Parametric curve-based representation of piecewise linear masks

Abstract

Background Efficient data representation of piecewise linear masks is essential for balancing mask accuracy and data compression. Traditional mask representations rely on piecewise-linear polygons, but parametric representations, such as Bezier and B-spline curves, offer potential advantages in reducing data size while maintaining accuracy. Previous studies have primarily focused on cubic Bezier and B-spline curves, limiting the exploration of higher-degree curves and different parameterization methods.

Aim We aim to investigate a broader range of Bezier and B-spline curve degrees and parameterization methods to identify the optimal representation that minimizes both mask error and data size.

Approach Target designs are corrected using the inverse lithography technique (ILT), and the least squares approximation is employed to fit Bezier and B-spline curves to the mask designs. Various curve degrees and parameterization methods are examined to optimize the approximation. The root mean square error (RMSE) and vertex reduction are assessed in comparison to the piecewise-linear representation. In addition, aerial image simulations are performed to evaluate the lithography errors introduced by mask conversion, specifically analyzing process variation bands and edge placement error distributions.

Results For logic metal designs, converting piecewise-linear polygons to seventh-degree Bezier curves resulted in an 87.5% vertex reduction with an RMSE of 0.5 nm. Similarly, second-degree B-spline conversion achieved an 86.2% vertex reduction with a lower RMSE of 0.1 nm. For logic VIA designs, the same conversion methods exhibited comparable trends. The seventh-degree Bezier representation yielded an 81.3% vertex reduction with an RMSE of 0.1 nm, whereas the second-degree B-spline achieved a similar reduction of 81.2% with the same RMSE. The aerial image simulations for Logic metal indicate that the errors introduced by both Bezier and B-spline representations are negligible.

Conclusions The results demonstrate that seventh-degree Bezier and second-degree B-spline curves provide significant vertex reduction while maintaining accuracy, making them viable alternatives to piecewise-linear representations. We highlight the potential of parametric representations in optimizing mask data size while maintaining comparable lithographic performance.