# Fast Simulation Methods for Non-Planar Phase and Multilayer Defects in DUV and EUV Photomasks for Lithography

### Michael Lam

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2005-28

December 19, 2005

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/EECS-2005-28.pdf

This dissertation develops rapid modeling methodologies for the printability and inspectability of various types of defects on photomasks in DUV and EUV lithography. Several fast and approximate methods for defect simulation are introduced and validated by comparing their results with Finite Difference Time Domain (FDTD) calculations of scattering from the same geometry. The common strategy is to decompose the electro-magnetic (EM) scattering into individual signal contributions by analyzing rigorous simulations, and then to develop efficient alternative models for each contribution. Two methods are introduced to calculate the observed scattering from DUV phase defects. First, the through focus behavior of an isolated defect can be used to extract two defect parameters, size and phase, which fully characterize the defect by means of an EM equivalent thin mask model. Post and void defects can also be differentiated based on the side of defocus that their peak signal occurs. Second, a defect projector methodology is introduced that allows results for an isolated defect and a defect-free pattern to be combined to predict their interaction for any defect location. The defect projector is four orders of magnitude faster than 3D FDTD simulation, and can correctly predict the defect induced dimension change to within 30% for worst case. The main emphasis of this dissertation is on scattering from non-planar multilayer structures to understand the printability of buried defects inside of EUV mask blanks. A new method based on ray tracing is developed by exploiting the small non-specular forward angular scattering of individual bilayers, which is 10X smaller than the back scatter, and its approximation as zero allows a new and tractable mathematical factoring. The method is tested for various deposition strategies, defect sizes, defect shapes, as well as various illumination angles of incidence and polarization. Smoothing of the defect shape during deposition is confirmed to help mitigate isolated defect printability to a size less than about 70nm for 3D defects. The method is 4 to 5 orders of magnitude faster than FDTD simulation, takes 40X less memory, and still achieves equivalent accuracy. FDTD results for resonant multilayers were also found to suffer from convergence lulls and reflection errors at angles >10º due to small wavelength shifts from numerical dispersion. The new methodology is then extended to model the interaction between absorber features and buried defects by developing a new 2D thin mask model for features. FDTD studies of signal components from an isolated absorber edge show that the scattering can be approximated to first order by adding phased line sources to vertically propagating waves at the edges of the thin mask model. The ray tracing method was also extended to model the case of optical inspection of EUV masks. The Single Surface Approximation and ray tracing method produce nearly identical results, while FDTD suffers from numerical errors due to the abnormally high cell densities of 700 cells per wavelength.

BibTeX citation:

@phdthesis{Lam:EECS-2005-28, Author = {Lam, Michael}, Title = {Fast Simulation Methods for Non-Planar Phase and Multilayer Defects in DUV and EUV Photomasks for Lithography}, School = {EECS Department, University of California, Berkeley}, Year = {2005}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/EECS-2005-28.html}, Number = {UCB/EECS-2005-28}, Abstract = { This dissertation develops rapid modeling methodologies for the printability and inspectability of various types of defects on photomasks in DUV and EUV lithography. Several fast and approximate methods for defect simulation are introduced and validated by comparing their results with Finite Difference Time Domain (FDTD) calculations of scattering from the same geometry. The common strategy is to decompose the electro-magnetic (EM) scattering into individual signal contributions by analyzing rigorous simulations, and then to develop efficient alternative models for each contribution. Two methods are introduced to calculate the observed scattering from DUV phase defects. First, the through focus behavior of an isolated defect can be used to extract two defect parameters, size and phase, which fully characterize the defect by means of an EM equivalent thin mask model. Post and void defects can also be differentiated based on the side of defocus that their peak signal occurs. Second, a defect projector methodology is introduced that allows results for an isolated defect and a defect-free pattern to be combined to predict their interaction for any defect location. The defect projector is four orders of magnitude faster than 3D FDTD simulation, and can correctly predict the defect induced dimension change to within 30% for worst case. The main emphasis of this dissertation is on scattering from non-planar multilayer structures to understand the printability of buried defects inside of EUV mask blanks. A new method based on ray tracing is developed by exploiting the small non-specular forward angular scattering of individual bilayers, which is 10X smaller than the back scatter, and its approximation as zero allows a new and tractable mathematical factoring. The method is tested for various deposition strategies, defect sizes, defect shapes, as well as various illumination angles of incidence and polarization. Smoothing of the defect shape during deposition is confirmed to help mitigate isolated defect printability to a size less than about 70nm for 3D defects. The method is 4 to 5 orders of magnitude faster than FDTD simulation, takes 40X less memory, and still achieves equivalent accuracy. FDTD results for resonant multilayers were also found to suffer from convergence lulls and reflection errors at angles >10º due to small wavelength shifts from numerical dispersion. The new methodology is then extended to model the interaction between absorber features and buried defects by developing a new 2D thin mask model for features. FDTD studies of signal components from an isolated absorber edge show that the scattering can be approximated to first order by adding phased line sources to vertically propagating waves at the edges of the thin mask model. The ray tracing method was also extended to model the case of optical inspection of EUV masks. The Single Surface Approximation and ray tracing method produce nearly identical results, while FDTD suffers from numerical errors due to the abnormally high cell densities of 700 cells per wavelength.} }

EndNote citation:

%0 Thesis %A Lam, Michael %T Fast Simulation Methods for Non-Planar Phase and Multilayer Defects in DUV and EUV Photomasks for Lithography %I EECS Department, University of California, Berkeley %D 2005 %8 December 19 %@ UCB/EECS-2005-28 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/EECS-2005-28.html %F Lam:EECS-2005-28