Spanos EE290H F05 Control Charts for Attributes P-chart (fraction non-conforming) C-chart (number of defects) U-chart (non-conformities per unit) The rest of the “magnificent seven” Lecture 11: Attribute Charts 1
Spanos EE290H F05 Yield Control 100 100 80 80 60 60 Yield 40 40 20 20 0 0 0 10 20 30 0 10 20 30 Months of Production Lecture 11: Attribute Charts 2
Spanos EE290H F05 The fraction non-conforming The most inexpensive statistic is the yield of the production line. Yield is related to the ratio of defective vs. non-defective, conforming vs. non-conforming or functional vs. non- functional. We often measure: • Fraction non-conforming (P) • Number of defects on product (C) • Average number of non-conformities per unit area (U) Lecture 11: Attribute Charts 3
Spanos EE290H F05 The P-Chart The P chart is based on the binomial distribution: n-x x = 0,1,...,n P{D = X} = n x (1-p) x p mean np variance np(1-p) the sample fraction p=D n mean p variance p(1-p) n Lecture 11: Attribute Charts 4
Spanos EE290H F05 The P-chart (cont.) p must be estimated. Limits are set at +/- 3 sigma. m mean p Σ i=1 p i variance p(1-p) p = nm m (in this and the following discussion, "n" is the number of (in this and the following discussion, "n" is the number of samples in each group and "m" is the number of groups samples in each group and "m" is the number of groups that we use in order to determine the control limits) that we use in order to determine the control limits) Lecture 11: Attribute Charts 5
Spanos EE290H F05 Designing the P-Chart In general, the control limits of a chart are: UCL= µ + k σ LCL= µ - k σ where k is typically set to 3. These formulae give us the limits for the P-Chart (using the binomial distribution of the variable): p can be estimated: p i = D i i = 1,...,m (m = 20 ~25) n p(1-p) UCL = p + 3 n m Σ p(1-p) p i LCL = p - 3 n i=1 p = m Lecture 11: Attribute Charts 6
Spanos EE290H F05 Example: Defectives (1.0 minus yield) Chart 0.5 UCL 0.411 0.4 % non-conforming 0.3 0.232 0.2 0.1 LCL 0.053 0.0 0 10 20 30 Count "Out of control points" must be explained and eliminated before we "Out of control points" must be explained and eliminated before we recalculate the control limits. recalculate the control limits. This means that setting the control limits is an iterative process! This means that setting the control limits is an iterative process! Special patterns must also be explained. Special patterns must also be explained. Lecture 11: Attribute Charts 7
Spanos EE290H F05 Example (cont.) After the original problems have been corrected, the limits must be evaluated again. Lecture 11: Attribute Charts 8
Spanos EE290H F05 Operating Characteristic of P-Chart In order to calculate type I and II errors of the P-chart we need a convenient statistic. Normal approximation to the binomial (DeMoivre-Laplace): if n large and np(1-p) >> 1, then e- (x-np) 2 P{D = x} = n p x (1-p) (n-x) ~ 1 2np(1-p) 2 π np(1-p) x In other words, the fraction nonconforming can be treated as having a nice normal distribution! (with μ and σ as given). This can be used to set frequency, sample size and control limits. Also to calculate the OC. Lecture 11: Attribute Charts 9
Spanos EE290H F05 Binomial distribution and the Normal bin 10, 0.1 bin 100, 0.5 bin 5000 0.007 4.0 50 65 45 60 3.0 40 55 35 2.0 50 30 45 25 1.0 40 20 35 0.0 15 As sample size increases, the Normal approximation becomes reasonable... As sample size increases, the Normal approximation becomes reasonable... Lecture 11: Attribute Charts 10
Spanos EE290H F05 Designing the P-Chart Assuming that the discrete distribution of x can be approximated by a continuous normal distribution as shown, then we may: • choose n so that we get at least one defective with 0.95 probability. • choose n so that a given shift is detected with 0.50 probability. or • choose n so that we get a positive LCL. Then, the operating characteristic can be drawn from: β = P { D < n UCL /p } - P { D < n LCL /p } Lecture 11: Attribute Charts 11
Spanos EE290H F05 The Operating Characteristic Curve (cont.) The OCC can be calculated two distributions are equivalent and np= λ ). p = 0.20, LCL=0.0303, UCL=0.3697 Lecture 11: Attribute Charts 12
Spanos EE290H F05 In reality, p changes over time (data from the Berkeley Competitive Semiconductor Manufacturing Study) Lecture 11: Attribute Charts 13
Spanos EE290H F05 The C-Chart Sometimes we want to actually count the number of defects. This gives us more information about the process. The basic assumption is that defects "arrive" according to a Poisson model: p(x) = e -c c x x = 0,1,2,.. x! μ = c, σ 2 = c This assumes that defects are independent and that they arrive uniformly over time and space. Under these assumptions: UCL = c + 3 c center at c LCL = c - 3 c and c can be estimated from measurements. Lecture 11: Attribute Charts 14
Spanos EE290H F05 Poisson and the Normal poisson 2 poisson 20 poisson 100 130 10 30 120 8 110 6 100 20 4 90 80 2 10 70 0 As the mean increases, the Normal approximation becomes reasonable... As the mean increases, the Normal approximation becomes reasonable... Lecture 11: Attribute Charts 15
Spanos EE290H F05 Example: "Filter" wafers used in yield model Fraction Nonconforming (P-chart) 0.5 UCL 0.454 0.4 Fraction Nonconforming ¯ 0.306 0.3 0.2 LCL 0.157 0.1 0 2 4 6 8 10 12 14 Defect Count (C-chart) 200 Number of Defects UCL 99.90 100 Ý 74.08 LCL 48.26 0 0 2 4 6 8 10 12 14 Wafer No Lecture 11: Attribute Charts 16
Spanos EE290H F05 Counting particles Scanning a “blanket” monitor wafer. Detects position and approximate size of particle. y x Lecture 11: Attribute Charts 17
Spanos EE290H F05 Scanning a product wafer Lecture 11: Attribute Charts 18
Spanos EE290H F05 Typical Spatial Distributions Lecture 11: Attribute Charts 19
Spanos EE290H F05 The Problem with Wafer Maps Wafer maps often contain information that is very difficult to enumerate A simple particle count cannot convey what is happening. Lecture 11: Attribute Charts 20
Spanos EE290H F05 Special Wafer Scan Statistics for SPC applications • Particle Count • Particle Count by Size (histogram) • Particle Density • Particle Density variation by sub area (clustering) • Cluster Count • Cluster Classification • Background Count Whatever we use (and we might have to use more Whatever we use (and we might have to use more than one), must follow a known, usable distribution. than one), must follow a known, usable distribution. Lecture 11: Attribute Charts 21
Spanos EE290H F05 In Situ Particle Monitoring Technology Laser light scattering system for detecting particles in exhaust flow. Sensor placed down stream from valves to prevent corrosion. Laser chamber to pump Detector Assumed to measure the particle concentration in vacuum Lecture 11: Attribute Charts 22
Spanos EE290H F05 Progression of scatter plots over time The endpoint detector failed during the ninth lot, and was detected during the tenth lot. Lecture 11: Attribute Charts 23
Spanos EE290H F05 Time series of ISPM counts vs. Wafer Scans Lecture 11: Attribute Charts 24
Spanos EE290H F05 The U-Chart We could condense the information and avoid outliers by using the “average” defect density u = Σ c/n. It can be shown that u obeys a Poisson "type" distribution with: 2 = u μ u = u, σ u n so u + 3 u UCL = n u - 3 u LCL = n u where is the estimated value of the unknown u . The sample size n may vary. This can easily be accommodated. Lecture 11: Attribute Charts 25
Spanos EE290H F05 The Averaging Effect of the u-chart poisson 2 average 5 5.0 10 4.0 8 3.0 6 2.0 4 1.0 2 0 0.0 Quantiles Quantiles Moments Moments By exploiting the central limit theorem, if small-sample poisson variables By exploiting the central limit theorem, if small-sample poisson variables can be made to approach normal by grouping and averaging can be made to approach normal by grouping and averaging Lecture 11: Attribute Charts 26
Spanos EE290H F05 Filter wafer data for yield models (CMOS-1): 0.5 Fraction Nonconforming (P-chart) UCL 0.454 0.4 Fraction Nonconforming ¯ 0.306 0.3 0.2 LCL 0.157 0.1 0 2 4 6 8 10 12 14 200 Defect Count (C-chart) Number of Defects UCL 99.90 100 Ý 74.08 LCL 48.26 0 0 2 4 6 8 10 12 14 6 Defect Density (U-chart) 5 Defects per Unit 4 UCL 3.76 3 × 2.79 2 LCL 1.82 1 0 2 4 6 8 10 12 14 Wafer No Lecture 11: Attribute Charts 27
Spanos EE290H F05 The Use of the Control Chart The control chart is in general a part of the feedback loop for process improvement and control. Input Output Process Measurement System Detect Verify and assignable follow up cause Implement Identify root corrective cause of problem action Lecture 11: Attribute Charts 28
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