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List of Terms

cantilever a test structure that consists of afreestanding beam that is fixed atone end 1

fixed-fixed beam a test structure that consists of afreestanding beam that is fixed atboth ends 1

in-plane length (or deflection) measurementthe experimental determination ofthe straight-line distance betweentwo transitional edges in a MEMSdevice 1

interferometer a non-contact optical instrumentused to obtain topographical 3-Ddata sets 1

Volume 115, Number 5, September-October 2010Journal of Research of the National Institute of Standards and Technology

303

[J. Res. Natl. Inst. Stand. Technol. 115, 303-342 (2010)]

MEMS Young’s Modulus andStep Height Measurements With

Round Robin Results

Volume 115 Number 5 September-October 2010

Janet Marshall, RichardA. Allen, Craig D. McGray, andJon Geist

Semiconductor ElectronicsDivision,National Institute of Standardsand Technology,Gaithersburg, MD 20899-8120

[email protected]@[email protected]@nist.gov

This paper presents the results of a micro-electromechanical systems (MEMS)Young’s modulus and step height roundrobin experiment, completed in April2009, which compares Young’s modulusand step height measurement results at anumber of laboratories. The purpose of theround robin was to provide data for theprecision and bias statements of two \related Semiconductor Equipment andMaterials International (SEMI) standardtest methods for MEMS. The technicalbasis for the test methods on Young’smodulus and step height measurements arealso provided in this paper.

Using the same test method, the goalof the round robin was to assess therepeatability of measurements at onelaboratory, by the same operator,with the same equipment, in the shortestpractical period of time as well as thereproducibility of measurements withindependent data sets from uniquecombinations of measurement setups andresearchers. Both the repeatability andreproducibility measurements were doneon random test structures made of thesame homogeneous material.

The average repeatability Young’smodulus value (as obtained fromresonating oxide cantilevers) was64.2 GPa with 95 % limits of ± 10.3 %and an average combined standard

uncertainty value of 3.1 GPa. The averagereproducibility Young’s modulus valuewas 62.8 GPa with 95 % limits of± 11.0 % and an average combinedstandard uncertainty value of 3.0 GPa.

The average repeatability step heightvalue (for a metal2-over-poly1 step fromactive area to field oxide) was 0.477 μmwith 95 % limits of 7.9 % and an averagecombined standard uncertainty value of0.014 μm. The average reproducibility stepheight value was 0.481 μm with 95 %limits of ± 6.2 % and an average combinedstandard uncertainty value of 0.014 μm.

In summary, this paper demonstratesthat a reliable methodology can be used tomeasure Young's modulus and step height.Furthermore, a micro and nano technology(MNT) 5-in-1 standard reference material(SRM) can be used by industry to comparetheir in-house measurements using thismethodology with NIST measurementsthereby validating their use of thedocumentary standards.

Key words: interferometry; microelectro-mechanical systems; round robin; standardtest methods; step height; test structure;vibrometry; Young's modulus

Accepted: September 10, 2008

Available online: http://www.nist.gov/jres

1 Reprinted, with permission, from ASTM E 2444 Terminology Relating to Measurements Taken on Thin, Reflecting Films, copyright ASTMInternational, 100 Barr Harbor Drive, West Conshohocken, PA 19428.

residual strain in a MEMS process, the amount ofdeformation (or displacement) perunit length constrained within thestructural layer of interest afterfabrication yet before the con-straint of the sacrificial layer (orsubstrate) is removed (in whole orin part) 1

residual stress the remaining forces per unit areawithin the structural layer ofinterest after the original cause(s)during fabrication have beenremoved yet before the constraintof the sacrificial layer (or sub-strate) is removed (in whole or inpart) 2

step height the distance in the z-direction thatan initial, flat, processed surface(or platform) is to a final, flat,processed surface (or platform) 2

(residual) strain a through-thickness variation (ofgradient the residual strain) in the structur-

al layer of interest before it isreleased 1

(residual) stress a through-thickness variation (ofgradient the residual stress) in the structur-

al layer of interest before it isreleased 2

test structure a component (such as, a fixed-fixed beam or cantilever) that isused to extract information (suchas, the residual strain or the straingradient of a layer) about a fabri-cation process 1

thickness the height in the z-direction of oneor more designated thin-filmlayers 2

vibrometer an instrument for non-contactmeasurements of surface motion 2

Young’s modulus a parameter indicative of materialstiffness that is equal to the stressdivided by the strain when thematerial is loaded in uniaxialtension, assuming the strain issmall enough such that it does notirreversibly deform the material 2

List of Symbols

For Young’s modulus measurements and calculations:μ = viscosity of the ambient surround-

ing the cantilever. [SEMI MS4]ρ = density of the thin film layer. [SEMI MS4]E = calculated Young’s modulus value of the

thin film layer. [SEMI MS4]Eclamped = calculated Young’s modulus value ob-

tained from the average resonance fre-quency of a fixed-fixed beam assumingclamped-clamped boundary conditions. [SEMI MS4]

Einit = initial estimate for the Young’s modulusvalue of the thin film layer. [SEMI MS4]

Esimple = calculated Young’s modulus value ob-tained from the average resonance fre-quency of a fixed-fixed beam assumingsimply supported boundary conditionsat both supports. [SEMI MS4]

fcan = average undamped resonance frequen-cy of the cantilever. [SEMI MS4]

fffb = average resonance frequency of thefixed-fixed beam. [SEMI MS4]

Lcan = suspended cantilever length. [SEMI MS4]Lffb = suspended fixed-fixed beam length.

[SEMI MS4]Q = oscillatory quality factor of the

cantilever. [SEMI MS4]t = thickness of the thin film layer. [SEMI

MS4]Wcan = suspended cantilever width. [SEMI MS4]

For combined standard uncertainty calculations ofYoung’s modulus measurements:

σμ = one sigma uncertainty of the value of μ.[SEMI MS4]

σρ = one sigma uncertainty of the value of ρ.[SEMI MS4]

σfreq = one sigma uncertainty of the value offcan. [SEMI MS4]

σL = one sigma uncertainty of the value ofLcan. [SEMI MS4]

σthick = one sigma uncertainty of the value of t.[SEMI MS4]

σW = one sigma uncertainty of the value ofWcan. [SEMI MS4]

Emax = maximum Young’s modulus value asdetermined in an uncertainty calcula-tion. [SEMI MS4]

Volume 115, Number 5, September-October 2010Journal of Research of the National Institute of Standards and Technology

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2 Reprinted, with permission, from SEMI MS2, MS3, and MS4copyright Semiconductor Equipment and Materials International,Inc. (SEMI) © 2010, 3081 Zanker Road, San Jose, CA 95134,www.semi.org.

Emin = minimum Young’s modulus value as de-termined in an uncertainty calculation.[SEMI MS4]

fresol = frequency resolution for the given set ofmeasurement conditions. [SEMI MS4]

pdiff = estimated percent difference between the damped and undamped resonancefrequency of the cantilever. [SEMI MS4]

uρ = component in the combined standarduncertainty calculation for Young’smodulus that is due to the uncertaintyof ρ. [SEMI MS4]

uc = combined standard uncertainty value (that is, the estimated standard deviationof the result). [SEMI MS4]

ucE = the combined standard uncertainty for a Young’s modulus measurement.

udamp = component in the combined standard uncertainty calculation for Young’s modulus that is due to damping. [SEMIMS4]

uE = the standard uncertainty for a Young’s modulus measurement as obtained from a fixed-fixed beam.

ufreq = component in the combined standard uncertainty calculation for Young’smodulus that is due to the measurement uncertainty of fcan. [SEMI MS4]

ufresol = component in the combined standard uncertainty calculation for Young’smodulus that is due to fresol. [SEMI MS4]

uL = component in the combined standard uncertainty calculation for Young’s modulus that is due to the measurement uncertainty of Lcan. [SEMI MS4]

uthick = component in the combined standard uncertainty calculation for Young’smodulus that is due to the measurementuncertainty of t. [SEMI MS4]

For calibration of step height measurements:

σcert = the one sigma uncertainty of the cali-brated physical step height used forcalibration.

cert = the certified value of the double-sidedphysical step height used for calibration.[SEMI MS2]

zdrift = the calibrated positive differencebetween the average of the six calibra-tion measurements taken before the datasession (at the same location on the

physical step height used for calibra-tion) and the average of the six calibra-tion measurements taken after the data session (at the same location on thephysical step height used for calibra-tion). [SEMI MS2]

zperc = over the instrument’s total scan range,the maximum percent deviation fromlinearity, as quoted by the instrumentmanufacturer (typically less than 3 %).[SEMI MS2]

zrepeat = the maximum of two calibrated values;one of which is the positive calibrated difference between the minimum and maximum values of the six calibration measurements taken before the datasession (at the same location on thephysical step height used for calibra-tion) and the other is the positivecalibrated difference between theminimum and maximum values of thesix calibration measurements taken afterthe data session (at the same location on the physical step height used for calibra-tion). [SEMI MS2]

z– = calibrated average of the twelve calibra-tion measurements taken before and after the data session at the same loca-tion on the physical step height used for calibration. [SEMI MS2]

z–6 = the calibrated average of the six calibra-tion measurements from which zrepeat is found. [SEMI MS2]

For step height measurements and calculations:

σplatNXt = the calibrated standard deviation of the data from trace “t” on platNX. [SEMIMS2]

σroughNX = the calibrated surface roughness of platNX measured as the smallest of all the values obtained for σplatNXt ; however, if the surfaces of the platforms (includ-ing the reference platform) all haveidentical compositions, then it is meas-ured as the smallest of all the standard deviation values obtained from data traces “a,” “b,” and “c” along theseplatforms. [SEMI MS2]

platNrD = the average of the calibrated referenceplatform height measurements takenfrom multiple data traces on one step

Volume 115, Number 5, September-October 2010Journal of Research of the National Institute of Standards and Technology

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height test structure, where N is the test structure number (“1,” “2,” “3,” etc.),r indicates it is from a reference plat-form, and D directionally indicateswhich reference platform (using thecompass indicators “N,” “S,” “E,” or“W” where “N” refers to the referenceplatform designed closest to the top ofthe chip). [SEMI MS2]

platNrDt = a calibrated reference platform height measurement from one data trace, whereN is the test structure number (“1,” “2,”“3,” etc.), r indicates it is from a refer-ence platform, D directionally indicateswhich reference platform (using thecompass indicators “N,” “S,” “E,” or “W” where “N” refers to the reference platform designed closest to the top of the chip), and t is the data trace (“a,”“b,” “c,” etc.) being examined. [SEMIMS2]

platNX = the calibrated platform height measure-ment, where N is the test structurenumber (“1,” “2,” “3,” etc.) and X is the capital letter (or “r” is used if it is the reference platform) associated with the platform (“A,” “B,” “C,” etc.) aslettered starting with “A” for theplatform closest to platNrW or platNrS.[SEMI MS2]

platNXt = a calibrated platform height measure-ment from one data trace, where N is the test structure number (“1,” “2,” “3,”etc.), X is the capital letter associatedwith the platform (“A,” “B,” “C,” etc.)as lettered starting with “A” for theplatform closest to platNrW or platNrS,and t is the data trace (“a,” “b,” “c,” etc.)being examined. [SEMI MS2]

stepNX MY = the calibrated step height measurementtaken from two different step height test structures (N and M) on the same testchip, which is equal to the final platformheight minus the initial platform height,where the step is from the initial plat-form to the final platform and where X isthe capital letter associated with theinitial platform from test structurenumber N, and Y is the capital letter associated with the final platform fromtest structure number M. [SEMI MS2]

stepNXY = the average of the calibrated step height

measurements taken from multipledata traces on one step height test struc-ture, where N is the number associatedwith the test structure, X is the capitalletter associated with the initial plat-form (or “r” is used if it is the referenceplatform), Y is the capital letter associat-ed with the final platform (or “r” is used if it is the reference platform), and thestep is from the initial platform to thefinal platform. [SEMI MS2]

stepNXYt = a calibrated step height measurementfrom one data trace on one step height test structure, where N is the numberassociated with the test structure, X isthe capital letter associated with theinitial platform (or “r” is used if it is thereference platform), Y is the capitalletter associated with the final platform(or “r” is used if it is the reference plat-form), t is the data trace (“a,” “b,” “c,”etc.) being examined, and the step isfrom the initial platform to the finalplatform. [SEMI MS2]

For combined standard uncertainty calculations of stepheight measurements:

uc = the combined standard uncertainty value(i.e., the estimated standard deviation ofthe result). [SEMI MS2]

ucert = the component in the combined standard uncertainty calculation that is due to the uncertainty of the value of the physicalstep height used for calibration. [SEMIMS2]

ucSH = the combined standard uncertainty or astep height measurement.

udrift = the uncertainty of a measurement due tothe amount of drift during the datasession.

ulinear = the uncertainty of a measurement due tothe deviation from linearity of the datascan. [SEMI MS2]

uLplatNX = the component in the combined standard uncertainty calculation for platformheight measurements that is due to the measurement uncertainty across the length of platNX, where the length ismeasured perpendicular to the edge ofthe step. [SEMI MS2]

uLstep = the component in the combined standard

306

uncertainty calculation for step heightmeasurements that is due to themeasurement uncertainty of the stepheight across the length of the step, where the length is measured perpendi-cular to the edge of the step. [SEMIMS2]

uplatNX = the component in the combined standard uncertainty calculation or step height measurements obtained from two step height test structures that is due to theuncertainty of the platform heightmeasurement for platNX. [SEMI MS2]

urepeat = the uncertainty of a measurement due to the repeatability of a measurement.[SEMI MS2]

uWplatNX = the component in the combined standarduncertainty calculation for platformheight measurements that is due to the measurement uncertainty across the width of platNX, where the width ismeasured parallel to the edge of thestep. [SEMI MS2]

uWstep = the component in the combined standard uncertainty calculation for step heightmeasurements that is due to the meas-urement uncertainty of the step heightacross the width of the step, where thewidth is measured parallel to the edge ofthe step. [SEMI MS2]

For residual stress and stress gradient calculations:

ε r = residual strain of the thin film layer.[SEMI MS4]

σg = stress gradient of the thin film layer.[SEMI MS4]

σgmax = maximum stress gradient value as deter-mined in an uncertainty calculation.[SEMI MS4]

σgmin = minimum stress gradient value as deter-mined in an uncertainty calculation.[SEMI MS4]

σr = residual stress of the thin film layer.[SEMI MS4]

σrmax = maximum residual stress value as deter-mined in an uncertainty calculation.[SEMI MS4]

σrmin = minimum residual stress value as deter-mined in an uncertainty calculation.[SEMI MS4]

sg = strain gradient of the thin film layer.[SEMI MS4]

uε r(σ r) = component in the combined standard uncertainty calculation for residualstress that is due to the measurementuncertainty of ε r. [SEMI MS4]

ucε r = combined standard uncertainty value forresidual strain.

ucσ g = combined standard uncertainty value forstress gradient.

ucσ r = combined standard uncertainty value forresidual stress.

ucsg = combined standard uncertainty value forstrain gradient.

uE(σ g) = component in the combined standarduncertainty calculation for stress gradi-ent that is due to the measurement uncertainty of E. [SEMI MS4]

uE(σ r) = component in the combined standarduncertainty calculation for residualstress that is due to the measurement uncertainty of E. [SEMI MS4]

usg(σ g) = component in the combined standard uncertainty calculation for stress gradi-ent that is due to the measurement uncertainty of sg. [SEMI MS4]

For round robin measurements:

n = the number of reproducibility or repea-tability measurements.

ucave = the average combined standard uncer-tainty value for the reproducibility orrepeatability measurements. It is equal to the sum of the uc values divided by n.

307

1. Introduction

Microelectromechanical systems (MEMS) foundariesare emerging from the economic downturn in a strongposition [1]. Applications for MEMS demand robustreliability [2], which necessitates the consideration ofreliability at the earliest stages of product development[3]. The U.S. Measurement System (USMS) stated thatthere are insufficient measurement data and measure-ment methods to generate that data to adequatelycharacterize a MEMS device and its dependence onfabrication processes [4]. In response to this, SEMI haspublished two MEMS standard test methods; one formeasuring Young’s modulus [5] and one for measuringstep heights [6]. This paper presents the technical basisfor the standards and the data from a MEMS Young’smodulus and step height round robin experiment,which was used to validate these standards. Thesestandards are expected to facilitate commerce inMEMS technologies and improve manufacturing yieldsby decreasing interlaboratory differences in thesemeasurements.

The first MEMS standards were approved by ASTMin 2002 then validated in 2005 [7-9] with round robinprecision and bias data. These ASTM test methods areused for measuring in-plane lengths or deflections,residual strain, and strain gradient. They employ a non-contact measurement approach using an optical inter-ferometer. 3

The SEMI test method for measuring Young’smodulus that is reported in this paper uses an opticalvibrometer, stroboscopic interferometer, or comparableinstrument, and the SEMI test method for step heightmeasurements also reported in this paper uses anoptical interferometer or comparable instrument.Therefore, a stroboscopic interferometer can be usedfor all five standard test methods for MEMS (the threeASTM test methods and the two SEMI test methods).

Test structures for use with all five standard testmethods are incorporated on the MEMS Young’sModulus and Step Height Round Robin Test Chip,which is shown in Fig. 1. This chip is also the prototypefor a micro and nano technology (MNT) 5-in-1standard reference material (SRM) [10]. Using theMNT 5-in-1 SRM, companies will be able to comparetheir in-house measurements taken on the SRM withNIST measurements thereby validating their use of thedocumentary standards.

The first test method (SEMI MS4) [5] to be present-ed in this paper is on Young’s modulus measurementsof thin, reflecting films. As presented in [11], there aremany Young’s modulus methods with values that havebeen reported in the literature. The method reportedhere is based on the textbook treatment [12] of the aver-age resonance frequency of a single-layered cantileveroscillating out-of-plane (recommended and shown inFig. 2) or based on the average resonance frequency ofa single layered fixed-fixed beam oscillating out-of-plane. The resonance frequency method was chosendue to a) its wide acceptance in the community, b) theinvolvement of simple, non-contact, non-destructivemeasurements, and c) its ability to obtain measure-ments from test structures on a single test chip along-side other test structures. A piezoelectric transducer(PZT) is used to create the out-of-plane excitations;however, a PZT is not the only means available forexcitation (for example, thermal excitation [13-15] ispossible). Given the value for Young’s modulus,residual stress and stress gradient calculations arepossible.

Young’s modulus measurements are an aid in thedesign and fabrication of MEMS devices and inte-grated circuits (ICs). For example, high values ofresidual stress can lead to failure mechanisms inICs such as electromigration, stress migration, anddelamination. So, methods for its characterizationare of interest for IC process development and monitor-ing in order to improve the yield in complementarymetal oxide semiconductor (CMOS) fabricationprocesses [16].

The second test method (SEMI MS2) [6] to be pre-sented in this paper is for step height measurements ofthin films. A step height test structure, such as shown inFig. 3(a) with its cross section given in Fig. 3(b), isused for these measurements. Multiple 2-D data tracesalong the top of this test structure, such as traces “a,”“b,” and “c” in Fig. 3(a), are obtained. A sample 2-Ddata trace is given in Fig. 3(c). Data averages andstandard deviations from the 2-D data traces along thepertinent platforms are recorded.

For data obtained from one step height test structure,the difference in the platform heights involved in the stepfor each 2-D data trace is calculated. The step height,stepNXY , [or step1AB as shown in Fig. 3(b)] is the averageof these values from the different 2-D data traces.

Step height measurements can also be obtained fromtwo step height test structures. The calculations areslightly different, and this approach is not recommend-ed due to higher resulting combined standarduncertainty values [17].

308

3 In this paper, commercial equipment, instruments, or processesmay be identified. This does not imply recommendation or endorse-ment by NIST, nor does it imply that the equipment, instruments, orprocesses are the best available for the purpose.

309

Fig. 1. The MEMS Young’s Modulus and Step Height Round Robin Test Chip.

Fig. 2. For a cantilever test structure on the test chip shown in Fig. 1 (a) a design rendition and (b) a cross section. In Fig. 2(a), the centralstructure (silicon dioxide and silicon nitride atop the silicon substrate) will become an all oxide cantilever after the silicon nitride is removed andthe exposed raw silicon on three of the structures sides and beneath it is etched, as shown in Fig. 2(b). The dimensional markers are typically madeof polysilicon or metal encapsulated in oxide and can be used to obtain a more accurate measurement of the cantilever length after thepost-processing etch.

Step height measurements can be used to determinethin film thickness values [18,19]. Thickness measure-ments are an aid in the design and fabrication of MEMSdevices and can be used to obtain thin film materialsparameters, such as Young’s modulus [5,16].

The SEMI test methods for Young’s modulus and stepheight measurements were used in the MEMS Young’smodulus and step height round robin experiment todetermine the repeatability of a measurement as well asto see if independent laboratories could reproduce thesemeasurements without introducing a bias.

For the round robin, test chips were fabricated on thesame processing run. One test chip (the design of whichis shown in Fig. 1) was delivered to each laboratory,and measurements were taken using the procedures inthe SEMI test methods. The data from these measure-ments were analyzed using a NIST web-based program[10] that also verifies the data. Five laboratories(including NIST) participated in the round robin,resulting in eight independent data sets (due to differentequipment setups) for Young’s modulus and sevenindependent data sets (due to different equipment

310

Fig. 3. For the step height test structure, on the test chip shown in Fig. 1, with a 0°orientation (a) a design rendition showing 2-D data traces “a,” “b,” and “c,” (b) its crosssection, (where aa indicates active area, fox indicates field oxide, pl indicates polyl, m2indicates metal2, and gl indicates glass), and (c) a sample 2-D data trace.

setups) for step heights. The results from this roundrobin are reported in this paper.

Section 2 presents the packaged Round RobinTest Chip. Section 3 presents the technical basis forthe Young’s modulus and step height standard testmethods. Section 4 presents the uncertainty equations.And, Sec. 5 presents the results of the MEMS Young’smodulus and step height round robin experimentfollowed by the conclusions in Sec. 6. Instrumentspecifications are given in Appendix A (Sec. 7).Supplemental material for Young’s modulus measure-ments can be found in Appendix B (Sec. 8), and supple-mental material for step height measurements can befound in Appendix C (Sec. 9).

2. Packaged Round Robin Test Chip

The MEMS Round Robin Test Chip was fabricatedon a 1.5 μm CMOS process available through MOSIS[20]. The design for this test chip is depicted in Fig. 1.The design file (in GDS-II format) for this chip can bedownloaded from the NIST Semiconductor ElectronicsDivision (SED) MEMS Calculator website [10].

For the round robin chip design shown in Fig. 1, in anumber of places one mechanical layer is fabricatedinto suspended structures such as cantilevers andfixed-fixed beams. This layer consists of all oxide;namely, the field oxide, the deposited oxide before andafter the first metal (m1) deposition, and the glass (gl)layer. [The nitride cap (present atop the glass layerwhen the chips are received from MOSIS) wasremoved after fabrication using a CF4 + O2 etch beforea post-processing XeF2 etch that released the beams.]

As seen in Fig. 1, the test chip contains six groupingsof test structures with the following labels:

1. Young’s Modulus,2. Residual Strain,3. Strain Gradient,4. Step Height,5. In-Plane Length, and6. Certification Plus.

However, for the MEMS Young’s modulus andstep height round robin experiment, we were only con-cerned with the first and fourth groupings of test struc-tures, for Young’s modulus and step height measure-ments, respectively.

In the first grouping of test structures on the RoundRobin Test Chip shown in Fig. 1, Young’s modulusmeasurements were made. Cantilever and fixed-fixed

beam test structures were provided for this purpose with25 cantilevers grouped above 25 fixed-fixed beams onthis chip; however, for the round robin we were onlyconcerned with the cantilevers, such as that shown inFig. 2. Configurations for the cantilevers on the chipshown in Fig. 1 are given in Table 1. (See Sec. 3.1 for therationale for the chosen cantilever dimensions.)

As shown in Table 1, the cantilever design lengthsare 200 μm, 248 μm, 300 μm, 348 μm, and 400 μm.The length of the cantilever (in micrometers) is given atthe top of each column of cantilevers in Fig. 1 follow-ing the column number (i.e., 1 to 5). The beams aredesigned at only a 0° orientation.4 There are fivecantilevers designed at each length. Therefore, there are25 oxide cantilevers with a 0° orientation.

In the fourth grouping of test structures on the RoundRobin Test Chip shown in Fig. 1, step height measure-ments are made. Step height test structures, such asshown in Fig. 3, are provided for this purpose for ametal2-over-poly1 (m2-over-p1) step from active area(aa) to field oxide (fox). The surrounding referenceplatform consists of the deposited oxides and metal2over active area. The metal2 thickness is approximately1.0 μm.

There are four orientations (0°, 90°, 180°, and 270°)of the step height test structure shown in Fig. 3(a), andthese orientations are grouped in quads. For the RoundRobin Test Chip in Fig. 1 there are three quads, one ofwhich is shown in Fig. 4. The quad number is given inthe center of the quad. Therefore, quad “2” (out of 3) isshown in Fig. 4.

After the chips were received from MOSIS, theywere post-fabricated in a class 100 clean room (to remove the nitride cap with a CF4 + O2 etch and to

311

Table 1. Cantilever configurations for Young’s modulus measure-ments

Test Structure Cantilevers

Width (in μm) 28Length (in μm) 200, 248, 300, 348, and 400Orientation 0 °Mechanical Layer OxideQuantity of Beams 5 of each length (or 25 beams)

4 A 0° orientation implies that the length of the beam is parallel tothe x-axis of the test chip, the axes of which are shown in Fig. 1 andagain in Fig. 2, with the connection point of the cantilever havinga smaller x-value than the x-values associated with the suspendedportion of the cantilever.

release the beams with a XeF2 etch), packaged in alaboratory environment, then stored in a dust-free N2atmosphere.

Each round robin participant received a packagedRound Robin Test Chip, as shown in Fig. 5. The pack-aged part was put together in the following way:

1. Starting with a hybrid package with a pinarrangement similar to that shown in Fig. 5, thePZT was secured to the top of the chip cavityusing two thin layers of low stress, non-conduct-

ing epoxy. (The first layer of epoxy ensures thatthere will not be a conducting path between thepackage and the PZT. )

2. The PZT has the following properties:a. The dimensions of the PZT are approxi-

mately 5 mm by 5 mm and 2 mm in height,b. It is provided with a red and a black wire,c. It can achieve a 2.2 μm (± 20 %) displace-

ment at 100 V,d. It has an electrical capacitance of 250 nF

( ± 20 %), ande. It has a resonance frequency greater than

300 kHz, at which or above which it shallnot be operated because it could damage the PZT.

3. The two PZT wires were secured to their respec-tive package connections.

4. The Round Robin Test Chip was secured to thetop of the PZT using two thin layers of a lowstress non-conducting epoxy. (The first layer ofepoxy ensures that there will not be a conductingpath between the PZT and the test chip. )

5. The lid (or can) was placed on top of the pack-age to protect the chip, and the can was securedto the package with tape before shipment.

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Fig. 4. A design rendition of quad “2” from the step height groupingof test structures in Fig. 1.

Fig. 5. The packaged Round Robin Test Chip.

To take measurements on the Round Robin TestChip, the can is carefully removed. Young’s modulusand step height measurements can now be taken. ForYoung’s modulus measurements, to operate the PZT,the red wire is driven with a voltage that is positive rel-ative to the black wire. To ensure that you have suc-cessfully connected to the PZT, when activated at 10 Vand 7000 Hz, the resulting PZT vibration is barelyaudible.

3. Young’s Modulus and Step HeightMeasurements

The technical basis for the standard test method onYoung’s modulus measurements is given in Sec. 3.1,and the technical basis for the standard test method onstep height measurements is given in Sec. 3.2.

3. 1. Young’s Modulus Measurements

The Young’s modulus of a single layer is obtainedfrom resonance frequency measurements of a can-tilever comprised of that layer (such as shown in Fig. 2)or from resonance frequency measurements of a fixed-fixed beam comprised of that layer. To determine anestimate for the fundamental resonance frequency of acantilever, fcaninit , the following equation (as derived inSec. 8.1) is used:5

(1)

where Einit is the initial estimate for the Young’s modu-lus value of the thin film layer, t is the thickness, ρ isthe density, and Lcan is the suspended length of thecantilever.

Measurements are taken at frequencies whichencompass fcaninit , and an excitation-magnitude versusfrequency plot is obtained from which the resonancefrequency is found. For a given cantilever, three meas-urements of resonance frequency are obtained (namely,fmeas1, fmeas2, and fmeas3). If these are undamped measure-ments (e.g., if they are performed in a vacuum) theyare called fundamped1, fundamped2, and fundamped3, respectively.

If these are damped measurements they are calledfdamped1, fdamped2, and fdamped3, respectively. For eachdamped frequency (fdamped1, fdamped2, and fdamped3), a corre-sponding undamped frequency (fundamped1, fundamped2, andfundamped3, respectively) is calculated using the equationbelow:

(2)

where the n in the subscript of fdampedn and fundampedn is 1,2, or 3 and where Q is the oscillatory quality factor ofthe cantilever as given by the following equation [21]:

(3)

where Wcan is the suspended cantilever width and μ isthe viscosity (in air, μ = 1.84 × 10–5 Ns/m2 at 20 °C).

The cantilever dimensions for the Round Robin TestChip in Fig. 1, as specified in Table 1, were chosensuch that 5 μm ≤ Wcan ≤ 40 μm, Wcan > t, and Lcan >> twhere t = 2.743 μm, as determined by the electro-physical technique [18]. Data Sheet T.1 [10] can beused to calculate t. In addition, the cantilever dimen-sions were chosen to achieve a) an estimated reso-nance frequency between 10 kHz and 75 kHz usingEq. (1) with the assumptions that Einit = 70 GPa andρ = 2.2 g/cm3, b) a Q value above 30 using Eq. (3), andc) a value less than 2 % for pdiff as given by the follow-ing equation:

(4)

See Table 2 for the calculations of fcaninit, Q, and pdiff forthe chosen dimensions.

313

5 By inserting the inputs into the correct locations on the appropri-ate NIST Web page (http://www.eeel.nist.gov/812/test-structures/MEMSCalculator.htm), the calculations can be performed on-line ina matter of seconds.

2

4 ,38.330init

caninitcan

E tf

Lρ=

dampedundamped

2

,11

4

nn

ff

Q

=−

2

,24

can init

can

W E tQL

ρμ

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟

⎢ ⎥ ⎝ ⎠⎣ ⎦

damped2

undamped

11 100% 1 1 100% .4

ndiff

n

fp

f Q⎛ ⎞ ⎛ ⎞

= − = − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

Table 2. Calculations of fcaninit , Q, and pdiff for the designedcantilever lengths (Wcan = 28 μm)

Lcan fcaninit Q pdiff(μm) (kHz) (%)

200 62.5 148.0 0.0006248a 40.6 96.3 0.0013300 27.8 65.8 0.0029348a 20.6 48.9 0.0052400 15.6 37.0 0.0091

a These values were chosen in order to design on a 0.8 μm grid tosimplify the interface with MOSIS and the fabrication facility.

Also, to ensure that the resonance frequency of thecantilever is not altered by squeeze film or other damp-ing phenomena,6 the cantilever should be suspendedhigh enough above the underlying layer such that itsmotion is not altered by the underlying layer.7 In otherwords, the gap, d, between the suspended cantileverand the underlying layer should adhere to the followingequation [22]:

(5)

Therefore, if Wcan = 28 μm, the gap between thesuspended cantilever and the underlying layer shouldbe at least 9.3 μm.

The average undamped resonance frequency, fcan, iscalculated from the three undamped resonance frequen-cies using the following equation:

(6)

Given the measured value for fcan, the Young’s modulusvalue, E, is calculated as follows:

(7)

which assumes clamped-free boundary conditions andno undercutting of the beam. The derivation of thisequation is presented in Sec. 8.1. The combinedstandard uncertainty for E, or ucE , is given in Sec. 4.2.Residual stress and stress gradient equations for thisthin film layer can be found in Sec. 8.2. ConsultSec. 8.3 for Young’s modulus measurements obtainedfrom fixed-fixed beams.

3.2 Step Height Measurements

Step height measurements can be taken from eitherone or two step height test structures; however,measurements from one step height test structure are

recommended due to lower resulting values for thecombined standard uncertainty, ucSH [17].

If one step height test structure is used to obtain astep height measurement, three 2-D data traces [“a,”“b,” and “c,” as shown in Fig. 3(a)] are taken along thetop of the test structure, a cross section of which isgiven in Fig. 3(b). A sample 2-D data trace is shown inFig. 3(c). All height measurements are with respect tothe height of the surrounding reference platform that isused to level and zero the data. For generic test struc-ture “N” with platforms labelled “X” and “Y,” the indi-vidual platform height measurements (namely,platNXa, platNXb, platNXc, platNYa, platNYb, andplatNYc) and the standard deviations (σplatNXa, σplatNXb,σplatNXc , σplatNYa , σplatNYb , and σplatNYc) from the two plat-forms involved in the step in data traces “a,” “b,” and“c” are recorded.8 If the test structure in Fig. 3(a) iscalled test structure “1,” then for the step in test struc-ture “1” from platform “A” to platform “B,” the plat-form height measurements would be plat1Aa, plat1Ab,plat1Ac, plat1Ba, plat1Bb, and plat1Bc, and thestandard deviations would be σplat1Aa , σplat1Ab , σplat1Ac ,σplat1Ba , σplat1Bb , and σplat1Bc . Therefore, 12 measurementsare obtained (6 from platform “A” and 6 from platform“B”).

For each 2-D data trace, the difference in the plat-form heights involved in a step (in general, stepNXYt)9is calculated using the following equation:

(8)

where t is the data trace (“a,” “b,” “c,” etc.) beingexamined. For the step shown in Fig. 3(b) fromplatform “A” to platform “B,” the equations are:

(9)

(10)

and

(11)

314

6 Squeeze film and other damping phenomena routinely lead to amplitude dependent resonance frequencies and shifts in the natural frequencyof the system, which may limit the accuracy of the technique.7 The damping may not be present in bulk-micromachining processes because it is dependent upon the depth of the cavity and the vicinity of the

sides of the cavity to the beam. (The damping phenomena are expected to be present in surface micromachining processes without the use of abackside etch, unless the measurement is performed in a vacuum.)8 Consult the List of Symbols for the nomenclature used for platNXt and σplatNXt .9 Consult the List of Symbols.

.3canWd ≥

undamped1 undamped2 undamped3 .3can

f f ff

+ +=

2 4 2 42

4 2 2

38.33048 ,1.875

can can can canf L f LEt t

ρ ρπ= =

XYtstepN platNYt platNXt= −

ABa1 1Ba 1Aa ,step plat plat= −

ABb1 1Bb 1Ab ,step plat plat= −

ABc1 1Bc 1Ac .step plat plat= −

The step height, stepNXY, is the average of thevalues from the different 2-D data traces as givenbelow:

(12)

or for the step shown in Fig. 3(b), the step height,step1AB, is:

(13)

The combined standard uncertainty, ucSH, for stepNXYis given in Sec. 4.3. Consult Sec. 9. 1 for a step heightmeasurement obtained from two step height teststructures.

4. Uncertainty Equations

In this section, the equations used to determine thevalues of the combined standard uncertainty [17], uc ,are presented. Sec. 4.1 presents the basic combinedstandard uncertainty equation, and Sec. 4.2 and Sec 4.3present the more specific uncertainty equations forYoung’s modulus and step height measurements,respectively.

4.1 Combined Standard Uncertainty Equation

The combined standard uncertainty, uc , [17] is calcu-lated as the estimated standard deviation of the result. Itis equal to the square root of the sum of the squares ofthe uncertainty components where each componentmust have the same units as uc (e.g., GPa).

For the case of three sources of uncertainty, theuncertainty equation would be as follows:

(14)

where u1 is the uncertainty component due to the firstsource of uncertainty, u2 is the uncertainty componentdue to the second source of uncertainty, and u3 is due tothe third source of uncertainty. Additional terms may beadded under the square root sign for additional sourcesof uncertainty.

4.2 Young’s Modulus Uncertainty Equations

In this section, a combined standard uncertaintyequation is presented for use with Young’s modulusmeasurements (ucE) as obtained from resonance fre-quency measurements from a cantilever. For thesemeasurements, six sources of uncertainty are identifiedwith all other sources considered negligible. The sixsources of uncertainty are the uncertainty of the thick-ness (uthick), the uncertainty of the density (uρ), theuncertainty of the cantilever length (uL), the uncertain-ty of the average resonance frequency (ufreq), the uncer-tainty due to the frequency resolution (ufresol), and theuncertainty due to damping (udamp). As such, the com-bined standard uncertainty equation for ucE (as calculat-ed in SEMI Test Method MS4 [5] and in Data AnalysisSheet YM. 1 [10]) with six sources of uncertainty is asfollows:

(15)

In determining the combined standard uncertainty, aType B evaluation [17] (i.e., one that uses means otherthan the statistical Type A analysis) is used for eachsource of error. Table 3 gives sample values for each ofthese uncertainty components.

The uncertainty for uthick is determined from calculat-ed minimum and maximum Young’s modulus values(namely, Emin and Emax , respectively). It is derived usingthe extremes of values expected for the cantileverthickness as given below:

(16)

and

(17)

where σthick is the one sigma uncertainty of the value oft, which is found using the electro-physical technique[18]. Data Sheet T.1 [10] can be used to calculate σthick .With 99.7 % confidence, assuming a Gaussian distribu-tion (and assuming uρ , uL , ufreq , ufresol , and udamp equalzero), the value for E lies between Emin and Emax.Therefore, uthick is calculated as follows:

(18)

Volume 115, Number 5, September-October 20108Journal of Research of the National Institute of Standards and Technology

315

a b c ,3

XY XY XYXY

stepN stepN stepNstepN

+ +=

ABa ABb ABcAB .3

step1 step1 step1step1

+ +=

2 2 21 2 3cu u u u= + +

2 2 2 2 2 2 .cE thick L freq fresol dampu u u u u u uρ= + + + + +

( )2 4

min 2

38.3303

can can

thick

f LE

tρσ

=+

( )2 4

max 2

38.330,

3can can

thick

f LE

tρσ

=−

max min .6thick

E Eu

−=

In the same way, to determine the uncertainty com-ponent, uρ , the calculated minimum and maximumYoung’s modulus values (namely, Emin and Emax , respec-tively) are given below:

(19)

and

(20)

where σρ is the estimated one sigma uncertainty of thevalue of ρ. The uncertainty component, uρ , is thereforecalculated in the same way as before:

(21)

The uncertainty equation for uL is determined fromthe minimum and maximum Young’s modulus values(namely, Emin and Emax , respectively) as given below:

(22)

and

(23)

where σL is the estimated one sigma uncertainty of thevalue of Lcan . Therefore, uL is calculated as follows:

(24)

316

Table 3. Sample Young’s modulus uncertainty values (assuming Einit = 70 GPa)

uncertainty Type A or Type B value (in GPa)

uthick Type B 2.8(using t = 2.743 μm and σthick = 0.058 μm)

uρ Type B 1.5(using ρ = 2.2 g / cm3 and σρ = 0.05 g/cm

3)

uL Type B 0.17(using Lcan = 300 μm and σL = 0.2 μm)

ufreq Type B 0.027(using fmeas1 = 26.82625 kHz,fmeas2 = 26.8351 kHz,and fmeas3 = 26.8251 kHz)

ufresol Type B 0.0018(using fresol = 1.25 Hz)

udamp Type B 0.0004(using Wcan = 28 μm and σW = 0.1 μmand using μ = 1.84 × 10–5 Ns / m2 and σμ = 0.01 × 10

–5 Ns / m2)

ucE 3.2

3ucEa 9.5

aThis 3ucE uncertainty is plotted in Fig. 6 with the repeatability data point corresponding to the first cantilever with length of 300 μm.

2 2 2 2 2 2thick freq fresol damp= Lu u u u u uρ+ + + + +

( ) 2 4min 2

38.330 3 can canf LEt

ρρ σ−=

( ) 2 4max 2

38.330 3,can can

f LE

tρρ σ+=

max min .6

E Euρ

−=

( )42min 2

38.330 3can can Lf LEt

ρ σ−=

( )42max 2

38.330 3,can can L

f LE

tρ σ+

=

max min .6L

E Eu

−=

The uncertainty equation for ufreq is determined fromthe minimum and maximum Young’s modulus values(namely, Emin and Emax , respectively) as given below:

(25)

and

(26)

where σfreq is the one sigma uncertainty of the value offcan . Therefore, ufreq is calculated as follows:

(27)

The uncertainty equation for ufresol is determined fromthe minimum and maximum Young’s modulus values(namely, Emin and Emax , respectively) as given below:

(28)

and

(29)

where fresol is the frequency resolution for the given setof measurement conditions. Assuming a uniform (i.e.,rectangular) probability distribution (and assuminguthick , uρ , uL , ufreq , and udamp equal zero), the value forE lies between Emin and Emax . Therefore, ufresol is calcu-lated as follows:

(30)

If undamped resonance frequencies (e.g., if themeasurements were performed in a vacuum) wererecorded as fmeas1 , fmeas2 , and fmeas3 , then udamp is set equalto 0.0 Pa. For damped resonance frequencies (i.e., iffmeas1 , fmeas2 , and fmeas3 were damped measurements), theuncertainty equation for udamp is determined from the

minimum and maximum Young’s modulus values(namely, Emin and Emax , respectively) as given below:

(31)

and

(32)

where the calculation for σfQ is given in Sec. 8.6.Therefore, udamp is calculated as follows:

(33)

4.3. Step Height Uncertainty Equations

In this section, a combined standard uncertaintyequation is presented for use with step height measure-ments (ucSH). Six sources of uncertainty are identifiedwith all other sources of uncertainty considerednegligible. The six sources of uncertainty are the uncer-tainty of the measurement across the length of the step(uLstep) where the length is measured perpendicular tothe edge of the step, the uncertainty of the measurementacross the width of the step (uWstep) where the width ismeasured parallel to the edge of the step, the uncertain-ty of the value of the physical step height used forcalibration (ucert ), the uncertainty of the measurementdue to the repeatability (urepeat ), the uncertainty due tothe amount of drift during the data session (udrift ), andthe uncertainty of a measurement due to the deviationfrom height linearity of the data scan (ulinear ). As such,the combined standard uncertainty equation (as calcu-lated in SEMI Test Method MS2 [6] and in DataAnalysis Sheet SH.1 [10]) with six sources of uncer-tainty is as follows:

(34)

In determining the combined standard uncertainty, aType B evaluation [17] (i.e., one that uses means otherthan the statistical Type A analysis) is used for each TypeB source of error, except where noted. Table 4 givessample values for each of these uncertainty components.

317

( )2 4min 2

38.330 3can freq canf LEt

ρ σ−=

( )2 4max 2

38.330 3,can freq can

f LE

t

ρ σ+=

max min .6freq

E Eu

−=

24

min 2

38.3302

resolcan can

ff LE

t

ρ ⎛ ⎞−⎜ ⎟⎝ ⎠=

24

max 2

38.3302 ,

resolcan can

ff LE

t

ρ ⎛ ⎞+⎜ ⎟⎝ ⎠=

max min .2 3fresol

E Eu

−=

( )2 4min 2

38.330 3can fQ canf LEt

ρ σ−=

( )2 4max 2

38.330 3,can fQ can

f LE

t

ρ σ+=

max min .6damp

E Eu

−=

2 2 2 2 2 2cSH step step repeat drif linear .L W cert tu u u u u u u= + + + + +

The uncertainty equation for uLstep assuming aGaussian distribution is as follows:

(35)

where σroughNX and σroughNY are the surface roughnessesof platNX and platNY, respectively, and are calculatedfrom the smallest of all the calibrated values obtainedfor σplatNXt and σplatNYt , respectively. However, if thesurfaces of platNX (defined in the List of Symbols),platNY, and platNr all have identical compositions,then σroughNX equals σroughNY, which equals the smallestof all the values obtained for σplatNXt , σplatNYt ,

and σplatN rDt .10 Also, in the above equation, σplatNXave andσplatNYave are calculated using the following equations:

(36)

and

(37)

The derivation of uLstep is given in Sec. 9.3.

318

uLstep Type B 0.011(using σplatNXave = 0.0118 μm,σplatNYave = 0.0102 μm,and σroughNX =σroughNY = 0.0036 μm)

uWstep Type A 0.0073(using stepNXYa = 0.4928 μm,stepNXYb = 0.4814 μm,and stepNXYc = 0.4949 μm)

ucert Type B 0.0041(using cert = 9.887 μm,σcert = 0.083 μm,and stepNXY = 0.490 μm)

urepeat Type B 0.00034(using zrepeat = 0.024 μmand Z–6 = 9.879 μm)

udrift Type B 0.00023(using zdrift = 0.016 μmand Z– = 9.887 μm)

ulinear Type B 0.0028(using zperc = 1.0 %)

ucSH0.014

3ucSHa 0.041

aThis 3ucSH uncertainty is associated with the first repeatability data point plotted in Fig. 7 (for TS1 in quad 1).

uncertainty Type A or Type B value (in μm)

2 2 2 2 2 2repeat drift linearLstep Wstep certu u u u u u= + + + + +

Table 4. Sample step height uncertainty values

( ) ( )2 2step ave rough ave rough ,L platNX NX platNY NYu σ σ σ σ= − + −

10 Consult platNrDt in the List of Symbols for the nomenclatureused in the subscript of σplatNrDt .

a b cave 3

platNX platNX platNXplatNX

σ σ σσ

+ +=

a b cave .3

platNY platNY platNYplatNY

σ σ σσ

+ +=

The uncertainty equation for uWstep is determinedfrom σWstep , the one sigma standard deviation of the stepheight measurements stepNXYa , stepNXYb, and stepNXYc ,using the following equation:

(38)

This is a statistical Type A component.The uncertainty equation for ucert is determined

from the minimum and maximum step height values(namely, stepNXYmin and stepNXYmax , respectively). Theuncertainty of the measured step height is assumed toscale linearly with height. With 99.7 % confidence,assuming a Gaussian distribution (and assuming uLstep ,uWstep , urepeat , udrift , and ulinear equal zero), the value for|stepNXY| lies between stepNXYmin and stepNXYmax . Assuch, ucert can be calculated using the following equa-tion:

(39)

where cert is the certified value of the double-sidedphysical step height used for calibration and σcert is theone sigma uncertainty of the calibrated physical stepheight.

The uncertainty equation for urepeat is determinedfrom the minimum and maximum step height values(namely, stepNXYmin and stepNXYmax , respectively) asgiven below:

(40)

and

(41)

where zrepeat is the maximum of two calibrated values;one of which is the positive calibrated differencebetween the minimum and maximum values of the sixcalibration measurements taken at a single location onthe calibration step before the data session and the otherof which is the positive calibrated difference betweenthe minimum and maximum values of the six calibra-tion measurements taken at the same location on thecalibration step after the data session and where z–6 is thecalibrated average of the six calibration measurementsfrom which zrepeat is found. The uncertainty of the meas-ured step height is assumed to scale linearly withheight. Assuming a uniform distribution (and assuming

uLstep , uWstep , ucert , udrift , and ulinear equal zero), the valuefor |stepNXY| lies between stepNXYmin and stepNXYmax .Therefore, urepeat is calculated as follows:

(42)

which simplifies to the following equation:

(43)

The uncertainty equation for udrift is determined fromthe minimum and maximum step height values (name-ly, stepNXYmin and stepNXYmax , respectively) as givenbelow:

(44)

and

(45)

where zdrift is the calibrated positive difference betweenthe averages of the 6 calibration measurements takenbefore and after the data session (at the same locationon the physical step height used for calibration) andwhere z– is the calibrated average of all 12 calibrationmeasurements. The uncertainty of the measured stepheight is assumed to scale linearly with height.Assuming a uniform distribution (and assuming uLstep ,uWstep , ucert , urepeat , and u linear equal zero), the value for|stepNXY| lies between stepNXYmin and stepNXYmax .Therefore, udrift is calculated as follows:

(46)

which simplifies to the following equation:

(47)

The uncertainty equation for ulinear is calculatedfrom the minimum and maximum step height values(namely, stepNXYmin and stepNXYmax , respectively) asgiven:

319

.Wstep Wstepu σ=

certcert XYu stepNcert

σ=

repeatmin

62XY XY XY

zstepN stepN stepN

z= −

repeatmax

6

,2XY XY XY

zstepN stepN stepN

z= +

max min ,2 3

XY XYrepeat

stepN stepNu

−=

6

.2 3

repeatrepeat XY

zu stepN

z=

driftmin 2XY XY XY

zstepN stepN stepN

z= −

drifmax ,2

tXY XY XY

zstepN stepN stepN

z= +

max min ,2 3

XY XYdrift

stepN stepNu

−=

.2 3

driftdrift XY

zu stepN

z=

(48)

and

(49)

where zperc is the maximum percent deviation fromlinearity over the instrument’s total scan range, asquoted by the instrument manufacturer. The uncertain-ty of the measured step height is assumed to scalelinearly with height. Assuming a uniform distribution,ulinear can be calculated using the following equation:

(50)

Consult Sec. 9.2 for the uncertainty calculationswhen two step height test structures are used for a stepheight measurement.

5. Round Robin Results

The round robin repeatability and reproducibilityresults are given in Sec. 5.1 for Young’s modulusmeasurements and in Sec. 5.2 for step height measure-ments.

The repeatability measurements are performed usingthe same test method, in the same laboratory (NIST),by the same operator, with the same equipment, in theshortest practicable period of time. These measure-ments are done on random test structures.

For the reproducibility measurements, at least sixindependent data sets (each using a different piece ofequipment or equipment setup) must be obtainedfollowing the same test method before the results canbe recorded in the precision and bias statement of aSEMI standard test method. These measurements aredone on random test structures, as described below.

5.1 Young’s Modulus Round Robin Results

For the Young’s modulus portion of the MEMSYoung’s modulus and step height round robin experi-ment, both repeatability and reproducibility data weretaken.

The repeatability data were taken at one laboratoryusing a dual beam vibrometer (see Sec. 7.1 for specifics

associated with the vibrometer). Young’s modulusvalues were found from 12 different cantilevers4 times, with each Young’s modulus value determinedfrom the average of 3 resonance frequency measure-ments. Therefore, 48 Young’s modulus values wereobtained. Of these values, 16 were from 4 differentcantilevers with L = 200 μm, 16 from 4 differentcantilevers with L = 300 μm, and 16 from 4 differentcantilevers with L = 400 μm.

For the reproducibility data, 8 participants wereidentified.11 Each participant was supplied with aRound Robin Test Chip and asked to obtain a Young’smodulus value from 3 oxide cantilevers with designlengths of 200 μm, 300 μm, and 400 μm. (The partici-pant could choose to measure any one of five can-tilevers of the given length that were available on thetest chip as long as it passed a visual inspection.) EachYoung's modulus value was determined from the aver-age of three resonance frequency measurements fromthe cantilever as specified in Sec. 3.1, using an instru-ment that meets the manufacturer’s alignment andcalibration criteria. Following SEMI MS4 for Young’smodulus measurements, the measurements wererecorded on Data Analysis Sheet YM.1 [10].

The eight participants used a variety of instruments(consult Sec. 7.1 for details associated with the instru-ments) to obtain Young’s modulus. These included asingle beam vibrometer, a dual beam vibrometer, and astroboscopic interferometer. In addition, thermal exci-tation measurements are included for comparison withPZT excitation measurements on the same chip.

The repeatability and reproducibility data forYoung’s modulus and for the combined standard uncer-tainty is presented in Table 5 and Table 6, respectively,where n indicates the number of calculated Young’smodulus values. The average of the repeatability orreproducibility data (namely Eave) is listed next,followed by the 95 % limits for E that is calculated asfollows: (a) the standard deviations were found,(b) these values were multiplied by 2.0 (assuming aGaussian distribution) [17], and (c) the resulting valueswere reported as percents. Below this, the average ofthe combined standard uncertainty [17] values (ucave)and the 95 % limits for ucE are presented.

320

minXY XY XY percstepN stepN stepN z= −

max ,XY XY XY percstepN stepN stepN z= +

.3

perclinear XY

zu stepN=

11 The term participant refers to a single data set from a unique com-bination of measurement setup and researcher. That is, a singleresearcher equipped with multiple, unique instruments (e.g., a dualbeam vibrometer and a single beam vibrometer) or different forms ofexcitation (e.g., PZT and thermal excitation) could serve as multiple“participants” in this round robin.

The Young’s modulus round robin results are plottedin Fig. 6, where both the repeatability and repro-ducibility data are plotted. The average Young’smodulus value for the repeatability data is specified atthe top of Fig. 6 along with the average 3ucE uncertain-ty bars for this value.12 These quantities are plotted inthis figure with both the repeatability and reproducibil-ity data. As an observation, all of the reproducibilityresults fall comfortably between the repeatabilitybounds of Eave plus or minus 3ucave .

In Fig. 6, the repeatability data are grouped accord-ing to the cantilever length with the L = 200 μm dataplotted first, followed by the L = 300 μm data, then theL = 400 μm data. In like manner with the reproducibil-ity data, for each participant, the L = 200 μm data areplotted first, followed by the L = 300 μm data, then theL = 400 μm data. The repeatability data and the repro-ducibility data both indicate a length dependency. Therepeatability data in Fig. 6 show a clustering of the dataat each length. In other words, the 95 % limits for E ateach length (which are plotted in this figure along withEave for each length) are all less than 1.5 %, which ismuch less than the 10.3 % value (as given in Table 5)when all the lengths are considered. This suggests thatwhen Young’s modulus values extracted by differentmeasurement instruments or excitation methods arecompared, the cantilevers should have the same length.

321

Table 5. Young’s modulus repeatability data (1 participant, 1 laboratory, 1 instrument, 1 chip, 12 differentcantilevers)

200 μm 300 μm 400 μm 200 μm to 400 μmlength length length lengths

n 16 16 16 48

Eave (in GPa) 59.8 65.4 67.5 64.2

95 % limits for E ± 1.4 % ± 0.5 % ± 1.1 % ± 10.3 %

ucave (in GPa) 2.9 3.2 3.3 3.1(4.9 %) (4.8 %) (4.8 %) (4.8 %)

95 % limits for ucE ± 1.4 % ± 0.5 % ± 1.1 % ± 10.1 %

Table 6. Young’s modulus reproducibility data (eight participants, five laboratories, seven instruments, fourchips)

n 8 8 8 24

Eave (in GPa) 58.7 63.7 66.0 62.8

95 % limits for E ± 4.4 % ± 5.5 % ± 4.4 % ± 11.0 %

ucave (in GPa) 2.8 3.1 3.2 3.0(4.9 %) (4.8 %) (4.9 %) (4.9 %)

95 % limits for ucE ± 4.4 % ± 5.5 % ± 4.5 % ± 11.0 %

200 μm 300 μm 400 μm 200 μm to 400 μmlength length length lengths

12 Table 3 specifies the value of each of the uncertainty componentscomprising the 3ucE uncertainty bars for the repeatability data pointcorresponding to the first cantilever in Fig. 6 with a length of300 μm.

This length dependency can be due to a number ofthings including debris in the attachment corners of thecantilevers to the beam support, which would causelarger errors for shorter length cantilevers. This can bea topic for future investigation where a) the physicalform and chemical composition of the cantilever ischecked to see if it matches the assumptions used inthe calculations and b) finite element methods areused to determine if the length dependency is dueto the attachment conditions. Therefore, at this point,we can only state that, given the existing canti-levers, we can only report an “effective” value forYoung’s modulus.

Round robin participant #1, participant #2, andparticipant #3 took measurements on the same chip(chip #1) using a dual beam vibrometer, a single beamvibrometer, and a stroboscopic interferometer, respec-tively. The results given in Fig. 6 indicate that compa-rable results were obtained from these instruments.

Round robin participant #4, participant #5, andparticipant #6 took data from the same chip (chip #2);however, round robin participant #5 used thermal

excitation to obtain the required data while participant#4 and participant #6 used PZT excitation. No signifi-cant difference in the results for these measurements isseen in Fig. 6.

No information can be presented on the bias of theprocedure in the test method for measuring Young’smodulus because there is not a certified MEMS materi-al for this purpose. Many values for Young’s modulusfor various materials have been published with anattempt to consolidate this information in [23]. For asilicon dioxide film, the Young’s modulus valuesreported in [23] range from 46 GPa to 92 GPa. Theaverage repeatability value reported in Table 5 of64.2 GPa falls comfortably within this range.

The Young’s modulus results are reported as follows[17]: Since it can be assumed that the possible estimat-ed values are either approximately uniformly distrib-uted or Gaussian (as specified in Sec. 4.2) with approx-imate standard deviation ucE , the Young’s modulusvalue is believed to lie in the interval E ± ucE with alevel of confidence of approximately 68 % assuming aGaussian distribution.

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Fig. 6. Young’s modulus round robin results.

5.2 Step Height Round Robin Results

For the step height portion of the MEMS Young’smodulus and step height round robin experiment, bothrepeatability and reproducibility data were taken.

The repeatability data were taken at one laboratoryusing a stroboscopic interferometer operated in thestatic mode (see Appendix A). Four step height meas-urements were taken from each of the four test struc-tures in each of the three quads shown in Fig. 1.Therefore, 48 step height measurements were obtainedwith 1 step height measurement defined as the averageof 3 measurements taken from different positionssomewhat evenly spaced along the step as specified inSec. 3.2.

For the reproducibility data, seven participants wereidentified. Each participant was supplied with a RoundRobin Test Chip and asked to obtain the step heightfrom any two test structures in the first of the threequads of step height test structures. Following SEMIMS2 [6] for step height measurements, the raw, uncal-ibrated measurements were recorded on Data AnalysisSheet SH.1 [10].

The repeatability and reproducibility data for stepheight and for the combined standard uncertainty ispresented in Table 7, where n indicates the number ofstep height measurements. The average of the repeat-ability or reproducibility data (namely |stepNABave|) islisted next followed by the 95 % limits for |stepNAB| thatis calculated as follows: (a) the standard deviationswere found, (b) these values were multiplied by 2.0(assuming a Gaussian distribution) [17], and (c) theresulting values were reported as percents. Below this,the average of the combined standard uncertainty [17]values (ucave) and the 95 % limits for ucSH arepresented. (It is interesting in comparing the 95 %

limits for |stepNAB| that the repeatability limits arelarger than the reproducibility limits. The reason forthis anomaly is not known.)

The step height round robin results are plotted inFig. 7 and Fig. 8. In each of these figures, the repeata-bility data are plotted first, followed by the results fromthe seven participants.13 The absolute value of the aver-age step height for the repeatability data is specified at the top of Fig. 7 and Fig. 8 along with the 3ucave uncer-tainty bars for this value, as obtained or derived fromTable 7.14 These quantities are plotted in each figurewith both the repeatability and reproducibility data. Asan observation, all of the reproducibility results fallcomfortably between |stepNABave| plus or minus 3ucave asobtained from the repeatability results.

Figure 7 groups the repeatability results by quadnumber with the results from quad “1” plotted first, fol-lowed by the results from quad “2,” then the resultsfrom quad “3.” The results within each quad aregrouped according to test structure number15 with theresults from test structure “1” plotted first, followed bythe results from test structure “2,” etc. The average stepheight value and the 95 % limits for this value foreach quad are given at the bottom of Fig. 7 and also inTable 8. These results reveal comparable values for thestep height measurements and comparable values forthe 95 % limits. This implies there are no discernablevariations in the step height value between neighboringquads.

Figure 8 groups the repeatability results by test struc-ture number with the results from test structure “1”(TS1) plotted first, followed by the results from teststructure “2” (TS2), followed by the results from teststructure “3” (TS3), then test structure “4” (TS4). Theresults for each test structure number are groupedaccording to quad with the results from quad “1”

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13 Participant #2 provided stylus profilometer results, and thatinstrument did not provide standard deviation values; however it didprovide average roughness values. Although this test method usesstandard deviation values, in this analysis for that laboratory averageroughness values were inserted into the data sheet for analysis asopposed to standard deviation values.14 Table 4 specifies the value of each of the uncertainty componentscomprising the 3ucSH uncertainty bars for the first repeatability datapoint plotted in Fig. 7 (for TS1 in quad 1).15 The upper left hand step height test structure in a quad, such asshown in Fig. 4, is called test structure “1” and it has a 0° orientation.Test structure “2” (the upper right test structure) has a 270° orienta-tion. Test structure “3” (the bottom right test structure) has a180° orientation. And test structure “4” (the bottom left test structure)has a 90° orientation.

Table 7. Step height measurement results

Repeatability Reproducibilityresults results

n 48 14

|stepNABave| (in μm) 0.477 0.481

95 % limits for |stepNAB| ± 7.9 % ± 6.2 %

ucave (in μm) 0.014 0.014(3.0 %) (3.0 %)

95 % limits for ucSH ± 70 % ± 80 %

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Fig. 7. Step height round robin results with the repeatability results grouped according to quad

Fig. 8. Step height round robin results with the repeatability results grouped according to test structure number.

plotted first, followed by the results from quad “2,”then the results from quad “3.” As in Fig. 7, the averagestep height value and the 95 % limits for this value foreach test structure are given at the bottom of Fig. 8 andalso in Table 9. These results reveal that TS1 and TS3(which are rotated ± 90° with respect to TS2 and TS4)have comparable 95 % limits as do TS2 and TS4;however the 95 % limits for TS1 and TS3 are slightlyless than the 95 % limits for TS2 and TS4 when theyshould be comparable. There are also more variationsin the average step height value between rotated teststructures (as shown in Fig. 8 and Table 9) than varia-tions in this value between quads (as shown in Fig. 7and Table 8).

The platform surfaces involved in the step were notideal surfaces. Oftentimes they were tilted (even thoughthe data were leveled with respect to the reference

platform) and the data jagged. Therefore, the preciseselection of the analysis regions (including the numberof data points within these regions) affects the standarddeviations obtained. An averaging capability incorpo-rated in most non-contact instruments can have theeffect of smoothing the data; however, a more compre-hensive determination of the length and width varia-tions may be necessary when dealing with tilt.Repeatability might also be improved by calculatingthe step height from fitted straight lines as described forNIST step height calibrations [24] and outlined inASTM E2530 [25]. As given in [24], “For step heightmeasurements, one of several algorithms may be used.For single-sided steps, a straight line is fitted by themethod of least squares to each side of the step transi-tion, and the height is calculated from the relative posi-tion of these two lines extrapolated to the step edge.”

The interferometer or comparable instrument is cali-brated in the out-of-plane z-direction. If the calibrationis not done, a bias to the measurements is expected. Thedirection and degree of the resulting bias is different foreach magnification of each instrument. As such, cali-bration of the interferometer or comparable instrumentis considered mandatory for step height measurements.

The step height results are reported as follows [17]:Since it can be assumed that the possible estimatedvalues are either approximately uniformly distributedor Gaussian (as specified in Sec. 4.3) with approximatestandard deviation ucSH, the step height is believedto lie in the interval stepNXY ± ucSH with a level of con-fidence of approximately 68 % assuming a Gaussiandistribution.

6. Conclusions

The technical basis for the two SEMI standard testmethods on Young’s modulus measurements [5] andstep height measurements [6] was presented, alongwith the data obtained from the MEMS Young’s modu-lus and step height round robin experiment. These datawere incorporated into the precision and bias state-ments for the applicable SEMI standards.

For Young’s modulus measurements, data obtainedfrom a single beam vibrometer, a dual beam vibrome-ter, and a stroboscopic interferometer yielded compara-ble results. Also, PZT excitation and thermal excitationyielded comparable results. The repeatability data andthe reproducibility data both indicate a length depend-ency. In other words, the 95 % limits for E at eachlength are all less than ±1.5 %, which is much less thanthe ±10.3 % value when all the lengths from 200 μm to

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Table 8. Step height repeatability data grouped by quad

Q1 Q2 Q3

n 16 16 16

|stepNABave| 0.479 0.473 0.478(in μm)

95 % limits for ± 8.4 % ± 7.2 % ± 8.3 %|stepNAB|

ucave (in μm) 0.015 0.015 0.013(3.1 %) (3.2 %) (2.8 %)

95 % limits for ± 200 % ± 200 % ± 200 %ucSH

Table 9. Step height repeatability data grouped by test structure

TS1 TS2 TS3 TS4

n 12 12 12 12

|stepNABave| 0.486 0.469 0.474 0.478(in μm)

95 % limits ± 6.4 % ± 8.7 % ± 7.0 % ± 8.5 %for |stepNAB|

ucave (in μm) 0.014 0.015 0.011 0.018(2.8 %) (3.1 %) (2.4 %) (3.7 %)

95 % limits ± 28.3 % ± 71.6 % ± 27.7 % ± 81.2 %for ucSH

400 μm are considered. This could be due to debris inthe attachment corners of the cantilevers to the beamsupport that would cause larger errors for shorter lengthcantilevers. Additional research can ascertain if thephysical form and chemical composition of thecantilever matches the assumptions used in the calcula-tions and if the length dependency is due to the attach-ment conditions. Therefore, at this point, we can onlystate that, given the existing cantilevers, we can onlyreport an “effective” value for Young’s modulus. Also,when Young’s modulus values extracted by differentmeasurement instruments or excitation methods arecompared, the cantilevers should have the same length.

For step height measurements, it is currently notunderstood why the repeatability limits (± 7.9 %) arelarger than the reproducibility limits (± 6.2 %). Theplatform surfaces involved in the step are not idealsurfaces, such that the precise selection of the analysisregions (including the number of data points withinthese regions) affects the standard deviations obtained.Also, the repeatability data indicate there are nodiscernable variations in the step height value betweenneighboring quads. And, there are more variations inthe average step height value between rotated teststructures than variations in this value between quads.For one-sided steps, the large uncertainties can beimproved by calculating the step height from fittedstraight lines (using the method of least squares) withthe height calculated from the relative position of thetwo lines extrapolated to the step edge. This approachis typically used for step height measurements andcalibrations [24,25].

It is expected that these standards will be instrumen-tal in reducing the interlaboratory differences in theparametric measurements. The following guidelinesshould be of value to the MEMS industry, whencommunicating data obtained using these SEMIstandard test methods:

(a) To record the data and perform the calculations,use the data analysis sheets which are accessiblevia the NIST SED MEMS Calculator website[10],

(b) Make sure the data have passed the verificationcheck given at the bottom of the data analysissheets, and

(c) Ask if all the verification checks were passed,when communicating with others concerningSEMI standard data.

Consult Ref. [10] for information associated with theMNT 5-in-1 SRM that enables companies to comparetheir in-house measurements taken on this SRM withNIST measurements thereby validating their use of thedocumentary standards.

7. Appendix A.Instrument Specifications

An optical vibrometer, stroboscopic interferometer,or comparable instrument is required for Young’smodulus measurements, as specified in Sec. 7.1. Forstep height measurements, an optical interferometer orcomparable instrument is used as specified in Sec. 7.2.

7.1 Instrument Specifications for Young’sModulus Measurements

For Young’s modulus measurements, an opticalvibrometer, stroboscopic interferometer, or an instru-ment comparable to one of these is required that iscapable of non-contact measurements of surfacemotion. This section briefly describes the operation andspecifications for a typical single beam laser vibro-meter, a dual beam laser vibrometer, and a stroboscop-ic interferometer. The specifications can be applied tocomparable instruments, if appropriate.

For a single beam laser vibrometer, a typicalschematic is given in Fig. 9. A signal generator pro-vides an excitation signal, which excites the sample viaa piezoelectric transducer (PZT). The measurementbeam is positioned to a scan point on the sample (bymeans of mirrors) and is reflected back. The reflectedlaser light interferes with the reference beam at thebeam splitter (BS). A photodetector (PD) records theinterference signal, converting it into an electricalsignal. The frequency difference between the beams isproportional to the instantaneous velocity of the vibra-tion parallel to the measurement beam. (The Bragg cellis instrumental in determining the sign of the velocity.)The velocity decoder provides a voltage proportional tothe instantaneous velocity.

A dual beam laser vibrometer incorporates twobeams. The measurement beam is positioned to a scanpoint on the sample (for example, positioned near thetip of a cantilever). The reference beam emanates fromthe beam splitter (BS) shown in Fig. 9 and is positionedto a second point on the sample (for example, posi-tioned on the support region at the base of the can-tilever). The two scattered beams optically combine at

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the beam splitter where the reference beam is used todirectly eliminate any movement of the sample alsoexperienced by the measurement beam.

For a stroboscopioc interferometer, a simplifiedschematic of a typical setup is shown in Fig. 10(a).When operated in the static mode, the interferometer isused to determine surface profiles. The incident lighttravels through the microscope objective to the beamsplitter. Half of the light travels to the sample surfaceand then back to the beam splitter. The other half isreflected to a reference surface and then back to thebeam splitter. These two paths of light recombine at thebeam splitter to form interference light fringes. As theinterferometer scans downward, an intensity envelope

incorporating these fringes is determined by the soft-ware [see Fig. 10(b)]. The peak contrast of the fringes, phase, or both are used in determining the sampleheight at that pixel location. The surface profile isfound by collecting sample height data for each pixelwithin the field of view (FOV). When operated in thedynamic mode, the incident light is strobed at the samefrequency as that used to actuate the device. Thesample is actuated, for example, after securing it to thetop of a PZT. The phase, frequency, and drive signal tothe strobe and PZT are varied, performing a downwardscan as is done for static devices at each combination toobtain successive 3D images as the sample cyclesthrough its range of motion.

327

Fig. 9. Schematic of a typical setup for a single beam laser vibrometer. (PBS indicates a polarizing beamsplitter, BS indicates a beam splitter, P indicates a prism, and PD indicates a photodetector.)

Fig. 10. For a typical stroboscopic interferometer (a) a schematic and (b) an intensity envelope used to obtaina pixel’s sample height.

Specifications for the (previous) above instruments or acomparable instrument are as follows:

1. The microscope objective or objectives shouldbe chosen to allow for sufficient resolution of thecantilever or fixed-fixed beam and a portion of thesurrounding sample. The objective(s) should have aFOV that can encompass at least half of the length ofthe cantilever or fixed-fixed beam being measured.Typically, a 4× and a 20× objective will suffice.

2. The signal generator should be able to produce awaveform function (such as a periodic chirp function16or a sine wave function17) if applicable, such that fromits use, a reproducible resonance frequency can beobtained and good 3-D oscillating images can beobtained such that it is obvious by inspection that thebeam is in resonance.

3. The instrument shall be capable of producing amagnitude versus frequency plot from which theresonance frequency can be obtained.

4. The instrument should be capable of obtaining 3-Dimages of beam oscillations in order to ascertain if thecorrect frequency peak has been chosen as the beam’sresonance frequency.

5. An estimate for the maximum frequency of theinstrument needed for a resonating cantilever, fcaninit , isat least the value calculated using the followingequation:18

(51)

as derived in Sec. 8.1. An estimate for the maximumfrequency of the instrument needed for a resonatingfixed-fixed beam, fffbinithi , is at least the value calculatedusing the following equation:

(52)

as derived in Sec. 8.5.

6. An instrument that can make differential measure-ments (e.g., with the use of two laser beams) is recom-mended for use with fixed-fixed beams. It is alsorecommended for use with cantilevers, especially forestimated resonance frequencies less than 10 kHz andalso if the value for pdiff as calculated in the followingequation is greater than or equal to 2 %:

(53)

For a cantilever, the Q-factor, Q, in the above equationcan be estimated using the following equation:

(54)

7.2 Instrument Specifications for Step HeightMeasurements

For step height measurements, an optical interfero-metric microscope or comparable instrument is usedthat is capable of obtaining topographical 2-D datatraces. (The stroboscopic interferometer operated in thestatic mode, as described in Sec. 7.1, can be used forthese measurements.) Figure 11 is a schematic of a typ-ical optical interferometric microscope that uses themethod of coherence scanning interferometry [26], alsocalled vertical scanning interferometry or scanningwhite light interferometry, for these measurements.However, any calibrated topography measuring instru-ment that has pixel-to-pixel spacings or samplingintervals as specified in Table 10 and that iscapable of performing the test procedure with a verticalresolution less than 1 nm is permitted. The opticalinterferometric microscope or comparable instrumentmust be capable of measuring step heights to at least 5μm higher than the step heights to be measured andmust be capable of extracting standard deviations.

328

16 The periodic chirp function enables quick results without averag-ing. For the periodic chirp function, sinusoidal signals (within theselected frequency range and of the same amplitude) are emittedsimultaneously for all fast Fourier transform (FFT) lines. The period-ic chirp function is periodic within the time window, and the phasesare adapted to maximize the energy of the resulting signal.17 The periodic chirp function produces a reproducible resonancefrequency. A sine wave sweep function produces a resonancefrequency that can be affected by the direction of the sweep if thereis not a sufficient amount of time between measurements.18 By inserting the inputs into the correct locations on the appropri-ate NIST Web page [10], the given calculation can be performedon-line in a matter of seconds.

2

4 ,38.330init

caninitcan

E tf

Lρ=

2

4 ,0.946init

ffbinithiffb

E tf

Lρ=

2

11 1 100% .4diff

pQ

⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠

2

.24

can init

can

W E tQL

ρμ

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟

⎢ ⎥ ⎝ ⎠⎣ ⎦

8. Appendix B.Supplemental Material for Young’sModulus Measurements

This appendix contains supplemental material forYoung’s modulus measurements for reference only. InSec. 8.1, the derivation of the Young’s modulus equa-tion from a resonating cantilever is presented. InSec. 8.2, the residual stress and stress gradient equa-tions are given. The equations for Young’s modulusmeasurements as obtained from a fixed-fixed beam are given in Sec. 8.3 followed by two derivations. The

deriviation of the Young’s modulus equation from aresonating fixed-fixed beam assuming simply support-ed boundary conditions is given in Sec. 8.4, and thederivation of the Young’s modulus equation from aresonating fixed-fixed beam assuming clamped-clamped boundary conditions is given in Sec. 8.5.Lastly, Sec. 8.6 presents the calculation of σfQ asdiscussed in Sec. 4.2.

8.1 Derivation of Young’s Modulus EquationFrom a Resonating Cantilever

In this section, the Young’s modulus equation is pre-sented as derived, in a manner similar to that presentedin [12], from measurements taken on a resonatingcantilever. Clamped-free boundary conditions areassumed, as given by the following equations:

(55)

(56)

(57)

and

Volume 115, Number 5, September-October 20108Journal of Research of the National Institute of Standards and Technology

329

Fig. 11. Schematic of an optical interferometric microscope operating in the Mirau configuration where thebeam splitter and the reference surface are between the microscope objective and the sample.

Table 10. Interferometer pixel-to-pixel spacing requirementsa

Magnification, × Pixel-to-pixel spacing, μm

5 < 1.5710 < 0.8320 < 0.3940 < 0.2180 < 0.11

a This table does not include magnifications at or less than 2.5 × foroptical interferometry because the pixel-to-pixel spacings will be toolarge for this work an