Monte Carlo method for highly efficient and accurate statistical lithography simulations

Recent years have shown a strong increase in the use of statistical lithography error analysis for process tuning and in making technology choices. Simulation has shown it can play an important role in this area by accurately predicting experimental critical dimension (CD) distributions. Earlier statistical lithography simulation work was based on the Response Surface Methodology. The response surface is built by simulating CD dependence on input lithography process variables of interest such as focus, dose, mask CD, resist thickness, etc. The process parameters are then sampled from the Gaussian distribution to generate the distribution of the resulting resist CDs. When a large number of input parameters are being considered in order to describe the important experimental variations, the computational runtime is rapidly increased due to the requirements to fully simulate an (N+1)-dimensional response surface, where N is the number of input parameters. The work we present here has improved the speed of statistical lithography simulations through the use of Monte Carlo technique. With this technique, the runtime of the simulations is independent of the number of input parameters. The technique can be used for 1D or 2D simulations. We present results benchmarked with 130 nm process data showing the usefulness, runtime improvements and accuracy of this method. We have also used Variable Threshold Resist model (VTRM) in conjunction with the Monte Carlo technique. VTRM was calibrated against experimental focus-exposure matrices at varying line width and pitch. The use of VTRM greatly improves the accuracy of the statistical results by the virtue of establishing a good fit to the experimental data, which can be quantified by the root mean squares of residuals. VTRM also significantly speeds up the computation, since it uses only aerial image calculation as opposed to full resist modeling. Simulation results produced by using VTRM closely match the experimental results through a range of pitches, mask line widths and various illumination conditions.