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SEMI MF1390-1104 © SEMI 2004 4 between the warp val ue ob tained by the measuring equipment under test and the refe rence warp value (see Section 9). 8.3 Representative Wafer — required only if the Representati v e Wafer…

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SEMI MF1390-1104 © SEMI 2004 3
6.2 The wafer is supported by a small-area chuck with
front surface up.
6.3 Both external surfaces are simultaneously scanned
along a prescribed pattern by an opposed pair of probes
to obtain a set of values of the distances between each
surface and the nearest probe. In each case, both
members of the pair of distances is taken at the same
value of the x and y coordinates.
6.4 The paired displacement values are used to
construct the median surface.
6.5 A correction for gravity effects on the median
surface is made either by subtracting a gravity
correction obtained (1) from measurements on a
representative wafer or (2) from theoretical
considerations or by repeating the scan with the wafer
inverted.
6.6 A least-squares reference plane is constructed from
the corrected median surface.
6.7 The reference plane deviation (RPD) is calculated at
each measured pair of points.
6.8 Warp is reported as the algebraic difference
between the most positive RPD and the most negative
RPD.
7 Apparatus
7.1 Measuring Equipment, consisting of wafer holding
device, multiple-axis transport mechanism, probe
assembly with indicator, and system
controller/computer, including data processor and suit-
able software.
7.1.1 The equipment shall be direct reading with all
necessary calculations performed internally and
automatically as outlined in Section 11.2.
7.1.2 The equipment shall be equipped with an over-
range signal.
7.1.3 Instrument data reporting resolution shall be 100
nm or smaller.
7.1.4 The measuring equipment contains the following
subsystems:
7.1.4.1 Wafer-holding device, for example a chuck
whose face is perpendicular to the measurement axis,
and on which the wafer is placed for the measurement
scan. The nature and size of the wafer holding device
shall be agreed upon between the parties to the test.
7.1.4.2 Multiple-axis transport mechanism, which
provides a means for moving the wafer-holding device,
or the probe assembly, perpendicularly to the
measurement axis in a controlled fashion in several
directions. This motion must permit data gathering
over a prescribed scan pattern covering the entire fixed
quality area. Data point spacing to be used shall be
agreed upon between the parties to the test.
Figure 1
Schematic View of Wafer, Probes, and Fixture
7.1.4.3 Probe assembly with paired non-contacting
displacement-sensing probes, probe supports, and
indicator unit (see Figure 1).
7.1.4.3.1 The probes shall be capable of independent
measurement of the distances a and b between the
probed site on each surface of the sample wafer and the
nearest probe surface.
7.1.4.3.2
The probes shall be mounted above and
below the wafer in a manner so that the probed site on
one surface of the wafer is opposite the probed site on
the other.
7.1.4.3.3 The common axis of these probes is the
measurement axis.
7.1.4.3.4 The probe separation D shall be kept constant
during calibration and measurement.
7.1.4.3.5 Displacement resolution shall be 100 nm or
smaller.
7.1.4.3.6 The probe sensor size shall be 4 mm by 4
mm, or other value to be agreed upon between the
parties to the test.
7.1.4.3.7 Measuring equipment employing either the
Representative Wafer Inversion Method or the Sample
Wafer Inversion Method for gravity compensation must
provide precise positioning in both measurement
orientations so that measurements are taken at identical
locations for each orientation of the sample.
8 Materials
8.1 Set-up Masters — suitable to accomplish
calibration and standardization as recommended by the
equipment manufacturer.
8.2 Reference Wafer — with a reference warp value
20 m that is used to determine the level of agreement
SEMI MF1390-1104 © SEMI 2004 4
between the warp value obtained by the measuring
equipment under test and the reference warp value (see
Section 9).
8.3 Representative Wafer — required only if the
Representative Wafer Inversion Method is used for the
gravity correction. A representative wafer shall be
identical in nominal diameter, nominal thickness,
fiducials, composition and crystalline orientation to
those being measured. Its warp need not be known.
9 Suitability of Measuring Equipment
9.1 Determine the suitability of the measuring
equipment with the use of a reference wafer and its
reference warp value in accordance with the
procedures of Sub-section 9.2, or by performance of a
statistically-based instrument repeatability study to
ascertain whether the equipment is operating within the
manufacturer's stated specification for repeatability.
9.1.1 The reference warp value is the average of a
number of values obtained for that wafer over a number
of “passes” (repeat measurements). The reference
wafer is measured on the measuring equipment under
test and its reference warp value is compared against
the measured warp value. The acceptable level of the
agreement between the reference and measured warp
values is to be agreed upon by the parties to the test.
9.2 Procedure
9.2.1 Select a reference wafer of appropriate criteria,
together with its associated reference warp value.
9.2.2 Measure the reference wafer on the measuring
equipment under test to obtain a sample warp value.
9.2.3 Subtract the two warp values to obtain the
difference:
samplereferencewarp
WarpWarp
(1)
9.2.4 The metric to be used to determine acceptability
is difference,
warp.
Accept the measuring equipment as
suitable for use if this difference is less than a value that
is agreed upon between the parties to the test.
NOTE 4: If the measuring equipment is to be used to
measure other parameters, such as flatness and thickness
variation in addition to warp, the reference and sample warp
values may be included in the reference and sample data sets
specified in SEMI MF1530, but this is not necessary if only
warp measurements are to be made.
10 Sampling
10.1 This test method is nondestructive and may be
used on either 100% of the wafers in a lot or on a
sampling basis.
10.1.1 If samples are to be taken, procedures for
selecting the sample from each lot of wafers to be tested
shall be agreed upon between the parties to the test, as
shall the definition of what constitutes a lot.
11 Calibration and Standardization
11.1 Calibrate the measuring equipment in accordance
with the manufacturer's instructions.
11.2 When using the Representative Wafer Inversion
Method for correcting the gravity-induced deflection,
determine z
gravity
, the deflection due to gravity and
machine effects on the representative wafer, in
accordance with Section 12 and Section 13 through
paragraph 13.7.
12 Procedure
12.1 Prepare the apparatus for measurement of wafers,
including selection of diameter, peripheral fiducials,
scan area and data display/output functions. Also select
the gravitational correction method from one of the
following:
Reference Wafer Inversion Method (see Note 3),
Sample Wafer Inversion Method, or
Theoretical Modeling Method.
12.2 Select the fixed quality area (FQA) by specifying
the nominal edge exclusion (EE).
12.3 Introduce the test specimen into the measurement
mechanism with the front surface upward and initiate
the measurement sequence to determine and record the
distances between each probe and the nearest wafer
surface in pairs, a and b, at each measurement position.
Proceed directly to Section 13 unless (1) the Sample
Wafer Inversion Method is being used to correct for
effects of distortion due to gravity or (2) a
representative wafer is being measured to obtain the
gravity correction for use in the Representative Wafer
Inversion Method (see Note 3).
12.4 Repeat Section 12.2 with the wafer inverted (front
surface downward).
13 Calculations
13.1 The following calculations are performed
automatically within the instrument. An outline of the
calculation structures is provided here to indicate the
nature of the procedure.
13.2 Determine the displacements (distances) between
each probe and the nearest surface of the wafer (in
pairs) at intervals along the scan pattern.
SEMI MF1390-1104 © SEMI 2004 5
NOTE 5: From Figure 1, note that the distance between
Probe A and the nearest surface of the wafer is displacement
value a and the distance between Probe B and the nearest
surface of the wafer is displacement value b.
13.3 Set the origin of the z-axis at the midpoint between
the two probes, A and B.
13.4 Find the distance, z
m
, of the median surface from
the z-axis origin at each point. From Figure 1,
22
t
a
D
z
m
(2)
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 6: This cancels the effect of the representative wafer's
shape while retaining the effect of gravity.
13.8 Determine the gravity compensated median sur-
face as follows:
13.8.1 Representative Wafer Inversion Method —
Subtract z
gravity
from z
nor
to produce z
com
at each
measurement point.
NOTE 7: 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 Construct a reference plane that is a least-squares
fit to the gravity-compensated median surface data at all
the points of the of the scan pattern. The reference
plane is of the form:
RRRref
cybxaz
(8)
where a
R
, b
R
, and c
R
are constants selected so that
yx
RRRcom
cybxayxz
,
2
)](),([ (9)
is minimized over the FQA (see Section 12.2).
13.10 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:
refcom
zzRPD
(10)
13.11 Calculate the warp of the wafer as the difference
between the maximum (most positive) and minimum
(most negative) RPD:
minmax
warp RPDRPD
(11)
13.12 Record the calculated warp value.
13.13 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.