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`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995
`
`Modeling of Chemical-Mechanical Polishing:
`A Review
`Gerd Nanz and Lawrence E. Camilletti
`
`Abstract-This paper gives a survey of the status of today’s
`modeling of chemical-mechanical polishing (CMP). Most existing
`models describe specific aspects of CMP, such as the Row of the
`slurry or the bending of the polishing pad. However, as yet no
`model describes the entire available process. This paper critically
`reviews existing models with respect to generality. In particular,
`the different assumptions of the models are investigated. Further-
`more, the models are compared and the controversial treatment
`of physical effects is discussed.
`
`I. INTRODUCTION
`NOVEL technique for planarization is of growing in-
`
`A terest, since conventional planarizing methods such as
`
`flowing oxide layers do not give the required global planarity
`for advanced processes. Multilevel interconnects and the use
`of 3-D packaging require sophisticated methods to planarize
`the surfaces of wafers for subsequent device processing. For
`five or more layers of a logic device at least one layer should
`be perfectly planar [14]. Lack of planarity may lead to severe
`problems for lithography (insufficient focus depth) and dry
`etching in sub 0.5-pm IC processes [7].
`Several methods are known to achieve a higher level of
`planarization: (chemical-mechanical) polishing, laser reflow,
`coating with spin-on glasses, polymers, and resists, thermally
`reflowing materials, dielectric deposition [ 141, and flowable
`oxides.
`In the following sections the present status of modeling
`CMP available from the literature is discussed. First, the basic
`ideas of CMP are described. Then an overview of existing
`models for CMP is given, including a discussion of their
`capabilities. Finally, the models are compared pointing out
`some controversial approaches to describe physical effects.
`
`A. CMP
`A schematic of a CMP machine is shown in Fig. 1 (view
`from top and cross section). A CMP machine uses orbital,
`circular and lapping motions. The wafer is held on a rotating
`carrier (holder) while the face being polished is pressed against
`a resilient polishing pad attached to a rotating platen disk.
`For oxide or silicon polishing, an alkaline slurry of colloidal
`silica (a suspension of Si02 particles) is used as the chemical
`
`Manuscript received December 10, 1994; revised March 23, 1995.
`G. Nanz is with Digital Equipment Corporation, Favoritenstrasse 7 (CEC
`Vienna), A-1040 Vienna, Austria.
`L. E. Camilletti is with Digital Equipment Corporation, Hudson, MA 01749
`USA.
`IEEE Log Number 9414529.
`
`Fig. 1. Schematic of chemical-mechanical polishing technique.
`
`abrasive. The size of the particles varies in literature between
`100 8, [19] and 3 pm [2]. According to [15], the size of the
`particles is between 600 8, and 800 A forming agglomerates of
`a size of 2500 8, in diameter. The slurry is carried to the wafer
`by the porosity of the polishing pad. This slurry chemically
`attacks the wafer surface, converting the silicon top layer to
`a hydroxilated form (with the OH- radical) which is more
`easily removed by the mechanical abrasive. The details of the
`formation of this top layer are not yet well understood [SI,
`(231.
`Gross mechanical damage of the surface is prevented by
`the fact that the colloidal silica particles in the slurry are not
`harder than the oxide being removed [8]. Otherwise the quality
`of the surface planarity would be limited by the diameter of
`the silica particles.
`CMP needs fewer steps compared to depositiodetchback
`[6]. Furthermore, CMP uses nontoxic substances, has a good
`removal selectivity, and a good rate control. Typical values
`of some essential parameters in the CMP process are given
`in Table I.
`Another advantage of CMP lies in the global planarization.
`Since the sizes of flat areas on a chip become smaller the
`quality of local planarization (in a global sense) becomes
`worse. Additional difficulties may arise for the filling of
`small holes. CMP reduces defect density, according to [15].
`Shortddefects due to residual metal can be significantly re-
`duced, as reported in [4]. However, real-life processes, if not
`controlled properly, will add a significant number of defects.
`For instance, as the pressure (down force) is increased, the
`
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`NANZ AND CAMILLETTI: MODELING OF CHEMICAL-MECHANICAL POLISHING
`
`TABLE I
`TYPICAL VALUES FOR CMP PARAMETERS
`Reference I
`
`value
`
`Quantitr
`
`PolihinqPad
`n
`
`~
`
`383
`
`&mcnnl of Si02 (thermal)
`R e m d of Si02 (LPCVD)
`Polishing time
`Pressure (pad/wafff)
`Velocity of pad
`velacity of wafer
`
`risks also increase. There is a greater propensity for dislocation
`and defect generation and the possibility of diffusion or
`penetration of the slurry contaminants below the surface exists.
`CMP is applied for several types of structures: Bare silicon
`before any processing starts, metallization, the intermetal
`layer dielectric (ILD), and process silicon with dielectrics
`and metals. In the latter two cases the selective removal of
`materials such as aluminum, tungsten, and Si02 is necessary.
`
`11. MODELING AND SIhlLJLATION
`The main feature of CMP, namely the removal of
`material, is described by Preston’s equation:
`N ds
`d T
`-=K.-.-
`A
`d t
`d t
`where T denotes the thickness of the wafer, N / A denotes
`the pressure caused by the normal force N on the area A,
`s is the total distance traveled by the wafer, and t denotes
`the elapsed time. This means that the removal of the wafer
`material is proportional to the pressure and the velocity of the
`rotation. Any physical considerations are put into Preston’s
`constant K, which often is considered a proportionality con-
`stant (independent of pressure and velocity), but may also
`contain advanced physics and include the effects caused by
`the chemical reactions. However, the removal rate tends to
`dominate in real-life situations.
`Some models recently published are described below and
`discussed with respect to their limitations and their range of
`applicability.
`
`A. Model by Sivaram
`Sivaram et al. [16], [17] have developed a model which
`describes the removal according to (1) taking also into account
`the bending of the polishing pad.
`As an important physical effect CMP has to deal with
`bending [4]. A schematic of the pad and the wafer is drawn
`in Fig. 2. Due to the pressure on top of the polishing pad, the
`pad behaves locally like a beam which is supported by the
`blocks of material 2. Assuming that material 2 is inelastic, the
`deflection of the pad can be easily calculated. The solution w
`of the differential equation (2) describes the deflection of a
`beam with length 1
`
`d4 v
`E . I. - = W(.)
`dx4
`
`Fig. 2. Pad bending during CMP.
`
`(3)
`
`where E and I are Young’s modulus (elasticity modulus) and
`the moment of inertia, respectively, and w(z) denotes the load
`on the beam.
`For a uniform load W(Z) = WO and for the boundary
`conditions ~ ( z ) = 0 and M ( z ) = 0 for z E {O,Z), where
`M denotes the momentum (second derivative of w(z)), one
`obtains (3) as the solution of (2)
`- (z4 - 2 . 1 . z3 + i3. .).
`WO
`w(x) =
`2 4 . E . I
`Due to the deflection of the pad it may occur that material
`1 is affected by the pad. Therefore (3) is also a measure for
`the planarity which can be achieved.
`Discussion: The treatment of the deflection of the polishing
`pad is important since the quality of the planarization is
`affected by the bending. However, Sivaram ef al. consider only
`the effects of the bending between two neighboring peaks on
`the wafer. For a rigorous treatment of the deflection of the
`pad a wider range of the wafer must be considered requiring
`the calculation of the deflection of a multiply-supported beam.
`Additionally, the model assumes that the product E
`I for
`the polishing pad is the same or higher than for the platen
`(holding the polishing pad). This is usually not true. If the
`polishing pad is softer than the platen, the pad is compressed
`while the platen stays nearly flat. This effect is not included
`in the approach for the calculation of the beam bending.
`This model deals only with two-dimensional cross sections
`of the structure and neglects the slurry flow-thus
`the appli-
`cation is limited. For modeling purposes of the entire CMP
`process, a completely three-dimensional simulation would be
`necessary.
`
`1
`
`B. Model by Burke
`A model dealing with the polishing rate depending on
`the degree of nonplanarity has been proposed by Burke [l].
`The model has two stages: An analytical model which is
`based on the closed solution of a simple ordinary differential
`equation and a more complex model which iteratively adapts
`the polishing rate to the actual nonplanarity.
`DO denotes the percent polish rate of areas which are low
`compared to the polishing rate for the blanket-wafer (“down”
`polishing rate). This means that for small DO planarization
`is good (ideal for 0), and planarization stops for DO close
`to 1, since then the lower areas are polished in the same
`way as the higher areas (“up”). SO denotes the initial step
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`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995
`
`height associated with DO. Burke observes that for DO > 0.3
`the ‘down’ polishing rate is linear and for DO < 0.3 it is
`logarithmic.
`For DO < 0.3, and a constant polishing rate U for ‘up’ areas
`the polishing rate D for ‘down’ areas, is given by (4), where
`S is the actual step height, i.e., the amount of nonplanarity
`(
`D = l - ( l - D o ) - - S ) .U*
`SO
`The governing differential (5) looks similar to Preston’s
`equation (l), but it is different
`1-Do
`d S
`-
`SO
`dt
`where U has the dimension of a velocity similar to the term
`d s / d t on the right side of (1). However, U is a removal rate
`and not the relative velocity between the pad and the wafer.
`Burke integrates (5) and obtains the solution (6)
`
`s . u
`
`(4)
`
`S large
`Fig. 3. Arbitrary surface with ‘effects on the factors of the polishing rate.
`
`There is also a direct reciprocity between the Ai and Si at all
`n points, which is given by (8). It includes the assumption that
`the polishing rate is linearly proportional to the pressure
`
`si = exp (2)
`
`(9)
`
`Si is given by (9), where zo is a scaling factor for the vertical
`length scale and
`describes how much the surrounding
`topography protrudes above point i. The exponential function
`has been chosen to account for the rough nature of the pad.
`However, Warnock agrees that an improved formulation could
`be found, even though he claims that the model is not very
`sensitive with respect to the formulation
`
`z(r, 0) . W ( T ) dr dt? (10)
`
`W’(r,) dB
`
`(11)
`
`aZ, = & . /o
`.1 .(.,,e).
`1
`2.T
`aZ, M -
`2 . T
`
`/o
`
`Burke also proposes a more advanced differential model.
`The surface of the wafer is represented in the computer
`program as a two-dimensional topography given by the user.
`Then the polishing rate is adapted at each point according
`to the neighboring points similarly to the analytical model.
`Burke claims that corner rounding is predicted with sufficient
`accuracy.
`Discussion: This model takes into account the type of the
`nonplanarity of the wafer surface and adjusts the polishing
`rate accordingly. The model does not deal with pad bending,
`asperities or the fluid flow. Furthermore, the model is empirical
`and does not address the dependence of the polishing rate on
`the pressure. Even though it takes into account the entire wafer
`(and not only two-dimensional cross sections) it covers only
`a small part of the entire CMP process.
`
`C. Model by Wamock
`A model which allows the quantitative analysis of the abso-
`lute and the relative polish rate for different sizes and pattern
`factors has been presented by Warnock [22]. It is a microscopic
`mathematical model which is completely phenomenological.
`Due to the nonlocal nature of CMP, different length scales are
`necessary which are determined by the flexibility and hardness
`of the polishing pad. The horizontal length scale is determined
`by the pad deformation and the vertical length scale by the pad
`roughness. In this model, the polishing rate Pi at each point
`i of the wafer is given by (7)
`
`-
`Azi is obtained by integration over the surrounding topogra-
`phy of point i. In (lo), .(.,e)
`is the vertical height at the
`coordinates r and t? with respect to point i, and W ( r ) is a
`weighting function describing the horizontal length scale over
`which the pad deforms. Under reasonable assumptions (IO)
`can be simplified to (ll), where rm is the value of T which
`maximizes W ( T ) (as a function of e). W’(T,) as chosen by
`Warnock is given in (12). Thus, the deformation length scale
`is TO.
`The A; can be determined in an iterative process from the
`Si. Warnock claims that uniqueness is given and that a solution
`is obtained ‘after a large enough number of iterations.’
`The Ki are determined by calculating an effective vertical
`component of the horizontal polish rate. Thus, K; = 1 + KO .
`tan@,, where KO is a model parameter, and a; is the local
`(7)
`angle between the horizontal and the surface.
`where Ki is the kinetic factor (horizontal component), Ai is
`Discussion: This model gives a reasonable approach for
`the accelerating factor (higher points on the wafer), and Si is
`defining the dependence of the polish rate on the wafer shape,
`even though it is completely phenomenological. In particular
`the shading factor (lower points on the wafer). A schematic
`surface demonstrating the meaning of the coefficients Ki , Ai,
`this model takes into account all geometrical cases; therefore,
`and Si is shown in Fig. 3. In lower regions Si is large, thus
`it might be general enough to cover a part of the simulation
`reducing Pi; in higher regions A, is large, thus increasing Pi.
`of the entire CMP process. Furthermore, it is not limited to
`Ki may be large on a sloped surface depending on the slope.
`the two-dimensional case.
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`385
`
`D. Model by Yu
`The model by Yu et al. [23] deals with the dependence
`of the removal rate on the asperity of the polishing pad.
`From measurements asperities can be observed all over the
`polishing pad. The surface height variation is reported to be
`more than 100 pm on a 200 pm x 200 pm polishing pad. The
`asperities are assumed to be spherical at the summit and that
`the variations in height and radius are Gaussian-distributed in
`the model. Then Preston's constant K in (1) can be modified
`according to the new model, splitting it into three parts: One
`is a constant only determined by the pad roughness and the
`elasticity, one is a factor of surface chemistry and abrasion
`effects, and one is related to the contact area thus accounting
`for the asperity [ 151, i.e., the contact properties of the asperities
`depend on the width of a trench.
`The behavior of the pad is assumed to be viscoelastic in the
`model, which means that the deformation of the pad can be
`taken into account.
`Discussion: The model gives an approach how to deal with
`the asperities of the pad. However, it is not clear whether or
`how the asperities affect the global quality of planarization. A
`global planarization quantity of 200 8, over a distance of 0.5
`cm has been reported [20]. This makes further investigations
`necessary as to how this approach fits into a general CMP
`simulation model. Additionally, the validity of the assumptions
`for the radius and the height of the asperities seem to need
`further research.
`The reported variation in height of the polishing pad cer-
`tainly overshadows the nonplanarities of the wafer. It is even
`comparable with the thickness of the fluid layer between the
`pad and the wafer. This question is neither treated by the model
`nor discussed.
`
`E. Models by Runnels
`Runnels et al. [ 1014 121 propose several models accounting
`for the stress in the polishing pad and the fluid flow as well
`as the removal of material by erosion.
`I ) Flow ofthe Slurry: Runnels et al. propose a model ac-
`counting for the fluid flow between the wafer and the pad [ 111.
`A wafer of radius 10 cm and spherical curvature rotates
`about its axis of symmetry, which is approximately 30 cm
`from the pad's rotational axis. The wafer glides at an angle of
`attack 8 upon the slurry film whose thickness is denoted by
`h. The wafer carrier is mounted on a gimbal mechanism to
`prevent the wafer snagging on the pad. The model focuses on
`the flow of the slurry between the wafer and the pad. The flow
`simulation is embedded in an iterative scheme for determining
`h and 8. For the flow simulation, it is assumed that the wafer
`and the pad are rigid and smooth. Therefore, both the pad and
`the wafer can be described by boundary conditions for the flow
`of the slurry. Even though the slurry contains particles (with
`a magnitude of approximately 1 pm) the flow of the slurry is
`assumed to be Newtonian with a constant viscosity. Thus, the
`flow is given by the steady-state incompressible Navier-Stokes
`equations in three space dimensions (1 3)
`
`where p denotes the density, p is the dynamic viscosity, p is the
`pressure, and U' is the vector-valued function of the velocity
`at any point in the flow.
`The simulation domain consists of a thin disk bounded by
`the surfaces of the wafer and the pad, respectively, and a ring,
`representing the flow around the outside of the pad. For the
`wafer, the pad, and the sidewalls of the ring no-slip boundary
`conditions are applied-thus
`giving the fluid the velocity of
`the wafer, the pad, and the flow in the ring. For the remaining
`surfaces (mainly of the ring) stress-free boundary conditions
`are applied, thus allowing the fluid to enter and leave freely.
`Two conditions have to be fulfilled for the determination
`of the fluid layer. First the fluid layer must support the wafer
`carrier, including the applied load during polishing. The force
`F on the wafer surface from the fluid flow is given by (14),
`where o is the stress tensor related to the flow field by (15).
`Sij denotes the Kronecker symbol
`
`F = J
`U - f i d A
`wafer surface
`
`(14)
`
`The second condition is that the fluid is stable. This requires
`the moment of the force caused by the fluid film to have
`components which vanish in the plane perpendicular to the
`carrier's axis of rotation. The moment M f about the gimbal
`point from the fluid flow is given by (16), where
`denotes
`the distance of each point on the wafer measured from the
`gimbal point
`
`Mf =/
`
`(16)
`
`gg x a . n ' d A .
`wafer surface
`The thickness of the fluid layer h and the angle of attack
`0 are calculated by an iterative scheme satisfying that M f
`vanishes.
`Discussion: This model analyzes fluid flow. It turns out
`that stringent assumptions are necessary to put this complex
`problem into a mathematical model. The wafer surface is
`assumed to be spherical with a large radius in the model,
`which means that all questions about the polishing mechanism
`and the structure of the wafer surface are neglected. Further-
`more, Runnels admits that the validity of their assumptions
`is not completely clear and that their model can give only a
`qualitative estimation of the fluid layer thickness.
`The main result of this model is the thickness of the fluid
`layer between the wafer and pad. The simulation must be made
`for the entire wafer and be posed as a fully three-dimensional
`problem in order to determine this thickness correctly. The
`solution of the Navier-Stokes equations can be used for the
`modeling of the material removal rate.
`To the authors' knowledge this is the first work dealing with
`the slurry flow, which is an essential factor in modeling CMP.
`2) Removal by Erosion: Runnels proposes a model which
`calculates the removal as a consequence of erosion due to the
`slurry flow [lo].
`This two-dimensional feature scale model, which takes
`into account cross sections of the CMP machine, is mainly
`phenomenological.
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`
`ux=y.uy=o 9
`I
`
`duny now
`
`Y
`
`inflow
`
`wafer
`
`nonplanarity of
`-...._
`wafer
`.....
`
`a...
`
`oumow
`
`...-
`
`....e-- ...-
`
`.....
`uz=o.uy=o
`
`*e..
`
`....a-
`
`Fig. 4. Geometry of erosion simulation domain.
`
`The fluid flow around a nonplanarity on the wafer (see
`Fig. 4) is given by the solution of the two-dimensional
`Navier-Stokes equations (13). The boundary conditions are
`also given in Fig. 4, where uz and uy denote the unknown
`velocities in 2- and y-direction, and U is the velocity of the
`pad. Again, the surface stress tensor is calculated by (15).
`The erosion rate V, in the normal direction is given by (17)
`as a function f of the time-dependent tangential and normal
`stresses Ut and U,
`
`Equation (17) is integrated in time and gives the shape of
`the wafer surface for any time.
`Runnels relates V, to Preston’s equation by an approxima-
`tion following heuristic arguments with formulas (1 8)-(22).
`The constant C represents microfracturing and chemical as-
`pects. A is the area of the wafer and P is the bulk pressure.
`An estimation of V, is given in (18). Equation (19) together
`with the approximation for slider bearings (20) leads to the
`formulation (21) which can be rewritten by (22), which is the
`same as Preston’s equation (1)
`
`Runnels also includes the stress dependence of the chemical
`reaction and an approximation for the rotation of the pad. This
`increases the complexity of the expression for V, and is still
`a matter of research and further modeling work.
`Discussion: The model is mainly based on the solution of
`the two-dimensional Navier-Stokes equations thus simulating
`the flow of the slurry in a cross section. The simulation setup
`is significantly more reasonable than in Runnels’ previous
`
`model. Since the Reynold‘s number seems to be small for
`this type of application no major numerical problems should
`be encountered. Therefore, the model is a good base for further
`extensions such as accounting for the pad deflection. Runnels
`leaves several questions about the physical modeling of the
`erosion law open and uses completely heuristic arguments to
`fill gaps in his model. The influence of the fluid flow on the
`erosion rate is only given by the stress tensor.
`The question of the boundary conditions for the Navier-
`Stokes equations along the wafer surface (vanishing velocities
`in beth coordinate directions) should be reconsidered.
`3) Deformation of Pad: Runnels et al. propose a model
`which accounts for the deflection of the polishing pad at the
`wafer edges and the resulting stress distribution [12].
`Preston’s equation (1) is reformulated leading to (23), where
`R is the removal rate
`
`R = K . P . 1 1 ~ 1 1 .
`(23)
`Runnels er al. claim that llwll should be replaced by IIITII,
`where U is the vector-valued shearing stress acting in the plane
`of the wafer surface while P can be interpreted as a stress
`perpendicular to the plane of the wafer.
`Several assumptions are made: 1) The transfer of stresses
`between wafer and pad are neglected, 2) the pad is assumed to
`be elastic, even though it is known to be viscoelastic-Runnels
`et al. claim that for the typical speeds of rotation an elastic
`representation is suitable, 3) the slurry flow is neglected.
`Therefore, two extremes are considered: Once the pad and
`the wafer adhere to each other, and once they are allowed to
`slide freely without stresses.
`The boundary of a planar wafer is considered for the
`analysis. The governing equations for the axisymmetric case
`(in polar coordinates) are given by (24), where U, and U,
`are the normal stresses in the radial and vertical directions,
`respectively, and T,., is the shear stress
`aTr,
`aa,
`-+-
`- U6
`aT
`az
`+-=O
`T
`a ~ z
` Trz
`aTrz
`ar
`-+-+--0.
`dz
`T
`The deflection of the pad is then related to the stress through
`Hooke’s law and the kinematic definitions of strain [31, [21].
`
`(24)
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`NANZ AND CAMILLE7TI: MODELING OF CHEMICAL-MECHANICAL POLISHING
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`From (23), where IlwII is replaced by the vector 1 1 ~ 1 1 of
`the stresses, it becomes clear that the removal rate is strongly
`affected by stresses. Runnels et al. have performed an analysis
`of the stress distribution close to the edge of the wafer and the
`curvature of the pad, and thus obtained a qualitative idea of
`the influence of these mechanical effects and compared the
`results with experimental data.
`Discussion: The replacement of
`IlwII by //all does not
`provide anything new. It just takes the effects of the stresses
`out of K and places them explicitly in the formulation. But
`the new formulation clearly shows that the stresses strongly
`influence the removal rate. The model seems to be well suited
`to perform a qualitative analysis of the influence of stresses
`on the removal rate. However, it is only applicable to the
`axisymmetric case of a planar wafer and does not give any
`information about the interior of a wafer. Furthermore, the
`different elastic behavior of the platen and the pad is neglected.
`Since the slurry flow and the velocity are neglected-in this
`model it is only applicable for the calculation of extreme cases
`such as adhesion and free slip between pad and wafer.
`
`F. Model by Cook
`Cook's model [2] is applicable to CMP for bare silicon
`wafers. However, many ideas are also applicable to the more
`general case of CMP for ILD.
`Cook starts from Preston's equation (1). The slurry is
`assumed to be a viscous Newtonian fluid with a viscosity of
`around 10gP with particles in it. The mechanical part of the
`interaction between polishing particles and the wafer surface
`can be described by a model with a spherical particle of
`diameter a, that penetrates the surface with force F, under the
`uniform load N . For a standard Hertzian penetration Preston's
`constant in (1) becomes (2 . E)-', where E denotes Young's
`modulus (modulus of elasticity). The surface roughness is the
`penetration depth R, given by (25), where k is the particle
`concentration (unity for a fully-filled closed packing) and
`P = N / A the pressure
`
`TABLE II
`PHYSICAL EFFECTS IN THE PRESENT MODELS
`
`Physical effect
`
`lark
`
`Pad bending 1D
`Pad deformation
`Asperity of pad
`Stress in pad
`Removal (erosion)
`Removal (abrasion)
`Slurry flow 2D
`Slurry flow 3D
`Nonplanar de^
`Chemical reaction
`
`r,=
`
`strength os of network bonds
`gt =0.5. (1 - 2 . U ) 'PO
`( 1 - 2 I 1-/2))
`( 3
`1 / 3
`- . N . - . - (27)
`E
`E'
`4
`2
`E
`Ffr = K ' r," . fb . -
`10'
`An extensive study of the chemical part is presented in [2].
`However, a discussion of it is considered out of the scope of
`this review.
`Discussion: Cook's model is the most elaborate modeling
`work for polishing. In particular, it deals with the mechanics
`of the polishing particles and with the chemical reactions. It
`covers almost all interesting topics and the method is clearly
`explained by an example (Si02 polished by SiOz-particles).
`This model is based on a different feature length compared
`to the models discussed previously. While Runnels et aZ.
`consider slurry as a fluid, Cook also deals with the particles
`and the particle size in this fluid. Therefore, additional work
`is necessary to combine Cook's model with other models in
`order to get a sufficiently general model for the entire CMP
`(25) process.
`
`Impingement of particles canied in the turbulent liquid
`leads to Hertzian penetration of the surface, converting ki-
`netic energy into strain energy. Local bonding during contact
`leads to weakening of binding forces at the surface, which
`allows atomic removal to occur without introducing lattice
`dislocations. For the case of a static Hertzian spherical indenter
`the maximum tensile stress at (26) and the friction force
`F f r (28) can be determined and used for the calculation of
`Preston's constant, where U denotes Poisson's constant for
`the polished material, PO = N / ( K . r,") is the mean contact
`pressure over a circular zone of radius T, (27), and U' and E'
`are Poisson's ratio and Young's modulus (elasticity modulus)
`of the polishing particle, respectively. K . r," is the contact
`area between the polishing particle and the polished material
`surface, fb is the fraction of the contact area in which bonding
`occurs, and E/10 is an estimate for the theoretical fracture
`
`111. COMPARISON OF THE MODELS
`A variety of effects is taken into account or neglected in
`the existing models. Some of the physical effects are listed
`in Table 11. Listing a reference does not mean that the model
`treats the problem rigorously. In particular, hardly any rigorous
`formulation of the dependence of Preston's constant K on the
`various effects can be found in literature.
`It becomes also clear from Table I1 that physical effects
`are treated in a controversial way by the different authors.
`For example, the treatment of the deformation of the pad has
`two different approaches: Elastic or viscoelastic deformation.
`While Runnels claims that an elastic formulation to describe
`the pad bending is sufficient for the typical speed of rotation,
`Yu explains some observations with the viscoelasticity of the
`pad. To get a more accurate description of the real behavior of
`the polishing pad and the platen a two-dimensional approach
`
`t
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`
`consisting of the biharmonic equation [3J (two-dimensional
`formulation of (2)) for the platen and the corresponding
`equation for the viscoelastic deformation of the pad would
`be necessary. According to [18] the simple beam bending
`model can give an idea of important parameters that affect
`dishing. Nevertheless for the simulation of the deformation of
`the wafer and the pad a viscoelastic model (as proposed by Yu
`[23]) seems to be superior and more accurate than the elastic
`approach, even though many questions are still open.
`The nonplanarities of the wafer are treated in a very
`empirical way. Both Burke [l] and Warnock [22] use the full
`information about the nonplanarities of the entire wafer while
`the other authors (Sivaram [16] and Runnels [ll]) consider
`nonplanarities only locally. For a phenomenological descrip-
`tion of the effects caused by the nonplanarities Warnock‘s
`model see [22], which seems to be the most reasonable and
`probably needs some extensions.
`Another major contradiction can be seen in the explanation
`of the material removal: Erosion or abrasion. In most models
`abrasion is assumed to take place and the chemical effects are
`somehow included in Preston’s constant I(. It is also possible
`that both erosion and abrasion take place. Erosion of Si02
`has been reported and analyzed in combination with copper
`dishing [18]. Obviously the effects leading to the material
`removal are not yet completely understood.
`The slurry flow is only treated by Runnels. None of his
`proposed models are satisfying in their present form, since
`too many details are not yet fully understood. It is not clear
`whether the reason for neglecting the slurry flow by the other
`authors lies in the computational complexity of solving the
`three-dimensional Navier-Stokes equations or in the lack of
`information about the slurry layer even though the slurry flow
`is considered of critical importance [15]. Furthermore, the
`effects caused by particles contained in the slurry are not clear
`and need additional investigation.
`
`IV. CONCLUSION
`This review deals with the present status of modeling CMP.
`It tums out that only a few models exist, and that almost none
`of them are applicable in a general sense. The most important
`topics for modeling of CMP include the removal rate, which
`consists of a mechanical as well as a chemical contribution, the
`bending and the stresses of the pad, which strongly influence
`the quality of the planarization, and the transport and the
`flow of the slurry. Whether the primary removal mechanism
`is mechanical or chemical will also depend on the layer
`being removed and must be modeled accordingly. From the
`modeling point of view the flow of the slurry seems to be the
`most difficult question since the behavior of the fluid and the
`mechanical effects are not completely understood.
`CMP is in a very early stage of modeling since even the
`physical effects are not yet completely clear. Programs wh