Reversible magnetization processes in scalar Preisachtype models of hysteresis

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1 JOURNAL O OPTOELECTRONIC AND ADVANCED ATERIAL Vol. 8, No. 5, Octoer 26, p Reversile mgnetiztion processes in sclr Preischtype models of hysteresis L. TOLERIU *, A. TANCU Deprtment of olid tte nd Theoreticl Physics, culty of Physics, Alexndru Ion Cuz University, Blvd. Crol I, 11, Isi, 7576, Romni The ersile prt of mgnetiztion is one of the purely phenomenologicl defined components of the Preisch-type models. The rticle presents new type of ersile distriution sed on physicl oservtions nd cple to descrie the vrition of the ersile component with the ngle of the pplied mgnetic field. It lso presents new phenomenologicl pproch for the description of toner-wohlfrth type mgnetiztion loops using Preisch-type models. (Received eptemer 5, 26; eptemer 13, 26) Keywords: Reversile mgnetiztion phenomen, Preisch model 1. Introduction Any mgnetiion process implies oth irersile nd ersile processes. The irersile processes re usully ssocited with the dissiption of energy nd with the switching of the mgnetic moment from one equilirium position to nother while the ersile mgnetic processes re essentilly qusisttic processes. Both of these processes re involved, for exmple, in the mgnetic recording process. During recording, in the writing process, the mgnetic hed rings the medium into the desired mgnetized stte. The im of this process is to induce irersile chnges in the mgnetic stte of the medium using mgnetic field strong enough to switch the mgnetic moments from certin region of the medium in order to ring them in the desired mgnetic stte. When the hed leves certin re, its mgnetic field cts no more on the medium nd remnence mgnetiztion is otined. While the reding process is ting plce in the sence of n externl mgnetic field, the remnent mgnetiztion is more relevnt for the recording performnce thn the mgnetiion during the ppliction of the field. The difference etween the vlues of the two mgnetiztions in-field nd remnent is minly due to ersile processes. In mny cses the weight of the ersile prt of the mgnetiztion is considerle nd, s consequence, it is very importnt to descrie it correctly in the hysteresis models. Ting into considertion tht the physicl root of the ersile mgnetiztion processes is the ersile rottion of the mgnetic moment, one my oserve tht this processes fit well in the frme of the 3D vector mgnetiztion models while the sclr models re usully trying to descrie them in phenomenologicl mnner. In this pper we re proposing new pproch for the description of the ersile mgnetiztion processes in phenomenologicl sclr Preisch-type models, n pproch which is sed on the nlysis of the physicl cground of the mgnetiztion processes of single domin ferromgnetic entity. 2. odels for mgnetic ersiility The Clssicl Preisch model (CP) [1] is sclr model which ws initilly developed to descrie systems of perfectly ligned single-domin ferromgnetic prticles. The externl mgnetic field is considered to ct on the esy xis of the prticles. In this cse, the hysteresis loop of ech prticle is rectngulr nd cn e completely chrcterized mthemticlly y the vlue of the switching fields nd the urtion mgnetic moment. In system, one considers tht the switching fields re distriuted which leds to the ide of Preisch distriution which is, in this cse, deling only with the irersile mgnetiztion processes. The Preisch distriution lso includes distriution of the interction fields in the system. A simple solution for ting into ccount the ersile mgnetiztion processes ws the ppending of new, fully independent distriution of completely ersile hysterons or step opertors in the Generlised Preisch odel (GP) [2], [3] (ig. 1). The most common shpe of the ersile distriution is doule-exponentil: γ f ( ) = exp γ 2 (1) c c c is the most prole vlue of the coercive field, is the urtion mgnetiztion of the system, r is the vlue of the remnent mgnetiztion, is the rectngulrity of r the mjor loop = nd γ is fit prmeter. The lgorithm for the ersile prt identifiction is simple nd involves fitting the segment etween urtion nd remnence on mjor hysteresis rnch with f ( ) d.

2 Reversile mgnetiztion processes in sclr Preisch-type models of hysteresis 1711 ig. 1. Typicl irersile () nd ersile () hysterons in GP. A more ccurte technique of introducing ersile mgnetiztion in Preisch-type models ws presented y Dell Torre in the series of models DOK [4], VD-2 [5] nd C [6]. In these models the ersile component is not independent of the irersile distriution nd insted to ech point of the Preisch plne is ssocited nonrectngulr hysteresis loop (ig. 2()) resulting ersile component tht is distriuted ll over the plne (while in GP is one-dimensionl distriution). 1- prticles in the CP. As test, we re using the mgnetiztion processes of the well nown toner- Wohlfrth single domin ferromgnetic prticles [7]. The first prolem is to find the Preisch representtion of the toner-wohlfrth (W) hysteresis loop. As cn e oserved in ig. 3 the W loop contins irersile switches nd ersile moment vritions. This llows first decomposition of the W loop into rectngulr one with the height given y the vrition of the moment during the switch (ig. 3()) nd nother hysteresis loop which contins the ersile vrition of the totl moment (ig. 3(c)) computed y simply extrcting the irersile component from the totl moment. It is worthy to sy tht the ersile prt is not represented y mthemticl (single vlued) ersile function, s expected, ut y hysteretic function s the irersile component. A ir. ir. 1 B c ig. 3. A single domin prticle s hysteresis loop () nd the proposed irersile () nd ersile (c) components. Esy xis ersile c ig. 2. Typicl totl (), irersile () nd ersile (c) hysterons in Preisch-type models with distriuted ersile prt. irersile gnetic field direction These Preisch-type models hve tried to give more correct equivlence etween the rel ersile mgnetiztion processes nd the chrcteristic Preisch description of the totl mgnetic moment of the smple. Nevertheless, it ws ccepted tht the ersile processes re not completely given y (1) nd tht the effects re more complex. In this pper we re nlyzing the possiility to completely nd ccurtely include the ersile mgnetiztion processes of the component ig. 4. Trjectory of the mgnetic moment vector for single domin prticle. This pproch is vlidted y the nlysis of the trjectory of the mgnetic moment when pssing from

3 1712 L. toleriu, A. tncu positive to negtive urtion. While the irersile contriution ppers only during the switch, one cn ssocite to this component vlue given y the difference etween the projections of the mgnetic moment in A nd B positions (ig. 4). 3. Preisch representtion of the W Loop In order to represent the mgnetiztion loops of toner-wohlfrth single domin ferromgnetic prticle using Preisch-type model one must identify the irersile nd irersile distriution of the phenomenologicl model. or the irersile prt it is strightforwrd to ssocite singulr distriution in the Preisch plne for the rectngulr hysteresis loop (loop () in ig. 3). owever, it is oviously impossile to find stle distriution for the ersile component. As solution to this prolem we propose to define new type of ersile component, lso distriuted on the first isector of the Preisch plne, s it is usully done for the ersile processes (ig. 5). This distriution cn e siclly represented with comintion of exponentil functions nd it switches etween the nd shpes (represented with solid nd dshed line respectively) s function of the stte of the irersile component (loop () in ig. 3). By including the ngle of the externl pplied field s prmeter in the lgeric expression of the ersile function one cn simulte the mgnetiztion curves for ny ngle y chnging only this prmeter. or the simultions we considered two expressions for the nd sttes given y (2) nd (3): exp r ( ) σ f (, θ, ) = ( θ) (2) ( θ ) = ( θ) f,, σr σr ( ) exp σ r ( ) ( ) where is function of θ, the ngle etween the esy xis nd the direction of the pplied field nd σr is function of the nisotropy field. f (3) integrted lie in stndrd Preisch-type model so the mgnetiztion on the per nd lower rnch will e: (, θ, ) = 1 f (, θ, ) d (4) (, θ, ) = 1+ f (, θ, ) d (5) where is the vlue of the externl pplied mgnetic field for which the mgnetiztion is clculted. One cn oserve tht the response of the system is function of severl prmeters, which re the sme prmeters s for the toner-wohlfrth model: the vlue of the externl field, the ngle etween the field nd the esy xis, the vlue of the nisotropy field. Due to the presence of the singulr irersile distriution the concept of the mgnetic history of the system which is often ssocited with the Preisch plne is reduced to only two distinct sttes: nd. This lst prmeter decides which one of the two expressions (4) or (5) must e used. ig. 6 presents the result of the Preisch simultion of the descending mjor hysteresis rnches for severl ngles of the externl mgnetic field pplied to toner- Wohlfrth-type prticle for certin vlue of the nisotropy field. /s (.u.) ig. 6. The mjor hysteresis rnches for different ngles of the externl pplied field. 4. Preisch representtion of system mde of mny W prticles ig. 5. The modified ersile function used to simulte toner-wohlfrth-type ehvior in Preisch-type models. In order to compute the mgnetic response of the system the ersile functions (2) nd (3) must e There re severl configurtions worth ting into ccount: - systems mde of ligned prticles with switching fields distriution, - systems mde of prtilly ligned prticles with switching field distriution, - systems mde of rndom oriented prticles with switching field distriution.

4 Reversile mgnetiztion processes in sclr Preisch-type models of hysteresis 1713 or ech type of systems, the ersile distriution will e otined y dding the ersile functions for ll prticles. or continuous distriutions, the ersile distriution is sum of the ersile contriutions of ll of the prticles, computed s integrls: ( ) ( ) ( ) ( ) θ = f θ f f, θ, d dθ θ ( ) ( ) ( ) ( ) θ = f θ f f, θ, d dθ θ ( ) = ( ) + ( ) where f θ (θ) is the esy xis distriution nd f ( ) is the nisotropy field distriution. In (6), refers to the ersile contriution of the positive urted zones of the Preisch plne nd ( ) ( ) (6) refers to the ersile contriution of the negtive urted zones of the Preisch plne. This mens tht the profile of the ersile distriution () is stte-dependent nd difficult to identify for complicte sttes of the Preisch plne. ig. 7 presents the evolution of the profile of the ersile distriution () computed using (6) for the cse of the descending rnch of the mjor hysteresis loop for system mde of prtilly ligned prticles (θ σ =2º) with switching field distriution. We hve lso computed the ersile distriution for the other types of prticulte systems mentioned ove nd we hven t found significnt differences etween the three cses. () = 1. =.8 =.4 = -.5 = -.4 = -.8 = (.u.) ig. 7. The ersile distriution for different points on the descending rnch of L for system with θ nd distriutions. The trend of ersile distriution when chnges loos similr with the dynmics of the interction field distriution for the P 2 model presented in [8] nd they hve the sme physicl cground the switching of the mgnetic stte. This llows us to propose s good pproximtion simpler lgorithm for the identifiction of the ersile distriution. + ( ) ( ) If one notes with nd the profile of the ersile distriution for positive nd negtive urtion of the irersile prt, then, for n intermedite stte chrcterized y vlue ir of the irersile mgnetiztion the ersile distriution is: + (ir ( ) ( ) ) = + ( ) ir ir where is the urtion vlue of the irersile component of mgnetiztion. One must mention tht vlue is different form the totl urtion mgnetiztion. or exmple, for one-qudrnt medium medium for which the entire Preisch distriution is locted in the third qudrnt of the Preisch plne the totl urtion nd remnence mgnetiztion vlues re: where nd = ir +, = ir + = = (7) (8) signify the urtion vlues of totl nd ersile mgnetiztion, while = = nd denote the corresponding remnent vlues. In n even more simpler pproch, the ( ) + ( ) nd functions cn e pproximted with guss or log-norml functions ming the evlution of (7) very strightforwrd. 5. Conclusions trting from physicl considertions we designed new type of ersile distriution for Preisch-type model cple to correctly descrie the shpe of the mgnetic hysteresis loop of toner-wohlfrth single domin prticle for ny ngle of the pplied mgnetic field. Using s strting point the results for one prticle we clculted the ersile distriution of system mde of mny prticles nd, following the oserved dynmics of the distriution, we proposed n esy wy to mimic it voiding hevy clcultions. This study provides method to void more complex representtion of the ersile component s distriution in the Preisch plne, lie in the C model. While this method involves only chnges of the ersile prt it cn e fitted to ny Preisch-type model in order to descrie more complicted systems, lie the P 2 model for strongly intercting prticles [8]. Acnowledgement This wor is sported y ET grnt of Romnin CNCI entitled Theoreticl nd experimentl study of

5 1714 L. toleriu, A. tncu physicl processes with hysteresis in ferromgnetic, ferroelectric nd spin trnsition mterils. References [1] P. Preisch, Üer die mgnetische Nchwirung, Zeitschrift für Physi, vol. 94, pp , [2] I. D. yergoyz, G. riedmn, Generlized Preisch model of hysteresis, IEEE Trns. gn. 24(1), (1988). [3] I. D. yergoyz, themticl models of hysteresis, pringer-verlg, 199. [4] E. Dell Torre, J. Oti, G. Kdr, Preisch modelling nd ersile mgnetiztion, IEEE Trns. gn. 26, (199). [5]. Vjd, E. Dell Tome, esurements of outputdependent Preisch functions, IEEE Trns. gn.. 27, (1991). [6]. Vjd nd E. Dell Torre, Chrcteristics of gnetic edi odels, IEEE Trns. gn. 28(5), (1992). [7] E. C. toner, E. P. Wohlfrth, A mechnism of mgnetic hystersis in heterogeneous lloys, Phil. Trns. Roy. oc. London 24, (1948). [8] Al. tncu, L. toleriu, P. Postolche,. Cerchez, gnetic chrcteriion of strongly correlted mgnetic prticulte systems, J. Optoelectron. Adv. ter. 5(4), (23). * Corresponding uthor: lstoler@uic.ro