Stiction model for a head-disc interface of a rigid disc drive

A quantitative stiction model for HDI has been proposed, which takes into account local redistribution of lubricant and its effect on meniscus formation. Equations for calculating stiction have been obtained for three distinct situations, depending on the ratio of the interface separation to the lubricant film thickness and the interactions among contacting asperities. The model has been applied to HDIs consisting of either regular or random texture. The model predicts that the relationship between stiction and the real area of contact is not necessarily monotonic. Instead, stiction exhibits a broad minimum for intermediate values of true contact. This prediction is consistent with some of the experimental results reported in the literature. The model also suggests that the role of various roughness parameters on stiction are not independent of one another. To minimize stiction and maximize reliability under a given flyability constraint, the asperity height, radius and density need to be considered simultaneously.

This application is a continuation of Provisional patent application Ser. 
No. 60/003,319 filed Sep. 6, 1995 entitled, "Stiction Model for a 
Head-Disc Interface of a Rigid Disc Drive" and Provisional patent 
application Ser. No. 60/001,975 filed Jul. 28, 1995 entitled "Optimization 
of the Bump Spacing on Patterned Texture of Thin Film Media". 
BACKGROUND OF THE INVENTION 
In a hard disc drive, the disc is rotated at a constant speed by a spindle 
motor which supports the disc at its center, the rotation of the disc 
creating a film of air over the surface of the disc. An actuator is 
provided adjacent the disc, supporting a transducer which reads and writes 
data on the surface of the disc. The transducer, which is incorporated 
into or supported on a slider, "flies" over the surface of the disc, 
supported on this thin film of air. The data is stored on defined 
concentric tracks on the surface of the disc. The ability to maximize 
storage of data on the disc is a function, in part, of the ability to fly 
the transducer very close to the surface of the disc so that the 
dispersion of the interaction between transducer and disc surface at each 
data transition point is minimized. 
In order to allow lower and lower flying heights, the disc surface must be 
made smoother and smoother. However, the disc surface cannot be made 
perfectly smooth because of the need, among other reasons, to avoid what 
is termed "stiction". For a head disc interface (HDI), stiction can be 
defined as the lateral force that needs to be overcome to separate the 
slider from the surface of the disc. For example, in most disc drives, 
when the disc stops rotating because the disc drive is out of use, then 
the slider is no longer supported on the film of air and lands on the 
surface of the disc. If the disc surface is too perfectly smooth, then 
because of the affect of stiction, when the spindle motor is to be started 
up again so that the transducer resumes flying over the surface of the 
disc, the head or transducer cannot be broken free from the surface of the 
disc but remains "stuck" to the surface of the disc. 
In the presence of liquid lubricant on the disc which is introduced mainly 
to reduce wear during the contact start/stop (CSS), stiction is even 
higher than when no liquid is present at the surface. Consequently, much 
of the challenge for the tribology of the head disc interface lies in 
defining an interface that will show both low wear from the head landing 
on or periodically contacting the surface of the disc and low stiction so 
that the rotation of the disc can be restarted after the disc has been 
stopped. 
In the past decade or so, the magnetic storage industry has made tremendous 
progress in defining the interface in order to keep up with the pace of 
the continuing trend toward higher storage capacity through lowering the 
flying heights of read/write heads over discs. However, as the disc 
surface becomes smoother, the task of defining the head disc interface 
becomes more difficult. Thus, the need for a better understanding of the 
stiction phenomenon on a fundamental ground, and corresponding 
quantitative (instead of qualitative) models for defining Head Disc 
Interface (HDI) stiction becomes more ever pressing. 
In the literature, there exist abundant experimental studies on the HDI 
stiction phenomenon. Factors that affect stiction of an HDI include head 
load, slider size, surface roughness, geometrical conformity (as measured 
by crown) between slider and disc, physical and chemical properties of 
lubricants and lubricant film thickness, physical and chemical properties 
of slider and disc overcoat materials, and environmental factors such as 
temperature and humidity. Relatively speaking, theoretical studies on HDI 
stiction are rare, and are mainly focused on roughness aspects. The 
concepts of disjoining and capillary pressure have been applied by C. M. 
Mate, J. Appl. Phys. 72(7), 3084 (1992), to the analysis of how liquid 
lubricants behave on magnetic disc surfaces, but a quantitative stiction 
model that incorporates such key concepts has yet to emerge. 
Several studies have contributed to the understanding of stiction as a 
contact phenomenon at head-disc interfaces. It is now understood that the 
increase in stiction in the presence of a liquid lubricant is a direct 
result of the meniscus effect of the liquid. When two surfaces, separated 
by a layer of liquid lubricant film, are brought into contact, the 
lubricant tends to wet both surfaces, forming menisci around contacting 
points due to surface energy effect. The pressure inside a meniscus is 
lower than outside, which results in an additional normal force, the 
meniscus force, pulling two mating surfaces closer together. Stiction 
typically shows a significant time dependence, the longer the head rests 
in the disc, the higher the stiction. 
To calculate the stiction arising from the meniscus force for an HDI, 
several models have been proposed. A good summary on these models may be 
found in a recent study by C. Gao, X. Tian, and B. Bhushan, Tribol. 
Trans., vol. 38, 2nd ed., p.201 (1995). All these models employed roughly 
the same treatment a model to calculate the adhesive force for a single 
isolated meniscus under specific geometrical constraints, plus an 
incorporation of the Greenwood-Williamson's statistical treatment of 
surface roughness, J. A. Greenwood and J. P. B. Williamson, Proc. Roy. 
Soc. London A295, 300 (1966). The differences among these various studies 
lie largely on the models adapted or proposed for the calculation of 
adhesive force for a single meniscus. For example, the models proposed by 
F. P. Bowden and D. Tabor, "The Friction and Lubrication of Solids," 
Clarendon Press (1986), for an isolated contacting asperity, and by J. N. 
Israelachvili, "Intermolecular and Surface Forces," Academic Press (1985) 
for an isolated non-contacting asperity have been used by Y. Li and F. 
Talke, Tribol. Mech. Magn. Stor. Syst. Vol. VII, STLE Special Publ. SP-29, 
79 (1990) and by N. V. Gitis, L. Volpe, and R. Sonnenfeld, Adv. Inf. Stor. 
Syst. 3, 91 (1991). A model based on lubricant displacement around the 
contacting asperity has been proposed by H. Tian and T. Matsudaira, J. 
Tribol. 115(1), 28 (1993). More recently, C. Gao, X. Tian, and B. Bhushan, 
Tribol. Trans. (cited above), have proposed another model for calculating 
the adhesive force for a single meniscus between a lubricated flat and an 
unlubricated sphere based on surface energy considerations. Their result 
is different from that obtained by Tian and Matsudaira by a factor of two, 
but agrees with Bowden and Tabor's experiment results for a single 
asperity, F. P. Bowden and D. Tabor, "The Friction and Lubrication of 
Solids," Clarendon Press (1986). 
Intentionally or not, two assumptions are invariably found in all the 
aforementioned models. First, the lubricant film thickness in the 
non-contacting area remains constant, and is always equal to the original 
lubricant film thickness. Second, the geometry of a single meniscus is 
always treated as a sphere against a flat, as shown in FIG. 1A. 
Unfortunately, neither of these two assumptions may be representative of 
the situation within an HDI. The studies by Mate have clearly demonstrated 
that whenever head-disc contact occurs, lubricant redistribution has to 
occur in order to maintain equilibrium between the axillary pressure 
inside menisci and the disjoining pressure of the lubricant film outside 
menisci. The driving force for liquid lubricant to be drawn from 
surrounding area to join the meniscus can be enormous. For example, as 
pointed out by Mate, if a disc is to be lubricated with only 2 nm of a 
perfluoropolyether (PFPE) type of liquid lubricant, there exists a large 
thermodynamic driving force for the lubricant to fill the gap between the 
head and the disc. As a result of such local lubricant redistribution, the 
growth of a meniscus is likely to go beyond the point of the total 
immersion of the contacting asperity, as shown in FIGS. 1B and 1C. Under 
such circumstances, the geometry of a meniscus can no longer be treated as 
a sphere against a flat. Following Mathewson and Mamin, it is convenient 
to consider stiction for three distinctly different regimes, i.e., "toe 
dipping", "pill-box" and "flooded" as shown in FIGS. 1A, 1B and 1C 
respectively, depending on the ratio of the mean interplanar separation of 
the interface to lubricant thickness. Evidently, only within the "toe 
dipping" regime, could one legitimately treat an individual meniscus as 
that to exist between a sphere and a flat. As will be discussed later, in 
today's disc drives, most HDI's are operating within the "pill-box" regime 
in order to achieve a good compromise between wear and stiction. 
Nevertheless, none of the existing stiction models is adequate for 
describing the stiction within this regime. 
SUMMARY OF THE INVENTION 
A quantitative stiction model for HDI has been proposed, which takes into 
account local redistribution of lubricant and its effect on meniscus 
formation. Equations for calculating stiction have been obtained for three 
distinct situations, depending on the ratio of the interface separation to 
the lubricant film thickness and the interactions among contacting 
asperities, namely the "toe dipping" regime, and the "pillbox" regime with 
or without interactions among asperities. The model has been applied to 
HDIs consisting of either regular or random texture. An important 
prediction from this model is that the relationship between stiction and 
the real area of contact is not necessarily monotonic. Instead, stiction 
exhibits a broad minimum for intermediate values of true contact. This 
prediction is consistent with some of the experimental results reported in 
the literature. The model also suggests that the role of various roughness 
parameters on stiction are not independent of one another. To minimize 
stiction and maximize reliability under a given flyability constraint, the 
asperity height, radius and density need to be considered simultaneously. 
In the case of a random texture, under the same flyability constraint 
which limits the maximum allowable asperity height, a reduction in the 
standard deviation of the asperity height distribution leads to a 
reduction in stiction. Consequently, under the same constraint a regular 
texture should always be superior to a random texture in terms of 
stiction. 
The model has also been used to examine the effects of lubricants, the disc 
overcoat the interactions between them. To ensure low stiction, a good 
approach is to maintain the HDI within the "toe dipping" regime. But this 
approach may not always be possible because of other tribology constraints 
such as wear durability. Lubricant film thickness needs to be optimized to 
minimize stiction while maintaining good wear resistance. In addition, the 
physical and chemical properties of a lubricant, as well as its 
interactions with the disc overcoat and slider materials, play very 
crucial roles in determining stiction. Since stiction results mainly from 
meniscus build-up around contacting asperities, it should depend strongly 
on the speed of lubricant migration. Consequently, reducing the lubricant 
mobility should be an effective way to suppress stiction. This model can 
also be applied to study the effects of the head variables on stiction. 
These effects may include form factor, crown, and the head pre-load.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
The following is a description of the critical elements of an analytical, 
quantitative algorithm for deferring stiction in a disc device as a 
function of various other parameters which may be modified by the disc 
drive manufacturers and other parameters such as available motor torque 
are defined by the designers of the disc drive; they are typically not 
subject to modification by the disc manufacturers, but are accounted for 
in this algorithm, which thereby accurately represents the overall system. 
Ignoring the van der Waals force between two contacting solid surfaces 
(negligibly small compared to either the meniscus force or the applied 
load), the pseudo-equilibrium normal force for the slider, after 
sufficiently long rest, is 
EQU P=F.sub.m +W (1) 
where P is the deformation response force, F.sub.m is the meniscus force, 
and W is the applied load. 
In its most general form 
##EQU1## 
where N is the total number of apserities under a slider, h is the 
slider-disc separation, .phi.(z) is the asperity height distribution, 
meaning that the probability that a particular asperity has a height 
between z and z+dz above some reference plan will be .phi.(z)dz, and p(z) 
is the deformation response from a single contacting asperity, namely, 
##EQU2## 
where .sigma..sub.z is the normal stress on the contacting asperity, and S 
is the contacting area. If P.sub.m is the meniscus pressure on a single 
contacting asperity and A.sub.m is the area of the meniscus, the total 
meniscus force F.sub.m is 
##EQU3## 
where d is the lubricant film thickness at equilibrium. Substituting Eqs 
(2) and (4) into Eq (1) results in 
##EQU4## 
Both h and d are changing during the loading process. When the applied 
load brings the slider and disc into contact, menisci start to form at 
contacting points. Accompanying the meniscus formation is an additional 
loading resulted from the meniscus force. This additional loading in turn 
brings more asperities into contact, hence resulting in further meniscus 
formation and higher loading. This seemingly perpetual process will 
eventually come to a halt because the lubricant supply is finite. To form 
menisci, the lubricant has to be drawn from the area surrounding the 
contacting asperities. This process results in thinning of the lubricant 
films around the contacting asperities. As the lubricant film thins, the 
disjoining pressure associated with the lubricant film increases, which 
makes it more and more difficult for a meniscus to draw lubricant from its 
surrounding area. Eventually, it reaches a (quasi-)equilibrium state at 
which the disjoining pressure of the film equals the capillary pressure of 
the meniscus, i.e., 
##EQU5## 
where A.sub.H is Hamaker constant, .gamma. is the surface tension of the 
lubricant, and r is the radius of the meniscus. In Eq (6), it has been 
assumed that the contact angle between the lubricant and either solid is 
zero, which is a good approximation for most PFPE-type of lubricants on 
either carbon or common head materials such as Al.sub.2 O.sub.3 /TiC. We 
will continue to use this approximation unless otherwise specified. 
Yet, there is another constraint that the total volume of the lubricant has 
to be conserved during the process of meniscus formation. Assuming that 
v.sub.m is the volume of the lubricant forming a meniscus, A is the total 
area of the slider, and D.sub.o is the original lubricant film thickness, 
one can express this condition as 
##EQU6## 
Eqs (5) to (7) are three fundamental equations for calculating the total 
normal force P. Once P is known, the stiction, F, is simply 
EQU F=.mu.P=.mu.(F.sub.m +W) (8) 
where .mu. is the coefficient of friction. 
To carry out the calculation, one needs to make assumptions on the contact 
geometries and the distribution of the asperities. 
To create a random texture, one must make certain assumptions on the shape 
and height distributions of the texture. Following Greenwood and 
Williamson's model, asperity summits can be treated as a Gaussian 
distribution of semi-spheres of same radius R, 
##EQU7## 
Based on their model, the deformation response force on a single contacting 
asperity can be readily obtained 
##EQU8## 
where E is the composite Young's modulus of the two solids. 
When the roughness of a head-disc interface is sufficiently high and/or the 
lubricant film thickness is sufficiently low such that 
h&gt;(12.pi..gamma.d.sup.3 /H.sub.H) (ref. to Eq(6)), the junction is in the 
so-called "toe dipping" regime, as shown schematically in FIG. 1A. In this 
regime, a lubricant film cannot bridge across the gap between the head and 
the disc. For a PFPE type of lubricant film of 2 nm in thickness, h should 
be in excess of 75 nm, which is much rougher than most of the advanced 
head-disc interfaces. 
In this regime, P.sub.m, A.sub.m and v.sub.m can be calculated from the 
geometry of the asperity, as shown in FIG. 2A, using the following 
relationships, for R&gt;&gt;h, h.sub.1, 
EQU r.sub.m.sup.1 =2Rh.sub.1 +(z-h)! 
EQU r.sup.2 =2R(z-h) (11) 
where h.sub.1 is the height of the meniscus, and 
##EQU9## 
where a=12.pi..gamma./A.sub.H. Therefore, 
##EQU10## 
EQU A.sub.m (z)=.pi.(r.sub.m.sup.2 -r.sup.2)=2.pi.Rh.sub.1, (14) 
and 
##EQU11## 
One may notice that P.sub.m, A.sub.m and v.sub.m are independent of z. In 
other words, the capillary pressure, the area and the volume of a meniscus 
are all independent of the penetration depth of a contacting asperity. 
Substituting Eqs (9), (10) and Eqs (12)-(15) into Eqs (4)-(7), one has 
##EQU12## 
and 
EQU A(d.sub.0 -d)=.pi.NRad.sup.3 (ad.sup.3 -2d).PHI..sub.0 
(h-d)-2.pi.NR.phi..sub.1 (h-d) (17) 
where the function .PHI..sub.n (x) is defined as 
##EQU13## 
The Eqs (16) and (17) can be solved simultaneously to obtain h and d. Once 
h and d are known, Eq (8) is used to calculate stiction. It is interesting 
to note that Eq (16) has essentially the same form as Gao et al.'s model 
except that in their expression the lubricant film thickness, d, is a 
constant, namely, the original lube thickness, instead of a variable. 
In the "pillbox" regime, a lubricant film is thick enough to bridge across 
the gap between the head and the disc, causing local "flooding" around 
contacting asperities. The only reason that the entire gap is not all 
flooded is that the total volume of the lubricant film is limited. Within 
this regime, there can be two different situations that require different 
treatments. In one case, contacting asperities are well separated such 
that the formation of a meniscus around one asperity has no influence on 
the formation of a meniscus around another asperity. In other words, 
menisci are all independent of each other. In the other case, contacting 
asperities are close enough such that they interfere with each other 
during meniscus formation by competing for lubricant supply. 
In the case of no interference, each meniscus will grow to its full 
potential. The size of a meniscus is determined solely by the separation 
of the gap, H, and the lubricant film thickness, D, both of which should 
be the same everywhere. Hence, the size of a meniscus should be the same 
for all the contacting asperities. Furthermore, since the volume of a 
meniscus is much larger than that of an asperity, as shown in FIG. 2B, to 
a good approximation one has .nu..sub.m V A.sub.m h-.pi.Rh.sup.2. Let us 
define 
##EQU14## 
where A.sub.M and V.sub.M are the total area and volume of menisci, 
respectively. Obviously, in the "pillbox" regime, the capillary radius is 
simply one half of the head-disc separation, i.e., 
##EQU15## 
The substitution of Eqs (19) to (21) into Eqs (4) to (7) yields 
##EQU16## 
Combining Eqs (22), (23) and (24) gives 
##EQU17## 
For a given lubricant thickness, h can be obtained by solving Eq (25), 
which can then be used to calculate the stiction using Eq (8). 
In the case of interference, contacting asperities are sufficiently close 
to each other such that they compete for lubricant during the meniscus 
formation. The higher an asperity is, the earlier it gets into contact, 
thus, the earlier it starts to draw lubricant to form a meniscus around 
itself. When a neighboring asperity of lower height starts to get into 
contact, the lubricant film thickness is already lower than the original 
value. Therefore, only a smaller meniscus will be formed around this 
asperity. Again, the formation of a meniscus around this contacting 
asperity will further deplete the supply of lubricant within the area, 
leading to an even smaller meniscus around a later contact. Under such a 
situation, the size of a meniscus depends on the state of contact. The 
larger the contact area, the larger the meniscus. Let us assume that the 
radius of a meniscus forming around a contacting asperity is directly 
proportional to the radius of the contact area, namely 
EQU r.sub.m =.alpha.r (26) 
where .alpha. is a constant. With this assumption, A.sub.m (z) and v.sub.m 
(z) can be readily found based on the geometries of the asperity and the 
meniscus, as shown in FIG. 2C. 
EQU A.sub.m (z)=.pi.(.alpha..sup.2 -1)r.sup.2 =2.pi.R(.alpha..sup.2 -1)(z-h) 
(27) 
and 
##EQU18## 
Substituting Eqs (27) and (28), plus Eq (21) into Eqs (4) to (7), one 
obtains a set of equations of solving h and d. 
##EQU19## 
EQU A(d.sub.0 -d)=2.pi.NR.alpha..sup.2 (h-d).PHI..sub.1 (h-d)-.pi.NRh.psi.(h), 
(30) 
and 
##EQU20## 
where the function .psi.(x) is defined as 
##EQU21## 
Combining Eqs (29), (30) and (31) gives 
##EQU22## 
For a given applied lubricant film thickness, Eq (33) can be used to solve 
h, which then can be used to calculate the stiction. 
A regular texture is defined as arrays of bumps with the same shape and 
same height, H. The asperity peak height distribution in this case can be 
closely approximated by a delta function, i.e., 
EQU .phi.(z)=.delta.(z-H). (34) 
With this kind of distribution, the calculation is greatly simplified. 
Again, each lube regime will be treated separately. 
In the "toe dipping" regime, assuming that the bump summits are spherical, 
the total meniscus force, F.sub.m, is simply 
EQU F.sub.m =4.pi.NR.gamma., (35) 
which is directly proportional to the radius of the asperities, and to the 
total number of asperities under the slider. However, it is independent of 
the asperity height and the applied lubricant thickness, provided that the 
lubricant film is sufficiently thin and the asperity is sufficiently high 
such that the assumption that the interface is inside the "toe dipping" 
regime holds true. Assuming .gamma.=25 mN/m, R=25 .mu.m, and N=5000 
(10.times.10 .mu.m spacing for a slider area of 0.5 mm.sup.2), the total 
meniscus force is only 39 mN, which is comparable to, or, in some cases, 
less than the applied load. By comparing Eq (35) and Eq (16), one can also 
see that, within the "toe dipping" regime, the meniscus force is always 
higher for a regular texture than it is for a random texture, provided 
that the density of asperities is the same. In fact, within this regime, 
total meniscus force is determined by the total number of contacting 
asperities. As regular texture experiences a higher meniscus force because 
all the asperities are in contact. Yet, based on the above estimation, the 
total meniscus force is quite limited even for the case of a regular 
texture. One may conclude that in the "toe dipping" regime, the meniscus 
contribution to stiction is not very significant. 
In the "pillbox" regime, for semi-spherical bumps, one can substitute Eq 
(34) directly into Eq (25). The equations to solve h are reduced to 
##EQU23## 
In the literature, quantitative experimental stiction data for a regular 
texture are available only for square-shaped bumps. It is, therefore, 
desirable to derive equations applicable to this particular geometry. For 
square bumps, Eq (25) needs to be modified because p(z) has a different 
form from Eq (10). Assuming the width of a square bump is b, and the 
separation between bumps is B, 
##EQU24## 
and the equation for solving h becomes 
##EQU25## 
where .beta.=b.sup.2 /B.sup.2. 
In the following, numerical results based on the above equations will be 
derived. Texture parameters will be investigated in the following ranges: 
asperity height H=5-35 nm; bump spacing 5-70 .mu.m; asperity radius of 
curvature R=10-100 .mu.m. The materials used will be as follows: overcoat 
Young's modulus E=160 Gpa; the coefficient of friction .mu.=0.2; lubricant 
surface tension .gamma.=25 mJ/m.sup.2 and Hamaker constant A.sub.H 
=10.sup.-19. Slider effective area of contact will be taken as A=0.5 
mm.sup.2. This value is slightly lower than for a micro-slider (50% form 
factor) which has a typical value in the range 0.8-1.0 mm.sup.2. This 
"effective" value was arbitrarily chosen to take into account long 
wavelength geometrical effects such as crown, camber, twist and waviness. 
As discussed earlier, the risk for high stiction is low when the interface 
is in the toe-dipping regime, and high when in the pill-box regime. The 
case of the flooded regime is actually a limit of the pill-box regime in 
the ultra-low separation and/or high lubricant thickness ranges. Although 
often overlooked, this particular regime may take place when the extent of 
elastic deformation is so large that head-disc separation becomes 
comparable to the lubricant thickness. When such a phenomenon takes place, 
the stiction force is entirely governed by the slider area of contact, and 
it can be as high as 1.25N. However, when the whole slider is immersed in 
liquid (FIG. 1D), the resulting increase in meniscus radius will lead to a 
dramatic decrease in adhesion/stiction forces. Such head-disc interface 
scheme has been used to decrease head-disc separation for increasing 
storage capacity. 
According to Eq (6), and as discussed earlier, the boundary between the 
toe-dipping and pillbox regimes can be numerically defined in the 
interface separation h/lubricant thickness d space, as reproduced in FIG. 
3. In today's disc drives, head flying heights are below ca. 50 nm, and 
roughness peak-to-valley is in the 10 to 30 nm range. It is, therefore, 
immediately apparent that the toe-dipping regime can only be achieved for 
lubricant thicknesses well below 2 nm. This is less than typical values in 
the 2-3 nm range widely used in the industry. As a result, it is fair to 
say that, except in a few isolated cases, most of today's head-disc 
interfaces lie in the pillbox regime. Therefore, in the following 
discussion, the scope will be limited to this particular regime with no 
interference. 
Stiction versus Asperity Height 
FIG. 4 shows the relationships between stiction and the texture bump height 
at various lube thicknesses. The texture is assumed to consist of 
spherical bump arrays. As expected, stiction increases as the bump height 
decreases. The functional relationship between stiction and the bump 
height is a non-linear one. Stiction increases only mildly with decreasing 
bump height when the bump height is relatively large, but rises sharply as 
the bump height becomes very low. The onset of this sharp rise depends on 
lube thickness. The thicker the lube, the earlier the onset. As bump 
height decreases, not only can each meniscus grow bigger, the capillary 
pressure within each meniscus also becomes high; both of these changes 
lead to higher stiction. It is worth noting that for a regular texture the 
real area of contact does not change in any significant way as the bump 
height decreases. In FIG. 5, both the contact radius and the meniscus 
radius at a single bump, as well as the ratio between these two radii, are 
plotted against the bump height. From these curves is should immediately 
become obvious that the meniscus area plays a much more important role in 
determining stiction than does the actual contacting area. The meniscus 
area can be one or two orders of magnitude larger than the contacting 
area, and it changes with the asperity height in a much more sensitive way 
than the contacting area does. For a better visualization, a top view, as 
well as a side view, of an interface is schematically plotted in FIGS. 6A 
and 6B. This interface consists of regular bumps of 25 .mu.m in radius, 28 
nm in height, and 15 .mu.m bump-to-bump separation, as well as an original 
lubricant film thickness of 2.5 nm. The top view. FIG. 6A, is drawn to 
proportion. The gray circles are menisci. The black dots in the centers of 
gray circles represent the asperity contacting areas. Obviously, the side 
view, FIG. 6B, is not drawn to proportion, with the vertical scale being 
magnified to allow a clear view. 
Tanaka et al. have reported experimental studies on regular dot array 
square-shaped bump texture. It is convenient to compare the results 
calculated based on this model with their experimental results since most 
of the geometrical parameters needed for the calculation are available 
from their paper. FIG. 7 shows the relationship between stiction and the 
bump height of discs with 2% of the area ratio of contact (.beta.=b.sup.2 
/B.sup.2). The good agreement between values predicted by this model and 
Tanaka's experimental results is achieved by assuming an effective slider 
area equal to 50% of the apparent area A (see above discussion). 
Stiction versus Asperity Density 
The asperity density relates directly to the bearing ratio, which describes 
the real area of contact between two solid surfaces. Holding the radius of 
the asperity the same, the higher the asperity density, the higher the 
bearing ratio. Intuitively, one could guess that stiction decreases as 
asperity density decreases, or as the distance between asperities 
increases, because low asperity density means fewer contacting points. 
According to this model, however, this intuition is true only when 
asperity density is relatively high. As shown in FIG. 8, stiction may 
decrease, or be relatively independent of, or even increase with 
decreasing asperity density, depending on whether the density is high, 
medium or low. From this model, it is easy to understand this kind of 
behavior. Stiction depends on the total meniscus area, and the head-disc 
separation. Larger meniscus area and smaller head-disc separation both 
result in higher stiction. For a fixed asperity height, the head-disc 
separation is determined by the extent of elastic penetration. When 
asperity density is high, elastic penetration is small, stiction is 
dominated by the contacting area, and hence, stiction will decrease as the 
asperity density decreases. On the other hand, when asperity density is 
low, elastic penetration dominates stiction. Lower asperity density means 
less support, thus larger penetrating, leading to smaller head-disc 
separation, and higher stiction. Between the two extremes, there is a 
region where both effects are important and they tend to cancel each 
other. Within this intermediate region, stiction is relatively independent 
of the asperity density. FIG. 9 shows the dependence of the meniscus area, 
as well as the elastic penetration, on asperity density. One may notice 
that at very low asperity density, the meniscus area starts to increase as 
the density decreases. This increase is caused by the deformation-induced 
growth of individual menisci, as shown in FIG. 10. When such growth 
outweighs the reduction of total number of menisci, the total meniscus 
area starts to increase and the asperity density decreases. 
Ishihara et al. recently reported their contact start/stop (CSS) lifetime 
results on media with well-defined pattern texture. They have observed 
stiction failures at zero cycle for media with either very low bump 
density or very high bump density. Although their results cannot be used 
quantitatively for a direct comparison, the overall trend qualitatively 
agrees with this method prediction. 
The width of the flat region, or stiction window where the stiction is low 
and relatively independent of the asperity density, depends on the size of 
individual asperities, as shown in FIG. 8. Larger asperities tends to have 
a wider stiction window because they are more resistant to elastic 
penetration. From a head-disc interface design point of view, a wide 
section window is very desirable because it means higher tolerance to 
texture variation. A trade-off for using broader bumps is that the minimum 
stiction is higher. Nevertheless, such increase in the minimum stiction is 
not very significant as long as the asperity radius is not very large. 
Stiction versus Asperity Radius 
In the discussion in the previous section, it is stressed that the effects 
of asperity density and asperity radius are intimately related. FIG. 11 
further demonstrates such interrelationship. Depending on the spacing 
between asperities, stiction can exhibit completely different dependence 
on the asperity radius. If the spacing between bumps is small, stiction 
increases linearly with increasing asperity radius, whereas if the spacing 
is large, stiction decreases with increasing asperity radius, in a rather 
non-linear fashion. However, if the spacing is within some intermediate 
region, stiction is low and almost independent of asperity radius. The 
reason for such complicated relationships is exactly the same as discussed 
in the last section, namely, the competition between the meniscus area and 
the elastic penetration. This result provides a useful texture design 
guideline, i.e., if the spacing between bumps is properly selected and 
controlled, one could have a texture which is basically insensitive to 
bump radius variation. 
For square-shaped bumps, Eq (38) predicts that stiction is independent of 
the bump size as long as the bump height and apparent bearing ratio .beta. 
remains the same. This result is in good agreement with Tanaka et al.'s 
experimental observation. 
Random Texture 
The above discussion about the relationships between stiction and texture 
parameters was based on the equations derived for a regular texture. From 
this model, it can be shown that the trends are fundamentally the same for 
a random texture, too. However, for a random texture, in addition to those 
texture parameters discussed in the last three sections, stiction can also 
depend very critically on the standard deviation of asperity height 
distribution. 
Based on true surface of contact calculation, one expects that stiction 
decreases as the standard deviation increases; however, this is true only 
under certain conditions. For example, as shown in FIG. 12, if the average 
asperity height, i.e., Z.sub.o in Eq (9), is kept constant, stiction will 
decrease with increasing standard deviation. Because, for a fixed average 
asperity height, the texture with larger standard deviation is able to 
maintain a larger separation between the head and the media than can the 
texture with a smaller standard deviation, the larger standard deviation 
leads to lower stiction. However, trying to make a rule out of this 
special case by generalizing the connection between low stiction and high 
standard deviation can be misleading, especially for a head-disc 
interface. For a head-disc interface, the maximum allowable asperity 
height is dictated by the glide requirement, which demands that any 
protrusion on a disc surface is not to be higher than a predetermined 
value. Therefore, one does not have the freedom to have asperities of 
arbitrary height on the surface. In fact, according to Marchon et al.'s 
recent study on the relationship between the asperity distribution and the 
glide avalanche point, the maximum asperity height, H.sub.max, can be 
approximated to the average asperity height plus roughly 6 times the 
standard deviation of the asperity height distribution. When the maximum 
allowable asperity height is fixed, the increase of standard deviation has 
to come at a price of lowering the average asperity height. Therefore, the 
relationship between standard deviation and stiction based on a fixed 
average asperity height is no longer applicable. As a matter of fact, in 
this case, stiction increases as standard deviation increases, as shown in 
FIG. 13. This is because a higher standard deviation in this case means 
lower average asperity height, z.sub.o, fewer contact points, and hence 
larger elastic deformation, leading to smaller separation between the head 
and the disc, and thus higher stiction. The results in FIG. 13 implies 
that, in order to minimize stiction while maintaining a good glide 
capability for a head-disc interface, one should make an effort to reduce 
the standard deviation while maintaining the proper asperity height. As a 
result, a regular texture, which has a near-zero standard deviation in 
asperity height, should always be superior in terms of stiction and glide 
capability to a random texture, which has a finite standard deviation. 
Since the contacting stress at each contacting point is lower for a 
regular texture than it is for a random texture, the regular texture 
should also have better performance in wear, as well. 
Influence of Lubricant and Overcoat on Stiction 
Stiction versus Lube Thickness 
Within the "pillbox" regime, stiction always increases with like thickness. 
The sensitivity of such dependence varies dramatically with surface 
morphology. FIG. 14 shows the relationships between stiction and lube 
thickness for different asperity height. The "stiction wall", which is 
defined by a sharp rise in stiction, occurs at lower lube thickness for 
lower asperities. This phenomenon is commonly observed at the head-disc 
interface. The occurrence of this "stiction wall" represents a fundamental 
challenge in the tribology of the head disc interface, as the roughness of 
the interface continues to decrease. On one hand, to prevent wear, the 
amount of lubricant has to be higher than a certain critical value so that 
a sufficient lube supply can be maintained for replenishment wherever the 
lubricant is removed during sliding contact. On the other hand, the amount 
of lubricant applied has to be lower than a certain critical value to 
prevent overwhelming stiction. The curves in FIG. 14, which are based on 
the assumption of a regular texture of spherical bumps, can serve as a 
useful guideline in determining tribological parameters for a head-media 
interface under a certain glide requirement. From these curves, it is 
conceivable to have a disc that glides at 15 nm without encountering 
serious reliability problems. Yet, these curves are affected by other 
factors such as the shape and size of the asperity, the density of the 
asperity, and the physical and the chemical properties of lubricant. 
The dependence of stiction on lube thickness is affected by the asperity 
density, as shown in FIG. 15. For a texture of high asperity density, 
stiction is generally high, and increases linearly with lube thickness. 
For a texture of low asperity density, stiction is low at low lube 
thickness, but increases more rapidly with lube thickness, exhibiting a 
clear "stiction wall". For a texture with an intermediate asperity 
density, stiction is low and increases linearly with lube thickness. The 
role the asperity radius plays in altering the relationship between 
stiction and lube thickness is rather complicated. This complexity is 
illustrated in FIGS. 16A, B, and C. At high asperity density (16A), the 
asperity radius does not change the rate of stiction increase with lube 
thickness, but changes the absolute stiction value at any fixed lube 
thickness by shifting the whole curve upward or downward. The larger the 
asperity radius, the higher the stiction. At low asperity density (16B), 
the opposite is true. The asperity radius changes the rate of stiction 
increase with lube thickness. The smaller the asperity radius is, the 
faster the stiction increases. At some intermediate asperity density 
(16C), the relationship between stiction and lube thickness is essentially 
not affected by the asperity radius. 
In the above discussions, the importance of asperity density has been 
discussed repeatedly. Stiction could be high if the asperity density is 
either too high or too low. At the optimum density, however, not only is 
stiction low, but also is the tolerance for variations in asperity radius 
and lubricant thickness high. Better yet, varying the asperity density 
does not impact the intrinsic glide capability which is dictated by the 
asperity height. In addition, in the case of a regular texture, the 
spacing is generally easier to control than the bump radius. 
The physical and chemical properties of a lubricant play an important role 
in determining stiction. In this model, physical properties of a lubricant 
such as the Hamaker constant and surface tension are directly involved in 
the stiction equations. FIG. 17 shows the effect of the Hamaker constant 
on stiction. The stiction-lube thickness curve shifts down as the hamaker 
constant increases. The Hamaker constant is a measure of the strength of 
the chemical interactions between a lubricant and a solid surface. A 
higher Hamaker constant means stronger interaction between the surface and 
the lubricant film, thus higher disjoining pressure for the lubricant 
film. When the disjoining pressure of a lubricant film is high, it is more 
difficult to draw lubricant from the film. Strictly speaking, the Hamaker 
constant is a property of an interface, rather than that of a lubricant 
film alone. For a magnetic medium, this interface includes a lubricant and 
an overcoat. For the same lubricant, depending on the chemical nature of 
the overcoat, the Hamaker constant could be different. Even for the same 
type of overcoat, the Hamaker constant could still be different is the 
density of the overcoat is different because the Hamaker constant scales 
with the densities of both the lubricant and the solid surface. However, 
one should realize that stiction does not depend on the Hamaker constant 
very sensitively. An order of magnitude change in the Hamaker constant is 
needed in order to have a major impact on stiction. 
In contrast with the effect of the Hamaker constant, the effect of the 
surface tension of a lubricant on stiction is much more pronounced. This 
dramatic effect can be seen clearly from FIG. 18. A significant change in 
stiction can be brought about with only a modest change in the surface 
tension. From this model, it is not difficult to understand why stiction 
depends on surface tension more sensitively than it depends on the Hamaker 
constant. In addition to having a similar influence on the extent of 
meniscus formation as the Hamaker constant does, the surface tension of a 
lubricant directly affects the capillary pressure of a meniscus. A 
lubricant of high surface tension not only creates bigger menisci, but 
also produces a higher pressure inside each meniscus, both of which lead 
to higher stiction. Unlike the Hamaker constant, surface tension is 
strictly a property of a lubricant. Therefore, regardless of the physical 
and chemical properties of an overcoat, a lubricant with high surface 
tension always produces high stiction, and vice versa. 
Besides the Hamaker constant and surface tension, the physical and chemical 
properties of a lubricant has another important, although maybe less 
apparent, effect on stiction. Namely, the surface mobility of a lubricant 
does not appear in the equations explicitly, the effect of the surface 
mobility can be included through the effective apparent contact area of 
the slider, A. If the surface mobility is high, the lubricant can be drawn 
from an area that is larger than the area of a slider surface, to form 
menisci. In this case, the effective apparent contact area, A, should be 
larger than the slider surface area. Since stiction basically scales with 
this area, it will be higher for higher lubricant surface mobility, and 
vice versa. Novotny measured the surface diffusion constant on the order 
of 10.sup.-12 m.sup.2 /s for polyperfluoropropylene oxide on a smooth 
silicon dioxide surface. According to this value, the lubricant migrates 1 
.mu.m in 1 second, and 0.25 mm in roughly 17 hours. It is well known that 
the surface mobility of a lubricant can vary by many orders of magnitude 
depending on the physical and chemical properties of both the lubricant 
and the surface. Even with the same lubricant on the same surface, the 
surface mobility of the lubricant can still be dramatically different 
depending on kinetics of its interaction with the disc surface. 
Consequently, to be more rigorous, the effective apparent contact area 
should be determined from the knowledge of the surface mobility of the 
lubricant under specific conditions. Compared to either the Hamaker 
constant or the surface tension, the effect of surface mobility on 
stiction provides a much more practical way to control stiction. For a 
given choice of a lubricant and an overcoat, there is very little one can 
do to change either the Hamaker constat or the surface tension, but one 
can definitely alter the surface mobility of the lubricant through various 
treatments to promote interactions between the lubricant and the surface. 
Stiction versus the Mechanical Property of Overcoat 
Besides the chemical properties of the overcoat, discussed in the previous 
section, the mechanical properties of the overcoat also affect stiction 
directly. FIG. 19 shows stiction versus the composite Young's modulus, E, 
of an interface. Within a wide range, stiction only weakly depends on the 
elastic properties of the interface. However, stiction can rise very 
quickly when the Young's modulus of the interface is lower than a certain 
limit. From the model it is clear that stiction depends on the stiffness 
(or compliance) of the interface. A weak (i.e., a more compliant) 
interface is more likely to collapse under load, resulting in high 
stiction. From Eq (10), the compliance of an interface is inversely 
proportional to the composite Young's modulus of the interface. 
Some examples of the use of the above theoretical model for the 
optimization of stiction levels in a given disc drive have been developed 
and will be described with references to FIGS. 20-24. In applying this 
model, certain parameters such as the glide height of the slider over the 
surface of the disc and the available motor torque of the spindle motor 
are defined by disc drive designers, either exactly or within narrow 
limits. Therefore, the challenge and application of the model becomes to 
operate within the constraints imposed. For example, since the glide 
height is typically predefined in order to optimize bit-to-bit spacing, 
then this glide height also defines the maximum height of any asperity 
which must be less than the glide height and leave a reasonable safety 
margin. The slider area and lubricant thickness may also be predefined. 
Thus, to take into account the constraints imposed by the design 
limitations, a commercially available piece of equation solving software 
such as Math Cab which is available from MathSoft Corporation is 
programmed with the five basic equations identified in the detailed 
theoretical analysis above as equation (2), which establishes the normal 
force for the slider; equation (4), which defines the total meniscus 
force; equation (5), which defines the relationship between equations (2) 
and (4); equation (6), which defines the quasi-equilibrium state for the 
disjoining pressure of the film; equation (7), which describes the total 
volume of lubricant which is conserved during meniscus formation; and 
equation (8), which calculates the stiction force. The equations may be 
solved for the numerical value of the stiction given any other set of 
physical parameter including the shape and size of the bumps, the contact 
area of the slider, the height of the bumps, the surface tension of the 
assumed lubricant and the necessary constants such as the Hamaker constant 
and Young's modulus of the overcoat. 
The shape of the bump may be preferably defined as spherical or 
cylindrical; in many preferred embodiments, the bump shape is ring or 
volcano-like shape, i.e., with the center being open or hollow. An 
additional constant would be the radius of curvature of the bump. The head 
load and slider area are typically also defined. It will be seen from the 
figures to follow that, primarily, the design is optimized by changing the 
values of the bump spacing and bump height to optimize the value of the 
stiction. It has been found based on the analysis above, that it is the 
variations in these parameter which can have a primary influence on 
optimization of the stiction, especially in view of the fact that as noted 
above, most other parameters are established by the designers of the 
overall disc drive rather than the fabricators of the rotating hard disc. 
Referring next to the FIGS. 20-24, the results which may be produced with 
this analytic approach are demonstrated. For example, referring first to 
FIG. 20, the relationship between bump spacing and stiction is explored 
for two different types of bumps. That is, the higher pair of lines that 
is representing a greater level of stiction, and marked 100 and 102, both 
utilize a lubrication thickness of 40 .ANG.. In thicker line 100, each 
bump is the shape of a cylinder or more particularly, a ring or 
volcano-like structure. The narrower line 102, uses bumps on the disc 
having a spherical geometry. It can be seen at the optimum point, a bump 
spacing of about 30 .mu.m; and as the spacing approaches 60, the stiction 
increases so rapidly as to make the disc effectively non-functional. The 
lower pair of lines, labelled 106, 108 used a thinner lubrication layer of 
approximately 20 .ANG.. For this, the bump spacing optimizes approximately 
50 mm and, in fact, remains fairly constant through the range of bump 
spacing of interest. It is interesting to note that it has been surmised 
that performance in terms of reduction of stiction could be optimized for 
maximizing the number of bumps; that is, packing them as closely together 
as possible. In fact, there is a lower limit to effective bump spacing at 
about 30 mm for both spherical and cylindrical geometries, which is 
clearly demonstrated to reside at about 30 to 35 mm by this model. It is 
worth referring at this point back to the theoretical development found in 
this paper. The curve shown in FIG. 20 and the Figures to follow, are 
based primarily on five key equations, which are discussed above, and 
which are programmed into the computer to analyze the constants and 
variables imposed by the disc drive design. The first of these is equation 
(2), wherein the elastic deformation of the bump responds to applied force 
is analyzed. The normal load on the bump comprises the head load imposed 
on the bump together with the meniscus force. That is, the bump will 
actually deform slightly in response to a load. 
The next equation to be solved is equation (4) which defines the meniscus 
force which is the interaction between the lubricant and the surface which 
takes into account the fact that as a meniscus form when the applied load 
brings the slider disc into contact. Menisci form at contacting points, 
resulting in additional loading which, in turn, brings more asperities 
into contact with the slider resulting in further meniscus formation and 
higher loading. The next equation is equation (7) which is a substitution 
of equations (2) and (4) in paragraph 1 and essentially takes into account 
that the supply of lubricant on the surface of the disc is constant as 
there is little or no evaporation of the lubricant. 
The next equation is equation (6), which further takes into account the 
fact that as a meniscus forms, as described above in the discussion of 
equation (4), it must be formed by drawing some lubricant from the 
surrounding area so that eventually an equilibrium state is achieved at 
which the disjoining pressure of the film, which is what makes it more and 
more difficult for meniscus to draw lubricant from the surrounding area, 
equals the capillary pressure of meniscus. 
The final equation, of course, is equation (8) which utilizes the other 
forces already calculated to calculate the stiction. 
It should be further noted that relative to work done previously in the 
field, work is concentrated on what is shown in FIG. 1 at section A which 
is the "toe dipping" regime. This "toe dipping" regime did not accurately 
represent the actual operational state of a disc drive slider relative to 
a disc. The "pillbox" regime studied in this paper and shown at section B 
is, in fact, much more representative of actual operation of a disc drive. 
Section C of FIG. 1, in fact, guarantees failure of the disc drive. By 
"pillbox" it is meant that the individual asperities are submerged in the 
lubricant; however, each individual meniscus has not joined each other and 
is separately defined across the surface of the disc. It is also 
noticeable that this model assumes that the lubricant flows over the 
surface of the disc. This invention takes account of the fact that the 
lubricant when supplied to the surface of the disc will redistribute 
itself to satisfy the laws of physics. The previous models failed to take 
account of this issue, either because of a lack of recognition of the 
issue or the difficulty engendered by accounting for it. 
Turning next to FIG. 21, this is a graph of experimental data based on 
certain fixed parameters and measurements of stiction for different bump 
heights. The bump height has a very significant parameter because current 
trends in disc drive technology where efforts are directed to storing 
increased amounts of data on the same disc seek to achieve this goal by 
flying the transducer as closely as possible to the disc. Obviously, this 
limits the bump height which can be utilized to texture the surface of the 
disc. In FIG. 21, the bumps of the volcano shape described above, and have 
a diameter of 5.35 microns; the lubrication thickness is 25 angstroms; the 
spacing of the bumps is 35 microns; and the slider is a 50% slider. The 
way to use this graph would be to begin with the motor torque, which is a 
known value specified by the disc drive designer and is the torque 
available to start the disc and break the disc free from the slider 
resting on the disc, and which therefore is the torque available to 
overcome the stiction. In this example if the torque available allowed a 
maximum stiction of about 5 grams, then the bump height could be no less 
than amount 120 angstroms. 
Turning next to FIG. 22, a number of curves developed using the model 
described above are shown herein. One of the curves, 126 is the same curve 
as shown on the previous page, and as indicated includes a diameter of 
5.35 microns for the bump. As noted, for this curve 126, the spacing was 
35 microns. The adjacent curve 128 indicates that by reducing the diameter 
of the bumps and maintaining the same spacing of 35 microns, that the 
stiction margin is reduced for bumps of the same height, therefore 
indicating that this is an undesirable approach. 
The remaining curves 120, 122, and 124 are for different spacings, all the 
curves being done for an assumed diameter of 5.35 microns. Thus for 
example the series in curves in FIG. 22 that in selecting an optimum 
diameter of 5.35 microns for the bumps, that performance in terms of 
stiction versus bump height where there is a fixed upper limit on stiction 
could be improved by going to a smaller spacing SM such as shown by curve 
122. However, for practical purposes it is difficult to go beyond curve 
122 where the spacing is 25 microns, to the spacing of the curve 120 which 
is 10 microns. This would have a dramatic effect on throughput; the moving 
the spacing from 25 microns to 10 microns would require making two and a 
half times as many bumps on a disc, dramatically slowing the rate of 
manufacture of the discs. 
Further research not shown in these Figures indicates that more advanced 
discs could be made with an average bump height of about 180 Angstroms; 
bump diameter of 5.3 microns; spacing of 25.times.32 microns; and a 
lubrication thickness of 25 Angstroms, .+-.5 Angstroms. Further 
development yields an average bump height of 160 Angstroms and lubrication 
thickness of 10 Angstroms .+-.3 Angstroms. 
This fact also demonstrates one of the attractions of this mode of 
theoretical analysis. Since the motor torque and glide height are 
predefined, the model can be replicated to optimize stiction against a 
target level number, while continuing to take into account other factors 
which cannot be built into the model such as the through put and other 
costs of building the disc defined by any one of these curves.