Effect of the second-order anistropy constant on the transverse susceptibility ofuniaxial ferromagnetsL. Spinu, Al. Stancu, C. J. O’Connor, and H. Srikanth Citation: Applied Physics Letters 80, 276 (2002); doi: 10.1063/1.1428626 View online: http://dx.doi.org/10.1063/1.1428626 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/80/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamics and collective state of ordered magnetic nanoparticles in mesoporous systems J. Appl. Phys. 112, 094309 (2012); 10.1063/1.4764018 Reversal time of the magnetization of single-domain ferromagnetic particles with cubic anisotropy in thepresence of a uniform magnetic field J. Appl. Phys. 101, 093909 (2007); 10.1063/1.2728783 Investigation of the magnetic susceptibility of nanocomposites obtained in zero-field-cooled conditions J. Vac. Sci. Technol. B 24, 321 (2006); 10.1116/1.2162571 Temperature dependence of ferromagnetic resonance in granular Cu–Co alloy J. Appl. Phys. 88, 368 (2000); 10.1063/1.373669 Temperature dependence of magnetic viscosity and irreversible susceptibility for aligned Stoner–Wohlfarthparticle J. Appl. Phys. 87, 5708 (2000); 10.1063/1.372497

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Effect of the second-order anistropy constant on the transversesusceptibility of uniaxial ferromagnets

L. Spinu,a) Al. Stancu,b) and C. J. O’ConnorAdvanced Materials Research Institute, University of New Orleans, New Orleans, Louisiana 70148

H. SrikanthDepartment of Physics, University of South Florida, Tampa, Florida 33620

~Received 26 June 2001; accepted for publication 29 October 2001!

A generalized theoretical approach of the transverse susceptibility,xT , in the case of uniaxialferromagnets is presented. This advances the classical model of transverse susceptibility where onlythe first-order anisotropy constant,K1 , is taken into account and makes possible the study of theinfluence of the second-order anisotropy constant,K2 on thexT curves. It is shown that additionalBarkhausen jumps driven by higher-order anisotropy constants emerge more evidently in the fielddependence ofxT making their detection much easier than from the hysteresis loop. Based on theobtained results, we propose a simple method to determine both,K1 andK2 independently fromxT

experiments. ©2002 American Institute of Physics.@DOI: 10.1063/1.1428626#

With the demand for new magnetic materials capable ofhigh data density storage, there is a need to establish theexact relationship between recorded information stability andfactors that can limit the achievable data density, such asparticle size and anisotropy. For particulate media, the mostwidely used experimental techniques to determine the anisot-ropy constants are the switching methods, such as rotationalhysteresis1 and stiffness methods such as singular point de-tection techniques,2 utilization of the law of approach tosaturation,3 and transverse susceptibility (xT).4 The main ad-vantage of these methods is provided by the possibility ofdetermining the anisotropy constant in polycrystallinesamples under quite general conditions, disregarding the ori-entation or size distributions of the particles. However, this iscounterbalanced by the fact that these methods are not suit-able for measuring the anisotropy constants in materialswhere the higher-order terms~HT! of the crystalline anisot-ropy energyWK5K1 sin2 u1K2 sin4 u1 . . . , are notnegli-gible. The anisotropy constant is determined by measuringthe anisotropy field,HK , which in general depends onhigher-order anisotropy constants asHK52K1 /MS( i ik i ,whereki5Ki /K1 andMs is the saturation magnetization ofthe material. It is obvious that determining anisotropy con-stantK1 by neglecting the contribution of the higher-orderterms leads to important systematic errors, overestimatingthe anisotropy constant value.

The experimental method of transverse susceptibility isextensively used owing to the fact that it offers a versatileand simple method of direct evaluation of the anisotropyfield HK. Uniaxial systems with random orientation in mag-netic easy axes show sharp peaks located at the anisotropyand switching fields~HK andHS , respectively!.4,5

In this letter, we present a generalized theoretical ap-proach of the transverse susceptibility in the case of uniaxialferromagnets. This advances the classical model ofxT due to

Aharoni5 where only the first-order anisotropy constant istaken into account, and makes possible the study of the in-fluence of the second-order anisotropy constant on thecurves. Also, based on our results, we suggest a method todetermine both,K1 and K2 independently, fromxT experi-ments.

In the past, the importance of the HT for a coherent andcorrect analysis of various experimental data has beenpointed out.6–9 For xT experiments, the influence of thehigher-order terms of the anistropy energy is even more criti-cal. The approximationWK

1 5K1sin2 u, which is usually con-sidered, is accurate only when the angle between the easyaxis and the magnetization is sufficiently small. InxT experi-ments, it has been shown that the particles with the easy axisoriented near 90° to the dc field direction are responsible forthe peaks located at6HK .4 For such particles, when thefield is nearHK , the angle between their easy axis and themagnetization is close to 90°. Therefore, neglecting the HTin the xT expression is a major source of errors. Moreover,for materials with important positive values ofK2 , or nega-tive values of K2 ~as it is observed for Co at hightemperatures!,10 this error is compounded significantly.

In xT experiments, one simultaneously applies two fieldson the sample: a dc field,Hdc and a very small amplitude acfield, Hac, perpendicular to the dc field direction. Maintain-ing the dc field and the amplitude of the ac field constant,with an appropriate coil system, one detects the variation ofthe total magnetization projection on the ac field direction.

Using a coordinate system withHdc lying along thezaxis andHac along thex axis, one may express the transversesusceptibility asxT5dMx /dHx with HY50 andHZ5Hdc.

If the easy axis of the single-domain particle orientationin this system is given by the spherical coordinates (uK ,wK)and the orientation of the magnetic moment is given by(uM ,wM), using a similar perturbation approach (Hac→0)as in Ref. 5, one obtains the followingk2 dependent expres-sion for thexT

a!Electronic mail: lspinu@uno.edub!Permanent address: Faculty of Physics, ‘‘Al.I. Cuza’’ University, Iasi 6600,

Romania.

APPLIED PHYSICS LETTERS VOLUME 80, NUMBER 2 14 JANUARY 2002

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xT53

2x0Fcos2 wK

cos2 uM

h cosuM1cos 2u1k2~cos 2u2cos 4u!

2sin2 wK

sinu

h sinuKG , ~1!

valid for anyK1 andK2 values, positives or negatives, whereh5Hdc/HK

1 with HK1 52K1 /MS , k25K2 /K1 . and u

5uMmuk.

It can be easily seen that fork250, one obtains thexT aspresented in Ref. 5. In this case, thexT evaluation is simplerdue to the fact that the total free energy,W, could only havetwo minima uM . When higher-order termK2 is present inthe uniaxial anisotropy energy,WK , the nature of the anisot-ropy energy ground state can change from easy axis (K1

.0) or easy plane (K1,0) to easy cone depending on thevalues of ratiok25K2 /K1 . These additional equilibrium po-sitions for the magnetization vector can generate a very com-plicated energy landscape, where metastable states can coex-ist with the global minimum. The external field may inducetransitions between the additional minima that will be re-flected in the major hysteresis loop by apparition of smallBarkhausen jumps.11 In the conditions of a complex energyprofile generated by the presence of higher-order anisotropyconstants, finding the field variation ofxT using Eq. ~1!,which depends on the equilibrium positionuM , can be adifficult task. The use of the critical-curves formalism in thiscase is a helpful tool.12 The critical curve separates the(h' ,hi) plane in regions where the magnetization vector hasa single stable equilibrium position~outer critical curve! ormore than one stable equilibrium position~region inner criti-cal curve!. h' andhi are the field components perpendicularand parallel to the easy axis direction, respectively.

Using Eq.~1! and selecting the appropriate stable orien-tation of the magnetization vector,uM , thexT can be calcu-lated. First, we consider the case of a single domain particlewith a uniaxial magnetocrystalline anisotropy free energywith K1.0, and different positive values fork2 . In this case,the anisotropy axis is an easy magnetization direction, anddifferent values ofk2 will affect only the shape of the criticalcurve. Hence, the critical curve is not a symmetric astroidanymore~as fork250!, but a curve extended along the hardaxis direction and with extra cusps along easy direction fork2.1/4.8 The calculatedxT curves for different orientationsuK of the applied field with respect to the easy axis, as thefield is swept from positive saturation to negative one, arepresented in the panel of Fig. 1. ForuK50°, we notice thatxT is the same for all values ofk2P(0,1/4), giving the ex-pected Barkhausen jump ath521. Only fork2.1/4,xT pre-sents an additional singularity located at a reduced fieldvalue depending onk2 , h(k2)54k2(112k2/6k2)3/2, due tothe appearance of the cusps in the easy axis direction. ForuK590°, xT is symmetric with two singularities located ath(k2)56(112k2). For all the other orientations of the ap-plied field, the effect of a positivek2 is a displacement of theanisotropy and switching peaks, according to the shapechange of the modified astroid.

Figure 2 summarizes the results obtained forK1.0 anddifferent negative values fork2 . For uK50°,xT curve doesnot depend onk2 , and is identical to that fork250. At uK

590° andk2P(21/6,0) symmetric peaks located ath(k2)

56(112k2) are observed. Fork2,21/6 we have two dif-ferent types of behavior: fork2P(22/3,21/6) one obtainstwo asymmetric peaks located ath(k2)51(112k2) andh(k2)524k2(2 1/6k2)3/2, and fork2,22/3, xT presents asingle central peak. ForuK530° andk2 sufficiently nega-tive, instead of one switching peak per sweep one obtainstwo switching peaks. The additional Barkhausen jump is thedirect consequence of the second anisotropy constant. Thepanel of Fig. 3 presents in detail this case, fork2521 show-ing in addition to thexT curve, the hysteresis loop and thecritical curve. Note the points where switching occurs in allthe curves. We mention that in Ref. 8, the occurrence of thetwo switching peaks was not remarked in this particular case~Fig. 7, in Ref. 8!. This complex switching behavior wasconfirmed by micromagnetic calculations of both hysteresisloop andxT , based on a Landau–Lifshitz–Gilbert~LLG!approach.13 The LLG results are displayed in Fig. 3 by sym-bols. At uK560°, we notice only the reversible behavior onthe xT curve fork2,22/3. This reversible behavior occursin a wider angular domain foruK than presented by Thiavilleet al. in Ref. 12, as confirmed by micromagnetic LLG calcu-lations. We remark that for significant negative values ofk2

FIG. 1. The theoretical curves ofxT for a monodomain particle having auniaxial anisotropy energy withK1.0, and different positive values fork2 .The different figures in the panel corresponds to different values foruK .

FIG. 2. The theoretical curves ofxT for a monodomain particle having auniaxial anisotropy energy withK1.0, and different negative values fork2 .The different figures in the panel corresponds to different values foruK .

277Appl. Phys. Lett., Vol. 80, No. 2, 14 January 2002 Spinu et al.

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the use of the two-dimensional~2D! critical curves is notsufficient for the proper selection of the stable equilibriumstatesuM and a three-dimensional~3D! approach, as the 3Dcritical curves method14 or magnetization dynamics usingLLG equation is necessary.

The easy-plane and easy-cone symmetries for magneto-crystalline free energy can be treated with the same uniaxialanisotropy approach using a negative value forK1 . Hencefor K1,0 andk2,1/2, the system has an easy plane and areversible and symmetric behavior is obtained onxT curves.For k2.1/2, an easy cone replaces the easy plane and a morecomplex peak structure forxT is observed.

For an assembly of noninteracting single domain par-ticles thexT response is given by the integral of the trans-verse susceptibility of each particle over the easy axis distri-bution f (uK ,wK). Using appropriate texture functions,f (uK ,wK), the transverse susceptibility can be calculated indifferent cases as uniform in plane@ f (uK ,wK)52p#, ran-domly @ f (uK ,wK)52p sinuK# or preferential distribution15

of the easy axis orientation. Figure 4 displays the resultsobtained for a randomly oriented system for different valuesof k2 . In this case, the well-knownxT curve, with anisotropypeaks located athK561 and switching field peak located at

hS520.5 is replaced by a curve with a more complicatedshape. The anisotropy peaks are located now athK(k2)56(112k2) and the switching peak location and heightdepends onk2 value. It can be shown that for a randomlyoriented system the switching peak field position is done byhS(k2)5min(Ah'

2 (k2)1hi2(k2)) whereh' andhi are thek2

dependent critical curve coordinates.This general theoretical analysis of the transverse sus-

ceptibility of uniaxial ferromagnet makes it possible to applythe xT experiments as a tool for a proper evaluation of theanisotropy constants. In the case of oriented particulate mag-netic materials, theK1 and K2 values can be determinedgenerally by two different measurements performed atuK

50° and uK590°. From the measurement performed atuK50°, we determine directly theK1 constant~the peak islocated at 2K1 /Ms!. At uK590°, the peaks are located ath(K1 ,K2)56(112K2 /K1 ) from which one may deter-mine the value ofK2 . For a high enough value ofK2 , asdescribed previously, an additional peak appears foruK

50°. From the positions of those peaks, we can determinebothK1 andK2 . Therefore, only one measurement is enoughto determine both anisotropy constants in this case.

Also, in the case of a randomly oriented particulate mag-netic system, the anisotropy constants, bothK1 andK2 , canbe determined fromxT curves. As it was presented in thisstudy, thek2 dependent anisotropy and switching peaks pro-vide the two necessary and sufficient relationships betweenK1 andK2 for their proper determination.

In summary, we have presented a comprehensive theo-retical approach of the transverse susceptibility, which ac-counts for the higher terms,K1 andK2 , in the uniaxial an-isotropy energy expression. Taking into account thegeneralized approach of transverse susceptibility presented inthis letter, makes possible the expansion of the range of ap-plicability of transverse susceptibility experiment as amethod for determining the anisotropy in magnetic materials.

The work at AMRI was supported through DARPAGrant No. MDA 972-97-1-0003.

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FIG. 3. The calculated hysteresis andxT curves fork2521, K1.0, anduK530°. The right-hand side panel shows the critical curve and the direc-tion of the applied field in the same conditions

FIG. 4. The meanxT of an assembly of randomly oriented monodomainparticles, having an uniaxial anisotropy energy withxT , and different valuesfor k2 .

278 Appl. Phys. Lett., Vol. 80, No. 2, 14 January 2002 Spinu et al.

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