TDA applied to wafer defects

Topological data analysis (TDA) typically shines on data that has a shape and structure. In nature, such patterns arise constantly. Proteins have cavities and tunnels, galaxies form filaments, and porous materials are characterized by their network of interconnected channels. Although these kinds of structure do contain some amount of randomness (i.e. a stochastic process that underlies the formation), it seems that Nature prefers shape over uniformity. Only through shape can the intricate structures that dictate our biology even be formed.

But shape not only arises in our own chemistry, it also comes in one of humanity’s most complex creations: semiconductor manufacturing. In semiconductor manufacturing, electronic circuits are built directly onto a thin slice of crystalline silicon called a wafer. In this process, the wafer is coated with a light-sensitive chemical and is then exposed to light through a patterned mask. This transfers an incredibly fine circuit design onto the surface. These are essentials of a process called lithography.

Defects in lithography#

One of the biggest challenges in lithography is the occurrence of defects. Defects appear on a chip-by-chip basis across the wafer, together forming a defect pattern. As a topologist, I see patterns as a sign of structure; and where there is structure, topology is the right tool to invoke. The ability to classify these defect patterns, and to connect them to their corresponding root causes, has real value. This is because wafers with similar defect distributions may point to a common problem in a particular step of the process.

In a paper by Seungchan Ko, Dowan Koo [1], tools from topology were used for defect pattern classification. Quite crucially, the authors point out that the usage of convolutional neural networks (CNNs) has some drawbacks in the task of pattern classification. Among the drawbacks they mention that:

• CNN-based models contain a large number of parameters and are computationally expensive
• the performance of classifiers highly depends on the amount and quality of the training data because they are based on the supervised learning algorithm
• the training data might not be balanced.

The last point is rather important, for the following reason: defects occur as rare events. Many machine learning methods implicitly assume reasonably balanced classes, but when trained on skewed data they tend to optimize for the common case. In practice, this means that a model can achieve high accuracy simply by predicting “no defect” almost every time, while failing at the actual task of catching the rare defects that actually matter. To make matters worse, defective patterns stem from different causes in the manufacturing process. Hence, the dataset is unbalanced to begin with.

To overcome these obstacles, it could be advantageous to represent the initial data in a completely different way. In [1], the authors suggest to extract a topological feature from the given defect pattern on a wafer, represent it as a so-called persistence image, and only then use these PIs as inputs of neural networks for classification.

Persistence image#

In this post, I will assume that you know how to construct a persistence diagram (PD) from a multiset of points in $\mathbb{R}^2$ . Recall that in a PD, each point is given by a pair $(b,d)$ standing for (birth, death). The authors define a map $T(x,y) = (x,y-x)$ . Any point will now be of the form $(b,d-b)$ . This means that points close to the horizontal axis are short-lived noise, while points higher up are more robust or persistent.

This is just a coordinate transformation; nothing has happened yet to make machine interpretability easier. The authors then turn the point set into a continuous surface. Instead of having one “spike” for every point, the authors blur out points by using Gaussians. But not each Gaussian should be as tall: persistence should be emphasized, while noise should be close to flat. This is where a weighting function $f(u)$ comes in. The chosen $f$ is zero on the horizontal axis and grows with persistence. Assuming $\phi_u(z)$ to be a Gaussian bump for a point $u$ , the persistence surface is given by the weighted sum

This produces a smooth surface, but to be able to put this into a machine learning model, the surface should be discretized. The authors do this using integration.

The result is a persistence image, which is then represented as a vector, used as the input of a machine learning model. The entire pipeline has been illustrated in Figure 1, which is the original image from [1].

Testing in experiments#

To test the performance of the proposed method, the authors used simulated datasets based on real data that resulted from experiments performed in other research papers. The authors are able to classify defects with 99% accuracy, improving over CNN-based models that resulted in 93.8% accuracy. While this difference is modest, the authors point out that their model outperforms in a real manufacturing process because their model:

• is much faster for real-time classification
• is much more efficient, allowing for fast retraining in accordance with the manufacturing situation and the engineers’ opinions.

What to learn from this#

Machine learning heavily relies on the data that is inserted. By using techniques from topology before training, much of the unnecessary data is removed from the system, so that the model can be trained on the type of data that really matters. This reduces time and improves efficiency. Topology saves the day!

References#

1. S. Ko and D. Koo, A novel approach for wafer defect pattern classification based on topological data analysis, Expert Systems with Applications, vol. 231, p. 120765, 2023. doi:10.1016/j.eswa.2023.120765