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SEMI M40-0200 © SEMI 2000 24 Table R1-11 S/N R atios for the 2 nd Order Effects of Table R1-9 c1/e1 c1/ c2 c1 /e2 c1/ sym e1/c2 e1 /e2 e1/sym c2 /e2 c2/s ym e2/sym ravg1 7 .7045 10.7110 7 .7045 19.2237 10.5897 7 .8732 19…

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

SEMI M40-0200 © SEMI 2000 24

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.

R1-6 Related Documents

ASTM E 1392 — Practice for Angle Resolved Optical Scattering Measurements on Specular or Diffuse Surfaces

ASTM F 1048 — Test Method for Measuring the Effective Surface Roughness of Optical Components by Total

Integrated Scattering

J.M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering, Optical Society of America,

Washington, D.C., 1989

G.E.P. Box, W.G. Hunter, J.S. Hunter, Statistics for Experiments, An Introduction to Design, Data Analysis, and

Model Building, John Wiley, N.Y.

ISO 1879 — Instruments for the measurement of surface roughness by the profile method -- Vocabulary

ISO 3274 — Instruments for the measurement of surface roughness by the profile method – Contact (stylus)

instruments of consecutive profile transformation – Contact profile meters, system M

ISO 4287/1 — Surface roughness – Terminology – Part 1: Surface and its parameters

J.A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, IOP Publishing, Bristol, 1991

J.C. Stover, Optical Scattering, Measurement and Analysis, Second Edition, McGraw-Hill, Inc., N.Y. 1995

P. Wagner, H.A. Gerber, in Particles, Haze and Microroughness on Silicon Wafers, SEMICON Europe 1995, W.

Baylies, P. Wagner, Eds.

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