Williams-Comstock model with finite-length transition functions

1070 IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 2, MARCH 1999

Williams–Comstock Model withFinite-Length Transition Functions

Erich P. Valstyn and Charles R. Bond

Abstract—The theory of the third-order-polynomial (TOP)and fifth-order-polynomial (FOP) magnetization transitions ispresented. These transitions have a finite length, rather thanan asymptotic approach to�Mr:, which is the case with somewidely-used transition functions. The Williams–Comstock modelis used to obtain the transition parameters, which are equal tohalf the transition lengths, resulting in quadratic equations andsimple expressions. In this analysis, the write-field gradient ismaximized with respect to the deep-gap field, as well as withrespect to the distance of the transition from gap center, whichresults in a higher gradient than is obtained with the originalWilliams–Comstock approach, at the expense of a higher writecurrent. Analytic expressions are obtained for the read pulses ofinductive and shielded magnetoresistive heads, and equations fornonlinear transition shift are derived for the arctangent and theTOP transitions. The results are compared with those obtainedusing arctangent and tanh transitions and with experiment. Inaddition, certain aspects of the write-processQ function and theoptimum deep-gap field are discussed.

Index Terms—Longitudinal recording, magnetic head-mediuminterface, magnetic recording, magnetization transition, nonlin-ear transition shift (NLTS), read and write theory, read-pulsewidth (PW50).

I. INTRODUCTION

T HE original Williams–Comstock model [1] assumes anarctangent magnetization transition. However, the same

model can be used assuming other transitions. In this paper,two finite-length transitions, the third-order-polynomial (TOP),and the fifth-order-polynomial (FOP) transitions are discussed.Finite-length transitions are more realistic, as various imagingtechniques show, and are better suited for modeling high-density magnetic recording, especially with regard to nonlineartransition shift (NLTS). It is shown that for both the TOP andthe FOP transitions, the Williams–Comstock approach leadsto quadratic equations and to simple expressions for theirtransition parameters, which are equal to half the transitionlengths. In this analysis, the write-field gradient is maximizedwith respect to the deep-gap field as well as with respect tothe location at which the transition is written. It is shown thatthis leads to a higher write-field gradient than obtained in [1],at the expense of a higher write current.

Manuscript received March 13, 1998; revised November 6, 1998.E. P. Valstyn was with Read-Rite Corp., Milpitas, CA 95035 USA.

He is now with Valmag Consulting, Los Gatos, CA 95032 USA (e-mail:[email protected]).

C. R. Bond is with Test Technology, Milpitas, CA 95035 USA (e-mail:[email protected]).

Publisher Item Identifier S 0018-9464(99)02018-X.

Analytic expressions are obtained for the read pulses ofinductive and shielded magnetoresistive heads, the formerby convolving the Karlqvist [2] sensitivity function with thederivative of the transition function, the latter by convolu-tion of the transition function with the Potter [3] sensitivityfunction. Equations for NLTS of the arctangent transitionand the TOP transition are derived. The best agreement withexperiment is achieved with the TOP transition.

II. THEORY OF TOP AND FOP TRANSITIONS

A. TOP Transition

If the transition length is 2, the equation for the TOPtransition is

sgn (1)

This is an odd function, so that its parametercan bedetermined by using the Williams–Comstock equation [1]

(2)

where is the head field and is the demagnetizing fieldof the transition. It is shown in Appendix A that, if the mediumthickness is small compared to, then at the center of thetransition

(3)

According to (1), for

(4)

and, as in [1]

and (5)

where is the medium coercivity, is its coercive square-ness, and , being the head-to-medium spacing.is the maximumhead-field gradient, normalized with respectto , where and is therecoil susceptibility.

Substituting (3)–(5) into (2) and solving for, we obtain ,the parameter for the first step of the Williams–Comstockanalysis

(6)

0018–9464/99$10.00 1999 IEEE

VALSTYN AND BOND: WILLIAMS–COMSTOCK MODEL 1071

In the second step, the transition moves away from the headand expands under the influence of its demagnetizing field, sothat its length increases from 2to 2 where is found bysolving the following [1]:

(7)

The solution is

(8)

B. FOP Transition

For a transition length of 2, the FOP transition is

sgn (9)

In a way similar to the one used for the TOP transition, we findthe field gradient at the center of the FOP transition for ,

and from (2), the transition parameter

(10)

where

(11)

C. The Function

In [1], it is assumed that the write-field gradientis maximized. The equations derived there contain the so-called function, a function of , which is defined asthe maximum write-field gradient normalized with respect to

is derived from the Karlqvist equation [2] for thecomponent write field

(12)

where is the deep-gap field. It is assumed that isgiven, and the equation is solved for ,arriving at a value (0.51 in Fig. 1). The is then foundby substituting this value in the expression for andnormalizing with respect to In order to write at thelocation , the deep-gap field, which is determined bythe write current, must be

(13)

However, the obtained in this way does not correspond tothe maximum field gradient. The interdependence of, ,

Fig. 1. The x component of the write field as a function ofx=g fory=g = 0:3 and various values of the deep-gap fieldHg : Location of the max-imum field gradient according to Williams and Comstock(x=g)W = 0:51,location of the true maximum gradient(x=g)B = 0:66: For a coercivityHc = 2000 Oe, the corresponding deep-gap fields are 5000 and 7600 Oe,respectively.

Fig. 2. Location of the maximum field gradient according to Williams andComstock(x=g)W and location of the true maximum gradient(x=g)B andthe correspondingQ functionsQW andQB as functions ofy=g:

and , which is the basis of (13), must be taken into accountbeforethe maximization step, which results in a higher[4],[5]. Thus, from (12)

(14)

Deriving from (14) and setting it equal to zero,we arrive at

(15)

which must be solved numerically for The result is avalue (0.66 in Fig. 1). Fig. 1 shows that the gradient at

is higher than at , but the deep-gap field requiredto achieve this higher gradient is also considerably higher.

and , and the corresponding-functions and, are plotted as functions of in Fig. 2. The optimum

deep-gap field arrived at in this way is the one which results inthe maximum write-field gradient. It may, however, not be the


Fig. 3. Normalized distance from gap centerx=g at which a tran-sition is written, and corresponding normalized write-field gradient(@Hx=@x)� (y=Hc) as a function of the normalized deep-gap fieldHg=Hc

for various values ofy=g:

optimum for other reasons. Referring to Fig. 3, one family ofcurves is the normalized field gradient as a function offor various values of , the other one is obtained by plottingthe location for which as a function of ,for a range of values. The curves of the latter familycross over in a small region near This meansthat, if the deep-gap field is maintained at this value, the writelocation is virtually insensitive to changes in gap geometry

This is of importance for all systems which are sensitiveto small transition shifts, such as partial response, maximumlikelihood systems. However, this value of the deep-gap fielddoes not correspond to the maximum write-field gradient, sothat the choice of is a compromise and depends on thetype of system in which the head is to be used. Thewhichresults in maximum field gradient is about 20% higher thanthe which results in minimum fluctuation of the writelocation. Overwrite requirements add another dimension tothis compromise.

For the calculated results presented in this paper, we alwaysassume that the write current is chosen to give the maximumwrite-field gradient, i.e., we choose the value that cor-responds to in Fig. 1. In the experiments, the writecurrent was adjusted so that the minimum PW50 was obtained.

D. Transition Fields

It is shown in Appendix A that the field of the TOPtransition is given by

sgn (16)

where

Fig. 4. Four transitions and their fields based on the following systemparameters:Mr = 480 emu/cm3, gap lengthg = 0:25 �m, head-mediumspacingd = 0:06 �m, medium thicknesst = 0:0125 �m, coercivityHc = 2500 Oe, coercive squarenesss� = 0:85, and recoil susceptibility� = 0:1: The TOP transition is obtained from (6) and (8), the FOPtransition from (10) and (11), the arctangent transition from the originalWilliams–Comstock equations [1], and the tanh transition by using theWilliams–Comstock model and the field according to [6]. Also shown arethe tanh transition and its field, based on the assumption thats� = 1 and� = 0 [6].

and

Fig. 4 shows five transitions and their fields. Four of them,the TOP and FOP transitions, the arctangent transition, andthe tanh transition are based on the same system parameters.A tanh transition based on assuming a medium [6] withand , as well as its field, are also shown. It can be seenthat up to about 300 kfci m) all transitions,except the arctangent transition, will predict about the sameNLTS, while for higher linear densities the NLTS predictedby the TOP transition will be higher. Fig. 4 also shows that,although the shapes of some transitions do not differ much,their fields do.

III. READ PULSES

A. Inductive Head

The voltage pulse produced by an inductive head sensingan isolated transition is obtained by convolving the derivativeof the transition function with the Karlqvist [2] sensitivityfunction (Appendix B).

Defining and , where is thegap length and is the displacement of the transition centerwith respect to the gap center, the voltage pulse is given by

(17)


and

(18)

where is the number of turns, is the medium velocity,is the track width, is the gap length, and is the

transfer function or complex efficiency, which is a functionof frequency. For to be in volts, all dimensions in (17)and (18) must be in meters, and , the remanent flux densityof the medium, must be in tesla. An of one emu/cmcorresponds to a of 0.001 26 tesla.

B. Symmetrical Shielded MR Head

The voltage pulse generated by a symmetrical shielded MRhead, sensing an isolated transition, is obtained by convolvingthe transition function with the Potter sensitivity function(Appendix C). The result is

(19)

where is the transfer function of the MR head (involts per weber of flux entering the stripe) and

, where is the thickness of the MR stripe andisthe separation between stripe and shields, and where we havemade the following definitions:

Fig. 5. PW50 and NLTS of TOP and arctangent transitions as a function ofhead-medium spacing for an MR head, based on the following system param-eters. Write-head gap lengthg = 0:398 �m. Read element: stripe-to-shieldspacinggr = 0:081 �m, stripe thicknesss = 0:03 �m. Bit-cell lengthB = 0:11 �m (230 kfci). Medium parameters are the same as in Fig. 4,except thatHc = 2650 Oe. The experimental points and their error bars arethe averages and their standard deviations for six heads.

PW50 is computed numerically from (17) and (19).

IV. NONLINEAR TRANSITION SHIFT

It is shown in Appendix D, that NLTS, normalized withrespect to half the bit length for the arctangent transitionis given by

(20)

For the TOP transition (Appendix E),

(21)

V. RESULTS

In Fig. 5, computed PW50 and NLTS values are plottedas a function of effective head-medium spacing for TOP andarctangent transitions and are compared with experimentalresults. As was mentioned at the end of Section II-C, for the


calculations, we chose the deep-gap field which gives themaximum write-field gradient, and in the experiments, thewrite current was adjusted so that the minimum PW50 wasobtained. Fig. 5 shows that, for the same system parameters,TOP transitions have higher NLTS and lower PW50 thanarctangent transitions. The experimental data represent theaverages and standard deviations of six heads. The averageflying height was measured as 0.031m and the pole-tiprecession as 0.014m. The nominal overcoat thicknesses were0.008 m for the disk and 0.007m for the slider. The effect ofpole-face roughness was taken into account by adding another0.020 m to the effective spacing. In a series of experimentsit was shown that good agreement with experiments can beobtained only if the effects of various lapping techniques,which determine the pole-face roughness and possibly thethickness of a magnetic dead layer, are included in thisway. Good agreement between computed and experimentalresults for the total effective head-medium spacing of 0.08mwas obtained. It is obvious that computations based on thearctangent transition cannot predict the experimental results,no matter what correction for pole-face conditions is applied.

VI. CONCLUSIONS

Two finite-length transition functions, a TOP and a FOP,have been discussed using the Williams–Comstock approach[1]. In the course of this discussion, it has been pointed outthat the write-field gradient used in [1] is not the maximumwrite-field gradient (Figs. 1 and 2). It has, moreover, beenshown that the choice of write current, which determines thedeep-gap field, is a compromise between achieving the max-imum write-field gradient, providing adequate overwrite, andobtaining minimum fluctuation of the write location (Fig. 3).All calculated results in this paper are based on choosing thedeep-gap field that gives the maximum write-field gradient.

The polynomial-function transitions discussed here have theadvantage that, in contrast to tanh and exponential functions,inductive and MR read pulses can be obtained in analyticalform, which reduces computation time and provides betterinsight. The same is true for the demagnetizing fields of thesetransitions. Moreover, the Williams–Comstock theory leads tosimple expressions for the transition parameters.

Computed PW50 and NLTS for TOP and arctangent tran-sitions have been compared with experiment (Fig. 5). Goodagreement was obtained for the TOP transition when the effectof pole-face roughness was taken into account. The magnitudeof this effect depends on the lapping conditions and can varyfrom 0.005–0.035 m.

It has also been found that including the effect of imaging ofthe transitions in the pole faces leads to poorer agreement withexperiment. This is true for all types of transitions. The reasonfor this is probably that, during writing, the pole faces arenear saturation, some of their parts closer than others [9], andtheir permeability is therefore low and not uniform. Moreover,domain walls have been observed on the pole faces [10]. Allthis seems to prevent the pole faces from functioning likemagnetic mirrors.

Referring to Fig. 4, it can be seen that, although sometransitions may not differ from each other very much, theirdemagnetizing fields do, e.g., TOP and FOP. The shape ofthese two curves indicates that for very high linear densities(e.g., 500 kfci, m) the FOP transition may givebetter agreement with experimental NLTS results than the TOPtransition.

Fig. 4 also shows that the fields of transitions obtained byassuming and are significantly different fromthose based on measured medium parameters. This can be seenby comparing the fields of the two tanh transitions. At 400 kfcithe difference is about 15%; at 500 kfci, it is 30%.

Also note that not performing the second step of theWilliams–Comstock procedure [(6) and (8)] leads to an errorin transition length of 20–25%. This, in turn, causes an error inPW50 of about half that percentage. This is true for arctangenttransitions as well as finite-length transitions.

APPENDIX AFIELD OF THE THIRD-ORDER-POLYNOMIAL

MAGNETIZATION TRANSITION

The component field in the center plane of the medium is

where is the volume occupied by the transition. Assumingthat the track width dimension) of the transition is largecompared to the other pertinent dimensions

The inverse tangent is multivaluedwhere is any positive or negative integer or

zero which must be chosen in such a way that the result-ing expression has physical meaning. Also,

Hence, for the region

Carrying out the integrations, we find that

(A-1)


where

For the region

sgn

so that

sgn (A-2)

From (A-1), we obtain the derivative of at

(A-3)

For , we get (3)

APPENDIX BVOLTAGE PULSE OF AN INDUCTIVE HEAD

The voltage pulse is obtained by convolving the first deriva-tive of the transition function with the Karlqvist sensitivityfunction

where

and , where is the head-medium spacing andis the medium thickness. We therefore have

carrying out the integration results in (17). An expression foris given in (18).

APPENDIX CVOLTAGE PULSE OF A SYMMETRIC

SHIELDED MAGNETORESISTIVE HEAD

The pulse is obtained by convolving the transition functionwith the Potter [3] sensitivity function

where

being the thickness of the MR stripe andthe spacingbetween stripe and shields. According to (1)

sgn

so that

Carrying out the integration, we arrive at (19).

APPENDIX DNLTS OF THE ARCTANGENT TRANSITION

The nonlinear shift of the second transition in a dibit pattern,located at , is equal to [7]

(D-1)

where is the magnetostatic field of the first transition,is the writing separation of the two transitions, and is

the head field. For an arctangent transition [8]1

Imaging in the pole face is neglected here because, at the timethe transition is written, the pole face is near saturation andthe linear (reversible) part of its permeability is low.

Assuming that , we have

(D-2)

1 Bertram uses mks units, whereas cgs units are used here (so that the fieldswill be in oersteds); hence, the equation in the reference must be multipliedby 4�:


If the transitions are written with the maximum write-fieldgradient, then [1]

(D-3)

Substituting (D-2) and (D-3) in (D-1), we obtain the nonlineartransition shift

Normalizing this with respect to half the bit-cell length

APPENDIX ENLTS OF THE TOP TRANSITION

For the TOP transition, is found from (16) bysetting , multiplying by ( 1), because the field isgenerated by a negative magnetic charge, and again assuming

and therefore, from (D-1) and (D-3)

ACKNOWLEDGMENT

The authors thank T. Tran of Read-Rite Corporation forproviding the experimental results and for useful discussions.

REFERENCES

[1] M. L. Williams and R. L. Comstock, “An analytical model of the writeprocess in digital magnetic recording,”17th Annu. AIP Conf. Proc., vol.5, pp. 738–742, 1971.

[2] O. Karlqvist, “Calculation of the magnetic field in the ferromagneticlayer of a magnetic drum,”Trans. Roy. Inst. Technol. Stockholm, vol.86, pp. 3–27, 1954.

[3] R. I. Potter, “Digital magnetic recording theory,”IEEE Trans. Magn.,vol. 10, pp. 502–508, Sept. 1974.

[4] E. P. Valstyn, E. Packard, and V. G. Kelley, “Optimization of ferriteheads for thin media,”IEEE Trans. Magn., vol. 22, pp. 847–849, Sept.1986.

[5] H. Neal Bertram,Theory of Magnetic Recording. Cambride, U.K.:Cambridge Univ. Press, 1994, p. 216.

[6] Y. Zhang and H. N. Bertram, “A theoretical study of nonlinear transitionshift,” IEEE Trans. Magn., vol. 34, pp. 1955–1957, July 1998.

[7] H. Neal Bertram,Theory of Magnetic Recording. Cambride, U.K.:Cambridge Univ. Press, 1994, p. 246.

[8] , Theory of Magnetic Recording. Cambride, U.K.: CambridgeUniv. Press, 1994, p. 98.

[9] M. R. Freeman, A. Y. Elezzabi, and J. A. H. Stotz, “Current dependenceof the magnetization rise time in thin film heads,”J. Appl. Phys., vol.81, no. 8, pp. 4516–4518, 1997.

[10] X. Shi, F. H. Liu, Y. Li, and M. H. Kryder, “Dynamic response ofdomain walls on the air-bearing surface of thin-film heads,”J. Appl.Phys., vol. 75, no. 10, pp. 6394–6396, 1994.

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Williams-Comstock model with finite-length transition functions