2.5 A Novel Multi-Sensor Spectral Alignment Procedure
2.5.2 PSO Calibration: Examples and Applications
Two simulated examples are presented in order to better illustrate the prob- lem and the solution methodology. These two examples are designed in order to match as closely as possible the characteristics of real spectra. As such the generated signals are characterised by having several peaks. The misalign- ment of these peaks between two spectrometers generates a cost function with several local minima. In the first example two signals with linear distortion are considered:
Example 2.5.1 (Linear Distortion). Consider two simulated signals f and g, with g(i) = f (s(i)) + , where is measurement noise and s(i) = 1.01i + 1.1, as depicted in Figure 2.20. We can construct f (t) and g(t), using the aforementioned spline interpolation procedure given by equations 2.38 and 2.39 respectively. Here, the associated cost function is given by
c(a0, a1) = X
k
|f (a1tk+ a0) − g(tk)|2+ Φ. (2.43)
By plotting the additive and multiplicative coefficients, as illustrated in Fig- ure 2.21 and Figure 2.22, it is observed that both have numerous local min- ima. Using PSO [53], it is possible to estimate the calibration function, ˆs(t), required to align f (t) and g(t), to a mean squared error (MSE) accuracy of
0.1058. This compares favourably to an initial error of 249.4048. The aligned signals are depicted in Figure 2.23.
50 100 150 200 250 300 350 400 450 −20 0 20 40 60 80 100 120 Channel Id Intensity f g
Figure 2.20: Simulated signals f and g, where signal offset is based on a linear function as described in Example 2.5.1.
Typically, the alignment characteristics between two non calibrated OES sen- sors exhibits a non-linear relationship. To model this behaviour two signals with quadratic distortion are used in the next example.
Example 2.5.2 (Quadratic Distortion). Consider the two signals as de-
scribed in Example 2.5.1. This signal f (i) is altered such that g(i) =
f (s(i)) + where s(i) = 0.00001i2+ 1.0008i + 3. As before, f (t) and g(t) may be constructed using the spline interpolation procedure given by equation 2.38 and equation 2.39 respectively. f (t) and g(t) are illustrated in Figure 2.24. Applying PSO to solve ˆs(t) as above, it is possible to align f (t) and g(t) with a MSE of 0.0879. This is compared with an initial MSE of 297.8234. The aligned signals are depicted in Figure 2.25, while Figure 2.26 illustrates the estimated calibration function.
To demonstrate the effectiveness of the proposed retrospective calibration methodology for OES, calibration is performed on a sample dataset from an industrial plasma etch chamber as a case study.
Example 2.5.3 (OES Calibration). The dataset, which consists of 60 time samples × 2048 OES channels (wavelengths), was collected from the cham- ber exhaust from two etching tools. Before retrospective calibration can be
−500 −40 −30 −20 −10 0 10 20 30 40 50 200 400 600 800 1000 1200 a0 cost
Figure 2.21: The cost function in Example 2.5.1 as a function of the additive coefficient. The multiplicative coefficient has been set equal to its true value.
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 200 400 600 800 1000 1200 1400 a1 cost
Figure 2.22: The cost function defined in Example 2.5.1 as a function of the multiplicative coefficient. The additive coefficient has been set equal to its true value.
50 100 150 200 250 300 350 400 450 0 20 40 60 80 100 120 Channel Id Intensity f (t) g(ˆs(t))
Figure 2.23: Calibrated f (t) and g(t) on the problem described in Example 2.5.1. The calibration function is estimated using PSO.
50 100 150 200 250 300 350 400 450 −20 0 20 40 60 80 100 120 Channel Id Intensity f (t) g(t)
Figure 2.24: Simulated signals f (t) and g(t), where signal offset is based on a quadratic function as described in Example 2.5.2.
50 100 150 200 250 300 350 400 450 0 20 40 60 80 100 120 Channel Id Intensity f (t) g(ˆs(t))
Figure 2.25: Calibrated f (t) and g(t) on the problem described in Example 2.5.2. The calibration function is estimated using PSO.
0 50 100 150 200 250 300 350 400 450 2.5 3 3.5 4 4.5 5 5.5 Channel Id Intensity
Real Calibration Curve Estimated Calibration Curve
Figure 2.26: The estimated calibration curve ˆs(t) obtained using PSO in Example 2.5.2.
applied between the two sensors, suitable time series signals need to be ex- tracted for each etching step. Using one sensor as a reference, a suitable time series signal for an individual step is extracted based on the statistical distance of each temporal sample within X, to the centre of all temporal samples of X for that particular step. This distance is calculated using the Mahalanobis distance [54]. The centrally distributed signal, ρ, corresponds to the minimum distance signal. This method is used to find the reference time series signal for each step, because the centrally distributed signal will correspond to the average behaving time series signal in that step. Once a centrally distributed signal, ρ, is found for a given step, α, the equivalent signal from the second sensor, σ, is given by
σ = argmin σ∈Xα p X q=1 |ρ(q) − σ(q)|2 . (2.44)
where Xα is the set of OES recordings for step α, on the second sensor. For each set of matched signals, ρ and σ, the calibration function ˆs(t) is estimated using the methodology described in Section 2.5.1. The estimated calibration curve , ˆs(t), for each process is depicted in Figure 2.27. When compared to the actual calibration curve, it is clear that the estimated calibration curve produced for step 2 is a better approximation compared to the one estimated for step 1. Applying the calibration curve produced for step 2, alignment across each step is achieved, as illustrated by Figure 2.28. The difference between ˆs(t) for step 1 and 2 highlights the impact of etching species on calibration estimation. Therefore, in a multi-step etching process where the etching species varies, certain spectral regions for each step may be more suited for alignment proposes.
The proposed methodology consists of several stages summarised in the fol- lowing algorithm:
Algorithm 2.5.1. OES alignment
1. Select one OES sensor as a reference sensor, and one as the target sensor.
2. For each process step, find the centrally distributed temporal OES recording from the reference sensor, ρ.
3. For each ρ, find the matching measurement from the target sensor, σ. 4. For each pair of matched signals, ρ and σ, construct a calibration curve
ˆ
200 400 600 800 1000 1200 1400 1600 1800 2000 0 1 2 3 4 5 6 Channel Id Intensity
Real Calibration Curve
Estimated Calibration Curve Step 1 Estimated Calibration Curve Step 2
Figure 2.27: The estimated calibration curve ˆs(t) obtained using PSO for each process step.
Figure 2.28: A portion of calibrated spectrum from two OES sensors, where the calibration function is estimated using PSO.
5. Construct a global calibration curve by combining suitable subsections from different process step calibration curves.
The etching species of a given step can have a significant impact on calibra- tion performance. Consequently, to achieve alignment across each process step it may be necessary to construct a global calibration curve from subsec- tions of individual step calibration curves.
2.6
Conclusion
This chapter provided the background material for the whole thesis. A basic introduction to the physics behind plasma etching is provided and the pro- cess behind OES measurements is described. It is shown how the raw data collected during production needs to be cleaned as wafer measurements are not properly aligned. After that the OES time series is formally described as a 3-dimensional matrix and various two dimensional representations are discussed. Some techniques commonly used in the industry for dimension- ality reduction of OES time series are described and some of the datasets that will be used in the rest of the thesis are introduced. It is shown that slot variation can be tracked with the W matrix and the idea to partition the W matrix into time intervals is proposed. This will be very useful in the anomaly detection application in Chapter 6 as it allows an high dimen- sional matrix to be split into a set of matrices of lower dimension. In the last part of the chapter the multi-sensor matching problem is investigated. A matching procedure based on the Particle Swarm Optimization algorithm is proposed. Results indicate that good alignment is possible given suitably matched signals.