Control Charts for Attributes P-chart (fraction non-conforming) - - PowerPoint PPT Presentation

Lecture 11: Attribute Charts

Spanos

EE290H F05

1

P-chart (fraction non-conforming) C-chart (number of defects) U-chart (non-conformities per unit) The rest of the “magnificent seven”

Control Charts for Attributes

Lecture 11: Attribute Charts

Spanos

EE290H F05

2

Yield Control

30 20 10 20 40 60 80 100 Months of Production 30 20 10 20 40 60 80 100 Yield

Lecture 11: Attribute Charts

Spanos

EE290H F05

3

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

Spanos

EE290H F05

4

The P-Chart

P{D = X} = n x p

x(1-p) n-x x = 0,1,...,n

mean np variance np(1-p) the sample fraction p=D n mean p variance p(1-p) n

The P chart is based on the binomial distribution:

Lecture 11: Attribute Charts

Spanos

EE290H F05

5

The P-chart (cont.)

p =

m

Σ

i=1 pi

m mean p variance p(1-p) nm (in this and the following discussion, "n" is the number of samples in each group and "m" is the number of groups that we use in order to determine the control limits) (in this and the following discussion, "n" is the number of samples in each group and "m" is the number of groups that we use in order to determine the control limits) p must be estimated. Limits are set at +/- 3 sigma.

Lecture 11: Attribute Charts

Spanos

EE290H F05

6

Designing the P-Chart

p can be estimated: pi = Di n i = 1,...,m (m = 20 ~25) p =

m

Σ

i=1

pi m 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): UCL = p + 3 p(1-p) n LCL = p - 3 p(1-p) n

Lecture 11: Attribute Charts

Spanos

EE290H F05

7

Example: Defectives (1.0 minus yield) Chart

"Out of control points" must be explained and eliminated before we recalculate the control limits. This means that setting the control limits is an iterative process! Special patterns must also be explained. "Out of control points" must be explained and eliminated before we recalculate the control limits. This means that setting the control limits is an iterative process! Special patterns must also be explained.

30 20 10 0.0 0.1 0.2 0.3 0.4 0.5 Count LCL 0.053 0.232 UCL 0.411 % non-conforming

Lecture 11: Attribute Charts

Spanos

EE290H F05

8

Example (cont.)

After the original problems have been corrected, the limits must be evaluated again.

Lecture 11: Attribute Charts

Spanos

EE290H F05

9

Operating Characteristic of P-Chart

if n large and np(1-p) >> 1, then

P{D = x} = n x px(1-p) (n-x) ~ 1 2πnp(1-p) e- (x-np) 2 2np(1-p)

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): 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

Spanos

EE290H F05

10

Binomial distribution and the Normal

bin 10, 0.1 0.0 1.0 2.0 3.0 4.0 bin 100, 0.5 35 40 45 50 55 60 65 bin 5000 0.007 15 20 25 30 35 40 45 50

As sample size increases, the Normal approximation becomes reasonable... As sample size increases, the Normal approximation becomes reasonable...

Lecture 11: Attribute Charts

Spanos

EE290H F05

11

Designing the P-Chart

β = P { D < n UCL /p } - P { D < n LCL /p }

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.

• r
• choose n so that we get a positive LCL.

Then, the operating characteristic can be drawn from:

Lecture 11: Attribute Charts

Spanos

EE290H F05

12

The Operating Characteristic Curve (cont.)

p = 0.20, LCL=0.0303, UCL=0.3697 The OCC can be calculated two distributions are equivalent and np=λ).

Lecture 11: Attribute Charts

Spanos

EE290H F05

13

In reality, p changes over time

(data from the Berkeley Competitive Semiconductor Manufacturing Study)

Lecture 11: Attribute Charts

Spanos

EE290H F05

14

The C-Chart

UCL = c + 3 c center at c LCL = c - 3 c p(x) = e-c cx x! x = 0,1,2,.. μ = c, σ2 = c 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: This assumes that defects are independent and that they arrive uniformly over time and space. Under these assumptions: and c can be estimated from measurements.

Lecture 11: Attribute Charts

Spanos

EE290H F05

15

Poisson and the Normal

poisson 2 2 4 6 8 10 poisson 20 10 20 30 poisson 100 70 80 90 100 110 120 130

As the mean increases, the Normal approximation becomes reasonable... As the mean increases, the Normal approximation becomes reasonable...

Lecture 11: Attribute Charts

Spanos

EE290H F05

16

Example: "Filter" wafers used in yield model

14 12 10 8 6 4 2 0.1 0.2 0.3 0.4 0.5

Fraction Nonconforming (P-chart)

Fraction Nonconforming

LCL 0.157 ¯ 0.306 UCL 0.454

14 12 10 8 6 4 2 100 200

Defect Count (C-chart)

Wafer No Number of Defects

LCL 48.26 Ý 74.08 UCL 99.90

Lecture 11: Attribute Charts

Spanos

EE290H F05

17

Counting particles

Scanning a “blanket” monitor wafer. Detects position and approximate size of particle. x y

Lecture 11: Attribute Charts

Spanos

EE290H F05

18

Scanning a product wafer

Lecture 11: Attribute Charts

Spanos

EE290H F05

19

Typical Spatial Distributions

Lecture 11: Attribute Charts

Spanos

EE290H F05

20

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

Spanos

EE290H F05

21

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 than one), must follow a known, usable distribution. Whatever we use (and we might have to use more than one), must follow a known, usable distribution.

Lecture 11: Attribute Charts

Spanos

EE290H F05

22

In Situ Particle Monitoring Technology

Laser light scattering system for detecting particles in exhaust flow. Sensor placed down stream from valves to prevent corrosion. chamber Laser Detector to pump Assumed to measure the particle concentration in vacuum

Lecture 11: Attribute Charts

Spanos

EE290H F05

23

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

Spanos

EE290H F05

24

Time series of ISPM counts vs. Wafer Scans

Lecture 11: Attribute Charts

Spanos

EE290H F05

25

The U-Chart

We could condense the information and avoid outliers by using the “average” defect density u = Σc/n. It can be shown that u obeys a Poisson "type" distribution with: where is the estimated value of the unknown u. The sample size n may vary. This can easily be accommodated.

μu = u, σu

2 = u

n so UCL = u + 3 u n LCL = u - 3 u n

u

Lecture 11: Attribute Charts

Spanos

EE290H F05

26

The Averaging Effect of the u-chart

poisson 2 2 4 6 8 10 Quantiles Moments average 5 0.0 1.0 2.0 3.0 4.0 5.0 Quantiles Moments

By exploiting the central limit theorem, if small-sample poisson variables can be made to approach normal by grouping and averaging By exploiting the central limit theorem, if small-sample poisson variables can be made to approach normal by grouping and averaging

Lecture 11: Attribute Charts

Spanos

EE290H F05

27

Filter wafer data for yield models (CMOS-1):

14 12 10 8 6 4 2 0.1 0.2 0.3 0.4 0.5

Fraction Nonconforming (P-chart)

Fraction Nonconforming

LCL 0.157 ¯ 0.306 UCL 0.454

14 12 10 8 6 4 2 100 200

Defect Count (C-chart)

Number of Defects

LCL 48.26 Ý 74.08 UCL 99.90

14 12 10 8 6 4 2 1 2 3 4 5 6

Defect Density (U-chart)

Wafer No Defects per Unit

LCL 1.82 × 2.79 UCL 3.76

Lecture 11: Attribute Charts

Spanos

EE290H F05

28

The Use of the Control Chart

The control chart is in general a part of the feedback loop for process improvement and control. Process Input Output

Measurement System

Verify and follow up Implement corrective action Detect assignable cause Identify root cause of problem

Lecture 11: Attribute Charts

Spanos

EE290H F05

29

Choosing a control chart...

...depends very much on the analysis that we are

• pursuing. In general, the control chart is only a small

part of a procedure that involves a number of statistical and engineering tools, such as:

• experimental design
• trial and error
• pareto diagrams
• influence diagrams
• charting of critical parameters

Lecture 11: Attribute Charts

Spanos

EE290H F05

30

The Pareto Diagram in Defect Analysis

figure 3.1 pp 21 Kume

Typically, a small number of defect types is responsible for the largest part of yield loss. The most cost effective way to improve the yield is to identify these defect types.

Lecture 11: Attribute Charts

Spanos

EE290H F05

31

Pareto Diagrams (cont)

Diagrams by Phenomena

• defect types (pinholes, scratches, shorts,...)
• defect location (boat, lot and wafer maps...)
• test pattern (continuity etc.)

Diagrams by Causes

• operator (shift, group,...)
• machine (equipment, tools,...)
• raw material (wafer vendor, chemicals,...)
• processing method (conditions, recipes,...)

Lecture 11: Attribute Charts

Spanos

EE290H F05

32

Example: Pareto Analysis of DCMOS Process

evious layer ss problems s scratches

• ntamination

sed contacts ern bridging se particles

• thers

20 40 60 80 100

• ccurence

cummulative

DCMOS Defect Classification

Percentage

Though the defect classification by type is fairly easy, the classification by cause is not...

Lecture 11: Attribute Charts

Spanos

EE290H F05

33

Cause and Effect Diagrams

figure 4.1 pp 27 Kume

(Also known as Ishikawa,fish bone or influence diagrams.) Creating such a diagram requires good understanding of the process.

Lecture 11: Attribute Charts

Spanos

EE290H F05

34

An Actual Example

Lecture 11: Attribute Charts

Spanos

EE290H F05

35

Example: DCMOS Cause and Effect Diagram

Past Steps Parametric Control Particulate Control Operator Handling Contamination Control inspection

• rec. handling

transport loading chemicals utilities cassettes equipment cleaning vendor Wafers Defect skill experience vendor calibration SPC SPC boxes shift monitoring automation filters

Lecture 11: Attribute Charts

Spanos

EE290H F05

36

Example: Pareto Analysis of DCMOS (cont)

equipmnet utilities loading inspection smiff boxes

• thers

20 40 60 80 100

• ccurence

cummulative

DCMOS Defect Causes

percentage

Once classification by cause has been completed, we can choose the first target for improvement.

Lecture 11: Attribute Charts

Spanos

EE290H F05

37

Defect Control

In general, statistical tools like control charts must be combined with the rest of the "magnificent seven":

• Histograms
• Check Sheet
• Pareto Chart
• Cause and effect diagrams
• Defect Concentration Diagram
• Scatter Diagram
• Control Chart

Lecture 11: Attribute Charts

Spanos

EE290H F05

38

Logic Defect Density is also on the decline

Y = [ (1-e-AD)/AD ]2

Lecture 11: Attribute Charts

Spanos

EE290H F05

39

What Drives Yield Learning Speed?