Semiconductor process quality control assesment

Sample size determination for hypothesis testing

The target gate critical dimension (CD) is 32.0 nm

target=32.0

We would like to perform this test at an alpha=0.05 level of significance, and for the test to have a power of 1-beta = 0.85 at detecting a mean shift of 0.1nm from the target.

alpha = 0.05
power = 0.85
beta = 1-power
delta=0.1

A collection of n=10 presamples (in nm) yielded the following data: 31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32.0, 31.5, 31.9, 31.7

presamples= c(31.6, 32.1, 31.9, 31.7, 32.2, 31.8, 32.0, 31.5, 31.9, 31.7)

Determine how many samples must be collected to achieve the desired alpha and beta errors.

power.t.test(n=NULL, delta = delta, sd=sd(presamples), sig.level=alpha, power=power,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

Given the requirements, 47 samples are needed to achieve the given alpha and beta errors.

Following the determination of and the collection of data, the following data was obtained.

real_samples=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 )

Plot a histogram and boxplot of the data. Perform the appropriate t-test and state conclusions.

# boxplot
boxplot(real_samples,
        main = "Boxplot of gate CD measurements (Tool 1)",
        ylab = "Gate CD (nm)",
        col = "lightblue",
        border = "darkblue")

grid(nx = NA, ny = NULL)

# histogram
hist(real_samples,
     main = "Histogram of gate CD measurements (Tool 1)",
     xlab = "Gate CD (nm)",
     col = "lightgreen",
     border = "darkgreen",
     breaks = 10)

grid(nx = NA, ny = NULL)

t.test(real_samples, mu=target,)
## 
##  One Sample t-test
## 
## data:  real_samples
## 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

Given the results of the t test, the process mean deviates from the 32 nm target. This was deduced because p<alpha, so we reject the Null hypothesis. The process mean has statistically significant difference from the 32nm target.

Lithography tools

A fab is evaluating two different lithography tools to determine whether they produce wafers with the same mean gate CD.

alpha=0.05
power=0.9
delta=0.2


power.t.test(n=NULL, delta = delta, sd=sd(real_samples), sig.level=alpha, power=power,type='two.sample',)
## 
##      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

From our testing, 24 samples should be collected from the second population (Tool 2) and used in the two-sample t-test alongside the existing Tool 1 data.