THEORY OF
`MAGNETIC RECORDING
`
`H. NEAL BERTRAM
`University of California at San Diego
`
`..... ~ ..... CAMBRIDGE
`::: UNIVERSITY PRESS
`
`UMN EXHIBIT 2028
`LSI Corp. et al. v. Regents of Univ. of Minn.
`IPR2017-01068
`
`
`Page 1 of 46
`
`

`

`Published by the Press Syndicate of the University of Cambridge
`The Pitt Building, Trumpington Street, Cambridge CB2 IRP
`40 West 20th Street, New York, NY l0011-421 I, USA
`10 Stamford Road, Oakleigh, Melbourne 3166, Australia
`
`© Cambridge University Press 1994
`
`First published 1994
`
`A catalogue record for this book is available from the British Library
`
`Library of Congress cataloguing in publication data
`Bertram, H. Neal.
`Theory of magnetic recording / H. Neal Bertram.
`p. cm.
`Includes bibliographical references and index.
`ISBN 0-521-44973-l (pbk.)
`ISBN 0-521-44512-4. -
`I. Title
`1. Magnetic recorders and recording.
`TK7881.6.B47 1994
`621.382'34-dcZ0 93-29978 CIP
`
`ISBN 0 521 44512 4 hardback
`ISBN 0 521 44973 I paperback
`
`Transferred to digital printing 2003
`
`KW
`
`
`Page 2 of 46
`
`

`

`Contents
`
`Preface
`1 Overview
`Materials and magnetization processes
`The magnetic recording channel
`Units
`2 Review of magnetostatic fields
`Introduction
`Basic field expressions
`Demagnetizing factors
`Magnetostatic fields from flat surfaces
`Two-dimensional fields
`Imaging
`Vector and scalar potentials
`Fourier and Hilbert transforms
`Integral relations for free space fields
`Problems
`3 Inductive head fields
`Introduction
`Head efficiency and deep-gap field
`Fields due to a finite gap
`Far field approximation
`Medium range approximation ( Karlqvist field)
`Near-field expressions
`Near-field analytic approximation
`Finite length heads - thin film heads
`Keepered heads
`Concluding remarks
`
`lX
`
`page xiii
`1
`2
`10
`15
`17
`17
`18
`24
`26
`28
`31
`35
`38
`44
`47
`48
`48
`52
`56
`56
`59
`63
`73·
`75
`81
`85
`
`
`Page 3 of 46
`
`

`

`ontents
`
`Contents
`
`86
`89
`89
`89
`98
`100
`105
`
`107
`107
`108
`112
`117
`119
`119
`124
`125
`127
`133
`139
`139
`139
`141
`146
`147
`148
`152
`154
`155
`
`157
`159
`161
`164
`166
`166
`169
`177
`182
`186
`
`>r longitudinal magnetization
`
`neral concepts and single
`
`voltage
`
`:xpressions
`
`!xample
`
`ltiple transitions
`
`compared - simplified D 50
`
`re
`
`Fourier transforms and accurate pulse shapes
`Dual MR heads
`Off-track response - finite track widths
`MR resistance noise and SNR
`Problems
`8 Record process: Part 1 - Transition models
`Introduction
`The magnetic recording process
`Models of longitudinal recording
`Voltage-current relations
`Demagnetization limits
`Thick media tape recording
`Models of perpendicular recording
`Thin medium model
`Thick medium modeling
`Problems
`9 Record process: Part 2 - Non-linearities and overwrite
`Introduction
`Non-linear bit shift
`Non-linear amplitude loss
`Overwrite
`Problems
`10 Medium noise mechanisms: Part I - General concepts,
`modulation noise
`Introduction
`Noise formalism
`Tape density fluctuations
`Tape surface roughness and asperities
`Problems
`11 Medium noise mechanisms: Part 2 - Particulate noise
`Introduction
`Single particle replay expressions
`General particulate noise expression
`Uncorrelated noise power
`Correlation terms
`Signal to particulate noise ratios
`Case 1: Thick tape .
`Case 2: Thin films
`Erased noise spectra and correlations
`Problems
`
`Xl
`
`192
`194
`199
`202
`203
`205
`205
`207
`213
`219
`223
`224
`230
`232
`235
`240
`245
`245
`245
`249
`252
`260
`
`261
`261
`265
`266
`276
`280
`283
`283
`284
`288
`290
`292
`293
`· ,293
`294
`296
`302
`
`
`Page 4 of 46
`
`

`

`X
`
`Contents
`
`Problems
`4 Medium magnetic fields
`Introduction
`Single transitions
`Field maxima and gradients for longitudinal magnetization
`Fourier transforms
`Problems
`5 Playback process: Part 1 - General concepts and single
`transitions
`Introduction
`Direct calculation of playback voltage
`The reciprocity principle
`Head definitions
`Summary comments
`Generalized playback voltage expressions
`Isolated transitions
`Sharp transitions
`Broad transitions - arctangent example
`Problems
`6 Playback process: Part 2 - Multiple transitions
`Introduction
`Linear superposition
`Square wave recording
`'Roll-off' curve and D 50
`Thin film head
`Spectral analysis
`,
`The 'spectrum
`Wallace factor or 'thickness loss'
`Analysis of spectra
`'Roll-off curve and spectrum compared - simplified Dso
`analysis
`Transfer function
`Linear bit shift
`Problems
`7 Magnetoresistive heads
`Introduction
`Magnetization configurations
`Reciprocity for MR heads
`Application to shielded heads
`Evaluation of the playback voltage
`
`86
`89
`89
`89
`98
`100
`105
`
`107
`107
`108
`112
`117
`119
`119
`124
`125
`127
`133
`139
`
`i
`t •
`i ;)
`!:
`i i
`~
`I
`l
`I
`I
`',i I
`"'
`'ii
`I
`139 I
`
`139
`141
`146
`147
`148
`152
`154
`155
`
`157
`159
`161
`164
`166
`166
`169
`177
`182
`186
`
`
`Page 5 of 46
`
`

`

`xii
`
`Contents
`
`12 Medium noise mechanisms: Part 3 - Transition noise
`Introduction
`Transition position and voltage amplitude jitter models
`Spectral analysis
`Comments on track-width dependence
`lnterbit interactions
`Generalized microscopic formulation
`Uniformly magnetized media
`Noise in the presence of a recorded transition
`Problems
`References and bibliography
`Index
`
`306
`306
`310
`312
`315
`318
`322
`326
`329
`335
`337
`353
`
`!
`
`1:
`
`:i;;:
`Jc
`
`' ' £
`1 i:
`!i i
`I ¥
`!f 1
`.. ;$
`j
`
`Magn,
`steady
`the la
`increa
`OCCUfl
`media
`
`growt
`heads
`
`not 01
`comp,
`time-c
`difficl
`thorn
`itudin
`In
`funda
`boob
`specif
`the b
`provi,
`The I
`incluc
`the n
`noise.
`be de
`result
`
`I of the
`I menta
`I
`
`I
`
`.'.,!
`
`
`Page 6 of 46
`
`

`

`!eads
`
`ation due to a permanent
`taut bias field yields a
`>th into the medium. Show
`stance with field holds until
`1/hat is the average angle at
`nd this critical point, the
`the element surfaces with a
`vely increasings. Develop a
`by examining the energy in
`:sistance versus field.
`
`>ility the right hand side in
`u, where µ, is the relative
`
`ltage and Fourier transform
`with an off-center element.
`both the voltage and its
`
`>f the effect on the Fourier
`lement of a surface gap field
`1etric cusps.
`
`s in Thompson (1975) to
`efficiency of a shielded MR
`
`Fourier transform of the
`~ the simple linear potential
`:e that the isolated pulse
`differential head at long
`tre the response minima?
`
`1tial and surface field (Fig.
`~eper is brought close to the
`
`8
`Record process:
`Part 1 - Transition models ·
`
`Introduction
`In this chapter, models will be discussed that give insight into the
`recording process. In contrast to the playback process, the recording of
`magnetization patterns is a non-linear phenomenon. Thus, except for
`special cases, analysis must be by computer simulation. It is the long(cid:173)
`range magnetostatic fields that cause computer simulations to be iterative
`and extremely time consuming. There are two philosophical approaches
`to computations of the record process. One is to neglect the fine details
`and develop reasonably approximate models that capture the main
`physical features of the process. Such simplified models allow for analytic
`solutions of magnetization patterns or solutions that require a minimum
`of computer simulation. This simplified approach is useful for developing
`guidelines in media development (e.g. the effect of coercivity or
`remanence changes) or in head-media interface geometry development
`(effect of head-medium spacing, record gap, medium thickness).
`Parameters can be easily changed in simplified models.
`The other approach is to develop full numerical micromagnetic
`models. These are required in order to compute detailed, or second
`order effects, of the recording process, such as noise, edge-track writing,
`and non-linear amplitude loss and thus, for example, give fundamental
`input to error-rate calculations. Simulations can give detailed informa(cid:173)
`tion about pulse asymmetry in tape recording (Bertram, et al., 1992) or
`the fluctuations of the transition center across the track that leads to
`position jitter or transition noise in thin film media (Zhu, 1992). For
`example, in Fig. 8.1
`the vector magnetization distribution from
`micromagnetic simulation of a recorded transition in thin film media is
`shown. The medium is presumed to be polycrystalline, as in Fig. 1.2, with
`
`205
`
`
`Page 7 of 46
`
`

`

`206
`
`Record process: Part 1 - Transition models
`
`Fig. 8.1.
`
`(b)
`thin film from
`in a polycrystalline
`(a) Grain magnetizations
`micromagnetic simulation of a recorded transition. (b) Gray scale plot of the
`component of magnetization along the recording direction. The vectors at the
`right of the figures denote the average position of the transition so that the cross(cid:173)
`track direction is from left to right. For the simulation the grain crystalline
`anisotropy axes were randomly oriented in 3D and no intergranular exchange
`coupling was assumed. Taken from Bertram & Zhu (1992).
`individual grain anisotropy axes randomly oriented. In Fig. 8. l(a) the
`magnetization vector orientation is shown corresponding to each grain
`(simulated by a hexagonal lattice). On either side of the transition the
`magnetization is in the remanent state (±Mr): the magnetizations are
`approximately parallel to the track direction except for a small ripple
`pattern. In the transition region the equilibrium magnetizations of each
`grain rotate away from the recording direction to form localized vortex
`
`
`Page 8 of 46
`
`

`

`,ition models
`
`ycrystalline thin film from
`1. (b) Gray scale plot of the
`direction. The vectors at the
`1e transition so that the cross(cid:173)
`mlation the grain crystalline
`1d no intergranular exchange
`u (1992).
`
`,riented. In Fig. 8.l(a) the
`1rresponding to each grain
`· side of the transition the
`·,): the magnetizations are
`, except for a small ripple
`1m magnetizations of each
`m to form localized vortex
`
`The magnetic recording process
`
`207
`
`patterns. The magnetization configuration of the transition results from a
`reduction of magnetostatic energy against the random crystalline
`anisotropy. The magnetization averaged across the track width plotted
`versus track position will show a general antisymmetric shape that can be
`approximated, for example, by the functional forms in Table 4.1. Note
`that at the transition center where the average component of
`magnetization along the track direction vanishes, the grain magnetiza(cid:173)
`tions are correlated. The magnetization directions, however, are equally
`distributed in the film plane (Bertram, et al., 1993). Even though the
`anisotropy axes are randomly oriented, magnetostatic fields from typical
`Co based media force the grain magnetizations to lie in the film plane.
`In Fig. 8.l(b) a gray scale picture of Fig. 8.l(a) is shown in which the
`component of magnetization along the track direction varying between
`±Mr is represented by a variation from black to white. The approximate
`'zigzag' wall form separating regions of the two different remanent
`magnetization directions is readily seen. Simplified modeling assuming
`zigzag walls can give reasonable analytic models of transition shape as
`well as noise (Minnaja & Nobile, 1972; Freiser, 1979; Muller & Murdock,
`1987; Middleton & Miles, 1991; Semenov, et al., 1991).
`In this chapter simplified, macroscopic models of the recording process
`are discussed. The essential simplification is to assume a (possible vector)
`hysteresis loop and then calculate the effect of the net field including the
`external field and the macroscopic magnetostatic field (2.16). Here only
`2D models are discussed, so that magnetization variations are restricted
`to the (x, y) plane. First an overview of the basic writing process is given,
`followed by transition modeling for thin longitudinal media. Comments
`about problems in simplified modeling of thick tape media are included.
`The chapter also deals with modeling of perpendicular media and
`demagnetization limits of longitudinal transition shapes.
`
`The magnetic recording process
`Here, a qualitative discussion of the writing process in longitudinal
`recording is given, including the formation of 'hard' and 'easy'
`transitions. The essence of the recording process is to produce reversals
`of magnetization according to the input record-head current. The writing
`process yields alternately 'hard' and 'easy' transitions. In Fig. 8.2, a
`positive record-head field is sketched that is oppositely directed to .ap
`assumed initial 'DC' reversed magnetization (at remanent level-M,). In a
`region (Hh > He) near the gap, the head field exceeds the coercivity, and
`
`
`Page 9 of 46
`
`

`

`208
`
`Record process: Part 1 - Transition models
`
`-----..tt-••··········· --
`
`_ _ _ :._::,•..:.::.:.•..!'!-".L•.,1.._ _ _ _ _ X
`
`,------- - - - - - - M ,
`
`I
`I
`I
`I
`I
`
`Fig. 8.2. Writing o f a ' h a r d ' transition against uniformly saturated media. This
`case forms transitions a t either side o f the gap the instant the head current is
`applied. The magnetization is solid and the head field is dashed.
`the magnetization is reversed in t h a t region (and saturated t o Ms)- I n this
`case, as illustrated i n the figure, two transitions are written, one o n either
`side o f the gap. As the medium moves i n the presence o f the constant
`head field (fixed current), the 'upstream' o r right transition i n the figure
`moves with the medium while the left transition remains fixed to the gap
`edge continuously formed by new media moving into the gap region a n d
`becoming saturated. The right transition is the recorded transition t h a t
`will eventually be p a r t o f a d a t a pattern. This written transition is termed
`a ' h a r d ' transition and is formed simultaneously with a transition a t t h e
`I n Fig. 8.3 the magnetization p a t t e r n is shown after a time interval
`opposite gap edge.
`when the medium has moved with respect to t h e configuration formed a t
`the instant o f field reversal a n d transition formation. I t is assumed t h a t
`the recording current is held constant during this motion. The recorded
`o r right h a n d transition moves with the medium while the left h a n d
`transition remains fixed to the gap. N o t e t h a t the recorded transition
`after medium m o t i o n relaxes somewhat away from the presence o f the
`head fields, so t h a t the remanent transition magnetization varies from
`- M , to + Mr. I n Fig. 8.4 the instant o f writing a second transition is
`shown. The (negaiive) head field tha t writes this transition is now in the
`' D C ' m a g n e t i z e d m e d i u m a n d ,
`approximation, only the recorded o r right hand transition is written.
`same d i r e c t i o n as
`The head field saturates the gap region in the same direction as the
`
`to g o o d
`
`the
`
`
`Page 10 of 46
`
`

`

`nsition models
`
`The magnetic recording process
`
`209
`
`v-
`
`------,
`
`I
`I
`I
`I
`
`Fig. 8.3. Magnetization configuration as time passes with head current on. The
`'upstream' transition is fixed to the medium and the left edge transition is fixed to
`the gap.
`
`••••··••••.J,.· ·---- X
`
`-------M,
`
`11iformly saturated media. This
`he instant the head current is
`field is dashed.
`
`md saturated to M 5 ). In this
`ns are written, one on either
`1e presence of the constant
`ight transition in the figure
`.on remains fixed to the gap
`ring into the gap region and
`the recorded transition that
`written transition is termed
`usly with a transition at the
`
`;hown after a time interval
`the configuration formed at
`rmation. It is assumed that
`i this motion. The recorded
`tedium while the left hand
`hat the recorded transition
`ry from the presence of the
`magnetization varies from
`iting a second transition is
`this transition is now in the
`l medium and, to good
`hand transition is written.
`the same direction as the
`
`-m~....,--.-:-=-=-----t--i,-,--_:-:;_;-;;:_-.._~m,.....,,---.---x (x)
`Hh(x) ~~,,
`\
`\
`I
`----r--+--
`, ... .._,
`
`'
`\
`\
`_______ ...,_
`
`------,
`
`I
`I
`I
`I
`
`v-
`
`I
`I
`I
`I
`
`Fig. 8.4. Writing of an 'easy' transition in the direction of incoming
`magnetization when the head current is reversed. Left edge transition does not
`occur or is minimal. The writing transition spacing between the first and secoqd
`transition in this 'dibit' is B = v/T where Tis the time interval between head.(cid:173)
`current reversal.
`
`
`Page 11 of 46
`
`

`

`210
`
`t r a n s i t i o n
`
`is
`
`in
`
`(8.1)
`
`t h a t d e p e n d o n
`
`t h e
`
`R e c o r d process: Par t I - Transition models
`left edge
`incoming magnetization, a n d no apprec iab le
`written. This second t r a n s i t i o n w r i t t e n i n t h e m e d i u m is termed a n ' e a sy '
`t r a n s i t i o n . Subsequent transitions a l t e r n a t e between ' h a r d ' a n d 'easy ' .
`T h e terms ' h a r d ' a n d ' e a sy ' arise f r o m the effect o f demagnetization
`fields on t h e recording process. As will be discussed i n this chap te r , t h e
`' h a r d ' t ran s i t ion requires m o r e head field t o s a t u r a t e t h e magne t iza t ion in
`t h e gap region du e t o increased d em agn e t i z a t ion fields (Bloomberg, e t al.,
`1979). I n add i t ion , non-linear bit shift occurs i n t h e writing o f a ' h a r d '
`t r a n s i t i o n t h a t affects t h e overwrite process, even i f t h e record c u r r e n t is
`trans i t ions , as discussed
`sufficient to s a t u r a t e b o t h h a r d a n d easy
`C h a p t e r 9. The p a t t e r n o f h a r d a n d easy t r a n s i t i o n s is complicated i f t h e
`the h e a d region con ta in s a previously w r i t t e n d a t a
`medium entering
`Utilizing simplified models t h a t give averaged proper t ies , t h e s h a p e o f
`p a t t e r n .
`a t r a c k width averaged recorded t r a n s i t i o n depends o n t h e head field
`d u r i n g recording, the m a g n e t o s t a t i c fields associated with the t r a n s i t i o n s ,
`a n d the medium M - H (vector) hysteresis loop . Generally the magnetiza(cid:173)
`t i o n a t a n y position a long the m e d i u m (x), o r a t a n y d e p t h i n t o t h e
`medium (y) is given by:
`M ( x , y ) = Fioop(Hh(x,y) + H a ( x , y ) )
`I n (8.1) Fioop is a generalized vector hysteretic function t h a t relates the
`t o t a l vector fields to t h e resu l tan t vector m a g n e t i z a t i o n a t each p o i n t i n
`t h a t includes
`the medium. A vector M - H l o o p r e p r e s e n t a t i o n for F 1 0 0P
`remanen t as well as reversible p rope r t ie s h a s been developed using the
`(e.g., B h a t t a c h a r y y a , e t al., 1989) o r particle
`i n general, a non - l inea r
`f o r m u l a t i o n
`Preisach
`assemblies (Beardsley, 1986). Since F 1 0 0p
`function, as well as hysteretic a n d field d i r e c t i o n dependent, numerical
`analysis m u s t be utilized t o solve (8.1 ). E a c h o f N discretization cells is
`represented by (a possibly s p a t i a l varying) F 1 0 0 p , a n d the set o f N coupled
`equ a t ion s is solved iteratively. The p r o b l e m is numerically intensive d u e
`m agn e t i z a t ion p a t t e r n (2.8, 2.22). C o m p u t a t i o n time varies as N 2 f o r
`t o c o m p u t a t i o n o f
`direct field evaluation, b u t only as N l n N when F a s t F o u r i e r t r a n s f o r m
`techniques are utilized ( M a n s u r i p u r , 1989; Y u a n & B e r t r a m , 1992a).
`t r a c k edge r e c o r d i n g (z).
`this f o rm u l a t i o n utilizes average m a g n e t i z a t i o n properties a t
`E q u a t i o n (8.1) c a n be extended
`each p o i n t i n the medium, m e d i um noise does n o t result. T h u s , (8.1) is
`Because
`convenient t o s t u d y overwrite ( B h a t t a c h a r y y a , e t al., 1991) a n d n o n(cid:173)
`linearities arising from l o o p hysteresis (Simmons & D a v i d s o n , 1992).
`
`t h e m a g n e t o s t a t i c
`
`is,
`
`fields
`
`to analyze
`
`
`Page 12 of 46
`
`

`

`I
`\!,':'
`
`I'
`i'i: !Ii'
`
`11 1
`i'
`
`tion models
`
`left edge transition is
`'
`'
`!dium is terme an easy
`d
`Neen 'hard' and 'easy'•
`ffect of demagnetization
`ussed in this chapter, the
`trate the magnetization in
`t fields (Bloomberg, et al.,
`in the writing of a 'hard'
`c:n if the record current is
`.nsitions, as discussed in
`tions is complicated if the
`previously written data
`
`d properties, the shape of
`epends on the head field
`;iated with the transitions,
`Generally the magnetiza(cid:173)
`or at any depth into the
`
`(8.1)
`
`~ function that relates the
`1etization at each point in
`.on for Fioop that includes
`been developed using the
`et al., 1989) or particle
`in general, a non-linear
`tion dependent, numerical
`of N discretization cells is
`and the set of N coupled
`,p,
`; numerically intensive due
`lds that depend on the
`.ion time varies as N2 for
`en Fast Fourier transform
`Yuan & Bertram, 1992a).
`track edge recording (z).
`1agnetization properties ~t
`~s not result. Thus, (8.1) is
`ya, et al., 1991) and non(cid:173)
`ons. & Davidson, 1992).
`
`The magnetic recording process
`
`211
`For thin film recording media where the magnetization does not vary
`with depth into the medium and a single component of magnetization is
`of interest, (8 .1) simplifies to:
`
`(8.2)
`
`where M(x) is the magnetization component of interest and ii represents
`the field component along the magnetization direction at each position x
`averaged through the depth y of the medium. The M-H loop appropriate
`to F100P corresponds to the conventional unidirectional form illustrated in
`Fig. 1.3. In general, numerical analysis simulates the writing of a
`transition sequence, including reversible and irreversible magnetization
`changes that occur during medium motion.
`The writing of a single transition against a reversed background
`remanent magnetization (Fig. 8.2) involves utilization of only half of
`the major or outer loop. For media magnetized initially negatively, the
`loop portion illustrated in Fig. 8.4 (solid) applies. Application of a
`spatially varying, but generally positive field, changes the magnetization
`along this curve. As the recording process proceeds by the increase of
`the head field to its final value, demagnetization fields form
`corresponding to the changing magnetization pattern. Even if the
`head field has regions of negative value (as in perpendicular recording
`with a ring head) or the magnetostatic field is opposite to and exceeds
`the head field, the development of the magnetization will be along the
`monotonic curve in Fig. 8.4. Changes along minor loops will not occur.
`Thus, in an iterative solution to (8.2) in which the magnetization at
`each position x might oscillate due to the iteration procedure, a
`decrease of the magnetization will follow the major loop and not a
`minor loop. Decreases in magnetization from a given stage in the
`iteration is a mathematical, not a physical phenomenon. The writing of
`a single transition, in essence, involves a monotonic change of
`magnetization away from an initial remanent state. Simple approxima(cid:173)
`tions to the major loop are shown in Fig. 8.4 of a linear slope (dashed)
`and a perfectly square loop ( dotted).
`Examples of the iterative solution of (8.1) are given in (Potter &
`Schmulian, 1971; Barany, 1989; Speliotis & Chi, 1978) for thin
`longitudinal media and (Beusekamp & Fluitman, 1986; Zhu &
`Bertram, 1986) for perpendicular recording. If a perfectly square·
`medium is assumed, Fig. 8.4 (dotted), then the magnetization transition
`shape that develops is one in which the magnetostatic field, at least in the
`
`
`Page 13 of 46
`
`

`

`212
`
`I n
`
`p a t t e r n
`
`s o l u t i o n
`
`fo r
`
`Record process: Part 1 - Transition models
`c en t r a l p o r t i o n o f the transition, j u s t balances t h e h e a d field:
`T h i s f o r m m a y be inverted analytically t o solve fo r J h e m a g n e t i z a t i o n
`t h i s c h a p t e r a n a l y t i c a l a n a l y s i s
`p e r p e n d i c u l a r recording is discussed; however, a t t e m p t s a t a n analytic
`i n m a n y cases.
`long i tud in a l recording have been m a d e ( v a n H e r k &
`Wesseling, 1974). F o r longitudinal r e c o r d i n g the a s s u m p t i o n o f (8.3) is
`n o t valid over t h e regions o f n e a r s a t u r a t i o n where the demagnetizing
`field vanishes. W i t h the a s s u m p t i o n o f a s q u a r e l o o p with n o reversible
`susceptibility, t h e effects o f m e d i u m m o t i o n a n d s u b s e q u e n t field reversal
`a r e readily included. A s the m e d i u m moves away f r o m the g a p region (as
`i n Fig. 8.3), t h e h e a d fields decrease. T h e m a g n e t i z a t i o n remains fixed as
`the net field decreases, until a t some s t ag e t h e m a g n e t o s t a t i c field reaches
`a value o f - H e - A t this p o i n t (8.2) m u s t be solved, b u t the i t e r a t i o n is
`m o r e c om p l i c a t e d t h a n simple u t i l i z a t i o n o f a m o n o t o n i c t r a n s f e r curve,
`as i l l u s t r a t e d i n Fig. 8.5.
`E q u a t i o n (8.3) can also be w r i t ten i n a derivative form:
`( d H h ( x ) + dHct(x))
`d M ( x ) = d M
`d x
`d x
`d H 1 o o p
`d x
`
`(8.3)
`
`f o r
`
`(8.4)
`
`M
`
`- M , - - - - - - + - ' " " -
`
`Fig. 8.5 Sketch o f ma jo r loop f r o m Fig. 1.3, which is the transfer curve for the
`writing o f a transition from a medium initially s a t u r a t e d a t - M r . The dashed
`curve is a linear slope approx ima t ion where the magnetization varies from - M ,
`t o +M, w i th no distinction between M , a n d M 5 • The dotted curve represents a
`
`square-loop material.
`
`
`Page 14 of 46
`
`

`

`rzsition models
`
`:s the head field:
`
`r)
`
`(8.3)
`
`;olve for the magnetization
`r analytical analysis for
`'er, attempts at an analytic
`been made (van Herk &
`" the assumption of (8.3) is
`~n where the demagnetizing
`1are loop with no reversible
`md subsequent field reversal
`rway from the gap region (as
`tgnetization remains fixed as
`e magnetostatic field reaches
`~ solved, but the iteration is
`a monotonic transfer curve,
`
`:rivative form:
`
`(8.4)
`
`I
`
`.
`
`L +M,
`
`H
`
`iHc
`!
`!
`!
`
`,hich is the transfer curve for the
`' saturated at -M,. The dashed
`: magnetization varies from - M,
`rs. The dotted curve represents a
`
`Mode ls of longitudinal recording
`
`213
`Equation (8.4) can be solved iteratively in terms of the transition
`derivative. dM / dH1oop is simply the field derivative of the Fioop or the M(cid:173)
`H loop derivative and is a function of the total field at each position
`along the medium. Corresponding to (8.3), if the magnetization changes
`are confined to the region of the loop where H = He, then the transition
`shape may be determined, assuming a square loop, by solving:
`
`(8.5)
`
`Note that in a macroscopic formulation, the magnetization at each point
`in the medium is a function of the net field via the M-H loop. The net
`field in magnetic recording is due to the sum of the applied recording field
`and the magnetostatic field. If the spatial change of the fields is on the
`order of the film grain size or the spatial scale of whatever microscopic
`phenomena that determines the bulk M-H loop, then rnicromagnetic
`analysis must be utilized in the determination of recorded magnetization
`patterns.
`
`Models of longitudinal recording
`In the first part of this section a simplified analytic model for the
`longitudinal recording in thin media will be discussed in detail (Williams
`& Comstock, 1971; Maller & Middleton, 1973). The model gives great
`insight into the longitudinal recording process and provides a simple
`analytic formula for the transition length. The magnetization is assumed
`to be longitudinal, which is a good approximation for high-moment thin
`film media and the differential form (8.4) will be utilized with only
`longitudinal field components. The simplification arises from the a priori
`assumption of the transition shape. In the simplest form of the model the
`magnetization is assumed to vary between ±Mr in a functional form
`where the position xis scaled by the transition parameter 'a' (4.17). The
`transition parameter is taken to be an unknown to be determined. Here
`the transition shape wil be left arbitrary, but typical shapes are listed in
`Table 4.1. For an arctangent transition the form:
`M(x) = 2Mr tan-Ix - xo
`a
`is utilized where x 0 denotes the location of the center of the transitipn.
`The demagnetization fields associated with ( 4.6) or any antisymmetric
`transition have the form shown in Fig. 4.6(a).
`
`(8.6)
`
`7r
`
`
`Page 15 of 46
`
`

`

`214
`
`Record process: Part 1 - Transition models
`
`M ( x )
`
`- ,
`\
`
`I
`, '
`
`I
`I
`I
`I
`I
`I
`I
`
`\
`
`\
`
`\
`
`--\\::
`--/---
`,..-_"-'_"-'_"-'_"--'_:..:_::... ____ , _ _ _ ---+-----Jl.'------_-_-....;-:..:-:..:-c:-~-c.,,-~---~ X
`
`Hn(X) , /
`~
`
`: \ Hnet
`' ,
`
`I
`
`Ho
`
`------,
`
`I
`I
`I
`I
`
`Fig. 8.6. Magnetization transition, head field and demagnetizing field near the
`transition center (x = x 0 ) for a simplified analysis o f longitudinal recording.
`A simple view o f a recorded transition without regard to ' h a r d ' o r
`'easy' effects is shown in Fig. 8.6. A f t e r the h e a d field is applied a n d
`before medium m o t i o n is considered, a transition is written with center
`l o c a t i o n a t xo. A s s um i n g a p e r f e c t l y a s y m m e t r i c
`magnetostatic field vanishes a t the transition center so t h a t the center
`the position where
`t o first order, by
`location is given,
`averaged) head-field magnitude equals the med ium coercivity:
`T h e magnetostatic fields a t the transition center are sketched in Fig. 8.6,
`a n d it is seen t h a t the slope o f the n e t field a t the transition center is
`reduced. I f a transition shape is specified with one u n k n o w n parameter,
`t h e n (8.4) c a n be utilized only a t one location t o solve f o r the unknown. It
`is assumed in this model t h a t the significant region for specifying ' a ' is a t
`the transition center. A t high densities, when transitions overlap, the
`t r a n s i t i o n c e n t e r p r o v i d e s
`contribution to the replay voltage (6.35); a t low densities, the transition
`m a g n e t i z a t i o n slope a t
`region is i m p o r t a n t for a n accurate determination o f the p e a k voltage
`A general transition shape is assumed (Table 4.1) with center slope
`
`(5.36).
`
`given by:
`
`2Mr
`d M
`- = - -
`1C'a
`d x
`
`(8.8)
`
`t r a n s i t i o n
`
`t h e
`
`the (depth(cid:173)
`
`(8.7)
`
`t h e
`
`t h e m a j o r
`
`The
`equ
`
`The
`thir
`
`whi
`sha
`sha
`1
`apJ
`y (:
`
`( T
`H(
`X =
`
`wl
`
`N
`( }
`F
`A
`fr
`
`
`Page 16 of 46
`
`

`

`nsition models
`
`::..:• -:..::-e..::-c=--=--=-•~•-------- X
`
`nd demagnetizing field ~ear the
`is oflongitudinal recordmg.
`
`without regard to 'hard' or
`lie head field is applied and
`nsition is written with center
`asymmetric transition the
`on center so that the center
`position where the (depth(cid:173)
`medium coercivity:
`
`(8.7)
`
`~nter are sketched in Fig. 8.6,
`:ld at the transition center is
`,ith one unknown parameter,
`n to solve for the unknown. It
`t region for specifying 'a' is at
`1/hen transitions overlap, the
`center provides the major
`Lt low densities, the transition
`mination of the peak voltage
`
`(Table 4.1) with center slope
`
`(8.8)
`
`Mode ls of longitudinal recording
`
`215
`The M-H loop derivative at the transition center where the net field
`equals the medium coercivity may be expressed as:
`
`dM
`Mr
`dH1aop He(l - S*)
`
`(1.2)
`
`The gradient of the magnetostatic field at the transition center is given for
`thin media (8 «: a) by (4.19):
`
`dH~ _ Mr81 00 ds d2.f{s) _ Mr8I
`dx -
`- 1ra2
`1ra2
`-; ~ -
`-
`0
`where I denotes the integral. From Table 4.1 I depends on the transition
`shape, but is close to the value of I = 1 for an arctangent transition
`shape.
`The head-field gradient may be determined utilizing the Karlqvist
`approximation for the longitudinal field (3. 16). Inverting (8.7) at spacing
`y (assuming thin media) yields:
`
`(8.9)
`
`xo = ±(g/2)✓1 - (2y/g) 2 + (4y/gtan-1(1rHe/Ho))
`x2 + y2 > (g/2)2
`
`(8.10)
`
`xo = ±(g/2)✓1 -(2y/g)2 -(4y/gtan- 1 1r(l -Hc/Ho))
`x2 + y2 < (g/2)2
`
`(The transition between the two regimes in (8.10) occurs when
`Ho = 2Hc.) Taking the derivative of the field (3. 16) with evaluation at
`x = x0 given by (8.10) yields:
`
`where Q is given by:
`
`dHh __ x
`dx
`
`QHe
`y
`
`Q _ 2xoHo
`-
`1rgHc
`
`. 2 ( H. /H. )
`Sln7fe
`O
`
`(8.11)
`
`(8.12)
`
`Note that Q depends on the deep-gap field relative to the coercivity
`(