Determination of Planarization Length

A key component in CMP process characterization is the choice of test layout mask. For planarization length extraction, the test mask should 

The next stage is to discretize the local density into small cells to facilitate profile simulation. The mask is divided into small square cells in a regular 

The final density calculation step is the stage at which the 2-D weighting density window function is first employed. The 2-D filter of the correct 

The actual limit of the summation is the extent of the weighting filter. Zero padding is used to ensure that the discretized matrices have sizes which are a power of two so that the computation can be done in the frequency domain using fast fourier transform (FFT) techniques. The effective discretized density, pin, Wj), is then given by 

The use of the FFT reduces the number of computations significantly. If the number of cells is N x N, and the discretized filter is of size M x M, Eq. (28) requires computations, while Eq. (29) requires NMlo 2 NM) 

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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. 

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