MEMS Length and Strain

Round Robin Results

with Uncertainty Analysis

Janet C. Marshall

Robert I. Scace

Winthrop A. Baylies

NISTIR 7291
 

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 of Standards and Technology

Gaithersburg, MD 20899-8120

Robert I. Scace

President, Klaros Corporation

Beallsville, MD 20839-3314

Winthrop A. Baylies

President, BayTech Group

Weston, MA 02493

January 2006

U.S. DEPARTMENT OF COMMERCE

Carlos M. Gutierrez, Secretary

TECHNOLOGY ADMINISTRATION

Michelle O’Neill, Acting Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

William Jeffrey, Director
 ii
 iii

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 ucave in Data Sheet A . . . . . . . . 13 

11. Comparing reproducibility and repeatabiltiy results for ucave in Data Sheet B . . . . . . . . 13 

12. A plot of –εr versus orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

13. A plot of –εr versus length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

14. A plot of sg versus orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

15. A plot of sg versus length for two different orientations . . . . . . . . . . . . . . . . . 15 
 iv

LIST OF TABLES 

Page 

1. Interferometer Pixel-to-Pixel Spacing Requirements . . . . . . . . . . . . . . . . . . 2 

2. Quadrant, Measurement, and Associated Test Structures for the MEMS Round Robin Test 

 Chip in Figure 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

3. Determination of the Residual Strain Uncertainty Components . . . . . . . . . . . . . . 6 

4. Seven Sets of Inputs for Residual Strain Calculations to Determine uzres, usamp, uRave, and unoise . 7 

5. Three Sets of Inputs for Residual Strain Calculations to Determine uzcal, ucert, urepeat, udrift, 

 and ulinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

6. Seven Sets of Inputs for Residual Strain Calculations to Determine uxres . . . . . . . . . . 8 

7. Two Sets of Inputs for Residual Strain Calculations to Determine uxresL . . . . . . . . . . 8 

8. Determination of the Strain Gradient Uncertainty Components . . . . . . . . . . . . . . 10 

9. Three Sets of Inputs for Strain Gradient Calculations to Determine uzres, usamp, uRave and unoise . . 10 

10. Three Sets of Inputs for Strain Gradient Calculations to Determine uzcal, ucert, urepeat, udrift, 

 and ulinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 

11. Seven Sets of Inputs for Strain Gradient Calculations to Determine uxres . . . . . . . . . . 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 Face 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: 

μ = the mean value of the step-height measurements (on the double-sided step height 

standard) used to calculate calz. 

μ6 = the average of the six calibration measurements from which zrepeat was found. 

σcert = the certified one sigma uncertainty of the certified step height standard.

σxcal = the standard deviation in a ruler measurement in the interferometer’s x-direction for the 

given combination of lenses. 

σzcal = 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. 

calavebefore = the average of the six calibration measurements taken before the data session.

calx = the x-calibration factor of the interferometer for the given combination of lenses. 

calxmax = the maximum x-calibration factor. 

calxmin = the minimum x-calibration factor.

calz = the z-calibration factor of the interferometer for the given combination of lenses.

cert = the certified value of the double-sided step height standard. 
 v

interx = the interferometer’s maximum field of view in the x-direction for the given combination 

of lenses. 

rulerx = the interferometer’s maximum field of view in the x-direction for the given combination 

of lenses as measured with a 10 μm grid (or finer grid) ruler. 

For in-plane length, residual strain, and strain gradient measurements:

εr = 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. 

x1lower = the x-data value along Edge “1” locating the lower part of the transitional edge. 

x1max = the smaller of the two x values (x1lower or x1upper) used to calculate Lmax. 

x1min = the larger of the two x values (x1lower or x1upper) used to calculate Lmin. 

x1upper = the x-data value along Edge “1” locating the upper part of the transitional edge. 

x2lower = the x-data value along Edge “2” locating the lower part of the transitional edge. 

x2max = the larger of the two x values (x2lower or x2upper) used to calculate Lmax. 

x2min = the smaller of the two x values (x2lower or x2upper) used to calculate Lmin. 

x2upper = the x-data value along Edge “2” locating the upper part of the transitional edge. 

xlower = the x-data value along the transitional edge of interest locating the lower part of the 

transition. 

xupper = the x-data value along the transitional edge of interest locating the upper part of the 

transition. 

For combined standard uncertainty calculations: 

εr-high = in determining the combined standard uncertainty value for the residual strain 

measurement, the highest value for εr given the specified variations. 

εr-low = in determining the combined standard uncertainty value for the residual strain 

measurement, the lowest value for εr given the specified variations. 

σnoise = the standard deviation of the noise measurement, calculated to be one-sixth the value of 

Rtave minus Rave.

σRave = the standard deviation of the surface roughness measurement, calculated to be one-sixth 

the value of Rave.

σsamp = the standard deviation in a height measurement due to the sample’s peak-to-valley surface 

roughness as measured with the interferometer. 

Lmaxres = the maximum length as determined from the resolution of the interferometer in the xdirection.

Lmaxxcal = the maximum length as determined from the maximum x-calibration factor. 

Lminres = the minimum length as determined from the resolution of the interferometer in the xdirection.

Lminxcal = the minimum length as determined from the minimum x-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. 

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. 
 vi

sg-high = in determining the combined standard uncertainty value for the strain gradient 

measurement, the highest value for sg given the specified variations. 

sg-low = in determining the combined standard uncertainty value for the strain gradient 

measurement, the lowest value for sg given the specified variations. 

uc = the combined standard uncertainty value (that is, the estimated standard deviation of the 

result). 

ucert = the component in the combined standard uncertainty calculation that is due to the 

uncertainty of the value of the step-height standard. 

udrift = the component in the combined standard uncertainty calculation that is due to the amount 

of drift during the data session. 

uL = the component in the combined standard uncertainty calculation that is due to the 

measurement uncertainty of L. 

ulinear = the component in the combined standard uncertainty calculation that is due to the 

deviation from linearity of the data scan. 

unoise = the component in the combined standard uncertainty calculation that is due to 

interferometric noise. 

uRave = the component in the combined standard uncertainty calculation that is due to the 

sample’s surface roughness as measured with the interferometer. 

urepeat = the component in the combined standard uncertainty calculation that is due to the 

repeatability of a measurement. 

usamp = 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. 

uW = the component in the combined standard uncertainty calculation that is due to the 

measurement uncertainty across the width of the beam. 

uxcal = the component in the combined standard uncertainty calculation that is due to the 

uncertainty of the calibration in the x-direction. 

uxres = the component in the combined standard uncertainty calculation that is due to the 

resolution of the interferometer in the x-direction. (For residual strain calculations, this is 

restricted to the chosen data points along the beam.) 

uxresL = the component in the combined standard uncertainty calculation for residual strain that is 

due to the resolution of the interferometer in the x-direction as pertains to the in-plane 

length measurement. 

uzcal = the component in the combined standard uncertainty calculation that is due to the 

uncertainty of the calibration in the z-direction. 

uzres = the component in the combined standard uncertainty calculation that is due to the 

resolution of the interferometer in the z-direction. 

w1/2 = the half width of the interval from εr-low to εr-high or the half width of the interval from sglow to sg-high.

xres = the resolution of the interferometer in the x-direction. 

zdrift = the positive difference between calavebefore and calaveafter.

zlinear = the difference in height between two points times zperc and divided by 100.

zperc = 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.

zrepeat = 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.

zres = the resolution of the interferometer in the z-direction. 
 vii

For round robin measurements: 

∆L = for the given value of Ldes, Lave minus Ldes. 

∆Lave = the average value of ∆L over the given range of Ldes values. 

εrave = the average residual strain value for the reproducibility or repeatability measurements. It 

is equal to the sum of the εr values divided by n. 

Lave = the average in-plane length value for the reproducibility or repeatability measurements. It 

is equal to the sum of the L values divided by n. 

Ldes = the design length. 

mag = the magnification used for the measurement. 

n = the number of reproducibility or repeatability measurements. 

sgave = the average strain gradient value for the reproducibility or repeatability measurements. It 

is equal to the sum of the 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. 
 viii

MEMS Length and Strain Round Robin 

Results with Uncertainty Analysis 

Janet C. Marshall, Robert I. Scace,* 

and Winthrop A. Baylies** 

Semiconductor Electronics Division 

National Institute of Standards and Technology 

Gaithersburg, MD 20899-8120 

* Klaros Corporation 

Beallsville, MD 20839-3314 

** BayTech Group 

Weston, MA 02493 

Abstract 

A microelectromechanical systems (MEMS) Length 

and Strain Round Robin Experiment was completed in 

January 2005 to compare results of in-plane length 

measurements, residual strain measurements, and strain 

gradient measurements at a number of laboratories. This 

paper1

 presents the results from this MEMS Length and 

Strain Round Robin Experiment and the uncertainties 

associated with the measurements. The final results satisfy 

the requirements for the formulation of precision and bias 

statements in three related American Society for Testing 

and Materials (ASTM) standard test methods for MEMS. 

The goal of the round robin was to assess the 

reproducibility of measurements performed using the same 

test method at independent laboratories as well as the 

repeatability of measurements performed using the same 

test method, in the same laboratory, by the same operator, 

with the same equipment, in the shortest practicable period 

of time. Both the reproducibility and repeatability 

measurements were done on essentially random test units 

made of the same homogeneous material. 

The outcome of this round robin is a step towards 

assuring high quality data, which will help reduce the 

laboratory-to-laboratory differences in the parametric 

measurements. 

1. Introduction 

 Microelectromechanical systems (MEMS) is a rapidly 

growing complement to the semiconductor industry. 

Applications for MEMS demand high performance and 

reliability. The first MEMS standards [1-3] were approved 

by ASTM in 2002. These standards are expected to 

facilitate commerce in MEMS technologies and improve 

manufacturing yields by decreasing interlaboratory 

differences in measurement. 

 1

 Contribution of the National Institute of Standards and Technology 

(NIST); not subject to copyright in the U.S. 

The ASTM test methods are used for measuring inplane lengths or deflections,2

 residual strain,3 and strain 

gradient.4

 They employ a non-contact measurement 

approach using an optical interferometer. These test 

methods were used in a round robin to determine their 

repeatability as well as to see if independent laboratories 

could reproduce these measurements without introducing a 

bias. 

In addition, for each measurement, the combined 

standard uncertainty [4] (which is comparable to the 

estimated standard deviation of the result) is determined. 

Individual uncertainty components are specified in this 

report and included in the analysis. 

Figure 1. The round robin test chip. 

In the round robin, a test chip (the design of which is 

shown in Figure 1) was passed from laboratory to 

laboratory, and measurements were taken on random test 

units using the procedures of the ASTM test methods. 

Data (i.e., x or y and/or z values) from these measurements 

were analyzed using a NIST web-based program that also 

verifies the data before submitting the results to NIST [5]. 

Eight laboratories (NIST, Kavlico, Motorola, Lucent 

Technologies, Veeco Metrology, Zygo, FLX Micro, and 

Delphi) participated in the round robin. 

 2

 An in-plane length (or deflection) measurement is defined as the 

experimental determination of the straight-line distance between two 

transitional edges in a MEMS device. 

3

 Residual strain is defined in a MEMS process as the amount of 

deformation (or displacement) per unit length constrained within the 

structural layer of interest after fabrication yet before the constraint of the 

sacrificial layer (or substrate) is removed (in whole or in part). 

4

 Strain gradient is defined as a through-thickness variation (of the 

residual strain) in the structural layer of interest before it is released. 

1 
 2

The participants were instructed to use the three ASTM 

standard test methods in performing their measurements. 

The first test method (E 2244) [6] is for measuring in-plane 

lengths. This test method is used when measuring in-plane 

deflections and when measuring lengths in fabrication 

processes that are being developed. Two transitional edges 

(such as Edges “1” and “2” in Figures 2 and 3) are chosen 

that define the in-plane length measurement. (A 

transitional edge is an edge of a MEMS structure that is 

characterized by a distinctive out-of-plane displacement.) 

At each transitional edge defining the in-plane length 

measurement, values defining a minimum and maximum 

length measurement are obtained. The minimum and 

maximum lengths are calculated, and the in-plane length is 

the average of these two values. [Alternatively, if the 

transitional edges that define the in-plane length 

measurement face the same way and have similar slopes 

and magnitudes (such as Edges “1” and “5” in Figures 2 

and 3), a different approach can be taken. Here, an x-value 

is obtained at the lower (or upper) portion of each 

transitional edge. The smaller x value is subtracted from 

the larger x value to find the in-plane length.] 

Figure 2. Top view of a fixed-fixed beam test structure. 

Data along Trace "a" or "e" in Figure 2

-1

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2

x (mm)

z (μm)

Figure 3. An example of a 2-D data trace taken between 

the anchors of a fixed-fixed beam test structure. 

The second test method (E 2245) [6] measures the 

residual strain. This value is important to aid in the design 

and fabrication of MEMS devices. Three data points are 

obtained to define each function that characterizes the outof-plane shape of fixed-fixed beams. Two cosine functions 

are used to model the shape. Therefore, five independent 

data points are required, since one data point is common to 

both functions. The residual strain is calculated after the 

appropriate lengths are determined. 

The third test method (E 2246) [6] measures the strain 

gradient. Given this value, and the definition of a “flat” or 

“straight” cantilever for the given application, the 

maximum length of such a cantilever can be determined.

Three data points are obtained to define an arc of a circle 

that is used to model the out-of-plane shape of cantilevers. 

The strain gradient is calculated from the radius of the 

circle. 

2. Interferometer Specifications and Round Robin Test 

Chip 

2.1 Interferometer Specifications

The tool required for the round robin was a non-contact 

optical interferometer capable of obtaining a topographical 

3-D data set and exporting a 2-D data trace. Figure 4 is a 

schematic of such an interferometer. This non-contact 

optical interferometer must have pixel-to-pixel spacings as 

specified in Table 1 and a vertical resolution better than 1 

nm. This resolution may be obtained by averaging 

multiple measurements. The interferometer must be 

capable of measuring step heights to at least 5 μm higher 

than the step height to be measured. 

Figure 4. Schematic of an optical interferometer. 

Table 1. Interferometer Pixel-to-Pixel Spacing 

Requirements 

Magnification, × Pixel-to-pixel spacing, μm 

5 < 1.57 

10 < 0.83 

20 < 0.39 

40 < 0.21 

80 < 0.11 

detector 

reference

surface 

light 

source

sample

microscope

objective 

filter 

beam 

splitter x

z

a 

e 

b 

c 

d 

L

Edge 1 Edge 2 

x

y

Edge 5 

g h 

Edge 1 Edge 2 Edge 5 

g

h 

x1upper

x1lower
 3

2.2 Round Robin Test Chip

The test chip used for this experiment was fabricated on 

the Multi-User MEMS Processes (MUMPs)5 process run 

46. The design for this test chip is depicted in Figure 1. 

The design file (in GDS-II format) for this chip can be 

downloaded from the MEMS Length and Strain Calculator 

web site [7]. 

For the MUMPs, there are three mechanical layers 

(poly1, poly2, and a combined thickness of poly1 and 

poly2). Within the design shown in Figure 1, structures in 

these mechanical layers are specified with a “P1” (poly1), a 

“P2” (poly2), or a “P12” (combined poly1 and poly2). 

The test chip is divided into four quadrants with 

measurement-specific test structures for each of the 

mechanical layers included in each quadrant as listed in 

Table 2. All test structures are provided in both 0 degree 

and 90 degree orientations. 

Table 2. Quadrant, Measurement, and Associated Test 

Structures for the MEMS Round Robin Test Chip 

in Figure 1 

Quad Measurement Associated Test Structures 

Q1 

(upper 

left) 

In-plane length Two ends anchored: Fixed-fixed beams 

One end anchored: Cantilevers 

Two ends unanchored*: Rings 

Q2 

(lower 

left) 

In-plane deflection Released part to released part: 

 Bow-ties 

Released part to fixed location: 

 Pointers 

Q3 

(upper 

right) 

Residual strain Fixed-fixed beams 

Q4 

(lower 

right) 

Strain gradient Cantilevers 

*A structure with two ends unanchored refers to a MEMS structure that is 

not anchored along the direction of the measurement. 

3. Round Robin Measurement Procedure

The procedures for measurement of in-plane length are 

given in subsection 3.1, the residual strain in subsection 

3.2, and the strain gradient in subsection 3.3. 

3.1 In-Plane Length Measurements

There are three classes of in-plane length test structures 

(given in Table 2) as determined by their end conditions. 

However, for the round robin, measurements were made in 

only the first Q1 subdivision on fixed-fixed beam test 

structures, which have two ends anchored, as shown in 

Figures 1 and 2. 

The in-plane length measurements are made with an 

optical interferometer. Many interferometers are purchased 

with five magnifications (5×, 10×, 20×, 40×, and 80×). 

Therefore, for each magnification, a fixed-fixed beam is 

provided. The design lengths (Ldes) of 1000 μm, 500 μm, 

 5

 In this guide, commercial equipment, instruments, or processes may be 

indentified. This does not imply recommendation or endorsement by the 

National Institute of Standards and Technology (NIST), nor does it imply 

that the equipment, instruments, or processes are the best available for the 

purpose. 

200 μm, 80 μm, and 25 μm, respectively, are included for 

each mechanical layer and for each orientation. For the 

round robin, the participant was informed which 

orientation of five poly1 fixed-fixed beams to measure in a 

letter accompanying the round robin test chip. 

In this Q1 subdivision, two different data analysis 

sheets can be used. Measurements for both Data Analysis 

Sheet A and Data Analysis Sheet B were requested. Sheet 

A is for in-plane length measurements when the 

transitional edges are facing each other. For this sheet, 

measurements were requested from Edges “1” and “2,” as 

shown in Figures 2 and 3, using Trace “a” or “e.” 

Given a transitional edge, such as Edge “1” in Figures 2 

and 3, the lower transitional x-value, xlower, is found as 

follows. Going from Point “g” to Point “h,” the out-ofplane z-values are examined one-by-one. The data points 

are skipped over until a z value is obtained that is less than 

75 nm. (This assumes of course that the data were 

properly leveled with respect to the underlying layer.) This 

criterion may need to be modifed for peculiarities in the 

sample and/or the 2-D data trace being examined. The x

value associated with the newly found z value is x1lower, 

where the number “1” is indicative of the specific 

transitional edge being examined, which in this case is 

Edge “1.” 

The upper transitional x-data value, xupper, is found as 

follows. The x values are examined one-by-one going 

from Point “h” to Point “g” in Figures 2 and 3. Along the 

upper half of the transition, the x value associated with the 

first z value, which is less than 300 nm from the next z

value, is called xupper, or x1upper in this case. This criteria 

may need to be modifed for peculiarities in the sample 

and/or the 2-D data trace being examined. 

If an in-plane length measurement is determined 

between transitional edges that face each other, such as 

Edges “1” and “2” in Figures 2 and 3, then the in-plane 

length, L, the minimum length, Lmin, and the maximum 

length, Lmax, are calculated as follows: 

( )/ 2 L = Lmin + Lmax (1) 

where 

min min 1min L = x2 − x (2) 

and 

max max 1max L = x2 − x . (3) 

The “min” subscript refers to the transitional x value (xlower

or xupper) that yields a minimum length. The “max” 

subscript refers to the transitional x value (xlower or xupper) 

that yields a maximum length. 

Data Analysis Sheet B is for in-plane length 

measurements when the transitional edges are oriented in 

the same direction. For this sheet, measurements are 

requested from Edges “1” and “5,” as shown in Figures 2 

and 3, using Trace “a” or “e.” The design lengths (Ldes) for 
 4

these measurements are 1035 μm, 535 μm, 235 μm, 

115 μm, and 60 μm. The equations for Lmin and Lmax are 

Lmin = L − 2* sep (4) 

and 

Lmax = L + 2*sep (5) 

where 

lower 1lower L = x2 − x (6) 

or 

upper 1upper L = x2 − x (7) 

as determined from the 2-D data trace being examined, and 

sep is the average calibrated separation between two 

interferometric pixels.

Following ASTM E 2244, the raw, uncalibrated 

measurements were recorded on Data Analysis Sheet A or 

B, as appropriate. A summary of the round robin results 

from Data Sheets A and B is presented in subsection 5.1. 

3.2 Residual Strain Measurements 

In Q3, the measurements for residual strain, εr, are 

made. Fixed-fixed beams are provided for this purpose as 

shown in Figures 1 and 2. The fixed-fixed beam design 

lengths are 400 μm, 450 μm, 500 μm, 550 μm, 600 μm, 

650 μm, 700 μm, 750 μm, and 800 μm. There are three 

fixed-fixed beams designed at each length. Thus, there are 

twenty-seven fixed-fixed beams for each mechanical layer 

and for each orientation. 

Measurements from two poly1 fixed-fixed beams on 

the MUMPs chip were requested using Data Analysis 

Sheet G. Measurements are requested from Edges “1” and 

“2,” as shown in Figure 2, using Trace “a” or “e,” depicted 

in Figure 3. Data points along the fixed-fixed beam, as 

shown in Figure 5, namely (x1F, z1F), (x2F, z2F), (x3F=x1S, 

z3F=z1S), (x2S, z2S), and (x3S, z3S), are also requested from 

Traces “b,” “c,” and “d.” 

Figure 5. Two data sets derived from an abbreviated data 

trace along a fixed-fixed beam. The vertical scale in the 

figure above has been exaggerated. 

Following ASTM E 2245 for measuring residual strain, 

the raw, uncalibrated measurements were recorded on Data 

Analysis Sheet G. A summary of the round robin results 

from these data analysis sheets is presented in subsection 

5.2. 

3.3 Strain Gradient Measurements

In Q4, the measurements for strain gradient, sg, are 

made. Cantilevers are provided for this purpose as shown 

in Figures 1 and 6. The cantilever design lengths are 400 

μm, 450 μm, 500 μm, 550 μm, 600 μm, 650 μm, 700 μm, 

750 μm, and 800 μm. There are three cantilevers designed 

at each length. Thus, there are twenty-seven cantilevers for 

each mechanical layer and for each orientation. 

Measurements from two poly1 cantilevers on the 

MUMPs chip were requested using Data Analysis Sheet H. 

Measurements were requested from Edge “1,” as shown in 

Figure 6, using Trace “a” or “e,” depicted in Figure 7. 

Data points along the cantilever, as shown in Figure 8, 

namely (x1, z1), (x2, z2), and (x3, z3), were also requested 

from Traces “b,” “c,” and “d.” 

Following ASTM E 2246 for strain gradient 

measurements, the raw, uncalibrated measurements were 

recorded on Data Analysis Sheet H. A summary of the 

round robin results from these data analysis sheets is 

presented in subsection 5.3. 

Figure 6. Top view of a cantilever test structure. 

Data along Trace "a" or "e" in Figure 6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1 0.3 0.5 0.7 0.9 1.1

x (mm)

z (μm)

Figure 7. An example of a 2-D data trace adjacent to a 

cantilever. 

Edge 2

a 

b 

c 

d 

e 

L

Edge 1 

Edge 3

y

x

z

x

x1F x2F x3F = x1S x2S x3S

x1ave L x2ave

Data Derived from Trace “b,” 

“c,” or “d” in Figure 2 

Edge 1 
 5

Data along Trace "b," "c," or "d"

 in Figure 6

0

2

4

6

8

0.1 0.3 0.5 0.7 0.9 1.1

x (mm)

z (μm)

Figure 8. A 2-D data trace used to find three data points. 

4. Uncertainty Calculations 

In this section, the equations used to determine the 

values of the combined standard uncertainty [4], uc, are 

presented. The first subsection presents the basic 

combined standard uncertainty equation, and the second 

through fourth subsections present the more specific 

uncertainty equations for the in-plane length 

measurements, the residual strain measurements, and the 

strain gradient measurements, respectively. 

4.1 The Combined Standard Uncertainty Equation

The combined standard uncertainty [4] is comparable to 

the estimated standard deviation of the result. It is equal to 

the square root of the sum of the squares of the uncertainty 

components. 

For the case of three sources of uncertainty, the 

uncertainty equation would be as follows: 

2

3

2

2

2

uc = u1 + u + u (8) 

where u1 is the uncertainty component due to the first 

source of uncertainty, u2 is the uncertainty component due 

to the second source of uncertainty, and u3 is due to the 

third source of uncertainty. Additional terms may be added 

under the square root sign in (8) for additional sources of 

uncertainty. 

4.2 In-Plane Length Uncertainty Calculations

In this subsection, a combined standard uncertainty 

equation is presented for use with in-plane length 

measurements. For these measurements, three sources of 

uncertainty are identified with all other sources considered 

negligible. The three sources of uncertainty are the 

uncertainty of the in-plane length measurement (uL), the 

uncertainty of the calibration of the interferometer in the xdirection (uxcal), and the uncertainty due to the resolution 

of the interferometer in the x-direction (uxres). As such, the 

combined standard uncertainty equation [8] can be written 

as follows: 

2 2 2

uc = uL + uxcal + uxres . (9) 

 

The uncertainty of uL is determined from the minimum 

and maximum length values (namely, Lmin and Lmax) as 

given in (2) and (3), respectively, or (4) and (5), 

respectively. With 99.7 % confidence, assuming a 

Gausssian distribution (and assuming uxcal and uxres equal 

zero), the value for L lies between Lmin and Lmax. 

Therefore, uL is calculated as follows: 

uL = (Lmax − Lmin ) 6 . (10) 

If the transitional edges defining the in-plane length 

measurement face the same direction and have similar 

slopes and magnitudes, the equations for Lmin and Lmax are 

given by (4) and (5), and (10) can be reduced to the 

following equation: 

u 2*sep 3 L = (11) 

where sep is the average calibrated separation between two 

interferometric pixels. 

The uncertainty equation for uxcal can be found from 

the minumum and maximum length values (namely, Lminxcal

and Lmaxxcal) as determined from the mininum and 

maximum calibration factors (namely, calxmin and calxmax). 

Therefore, 

minxcal xmin x L = L*cal / cal (12) 

and 

maxxcal xmax x L = L*cal / cal (13) 

where 

xmin x xcal x cal = cal −3*σ inter , (14) 

x max x xcal x cal = cal + 3*σ inter , (15) 

and 

x x x cal = ruler inter (16) 

 

with L given in (1), (6), or (7). Here, calx is defined as the 

x-calibration factor of the interferometer for the given 

combination of lenses; rulerx is defined as the 

interferometer’s maximum field of view in the x-direction 

for the given combination of lenses as measured with a 

10 μm grid (or finer grid) ruler; σxcal is the standard 

deviation in a ruler measurement in the interferometer’s xdirection for the given combination of lenses; and interx is 

the interferometer’s maximum field of view in the xdirection for the given combination of lenses. With 99.7 % 

confidence assuming a Gaussian distribution (and 

assuming uL and uxres equal zero), the value for L lies 

between Lminxcal and Lmaxxcal. Therefore, uxcal is calculated as 

follows: 

Edge 3 

Edge 2 

3 chosen 

data points 
 6

( * ) ( * )

( * * ) (6* )

( ) 6

xcal x x

xmax xmin x

xcal maxxcal minxcal

L inter cal

L cal L cal cal

u L L

= σ

= −

= −

 . (17) 

And lastly, the uncertainty equation for uxres, is found 

from the minimum and maximum length values (namely, 

Lminres and Lmaxres) defined as follows: 

minres res L = L − x (18) 

and 

maxres res L = L + x (19) 

where xres is the resolution of the interferometer in the xdirection. Assuming a uniform distribution (and assuming 

uL and uxcal equal zero), the value for L lies between Lminres

and Lmaxres. Therefore, uxres is calculated as follows: 

( ) (2 3) 1.732 xres maxres minres res u = L − L = x . (20) 

4.3 Residual Strain Uncertainty Calculations

In this subsection, a combined standard uncertainty 

equation is presented for use with residual strain 

measurements. Eight sources of uncertainty are indentified 

with all other sources of uncertainty considered negligible. 

Towards the end of this subsection, the uncertainty 

equation will be expanded to include twelve sources of 

uncertainty. Knowledge of the additional sources of 

uncertainty can be a tremendous benefit when taking and 

analyzing other interferometeric measurements (such as, 80 

μm step heights). 

The residual strain combined standard uncertainty 

equation (as calculated in Data Analysis Sheet G [7] and in 

ASTM Test Method E 2245-05 [9]) with eight sources of 

uncertainty is as follows: 

2 2 2 2

2 2 2 2

zcal xcal xres xresL

W L zres samp

c

u u u u

u u u u

u

+ + +

+ + + + = (21) 

where uW is the uncertainty due to variations across the 

width of the beam, uzres is the uncertainty due to the 

resolution of the interferometer in the z-direction, usamp is 

the uncertainty due to the sample’s peak-to-valley surface 

roughness as measured with the interferometer, uzcal is the 

uncertainty of the calibration in the z-direction, and uxresL

is the uncertainty due to the resolution of the interferometer 

in the x-direction as pertains to the in-plane length 

measurement. Calculations for each uncertainty 

component will be presented in sequence, with Table 3 

giving a brief tabular summary of how each uncertainty 

component is obtained. This table can be referenced as 

each component is discussed. 

Table 3. Determination of the Residual Strain Uncertainty 

Components* 

Uncertainty 

Component

Varied 

From 

Varied 

To 

How 

Varied

** 

G 

or 

u 

*** 

1. uW max εr value 

from Trace b, 

c, and d 

min εr value 

from Trace b, 

c, and d 

− u 

2. uL with L=Lmax with L=Lmin − G 

3. uzres +(1/2)zres −(1/2)zres 4 u 

4. usamp +3σsamp −3σsamp 4 G 

5. uzcal +d −d 5 G 

6. uxcal with 

calx=calxmax

with 

calx=calxmin

− G 

7. uxres +(1/2)xres −(1/2)xres 6 u 

8. uxresL +(1/2)xres −(1/2)xres 7 u 

9. uRave +3σRave −3σRave 4 G 

10. unoise +3σnoise −3σnoise 4 G 

11. ucert +d −d 5 G 

12. urepeat +d −d 5 u 

13. udrift +d −d 5 u 

14. ulinear +d −d 5 u 

* In this table, d=3(zxx−z1F)σzcal /cert to determine uzcal where zxx is the 

appropriate column heading in Table 5, d=3(zxx−z1F)σcert /cert to determine 

ucert, d=(zxx−z1F)zrepeat /(2*μ6) to determine urepeat, d=(zxx−z1F)zdrift /(2*μ) to 

determine udrift, and d=zlinear to determine ulinear. 

** The entries in this column specify the table number where the input 

variations can be found, if applicable. 

*** “G” indicates a Gaussian distribution and “u” indicates a uniform 

distribution. 

The uncertainty equation for uW is found using the 

residual strain results from at least two data traces across 

the width of the beam. The two extreme residual strain 

values (εr-low and εr-high) are subtracted and divided by 

two as follows: 

w1/2 = (ε r−high −ε r−low ) 2 (22) 

to determine the half width of the interval, w1/2. Assuming 

a uniform probability distribution, uW is calculated using 

the following formula: 

uW = w1/2 1.732 . (23) 

The uncertainty equation for uL is found after 

calculating the residual strain in two different ways. First, 

the residual strain is found assuming that Lmin is the in-
 7

plane length of the fixed-fixed beam and second, assuming 

that Lmax is the in-plane length of the fixed-fixed beam. 

Both of these assumptions will precipitate a recalculation 

of the curved length of the fixed-fixed beam. Then, w1/2 is 

calculated using (22), and uL is calculated assuming a 

Gaussian distribution with the following equation: 

uL = w1/2 3. (24) 

The uncertainty equation for uzres is found from the 

residual strain calculations using the different sets of inputs 

given in Table 4. Here, the inputed z-values along the top 

of the beam are varied plus or minus half zres, where zres is 

the resolution of the interferometer in the z-direction. The 

two residual strain values (εr-low and εr-high) are identified. 

Then, w1/2 is calculated using (22), followed by the 

determination of uzres, assuming a uniform distribution, 

using the following equation: 

uzres = w1/2 1.732 . (25) 

The uncertainty equation for usamp is found from a 

determination of 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. Then, the standard deviation, σsamp, of this 

measurement is calculated to be one-sixth the value of 

Rtave. The data points obtained along the top of the beam 

are then varied as specified in Table 4 (with d=3σsamp) and 

the residual strain determined for the different sets of 

inputs. Given the resulting residual strain values, εr-low and 

εr-high are identified. The interval from εr-low to εr-high is 

assumed to encompass 99 % of the measurements. Then, 

w1/2 is calculated using (22), followed by the determination 

of usamp, assuming a Gaussian distribution, using the 

following equation: 

usamp = w1/2 3. (26) 

The method of calibration of the interferometer in the zdirection affects the determination of the uncertainty 

component uzcal. (Later in this subsection, using a different 

calibration method, uzcal is divided into four separate 

components.) To determine uzcal, the z-calibration of the 

interferometer is done as follows: twelve step height 

measurements are taken on the double-sided step height 

standard with six measurements taken before the data 

session and six after the data session. Three of the six 

measurements are spread out evenly along one side of the 

step height standard, and the other three are spread out 

evenly along the other side of the step height standard with 

one 3-D data set obtained for each measurement. The zcalibration factor (calz) is determined by dividing the 

certified value of the step-height standard (cert) by the 

mean value of the twelve measurements (μ). All of the 

data values obtained during the data session are multiplied 

by calz to obtain calibrated z-data values. 

Table 4. Seven Sets of Inputs for Residual Strain 

Calculations to Determine uzres, usamp, uRave, and unoise* 

z1F z2F z3F=z1S z2S z3S

1 z1F z2F z3F z2S z3S

2 z1F+d z2F z3F−d z2S z3S+d

3 z1F−d z2F z3F+d z2S z3S−d

4 z1F+d z2F+d z3F−d z2S+d z3S+d

5 z1F+d z2F−d z3F−d z2S−d z3S+d

6 z1F−d z2F+d z3F+d z2S+d z3S−d

7 z1F−d z2F−d z3F+d z2S−d z3S−d

* In this table, d=(1/2)zres to determine uzres, d=3σsamp to determine usamp, 

d=3σRave to determine uRave, and d=3σnoise to deterimine unoise. 

In view of the above method of calibration, the 

uncertainty equation for uzcal is found from the residual 

strain calculations using the three different sets of inputs 

given in Table 5 [with d=3(zxx−z1F)σzcal /cert]. Here, σzcal, 

the standard deviation of the twelve step height 

measurements on the double-sided step height standard, is 

assumed to scale linearly with height. Due to the fact that 

the difference in height being measured for residual strain 

measurements is small, uzcal is almost negligible. Also, due 

to the method of calibration and other factors, uzcal is very 

much considered an estimate. 

Given the three different residual strain values for the 

inputs specified in Table 5, εr-low and εr-high are identified. 

The interval from εr-low to εr-high is assumed to encompass 

99 % of the measurements. Then, w1/2 is calculated using 

(22), followed by the determination of uzcal, assuming a 

Gaussian distribution, using the following equation: 

uzcal = w1/2 3 . (27) 

Table 5. Three Sets of Inputs for Residual Strain 

Calculations to Determine uzcal, ucert, urepeat, 

udrift, and ulinear* 

z1F z2F z3F=z1S z2S z3S

1 z1F z2F z3F z2S z3S

2 z1F z2F+d z3F+d z2S+d z3S+d

3 z1F z2F−d z3F−d z2S−d z3S−d

* In this table, d=3(zxx−z1F)σzcal /cert to determine uzcal where zxx is the 

column heading, d=3(zxx−z1F)σcert /cert to determine ucert, 

d=(zxx−z1F)zrepeat /(2*μ6) to determine urepeat, d=(zxx−z1F)zdrift /(2*μ) to 

determine udrift, and d=zlinear to determine ulinear. 
 8

The uncertainty equation for uxcal is found by finding 

the residual strain value with calxmin from (14) as the xcalibration factor and then finding the residual strain value 

with calxmax from (15) as the x-calibration factor. The two 

residual strain values (εr-low and εr-high) are identified. 

Then, w1/2 is calculated using (22), followed by the 

determination of uxcal, assuming a Gaussian distribution, 

using the following equation: 

uxcal = w1/2 3 . (28) 

The uncertainty equation for uxres is found from the 

residual strain calculations using the seven different sets of 

inputs given in Table 6. Here, the inputted x-values along 

the top of the beam are varied plus or minus half xres. The 

two residual strain values (εr-low and εr-high) are identified. 

Then, w1/2 is calculated using (22), followed by the 

determination of uxres, assuming a uniform distribution, 

using the following equation: 

uxres = w1/2 1.732 . (29) 

Table 6. Seven Sets of Inputs for Residual Strain 

Calculations to Determine uxres* 

x1F x2F x3F=x1S x2S x3S

1 x1F x2F x3F x2S x3S

2 x1F+d x2F x3F x2S x3S−d

3 x1F−d x2F x3F x2S x3S+d

4 x1F+d x2F+d x3F x2S−d x3S−d

5 x1F+d x2F−d x3F x2S+d x3S−d

6 x1F−d x2F+d x3F x2S−d x3S+d

7 x1F−d x2F−d x3F x2S+d x3S+d

* In this table, d = (1/2)xres. 

The uncertainty equation for uxresL is found from the 

residual strain calculations from the two sets of inputs 

given in Table 7 for x1max, x1min, x2min, and x2max. In this 

table, x1upper is the original value for x1max, x1lower is the 

original value for x1min, x2lower is the original value for 

x2min, and x2upper is the original value for x2max. As can 

be seen in this table, the minimum and maximum in-plane 

length endpoints are varied plus or minus half xres. Then, 

εr-low and εr-high are determined and w1/2 is calculated using 

(22), followed by the determination of uxresL, assuming a 

uniform distribution, using the following equation: 

uxresL = w1/2 1.732 . (30) 

Table 7. Two Sets of Inputs for Residual Strain 

Calculations to Determine uxresL

 1 2 

x1max x1upper−(1/2)xres x1upper+(1/2)xres

x1min x1lower−(1/2)xres x1lower+(1/2)xres

x2min x2lower+(1/2)xres x2lower−(1/2)xres

x2max x2upper+(1/2)xres x2upper−(1/2)xres

The combined standard uncertainty equation for 

residual strain measurements (as calculated in Data 

Analysis Sheet I [7]) with twelve uncertainty components 

is as follows: 

2 2 2

2 2 2

2 2 2

2 2 2

repeat drift linear

Rave noise cert

xcal xres xresL

W L zres

c

u u u

u u u

u u u

u u u

u

+ +

+ + +

+ + +

+ + +

= (31) 

where uRave is the uncertainty of the measurement due to 

the sample’s surface roughness, unoise is the uncertainty of 

the measurement due to interferometric noise, ucert is the 

component in the combined standard uncertainty 

calculation that is due to the uncertainty of the value of the 

step-height standard, urepeat is the uncertainty of a 

measurement due to the repeatability of a measurement, 

udrift is the uncertainty of a measurement due to the amount 

of drift during the data session, and ulinear is the uncertainty 

of a measurement due to the deviation from linearity of the 

data scan. In the above equation, the uncertainty 

components uRave and unoise replace the previous component 

usamp. Also, the uncertainty components ucert, urepeat, udrift, 

and ulinear replace the previous component uzcal. 

As mentioned in the previous paragraph, usamp is now 

replaced with uRave and unoise. Rtave was the measurement 

used to obtain σsamp. Now, Rtave is being divided into Rave

for the determination of uRave and Rtave minus Rave for the 

determination of unoise. Rave is defined as 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. 

The uncertainty equation for uRave is found from a 

determination of σRave, where σRave is calculated to be onesixth the value of Rave. The data points obtained along the 

top of the beam are then varied as specified in Table 4 

(with d =3σRave) and the residual strain determined for the 

different sets of inputs. The two residaul strain values (εrlow and εr-high) are then identified. The interval from εr-low 

to εr-high is assumed to encompass 99 % of the 

measurements. Then, w1/2 is calculated using (22), 

followed by the determination of uRave, assuming a 

Gaussian distribution, using the following equation: 
 9

uRave = w1/2 3. (32) 

The uncertainty equation for unoise is found from a 

determination of σnoise, where σnoise is the standard 

deviation of the noise measurement, calculated to be onesixth the value of Rtave minus Rave. The data points 

obtained along the top of the beam are then varied as 

specified in Table 4 (with d =3σnoise) and the residual strain 

determined for the different sets of inputs. The two residual 

strain values (εr-low and εr-high) are then identified. The 

interval from εr-low to εr-high is assumed to encompass 

99 % of the measurements. Then, w1/2 is calculated using 

(22), followed by the determination of unoise, assuming a 

Gaussian distribution, using the following equation: 

 unoise = w1/2 3. (33) 

To use the uncertainty equation in (31), the calibration 

is done differently. Instead of the method presented earlier 

in this subsection, six measurements should be taken at the 

same location on the step height standard before the data 

session and six measurements taken at this same location 

after the data session with the average of these twelve 

measurements being called the mean, or μ. Then, calz is 

calculated to be cert divided by μ, and all the data values 

obtained during the data session are multiplied by calz to 

obtain calibrated z-data values. If this is done, the previous 

component, uzcal, gets replaced with ucert, urepeat, udrift, and 

ulinear to be described next. 

The uncertainty equation for ucert is found from the 

residual strain calculations using the inputs in Table 5 

[with d=3(zxx−z1F)σcert /cert where σcert is the certified one 

sigma uncertainty of the certified step height standard]. As 

with σzcal, σcert is assumed to scale linearly with height. 

Given the three different residual strain values for the 

inputs specified in Table 5, εr-low and εr-high are identified. 

The interval from εr-low to εr-high is assumed to encompass 

99 % of the measurements. Then, w1/2 is calculated using 

(22), followed by the determination of ucert, assuming a 

Gaussian distribution, using the following equation: 

ucert = w1/2 3. (34) 

The uncertainty equation for urepeat is found from the 

residual strain calculations using zrepeat, which is 

calculated to be 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. As with 

σzcal, zrepeat is assumed to scale linearly with height. The 

input values to the residual strain calculations are then 

varied as specified in Table 5 [with 

d=(zxx−z1F)zrepeat /(2*μ6) where μ6 is the average of the six 

calibration measurements from which zrepeat was found]. 

For the three different input combinations, εr-low and εr-high

are identified. Then, w1/2 is calculated using (22), followed 

by the determination of urepeat, assuming a uniform 

distribution, using the following equation: 

urepeat = w1/2 1.732 . (35) 

The uncertainty equation for udrift is found from the 

residual strain calculations using zdrift, which is calculated 

as follows: the average of the six calibration measurements 

before the data session (calavebefore) is determined, and the 

average of the six calibration measurements after the data 

session (calaveafter) is determined. Then, zdrift is calculated 

as the positive difference of these two values. As with 

σzcal, zdrift is assumed to scale linearly with height. The 

input values to the residual strain calculations are then 

varied as specified in Table 5 [with d=(zxx−z1F)zdrift /(2*μ)]. 

For the three different input combinations, εr-low and εr-high

are identified. Then, w1/2 is calculated using (22), followed 

by the determination of udrift, assuming a uniform 

distribution, using the following equation: 

udrift = w1/2 1.732 . (36) 

The uncertainty equation for ulinear is found from the 

residual strain calculations using zlinear, which is the 

difference in height between two points times zperc/100 

where zperc is 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 zperc (typically less than 3) is unitless. The input values 

to the residual strain calculations are then varied as 

specified in Table 5 with d=zlinear where zlinear is given by 

the following equation: 

100 ( ) perc

linear xx 1F

z

z = z − z (37) 

with zxx being the column heading in the table. For the 

three different input combinations, εr-low and εr-high are 

identified. Then, w1/2 is calculated using (22), followed by 

the determination of ulinear, assuming a uniform 

distribution, using the following equation: 

ulinear = w1/2 1.732 . (38) 

4.4 Strain Gradient Uncertainty Calculations

In this subsection, a combined standard uncertainty 

equation is presented for use with strain gradient 

measurements. Six sources of uncertainty are identified 
 10

with all other sources of uncertainty considered negligible. 

This uncertainty equation is also expanded to include ten 

sources of uncertainty. Knowledge of the additional 

sources of uncertainty can be a tremendous benefit when 

taking and analyzing other interferometeric measurements 

(such as 80 μm step heights). 

Table 8. Determination of the Strain Gradient Uncertainty 

Components* 

Uncertainty 

Component

Varied 

From 

Varied 

To 

How 

Varied

** 

G 

or 

u 

*** 

1. uW max sg value 

from Trace b, 

c, and d 

min sg value 

from Trace b, 

c, and d 

− u 

2. uzres +(1/2)zres −(1/2)zres 9 u 

3. usamp +3σsamp −3σsamp 9 G 

4. uzcal +d −d 10 G 

5. uxcal with 

calx=calxmax

with 

calx=calxmin

− G 

6. uxres +(1/2)xres −(1/2)xres 11 u 

7. uRave +3σRave −3σRave 9 G 

8. unoise +3σnoise −3σnoise 9 G 

9. ucert +d −d 10 G 

10. urepeat +d −d 10 u 

11. udrift +d −d 10 u 

12. ulinear +d −d 10 u 

* In this table, d =3(zx−z1)σzcal /cert to determine uzcal where zx is the 

appropriate column heading in Table 10, d=3(zx−z1)σcert /cert to determine 

ucert, d=(zx−z1)zrepeat /(2*μ6) to determine urepeat, d=(zx−z1)zdrift /(2*μ) to 

determine udrift, and d=zlinear to determine ulinear. 

** The entries in this column specify the table number where the input 

variations can be found, if applicable. 

*** “G” indicates a Gaussian distribution and “u” indicates a uniform 

distribution. 

With a few variations, the combined standard 

uncertainty equations and the calculations for each 

uncertainty component are similar to those presented for 

residual strain measurements. Therefore, refer to the 

previous subsection for the general approach. More 

specifically, refer to Tables 8, 9, 10, and 11 where mention 

is made of Tables 3, 4, 5, and 6, respectively. In addition, 

replace the words “residual strain” with the words “strain 

gradient,” replace εr-low with sg-low, εr-high with sg-high, zxx

with zx, and replace (21), (22), (31), and (37) with (39), 

(40), (41), and (42), respectively, given by the following 

equations: 

2 2 2

2 2 2

zcal xcal xres

W zres samp

c

u u u

u u u

u

+ +

+ + + = , (39) 

w1/2 = (sg−high − sg−low ) 2 , (40) 

2 2 2

2 2 2

2 2 2 2

repeat drift linear

Rave noise cert

W zres xcal xres

c

u u u

u u u

u u u u

u

+ +

+ + +

+ + + +

= , (41) 

and 

100 ( ) perc

linear x 1

z

z = z − z . (42) 

Note that (39) and (41) are the strain gradient combined 

standard uncertainty equations with six sources and ten 

sources of uncertainty, respectively. Data Analysis Sheet 

H [7] and ASTM Test Method E 2246-05 [10] use (39) and 

Data Analysis Sheet J uses (41) [7]. 

Table 9. Three Sets of Inputs for Strain Gradient 

Calculations to Determine uzres, usamp, uRave and unoise* 

z1 z2 z3

1 z1 z2 z3

2 z1+d z2−d z3+d

3 z1−d z2+d z3−d

* In this table, d=(1/2)zres to determine uzres, d=3σsamp to determine usamp, 

d=3σRave to determine uRave, and d=3σnoise to determine unoise. 

Table 10. Three Sets of Inputs for Strain Gradient 

Calculations to Determine uzcal, ucert, urepeat, 

udrift, and ulinear* 

z1 z2 z3

1 z1 z2 z3

2 z1 z2+d z3+d

3 z1 z2−d z3−d

* In this table, d=3(zx−z1)σzcal /cert where zx is the column heading to 

determine uzcal, d=3(zx−z1)σcert /cert to determine ucert, 

d=(zx−z1)zrepeat /(2*μ6) to determine urepeat, d=(zx−z1)zdrift /(2*μ) to determine 

udrift, and d=zlinear to determine ulinear. 

Table 11. Seven Sets of Inputs for Strain Gradient 

Calculations to Determine uxres

x1 x2 x3

1 x1 x2 x3

2 x1+(1/2)xres x2 x3−(1/2)xres

3 x1−(1/2)xres x2 x3+(1/2)xres

4 x1+(1/2)xres x2+(1/2)xres x3−(1/2)xres

5 x1+(1/2)xres x2−(1/2)xres x3−(1/2)xres

6 x1−(1/2)xres x2+(1/2)xres x3+(1/2)xres

7 x1−(1/2)xres x2−(1/2)xres x3+(1/2)xres
 11

5. Round Robin Results 

The reproducibility and repeatability results for inplane length are given in subsection 5.1, for residual strain 

in subsection 5.2, and for strain gradient in subsection 5.3. 

The reproducibility measurements were performed 

using the same test methods at independent laboratories. 

At least six independent laboratories must submit 

“validated” measurement results before the results can be 

recorded in the precision and bias statement of an ASTM 

standard test method. 

The repeatability measurements were performed using 

the same test method, in the same laboratory (NIST), by 

the same operator, with the same equipment, in the shortest 

practicable period of time. These measurements were done 

on random test structures. 

In each subsection one or two tables will be presented, 

either one for the reproducibility results and one for the 

repeatability results, or one for both the reproducibility and 

repeatability results, with the precision data appearing in 

bold print. In each table, the number of collected 

measurements (n) is specified. The tables also include the 

average (i.e., Lave, εrave, or sgave) of the reproducibility 

(and/or repeatability) measurement results and the 95 % 

limits (as given by 2.8 times the standard deviation [11] in 

the form of a percent). Below this, the average of the 

reproducibility (and/or repeatability) combined standard 

uncertainty [4] values (ucave) and the 95 % limits for uc are 

presented. 

In the tables for in-plane length measurements, the 

magnifications (mag) are also included along with the 

values for ∆L (where ∆L = Lave – Ldes). 

5.1 In-Plane Length Results

This subsection includes in-plane length measurements 

with transitional edges facing each other (such as Edges 

“1” and “2,” as shown in Figures 2 and 3) and 

measurements with transitional edges facing the same 

direction (such as Edges “1” and “5,” as shown in Figures 

2 and 3). Tables 12 and 13 present the reproducibility and 

repeatability results, respectively, for the in-plane length 

measurements with transitional edges facing each other, 

and Tables 14 and 15 present similar results for in-plane 

length measurements with transitional edges facing the 

same direction. 

In particular, note in Tables 12 and 14 the 

magnifications used for the specified values of the design 

length (Ldes). Not all laboratories had the same 

magnifications available; therefore, each laboratory was 

instructed to use the highest magnification available for the 

given measurement. The underlined value was the 

magnification used for the repeatability measurements. 

Table 12 includes the in-plane length results from the 

650 μm long fixed-fixed beams from the residual strain 

measurements. Similar 650 μm long measurements are not 

included in Table 13 since repeatability measurements at a 

magnification of 5× are already available with the 1000 μm 

long measurements. 

Table 12. Reproducibility of Round Robin Measurement 

Results for In-Plane Length Measurements When the 

Transitional Edges Face Each Other 

Ldes

25 

μm 

Ldes

80 

μm 

Ldes

200 

μm 

Ldes

500 

μm 

Ldes

650 

μm 

Ldes

1000 

μm 

n 7 7 6 6A 7 6A

mag

(×) 

100, 80,

C

50, 39, 

20, 10, 

uD

50, 40, 

25, 25, 

10, 10, u

25, 20.4, 

20, 10, 

10, 10 

10.2, 10, 

10, 10, 

5, 5 

25, 7.8, 

5, 5, 5, 

5, u

5, 5, 5, 

5, 5, 5 

Lave

(μm) 

24.91 79.70 200.61 497.8 651.4 999.8 

95 % 

limits 

for L

±23.9 % ±9.3 % ±2.8 % ±1.1 % ±1.1 % ±0.7 % 

ucave

(μm)B

0.60 

(2.4 %) 

0.71 

(0.9 %) 

0.86 

(0.4 %) 

1.5 

(0.3 %) 

1.6 

(0.2 %) 

2.5 

(0.3 %) 

95 % 

limits 

for uc

B

±257 % ±172 % ±166 % ±170 % ±203 % ±63 % 

∆L

(μm) 

−0.09 −0.30 0.61 −2.2 1.4 −0.2 

A

 Three of these measurements were taken from the same instrument by 

two different operators. 

B

 As determined in Test Method E 2244−02 [1] using (9) with uxcal=uxres=0. C

 Underlined values correspond to the magnifications used for the 

repeatability measurements. 

D The symbol “u” stands for “unknown.” The magnification was not 

reported by the round robin participant. 

Table 13. Repeatability of NIST Measurement Results for 

In-Plane Length Measurements When the 

Transitional Edges Face Each Other 

Ldes

25 

μm 

Ldes

80 

μm 

Ldes

200 

μm 

Ldes

500 

μm 

Ldes

1000 

μm 

n 48 48 48 48 48 

mag

(×) 

80 40 20 10 5 

Lave

(μm) 

24.37 79.76 199.10 495.0 995.5 

95 % 

limits 

for L

± 1.1 % ± 0.3 % ± 0.2 % ± 0.5 % ± 0.7 % 

ucave

(μm) A

0.33 

(1.34 %) 

1.11 

(1.4 %) 

1.08 

(0.5 %) 

1.9 

(0.4 %) 

3.6 

(0.4 %) 

95 % 

limits 

for uc

A

± 14 % ± 6 % ± 10 % ± 35 % ± 57 % 

ucave

(μm)B

0.23 

(0.95 %) 

1.05 

(1.3 %) 

0.95 

(0.5 %) 

1.7 

(0.3 %) 

3.2 

(0.3 %) 

95 % 

limits 

for uc

B

± 29 % ± 7 % ± 13 % ± 46 % ± 72 % 

∆L

(μm) 

−0.63 −0.24 −0.90 −5.0 −4.5 

A

 As determined in ASTM Test Method E 2244−05 [8] using (9). 

B

 As determined in ASTM Test Method E 2244−02 [1] using (9) with 

uxcal=uxres=0. 
 12

Table 14. Reproducibility of Round Robin Measurement 

Results for In-Plane Length Measurements When the 

 Transitional Edges Face the Same Direction 

Ldes

60 

μm 

Ldes

115 

μm 

Ldes

235 

μm 

Ldes

535 

μm 

Ldes

1035 

μm 

n 6A 6 6A 6B 6B

mag (×) 80, 80, 

50, 39, 

20, 10 

40, 25, 

25, 25, 

10, 10 

20.4, 20, 

20, 10, 

10, 5 

10.2, 10, 

10, 10, 

5, 5 

5.9, 5, 5, 

5, 5, 5 

Lave (μm) 59.68 115.34 235.79 533.8 1035.1 

95 % 

limits 

for L

± 2.9 % ± 6.1 % ± 2.4 % ± 1.5 % ± 0.7 % 

ucave

(μm)C

0.22 

(0.4 %) 

0.32 

(0.3 %) 

0.43 

(0.2 %) 

0.7 

(0.1 %) 

1.1 

(0.1 %) 

95 % 

limits 

for uc

C

± 331 % ± 193 % ± 217 % ± 93 % ± 16 % 

∆L (μm) −0.32 0.34 0.79 −1.2 0.1 

A

 Two of these measurements were taken from the same instrument by 

different operators. 

B

 Three of these measurements were taken from the same instrument by 

two different operators. 

C As determined in Test Method E 2244−02 [1] using (9) with 

uxcal=uxres=0. 

Table 15. Repeatability of NIST Measurement Results for 

In-Plane Length Measurements When the 

Transitional Edges Face the Same Direction 

Ldes

60 

μm 

Ldes 

115 

μm 

Ldes 

235 

μm 

Ldes

535 

μm 

Ldes

1035 

μm 

n 48 48 48 48 48 

mag

(×) 

80 40 20 10 5 

Lave

(μm) 

59.56 115.96 234.67 532.2 1035.0 

95 % 

limits 

for L

±0.6 % ±0.5 % ±0.3 % ±0.1 % ±0.2 % 

ucave

(μm)A

0.54 

(0.9 %) 

0.55 

(0.5 %) 

0.64 

(0.3 %) 

1.1 

(0.2 %) 

2.0 

(0.2 %) 

95 % 

limits 

for uc

A

±0.6 % ±0.4 % ±0.2 % ±0.1 % ±0.1 % 

ucave

(μm)B

0.07 

(0.1 %) 

0.14 

(0.1 %) 

0.26 

(0.1 %) 

0.5 

(0.1 %) 

1.1 

(0.1 %) 

95 % 

limits 

for uc

B

±0 % ±0 % ±0 % ±0 % ±0 % 

∆L

(μm) 

−0.44 0.96 −0.33 −2.8 0.0 

A

As determined in ASTM Test Method E 2244−05 [8] using (9). 

B

As determined in ASTM Test Method E 2244−02 [1] using (9) with 

uxcal=uxres=0. 

The test method for in-plane length measurements 

emphasizes two values, the in-plane length measurement, 

L, and the combined standard uncertainty [4], uc, of that 

measurement. The true length is believed to lie in the 

interval L +/− 3uc with a level of confidence of 

approximately 99.7 % assuming a Gaussian distribution 

[6]. 

Figure 9 illustrates reproducibility and repeatability 

bias data for L. This figure consists of four plots of ∆L

versus Ldes where ∆L = Lave – Ldes. These plots are for the 

reproducibility and repeatability measurements in Data 

Sheets A and B. The average ∆L value (∆Lave) for each 

plot is given. For the reproducibility and repeatability data 

from Data Sheet B and the reproducibility data from Data 

Sheet A, assuming the design lengths (Ldes) are the true 

length values, due to the very small values of ∆Lave in 

combination with at least two values of ∆L being greater 

than or equal to zero and at least two being less than or 

equal to zero for each curve (as given in Tables 12, 14, and 

15), it can be concluded that there is no obvious bias to this 

data. 

The repeatability data from Data Sheet A show the 

highest |∆Lave| value with all the measurements of Lave

being less than Ldes (as given in Table 13). Therefore, there 

may be a bias towards measuring lower values of L when 

using Data Sheet A at this laboratory, and the degree of the 

resulting bias varies with magnification. Even though 

these data were calibrated, it should be emphasized that 

calibration of the interferometer is considered mandatory 

for in-plane length measurements. The interferometer is 

calibrated in-plane using a 10 μm grid or finer grid ruler. 

If the calibration is not done at each magnification, a bias 

to the measurements is expected. The direction and degree 

of the resulting bias are different for each magnification of 

each interferometer. Then, calx is calculated using (16) and 

all x-values are multiplied by calx to obtain calibrated xdata values. 

Reproducibility and Repeatability 

Bias Data for L

0 250 500 750 1000

L des (μm)

∆L = Lave −Ldes

Figure 9. Reproducibility and repeatability bias data 

for L. 

Now, consider the trends associated with the average 

combined standard uncertainty value (ucave) as given in 

Figures 10 and 11 for Data Sheets A and B, respectively.6

 

In both of these figures, ucave increases with increasing 

length primarily due to the increase in the pixel-to-pixel 

spacings associated with the lower-powered objectives that 

are used for the longer length measurements. Also, these 

 6

 In this discussion, the equation for uc in (9) does not include uxcal and 

uxres. 

−2.25 μm 

−0.13 μm 

−0.52 μm 

−0.06 μm 

∆Lave

Reproducibility

Sheet A 

Repeatability 

Sheet A

Reproducibility

Sheet B 

Repeatability 

Sheet B 0 

0

0

0
 13

figures show that the reproducibility and repeatability 

measurements are somewhat comparable. For Data Sheet 

A, uc = (Lmax – Lmin) / 6, and if the data points determining 

Lmax and Lmin are chosen in the manner specified in the 

standard test method, this result is expected. For Data 

Sheet B, the equation for uc can be rewritten [6] to be uc = 

2sep/3 where sep is the pixel-to-pixel separation for the 

given magnification. Therefore, uc can be determined 

before the measurement is even taken. Therefore, the 

reproducibility and repeatability measurements should be 

comparable. 

Probing deeper into this, Figure 11 (in combination 

with the magnifications specified in Table 14) indicates 

that the interferometer used at NIST for the repeatability 

measurements has comparable values for sep for the 

Ldes=1035 μm measurement taken at 5× in comparison to 

the interferometers used by the other laboratories that 

participated in this round robin. However, this laboratory 

benefited by having 5 different magnifications with which 

to take measurements. Therefore, for the smaller values of 

Ldes, due to a relatively higher-powered objective that was 

used for the measurements in comparison to the other 

laboratories, this laboratory was able to achieve lower 

values for ucave. This could imply that the effective 

technical lifetime of an interferometer can be extended by 

purchasing multiple objectives. 

Comparing ucave in Figures 10 and 11, the values for 

ucave from Data Sheet B are considerably less than those 

from Data Sheet A. This implies that more precise in-plane 

length measurements are possible when the transitional 

edges face the same direction. 

Data Sheet A 

Reproducibility and Repeatability Results 

for u cave

0

1

2

3

4

0 250 500 750 1000

L des (μm)

u cave (μm)

Figure 10. Comparing reproducibility and repeatability 

results for ucave in Data Sheet A. 

Comments concerning the round robin data include the 

following: 

a) Precision: The precision data for L and the precision 

data for the combined standard uncertainty, uc, of that 

measurement, appear in bold print in Tables 12 through 15. 

The 95 % limits for L given in these tables indicate that the 

reproducibility results (e.g., ±2.8 % for Data Sheet A in 

Table 12 for Ldes=200 μm) are much poorer than the 

corresponding repeatability results (i.e., ±0.2 % in Table 

13). This can be due to different magnifications used for 

the measurements (especially for smaller values of Ldes), 

the calibrations of the instruments being slightly different, 

the use of different instruments, and a different person 

taking the measurements and analyzing the data. 

Data Sheet B 

Reproducibility and Repeatability Results 

for u cave

0

0.2

0.4

0.6

0.8

1

1.2

0 250 500 750 1000

L des (μm)

ucave (μm)

Figure 11. Comparing reproducibility and repeatability 

results for ucave in Data Sheet B. 

b) Bias: As mentioned previously, the data in Figure 9 

suggests no bias associated with the length data, but a 

tendency for repeatability Data Sheet A length data to be 

less than Ldes for this laboratory for all magnifications. The 

degree of the resulting bias is different for each 

magnification. 

5.2 Residual Strain Results

Table 16 presents the residual strain reproducibility and 

repeatability results. For the reproducibility measurements, 

the fixed-fixed beams measured had either a 0 degree or a 

90 degree orientation and lengths ranging from 550 μm to 

700 μm. For the repeatability measurements, the fixedfixed beams measured had different orientations and 

lengths ranging from 600 μm to 750 μm. All fixed-fixed 

beams were 10 μm wide. 

Comments concerning the round robin data include the 

following: 

a) Plots: In this round robin, random length fixedfixed beams were measured. As such, there were at least 

two variables (orientation and length) as discussed below: 

i) Orientation: Figure 12 is a plot of −εr versus 

orientation. In this plot, there is no obvious orientation 

dependence considering the values for εrave are 

approximately the same for the two different orientations. 

However, the 95 % limits for the data taken from the test 

structures with a 0 degree orientation are approximately 

half the 95 % limits for the data taken from the test 

structures with a 90 degree orientation. 

ii) Length: Figure 13 is a plot of −εr versus length. In 

this plot, the data indicate there is no obvious length 

dependence. 

b) Precision: The reproducibility and repeatability 

precision data appear in Table 16 in bold print. Note that 

for the 95 % limits for εr, the repeatability data (i.e., 

±15.9 %) are less than the reproducibility data (i.e., 

±27.8 %). This is due to the repeatability measurements 

Repeatability

Reproducibility 

higher mag used 

similar sep 

values 

Repeatability 

Reproducibility 
 14

being taken at the same laboratory using the same 

instrument by the same operator. 

Table 16. Residual Strain Measurement Results 

Reproducibility 

results 

Ldes=550 – 700 μm 

Repeatability 

results 

Ldes=600 – 750 μm 

n 6A 24 

εrave −44.02×10−6 −41.65×10−6

95 % limits 

for εr

± 27.8 % ± 15.9 % 

ucaveB

− 0.53×10−6 (1.3 %) 

95 % limits 

for uc

B − ± 79.1 % 

ucaveC

− 0.57×10−6 (1.4 %) 

95 % limits 

for uc

C − ± 55.8 % 

ucaveD

1.07×10−6 (2.4 %) 0.77×10−6 (1.8 %) 

95 % limits 

for uc

D ± 132 % ± 136 % 

A

 Two of these measurements were taken from the same instrument by 

different operators. 

B

 As determined in ASTM Test Method E 2245−05 [9] using (21). 

C

 As determined using (31). 

D As determined in ASTM Test Method E 2245−02 [2], the usamp and uzcal

components in the uc calculation were combined into one component. As 

such, for this component, the 99 % limits, assuming a uniform (that is, 

rectangular) probability distribution, were represented by a +/−20 nm 

variation in the z-value of the data points. Also in Test Method E 

2245−02, uzres= uxcal=uxres=uxresL=0. 

Residual Strain Data for Two Different 

Orientations of Fixed-Fixed Beams

35

40

45

50

-45 0 45 90 135

Orientation (degrees)

−εr (x10−6

)

Figure 12. A plot of –εr versus orientation. 

Residual Strain Data as a Function of Length

35

40

45

50

500 550 600 650 700 750 800

L des (μm)

−εr (x10−6

)

Figure 13. A plot of –εr versus length. 

c) Bias: No information can be presented on the bias of 

the procedure in Test Method E 2245 for measuring 

residual strain because no material having an accepted 

reference value is available. 

5.3 Strain Gradient Results

Table 17 presents the strain gradient reproducibility and 

repeatability results. For the reproducibility measurements, 

the cantilevers measured had either a 0 degree or a 90 

degree orientation and lengths ranging from 500 μm to 650 

μm. For the repeatability measurements, the cantilevers 

measured had different orientations and lengths from 

400 μm to 750 μm. All cantilevers were 10 μm wide. 

Table 17. Strain Gradient Measurement Results 

Reproducibilit

y results 

Ldes= 

500 – 650 μm

Repeatability 

results 

Ldes= 

500 – 650 μm

Repeatability 

results 

Ldes= 

400 – 750 μm

n 6 24 48 

sgave 4.67 m−1

 4.71 m−1 4.97 m−1

95 % 

limits 

for sg

± 51.6 % ± 35.5 % ± 56.1 % 

ucaveA

− 0.47 m−1 

(10.1 %) 

0.56 m−1

(11.2 %) 

95 % 

limits 

for uc

A

− ± 142 % ± 176 % 

ucaveB

− 0.44 m−1 

(9.5 %) 

0.52 m−1

(10.5 %) 

95 % 

limits 

for uc

B

− ± 151 % ± 182 % 

ucaveC

0.56 m−1

(12.0 %) 

0.73 m−1

(15.5 %) 

0.84 m−1

(16.9 %) 

95 % 

limits 

for uc

C

± 147 % ± 95.3 % ± 150 % 

A As determined in ASTM Test Method E 2246−05 [10] using (39). 

B

 As determined using (41).

C

 As determined in ASTM Test Method E 2246−02 [3], the usamp and uzcal

components in the uc calculation were combined into one component. For 

this component, assuming a uniform (that is, rectangular) probability 

distribution, the 99 % limits were represented by a +/−20 nm variation in 

the z-value of the data points. Also, in Test Method E 2246−02, 

uzres=uxcal=uxres=0. 

Comments concerning the round robin data include the 

following: 

a) Plots: In this round robin, random length 

cantilevers were measured. As such, there were at least 

two variables (orientation and length) as discussed below. 

i) Orientation: Figure 14 is a plot of sg versus 

orientation. In this plot, there is no obvious orientation 

dependence. 

ii) Length: Figure 15 is a plot of sg versus design 

length. In this plot, the data indicate a decrease in the 

strain gradient for increasing length (for Ldes = 400 μm to 

600 μm) that levels off (from Ldes=600 μm to 750 μm). 

εrave=−41.72×10−6

95 % limits for εr = ±10.1 % 

εrave=−41.58×10−6

95 % limits for εr = ±20.7 % 

εrave=−44.44×10-6

95 % limits for εr = ±11.8 % 
 15

Strain Gradient Data for Two Different 

Orientations of Cantilevers

0

1

2

3

4

5

6

7

8

-45 0 45 90 135

Orientation (degrees)

sg (m−1

)

Figure 14. A plot of sg versus orientation. 

Strain Gradient Data as a Function of Length 

0

2

4

6

8

300 400 500 600 700 800 900

L des (μm)

sg (m−1

) Figure 15. A plot of sg versus length for two different 

orientations. 

b) Precision: The reproducibility and repeatability 

precision data appear in Table 17 in bold print. Note that 

for the 95 % limits for sg, the repeatability data (i.e., 

±35.5 %) is less than the reproducibility data (i.e., 

±51.6 %) for the same span of design lengths. This is due 

to the repeatability measurements being taken at the same 

laboratory using the same instrument by the same operator. 

It is interesting to note that the ucave values for the 

repeatability measurements are slightly higher than the 

ucave values for the reproducibility measurements. 

c) Bias: No information can be presented on the bias 

of the procedure in Test Method E 2246 for measuring 

strain gradient because no material having an accepted 

reference value is available. 

6. Conclusions 

ASTM standard test method precision and bias data 

obtained from the MEMS Length and Strain Round Robin 

were presented for in-plane length, residual strain, and 

strain gradient measurements. The final results are 

incorporated in the ASTM standard test methods [8-10]. 

In addition to the precision and bias data presented in 

Tables 12 through 17, this paper presented the combined 

standard uncertainty equations [namely, (9), (21), and (39) 

for in-plane length, residual strain, and strain gradient 

measurements, respectively] used in the validated ASTM 

standard test methods [8-10]. The determination of each 

uncertainty component was discussed in detail. 

The combined standard uncerainty equations for 

residual strain and strain gradient were further expanded 

into (31) and (41), respectively, and each additional 

uncertainty component discussed in detail. Knowledge of 

the additional sources of uncertainty can be a tremendous 

benefit when taking and analyzing other interferometric 

measurements (such as, 80 μm step heights). 

One positive result of this round robin analysis is the 

reduction of interlaboratory differences in the parametric 

measurements due in part to the verification sections in the 

data analysis sheets. Some round robin results for 

reproducibility were not incorporated in the summary 

results because of lack of compliance with these quality 

control checks. 

And lastly, the following guidelines should be of value 

to the MEMS industry, when communicating data obtained 

using these ASTM standard test methods: 

a) To record the data and perform the calculations, use 

the data analysis sheets which are available on the MEMS 

Length and Strain Calculator web site [7], 

b) Make sure the data have passed the verification 

given at the bottom of the data analysis sheets, and 

c) Ask if all the verification checks were passed, when 

communicating with others concerning ASTM standard 

data. 

7. Acknowledgment 

First and foremost, we wholeheartedly acknowledge the 

many round robin participants whose combined efforts are 

displayed in many of the tables and figures throughout this 

paper. In the order of participation, we acknowledge Dr. 

Alexander Spivak (Kavlico), Andrew McNeil (Motorola), 

Dr. Herbert R. Shea (Lucent Technologies), Tom Stout and 

Misha Scepanovic (Veeco Metrology), Dr. Rognvald P. 

Garden and Eric Felkel (Zygo), Jeff Melzak and Chien Wu 

(FLX Micro), and Dan W. Chilcott (Delphi). Thank you! 

John Huang (NIST) and Erik Secula (NIST) are 

acknowledged for the taking of additional round robin data 

and for incorporating web pages onto the SED Web site, 

respectively. 

The following NIST people reviewed this paper: Dr. 

David L. Blackburn, Dr. David T. Read, Richard A. Allen, 

Erik M. Secula, and Dr. David G. Seiler. Thank you for 

the time you put into it. 

And lastly, an acknowledgment to Dr. Michael Gaitan 

(MEMS Project Leader) and the Office of Microelectronics 

Programs for supporting this work. 

8. References 

[1] ASTM E08, “E 2244 Standard Test Method for InPlane Length Measurements of Thin, Reflecting Films 

Using an Optical Interferometer,” Annual Book of ASTM 

Standards, Vol. 03.01, 2003. 

[2] ASTM E08, “E 2245 Standard Test Method for 

Residual Strain Measurements of Thin, Reflecting Films 

Using an Optical Interferometer,” Annual Book of ASTM 

Standards, Vol. 03.01, 2003. 

sgavex=5.05 m−1

95 % limits = ±56.0 % 

sgavey = 4.89 m−1

95 % limits= ±57.0 % 

sgave=4.26 m−1

95% limits for sg = ±27.7%

sgave=4.24 m−1

95% limits for sg = ±26.4%

sgave=5.71 m−1

95% limits for sg = ±42.4% 
 16

[3] ASTM E08, “E 2246 Standard Test Method for Strain 

Gradient Measurements of Thin, Reflecting Films Using an 

Optical Interferometer,” Annual Book of ASTM Standards, 

Vol. 03.01, 2003. 

[4] Taylor, B. N. and Kuyatt, C. E., “Guidelines for 

Evaluating and Expressing the Uncertainty of NIST 

Measurement Results,” NIST Technical Note 1297, 

National Institute of Standards and Technology, September 

1994. 

[5] Marshall, J. C., Secula, E. M., and Huang, J., “Round 

Robin for Standardization of MEMS Length and Strain 

Measurements,” SEMI Technology Symposium: 

Innovations in Semiconductor Manufacturing (STS: ISM), 

SEMICON West 2004, San Francisco, CA, July 12-14, 

2004. 

[6] Marshall, J. C., “MEMS Length and Strain 

Measurements Using an Optical Interferometer,” NISTIR 

6779, National Institute of Standards and Technology, 

August 2001. 

[7] The NIST, Semiconductor Electronics Division (SED) 

MEMS Length and Strain Calculator web site 

(http://www.eeel.nist.gov/812/teststructures/MEMSCalculator.htm) is accessible through the 

MEMS Project web page at 

http://www.eeel.nist.gov/812/44.htm. 

[8] ASTM E08, “E 2244 Standard Test Method for InPlane Length Measurements of Thin, Reflecting Films 

Using an Optical Interferometer,” Annual Book of ASTM 

Standards, Vol. 03.01, 2006. (Also available via 

http://www.astm.org.) 

[9] ASTM E08, “E 2245 Standard Test Method for 

Residual Strain Measurements of Thin, Reflecting Films 

Using an Optical Interferometer,” Annual Book of ASTM 

Standards, Vol. 03.01, 2006. (Also available via 

http://www.astm.org.) 

[10] ASTM E08, “E 2246 Standard Test Method for Strain 

Gradient Measurements of Thin, Reflecting Films Using an 

Optical Interferometer,” Annual Book of ASTM Standards, 

Vol. 03.01, 2006. (Also available via 

http://www.astm.org.) 

[11] American Society for Testing and Materials, “Form 

and Style for ASTM Standards,” ASTM, West 

Conshohocken, PA, March 2001.