A Digital Approach to via Edge Roughness Characterization and Quantification

3.1 Experiments Setup

A grayfield through silicon via (TSV) inspection system was constructed as shown in Fig. 2(a), which features enhanced sensitivity to scattered light and remarkable diffractive image contrast. A collimated laser (λ 450 nm) was utilized as the light source. The laser was further shaped into a donut shape by the two axicon lenses installed below it, as depicted in Fig. 2(b). To reduce the intensity saturation in the imaging device, a 12.5% neutral density (ND) filter was applied. The filtered donut-shaped beam was then formed into a spherical wavefront through the first objective lens. Once the light interacted with the edge of the TSV, the Charge-coupled Device (CCD) on the bottom could capture the diffraction images. In this study, all diffraction images were captured from φ100 μm TSVs fabricated on 210 μm thick undoped single-crystalline silicon wafers.

Fig. 2

Schematics of grayfield Si-via inspection system

3.2 Simulation Model

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 [21,22] and other devices and media, such as waveguide junctions [23], optical switches [24], optical fiber amplifiers [25] and atmosphere with turbulence [26]. The BPM assumes that the envelope of the optical wave would not change abruptly during the propagation [23]. Therefore, the transverse electrical field density of the optical wave traveling in z-direction is represented as:

where Ψ_t is the Slowly Varying Envelope (SVE) function, n₀ is the reference refractive index, and k₀ 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 [23]:

where n is the refractive index of the media. After giving the initial condition Ψ_t(z = 0) = Ψ₀(x,y), the Ψ_t cou.ld be solved for each z value by finite difference method in the alternative direction implicit (ADI) scheme. In this study, the initial condition was set as the Bessel-Gaussian function to represent the laser beam generated by passing a Gaussian beam through a pair of axicon lenses [27,28]:

where w₀ is the size of the beam waist, ρ=x2+y2 is the polar coordinate, A_m, m and a are constant values. By choosing different combinations of w₀, A_m, m and a, various types of laser beams, including Gaussian beam and Donut-Beam, could be represented.

The simulation of BPM started and ended at the top (z = 0) and bottom (z = h) of via sample, respectively. Once the field value U₀(x,y) = Ψ_t(z = h)·e^{−in₀k₀h} was obtained at the bottom, the scalar diffraction theory was used to compute the diffraction pattern received by the CCD. The expression for computing the diffraction filed after propagating a distance Z is written as [29]:

where ℑ and ℑ⁻¹ are Fourier transform and inverse Fourier transform, f_X and f_Y are frequency variables associated with x and y respectively. The diffraction pattern on the CCD could be computed as [29]:

where I(x,y) represents the irradiance of the light field, which is the time-averaged radiometric energy flux per unit area.

3.3 Training Set Preparation

Fig. 3 depicts the workflow of training datasets processing. After the acquisition of diffraction patterns through interferogram, image segmentation was conducted in a 10-degree interval. Subsequently, the measured diffraction pattern of each segment was extracted and compared to the BPM simulation fringe in ideal condition to obtain the similarity value (SI). Diffraction fringes of multiple edge conditions with different VER values were further simulated using BPM methods, which was used to compared with the ideal fringe. Thus, the relationship between VER and SI was obtained, and this calibration result was used to estimate the VER of experimental fringe pattern. This method was described in detail in a previous work [17].

Fig. 3

Schematics of the establishment of training datasets

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.

3.4 Model Optimization

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 Fig. 4, which consists of 2 convolutional layers, 2 max-pooling layers, and 1 fully connected dense layer. In corresponding sub boxes of Fig. 4, @ denotes the number of feature maps, and × represents the dimensions of the filter or pooling window. During the training process, the adaptive learning rate strategy was applied to reduce the training time needed to reach convergence.

Fig. 4

The final architecture of the CNN model

4.1 CNN Prediction Results

Fig. 5 shows the CNN prediction results of five experiments, allowing an intuitive evaluation of its performance. In each subplot, the blue points represented the relationship between the actual value and CNN’s predicted value, while the red dashed line represented the ideal prediction scenario where the predicted value equaled the actual value (Y = X). These charts visualized the prediction accuracy: the closer the data points were to the red line, the higher the accuracy of the predictions. As shown in Fig. 5, in all five scenarios, blue points were scattered in a region close to the red reference line along the whole VER range, indicating the CNN model was capable of making relatively precise VER predictions.

Fig. 5

Predicted vs actual scatter plot results of five datasets

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.

4.2 Additional Multilayer Perceptron (MLP) Results

The MLP, also a neural network-based model, is widely used to solve classification and prediction problems [30–32]. In the MLP model, all the layers, including the input, hidden, and output layers, are fully connected layers [33]. As shown in Fig. 6, the MLP consists of one input layer, one output layer and 4 hidden layers. The hyper parameters of the MLP were also tuned through a 10-fold cross-validation for achieving the best performance. This relatively simple model structure can make the model training and prediction process more efficient by reducing the number of parameters. However, this smaller number of parameters limits the model’s capability for complicated tasks such as complicated image and video processing [34].

Fig. 6

The final architecture of the MLP model

A comprehensive comparison of the prediction accuracy and computation time between CNN and MLP was conducted, as shown in Fig. 7. From the results, CNN showed a significant advantage in prediction accuracy: with a lower average RMSE and a better MADR. The average RMSE of MLP results was 0.0872, 19.78% higher than that of CNN. Similarly, the average MADR results for MLP was 2.507%, 10.249% higher than that of CNN. In semiconductor industry, the TSVs provide electrical connection among multiple layers, even a small increase in VER could result in attenuation and distortion of the signal being transferred, due to the skin effect. Therefore, a small difference in prediction performance can have huge impacts in engineering practice.

Fig. 7

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

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.