A statistical model for material removal prediction in polishing

This paper develops analytically a statistical model for predicting the material removal in mechanical polishing of material surfaces (MS). The model was based on the statistical theory and the abrasive–MS contact mechanisms. The pad-MS and pad-abrasive-MS interactions in polishing were characterised by contact mechanics. Two types of active abrasive particles in the polishing system were considered, i.e., Type I – the particles that can slide and rotate between the pad and MS, and Type II – those embedded in the pad without a rigid body motion. Accordingly, the material removal is considered to be the sum of the contributions from the two types of abrasive interactions. It was found that the mechanical properties and microstructure of the polishing pad and polishing conditions have a significant effect on the material removal rate, such as the porosity and elastic modulus of the pad, polishing pressure, volume brasion wear aterial removal tatistical theory hemo-mechanical polishing concentration of abrasives, particle size, pad asperity radius and pad roughness. It was also found that different types of active particles contribute quite differently to the material removal. When the mean particle radius is small, the material removal is mainly due to the Type II particles, but when the mean particle radius becomes large, the Type I particles remove more materials. The model predictions are well aligned with experimental results available in the literature and can be used for the material removal prediction in chemo-mechanical polishing if a proper treatment of the chemical effect is introduced. . Introduction Mechanical polishing using abrasive slurry is a key finishing proess in industry for producing a material surface (MS) of low surface oughness. Chemo-mechanical polishing (CMP) is a good examle that is based on mechanical polishing, but introduces chemical eaction by adding chemicals to abrasive slurry to promote the aterial removal rate. The technique has been widely used in polshing glass, silicon and ceramic surfaces as well as in planarizing urfaces of inter-level dielectrics or inter-metal dielectrics durng integrated circuit fabrication [1–5]. In a typical CMP process, rotating material surface (MS) attached to a carrier is pressed gainst a rotating polishing pad in the presence of liquid slurry hich contains abrasive particles with chemicals. The material emoval of the process is generally due to the combination of eroion and abrasion. It is known that many variables such as applied ormal force, relative velocity of the MS to the pad, pad properties elastic modulus, hardness, etc.) and slurry characteristics, have rofound influences on the material removal mechanically. The Please cite this article in press as: X.L. Jin, L.C. Zhang, A statistical m doi:10.1016/j.wear.2011.08.028 undamental mechanisms of the material removal in the process re very complicated and have not been well understood, because ∗ Corresponding author. Tel.: +61 2 93856078; fax: +61 2 9385 7316. E-mail address: [email protected] (L.C. Zhang). 043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved. oi:10.1016/j.wear.2011.08.028 © 2011 Elsevier B.V. All rights reserved. of the statistical nature of the surfaces in contact such as the random distributions of the surface asperities and abrasive particles. In the literature, the modelling of the material removal in CMP processes can be generally classified into two categories. One was based on fluid hydrodynamics. For example, Runels et al. [6,7] obtained the wear rate by numerically solving the Navier–Stockes equation. Sumdrarajian et al. [8] studied the removal rate based on the lubrication and mass transport models, in which slurry erosion was considered a main mechanism. The other group of the modelling methods was based on the theory of contact mechanics. Since this approach is more plausible to describe experimental observations, it has been widely accepted and investigated. Larsenbasse and Liang [9] concluded that material removal in CMP is due to particle abrasion. There are also some similar investigations. Luo and Dornfeld [10] investigated the abrasion mechanism in solid–solid mode of the CMP process based on a number of assumptions: plastic wafer–abrasive and pad–abrasive contacts, normal distribution of abrasive size and periodic roughness of pad surface. They extended the model as a function of the abrasive weight concentration [11] and then further used it to explain the effects of abrasive size distribution [12]. However, these models were based on the assumption of periodic roughness of pad surface. From the odel for material removal prediction in polishing. Wear (2011), perspective of pad modelling, abrasive behaviour and distribution effects of abrasive, Wang et al. [13] presented three models for material removal to try to understand how particle properties in conjunction with pad information influence material removal rate. ARTICLE IN PRESS G Model WEA-100114; No. of Pages 9 2 X.L. Jin, L.C. Zhang / Wear xxx (2011) xxx– xxx Nomenclature A0 nominal area of contact between pad and material surface A1 total contact area by Type I particles A2 total contact area by Type II particles Ac total contact area in a polishing process Ad total direct contact area between the pad and material surface Ã1 area of asperity contact due to Type I particles Ã2 area of asperity contact due to Type II particles d separation of the reference planes of Surface 1 and Surface 2 (pad) E2 elastic modulus of a pad E′ composite elastic modulus E′ = E2/(1 − 2 2) G1 wear volume of the material due to Type I particles G2 wear volume of the material due to Type II particles G̃ wear volume of the material by an individual active particle Hw hardness of a workpiece material K wear coefficient M1 material removal rate due to Type I particles M2 material removal rate due to Type II particles Mr total material removal rate p0 polishing pressure Rp average asperity radius of a pad P1 total contact force on Type I particles P2 total contact force on Type II particles Pd direct contact force between the pad and material surface P̃1 contact force of Type I particles P̃2 contact force of Type II particles r particle radius t polishing time ur mean particle radius V pad/material sliding velocity x particle volume concentration of slurry z2 asperity height of Surface 2 (pad) ̨ porous coefficient v number of particles per unit volume of slurry p surface density of asperity on Surface 2 (pad) r standard deviation of particle radius p standard deviation of pad asperity height r(r) probability density function of particle size (z ) probability density function of pad asperity height Z t w b S m t i t p o h r t [ p h 2 2 Poisson’s ratio of a pad hao and Chang [14] studied the material removal rate based on he elastic–plastic micro-contact mechanics and abrasion wear, here the chemical effect was claimed to have been formulated y introducing a density ratio of a chemical thin film. Oh and eok [15] proposed a model for silicon dioxide CMP based on a ulti-scale mechanical abrasion consideration and coupled with he effect of the slurry chemical diffusion. Bozkaya and Muftu [16] nvestigated the material removal with two-body pad–wafer and hree-body pad–abrasive–wafer contacts, and introduced a thin assivated layer on the wafer surface to take into account the effect f chemical reactions between slurry and wafer. Some researchers ave also studied the wear mechanisms and material removal ates in CMP processes based on the combination of the above Please cite this article in press as: X.L. Jin, L.C. Zhang, A statistical m doi:10.1016/j.wear.2011.08.028 wo approaches, i.e., contact mechanics and fluid hydrodynamics 17–20]. To our knowledge, most existing studies only consider the articles embedded in the pad as the active particles. In reality, owever, many active particles also slide and rotate between the Fig. 1. The contact model of Type I particles, where the dashed circles indicate particles and the solid curves stand for asperities. pad and wafer, as pointed out by Zhang and Tanaka [21]. Therefore, some critical questions naturally arise: How do the two types of active particles contribute to the material removal and how do their contributions vary with the change of polishing conditions when the particle size and distribution are random? This paper will try to answer these questions by developing a statistical model for predicting the material removal in mechanical polishing where both the polishing pad surface and the particle size are random. The investigation will be based on the abrasion wear and contact mechanics of pad–MS and pad–abrasive–MS interactions.