2415 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 5, SEPTEMBER 1990

A TRACK DENSITY MODEL FOR MAGNETORESISTIVE HEADS CONSIDERING ERASE BANDS

J. K . Lee and P. I . Bonyhard IBM General Products Division

5600 Cottle Road, San Jose, CA 95193, U.S.A.

Abstract - Equations are derived that describe the offtrack behavior of magnetic recording heads with inductive-write-wide, MR-read-narrow in the presence of squeeze and erase bands. The resulting model can be used to predict offtrack perform- ance as a function of trackpitch.

INTRODUCTION

The existence of erase bands along the edges of written tracks has been reported [1,2]. It is desirable to understand how the presence of erase bands affects the offtrack perform- ance and the limits of the track density in the presence of track misregistration (TMR) for write-wide-read-narrow magnetoresistive read (MR) heads. An analytical track density model will be developed here to predict offtrack per- formance as a function of trackpitch.

DERIVATION O F T H E MODEL

We define the offtrack k parameter such that, for a read head of width r, and finite erase bands e, a (kr-e/2) portion of the read width reads the side “old information” such that a specified error rate limit is reached [3]. The offtrack capa- bility (OTC) can then be characterized by the k parameter and the write-wide-read-narrow component. Hence, k and the offtrack capability (OTC) depend primarily on the system signal-to-noise ratio (SNR). We will derive the equation gov- erning OTC, and its dependance on r, e, and the written trackwidth (w).

We define the geometric SNR for the head as the ratio of current information read to old information read, and assume that, a t the point of offtrack failure, all other noises are negli- gible in comparison to old information. Assuming also that failure occurs at wme specific ratio, say, k’, of geometric noise (N) and head signal ( S ) , we can write

N/S = k’ ( 1 )

We consider this to be a reasonably good assumption for most high density recording systems where the interference from old information and adjacent tracks dominate the error rate at failure. The offtrack behavior may then be considered in five different regions of reducing track pitch, as discussed below.

Case 1: p 2 w + kr + 0.5e

In this case, shown in Figure l(a), where the trackpitch is large, the OTC can be expressed as the sum of the extra width of the write-wide-read-narrow and the genuine offtrack capability, that is

OTC(1) = OTC, = 0.5(w - r) + kr (2)

Noise in this case comes only from old information and is proportional to OTC-0.5 (w-r+e), whereas the signal is pro- portional to r-OTC+0.5 (w-r-e). Consequently,

N/S = (OTC - O..5(w - r + e))/(r - OTC + 0S(w - r - e)) (3)

It is useful to convert this expression for OTC by the intro- duction of a new parameter, k,, that is, the valuc of k corre- sponding to no erase band, i.e., e = 0. In this case, Equation (3) reduces to

W-OTC-O.l(w-r)

+ WOTC-O.B(w-r)’ ’

Adjacent Track

JcOTC-O.S(w-r)

Figure 1. Configurations of squeeze and crase bands for Cases 1 through V.

N/S = (O‘I’C - O.S(w - r))/(r - OTC + O.S(w - r))

or N/S = k,r/( I - k,)r = k,/( I - k,) (4)

From Equations ( I ) and (4)

(5) k’ = k,/( I - k,)

By using Equations (3) and (4), one can solve for OTC as

OTC(I) = O.S(w - r) + k,r + (0.5 - k,)e + / - d (6)

where, and hereafter, the parameter d is added to include the misalignment between the read and write head for all cases. ’

k in Equation (2) is then related to k, in Equation (6) as

k 7 k, + (0.5 - k,)(e/r) (7)

It is seen, in this isolated track case, that k depends on the ratio e/r and that the erase band increases the offtrack capa- bility by an amount (0.5 - k,)e, as the erase band reduces the interference from the old information band, although the signal decreases as c increases as well.

Case 11: w + kr - 0.Se + A < p < w + kr +0Se

OO18-9464/90/0900-2475$01 .OO 0 1990 IEEE

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In Case 11, shown in Figure I@), the adjacent track is writtcn close enough so that the head reads two erase bands, one from the data track and the other one from the adjacent track, and the old information band. The erase band reduces the readback signal proportionally while it contributes no additional noise except that the media-DC-erased noise, which is always there and is treated minimal according to our previous assumption. The noisc N and signal S are then:

N = ~ - w - c (84

S = r - OTC + 0.5(w - r) - 0 . 5 ~ (W Substituting Equations (sa) and (8b) into Equation ( I ) and

parametrizing as a function of k, with Equation ( 5 ) , one gets:

orc(r1) = - ___ ( I - ko) p + OS(w - r) + k,

1-k, (9)

(- )(w + e) + r - 0.5e + / - d k,

Equation (9) shows clearly that OTC, in this region, dcpends on the trackpitch p in a linear fashion with a steep slope of - (1 - k,)/k,. For example, as the tracks move closer together, OTC increases four times faster than p decreases, if k, = 0.2 . Decrease of the track pitch causes an increase of the OTC(l1) at a fast rate. Therefore, the offtrack parameter, k in the lower limit of this region, is also altered as p changcs. For this reason, the newly assigned value of k has to be included in the boundary limits. This can be done by using a parameter A to account for the difference, that is, A = OTC(I1) - OTC(I). A is calculated iteratively for every decrement of p. Region I 1 is terminated when the head reads interference from both old information and adjacent track.

Case I l l : w + e _< p < w + kr - 0.5e + A

As the adjacent track squeezes further into the data track, the read head reads not only two erase bands and old infor- mation, but also picks up the intcrferencc from the adjacent track, as shown in Figure I(c). The total noise is composed of two components, one is the old information, No;, and the other is the adjacent track, N,. One can thcn write Noi, N,, and S, respectively, as:

(10)

( 1 1)

(12)

The total noise, N, is the squarc root of the sum of the square of No, and N, componcnts, assuming they are inco- herent. Hence,

N,i = p - w - e

N, : (OX: - 0.5(w - r)) - NOi - 1.Se

S = r - (OTC - O.S(w - r)) - 0 . 5 ~

This is a second order equation in U and

OTC( I I I) = uw + / - d (15)

Since OTC uw, and U is in a quadratic form, O T C is quadratic in this region. This causes a bump in the plot of O T C vs p, which leads to the name “747 curve.”

Case IV: w 2 p < w + e

Figure I(d) shows that, in this case, the erasc band of the adjacent track has replaced thc old information. Therc is some overlap between the erase band of old information and that of the adjacent track. The only interference noise, N, from the adjacent track and S are:

N, = OI‘C - 0.5(w - r) - (p - w) - 0 . 5 ~

S = r - (OTC - O.S(w - r)) - 0.5e

(16)

(17)

Again, substituting Equations ( 1 6 ) and ( I 7) into Equation ( I ) , one obtains O T C as:

w + 0.5~ + k’r -

+ O.S(w - r) + / - d

The depcndcncc of O T C on p in Equation (18) is linear, similar to that in Case 11; thc O T C vs p i n this region is monotonically decrcasing with thc slope of 1 / ( 1 + k’) . Case V: w - OTCm < p < w

The interference in this casc, shown in Figure l(e), is similar to Case 1, where only one intcrfcrence is involved. However, the adjacent track moves into the data track so that part of the written data is wipcd out. The N, and S are represented by:

N, = OTT - 0.5(w - r) + (w - p - 0 . 5 ~ )

S = r - N, - e

(19)

(20)

Solving OTC by substituting Equations (19) and (20) into Equation ( I ) , OTC becomes

+ 0.5(w - r) + / - d

We then parametrize OTC, No,, r, and e as:

OK uw; NOi y o i w ; r = mw; e E yew (14) and also defining q = 0.5 (I-m).

Substituting Equations ( IO) , ( I I ) , (12), and (13), and employing the relationships of Equation (l4), and squaring Equation ( I ) on both sides, one can then factor out U as:

Au2 + BU + C = O

Equation (21) shows clearly that the slope of O T C vs p is always 1 and that the partially wiped-off data track results in reduction of OTC at the same rate, accordingly. A minimum trackpitch, p,,, bcyond which the on-track error [ate exceeds the limit, can then be determined by setting OTC = 0 in Equation (21). Then

pmin = - O.S(w - 1 + k‘( I - m + ye)

r ) + ( I + k 8 - 0.5ye)w

Adding Equations (2) and (22) using the relationship of Equation ( 5 ) yields:

oTcm + P”,,“ = w (23) where A = 1 - k’2

or Pmin = w - OTC,, which dctermines the limit of this region.

DISCUSSION

The plot of OTC vs p is broken into five regions, corre- sponding to the offtrack Equations (6), (9), (15), (18), and (21) in cases J to V. This is graphically represented in Figure 2 and is different from the traditional “747 curve” [4] where, if the erase band is not considered, only three regions exist.

Figure 2. The “747 curve” is broken to five regions when the erase band is considered. The slopes in linear Regions I, 11, IV, and V are indicated. The parameter A for the onset of Region 111 should be calculated by iter- ation method.

The inductive write-wide and MR-read-narrow head has an advantage over the single element head as indicated in Figure 3, in that the OTC is larger in most of the regions. More- over, the intercept of the curve with the abscissa, that is, pin is smaller for the case m < I , indicating that the hard error rate is also reduced. However, the 747 bump is not strong f o r m < 1 .

/

rn = 0.5-

e = .5 prn

/ /

I

I I I I I

8 10 12

Adjacent Track Pitch brn)

Figure 3. The offtrack capability vs adjacent track pitch for different ratio, m, of read width vs write width, ranging from 0.5 to 1 .

The erase bands tend to narrow region I and widen regions I I and IV, as shown in Figure 4 and, in general, OTC improves within reasonable limits with an increasing erase band width. Also, the onset of the bump in the “747” curve occurs earlier, and the bump is stronger as well. However, the leverage is diminished when small p is reached.

The improvement in offtrack performance due to an increased erase band width, as suggested in Figure 4, has a practical limit beyond which the OTC fails before that is pre- dicted by the model. While the presence of the erase bands postpones failure due to old information noise pickup as the

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tracks are squeezed, output signal is reduced as well. Since noise from other sources (channel electronics, head, media) cannot be ignored completely, especially for narrow

5 r Erase Band Width = 1.5 p r n q

’ \

/ 0 A’ I I 1

I I 8 10 12

Adjacent Track Pitch brn)

Figure 4. The offtrack capability vs adjacent track pitch for different erase bands.

trackwidth heads, an optimum erase band width exists for minimum error rate. The range of regions I 1 and IV are less pronounced for the narrow erase band case. The “747” bump may disappear as the erase bands widen enough so that the head never gets a chance to reach both old informa- tion and the adjacent track. This happens when e > 2/3 (kr + A).

The write-wide-read-narrow had has the advantage that the read head is farther away from the outside edge of the written track and, therefore, it may move off track farther before it starts picking up noise. Consequently, the OTC improves. However, the read-to-write-head misalignment must also be taken into account. Imperfect alignment between read and write cancels out to some extent the offtrack advantage. In addition, the read-narrow head reduces the head signal proportionally, and makes the had more sensitive to media defects and other system noise, which must be considered in the design of MR heads.

CONCLUSIONS

Equations have been derived to predict offtrack capability as a function of track spacing. I t has been shown that the offtrack behavior can be broken into five regions in the pres- ence of squeeze and erase bands. This treatment can be used for both MR and inductive heads.

REFERENCES

T. Lin et al., “Effects of Current and Frequency on Write, Read, and Erase Widths for Thin-Film Inductive and Magnetoresistive Heads,” IEEE Trans Magn., Vo1.25, pp

[2] J. Feng, “Spin Stand Measurements of Read, Write and Erase Widths,” Paper AA-03, 1989 lntermag Confer- ence.

[3] P. 1. Bonyhard and J. K . LCC, “Magnetoresistive Read Magnetic Recording Head Offtrack Performance Assessment,” Paper FR-01, 1990 lntermag Conference.

E. A. Cunningham and D. C. Palmer, “A Model for the Prediction of Disk File Performance from Basic Compo- nent Capabilities,” Paper JE-3, 1988 MMM-lntermag Con- ference.

[ I ]

710-715 (1989).

[4]