Name: Raunak Mane

IE4331

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#Question 1

In semiconductor manufacturing, gate lithography defines the critical dimension (CD) of transistor gates, directly impacting device performance, speed, and yield.
The target CD for this facility is 32.0 nm. Small deviations from this target can affect transistor behavior, increasing variability and reducing reliability.

This study aims to:

  1. Determine the required sample size to detect a mean shift of 0.1 nm with specified error rates.

  2. Collect and analyze production samples.

  3. Use statistical hypothesis testing to assess whether the mean CD deviates significantly from the target.

  4. Provide data visualization

##Data
data <- c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32.0, 31.5, 31.9, 31.7)

## Parameter
alpha <- 0.05
power <- 0.85     
delta <- 0.10        
sigma <- sd(data)    

## Z-value
z_alpha <- qnorm(1 - alpha/2)
z_beta  <- qnorm(power)

## Sample size 
n_required <- ((z_alpha + z_beta) * sigma / delta)^2
n_required
## [1] 44.29343
#Dataset2
data1 <- 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
)

## Histogram and boxplot
hist(data1, main = "Histogram of Gate CD", xlab = "CD (nm)", col = "purple", border = "black")

boxplot(data1, main = "Boxplot of Gate CD", xlab = "CD (nm)", horizontal = TRUE, col = "cyan")

## t-test
t_test_result <- t.test(data1, mu = 32.0, alternative = "two.sided")
t_test_result
## 
##  One Sample t-test
## 
## data:  data1
## t = -2.1047, df = 46, p-value = 0.04081
## alternative hypothesis: true mean is not equal to 32
## 95 percent confidence interval:
##  31.87512 31.99722
## sample estimates:
## mean of x 
##  31.93617
## Solution
c(n = length(data1), mean = mean(data1), sd = sd(data1),
  diff = mean(data1) - 32.0)
##           n        mean          sd        diff 
## 47.00000000 31.93617021  0.20791463 -0.06382979

Results

  • We needed about 44 samples, we collected 47, which is enough.

  • Average CD: 31.94 nm (target is 32.0 nm).

  • Standard deviation: 0.21 nm ( Process variation is low)

  • The average is about 0.06 nm lower than target.

Conclusions & Recommendations:

  • Recommendations:

    1. Check lithography tool calibration.

    2. Keep monitoring tool drift.

    3. If the downward shift continues, adjust focus to bring the mean back to 32.0 nm.

Conclusion: The process is stable and consistent, but it is running a little low. A quick calibration should fix the issue and prevent long-term drift.

#Question 2

#Question 2

## Tool 1
tool1 <- data1
n1 <- length(tool1)
sigma_2 <- sd(tool1)

## Parameter
alpha <- 0.05
power <- 0.90
delta <- 0.20

## Z-value
z_alpha <- qnorm(1 - alpha/2)
z_beta  <- qnorm(power)

## Solution
rhs <- (delta / ((z_alpha + z_beta) * sigma_2))^2
n2  <- 1 / (rhs - 1/n1)
(n2)
## [1] 14.9731

To confidently compare the two tools, we should collect at least 15 wafers from Tool 2. Together with the 47 wafers already measured from Tool 1, this will provide 90% power to detect a 0.20 nm difference in mean CD at the 5% significance level.