Preston’s Equation

In Section II, we focus first on wafer-scale models, including macroscopic or bulk polish models (e.g., via Preston s equation), as well as mechanistic and empirical approaches to model wafer-scale dependencies and sources of nonuniformity. In Section III, we turn to patterned wafer CMP modeling and discuss the pattern-dependent issues that have been examined we also discuss early work on feature-scale modeling. In Section IV, we focus on die-scale modeling efforts and issues in the context of dielectric planarization. In Section V, we examine issues in modeling pattern-dependent issues in metal polishing. Summary comments on the status and application of CMP modeling are offered in Section VI. 

Preston s equation indicates a pressure dependency and if the pressure distribution across the surface of the wafer is not uniform, one expects a wafer-level removal rate dependency. Runnels et al, for example, report a model incorporating pressure dependencies to account for wafer scale nonuniformity [42]. The distribution of applied force across the surface of the wafer is highly dependent on the wafer carrier design, and significant innovation in head design to achieve either uniform or controllable pressure distributions is an important area of development. 

Stine et al. proposed a simple analytic model for patterned feature removal incorporating an effective density determination step [54]. Figure 10 defines terms used in the model, which reformulates Preston s equation [Eq. (2)] as a function of blanket rate K and effective pattern density... 

A. Maury, D. Ouma, D. Boning, J. Chung, A Modification to Preston s Equation and Impact on Pattern Density Effect Modeling, Conf. for Advanced Metallization and Interconnect Systems for VLSI Applications, San Diego, CA, Oct. 1997. 

Preston coefficients may be readily measured for a polish process. The simplest method for measuring is to measure the polish rate and divide by the pressure and velocity. However, this method is sensitive to error in the polish rate measurement. A better measure of is obtained by measuring the polish rate over a range of velocities and pressures. If the polish rate behavior follows Preston s equation, i.e., if it is linear and intersects the origin, may then be obtained by differentiating Equation (4.1) with respect to either pressure or velocity. is then given by ... 

For the properly conditioned Suba IV pad, the values are all very consistent. For example, the values obtained from the velocity plots are within 4% of each other. Thus, changing pads or pressure does not appear to affect within the range of velocities used to calculate K, i.e., the range where the plots are linear. Note, however, that at the higher velocities, the plots are nonlinear and Preston s equation does not apply. 

The polish rate behavior deviates from the Preston equation at high velocities and pressure or if the pad is poorly conditioned. The apparent saturation in the polish rate observed at high velocities and pressures indicates that the polish rate enters a dissolution rate limited regime. In this regime, increasing the abrasion rate does not affect the polish rate because the slurry cannot dissolve any additional abraded material and the additional material simply redeposits. Thus, Preston s equation is only valid in an abrasion rate limited regime. 

Polishing of copper (Section 7.4) and follow Preston s equation,... 

An early understanding of material removal rate (MRR) from two surfaces in contact was developed by the glass polishing industry, where MRR became represented in what is known as Preston s equation MRR = kPv. In this empirical equation, the MRR is seen as a linear function of the pressure between the working and the worked surfaces (P) and the shear velocity between the two surfaces (v). This equation has stood the test of time and still forms the basis of approaches to and control of the CMP process. However, it can readily be shown that neither P nor v can be primary variables in the removal of material from a surface. Material is only accelerated (removed) in the direction of an applied force. The interface pressure P represents a force per unit area but is applied compressionally and normal... 

Figure 2.24 shows the removal rate variation as a function of pattern density for five dies with the positions shown in Figure 2.23a. This trend indicates that the removal rate decreases linearly as the pattern density increases. It can intuitively be explained by Preston s equation, R = k pv, where k, is the Preston coefficient, R is the material removal rate, p is pressure, and V is relative velocity. As pattern density increases, the effective...

The most famous and simplified CMP model for material removal is Preston s equation (Preston, 1927), which was originally developed to describe glass polishing... 

Once contact pressure p(x, y) is known on the feature, the instantaneous material removal can be obtained utilizing Preston s equation at a localized position as... 

Based on experimental observation, Stine et al. (1997) proposed a PD CMP model for oxide polishing. Figure 6.6 shows the wafer topography and the key variables defined in the model. This model takes a straightforward consideration of local PD in a single die to reformulate Preston s equation as... 

Modeling of the CMP process. To calculate the wafer topography evolution during the CMP process, PD p(x, y) needs to he extracted from the chip layout. With initial values of up area thickness and step height, the die-level pressure distrihution p x, y) can he obtained hy solving Eqn (6.27). Once p x, y) is solved, pu(x, y) and pi x, y) can he calculated hy Eqn (6.26). Then we utilize Preston s equation with local pressures pu(x, y) and pd(x, y) to calculate the instantaneous MRR of up area and down area as... 

As conditioning is primarily considered as a mechanical process characterized by a two-body abrasive wear mechanism [8], the classical Preston equation [62], originally used to model polishing of glass, has been widely used to describe material removal (polishing) rate in [61]. Considering the similarity between wafer—pad interaction and pad—conditioner interaction, the Preston s equation has been adopted by many to model pad wear caused by conditioning. The Preston equation states that MRR is proportional to the applied pressure P and the relative velocity V between the wafer and the pad and Kp is a constant, called Preston s coefficient. 

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