrs and upper Iclt.

0018-946419010900-1518$01.00 0 1990 IEEE

1519

(a) If = 400 Oe

. - cb) iI= 300 Oe

(c ) I1 = 100 Oe

T

(d) II= 0 Oe

(e) I1 = -300 Oe

Figure 3. Nucleation and subsequent behavior of the four domain pattern. (a) Two vortices formed near opposite, lower corners. (b) Both vortices have moved upwards. (c) The upward motion continucs and a new vortex accompanying the reversal of the lower middle area appears. (d The classical lour domain pattern. (e) The 180" wab has moved through the particlc and is near to leaving.

Four Domain Pattern, t' = 2. I7pm, H 11 ( I , 0,O) The first case we shall consider is that with the field applied

along the long axis of a particle orlcngth 2.17pm. The gradual dc- velopment of the patterns away from high field is shown in Fig. 2. The scheme used in illustrations of magnetization patterns is dc- scribed in Ref. 2 and Rererenccs contained therein. The high field outward bending is shown in Fig. 2a and the developing tendency toward vortex Formation can be seen in Fig. 2b. Vortices have not quite formed in Pig. 2c: none of the magnetization vectors has a component directed opposite to the original direction of N, which is to the right.

With further reduction in thc applied field, vortices form a t the corners, the corner vortices move and eventually combine to form domain walls, which are quite stable once formed. This proccss is illustrated in Fig. 3. The early two vortex patterns in Figs. 3aJb develop directly from the pattern of Fig. 2c. The four domain at tern evolves from these in a rather straightrorward way, except txac Tor this pattern to appear, the middle central region of Fig. 3b has t o reverse. The appearance of the vortex a t the lower edge in Fig. 3c provides a possible mechanism for doing so. It is easy to imaginc that downward motion of this vortex produces the classical four domain pattern wen in Fig. 3d, but we havc no proof of this. Fur- ther reversal is accompanied by the smooth motion of the wall through the particle. The wall is still well defined when it is close t o exiting, as seen in Fig. 3e.

Vortex Wall, The 180" wall sccn in Figs. 3d, 3e docs not appear to be a

Bloch or Necl wall. A cross section view confirms that it is a vortex wall. Its nucleation and dcvelopment are shown in Fig. 4 at the same applied field values used in Fig. 3. The vorticity at the upper edge of Fig. 3a is brought out clearly in Fig 4a. The sltilting, com- plex central pattcl-ns observablc in Figs. 3b, 3c are rcflected in pat- terns of Figs. 4b, &:, A symmetrical, well defined vortex wall is seen in the picture of Ihc remanent state shown in Fig. 4d. The vortex is somewhat elliptical, probably because the particle is thin. The vortex remains wcll defined as it moves out, as can he sccn in Fig. 4e.

= 2. I7pm, H (1 (1 .0 ,O)

(a) II = 400 Oe

(b) H = 300 Oe

(c) If = 100 Oe

(d) if = 0 Oe

( e ) N = -300 Oe

Figure 4. Nucleation and subscqucnt bchavior of the vortex wall. The upper region in Fig. 3 is toward the right. (a)-(c) Vortex like and more complex behavior associated with complcx looking rcgions in Fig. 3. (d) The well defined vortex wall in the remanent state. (e) The vortex wall near to exiting.

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Four Domain Pattern, t' = 0.437pm, H 11 (0, I , 0) The nuclcal.ion of thc four domain pattern with the field ap-

plied along the short axis of a particle of length 0.437pm is illus- tratcd in Fig. 5. !;incc the applicd ficld and preferred niagneti7,ation dircction arc in cornpctition herc, thcrc is more deviation from uni- form magneti7ation than in thc previous example and thc vortex nucleates at higher applied field. Thc magnetization patterns at thc field steps immec5atcly hcfore and altcr nucleation are shown in Figs. 5a, Sb rcspcxtively. This was the only case wherc only a sin- gle vortex nuclcatctl. Although there is not enough spacc to show it, the 180" (vortex) wall in the final four domain statc grows from the single vortex 01. Fig. Sh rnuch in the manner of an arnocha cx- tending a pseudopod. Seven Domain Pntterii, f = 1.37pm, tl 11 ( I , I , 0)

The nucleation of the seven domain pattern witli transverse, in plane ficld applied to a particlc of Icngth 1.37pm is illustrated il- lustrated in Fig. 6. I?erc, as in the previous example, the compcti- tion betwecn thc casy axis and the applied field dircction has a considerable influtiice on the pattern at high field. In Pig. 6a, the magnetization in the ccntral region of the particle is rather uniform and balanced hctwccn the easy and applied field directions. Ilow- cver, thc applied Geld enhances the dcmagnctizing field at the cor- ners. In contrast to 1;igs. 2a, 2h, thc magnetization along the short edges in I3g. 6a i r nearly all in same dircction and requires only a small rotation to he aligned appropriately for the scvcn domain pattern. Thi? rot;ition l in t leads to thc just Formed vortices shown in Fig. 6b, which m o w toward thc long axis, rcvcrsing the magnct- ization in their \wkc and ultimately producing the scvcn domain pattern sccn in Fig 6c. A distortctl, hut well defined, vcrsion of the pattcrn movcs tiirorigh thc particlc for 500 O e z / e -500 Oc.

(a) II = 900 OP

(h) / I = 800 Oe

Figure 5. Nucleation of vnrtcx at midpoint in cdgc of particle for H /I (0, I , 0) (a) Just hcfore nucleation. (h) Just after nucleation

DISC( ISSUON We have gi1ic.n a dctailed cxposition of selected results for

simulations of magnetic revcrsal in rectangular permalloy particlcs with aspcct ratios. I O x 5 x 1 Wic results support a consistcnt pic- ture of revcrsal doniinated by thc nucleation and subsequcnt mo- tion of vortices. What was shown here is consistcnt with the results of many more simulations of the same sysl.em except as noted i n the next paragraph. in general, variation in particle size did not aKect the remanent state until the particles became too small to support any domain struclure.

The mesh uscd in all of thc runs discussed here had fairly uni- formly sized elcmcnts with good aspect ratios. An earlier mesh, produccd by fillirig the sample with similarly meshcd hexahcdra, introduced a bias such that the short cdgcs of I?g. 2b hegan to rc- semble those of rig. 6a. Although the bias was slight, this always

(a) N = 1200 Oe

(b) I f = 600 Oe

(c) 1.1 = 0 Oe

Figure 6. Nuclcalion of scvcn domain pattcrri for 14 I( ( I , 1 ,O) pat- tern. (a ) Anglc of applied ficld has oriented magnctiza- tion at short edgcs of particlc favorably for lormation of thc scvm domain pattcrn. (bj Vortices have formcd consistcnt with pattern of (a). (c) Final scvcn domain pattern is topologically conristent with (b).

led to the sevcn domain pattern for all particle sizcs with a ficld applied along thc long axis, an indication of a delicatc encrgy bal- ance between the two patterns. Another indication was found in one run for El 11 (1,0,0) and another for €I 11 (l,l,0) in which the seven and four domain pattern respectivcly appeared. These runs were with an applied field step s i x of IO0 OE When thc step sirc was reduced to 5 OP, thc usual result was obtained.

A further iswc is the ultimate comparison of simulations with relevant expcrimcnts. In the simulations reported here, the particlcs are perfectly rcctangular, stress free and of uniform composition, and the applied field is perfectly uniform and aligned with a choscn direction. This would not bc thc case for an cxpcrimental particle. Furthermore, imperfections in a experimental particle in this size range would be diflicult to diagnose We plan to study the erect of plausible dcviations fiom pcrfcction, such as random pinning sites, in the futurc. Ilowever, some of thc numerical erects men- tioned in thc prcvicius paragraph probably reflcct impcrfcctions and fluctuations found in nature.

REFER EN C ES 1. D. R. Fredkin arid 'I'. R. Kochler,"Ab Initio Micromagnctic Cal-

culations for Par1 icles", presented at 34th M M M Conference and to appear in J. Appl. Phys., May, 1990.

2 We wish to acknowledgc that thc motivation for rnuch of thc work discusscd here was provided by private communication of experimental results on similar systcms by I . Chapman, I . McFaydcn and S. McVitic and by S. Schultz and 1 . Smyth. A comparison of ou r rcsults with experimcnt will be made elsc- where.

3 A. E. La Igontr, .I. Appl. Phys. 44. 2450 (1969). 4 A. frubert, Phiis. Status Solidi 32, 519 (1969). S A. Aharoni, .I. Appl. Phys. 46, 1783 (1975).