DUV lithography simulation of aerial images using FDTD Solutions

Step 1. Lithography simulation setup in the Layout Editor

As optical lithography techniques have continued to improve, so too have lithography simulation techniques improved. FDTD Solutions uses the finite difference time domain technique to rigorously solve for the object fields at the mask. All diffraction, refraction, interference, absorption and polarization effects are calculated in the near field of the mask without approximation. FDTD Solutions also incorporates a graded mesh, which greatly reduces memory requirements and time per simulation. By post-processing the FDTD simulation data, the aerial image at the wafer may be calculated. Several examples of how to do this are shown in what follows.

A chrome binary mask is shown as constructed in the Layout Editor of FDTD Solutions. The mask modelled consists of a periodic array of cross-shaped openings with CD = 2λ . The Layout Editor provides a comprehensive view of the structure to be modelled and the sources and data monitors used to perform the calculation.

Three-dimensional FDTD Solutions model of a chrome binary mask on a quartz substrate, making use of pre-defined dispersive material models from the material database.

Step 2. Examine the object field intensity as calculated in FDTD Solutions

Here, the graded mesh technology in FDTD Solutions provides nearly a 40x improvement in memory requirements and simulation time needed, compared to an equivalent uniform mesh. The rigorously calculated object field is shown below 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, as the feature sizes on the mask are on the order of the illumination wavelength and the thickness of the chrome layer itself (~100nm) is not optically thin relative to the wavelength. Clearly a scalar, thin-mask model will not accurately describe many of the types of masks used in DUV lithography.

Electric field intensity just below the chrome mask, inside the quartz.

Using the above object intensity profile, the aerial image intensity profile can be calculated for lithography parameters M=1, σ =0 with an imaging objective of NA= 0.85.

Even when imaging photomasks with no reduction when the CD is at 2λ , there is significant corner rounding and some line shortening in the aerial image.

Step 3. Re-calculate for a M=4 projection lithography system

Determining the aerial image for different projection settings does not involve re-running the FDTD simulation and instead the existing simulation results can be easily re-analyzed for different imaging settings.  Post-processing the data for a 4x reduction system produces the plot below. Note that both aerial images (above, and below) 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 and significant line shortening and corner rounding, the images 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.

Aerial image at 4X reduction, showing extreme line shortening and corner rounding.

As shown, FDTD Solutions uses rigorous electromagnetic simulation to accurately predict the aerial images produced by masks used in DUV lithography. Using the graded-mesh algorithms incorporated within FDTD Solutions, substantial memory savings can be realized in performing simulations of lithographic systems. These memory savings can be exploited to rapidly prototype smaller fields of view within the mask. Alternatively, these memory savings can be used to accurately simulate a much larger structure than would otherwise be feasible using a uniform mesh simulation. Based on these considerations, FDTD Solutions provides an expedient and accurate process by which aerial images can be calculated and optimized.