Computational Lithography: Forward Optical Imaging Simulation and Inverse Source Mask Optimization

193nm ArF  immersion lithography is widely used in the semiconductor industry to  manufacture integrated circuits whose critical dimensions are much smaller than  the source wavelength. The manufacturing process is similar to creating thin  lines with a broad brush, which is physically beyond the resolution limit.  Computational lithography (CL) enables this process and guarantees high volume  production as well. It has been a driving force of Moore's Law. Computational  lithography models the lithography process, including the imaging system  (illumination source, mask, and projection lens) and the manufacturing process  (such as exposing and etching), and mathematically performs mask division and  correction, source optimization, and lens aberrations adjustments to enhance  the resolution. The success of computational lithography relies heavily on two  main procedures involved, namely, the fast and accurate forward optical imaging  simulation and the fast and robust inverse source mask optimization strategy. We  have carried out a series investigations to address these issues.


(1) Fast and accurate forward optical imaging  simulations

Fast optical image simulation by transmission cross-coefficient  decomposition with analytical kernels. Optical image simulation is one of the key parts in a  model-based optical proximity correction (OPC) technique. To improve its  computational efficiency, we propose a fast simulation method by decomposing  the transmission cross-coefficient (TCC) into analytical kernels. The TCC  matrix is projected onto a function space whose basis consists of analytical  circle-sampling functions (CSFs) and is then converted into a much smaller  projected matrix. By performing singular value decomposition (SVD) to the  projected matrix, its eigenvectors, together with the CSFs, are used to  generate a set of analytical kernels. This method avoids the need to directly  perform SVD to the large TCC matrix, making it much more runtime efficient.  Furthermore, the grid size of the kernels can be flexibly set to any desired  value in optical image simulations.

Fast optical image simulation using one basis mask  pattern. We propose an optical  image simulation method by using one basis mask pattern to generate a lookup  table, where the convolutions of the basis pattern with the kernels are  pre-calculated and stored. A rectilinear polygon mask pattern used in  integrated circuit layouts can be decomposed into several shifted basis  patterns. Its convolutions with kernels for use in optical image calculation  can then be quickly obtained from the pre-calculated lookup table by applying  the translation-invariant property of two-dimensional convolution. Simulations  conducted by using the proposed approach have demonstrated superior quality in  the fields of optical image calculation and OPC.


(2) Vector imaging simulations using the convolution-variation  separation method

Efficient optical imaging simulation with process  variations. We propose a  new method called convolution-variation separation (CVS) to enable efficient  optical imaging calculations without sacrificing accuracy when simulating  images for a wide range of process variations. The CVS method is derived from  first principles using a series expansion, which consists of a set of  predetermined basis functions weighted by a set of predetermined expansion  coefficients. The basis functions are independent of the process variations and  thus may be computed and stored in advance, while the expansion coefficients  depend only on the process variations. Optical image simulations for defocus  and aberration variations with applications in robust inverse lithography  technology and lens aberration metrology have demonstrated the main concept and  advantage of the CVS method.

Vector imaging simulation based on efficient  representation of mask transmittance function. We propose a generalized method to efficiently  represent the incident-angle-dependent mask transmittance function (MTF) of a  thick mask. This method expands the MTF into a series expansion, which consists  of a set of predetermined basis functions weighted by a set of predetermined  expansion coefficients. The predetermined basis functions are independent of  the incident angles and thus may be computed offline and stored, while the  expansion coefficients depend only on the incident angles and can be rapidly  calculated online. Near field and optical image simulations of thick masks have  demonstrated the excellent accuracy and superior speed performance of the  proposed method. We believe this work will present a new perspective aimed at  significantly improving the speed of thick mask simulations without serious  loss of accuracy, which has great potential for applications in optical  approximation correction (OPC) and inverse lithography technology (ILT).


(3) Fast and robust inverse source mask  optimization algorithms

Cascadic  multigrid algorithm for fast inverse mask synthesis. We propose a cascadic multigrid (CMG)  algorithm for fast inverse mask synthesis, which starts from a relatively  coarse mask grid and refines it iteratively in stages, so as to achieve a significant  increase in speed without compromising numerical accuracy. As a result, our  algorithm achieves more than four times the increase in speed over conventional  methods that synthesize a mask on a fixed fine grid.

     Mask  manufacturability enhancement methods. We propose a level-set method to solve the inverse  mask synthesis problem where the boundary of the mask pattern is iteratively  evolved instead of the mask itself.  It  results in a smooth mask contour. In addition, we propose a new regularization  framework for inverse lithography that regularizes masks directly by applying a  mask filtering technique to improve computational efficiency and to enhance  mask manufacturability. This technique is different from the conventional regularization  method that regularizes a mask by incorporating various penalty functions to  the cost function.

     Statistical  strategy for robust inverse mask synthesis. As critical dimensions shrinks, the pattern  density of integrated circuits becomes much denser, and lithographic process  variations become more pronounced. In order to synthesize masks that are robust  to process variations, the average wafer performance with respect to process  fluctuations is optimized. This approach explicitly takes into account process  variations. Furthermore, we investigate the impacts of arbitrary statistical  distribution of process variations on the synthesized mask patterns.

     Derivative-free  optimization for source optimization under a rigorous simulation model. We perform source optimization (SO) in  optical lithography under a rigorous simulation model which considers critical  effects, such as the vector nature of light and mask topography. We propose a  new source pattern representation method, which has moderate parameter  variations but remains complete in the solution space. Then, we develop a  derivative-free optimization (DFO) method to optimize these parameters under a  rigorous simulation model. Unlike gradient-based techniques, DFO methods do not  require a closed-form formulation of the model and are independent of the form  of cost function.

Imaging of a binary grating object in a lithographic tool

Fig.  1. First six kernels derived by the proposed TCC decomposition method.

The impact of aberrations on imaging

Fig. 2.  Decomposition of any rectilinear polygon mask pattern into the sum of several  shifted basis patterns according to their contributions.

Correlation plots between simulated and calculated shifts for all the inputaberrated wavefronts under σ = 0.31. (a) Plot of predicted phase shifts versus simulated phase shifts.(b) Plot of predicted axial shifts versus simulated axial shifts

Fig. 3. Separation and efficient simulation of image  intensity under lens aberrations.

Simulation result of the measurement errors of Zernike coefficients for all the input aberrated wavefronts

Fig. 4.  Simulation results of the optical image intensity and its decomposition into  different terms by using the method of incident-angle-dependent mask transmittance function (MTF).

>Agreement between the input and measured aberrated wavefronts

Fig. 5. Simulation  results of the cascadic multigrid algorithm. (a) Desired pattern. (b) The  synthesized mask pattern on the coarsest space. (c) The interpolated mask  pattern of (b) to the less coarse space. (d) The synthesized mask pattern on  the less coarse space. (e) The interpolated mask of (d) to the finest space.  (f) The synthesized mask pattern on the finest space.

Characterization of analytical kernels for measuring <br /> odd aberrations under a smooth conventional illumination<br /> (n_odd=2, 3, 7, 8, 10, 11, 14, 15, 19, 20, 23, 24, 26, 27, 30, 31, 34, 35)

Fig. 6.  Some intermediate results obtained in the iteration process using the mask filtering  technique.

Characterization of analytical kernels for measuring <br />even aberrations under the smooth conventional illumination<br /> (n_even=4, 5, 6, 9, 12, 13, 16, 17, 18, 21, 22, 25, 28, 29, 32, 33, 36, 37)

Fig. 7. Simulation  results under different process distribution. (a) The synthesized mask pattern  under the exposure distribution (b) and defocus distribution (c), (d) is its  E-D window. (e) The synthesized mask pattern under the exposure distribution  (f) and defocus distribution (g), (h) is its E-D window.

Characterization of analytical kernels for measuring <br />even aberrations under the smooth conventional illumination<br /> (n_even=4, 5, 6, 9, 12, 13, 16, 17, 18, 21, 22, 25, 28, 29, 32, 33, 36, 37)

Fig. 8. Simulation  results for a brick array pattern using the derivative-free optimization method.  (a) Regular brick array. (b) Initial source. (c) The optimized source. (d) The  contour comparison between the annular source (b) and the optimized source (c)  on the wafer side. (e)The process latitude of the optimized source (c). Process  latitude is unavailable with the annular source (b). (f) Convergence history.

Selected papers

  1.                 W.  Lv, S. Y. Liu, X. F. Wu, and E. Y. Lam, "Illumination source optimization in optical lithography via derivative-free optimization,"  J.  Opt.  Soc.  Am. A,  31(12), B19-B26 (2014). (URL, PDF)

  2.                 X. J.  Zhou, C. W.  Zhang, H. J. Jiang, H. Q. Wei, and S. Y. Liu, "Efficient representation of mask transmittance functions for vectorial lithography simulations," J.  Opt.  Soc.  Am. A,  31(12), B10-B18 (2014). (URL, PDF)

  3.                 W. Lv, E. Y. Lam, H. Q. Wei, and S. Y. Liu, "Cascadic  multigrid algorithm for robust inverse mask synthesis in optical lithography," J. Micro/Nanolith. MEMS MOEMS 13(2), 023003  (2014). (URL, PDF)

  4.                 S. Y. Liu, X. J. Zhou, W. Lv, S. Xu, and H. Q.  Wei, "Convolution-variation separation method for efficient modeling of optical  lithography," Opt. Lett. 38(13),  2168-2170 (2013). (URL, PDF)

  5.                 W. Lv, Q. Xia, and S. Y. Liu, "Mask-filtering-based  inverse lithography," J. Micro/Nanolith.  MEMS MOEMS 12(4), 043003 (2013). (URL, PDF)

  6.                 W. Lv, S. Y. Liu, Q. Xia, X. F. Wu, Y. Y. Shen,  and E. Y. Lam, "Level-set-based inverse lithography for mask synthesis using  the conjugate gradient and an optimal time step," J. Vac. Sci. Technol. B 31(4), 041605 (2013). (URL, PDF)

  7.                 P. Gong, S. Y. Liu, W. Lv, and X. J. Zhou, "Fast aerial image  simulations for partially coherent systems by transmission cross coefficient  decomposition with analytical kernels," J.  Vac. Sci. Technol. B 30(6), 06FG03 (2012). (URL, PDF)

  8.                 S.  Y. Liu, W. Liu, X. J. Zhou, and P. Gong, "Kernel-based parametric analytical  model of source intensity distributions in lithographic tools," Appl. Opt. 51(10), 1479-1486 (2012). (URL, PDF)

  9.                 S. Y. Liu, X. F. Wu, W. Liu, and C. W. Zhang, "Fast aerial image  simulations using one basis mask pattern for optical proximity correction," J. Vac. Sci. Technol. B 29(6), 06FH03 (2011). (URL, PDF)