ASTM E2246-11(2018)
(Test Method)Standard Test Method for Strain Gradient Measurements of Thin, Reflecting Films Using an Optical Interferometer (Withdrawn 2023)
Standard Test Method for Strain Gradient Measurements of Thin, Reflecting Films Using an Optical Interferometer (Withdrawn 2023)
SIGNIFICANCE AND USE
5.1 Strain gradient values are an aid in the design and fabrication of MEMS devices.
SCOPE
1.1 This test method covers a procedure for measuring the strain gradient in thin, reflecting films. It applies only to films, such as found in microelectromechanical systems (MEMS) materials, which can be imaged using an optical interferometer, also called an interferometric microscope. Measurements from cantilevers that are touching the underlying layer are not accepted.
1.2 This test method uses a non-contact optical interferometric microscope with the capability of obtaining topographical 3-D data sets. It is performed in the laboratory.
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
1.4 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
WITHDRAWN RATIONALE
This test method covers a procedure for measuring the strain gradient in thin, reflecting films. It applies only to films, such as found in microelectromechanical systems (MEMS) materials, which can be imaged using an optical interferometer, also called an interferometric microscope. Measurements from cantilevers that are touching the underlying layer are not accepted.
Formerly under the jurisdiction of Committee E08 on Fatigue and Fracture, this test method was withdrawn in November 2023. This standard is being withdrawn without replacement due to its limited use by industry.
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This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation:E2246 −11 (Reapproved 2018)
Standard Test Method for
Strain Gradient Measurements of Thin, Reflecting Films
Using an Optical Interferometer
This standard is issued under the fixed designation E2246; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope Atomic Force Microscope at Subnanometer Displacement
Levels Using Si(111) Monatomic Steps (Withdrawn
1.1 This test method covers a procedure for measuring the
2015)
strain gradient in thin, reflecting films. It applies only to films,
such as found in microelectromechanical systems (MEMS) 2.2 SEMI Standard:
MS2Test Method for Step Height Measurements of Thin
materials,whichcanbeimagedusinganopticalinterferometer,
Films
also called an interferometric microscope. Measurements from
cantilevers that are touching the underlying layer are not
3. Terminology
accepted.
3.1 Definitions:
1.2 This test method uses a non-contact optical interfero-
3.1.1 The following terms can be found in Terminology
metricmicroscopewiththecapabilityofobtainingtopographi-
E2444.
cal 3-D data sets. It is performed in the laboratory.
3.1.2 2-D data trace, n—a two-dimensional group of points
1.3 This standard does not purport to address all of the
that is extracted from a topographical 3-D data set and that is
safety concerns, if any, associated with its use. It is the
parallel to the xz-or yz-plane of the interferometric micro-
responsibility of the user of this standard to establish appro-
scope.
priate safety, health, and environmental practices and deter-
3.1.3 3-D data set, n—a three-dimensional group of points
mine the applicability of regulatory limitations prior to use.
with a topographical z-value for each (x, y) pixel location
1.4 This international standard was developed in accor-
within the interferometric microscope’s field of view.
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the
3.1.4 anchor, n—in a surface-micromachining process, the
Development of International Standards, Guides and Recom-
portion of the test structure where a structural layer is inten-
mendations issued by the World Trade Organization Technical
tionally attached to its underlying layer.
Barriers to Trade (TBT) Committee.
3.1.5 anchor lip, n—in a surface-micromachining process,
the freestanding extension of the structural layer of interest
2. Referenced Documents
around the edges of the anchor to its underlying layer.
2.1 ASTM Standards:
3.1.5.1 Discussion—In some processes, the width of the
E2244Test Method for In-Plane Length Measurements of
anchor lip may be zero.
Thin, Reflecting Films Using an Optical Interferometer
3.1.6 bulk micromachining, adj—a MEMS fabrication pro-
E2245Test Method for Residual Strain Measurements of
cess where the substrate is removed at specified locations.
Thin, Reflecting Films Using an Optical Interferometer
3.1.7 cantilever, n—a test structure that consists of a free-
E2444Terminology Relating to Measurements Taken on
standing beam that is fixed at one end.
Thin, Reflecting Films
E2530Practice for Calibrating the Z-Magnification of an 3.1.8 fixed-fixed beam, n—a test structure that consists of a
freestanding beam that is fixed at both ends.
3.1.9 in-plane length (or deflection) measurement, n—the
This test method is under the jurisdiction ofASTM Committee E08 on Fatigue
experimental determination of the straight-line distance be-
and Fracture and is the direct responsibility of Subcommittee E08.05 on Cyclic
Deformation and Fatigue Crack Formation. tween two transitional edges in a MEMS device.
Current edition approved May 1, 2018. Published May 2018. Originally
ɛ1
approved in 2002. Last previous edition approved in 2011 as E2246–11 . DOI:
10.1520/E2246–11R18
2 3
For referenced ASTM standards, visit the ASTM website, www.astm.org, or The last approved version of this historical standard is referenced on
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM www.astm.org.
Standards volume information, refer to the standard’s Document Summary page on For referenced Semiconductor Equipment and Materials International (SEMI)
the ASTM website. standards, visit the SEMI website, www.semi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2246−11 (2018)
3.1.9.1 Discussion—This length (or deflection) measure- 3.2 Symbols:
ment is made parallel to the underlying layer (or the xy-plane
3.2.1 ForCalibration:σ =themaximumoftwouncali-
6same
of the interferometric microscope).
brated values (σ and σ ) where σ is the standard
same1 same2 same1
deviation of the six step height measurements taken on the
3.1.10 interferometer, n—a non-contact optical instrument
physical step height standard at the same location before the
used to obtain topographical 3-D data sets.
data session and σ is the standard deviation of the six
3.1.10.1 Discussion—The height of the sample is measured same2
measurementstakenatthissamelocationafterthedatasession
along the z-axis of the interferometer. The x-axis is typically
σ =the certified one sigma uncertainty of the physical
aligned parallel or perpendicular to the transitional edges to be
cert
step height standard used for calibration
measured.
σ =the standard deviation in a ruler measurement in the
xcal
3.1.11 MEMS, adj—microelectromechanical systems.
interferometric microscope’s x-direction for the given combi-
3.1.12 microelectromechanical systems, adj—in general,
nation of lenses
this term is used to describe micron-scale structures, sensors,
σ =the standard deviation in a ruler measurement in the
ycal
actuators,andtechnologiesusedfortheirmanufacture(suchas,
interferometric microscope’s y-direction for the given combi-
silicon process technologies), or combinations thereof.
nation of lenses
3.1.13 residual strain, n—in a MEMS process, the amount
cal =the x-calibration factor of the interferometric micro-
x
of deformation (or displacement) per unit length constrained
scope for the given combination of lenses
within the structural layer of interest after fabrication yet
cal =the y-calibration factor of the interferometric micro-
y
before the constraint of the sacrificial layer (or substrate) is
scope for the given combination of lenses
removed (in whole or in part).
cal =the z-calibration factor of the interferometric micro-
z
3.1.14 sacrificial layer, n—a single thickness of material
scope for the given combination of lenses
that is intentionally deposited (or added) then removed (in
cert=the certified (that is, calibrated) value of the physical
whole or in part) during the micromachining process, to allow
step height standard
freestanding microstructures.
ruler =the interferometric microscope’s maximum field of
x
3.1.15 stiction, n—adhesion between the portion of a struc-
view in the x-direction for the given combination of lenses as
turallayerthatisintendedtobefreestandinganditsunderlying
measured with a 10-µm grid (or finer grid) ruler
layer.
ruler =the interferometric microscope’s maximum field of
y
3.1.16 (residual) strain gradient, n—a through-thickness view in the y-direction for the given combination of lenses as
variation (of the residual strain) in the structural layer of measured with a 10-µm grid (or finer grid) ruler
interest before it is released.
scope =the interferometric microscope’s maximum field of
x
3.1.16.1 Discussion—If the variation through the thickness
view in the x-direction for the given combination of lenses
inthestructurallayerisassumedtobelinear,itiscalculatedto
scope =the interferometric microscope’s maximum field of
y
be the positive difference in the residual strain between the top
view in the y-direction for the given combination of lenses
andbottomofacantileverdividedbyitsthickness.Directional
x =the calibrated resolution of the interferometric micro-
res
information is assigned to the value of “s.”
scope in the x-direction
3.1.17 structural layer, n—a single thickness of material
z¯ =the uncalibrated average of the six calibration
6same
present in the final MEMS device.
measurements from which σ is found
6same
3.1.18 substrate, n—thethick,startingmaterial(oftensingle z =the uncalibrated positive difference between the av-
drift
erageofthesixcalibrationmeasurementstakenbeforethedata
crystalsiliconorglass)inafabricationprocessthatcanbeused
to build MEMS devices. session (at the same location on the physical step height
standard used for calibration) and the average of the six
3.1.19 support region, n—in a bulk-micromachining
calibration measurements taken after the data session (at this
process,theareathatmarkstheendofthesuspendedstructure.
same location)
3.1.20 surface micromachining, adj—a MEMS fabrication
z =over the instrument’s total scan range, the maximum
lin
process where micron-scale components are formed on a
relative deviation from linearity, as quoted by the instrument
substratebythedeposition(oraddition)andremoval(inwhole
manufacturer (typically less than 3%)
or in part) of structural and sacrificial layers.
z =the calibrated resolution of the interferometric micro-
res
3.1.21 test structure, n—acomponent(suchas,afixed-fixed
scope in the z-direction
beamorcantilever)thatisusedtoextractinformation(suchas,
z¯ =the average of the calibration measurements taken
ave
the residual strain or the strain gradient of a layer) about a
alongthephysicalstepheightstandardbeforeandafterthedata
fabrication process.
session
3.1.22 transitional edge, n—the side of a MEMS structure
3.2.2 For Strain Gradient Calculations: α=the misalign-
that is characterized by a distinctive out-of-plane vertical
ment angle
displacement as seen in an interferometric 2-D data trace.
a=the x- (or y-) coordinate of the origin of the circle of
3.1.23 underlying layer, n—the single thickness of material radius R .An arc of this circle models the out-of-plane shape
int
directly beneath the material of interest.
in the z-direction of the surface of the cantilever that is
3.1.23.1 Discussion—This layer could be the substrate. measured with the interferometric microscope
E2246−11 (2018)
b=the z-coordinate of the origin of the circle of radius R . u =the component in the combined standard uncertainty
int drift
An arc of this circle models the out-of-plane shape in the calculation for strain gradient that is due to the amount of drift
z-directionofthesurfaceofthecantileverthatismeasuredwith during the data session
the interferometric microscope
u =thecomponentinthecombinedstandarduncertainty
linear
L=the in-plane length measurement of the cantilever calculation for strain gradient that is due to the deviation from
linearity of the data scan
n1 =indicative of the data point uncertainty associated with
t
the chosen value for x1 , with the subscript “t” referring to
u =the component in the combined standard uncertainty
uppert
noise
the data trace. If it is easy to identify one point that accurately calculation for strain gradient that is due to interferometric
locates the upper corner of Edge 1, the maximum uncertainty
noise
associated with the identification of this point is n1x cal ,
u =the component in the combined standard uncertainty
t res x
Rave
where n1=1.
t calculation for strain gradient that is due to the sample’s
R =the radius of the circle with an arc that models the surface roughness
int
shape of the surface of the cantilever that is measured with the
u =the component in the combined standard un-
repeat(samp)
interferometric microscope
certainty calculation for strain gradient that is due to the
s=equals1forcantileversdeflectedintheminus z-direction
repeatability of measurements taken on cantilevers processed
oftheinterferometricmicroscope,andequals–1forcantilevers
similarly to the one being measured
deflected in the plus z-direction
u =the component in the combined standard uncer-
repeat(shs)
s =the strain gradient as calculated from three data points
tainty calculation for strain gradient that is due to the repeat-
g
s =the strain gradient when the residual strain equals zero ability of measurements taken on the physical step height
g0
standard
s =the strain gradient correction term for the given
gcorrection
design length u =the component in the combined standard uncertainty
W
calculation for strain gradient that is due to the measurement
t=the thickness of the suspended, structural layer
uncertainty across the width of the cantilever
x1 =the calibrated average of x1 and x1
ave uppera uppere
u =the component in the combined standard uncertainty
x1 =the calibrated x-value along Edge 1 locating the
xcal
uppert
calculation for strain gradient that is due to the uncertainty of
upper corner of the transitional edge using Trace t
the calibration in the x-direction
x2 =the calibrated x-value along Edge 2 locating the
uppert
u =the component in the combined standard uncertainty
upper corner of the transitional edge using Trace t xres
calculationforstraingradientthatisduetotheresolutionofthe
y =the calibrated y-value associated with Trace t
t
interferometric microscope in the x-direction
3.2.3 For Combined Standard Uncertainty Calculations:
u =the component in the combined standard uncertainty
σ =the relative strain gradient repeatability standard zres
repeat(samp)
calculationforstraingradientthatisduetotheresolutionofthe
deviation as obtained from cantilevers fabricated in a process
interferometric microscope in the z-direction
similar to that used to fabricate the sample
3.2.4 For Round Robin Measurements: L =the design
R =the calibrated surface roughness of a flat and leveled
des
ave
length of the cantilever
surface of the sample material calculated to be the average of
three or more measurements, each measurement taken from a n=the number of repeatability or reproducibility measure-
ments
different 2-D data trace
R =the calibrated peak-to-valley roughness of a flat and s =the average strain gradient value for the repeatability
gave
tave
leveled surface of the sample material calculated to be the or reproducibility measurements that is equal to the sum of the
average of three or more measurements, each measurement s values divided by n
g
taken from a different 2-D data trace
u =theaveragecombinedstandarduncertaintyvaluefor
cgave
s =in determining the combined standard uncertainty thestraingradientmeasurementsthatisequaltothesumofthe
g
...
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E2246 − 11 (Reapproved 2018)
Standard Test Method for
Strain Gradient Measurements of Thin, Reflecting Films
Using an Optical Interferometer
This standard is issued under the fixed designation E2246; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope Atomic Force Microscope at Subnanometer Displacement
Levels Using Si(111) Monatomic Steps (Withdrawn
1.1 This test method covers a procedure for measuring the
2015)
strain gradient in thin, reflecting films. It applies only to films,
2.2 SEMI Standard:
such as found in microelectromechanical systems (MEMS)
MS2 Test Method for Step Height Measurements of Thin
materials, which can be imaged using an optical interferometer,
Films
also called an interferometric microscope. Measurements from
cantilevers that are touching the underlying layer are not
3. Terminology
accepted.
3.1 Definitions:
1.2 This test method uses a non-contact optical interfero-
3.1.1 The following terms can be found in Terminology
metric microscope with the capability of obtaining topographi-
E2444.
cal 3-D data sets. It is performed in the laboratory.
3.1.2 2-D data trace, n—a two-dimensional group of points
1.3 This standard does not purport to address all of the
that is extracted from a topographical 3-D data set and that is
safety concerns, if any, associated with its use. It is the
parallel to the xz- or yz-plane of the interferometric micro-
responsibility of the user of this standard to establish appro-
scope.
priate safety, health, and environmental practices and deter-
3.1.3 3-D data set, n—a three-dimensional group of points
mine the applicability of regulatory limitations prior to use.
with a topographical z-value for each (x, y) pixel location
1.4 This international standard was developed in accor-
within the interferometric microscope’s field of view.
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the
3.1.4 anchor, n—in a surface-micromachining process, the
Development of International Standards, Guides and Recom-
portion of the test structure where a structural layer is inten-
mendations issued by the World Trade Organization Technical
tionally attached to its underlying layer.
Barriers to Trade (TBT) Committee.
3.1.5 anchor lip, n—in a surface-micromachining process,
the freestanding extension of the structural layer of interest
2. Referenced Documents
around the edges of the anchor to its underlying layer.
2.1 ASTM Standards:
3.1.5.1 Discussion—In some processes, the width of the
E2244 Test Method for In-Plane Length Measurements of
anchor lip may be zero.
Thin, Reflecting Films Using an Optical Interferometer
3.1.6 bulk micromachining, adj—a MEMS fabrication pro-
E2245 Test Method for Residual Strain Measurements of
cess where the substrate is removed at specified locations.
Thin, Reflecting Films Using an Optical Interferometer
3.1.7 cantilever, n—a test structure that consists of a free-
E2444 Terminology Relating to Measurements Taken on
standing beam that is fixed at one end.
Thin, Reflecting Films
E2530 Practice for Calibrating the Z-Magnification of an 3.1.8 fixed-fixed beam, n—a test structure that consists of a
freestanding beam that is fixed at both ends.
3.1.9 in-plane length (or deflection) measurement, n—the
This test method is under the jurisdiction of ASTM Committee E08 on Fatigue
experimental determination of the straight-line distance be-
and Fracture and is the direct responsibility of Subcommittee E08.05 on Cyclic
tween two transitional edges in a MEMS device.
Deformation and Fatigue Crack Formation.
Current edition approved May 1, 2018. Published May 2018. Originally
ɛ1
approved in 2002. Last previous edition approved in 2011 as E2246 – 11 . DOI:
10.1520/E2246–11R18
2 3
For referenced ASTM standards, visit the ASTM website, www.astm.org, or The last approved version of this historical standard is referenced on
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM www.astm.org.
Standards volume information, refer to the standard’s Document Summary page on For referenced Semiconductor Equipment and Materials International (SEMI)
the ASTM website. standards, visit the SEMI website, www.semi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2246 − 11 (2018)
3.1.9.1 Discussion—This length (or deflection) measure- 3.2 Symbols:
ment is made parallel to the underlying layer (or the xy-plane
3.2.1 For Calibration: σ = the maximum of two uncali-
6same
of the interferometric microscope).
brated values (σ and σ ) where σ is the standard
same1 same2 same1
deviation of the six step height measurements taken on the
3.1.10 interferometer, n—a non-contact optical instrument
physical step height standard at the same location before the
used to obtain topographical 3-D data sets.
data session and σ is the standard deviation of the six
3.1.10.1 Discussion—The height of the sample is measured same2
measurements taken at this same location after the data session
along the z-axis of the interferometer. The x-axis is typically
σ = the certified one sigma uncertainty of the physical
aligned parallel or perpendicular to the transitional edges to be
cert
step height standard used for calibration
measured.
σ = the standard deviation in a ruler measurement in the
xcal
3.1.11 MEMS, adj—microelectromechanical systems.
interferometric microscope’s x-direction for the given combi-
3.1.12 microelectromechanical systems, adj—in general,
nation of lenses
this term is used to describe micron-scale structures, sensors,
σ = the standard deviation in a ruler measurement in the
ycal
actuators, and technologies used for their manufacture (such as,
interferometric microscope’s y-direction for the given combi-
silicon process technologies), or combinations thereof.
nation of lenses
3.1.13 residual strain, n—in a MEMS process, the amount
cal = the x-calibration factor of the interferometric micro-
x
of deformation (or displacement) per unit length constrained
scope for the given combination of lenses
within the structural layer of interest after fabrication yet
cal = the y-calibration factor of the interferometric micro-
y
before the constraint of the sacrificial layer (or substrate) is
scope for the given combination of lenses
removed (in whole or in part).
cal = the z-calibration factor of the interferometric micro-
z
3.1.14 sacrificial layer, n—a single thickness of material
scope for the given combination of lenses
that is intentionally deposited (or added) then removed (in
cert = the certified (that is, calibrated) value of the physical
whole or in part) during the micromachining process, to allow
step height standard
freestanding microstructures.
ruler = the interferometric microscope’s maximum field of
x
3.1.15 stiction, n—adhesion between the portion of a struc-
view in the x-direction for the given combination of lenses as
tural layer that is intended to be freestanding and its underlying
measured with a 10-µm grid (or finer grid) ruler
layer.
ruler = the interferometric microscope’s maximum field of
y
3.1.16 (residual) strain gradient, n—a through-thickness view in the y-direction for the given combination of lenses as
variation (of the residual strain) in the structural layer of measured with a 10-µm grid (or finer grid) ruler
interest before it is released.
scope = the interferometric microscope’s maximum field of
x
3.1.16.1 Discussion—If the variation through the thickness
view in the x-direction for the given combination of lenses
in the structural layer is assumed to be linear, it is calculated to
scope = the interferometric microscope’s maximum field of
y
be the positive difference in the residual strain between the top
view in the y-direction for the given combination of lenses
and bottom of a cantilever divided by its thickness. Directional
x = the calibrated resolution of the interferometric micro-
res
information is assigned to the value of “s.”
scope in the x-direction
3.1.17 structural layer, n—a single thickness of material
z¯ = the uncalibrated average of the six calibration
6same
present in the final MEMS device.
measurements from which σ is found
6same
z = the uncalibrated positive difference between the av-
3.1.18 substrate, n—the thick, starting material (often single
drift
crystal silicon or glass) in a fabrication process that can be used erage of the six calibration measurements taken before the data
session (at the same location on the physical step height
to build MEMS devices.
standard used for calibration) and the average of the six
3.1.19 support region, n—in a bulk-micromachining
calibration measurements taken after the data session (at this
process, the area that marks the end of the suspended structure.
same location)
3.1.20 surface micromachining, adj—a MEMS fabrication
z = over the instrument’s total scan range, the maximum
lin
process where micron-scale components are formed on a
relative deviation from linearity, as quoted by the instrument
substrate by the deposition (or addition) and removal (in whole
manufacturer (typically less than 3 %)
or in part) of structural and sacrificial layers.
z = the calibrated resolution of the interferometric micro-
res
3.1.21 test structure, n—a component (such as, a fixed-fixed
scope in the z-direction
beam or cantilever) that is used to extract information (such as,
z¯ = the average of the calibration measurements taken
ave
the residual strain or the strain gradient of a layer) about a
along the physical step height standard before and after the data
fabrication process.
session
3.1.22 transitional edge, n—the side of a MEMS structure
3.2.2 For Strain Gradient Calculations: α = the misalign-
that is characterized by a distinctive out-of-plane vertical
ment angle
displacement as seen in an interferometric 2-D data trace.
a = the x- (or y-) coordinate of the origin of the circle of
3.1.23 underlying layer, n—the single thickness of material
radius R . An arc of this circle models the out-of-plane shape
int
directly beneath the material of interest. in the z-direction of the surface of the cantilever that is
3.1.23.1 Discussion—This layer could be the substrate. measured with the interferometric microscope
E2246 − 11 (2018)
b = the z-coordinate of the origin of the circle of radius R . u = the component in the combined standard uncertainty
int drift
An arc of this circle models the out-of-plane shape in the calculation for strain gradient that is due to the amount of drift
z-direction of the surface of the cantilever that is measured with during the data session
the interferometric microscope
u = the component in the combined standard uncertainty
linear
L = the in-plane length measurement of the cantilever calculation for strain gradient that is due to the deviation from
linearity of the data scan
n1 = indicative of the data point uncertainty associated with
t
the chosen value for x1 , with the subscript “t” referring to u = the component in the combined standard uncertainty
uppert noise
the data trace. If it is easy to identify one point that accurately
calculation for strain gradient that is due to interferometric
locates the upper corner of Edge 1, the maximum uncertainty noise
associated with the identification of this point is n1 x cal ,
t res x u = the component in the combined standard uncertainty
Rave
where n1 =1.
calculation for strain gradient that is due to the sample’s
t
R = the radius of the circle with an arc that models the
surface roughness
int
shape of the surface of the cantilever that is measured with the
u = the component in the combined standard un-
repeat(samp)
interferometric microscope
certainty calculation for strain gradient that is due to the
s = equals 1 for cantilevers deflected in the minus z-direction
repeatability of measurements taken on cantilevers processed
of the interferometric microscope, and equals –1 for cantilevers
similarly to the one being measured
deflected in the plus z-direction
u = the component in the combined standard uncer-
repeat(shs)
s = the strain gradient as calculated from three data points
tainty calculation for strain gradient that is due to the repeat-
g
s = the strain gradient when the residual strain equals zero ability of measurements taken on the physical step height
g0
standard
s = the strain gradient correction term for the given
gcorrection
design length u = the component in the combined standard uncertainty
W
calculation for strain gradient that is due to the measurement
t = the thickness of the suspended, structural layer
uncertainty across the width of the cantilever
x1 = the calibrated average of x1 and x1
ave uppera uppere
u = the component in the combined standard uncertainty
x1 = the calibrated x-value along Edge 1 locating the xcal
uppert
calculation for strain gradient that is due to the uncertainty of
upper corner of the transitional edge using Trace t
the calibration in the x-direction
x2 = the calibrated x-value along Edge 2 locating the
uppert
u = the component in the combined standard uncertainty
upper corner of the transitional edge using Trace t
xres
calculation for strain gradient that is due to the resolution of the
y = the calibrated y-value associated with Trace t
t
interferometric microscope in the x-direction
3.2.3 For Combined Standard Uncertainty Calculations:
u = the component in the combined standard uncertainty
σ = the relative strain gradient repeatability standard
zres
repeat(samp)
calculation for strain gradient that is due to the resolution of the
deviation as obtained from cantilevers fabricated in a process
interferometric microscope in the z-direction
similar to that used to fabricate the sample
3.2.4 For Round Robin Measurements: L = the design
R = the calibrated surface roughness of a flat and leveled des
ave
length of the cantilever
surface of the sample material calculated to be the average of
n = the number of repeatability or reproducibility measure-
three or more measurements, each measurement taken from a
different 2-D data trace ments
R = the calibrated peak-to-valley roughness of a flat and s = the average strain gradient value for the repeatability
tave gave
leveled surface of the sample material calculated to be the or reproducibility measurements that is equal to the sum of the
average of three or more measurements, each measurement s values divided by n
g
taken from a different 2-D data trace
u = the av
...
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