Control of magnetic vortex chirality in square ring micromagnets

JOURNAL OF APPLIED PHYSICS 98, 083904 �2005�

Control of magnetic vortex chirality in square ring micromagnetsA. Libála�

Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 and Materials ScienceDivision, Argonne National Laboratory, Argonne, Illinois 60439

M. GrimsditchMaterials Science Division, Argonne National Laboratory, Argonne, Illinois 60439

V. MetlushkoDepartment of Electrical and Computer Engineering, University of Illinois at Chicago,Chicago, Illinois 60607

P. VavassoriIMFM-National Center S3 and Dipartimento di Fisica, Universita di Ferrara, Italy

B. JankóDepartment of Physics, University of Notre Dame, Notre Dame, Indiana 46556

�Received 4 May 2005; accepted 16 September 2005; published online 24 October 2005�

We investigate the effect of a deliberately introduced shape asymmetry on magnetization reversal insmall, square-shaped, magnetic rings. The magnetization reversal process is investigated using thediffracted magneto-optical Kerr effect combined with micromagnetic simulations. Experimentallywe find that the reversal path is sensitive to small �±1° � changes in the direction of the applied field.Micromagnetic simulations that reproduce the measured zeroth- and first-order loops allow us toidentify the reversal mechanisms as due to different intermediate states, namely, the so-called vortexand horseshoe states. Based on our results we are now able to prescribe a methodology for writinga vortex state with specific chirality in these asymmetric rings. Such control will be necessary ifpatterned arrays of this kind are to be used as magnetic storage elements. © 2005 American Instituteof Physics. �DOI: 10.1063/1.2113407�

I. INTRODUCTION

One of the challenges facing magnetic data storage tech-nology is how to reduce the size of the individual data stor-age elements. Replicating Moore’s law for magnetic storagehas so far been successful—magnetic storage capacity isdoubling every two years. There are, however, physical lim-its to how far the technology can ultimately go. Below somecritical size individual grains or particles will reach the su-perparamagnetic limit and will no longer store information.Based on some new technologies that are already beingimplemented to keep pace with development �such as heatassisted magnetic recording—HAMR�, it seems clear that weare fast approaching the limit of what is possible with con-ventional magnetic storage and that new approaches will benecessary to achieve the demand for further miniaturization.

One of the ideas that has been proposed to extend thelimits of magnetic storage is to use ring-shaped particles in avortex state.1,2 Such particles produce much lower dipolarfringe fields that greatly reduce the interaction with theirneighbors, and consequently make the stored informationmore stable. The vortex state is also intrinsically quite stableespecially if formed in a ringlike structure that eliminates thevortex core.3 If such an application were to evolve it wouldrequire the information to be stored in the chirality of thevortex. In this context understanding magnetic domainformation,4,5 and controlling the vortex chirality are impor-

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tant issues. From a more fundamental viewpoint understand-ing the driving forces that lead to vortex nucleation, annihi-lation, and determine its chirality are still not fullyunderstood.

It is known that particle shape is a key factor determin-ing the vortex chirality.3,6 In asymmetric disks it was foundthat, depending on field direction, vortices with oppositechirality are nucleated.7 In arrays of nominally circular disksand square rings the coherent chirality in the arrays was at-tributed to subtle shape deviations produced duringfabrication.6,8 In the latter investigation we found that themeasured diffracted magneto-optical Kerr effect �D-MOKE�loops were not reproduced by micromagnetic simulations of

FIG. 1. A scanning electron microscope �SEM� image of the fabricatedpermalloy open-slit square ring lattice. The size of the rings is 1.1�
�1.1�, the rings are 2� apart and the width of each segment is 200 nm.
© 2005 American Institute of Physics4-1

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http://dx.doi.org/10.1063/1.2113407

http://dx.doi.org/10.1063/1.2113407

083904-2 Libál et al. J. Appl. Phys. 98, 083904 �2005�

a perfect square ring.6 In order to achieve agreement wefound it necessary to introduce an asymmetry in the simu-lated rings. The introduction of this asymmetry led to excel-lent agreement with experiment and enabled the identifica-tion of two unexpected metastable states appearing duringthe reversal process. Although the agreement between ex-periment and simulation makes it unlikely, it can always beargued that some other effect, e.g., anisotropy, field direction,magnetization history, convergence criteria, etc., could alsoexplain the experimental results. To address this issue herewe have investigated magnetization reversal in square ringsin which a well-defined anisotropy is present.

State of the art electron-beam lithography techniques en-able fabrication of nanoscale-size magnetic particles of anydesired shape.9 With this shape control we fabricated ringswith a small �50 nm wide� gap at one of the corners. Ascanning electron microscope �SEM� image of the sample isshown in Fig. 1. We studied magnetization reversal in thissample using D-MOKE combined with micromagnetic simu-lations. We find that the simulations provide an excellentdescription of the experimental results and even account forthe dramatic changes in the hysteresis loops measured forsmall, 1°, changes in the direction of the applied fields. Fromthe comparison of experiment and simulations we are able toidentify the intermediate states during reversal and thus toprescribe how to write a vortex, with a specific chirality, intothe magnetic ring.

II. D-MOKE RESULTS

The rings were fabricated with electron-beam lithogra-phy and lift-off techniques.10,11 The individual permalloy

FIG. 2. The angular notation used in this article. If we rotate the ring in theclockwise direction we get to the successive positions of the ring. The “�”and “�” labels respect this notation, the � means that the ring’s position isrotated slightly more in the clockwise direction.


rings are ordered on a square lattice, with a lattice constant of2.0�. The outside size of the rings is 1.1��1.1� and thewidth of each segment is 200 nm. The film thickness is25 nm. The most important feature of these rings is the de-liberately introduced shape asymmetry in a form of a small50 nm slit at one of the corners �see Fig. 1�. The D-MOKEsetup used to investigate the rings is described in.6,12,13 Itrelies on the transverse-MOKE geometry with a p-polarized,532 nm laser-beam incident on the array and the field appliedperpendicular to the plane of incidence.

The D-MOKE hysteresis loops are the field induced in-tensity changes in the zeroth �reflected� and higher-order dif-fracted beams. We define the in-plane orientation of thesample by the angle that one of the edges subtends with theapplied field �see Fig. 2�. For fields applied along an edge wefind that the loops depend very critically on the exact direc-tion of the field. In Figs. 3 and 4 we show the zeroth- andfirst-order loops measured for two angles straddling the fieldalong the edges, viz., 0±, 90±, 180±, and 270± where the “�”means a small tilt of about 1°. The dramatic changes in thefirst-order loops are an indication that the reversal path isvery different for these small changes in the direction of thefield.

For fields applied along the diagonals of the square wefind that the loops are insensitive to small changes in direc-tion. In Fig. 5 we show the loops measured at 45°, 135°,225°, and 315°.

III. THEORY AND MICROMAGNETIC SIMULATIONS

The D-MOKE loops in the nth diffracted order havebeen shown to depend on the magnetic form factor �fn

m� de-fined by

fnm =� � my exp�inG · r�dxdy , �1�

where G is the reciprocal lattice vector in the plane of inci-dence associated with the square mesh of the ring array, r is

FIG. 3. Measured D-MOKE hysteresiscurves both for zeroth- and first-orderdiffracted beams for two orientations�0 and 90� of the slit.



the position vector on the ring, and my is the y component ofthe magnetization �i.e., the component perpendicular to theplane of incidence�, and we integrate over the surface of asingle ring �we integrate over one primitive cell of the ringlattice�. In previous papers,13 it was shown that the hysteresisloops could be obtained from the form factors using the fol-lowing formula:

I � Re�fnm� + An Im�fn

m� , �2�

where An is a number that depends on the nonmagnetic formfactor and the Fresnel coefficients. In the present case, wherethe nonmagnetic form factor �Eq. �1� with my =1� is alsocomplex, �i.e., the rings do not have a center of symmetry�An is orientation and diffraction-order dependent. From apractical standpoint, however, since the small gap in the ringleads only to small imaginary components of the nonmag-netic form factor, we found that we could use the same valueof An independent of the location of the gap. Our spectra


were fit with An=−1.3 when the field is along and edge andAn=−0.5 for fields along the diagonal.

From Eqs. �1� and �2� it is easy to see that the zeroth-order hysteresis loop is proportional to the average magneti-zation �thus the 0th order loops are equivalent to the hyster-esis loops measured by conventional techniques such asvibrating-sample magnetometer �VSM� or superconductingquantum interference device �SQUID� measurements�. Thefirst-order loop contains information about the first Fouriercomponent of the magnetization. We can say that while thezeroth-order hysteresis loop provides the resultant magneti-zation within the coherence length of the laser beam, thefirst-order loop is sensitive to the differences in the magne-tization within the four segments �and thus provides addi-tional information on a smaller scale related to the magneti-zation distribution�.

In order to calculate the magnetic form factor in Eq. �1�it is necessary to know the magnetization distribution withinthe ring. As described in previous work, this can be obtained

FIG. 4. Measured D-MOKE hysteresiscurves both for zeroth- and first-orderdiffracted beams for the other two ori-entations �180 and 270� of the slit.

FIG. 5. Measured D-MOKE hysteresiscurves for fields along the diagonal.



from micromagnetic simulations. Here, micromagnetic simu-lations were performed with the object-oriented micromag-netic framework,14 �OOMMF� package from the National In-stitute of Standards and Technology �NIST�. The simulationswere performed on a single ring with the same dimensions asthe experiment �1.1��1.1�, 200 nm branch widths, and25 nm sample thickness�. The ring was divided into 10�10 nm,2 cells yielding a total of 7100 simulation cells. Toconfirm that the results do not depend on the size of themicromagnetic cell used in the simulations, for one case wehave performed the simulations with both 5 and 10 nm cells.We find that the overall shape of the loops are unchanged ongoing to the smaller cell, the only differences are the exactvalues of the nucleation and annihilation fields. The materialparameters used in the simulation for the permalloy were thesaturation magnetization Ms=860�103 A/m and the ex-change constant 13�1012 J /m. The experimental values ofthe field were used in the simulation. From the magnetizationconfiguration at each field the form factors were calculated


by using Eq. �1� and the hysteresis loops via Eq. �2�. Theangular notation used for the simulation is shown in Fig. 2.

The calculated loops for the field along the edges and thediagonals are shown in Figs. 6–8, respectively. A comparisonof Figs. 3 and 6 and of Figs. 4 and 7 shows that, qualita-tively, the simulated loops reproduce the measured ones. Westress that once the value of An has been chosen �−1.3 and−0.5 for fields along the edges and diagonals, respectively�,there is no other adjustable parameter. In particular, the cal-culations reproduce the sign of the spikes in all first-orderloops. Since the spikes in the first-order loops can be tracedto a large imaginary part of the magnetic form factor, andsince the sign of the imaginary part is determined by the“chirality” of the left and right branches of the ring, theabove agreement indicates that the simulations correctly pre-dict the chirality of the intermediate state for all orientations.Less good is the agreement between the exact fields at which

FIG. 6. Numerical simulation resultsfor the hysteresis curves for the paral-lel cases �0 and 90�.

FIG. 7. Numerical simulation resultsfor the hysteresis curves for the paral-lel cases �180 and 270�.



the spikes occur. This we ascribe to other more subtle factorsthat affect the nucleation fields for each step of the reversalprocess.

An interesting feature is the difference in the width ofthe spikes observed for the � and � loops at 0 and 180° inFigs. 3 and 4 and that this effect is reproduced in the calcu-lated loops in Figs. 6 and 7. Since we know that the peaks inthe first-order loops appear when the magnetization in theleft and right branches of the ring point in opposite direc-tions, a wider peak means that there is a larger interval ofapplied field where the magnetization in the two branchespoint in opposite directions. The origin of this can be ex-tracted from the simulations. In Fig. 9 we show the calcu-lated reversal process for �=0±0.5°.

For the 0− case we see that reversal occurs via a vortexstate. Since this state is energetically more stable than thehorseshoe state observed for 0+, the system remains in thisstate over a wider range of applied field. There are threemain driving forces that set the magnetization direction ofthe top and bottom branches. One is the small tilt in the field:this leads to the magnetic moments in the top and bottombranches aligning along the direction of the field tilt. Thesecond effect is the exchange energy that favors a continuousmagnetization profile at each corner of the ring. The thirdeffect is the demagnetizing energy at the open end that favorsa closure state. If a branch has two shared corners �upperbranches in all diagrams in Fig. 9� the two latter contribu-tions do not favor any specific orientation so that the mag-netization follows the direction of the tilt. However, in abranch with a gap at one end �bottom branches in all dia-grams Fig. 9� the competing forces can lead to a complexnonuniform state. When the first two driving forces are par-allel the horseshoe state is formed �lower diagrams� butwhen they are antiparallel, the vortex state is generated �up-per diagrams�. It is this frustration that enables the energeti-cally more favorable vortex state to be achieved. We can nowprescribe a way to write a stable vortex state with a desiredchirality into our magnetic storage element. We know the
field tilt and the directions necessary for a stable vortex are

0− and 180−. From the simulation we also know the chiralityof the intermediate vortex state, as shown in Fig. 10.

The measured and calculated loops match for the �0 and180� orientations but there is some discrepancy for the �90and 270� cases. The simulation always reproduces the correctswitching order of the vertical branches �the peaks in thefirst-order loops are always in the correct order�, but theshape of the curves is different. For some reason the ex-pected vortex transition does not appear in two of the mea-sured data. We think that this is related to the fact that inthese cases the shorter branch �the one with the slit� is theone that is going into a magnetically frustrated state. Weexpect that these subtle differences could be explained bytaking into consideration lithography errors such as slit size,branch thickness, possible rounding effect at the corners, ordeviations in the applied field directions. However, it is un-likely that any greater insight would be gained from suchsimulations.

An important direction for future projects would be toreduce the size of the fabricated magnetic ring.2,5 This iscrucial in order to actually use this configuration in magnetic

FIG. 9. The two possible ways of switching for the magnetization: vortexstate �top three slides� and the horseshoe state �bottom three slides�. In both

FIG. 8. Numerical simulation resultsfor the diagonal orientations.

cases we are approaching the switching from large negative fields.



memories. The challenge is a complex one because, not onlydo we need to reduce the size of the rings, but we must alsoretain control over the shape of the sample, in particular, thesmall gap in the ring. At present, electron beam-lithographylimitations are unlikely to permit dramatic further reductionin size.

IV. CONCLUSIONS

We have manufactured arrays of open-slit square-ringmicromagnets and we investigated these rings with the use ofthe D-MOKE technique. We found that the magnetic switch-ing process in a changing exterior magnetic field criticallydepends on the position of the introduced shape asymmetry�slit� and the field orientation. We investigated the switchingprocess with computer simulation as well, and we found thatthe switching can occur in two different ways, one charac-terized by the appearance of a vortex state of magnetization,the other process going trough an intermediate horseshoestate. Thus, we can predict the cases of field and samplearrangements when a vortex state with a known chirality iscreated in our rings. Furthermore, we can prescribe a way towrite in a vortex with a given chirality by controlling thedirection of the applied field. Understanding the switchingand predicting the switching order of the branches is a keyissue if we desire to build controllable magnetic elements formemory purposes in the future.


ACKNOWLEDGMENTS

This work was supported by NSF Grant No. DMR-0210519, DOE-BES Contract No. W-7045-ENG-36, DOE-BES Grant No. W-31-109-ENG-38, and the Alfred P. SloanFoundation. One of the authors �P.V.� acknowledges supportfrom the projects FIRB RBNE017XSW and COFIN 2003 ofMinistero dell Istruzione, dell Universita e della Ricerca.

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FIG. 10. The first diffracted order forthe 0− direction. The peaks in the fullhysteresis loop correspond to vortexstates with different chirality. The di-rection of the applied magnetic field toarchive a desired chirality is alsoshown below the magnetization slides.


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Control of magnetic vortex chirality in square ring micromagnets