This page shows how to use FDTD to accurately predict the aerial images produced by masks used in DUV lithography.
A binary mask consists of a transparent plate covered with a patterned film of opaque material. The transmission characteristic is "binary" in the sense that the field transmitted is approximately "1" in the transparent region and "0" in the opaque region. This is an approximate description known as the scalar, thin-mask model. FDTD allows for the exact calculation of the transmitted field, and we can determine to what extent the scalar, thin-mask approximation is valid by examining the transmitted field intensity across the mask. DUV lithography masks are typically constructed from fused quartz for the transparent plate (or "blank") and a thin chrome layer for the opaque material.
We first consider the simulation of a chrome binary mask that contains a periodic array of cross shapes with CD = 2λ. Load the simulation project file litho_cross.fsp to review the setup of a chrome binary mask in the Layout Editor as above.
A plane wave source at a wavelength of λ=193 nm is incident on the mask at normal incidence. The chrome mask layer is 100 nm thick, on top of a fused quartz layer with n = 1.55. The material properties for chrome are defined in the material database (n=0.85; k=1.63 at λ=193 nm). The mask is periodic in x and y, with a period of 10λ. The FDTD simulation region has periodic boundary conditions in x and y, and absorbing boundaries/PML in z. A graded mesh is used with conformal variant 1 to very accurately resolve the extent of the chrome mask. A profile monitor is used to record the field transmitted through the mask as this is the object field to be imaged onto the wafer. The sweep module contains sweep settings so that both polarizations can be performed concurrently.
For this particular mask and its symmetric pattern, the memory requirements can be further reduced by a factor of 4 by using the appropriate symmetric boundary conditions in conjunction with periodic boundary conditions. Keep in mind that the symmetry settings must be changed between source polarizations.
Load the script file plot_incoherent.lsf and change the variable 'sn' to the appropriate sweep name. Set the magnification M=1 and run the script file to run the sweep and plot the results.
Object field intensity
After running the simulation we can look at the object field intensity at the mask. The rigorously calculated object field is shown here for x polarized incident illumination. Note that there is significant variation across the cross-shaped opening in the chrome mask layer. This is due to two common issues in DUV lithography, the feature sizes on the mask are on the order of the illumination wavelength and the thickness of the chrome layer itself (~100nm) is no longer "thin" relative to the wavelength. A scalar, thin-mask model may not accurately describe many of the types of masks used in DUV lithography.
Calculate the aerial image intensity for M=1, σ=0, NA= 0.85
Using the complementary simulation for the plane wave polarized in the y direction, the aerial image is calculated just above the wafer in air using a simple Fourier optics model of the lithography projection system. The script file performs the necessary analysis to calculate the aerial image for illumination with partial coherence σ=0 (coherent) and imaging optics with numerical aperture NAo= 0.85 and demagnification M=1. The resulting aerial image is plotted in the figure below.
One can see that even when imaging photomasks with no reduction (at M=1) when the CD is at 2λ, there is significant corner rounding and some line shortening in the aerial image.
Aerial image intensity, M=4
Many projection lithography systems use a reduction factor of 4 times from the object to the image. If we re-run the analysis script with a demagnification factor M=4, then we obtain the following aerial image for incoherent illumination.
Note that images for M=1 and M=4 are plotted on the same scale. While we can see the 4x reduction in the aerial image (i.e. four bright spots in each field in both the x and y directions), it does not faithfully reproduce the mask object; due to diffraction, the line shortening and corner rounding are extreme and the spots in the aerial image are round rather than cross-shaped. In addition, interference and proximity effects lead to non-zero intensity between the bright intensity spots. Clearly the cross-shape with CD = 2λ (on mask) is beyond the resolution limit of a binary mask in this type of 4X reduction project lithography system. This is because with 4X reduction, the CD feature size at the wafer is only λ/2.
In the example Lithography using alternating phase shift mask, we consider a different type of mask that can produce these small feature sizes.
Simplified Microscopic Imaging
Photolithographic projection to photoresist
Lithography using alternating phase shift mask