Thickness Evolution

The equation is then solved for the oxide thickness z under the assumption that no down area polishing occurs until the local step, Zj, has been removed, after which the local rate is equal to the blanket polish rate. This is captured by expressing the effective density as 

Before local planarity is achieved (i.e., while local step height still exists), the final film thickness is inversely proportional to the effective local density. The film is assumed to polish linearly at the blanket rate afterward. The key 

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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 ... 

Fig. 9. Polish thickness evolution profiles illustrating the significance of layout pattern density.

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. 

The calibration phase focuses on the determination of the planarization length itself. This is a crucial characterization phase since once the planarization length is determined, the effective density, and thus the thickness evolution, can be determined for any layout of interest polished under similar process conditions. The determination of planarization length is an iterative process. First, an initial approximate length is chosen. This is used to determine the effective density as detailed in the previous subsection. The calculated effective density is then used in the model to compute predicted oxide thicknesses, which are then compared to measured thickness data. A sum of square error minimization scheme is used to determine when an acceptably small error is achieved by gradient descent on the choice of planarization length. 

For a= 1, soot in the catalytic layer is oxidized fast leaving the soot in the thermal layer unreacted. This has been observed with some early catalytic filters. As a decreases the soot from the top layer replaces more rapidly the soot oxidized in the catalytic layer increasing the global oxidation rate. The corresponding soot layer thickness evolution is shown in Fig. 22. For values of a close to 1 (e.g. 0.9) the catalytic layer is totally depleted from soot at some instances, followed by sudden penetration events from the soot of the thermal layer. These events are clearly shown in the thickness evolution for oc = 0.9 in... 

Fig. 22. Normalized soot deposit thickness evolution for different values of the microstructural parameter a.

Feasibility tool-Thickness evolution on a 104A seed USER-DEF.D X-Y... 

The classical expression for the striation thickness evolution vith time, 5 = 5oexp(—Tit), where Ti is the Lyapounov exponent, shows that f and n values are related to Ti. 

Figure 6.12 Pattern density (PD) dependence of CMP process (Fan, 2012) (a) monitor sites in STEP arrays of MIT standard layout (b) up area thickness evolution (c) step height evolution.

Fig.6 - Langmuir-Blodgett film made up of 100 layers (3000 A thick). Evolution of the second harmonic intensity under laser pulses (1 mJ/pulse) focal length 50 cm pulse frequency f ...

To determine the prevailing termination mechanism in SI-PMP over a broad range of relevant reaction conditions, experimental data on film thickness evolution such as that shown in Figure 12.4 were fit in the brush regime (transitions marked by arrows in Figure 12.4) by the kinetic models that incorporate one or more termination mechanisms. For example, Rahane et al. combined Equation 12.1 with expressions for STR based on either bimolecular termination or chain transfer to monomer to develop models for how layer thickness should evolve as a function of exposure time. These models, shown as Equations 12.3 and 12.4, respectively, can be compared to experimental data of polymer layer thickness as a function of time to deduce which irreversible termination mechanisms are prevalent. 

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