np Control Chart for Wafer Defects

Introduction

Silicon wafer fabrication requires strict quality control because even small surface defects can make an integrated circuit unusable. During the photolithography stage, wafers are inspected for defects such as particles, scratches, or pattern misalignment.

In this analysis, 200 wafers are inspected from each production lot. The number of defective wafers is recorded for 50 consecutive lots believed to represent a stable process. An np control chart will be constructed to evaluate whether the process appears to be in statistical control.

Objective

The goal of this report is to construct an np control chart using R and the qcc package to verify process stability and establish control limits for monitoring wafer defects.

Data

defects <- c(8, 7, 11, 3, 6, 11, 4, 4, 5, 8, 8, 6, 7, 6, 10, 7, 9, 5, 6, 8,
             8, 10, 4, 11, 4, 11, 6, 6, 9, 5, 4, 7, 5, 7, 8, 5, 10, 9, 7, 9,
             4, 9, 4, 7, 11, 8, 10, 8, 7, 5)

n <- 200
k <- length(defects)

k

## [1] 50

sum(defects)

## [1] 357

mean(defects)

## [1] 7.14

Control Chart Formulas

For an np chart, the estimated fraction defective is

\[ \bar{p} = \frac{\sum d_i}{kn} \]

where \(\sum d_i\) is the total number of defective wafers, \(k\) is the number of lots, and \(n\) is the sample size per lot.

The center line of the np chart is

\[ \text{CL} = n\bar{p} \]

The upper and lower control limits are

\[ UCL = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})} \]

\[ LCL = n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})} \]

If the lower control limit is negative, it is set equal to 0 because the number of defective wafers cannot be negative.

Calculations

pbar <- sum(defects) / (k * n)
center_line <- n * pbar
ucl <- center_line + 3 * sqrt(n * pbar * (1 - pbar))
lcl <- center_line - 3 * sqrt(n * pbar * (1 - pbar))
lcl_adjusted <- max(0, lcl)

pbar

## [1] 0.0357

center_line

## [1] 7.14

ucl

## [1] 15.01184

lcl

## [1] -0.7318434

lcl_adjusted

## [1] 0

np Control Chart

library(qcc)

## Package 'qcc' version 2.7

## Type 'citation("qcc")' for citing this R package in publications.

wafer_chart <- qcc(defects,
                   type = "np",
                   sizes = rep(n, k),
                   title = "np Chart for Defective Wafers per Lot",
                   xlab = "Lot Number",
                   ylab = "Number of Defective Wafers")

## Interpretation

The np control chart shows that the number of defective wafers observed in each of the 50 lots falls within the calculated control limits. The center line is approximately 7.14 defective wafers per lot, while the upper control limit is about 15.01 and the lower control limit is effectively 0.

Since no points fall outside the control limits and there are no unusual patterns in the chart, the process appears to be in statistical control. This suggests that the variation in the number of defective wafers is due to common causes inherent in the manufacturing process rather than special causes.

Therefore, these control limits can be used as a baseline for monitoring future wafer production.