Semiconductor Manufacturing - Sample Size Determination and Hypothesis Testing

Introduction

In semiconductor manufacturing, precise control of the gate critical dimension (CD) is essential to ensure chip performance, functionality, and yield. The current manufacturing target for the gate CD is 32.0 nm.

This report evaluates the gate lithography process with a focus on statistical hypothesis testing and sample size determination. The engineering team aims to:

• Determine whether the process mean gate CD significantly deviates from the 32.0 nm target.

• Calculate the sample sizes required for a one-sample t-test (detecting a 0.10 nm shift) and a two-sample t-test (detecting a 0.20 nm difference between lithography tools).

Methodology

A one-sample t-test power analysis was performed to determine the required sample size for detecting a potential shift in the mean gate CD from the target of 32.0 nm. The following parameters were used in the calculation:

• Null hypothesis (H₀): Mean CD = 32.0 nm

• Alternative hypothesis (Ha): Mean CD ≠ 32.0 nm

• Level of significance α = 0.05

• Power: 1 - β = 0.85

• Detectable mean shift: 0.1 nm

• Sample standard deviation: Based on presample measurements

Sample Size Determination: One-sample t-test

pre_samples <-c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32.0, 31.5, 31.9, 31.7)

power.t.test(n=NULL,
             delta=0.1,
             sd=sd(pre_samples),
             sig.level = 0.05,
             power=0.85,
             type="one.sample")

## 
##      One-sample t test power calculation 
## 
##               n = 46.25362
##           delta = 0.1
##              sd = 0.2221111
##       sig.level = 0.05
##           power = 0.85
##     alternative = two.sided

Results

The power analysis for the one-sample t-test indicated that 47 samples are required to achieve a significance level of α = 0.05 and a power of 0.85 to detect a 0.10 nm mean shift from the target CD of 32.0 nm.

 One-sample t test power calculation 

          n = 46.25362
      delta = 0.1
         sd = 0.2221111
  sig.level = 0.05
      power = 0.85
alternative = two.sided

Data Visualization of Gate CD Measurements

Following the sample size determination, 47 wafer CD measurements were collected from the lithography process.

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)

A histogram and a boxplot were created to summarize the distribution of the measurements:

The histogram illustrates the frequency of measurements across defined intervals, allowing visual assessment of overall distribution and symmetry.

# Plot histogram
hist(dat,main="Histogram of Collected Gate CD Data", 
     xlab="Observations",ylab="Frequency",col="blue")

The box plot highlights the median, spread, and potential outliers, providing a concise summary of process variability.

# Plot box plot
boxplot(dat,main="Box Plot of Collected Gate CD Data",col='red')

One-sample t-test

A one-sample t-test was conducted to determine whether the mean gate critical dimension (CD) of the 47 collected wafer measurements significantly differs from the target of 32.0 nm.

t.test(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, 
         3.7, 32.1, 31.9, 31.9, 31.9, 32.4, 
         32.0, 31.8, 32.2, 31.7, 31.9)
       ,alternative = "two.sided",mu=32)

## 
##  One Sample t-test
## 
## data:  c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32, 31.5, 31.9, 31.7, 32.1, 32, 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, 32, 31.7, 31.9, 32, 3.7, 32.1, 31.9, 31.9, 31.9, 32.4, 32, 31.8, 32.2, 31.7, 31.9)
## t = -1.0963, df = 46, p-value = 0.2786
## alternative hypothesis: true mean is not equal to 32
## 95 percent confidence interval:
##  30.12942 32.55143
## sample estimates:
## mean of x 
##  31.34043

Results

Since the p-value = 0.04081 is less than α = 0.05 , we reject the null hypothesis.

    One Sample t-test

data:  c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32, 31.5, 31.9, 31.7, 32.1, 32, 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, 32, 31.7, 31.9, 32, 31.7, 32.1, 31.9, 31.9, 31.9, 32.4, 32, 31.8, 32.2, 31.7, 31.9)

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 

Sample Size Determination: Two-Sample t-test

A power analysis was conducted for a two-sample t-test to determine the number of measurements required from each lithography tool to detect a mean difference of 0.20 nm with 90% power at a 0.05 significance level. The calculation used the estimated standard deviation from the presample measurements.

power.t.test(n=NULL,
             delta=0.2,
             sd=sd(pre_samples),
             sig.level = 0.05,
             power=0.90,
             type="two.sample")

## 
##      Two-sample t test power calculation 
## 
##               n = 26.91597
##           delta = 0.2
##              sd = 0.2221111
##       sig.level = 0.05
##           power = 0.9
##     alternative = two.sided
## 
## NOTE: n is number in *each* group

Results

Two-sample t test power calculation
          n = 26.91597
      delta = 0.2
         sd = 0.2221111
  sig.level = 0.05
      power = 0.9
alternative = two.sided

This analysis indicates that 27 samples are needed to achieve the desired significance level and power for detecting a 0.2 nm mean shift.

Conclusion

We determined the required sample sizes for both a one-sample and a two-sample t-test based on specified power and shift values.

• For the one-sample t-test (power = 0.85, shift = 0.10 nm), the required sample size was n = 47.

  For the two-sample t-test (power = 0.90, shift = 0.20 nm), the required sample size was n = 27 per group.

The one-sample t-test analysis conclusion was to reject the null hypothesis, with the decision based on the calculated p-value.