Singular value decomposition of near-field electromagnetic data for compressing and accelerating deep neural networks in the prediction of geometric parameters for through silicon via array

Abstract

In this paper, we propose a singular value decomposition-based deep learning model to investigate the inverse problem between simulated near field electromagnetic data and the geometric parameters of through silicon via array. This is of great importance for predicting the critical dimensions of through silicon via in the semiconductor industry, and it becomes more challenging due to the decreasing size of through silicon via. Simulation of electromagnetic field data for various through silicon via arrays is used by the finite-difference time-domain method. We analyze the near-field electromagnetic intensity distribution of different geometric parameters, including critical dimensions such as depth, top diameter, bottom diameter, sidewall roughness, and bottom ellipsoid radius. Due to the sub-micron scale of the critical dimensions and the high aspect ratios, single-wavelength electric field data is insufficient for accurate predictions. However, due to its size, multi-wavelength electric field data presents a significant computational challenge. We employ singular value decomposition to compress the multi-wavelength electric field data to overcome this. By analyzing the dominant singular value components, we reduce the data volume to 4.56 % of its original size while preserving predictive accuracy. The compressed data is subsequently integrated with deep learning models for critical dimension prediction. We compare three model architectures and demonstrate that utilizing the largest singular values from 30-wavelength electric field data substantially improves the prediction of vertical critical dimensions, such as through silicon via depth and bottom ellipsoid depth. Specifically, the singular value decomposition-based deep learning model, which incorporates the largest singular values from 5-wavelength electric field data, reduces computation time by 34.88 % and decreases the mean absolute percentage error for through silicon via depth and bottom ellipsoid depth by 2.78 % and 6.60 %, respectively. The singular value decomposition based deep learning model, which uses the largest singular values from 30-wavelength data, further reduces the mean absolute percentage error for the depth and bottom ellipsoid depth of through silicon via by 2.86 % and 10.60 %. These findings underscore the efficacy of singular value decomposition-based multi-wavelength electric field data compression combined with deep learning, offering an efficient approach for managing large-scale electromagnetic simulations in through silicon via design. Our source code is available at https://github.com/AOI-Laboratory/EMDataSVD.