- Email: [email protected]
Coercivity and fluctuation field in granular recording media
Journal of Magnetism and Magnetic Materials 242–245 (2002) 434–436
Coercivity and fluctuation field in granular recording media V. Karanasos*, I. Panagiotopoulos, D. Niarchos IMS, NCSR, Demokritos, Ag. Paraskevi, Attiki 153 10, Greece
Abstract Fluctuation field as a function of applied field and coercivity data derived from magnetic measurements on two wellcharacterized CoPt/B (1 1 1) textured films is presented and compared with the predictions of a simple model based on Arrhenius–N!eel type of thermal activation of Stoner–Wohlfarth particles that takes into account the specifics of the easy-axis distribution in hand. The model gives a good qualitative description of the results. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Fluctuation field; Coercivity; Information storageFhigh density recording; Granular systems
Recently, granular magnetic films consisting of FePt or CoPt particles with the L10 high anisotropy phase, embedded in a non-magnetic matrix such as C [1–4], Ag [5,6], Si3N4 [7], Al2O3 [8], SiO2 [9–12], BN [13], and B [14,15], have been proposed as possible candidates for a new generation of longitudinal rigid disk media. Laboratory demonstrations have confirmed their capability for recording densities to be as high as 5.3 kfc/mm [2]. The fluctuation field Hf [16,17] defined as the ratio of the magnetic viscosity S to the irreversible susceptibility wirr : Hf ¼ S=wirr
ð1Þ
is a parameter commonly used to describe the thermal effects, which are of decisive importance to the stability of the magnetic recording media, especially as the recording densities tend to increase [18]. Barbier [19] gave a log–log plot of Hf against Hc for several magnetic materials showing that the data fall on a straight line. Ferguson et al. extended the Barbier’s plots by presenting the Hf ; Hc values for a NdFeB sample measured at different temperatures [20]. The field and temperature dependence of S; wirr and Hf can be obtained for a system of Stoner–Wolfrarth particles [21,22] by introducing time phenomena in the framework of the N!eel–Arrhenius law predicting ex*Corresponding author. Tel.: +30-1-650-3321; fax: +30-1651-9430. E-mail address: [email protected] (V. Karanasos).
ponential thermal relaxation over an energy barrier of height EB with relaxation time: 1 EB ; ð2Þ t ¼ exp f0 kT where f0 is the thermal attempt frequency. In what follows, we work out the Hf dependence on H and HC on the basis of a similar model and compare the theoretical curves for different easy-axis distributions to experimental data obtained in magnetron-sputtered CoPt/B nanocomposite films [14,15] with (1 1 1) texture. The data were obtained in a SQUID magnetometer with the applied field in the film plane. According to Pfeifer [23], in the general case of a fieldeasy axis misalignment by an angle c; the energy barrier has the following form: aðcÞ H EB ðH; cÞ ¼ KV 1 ; ð3Þ HK hðcÞ where HK is the anisotropy field 2K=MS ; K the uniaxial anisotropy constant, V the particle volume, MS the saturation magnetization and hðcÞ; aðcÞ are two functions given by hðcÞ ¼ ½cos2=3 cþsin2=3 c3=2 ; and aðcÞ ¼ 0:86 þ 1:14hðcÞ: For (1 1 1) textured films, the easy-axis direction (0 0 1) lies on a cone forming an angle of 361 with the film plane. If the field is applied along the film plane, y is the angle between the field direction and the in-plane projection of easy axis, and a uniform distribution f ðyÞ ¼ 2=p is assumed, then cos c=cos 361 cos y with y ranging between 01 and 901.
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 1 4 3 - X
V. Karanasos et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 434–436
0
where j1 and j2 are the equilibrium angles between magnetization and field when the moment lies parallel or antiparallel to the applied field, respectively. In the absence of the field, the j1 ; j2 coincide with c and 180 þ c: The paramagnetic volume is defined as Vp ðH; T; cÞ ¼
kB T lnðf0 tÞ aðcÞ : H K 1 HK hðcÞ
ð5Þ
(a)
0.016
0.012
Hf /Hk
The contribution of the fraction of particles with angle y within y; y þ dy to the total magnetization along field direction is Z N 2 MðyÞ ¼ dyMs ðcosðf1 Þ f ðV Þ dV p Vp Z Vp þ cosðf2 Þ f ðV Þ dV Þ; ð4Þ
435
0.008 (b) 0.004
(c) Hf =f(Hc)
(d)
0.000 0.1
0.2
0.3
0.4
0.5
H/Hk
Simple summations over y and differentiations with respect to time and field, give the following expressions for remanent magnetization Mr ; viscosity S and irreversible susceptibility wirr : Z Z Vp 2 cos 361 p=2 Mr =Ms E cos y 1 2 f ðV Þ dV dy; p 0 0
Fig. 1. Fluctuation field Hf vs. reduced applied field h ¼ H=HK for: (a) aligned particles, (b) completely randomly oriented particles, (c) (1 1 1) textured particles with uniform in-plane projection distribution and (d) aligned particles at an angle of 361 with the field.
ð6Þ SðH; TÞ=MS ¼
2 p lnðf0 tÞ
Z
xirr ðH; TÞ=MS ¼
4 cos 361 p
p=2
ðcos f1 cos f2 Þf ðVp ÞVp dy;
ð7Þ
0
Z 0
p=2
aðcÞ f ðVp ÞVp dy: ð8Þ cos y ðHK hðcÞ HÞ
For a hypothetical case of perfect alignment, f ðyÞ ¼ dð0Þ and c ¼ 361; Hf reduces to the common expression [19,20] Hf ¼
cos f1 cos f2 0:5HK H 0:5HK H E 1:44 lnðtf0 Þ 1:44 lnðtf0 Þ 2 cos 361
ð9Þ
predicting a linear decrease of Hf with H: An important characteristic of Eq. (9) is the temperature and particlevolume independence of Hf ; which allows the calculation of Barbier plots by simply setting H ¼ HC : Hf was calculated as a function of H for the following four cases: (a) perfect alignment of easy axes along the field, (b) random distribution of easy axes, (c) (1 1 1) texturing, by numerical integration of Eqs. (7) and (8) and (d) perfect easy-axis alignment at an angle of 361 with the field. Several temperatures and mean particle volumes in the range of our interest were used. The results for a log-normal particle-volume distribution with standard deviation w ¼ 0:2 at room temperature are presented in Fig. 1. Hf decreases linearly with the field apart from a field region above HC where a small curvature is observed. This plot is in very good
Fig. 2. The Hf vs. H for the 50 vol% B CoPt/B sample for T ¼ 50; 150 and 250 K. The arrows indicate HC for each temperature.
agreement with that of DeWitte et al. [21]. The linear part of this plot changes slightly under temperature and particle-volume changes, and has a slope almost equal to that of the perfect alignment case of Eq. (9). The deviation from linearity is restricted to a convex part that shifts at higher fields as temperature decreases or particle-volume increases in a trend similar to that of HC : This feature is also observed in the experimental data obtained in CoPt/B films (Fig. 2). The estimation of HC has been performed based on Eq. (6) for the temperature range of our interest. The
V. Karanasos et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 434–436
436
in a complete way the real reversal mechanism. However, the theoretical predictions are in qualitative agreement with experimental data as the reduction of Hf with HC is experimentally observed and Hf varies in the same way for both samples independently of particle volume.
(a)
0.005
(b)
Hf /Hk
0.004
0.003
References
0.002
0.001 0.30
0.35
0.40
0.45
Hc /Hk Fig. 3. Calculated Hf vs. Hc values: (a) for randomly oriented particles and (b) for (1 1 1) textured particles.
40
(50 vol% B)
35
300 250
Hf (Oe)
30
200
25
(20 vol% B) 20
150
300 250
100
200 150
15 10
50
100 50
5 1
2
3
6
7
8
9
HC(kOe) Fig. 4. The Hf vs. Hc for two CoPt/B samples with 20 and 50 vol% B. The labels by each data point indicate the temperature of the measurement (in K).
predicted Hf vs. HC curves are presented in Fig. 3. The linear dependence of Hf on HC may be viewed as a result of the fact that HC lies in the temperature and volume invariant linear part of Hf vs. H plot. In Fig. 4, the experimental correlation of HC ; Hf for two films of CoPt/B [14,15] with different microstructures and anisotropy constants is presented. Despite their differences, the two samples have similar Hf vs. HC curves which are curved slightly downwards. The divergence from the theoretically predicted linearity can be attributed to particle interactions and the fact that the above form of the energy barrier cannot explain
[1] I. Panagiotopoulos, S. Stavroyiannis, D. Niarchos, J.A. Christodoulides, G.C. Hadjipanayis, J. Appl. Phys. 87 (2000) 2740. [2] M. Yu, Y. Liu, A. Moser, D. Weller, D.J. Sellmyer, Appl. Phys. Lett. 75 (1999) 3992. [3] J.-J. Delaunay, T. Hayashi, M. Tomita, S. Hirono, S. Umemura, Appl. Phys. Lett. 71 (1997) 3427. [4] M. Yu, Y. Liu, D.J. Sellmyer, J. Appl. Phys. 87 (2000) 6959. [5] S. Stavroyiannis, I. Panagiotopoulos, D. Niarchos, J.A. Christodoulides, Y. Zhang, G.C. Hadjipanayis, Appl. Phys. Lett. 73 (1998) 3453. [6] S. Stavroyiannis, I. Panagiotopoulos, D. Niarchos, J.A. Christodoulides, Y. Zhang, G.C. Hadjipanayis, J. Magn. Magn. Mater. 193 (1999) 181. [7] Chih-Ming Kuo, P.C. Kuo, J. Appl. Phys. 87 (2000) 419. [8] Bo-Bian, Kazuhisa Sato, Yoshikiko Hirotsu, Akihiro Makino, Appl. Phys. Lett. 75 (1999) 3686. [9] C.P. Luo, D.J. Sellmyer, Appl. Phys. Lett. 75 (1999) 3162. [10] C.P. Luo, S.H. Liou, D.J. Sellmyer, J. Appl. Phys. 87 (2000) 6941. [11] C. Chen, O. Kitakami, S. Okamoto, Y. Shimada, J. Appl. Phys. 87 (2000) 6947. [12] A. Kikitsu, A. Murayama, K. Hyomi, C.M. Falco, J. Appl. Phys. 87 (2000) 6944. [13] J.A. Christodoulides, Y. Huang, Y. Zhang, G.C. Hadjipanayis, I. Panagiotopoulos, D. Niarchos, J. Appl. Phys. 87 (2000) 6938. [14] V. Karanasos, I. Panagiotopoulos, D. Niarchos, H. Okumura, G.C. Hadjipanayis, J. Appl. Phys. 88 (2000) 8979. [15] V. Karanasos, I. Panagiotopoulos, D. Niarchos, H. Okumura, G.C. Hadjipanayis, J. Magn. Magn. Mater. 236 (2001) 234. [16] E.P. Wohlfarth, J. Phys. F 14 (1984) L155. [17] A. Lyberatos, R.W. Chantrell, J. Phys: Condens. Matter. 9 (1997) 2623. [18] M.P. Sharrock, IEEE Trans. Magn. 35 (1999) 4414–4422. [19] J.C. Barbier, Ann Phys. (Leipzig) 9 (1954) 84. [20] G.B. Ferguson, K. O’Grady, J. Popplewell, R.W. Chantrell, J. Appl. Phys. 69 (1991) 5495. [21] A.M. deWitte, K. O’Grady, G.N. Coverdale, R.W. Chantrell, J. Magn. Magn. Mater. 88 (1990) 183–193. [22] A. Lyberatos, R.W. Chantrell, K. O’Grady, in: G.C. Hadjipanayis, R.W. Siegel (Eds.), NATO ASI Series E 260, Nanophase Materials, Vol. 653, Kluwer Academic Publishers, Dordrecht, 1994. [23] H. Pfeiffer, Phys. Stat. Sol. 118 (1990) 295.
Coercivity and fluctuation field in granular recording media V. Karanasos*, I. Panagiotopoulos, D. Niarchos IMS, NCSR, Demokritos, Ag. Paraskevi, Attiki 153 10, Greece
Abstract Fluctuation field as a function of applied field and coercivity data derived from magnetic measurements on two wellcharacterized CoPt/B (1 1 1) textured films is presented and compared with the predictions of a simple model based on Arrhenius–N!eel type of thermal activation of Stoner–Wohlfarth particles that takes into account the specifics of the easy-axis distribution in hand. The model gives a good qualitative description of the results. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Fluctuation field; Coercivity; Information storageFhigh density recording; Granular systems
Recently, granular magnetic films consisting of FePt or CoPt particles with the L10 high anisotropy phase, embedded in a non-magnetic matrix such as C [1–4], Ag [5,6], Si3N4 [7], Al2O3 [8], SiO2 [9–12], BN [13], and B [14,15], have been proposed as possible candidates for a new generation of longitudinal rigid disk media. Laboratory demonstrations have confirmed their capability for recording densities to be as high as 5.3 kfc/mm [2]. The fluctuation field Hf [16,17] defined as the ratio of the magnetic viscosity S to the irreversible susceptibility wirr : Hf ¼ S=wirr
ð1Þ
is a parameter commonly used to describe the thermal effects, which are of decisive importance to the stability of the magnetic recording media, especially as the recording densities tend to increase [18]. Barbier [19] gave a log–log plot of Hf against Hc for several magnetic materials showing that the data fall on a straight line. Ferguson et al. extended the Barbier’s plots by presenting the Hf ; Hc values for a NdFeB sample measured at different temperatures [20]. The field and temperature dependence of S; wirr and Hf can be obtained for a system of Stoner–Wolfrarth particles [21,22] by introducing time phenomena in the framework of the N!eel–Arrhenius law predicting ex*Corresponding author. Tel.: +30-1-650-3321; fax: +30-1651-9430. E-mail address: [email protected] (V. Karanasos).
ponential thermal relaxation over an energy barrier of height EB with relaxation time: 1 EB ; ð2Þ t ¼ exp f0 kT where f0 is the thermal attempt frequency. In what follows, we work out the Hf dependence on H and HC on the basis of a similar model and compare the theoretical curves for different easy-axis distributions to experimental data obtained in magnetron-sputtered CoPt/B nanocomposite films [14,15] with (1 1 1) texture. The data were obtained in a SQUID magnetometer with the applied field in the film plane. According to Pfeifer [23], in the general case of a fieldeasy axis misalignment by an angle c; the energy barrier has the following form: aðcÞ H EB ðH; cÞ ¼ KV 1 ; ð3Þ HK hðcÞ where HK is the anisotropy field 2K=MS ; K the uniaxial anisotropy constant, V the particle volume, MS the saturation magnetization and hðcÞ; aðcÞ are two functions given by hðcÞ ¼ ½cos2=3 cþsin2=3 c3=2 ; and aðcÞ ¼ 0:86 þ 1:14hðcÞ: For (1 1 1) textured films, the easy-axis direction (0 0 1) lies on a cone forming an angle of 361 with the film plane. If the field is applied along the film plane, y is the angle between the field direction and the in-plane projection of easy axis, and a uniform distribution f ðyÞ ¼ 2=p is assumed, then cos c=cos 361 cos y with y ranging between 01 and 901.
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 1 4 3 - X
V. Karanasos et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 434–436
0
where j1 and j2 are the equilibrium angles between magnetization and field when the moment lies parallel or antiparallel to the applied field, respectively. In the absence of the field, the j1 ; j2 coincide with c and 180 þ c: The paramagnetic volume is defined as Vp ðH; T; cÞ ¼
kB T lnðf0 tÞ aðcÞ : H K 1 HK hðcÞ
ð5Þ
(a)
0.016
0.012
Hf /Hk
The contribution of the fraction of particles with angle y within y; y þ dy to the total magnetization along field direction is Z N 2 MðyÞ ¼ dyMs ðcosðf1 Þ f ðV Þ dV p Vp Z Vp þ cosðf2 Þ f ðV Þ dV Þ; ð4Þ
435
0.008 (b) 0.004
(c) Hf =f(Hc)
(d)
0.000 0.1
0.2
0.3
0.4
0.5
H/Hk
Simple summations over y and differentiations with respect to time and field, give the following expressions for remanent magnetization Mr ; viscosity S and irreversible susceptibility wirr : Z Z Vp 2 cos 361 p=2 Mr =Ms E cos y 1 2 f ðV Þ dV dy; p 0 0
Fig. 1. Fluctuation field Hf vs. reduced applied field h ¼ H=HK for: (a) aligned particles, (b) completely randomly oriented particles, (c) (1 1 1) textured particles with uniform in-plane projection distribution and (d) aligned particles at an angle of 361 with the field.
ð6Þ SðH; TÞ=MS ¼
2 p lnðf0 tÞ
Z
xirr ðH; TÞ=MS ¼
4 cos 361 p
p=2
ðcos f1 cos f2 Þf ðVp ÞVp dy;
ð7Þ
0
Z 0
p=2
aðcÞ f ðVp ÞVp dy: ð8Þ cos y ðHK hðcÞ HÞ
For a hypothetical case of perfect alignment, f ðyÞ ¼ dð0Þ and c ¼ 361; Hf reduces to the common expression [19,20] Hf ¼
cos f1 cos f2 0:5HK H 0:5HK H E 1:44 lnðtf0 Þ 1:44 lnðtf0 Þ 2 cos 361
ð9Þ
predicting a linear decrease of Hf with H: An important characteristic of Eq. (9) is the temperature and particlevolume independence of Hf ; which allows the calculation of Barbier plots by simply setting H ¼ HC : Hf was calculated as a function of H for the following four cases: (a) perfect alignment of easy axes along the field, (b) random distribution of easy axes, (c) (1 1 1) texturing, by numerical integration of Eqs. (7) and (8) and (d) perfect easy-axis alignment at an angle of 361 with the field. Several temperatures and mean particle volumes in the range of our interest were used. The results for a log-normal particle-volume distribution with standard deviation w ¼ 0:2 at room temperature are presented in Fig. 1. Hf decreases linearly with the field apart from a field region above HC where a small curvature is observed. This plot is in very good
Fig. 2. The Hf vs. H for the 50 vol% B CoPt/B sample for T ¼ 50; 150 and 250 K. The arrows indicate HC for each temperature.
agreement with that of DeWitte et al. [21]. The linear part of this plot changes slightly under temperature and particle-volume changes, and has a slope almost equal to that of the perfect alignment case of Eq. (9). The deviation from linearity is restricted to a convex part that shifts at higher fields as temperature decreases or particle-volume increases in a trend similar to that of HC : This feature is also observed in the experimental data obtained in CoPt/B films (Fig. 2). The estimation of HC has been performed based on Eq. (6) for the temperature range of our interest. The
V. Karanasos et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 434–436
436
in a complete way the real reversal mechanism. However, the theoretical predictions are in qualitative agreement with experimental data as the reduction of Hf with HC is experimentally observed and Hf varies in the same way for both samples independently of particle volume.
(a)
0.005
(b)
Hf /Hk
0.004
0.003
References
0.002
0.001 0.30
0.35
0.40
0.45
Hc /Hk Fig. 3. Calculated Hf vs. Hc values: (a) for randomly oriented particles and (b) for (1 1 1) textured particles.
40
(50 vol% B)
35
300 250
Hf (Oe)
30
200
25
(20 vol% B) 20
150
300 250
100
200 150
15 10
50
100 50
5 1
2
3
6
7
8
9
HC(kOe) Fig. 4. The Hf vs. Hc for two CoPt/B samples with 20 and 50 vol% B. The labels by each data point indicate the temperature of the measurement (in K).
predicted Hf vs. HC curves are presented in Fig. 3. The linear dependence of Hf on HC may be viewed as a result of the fact that HC lies in the temperature and volume invariant linear part of Hf vs. H plot. In Fig. 4, the experimental correlation of HC ; Hf for two films of CoPt/B [14,15] with different microstructures and anisotropy constants is presented. Despite their differences, the two samples have similar Hf vs. HC curves which are curved slightly downwards. The divergence from the theoretically predicted linearity can be attributed to particle interactions and the fact that the above form of the energy barrier cannot explain
[1] I. Panagiotopoulos, S. Stavroyiannis, D. Niarchos, J.A. Christodoulides, G.C. Hadjipanayis, J. Appl. Phys. 87 (2000) 2740. [2] M. Yu, Y. Liu, A. Moser, D. Weller, D.J. Sellmyer, Appl. Phys. Lett. 75 (1999) 3992. [3] J.-J. Delaunay, T. Hayashi, M. Tomita, S. Hirono, S. Umemura, Appl. Phys. Lett. 71 (1997) 3427. [4] M. Yu, Y. Liu, D.J. Sellmyer, J. Appl. Phys. 87 (2000) 6959. [5] S. Stavroyiannis, I. Panagiotopoulos, D. Niarchos, J.A. Christodoulides, Y. Zhang, G.C. Hadjipanayis, Appl. Phys. Lett. 73 (1998) 3453. [6] S. Stavroyiannis, I. Panagiotopoulos, D. Niarchos, J.A. Christodoulides, Y. Zhang, G.C. Hadjipanayis, J. Magn. Magn. Mater. 193 (1999) 181. [7] Chih-Ming Kuo, P.C. Kuo, J. Appl. Phys. 87 (2000) 419. [8] Bo-Bian, Kazuhisa Sato, Yoshikiko Hirotsu, Akihiro Makino, Appl. Phys. Lett. 75 (1999) 3686. [9] C.P. Luo, D.J. Sellmyer, Appl. Phys. Lett. 75 (1999) 3162. [10] C.P. Luo, S.H. Liou, D.J. Sellmyer, J. Appl. Phys. 87 (2000) 6941. [11] C. Chen, O. Kitakami, S. Okamoto, Y. Shimada, J. Appl. Phys. 87 (2000) 6947. [12] A. Kikitsu, A. Murayama, K. Hyomi, C.M. Falco, J. Appl. Phys. 87 (2000) 6944. [13] J.A. Christodoulides, Y. Huang, Y. Zhang, G.C. Hadjipanayis, I. Panagiotopoulos, D. Niarchos, J. Appl. Phys. 87 (2000) 6938. [14] V. Karanasos, I. Panagiotopoulos, D. Niarchos, H. Okumura, G.C. Hadjipanayis, J. Appl. Phys. 88 (2000) 8979. [15] V. Karanasos, I. Panagiotopoulos, D. Niarchos, H. Okumura, G.C. Hadjipanayis, J. Magn. Magn. Mater. 236 (2001) 234. [16] E.P. Wohlfarth, J. Phys. F 14 (1984) L155. [17] A. Lyberatos, R.W. Chantrell, J. Phys: Condens. Matter. 9 (1997) 2623. [18] M.P. Sharrock, IEEE Trans. Magn. 35 (1999) 4414–4422. [19] J.C. Barbier, Ann Phys. (Leipzig) 9 (1954) 84. [20] G.B. Ferguson, K. O’Grady, J. Popplewell, R.W. Chantrell, J. Appl. Phys. 69 (1991) 5495. [21] A.M. deWitte, K. O’Grady, G.N. Coverdale, R.W. Chantrell, J. Magn. Magn. Mater. 88 (1990) 183–193. [22] A. Lyberatos, R.W. Chantrell, K. O’Grady, in: G.C. Hadjipanayis, R.W. Siegel (Eds.), NATO ASI Series E 260, Nanophase Materials, Vol. 653, Kluwer Academic Publishers, Dordrecht, 1994. [23] H. Pfeiffer, Phys. Stat. Sol. 118 (1990) 295.