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Modeling of the iron irradiation effect on the hysteresis in nanostructured thin films
ARTICLE IN PRESS
Physica B 343 (2004) 384–389
Modeling of the iron irradiation effect on the hysteresis in nanostructured thin films D. Bajalana, H. Hausera,b,*, P.L. Fulmeka,c a
Institute of Industrial Electronics and Material Science, Vienna University of Technology, Gusshausstrasse 27–29/366, A-1040 Vienna, Austria b Siemens AG, Corporate Technology, Materials and Manufacturing, Paul Gossen Strasse 100/CTMM1, D-91052 Erlangen, Germany c Christian Doppler Laboratory for Automotive Measurement Research, Gusshausstrasse 27–29, A-1040 Vienna, Austria
Abstract Elongated ferromagnetic nanoparticles have been formed by heavy ion irradiation of paramagnetic YCo2 thin films. The fluence affects both anisotropy and spontaneous magnetization. The measured effect on the shape of the hysteresis curve, for example initial susceptibility and coercivity, is predicted by the energetic model. r 2003 Elsevier B.V. All rights reserved. PACS: 75.60.d; 75.60.Ej; 75.75.+a Keywords: Nanoparticles; Ion irradiation; YCo2 thin film; Hysteresis model; Spontaneous magnetization
1. Introduction Magnetic nanostructures are subjects of growing interest because of their potential applications in high-density magnetic recording media and their original magnetic properties [1]. Multilayer thin films (like Co/Pt) are well known for their high magnetic anisotropy, and the origin of this high magnetic anisotropy has been the subject of interest for many researchers [2]. Demands for the continuous increase in the data storage density bring the challenge to overcome physical limits for currently used magnetic recording media [3]. Patterned magnetic media could be a way of realizing ultrahigh density storage media. Recently, *Corresponding author. Tel.: +43-1-58801-36614; fax: +431-58801-36695. E-mail address: [email protected] (H. Hauser).
demonstrations of areal recording density over 60 Gb=in2 in both longitudinal and perpendicular magnetic recordings have been successfully made [4]. Determining the properties of small magnetic structures is extremely important for the development of data storage devices [5]. Better understanding of the micromagnetic processes in magnetic recording media is essential for developing novel materials for future ultrahigh density recording [6]. Good understanding of the noise mechanism in magnetic recording is required for developing heads and media for future applications [7]. In perpendicular recording, the magnetization pattern corresponding to the bits is provided perpendicular to the plane of the medium. The information is being stored in vertical domains or other structures of uniform magnetization [8]. The magnetic properties of an ultrathin multilayer can
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.074
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
385
be patterned by controlled ion beam irradiation [9]. The basic step in this technique is to control the changes in the magnetic properties induced by the irradiation process. In magnetic materials two characteristic length scales have to be considered [10]: *
*
at the atomic level, nearest-neighbour exchange interaction is dominating, at a mesoscopic level, the domain wall width is the characteristic length dominating the magnetization reversal.
When the physical dimensions of a system become comparable to the interatomic spacing, strong modifications of the intrinsic magnetic properties (ordering temperature, magnetic anisotropy, spontaneous magnetization) are expected. Micromagnetic modelling of the behaviour of a nanostructured film beautifully describes the magnetization process, but requires a high calculational effort and long computation times. Furthermore, it is difficult to predict changes of the macroscopic physical behaviour due to variation of parameters. Phenomenological models, on the other hand, are very useful to simulate the behaviour of the magnetic material under the influence of varying parameters, especially when the parameters are based on physical constants.
Fig. 1. Hysteresis of YCo2 thin films at T ¼ 10 K perpendicular (solid —) and parallel (dashed - -) to the film plane after U ion irradiation with 1012 (above), 5 1012 (middle), and 2 1013 ions=cm2 (below).
2. Experiments An assembly of ferromagnetic amorphous nanoparticles has been prepared by heavy ions irradiation of paramagnetic YCo2 thin films [11,12]. Several irradiation experiments carried out on YCo2 samples have shown that fluences on the order 1012 U ions=cm2 causes changes in magnetic properties of the samples [12]. Important changes are reported to take place after the irradiation: *
*
change of spontaneous magnetization, coercivity and initial susceptibility [12], and a distinct change of the anisotropy perpendicular to the film plane [11].
Fig. 1 shows the measured hysteresis loops of three YCo2 thin film samples at T ¼ 10 K perpendicular and parallel to the film plane after U ion irradiation with different fluences [11].
3. Energetic model The magnetic behaviour of magnetic moments is mainly described by the well-known equations of . Schrodinger and Heisenberg (exchange interaction) and Landau, Lifshitz, and Gilbert (dynamics of magnetization reversal). Above this fundament
ARTICLE IN PRESS 386
D. Bajalan et al. / Physica B 343 (2004) 384–389
is the shell of the physical constants describing spontaneous magnetization, anisotropy, magnetostriction, etc. The energetic model (EM) is designed as an interface between this shell and the macroscopic hysteresis phenomenon, able to predict many magnetic properties due to the relation of the parameters with the physical constants. The EM has been applied for different magnetization processes and materials [13–17]. The hysteresis of the magnetization M depending on the applied field H is described by the following equations, with the spontaneous magnetization Ms ; the geometrical demagnetizing factor Nd ; and the following phenomenological parameters: 1. g (1) related to anisotropy, reversible processes; 2. h (A/m) related to saturation field Hs ; reversible processes; 3. k ðJ=m3 Þ related to static hysteresis loss, irreversible processes; 4. q (1) related to pinning site density, irreversible processes. In the cases of large domains, the microscopic constant cr describes the influence of reversal speed. The sgnðxÞ function provides the correct four-quadrant calculation (with the related magnetization m ¼ M=Ms ): H ¼ Hd þ sgnðmÞHR þ sgnðm m0 ÞHI :
ð1Þ
The first term of Eq. (1) describes linear material behaviour, using the demagnetizing field Hd ¼ Nd Ms m;
For the initial magnetization, beginning with M ¼ 0; H ¼ 0; we set m0 ¼ 0 and k ¼ 1: The function k describes the influence of the total magnetic state at points of magnetization reversal. Therefore, k (previous value k0 ) depends on the unit magnetization reversals s ¼ jm m0 j up to this point of field reversal (m0 is the starting value of m at the last field reversal) with the simplification eq 51: q k ¼ 2 k0 exp jm m0 j : ð5Þ k0 The calculation always starts with the initial magnetization curve and m is increased stepwise (the stepwidth determines the desired resolution of the calculation), which gives the corresponding field by Eq. (1). At a point of field, reversal k is calculated by Eq. (5) and m0 is set to the actual value of m at this point. Then m is decreased stepwise until the next reversal point, etc. 3.1. Identification The identification of the EM with measurements or data sheets can be done easily. At given Ms and Nd the parameters are directly calculated from special points of the hysteresis loop. Considering reference conditions, the index 0 is to indicate that the identification is done at a temperature T ¼ T0 without applied mechanical stress s; using the following equations: k0 ¼ m0 Ms Hc ;
ð6Þ
ð2Þ M s 1 N d w0 : Hc w0
the second term represents non-linear behaviour using the reversible field
q0 ¼
HR ¼ h½ðð1 þ mÞ1þm ð1 mÞ1m Þg=2 1;
If w0 is not available R þM one can also R Muse the total static losses wl ¼ Mss H dM þ þMss H dM corresponding to the area of the closed major loop (upper and lower branch of hysteresis) 2 wl ¼ 4k 1 ð8Þ q
ð3Þ
including saturation at a field Hs ðMs Þ; and the third term describes hysteresis effects like initial susceptibility w0 ; remanence Mr ; coercivity Hc ; static losses, and accommodation, using the irreversible field k HI ¼ þ cr H R m0 M s h q i 1 k exp jm m0 j : ð4Þ k
ð7Þ
and we can write the equation for q0 as q0 ¼
8m0 Ms Hc : 4m0 Ms Hc wl
ð9Þ
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
These relations allow even an estimation of Ms (at cr E0), using Eqs. (6), (7), and (9) to Ms ¼
2w0 Hc wl þ : 1 Nd w0 4m0 Hc
ð10Þ
Furthermore, q0 can also be determined by the reduced remanence mr of the upper branch of a loop with the measured reduced maximum magnetization mm : Using fq as a factor related to mr ; fq ¼ ½ð1 þ mr Þ1þmr ð1 mr Þ1mr g0 =2 1;
ð11Þ
we identify q0 as 2 2Hc q0 ¼ ln : mm mr Hc h0 fq þ Nd Ms mr
ð12Þ
By using fg as a factor related to mg which is the reduced magnetization at H ¼ Hg ; in the knee of the lower branch of the hysteresis, fg ¼
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1þmg ln ð1 þ mg Þ ð1 mg Þ1mg ln 2
ð13Þ
hence g0 is g0 ¼ fg ln
Hg Hc Nd M s mg : Hs Hc Nd M s
Using fc as a factor related to mr and mm ; h m m i r m fc ¼ 1 2 exp q0 ; 2
ð15Þ
ð16Þ Finally, the identification equation of h0 is H s H c N d Ms : ðcr þ 1Þðexp½g0 ln 2 1Þ
we find Hs ¼ ðHm Hc Nd Ms mm Þ Hm Hc Nd Ms mm fh : Hg Hc Nd Ms mg
ð19Þ
If Nd of the experimental arrangement is unknown then it can be estimated roughly by the differential susceptibility wc at coercivity of a measured hysteresis loop:
1
Nd E : ð20Þ wc H¼Hc If Nd of the sample is rather large so that the magnetization curve is strongly sheared (Mr Nd > Hc ), then it can be necessary to identify g0 and cr by the backsheared curve (Nd ¼ 0). 3.2. Parameter dependence on anisotropy
fq ðHs Hc Nd Ms Þ=Ms exp g0 ln 2 fc ðHc =Ms Þ þ Nd mr : ðfq ðHs Hc Nd Ms Þ=Ms exp g0 ln 2 þ fc ðHc =Ms ÞÞ Nd mr
h0 ¼
ð18Þ
ð14Þ
the microscopic constant describing the domain (grain) geometry ratio becomes cr ¼
factor related to mg and mm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ð1 þ mm Þ1þmm ð1 mm Þ1mm ln 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; fh ¼ ð1 þ mg Þ1þmg ð1 mg Þ1mg ln ð1 þ mm Þ1þmm ð1 mm Þ1mm
387
ð17Þ
If Hs is not available, one can estimate Hs from the measured maximum field Hm at mm using the approximation Hs bHc þ Nd Ms : Using fh as a
It is possible to relate the model parameters to anisotropy (e.g. magnetocrystalline, stress). Depending on the inner structure of the considered material, these relations will differ from case to case. The following calculations are general and based on classical principles, basically indicating the possibilities of the EM. The law of approach to saturation (LAS) a b M ¼ Ms 1 2 y ð21Þ H H describes the rotation of the domain magnetization against the anisotropy energy at strong fields, for cubic anisotropy (constant K1 > 0) as an example with 8 K1 2 ¼ eHk2 ð22Þ b¼ 105 m0 Ms and a ¼ 0; where Hk ¼ 2K1 =m0 Ms is the fictive anisotropy field. To find a relation between LAS
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
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and EM, the reciprocal differential susceptibility at H ¼ Hk dH 1 Hk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ dM 2e 3 2 b=ð1 mÞ
ð23Þ
is set equal to the derivation of Eq. (1) near saturation with m ¼ 1 e; HR bHI þ Hd : dHR gh 2e g 2 EHs ln : ð24Þ ¼ ½ð2 eÞ2e ee g=2 ln 2 e 2 e dM With Eq. (17) and the approximations Hs bHc þ Nd Ms and 2g b1 which are valid for most magnetic materials, the result is g¼
2K1 ln 2=e : m0 Ms Hs e
ð25Þ
Using the microscopic constants ch ¼
Hs Ms
ð26Þ
2ch 2 ln ; e e
ð27Þ
Fig. 2. Calculated hysteresis loops for YCo2 thin films with ion irradiation perpendicular to film plane, related to Ms;0 ¼ 40 kA=m (—), maximum field Hm ¼ 160 kA=m; the values Ms ¼ 60 kA=m (- - -) and Ms ¼ 20 kA=m (– –) have been used to calculate the other loops without any other parameter change.
and cg ¼
we find that Dwk g ¼ cg m0 Ms2
ð28Þ
is proportional to the anisotropy energy difference Dwk which has to be overcome during magnetization reversal. Other calculations [19] are related to the differential susceptibilities of LAS and EM at H-Hs which are valid in a narrow range only. 3.3. Calculation The calculations have been done as following: At a given Nd ¼ 0:47; the parameters g0 ¼ 5:24; h0 ¼ 2:79 kA=m; k0 ¼ 1:10 kJ=m3 ; and q0 ¼ 8:79 are identified for the perpendicular hysteresis at F ¼ 5 1012 ions=cm2 with Ms ¼ 20 kA=m (see Fig. 1, middle). In the next step we vary only Ms in order to calculate the hysteresis of the other irradiation cases. Using Eqs. (6) and (7), we find the dependencies Hc ¼
k0 m0 Ms
ð29Þ
Table 1 Macroscopic hysteresis features depending on Ms ; measured (M) and modelled (EM) values F Ms w0 (EM)
ðions=cm2 Þ (kA/m) (1)
1012 20 0.157
5 1012 40 0.199
2 1013 60 0.536
Hc (M) Hc (EM)
(kA/m) (kA/m)
50.3 43.6
17.5 21.8
9.2 13.7
Mr (M) Mr (EM)
(kA/m) (kA/m)
15.6 14.9
22.2 22.5
24.7 25.9
Mm (M) Mm (EM)
(kA/m) (kA/m)
19.7 19.6
39.9 39.6
60.0 60.0
and w0 ¼
m0 Ms2 ; k0 q0 þ m0 Ms2 Nd
ð30Þ
which strongly affects the shape of the hysteresis as shown in Fig. 2. No other change of the parameter values has been done. The calculation of the hysteresis under highest fluence fits the parallel
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
case better than the perpendicular because of a change of anisotropy due to fluence. The parameter dependence on anisotropy has not been considered in these calculations. Table 1 compares some modelled perpendicular macroscopic hysteresis features with the measurements. The modelling of the parallel hysteresis— considering only the different anisotropy energy for the parameter identification—is currently under progress.
4. Conclusions The rapid development of magnetic recording leads to a large increase of the bit density. Multilayer thin films with perpendicular magnetic anisotropy devices may play an active role in the development and establishment of future storage technologies. Patterning magnetic media is a potential solution for ultrahigh density magnetic recording [18]. Ion beam modification of magnetic layers may be the possible future of ultrahigh density magnetic recording media. After ion irradiation of YCo2 thin films with different fluence values, the measured magnetization curves clearly show a perpendicular shape anisotropy [11]. The shape of the hysteresis loops depends strongly on Ms ; which is predicted by the EM. It turns out that Hc is inversely proportional to Ms and w0 is proportional to Ms2 ; if Nd is neglected. As the EM parameters are also related to anisotropy it will be possible also to calculate the direction of dependence of these magnetization curves, which is subject to further work.
389
References [1] M. Jamet, W. Wernsdorfer, C. Thirion, D. Mailly, V. Dupuis, P. M!elinon, A. P!erez, Phys. Rev. Lett. 86 (2001) 4676. [2] Y. Yamada, T. Suzuki, H. Kanazawa, J.C. Osterman, J. Appl. Phys. 85 (1999) 5094. [3] A. Lapicki, K. Kang, T. Suzuki, IEEE Trans. Magn. 38 (2002) 2589. [4] C. Papusoi, T. Suzuki, J. Magn. Magn. Mater. 240 (2002) 568. [5] K.J. Kirk, J.N. Chapman, C.D.W. Wilkinson, J. Appl. Phys. 85 (1999) 5237. [6] H. Kisker, N. Abarra, Y. Yamada, P. Glijer, T. Suzuki, J. Appl. Phys. 81 (1997) 3937. [7] G.N. Phillips, T. Suzuki, K. Takano, M. Takahashi, J. Magn. Magn. Mater. 193 (1999) 434. [8] T. Suzuki, Magneto-optic recording thin films, in: I.M.H. Francombe (Ed.), Handbook of Thin Film Devices Magnetic, Vol. 4, Magnetic Thin Film Devices, Academic Press, New York, 2000. [9] J. Ferre, C. Chappert, H. Bernas, J.-P. Jamet, P. Meyer, O. Kaitasov, S. Lemerle, V. Mathet, F. Rousseaux, H. Launois, J. Magn. Magn. Mater. 198 (1999) 191. [10] J.P. Nozi!eres, M. Ghidini, N.M. Dempsey, B. Gervais, D. Givord, G. Suran, J.M.D. Coey, Nucl. Instr. and Meth. B 146 (1998) 250. [11] D. Givord, J.P. Nozi!eres, M. Ghidini, B. Gervais, Y. Otani, J. Magn. Magn. Mater. 148 (1995) 253. [12] M. Solzi, M. Ghidini, G. Asti, Magn. Nanostruct. 4 (2002) 123. [13] D.R. Cornejo, A. Azevedo, S.M. Rezende, J. Appl. Phys. 93 (2003) 6623. [14] P.L. Fulmek, H. Hauser, J. Magn. Magn. Mater. 157 (1996) 361. . [15] H. Hauser, R. Grossinger, J. Appl. Phys. 85 (1999) 5133. [16] P. Andrei, L. Stoleriu, H. Hauser, J. Appl. Phys. 87 (2000) 6555. . [17] R. Grossinger, H. Hauser, M. Dahlgren, J. Fidler, Physica B 275 (2000) 248. [18] T. Devolder, C. Chappert, Y. Chen, E. Cambril, H. Launois, H. Bernas, J. Ferre, J.P. Jamet, J. Vac. Sci. Technol. B 17 (1999) 3177. [19] H. Hauser, Sensors and Actuators 106 (2003) 84.
Physica B 343 (2004) 384–389
Modeling of the iron irradiation effect on the hysteresis in nanostructured thin films D. Bajalana, H. Hausera,b,*, P.L. Fulmeka,c a
Institute of Industrial Electronics and Material Science, Vienna University of Technology, Gusshausstrasse 27–29/366, A-1040 Vienna, Austria b Siemens AG, Corporate Technology, Materials and Manufacturing, Paul Gossen Strasse 100/CTMM1, D-91052 Erlangen, Germany c Christian Doppler Laboratory for Automotive Measurement Research, Gusshausstrasse 27–29, A-1040 Vienna, Austria
Abstract Elongated ferromagnetic nanoparticles have been formed by heavy ion irradiation of paramagnetic YCo2 thin films. The fluence affects both anisotropy and spontaneous magnetization. The measured effect on the shape of the hysteresis curve, for example initial susceptibility and coercivity, is predicted by the energetic model. r 2003 Elsevier B.V. All rights reserved. PACS: 75.60.d; 75.60.Ej; 75.75.+a Keywords: Nanoparticles; Ion irradiation; YCo2 thin film; Hysteresis model; Spontaneous magnetization
1. Introduction Magnetic nanostructures are subjects of growing interest because of their potential applications in high-density magnetic recording media and their original magnetic properties [1]. Multilayer thin films (like Co/Pt) are well known for their high magnetic anisotropy, and the origin of this high magnetic anisotropy has been the subject of interest for many researchers [2]. Demands for the continuous increase in the data storage density bring the challenge to overcome physical limits for currently used magnetic recording media [3]. Patterned magnetic media could be a way of realizing ultrahigh density storage media. Recently, *Corresponding author. Tel.: +43-1-58801-36614; fax: +431-58801-36695. E-mail address: [email protected] (H. Hauser).
demonstrations of areal recording density over 60 Gb=in2 in both longitudinal and perpendicular magnetic recordings have been successfully made [4]. Determining the properties of small magnetic structures is extremely important for the development of data storage devices [5]. Better understanding of the micromagnetic processes in magnetic recording media is essential for developing novel materials for future ultrahigh density recording [6]. Good understanding of the noise mechanism in magnetic recording is required for developing heads and media for future applications [7]. In perpendicular recording, the magnetization pattern corresponding to the bits is provided perpendicular to the plane of the medium. The information is being stored in vertical domains or other structures of uniform magnetization [8]. The magnetic properties of an ultrathin multilayer can
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.074
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
385
be patterned by controlled ion beam irradiation [9]. The basic step in this technique is to control the changes in the magnetic properties induced by the irradiation process. In magnetic materials two characteristic length scales have to be considered [10]: *
*
at the atomic level, nearest-neighbour exchange interaction is dominating, at a mesoscopic level, the domain wall width is the characteristic length dominating the magnetization reversal.
When the physical dimensions of a system become comparable to the interatomic spacing, strong modifications of the intrinsic magnetic properties (ordering temperature, magnetic anisotropy, spontaneous magnetization) are expected. Micromagnetic modelling of the behaviour of a nanostructured film beautifully describes the magnetization process, but requires a high calculational effort and long computation times. Furthermore, it is difficult to predict changes of the macroscopic physical behaviour due to variation of parameters. Phenomenological models, on the other hand, are very useful to simulate the behaviour of the magnetic material under the influence of varying parameters, especially when the parameters are based on physical constants.
Fig. 1. Hysteresis of YCo2 thin films at T ¼ 10 K perpendicular (solid —) and parallel (dashed - -) to the film plane after U ion irradiation with 1012 (above), 5 1012 (middle), and 2 1013 ions=cm2 (below).
2. Experiments An assembly of ferromagnetic amorphous nanoparticles has been prepared by heavy ions irradiation of paramagnetic YCo2 thin films [11,12]. Several irradiation experiments carried out on YCo2 samples have shown that fluences on the order 1012 U ions=cm2 causes changes in magnetic properties of the samples [12]. Important changes are reported to take place after the irradiation: *
*
change of spontaneous magnetization, coercivity and initial susceptibility [12], and a distinct change of the anisotropy perpendicular to the film plane [11].
Fig. 1 shows the measured hysteresis loops of three YCo2 thin film samples at T ¼ 10 K perpendicular and parallel to the film plane after U ion irradiation with different fluences [11].
3. Energetic model The magnetic behaviour of magnetic moments is mainly described by the well-known equations of . Schrodinger and Heisenberg (exchange interaction) and Landau, Lifshitz, and Gilbert (dynamics of magnetization reversal). Above this fundament
ARTICLE IN PRESS 386
D. Bajalan et al. / Physica B 343 (2004) 384–389
is the shell of the physical constants describing spontaneous magnetization, anisotropy, magnetostriction, etc. The energetic model (EM) is designed as an interface between this shell and the macroscopic hysteresis phenomenon, able to predict many magnetic properties due to the relation of the parameters with the physical constants. The EM has been applied for different magnetization processes and materials [13–17]. The hysteresis of the magnetization M depending on the applied field H is described by the following equations, with the spontaneous magnetization Ms ; the geometrical demagnetizing factor Nd ; and the following phenomenological parameters: 1. g (1) related to anisotropy, reversible processes; 2. h (A/m) related to saturation field Hs ; reversible processes; 3. k ðJ=m3 Þ related to static hysteresis loss, irreversible processes; 4. q (1) related to pinning site density, irreversible processes. In the cases of large domains, the microscopic constant cr describes the influence of reversal speed. The sgnðxÞ function provides the correct four-quadrant calculation (with the related magnetization m ¼ M=Ms ): H ¼ Hd þ sgnðmÞHR þ sgnðm m0 ÞHI :
ð1Þ
The first term of Eq. (1) describes linear material behaviour, using the demagnetizing field Hd ¼ Nd Ms m;
For the initial magnetization, beginning with M ¼ 0; H ¼ 0; we set m0 ¼ 0 and k ¼ 1: The function k describes the influence of the total magnetic state at points of magnetization reversal. Therefore, k (previous value k0 ) depends on the unit magnetization reversals s ¼ jm m0 j up to this point of field reversal (m0 is the starting value of m at the last field reversal) with the simplification eq 51: q k ¼ 2 k0 exp jm m0 j : ð5Þ k0 The calculation always starts with the initial magnetization curve and m is increased stepwise (the stepwidth determines the desired resolution of the calculation), which gives the corresponding field by Eq. (1). At a point of field, reversal k is calculated by Eq. (5) and m0 is set to the actual value of m at this point. Then m is decreased stepwise until the next reversal point, etc. 3.1. Identification The identification of the EM with measurements or data sheets can be done easily. At given Ms and Nd the parameters are directly calculated from special points of the hysteresis loop. Considering reference conditions, the index 0 is to indicate that the identification is done at a temperature T ¼ T0 without applied mechanical stress s; using the following equations: k0 ¼ m0 Ms Hc ;
ð6Þ
ð2Þ M s 1 N d w0 : Hc w0
the second term represents non-linear behaviour using the reversible field
q0 ¼
HR ¼ h½ðð1 þ mÞ1þm ð1 mÞ1m Þg=2 1;
If w0 is not available R þM one can also R Muse the total static losses wl ¼ Mss H dM þ þMss H dM corresponding to the area of the closed major loop (upper and lower branch of hysteresis) 2 wl ¼ 4k 1 ð8Þ q
ð3Þ
including saturation at a field Hs ðMs Þ; and the third term describes hysteresis effects like initial susceptibility w0 ; remanence Mr ; coercivity Hc ; static losses, and accommodation, using the irreversible field k HI ¼ þ cr H R m0 M s h q i 1 k exp jm m0 j : ð4Þ k
ð7Þ
and we can write the equation for q0 as q0 ¼
8m0 Ms Hc : 4m0 Ms Hc wl
ð9Þ
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
These relations allow even an estimation of Ms (at cr E0), using Eqs. (6), (7), and (9) to Ms ¼
2w0 Hc wl þ : 1 Nd w0 4m0 Hc
ð10Þ
Furthermore, q0 can also be determined by the reduced remanence mr of the upper branch of a loop with the measured reduced maximum magnetization mm : Using fq as a factor related to mr ; fq ¼ ½ð1 þ mr Þ1þmr ð1 mr Þ1mr g0 =2 1;
ð11Þ
we identify q0 as 2 2Hc q0 ¼ ln : mm mr Hc h0 fq þ Nd Ms mr
ð12Þ
By using fg as a factor related to mg which is the reduced magnetization at H ¼ Hg ; in the knee of the lower branch of the hysteresis, fg ¼
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1þmg ln ð1 þ mg Þ ð1 mg Þ1mg ln 2
ð13Þ
hence g0 is g0 ¼ fg ln
Hg Hc Nd M s mg : Hs Hc Nd M s
Using fc as a factor related to mr and mm ; h m m i r m fc ¼ 1 2 exp q0 ; 2
ð15Þ
ð16Þ Finally, the identification equation of h0 is H s H c N d Ms : ðcr þ 1Þðexp½g0 ln 2 1Þ
we find Hs ¼ ðHm Hc Nd Ms mm Þ Hm Hc Nd Ms mm fh : Hg Hc Nd Ms mg
ð19Þ
If Nd of the experimental arrangement is unknown then it can be estimated roughly by the differential susceptibility wc at coercivity of a measured hysteresis loop:
1
Nd E : ð20Þ wc H¼Hc If Nd of the sample is rather large so that the magnetization curve is strongly sheared (Mr Nd > Hc ), then it can be necessary to identify g0 and cr by the backsheared curve (Nd ¼ 0). 3.2. Parameter dependence on anisotropy
fq ðHs Hc Nd Ms Þ=Ms exp g0 ln 2 fc ðHc =Ms Þ þ Nd mr : ðfq ðHs Hc Nd Ms Þ=Ms exp g0 ln 2 þ fc ðHc =Ms ÞÞ Nd mr
h0 ¼
ð18Þ
ð14Þ
the microscopic constant describing the domain (grain) geometry ratio becomes cr ¼
factor related to mg and mm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ð1 þ mm Þ1þmm ð1 mm Þ1mm ln 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; fh ¼ ð1 þ mg Þ1þmg ð1 mg Þ1mg ln ð1 þ mm Þ1þmm ð1 mm Þ1mm
387
ð17Þ
If Hs is not available, one can estimate Hs from the measured maximum field Hm at mm using the approximation Hs bHc þ Nd Ms : Using fh as a
It is possible to relate the model parameters to anisotropy (e.g. magnetocrystalline, stress). Depending on the inner structure of the considered material, these relations will differ from case to case. The following calculations are general and based on classical principles, basically indicating the possibilities of the EM. The law of approach to saturation (LAS) a b M ¼ Ms 1 2 y ð21Þ H H describes the rotation of the domain magnetization against the anisotropy energy at strong fields, for cubic anisotropy (constant K1 > 0) as an example with 8 K1 2 ¼ eHk2 ð22Þ b¼ 105 m0 Ms and a ¼ 0; where Hk ¼ 2K1 =m0 Ms is the fictive anisotropy field. To find a relation between LAS
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and EM, the reciprocal differential susceptibility at H ¼ Hk dH 1 Hk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ dM 2e 3 2 b=ð1 mÞ
ð23Þ
is set equal to the derivation of Eq. (1) near saturation with m ¼ 1 e; HR bHI þ Hd : dHR gh 2e g 2 EHs ln : ð24Þ ¼ ½ð2 eÞ2e ee g=2 ln 2 e 2 e dM With Eq. (17) and the approximations Hs bHc þ Nd Ms and 2g b1 which are valid for most magnetic materials, the result is g¼
2K1 ln 2=e : m0 Ms Hs e
ð25Þ
Using the microscopic constants ch ¼
Hs Ms
ð26Þ
2ch 2 ln ; e e
ð27Þ
Fig. 2. Calculated hysteresis loops for YCo2 thin films with ion irradiation perpendicular to film plane, related to Ms;0 ¼ 40 kA=m (—), maximum field Hm ¼ 160 kA=m; the values Ms ¼ 60 kA=m (- - -) and Ms ¼ 20 kA=m (– –) have been used to calculate the other loops without any other parameter change.
and cg ¼
we find that Dwk g ¼ cg m0 Ms2
ð28Þ
is proportional to the anisotropy energy difference Dwk which has to be overcome during magnetization reversal. Other calculations [19] are related to the differential susceptibilities of LAS and EM at H-Hs which are valid in a narrow range only. 3.3. Calculation The calculations have been done as following: At a given Nd ¼ 0:47; the parameters g0 ¼ 5:24; h0 ¼ 2:79 kA=m; k0 ¼ 1:10 kJ=m3 ; and q0 ¼ 8:79 are identified for the perpendicular hysteresis at F ¼ 5 1012 ions=cm2 with Ms ¼ 20 kA=m (see Fig. 1, middle). In the next step we vary only Ms in order to calculate the hysteresis of the other irradiation cases. Using Eqs. (6) and (7), we find the dependencies Hc ¼
k0 m0 Ms
ð29Þ
Table 1 Macroscopic hysteresis features depending on Ms ; measured (M) and modelled (EM) values F Ms w0 (EM)
ðions=cm2 Þ (kA/m) (1)
1012 20 0.157
5 1012 40 0.199
2 1013 60 0.536
Hc (M) Hc (EM)
(kA/m) (kA/m)
50.3 43.6
17.5 21.8
9.2 13.7
Mr (M) Mr (EM)
(kA/m) (kA/m)
15.6 14.9
22.2 22.5
24.7 25.9
Mm (M) Mm (EM)
(kA/m) (kA/m)
19.7 19.6
39.9 39.6
60.0 60.0
and w0 ¼
m0 Ms2 ; k0 q0 þ m0 Ms2 Nd
ð30Þ
which strongly affects the shape of the hysteresis as shown in Fig. 2. No other change of the parameter values has been done. The calculation of the hysteresis under highest fluence fits the parallel
ARTICLE IN PRESS D. Bajalan et al. / Physica B 343 (2004) 384–389
case better than the perpendicular because of a change of anisotropy due to fluence. The parameter dependence on anisotropy has not been considered in these calculations. Table 1 compares some modelled perpendicular macroscopic hysteresis features with the measurements. The modelling of the parallel hysteresis— considering only the different anisotropy energy for the parameter identification—is currently under progress.
4. Conclusions The rapid development of magnetic recording leads to a large increase of the bit density. Multilayer thin films with perpendicular magnetic anisotropy devices may play an active role in the development and establishment of future storage technologies. Patterning magnetic media is a potential solution for ultrahigh density magnetic recording [18]. Ion beam modification of magnetic layers may be the possible future of ultrahigh density magnetic recording media. After ion irradiation of YCo2 thin films with different fluence values, the measured magnetization curves clearly show a perpendicular shape anisotropy [11]. The shape of the hysteresis loops depends strongly on Ms ; which is predicted by the EM. It turns out that Hc is inversely proportional to Ms and w0 is proportional to Ms2 ; if Nd is neglected. As the EM parameters are also related to anisotropy it will be possible also to calculate the direction of dependence of these magnetization curves, which is subject to further work.
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