Micromagnetic and Preisach analysis of the First Order Reversal Curves (FORC)diagramAlexandru Stancu, Christopher Pike, Laurentiu Stoleriu, Petronel Postolache, and Dorin Cimpoesu Citation: Journal of Applied Physics 93, 6620 (2003); doi: 10.1063/1.1557656 View online: http://dx.doi.org/10.1063/1.1557656 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/93/10?ver=pdfcov Published by the AIP Publishing

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JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003

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Micromagnetic and Preisach analysis of the First Order Reversal Curves„FORC… diagram

Alexandru StancuFaculty of Physics, ‘‘Alexandru Ioan Cuza’’ University, Iasi, 6600, Romania

Christopher PikeDepartment of Geology, University of California, Davis, California 95616

Laurentiu Stoleriu, Petronel Postolache, and Dorin CimpoesuFaculty of Physics, ‘‘Alexandru Ioan Cuza’’ University, Iasi, 6600, Romania

~Presented on 12 November 2002!

The First Order Reversal Curve~FORC! diagrams of interacting single-domain ferromagneticparticle systems have been found experimentally to contain negative regions. In this paper, we usemicromagnetic and phenomenological~Preisach-type! models to help explain the occurrence ofthese negative regions. In Preisach-type modeling, the position of the negative region is correlatedwith the sign of the mean-field interactions. In micromagnetic modeling, the position of the negativeregion is correlated with the spatial arrangement of the particles in the model. ©2003 AmericanInstitute of Physics.@DOI: 10.1063/1.1557656#

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INTRODUCTION

Recently, the First Order Reversal Curves~FORC! ~Ref.1! diagrams were proposed as an experimental methodthe characterization of interparticle interactions in particulferromagnetic systems. FORC diagrams are more conventhan other similar experimental methods because they dorequire that the magnetization be measured in a remastate and because they do not require an ac demagnestate, as is the case in the well-knownDM experimentalprocedure. In previous studies with FORC diagrams, bpositive and negative regions have been observed on theperimental diagrams. To better understand the source of tnegative regions, we have simulated particulate magnmedia with both micromagnetic~a bidimensional system osingle-domain particles with magnetostatic interactions wthe magnetic moment dynamics described by the LandLifshitz–Gilbert equation! and phenomenological Preisacmodels. We propose an explanation for why the negaregion occurs and what parameters are influencing it.

THE FORC DIAGRAM

A first order reversal curve starts from the descendbranch of the Major Hysteresis Loop~MHL ! at a reversalfield Hr , as illustrated in Fig. 1. The FORC evaluated afunction of the applied fieldH, whereH.Hr is denoted bymFORC

2 (Hr ,H). The FORC distribution is defined as thmixed second derivative of the set of first order revercurves,

r~Hr ,H !521

2

]2mFORC2 ~Hr ,H !

]Hr]H. ~1!

There is a strong similarity between the definition of tFORC diagram and the experimental method for the demination of the Preisach distribution for a system obeyingthe deletion and congruency properties, as presented

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Mayergoyz.2 The FORC diagram and the Preisach distribtion will be identical for a system correctly described by tClassical Preisach Model~CPM!. But with FORC diagrams,we apply this nonparametric identification procedure directo the experimental data of any system, regardless of wheit is correctly described by the CPM. As presented in preous papers, the FORC diagram can have strange patwhen measured for various magnetic1,3 samples. To help usunderstand the causes of these patterns, we have perfoFORC diagram analysis of both micromagnetic aPreisach-type models, as presented in the following secti

MICROMAGNETIC MODEL

We have simulated the hysteresis behavior of a systemuniaxial single-domain ferromagnetic particles with magntostatic inter-particle interactions. The particles are arranin a 2D lattice like the one presented in the Fig. 2, and h

FIG. 1. MHL and FORC curve starting from the descending branch ofMHL.

0 © 2003 American Institute of Physics

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6621J. Appl. Phys., Vol. 93, No. 10, Parts 2 & 3, 15 May 2003 Stancu et al.

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a distribution of easy axes orientations and volumes. Ttypes of 2D geometries were used~see notation on Fig. 2!. Ingeometry A we letz0.x0 ~typical for systems of orientedacicular particles! and in geometryP we let z0,x0 ~typicalfor systems of platelets particles, like barium ferrite!. In oursimulations, the external field is applied on the Oz directiThe evolution of the magnetization was calculated withwell-known full 3D Landau–Lifshitz–Gilbert4,5 equation.Continuity conditions were considered, and in each simution a complete analysis of the interparticle magnetostinteractions was performed. These interactions are disuted and both the average value and the variance are dedent on the magnetic state of the system. If the interacfield distribution average becomes positive when the tomagnetic moment of the system is positive and becomnegative when the moment is negative, the system is refeto as a system with a positive mean interaction field. Ifaverage is positive when the moment is negative and ntive when the moment is positive, the system is referred toa system with a negative mean interaction field. In our anasis we have found that theP-type geometry is associatewith positive mean field interactions, while geometry Aassociated with negative mean field interactions. In Fig. 3~a!we present a simulated FORC diagram generated withmicromagnetic code for a ‘‘P’’-type sample and in Fig. 3~b!we present a FORC diagram for a ‘‘A’’-type sample. Whilethe ‘‘P’’-type system has a FORC diagram with the mamum value above the245° line defined byHb52Ha , thesame maximum is shifted below this245° line for the ‘‘A’’-type systems. Both FORC diagrams have negative valuegions but in different places. To better understand thesegrams we next use a Preisach approach.

PREISACH SIMULATION OF THE FORC DIAGRAM

With a Generalized Moving Preisach model6 one obtainsFORC diagrams like those presented in Figs. 4~a! and 4~b!~for positive and negative moving parameter, respective!,that are in excellent agreement with the micromagneticculations presented in the previous section. As shownDella Torre and Vajda,7 the systems with linear mean fielinteractions can be analyzed in a straightforward way inoperative field planein which on the field axis one useinstead of the applied field,Happl, the operative field,Hop,

FIG. 2. Particulate system used in the micromagnetic simulation. For eparticle the orientation of the easy axis is shown with solid line.

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given byHop5Happl1am, wherea is the moving constantandm is the value of the magnetic moment normalized tosaturation magnetic moment of the sample. The experimemagnetization loops, can be transferred into the operafield plane by performing a simple transformation with rspect to them50, Happl50 point clockwise for positivemoving constant and counterclockwise for negative movconstant. The mixed double derivative of the FORCs,~1!,have a simple geometric2 interpretation. Since the angle between the tangent to a FORC and the field axis is given

u~Hr ,H !5arctanF]mFORC2 ~Hr ,H !

]H G , ~2!

~1! can be written as

r~Hr ,H !521

2

]@ tanu~Hr ,H !#

]Hr. ~3!

The FORC diagram is positive when tan@u(Hr ,H)# is a mono-tonically decreasing function ofHr for a fixed value ofH.This can be observed easier on a plot of the curves chaterized by the same value of the susceptibility on the FOcurves described by the equationsu(Hr ,H)5constant~see

ch

FIG. 3. ~a! FORC diagram for micromagnetic calculation and geometryi.e.,z0.x0 , where the applied field is along thez axis. The mean interactionfield, in this case, is positive.~b! FORC diagram for micromagnetic calculation and geometry A, i.e.,z0,x0 , where the applied field is along thezaxis. The mean interaction field, in this case, is negative.

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6622 J. Appl. Phys., Vol. 93, No. 10, Parts 2 & 3, 15 May 2003 Stancu et al.

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Fig. 5!. To have a positive FORC diagram the lines parato them axis in the~H,m! plane~experimental plane! shouldnot intersect twice any equal susceptibility curve. The mixderivatives for the FORC diagram are calculated along liparallel to them axis in the experimental plane. These linein the operative field plane, have the slope 1/a and are re-ferred to as the transformation lines. If the transformatlines intersect twice the equal susceptibility curves inoperative field plane, that will generate negative regionsthe FORC diagram. In Fig. 5 one can also observe thatnegative region on the FORC diagram for positive mean fiinteractions is obtained atH fields less than the field corresponding to the FORC diagram maximum (H0) and forHr

in the vicinity of 2H0 . The case of negative mean fie

FIG. 4. ~a! FORC diagrams calculated with the Preisach model for posimoving parameter.~b! FORC diagrams calculated with the Preisach mofor negative moving parameter.

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interactions is not so obvious. The negative region is in tcase given by the FORCs starting on the descending braof MHL near the negative saturation and forH fields near thepositive saturation.

CONCLUSIONS

Our micromagnetic calculations show that systems wdifferent geometrical structures will have negative regionsdifferent locations on a FORC diagram. When the negatregion occurs at fields higher than the field correspondingthe FORC diagram maximum, one can conclude that thetem has a geometry similar to an oriented acicular particusystem, where the mean field interactions will have a demnetizing effect. When the negative region is obtainedlower fields, this implies that the system geometry is likeensemble of platelet particles, and that the mean interacfield has a magnetizing effect.

ACKNOWLEDGMENT

Work was supported by Romanian CNCSIS undGrants Nos. A/2002 and D42.

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3C. R. Pike, A. P. Roberts, M. J. Dekkers, and K. L. Verosub, Phys. EPlanet. Inter.126, 11 ~2001!.

4Al. Stancu, L. Stoleriu, and M. Cerchez, J. Magn. Magn. Mater.225, 411~2001!.

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7E. Della Torre and F. Vajda, IEEE Trans. Magn.30, 4987~1994!.

el

FIG. 5. Curves of equal susceptibility~x! in the ~H,m! plane.

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