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Analysis of magnetization transition in perpendicular magnetic recording for narrow transition width
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Analysis of magnetization transition in perpendicular magnetic recording for narrow transition width Takeshi Kato, Kenji Miura, Hajime Aoi, Hiroaki Muraoka , Yoshihisa Nakamura RIEC, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
a r t i c l e in f o
a b s t r a c t
Available online 6 August 2008
In order to achieve high linear density recording, a narrow transition width or small transition parameter, a, is imperative. Reducing the grain size improves transition jitter and also reduces the transition width. The correlation between transition width and transition jitter was examined. The observed discrepancy from simple, analytical expressions introduces more realistic magnetization transitions with island magnetization reversals. It was found that island reversals increase the transition width, rather than the transition jitter, which must be suppressed to attain high linear density. & 2008 Elsevier B.V. All rights reserved.
Keywords: Perpendicular magnetic recording High linear density Transition width Transition jitter
1. Introduction Recent progress of areal density tends to be attained by increases in track density rather than linear density. However, for future high-density recording at more than 1 Tbit/in2, linear densities greater than 1500–2000 kBPI are unavoidable. Besides, a high linear density is advantageous for fast data transfer, which is an important performance issue for large capacity disk drives. At such high linear densities, where the bit length is less than 15–20 nm, a very narrow transition width is indispensable. The transition parameter, which is referred to as ‘a’, cannot be more than a few nanometers. Careful studies to realize these extremely narrow transitions are needed. In the granular medium structure a reduction of the grain size is required. However, experimental transition widths are usually much larger than one would expect from the grain size. In this paper, analyses based on granular perpendicular media are carried out.
A simple transition model for granular media has already been formulated in previous papers [1–4]. Transition jitter is governed by the transition parameter, the cross-track correlation length and the read track width. A coefficient that depends upon transition shape is also a parameter. The relationship between transition jitter, sjitter, and magnetic transition width, pa, was derived from a micro-track model by Caroselli et al. [1,2]:
p4 a2 sc 48W r
Corresponding author. Tel.: +81 22 217 5456; fax: +81 22 217 5496.
E-mail address: [email protected] (H. Muraoka). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.08.015
s2jitter ¼
pa2 sc 2W r
(2)
The relationships between sc, pa and the cluster size to the averaged grain size, /DgrainS, were applied to the above equation to eliminate sc:
(3) a ¼ 0:44 Dgrain ; sc ¼ 1:6 Dgrain This procedure ended up with the following relations:
sjitter / a3=2 or a / s2=3 jitter
(4)
These equations indicate that the transition width is proportional to the two-thirds power of the transition jitter.
2. Theoretical analysis
s2jitter ¼
where sc is the cross-track correlation length and Wr is the read track width. The averaged transition shape was assumed to obey a hyperbolic tangent function. Shimizu gave a similar formula with 96 in the denominator of Eq. (1), instead of 48 [3]. Bertram et al. [4] have developed the same relationship between transition width and transition jitter by a similar analytical derivation, but they assumed an error function for the transition shape. Their equation is
(1)
3. Experimental setup A merged single-pole writer with a GMR reader was used for spin-stand measurements. The track width of the write head was 180 nm, and the track width and shield gap of the GMR head were 140 and 55 nm, respectively. A shielded, single-pole writer was also used to test for the influence of head field gradient. CoPtCr–SiO2 perpendicular media and the coupled granular and continuous (CGC) perpendicular media [5] were used. Media #A were granular, CoPtCr–SiO2 media with various SiO2 contents and a recording layer thickness of 10 nm. The SiO2 content increases with sample number. Other specifications of media #A are listed
ARTICLE IN PRESS T. Kato et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
2901
25
Table 1 Specifications of media (media #A) SiO2 content (vol%)
Mr/Ms
KuV/kT
H0 (kOe)
a
#A1 #A2 #A3 #A4 #A5
20.53 23.99 27.17 30.09 32.79
0.95 0.95 0.95 0.90 0.92
254 173 123 93 74
7.74 8.80 9.59 9.22 8.43
4.40 3.02 2.14 2.01 2.14
in Table 1. Media #B were CGC media with various continuous layer thicknesses. Media #C were CoPtCr–SiO2 media with various SiO2 content and a 12-nm recording layer thickness. The magnetic transition width, pa, was estimated from measured readback waveforms. Readback loss was compensated according to the procedure described in Ref. [6]. First, the reproduced spectrum (50 kFCI) was sampled with a spectrum analyzer. Then, the head’s sensitivity function was calculated for the parameters of media and head used in the experiment. The sensitivity function was typically a product of the gap loss and spacing loss. The amplitude of each harmonic in the readback spectrum was corrected according to the transfer function, and inverse Fourier transformation provided a compensated magnetization distribution to estimate the transition width. The transition parameter, a, was calculated for this estimated magnetization distribution. Transition jitter, sjitter, was calculated by time-domain analysis.
20 Transition parameter a (nm)
Medium
15
a = 3.6·σ2/3
10
Medium #A1–#A5 Medium #B1–#B6
5
Medium #C1–#C4 Ref.
0 0
2
4 6 8 Transition jitter σjitter (nm)
10
12
Fig. 1. Analytical correlation between transition parameter and transition jitter.
4. Calculation of Voronoi cell model Granular media consisting of grains with random shapes and positions were modeled using Voronoi cells [7]. Magnetic transitions were represented by the boundaries of the Voronoi cells. The applied head field was approximated by a linear distribution in the vicinity of the transition. The head field strength at the center of gravity of each cell was compared to the switching field to determine the cell’s magnetization direction. The transition model was numerically sampled by dividing the media into micro-tracks. Finally, the readback waveform was obtained by reciprocity with a three-dimensional head sensitivity function [8]. Transition parameter, a, and transition jitter, sjitter, were then calculated. The mean grain size was varied from 10 to 70 nm. The coercivities of the cells were randomly assigned according to a normal distribution with a standard deviation sHsw. The normalized switching field distribution (SFD), sHsw//HswS, was varied between 0 and 0.5. Here, the SFD is considered to include the demagnetization effect during writing. Further discussion is required to know the quantitative relation to the measured SFD, in which the demagnetization field is eliminated [9]. The SFD of the actual samples was not measured. The head field gradient varied from 150 to 300 Oe/nm. The relationship between transition width and transition jitter for the Voronoi cell model in comparison with three analytical expressions is shown in Fig. 1. The writing condition was indicated in the figure. The thick solid line is a two-thirds power function fit to the Voronoi calculation. The transition width is proportional to the two-thirds power of transition jitter, not only for the analytical expressions but also for the Voronoi cell model calculation with zero SFD.
5. Relationship between transition width and jitter The correlation between transition width and transition jitter in experimental measurements is shown in Fig. 2. The data points
Fig. 2. Correlation between the measured transition parameter and transition jitter.
were scattered and did not obey the analytical prediction. The transition parameters tended to be larger than the Voronoi estimation. Since neither the simple cluster model nor the analytical formula can explain the experiments, any more realistic model should be introduced to simulate the transitions in the experimental media. A possible source of the large transition width is the generation of island reversals, that is, some grains are separately switched from neighboring grains in the vicinity of magnetization transitions, as shown in Fig. 3(a). A wide distribution of reversed islands increases the width of the averaged magnetization distribution. The transition jitter may not increase because of the small crosscorrelation length due to the small grain size. Fig. 3 demonstrates the influence of the island structure. In this transition width and transition jitter diagram, a pair of magnetization distributions that gives the same transition jitter of 7 nm
ARTICLE IN PRESS 2902
T. Kato et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
Transition parameter a (nm)
16 Lower write current
Lower write current
12
8
Shielded head
Head #1, medium #B6
4
Head #1, medium #B5 Head #2, medium #B6 Head #2, medium #B5
0 0
2
4 6 Transition jitter σjitter (nm)
8
Fig. 4. Correlation between the measured transition parameter and transition jitter for various head field gradients.
The results of the measurements and calculations show that island reversals are caused by insufficiently small head field gradients and large SFDs of the media. The transition width is largely influenced by the island reversals, especially for smaller grains; this prevents a reduction of transition width simply by reducing the grain size. High linear density performance will therefore suffer as a result of the magnetization distribution.
6. Suppression of the island reversals
Fig. 3. Influence of island reversals. The magnetization distribution of A has island reversals as shown in the bottom Voronoi modeling, but no island for B.
are shown. The ‘A’ condition has the island reversals generated by a large SFD of 50%, while the B condition gives the same jitter by making the grain size larger (40 nm) and eliminating the island reversals using a small SFD. The island structure enlarged the transition width or transition parameter by approximately 50% in spite of the small grain size of 20 nm. This result can explain well the large scatter in the experimental data. The island reversals should also depend upon the head field gradient. This influence was examined by measurements. In order to vary the head field gradient, the recording current was decreased. The results are plotted with closed symbols in Fig. 4. When the writing current is too large, the coercivity point moves to the foot of the head field distribution, which results in switching in a lower head field gradient. Reducing the current improved the head field gradient. A shielded, single-pole head was also used to improve the head field gradient. Both transition width and transition jitter were improved, as shown in Fig. 4 by the hollow symbols. The results of these experiments suggested that the steeper head field gradient reduced the number of island domains. A steep head field gradient led to narrower transition width and lower transition jitter. However, other factors that may be responsible for transition broadening should be carefully examined.
The generation of island reversals deteriorates transition width and transition jitter. The criterion not to generate the island reversals can be formulated as follows. Assuming the grains are circles with a diameter of D, and for a head field gradient represented as dHhy/dx, as shown in Fig. 5, the necessary condition for the formation of no island reversals is 3sHsw p
dHy D dx 2
(5)
Here, the SFD is assumed to be a Gaussian distribution with a standard deviation of s. The 3sHsw on the left-hand side of Eq. (5) means 99.7% of the grains will switch for the condition described on the right-hand side, i.e. the difference of the magnetic field strength between two neighboring grains. In the case that the right side is larger, and the island reversals are almost suppressed, the simple analytical expression can be used. As long as this condition is satisfied, a small grain size directly contributes to minimizing the transition width. Therefore, a narrower SFD or a steeper head field gradient is needed to suppress the formation of island domains. When the grain size is larger, the requirement on the SFD is not tight. For example, for a grain diameter of 20 nm, the necessary sHsw corresponds to 1 kOe in order to satisfy the condition represented in Eq. (5). The normalized standard deviation sHsw//HswS is 9.1% for a switching field of 11 kOe. However, this limitation becomes severe for a small grain size. If the magnetic grain size must be 6.5 nm to attain 1-Tbit/in2 recording, 3sHsw should be less than 975 Oe even for the head field gradient of 300 Oe/nm. This SFD corresponds to a normalized standard deviation, sHsw//HswS, of only 3% for an 11 kOe average switching field.
ARTICLE IN PRESS T. Kato et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
2903
observed in experiments was explained by this extended modeling. The transition width was increased by the island reversal, for the same transition jitter. The island reversal was generated due to insufficient head field gradients and broad switching field distributions. Therefore, in order to decrease the formation of island domains, steeper head field gradients or lower switching field distributions are necessary. In order to obtain a narrow transition width or a high linear density, grain size reduction is necessary. However, improvement of the switching field distribution and head field gradient must be simultaneously be attained alongside the grain size reduction, otherwise high linear density recording will not be realized.
Acknowledgments This work was partly supported by the IT-Program (RR2002) and Research and Development for Next-Generation Information Technology from MEXT, Japanese Government and Storage Research Consortium. The author would like to thank Dr. M. Hashimoto for his helpful discussions. References
Fig. 5. Schematics for the necessary condition not to form island reversals for head field gradient, grain size and switching field distribution.
7. Conclusion Analytical expressions in which the transition width is proportional to the two-thirds power of transition jitter were extended by the introduction of island grain reversals. The relationship between transition width and transition jitter
[1] [2] [3] [4] [5] [6] [7] [8] [9]
J. Caroselli, J.K. Wolf, IEEE Trans. Magn. 32 (1996) 3917. K.-Z. Gao, et al., IEEE Trans. Magn. 39 (2003) 704. Y. Shimizu, et al., IEEE Trans. Magn. 39 (2003) 1846. H.N. Bertram, et al., IEEE Trans. Magn. 42 (2006) 1620. Y. Sonobe, et al., IEEE Trans. Magn. 38 (2002) 2006. M.L. Mallary, et al., IEEE Trans. Magn. 40 (2004) 2567. K. Miura, et al., J. Magn. Magn. Mater. 287 (2005) 133. Y. Suzuki, et al., J. Magn. Magn. Mater. 287 (2005) 138. I. Tagawa, et al., IEEE Trans. Magn. 27 (6) (1991) 4975.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Analysis of magnetization transition in perpendicular magnetic recording for narrow transition width Takeshi Kato, Kenji Miura, Hajime Aoi, Hiroaki Muraoka , Yoshihisa Nakamura RIEC, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
a r t i c l e in f o
a b s t r a c t
Available online 6 August 2008
In order to achieve high linear density recording, a narrow transition width or small transition parameter, a, is imperative. Reducing the grain size improves transition jitter and also reduces the transition width. The correlation between transition width and transition jitter was examined. The observed discrepancy from simple, analytical expressions introduces more realistic magnetization transitions with island magnetization reversals. It was found that island reversals increase the transition width, rather than the transition jitter, which must be suppressed to attain high linear density. & 2008 Elsevier B.V. All rights reserved.
Keywords: Perpendicular magnetic recording High linear density Transition width Transition jitter
1. Introduction Recent progress of areal density tends to be attained by increases in track density rather than linear density. However, for future high-density recording at more than 1 Tbit/in2, linear densities greater than 1500–2000 kBPI are unavoidable. Besides, a high linear density is advantageous for fast data transfer, which is an important performance issue for large capacity disk drives. At such high linear densities, where the bit length is less than 15–20 nm, a very narrow transition width is indispensable. The transition parameter, which is referred to as ‘a’, cannot be more than a few nanometers. Careful studies to realize these extremely narrow transitions are needed. In the granular medium structure a reduction of the grain size is required. However, experimental transition widths are usually much larger than one would expect from the grain size. In this paper, analyses based on granular perpendicular media are carried out.
A simple transition model for granular media has already been formulated in previous papers [1–4]. Transition jitter is governed by the transition parameter, the cross-track correlation length and the read track width. A coefficient that depends upon transition shape is also a parameter. The relationship between transition jitter, sjitter, and magnetic transition width, pa, was derived from a micro-track model by Caroselli et al. [1,2]:
p4 a2 sc 48W r
Corresponding author. Tel.: +81 22 217 5456; fax: +81 22 217 5496.
E-mail address: [email protected] (H. Muraoka). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.08.015
s2jitter ¼
pa2 sc 2W r
(2)
The relationships between sc, pa and the cluster size to the averaged grain size, /DgrainS, were applied to the above equation to eliminate sc:
(3) a ¼ 0:44 Dgrain ; sc ¼ 1:6 Dgrain This procedure ended up with the following relations:
sjitter / a3=2 or a / s2=3 jitter
(4)
These equations indicate that the transition width is proportional to the two-thirds power of the transition jitter.
2. Theoretical analysis
s2jitter ¼
where sc is the cross-track correlation length and Wr is the read track width. The averaged transition shape was assumed to obey a hyperbolic tangent function. Shimizu gave a similar formula with 96 in the denominator of Eq. (1), instead of 48 [3]. Bertram et al. [4] have developed the same relationship between transition width and transition jitter by a similar analytical derivation, but they assumed an error function for the transition shape. Their equation is
(1)
3. Experimental setup A merged single-pole writer with a GMR reader was used for spin-stand measurements. The track width of the write head was 180 nm, and the track width and shield gap of the GMR head were 140 and 55 nm, respectively. A shielded, single-pole writer was also used to test for the influence of head field gradient. CoPtCr–SiO2 perpendicular media and the coupled granular and continuous (CGC) perpendicular media [5] were used. Media #A were granular, CoPtCr–SiO2 media with various SiO2 contents and a recording layer thickness of 10 nm. The SiO2 content increases with sample number. Other specifications of media #A are listed
ARTICLE IN PRESS T. Kato et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
2901
25
Table 1 Specifications of media (media #A) SiO2 content (vol%)
Mr/Ms
KuV/kT
H0 (kOe)
a
#A1 #A2 #A3 #A4 #A5
20.53 23.99 27.17 30.09 32.79
0.95 0.95 0.95 0.90 0.92
254 173 123 93 74
7.74 8.80 9.59 9.22 8.43
4.40 3.02 2.14 2.01 2.14
in Table 1. Media #B were CGC media with various continuous layer thicknesses. Media #C were CoPtCr–SiO2 media with various SiO2 content and a 12-nm recording layer thickness. The magnetic transition width, pa, was estimated from measured readback waveforms. Readback loss was compensated according to the procedure described in Ref. [6]. First, the reproduced spectrum (50 kFCI) was sampled with a spectrum analyzer. Then, the head’s sensitivity function was calculated for the parameters of media and head used in the experiment. The sensitivity function was typically a product of the gap loss and spacing loss. The amplitude of each harmonic in the readback spectrum was corrected according to the transfer function, and inverse Fourier transformation provided a compensated magnetization distribution to estimate the transition width. The transition parameter, a, was calculated for this estimated magnetization distribution. Transition jitter, sjitter, was calculated by time-domain analysis.
20 Transition parameter a (nm)
Medium
15
a = 3.6·σ2/3
10
Medium #A1–#A5 Medium #B1–#B6
5
Medium #C1–#C4 Ref.
0 0
2
4 6 8 Transition jitter σjitter (nm)
10
12
Fig. 1. Analytical correlation between transition parameter and transition jitter.
4. Calculation of Voronoi cell model Granular media consisting of grains with random shapes and positions were modeled using Voronoi cells [7]. Magnetic transitions were represented by the boundaries of the Voronoi cells. The applied head field was approximated by a linear distribution in the vicinity of the transition. The head field strength at the center of gravity of each cell was compared to the switching field to determine the cell’s magnetization direction. The transition model was numerically sampled by dividing the media into micro-tracks. Finally, the readback waveform was obtained by reciprocity with a three-dimensional head sensitivity function [8]. Transition parameter, a, and transition jitter, sjitter, were then calculated. The mean grain size was varied from 10 to 70 nm. The coercivities of the cells were randomly assigned according to a normal distribution with a standard deviation sHsw. The normalized switching field distribution (SFD), sHsw//HswS, was varied between 0 and 0.5. Here, the SFD is considered to include the demagnetization effect during writing. Further discussion is required to know the quantitative relation to the measured SFD, in which the demagnetization field is eliminated [9]. The SFD of the actual samples was not measured. The head field gradient varied from 150 to 300 Oe/nm. The relationship between transition width and transition jitter for the Voronoi cell model in comparison with three analytical expressions is shown in Fig. 1. The writing condition was indicated in the figure. The thick solid line is a two-thirds power function fit to the Voronoi calculation. The transition width is proportional to the two-thirds power of transition jitter, not only for the analytical expressions but also for the Voronoi cell model calculation with zero SFD.
5. Relationship between transition width and jitter The correlation between transition width and transition jitter in experimental measurements is shown in Fig. 2. The data points
Fig. 2. Correlation between the measured transition parameter and transition jitter.
were scattered and did not obey the analytical prediction. The transition parameters tended to be larger than the Voronoi estimation. Since neither the simple cluster model nor the analytical formula can explain the experiments, any more realistic model should be introduced to simulate the transitions in the experimental media. A possible source of the large transition width is the generation of island reversals, that is, some grains are separately switched from neighboring grains in the vicinity of magnetization transitions, as shown in Fig. 3(a). A wide distribution of reversed islands increases the width of the averaged magnetization distribution. The transition jitter may not increase because of the small crosscorrelation length due to the small grain size. Fig. 3 demonstrates the influence of the island structure. In this transition width and transition jitter diagram, a pair of magnetization distributions that gives the same transition jitter of 7 nm
ARTICLE IN PRESS 2902
T. Kato et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
Transition parameter a (nm)
16 Lower write current
Lower write current
12
8
Shielded head
Head #1, medium #B6
4
Head #1, medium #B5 Head #2, medium #B6 Head #2, medium #B5
0 0
2
4 6 Transition jitter σjitter (nm)
8
Fig. 4. Correlation between the measured transition parameter and transition jitter for various head field gradients.
The results of the measurements and calculations show that island reversals are caused by insufficiently small head field gradients and large SFDs of the media. The transition width is largely influenced by the island reversals, especially for smaller grains; this prevents a reduction of transition width simply by reducing the grain size. High linear density performance will therefore suffer as a result of the magnetization distribution.
6. Suppression of the island reversals
Fig. 3. Influence of island reversals. The magnetization distribution of A has island reversals as shown in the bottom Voronoi modeling, but no island for B.
are shown. The ‘A’ condition has the island reversals generated by a large SFD of 50%, while the B condition gives the same jitter by making the grain size larger (40 nm) and eliminating the island reversals using a small SFD. The island structure enlarged the transition width or transition parameter by approximately 50% in spite of the small grain size of 20 nm. This result can explain well the large scatter in the experimental data. The island reversals should also depend upon the head field gradient. This influence was examined by measurements. In order to vary the head field gradient, the recording current was decreased. The results are plotted with closed symbols in Fig. 4. When the writing current is too large, the coercivity point moves to the foot of the head field distribution, which results in switching in a lower head field gradient. Reducing the current improved the head field gradient. A shielded, single-pole head was also used to improve the head field gradient. Both transition width and transition jitter were improved, as shown in Fig. 4 by the hollow symbols. The results of these experiments suggested that the steeper head field gradient reduced the number of island domains. A steep head field gradient led to narrower transition width and lower transition jitter. However, other factors that may be responsible for transition broadening should be carefully examined.
The generation of island reversals deteriorates transition width and transition jitter. The criterion not to generate the island reversals can be formulated as follows. Assuming the grains are circles with a diameter of D, and for a head field gradient represented as dHhy/dx, as shown in Fig. 5, the necessary condition for the formation of no island reversals is 3sHsw p
dHy D dx 2
(5)
Here, the SFD is assumed to be a Gaussian distribution with a standard deviation of s. The 3sHsw on the left-hand side of Eq. (5) means 99.7% of the grains will switch for the condition described on the right-hand side, i.e. the difference of the magnetic field strength between two neighboring grains. In the case that the right side is larger, and the island reversals are almost suppressed, the simple analytical expression can be used. As long as this condition is satisfied, a small grain size directly contributes to minimizing the transition width. Therefore, a narrower SFD or a steeper head field gradient is needed to suppress the formation of island domains. When the grain size is larger, the requirement on the SFD is not tight. For example, for a grain diameter of 20 nm, the necessary sHsw corresponds to 1 kOe in order to satisfy the condition represented in Eq. (5). The normalized standard deviation sHsw//HswS is 9.1% for a switching field of 11 kOe. However, this limitation becomes severe for a small grain size. If the magnetic grain size must be 6.5 nm to attain 1-Tbit/in2 recording, 3sHsw should be less than 975 Oe even for the head field gradient of 300 Oe/nm. This SFD corresponds to a normalized standard deviation, sHsw//HswS, of only 3% for an 11 kOe average switching field.
ARTICLE IN PRESS T. Kato et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 2900–2903
2903
observed in experiments was explained by this extended modeling. The transition width was increased by the island reversal, for the same transition jitter. The island reversal was generated due to insufficient head field gradients and broad switching field distributions. Therefore, in order to decrease the formation of island domains, steeper head field gradients or lower switching field distributions are necessary. In order to obtain a narrow transition width or a high linear density, grain size reduction is necessary. However, improvement of the switching field distribution and head field gradient must be simultaneously be attained alongside the grain size reduction, otherwise high linear density recording will not be realized.
Acknowledgments This work was partly supported by the IT-Program (RR2002) and Research and Development for Next-Generation Information Technology from MEXT, Japanese Government and Storage Research Consortium. The author would like to thank Dr. M. Hashimoto for his helpful discussions. References
Fig. 5. Schematics for the necessary condition not to form island reversals for head field gradient, grain size and switching field distribution.
7. Conclusion Analytical expressions in which the transition width is proportional to the two-thirds power of transition jitter were extended by the introduction of island grain reversals. The relationship between transition width and transition jitter
[1] [2] [3] [4] [5] [6] [7] [8] [9]
J. Caroselli, J.K. Wolf, IEEE Trans. Magn. 32 (1996) 3917. K.-Z. Gao, et al., IEEE Trans. Magn. 39 (2003) 704. Y. Shimizu, et al., IEEE Trans. Magn. 39 (2003) 1846. H.N. Bertram, et al., IEEE Trans. Magn. 42 (2006) 1620. Y. Sonobe, et al., IEEE Trans. Magn. 38 (2002) 2006. M.L. Mallary, et al., IEEE Trans. Magn. 40 (2004) 2567. K. Miura, et al., J. Magn. Magn. Mater. 287 (2005) 133. Y. Suzuki, et al., J. Magn. Magn. Mater. 287 (2005) 138. I. Tagawa, et al., IEEE Trans. Magn. 27 (6) (1991) 4975.