Introduction

In semiconductor manufacturing, maintaining control over the gate critical dimension (CD) is essential for product performance and yield. The target CD in the gate lithography step is 32.0 nm. This report investigates:

  1. Whether the process mean significantly deviates from the target (one-sample t-test).
  2. How many samples are required to compare two lithography tools with high statistical power (two-sample t-test design).

All code, calculations, and plots are included to support conclusions. The intended audience is the Production Manager, so interpretations emphasize practical implications.

Problem 1: Testing Against the Target

Context

The engineering team is concerned about possible deviations from the 32.0 nm target. A presample of 10 wafers was measured to estimate process variability.

Placeholder: Explain why small deviations in CD matter in manufacturing (impact on yield, performance, costs).

Sample Size Determination

Based on the presample standard deviation, we determined the sample size needed to detect a 0.1 nm shift with α = 0.05 and 90% power.

presamples<-c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32.0, 31.5, 31.9, 31.7)
mean(presamples)
## [1] 31.84

Sample Size Determination

Based on the presample standard deviation, we determined the sample size needed to detect a 0.1 nm shift with α = 0.05 and 90% power.

sd(presamples)
## [1] 0.2221111
sd0<-sd(presamples)
power.t.test(delta=0.1,sd=sd0,sig.level = 0.05,power=0.9,
             type="one.sample",alternative="two.sided")
## 
##      One-sample t test power calculation 
## 
##               n = 53.79419
##           delta = 0.1
##              sd = 0.2221111
##       sig.level = 0.05
##           power = 0.9
##     alternative = two.sided
z_alpha<-qnorm((1-0.05)/2)
z_beta<-qnorm(0.9)
n_req<-((z_alpha+z_beta)*(sd0/0.1)^2)
n_req
## [1] 6.012968

Collected Data

After determining the required sample size, the following wafers were measured:

dat<-c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32.0, 31.5, 31.9, 31.7,
       32.1, 32.0, 32.1, 32.3, 31.9, 31.9, 32.3, 32.2, 31.9, 32.1,
       31.9, 31.9, 32.1, 31.6, 31.6, 31.9, 31.8, 32.1, 31.8, 31.7,
       32.3, 32.0, 32.0, 31.7, 31.9, 32.0, 31.7, 32.1, 31.9, 31.9,
       31.9, 32.4, 32.0, 31.8, 32.2, 31.7, 31.9)
length(dat);mean(dat);sd(dat)
## [1] 47
## [1] 31.93617
## [1] 0.2079146

Visualization

hist(dat,main="Histogram Of Collected Data",xlab="observations",ylab="Frequency",col="pink",ylim = c(0,15))

boxplot(dat,main="Boxplot of data",col="purple")

Hypothesis Test

t.test(dat,nu=32,alternative = "two.sided")
## 
##  One Sample t-test
## 
## data:  dat
## t = 1053, df = 46, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  31.87512 31.99722
## sample estimates:
## mean of x 
##  31.93617

Problem 2: Comparing Two Lithography Tools

Context

Management wants to compare Tool 1 (data from Problem 1) and Tool 2 (new tool). The goal is to detect whether the mean CD differs between tools.

Placeholder: Explain why comparing tools matters (equipment qualification, consistency, yield risk).

Sample Size Determination

We want to detect a mean difference of 0.20 nm at α = 0.05 with 90% power. The required sample size for Tool 2 is:

# Problem 2
dat
##  [1] 31.6 32.1 31.9 31.7 32.2 31.8 32.0 31.5 31.9 31.7 32.1 32.0 32.1 32.3 31.9
## [16] 31.9 32.3 32.2 31.9 32.1 31.9 31.9 32.1 31.6 31.6 31.9 31.8 32.1 31.8 31.7
## [31] 32.3 32.0 32.0 31.7 31.9 32.0 31.7 32.1 31.9 31.9 31.9 32.4 32.0 31.8 32.2
## [46] 31.7 31.9
sd1<-sd(dat)
sd1
## [1] 0.2079146
power.t.test(delta = 0.2,sd=sd1,sig.level = 0.05,
             power=0.9,type = "two.sample",
             alternative = "two.sided")
## 
##      Two-sample t test power calculation 
## 
##               n = 23.71388
##           delta = 0.2
##              sd = 0.2079146
##       sig.level = 0.05
##           power = 0.9
##     alternative = two.sided
## 
## NOTE: n is number in *each* group