APPENDIX A EALY MODELS OF OXIDE CMP Over the past decade ad a half several process models have bee proposed to elucidate the mechaism ad material removal rate i CMP. Each model addresses a specific aspect of the process. Some of these are briefly described below to focus o the mechaisms of the plaarizatio process ad to idetify the research tasks. It is ot the objective of this review to be exhaustive; oly a few importat process models are reviewed. Brow, N.J., et al. [1981] These authors preset a model for optical polishig ad derive the Presto equatio from the elastic model. The Presto equatio may be writte as d ξ = dt k p pv (A.1) where ξ is the thickess of the layer removed, t the polishig time, p the omial pressure, v the relative velocity, ad k p a costat kow as the Presto costat. They assume that the volume of the material removed by each abrasive particle is equal to the product of the area of the side view projectio of the peetratio ad the distace traveled. For the closest particle packig, the polishig rate, dξ/dt or i, is give by 1 = (A.2) 2E i pv where E is the Youg's modulus of the material beig polished. Comparig with the Presto equatio, the Presto coefficiet, k p, is 1/2E. Cook, L.M. [1990] Chemical processes durig the glass polishig are reviewed. (Silico oxides used for IC are a form of silicate glass.) The author poits out that the primary chemical process durig polishig is the iteractio of both the glass surface ad of the polishig particle with water. I order to explai the lower polishig rates observed i experimets tha that predicted by the Hertzia idetatio wear model [Brow, et al., 1981)], 275
he proposes that the material removal durig glass polishig is a chemical process, such as dissolutio, rather tha mechaically produced particle geeratio. The presece of water is critical to glass polishig. The reactios betwee siloxae bods (Si-O-Si) ad water primarily determie the behavior of silicate glass surfaces durig polishig, as attackig the siloxae etwork will cotrol the rate of surface removal. The mass trasport is determied by the relative rates of the followig processes: (a) the rate of molecular dissolutio ad water diffusio ito the glass surface, (b) subsequet glass dissolutio uder the load imposed by the polishig particle, (c) the adsorptio rate of dissolutio products oto the surface of the polishig grai, (d) the rate of silica re-depositio back oto the glass surface, ad (e) the aqueous corrosio rate betwee particle impacts. It is suggested that both elastic ad plastic idetatio wear caot occur uless the et material trasport off the wor surface is positive. Otherwise, the material behid the travelig ideter will simply sprig back to its origial positio (elastic effects) or chage it topography (plastic effects). I this model, the relative polishig activity of compouds as well as the requisite chemical eviromet at the glass surface for optimal polishig rate are predicted. However, a great deal of experimetal work remais to characterize the effects of particle size distributio, surface area ad surface activity of the polishig compouds o the polishig rate. I ay case, eve if this model ca be applied to glass polishig, its applicability to metals ad polymers is questioable. Warock, J. [1991] This paper presets a pheomeological model for the CMP process. This model allows quatitative predictios to be made o both relative ad absolute polishig rate of arrays of features with differet sizes ad patter desity. For each surface poit i, the polishig rate, i, is defied as i ~ K i A i S i (A.3) where K i is a kietic factor, A i the acceleratig factor associated with poit i which protrudes above their eighbors, ad S i the shadig factor which describes the decrease of 276
polishig rate by the effect of eighborig poits protrudig poit i. I geeral, K i, A i,ad S i are defied such that they are greater tha or equal to 1. This model assumes that the polishig rate decrease at poit i will be compesated for by a correspodig polishig rate icrease at poit j, i.e. A i = (A.4) S i i=1 The mathematical forms for S i ad K i are chose by cosiderig the expected chages i features associated with the sprig-like properties of the rough pad. Oce the set of S i is chose, the set of A i ca be determied by the reciprocal relatio, Equatio (A.4). The polishig rates from experimetal measuremets supposedly show good agreemet with those predicted by this model. But the tribological mechaisms of plaarizatio ad polishig are left uawered. Yu, T.K., et al. [1993] A physical CMP model that icludes the effects of polishig pad roughess ad dyamic iteractio betwee pad ad wafer is preseted by these authors. Two assumptios have bee made: (i) the pad asperity is spherical at its summit, ad (ii) the variatios i asperity height, z, ad radius, β, are Gaussia distributio Φ z (µ z,σ z ) ad Φ β (µ β,σ β ), where µ x ad σ x are the mea ad stadard deviatio of x. The asperity is defied such that µ z =0. The cotact area, a, ad load, l, o each asperity are kow from Hertz's equatios. The the total cotact area, A co, ad the load, L, over the omial pad area, A, are obtaied as: A co = ηa aφ β Φ z dβ dz (A.5) d 0 L = ηa lφ β Φ z dβ dz (A.6) d 0 277
where η is the asperity desity. Measuremets give η, µ β, σ z, ad σ β. The results show that the cotact pressure p co is idepedet of the omial pressure p. This leads to the relatio: A co / A = k 1 p (A.7) where k 1 is a costat solely determied by the pad roughess ad elasticity. The authors try to coect the above result with the Presto equatio. For a upattered wafer, if the material removal rate,, is proportioal to the area wiped by the pad per uit time: = k 2 A co v (A.8) where k 2 is a experimetal costat ad v the pad velocity. By relatig Eqs. (A.7) ad (A.8), a form similar to the Presto equatio ca be obtaied. = 2 k1 k pv = k p pv (A.9) Surface chemistry ad abrasio effects are combied ito costat, k 2, ad may be de-coupled from the wafer-pad cotact problem. uels, S. ad Eyma, L.M. [1994] The authors give a wafer-scale model to demostrate that hydroplaig is possible durig the stadard CMP process. A feature-scale erosio model is employed to calculate the stresses iduced by the flowig slurry ad the polishig rate o the feature surfaces. For the wafer-scale model, they assume that both the pad ad the wafer are rigid. The pad surface is flat ad the wafer surface is smooth with a give curvature. A Newtoia fluid assumptio ad the two-dimesioal Navier-Stokes equatios are used to describe the flow field ad pressure at the wafer-pad iterface. Three parameters are itroduced to describe this fluid film: the miimum thickess of the film (t), the wafer agle of attack (θ ), ad the radius of curvature of the wafer ( w ). Oce h is foud for the give pad velocity ad wafer curvature, it ca be used i a feature-scale model with the 278
real feature shape as the flow boudary to calculate the stress distributio o the feature surface. To estimate the polishig rate, a erosio model of the followig form is assumed: = f [ σ ( t), σ ( t)] (A.10) t where is the erosio rate i the ormal directio at a poit o the surface, f the law empirically relatig the chemistry ad mechaics to erosio, ad σ t ad σ are ormal ad shear stress o the feature surface. The authors estimate i the form: = Cσ (A.11) 2 t They suggest this form because the approximatio σ t µv h (A.12) is similar to the tribological behavior for slider bearigs: µ v h (A.13) pa (where A is the area of wafer, ad p the average pressure). By usig Eqs. (A.11), (A.12), ad (A.13), becomes (µ A) pv (A.14) which is idetical to the Presto equatio. Compared with the experimetal results, the predicted profiles show good correlatio with the shape of the erodig features. The authors claim that the discrepacy betwee the experimetal data ad the predicted curve is due to measuremet iaccuracy, pad feature size, pad deformatio ad to two-dimesioal modelig. 279
I additio, the experimetal data were obtaied for a poorly characterized pad-wafer iterface, ad thus the validatio of the hydroplaig model is questioable. Nomeclature A = wafer area (m 2 ) A co = total cotact area of the pad (m 2 ) A i = acceleratig factor at poit i a = cotact area of a pad asperity (m 2 ) C = proportioality costat E = Youg s modulus (N/m 2 ) h = slurry film thickess (m) K i = kietic factor at poit i k p = Presto costat (m 2 /N) L = load o the wafer (N) l = load carried by a pad asperity (N) = umber of poit cosidered o the wafer surface p = ormal pressure o wafer (N/m 2 ) t = polishig duratio (s) = material removal rate (m/s) S i = shadig factor at poit i v, v = magitude of the relative velocity (m/s) x, y = Cartesia coordiates z = asperity height (m) β = radius of the asperity (m) Φ = ormalized Gaussia distributio fuctio η = asperity desity µ = dyamic viscosity of the slurry (Pa s) σ, σ t = ormal ad shear stresses o the feature surface (N/m 2 ) ξ = thickess of the material removed o wafer surface (m) 280
efereces Brow, N.J., Baker, P.C., ad Maey,.T., 1981, Optical Polishig of Metals, Proc. SPIE, Vol. 306, pp. 42-57. Cook, L.M., 1990, Chemical Processes i Glass Polishig, J. No-Crystallie Solids, Vol. 120, pp. 152-171. Presto, F.W., 1927, The Theory ad Desig of Plate Glass Polishig Machies, J. Soc Glass Techology, Vol. 11, pp. 214-256. uels, S.., 1994, "Feature-Scale Fluid-Based Erosio Modelig for Chemical- Mechaical Polishig," J. Electrochem. Soc., Vol. 141, pp. 1900-1904. uel, S.., ad Eyma, L.M., 1994, "Tribology Aalysis of Chemical-Mechaical Polishig," J. Electrochem. Soc., Vol. 141, pp. 1698-1701. Warock, J., 1991, A Two-Dimesioal Process Model for Chemimechaical Polishig Plaarizatio, J. Electrochem. Soc., Vol. 138, pp. 2398-2402. Yu, T.-K., Yu, C.C., ad Orlowski, M., 1993, "A Statistical Polishig Pad Model for Chemical-Mechaical Polishig," Proc. 1993 IEEE It. Electro Dev. Mfg., pp. 865-868. 281