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The demand for smaller, faster and lower power semiconductor devices continues to drive improvements in optical lithography. Currently very high numerical aperture (NA) exposure tools combined with resolution enhancement techniques (RET) are used to produce state of the art devices with critical dimensions (CD) less than 100 nm. For example, at the 45 nm node, some of the features to be imaged are less than a quarter of the wavelength of the 193 nm light source used, requiring the use of alternating phase shift masks (APSM). The associated pitches are sub-wavelength (~130 nm), which leads to severe proximity effects requiring optical proximity correction (OPC). These effects need to be understood using lithography simulation so that they may be taken into account during reticle design in order to achieve a predictable and reliable process. Lithography simulation can assist with improving device yields and reducing the number of reticle iterations, allowing a fabrication house to ramp products faster and save substantially in production costs.
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"As optical lithography pushes to the 22 nm node these types of tools are absolutely necessary. There would no way I could do the problems I'm presently doing without FDTD Solutions' multi-processor capability... I found it very easy to get parallel FDTD Solutions started. It actually worked so quickly at first that I thought I didn't have it set up properly!
B. Thorenson, ASML
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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.
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Lithography simulation setup in the Layout Editor
 Figure 2. Chrome binary mask shown in FDTD Solutions Layout Editor with graded mesh used for simulation |
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:
| • | The material properties of chrome (silver) and the quartz blank (blue) are defined in the built-in materials database. |
| • | The simulation region (orange) is shown along with the graded mesh used for the calculation. |
| • | A plane wave source is normally incident (purple arrow), polarized in the x dierction (blue arrow). |
| • | A field profile monitor (yellow) records the object field distribution transmitted through the mask. |
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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.
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Examine the object field intensity as calculated in FDTD Solutions
 Figure 3. Object field intensity transmitted through binary mask rigorously simulated using FDTD Solutions. |
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. Clearly a scalar, thin-mask model will not accurately describe many of the types of masks used in DUV lithography.
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Calculate the aerial image intensity for M=1, σ
=0, NA= 0.85
 Figure 4. Aerial image intensity for cross binary mask.
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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.
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Re-calculate for a M=4 projection lithography system
 Figure 5. Aerial image intensity for cross binary mask, M=4
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Determining the aerial image for different projection settings does not involve re-running the FDTD simulation. Post-processing the data for a 4x reduction system produces the plot to the left.
Note that Figure 5 and Figure 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.
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For more information on how FDTD Solutions can be used for lithography simulation, please see our white paper which includes an analysis of Alternating Phase Shift Masks.
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Summary
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.
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See how easily FDTD Solutions can assist you with your design efforts! Download a free 30 day trial and request that a technical expert contact you.
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