Wafer-scale models

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. 

Cho, C. H., Park, S. S., andAhn, Y, Three-Dimensional Wafer Scale Hydrodynamic Modeling for Chemical Mechanical Polishing," Thin Solid Films, Vol. 389,2001, pp. 254-260. 

Sundararajan, S. andXhakurta, D., Two-Dimensional Wafer-Scale Chemical-Mechanical Planarization Models Based on Lubrication Theory and Mass Transport, Journal of the Electrochemical Society, Vol. 146, No. 2, 1999, pp. 761-766. 

The modeling of polishing effects in CMP begins with two key issues what are the process-related dependeneies in the rate of removal of exposed surface material during polishing, and on what does the wafer-scale uniformity of that polish depend In this section, we begin with the modeling of polish or removal rate, and then consider models for the effects that impact the commonly observed nonuniformity in polish across the wafer. 

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. 

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 ... 

The desire to improve wafer-level uniformity continues to drive the study of wafer-scale CMP dependencies. Effective and practical means now exist for empirical simulation of wafer-level polish rate across the entire wafer however, substantial enhancement remains necessary to develop predicitive models with sufficient detail and physical bases for exploratory tool design and optimization. 

D. Ouma, B. Stine, R. Divecha, D. Boning, J. Chung, G. Shinn, I. Ali, and J. Clark, Wafer-Scale Modeling of Pattern Effect in Oxide Chemical Mechanical Polishing, Proc. SPIE Microelectronic Man. Conf., Austin, TX, Oct. 1997. 

A number of models for wafer-scale nonuniformity have been published. While of significance for understanding the CMP process, they have limited relevance for this present chapter as, for the most part, pad effects are neglected. The reader is referred to the primary references listed here by focus area ... 

Fu G, Chandra A. A model for wafer scale variation of material removal rate in chemical mechanical polishing (CMP) based on viscoelastic pad deformation. J Electron Mater 2002 31(10) 1066-1073. 

Thakurta DG, Schwendeman DW, Gutmann RJ, Shankar S, Jiang L, Gill WN. Three-dimensional wafer-scale copper chemical-mechanical planarization model. Thin Solid Films 2002 414(l) 78-90. 

The flow of a slurry in a CMP process has been investigated. This wafer-scale model provides the three dimensional flow field of the slurry, the spatial distribution of the local shear rate imposed on the wafer surface, and the streamline patterns which reflect the transport characteristics of the slurry. 

Modeling of the CMP process is often classified into two categories wafer-scale model and feature-scale model. The characteristic length scale of the wafer-scale model is the gap between the pad and wafer which is in the order of 50 pm, and it attempts to describe the overall removal rate of the CMP process. The feature-scale model is for the length scale of typical device features on the wafer which is in the order of a few micrometers, and focuses on the local removal rate rather than the overall removal rate. 

The present study is a wafer-scale modeling which provides the three dimensional flow field of the slurry in the gap between the wafer and the pad which are assumed to be parallel to one another. 

In this section, three example applications intimately related to the pattern dependent behavior of CMP are briefly discussed. First, we note the relative importance of die-level effects with respect to typical wafer-scale nonuniformity. Second, we describe recent application of the integrated density and step height model to the run by run control of ILD CMP. Finally, we summarize issues in the application of density models to prediction of shallow trench isolation. 

Abstract Besides the dominant monolithic VLSI integration paradigm, many non-monolithic schemes have already been developed in the past. Typical such schemes include wafer scale integration or multi-reticle wafer, multi-chip module, and 3-D integration. In this chapter we compared these different schemes in a unified cost analysis framework. Our model takes a few parameters extracted from representative fabrication and evaluates the cost efficiency. Our analysis proves that the proposed 2.5-D out significantly outperform other integration paradigms from a cost perspective. 

Thakurta, D.G., Schwendeman, D.W., Gutmann, R.J., et al., 2002. Three-dimensional wafer-scale copper chemical—mechanical planarization model. Thin Solid Films 414, 78—90. 

Fig. 8. Feature-scale geometry illustrating fluid flow near the wafer surface in Runnels model [40]. Reproduced by permission of the Electrochemical Society, Inc.

As understanding of the CMP process improves, one can expect a great deal of work in all aspects of CMP modeling and simulation. These improvements are likely to extend over many length scales spanning wafer-level polish and uniformity concerns to die-level prediction to microscopic feature, chemical, and mechanical interactions. 

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