MEMS length and strain round robin results with uncertainty analysis

2.5 MB·English

by  Marshall, Janet C.

NIST
PUBLICATIONS
AlllOt. 4TS0L7
NISTIR
7291
MEMS
Length and
Strain
Round Robin
Results
with Uncertainty Analysis
Janet C. Marshall
Robert Scace
I.
Winthrop A. Baylies
QC- NIST
too
National Institute of Standards and Technology
Technology Administration, U.S. Department of Commerce
^00
{o
[ia
/!
1
i
NISTIR 7291
MEMS
Length and
Strain
Round Robin
Results
with Uncertainty Analysis
Janet C. Marshall
Semiconductor Electronics Division
Electronics and Electrical Engineering Laboratory
National Institute ofStandards and Technology
MD
Gaithersburg, 20899-8120
Robert Scace
1.
President, Klaros Corporation
MD
Beallsville, 20839-3314
Winthrop A. Baylies
President, BayTech Group
MA
Weston, 02493
January 2006
DEPARTMENT OE COMMERCE
U.S.
Carlos M. Gutierrez, Secretary'
TECHNOLOGY ADMINISTRATION
Michelle O'Neill, Acting Under Secretary ofCommercefor Technology
NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY
William Jeffrey, Director
-"-4 '
t
at
uau
Ecv^
' -: + -2
. ,
iM*•. ' \ f^i/« .» »•>/ihI
,t\ > >\‘ 'V '',' *' iO ^«'
/ .*. ' A \ . \''. i VV ' . *> \\, ..*.'•i'. . 'MjA'
<j. '•’* • .4a>^'.r \k .U4M)' «\ ^ ;s'»4‘i>
r , v', . .,i\V
r
,
TABLE OF CONTENTS
Page
Abstract 1
1. Introduction . 1
2. Interferometer Specifications and Round Robin Test Chip 2
2.1 Interferometer Specifications 2
2.2 Round Robin Test Chip 3
3. Round Robin Measurement Procedure 3
3.1 In-Plane Length Measurements 3
3.2 Residual Strain Measurements 4
3.3 Strain Gradient Measurements 4
4. Uncertainty Calculations 5
4.1 The Combined Standard Uncertainty Equation 5
4.2 In-Plane Length Uncertainty Calculations 5
4.3 Residual Strain Uncertainty Calculations 6
4.4 Strain Gradient Uncertainty Calculations 9
5. Round Robin Results 11
5.1 In-Plane Length Results 11
5.2 Residual Strain Results 13
5.3 Strain Gradient Results 14
6. Conclusions 15
.
7. Acknowledgment 15
8. References 15
LIST OF FIGURES
Page
1. The round robin test chip 1
2. Top view of a fixed-fixed beam test structure 2
3. An example of a 2-D data trace taken between the anchors of a fixed-fixed beam test structure 2
.
4. Schematic of an optical interferometer 2
5. Two data sets derived from an abbreviated data trace along a fixed-fixed beam 4
6. Top view of a cantilever test structure 4
7. An example of a 2-D data trace adjacent to a cantilever 4
8. A 2-D data trace used to find three data points 5
9. Reproducibility and repeatability bias data for L
12
.
10. Comparing reproducibility and repeatability results for in Data Sheet A 13
11. Comparing reproducibility and repeatabiltiy results for Ucave in Data Sheet B . 13
A
12. plot of-€r versus orientation 14
A
13. plot of versus length 14
A
14. plot ofSg versus orientation 15
A
15. plot ofSg versus length for two different orientations 15
hi
LIST OF TABLES
Page
1. Interferometer Pixel-to-Pixel Spacing Requirements 2
MEMS
2. Quadrant, Measurement, and Associated Test Structures for the Round Robin Test
Chip in Eigure 1 3
3. Determination of the Residual Strain Uncertainty Components 6
4. Seven Sets of Inputs for Residual Strain Calculations to Determine u-res, ihamp, HRave, ^ind Unoise • 7
5. Three Sets of Inputs for Residual Strain Calculations to Determine u~cai, Hcen, repeat, Adrift,
and U 7
linear
6. Seven Sets of Inputs for Residual Strain Calculations to Determine u^res 8
7. Two Sets of Inputs for Residual Strain Calculations to Determine ihrest 8
8. Determination of the Strain Gradient Uncertainty Components 10
9. Three Sets of Inputs for Strain Gradient Calculations to Determine u-res, ihamp, HRave ^tnd Unoise • • 10
10. Three Sets of Inputs for Strain Gradient Calculations to Determine u-raj, Ucen, ^repeat, Adrift,
and 10
Ulinear
1 1. Seven Sets of Inputs for Strain Gradient Calculations to Determine u^res 10
12. Reproducibility of Round Robin Measurement Results for In-Plane Length Measurements
When the Transitional Edges Face Each Other 11
13. Repeatability of NIST Measurement Results for In-Plane Length Measurements When the
Transitional Edges Face Each Other 11
14. Reproducibility of Round Robin Measurement Results for In-Plane Length Measurements
When the Transitional Edges Pace the Same Direction 12
15. Repeatability of NIST Measurement Results for In-Plane Length Measurements When the
Transitional Edges Face the Same Direction 12
16. Residual Strain Measurement Results 14
17. Strain Gradient Measurement Results 14
LIST OF SYMBOLS
For calibration.
A = the mean value of the step-height measurements (on the double-sided step height
standard) used to calculate cal-.
M6 = the average of the six calibration measurements from which Zrepeat was found.
= the certified one sigma uncertainty of the certified step height standard.
(^ceri
^xcal = the standard deviation in a ruler measurement in the interferometer’s v-direction for the
given combination of lenses.
Cfzcal = the standard deviation of the step height measurements (on the double-sided step height
standard).
calaveafter = the average of the six calibration measurements taken after the data session.
CalaVC = the average of the six calibration measurements taken before the data session.
before
cal^ = the jc-calibration factor of the interferometer for the given combination of lenses.
cal^ = the maximum jc-calibration factor.
caL = the minimum v-calibration factor.
cal- = the “-calibration factor of the interferometer for the given combination of lenses.
IV
cert = the certified value ofthe double-sided step height standard.
interx = the interferometer’s maximum field of view in the v-direction for the given combination
of lenses.
rulevx = the interferometer’s maximum field of view in the v-direction for the given combination
pm
of lenses as measured with a 10 grid (or finer grid) ruler.
For in-plane length, residual strain, and strain gradient measurements:
=
£r the residual strain.
L = the in-plane length measurement.
Lmax = the maximum in-plane length measurement.
Lmin = the minimum in-plane length measurement.
sep = the average calibrated separation between two interferometric pixels as applies to a given
measurement.
Sg = the strain gradient as calculated from three data points.
xllower = the x-ddXdL value along Edge “1” locating the lower part of the transitional edge.
A'fmax = the smaller of the two x values {xlumer orxlupper) used to calculate Lmax-
ximin = the larger ofthe two jc values {xllower orxlupper) used to calculate Lmn-
xlupper = the A'-data value along Edge “1” locating the upper part of the transitional edge.
x2iower = the A'-data value along Edge “2” locating the lower part of the transitional edge.
A'2max = the larger of the two a values {xliower or x2uppe,) used to calculate Z^iax-
A2min = the smaller of the two a values {x2lower orx2upper) used to calculate Lmin-
x2upper = the A'-data value along Edge “2” locating the upper part of the transitional edge.
xiower = the A-data value along the transitional edge of interest locating the lower part of the
transition.
Xupper = the A-data value along the transitional edge of interest locating the upper part of the
transition.
For combined standard uncertainty calculations:
Er-high = in determining the combined standard uncertainty value for the residual strain
measurement, the highest value for e, given the specified variations.
Er-iow = in determining the combined standard uncertainty value for the residual strain
measurement, the lowest value for £, given the specified variations.
Guoise = the standard deviation of the noise measurement, calculated to be one-sixth the value of
minus
Rtave Rave^
cjRave = the standard deviation of the surface roughness measurement, calculated to be one-sixth
the value ofRave-
(Jsamp = the Standard deviation in a height measurement due to the sample’s peak-to-valley surface
roughness as measured with the interferometer.
Unaxres = the maximum length as determined from the resolution of the interferometer in the a-
direction.
Rmaxxcal = the maximum length as determined from the maximumA-calibration factor.
Fniinres = thc minimum length as determined from the resolution of the interferometer in the a-
direction.
Unittxcai = the minimum length as determined from the minimumA-calibration factor.
Rave = the surface roughness of a flat and leveled surface of the sample material calculated to be
the average of three or more measurements, each measurement of which is taken from a
different 2-D data trace.
V
Rtave = the peak-to-valley roughness of a flat and leveled surface of the sample material
calculated to be the average of three or more measurements, each measurement of which is
taken from a different 2-D data trace.
== in determining the combined standard uncertainty value for the strain gradient
measurement, the highest value for given the specified variations.
= in determining the combined standard uncertainty value for the strain gradient
measurement, the lowest value for given the specified variations.
lie = the combined standard uncertainty value (that is, the estimated standard deviation of the
result).
IIcert = the component in the combined standard uncertainty calculation that is due to the
uncertainty of the value of the step-height standard.
IIdrift = the component in the combined standard uncertainty calculation that is due to the amount
of drift during the data session.
llL = the component in the combined standard uncertainty calculation that is due to the
measurement uncertainty ofL.
IIlinear = the component in the combined standard uncertainty calculation that is due to the
deviation from linearity of the data scan.
IInoise = the component in the combined standard uncertainty calculation that is due to
interferometric noise.
IIRave = the component in the combined standard uncertainty calculation that is due to the
sample’s surface roughness as measured with the interferometer.
IIrepeat = the component in the combined standard uncertainty calculation that is due to the
repeatability of a measurement.
IIsamp = the component in the combined standard uncertainty calculation that is due to the
sample's peak-to-valley surface roughness as measured with the interferometer.
U\v = the component in the combined standard uncertainty calculation that is due to the
measurement uncertainty across the width of the beam.
ll.xcal = the component in the combined standard uncertainty calculation that is due to the
uncertainty of the calibration in the A-direction.
= the component in the combined standard uncertainty calculation that is due to the
Il.xres
resolution of the interferometer in the A-direction. (For residual strain calculations, this is
restricted to the chosen data points along the beam.)
ll.xresL = the component in the combined standard uncertainty calculation for residual strain that is
due to the resolution of the interferometer in the A-direction as pertains to the in-plane
length measurement.
llzcal = the component in the combined standard uncertainty calculation that is due to the
uncertainty of the calibration in the z-direction.
llzres = the component in the combined standard uncertainty calculation that is due to the
resolution of the interferometer in the ::-direction.
^^'1/2 = the half width of the interv'al from £r.iow to or the half width of the interval from s^.
low to
= the resolution of the interferometer in the A-direction.
res
adrift = the positive difference between calavebejore tmd calaveafter-
^linear = the difference in height between two points times Zperc tmd divided by 100.
^perc = the percent quoted by the interferometer manufacturer for the maximum deviation from
%
linearity of the data scan over the total scan range divided by 1 such that it is unitless.
^repeat = the maximum of two values; one of which is the positive difference between the
minimum and maximum values of the six calibration measurements taken before the data
session and the other is the positive difference between the minimum and maximum values
of the six measurements taken after the data session.
= the resolution ofthe interferometer in the z-direction.
Zres
For round robin measurements:
AT. = for the given value ofLdes, Lave minus Ldes^
ALave = the average value of AL over the given range ofLdes values.
£rave = the average residual strain value for the reproducibility or repeatability measurements. It
is equal to the sum ofthe £r values divided by n.
Lave = the average in-plane length value for the reproducibility or repeatability measurements. It
is equal to the sum ofthe L values divided by n.
Ldes = the design length.
mag = the magnification used for the measurement.
n = the number ofreproducibility or repeatability measurements.
Sgave = the average strain gradient value for the reproducibility or repeatability measurements. It
is equal to the sum ofthe Sg values divided by n.
Ucave = the average combined standard uncertainty value for the reproducibility or repeatability
measurements. It is equal to the sum of the Uc values divided by n.
vii
,
, If, /
"• •’' 'i-i*t V-'> 1 ujuLPi 'i -m3 v'*f,'’^lriflO’:!:
%ffl •V.vf
M
"V’
* iMiJ* ' ^ '
il .Jf*f>Vi'.»'rr . >rt* .> * • ’ » * '*V»'jir ‘ 'J •^ pwjB
:'V v1
r* *• . .r
ii :' !f‘
r
•j • ' „. . \ fi!
li I . ;1 ,f ••If.” .. ,r'i -i? ^
>* \1 v|j[) <i. ''\ • \ '
'^
»#i.. li fi • '* ‘•V. f^.'r . 'v.'i,, /.Tt)-^t'^\\^il ,'4?^ '
<lifj^l^ru
'•ti< i
f
4'
a
>^
»
i