*Proceedings of the*

The 33rd European Safety and Reliability Conference (ESREL 2023)

3 – 8 September 2023, Southampton, UK

# Two Algorithms for Defect Detection in Wafer Fabrication

### Institute of Mathematical Stochastics, Technische Universität Dresden, Germany.

#### ABSTRACT

We propose two algorithms for defect detection in high-dimensional measurement data from wafer fabrication. The measurement data may be taken from different measurement steps and may include continuous values like electrical quantities and values from discrete value ranges like states of count registers.

The first of these algorithms is based on computing a similarity indicator of the form (\dfrac〈x,y〉H(x)) with several possibilities for the operator op, where x (device under test) and y (some sample chip of a randomly selected set) are obtained by scaling, followed by applying one of several thresholding methods.

The second of these algorithms is based on analyzing mode positions of the conditional distributions of positive and negative objects found in some random sample set. Based on this analysis, measurements can be ranked according to some specific definition of relevancy, which also implies a method for dimensional reduction. Then every non-sample chip is evaluated by counting for how many measurements of the selection the mode criterion is satisfied.

Both algorithms are designed in order to depend on only few variables to optimize over which makes finding an approximation of a global optimum, not just some local optimum, easy. For each of the two algorithms, we present an algorithmic method for dimensional reduction.

The implementations of these algorithms have been applied to measurement data of more than 100,000 chips from real-world wafer fabrication of different semiconductor products in course of the iRel40 project. A selection of results can be presented, where we are using Cohen's Kappa value, accuracy, sizes of sample sets and the necessary number of measurements as measures for success.

One ability of the second algorithm is learning to detect defects in certain lots of front-end measurement data, whereby optimizing for minimum sample set size and minimum number of measurements by dimensional reduction may show surprisingly small numbers. The time complexity of both algorithms is essentially multi-linear up to logarithmic factors.

Furthermore, we can give some stochastic analysis of the defect indicator underlying the first algorithm, whereby the subject of analysis is the quotient of two integer random variables, leading to approximation formulas consisting of summed exponentials. We also defined a modified indicator allowing for somewhat simpler expressions in analysis.

*Keywords: *Wafer fabrication, Data analysis, Defect detection, Dimensional reduction, Highdimensional, Mode analysis.