This paper introduces a new digital integration that combines edge diffractometry with convolutional neural networks (CNN) for via metrology and inspection. The beam propagation method (BMP) was used to simulate the interferogram generated by edge diffractometry to characterize via edge roughness (VER). A comprehensive database was established to link different fringe patterns to VER for CNN training. The well-trained CNN-based methodology provided a fast and accurate assessment of VER, with a root mean squared error (RMSE) of 0.073 and an average mean absolute deviation ratio (MADR) of 2.274%. In addition, the proposed digital approach was compared to the multilayer perceptron machine (MLP) in terms of computational efficiency and predictive accuracy. The proposed digital integration greatly improved the accuracy and speed of VER measurement, characterization, and quantification, potentially enhancing device yield and reliability. The successful application of this digital approach could open up possibilities for various types of via or pattern metrology.

Via metrology and inspection technology are vital to verify that the vias produced through machining, laser drilling, or chemical etching meet their designated specifications such as diameter, circularity, roundness, and edge roughness conditions [

With the advancement of artificial intelligence (AI), Convolutional Neural Networks (CNNs) can automatically conduct feature extraction from images and predict performance of semiconductor component [

Inspired by previous studies [

CNN is widely used in computer vision and image processing since it simulates the cognitive pattern of the human brain [

The proposed digital approach consists of two major steps: the first step involves the establishment of a database that links distinct fringe patterns with their corresponding edge roughness conditions. The second step focuses on the setup of neural networks, including the training of models and optimization of hyperparameters. Details of each step will be discussed in the following sections.

A grayfield through silicon via (TSV) inspection system was constructed as shown in

The BPM was used to simulate the laser propagation inside the via sample. This method is widely applied for simulating electromagnetic (EM) wave propagation through waveguides [

where _{t}_{0} is the reference refractive index, and _{0} is the wave number of the laser in the vacuum.

By substituting this expression into the Maxwell equation and applying the paraxial approximation, the governing equation for a scalar wave propagation can be written as [

where n is the refractive index of the media. After giving the initial condition _{t}_{0}(_{t}

where _{0} is the size of the beam waist, _{m}_{0}, _{m}

The simulation of BPM started and ended at the top (z = 0) and bottom (z = h) of via sample, respectively. Once the field value _{0}_{t}^{−}^{in}^{0}^{k}^{0}^{h}

where ℑ and ℑ^{−1} are Fourier transform and inverse Fourier transform, _{X}_{Y}

where

After performing the same operation on each fringe segment, a dataset containing diffractive images and its associated VER was established, consisting of 1,872 entries total. The dataset served as input for training the CNN model, aiming for accurate edge roughness prediction based on precise pattern recognitions. In this study, five experiments were conducted: in each experiment, 100 samples were randomly selected as the testing set while the remaining 1,772 samples served as training sets.

In order to improve the prediction accuracy and obtain the best CNN model, the hyperparameters, including the number and properties of different types of layers, were optimized in this study. Both perdition accuracy (RMSE and MADR) and computational burden (training time) were taken into consideration for choosing the best model. The final architecture of the CNN is shown in

Two evaluation metrics were further used to numerically evaluate the results: the root mean square error (RMSE), and the mean absolute deviation ratio (MADR) which is computed as mean absolute deviation divided by average. The average RMSE for all testing sets was 0.073, with the first and second sets having values less than 0.069 while the highest, 0.082, was from the fourth set. Considering the average VER was more than 2.480 μm, all RMSEs were considered relatively small and reflected the precision of the predictions. Upon further calculation of the MADR, the average value obtained was only 2.274%, with the fourth set having the highest value at just 2.767%. This further demonstrated that CNN’s average predictive error is below 2.300%, being considered highly accurate.

The MLP, also a neural network-based model, is widely used to solve classification and prediction problems [

A comprehensive comparison of the prediction accuracy and computation time between CNN and MLP was conducted, as shown in

However, due to the simpler network structure of MLP, 23.144% less training time was needed, making it a more efficient inspection method in industrial applications. It is noteworthy that by comparing the standard deviation, CNN showed a more robust predictive performance, while MLP’s performance was significantly influenced by the training sets itself.

The choice between utilizing CNN or MLP should be guided by the specific industrial requirements, depending on whether the priority lies with achieving the highest accuracy possible or with efficiency and speed of computation.

Through integration of experimental and theoretical edge diffractometry with CNN, a novel digital approach to automatic via metrology and inspection was proposed in this paper. The electromagnetic wave propagation-based computational model of the via created a database of simulated fringe profiles, the experiment by the grayfield edge diffractometry was conducted to obtain the fringe pattern dataset to further compute VER. The CNN model was then trained to analyze various via diffractive profiles and their corresponding VER, automatically extracting subtle patterns from these fringe images. A further correlation between the fringe image features and VER showed the linear characteristics.

The proposed CNN-based digital approach showed a 0.073 RMSE and a 2.274% MADR in VER prediction results, proving its capacity in complex fringes analysis, offering a robust solution for the semiconductor industry’s demand for rapid and precise via inspection. An additional study on an MLP-based model offered a trade-off between prediction efficiency and accuracy.

In summary, the implementation of the proposed digital approach in via metrology is promised to improve the yield and reliability of semiconductor devices, further supporting the technological advancements in the industry. Future research could focus on further optimizing the AI models and exploring potential applications in other areas of precision engineering.

This research has been supported by the National Science Foundation (CMMI #212499).

All authors equally contributed to this study: design, experiment, and data analysis.

The data that supports the findings of this study are available from the corresponding author upon reasonable request.

The authors declare no conflicts of interest.

Comparison between AI-based digital approach and AI-IS hole inspection method

Schematics of grayfield Si-via inspection system

Schematics of the establishment of training datasets

The final architecture of the CNN model

Predicted vs actual scatter plot results of five datasets

The final architecture of the MLP model

Result comparisons: (a) average RMSE, (b) average MADR, and (c) average training time