Method and apparatus for estimating model parameters of and controlling a lithographic apparatus by measuring a substrate property and using a polynomial model
System and methods estimate model parameters of a lithographic apparatus and control lithographic processing by a lithographic apparatus. An exposure is performed using a lithographic apparatus across a wafer. A set of predetermined wafer measurement locations is obtained. Discrete orthonormal polynomials are generated using the predetermined substrate measurement locations. The overlay errors arising from the exposure are measured at the predetermined locations to obtain overlay measurements. The estimated model parameters of the lithographic apparatus are calculated from the overlay measurements by using the discrete orthogonal polynomials as a basis function to model the overlay across the wafer. Finally, the estimated model parameters are used to control the lithographic apparatus in order to provide corrected overlay across the wafer.
1. A method comprising:
performing a lithographic process using a lithographic apparatus across a substrate;
using predetermined substrate measurement locations to generate discrete orthogonal polynomials;
measuring a substrate property arising from the lithographic process at locations of the substrate corresponding to the predetermined measurement locations to obtain substrate property measurements;
calculating estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate; and
controlling lithographic processing by the lithographic apparatus using the estimated model parameters.
2. The method according to claim 1 , wherein the discrete orthogonal polynomials are orthonormal.
3. The method according to claim 1 , wherein the discrete orthogonal polynomials comprise discrete Chebyshev polynomials.
4. The method according to claim 1 , wherein the calculating estimated model parameters of the lithographic apparatus comprises:
constructing a matrix using the generated discrete orthogonal polynomials and the predetermined substrate measurement locations; and
inverting the matrix.
5. The method according to claim 1 , wherein the predetermined substrate measurement locations are optimized to increase accuracy of the calculated estimated model parameters.
6. An apparatus for estimating model parameters of a lithographic apparatus used to perform a lithographic process across a substrate, the apparatus comprising a processor configured to:
use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations; and
calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate.
7. The apparatus according to claim 6 , wherein the discrete orthogonal polynomials are orthonormal.
8. The apparatus according to claim 6 , wherein the discrete orthogonal polynomials comprise discrete Chebyshev polynomials.
9. The apparatus according to claim 6 , wherein the processor is configured to calculate estimated model parameters of the lithographic apparatus by:
constructing a matrix using the generated discrete orthogonal polynomials and the predetermined substrate measurement locations; and
inverting the matrix.
10. The apparatus according to claim 6 , wherein the processor is configured to generate the predetermined substrate measurement locations such that they are optimized to increase accuracy of the calculated estimated model parameters.
11. An apparatus for controlling lithographic processing by a lithographic apparatus used to perform a lithographic process across a substrate, the apparatus comprising a processor configured to:
use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations;
calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate; and
control lithographic processing by the lithographic apparatus using the estimated model parameters.
12. The apparatus according to claim 11 , wherein the discrete orthogonal polynomials are orthonormal.
13. The apparatus according to claim 11 , wherein the discrete orthogonal polynomials comprise discrete Chebyshev polynomials.
14. The apparatus according to claim 11 , wherein the processor is configured to calculate estimated model parameters of the lithographic apparatus by:
constructing a matrix using, the generated discrete orthogonal polynomials and the predetermined substrate measurement locations; and
inverting the matrix.
15. The apparatus according to claim 11 , wherein the processor is configured to generate the predetermined substrate measurement locations such that they are optimized to increase accuracy of the calculated estimated model parameters.
16. A non-transitory computer readable medium having instructions stored thereon for estimating model parameters of a lithographic apparatus used to perform a lithographic process across a substrate, the instructions when executed by one or more processors, cause the one or more processors to:
use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations; and
calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate.
17. A non-transitory computer readable medium having instructions stored thereon for controlling lithographic processing by a lithographic apparatus used to perform a lithographic process across a substrate, the instructions when executed by one or more processors, cause the one or more processors to:
use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations;
calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate; and
control lithographic processing by the lithographic apparatus using the estimated model parameters.