Accelerating aerial image simulations for optimal lithography

Newswise — Lithography, a crucial procedure in the semiconductor field, serves as the foundation for manufacturing technology in today's electronics. Essentially, it is a printing method that utilizes light to precisely define and shape the intricate structures of a circuit. The remarkable nanoscale precision achieved through lithography enables the creation of advanced integrated chips containing billions of transistors.

Nevertheless, attaining such high precision demands an exquisitely calibrated and optimized lithography system, comprising meticulously crafted masks and optical components. This is where the significance of aerial image simulations becomes apparent. These simulations offer engineers a valuable approximation of the pattern that must be projected onto the silicon substrate as light traverses through masks and lenses. By utilizing these simulations, engineers can ensure that the final product will be devoid of manufacturing defects when produced in large quantities. However, a notable challenge associated with these simulations is their computational complexity and time-consuming nature.

Great news! A team of researchers from Taiwan has recently discovered an ingenious and highly effective resolution to overcome this obstacle. In their most recent publication in the esteemed Journal of Micro/Nanopatterning, Materials, and Metrology, they unveil a remarkably simple approach to significantly accelerate aerial image simulations, all without any noticeable drawbacks. This groundbreaking study was spearheaded by Professor Tsai-Sheng Gau, hailing from Taiwan's prestigious National Tsinghua University.

To comprehend the essence of their approach, it is beneficial to grasp the concept of a Fourier transform (FT). In basic terms, the FT is a mathematical operation that converts a signal in space into a signal in the spatial frequency domain, or vice versa. When a signal is transformed into the frequency domain, the information from the original signal in space is represented as a combination of sinusoidal waves with varying frequencies. In the context of aerial image simulations, the FT is utilized to simplify the calculation of how light interacts with the lithography system. By applying the effects of the system to the transformed function and subsequently performing an inverse FT, one can obtain the simulated aerial image in the space domain.

The researchers encountered a significant challenge when it came to the computation of the Fourier transform (FT) due to the time-consuming nature of the calculations involved. The conventional FT approach often necessitates several days for aerial image simulations. In order to overcome this obstacle, the researchers explored the potential of utilizing the fast Fourier transform (FFT) instead. In simple terms, the FFT algorithm performs the FT calculation in a significantly reduced number of steps. However, there is a limitation to its applicability: the input data size must be a power of two (e.g., 64, 128, 256, etc.). Unfortunately, in the context of aerial image simulations, the wavelength of the light employed in lithography does not conform to this requirement, rendering the use of FFT impractical.

Fascinatingly, the research team discovered a workaround to address this problem. By conducting mathematical analysis, they derived a scaling factor that, when applied to the lithography mask, effectively reduces the wavelength to a power of two. This ingenious technique enables the utilization of FFT despite the initial mismatch. Once the necessary calculations are carried out, the inverse FFT is computed, and the resulting aerial image is appropriately scaled back to restore the original wavelength. This innovative approach allows for the efficient application of FFT in aerial image simulations, significantly reducing computation time while preserving the accuracy of the results.

By employing this novel approach, the researchers achieved a remarkable advancement in computation speed. Professor Gau expressed, "When comparing it to the conventional FT method, we observed a staggering improvement in computation speed by a factor of 4000–5000, accompanied by a marginal intensity deviation of approximately 3%." Furthermore, the team extensively tested the performance of their algorithm across various test cases, consistently obtaining exceptional results in terms of both speedup and error reduction. This underscores the reliability and effectiveness of their innovative solution.

The impact of their work on lithography, both in the realm of research and industrial applications, is substantial. Professor Gau emphasized the practicality of their algorithm, stating, "Our algorithm is easy to implement on commonly used commercially available platforms." He further added, "This paper presents a powerful algorithm for converting traditional FT to FFT, offering a cost-saving solution for schools, research organizations, and industrial settings with limited resources for high-speed computing equipment or expensive simulation packages." These findings provide a valuable tool that enables more accessible and efficient computational capabilities, benefiting a wide range of users and facilitating advancements in the field of lithography.

Undoubtedly, the fundamental role of aerial image simulation in lithography, and its broader impact on the field of electronics, cannot be overstated. The groundbreaking study conducted by the research team holds immense potential in paving the way for numerous advancements. The improved efficiency and speed of aerial image simulations achieved through their innovative approach can have far-reaching implications. These implications include the potential for the development of superior devices, reduced manufacturing costs, and even breakthroughs in manufacturing technology. By enabling more accurate and efficient simulations, this study contributes to pushing the boundaries of what is achievable in lithography and electronics as a whole, opening doors to new possibilities and driving progress in the field.

Read the paper by Gau et al., “Ultra-fast aerial image simulation algorithm using wavelength scaling and fast Fourier transformation to speed up calculation by more than three orders of magnitude,” J. Micro/Nanopattern. Mater. Metrol. 22(2) 023201 (2023). doi: 10.1117/1.JMM.22.2.023201.