- Email: [email protected]
Novel simulation model for perpendicular magnetic recording
Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
Novel simulation model for perpendicular magnetic recording K.J. Leea,*, Y.H. Ima, Y.S. Kima, K.M. Leea, J.W. Kima, N.Y. Parka, G.S. Parkb, T.D. Leec a
MEMS Laboratory, Samsung Advanced Institute of Technology (SAIT), P.O. Box 111, Suwon 440-600, South Korea b Department of Electrical Eng., Korea Maritime University, Pusan, South Korea c Department of Mater. Science and Eng., KAIST, Taejon 305-701, South Korea
Abstract Novel simulation model for perpendicular recording is developed. The Preisach model merged with the micromagnetic model is used to simulate the perpendicular media. Moreover, a simple method to calculate the MR sensitivity function is also provided. Simulation results of the model are well agreed with experimental measurements. Precise modeling method and recording mechanism of the perpendicular recording with a ring head writing is discussed in detail. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Perpendicular magnetic recording; Preisach modeling; Mr sensitivity function
The perpendicular magnetic recording (PMR) is a promising candidate beyond the longitudinal magnetic recording (LMR) because it has better thermal stability and can be used with smaller writing field than the LMR [1–3]. Especially, a ring head application with a single layered (SL) medium or a double layered (DL) medium is attractive since it can be easily applied to the hard disk drives without drastic changes of the current ring head technologies. However, a systematic research of recording physics for comparisons of the SL and the DL medium with the ring head writing has not been reported. An appropriate medium modeling is essential to understand the recording physics of the PMR. Although the micromagnetic modeling of medium have suggested successful predictions in the LMR, a plausible micromagnetic modeling of the PMR *Corresponding author. Fax: +82-31-280-6955. E-mail address: [email protected] (K.J. Lee).
medium is hardly achieved since it must include obscure effects such as initial growth layer, surface nucleation, domain pinning and/or very thick soft back layer [4–6]. In this paper, a novel medium model of the PMR is provided. The Preisach model [7,8] merged with the micromagnetic model was used to simulate PMR media. Moreover, a simple method to calculate the MR sensitivity function is provided also. The simulation results were compared with experimental measurements and validity of the modeling was proved. Recording mechanisms of the SL and the DL medium with a ring head were analyzed along the head field gradients and waveforms. For experimental measurements, disk samples of CoCrNbPt(50 nm)/Ti(70 nm) and CoCrNbPt(50 nm)/Ti(5 nm)/NiFe(500 nm) were prepared. Magnetic properties of the CoCrNbPt medium are the coercivity of 2500 Oe, the squareness of 0.6 and the saturation magnetization of
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 3 9 6 - 1
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
250 emu/cm3. Read/write characteristics were investigated by using GUZIK RWA2585. In the GUZIK test, a ring head with track width of 0.75 mm and write gap of 0.19 mm was used for writing. Flying height and head-medium velocity were 23 nm and 10 m/s, respectively. Merged MR head with shield to shield spacing of 0.14 mm and read track width of 0.55 mm was used to measure readback signal and noise. A schematic diagram of medium modeling is shown in Fig. 1. A hard magnetic layer was divided into tetragonal cells. Hysteresis characterization was made in the context of generalized Preisach operators [8,9]. The Preisach density distribution was obtained from the sets of M–H curves [8,9]. In the recording process, the external head field and the internal interaction field were applied to each cell. Since the Preisach model contains magnetic interactions within the cell only, the interaction field among cells was taken from the micromagnetic modeling. It consists of the magnetostatic field from all other cells and the exchange field from two nearest neighbor cells. The interac~i exposed on the ith cell is given by tion field, H N 2 X 8A * X Mj ~i ¼ H Nij Mj þ 2 ; ð1Þ 2 M S j¼n:n: ðDi þ Dj Þ j¼N 2
where Nij is the demagnetization tensor, A * is the effective exchange constant, Di is the edge length of ith cell along the film plane, Mj is the magnetization of jth cell, and MS is the saturation magnetization. The relation between the effective exchange constant and the loop slope was obtained by using micromagnetic simulations and is given by [10] A * ¼ ðDMS Þ2 1:217 tan1 0:579
4pMS 0:579 ðHC þ Hn Þ
;
ð2Þ where HC is the medium coercivity. Hn is the nucleation field to nucleate reversal domain and is defined a minus value when the nucleation occurs in the second quadrature of hysteresis loop. Eq. (2) is valid only if A is a positive value. By using the measured loop shape and the Eq. (2), we obtained the exchange constant less than zero. Since the
399
Fig. 1. A schematic diagram of modeling method. Each cell has the respective Preisach plane.
exchange constant in ferromagnetic materials cannot have a minus value, the exchange constant in this paper was set to zero. Finite element methods (FEM) were used to calculate writing head fields. The reading process was simulated using the reciprocity theorem. MR sensitivity function for reciprocity calculation is defined as the field that is applied on the medium from the MR head with a single turn coil around the MR element [11]. The 3D MR head sensitivity function can be obtained simply by superposing the Lindholm fields [12] for a finite width head and is given by ~G ðx; y; zÞ ¼ HGB h~GB ðx þ ðGT þ tMR Þ=2; y; zÞ H MR Lind GT HGT h~Lind ðx ðGB þ tMR Þ=2; y; zÞ;
ð3Þ where GT and GB are the distance from the free layer to the top shield and from the bottom shield to the free layer, respectively. HG is the deep gap and tMR is the thickness of the free layer. hG Lind is the normalized Lindholm field of a ring head with the head gap of G and the track width of W and is given as followings [12]: ~1 x; y; z w h~G ¼ h Lind 2 w ~ 0 ~ ð4Þ þ h 1 x; y; z þ h2 ðx; y; zÞ; 2 1 ~ 1 x þ g=2 ~ h1 ¼ F 1 sinh p y2 þ z 2 1 x g=2 ~ F 1 sinh g; ð5Þ y2 þ z 2
400
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
where
sinð2u=3Þ ; sinð2j=3Þ u j coshðu=3Þ cos þ ctnh1 ; F1y ðuÞ ¼ 4 cosh 3 3 cosðj=3Þ u j coshðu=3Þ sin þ ctnh1 F1z ðuÞ ¼ 4 cosh 3 3 sinðj=3Þ
F1x ðuÞ ¼ tan1
ð6Þ and j¼
(
p tan1 ðy=zÞ; zX0 tan1 ðy=zÞ; zo0
:
ð7Þ
0 h~1 is same with h~1 except j that should be replaced by j0 : ( tan1 ðy=zÞ; zX0 j0 ¼ ; ð8Þ p þ tan1 ðy=zÞ; zo0
1 ~ x þ g=2 x g=2 ~ ~ F2 h2 ¼ F2 ; p y y
ð9Þ
F2x ðuÞ ¼ tan1 ½u ; F2y ðuÞ ¼ ð1=2Þ lnðu2 þ 1Þ; F2z ðuÞ ¼ 0:
ð10Þ
In Fig. 2, contours of z-component (film thickness direction) of the MR head sensitivity function obtained from the Eq. (3) and FEM calculation
are compared with each other. Except track edge region of MR sensor, the MR sensitivity function of Eq. (3) is well agreed with the FEM results. In the DL medium, the MR head sensitivity function was obtained from a mirror imaging of the high permeable soft back layer and the head surface [13]. Fig. 3a shows the calculated MR sensitivity functions of the SL and the DL medium. The MR sensitivity function of the DL medium at read gap center is higher that that of the SL medium because of effects of soft back layer. From the Fourier transformation of the MR head sensitivity function to frequency domain, it was found that the MR sensitivity function of the DL medium acts like a low-pass filter (Fig. 3b) [14]. Therefore, the noise spectra of the DL medium at low frequency are higher than that of the SL medium as shown in Fig. 3c. These mean that the magnetization positioned far from the MR sensor can affect the reading signal through the high permeable soft back layer. Distributions of z-component of head field and gradient for the SL and the DL medium are shown in Fig. 4. Soft back layer of the DL medium enlarges field gradient and head field amplitude. Sharpness of the magnetization transition is determined by the field gradient at the region indicated by an arrow A. In Fig. 5a, magnetization profiles of recording layer at 10 KFCI are shown. It should be noted that the magnetization profiles
Fig. 2. Contours of the MR sensitivity function; (a) FEM calculation, and (b) superposition of the Lindholm field, where Tw is read track width.
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
401
Fig. 4. Head field distribution and gradient of the single and double layered medium; (a) head field distribution, and (b) field gradient.
Fig. 3. Comparisons of the MR sensitivity function in the single and the double layered medium; (a) the calculated MR sensitivity function, (b) the calculated transfer function of the MR sensitivity function, and (c) the measured noise spectra at 50 KFCI.
in the PMR medium are largely deviated from the arctangent-like function. Such the asymmetric profiles result from the demagnetization field of neighboring bit and the head field gradient. Due to the smaller field gradient, the SL medium shows smaller remanent magnetization and more asym-
metric magnetization profile than those of the DL medium. Simulated waveforms are compared with measured waveforms (Fig. 5b and c). The simulation results almost duplicate the experimental measurements except signal amplitude of the DL medium that may result from an overestimation of soft back layer’s mirror effects in the calculation of the MR sensitivity function. Distorted magnetization profile and waveform (indicated by arrow B) are obtained in the simulation results of the SL medium, which is similar to the earlier experimental report and arises from the positive nucleation field [15]. The negative nucleation field can endure the reverse head field of trailing edge [16]. Therefore, the nucleation field is one of the key factors to determine read/write characteristics of the PMR in the ring head applications. When the nucleation field has a minus value, the signal outputs for all linear recording densities are much enhanced from those of the positive nucleation field (Fig. 6). Furthermore, the current saturation
402
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
waveforms of the single and the double layered medium were well agreed with the measurements. Importantly, profiles of the recorded magnetization were deviated from the arctangent-like function even in the double layered medium. In the ring head application, the sharp head field gradient and the negative nucleation field of the medium are essential for better read/write characteristics. The developed model has a big potential to understand the recording mechanism of the perpendicular recording and therefore can be used as an optimization tool of the head and the medium.
Acknowledgements The authors are grateful to Dr. K. Ouchi of AIT for assisting the media preparation. Fig. 5. Comparisons of calculated waveform and measured waveform in the single and the double layered medium at 10 KFCI; (a) Calculated magnetization profiles, (b) calculated waveform and measured waveform in the single layered medium, and (c) calculated waveform and measured waveform in the double layered medium.
Fig. 6. Roll-off curves of the single and double layered medium with varying the nucleation field.
curve is easily saturated when the medium has the negative nucleation field [16]. The developed model was examined in comparison with experimental measurements. Calculated
References [1] S. Yanase, T. Kiya, N. Honda, K. Ouchi, Technical Report of IEICE, Vol. 95–14, 1995, p. 15. [2] M. Futamoto, Y. Honda, Y. Hirayama, K. Itoh, H. Ide, Y. Maruyama, IEEE Trans. Magn. 32 (1996) 3789. [3] N. Honda, K. Ouchi, IEEE Trans. Magn. 33 (1997) 3097. [4] E. Miyashita, Y. Yoneda, J. Numazawa, J. Magn. Soc. Japan 21–S2 (1997) 31. [5] S. Saito, Y. Hatta, F. Hoshi, M. Takahashi, Digests of PMRC 2000, 23pA-26, 2000, p. 56. [6] R. Wood, Y. Sonobe, Digests of PMRC 2000, 23aA-01, 2000, p 1. [7] F. Preisach, Zeitschrift fuer Physik 94 (1935) 277. [8] G.S. Park, Ph. D. Thesis, Seoul National University, 1992. [9] C.D. Mee, E.D. Daniel, Magnetic recording technology, McGraw-Hill, New York, 1996, Chapter 2, p. 9. [10] K.J. Lee, T.D. Lee, Digests of CompuMAG 2001, OH-02, 2001. [11] H.N. Bertram, Theory of magnetic recording, Cambridge press, Cambridge, 1994, p. 178. [12] D.A. Lindholm, IEEE Trans. Magn. MAG-13 (1977) 1460. [13] H. Muraoka, Y. Sugita, Y. Nakamura, IEEE Trans. Magn. 35 (1999) 2235. [14] H. Muraoka, Y. Nakamura, Digests of Fifth ASIST, MR2000-52, 2000, p. 103. [15] A. Takeo, S. Oikawa, T. Hikosaka, Y. Tanaka, Digests of Intermag 2000, 2000, BB-03. [16] T.D. Lee, M.S. Hwang, K.J. Lee, Digests of PMRC 2000, 25aA-05, 2000, p. 147.
Novel simulation model for perpendicular magnetic recording K.J. Leea,*, Y.H. Ima, Y.S. Kima, K.M. Leea, J.W. Kima, N.Y. Parka, G.S. Parkb, T.D. Leec a
MEMS Laboratory, Samsung Advanced Institute of Technology (SAIT), P.O. Box 111, Suwon 440-600, South Korea b Department of Electrical Eng., Korea Maritime University, Pusan, South Korea c Department of Mater. Science and Eng., KAIST, Taejon 305-701, South Korea
Abstract Novel simulation model for perpendicular recording is developed. The Preisach model merged with the micromagnetic model is used to simulate the perpendicular media. Moreover, a simple method to calculate the MR sensitivity function is also provided. Simulation results of the model are well agreed with experimental measurements. Precise modeling method and recording mechanism of the perpendicular recording with a ring head writing is discussed in detail. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Perpendicular magnetic recording; Preisach modeling; Mr sensitivity function
The perpendicular magnetic recording (PMR) is a promising candidate beyond the longitudinal magnetic recording (LMR) because it has better thermal stability and can be used with smaller writing field than the LMR [1–3]. Especially, a ring head application with a single layered (SL) medium or a double layered (DL) medium is attractive since it can be easily applied to the hard disk drives without drastic changes of the current ring head technologies. However, a systematic research of recording physics for comparisons of the SL and the DL medium with the ring head writing has not been reported. An appropriate medium modeling is essential to understand the recording physics of the PMR. Although the micromagnetic modeling of medium have suggested successful predictions in the LMR, a plausible micromagnetic modeling of the PMR *Corresponding author. Fax: +82-31-280-6955. E-mail address: [email protected] (K.J. Lee).
medium is hardly achieved since it must include obscure effects such as initial growth layer, surface nucleation, domain pinning and/or very thick soft back layer [4–6]. In this paper, a novel medium model of the PMR is provided. The Preisach model [7,8] merged with the micromagnetic model was used to simulate PMR media. Moreover, a simple method to calculate the MR sensitivity function is provided also. The simulation results were compared with experimental measurements and validity of the modeling was proved. Recording mechanisms of the SL and the DL medium with a ring head were analyzed along the head field gradients and waveforms. For experimental measurements, disk samples of CoCrNbPt(50 nm)/Ti(70 nm) and CoCrNbPt(50 nm)/Ti(5 nm)/NiFe(500 nm) were prepared. Magnetic properties of the CoCrNbPt medium are the coercivity of 2500 Oe, the squareness of 0.6 and the saturation magnetization of
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 3 9 6 - 1
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
250 emu/cm3. Read/write characteristics were investigated by using GUZIK RWA2585. In the GUZIK test, a ring head with track width of 0.75 mm and write gap of 0.19 mm was used for writing. Flying height and head-medium velocity were 23 nm and 10 m/s, respectively. Merged MR head with shield to shield spacing of 0.14 mm and read track width of 0.55 mm was used to measure readback signal and noise. A schematic diagram of medium modeling is shown in Fig. 1. A hard magnetic layer was divided into tetragonal cells. Hysteresis characterization was made in the context of generalized Preisach operators [8,9]. The Preisach density distribution was obtained from the sets of M–H curves [8,9]. In the recording process, the external head field and the internal interaction field were applied to each cell. Since the Preisach model contains magnetic interactions within the cell only, the interaction field among cells was taken from the micromagnetic modeling. It consists of the magnetostatic field from all other cells and the exchange field from two nearest neighbor cells. The interac~i exposed on the ith cell is given by tion field, H N 2 X 8A * X Mj ~i ¼ H Nij Mj þ 2 ; ð1Þ 2 M S j¼n:n: ðDi þ Dj Þ j¼N 2
where Nij is the demagnetization tensor, A * is the effective exchange constant, Di is the edge length of ith cell along the film plane, Mj is the magnetization of jth cell, and MS is the saturation magnetization. The relation between the effective exchange constant and the loop slope was obtained by using micromagnetic simulations and is given by [10] A * ¼ ðDMS Þ2 1:217 tan1 0:579
4pMS 0:579 ðHC þ Hn Þ
;
ð2Þ where HC is the medium coercivity. Hn is the nucleation field to nucleate reversal domain and is defined a minus value when the nucleation occurs in the second quadrature of hysteresis loop. Eq. (2) is valid only if A is a positive value. By using the measured loop shape and the Eq. (2), we obtained the exchange constant less than zero. Since the
399
Fig. 1. A schematic diagram of modeling method. Each cell has the respective Preisach plane.
exchange constant in ferromagnetic materials cannot have a minus value, the exchange constant in this paper was set to zero. Finite element methods (FEM) were used to calculate writing head fields. The reading process was simulated using the reciprocity theorem. MR sensitivity function for reciprocity calculation is defined as the field that is applied on the medium from the MR head with a single turn coil around the MR element [11]. The 3D MR head sensitivity function can be obtained simply by superposing the Lindholm fields [12] for a finite width head and is given by ~G ðx; y; zÞ ¼ HGB h~GB ðx þ ðGT þ tMR Þ=2; y; zÞ H MR Lind GT HGT h~Lind ðx ðGB þ tMR Þ=2; y; zÞ;
ð3Þ where GT and GB are the distance from the free layer to the top shield and from the bottom shield to the free layer, respectively. HG is the deep gap and tMR is the thickness of the free layer. hG Lind is the normalized Lindholm field of a ring head with the head gap of G and the track width of W and is given as followings [12]: ~1 x; y; z w h~G ¼ h Lind 2 w ~ 0 ~ ð4Þ þ h 1 x; y; z þ h2 ðx; y; zÞ; 2 1 ~ 1 x þ g=2 ~ h1 ¼ F 1 sinh p y2 þ z 2 1 x g=2 ~ F 1 sinh g; ð5Þ y2 þ z 2
400
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
where
sinð2u=3Þ ; sinð2j=3Þ u j coshðu=3Þ cos þ ctnh1 ; F1y ðuÞ ¼ 4 cosh 3 3 cosðj=3Þ u j coshðu=3Þ sin þ ctnh1 F1z ðuÞ ¼ 4 cosh 3 3 sinðj=3Þ
F1x ðuÞ ¼ tan1
ð6Þ and j¼
(
p tan1 ðy=zÞ; zX0 tan1 ðy=zÞ; zo0
:
ð7Þ
0 h~1 is same with h~1 except j that should be replaced by j0 : ( tan1 ðy=zÞ; zX0 j0 ¼ ; ð8Þ p þ tan1 ðy=zÞ; zo0
1 ~ x þ g=2 x g=2 ~ ~ F2 h2 ¼ F2 ; p y y
ð9Þ
F2x ðuÞ ¼ tan1 ½u ; F2y ðuÞ ¼ ð1=2Þ lnðu2 þ 1Þ; F2z ðuÞ ¼ 0:
ð10Þ
In Fig. 2, contours of z-component (film thickness direction) of the MR head sensitivity function obtained from the Eq. (3) and FEM calculation
are compared with each other. Except track edge region of MR sensor, the MR sensitivity function of Eq. (3) is well agreed with the FEM results. In the DL medium, the MR head sensitivity function was obtained from a mirror imaging of the high permeable soft back layer and the head surface [13]. Fig. 3a shows the calculated MR sensitivity functions of the SL and the DL medium. The MR sensitivity function of the DL medium at read gap center is higher that that of the SL medium because of effects of soft back layer. From the Fourier transformation of the MR head sensitivity function to frequency domain, it was found that the MR sensitivity function of the DL medium acts like a low-pass filter (Fig. 3b) [14]. Therefore, the noise spectra of the DL medium at low frequency are higher than that of the SL medium as shown in Fig. 3c. These mean that the magnetization positioned far from the MR sensor can affect the reading signal through the high permeable soft back layer. Distributions of z-component of head field and gradient for the SL and the DL medium are shown in Fig. 4. Soft back layer of the DL medium enlarges field gradient and head field amplitude. Sharpness of the magnetization transition is determined by the field gradient at the region indicated by an arrow A. In Fig. 5a, magnetization profiles of recording layer at 10 KFCI are shown. It should be noted that the magnetization profiles
Fig. 2. Contours of the MR sensitivity function; (a) FEM calculation, and (b) superposition of the Lindholm field, where Tw is read track width.
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
401
Fig. 4. Head field distribution and gradient of the single and double layered medium; (a) head field distribution, and (b) field gradient.
Fig. 3. Comparisons of the MR sensitivity function in the single and the double layered medium; (a) the calculated MR sensitivity function, (b) the calculated transfer function of the MR sensitivity function, and (c) the measured noise spectra at 50 KFCI.
in the PMR medium are largely deviated from the arctangent-like function. Such the asymmetric profiles result from the demagnetization field of neighboring bit and the head field gradient. Due to the smaller field gradient, the SL medium shows smaller remanent magnetization and more asym-
metric magnetization profile than those of the DL medium. Simulated waveforms are compared with measured waveforms (Fig. 5b and c). The simulation results almost duplicate the experimental measurements except signal amplitude of the DL medium that may result from an overestimation of soft back layer’s mirror effects in the calculation of the MR sensitivity function. Distorted magnetization profile and waveform (indicated by arrow B) are obtained in the simulation results of the SL medium, which is similar to the earlier experimental report and arises from the positive nucleation field [15]. The negative nucleation field can endure the reverse head field of trailing edge [16]. Therefore, the nucleation field is one of the key factors to determine read/write characteristics of the PMR in the ring head applications. When the nucleation field has a minus value, the signal outputs for all linear recording densities are much enhanced from those of the positive nucleation field (Fig. 6). Furthermore, the current saturation
402
K.J. Lee et al. / Journal of Magnetism and Magnetic Materials 235 (2001) 398–402
waveforms of the single and the double layered medium were well agreed with the measurements. Importantly, profiles of the recorded magnetization were deviated from the arctangent-like function even in the double layered medium. In the ring head application, the sharp head field gradient and the negative nucleation field of the medium are essential for better read/write characteristics. The developed model has a big potential to understand the recording mechanism of the perpendicular recording and therefore can be used as an optimization tool of the head and the medium.
Acknowledgements The authors are grateful to Dr. K. Ouchi of AIT for assisting the media preparation. Fig. 5. Comparisons of calculated waveform and measured waveform in the single and the double layered medium at 10 KFCI; (a) Calculated magnetization profiles, (b) calculated waveform and measured waveform in the single layered medium, and (c) calculated waveform and measured waveform in the double layered medium.
Fig. 6. Roll-off curves of the single and double layered medium with varying the nucleation field.
curve is easily saturated when the medium has the negative nucleation field [16]. The developed model was examined in comparison with experimental measurements. Calculated
References [1] S. Yanase, T. Kiya, N. Honda, K. Ouchi, Technical Report of IEICE, Vol. 95–14, 1995, p. 15. [2] M. Futamoto, Y. Honda, Y. Hirayama, K. Itoh, H. Ide, Y. Maruyama, IEEE Trans. Magn. 32 (1996) 3789. [3] N. Honda, K. Ouchi, IEEE Trans. Magn. 33 (1997) 3097. [4] E. Miyashita, Y. Yoneda, J. Numazawa, J. Magn. Soc. Japan 21–S2 (1997) 31. [5] S. Saito, Y. Hatta, F. Hoshi, M. Takahashi, Digests of PMRC 2000, 23pA-26, 2000, p. 56. [6] R. Wood, Y. Sonobe, Digests of PMRC 2000, 23aA-01, 2000, p 1. [7] F. Preisach, Zeitschrift fuer Physik 94 (1935) 277. [8] G.S. Park, Ph. D. Thesis, Seoul National University, 1992. [9] C.D. Mee, E.D. Daniel, Magnetic recording technology, McGraw-Hill, New York, 1996, Chapter 2, p. 9. [10] K.J. Lee, T.D. Lee, Digests of CompuMAG 2001, OH-02, 2001. [11] H.N. Bertram, Theory of magnetic recording, Cambridge press, Cambridge, 1994, p. 178. [12] D.A. Lindholm, IEEE Trans. Magn. MAG-13 (1977) 1460. [13] H. Muraoka, Y. Sugita, Y. Nakamura, IEEE Trans. Magn. 35 (1999) 2235. [14] H. Muraoka, Y. Nakamura, Digests of Fifth ASIST, MR2000-52, 2000, p. 103. [15] A. Takeo, S. Oikawa, T. Hikosaka, Y. Tanaka, Digests of Intermag 2000, 2000, BB-03. [16] T.D. Lee, M.S. Hwang, K.J. Lee, Digests of PMRC 2000, 25aA-05, 2000, p. 147.