Feature-scale model

Burke developed an empirical model which gives the polish rate of the down and up areas [7]. The down area polish rate D is given by 

The model is empirical and can be used to explain aspects of measured data. Characterization or extraction of parameters would include D°, which is obtained as the average rate over a raised area divided by the blanket rate. Equation (9) is then used to trace the actual thickness evolution in conjunction with step height reduction in Eq. (10). These equations define the feature polish characteristics for relatively large features. For a given array of patterned features, the up polish rates follows the following relationship  

Warnock proposed a phenomenological model that mathematically captures the polishing process but that did not directly seek to incorporate the physical phenomena in CMP [63], The model has three parameters that can be used to fit measured data. The surface is divided into n discrete points each with x, y, and z coordinates. For each point i in the set of n points, the polish rate P,- is defined as 

The analysis is accomplished by first obtaining an expression for S and then using the reciprocity condition to obtain A. Then K is obtained independently, and is assumed to be of the form  

A third model for feature-scale polish was proposed by Runnels [40], and focuses on stresses created by flowing slurry on feature surfaces under continuum mechanics. The model incorporates fracture mechanics and chemistry through empirical means. The geometry of a typical structure under study is shown in Fig. 8. 

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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. 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

Saxena R, Thakurta DG, Gutmann RJ, Gill WN. A feature scale model for chemical mechanical planarization of damascene structures. Thin Solid Films 2004 449 192-206. 

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. 

Zhang, D. Kushner, M.J. Investigations of surface reactions during C2F6 plasma etching of Si02 with equipment and feature scale models. J. Vac. Sci. Technol. 2001, A19, 524. 

In case experiments have been selected, the next step is to decide on the method visualization, full-scale measurements, or scale model experiments. Some features of these are listed in Table 12.1. 

Working within a similar scheme, DeBecker and West introduced a treatment of feature scale effects on the overall current distribution which they call the hierarchical model [138]. Rather than represent the features as a smoothly varying density of active area, they retain the features, but simplify their representation in the global model. An integral current for each feature is assigned to the geometric center of the feature to provide a simplified boundary condition for the secondary current distribution. This boundary condition captures a part of the ohmic penalty paid when current lines converge onto features. It thus contains more information than the active area approximation but still less than a fully matched current distribution on the two levels. 

All hydrocarbon fire mechanisms and estimates will be affected by to some extent of flame stability features such as varying fuel composition as lighter constituents are consumed, available ambient oxygen supplies, ventilation patterns, and wind effects. Studies into these effects have generally not progressed to the level where precise estimations can be made without scale model tests or on site measurements. 

Mislow and Bickart (258) have recently discussed the properties, and specified the limitations and essential features, of models that can be used for the prediction of chirality of a molecular system. In the simplified and idealized representation of molecular stracture, nonessential features are deliberately left out the model summarizes some selected aspects of the system and completely disregards or even falsifies, others. The model must be adequate to the time scale in which the phenomenon is observed. In particular, in mobile conformational systems it should refer to a time-averaged structure. In other words, the model can have a higher symmetry than that observed under static conditions (e.g., by X-ray diffraction in the crystalline state or by NMR under slow exchange conditions) (259). 

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

Models such as those proposed by Burke, Warnock, and Runnels can be effective in simulating the evolution of step height and film thicknesses around particular features. One limitation with feature based models is that they are difficult to apply over the die scale. In many cases, the features are so small that an attempt to trace their polish evolution is computationally expensive and may be difficult to apply to the entire die. In the next section, we focus on models which seek to address pattern dependencies observed over large regions of the chip or across the entire chip. 

S. R. Runnels, Feature-Scale Fluid-Based Erosion Modeling for Chemical-Mechanical Polishing, J. Electrochem.. Soc., vol. 141, no. 7, pp. 1900-1904, July 1994. 

The "variational type of multi-scale CFD, here, refers to CFD with meso-scale models featuring variational stability conditions. This approach can be exemplified by the coupling of the EMMS model (Li and Kwauk, 1994) and TFM, where the EMMS/matrix model (Wang and Li, 2007) at the subgrid level is applied to calculate a structure-dependent drag force. 

With respect to the Duality Principle, the first major accomplishment is coming to understand that one thing can stand for something other than itself. As discussed earlier, this idea appears to emerge quite early, as demonstrated by the 3-year-old child s ability to use a scale model (e.g., DeLoache, 1987) and maps (e.g., Bluestein Acredolo, 1979) in a stand-for relationship. However, it takes far longer for children to differentiate consistently between referential and incidental features of particular map components. 

I expect that SA of stochastic and multiscale models will be important in traditional tasks such as the identification of rate-determining steps and parameter estimation. I propose that SA will also be a key tool in controlling errors in information passing between scales. For example, within a multiscale framework, one could identify what features of a coarse-level model are affected from a finer scale model and need higher-level theory to improve accuracy of the overall multiscale simulation. Next a brief overview of SA for deterministic systems is given followed by recent work on SA of stochastic and multiscale systems. 

Runnels assumes the existence of a continuous fluid layer between the pad and the wafer and models planarization using a feature scale fluid-based-wear model. Runnels uses fluid mechanics to model the normal and shear stresses that are developed at the feature scale. Material removal rate is assumed to depend only upon the shear stress, Ot, according to ... 

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