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SEMI MF1451-1104 © SEMI 2004 6 13.15.1 Record sample st andard de viation and other statistical parameters as agreed upon between the parties to the test. 14 Report 14.1 Report the following information: 14.1.1 Date, tim…
SEMI MF1451-1104 © SEMI 2004 5
and
22
t
b
D
z
m
(3)
Therefore,
m
z
t
b
Dt
a
D
2
2222
(4)
and
2
ab
z
m
(5)
where
D = distance between Probes A and B,
a = distance between Probe A and the nearest
(top) wafer surface,
b = distance between Probe B and the nearest
(bottom) wafer surface, and
t = wafer thickness.
13.5 For measurements in the normal orientation (front
surface up), call the position of the median surface z
nor
.
13.6 For measurements in the inverse orientation (back
surface up), call the position of the median surface z
inv
.
13.7 For measurements on representative wafers or on
sample wafers when the Sample Wafer Inversion
Method is being used, determine the gravitational
correction to the median surface as follows:
2
invnor
gravity
zz
z
(6)
NOTE 5: This cancels the effect of the representative wafer's
shape while retaining the effect of gravity.
13.8 Determine the gravity compensated median
surface as follows:
13.8.1 Representative Wafer Inversion Method —
Subtract z
gravity
from z
nor
to produce z
com
at each
measurement point.
NOTE 6: The Representative Wafer Inversion Method deals
not only with first-order gravitational effects, but also with
other effects that may influence the measured value, such as
wafer-periphery effects, some machine-specific signature, etc.
13.8.2 Sample Wafer Inversion Method — Subtract
z
gravity
from z
nor
to produce z
com
at each measurement
point. Note that this is equivalent to taking the
difference between the normal and inverted
measurement values at each point:
22
invnorinvnor
norgravitynorcom
zzzz
zzzz
(7)
13.8.3 Theoretical Modeling Method — Apply
gravitational correction developed from a theoretical
model. Although a rigorous model is not known to
exist, approximate corrections have been calculated
3
(see Related Information 1).
13.9 Determine the thickness of the wafer at each point
from the following equation:
)( baDt
(8)
where the terms are defined after Equation (5).
13.10 Add one half the thickness to the gravity
compensated median surface z value at each point to
yield the gravity-compensated front surface:
2
t
zz
comfcom
(9)
13.11 Construct a reference plane that is a least-squares
fit to the gravity-compensated front surface data at all
the points of the of the scan pattern. The reference
plane is of the form:
RRRref
cybxaz
(10)
where a
R
, b
R
, and c
R
are constants selected so that
yx
RRRfcom
cybxayxz
,
2
)](),([ (11)
is minimized over the FQA (see Section 12.2).
13.12 Subtract the z-value of the reference plane (z
ref
)
from the compensated z-value, z
com
, at all the points of
the scan pattern to yield the reference plane deviation
(RPD) at each point:
reffcom
zzRPD
(12)
13.13 Calculate the sori of the wafer as the difference
between the maximum (most positive) and minimum
(most negative) RPD:
minmax
sori RPDRPD
(13)
13.14 Record the calculated sori value.
13.15 For referee or other measurements where the
wafer is measured more than once, calculate the
maximum, minimum, sample standard deviation,
average, and range of all measurements on the sample.
3 Application Note: “Gravitational Sag in Silicon Wafers,” ADE
Corporation, 80 Wilson Way, Westwood, MA 02090-1806, Nov. 20,
1991.
SEMI MF1451-1104 © SEMI 2004 6
13.15.1 Record sample standard deviation and other
statistical parameters as agreed upon between the
parties to the test.
14 Report
14.1 Report the following information:
14.1.1 Date, time, and temperature of test,
14.1.2 Identification of operator,
14.1.3 Location (laboratory) of test,
14.1.4 Identification of measuring instruments,
including wafer-holding device diameter, data point
spacing, sensor size, and gravitation-correction method,
14.1.5 Lot identification, including nominal diameter,
nominal center-point thickness, and nominal edge
exclusion (EE) specified.
14.1.6 Description of sampling plan, if any, and
14.1.7 Sori of each wafer measured.
14.2 For referee tests also include in the report the
standard deviation of each set of wafer measurements
and such other statistical parameters as have been
agreed to by the parties to the test.
15 Precision
15.1 An inter-laboratory experiment to establish the
precision of this test method has not been conducted.
15.2 In order to estimate the precision, an error analysis
was performed in accordance with ASTM Practice D
4356. This analysis indicates the best 1s precision
obtainable by this method is ±0.35 m.
15.3 This value is obtained as follows:
15.3.1 The instrument resolution is specified to be 0.1
m. Therefore, the values a and b cannot be known
closer than this during each of the six measurements
made during the determination of sori:
two measurements on the reference wafer in the
normal orientation,
two measurements on the reference wafer in the
inverted orientation, and
two measurements on the sample wafer in the
normal orientation.
15.4 Equation A1.1 in ASTM Practice D 4356 holds
that:
35.0
6)1.0(2
2e (1s)Precision
2
2
N
(14)
where:
e =
single measurement error (0.1 m), and
N = number of measurements made during the
determination of sori (6).
15.5 In practice, this precision is degraded by the
factors listed in Section 3.
16 Bias
16.1 Bias — No standards exist against which the bias
of this test method can be evaluated.
17 Keywords
non-contact measurement; semiconductor; shape;
silicon; sori; wafers
SEMI MF1451-1104 © SEMI 2004 7
RELATED INFORMATION 1
MEASUREMENT ERRORS DUE TO DIFFERENCES IN DIAMETER AND
THICKNESS BETWEEN A REPRESENTATIVE WAFER AND A WAFER
UNDER TEST
NOTICE: This related information is not an official part of SEMI MF1451. This related information was approved
for publication by full letter ballot on August 16, 2004.
R1-1 The sag, or deflection induced by gravity at the
edge of a wafer supported at its center, in m, has been
estimated
3
as:
2
4
2
48
32
)103(
t
KD
Et
kgdD
S
(R1-1)
where:
S =
deflection , in m,
k = geometrical constant (=0.5854),
g = gravitational constant (980 cm/s
2
),
d = density of silicon (2.329 g/cm
3
),
E =
Young’s modulus (~1.610
12
dyne/cm
2
,
D = nominal wafer diameter, in mm, and
t =
nominal wafer thickness, in m.
K, the constant of proportionality is therefore equal to
7.83 10
3
m
3
/mm
4
. Table R1-1 gives estimated
values of sag in micrometers for 100 mm through 300
mm diameter wafers with thickness and diameter as
specified in SEMI M1.
Table R1-1. Estimated Sag, in
m, of Wafers of
Nominal Diameter and Thickness
Diameter,
mm
Thickness,
m
SEMI M1 Reference Estimated Sag,
m
300 775 SEMI M1.15 105.6
200 725 SEMI M1.9 23.8
150 675 SEMI M1.8 8.7
150 625 SEMI M1.13 10.1
125 625 SEMI M1.7 4.9
100 525 SEMI M1.5 2.8
R1-2 For small variations about the nominal values of
diameter and thickness, the relative change of the
gravity effect is 4 times the relative change of the
diameter and 2 times the relative change of thickness:
2
3
4
t
Kd
d
S
(R1-2)
and
3
4
2
t
Kd
t
S
(R1-3)
Therefore the relative changes are as follows:
d
d
S
S
4 (R1-4)
and
t
t
S
S
2 (R1-5)
R1-3 Table R1-2 gives examples of worst-case gravity
effect errors (in micrometers), for 100 mm through 300
mm diameter wafers with thickness and diameter toler-
ances as specified in SEMI M1.