Background and Problem Context

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. Even a single defect can render a chip non-functional, making defect monitoring critical to maintaining yield.

A fabrication facility produces wafers in lots. At the end of the photolithography stage, quality engineers inspect n = 200 wafers per lot and record the number of defective wafers in each lot. Data were collected over k = 50 consecutive lots during a period believed to represent a stable, in-control process.

The objective of this report is to construct an np control chart using the qcc package in R to confirm process stability and establish control limits for ongoing monitoring.

Data and Setup

# Load the required package
library(qcc)
## Package 'qcc' version 2.7
## Type 'citation("qcc")' for citing this R package in publications.
# Defect 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)

# Sample size
n <- 200

np Chart Control Limits

The np control chart monitors the number of defective items in a sample of fixed size.

First, the fraction defective is calculated as:

\[ \bar{p} = \frac{\sum_{i=1}^{k} d_i}{k \cdot n} \]

where:

The control limits for the np chart are:

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

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

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

np Control Chart

np_chart <- qcc(defects,
                type = "np",
                size = n,
                title = "NP Control Chart for Wafer Defects")

Interpretation

The np control chart shows the number of defective wafers for each lot inspected. All observations fall within the calculated control limits, which indicates that the process is stable and operating under statistical control.

The center line represents the average number of defective wafers per lot. Because no points fall outside the control limits and no unusual patterns are present, there is no evidence of special-cause variation. Therefore, the wafer fabrication process appears to be consistent and predictable during the period of observation.

These control limits can now be used for future monitoring of the semiconductor fabrication process to quickly detect any deviations from normal operating conditions.

Complete R Code (Reference Only)

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(defects,
                type = "np",
                size = n,
                title = "NP Control Chart for Wafer Defects")