Magnetization processes in hard Co-rich Co–Pt films with perpendicularanisotropyM. Ghidini, G. Zangari, I. L. Prejbeanu, G. Pattanaik, L. D. Buda-Prejbeanu et al. Citation: J. Appl. Phys. 100, 103911 (2006); doi: 10.1063/1.2357869 View online: http://dx.doi.org/10.1063/1.2357869 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v100/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Magnetization processes in hard Co-rich Co–Pt films with perpendicularanisotropy

M. Ghidinia�

Department of Physics and CNISM, University of Parma, Viale G.P. Usberti 7/A, 43100 Parma, Italy

G. ZangariDepartment of Materials Science and Engineering and CESE, University of Virginia,Charlottesville, Virginia 22904

I. L. PrejbeanuSPINTEC, CEA-CNRS (URA 2512), 17 Avenue des Martyrs, 38054 Grenoble, France

G. PattanaikDepartment of Materials Science and Engineering and CESE, University of Virginia,Charlottesville, Virginia 22904

L. D. Buda-PrejbeanuSPINTEC, CEA-CNRS (URA 2512), 17 Avenue des Martyrs, 38054 Grenoble, France

G. Asti, C. Pernechele, and M. SolziDepartment of Physics and CNISM, University of Parma, Viale G.P. Usberti 7/A, 43100 Parma, Italy

�Received 28 April 2006; accepted 26 July 2006; published online 30 November 2006�

We present a study of the magnetic properties and magnetization processes in hard Co-Pt �Pt�20 at. % � films. Co-rich Co-Pt films, with thickness t ranging from 5 nm up to 2 �m, wereprepared by electrodeposition on �0001�-oriented Ru underlayers. All samples displayed strongperpendicular magnetic anisotropy and high coercivity. Virgin magnetic domain structures forvarying thickness were investigated by magnetic force microscopy �MFM�. The observed increaseof domain width with film thickness is well understood by full two-dimensional micromagneticcomputations with no adjustable parameters. The easy-axis magnetization process, as observed bymeasuring virgin curves by magnetometry and imaging the corresponding magnetizationconfigurations by MFM in variable field, consists of two stages separated by a well-defined criticalfield, marking the onset of domain wall propagation. A thorough analysis of the out-of-plane angulardependence of the switching field points out that unpinning of domain walls is the dominantmagnetization process. © 2006 American Institute of Physics. �DOI: 10.1063/1.2357869�

I. INTRODUCTION

Magnetic films with strong perpendicular magnetic an-isotropy �PMA� are attracting wide interest as viable mediafor perpendicular magnetic recording �PMR� systems1,2 aswell as for patterned magnetic media.3 In order to achievestate-of-the-art recording densities �102–103 Gbits/ in.2�, bitdimensions must be of the order of 80–25 nm. In conven-tional PMR this requires both a small grain size and a highanisotropy; in patterned recording, on the other hand, unifor-mity in the dimensions and magnetic properties of the islandarrays and strong exchange coupling within the single islandare most important. Binary alloys and intermetallic com-pounds of the Co–Pt and Fe–Pt systems are being exten-sively studied for this purpose. These materials in fact dis-play very high magnetic anisotropy, in the range of105–107 J /m3.4,5 Highly coercive Co–Pt and Fe–Pt films canbe prepared by sputtering,6,7 although suitable films must bedeposited at high temperature or undergo robust postdeposi-tion heat treatments. Electrodeposition of Co-rich Co–Pt

films �Pt�20 at. %� on the other hand yields polycrystallinefilms with a highly oriented hexagonal close packed �hcp�structure, exhibiting high coercivity and strong PMA in theas-deposited state.8 This possibility is particularly advanta-geous in view of the formation of nanostructures by pattern-ing procedures, which would require only additive processeswhen electrodeposition is used. Previous work has shownthat high coercivity and a wide range of magnetic propertiescan be achieved in these films using various substrates.9,10

However, detailed investigations and understanding of theirmagnetization processes, which are essential to optimize pat-terned structures and multi-component systems,11 are stillmissing. Here, we report a detailed study of the magneticproperties and magnetization processes in highly orientedCo–Pt films of thickness ranging from a few nanometers to afew micrometers, using magnetic force microscopy �MFM�imaging and magnetometry measurements, and also for vary-ing angles between the applied field and the easy axis. Withregard to the magnetization and switching processes, the fo-cus will be on films with thicknesses in the range of hun-dreds of nanometers, which are suitable for hard/soft layeredstructures and micromagnets and can be patterned by stan-dard lithographic techniques.

a�Author to whom correspondence should be addressed; electronic mail:massimo.ghidini@fis.unipr.it

JOURNAL OF APPLIED PHYSICS 100, 103911 �2006�

0021-8979/2006/100�10�/103911/10/$23.00 © 2006 American Institute of Physics100, 103911-1

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II. EXPERIMENTAL PROCEDURE

Co-rich Co–Pt films �Pt�20 at. % �, with thickness tranging from 5 nm up to 2 �m, were electrodeposited onto atemplate consisting of the following stacked structure:Si�011� /Cu�111� �200 nm� /Ru�0001� �20 nm�. Cu and Ruwere sequentially sputtered onto the hydrogen-terminatedSi.12 The �0001� Ru surface provides for an epitaxial tem-plate which favors the growth of hcp Co–Pt with the c axisperpendicular to the substrate. Co–Pt films were electrode-posited at constant current from an alkaline amino citrateelectrolyte, containing Pt as an amino nitrite complex and Coas sulfamate. The crystal structure and microstructure of thefilms were characterized by x-ray diffraction �XRD�, using aScintag diffractometer in the conventional �−2� geometry.The morphology of the films was studied by tapping modeatomic force microscopy �AFM�, using a Digital InstrumentNanoscope IIIa. Magnetic measurements �hysteresis loops atvarious angles, as well as virgin and recoil curves� wereperformed at room temperature using either a superconduct-ing quantum interference device �SQUID� or a vibrating

sample magnetometer �VSM�. Magnetic microstructures andmagnetization processes were observed by MFM in variablemagnetic fields, applied both in situ �up to 64 kA/m� and atremanence, ex situ, using either a Veeco Nanoscope IV or aDigital Instrument Nanoscope IIIa. All MFM measurementswere carried out using lift mode imaging. The frequencyshift of the tip, due to vertical magnetic force gradients, wasdetected by the phase detection method.

III. STRUCTURAL, MAGNETIC, AND MICROSCOPICCHARACTERIZATION

The Co–Pt films were found to be polycrystalline and toexhibit a highly oriented hcp structure, with the c axes of theindividual grains perpendicular to the substrate. X-ray dif-fraction spectra show in fact only peaks from the �00.2� and�00.4� reflections �Fig. 1�; a lattice constant c=4.14 Å can bederived from these data, which can be compared with c=4.22 Å observed in sputtered Co–Pt �Pt�25 at. % �.12 Allsamples displayed a homogeneous surface microstructurewith an apparent grain size—as determined by AFM—ofabout 50 nm, only slightly dependent on film thickness. Theapparent grain size distribution was remarkably sharp. rmsroughness, as determined over 5�5 �m2 scan areas, wasseen to increase with thickness t, from 0.6 nm for t=5 nm upto 6 nm in the t=1 �m sample. Figure 2 shows some suchimages collected on samples with different thicknesses.

The films show a PMA with high squareness S already ata thickness of 10 nm; S decreases with increasing thickness,and the corresponding loops are uniformly sheared �Fig. 3�.

FIG. 1. X-ray diffraction patterns of Co–Pt/Ru/Cu/Si structures, where theCo–Pt film thickness is varied. Only the CoPt �00.2� and �00.4� reflectionsare visible in addition to the corresponding Ru and Cu reflections from thetemplate structure.

FIG. 2. �Color online� Surface topography of CoPt films as observed bytapping mode AFM �scan area is 5�5 �m2 in all cases�.

FIG. 3. Out-of-plane hysteresis loops for the 10 and 2000 nm thick films.

FIG. 4. Thickness dependence of the coercive fields and of the squarenessS=Mr /Ms as deduced from out-of-plane loops in external field.

103911-2 Ghidini et al. J. Appl. Phys. 100, 103911 �2006�

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Perpendicular coercive fields and S as functions of thicknessare plotted in Fig. 4. It is worth noting that Hc displays asharp increase just above 10 nm, where S starts to decrease.We can only conjecture that the two effects are correlated,and that they could be due to a microstructural evolutionfrom strongly interacting grains at low thicknesses to weaklydecoupled grains. This microstructure change could justify amodification of the dominant reversal mechanism, namely,from nucleation to pinning.

Figure 5 shows in detail the in-plane and perpendicularhysteresis loops obtained for the 150 nm thick sample, rep-

resenting a typical behavior for all but the thinnest films.From positive saturation, the reduced magnetization is firstobserved to decrease slightly down to an external field valueof about 0.4 MA/m, where an abrupt slope change is ob-served. For decreasing fields, the magnetization then steadilydecreases. The inset of Fig. 5 shows the recoil curves anddemagnetization curves superimposed. The demagnetizationprocess is reversible down to the observed knee, while itbecomes almost completely irreversible below. This featureallows one to identify the external field at which the knee isobserved as the switching external field. The in-plane loopon the other hand shows small hysteresis and remanent mag-netization, related to the existence of an easy-axis distribu-tion function �see Sec. IV�. It also displays a significantlywide linear portion at intermediate fields, while at higherfields, but still far from saturation, a curvature is observed,which could hardly be attributed to a contribution from ahigher order anisotropy constant, since in this case oneshould not expect a linear behavior at low and intermediatefields. Rather it could be related to the effects due to thesurface domain pattern �see Sec. IV�. From the linear portionof the loop, we have extrapolated the value of the first orderanisotropy constant, K1=3.4�105 J /m3. From this value,one obtains a quality factor Q=K1 /Kd=1.5, where Kd

=�0Ms2 /2 is the stray field energy constant. The condition

Q�1 implies that the perpendicular magnetization directionis the preferred one both for the single-domain and for themultidomain state.13 MFM images of all the virgin magneticdomain structures were collected. Figure 6 reports selectedimages for the t=5, 50, 150, and 500 nm samples. The ap-parent domain size increases with film thickness. Differenttypes of magnetic structures are identified. For intermediatethicknesses, we observe the tendency to form stripe domainsorganized in labyrinthine patterns, while in the thinnest andthickest samples the magnetic microstructures follow a pat-tern with no clear order. However, in all samples, with theexception of those 5 and 10 nm thick, it was possible todetermine a prevalent wavelength of the magnetic structureby applying a bidimensional fast fourier transform �FFT� tothe MFM images. Figure 7 reports, as an example, the powerdensity spectrum obtained for the 150 nm thick film. Thepredominant periodicity thus obtained is plotted in Fig. 8 asa function of the sample thickness. We distinguish betweenthe samples which displayed labyrinthine structures, hereaf-ter named LS �black circles�, and those with a disorderedstructure, hereafter named DS �open triangles�. While the

FIG. 5. Out-of-plane �closed circles� and in-plane �crosses� loops for the150 nm thick sample. Inset: demagnetization and recoil curves as measuredwith the field applied perpendicular to the film plane.

FIG. 6. �Color online� Selected MFM images corresponding to the virginstate for different film thicknesses �scan area is 5�5 �m2 in all cases�.

FIG. 7. 2D power spectral density obtained by FFT of the MFM image forthe 150 nm thick sample in the virgin state.

103911-3 Ghidini et al. J. Appl. Phys. 100, 103911 �2006�

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observed widths are seen to increase in both cases with thick-ness, an evident spread is observed between the two datasets. Comparison with theory helps in elucidating such be-havior. A first estimate of the wavelengths W of the equilib-rium magnetic configurations and of their dependence on thelayer thickness can be obtained by conventional models14,15

for perpendicularly magnetized films and platelets with largeuniaxial anisotropies �Q�1�. Assuming ideal Bloch-type do-main walls, with an energy density �w=4�AexK1, Aex beingthe exchange constant and K1 the anisotropy constant, asquare root dependence of the equilibrium domain period onthickness is expected:

Weq = 2� �wt

�4.2�0Ms2/�3�

, �1�

The period W plotted in Fig. 8 �full line� is calculatedwithout adjustable parameters, using the experimentally de-termined quantities Ms=600 kA/m and K1=3.45�105 J /m3 and assuming the exchange constant of Co, Aex

=1.4�10−11 J /m.16 The resulting domain wall energy is�w=8.72�10−3 J /m2. There is a relatively good quantitativeagreement with both data sets. However, the thickness de-pendence of the DS data is better described. In fact, agree-ment with the DS data can be improved by taking explicitlyinto account the wall structure deviation from the ideal Blochtype, especially near the film surface, into a full bidimen-sional micromagnetic simulation �dashed line�, where strayfield, exchange, and magnetocrystalline anisotropy havebeen considered. A classical finite difference based two-dimensional �2D� micromagnetic solver17 was used to get theequilibrium magnetic state corresponding to the remanence.For the case of these thin films, the structure is supposed tobe periodic along one in-plane direction and invariant on the

other �Oz direction�. A mesh size of 2.5�2.5 nm2 was used,thus allowing an accurate description of the domain wallstructure. To find the stable state the total free energy of thesystem was simultaneously minimized with respect to themagnetization and the period of the system. Starting from thein-plane saturation the magnetization relaxes towards anequilibrium stripe domain structure as depicted in Fig. 9 for50, 100, and 200 nm thick samples.

The fact that reasonable quantitative agreement is foundbetween theory and DS data suggests that the observed LSmight be metastable structures. In fact, Fig. 11�a� shows theac demagnetized �maximum field=1.4 MA/m, step=72 kA/m, applied perpendicularly to the plane� state forthe 150 nm film. The LS has disappeared and a DS is ob-served, similar to the one observed in most of the virginsamples. The corresponding prevalent wavelength increasesfrom about 240 nm to almost 400 nm, a value on the DStrend.

Metastable structures may arise since the growth tem-perature of the layers is well below the Curie temperature ofthe alloy.18 Under these conditions, perpendicular magneticdomains could nucleate as soon as perpendicular magnetiza-tion sets in during growth. Their width therefore can be ex-pected to reflect the thickness at which they have nucleated,and thus should be smaller, for this particular case, with re-spect to the equilibrium one, as we do observe.

IV. PERPENDICULAR AND ANGLE DEPENDENTMAGNETIZATION AND SWITCHING PROCESSESIN SAMPLES OF INTERMEDIATE THICKNESS

To gain insight into the magnetic behavior in the inter-mediate thickness range, we focus on the 150 nm thicksample. Starting from the ac demagnetized state, initial mag-netization and demagnetization curves were measured withthe external field applied perpendicular to the film plane, asshown in detail in Fig. 10. In each measurement, the fieldwas increased to a varying maximum value and then returnedto zero. The minor loops thus obtained are superimposed

FIG. 8. Domain spatial period W as a function of the film thickness t: �a�prevalent period as determined by 2D FFT for the samples presenting adisordered domain structure �triangles� and for samples presenting a laby-rinth structure �circles�; �b� equilibrium spatial period as determined substi-tuting experimentally determined parameters in Eq. �1� �continuous line�; �c�equilibrium period as determined by full bidimensional micromagnetic com-putations �dashed line�. In all cases no adjustable parameters wereemployed.

FIG. 9. Cross-sectional view of the magnetic moment distribution as deter-mined by full 2D micromagnetic computations for different film thicknesses.

103911-4 Ghidini et al. J. Appl. Phys. 100, 103911 �2006�

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with the first quadrant major loop of Fig. 5. All the magne-tization curves display an extremely well defined criticalfield, Hex=Hcr

i , about 0.4 MA/m, at which an abrupt slopechange occurs. For Hex�Hcr

i , the magnetization process isreversible, displaying constant and almost negligible totalsusceptibility. For Hex�Hcr

i , irreversible processes becomedominant. The total susceptibility strongly increases but re-mains constant almost up to saturation.

Successively, we combine the information gathered frommagnetometry with MFM imaging in varying external fieldsstarting from the ac demagnetized state. In a first set of ex-periments, after carefully selecting a sample area withoutgross topographic features, thus representative for most partof the film, we have followed in situ the evolution of themicromagnetic structure of the same sample area by MFM inlow, varying, external magnetic fields �up to about 64 kA/m,at intervals of 8 kA/m� applied perpendicular to the filmplane by a coil under the microscope. No variation of themagnetic contrast with respect to the ac demagnetized state�Fig. 11�a�� could be observed in this set of experiments,consistent with the negligible initial susceptibility. We have

FIG. 10. Virgin magnetization and demagnetization curves performed on the150 nm thick film, starting from the ac demagnetized state, for different,increasing, maximum fields.

FIG. 11. �Color online� MFM images of �a� ac demag-netized state, �b� remanent state after magnetizing thesample in a 0.48 MA/m field, �c� remanent state aftermagnetizing the sample in a 0.8 MA/m field, and �d�remanent state after magnetizing the sample in a1.12 MA/m field �scan area is 2�2 �m2 in all cases�.Histograms and simplified domain patterns in black andwhite contrast are also shown.

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then imaged the remanent states obtained after following thevirgin magnetization curves at increasing maximum fields.Figures 11�b�–11�d� report MFM images �scan areas are 2�2 �m2� for such remanent states after application of 0.48,0.8, and 1.12 MA/m, respectively.

For each micrograph, we also report the correspondingpixel histogram, together with the simplified domain pattern,in black and white contrast. We recall that the sign of thephase shift indicates the orientation of the sample stray fieldgradient, and hence of the underlying magnetic moments,with respect to the magnetic dipole of the tip. In the case ofthe ac demagnetized sample, we repeated image acquisitionseveral times, in different parts of the sample, and we foundthat the number of pixels N+ with a positive phase shift, withrespect to the excitation reference signal, was equal to thenumber of pixels N− with a negative phase shift, within 10−3

accuracy. Correspondingly, as shown in Fig. 11�a�, an almostperfectly symmetric histogram is obtained, with zero meanvalue.

Modifications of the domain pattern are observed onlyafter application of fields greater than Hcr

i as demonstrated bythe histograms, which are progressively shifted towardsnegative phase shifts �Figs. 11�b�–11�d�� due to the progres-sive expansion of the areas magnetized parallel to the previ-ously applied field �antiparallel to the tip�. The effect is moreevident in the simplified domain patterns in black and whitecontrast. Figure 12 compares the applied field dependence ofthe measured Mr /Ms ratio and of �N /N= �N−−N+� / �N−

+N+� as deduced by MFM analysis. Although a significantqualitative agreement is found, quantitatively the data dis-agree, suggesting that the surface magnetization state differssomewhat from that in the sample inner volume. Strong de-viations between the bulk and surface magnetizations are awell-known phenomenon, yet not quantitatively investigatedeven in bulk uniaxial crystals,19 which can be due to theformation of closure domains. In fact, the stray field energyat the surface is strongly reduced by the formation of tinyopposite domains finely distributed at the surface �branchingof surface domains�. As a consequence, the bulk domain dis-tribution is hidden by a surface domain pattern, which pre-sents a peculiar reluctance to change configuration as an ex-ternal field is applied.

Angle dependent measurements are in principle a valu-able method to gain insight into the magnetization process,since they allow direct comparisons to be made with thepredictions about the switching field angle dependence basedon different fundamental magnetization mechanisms, such ascoherent rotation, curling, and domain wall processes, eithernucleation or pinning.

Let us first illustrate the general features of the hysteresisloops measured with the external field applied at varyingangles with respect to the film plane. Figure 13 displays theloops measured at various angles between the easy axisperpendicular to the film and the applied field positive direc-tion, with varying in the range of 0°–90°, by 10° steps.Detailed views of selected loops are displayed in Fig. 14. Acoherent evolution of the loops can be observed as the fielddirection varies from the easy out-of-plane to the hard in-plane direction. For increasing angles, all curves presentsimilarities in their main features. In particular, it seems thatdemagnetization develops into two distinct stages. In fact,

FIG. 12. Comparison between the maximum applied field dependence of thebulk Mr /Ms ratio �closed circles� and of �N /N �open circles� as deduced bythe MFM analysis.

FIG. 13. Hysteresis loops at various angles , varying in the range of0°–90° with 10° steps, between the perpendicular easy axis of the film andthe direction of the external field. Upper inset: angular dependence of thecoercive field. Lower inset: angular dependence of the switching field h2

=H2 /HA.

FIG. 14. Detailed view of the first quadrant part of selected hysteresis loopsmeasured at different angles . For a fixed angle, the same closed and opensymbols are used to indicate the demagnetization and remagnetization por-tions of the loop, respectively.

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coming down from saturation, a progressive change in slopeis observed around a certain critical field H1, followed by aquasilinear decrease down to a second critical field H2, atwhich dominant irreversible demagnetization processes start�inset of Fig. 5�. While H1 increases with tilting angle , theswitching field H2 can be seen to decrease monotonically. Itis remarkable that the slight curvature of the intermediatesection of the demagnetization curve comprised between H1

and H2 is constant over a wide field range and practicallyindependent of angle . For external fields Hex�H2, thedemagnetization curves are linear with a constant steeperslope. Concerning the field H1, we can only conjecture that itcould be connected with the nucleation of the surface closuredomain pattern. The second field H2 denotes the onset ofirreversible demagnetization processes probably due to irre-versible domain wall displacements, involving the sampleinner volume. We focus, therefore, the attention on the fieldH2, which can be extracted from each curve as the fieldcorresponding to maximum slope changes. Remarkably, themonotonic decrease of H2 with increasing angle �lower insetof Fig. 13� occurs in the first quadrant. The coercive fieldfollows a similar behavior �upper inset of Fig. 13�.

The angular variation of the Mr /Ms ratio is compared inFig. 15 with �Mr�0� /Ms�cos which obviously is the depen-dence we would expect if the remanent state would be thesame for all even in the presence of an easy-axis distribu-tion function.20 At angles in the range of 0°–25°, the rema-nent magnetization deviates from the cos dependence andremains almost constant. In the corresponding loops this im-plies that the same Mr value is reached after different pathsfrom saturation. This behavior is probably related to the pre-viously described branching of the surface domains, due toirreversible rotations or rearrangements of the closure do-main patterns, when magnetizing at angles different fromzero. Agreement is reasonable in the intermediate regionfrom 40° to 80°, while significant deviations are again ob-served at large angles up to =90°, the hard direction in thefilm plane, for which a remanence of about 9% is observed.The latter deviation from the cos dependence can be un-derstood as follows. Let us suppose that the easy axes of thecrystallites are distributed uniformly inside a double cone ofsemiaperture �0. In the demagnetized state, the magnetiza-tion of each crystallite will lie along one or the other of thetwo degenerate easy-axis directions, so that the two coneswill be equally populated. It can be argued that, after satura-

tion of the moments along the applied field, as the field isprogressively reduced towards zero, the magnetization insidethe individual crystallites will choose among the two degen-erate directions the one closer to the field direction. For ex-ample, when =90° �field in the film plane�, this assumptionimplies that half of the crystallites have reversed their mag-netization direction along the field orientation �y axis, as inFig. 16� with respect to the demagnetized state. For =90°the reduced remanent magnetization my can be calculated asfollows:

�my� = 20

�0�0 sin2 � sin d�d

0�0

�0 sin �d�d=

�0 − �1/2�sin 2�0

��1 − cos �0�, �2�

where and � are the azimuthal and polar angles, respec-tively, of the crystallite c axis in the cone, having chosen x asthe polar axis �Fig. 16�. By equating Eq. �2� to the experi-mentally determined remanence at =90°, one obtains �0

=6.1°. On the basis of the above explained effect, we expectthat deviations from the cos dependence will start at =90°−�084°, a value which turns out to be consistent withthe experimental results.

In the case of bulk hard materials, the analysis of theangular dependence of the switching field is a well-knownmethod for distinguishing between rotations and domain wallprocesses, either nucleation or pinning. Even in the presentcase dealing with hard thin films with a quite complex be-havior, we are at least interested in identifying the nature ofthe switching mechanism responsible for irreversible pro-cesses. In the case of perpendicular films, out-of-plane mea-surements are required, and large variations of the intensityand orientation of the internal field must be taken into ac-count as the operating angle varies. We have treated thisproblem for several classical models, developing suitableprocedures to reconcile the predicted angular variation in in-ternal field with the experimental data.21 Here we willpresent only the specific model used to interpret the measure-ments on the Co-Pt film under study.

As far as rotations are concerned, we have utilized amodified Stoner-Wohlfarth model, including a uniaxial mag-

FIG. 16. Schematic diagram illustrating the coordinate system and symbolsused in the text.

FIG. 15. Angular variation of the measured magnetization at remanence�open squares� compared with �Mr�0� /Ms�cos �closed squares�.

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netocrystalline anisotropy characterized by a constant K1.22

In this model, the switching field angular dependence is ex-pressed in terms of the external field and thus can be directlycompared with experiment. For a sample in the shape of anellipsoidal single-domain particle, with its c axis parallel tothe ellipsoid short axis, it can be shown that the followingexpression holds for the reduced external switching field:

hec =Hec

HA= −

1

�cos2/3 + sin2/3 �3/2 , �3�

where HA=2K1 /�0Ms, =1+ny −nx�1, ni=NiMs /HA

=Ni /Q with i=x ,y �Fig. 16�, and Nx=1 and Ny =0 for a film.It should be noted that �i� Eq. �3� differs from the Stoner-Wohlfarth model only for the presence of the multiplyingfactor which scales down the external switching field val-ues, due to the competition between shape and magnetocrys-talline anisotropy, and �ii� the magnetization is expected toswitch at negative external fields. From a comparison be-tween the predictions of this basic mechanism �Fig. 17� andour experiments, no evidence about a role played by rota-tions in the switching process can be extracted.

The situation is more complicated in the case of a do-main wall process. In fact, in this case, the effective criticalfield, nucleating an inverse domain or unpinning domainwalls, is expressed in terms of the internal field, and itsevaluation is generally very difficult, especially in thin filmsbecause the shape anisotropy is maximum and because of thestrong effect of the surface microstructure. In uniaxial mate-rials the classical Kondorsky relation is usually employed:

hp�� =hp�0�cos

, �4�

where by hp�� we denote the easy-axis component of theinternal switching field, which is obviously negative. This isthe dependence which is expected by the mechanisms ofnucleation/propagation of an inverse domain23 and ofpinning22,24 for the case �hp��1. In the case of perpendicularthin films, one would expect to observe directly in externalfield a behavior close to Eq. �4�, only for very hard materials�Q�1�. In the case of Q1, the effect of the demagnetizingfield is paramount and is expected to alter significantly theswitching field angular dependence in external representa-

tion. To understand to which extent the demagnetizing fieldcan modify Eq. �4� for the film under study, or in simpleterms, to figure out how Eq. �4� would look in terms of theexternal field, we express the component of the internal fieldalong the c axis under the simple assumption of a homoge-neous demagnetizing field:

H = H · x̂ = �He + Hd� · x̂

= �Hel̂ − �M · x̂�x̂� · x̂ = �He cos − Ms cos � ,

�5�

or, in reduced variables:

h = he cos −1

Qcos , �6�

where is the orientation angle of Ms, which turns out to bea function of he �Fig. 16�. We are interested in determiningfor each the critical value hep of he, corresponding to thepropagation internal field h=hp. For this purpose we needanother equation, which we obtain assuming reversal to takeplace in two steps. At first, starting from saturation and fordecreasing he, the magnetization rotates freely and revers-ibly, until a certain fixed value hp for the internal field isreached. At this point irreversible switching of the magneti-zation takes place, due to domain wall propagation.

To describe the rotation process we write the energy ofthe film as

E = K1 sin2 − �0MsHe cos� − � +1

2�0Ms

2 cos2 , �7�

or in reduced variables,

e =1

2sin2 − he cos� − � +

1

2Qcos2 . �7��

Imposing the condition �e /� =0 yields the stability equa-tions describing the rotation process at all angles, fromwhich the dependence of on he can be extracted:

�Q − 1

Q cos sin + he sin� − � = 0. �8�

Corresponding to the measured value H2 for =0, weestimate the internal field h2 to lie in the range of −0.25 to−0.37, considering the values of the parameters estimated inSec. III and the associated uncertainties. We therefore as-sume hp=−0.3 as the internal critical threshold for irrevers-ible switching.

As a result of the outlined calculation, we obtain theexternal propagation field hep for which the switching occurs.Repeating the calculation for different , we calculate theangular dependence of hep, which can then be compared withexperiment.

Figure 18 reports the calculated curves for Q=3, Q=1.5, and Q=1.05. The hep�0� / cos curve is also reportedfor reference for the case Q=3. As expected, for Q=3, theeffect of the demagnetizing field is small and therefore theangular dependence is in good agreement with the 1/cos dependence over a wide interval. The agreement can be fur-ther improved by increasing Q. As the Q is progressively

FIG. 17. Plot of the angular dependence of the switching field calculatedwith Eq. �3�, on the basis of the modified Stoner-Wohlfarth model with theexperimental parameters of the studied Co-Pt film.

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decreased, strong deviations take place. For Q=1.5, the ex-ternal critical field is almost constant for low angles, andafter reaching a maximum, sharply decreases. Only for Q=1.05 is a monotonic decrease obtained, qualitatively inagreement with experiment. The present model is certainlynot fully adequate for reproducing the complex features ofthe system, since, for example, while this two-step picturecorresponds well to the existence of a well-defined criticalfield H2, marking the onset of irreversible processes �inset ofFig. 5�, we have neglected the effect of the first demagneti-zation stage, taking place between the fields H1 and H2. Nev-ertheless it is worth noting that we reproduce various salientfeatures: �i� we obtain positive switching fields, �ii� we ob-tain reasonable absolute values of the switching field as com-pared with experiment, and �iii� we reproduce qualitativelythe angular dependence of the switching field using a Q pa-rameter value only slightly varied beyond the uncertaintyinterval of the value estimated experimentally.

If, as it can be expected, the quantitative discrepanciesare due to the presence of the robust demagnetization processinvolving surface domains discussed earlier and neglected bythe present model, it is likely that performing the same typeof analysis on thicker films, where surface effects should beless important, would yield even better agreement with ex-periment.

V. CONCLUSIONS

We have presented a study of the magnetic properties ofthe perpendicular and angle dependent magnetization andswitching processes in electrodeposited Co-rich Co-Pt films.Co–Pt films �Pt�20 at. % �, with thickness t ranging from5 nm up to 2 �m, were electrodeposited onto a suitablestacked structure. Crystalline structure and morphology wereinvestigated by XRD, scanning electron microscopy �SEM�,and AFM, respectively. The Co–Pt films were found to bepolycrystalline and to exhibit a highly oriented hcp structure,with the c axes of the individual grains perpendicular to thesubstrate. All samples displayed a fine grained surface mi-

crostructure with an apparent grain size of about 50 nm. Thesamples presented strong perpendicular magnetic anisotropyand coercivity.

The domain structure was investigated by MFM. Theequilibrium domain width dependence on film thickness wasdetermined. Full 2D micromagnetic computations allowed toreproduce the experimental results with no adjustable param-eters.

The magnetization and switching processes were inves-tigated by virgin and demagnetization curves, MFM in vari-able fields, and angle dependent hysteresis loop measure-ments. MFM in variable fields and virgin magnetizationcurves show that domain growth takes place only above acritical field Hcr

i , about 0.4 MA/m, below which a negligiblesusceptibility is observed. For Hex�Hcr

i , no modifications ofthe domain pattern, in the same sample area, could be ob-served. This clear experimental evidence, together with thecharacteristic two-stage feature of the virgin magnetizationcurve, with low initial susceptibility, typical of a pinning-type magnet,25 suggests that unpinning of domain walls isthe dominant process. The domain walls are likely to bepinned to microstructural heterogeneities or grain boundariesuntil the critical field value is reached, after which the mag-netization increases at constant rate. It is noteworthy that Hcr

i ,which is only slightly affected by the demagnetizing field, isfound to be in reasonable agreement with the internal switch-ing field h2, after which the demagnetization proceeds at con-stant rate. Our thorough, quantitative analysis of the angulardependence of the switching field allowed to exclude rota-tions, yielding results consistent with a pinning demagneti-zation mechanism.

ACKNOWLEDGMENTS

The present work has been supported by a FIRB projectof the Italian Ministry of Education and Research, entitled“Microsystems based on magnetic materials structured at thenanoscopic scale.” The extensive use of the scanning probemicroscope at the Centro Interfacoltà Misure of the Univer-sity of Parma is acknowledged. The work of two of the au-thors �G.Z. and G.P.� was supported by NSF, under GrantNo. DMR 0314233.

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FIG. 18. External propagation field hep vs angle deduced from Eq. �6�with hp=−0.3: �a� calculated angular dependence for Q=1.05 �crosses anddashed line as a guide to the eyes�; �b� calculated angular dependence forQ=1.5 �triangles and full line as a guide to the eyes�; �c� calculated angulardependence for Q=3 �closed circles� compared with hep�0� / cos �thick fullline�.

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