semi合集-English.pdf - 第5241页

SEMI M40-0200 © SEMI 2000 23 pronoun ced. In any case, the 5-point measurem ent provides the correct av erage ± 6%, the 9-point measurem e nt ± 2.5%. In a s imil ar way, the 5- point standard deviat io n is correct ± 1.6…

SEMI M40-0200 © SEMI 2000 22

R1-5.3.2 Wafer roughness maps with 1 mm

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

to 5

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

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

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

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

to 5

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

order) and 2

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

to 5

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

Order Effects (The variance as calculated from the 2

to 5

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

SEMI M40-0200 © SEMI 200023

pronounced. In any case, the 5-point measurement provides the correct average ± 6%, the 9-point measurement ±

2.5%. In a similar way, the 5-point standard deviation is correct ± 1.6% and the 9-point standard deviation ± 1%.

0.2

0.4

0.6

0.8

1.2

1.4

center1 lo

center1 hi

edge1 lo

edge1 h i

center2 lo

center2 hi

edge2 lo

edge2 hi

sym. rot.

sym. cyl

Variables

Effects

ravg1 ravg5 ravg9 stdabw5 stdabw9

Figure R1-5

Main Effects of the 2

Factorial Design for Models a and b

R1-5.3.7 The significance of numbers given in Table R1-8 and Table R1-9 can be estimated only in relation to the

noise or variance of the observables which is also displayed in both tables. The signal-to-noise ratio S/N is obtained

by using a logarithmic measure:

S/N = 10 log (effect

/variance)

R1-5.3.8 The corresponding S/N values for the main and for the second order effects are listed in Table R1-10 and

Table R1-11.

R1-5.3.9 A linear signal-to-noise ratio of 3:1 is commonly used to distinguish significant data from insignificant

data. This linear ratio corresponds to a S/N of about 10 in the present case of a logarithmic signal-to- noise ration.

The S/N ratios > 10 are shaded lightly gray in Table R1-10 and Table R1-11. The variables center1,2 and edge1,2

have a significant effect on the relative averages ravg1,5,9. Mainly the interactions of the model selected with the

other variables are significant for the relative averages ravg1,5,9 and the interaction of center1 and center2 for the

relative standard deviations rstdabw5,9. The other cases emphasized in Table R1-11 by shading have a S/N ratio

only slightly larger than 10.

Table R1-10 S/N Ratios for the Main Effects of Table R1-8

average c1 e1 c2 e2 Sym

ravg1 43.7817 36.8384 24.7251 30.8046 24.7251 -0.0364

ravg5 54.3215 35.4230 27.5959 25.8609 23.4801 2.9416

ravg9 53.5959 26.4387 10.1741 30.9124 24.6260 -26.2612

rstdabw5 16.8383 -13.6750 -11.4959 -8.0646 -7.3246 4.6154

rstdabw9 15.0934 -10.7996 -11.7656 -5.7926 -12.5745 -0.9359

Table R1-11 S/N Ratios for the 2

Order Effects of Table R1-9

c1/e1 c1/c2 c1/e2 c1/sym e1/c2 e1/e2 e1/sym c2/e2 c2/sym e2/sym

ravg1 7.7045 10.7110 7.7045 19.2237 10.5897 7.8732 19.0682 10.5897 18.5684 19.0682

ravg5 -2.3106 12.2743 6.3820 10.8291 10.5529 9.0162 10.4414 6.2417 21.3820 21.8517

ravg9 -17.0034 0.7278 8.7438 23.2512 7.3954 -3.1738 23.2558 -11.9623 12.8618 15.1361

Rstdabw5 -9.5587 19.9853 10.2575 -16.7484 7.1802 6.5178 -9.8910 5.1594 -32.1153 -16.1308

Rstdabw9 -8.8505 16.4782 10.2932 -40.5530 0.9730 12.1486 -37.1733 7.4191 -18.3336 -28.6846

R1-5.3.10 The interactions are discussed for two examples, ravg9 and rstdabw5 (Table R1-12). Going from sym lo

(model a, parabolic symmetry) to sym hi (model b, cylindrical symmetry) and keeping c1 fixed at the lo level

decreases ravg9 from 0.993 to 0.963 whereas it increases from 1.007 to 1.037 when c1 is kept fixed at the lo level.

This change in opposite directions indicate an interaction between c1 and sym. c1 and c2 interact in a similar way

with respect to rstdabw5.

Table R1-12 2

Order Effects or Interactions

C1 lo/sym hi 0.963 1.037 c1 hi/sym hi

ravg9

C1 lo/sym lo 0.993 1.007 c1 hi/sym lo

c1 lo/c2 hi 0.219 0.031 c1 hi/c2 hi

stdabw5

c1 lo/c2 lo 0.132 0.223 c1 hi/c2 lo

R1-5.4 Summary and Conclusions

R1-5.4.1 Three different models for surface roughness distribution (roughness maps) were investigated and three

different site patterns for measuring the roughness were applied to them. The parameters of the models – roughness

in the center and near the edge of the wafer – were used as variables in a factorial design and varied between two

levels. Average roughness and the corresponding standard deviations calculated for the site patterns were compared

with the “true” values obtained by evaluating all points of the roughness map.

R1-5.4.2 The patterns where the roughness is measured at five or nine points exhibit a standard deviation of 7 and 5

%, respectively, from the true average value for all possible combinations of the variables. The average of the

standard deviations of the roughness distribution of the single variable sets differs by about 13% from the true value

for the five- as well as nine-point site pattern. A standard deviation of the standard deviations of about 11% is

obtained by averaging over the 32 different variables sets.

R1-5.4.3 The evaluation of the factorial design outlines that the variation of the variables c1, c2, e1, and e2 has a

significant effect – with respect to “noise” – on the average roughness values but not on the corresponding standard

deviations. Varying the surface model does not significantly affect the averages and standard deviations. Some

second order effects or interactions are also significant but less pronounced than the main effects. A pronounced

interaction of the variables c1 and c2 occurs e.g. for the five point standard deviation rstdabw5. This interaction

could be reduced by introducing an additional site for measuring roughness in the center of the wafer with a

direction perpendicular to the present one.

R1-5.4.4 The goal of finding a site pattern which results only in non-significant effects is not completely achieved

by the five and nine point patterns utilized in the present work. However, they allow to measure the average

roughness of wide variety of surface roughness distributions with a one sigma deviation of 5-7 %, or a three sigma

deviation of 15-21 %. These values are certainly more than sufficient for present wafer surfaces.