Data analysis sheet for determining the
Young's modulus value of a thin film layer for use with the MEMS
5-in-1 RMs
a)
b)
Figure YM.3.1. For CMOS cantilever a) a design
rendition and b) a cross section
To obtain
the following measurements,
consult SEMI standard test
method MS4 entitled "Test
Method for Young's Modulus
Measurements of Thin, Reflecting
Films Based on the Frequency
of Beams in Resonance."
date (optional)
=
/
/
comments
(optional) =
Table 1 -
Preliminary
INPUTS
Description
1
temp
the temperature during measurement
(should be held constant)
2
relativehumidity
%
the relative humidity during
measurement (if not known, enter -1)
3
×
the
magnification
4
mat
the composition of the thin film layer
5*
ρ
g/cm3
the
density of the thin film layer
6
σρ
g/cm3
the
one sigma uncertainty of the value of
ρ
7*
μ
×10-5
Ns/m2
the
viscosity of the ambient surrounding the cantilever
8*
W
μm
the
suspended beam width
9*
t
μm
the
thickness of the thin film layer (as found using Data
Sheet T.1 or Data Sheet T.3)
10
σthick
μm
the
one sigma uncertainty of the value of t (as
found using Data
Sheet T.1 or Data Sheet T.3)
11
dgap
μm
the gap depth (distance between the
bottom of the suspended beam and the underlying layer)
12*
Einit
GPa
the initial estimate for the Young's
modulus value of the thin film layer
13
finstrument
MHz
used for calibrating the time base of the instrument:
the frequency setting for the calibration measurements
(or the manufacturer's specification for the clock
frequency)
14
fmeter
MHz
used for calibrating the time base of the instrument:
the calibrated average frequency of the calibration
measurements (or the calibrated average clock frequency)
taken with a frequency meter
15
smeter
Hz
used for calibrating the time base of the instrument:
the standard deviation of the frequency measurements taken with
the frequency meter
16
ucertf
Hz
used for calibrating the time base of the instrument:
the certified uncertainty of the frequency measurements
as specified on the frequency meter's certificate
* The five starred entries in this
table are required inputs for the calculations in the
Preliminary Estimates Table.
Table 2 - Cantilever
INPUTS
Description
17
name
the
cantilever name (optional)
18
the
orientation of the cantilever
19*
Lcan
μm
the
suspended cantilever length
20
indicates which cantilever on the test chip,
where "first" corresponds to the topmost cantilever in
the column or array that has the specified length?
21
σL
μm
the
one sigma uncertainty of the value of Lcan
22
fresol
Hz
the
uncalibrated
frequency resolution for the given set of measurement
conditions
23
fmeas1
kHz
the
first uncalibrated, damped resonance frequency
measurement
(or the first uncalibrated, undamped resonance frequency measurement, for
example, if the measurements were performed in a vacuum)
24
fmeas2
kHz
the
second uncalibrated, damped resonance frequency
measurement
(or the second uncalibrated, undamped resonance frequency measurement, for
example, if the measurements were performed in a vacuum)
25
fmeas3
kHz
the
third uncalibrated, damped resonance frequency
measurement
(or the third uncalibrated, undamped resonance frequency measurement, for
example, if the measurements were performed in a vacuum)
26
fcorrection
kHz
the
correction term for the cantilever's resonance
frequency
27
σsupport
kHz
the
uncertainty in the cantilever's resonance frequency
due to a non-ideal support (or attachment conditions)
28
σcantilever
kHz
the
uncertainty in the cantilever's resonance frequency
due to geometry and/or composition deviations from the ideal
* The starred entry in this table
is a required input for the calculations in the Preliminary
Estimates Table.
Table 3 - Fixed-Fixed Beam
INPUTS
(if cantilever not available)
Description
29
name2
the fixed-fixed beam name (optional)
30
the orientation of the fixed-fixed
beam
31*
Lffb
μm
the
suspended fixed-fixed beam length
32
indicates which fixed-fixed beam on the test
chip, where "first" corresponds to the topmost
fixed-fixed beam in the column or array that has the
specified length?
33
fffb
kHz
the average uncalibrated resonance frequency of the
fixed-fixed beam
* The starred entry in this table
is a required input for the calculations in the Preliminary
Estimates Table.
Table 4 -
Optional
INPUTS
For residual stress
calculations:
Description
34
εr
×10-6
the
residual strain
of the thin film layer
(as found using ASTM E 2245 and Data Sheet RS.3
for compressive residual strain)
35
ucεr
×10-6
the combined standard uncertainty
value for residual strain
(as found using Data Sheet RS.3
for compressive residual strain)
For stress gradient
calculations:
36
sg
m-1
the
strain gradient of the thin film layer
(as found using ASTM E 2246
and Data Sheet SG.3)
37
ucsg
m-1
the combined standard uncertainty value
for strain gradient
(as found using Data Sheet SG.3)
Table 5 - Preliminary
ESTIMATES*
Description
38
fcaninit
kHz
= SQRT[Einit t2
/ (38.330
ρ Lcan4)]
(the estimated
resonance frequency of the cantilever)
39
fffbinithi
kHz
= SQRT[Einit
t2 / (0.946 ρ
Lffb4)]
(the estimated
upper bound for the resonance frequency of the fixed-fixed
beam)
40
fffbinitlo
kHz
= SQRT[Einit
t2
/ (4.864 ρ Lffb4)]
(the estimated
lower bound for the resonance frequency of the fixed-fixed
beam)
41
Q
= Wt2
SQRT(ρ
Einit)
/ (24
μ
Lcan2)
(the estimated
Q-factor)
42
pdiff
%
={1-SQRT[1-1
/ (4 Q2)]}×100
% should be < 2 %
(the estimated
percent difference between the damped and undamped resonance
frequency of the cantilever)
* The seven starred inputs in the first three tables are required
for the calculations in this table.
OUTPUTS:
Table 6 -
Frequency calculations:
Description
43
calf
= fmeter /
finstrument (the calibration factor
for a frequency measurement)
44
fmeasave
kHz
= AVE [fmeas1, fmeas2,
fmeas3]calf
(the average calibrated damped resonance frequency of
the cantilever, fdampedave, or
the average calibrated undamped resonance frequency of the cantilever if,
for example, the measurements were performed in a vacuum)
45
fundamped1
kHz
= fdamped1/ SQRT[1-1/(4Q2)]
where fdamped1=fmeas1(calf)
(the first calibrated undamped resonance frequency
calculated from the cantilever's first damped resonance
frequency measurement, if applicable)
46
fundamped2
kHz
=fdamped2 / SQRT[1-1/(4Q2)] where fdamped2=fmeas2(calf)
(the second calibrated undamped resonance frequency
calculated from the cantilever's second damped resonance
frequency measurement, if applicable)
47
fundamped3
kHz
= fdamped3/ SQRT[1-1/(4Q2)] where fdamped3=fmeas3(calf)
(the third calibrated undamped resonance frequency
calculated from the cantilever's third damped resonance
frequency measurement, if applicable)
48
fundampedave
kHz
= AVE [fundamped1, fundamped2,
fundamped3]
(the average calibrated undamped resonance frequency
of the cantilever assuming fmeas1, fmeas2, and fmeas3 from
the second table are damped resonance frequencies)
49
σfundamped
= STDEV (fundamped1, fundamped2, fundamped3)
(the one sigma
uncertainty of the value of fundampedave
assuming fmeas1, fmeas2,
and fmeas3 from the second table
are damped resonance frequencies)
50
fcan
= fundampedave + fcorrection (the modified
resonance frequency of the cantilever for use if fmeas1, fmeas2, and fmeas3 from
the second table are damped resonance frequencies)
51
fmeasavenew
= fmeasave + fcorrection (the modified
resonance frequency of the cantilever for use if fmeas1, fmeas2, and fmeas3 from
the second table are undamped resonance frequencies)
1.
Young's modulus calculation
(as obtained from the cantilever assuming clamped-free
boundary conditions):
a.
E =
38.330 ρ fcan2 Lcan4
/ t2 =
GPa
(Use this value if fmeas1,
fmeas2, and fmeas3
in the second table are damped
resonance frequencies.)
b.
E =
38.330 ρ fmeasavenew2
Lcan4
/ t2 =
GPa
(Use this value if fmeas1,
fmeas2, and fmeas3
in the second table are undamped
resonance frequencies.)
c. ucE =
σE
= E SQRT[(σρ/ρ)2
+ 4(σfcan/fcan)2
+ 16(σL/Lcan)2
+ 4(σthick/t)2] =
*σE
/ E =
* σfcan/fcan
= SQRT[(σfundamped/fcan)2 +
(σfresol/fcan)2 + (σfreqcal/fcan)2+
(σsupport/fcan)2 + (σcantilever/fcan)2], σfresol= fresol
calf/ [2SQRT(3)], and
σfreqcal
= fundampedave[SQRT(σmeter2+
ucertf2) / fmeter] σρ/ρ
=
Type B σthick/t
=
Type B σL/Lcan
=
Type B σfundamped/fcan
=
*
Type A σfresol/fcan
=
Type B σfreqcal/fcan
=
Type B σsupport/fcan
=
Type B σcantilever/fcan
=
Type B *assumes fmeas1, fmeas2,
and fmeas3 in the second table are damped
resonance frequencies
UE = 2ucE = GPa
(expanded uncertainty)
3ucE = GPa
d. E -
UE
=
GPa
(a lower bound for E) E + UE
=
GPa
(an upper bound for E)
(assuming fmeas1, fmeas2,
and fmeas3 in the second table are damped
resonance frequencies)
e. Report the results as follows: If it is assumed that the estimated values
of the uncertainty components are
approximately Gaussianly distributed with approximate combined
standard uncertainty
ucE, the Young's modulus value is believed to lie in
the interval E ±
ucE
(expansion factor
k=1) representing a level of
confidence of approximately 68 %.
2. Young's
modulus calculation
(as obtained from a fixed-fixed beam...not
recommended):
a. Esimple = 4.864
ρ ( fffb calf )2 Lffb4
/
t2= GPa (as obtained from
the fixed-fixed beam assuming simply-
supported boundary conditions for both
supports) b. Eclamped = 0.946
ρ (
fffb calf)2
Lffb4 /
t2=
GPa (as obtained from
the fixed-fixed beam assuming
clamped-clamped boundary conditions)
c. E = (Esimple
+ Eclamped) / 2 =
(use this value, if must)
d. uE
= (Esimple - Eclamped)
/ 6 =
(as obtained from a Type B analysis)
e. Report the results as follows: If it is assumed that the
estimated value of the standard
uncertainty, uE, is approximately
Gaussianly distributed, the Young's modulus value is
believed to lie in
the interval E ±
uE (expansion factor k=1)
representing a level of
confidence of approximately 68 %.
Table 7 -
Optional
OUTPUTS
(using E and ucE
from the cantilever and assuming fmeas1,
fmeas2,
and fmeas3
in the second table are damped resonance frequencies)
For residual stress:
Description
52
σr
MPa
= E εr
(the residual
stress of the thin film layer)
53
ucσr
MPa
=
|σr|
SQRT[(σE
/ E)2+
(σεr
/ εr)2]
(the combined
standard uncertainty value for residual stress
where
σεr
is
equated with ucεr)
54
σσr
/
|σr|
where σσr
is equated with ucσr
55
σE
/ E
as obtained from this data sheet
56
σεr /
|εr|
where
σεr
is
equated with ucεr
and where
εr
and ucεr
were obtained from Data Sheet RS.3
57
2ucσr
MPa
= Uσr
the expanded uncertainty for residual
stress
58
3ucσr
MPa
three times the
combined
standard uncertainty for residual stress
59
σr-Uσr
MPa
a lower bound for
σr
60
σr+Uσr
MPa
an upper bound for
σr
For stress gradient:
61
σg
GPa/m
= E sg
(the stress
gradient of the thin film layer)
62
ucσg
GPa/m
=
σg SQRT[(σE
/ E)2+
(σsg
/ sg)2]
(the combined
standard uncertainty value for stress gradient where
σsg
is
equated with ucsg)
63
σσg
/σg
where σσg
is equated with ucσg
64
σE
/ E
as obtained from this data sheet
65
σsg /
sg
where
σsg
is
equated with ucsg
and where
sg
and ucsg
were obtained from Data Sheet SG.3
66
2ucσg
GPa/m
= Uσg
the expanded uncertainty for stress
gradient
67
3ucσg
GPa/m
three times the
combined
standard uncertainty for stress gradient
68
σg-Uσg
GPa/m
a lower bound for
σg
69
σg+Uσg
GPa/m
an upper bound for
σg
Modify the input data, given the information supplied in any flagged
statement below, if applicable, then recalculate:
1.
Please provide inputs to Tables 1 and 2 for calculations
using data from a cantilever.
2.
The value for temp
should be between 19.4 °C
and 21.6 °C,
inclusive.
3.
The value for relative humidity (if
known) should be
between 0 % and 60 %, inclusive.
4.
If applicable, please provide
inputs to Table 3, ρ,
W, t, and Einit for
calculations using data from a fixed-fixed beam.
5.
The value for mag should be greater than or equal to
20×.
6.
The value for
ρ
should be between 1.00 g/cm3 and 5.00 g/cm3.
7.
The value for
σρ
should be between 0.0 g/cm3 and 0.10 g/cm3.
8.
The value for
μ should be between
0.70×10-5
Ns/m2
and 3.0×10-5
Ns/m2.
9.
The value for W should be greater than t and
less than Lcan.
10.
If Lffb is inputted, the value for W should be greater than t and
less than Lffb.
11.
The value for t
should be between 0.000
μm and 10.000 μm.
12.
The value for
σthick
should be between 0.0
μm and 0.5
μm.
13.
Squeeze film damping expected for the
cantilever since dgap < W
/ 3.
14.
The value for Einit
should be between 10 GPa
and 300 GPa.
15.
The
values for
σmeter
and ucertf should be between
0.0 Hz and 25.0 Hz, inclusive.
16.
The value for Lcan
should be between 0
μm and 1000
μm.
17.
The value for
σL
should be between 0.0
μm
and 2.0
μm.
18.
The value for fresol
should be between 0 Hz
and 50 Hz.
19.
The values for fmeas1,
fmeas2,and
fmeas3should be between 5.00 kHz and 300.0
kHz.
20.
The value for fcorrection should be
between -10 kHz and 10 kHz, inclusive.
21.
The values for
σsupport
and
σcantilever
should be between 0 kHz
and
10 kHz, inclusive.
22.
If inputted, the value for
Lffb should be between 0
μm and 1000
μm.
23.
If inputted, the value for fffb
should be between 5.0 kHz and 1200 kHz.
24.
If inputted, the value for
εrshould be between -4500×10-6
and 4500×10-6
and not equal to 0.0.
25.
If inputted, the value for ucεrshould be between 0.0 and 300.0×10-6.
26.
If inputted, the value for sg
should be between 0.0 m-1
and 1500.0 m-1.
27.
If inputted, the value for ucsg
should be between 0.0 m-1
and 100.0 m-1.
28.
The values for fmeas1,
fmeas2, and fmeas3
are not within 20 kHz of fcaninit.
29.
If inputted, the value for fffb
should be between fffbinitlo and
fffbinithi.
30.
The value for pdiff
should be between 0 %
and 2 %.
31.
The value for calf should be
between 0.9990 and 1.0010.
32.
The value for
σfundamped
should be between 0.0 kHz and 0.5 kHz, inclusive.
33.
The value of E obtained from the
cantilever should be within 60 GPa of Einit.
34.
The value of E obtained from the cantilever should be
greater than
2ucE.
35.
If applicable, the value of E
obtained from the fixed-fixed beam should be within 70 GPa
of Einit.
36.
If applicable, the value of
uE
obtained from the fixed-fixed beam should be between 0 GPa
and 70 GPa.