Journal of The Electrochemical Society, 148 共10兲 G581-G586 共2001兲
G581
0013-4651/2001/148共10兲/G581/6/$7.00 © The Electrochemical Society, Inc.
Modeling and Experimental Analysis of the Material Removal
Rate in the Chemical Mechanical Planarization of
Dielectric Films and Bare Silicon Wafers
H. Hocheng,a,z H. Y. Tsai,a and Y. T. Sub
a
Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan
Department of Mechanical Engineering, National Chung Shan University, Kaoshiung, Taiwan
b
An analytic model of the material removal rate is proposed for chemical mechanical planarization 共CMP兲. The effects of the
applied pressure and the polishing velocity between the wafer surface and the pad surface are derived considering the chemical
reaction as well as the mechanical bear-and-shear process. The mechanism of microscopic material removal is presented. The
material removal rate is found less linearly correlated to the pressure and relative velocity between the pad and the wafer, which
was predicted by the frequently cited empirical Preston equation obtained from glass polishing. 关I. F. W. Preston, J. Soc. Glass.
Technol., 11, 214 共1927兲兴. It provides the analytical modification of the Preston equation in the dielectrics CMP. The experimental
results of authors and independent researchers are discussed.
© 2001 The Electrochemical Society. 关DOI: 10.1149/1.1401087兴 All rights reserved.
Manuscript submitted January 2, 2001; revised manuscript received June 1, 2001. Available electronically September 4, 2001.
Historically chemical mechanical planarization 共CMP兲 has been
used to polish optically flat surfaces. As the wiring density in highperformance chips increases and the device size scales down to submicrometers 共as shown in Table I兲, planarization technology becomes indispensable during both device fabrication and formation of
multilevel interconnects. CMP has emerged recently as a processing
technique for a higher degree of planarization in submicrometer
very large scale integration 共VLSI兲, and is widely accepted for planarizing interlevel dielectrics and selective removal of aluminum,
tungsten, copper, and titanium overburden following metal filling of
studs and interconnects.
A schematic diagram of a CMP is shown in Fig. 1 共view from top
and side兲. The wafer is held face down by a rotating carrier pressed
against the polishing pad, which is rotating as well. A down force is
applied onto the wafer. The polishing colloidal slurry, which consists
of chemical reagents, is dispensed onto the pad surface. The slurry is
distributed across the pad by the centrifugal force of the rotating pad
to wet the polishing pad and the wafer. The chemical reaction in
CMP softens the dielectric or hardens the metal surface material of
the wafer. The treated material is then removed mechanically by fine
abrasives. In balance between the chemical and mechanical actions,
the process removes material continuously.
Although quite a few papers discuss the experimental results and
the effects of the process parameters, the material removal mechanism is addressed less. Some authors consider the material removal
mechanism at the direct contact between the wafer and the pad,
while others consider that there is a thin fluid film between the wafer
and the pad.
Warnock presented the dependence of the polish rate on the wafer shape, though it is completely a phenomenological model.1 Yu
et al. presented a physical CMP model that includes the effects of
polishing pad roughness and dynamic interaction between pad and
wafer. Two new feature-scale-polishing mechanisms based on asperity theory are proposed and investigated experimentally. However, it
is not clear whether or how the asperity affects the global quality of
planarization.2 Warnock and Yu et al. both derive the reduction rate
of the step height assuming direct contact between pad and wafer
without considering the role of abrasive flow in between.
Liu assumes that the abrasive cuts the wafer under the loaded
pressure and the relative velocity between the wafer and the pad.
The model is capable of delineating the role of the mechanical properties of the slurry particles and the surface film to be polished. Liu
assumes that the deformed volume to be composed of three regions,
namely, the regions of elastic deformation, plastic deformation, and
microcutting. He also assumes the deformation volume during con-
z
E-mail: [email protected]
tact to be approximately equal to the volume of microcutting and
calculates the removal rate by the Hertzian elastic equation. The
results were also compared with the wear model of Cook,3 which is
derived from the glass polishing process.4 Cook and Liu both believe that the removal of wafer is primarily due to the direct contact
between abrasives and wafer, and the Hertzian contact is adopted.
Cook’s results did not reveal the detail of material removal rate, and
Liu did not characterize the role of flow.
Runnels presented two CMP models in 1994. One is the tribology analysis of CMP, and the other is the feature-scale fluid-based
erosion model for CMP. Runnels considers that the knowledge of
the stress distribution on the wafer surface is required in order to
explain the variation of material removal rate on a wafer during
CMP. This study analyzes the fluid film between the wafer and pad,
and demonstrates that hydroplaning is possible for standard CMP
processes. The importance of wafer curvature, slurry viscosity, and
rotation speed on the thickness of the fluid film is also demonstrated.
Although the analysis is novel, the model can only estimate the
thickness of the film between pad and wafer, and Runnels admitted
that the assumption of the spherical wafer needs modification.5 The
other physical-based feature-scale erosion model is critical to the
understanding of CMP, because it establishes the link between the
chemical effects at the particle scale and the process conditions at
the wafer scale utilizing a hydrodynamic slurry layer. Given some
assumptions, the fluid flow is modeled by the steady-state twodimensional Navier-Stokes equations for incompressible Newtonian
flow.6 Runnels later implemented Warnock’s phenomenological erosion model. The improved development allows for closer fits to the
experimental data and uniform discretization of the wafer feature.7
Although Runnels takes into account that it is the slurry flow that
erodes the pattern of wafer, his model does not account for the
effects of the abrasives in the slurry.
Tseng and Wang combined Runnels’ and Cook’s work without
physical interpretation. A model of removal rate was obtained to
account for the dependence of the removal rate on the down pressure and the rotation speed during the CMP process based on the
theory of elasticity.8
Although CMP has been widely practiced in semiconductor
manufacturing for a few years, the mechanism of material removal
remains to be explored. The macroscopic empirical Preston equation
obtained from glass polishing is often considered the major reference. The current paper presents a model based on individual abrasive material removal in a flow field. It considers that the abrasives
remove the material by a bear-and-shear process, while the slurry
flow between the wafer and the pad plays a role in transporting the
chemical means. Preston’s equation can be properly interpreted and
used with modification. The analysis and the independent experimental results are discussed in the following.
Journal of The Electrochemical Society, 148 共10兲 G581-G586 共2001兲
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Table I. Roadmap of semiconductor industry.
Year
Minimum feature size 共␮m兲
Maximum substrate diameter 共mm兲
DRAM
Numbers of metal levels
Contacts 共nm兲
MPU
Numbers of metal levels
Contacts 共nm兲
Depth of focus 共␮m兲
1999
2000
2001
2002
2003
2004
2005
0.18
200
0.18
200
0.15
300
0.13
300
0.13
300
0.13
300
0.1
300
3
200
3
185
3
170
3-4
150
4
145
4
140
4
130
6-7
230
0.7
6-7
210
0.7
7
180
0.7
7-8
160
0.6
8
145
0.6
8
130
0.6
8-9
115
0.5
Bear-and-Shear Process Model
Chemical and mechanical action.—First consider the chemical
reaction of silicon dioxide
SiO2 ⫹ 2H2O → Si共OH兲4
关1兴
There exists a rate constant in a chemical reaction. The rate constant often increases rapidly with the increasing temperature. A
rough rule valid for typical reactions in solution is that it doubles or
triples for each increase of 10°C in temperature. Arrhenius pointed
out that the constant k fits the expression9
k ⫽ Ce⫺Ea /RT
关2兴
where R is the gas constant, E a is the Arrhenius activation energy,
and C is the pre-exponential factor 共or frequency factor兲. The unit of
C is the same as k. The unit of E a is the same as RT, namely, energy
per mole 共kcal/mol or kJ/mol兲. C and E a characterize the reaction.
As mentioned above, the reaction rate is proportional to the rate
constant at the constant molecular concentration of water in CMP.
Therefore, the material removal rate in view of the chemical action
is proportional to the rate constant. Once the top surface is removed
mechanically, the silicon dioxide film exposed to the slurry will
keep on reacting chemically. These reactions are in sequence, that is,
the chemical reaction first and then the mechanical removal repeatedly. Besides, the following facts exist. The dissolution rate of glass
into water increases with compressive stress;10 therefore the hydrate
reaction rate of silica is accelerated by the pressure of the slurry
particle against the surface layer of the wafer. The compressive
stress decreases the activation energy and thus increases the rate
constant and the chemical reaction rate. If the chemical reaction
were the limiting step, the material removal rate would not increase
with the solid content in the slurry.11 Hence the mechanical removal
is considered to be the bottle neck in the material removal process
chain and is worthy of analysis. In other words, the material removal
is proportional to the mechanical removal rate.
Once the products 关 Si共OH兲4 兴 are produced by the chemical reaction, the slurry particles carried by the slurry flow remove the
products. The polishing by mechanical means in CMP is further
considered as follows.
1. The slurry film between pad and wafer supports the down
pressure, and the carrier 共i.e., the wafer兲 holds a nonparallel but
slightly inclined configuration relative to pad.
2. The slurry particles carrying kinetic energy approach and remove the molecules of the wafer surface material.
The major aspects in the process are described in the following.
Thrust bearing analogy.—The thrust load of rotary machinery is
often counterbalanced by self-acting or hydrodynamic bearings
shown in Fig. 2.12 A thrust plate attached to, or forming part of, the
rotating shaft is separated from the sector-shaped bearing pads by a
Figure 1. Schematic diagram of CMP.
Figure 2. Thrust bearing.12
Journal of The Electrochemical Society, 148 共10兲 G581-G586 共2001兲
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Figure 3. Simple pivoted pad bearing.12
Figure 4. Schematic of couette flow between two plates.13
lubricant film. The load-carrying capacity of the bearing arises entirely from the pressures generated by the geometry of the thrust
plate over the bearing pads.
The simplest form of the pivoted pad bearing provides straightline motion only and consists of a flat surface sliding over a pivoted
pad as shown in Fig. 3. If the pad is in equilibrium under a given set
of operating conditions, any change in these conditions 共such as
load, speed, or viscosity兲 will alter the pressure distribution and thus
momentarily shift the center of pressure. This event will create a
moment that causes the pad to change its inclination and shoulder
height (S h). A pivoted pad slider bearing is thus supported at a
single point so that the angle of the inclination becomes a variable
and has much better stability than a fixed inclined slider under varying conditions of operation.
This behavior is similar to the relationship between the gimbal
mechanism of the carrier and the pad during CMP, except that the
positions of the gimbal mechanism of the carrier and the pad are
reversed. The location of the shoe pivot can be found from the
equilibrium of moments acting on the shoe about the point. For all
practical purposes, the force due to friction in the pivot is ignored.
The minimum fluid film thickness in designing a pivoted pad thrust
bearing is found as
h0 ⬀
冉 冊
␮V
F
1/2
关3兴
Coutte flows in turbulence.—In Fig. 4, two infinite plates are 2h
apart, and the upper plate moves at speed V relative to the lower.
The fluid pressure p is assumed constant. The upper plate is held at
temperature T 1 and the lower plate at T 0 . These boundary conditions are independent of coordinates x or z 共called infinite plate兲;
hence it follows that u ⫽ u(y) and T ⫽ T(y). The reduced basic
equations are
Momentum equation
Energy equation
k
u ⫽ C1y ⫹ C2
⳵u
⫽0
⳵x
关4兴
d 2u
⫽0
dy 2
关5兴
冉 冊
du
d 2T
2 ⫹ ␮
dy
dy
2
⫽0
关6兴
关7兴
Using the no-slip boundary conditions, i.e., u(⫺h) ⫽ 0 and
u(⫹h) ⫽ V one obtains C 1 ⫽ V/2h and C 2 ⫽ V/2. Thus the velocity distribution is
u⫽
冉
V
y
1⫹
2
h
冊
关8兴
This is the exact steady-flow solution of the Navier-Stokes equation, called laminar flow, and has a smooth-streamline character. It is
well known that all laminar flows become unstable beyond a finite
value of a critical parameter 共Reynolds number兲. A different type of
flow then ensues on an entirely new fluctuating flow regime, called
turbulent.
The pattern changes into a randomly fluctuating flow, sketched
by Reichardt in 1956 as shown in Fig. 3, varies slightly with Reynolds number, and increases the wall shear by two orders of
magnitude.13 Therefore, the relationship between velocity and the
position from the middle of the gap to the upper wall is expressed as
u ⫽ ay n
where ␮ is the viscosity of the film, V is the velocity, and F is the
applied normal load. Although the exponential values other than 1/2
have been proposed for various applications, Bhushan and Runnels,
etc., considered it to be 1/2 in the CMP process.5,11
Continuity equation
where the continuity merely verifies the assumption that
u ⫽ u(y). Equation 5 can be integrated twice to obtain
关9兴
where n is a number larger than 100. The no-slip condition is
u ⫽ V/2 at y ⫽ h, or a ⫽ Vh ⫺n /2, hence the wall shear can be
determined
␶⫽␮
⳵u
␮nV
⫽ ␮nah n⫺1 ⫽
⳵y
2h
关10兴
Equation 10 tells us that the wall shear becomes two orders of magnitude larger. The wall shear stress produced by Couette flows is
applied to the derivation of the relative polishing velocity in CMP.
Energy balance.—When a slurry particle binds to the molecule
Si共OH兲4 of the wafer surface and removes the molecule, the shear
force must be larger than the binding force of the molecule, larger
energy than the surface energy of the generated new surface must be
transmitted.
Consider a slurry particle pulling the molecules off the wafer
surface, as shown in Fig. 5; the shear stress of slurry flow is assumed to be uniformly distributed around the particle. In order to
remove one molecule from surface, the following condition must be
satisfied14
Journal of The Electrochemical Society, 148 共10兲 G581-G586 共2001兲
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␶ ⬀ n 共 ␮V PA 兲 1/2
关15兴
where A is the area of the wafer.
Combining Eq. 13 and 15, one obtains the volumetric material
removal rate
Ṁ ⬀ f •
冉
2
nD 2p D m
12␲␥
冊
共 ␮A 兲 1/2共 PV 兲 1/2
关16兴
The removal rate in thickness per unit time is obtained by
Ḣ ⫽
Ṁ
A
关17兴
It shows the removal rate increases with the increasing polishing
speed and pressure to the power of 1/2. The experimental results are
discussed in the following.
Results and Discussion
Figure 5. Schematic of the chemical products carried by the slurry particle.
␶
冉 冊
␲D 2p
4
D m ⭓ 2␥A mN
关11兴
where ␥ is the surface energy between the surface and sublayer
molecule, D p is the diameter of the slurry particle, D m and A m are
the diameter and the cross-sectional area of the Si共OH兲4 molecule,
respectively, and N is the number of molecules to be removed.
In this case, N is considered quite large because the ratio of the
diameter of the molecule to the diameter of the abrasive is about
1/100. The left side of Eq. 11 is an approximation of the energy
transmitted to the wafer surface to move the particle for a distance
D m , which is defined as the molecule that will be sheared off once
the moving distance is larger than D m . The right side of Eq. 11 is
the surface energy of the newly generated surface, when one molecule is moved. Equation 11 can be rearranged as
N⫽
␶D 2p
The experiments are carried out on both the silicon dielectric film
and the bare silicon wafer by two polishers, IPEC372M and PeterWaltersAC1400, respectively.
IPEC 372M polisher.—The parameters of the material removal
experiments are listed in Table II. The experimental results of polishing silicon dioxide are shown in Fig. 6. The material removal rate
increases with the increasing velocity by the power of 0.65, slightly
larger than the predicted 0.5. The reason lies in that a small portion
of the wafer is polished in the direct contact mode as a result of a
not ideally conformed pad 共particularly at the wafer edge兲, where
the material removal rate is linearly proportional to the polishing
velocity.15 The empirical Preston equation based on glass polishing
also applies to this part of the process. In fact, the total material
removal rate consists of the contribution from the direct contact
mode as well as the noncontact mode, i.e.
Table II. Parameters of experiment of silicon dioxide.
Polisher
Pad type
Slurry type
关12兴
2␥D m
Slurry flow rate 共mL/min兲
Down pressure 共psi兲
Back pressure 共psi兲
Platen speed 共rpm兲a
Polishing time 共min兲
Equation 12 indicates that the number 共N兲 of removed Si共OH兲4 is
proportional to the wall shear stress. The role of the slurry particles
in the film between the pad and the wafer is to shear off the chemical products, meanwhile Si共OH兲4 is continuously produced.
Material Removal Rate
a
IPEC 372M polisher
Rodel IC1400
Cabot SS25 共1:1 dilute by DI water兲
pH 10.8
150
5, 8, 10
3
10, 20, 30, 40, 50, 60, 70, 80
1
Carrier speed is equal to platen speed.
The volumetric material removal rate 共Ṁ兲 is determined by the
volume 共ⵜ兲 of removed molecules of the chemical products 共Eq. 12兲
and the encounter frequency 共f 兲 of the slurry particles on the removed surface
Ṁ ⬀ N • ⵜ • f ⫽
3
␶D 2p ␲D m
2␥D m
6
f
关13兴
where f is a function of the solid content and the characteristics of
the turbulent flow.
One recalls that in Eq. 10 the wall shear is proportional to the
reciprocal of the film thickness. Substituting Eq. 3 into Eq. 10, one
obtains
␶⫽
冉 冊
␮nV
␮V
⬀ ␮nV
2h
F
⫺1/2
Since the force F is equal to PA, Eq. 14 can be reduced to
关14兴
Figure 6. Relationship between the removal rate of SiO2 and polishing velocity.
Journal of The Electrochemical Society, 148 共10兲 G581-G586 共2001兲
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Table III. Parameters of experiment of bare silicon.
Polisher
Pad type
Slurry type
Peter-Wolters ac1400-p polisher
Rodel-Nitta Suba 600
Rodel-Nitta Nalco 2350 共1:20 dilute by DI water兲
pH 10.3
750
1.42
15
8, 4, 0.1, ⫺4, ⫺8
⫺5.7/⫺20, ⫺5.7/⫺13, ⫺5.7/⫺5.7, ⫺5.7/1, ⫺5.7/8,
⫺16.7/⫺16.7, ⫺11.2/⫺11.2, ⫺22.2/⫺22.2, ⫺0.18/⫺0.18
10
Slurry flow rate 共mL/min兲
Down pressure 共psi兲
Sun gear speed 共rpm兲
Ring gear speed 共rpm兲
Upper/lower pad speed 共rpm兲
Polishing time 共min兲
Ḣ ⫽ ␣K 1 V 0.5 ⫹ ␤K 2 V 1.0 ⫽ K 3 V 0.65
关18兴
where ␣ and ␤ represent the fraction of the noncontact and the
contact area between wafer and pad during CMP, respectively, and
␣ ⫹ ␤ ⫽ 1. K 1 , K 2 , and K 3 are the proportionality constants between the removal rate and velocity.
As shown by Eq. 3, the minimum film thickness increases with
the velocity, thus the fraction of the contact area 共␤兲 decreases with
the increasing velocity. The effect of the polishing velocity in the
current experiment is indicated by its exponent of 0.65, which is
indeed between 0.5 and 1.0, close to 0.5.
Peter-Wolters ac 1400-P polisher.—The parameters of the material removal experiments are listed in Table III. The experimental
results of bare silicon wafer polishing are shown in Fig. 7. The
material removal rate increases with the increasing relative velocity
by the power of 0.45, slightly lower than the predicted 0.5. This is
attributed to the fact that the wafer is purposefully not held tightly
within the carrier of the polishing machine. Designed by the machine builder to avoid the danger of stress-induced cross-wafer
cracking, there is a clearance between wafer and inner rim of the
carrier. The wafer is thus allowed to rotate slightly within the carrier
due to the fluid drag, therefore the wafer velocity is slightly lower
than the carrier velocity. The actual relative velocity between wafer
and pad is less than the calculated velocity assuming full engagement. As a result, the effect of the polishing velocity contributes less
to the material removal rate than expected. The exponent of the
velocity is slightly less than 0.5 in use of this type of machine for
polishing a bare silicon wafer.
by the power between 0.44 and 0.56 for different dielectric thin
films. These results exactly fall into the prediction in the current
model of removal rate.
Case 2.—Tseng and Wang proposed a model to predict the removal
rate increasing with the velocity by the power of 0.5 on thermal
dioxide film.8,18 Their experimental results are redrawn in Fig. 9.
One notices that the polishing speed is in fact proportional to the
carrier speed.19 The power law of the velocity lies between 0.42 and
0.54, mostly 0.50. These experimental results agree well with the
current model.
Case 3.—Evans et al. published the experimental results of the removal rate of dielectrics as a function of the table speed in 1998.20
It is redrawn in Fig. 10. One also notices that the polishing speed is
proportional to the table speed. The oxide removal rate increases
with increasing velocity by the power of 0.52 and 0.55 for ceria
slurry and interlevel dielectric 共ILD兲 1300 slurry, respectively. These
results follow the prediction in the current study.
Independent experimental results.—Case 1.—Sun et al. published the experimental results of the removal rate of dielectrics as a
function of the platen rotation speed.16,17 These are redrawn in Fig.
8. The material removal rate increases with the increasing velocity
Figure 8. Relationship between the removal rate and polishing velocity in
Sun’s experiment.16
Figure 7. Relationship between the removal rate of bare silicon wafer and
polishing velocity.
Figure 9. Relationship between the removal rate and carrier speed in
Tseng’s experiment.18
G586
Journal of The Electrochemical Society, 148 共10兲 G581-G586 共2001兲
ticles and the wafer surface being attracted to each other will increase the encounter frequency and the removal rate as a result.
Conclusions
Figure 10. Relationship between the removal rate and table speed in Evans’
experiment.20
The clearance between the wafer surface and the pad is defined
by the minimum fluid film thickness determined by the parameters
including the load, the velocity, and the viscosity, similar to the
design of a thrust bearing. The abrasives in the slurry flow carry the
energy to take the chemically treated products off the wafer surface.
Based on the bear-and-shear process model, the material removal
rate increases with the polishing velocity by the power of approximately 1/2. This is verified by the polishing experiments of the
authors as well as independent researchers. The widely cited empirical Preston equation obtained from glass polishing can find an analytical ground for modification in the extended application of dielectrics CMP. The current model summarizes some representative
existing works and serves as a guide for the practical prediction of
material removal rate during CMP.
Acknowledgments
The paper was supported by the National Science Council under
contract 87-2212-E007-008.
National Tsung Hua University assisted in meeting the publication costs
of this article.
References
Figure 11. Relationship between the removal rate and polishing velocity in
Steigerwald’s experiment.17
Case 4.—In the case of metal CMP, Steigerwald et al. showed the
experimental results of the copper polishing rate vs. the polishing
velocity using Strassbaugh 6CU polisher, Suba500 pad, and
Suba550 pad. The results are redrawn in Fig. 11.17 The removal rate
increases with increasing polishing velocity by the power of 0.64
and 0.60 for Suba550 and Suba500, respectively. The results are
considered a positive reference to the prediction of the current
model.
Extended support.—The proposed model also predicts that the
removal rate increases with the encounter frequency, f, which is
determined by the solid particle content in the slurry. Bhushan’s
experimental results prove that the removal rate increases with the
increasing solid content.11 Huang’s work concluded that, when the
abrasives and the wafer surface are attracted to each other, the removal rate will be higher where the attraction is stronger.21 These
effects can be well explained by the current model. The slurry par-
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ULSI Multilevel Interconnection, T. E. Wade, Editor, p. 277, VMIC Proceedings
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20. D. R. Evans, B. D. Ulrich, and M. R. Oliver, in Chemical-Mechanical Planarization for ULSI Multilevel Interconnection, T. E. Wade, Editor, p. 347, VMIC Proceedings Series, Santa Clara, CA 共1998兲.
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