Lithography, a crucial process in the semiconductor industry, is at the core of modern electronics manufacturing. It involves using light to define and pattern intricate structures of a circuit, enabling the creation of integrated chips with billions of transistors. Achieving the necessary nanometric precision in lithography requires a finely tuned system with optimized masks and optical components. Aerial image simulations play a vital role in ensuring defect-free manufacturing during mass production. However, these simulations are computationally intensive and time-consuming.
A research team from Taiwan has recently addressed this challenge and developed an efficient solution with minimal drawbacks. Their study, led by Professor Tsai-Sheng Gau from National Tsinghua University, Taiwan, was published in the Journal of Micro/Nanopatterning, Materials, and Metrology.
To understand their approach, it’s helpful to know about the Fourier transform (FT), which maps a signal in space to a signal in the spatial frequency domain and vice versa. In aerial image simulations, the FT facilitates the calculation of light interactions with the lithography system. Applying an inverse FT to the transformed function yields the simulated aerial image in the space domain.
However, FT calculations are typically time-consuming due to their nature. Aerial image simulations can take several days to complete. To overcome this hurdle, the researchers explored the use of the fast Fourier transform (FFT), which calculates the FT in fewer steps. However, FFT requires the input data size to be in powers of two (64, 128, 256, etc.). Unfortunately, the wavelength of light used in lithography does not match this requirement, making FFT inapplicable.
Interestingly, the team found a workaround by mathematically analyzing a scaling factor. By applying this scaling factor to the lithography mask, the wavelength effectively scales down to a power of two, enabling the use of FFT. After the calculations are done, the inverse FFT is computed, and the obtained aerial image is scaled back to recover the original wavelength.
This approach yielded a significant improvement in computation speed. “Compared to the original FT, we noted a computation speed improvement of 4,000–5,000 times, with a slight intensity deviation of about 3%,” said Gau. The researchers also conducted tests on various cases, consistently obtaining speedup and error reduction with the new algorithm.
The implications of this work for lithography, research, and industrial applications are significant. “Our algorithm is simple and straightforward to implement on popular commercially available platforms,” Gau explained. “For schools and research organizations with limited resources for purchasing speedy computation equipment or industrial simulation packages, this paper provides a powerful algorithm for converting traditional FT to FFT, saving enormous computational costs.”
This study has the potential to revolutionize lithography, leading to improved devices, cost reductions, and advancements in manufacturing technology across the electronics industry.