1 Introduction

This report analyzes defect data from a semiconductor fabrication facility. Silicon wafers undergo a series of highly precise fabrication steps — including photolithography, etching, deposition, and chemical mechanical planarization (CMP) — before being diced into individual integrated circuits. At each stage, wafers are inspected for surface defects such as particles, scratches, and pattern misalignments.

Quality engineers inspect \(n = 200\) wafers per lot and record the number of defective wafers. Data were collected over \(k = 50\) consecutive lots during a period believed to represent a stable, in-control process. We will use the qcc package in R to construct an np control chart and confirm process stability.

#install.packages("qcc")
library(qcc)

2 np Control Chart

We would like to create an np chart for this data.

2.1 Formulas for np Chart

The np chart monitors the number of defective items in a sample of fixed size \(n\). The fraction defective is estimated as:

\[ \hat{p} = \frac{\sum_{i=1}^{k} D_i}{k \cdot n} \]

where \(D_i\) is the number of defectives in lot \(i\), \(k\) is the number of lots, and \(n\) is the sample size per lot.

The control limits for the np chart are:

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

\[ CL = n\hat{p} \]

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

If the computed \(LCL\) is negative, it is set to zero.

2.2 Data

We load the defect counts from 50 consecutive lots, each with \(n = 200\) wafers inspected.

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  # sample size per lot

2.3 QCC Output

The qcc object for the np chart is given below:

np_chart <- qcc(data = defects, type = "np", sizes = n, nsigmas = 3,
                title = "np Chart - Semiconductor Wafer Defects",
                xlab = "Lot Number", ylab = "Number of Defective Wafers")

3 Interpretation

The np control chart displays the number of defective wafers per lot across all \(k = 50\) lots. All points fall within the computed control limits — no points exceed the UCL or fall below the LCL. There are no obvious trends, runs, or patterns suggesting the process is out of statistical control.

This indicates that the photolithography stage was operating in a stable, in-control state during the data collection period. The estimated fraction defective \(\hat{p}\) and the corresponding control limits can therefore be used as a baseline for ongoing Phase II monitoring of future production lots. Any future lot with a defect count exceeding the UCL should trigger an investigation for assignable causes.

4 Complete R Code

#install.packages("qcc")
library(qcc)

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

np_chart <- qcc(data = defects, type = "np", sizes = n, nsigmas = 3,
                title = "np Chart - Semiconductor Wafer Defects",
                xlab = "Lot Number", ylab = "Number of Defective Wafers")
summary(np_chart)

np Chart for Semiconductor Wafer Defect Data

Marc Macias