3. Production Process Characterization
3.4. Data Analysis for PPC
3.4.3. Building Models

## Fitting Physical Models

Sometimes we want to use a physical model Sometimes, rather than approximating response behavior with polynomial models, we know and can model the physics behind the underlying process. In these cases we would want to fit physical models to our data. This kind of modeling allows for better prediction and is less subject to variation than polynomial models (as long as the underlying process doesn't change).
We will use a CMP process to illustrate We will illustrate this concept with an example. We have collected data on a chemical/mechanical planarization process (CMP) at a particular semiconductor processing step. In this process, wafers are polished using a combination of chemicals in a polishing slurry using polishing pads. We polished a number of wafers for differing periods of time in order to calculate material removal rates.
CMP removal rate can be modeled with a non-linear equation From first principles we know that removal rate changes with time. Early on, removal rate is high and as the wafer becomes more planar the removal rate declines. This is easily modeled with an exponential function of the form:

removal rate = p1 + p2 x exp  p3 x time
where p1, p2, and p3 are the parameters we want to estimate.
A non-linear regression routine was used to fit the data to the equation The equation was fit to the data using a non-linear regression routine. A plot of the original data and the fitted line are given in the image below. The fit is quite good. This fitted equation was subsequently used in process optimization work.