Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer

A method of approximating eigensolutions for use in determining the profile of a structure formed on a substrate, the method comprising:

• (a) obtaining a known set of eigenvectors and a known set of eigenvalues associated with a first section of a hypothetical profile of the structure;
• (b) obtaining a known characteristic matrix associated with a second section of a hypothetical profile of the structure;
• (c) determining an auxiliary matrix based on the known set of eigenvectors and the known characteristic matrix;
• (d) determining an approximated set of eigenvectors based on the auxiliary matrix;
• (e) determining an approximated set of eigenvalues based on the approximated set of eigenvectors and the auxiliary matrix;
• and
• (f) determining if a change between the known set of eigenvalues and the approximated set of eigenvalues meets a stop criterion;
• (i) when the stop criterion is not met, repeating steps (c) to (f) using the determined approximated set of eigenvectors as the known set of eigenvectors in repeating step (c) and the determined auxiliary matrix as the known characteristic matrix in repeating step (c) and the determined approximated set of eigenvalues as the known set of eigenvalues in repeating step (f), and
• (ii) when the stop criterion is met, using the approximated set of eigenvalues as an approximated set of eigenvalues for the second section.

A system to approximate eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer, the system comprising:

• a memory having:
• a known set of eigenvectors associated with a first section of a hypothetical profile of the structure,
• wherein the known set of eigenvectors is used to generate a simulated diffraction signal for the hypothetical profile, and
• a known characteristic matrix associated with a second section of a hypothetical profile;
• and
• a processor configured to determine an approximated set of eigenvalues for the second section based on the known set of eigenvectors associated with the first section and the known characteristic matrix associated with the second section.

A computer-readable storage medium containing computer executable instructions for causing a computer to approximate eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer, comprising instructions for:

• obtaining a known set of eigenvectors associated with a first section of a hypothetical profile of the structure,
• wherein the known set of eigenvectors is used to generate a simulated diffraction signal for the hypothetical profile;
• obtaining a known characteristic matrix associated with a second section of a hypothetical profile;
• and
• determining an approximated set of eigenvalues for the second section based on the known set of eigenvectors associated with the first section and the known characteristic matrix associated with the second section.