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SEMI MF1390-1104 © SEMI 2004 5 NOTE 5: From Figure 1, note that th e distance between Probe A and the nearest surface of the wafer is displac ement value a and the distance between Probe B and the nearest surface of the …

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

, of the median surface from

the z-axis origin at each point. From Figure 1,

 (2)

and

 (3)

Therefore,

2222

 (4)

and



 (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:

invnor

gravity



 (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

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:

invnorinvnor

norgravitynorcom

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

(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

, b

, and c

are constants selected so that





RRRcom

cybxayxz

)](),([ (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.

13.13.1 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, inclu-

ding 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 Warp 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 Twenty-three 200 mm diameter, single-side

polished silicon wafers were employed in a round-robin

experiment. These wafers represented three different

manufacturing processes. All three subsets were bare

on the front surface. Two subsets were bare on the

back surface and one subset had oxide on the back

surface.

15.2 Eight laboratories measured warp. Warp value

ranges for these three subsets are shown in Table 1.

Each of the twenty-three wafers was measured three

times in succession (in three cassette-to-cassette

“passes”), on a single day on automatic measurement

systems in accordance with this test method.

15.3 All measurement data were acquired with 3 mm

nominal edge exclusion.

15.4 The number of laboratories, samples, and

determinations in this study met the minimum

requirements for determining precision prescribed in

ASTM Practice E 691.

15.5 The ranges of 95% confidence interval, within

which two measurements are considered statistically to

be the same, for within-laboratory repeatability (r) and

between-laboratory reproducibility (R) are shown in

Table 1.

15.6 Figure 2 contains plots of the 95% confidence

intervals for repeatability (r) and reproducibility (R)

against mean value of warp.

15.7 For more details, refer to the Research Report.

Table 1 Summary Warp Measurement Statistics

Parameter Warp

Smallest 4.30

Mean, m

Largest 31.25

Smallest 0.184

r, m

Largest 0.695

Smallest 1.112

R, m

Largest 2.097

Figure 2

Warp Repeatability & Reproducibility

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; wafers; warp

4 Available on request from SEMI Headquarters, Publications

Department, San Jose, CA, Fax: 408-943-7015. Request International

Standards Research Report F01-1016, ASTM Interlaboratory Round

Robin Experiment on Measuring Warp on Silicon Wafers by

Automated Noncontact Scanning and Measuring Flatness, Thickness

and Thickness Variation of Silicon Wafers by Automated Noncontact

Scanning.

Mean Warp,

m

r and R, m

5 1015202530

0.0

0.5

1.0

1.5

2.0

2.5

Repeatability

Reproducibility