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A Mixed-Lubrication Approach to Predicting CMP FluidPressure Modeling and ExperimentsC. Fred Higgs III, a,c,z Sum Huan Ng,a Len Borucki,* ,b Inho Yoon,a

and Steven Danyluka

aSchool of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia, 30332, USAbIntelligent Planar, Phoenix, Arizona, USA

Chemical mechanical polishing~CMP! is a manufacturing process used to remove or planarize metallic, dielectric, or barrier layerson silicon wafers. During polishing, a wafer is mounted face up on a fixture and pressed against a rotating polymeric pad that isflooded with slurry. The wafer also rotates relative to the pad. The combination of load on the wafer fixture, relative speed ofrotation, slurry chemistry, and pad properties influences polishing rates. Prior work has shown that an asymmetrical subambientpressure, which exceeds that expected from the applied load, can develop at the interface between the fixture and an ungroovedpad. The spatial distribution of this pressure can be measured and then simulated using a specially designed fixture with water asthe slurry. A mixed-lubrication approach to modeling the fluid pressure was developed by including the contact stress, frictionalbehavior, and fluid film thickness. For a given fixture/pad separation, the contact stress can be determined using a Winkler modelapproximation. The film thickness can be approximated as the distance from the fixture surface to the mean asperity plane. Oncethe fluid film thickness is known, the fluid pressure can be determined from the two-dimensional polar Reynolds equation usingfinite-differencing. The theoretical pressure solution was found to match the experimental pressures when the system of forces andmoments were balanced. The iterative secant numerical method was employed to compute the appropriate fluid film thickness thataccommodates a balanced system of forces and moments produced by the fluid/solid interactions. After the fluid pressure isdetermined from an initially assumed separation, all shear and normal forces are computed from the solid contact stress andhydrodynamic fluid pressure. The results agree with the experiments.© 2005 The Electrochemical Society.@DOI: 10.1149/1.1855834# All rights reserved.

Manuscript submitted January 13, 2004; revised manuscript received July 28, 2004.

Chemical-mechanical planarization~CMP!, a surface processingstep used widely in integrated circuits~ICs! manufacturing, is cur-rently the leading nanomanufacturing process worldwide, with anannual economic impact well in excess of $1 billion.1 It is used toachieve planarization of various high-topography films on siliconwafers and in partially processed wafers as an interim step in ICmanufacturing.2 During polishing, a rotating wafer is pressedagainst a rotating polymeric pad that is flooded with a chemicallyreactive slurry. The slurry has nanoparticles in it which makes it a‘‘reverse-lubricant’’ that polishes or wears films by the combinedaction of chemical corrosion and mechanical removal. It has beenknown for some time that polishing is related to the load on thewafer, the relative speed between the wafer and pad, slurry chemis-try, and pad properties.3-5 Consequently, Preston’s equation predictspolishing~i.e., removal! rates to be proportional to the applied loadand speed.6 However, material removal rates on silicon wafers dur-ing CMP have not exhibited ‘‘Prestonian’’ behavior because addi-tional mechanical and chemical phenomena have beenobserved.2,7-10

Our work has focused on the mechanical interaction between thepad and the wafer surface. Our group measured the interfacial fluidpressure between the nonrotating fixture and pad and foundsuction.11 A one-dimensional model was developed to explain thesuction pressure observed in the experiments.8 In that model, theinterfacial fluid pressure was computed using the three-step processusually employed in elastohydrodynamic lubrication12 problems.These steps consist of solving for the:~i! contact stresss, (i i ) filmthicknessh, and (i i i ) fluid pressurep. The one-dimensional contactstresss was determined by considering the line loading of an elastichalf-space~i.e., the pad! produced by a flat, rigid, sliding punch~i.e.,the fixture or wafer!

s~x! 5P cospg

p~a2 2 x2!1/2 S a 1 x

a 2 xD g

@1#

whereP is the load per unit length,a is the fixture radius,x is the

position under the contact, andg is a function of the coefficient offriction. Assuming that the surface asperity heights obey an expo-nential distribution, the film thicknessh was determined fromGreenwood-Williamson13

h~x! 5 s lnS AphER1/2s3/2

~1 2 n2!s~x!D @2#

whereE is the elastic modulus of the pad,R is the asperity radius,sis the pad roughness andh is the asperity density. The resulting filmthickness was used in the one-dimensional Reynolds’ equation todetermine the fluid pressurep

d

dx S fxh3

dp

dx D 5 6mUdh

dx@3#

wherefx is the empirical pressure flow factor andm andU are theviscosity and velocity, respectively. The resulting fluid pressure wasasymmetrical and mostly subambient, which means it exhibited non-uniform suction. While this model showed that the mechanical phe-nomenon and basic physics of the CMP process were understood, itcould not predict the behavior of the fluid pressure in both the radialand tangential directions, areas of concern to the uniformity of thepolishing. It also modeled the pad as an elastic half-space, when inreality the pad is a thin elastic layer on a rigid surface. The rest ofthis paper presents a model that describes the predicted tangentialand radial fluid pressure.

In the past, several approaches were taken to predict the two-dimensional fluid pressure profiles. These models employed a finite-element modeling~FEM! approach to determine the contact stressthat produced the resulting film thickness. That film thickness wasthen used to determine the hydrodynamic fluid pressure. Shan14 usedan FEM analysis of the pad being indented by a rigid punch tocompute the fluid film thickness. He modeled the elastic pad withasperities as two media with asperities having an elastic modulus ofapproximately one-half that of the bulk pad. Kimet al.15 employeda hyperelastic asperity model to quantify the behavior of the asperi-ties relative to the bulk. In their ‘‘soft-hydrodynamic’’ approach,they were able to predict a film thickness that varied in two dimen-sions for a balanced system of forces and moments, which resultedin the prediction of the 2D fluid pressure. Both approaches obtained

* Electrochemical Society Active Member.c Present address: Carnegie Mellon University, Pittsburgh, PA 15213, USA.z E-mail: higgs@cmu.edu

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subambient pressures, but the models did not predict the positivepressure region seen in the experiments. Streator16 solved a similarmixed-lubrication problem while considering capillary effects inpredicting a subambient interfacial fluid pressure at an equilibriumconfiguration of balanced forces and moments.

This paper describes a method to determine the film thickness~i.e., separation! in 2D, accounting for the film shape that resultsfrom the normal load and tilting created by sliding friction. From thefilm thickness the interfacial fluid pressure can be determined. Forthe purpose of obtaining measurements and simplicity, the steel fix-ture ~i.e., wafer! is not rotating as it does in conventional CMP. TheWinkler model is used to determine the 2D contact stress on the pad.A recently developed FEM model confirms that Winkler sufficientlyapproximates the contact stress of the wafer at most points, exceptnear the wafer edge,17 excluding the high stress at the edge. Becauseboth hydrodynamic pressure and asperity contact pressure supportthe total load, an iterative, mixed lubrication approach must be em-ployed. This approach was suitable for predicting the tangential andradial components of the fluid pressures. The results compare favor-ably to fluid pressure experiments.

Experimental

Experiments were conducted using the apparatus shown in Fig.1a, which depicts a 150 mm pressure fixture that has electronicpressure transducers aligned along the tangential and radial direc-tions, relative to the center of the pad. This stainless steel fixture has20 membrane pressure sensors, eight surface mount thermisters, andeight capacitance probes respectively.14

The apparatus used was a Struers RotoPol-35 tabletop polishingmachine with a RotoForce-3 head. The turntable speed can be variedfrom 50 to 200 rpm. The pressure fixture is a 150 mm diam, 15 mmthick, stainless steel solid disk that is positioned on top of a plain-type, Rodel IC-1000 pad which was conditioned to have a roughnessof Ra 5 5 m. In the experiments, the rotating pad is flooded with

water while the nonrotating pressure fixture recorded the fluid pres-sure at the ‘‘wafer/pad’’ interface in both polar directions. The radiusof the turntabler pad is 305 mm ~12 in.!, and a peristaltic pumpdelivered slurry~i.e., water! to the pad at 500 mL/min.

While a conventional CMP process consists of a Newtonian-typefluid with suspended nanosized particles, these pressure experimentsutilized water as the ‘‘slurry’’ to prevent the pressure transducersfrom being clogged. Once the test was initiated, the fluid pressurewas measured in real-time, after steady state was reached. The pres-sure fixture was mechanically loaded with 50 N applied through aball joint ~see Fig. 1b!, while the platen speed was varied to 50, 100,150, and 200 rpm.

Modeling the Fluid Pressure

A model for the two-dimensional hydrodynamic fluid pressurerequires a suitable film thickness profile. The film thickness is ap-proximated as the separation, the distance from the fixture to themean asperity plane. We employ a superpositional approach10 fordetermining the separation~i.e., equivalent film thickness! that ac-commodates the sliding wafer on an elastic layer under mixed-lubrication. The film thickness profile is a function of the contactstress, sliding friction, and wafer geometry. This work assumes thatthe predominant factors for defining the film thickness are the con-tact stress, fluid pressure, and the tilt angles produced by friction.

The separation or ‘‘equivalent film thickness’’.—Figure 2 showsschematic diagrams of the asperity deformation and fixture tiltingand defines the tilt angles when a stiff fixture is loaded centrosym-metrically. Because the ball joint allows the fixture to rotate, the tiltangle at the equilibrium configuration can be obtained. The wafer isin the mixed-lubrication regime so that both the hydrodynamic pres-sure forceWH and the solid-solid asperity contact forceWC supportthe normal load. The total separationd of the fixture from the meanasperity plane is

Figure 1. 150 mm pressure fixture for measuring 2D fluid pressure:~a! photograph of top view and~b! schematic of cross-sectional view.

Figure 2. Fixture separation due to~a! normal loading,~b! y-tilt, and ~c! x-tilt.

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d~r , u, z! 5 dx 1 dy 1 d0 5 h~r , u, z! @4#

whereh is defined as the equivalent film thickness.The disk’s center is atx 5 hw in Fig. 2a and b. The center of the

fixture is displaced from the mean asperity plane by a distance ofdz 5 d0 ~Fig. 2a!. The resultant forces acting on the fixture createmoments that result in a tilt angle in each vertical plane. From Fig.2b, the separationdy due to the tiltb in the y2z plane is

dy~r , u! 5 ~y 2 kw!tan~b! 5 ~r sinu!tan~b! @5#

wherekw(50) is they-coordinate of the fixture’s center. From Fig.2c, the film thickness variationdx due to the tilta in thex2z planecan be described by

dx~r , u! 5 ~x 2 hw!tan~a! 5 ~r cosu 2 hw!tan~a! @6#

wherehw is thex-coordinate of the fixture’s center.

Solid contact pressure.—The solid contact pressure due to asper-ity contacts was obtained using the Winkler contact stress model18

s~d; r , u! 5Easp

DhaspE

d

1`

~z 2 d!f~z!dz @7#

whereDhasp is the maximum asperity height of 20mm, Easp is theelastic modulus of the asperities, andf is the distribution of asperityheights, which was assumed as exponential.8 Borucki et al. haveshown that exponential variation can occur on the wafer contact sideof f(z) as a result of conditioning.19 The pad is porous, so it can becharacterized by voids and solid material. The voids in the pad cre-ate a peak and valley topography which defines the locations ofasperities. Figure 3 shows the ‘‘preconditioned’’~i.e., no imposed

asperity height distribution! pad.9 A schematic of the physical padfrom Fig. 3 is shown in Fig. 4. More generally, the model is imple-mented for a conditioned pad, which means that the actual height ofthe asperities in Fig. 4 varies according the distributionf from Eq.7.

In the Winkler model, the compressible asperities of the pad aredefined as peaks in between voids, thus making them harder tocompress than the bulk pad. Consequently,Easp@ Epad,bulk. Thus,Easp ~>100 MPa! is the elastic modulus for solid polyurethane,20

which is an order of magnitude higher than the elastic modulus ofthe bulk pad. From Eq. 7, the Winkler spring stiffnessk is the ratioof Easp to Dhasp. From Fig. 4 it can be seen that the maximumasperity heightDhaspis equivalent to the average void radius, whichwas found to be 20mm.9

The 2D hydrodynamic fluid pressure.—Once the equivalent filmthicknessh from Eq. 4 is known, the hydrodynamic fluid pressurepcan be computed using a finite-differencing scheme shown in Fig.5b. We apply the 2D, polar Reynolds’ equation12 to the fluid flow inthe pad/wafer interface, where the fluid motion is prescribed by thepad’s rotation~Fig. 5a!.

]

]r S rh3]p

]r D 11

r

]

]u S h3]p

]u D 5 6m~rv!]h

]u@8#

Force, moment, and equilibrium equations.—For the CMP prob-lem, the forces acting on the wafer must be determined to obtain theequilibrium configuration during polishing. The shear stresst due tothe moving fluid is determined by

t 5mvu

h~r , u!1

h~r , u!

2

]p

]u@9#

where,vu 5 rv. The magnitude and direction of the force varieswith position on the pad. The shear forceFt of the fluid, which isnumerically negligible for this lubrication problem, can be deter-mined by integrating the shear stress over the entire wafer area using

Figure 3. Optical micrograph of pad cross section.

Figure 4. Diagram of pad cross section in Winkler nomenclature.

Figure 5. Wafer-on-pad schematic:~a! diagram of wafer on pad and~b! finite differencing mesh and coordinate system.

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Ft 5 Ft,xi 1 Ft,y j

Ft,x 5 2E t~r , u!~sinu!dA @10#

Ft,y 5 1E t~r , u!~cosu!dA

wheredA 5 rdrdu for Eq. 10-16. Likewise, the frictional forceFmdue to solid contact stress is

Fm 5 Fm,xi 1 Fm,y j

Fm,x 5 2mkE s~r , u!~sinu!dA @11#

Fm,y 5 1mkE s~r , u!~cosu!dA

whereFt and Fm are friction forces produced by the pad rotatingcounterclockwise fromu 5 0 to 2p. The negative and positivesigns in Eq. 10-11 ensure thatFt andFm act in the same direction asthe corresponding components of the pad velocity. The resultantforce WH due to the fluid pressure can be determined from

WH 5 E p~r , u!dA @12#

and the resultant contact forceWc on the asperities is obtained using

WC 5 E s~r , u!dA @13#

Using Eq. 12 and 13, the force balance yields the first equilibriumequation

( Fz 5 WH 1 WC 2 FN 5 0 @14#

whereFN is the normal load applied to the fixture. The resulting freebody diagram is shown in Fig. 6, where the center of the disk is atx 5 hw . In this figure, they-component center of pressure forp(r , u) ands(r , u) areyH andyc , respectively. Using the ‘‘right-hand rule’’ and summing the moments about thex-axis (y 5 0)yields the following equilibrium equation

( M ~x,y50! 5 ~Ft,y 1 Fm,y!dP 1 E ~p 1 s!~r sinu!dA 5 0

@15#

wheredp is the distance from the pad/fixture interface to the pivotand the variabley is expressed as ‘‘r sinu.’’ Summing the momentsabout the constantx 5 hw line yields the following equilibriumequation

( M ~x5hw ,y! 5 2~Ft,x 1 Fm,x!dP

2 E ~p 1 s!~r cosu 2 hw!dA 5 0

@16#

wherex is expressed as the variable ‘‘r cosu.’’ The fluid film thick-nessh must be known before the fluid pressure is found from theReynolds Eq. 8. However, it is a function of the center separationd0and the tilt angles. The angle that the fixture tilts during operation isb in the y 2 z planea in the x 2 z plane~see Fig. 2!. Therefore,the problem is to find the equilibrium configuration (a, b, d0) ofthe fixture that balances the system of forces and moments~Eq. 14-16!.

Numerical methodology.—Figure 5a shows the diagram of thewafer and the varying pad velocities. Figure 5b shows a coarsefinite-difference meshing scheme for the CMP problem using Eq. 8,where the tangential and radial distances between two nodes areduanddr, respectively. In Fig. 5b, the nodes at the edge and outside ofthe wafer boundary are set as ambient (p 5 0; h 5 s), wheres isthe rms surface roughness. However, the nodes on the irregularmesh are unlikely to touch the wafer boundary.

In the code, the polar mesh is automatically generated to encom-pass the wafer. This mesh is created as a function of the wafer radiusr w , the wafer’s center coordinate (hw , kw), and the distance be-tween the pad and wafer centersdcent (5hw). The mesh region isdefined by two lines, which go through the origin~i.e., pad center!and are tangent to the wafer geometry. The leading edge is in thesecond quadrant, with the pad rotating counterclockwise. It is as-sumed the wafer is positioned in the II and III quadrants.

The solution to this problem is obtained by the iterative, secantmethod to find the unknown center separationd0 and tilt anglesaandb that satisfy Eq. 17. Obtaining a good initial guess is essentialto solving this complicated three-parameter system. The flowchartfor the computation is shown in Fig. 7, whereg is a generic func-

Figure 6. Free-body diagram of fixture.

Figure 7. Flowchart of fluid pressure algorithm.

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tion. The equilibrium configuration must be obtained by balancingthe forces and moments in the wafer/pad interface. Because thefixture has three degrees of freedom, we can find the parameters thatsatisfy the following universal equation

( Mx 5 ( M y 5 ( Fz 5 0 @17#

where Eq. 17 represents a balanced system of forces and moments.Although Newton’s method is generally used to solve this type ofinvolved iterative system,21 a simpler, more robust Mathematicafunction was constructed which exploits the secant, iterativemethod. The function allows the user to define two initial guesses~e.g., guess 1 and guess 2! for each parameter, thus reducing thetedium of obtaining a good initial guess and reducing the time toconvergence.

Results and Discussion

Table I shows the values of the parameters used in the modelingsimulation. The results from the model are presented in Fig. 8. Themodel produced the results for an equilibrium configuration ofa5 47 mrad, b 5 2134 mrad, d0 5 76.3 mm, wherea and b arethe tilt angles with respect to the positivex andy axes, respectively~see Fig. 2!. The model predicted the contact stresss, film thicknessh, and the tangential and radial fluid pressures at the locations of thepressure sensors from Fig. 1a. Figure 8a-c corresponds with thetangential velocity of the pad, while Fig. 8d shows the fluid pressurealong the radius of the pad.

Because the traction from the pad causes the fixture to tilt down-ward, the Winkler model accurately predicts a rise in contact stressat the leading edge~Fig. 8a!. Correspondingly, Fig. 8b shows that

the film thickness is smaller at the leading edge due to the highercontact stress there. Here the pad is being treated like a sponge,where the high-contact stress compresses the fluid film. Figure 8cshows that the positive pressure is achieved at the trailing edge (u> 3.65 rad! due to the reduced contact stress and the tilted filmconfiguration.

The experimental fluid pressure was recorded at four differentpad rotation speedsvpad 5 50, 100, 150, and 200 rpm. It was mea-sured from the leading to trailing edge along an arc length of ap-proximately 165 mm. Figure 9 shows the pressure in the tangentialand radial directions. In Fig. 9a, the tangential pressure line ismostly subambient and varies with platen speed. The largest subam-bient pressure occurs approximately 70 mm from the leading edgeand a positive pressure region occurs near the trailing edge. Figure9b shows that the radial line pressure was entirely subambient and,as expected, the magnitude of the suction pressure increases withpad velocity. The fluid exhibited a positive pressure from 135 mm tothe trailing edge. The fluid pressure along the radial line was entirelysubambient. It is believed that there are regions of ‘‘wafer/pad’’contact and regions of ‘‘wafer/pad’’ separation at the interface. Thisstationary fixture experienced the largest suction pressure away fromthe center of the pad.

Figure 8. Model prediction of four parameters:~a! contact stress,~b! filmthickness,~c! tangential fluid pressure, and~d! radial fluid pressure.

Figure 9. 2D fluid pressure experimental results:~a! tangential and~b!radial.

Table I. List of parameters used for the model.

Parameters Values

Pad Type: Rodel IC 1000Epad,bulk5 12 MPaEpad,asp5 100 MPa

Surface roughness:s 5 8 mm ~rms!Maximum asperity height:Dhasp5 20 mm

Moment arm:dp 5 20 mmFluid Water

Viscosity: 0.001 Pa sRotational velocity 50 rpmNormal load 50 NCoefficient of friction 0.6

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Figure 10 shows the fluid pressure measured along the tangentialand radial direction with increasing velocity. The parametric resultsrun at the same experimental test conditions reveal a close matchwith experimental results shown in Fig. 9 for the tangential andradial directions. Figure 5b shows the locationr 5 0 in Fig. 10b,which is the edge of the fixture closest to the pad’s center.

This 2D model predicts the profiles for fluid pressure measuredalong a line in both the radial and tangential directions. While it candetermine pressure along a line fairly well, it is inadequate for de-termining the fluid pressure beneath the entire fixture area. However,the experimental film thickness values can be used as input in Eq. 8and correctly predict the fluid pressure under the entire fixture. Thissuggests that the fluid mechanics are understood, while treatments ofthe elastic contact need improvement.

Conclusion

The contact stress is determined by treating the pad asperity layeras a simple Winkler foundation. The model predicts the interfacialfluid pressure in both directions along a line. Experiments wereconducted to measure this fluid pressure in the gap between a pres-sure fixture~i.e., wafer! and rotating pad. The fluid pressure wasmeasured both in the tangential and radial directions simultaneouslyin real-time. The modeling approach employs an equilibrium analy-sis of the forces and moments acting on a wafer during CMP opera-tion. This model shows that the fluid mechanics are understood,while treatments of the elastic contact~i.e., the pad and/or asperities!appear less accurate.

Acknowledgments

The authors acknowledge the support of the NSF Center of Sur-face Engineering and Tribology, EKC, Rodel, Chemical ProductsCorporation, Ashland Corporation, Singapore Institute of Manufac-turing Technology, and Motorola.

Georgia Institute of Technology assisted in meeting the publication costsof this article.

References

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11. J. Levert, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA~1997!.12. B. J. Hamrock,Fundamentals of Fluid Film Lubrication, McGraw-Hill, New York

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~1966!.14. L. Shan, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA~2000!.15. A. Kim, J. Seok, C. Sukam, J. Tichy, and T. Cale, inAdvanced Metallization

Conference~2001!.16. J. Streator, inProceedings of the 28th Leeds-Lyon Symposium on Tribology—

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Figure 10. 2D fluid pressure theoretical results:~a! tangential and~b! radial.

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