semi合集-English.pdf - 第5239页

SEMI M40-0200 © SEMI 2000 21 Table R1-7 Normaliz ed values of Table R1-6 Variable Set # ravg1 ravg5 r avg9 rstdabw5 rstdabw9 1 1.0000 1.0000 1.0000 0.0000 0 .000 0 2 1.6000 1.1440 1.0160 0.1920 0 .136 0 3 0.8000 0.9360 1…

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SEMI M40-0200 © SEMI 2000 20
Table R1-6 Results of Roughness Simulation for Models a (variable = -1) and b (variable = 1)
Variable Set # c1 e1 c2 e2 Sym. avg1 avg5 avg9 avgtrue stdabw5 stdabw9 stdabwtrue
1 0.1 0.1 0.1 0.1 -1 0.1 0.1 0.1 0.1 0 0 0
2 0.2 0.1 0.1 0.1 -1 0.2 0.143 0.127 0.125 0.038 0.031 0.014
3 0.1 0.2 0.1 0.1 -1 0.1 0.117 0.129 0.125 0.021 0.032 0.014
4 0.2 0.2 0.1 0.1 -1 0.2 0.16 0.156 0.15 0.049 0.05 0
5 0.1 0.1 0.2 0.1 -1 0.1 0.123 0.116 0.125 0.028 0.018 0.014
6 0.2 0.1 0.2 0.1 -1 0.2 0.165 0.143 0.15 0.017 0.02 0.029
7 0.1 0.2 0.2 0.1 -1 0.1 0.14 0.144 0.15 0.021 0.021 0
8 0.2 0.2 0.2 0.1 -1 0.2 0.183 0.171 0.175 0.021 0.032 0.014
9 0.1 0.1 0.1 0.2 -1 0.1 0.117 0.129 0.125 0.021 0.032 0.014
10 0.2 0.1 0.1 0.2 -1 0.2 0.16 0.156 0.15 0.021 0.021 0
11 0.1 0.2 0.1 0.2 -1 0.1 0.135 0.157 0.15 0.017 0.02 0.029
12 0.2 0.2 0.1 0.2 -1 0.2 0.177 0.184 0.175 0.028 0.018 0.014
13 0.1 0.1 0.2 0.2 -1 0.1 0.14 0.144 0.15 0.049 0.05 0
14 0.2 0.1 0.2 0.2 -1 0.2 0.183 0.171 0.175 0.021 0.032 0.014
15 0.1 0.2 0.2 0.2 -1 0.1 0.157 0.173 0.175 0.038 0.031 0.014
16 0.2 0.2 0.2 0.2 -1 0.2 0.2 0.2 0.2 0 0 0
17 0.1 0.1 0.1 0.1 1 0.1 0.1 0.1 0.1 0 0 0
18 0.2 0.1 0.1 0.1 1 0.2 0.158 0.153 0.142 0.047 0.048 0.009
19 0.1 0.2 0.1 0.1 1 0.1 0.102 0.102 0.108 0.003 0.003 0.009
20 0.2 0.2 0.1 0.1 1 0.2 0.16 0.156 0.15 0.049 0.05 0
21 0.1 0.1 0.2 0.1 1 0.1 0.133 0.13 0.142 0.04 0.034 0.009
22 0.2 0.1 0.2 0.1 1 0.2 0.191 0.183 0.185 0.007 0.015 0.017
23 0.1 0.2 0.2 0.1 1 0.1 0.135 0.132 0.15 0.038 0.032 0
24 0.2 0.2 0.2 0.1 1 0.2 0.193 0.185 0.192 0.009 0.016 0.009
25 0.1 0.1 0.1 0.2 1 0.1 0.107 0.114 0.108 0.009 0.016 0.009
26 0.2 0.1 0.1 0.2 1 0.2 0.165 0.168 0.15 0.038 0.032 0
27 0.1 0.2 0.1 0.2 1 0.1 0.109 0.117 0.115 0.007 0.015 0.017
28 0.2 0.2 0.1 0.2 1 0.2 0.167 0.17 0.158 0.04 0.034 0.009
29 0.1 0.1 0.2 0.2 1 0.1 0.14 0.144 0.15 0.049 0.05 0
30 0.2 0.1 0.2 0.2 1 0.2 0.198 0.198 0.192 0.003 0.003 0.009
31 0.1 0.2 0.2 0.2 1 0.1 0.142 0.147 0.158 0.047 0.048 0.009
32 0.2 0.2 0.2 0.2 1 0.2 0.2 0.2 0.2 0 0 0
Average 0.15 0.15 0.1499 0.15 0.0241 0.0247 0.0066
Std.
Dev.
0.05 0.0335 0.0314 0.0303 0.0198 0.0179 0.0057
SEMI M40-0200 © SEMI 200021
Table R1-7 Normalized values of Table R1-6
Variable Set # ravg1 ravg5 ravg9 rstdabw5 rstdabw9
1 1.0000 1.0000 1.0000 0.0000 0.0000
2 1.6000 1.1440 1.0160 0.1920 0.1360
3 0.8000 0.9360 1.0320 0.0560 0.1440
4 1.3333 1.0667 1.0400 0.3267 0.3333
5 0.8000 0.9840 0.9280 0.1120 0.0320
6 1.3333 1.1000 0.9533 0.0800 0.0600
7 0.6667 0.9333 0.9600 0.1400 0.1400
8 1.1429 1.0457 0.9771 0.0400 0.1029
9 0.8000 0.9360 1.0320 0.0560 0.1440
10 1.3333 1.0667 1.0400 0.1400 0.1400
11 0.6667 0.9000 1.0467 0.0800 0.0600
12 1.1429 1.0114 1.0514 0.0800 0.0229
13 0.6667 0.9333 0.9600 0.3267 0.3333
14 1.1429 1.0457 0.9771 0.0400 0.1029
15 0.5714 0.8971 0.9886 0.1371 0.0971
16 1.0000 1.0000 1.0000 0.0000 0.0000
17 1.0000 1.0000 1.0000 0.0000 0.0000
18 1.4085 1.1127 1.0775 0.2676 0.2746
19 0.9259 0.9444 0.9444 0.0556 0.0556
20 1.3333 1.0667 1.0400 0.3267 0.3333
21 0.7042 0.9366 0.9155 0.2183 0.1761
22 1.0811 1.0324 0.9892 0.0541 0.0108
23 0.6667 0.9000 0.8800 0.2533 0.2133
24 1.0417 1.0052 0.9635 0.0000 0.0365
25 0.9259 0.9907 1.0556 0.0000 0.0648
26 1.3333 1.1000 1.1200 0.2533 0.2133
27 0.8696 0.9478 1.0174 0.0870 0.0174
28 1.2658 1.0570 1.0759 0.1962 0.1582
29 0.6667 0.9333 0.9600 0.3267 0.3333
30 1.0417 1.0313 1.0313 0.0313 0.0313
31 0.6329 0.8987 0.9304 0.2405 0.2468
32 1.0000 1.0000 1.0000 0.0000 0.0000
average 0.9968 0.9987 1.0001 0.1287 0.1254
standard deviation 0.2700 0.0682 0.0521 0.1106 0.1080
maximum 1.6000 1.1440 1.1200 0.3267 0.3333
minimum 0.5714 0.8971 0.8800 0.0000 0.0000
SEMI M40-0200 © SEMI 2000 22
R1-5.3.2 Wafer roughness maps with 1 mm
2
pixel size were generated using all possible 32 combinations of the
parameters in Table R1-5. Average roughness and standard deviation were calculated by using all pixels of a map
(avgtrue, stdabwtrue) as well as by using the three discrete site patterns displayed in Figure 1 of the main part of this
document (avg1, avg5, avg9, stdabw5, stdabw9). Corresponding results for comparing models a and b are displayed
in Table R1-6. These values were normalized for further evaluation, the averages with respect to avgtrue
(ravgi=avgi/avgtrue, i=1,5,9), the standard deviations with respect to their difference to the “true” value
((rstdabwi=stdabwi.-stdabwtrue) / stdabwtrue, i=5,9) (Table R1-7). Results for comparing models a and c are
similar and are not reported here in detail.
R1-5.3.3 The averages for ravg1,5,9 over the various variable sets differ by less than 1% from unity. The
corresponding standard deviations decrease from 27% to 5% going from ravg1 to ravg9, respectively, indicating
as one would expect that ravg9 is a much more precise value for the roughness of the entire surface as compared
to ravg1 or ravg5. The relative standard deviations rstdabw5,9 deviate in the average by about 12% from the true
value. The corresponding standard deviations differ not much for rstdabw5 and rstdabw9.
R1-5.3.4 More detailed information is obtained when the results are evaluated according to the factorial design
used. The 1
st
to 5
th
order effects of varying the variables were calculated by applying using a table of contrast
coefficients /10/ to the normalized results of the simulation. The 1
st
order or main effects are the difference of
the observations for both levels of one parameter and averages over all other observations. They measure the
average effect of a variable over all conditions of the other variables. The 2
nd
order effects are a measure for the
interaction of variables and are obtained by calculating one half of the difference of the average effect of variable 1
with variable 2 at level 1 and variable 1 with variable 2 at level 2. 3
rd
order and higher effects are not considered in
the present work. They are assumed to be negligible and are used to calculate the variance of an effect (=square root
of the average of the squares of 3
rd
to 5
th
order effects).
R1-5.3.5 The result of this evaluation is displayed in Table R1-8 and Table R1-9 for the main (1
st
order) and 2
nd
order effects, respectively.
Table R1-8 Average (over all sets of variables) and Main Effects of the Various Variables on the
Observables (The variance as calculated from the 2
nd
to 5
th
order effects is displayed in the last column.)
Average
c1 e1 c2 e2 Sym. Variance
ravg1 0.9968 0.4482 -0.1111 -0.2237 -0.1111 -0.0064 4.16E-05
ravg5 0.9987 0.1134 -0.0460 -0.0377 -0.0287 -0.0027 3.69E-06
ravg9 1.0001 0.0439 -0.0067 -0.0734 0.0356 -0.0001 4.37E-06
rstdabw5 0.1287 -0.0038 -0.0049 -0.0073 -0.0080 0.0315 3.43E-04
rstdabw9 0.1254 -0.0064 -0.0057 -0.0113 -0.0052 0.0198 4.87E-04
Table R1-9 2
nd
Order Effects (The variance as calculated from the 2
nd
to 5
th
order effects is displayed in the
last column.)
c1/e1 c1/c2 c1/e2 c1/sym e1/c2 e1/e2 e1/sym c2/e2 c2/sym e2/sym Variance
ravg1 -0.0157 -0.0221 -0.0157 -0.0590 0.0218 0.0160 0.0579 0.0218 -0.0547 0.0579 4.16E-05
ravg5 -0.0015 -0.0079 -0.0040 -0.0067 0.0065 0.0054 0.0064 0.0039 -0.0225 0.0238 3.69E-06
ravg9 -0.0003 0.0023 -0.0057 0.0304 0.0049 -0.0015 -0.0304 -0.0005 -0.0092 0.0119 4.37E-06
rstdabw5 -0.0062 -0.1848 -0.0603 -0.0027 -0.0423 -0.0392 0.0059 0.0335 -0.0005 0.0029 3.43E-04
rstdabw9 0.0080 -0.1471 -0.0722 0.0002 -0.0247 -0.0894 0.0003 0.0518 0.0027 0.0008 4.87E-04
R1-5.3.6 The behavior of the main effects is also illustrated in Figure R1-5, where the variation of ravg1,5,9 and
stdabw5,9 are plotted vs. the variables. The clear effect, on ravg1, of varying center1 between 0.1 and 0.2 is easy to
understand as ravg1 consists only of one measurement point in the center of the wafer surface. Similar but less
pronounced effects are observed for ravg5 and 9. Note that the opposite effect occurs for center 2 as this point is not
included in calculating ravg1,5 or 9. Also note that the influence of the variables edge1 or edge2 is much less