Pad modeling

In the linear regime, an incompressible pad model gives a step height h as a function of time... 

In the exponential regime, a compressible pad model results in a step height dependence as... 

Yu TK, Yu CC, Orlowski M. A statistical polishing pad model for chemical mechanical polishing. IEEE lEDM Washington DC, Dec 5-8 1993. pp 865-868. Seok J, Sukam CP, Kim AT, Tichy JA, Cale TS. Multiscale material removal modeling of chemical mechanical polishing. Wear 2003 254 307-320. 

Figure 7. Transition from the incompressible pad model to a compressible pad model as proposed by Grillaert et al. [5]. The transition occurs at some transition height h, or contact time. ...

DTncamps, B.L., C. Meunier, M.-L. Monnet, L. lami, and D. Zytnicki (1998). Reduction of presynaptic action potentials by pad model and experimental study. J. Comput. Neurosci. 5,141-156. 

FIGURE 9.32. Profile of a ball in the lul)ricated regime The tilted pad model. 

Luo and Domfeld [110] introduced a fitting parameter H , a d5mamical" hardness value of the wafer surface to show the chemical effect and mechanical effect on the interface in their model. It reflects the influences of chemicals on the mechanical material removal. It is found that the nonlinear down pressure dependence of material removal rate is related to a probability density function of the abrasive size and the elastic deformation of the pad. 

In all of the above discussions, the focus has been entirely on the local condition, such as the pressure exposed on the wafer and relative velocity between pad and wafer. A model is to be proposed to describe the entire available re-... 

The rate of metal complex formation is often modified (usually enhanced) by the presence of a charged interface in the aqueous phase. This may be provided by ionic micelles, e. g., SDS, microemulsions or polyelectrolytes. jjjg reactions of Ni + and Co with hydrophobic ligands pan, pap and pad 14-16 are popular ones for examining effects, since they are well characterized in the bulk water. The simple model (4.126)... 

Our single padded room soon proved inadequate. We needed additional space, as well as a proper nursing station. 1 went home and built a cardboard model of an ideal experimental ward. It would actually be a ward within a ward, with just enough space around the sides to let someone quietly pass by, if necessary. Inside would be a row of six identical cubicles, a well-equipped nursing station and an open area for meals, watching TV, and other forms of relaxation for... 

In a typical modeling approach, the material removal rate is modeled as a function of easily controlled process parameters. The most basic model is one that predicts the bulk rate of material removal in a macroscopic fashion. An empirical observation by Preston is widely used, in which the rate of material thickness reduction is proportial to the product of (a) the relative velocity between the wafer and the polish pad and (b) the pressure on the surface of the wafer ... 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

In the simplest case, a square area can be used to determine the effective density across the mask, as shown in Fig. 11. Density to be assigned to the coordinates at the center of the window is equal to the ratio of raised to total area of the square window. The length of each side of the square is then defined as the planarization length this square region approximates the deformation characteristics of the pad and process. The size of the square (or the planarization length) is determined experimentally by varying the square size until the effective density calculation results in predicted thickness values that best fit experimentally measured polish data when used in the thickness evolution model. 

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