In semiconductor processing, the photolithography process is used to transfer the circuit design to silicon wafers. In the first step of the process, a specified amount of a polymer solution, photoresist, is applied to a wafer as it spins at high speed on a turntable. The resulting photoresist thic

Question

In semiconductor processing, the photolithography process is used to transfer the circuit design to silicon wafers. In the first step of the process, a specified amount of a polymer solution, photoresist, is applied to a wafer as it spins at high speed on a turntable. The resulting photoresist thickness [latex]x[/latex] is a key process variable. Thickness data for 25 subgroups are shown in Table 21.2. Each subgroup consists of three randomly selected wafers. Construct [latex]\bar{x [/latex] and s control charts for these test data and critically evaluate the results.

Table 21.2 Thickness Data (in Å) for Example 21.2

No.		x Data		[latex]\bar{x}[/latex]	s
1	209.6	207.6	211.1	209.4	1.8
2	183.5	193.1	202.4	193	9.5
3	190.1	206.8	201.6	199.5	8.6
4	206.9	189.3	204.1	200.1	9.4
5	260	209	212.2	227.1	28.6
6	193.9	178.8	214.5	195.7	17.9
7	206.9	202.8	189.7	199.8	9
8	200.2	192.7	202.1	198.3	5
9	210.6	192.3	205.9	202.9	9.5
10	186.6	201.5	197.4	195.2	7.7
11	204.8	196.6	225	208.8	14.6
12	183.7	209.7	208.6	200.6	14.7
13	185.6	198.9	191.5	192	6.7
14	202.9	210.1	208.1	207.1	3.7
15	198.6	195.2	150	181.3	27.1
16	188.7	200.7	207.6	199	9.6
17	197.1	204	182.9	194.6	10.8
18	194.2	211.2	215.4	206.9	11.2
19	191.1	206.2	183.9	193.7	11.4
20	202.5	197.1	211.1	203.6	7
21	185.1	186.3	188.9	186.8	1.9
22	203.1	193.1	203.9	200	6
23	179.7	203.3	209.7	197.6	15.8
24	205.3	190	208.2	201.2	9.8
25	203.4	202.9	200.4	202.2	1.61

Accepted Answer

The following sample statistics can be calculated from the data in Table 21.2: [latex]\bar{\bar{x}}[/latex] = 19 .8 Å, [latex]\bar{s}[/latex]= 10.4 Å. For n = 3 the required constants from Table 21.1 are [latex]c_{4} [/latex] = 0.8862, [latex]B_{3}[/latex] = 0, and [latex]B_{4}[/latex] = 2.568. Then the [latex]\bar{x} [/latex] and s control limits can be calculated from Eqs.21-9 to 21-15. The traditional value of c = 3 is selected for Eqs. 21-9 and 21-10. The resulting control limits are labeled as the “original limits” in Fig. 21.5.

[latex]UCL ≜ T + c \hat{\sigma }_{ \bar{x}}[/latex] (21-9)

[latex]LCL ≜ T + c \hat{\sigma }_{ \bar{x}}[/latex] (21-10)

[latex]\bar{s} ≜ \frac{1}{N} \sum\limits_{i=1}^{N}{s_{i} }[/latex] (21-11)

[latex]s_{i} ≜\sqrt{\frac{1}{n-1}\sum\limits_{j=0}^{n}{(x_{ij}-\bar{x} _{i} )^{2} } }[/latex] (21-12)

[latex]\hat{\sigma }_{ \bar{x}} = \frac{1}{c_{4}\sqrt{n} }\bar{s}[/latex] (21-13)

[latex]UCL=B_{4} \bar{s}[/latex] (21-14)

[latex]LCL = B_{3} \bar{s}[/latex] (21-15)

Figure 21.5 indicates that sample #5 lies beyond the UCL for both the [latex]\bar{x}[/latex] and s control charts, while sample #15 is very close to a control limit on each chart. Thus, the question arises whether these two samples are “outliers” that should be omitted from the analysis. Table 21.2 indicates that sample #5 includes a very large value (260.0), while sample #15 includes a very small value (150.0). However, unusually large or small numerical values by themselves do not justify discarding samples; further investigation is required.

Suppose that a more detailed evaluation has discovered a specific reason as to why measurements #5 and #15 should be discarded (e.g., faulty sensor, data misreported, etc.). In this situation, these two samples should be removed and the control limits should be recalculated based on the remaining 23 samples. These modified control limits are tabulated below as well as in Fig. 21.5.

	Original Limits	Modified Limits (omit samples #5 and #15)
[latex]\bar{x}[/latex] Chart Control Limits
UCL	220.1	216.7
LCL	179.6	182.2
s Chart Control Limits
UCL	26.6	22.7
LCL	0	0

Question 21.2: In semiconductor processing, the photolithography process is...

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