Journal of The Electrochemical Society, 153 共6兲 K15-K22 共2006兲
`0013-4651/2006/153共6兲/K15/8/$20.00 © The Electrochemical Society
`Hydrodynamics of Slurry Flow in Chemical Mechanical
`Polishing
`A Review
`Elon J. Terrell and C. Fred Higgs IIIz
`
`Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh,
`Pennsylvania 15213-3890, USA
`
`K15
`
`Chemical mechanical polishing 共CMP兲 is a process that is commonly used to planarize wafer surfaces during fabrication. Although
`the complex interactions between the wafer, pad, and slurry make the CMP process difficult to predict, it has been postulated that
`the motion of the slurry fluid at the wafer–pad interface has an important effect on the wafer surface wear distribution. This paper
`thus serves as a review of past studies of the hydrodynamics of slurry flow during chemical mechanical polishing. The reviewed
`studies include theoretical and numerical models as well as experimental measurements.
`关DOI: 10.1149/1.2188329兴 All rights reserved.
`© 2006 The Electrochemical Society.
`
`Manuscript submitted August 30, 2005; revised manuscript received January 23, 2006. Available electronically April 19, 2006.
`
`in partial contact and have used a hybrid contact mechanics/fluid
`mechanics approach toward analyzing CMP. Finally, a set of studies
`have analyzed the CMP process solely using fluid mechanics, as-
`suming that the wafer and pad surfaces are completely separated by
`slurry. From their review, Nanz and Camilletti have indicated the
`importance of the slurry flowfield to the CMP process as well as the
`need for more in-depth understanding of slurry flow at the wafer–
`pad interface. Therefore this paper serves as a review of past studies
`in slurry hydrodynamics during CMP. These studies include film
`thickness and hydrodynamic pressure modeling, numerical fluid
`flow modeling, and experimental investigations.
`
`CMP Hydrodynamic Modeling
`In the modeling studies that are discussed in this paper, slurry
`hydrodynamic analysis was used to find expressions for a number of
`parameters, including the slurry pressure field, film thickness distri-
`bution, and shear rate.
`
`Slurry Film Thickness and Hydrodynamic Pressure Modeling
`Assuming that a thin slurry film separates the wafer and pad
`surfaces during CMP, the film thickness and pressure distribution of
`the slurry can be related using the Reynolds equation, shown in 1D
`form as follows
`
`关2兴
`
`冉h3dp
`
`冊 = 6␮U
`
`d d
`
`dh
`dx
`dx
`x
`where p is the hydrodynamic pressure, h is the local film thickness,
`␮ is the dynamic viscosity of the slurry, U is the relative velocity of
`the bottom surface, and x is the downstream distance. Analysis of
`the slurry pressure distribution is of importance in CMP hydrody-
`namic studies because it gives insight into how much the wafer and
`pad surfaces are being pushed away from each other 共positive pres-
`sure兲 or sucked toward each other 共negative pressure兲 along the
`length of the interface.
`A study by Sundararajan et al.2 involved the derivation of 2D
`wafer-scale lubrication and mass-transport models for an assumed
`hydrodynamic slurry interface during CMP. They started the model
`using the one-dimensional, steady-state Reynolds equation 共Eq. 2兲,
`assuming that the film thickness had a convex shape due to bending
`of the wafer. It is important to note the difference between a convex
`and concave wafer according to CMP terminology. As shown in
`Fig. 2, a wafer is termed convex if it is bent toward the pad in the
`middle and concave if it is bent away from the pad in the middle.
`Certain constant parameters in the film thickness expression were
`solved by assuming two constraints: 共i兲 that the integral of the pres-
`sure distribution across the length of the wafer is equal to the ap-
`plied load, and 共ii兲 that the movement of the forces around the center
`of the wafer is zero. After solving for the film thickness and pressure
`distributions, the authors calculated the slurry velocity distribution
`
`Chemical mechanical polishing 共CMP兲 is a manufacturing pro-
`cess that is used to planarize the surfaces of small-scale devices such
`as integrated circuits and hard disk read/write heads during fabrica-
`tion. CMP has emerged as a critical fabrication step due to the
`demand for faster and more complex small-scale devices with mul-
`tilevel interconnects. During CMP, the wafer containing the device
`is mounted face-down onto a rotating carrier and pressed against a
`rotating polishing pad that is flooded with chemically reactive slurry
`containing abrasive nanoparticles, as shown in Fig. 1. The mechani-
`cal and chemical interactions between the wafer, pad, and slurry
`cause the surface of the wafer to wear to atomically smooth levels.
`Although CMP is widely used in industry, much of the physics
`behind CMP is not known because of the complex phenomena at
`the wafer–pad interface. These complexities include 共i兲 the slurry
`flowfield and film thickness distribution in the wafer–pad interface,
`共ii兲 the material wear effects caused by interactions between contact-
`ing wafer and pad asperities, 共iii兲 the effects of wafer and pad
`surface roughness on the slurry flowfield, 共iv兲 the material wear
`effects caused by the nanoparticles, and 共v兲 the effect of the
`nanoparticles on the rheology of the slurry. The lack of detailed
`knowledge of these effects has reduced CMP optimization into a
`mostly empirical process. Thus, CMP modeling and experimentation
`has become critical for understanding CMP and minimizing the
`amount of trial-and-error schemes that are currently necessary for
`CMP optimization.
`A number of experimental and theoretical studies have been
`conducted in order to analyze different aspects of the CMP process.
`A great deal of CMP research involves analysis of the material
`removal rate 共MRR兲. A generalized expression for the wafer surface
`material removal rate is given by Preston’s wear equation, as
`follows
`
`MRR =
`
`kPU
`H
`
`关1兴
`
`where k is the nondimensional Preston’s wear coefficient, P is the
`wafer downforce, V is the relative velocity of the wafer–pad inter-
`face, and H is the hardness of the wafer surface. Preston’s wear
`equation has commonly been used as an approximation for global
`MRR.
`A number of more sophisticated wafer surface wear models have
`been developed to account for various physical phenomena that take
`place during CMP. Nanz and Camilletti1 provided a critical review
`of CMP models up until 1995. Several studies have taken a contact
`mechanics approach toward CMP analysis, assuming that the wafer
`and pad surfaces are in direct sliding contact during the CMP pro-
`cess. Additional studies have assumed that the wafer and pad are
`
`z E-mail: higgs@andrew.cmu.edu
`
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`Journal of The Electrochemical Society, 153 共6兲 K15-K22 共2006兲
`
`Figure 3. Diagram of 3D pad–surface model from from Nishioka et al.3
`Adapted with permission, © 1999 IEEE.
`
`Figure 1. Diagram of the CMP process.
`
`across the wafer using the expression for Couette/Poiseuelle flow,
`given as follows
`
`冊 −
`
`y h
`
`u共x,y兲 = U冉1 −
`
`h2
`dp
`2␮
`dx
`where U is the velocity of the pad surface, h is the film thickness,
`and p is the hydrodynamic pressure. From the results of their model,
`the authors found that certain conditions caused the slurry flow to
`have an unstable separation region. The results of their lubrication
`study were combined with mass-transport theory in order to predict
`the material removal rate distribution over the surface of the wafer.
`Nishioka et al.3 presented an analytical model for the slurry film
`thickness and wafer–pad coefficient of friction during CMP, ac-
`counting for pad surface roughness. The pad was modeled as a mov-
`ing 3D sinusoidal surface, while the wafer was modeled as a flat,
`stationary surface. The expression for the pad surface is given as
`follows
`
`关3兴
`
`冊
`
`y h
`
`冉1 −
`
`y h
`
`wafer surface was assumed to be fixed for simplification purposes.
`The slurry film thickness was estimated by assuming a contact stress
`distribution across the wafer and then solving for film thickness
`using the Greenwood and Williamson contact stress model.6 This
`resulted in a film thickness distribution that was smallest at the
`edges of the wafer and largest in the middle. From the film thickness
`distribution the authors used a finite-differencing algorithm on the
`1D Reynolds equation 共Eq. 2兲 to determine the predicted pressure
`distribution across the wafer. Their analysis showed that the slurry
`pressure distribution is subambient at certain locations along the
`length of the wafer, implying that the wafer is “sucked down” at
`those locations. Validation experiments were conducted using a
`commercial benchtop polisher that was fitted with preconditioned
`polishing pads. The pressure distribution was measured by outfitting
`the simulated wafer surface with a series of pressure taps whose data
`was acquired using an electronic pressure transducer. For these ex-
`periments, water was substituted for slurry as the interfacial fluid in
`order to prevent the possibility of the particles interfering with the
`pressure taps. The results of the validation experimentation also
`showed a region of subambient pressure and corresponded well with
`the predicted results.
`A study by Higgs III et al.7 expanded on the studies by Shan et
`al.4,5 by performing a 2D analysis of the entire wafer instead of a
`line of constant pad radius. This study, like the previous study, in-
`volved a combination of experimental pressure measurements and
`mathematical modeling. The mathematical model was created using
`the polar form of the Reynolds Equation, given as follows
`⳵p
`⳵p
`⳵r
`⳵␪
`r
`where r and ␪ are radial and tangential coordinates along the wafer–
`pad interface, respectively, and ␻ is the rotational speed of the pad.
`The film thickness, h, was found by assuming a given contact stress
`distribution and then finding the attack angle of the wafer by bal-
`ancing forces and moments about the pivot point. As a result of this
`study, the authors found yet again that a significant portion of the
`pressure distribution was subambient, as shown in Fig. 6. The vali-
`dation experiments were conducted using a tabletop polisher using
`
`冊 = 6␮共r␻兲 ⳵h
`
`⳵␪
`
`关5兴
`
`冉h3
`
`⳵ ⳵
`1 r
`
`␪
`
`冊 +
`
`冉rh3
`
`⳵ ⳵
`
`关4兴
`
`h共x,y兲 = h0 + RP cos
`
`cos
`
`2␲y
`2␲x
`␭y
`␭x
`where h0 is the mean line of the pad, RP is the peak roughness, and
`␭X and ␭Y are the wavelengths of the pad in the x and y directions,
`respectively. A diagram of the resultant sinusoidal surface is shown
`in Fig. 3. The parameters needed to define the sinusoidal surface
`共such as wavelength and roughness amplitude兲 were determined by
`performing roughness analysis on an actual pad. From this model,
`expressions for the mean hydrodynamic pressure and mean shear
`stress were derived as functions of the minimum film thickness.
`Their prediction of the pressure variation with film thickness is
`shown in Fig. 4. In order to validate their model, they compared the
`predicted coefficient of friction of their model to the measured co-
`efficient of friction from experimental CMP tests.
`Studies by Shan et al.4,5 have been used to analyze the slurry
`hydrodynamic pressure distribution across the wafer during CMP.
`Their analysis was conducted using a combination of mathematical
`modeling and validation experiments. The modeling aspect of their
`study involved the use of the one-dimensional Reynolds equation
`taken over a constant-pad-velocity line as shown in Fig. 5. The
`
`Figure 2. Diagram showing the difference between a convex and a concave
`wafer.
`
`Figure 4. Variation of mean pressure with minimum film thickness from
`Nishioka et al.3 Adapted with permission, © 1999 IEEE.
`
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`Journal of The Electrochemical Society, 153 共6兲 K15-K22 共2006兲
`
`
`
`K17K17
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`be a Newtonian, particle-free fluid, which provided the following
`governing equation
`
`−12␮ⵜ · 关共hw − s兲3ⵜp兴 +
`
`
`
`ⵜ · 共Uជw + Uជp兲
`
`
`
`共hw − s兲
`2
`
`
`
`
`· 共Uជp − Uជw兲 + Vw − Vp = 0
`
`关7兴
`
`共ⵜhw + ⵜs兲
`2
`where ␮ is the dynamic viscosity of the fluid, hw and s are the wafer
`and pad surface topographies, respectively, p is the hydrodynamic
`pressure, and Uw and Up are the velocities of the wafer and pad,
`respectively. The governing equation was solved using an iterative
`finite difference scheme by being subjected to a balance of forces
`and moments around the pivot point. From this analysis, the authors
`were able to calculate the slurry pressure field, flowfield, and film-
`thickness distribution. The pressure field from their study is shown
`in Fig. 8, showing concentric isobars that are greater than ambient
`everywhere in the flowfield. Figure 9 shows vector plots of the
`interfacial flowfield at different vertical “slice” locations inside the
`wafer–pad gap. As Fig. 9 shows, the slurry flowfield appears to
`closely follow the motion of the pad near the pad 共z* = 0兲, then
`transitions into following the motion of the wafer as vertical location
`of interest increases. In the vertical location directly next to the
`wafer 共z* = 1兲, the slurry flow approximately follows the motion as
`the wafer.
`Jeng and Tsai15 presented a CMP model which combines hydro-
`dynamic lubrication theory with granular flow analysis in order to
`account for the motion of the slurry with abrasive nanoparticles. The
`hydrodynamic aspect of this model was based on the macroscopic
`Navier–Stokes equations, while the granular flow aspect of this
`model was based on microscopic molecular theory from a separate
`study.16 These two approaches were combined and simplified into a
`set of governing equations which described particle-fluid motion.
`Their resultant model predicted that the material removal rate in-
`creases proportionately with particle size. They later expanded upon
`their previous model by accounting for pad roughness effects using
`a combination of flow factors from separate studies.17-20 The studies
`by Jeng and Tsai focused a significant amount of attention on the
`effect of the abrasive particles in CMP, which is an important aspect
`of CMP that is often neglected in literature. However, their studies
`assumed that
`the slurry was completely composed of particles,
`which is an exaggerated assumption because slurry is composed
`primarily of fluid and contains only a trace amount 共3–5 wt %兲 of
`particles.21,22
`Chen and Fang23 presented a mathematical model which predicts
`the slurry film thickness and pressure distribution during CMP. The
`wafer was assumed to have a convex shape while the pad was as-
`sumed to be completely flat and horizontal. The resultant pressure
`distribution was found by deriving the polar form of the Reynolds
`equation and solving it by expanding the pressure distribution into
`
`Figure 5. Diagram of constant pad velocity line that was analyzed in the 1D
`CMP hydrodynamic studies.
`
`+
`
`water as a substitute for slurry in a method similar to that of Shan et
`al.4,5 The predicted results matched up well with experimental data.
`More works related to the modeling of subambient hydrodynamic
`slurry pressure can be found in the literature8-12 but are omitted from
`this paper for the sake of brevity.
`Cho et al.13 presented a 2D mathematical model which predicts
`the slurry film thickness and pressure distribution during CMP. For
`this analysis the authors used the polar form of the Reynolds equa-
`tion. They assumed that both the wafer and pad surfaces were com-
`pletely flat, although the wafer was allowed to tilt from its center
`pivot point in order to balance forces and moments. The authors also
`assumed that the slurry was a particle-free, incompressible, Newton-
`ian fluid for their analysis. By coupling the tilt of the wafer 共film
`thickness distribution兲 with the balance of forces and moments on
`the wafer 共pressure distribution兲, the authors were able to solve for
`both using the numerical Newton–Raphson method.
`Thakurta et al.14 developed a model for slurry film thickness and
`velocity distribution during CMP and compared the predicted results
`with the results of experiment. Their model, outlined in Fig. 7, ac-
`counted for the porosity and deflection of the pad surface. The the-
`oretical model was created by first assuming a parabolic shape for
`the convex wafer as follows
`
`h共x,y兲 = h0 + SX冉 x
`
`冊 + SY冉 y
`
`冊 + ␦0冉 x2 + y2
`
`冊
`
`a2
`a
`a
`where h0 is the centerline height of the wafer, ␦0 is the wafer dome
`height, a is the radius of the wafer, and SX and SY are the horizontal
`and vertical slopes of the wafer due to its equilibrium angle on the
`gimbal. Additionally, the pad surface topography, given by s共x,y兲,
`accounted for elastic deformation of the pad, which was specified to
`be directly proportional to the slurry hydrodynamic pressure. The
`pad porosity was also accounted for by assuming that a certain
`amount of slurry seeps into the pad depending on the hydrodynamic
`pressure distribution. Lubrication approximations were then used to
`simplify the polar Navier–Stokes equations, assuming the slurry to
`
`关6兴
`
`Figure 6. Predicted 2D fluid pressure from Higgs III et al.7: 共a兲 tangential and 共b兲 radial.
`
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`Journal of The Electrochemical Society, 153 共6兲 K15-K22 共2006兲
`
`Figure 7. Diagram of modeling domain used in Thakurta et al.14 Reproduced
`by permission of The Electrochemical Society, Inc.
`
`Stoum–Liouville eigenfunctions. As a result of their work they
`found that the pressure distribution takes on a half-parabolic shape
`across the radial direction from the center of the wafer. All predicted
`pressures from this study were greater than atmospheric.
`
`Slurry shear rate modeling.— It is possible that the shear rate of
`the slurry is of great importance to CMP hydrodynamics. Runnels
`and Eyman24 have postulated that the wafer surface wear rate is
`directly proportional to the shear rate of the slurry in the hydrody-
`namic lubrication regime according to the following equation
`关8兴
`MRR = K␴␶
`where MRR is the material removal rate, K is the Preston coefficient,
`␴ is the normal stress, and ␶ is the shear stress of the slurry fluid.
`Equation 8 is proposed only for hydrodynamic lubrication. If the
`minimum film thickness is on the order of or less than the average
`surface roughness, then the effect of solid contact must be taken into
`account.14 Several solid contact CMP models are currently available
`in literature, if solid–solid contact is to be assumed. Details of the
`solid contact models are outside the scope of this paper.
`A study by Sohn et al.25 involved the derivation of expressions
`for the shear rate for slurry flow at the wafer–pad interface. Assum-
`ing that the slurry behaved as a particle-free, Newtonian fluid, the
`authors used a simplified version of the incompressible Navier–
`Stokes equations to model the flowfield. For this analysis it was
`
`Figure 8. Pressure distribution under rotating wafer from Thakurta et al.14
`Reproduced by permission of The Electrochemical Society, Inc.
`
`Figure 9. Slurry flowfield at different vertical locations inside the wafer–pad
`gap from Thakurta et al.14 Reproduced by permission of The Electrochemical
`Society, Inc.
`
`assumed that both the wafer and pad surfaces were rotating, no-slip
`walls. Additionally, the wafer and the pad were assumed to be per-
`fectly parallel to each other, which resulted in a constant slurry
`hydrodynamic pressure. The authors simplified the Navier–Stokes
`equations by assuming that the Reynolds number and aspect ratio
`are both negligibly small. From this analysis the authors were able
`to derive closed-form expressions for the slurry velocity field and
`shear rate. They found that when the wafer rotational speed is
`greater than the speed of the pad, the shear rate is greatest at the
`edges of the wafer. However, when the wafer rotational speed is the
`same as that of the pad, the shear rate is uniform throughout the
`wafer.
`
`discrep-
`Discussion of hydrodynamic modeling studies.— The
`ancy between the predicted results of each of these models appears
`to be rooted in the different assumptions of the wafer and pad sur-
`face geometries. The studies by Sundararajan et al.,2 Thakurta et
`al.,14 Jeng and Tsai,15,17 and Chen and Fang23 assumed that the
`wafer surface was slightly convex, which resulted in a pressure dis-
`tribution that was greater than ambient everywhere in the domain. In
`contrast, the studies by Shan et al.4,5 and Higgs III et al.7 incorpo-
`rated contact stress models into their analysis and thus ended with a
`film thickness that was smallest at the edges of the wafer and largest
`in the middle. As a result, the former group of authors predicted a
`pressure distribution that was greater than ambient everywhere in
`the domain, while the latter group predicted a region of subambient
`pressure. In order to determine the correct pressure distribution, one
`must analyze the film thickness distribution and account for wafer
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`
`Figure 10. Slurry flow domain and diagram of a real wafer–pad interface from Muldowney28 using Fluent 6.1, a commercial CFD solver. Reprinted with
`permission from Materials Research Society.
`
`bending and pad deflection. Numerous studies have been conducted
`which analyze the deflection of the wafer and pad during CMP,
`although a detailed discussion of these studies is omitted from this
`paper for the sake of brevity.
`Additionally, we must take note of some of the simplifications
`that are used in these models. The assumption of the slurry as a
`Newtonian fluid is the most widely used simplification in each of
`these models, with the exception of the studies by Jeng and Tsai,15,17
`who assumed that the slurry was composed completely of small
`“granular” particles.
`
`Numerical Studies in CMP Hydrodynamics
`A few studies have been conducted which use computational
`fluid dynamics 共CFD兲 to analyze the slurry flowfield in CMP. CFD
`solvers are advantageous for this analysis due to their ability to input
`complex flow domains and solve transport equations for multiphase
`flow and chemical reactions.
`One of the first CMP numerical studies was conducted by Run-
`nels and Eyman,24 who created a numerical model to predict the
`slurry film thickness and hydrodynamic pressure. Assuming that hy-
`drodynamic lubrication takes place between the wafer and pad sur-
`faces, they imposed a sample slurry flowfield domain into a numeri-
`cal code and solved it using a Galerkin finite element scheme. Both
`the wafer and pad surfaces were modeled as being rigid, smooth,
`no-slip walls. The pad was assumed to be flat while the wafer was
`designed to be convex with a specified radius of curvature. The
`wafer and pad walls were bounded by an additional wall which
`joined the two. This bounding wall was given a stress-free boundary
`condition in order to allow fluid to enter and exit the domain freely.
`The slurry film thickness was found by balancing the hydrodynamic
`forces with the applied load and pivoting the wafer such that the
`moment around the pivot point was zero. From the results of these
`simulations, the authors were able to find the amount of load that
`can be supported by the wafer as well as the minimum film thick-
`ness of slurry between the wafer and the pad.
`Fu and Chou26 used CFX-3D, a commercial numerical solver, to
`solve for the slurry flowfield in a CMP domain between the wafer
`and the pad. For their simulation, both the wafer and pad were
`modeled as being perfectly rigid, flat, and smooth, while the slurry
`was modeled as being a Newtonian, incompressible, particle-free
`fluid. The wafer–pad gap was fixed at a given input value, either 20
`or 40 ␮m. Both the wafer and pad walls were modeled as having
`no-slip boundary conditions, while the remaining surfaces were
`modeled as stress-free boundaries in order to allow the slurry to
`freely enter and exit the computational domain. The resultant slurry
`shear stress distribution from the simulation was used to estimate the
`material removal rate from the wafer surface.
`
`Yao et al.27 used the software package Fidap, a commercial nu-
`merical solver, to model the slurry flow pattern between two moving
`surfaces at different locations at the wafer/pad interface. They chose
`not to model the entire wafer/pad domain but rather modeled various
`geometries along the tangent of the wafer in order to conserve com-
`putational resources. Each of the geometries was square and fea-
`tured different scales of roughness. The boundary conditions for
`each of the geometries were dependent on the rotational movements
`of the pad and wafer. The slurry flow was modeled using the incom-
`pressible Navier–Stokes equations, assuming that the slurry exhib-
`ited nonNewtonian behavior due to the abrasive nanoparticles. They
`assumed that the material removal rate was directly proportional to
`the slurry shear stress as postulated in Runnels and Eyman, and then
`used Fidap’s time interval updating capability to change the slurry
`film thickness over time based on the predicted wafer surface wear.
`Using this method they were able to determine the amount of ma-
`terial removal that occurs to the surface roughness after a given
`amount of polishing time. They were also able to derive an empiri-
`cal model for the instantaneous polish rate with respect to time.
`Muldowney28 used the commercial numerical solver Fluent 6.1
`to model the slurry velocity field, thermal field, and chemical reac-
`tions between the pad and wafer during CMP. The CFD domain in
`this study is shown in Fig. 10. For this simulation, the wafer was
`modeled as being flat and smooth, while the pad was modeled as
`having a rough topography in the form of a series of concentric
`circular grooves that are separated by circular concentric “asperi-
`ties.” The gap between the “asperities” and the wafer surface was
`modeled as being porous in order to account for the porous flow of
`slurry through the asperities. The pad and the wafer surfaces were
`modeled as no-slip/no-penetration boundaries, while the slurry was
`assumed to be a Newtonian fluid. From this analysis, the author was
`able to show the resultant velocity profile of the slurry at the wafer–
`pad interface. Figure 11 shows the predicted velocity profile, which
`appears to have a stagnation/backflow region that is comparable to
`the midgap 共z* = 0.6兲 velocity field predicted by Thakurta et al.14
`共Fig. 9兲.
`Rogers et al.29 used a combination of numerical modeling and
`experimental testing to analyze the flow of slurry during CMP. For
`their numerical study they used Fluent, a commercial fluid flow
`solver, to analyze the flow of both the slurry and the surrounding air
`outside the wafer–pad interface. Their 2D flow domain consisted of
`the linearly moving pad surface, the surrounding air/slurry volume,
`and the fixed wafer surface, which was represented as a rigid punch.
`The motion of the air and slurry were analyzed using Fluent’s vol-
`ume of fluid 共VOF兲 solver, which modeled the air and slurry as two
`immiscible fluids. Both the wafer and pad walls were modeled as
`no-slip boundaries, while the side walls were modeled as cyclic
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`) unless CC License in place (see abstract).
`
`Raytheon2030-0005
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`K20K20
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`Journal of The Electrochemical Society, 153 共6兲 K15-K22 共2006兲
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`Figure 11. Resultant slurry velocity field from Muldowney et al.28 Reprinted
`with permission from Materials Research Society.
`
`boundaries. From their numerical study they found that the slurry
`hydrodynamic pressure was greater than ambient everywhere inside
`the wafer–pad gap and decreased linearly along the length of the
`wafer. These results are in contrast to the analytical results of Levert
`et al.30 共discussed in the Experimental section of this paper兲, who
`found a subambient pressure region in the wafer–pad gap. They
`attributed this discrepancy to the fact that Levert et al. operated in
`the asperity contact regime, while they worked in the hydrodynamic
`lubrication regime.
`
`numerical
`Discussion of numerical modeling studies.— These
`studies have shown that it is possible to analyze surface wear, pres-
`sure distribution, and chemical reactions in a CMP domain using a
`numerical solver. Each of these studies took into account various
`aspects of the CMP process, such as chemical reactions, the erosion
`of wafer roughness, the rotation of both the wafer and the pad, and
`the interaction of atmospheric air with the flow of the slurry. How-
`ever, the simulations described in each study necessitated the use of
`various assumptions in order to simplify the problem and minimize
`computational expense. Examples include the assumption of per-
`fectly flat, perfectly parallel wafer and pad surfaces in the study by
`Yao et al.27 and the 2D assumption in the study by Rogers et al.29
`Because CMP is a complex process involving several physical
`phenomena, it is desirable to have a CMP numerical model which
`accounts for as much of the CMP physical phenomena as possible. It
`is expected that CMP numerical simulations will become increas-
`ingly sophisticated and realistic as computing resources continue to
`improve.
`
`Experimental Studies in CMP Hydrodynamics
`Hydrodynamic experiments in CMP have primarily served to
`examine parameters such as the slurry pressure field, the slurry film
`thickness distribution, and the wafer coefficient of friction. The im-
`plications of the slurry pressure field and film thickness distribution
`are described in the previous section. The slurry coefficient of fric-
`tion provides insight into the amount of abrasive wear that the wafer
`experiences.
`Levert et al.31 conducted a series of experiments to determine the
`slurry pressure distribution, wafer–pad coefficient of friction, and
`the wafer surface wear during CMP. They performed two sets of
`tests: 共i兲 hydrodynamic CMP tests, which were conducted with a
`light load so that the wafer would hydroplane on top of the pad
`surface, and 共ii兲 commercial CMP tests, which were conducted with
`a heavier load such that the pad and wafer asperities touched. These
`experiments were conducted using a bench-top polisher that was
`outfitted with an overhead wafer carrier. The wafer carrier itself was
`attached to an array of capacitance probes which helped to measure
`the wafer surface wear. From these experiments it was found that the
`hydrodynamic regime causes the wafer surface to wear away at a
`rate which is three orders of magnitude lower than commercial CMP
`rates. The authors thus concluded that CMP must have contacting
`wafer–pad asperities in order to polish the wafer surface adequately.
`
`Figure 12. Slurry pressure test fixture from Zhou et al.34 Reprinted with
`permission from Elsevier, © 2002.
`
`Bullen et al.32 designed and performed a series of experiments in
`order to analyze the slurry pressure distribution during CMP. Their
`experimental facility consisted of a tabletop polisher which was
`pressed upon by a drill press which served as the wafer carrier. The
`wafer itself was outfitted with a series of pressure taps at various
`radii that were connected to a pressure transducer, which rotated
`with the wafer. Before each experiment, the pad surface was condi-
`tioned using a diamond-grit polisher and the wafer surface was pol-
`ished to a convex shape in order to prevent the possibility of a
`vacuum at the wafer–pad interface. A series of tests were conducted
`with both a rotating and nonrotating wafer surface. From the results
`of this study it was found that the pressure distribution in the non-
`rotating wafer had two peaks and two valleys but did not have a
`significantly large subambient pressure region, as predicted in Shan
`et al.4,5 It was in fact predicted that the applied load was supported
`by the center of the wafer, where the pressure was predicted to be
`the highest. It is possible that the discrepancy between the results of
`this study and Shan et al.’s study can be attributed to the convex
`shape of the wafer in this study, which causes a positive hydrody-
`namic pressure from lubrication theory. The authors in this study
`also investigated the dynamic pressure distribution for a rotating
`wafer. They found that although the dynamic pressure distribution
`still had a semblance of peaks and valleys like in the static case, the
`ampli