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SEMI M40-0200 © SEMI 2000 20 Table R1-6 Res ults of Roughness Simulat ion for Models a (v ariable = -1) and b (variabl e = 1) Variable Set # c1 e1 c2 e2 Sym. avg1 avg5 avg9 avgtr ue stdabw5 stdabw9 stdabwtrue 1 0.1 0.1 0…

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SEMI M40-0200 © SEMI 200019
R1-5 Models of Roughness Distribution and
their Evaluation with a Virtual Experiment of
Factorial Design
R1-5.1 Goal of the Simulation
R1-5.1.1 Roughness is measured on the entire wafer
surface only in rare cases. Therefore approximations
have to be found which represent the roughness of an
entire wafer surface with appropriate accuracy. Three
discrete site patterns were defined in section 9 of SEMI
M40, which are thought to provide such
approximations. Performing many measurements using
these site patterns is one approach to assess these site
patterns with respect to their validity. Another
approach is to simulate roughness variations across a
wafer surface and apply the site patterns to them. This
allows easy and systematic variations of the surface
roughness properties and maps. The assessment of the
results again has to be performed in a systematic way.
An appropriate tool for doing this is to utilize a factorial
design for the variables in the various surface models
used. The goal of finding a site pattern which
represents the entire surface is considered to be
achieved when the variations of the variables of the
different surface models result in a non-significant
effect in the evaluation of the factorial design.
R1-5.2 Simulation of Roughness Maps
R1-5.2.1 The measurement results that are reported in
section 3 provide some insight into the variation of
roughness that occurs across actual, typical wafer
surfaces. The variation observed on the different
wafers is related to the specific polishing technique
(wax mount polishing, wax free polishing) and to the
polishing parameters used. Examination of haze maps
reveals three basic patterns of variation:
a)
a circularly symmetric haze variation
b) a haze variation approximately symmetric with
respect to a diameter across the wafer
c) a linear gradient of haze from one wafer edge to
the opposite one.
R1-5.2.2 The following relations were used to simulate
the variation of roughness according to these three basic
models a)-c) with e and c being the roughness values
near the edge and at the center of a wafer, respectively:
model a: a parabolic relation
z(x,y) = (x
2
+y
2
) (e-c)/r
2
+ c (equ. 1)
model b: a semi-cylindrical relation
z(x,y) = (r
2
-(y cos a – x sin a)
2
)
1/2
(e-c)/r + e
(equ. 2)
model c: a linear gradient
z(x,y) = (y cos a - x sin a) (e-c)/r + c (equ. 3)
with wafer radius r and angle α corresponding to the
angle between the symmetry plane and the x-direction
(model b) or the direction of the gradient and the y-
direction (model c).
R1-5.2.3 Using equations 1-3, roughness maps of
wafer surfaces can be generated with a roughness value
assigned to each site. This was performed by using
MathCad® software and by assuming 200 mm wafers,
the area of each being partitioned into sites of 1 mm
2
size.
R1-5.2.4 Roughness is not necessarily an isotropic
property of a surface. Different roughness values are in
general obtained when e.g. two scans are performed at
the same spot on a wafer surface but in perpendicular
directions. Therefore two maps representing the
roughness anisotropy were generated in each case by
assuming two sets of the parameters e and c – e1, c1,
e2, and c2, per wafer surface.
R1-5.3 Factorial Design
R1-5.3.1 A factorial design at two levels was selected
in order to compare the various wafer maps generated
(10). The parameters ei and ci (i = 1,2) were used as
variables and were varied between two levels, 0.1 and
0.2, in arbitrary units. In addition the patterns (models
a – c) were also used as a variable and the values –1
and +1 were correspondingly assigned resulting in a 2
5
factorial design. The complete set of parameters used is
displayed in Table R1-5.
Table R1-5 Complete Set of Parameters Used in the 2
5
Factorial Design
Variable e1 c1 e2 C2 Model
High level 0.2 0.2 0.2 0.2 +1 (model b or c)
Low level 0.1 0.1 0.1 0.1 -1 (model a)
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