Scattering caused by the thickness of the absorber layer of the mask leads to deviations in deep ultraviolet (DUV) and extreme ultraviolet (EUV) lithography. Traditional lithography models are based on Hopkins' method and the thin mask approximation, wherein diffracted waves from the mask satisfy the Fourier transform of the mask patterns. As the aspect ratio of the absorbing layer on the mask increases, the mask thickness becomes a non-negligible factor in diffraction calculation. Thick absorbers change the diffracted waves, particularly along the edges of the pattern. To accurately predict the aerial image, a 3D mask model is proposed to correct the Hopkin's model using the mask near fields generated via rigorous electromagnetic simulation, such as the finite-difference time-domain (FDTD) method. The computational cost of rigorous electromagnetic simulations can be reduced using a fast mask near-field generation method based on rotation transformation and dimension reduction. By combining a 3D mask model with a fast near-field generation method, an accurate 3D mask lithographic model can be rapidly constructed. The 3D mask model offers the advantages of less runtime and high accuracy when dealing with cases involving complex source and any-angle mask patterns.

First, a rotation transformation was applied to any given case. The source was decomposed by polarization, and each component of the source was rotated with the incident edge in such a way that the incident angle of the light remained unchanged on the vertical or horizontal incident edge. Next, a 2D FDTD simulation was implemented by reducing the dimension of the simulation area along the incident edge. This 2D FDTD simulation generates a 1D accurate electromagnetic wave distribution near the mask absorbers, with the 1D accurate wave distribution describing the diffracted wave on the line perpendicular to the incident edge. Subsequently, the FDTD-generated 1D wave distribution was expanded along the incident edge and used to modify the thin mask approximated results to obtain a correct mask transmission function. Finally, the modified mask transmission function calculated the aerial images involving mask 3D effects.

Rigorous electromagnetic simulations are conducted via FDTD, wherein electromagnetic wave propagation is modeled by solving Maxwell's equations on Yee's grid. A 3D FDTD simulation is conducted with oblique incidence on a square with a 30-degree rotation angle (Fig.3). The mask absorber is set as MoSi with a 70 nm thickness and a refractive index of 2.2. The near fields after mask diffraction are calculated, with an emphasis on the fields on the marked line perpendicular to the edges. Subsequently, 3D FDTD simulations with a rotated source on Manhattan polygons are performed (Fig.4, 5). The incident angle of the wave on the observed edge is kept equal to that of the original case. Furthermore, near fields of y and z polarization components are calculated and weighted superposition is derived (Fig.6). Next, a series of 2D FDTD simulations with a rotated source and dimension reduction is conducted. Near fields along the line perpendicular to the observed edge are calculated, and superposition is applied. Finally, the 1D edge near fields extracted from 2D near fields generated via 3D FDTD is compared with the results generated directly via 2D FDTD (Fig.7). The agreement of the edge near fields demonstrates the applicability of the rotation and dimension reduction method. Furthermore, the 2D simulation shows advantages in runtime, taking 2 min, while the 3D simulation takes approximately 40 min.

A novel mask 3D imaging model for lithography is developed in this study. The principles of the Hopkin's model are analyzed to reveal that the mask transmission function can no longer be derived directly from the Fourier transform by considering scattering caused by the thickness of the absorber layers. The FDTD method is applied to calculate the near fields of absorbers to implement rigorous electromagnetic simulation. Experimental results show that the near fields on the line perpendicular to the observed edge can be calculated quickly with high accuracy by applying rotation transformation and dimension reduction. Any incident angle on an any-angle edge can be converted to a Manhattan case, and near fields can be computed rapidly using 2D FDTD simulation. These near fields are then used to derive a correct mask transmission function, and the modified Hopkin's approach calculates aerial images on the wafer. The method used in this paper has a shorter runtime when handling complex or even freeform illumination sources and masks with any-angle polygons. Furthermore, because near fields can be generated in advance and the corrected mask transmission function can be reused in the entire layout, this method becomes more practical in 3D mask image models for full-chip prediction. This helps foundries save time in the production flow.