The results of recent micromagnetic simulations of deep submicron supermalloy disks are presented. A recent experimental measurement of the hysteresis and magnetic domain structure in supermalloy disks with diameters ranging from 55 to 500 nm and thickness ranging from 6 to 15 nm has been reported. Our micromagnetic simulations show remarkable agreement with the experimental hysteresis loops. The simulation results show that for thin or small diameter disks a single magnetic domain exists with all spins aligned. The hysteresis loop represents free rotation of these spins. For larger diameter disks or as the thickness increases the hysteresis loops change shape due to the appearance of a single vortex state appearing at low applied fields

Cowburn, R. P.

Dao, NgocNga

Whittenburg, Scott L.

English

University of Montana

University of Montana ScholarWorks at University of Montana Chemistry and Biochemistry Faculty Publications Chemistry and Biochemistry 11-15-2001 Micromagnetics Simulation of Deep-Submicron Supermalloy Disks Scott L. Whittenburg University of Montana - Missoula, scott.whittenburg@umontana.edu NgocNga Dao University of New Orleans R. P. Cowburn University of Durham Follow this and additional works at: https://scholarworks.umt.edu/chem_pubs  Part of the Chemistry Commons, and the Physics Commons Let us know how access to this document benefits you. Recommended Citation Whittenburg, Scott L.; Dao, NgocNga; and Cowburn, R. P., "Micromagnetics Simulation of Deep-Submicron Supermalloy Disks" (2001). Chemistry and Biochemistry Faculty Publications. 42. https://scholarworks.umt.edu/chem_pubs/42 This Article is brought to you for free and open access by the Chemistry and Biochemistry at ScholarWorks at University of Montana. It has been accepted for inclusion in Chemistry and Biochemistry Faculty Publications by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact scholarworks@mso.umt.edu. Micromagnetics simulation of deep-submicron supermalloy disksN. Dao and S. L. Whittenburga)Advanced Materials Research Institute and Department of Chemistry, University of New Orleans,New Orleans, Louisiana 70148R. P. CowburnDepartment of Physics, University of Durham, Durham DH1 3LE, United Kingdom~Received 30 January 2001; accepted for publication 25 August 2001!The results of recent micromagnetic simulations of deep submicron supermalloy disks are presented.A recent experimental measurement of the hysteresis and magnetic domain structure in supermalloydisks with diameters ranging from 55 to 500 nm and thickness ranging from 6 to 15 nm has beenreported. Our micromagnetic simulations show remarkable agreement with the experimentalhysteresis loops. The simulation results show that for thin or small diameter disks a single magneticdomain exists with all spins aligned. The hysteresis loop represents free rotation of these spins. Forlarger diameter disks or as the thickness increases the hysteresis loops change shape due to theappearance of a single vortex state appearing at low applied fields. ©2001 American Institute ofPhysics. @DOI: 10.1063/1.1412838#I. INTRODUCTIONDot arrays of magnetic material have been studied bothexperimentally and theoretically in recent years as mediamanufactured using dot arrays may find a myriad of techno-logical applications such as magnetic sensors and ultrahighdensity storage media. There has been significant scientificinterest in the effect of size and shape of dot arrays on mag-netic characteristics.1–3 In particular, the circular shape hasreceived a lot of attention.2,4–6 In those studies,4–6 magneticproperties such as the magnetic domain structure as a func-tion of thickness as well as diameter of the dots were exam-ined.Understanding the formation and annihilation of mag-netic domains is important since domains consisting of vor-tex structures can destroy the information stored in magneticmemory. Also by allowing study of the magnetic domainstructure through each step in a hysteresis loop, micromag-netic simulations may provide a deeper understanding of themicroscopic changes in the magnetization of the material. Inthis article, we present the results of simulations on super-malloy dots and examine the influence of the particle’s di-ameter on nucleation and annihilation fields along with themagnetic domain structure.II. PROCEDUREThe object oriented microMagnetic framework~OOMMF!7 obtained from The National Institute of Stan-dards and Technology is an excellent tool for simulation ofmicromagnetic problems.8 The exchange energy is computedby an eight-neighbor dot product.9 The constant volumecharge method10 is used for the magnetostatic energy inwhich spins are on the corners of grids and with approxima-tion of the magnetic volume charger52div m. The mag-netization of the sample is followed as a function of timeusing the Laudau–Lifshitz–Gilbert equation of motion on atwo-dimensional~2D! grid with 3D magnetization spins.Experimental work on supermalloy (Ni80Fe14Mo5) cir-cular dots has recently been reported.5 In these experiments,supermalloy dots ranging in thickness from 6 to 15 nm anddiameters ranging from 55 to 500 nm were fabricated andstudied using a high sensitivity magneto-optical method. Oursimulation used a 232 array of dots for each diameter. Thesize and spacing of the dots was chosen to reproduce theexperimental dots. For larger dots, the spacing between thedots was equal to or greater than their diameter. However, fordots with diameters below 150 nm the spacing was two orthree times greater than their diameter. With this arrange-ment, the dots should be noninteracting. However, the pos-sible magnetostatic interaction between the dots is includedin the calculation.In these simulations, we used a saturation magnetizationof 8.03105 A/m and an exchange stiffness constant of 1.05310211 J/m. An anisotropy constantK1 of 3 J/m3 was takenas an approximation based on molybdenum permalloy11 within-plane anisotropy. This value ofK1 is close to 0 J/m3, theOOMMF default value for permalloy. It is unlikely that suchsmall differences in the value for the anisotropy constant canbe observed in the simulations, however, the value of 3 J/m3was used to conform to the experimentally obtained value.The default of damping constant of 0.5 was used. The samplewas discretized into a cell size of 10 nm. As a test, a 300 nmdisk was also simulated with a cell size of 5 nm and gavesimilar results. The roughness of the circular shape is prob-ably unimportant since the hysteresis loops are the same ir-respective of the in-plane magnetization direction.All of our results were performed on a 300 MHz DECAlpha workstation. The computation time for the largest dotwas approximately 1 day, and for the smallest one a fewhours. For all of the dots, there were 100 steps taken for eacha!Author to whom correspondence should be addressed; electronic mail:swhitten@uno.eduJOURNAL OF APPLIED PHYSICS VOLUME 90, NUMBER 10 15 NOVEMBER 200152350021-8979/2001/90(10)/5235/3/$18.00 © 2001 American Institute of PhysicsDownloaded 09 Aug 2010 to 137.30.216.119. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jspforward and reverse loop in the hysteresis loop with a maxi-mum external applied field of 100 mT.III. RESULTSIn general, our simulation results agree well with theexperimental hysteresis loops. There are two basic types ofhysteresis loops for these samples. One type shows a simpleloop corresponding to Stoner–Wohlfarth-like free rotation ofthe spins. The other type displays a more complex hysteresisloop. When the sample is thick or the diameter is large, thedemagnetizing field can overcome the exchange field, andcurl states may appear, producing an hysteresis loop withkinks. However, in the very thin or small diameter samples,because of the nature of dimensional confinement, the ex-change field is the dominant contribution. This gives rise tosingle domain structures and a free-rotation hysteresis loop.The shapes of the both type of loop are the same as thoseobserved experimentally. A measure of agreement for thoseloops with curl or vortex states is the value of the nucleationfield, the value of the field at which the vortex first appears.For all dot diameters the nucleation field obtained from thesimulation agrees with the experimental value to within 20Oe. At the largest diameter, 500 nm, the simulated hysteresisloop is exactly the same as the experimental loop; the nucle-ation field agrees within 2 Oe.The magnetic domain structures were mapped as a func-tion of applied field to observe how magnetic domainschange during the hysteresis loop. Samples with a diameterof 300 nm display a complex hysteresis loop as shown inFig. 1. The simulated hysteresis loop agrees very well withthe experimental hysteresis loop. At the saturation field, allmagnetization spins are in the direction of the applied field.The dramatic increase in the magnetization as a function ofapplied field at point~a! is due to the creation or nucleationof the vortex. At point~a! a vortex or curl state appears. Thelowest magnetization corresponds to a vortex state centeredin the sample@point ~b!#. The diagonal line in the hysteresisloop is the motion of vortex across the sample. This move-ment of the vortex across the dot continues until there is asudden increase in magnetization due to the annihilation ofthe vortex@point ~c!#.As an example of the samples with hysteresis loops dis-playing coherent rotation the loop for a dot with a diameterof 100 nm is given in Fig. 2. The simulated hysteresis loopand experimental loop show the same general shape; how-ever, the simulated coercive field is about six times higherthan the experimental coercive field. There is a kink in bothsimulated hysteresis loops. Point~a! corresponds to a domainpattern where the majority of spins are aligned in the samedirection. At the kink~b! some of the spins become slightlymore tilted relative to the applied field. This state is similarto the ‘‘S’’ state observed in other micromagnetic studies.All of the simulated hysteresis loops have the same gen-eral shape as the experimental hysteresis loops except forsamples with a diameter of 150 nm. For this sample, thesimulated hysteresis loop has a higher coercivity than theexperimental curve. Dots with a diameter of 150 nm are theborderline between supporting the vortex domain and thesingle domain.The various contributions to the energy of each dot areimportant in understanding the reversal mechanism. To sus-tain a vortex structure the demagnetizing energy decreases atthe expense of the exchange energy. In a vortex the spins atthe edge of the dot are aligned relative to the edge. As thevortex is annihilated, the exchange energy drops rapidly asthe spins become aligned~Fig. 3!.By combining the simulation results with those obtainedf om the experimentally measured hysteresis loops a mag-netic ‘‘phase’’ diagram may be obtained. Such a diagram isshown in Fig. 4. Our results suggest that the diameter andthickness of the dots determines whether the magnetic do-main structure will be single domain~SD!, a vortex state~VS!, or multidomain~MD!. The characteristic length scalein these systems is the exchange length,l ex5(2 A/moMs)1/2,because a dominant exchange interaction will lead to align-ment of the spins and result in a single domain state. Thus, aphase diagram may be obtained in terms of two dimension-less parameters, thickness/diameter, and diameter/l ex. TheFIG. 1. The simulated data~square! and the experimental data~circle! fordiameter of 300 nm and thickness of 10 nm:~a! nucleation of a vortex,~b!vortex structure, and~c! the annihilation state. Every third spin is shown.FIG. 2. The simulated hysteresis loop~square! and the experimental hyster-esis loop~circle! for diameter of 100 nm and thickness of 10 nm:~a! all ofmagnetization spins aligned in one direction, and~b! at the kink point in thisloop.5236 J. Appl. Phys., Vol. 90, No. 10, 15 November 2001 Dao, Whittenburg, and CowburnDownloaded 09 Aug 2010 to 137.30.216.119. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jspphase diagram for supermalloy given in Fig. 4 shows thatthere is a distinct line separating the SD and VS ‘‘phases.’’The functional form of this line is a hyperbola. For samplewith a diameter/l ex ratio below roughly 20 the magnetic do-main pattern is single domain regardless of the thickness ofthe dots, within the range of thickness studied here. Also, asthe thickness of the dot decreases the pattern is again singledomain for extremely small thickness/diameter ratios regard-less of the diameter of the dot. Presumably at sufficientlylarge diameters and thickness the magnetic domain patternswill be multidomain. This region of the phase diagram wasnot studied here and is suggested using question marks inFig. 4. This phase diagram is similar to the previously pub-lished diagram for permalloy.4IV. CONCLUSIONSOverall, the simulated hysteresis loops are in goodagreement with the experimental loops. For all dot diameterssupporting vortex formation the nucleation field obtainedfrom the simulation agrees with the experimental value towithin 20 Oe. At the largest diameter, 500 nm, the simulatedhysteresis loop is exactly the same as the experimental loop,the nucleation field agrees within 2 Oe. The size of thesample is critical. For thin or low diameter dots, the dotcannot sustain a vortex and a single domain pattern with allspins essentially aligned in one direction is obtained. Forthese dots, the large difference in coercive field is probablydue to imperfection in the experimental samples, leading topremature switching. It is also possible that these differencesare due to gridding artifacts in the simulations. For largediameter dots or thick samples, the hysteresis loops corre-sponds to the creation and annihilation of a vortex state. Asobserved in Fig. 4, the line separating the single domain‘‘phase’’ from the vortex state phase is hyperbolic. Dots witha diameter-to-exchange-length ratio below roughly 20 willbe single domain regardless of the thickness, within therange of thickness measured here.ACKNOWLEDGMENTSThe authors wish to acknowledge the support of DARPAvia DOD/DARPA MDA972-97-1-0003. The authors alsowish to acknowledge the Department of Engineering, Uni-versity of Cambridge, Cambridge.1J. F. Smyth, S. Schultz, D. Kern, H. Schmid, and D. Yee, J. Appl. Phys.63,4237 ~1988!.2A. Hubert and M. Ruhrig, J. Appl. Phys.69, 6072~1991!.3P. Vavassori, O. Donzelli, V. Metlushko, M. Grimsditch, B. Ilic, P. Neuzil,and R. Kumar, J. Appl. Phys.88, 999 ~2000!.4C. Miramond, C. Fermon, F. Rousseaux, D. Decanini, and F. Carcenac, J.Magn. Magn. Mater.165, 500 ~1997!.5R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M.Tricker, Phys. Rev. Lett.83, 1042~1999!.6T. Pokhil, D. Song, and J. Nowak, J. Appl. Phys.87, 6319~2000!.7http://math.nist.gov/oommf8N. Dao, S. R. Homer, and S. L. Whittenburg, J. Appl. Phys.86, 3262~1999!; C. T. Yu, H. Jiang, P. J. Flanders, and G. J. Mankey,ibid. 87, 6322~2000!; J. Raabe, R. Pulwey, R. Sattler, T. Schweinbock, J. Zweck, and D.Weiss,ibid. 88, 4437~2000!.9M. J. Donahue and R. D. McMichael, Physica B233, 272 ~1997!.10D. V. Berkov, K. Ramstock, and A. Hubert, Phys. Status Solidi A137, 207~1993!.11Concise Encyclopedia of Magnetic and Superconducting Materials, editedby J. E. Evetts~Elsevier Science, New York, 1992!, p. 356.FIG. 3. The demagnetizing energy~square! and the exchange energy~circle!for the 300 nm diam sample versus the normalized magnetization. Thesample thickness is 10 nm. The open symbols are for the increasing fieldsection of the hysteresis loop, while the filled symbols are for the decreasingfield section of the loop. The bend in the exchange energy curve atMx /Ms;0.8 results from vortex annihilation. The corresponding bend inthe curve due to vortex formation does not appear in the section of thehysteresis loop shown.FIG. 4. A phase diagram for supermalloy showing the regions where thesample is in the SD, VS, or MD state as a function of the thickness anddiameter of the supermalloy dots. Experimental and simulation results areshown.5237J. Appl. Phys., Vol. 90, No. 10, 15 November 2001 Dao, Whittenburg, and CowburnDownloaded 09 Aug 2010 to 137.30.216.119. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

Micromagnetics Simulation of Deep-Submicron Supermalloy Disks

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Micromagnetics Simulation of Deep-Submicron Supermalloy Disks

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