Method for modeling measurement data over a substrate area and associated apparatuses
Disclosed is a method for modeling measurement data over a substrate area and associated apparatus. The method comprises obtaining measurement data relating to a first layout; modeling a second model based on said first layout; evaluating the second model on a second layout, the second layout being more dense than said first layout; and fitting a first model to this second model according to the second layout.
1. A method for modeling measurement data over a substrate area relating to a substrate in a lithographic process, comprising:
obtaining measurement data relating to a first layout;
fitting a first model to the first layout, wherein the first model comprises an oblique subspace model or a generalized least squares model; and
performing a model mapping operation on the first model.
2. The method of claim 1 , wherein the first layout corresponds to mark locations from which the measurement data was measured.
3. The method of claim 1 , wherein the oblique subspace model is determined by performing an oblique fitting on a subspace matrix.
4. The method of claim 1 , wherein the modeling of the oblique subspace model comprises determining a bending energy minimum subspace by determining a subspace with minimum bending energy for the first layout.
5. The method of claim 4 , further comprising:
determining a bending energy minimum subspace matrix by performing operations comprising:
performing a bending energy orthonormalization of a first model matrix;
removing a linear part from the first model matrix;
calculating and truncating a single value decomposition of the first model matrix; and
calculating a subspace matrix corresponding to the truncated decomposed model matrix obtained by the previous step.
6. The method of claim 5 , wherein the performing the bending energy orthonormalization comprises:
determining a bending covariance matrix of model functions of the first model matrix based on a description of bending energy within the substrate; and
factorizing the bending covariance matrix.
7. The method of claim 1 , wherein the fitting is regularized by a Tikhonov regularization using bending energy.
8. The method of claim 7 , wherein the Tikhonov regularization is based on a bending covariance matrix based on a description of bending energy within the substrate.
9. The method of claim 1 , further comprising using the first model to determine a wafer grid for exposure and control of the lithographic process.
10. A non-transitory computer program carrier comprising a computer program comprising program instructions operable to perform a method for modeling measurement data over a substrate area relating to a substrate in a lithographic process, the program instructions configured to perform operations comprising:
obtaining measurement data relating to a first layout;
fitting a first model to the first layout, wherein the first model comprises an oblique subspace model or a generalized least squares model; and
performing a model mapping operation on the first model.
11. A processing arrangement comprising:
a non-transitory computer program carrier comprising a computer program comprising program instructions operable to perform a method for modeling measurement data over a substrate area relating to a substrate in a lithographic process, the program instructions configured to perform operations comprising:
obtaining measurement data relating to a first layout;
fitting a first model to the first layout, wherein the first model comprises an oblique subspace model or a generalized least squares model; and
performing a model mapping operation on the first model; and
a processor operable to run the computer program comprised on said non-transitory computer program carrier.
12. A lithographic apparatus comprising:
an alignment sensor;
a patterning device support configured to support a patterning device;
a substrate support configured to support a substrate;
a processing arrangement comprising a non-transitory computer program carrier comprising a computer program comprising program instructions operable to perform operations for modeling measurement data over a substrate area relating to a substrate in a lithographic process, the operations comprising:
obtaining measurement data relating to a first layout;
fitting a first model to the first layout, wherein the first model comprises an oblique subspace model or a generalized least squares model; and
performing a model mapping operation on the first model; and
a processor operable to run the computer program comprised on said non-transitory computer program carrier.
13. The lithographic apparatus of claim 12 , wherein the alignment sensor is operable to measure the substrate to obtain the measurement data.
14. The lithographic apparatus of claim 12 , wherein the processing arrangement is further operable to determine corrections configured to control the patterning device and/or substrate support based on the first model.