INVITED FEATURE PAPER Deformation and fracture …...INVITED FEATURE PAPER Deformation and fracture of single-crystal silicon theta-like specimens Michael S. Gaither,a) Frank W. DelRio,

$Page 1: INVITED FEATURE PAPER Deformation and fracture …...INVITED FEATURE PAPER Deformation and fracture of single-crystal silicon theta-like specimens Michael S. Gaither,a) Frank W. DelRio,$
INVITED FEATURE PAPER

Deformation and fracture of single-crystal silicontheta-like specimens

Michael S. Gaither,a) Frank W. DelRio, Richard S. Gates, and Robert F. CookNanomechanical Properties Group, Material Measurement Laboratory, National Institute of Standardsand Technology, Gaithersburg, Maryland 20899

(Received 6 July 2011; accepted 29 August 2011)

Single-crystal silicon test specimens, fabricated by lithography and deep reactive ion etching(DRIE), were used to measure microscale deformation and fracture properties. The mechanicalproperties of two specimen geometries, both in the form of a Greek letter H (theta), were measuredusing an instrumented indentation system. The DRIE process generated two different surfacestructures leading to two strength distributions that were specimen geometry independent: Onedistribution, centered about 2.1 GPa, was controlled by 35 nm surface roughness of scallops; thesecond distribution, centered about 1.4 GPa, was controlled by larger, 150 nm, pitting defects. Finiteelement analyses (FEA) converted measured loads into strengths; tensile elastic measurementsvalidated the FEA. Fractographic observations verified failure locations. The theta specimen andtesting protocols are shown to be extremely effective at testing statistically relevant (hundreds)numbers of samples to establish processing–structure–property relationships at ultrasmall scales andfor determining design parameters for components of microelectromechanical systems.

I. INTRODUCTION

Many advanced materials are intended for use insmall-scale applications, for example, microelectronics,1–3

microelectromechanical systems (MEMS),4,5 photonics,6–8

biotechnology,8 and magnetic storage,8–11 or may be avail-able only in small volumes, for example during materialsdevelopment. Developing or optimizing such materials andtheir processing methods thus requires measurements ofstructure and properties at small scales. A pervasive mea-surement requirement is that of measuring mechanicalproperties and relating them to processing and structure:To optimize manufacturing yield and operational perfor-mance, especially reliability,12 all materials and devicesmust maintain mechanical integrity, whether intended forprimarily mechanical applications, for example, MEMS, ornot, for example, microelectronics. However, establishingprocessing–structure–mechanical properties linkages atsmall scales is difficult13: Not only are the involved loadsand displacements small, making measurement difficult,but issues of specimen gripping and loading alignment,which are also often problematic at large scales,14 are mademore difficult aswell. In addition, posttest sample collectionand manipulation are difficult, which impedes the ability toidentify property-limiting structural defects during failureanalysis and thus hinders the capacity to alter processingprocedures for property optimization.

An experimental method that avoids many of thesedifficulties in measuring mechanical properties of materi-als at small scales is instrumented indentation testing(IIT).15–20 Commercial IIT instruments are well able tomeasure loads in the micronewton to 100s of millinewtonsrange and displacements in the nanometer to 10s ofmicrometers range, enabling small-scale mechanicaltesting. As the only IIT measurement requirement is themounting of a large specimen surface, typically milli-meters or more in dimension, perpendicular to the axis ofa probe loaded into the surface in compression, gripping,alignment, and manipulation difficulties are largely obvi-ated. In addition, examination of the resulting indentationson the large specimen surface is relatively easy, allowingfor failure analysis.21–25 Elastic modulus, hardness andyield stress, toughness, and viscosity are thus allmeasurable with IIT.15,16,20,26

Recently, a new experimental test specimen wasintroduced27–29 that allows “tensile strength” of brittlematerials to be measured at small scales and which utilizesmany of the advantages of IIT: The tensile test specimendeliberately does not attempt to replicate large-scale tensiletest specimen geometries with the attendant gripping andalignment difficulties, but instead integrates the “specimen”into a test “frame.” The integrated circular frame andspecimen cross-piece, or “web” segment, resemble theGreek letter H, and the overall specimen is known asa “theta” specimen.14,30,31 The specimen is easily tested incompression with an IIT device, resulting in tension in thespecimen web segment and thereby avoiding grippingissues. Precision microfabrication techniques lead toa well-defined alignment of the tensile specimen relative

a)Address all correspondence to this author.e-mail: [email protected]

This paper has been selected as an Invited Feature Paper.DOI: 10.1557/jmr.2011.319

J. Mater. Res., Vol. 26, No. 20, Oct 28, 2011 �Materials Research Society 2011 2575

to the loading axis, thereby minimizing alignment issues.Finally, a number of the microscale frames are incorporatedonto a macroscale strip for easymanipulation and recovery offailed parts, a feature that was included in a new design29

that improved on many aspects of the original concept.14

A major focus of the current work is the assessment ofthe utility of the new test specimen to determine the effectsof processing variations on strength. Such variations arehighlighted in Fig. 1, which shows a graph of measuredfracture strength of single-crystal silicon (Si) as a functionof the approximate stressed area for many sample andloading geometries (uniaxial and equibiaxial tension andbending) and surface processing methods.27,32–56 Twoclear trends are apparent in Fig. 1: (1) fracture strengthstend to decrease with increased stressed area, independentof processing method, consistent with the engineeringprinciple57 that fabrication methods tend to scale surfaceroughness, and therefore strength-limiting defect sizes,with component size; (2) fracture strengths tend to de-crease with increased stressed area, with a different de-pendence for a given processing method, consistentwith the physics principle58 that processing methods tendto generate a particular distribution of flaw potency andthat the probability of a component containing a morepotent defect increases with component size. The dashedlines on Fig. 1 indicate trend (2) for selected fabricationmethods. An implication of Fig. 1 is that if processingmethod and stressed area are invariant, then strengthshould not be altered by sample or component geometry.

Here, the tensile strength of the original and new thetaspecimen designs is compared directly, using the sameprocesses to fabricate samples of both sets of specimens,extending previous work that described the morerecent design.29 Section II describes the design and fabrica-

tion of the samples, the testing methodology, and themicroscopy methods used in sample characterization beforeand after testing. Section III describes finite element analyses(FEA) used to transform IIT sample load–displacementmeasurements into material stress–strain behavior as a func-tion of sample geometry. Section IV presents experimentalresults of tensile elastic deformation and fracture strengthsand strength distributions, surface topography measure-ments, and fractography of failed samples. Section Vdiscusses the implications of the results for establishingprocessing–structure–mechanical properties linkages formaterials at small scales using the theta specimen.

II. EXPERIMENTAL METHOD

A. Specimen design and fabrication

Figure 2 shows schematic diagrams of the theta testspecimens developed for small-scale testing. Both speci-men geometries are formed from a frame with a circularexterior that is attached to a macroscale strip at the base(not shown), include a hat structure at the top, andincorporate a web across the center of the specimen. Thespecimen geometry shown in Fig. 2(a) is based on theoriginal design by Durelli14,30,31 and consists of straightsections joined by tangential circular sections to define theframe interior. The specimen geometry shown in Fig. 2(b)is the new arch theta specimen29 and consists of a singlecircular arch to define the frame interior. Both specimendesigns use tangential circular sections to incorporate theweb and have the same diameter, D, of 250 lm and webwidth,w, of 8 lm. During testing, a load, P, is applied to thetop surface of the specimen and the load-point displace-ment, h, is measured. Loading these specimens in compres-sion generates a uniform tensile stress across the uniformcross section of the web. The arch theta design replaces thecomplex internal geometry of the original Durelli designwith an arch, thereby reducing the size and extent ofsecondary, non-web, stresses in the specimens on loading.

FIG. 1. Fracture strength as a function of approximate stressed areafor single-crystal silicon. Data were collected from a large number ofsources.27,32–56 Power law trend lines are shown to highlight theoverall behavior of the strength data. TMAH: tetramethylammoniumhydroxide; EDP: ethylenediamine pyrocatechol.

FIG. 2. Schematic diagrams of the (a) Durelli and (b) arch theta testspecimens. The outer ring of each specimen is diametrally compressedwith load P and displacement h, generating a uniaxial tensile stress statein the web segment. The diameter of the outer ring is D and the width ofthe web segment is w.

M.S. Gaither et al.: Deformation and fracture of single-crystal silicon theta-like specimens

J. Mater. Res., Vol. 26, No. 20, Oct 28, 20112576

The top hat structure is included to minimize loadingmisalignments and stress concentrations28,29 that also leadto large secondary stresses. Both of these design changesincrease the probability that sample failure will initiate, asintended, in the web. The stressed area of the web, intension, in both designswas approximately 6.25� 103 lm2,about the center of the area range of Fig. 1.

The fabrication sequence for both theta test sample setsis shown in Fig. 3. The process startedwith a single 100-mmdiameter (001) silicon-on-insulator (SOI) wafer consistingof a (25.0 6 0.5) lm thick single-crystal Si device layer,a (2.06 0.1) lm SiO2 isolation layer, and a (4006 10) lmSi handle wafer, Fig. 3(a), where the uncertainty valuesrepresent variations across the wafer as specified by themanufacturer. The SOI structure allowed for better control ofsample device thickness and more robust strips for manip-ulation and mounting of samples for testing than previousdesigns.27,59 The Si device layer and Si handle wafer layerwere patterned by front- and back-side photolithographicmasks and etched using deep reactive ion etching (DRIE) todefine the sample and strip features, Figs. 3(b) and 3(c); thefront-side mask included both Durelli and arch designs.The photolithographic mask designs were transferred to theSOI wafer such that the web segment of the test samples was

oriented along a ,110. direction with less than 0.5°misalignment.29 After Si etching, the SiO2 layer wasremoved with a buffered-oxide etch to create the free-standing samples, Fig. 3(d). Resulting Durelli and archtheta samples are shown in Figs. 3(e) and 3(f), respec-tively. Finally, each test strip, consisting of 10 thetasamples spaced along a 15 mm length, was removed fromthe wafer using a diamond scribe on notched regions ateach end of the strip.

B. Mechanical testing and characterization

The test strips were clamped across the full-wafer 427-lmthickness into a fixture that was then mounted into an IITdevice, such that the samples were upright and isolated fromthe surrounding clamp material. Each test sample was thendiametrally compressed “via” IIT using a 250-lm radiusspherical sapphire indenter and an IIT break-detectionroutine that withdrew the indenter on detection of samplefailure to minimize subsequent sample damage. Two sets oftest conditions were used for all samples. In the first, the IITdevice was operated in load control and set to load to a peakload of 200 mN, cycle five times between the peak load and100 mN with a 30 s peak hold each cycle, and then unload,using loading and unloading rates progressively increasingfrom 0.5 to 13.3 mN s�1. In the second, the IIT devicewas subsequently operated at a target displacement rateof 20 nm s�1 and loaded until the break-detection routinedetected a rapid increase in indenter velocity, associatedwith an increase in sample compliance and sample failure,and withdrew the indenter. Load and displacement wererecorded throughout the cyclic loading and sample failuremeasurements with data acquisition rates of 5 and 100 Hz,respectively.

Sample load and displacement data were converted intothe longitudinal stress, r, and strain, e, in the web usingFEA (see Section III); in particular, sample strength, rf,was calculated from the peak load at sample failure and thesample web dimensions (see Section II.C). The resultingstrength values were fit to a three-parameter Weibulldistribution

Pf ¼ 1� exp � rf � rth

rh

� �m� �; ð1Þ

where Pf is the cumulative probability of failure, rth is thethreshold strength, rh is a scaling stress (the “characteristicstrength” is rth + rh), and m is the Weibull modulus. Thecumulative failure probability was assigned to each strengthvalue by

Pf ¼ i� 0:5N

; ð2Þ

where i is the rank of the strength in an ascending-orderranked strength distribution and N is the total number of

FIG. 3. (a–d) The fabrication sequence for the theta test samples. (a) Theprocess began with a SOI wafer. (b) The Si device layer and (c) Si handlewafer were patterned by a photolithographic mask and etched using DRIEto define the device features. (d) The SiO2 layer was then removed locallywith a buffered-oxide etching solution to create the freestanding structures.(e, f) Field-emission scanning electron microscopy (FESEM) images ofcompleted (e) Durelli and (f) arch theta test samples.


J. Mater. Res., Vol. 26, No. 20, Oct 28, 2011 2577

strength values in the distribution. Weibull distributionparameter values were determined using a Levenberg–Marquardt fitting algorithm of Eq. (1) to the strength data.The strength values were also used to estimate the criticalflaw size leading to sample failure. The flaw sizes, cs, wereestimated assuming simple, nonresidually stressed flaws,

cs ¼ T

Wrf

� �2

; ð3Þ

where T is the material toughness, here taken as0.71 MPa m1/2 appropriate to the (110) plane of Si,24

and W is a crack geometry term, here taken as 1.12p1/2

appropriate to a linear surface flaw.60

C. Microscopy and sample characterization

The dimensions of the web cross section are critical indetermining the relationships between the load applied tothe sample, P, and stress, r, generated in the web, andthe load-point displacement, h, and web strain, e. Thedimension of the web in the section AA9 in Fig. 4(a), theweb “width,” w, controlled the P–r and h–e relationships,as the sample and web thickness were controlled by theSOI device layer thickness; lithographic and DRIE vari-ability generated variations in web width, but not web

thickness. High-contrast optical microscope images weredigitally recorded for every sample, as shown in the exampleFig. 4(a). The images were imported into an image-processing program, and the image intensity variation acrosshundreds of sections such asAA9 determined over the centralsection of the web. An example of an intensity variation isshown in Fig. 4(b). The minimum and maximum intensitiesin an image were then used to set a half-maximum intensityvariation, and aweb dimension at each sectionwas defined asthe full-width at half maximumof the intensity variation. Theweb dimensions from every section in an image were thencombined to generate a mean and standard deviation webdimension in pixels for that sample. Such oversamplingenabled super-resolution subpixel standard deviations to beobtained for the web widths. Calibration of the microscopeand image-processing program using a calibrated pitch grat-ing enabled the web width in micrometers to be calculatedfrom the pixel dimensions. Figure 4(c) shows histograms ofthe web widths of the 105 fabricated and tested Durelli andarch theta samples. Note that most samples had web widthsless than the target of 8 lm. Trials of the lithographic processshowed that the variability in web width resulting from theintended DRIE process was 0.7 lm and hence some other,unintended, process overetched the samples to generatesignificantly smaller web widths.

FIG. 4. (a) Grayscale optical image of an arch theta test sample. An optical image of each sample was imported into an image-processing routine priorto testing. (b) Vertical line scans of the pixel intensity were analyzed using a full-width half-maximum criteria to determine the web width. The webwidths from these line scans were averaged over the entire web segment, resulting in subpixel resolution for w. (c) Bar graph of the web widthdistribution for Durelli and arch theta samples at 0.5 lm intervals.



Field-emission scanning electron microscopy (FESEM)was utilized to examine fabrication etch quality and testedsample fracture surfaces; samples were sputter-coated withgold–palladium prior to examination. The samples exhi-bited two forms of surface etching, shown in Fig. 5. Theetch quality in Fig. 5(b) is consistent with the regular etchsteps known as scallops61 that are characteristic of DRIE.However, in Fig. 5(c), a clearly different etch surfacequality is apparent. The effect and likely origin of thisunintended etch surface, in the form of irregular pits, arediscussed in Sections IV and V, respectively. Fragmentsfrom tested samples were collected to analyze the fracturebehavior. Fracture surfaces were examined to identifyinitial fracture locations, crystallographic crack propagationplanes, and fracture mode. Fracture mirror radii,62–64 R, onthe fracture surfaces were estimated for comparisonwith theexpected mirror radii calculated from the Orr equation

rfR1=2 ¼ A ; ð4Þ

where rf is the stress at the origin at fracture and A is themirror constant for Si fracture, approximately 1.6 MPa m1/2

for the {110} tensile surface on the {110} fracture plane.65

Topography measurements of the etched surfaces weremade using atomic force microscopy (AFM) to determinesurface roughness. Intermittent contact mode AFM, ata line scan rate of 1 Hz, was used to scan 5.0 lm � 2.5 lmregions of sample sidewall surfaces, using a cantileverwith a 40 N m�1 spring constant, 325 kHz resonance

frequency, and 10 nm tip radius. Resulting scans were512 pixels � 256 pixels. Etch surface roughness wascompared to the calculated critical flaw sizes, Eq. (3).

III. FINITE ELEMENT ANALYSIS

For each specimen design, three-dimensional finiteelement analysis (FEA) simulations of loading were per-formed. Each specimen simulation utilized more than100,000 eight-node linear hexahedral mesh elements; thecritical specimen web segment had 36–66 elements in crosssection and approximately 2,000–10,000 total elementswithin the web segment, with more elements used forsmaller web widths. Silicon orthotropic elastic propertieswere used and oriented as in the fabrication sequence,aligning,110.with the theta web axis; the elastic stiffnessvalues were C11 5 165.773 GPa, C12 5 63.924 GPa, andC44 5 79.619 GPa.66 Simulated loads of 20–200 mN inincrements of 20 mN, along with a load of 2 N, were appliedto the top center of the theta specimens using a hemispherewith a 250-lm radius indenter and isotropic elastic propertyvalues approximating sapphire with a Young’s modulus of400 GPa and a Poisson’s ratio of 0.24.67

FEA images of the maximum principle stress for boththeta specimens with ideal dimensions are shown in Fig. 6.Both specimens have the same applied load. Themaximum principle stress is clearly greatest in the constantcross-section web segment; this stress is invariant withinthis region and is nearly equal for the two designs. The

FIG. 5. FESEM images of the sidewall etch surfaces for the theta test samples. A portion of the important web segment is indicated by the box in the tiltedtheta sample of (a). In (b), the sidewalls have regular etch features, known as “scallops”, which are expectedwith theDRIE process. In (c), the sidewalls haveirregular etch features, denoted here as “pits”, which result from unwanted etching of the Si device layer during the etching of the Si handle wafer.



stress fields were essentially uniaxial along the web axisfor most of the web length, with consequent longitudinaland Poisson-contracted lateral strain fields. Secondarystresses located at the top and bottom of the inside thetaregions are significantly less than the primary stress in theweb. Both the extent and magnitude of the secondarystress are reduced in the arch theta specimen comparedwith the Durelli design. In Fig. 6(a), the maximumsecondary stress is 0.65 of the maximum stress in theweb, and the secondary stress region in Fig. 6(b) hasa maximum stress of 0.62 of the stress in the web.

The simulations were used to translate the appliedindentation load, P, and load-point displacement, h,response into stress and strain behavior in the web segmentof the theta specimens of ideal dimensions. In all simu-lations, P and h were linearly related and specimencompliance, k, was determined by averaging over allsimulated loads. Stress, r, and strain, e, in the web segmentwere determined by averaging the principal stress and strainover the cross section of the center of the web, respectively.In all simulations,r and ewere linearly related and linearlyrelated to P and h, respectively. Equations for r and e asfunctions of P and h, theta diameter, D, and specimenthickness, t, were established by averaging responses overall simulated loads. For the Durelli theta specimen theseequations were

rD ¼ �14:367PDt

; ð5Þ

eD ¼ �0:563hD

; ð6Þ

where rD is in GPa, P is in mN, and D, t, and h are in lm;correlation coefficients for these linear relations wereR2 . 0.9999. The related specimen compliance for theDurelli design was kD 5 6.03 nm mN�1. The stress and

strain equations include negative signs as the load andload-point displacement were taken in the specimencompressive direction while the stress and strain weretaken in the web tensile direction. The original Durellispecimen equations had coefficients of 13.8 for stressand 0.585 for strain.14,30,31 Equations (5) and (6) containdifferent coefficients due to the changes in outer specimendesign geometry, namely the included top hat and spec-imen bottom attachment. For the arch theta specimens, thestress and strain in the web segment were determined to be

rA ¼ �14:239PDt

; ð7Þ

eA ¼ �0:639hD

; ð8Þ

with arch theta compliance of kA 5 5.27 nm mN�1 andunits as for Eqs. (5) and (6). (The change in coefficients inEqs. (7) and (8) compared to those cited previously29 is dueto a more detailed analysis of the finite element model.)Equations (5)–(8), describing the behavior of specimens ofideal geometry, provide the basis for determining the uncer-tainty arising from the use of the finite element interpolationequations describing the behavior of specimens with non-ideal web widths.

The variation in fabricated sample web widths, illus-trated by Fig. 4(c), was incorporated into the FEA simu-lations to accurately determine the mechanical behavior.Stress, strain, and compliance formulas as a function of webwidth, w, were developed for both theta specimens. Thegeometry of each specimen was altered by incrementallyperforming 0.5 lm offsets over the entire specimen planesurface, leaving the thickness unaltered, resulting in 1 lmchanges in web width. In particular, offsets were performedto create specimens with 3–9 lm web widths in 1 lmincrements. Simulated loads were applied and the resultingload-point displacements and web stress and strain weredetermined as described above for each simulated webwidth, and the coefficients relating these four parameterssimilarly determined. The relationships between theparameters maintained the same form with coefficients, K,which depended on w

r ¼ �KrP

Dt; ð9Þ

e ¼ �Keh

D; ð10Þ

k ¼ KkkI ; ð11Þwhere kI is the ideal, w 5 8 lm, compliance for a givenspecimen geometry. The coefficients K were found to bewell described by simple inverse dependencies on w, and

FIG. 6. Maximum principal stress distribution for the (a) Durelli and(b) arch theta test specimens subjected to loading with a sapphire sphere.The largestmaximumprincipal stress,rmax, occurs in the web segment, andthe largest secondary stress,rsec, is located at the top and bottomof the innertheta region. For the Durelli theta, the stress ratio rsec/rmax5 0.65. For thearch theta, the stress ratio rsec/rmax 5 0.62. Thus, for both geometries,initial fracture is expected to occur in the web segment.



averaging over the seven width simulations for eachspecimen generated the following expressions

Kr;D wð Þ ¼ 97:224=wþ 2:408 ; ð12Þ

Ke;D wð Þ ¼ 1:660=wþ 0:363 ; ð13Þ

Kk;D wð Þ ¼ 2:469=wþ 0:705 ; ð14Þ

for Durelli theta specimens, and

Kr;A wð Þ ¼ 86:001=wþ 3:751 ; ð15Þ

Ke;A wð Þ ¼ 1:670=wþ 0:439 ; ð16Þ

Kk;A wð Þ ¼ 2:309=wþ 0:725 ; ð17Þ

for the arch theta specimens. In Eqs. (9)–(17), Kr, Ke,and Kk are dimensionless and w is in lm. R2 values for theabove dependencieswere 0.98 or greater. Insertingw5 8 lminto the above interpolation equations reveals about 1%variation from the coefficients for stress and strain givenin Eqs. (5)–(8) and about 2% variation from the idealcompliance values.

IV. RESULTS

A. Elastic deformation

Figure 7 shows load–displacement responses for five ofeach of the fabricated Durelli and arch samples taken fromthe initial cyclic loading between a peak load of 200 and100 mN. Responses for sample widths from just less thanthe target width of 8 lm to slightly greater than 4 lm areshown, and only the first unloading response for each sampleis shown. The compliance of the samples for both geome-tries increases with decreasing web width, as highlightedby the best-fit straight lines passing through the extremes ofthe web widths. Best fits to all the responses generatedcompliance values that were all slightly greater than thecompliance values predicted from the FEA, Eqs. (11), (14),and (17). There was no trend of the increase in compliancewith sample web width, suggesting that the additionalinferred deformation and resulting compliance was associ-ated with indenter contact and test strip mounting. The meanand standard deviation of the additional test systemcompliance, determined from the 10 samples in Fig. 7 was(0.21 6 0.10) nm mN�1, approximately 3% of the averageexperimental compliance value for the 10 samples. De-formation associated with this test-system compliance wassubsequently subtracted from all presented displacementdata. The agreement between the measured and predictedcompliance values, notwithstanding the additional testsystem compliance, indicates that the FEA of the specimenswas accurate and that the dependencies of Eqs. (9)–(17) are

accurate for relating web stress and strain-to-indenter loadand displacement.

Figure 8 shows the complete cyclic load–displacementresponses for single example Durelli and arch samples.Equations (12), (13), (15), and (16) were used to generateweb stress and strain data from the sample load–displacementdata. An initial, recoverable, nonlinear response is visible forthe example arch sample, Fig. 8(b); this was typical of theinitial first-cycle response for all samples examined and isprobably associated with seating and unseating of the spher-ical indenter on the rough etched surface of the hat. Thisinitial nonlinear response was quite variable, and in somecases extended over several micrometers of displacement. Inother cases, this first-cycle nonlinearity was barely discern-ible, as shown in the example Durelli sample, Fig. 8(a). Afterthe initial loading, the data for all the loading cycles between100 and 200 mN for both geometries are indistinguishable,

FIG. 7. P–h data for the first unloading segment of five differentcyclically loaded (a) Durelli and (b) arch theta test samples with a 4–8 lmdistribution of web widths. As w decreases, the compliance, k, of the thetatest specimen increases, illustrated by the changing slope of the data sets.Best fits to all the responses generated compliance values that were allslightly greater than the compliance values predicted from the FEA due totest system deformation compliance.



indicating negligible subsequent hysteresis; this lack of cyclichysteresis was common to all samples tested, indicatinga completely elastic response after initial indenter seating andthat the sample mounting configuration was effective andessentially lossless. Best-fit straight lines to the cyclic loadingdata are shown in Fig. 8, giving rise to elastic moduli valuesof (1716 3) and (1676 3) GPa, for the example Durelli andarch samples, respectively, where the uncertainty includesthe measurement uncertainty and the finite element interpo-lation equations uncertainty. These values compare with168.9 GPa for the Young’s modulus of Si in the ,110.directions.68 The elastic responses and moduli agreementserve to validate the combined experimental and analyticalapproaches.

B. Strength

Figure 9 shows the load-to-failure responses for theexample Durelli and arch samples described above and inFig. 8. Once again, Eqs. (12), (13), (15), and (16) were usedto generate web stress and strain data from the sample load–displacement data; initial nonlinear responses are visible forthe samples, associated with the reseating of the indenterbetween the cyclic (Fig. 8) and load-to-failure measure-ments (Fig. 9). The increased scatter in the data prior to peakload and failure is associated with a change in the loadingmechanism of the instrumented indenter and decrease instability at large loads. Linear best fits to the data for 0.5%strain prior to failure (not shown) in Fig. 9, gave rise toelastic moduli values of (1696 3) and (1666 3) GPa, forthe example Durelli and arch samples, respectively, onceagain in good agreement with the expected value and values

FIG. 8. P–h and corresponding r–e data for (a) Durelli and (b) archtheta test samples subjected to five load–unload cycles with a maximumload of roughly one-fifth the typical fracture load. The traces are linearwith no discernable hysteresis, which suggests a secure and stable testplatform. E was determined from the slope of the linear portion of thetrace. For these particular samples, E 5 171 GPa for the Durelli thetaand E 5 167 GPa for the arch theta.

FIG. 9. P–h and r–e data for (a) Durelli and (b) arch theta test samplesloaded to failure. E was determined from the slope of the linear portionof the trace, and rf was the stress at which fracture occurred. For theseparticular samples, Ewas 169 and 166 GPa andrf was 2.32 and 2.29 GPafor the Durelli theta and arch theta, respectively.



determined from the cyclic loading measurements of Fig. 8.The strengths for these example samples were 2.32 and2.29 GPa, respectively, comparable to the values reportedin Fig. 1 for specimens of similar stressed area. In allsamples measured, the stress–strain response was linearprior to failure, at strengths up to 2.7 GPa and correspond-ing sample failure strains (excluding seating deformation)of approximately 1.8%.

As noted in Fig. 4(c), both Durelli and arch thetasamples had significant distributions in sample webwidths. The resulting strength,rf, values for the combinedset of Durelli and arch theta samples are plotted asa function of sample web width in Fig. 10(a). The idealweb width of w 5 8 lm is indicated by the vertical solidline and the normal variability in web width observed forDRIE lithography, 0.7 lm, is indicated by vertical dashedlines. Overall, samples with near-ideal web width hadgreater strengths than samples with reduced web widthsand exhibited less variability in the web width; the meanand standard deviation of the webwidth of each sample areindicated by the symbol and horizontal uncertainty bar inFig. 10(a). (The uncertainty in the strength arising from thefinite element interpolation equation and sample thicknessuncertainty is smaller than the symbol size.) This vari-ability was apparent when capturing the images for theweb width calculation image-processing routine; sampleswith smaller web widths had rough sample edges, in-cluding the web segment. Figure 10(b) is a plot of thecumulative failure probability, Pf, as a function ofstrength, rf, for the combined Durelli and arch thetasample data set. Examination of Fig. 10(b) suggests thatthe strength distribution is bimodal, as there appear to bethree inflection points in the distribution, with a strengthrange of approximately (1.4–1.9) GPa separating a lowstrength distribution from a high strength distribution.Comparison with Fig. 10(a) suggests that this was in factthe case, with the strength, and dominant flaw size, relatedto the web width. The picture that emerges, consideringboth Fig. 10 and Figs. 4 and 5, is that there are two flawpopulations governing the overall strength distribution:The first flaw population is associated with DRIE scallops[Fig. 5(b)], near-ideal web widths, and small surfaceroughness, giving rise to small flaws and large strengths.The second flaw population is associated with the pittedsurface [Fig. 5(c)], reduced web widths, and large surfaceroughness, giving rise to large flaws and small strengths. Thevertical dashed line in Fig. 10(a) at a web width of 7.5 lmindicates the boundary separating these width distributions.

In Fig. 10(c), the strength data from each distribution werefit to two different three-parameter Weibull distributions,Eq. (1). The strength data were separated into two groups on

FIG. 10. (a) Fracture strength as a function of web width. The data canbe divided into two groups associated with the sample surfacemorphology as determined by the etching process as indicated by thedashed line atw5 7.5 lm. Forw, 7.5 lm,rf varied from 0.8 to 1.9 GPa.For w . 7.5 lm, rf varied from 1.4 to 2.7 GPa. The smaller web widthsamples had greater web width standard deviations indicated by the dataerror bars. (b) The total strength distribution for the combined Durelli andarch theta test samples is bimodal, as there appear to be three inflectionpoints in the distribution. (c) Weibull failure probability plots for the twodistributions in (b) split atw5 7.5lm.Forw. 7.5lm,m,rh, andrth were4.57, 0.90GPa, and 1.32GPa, respectively. Forw, 7.5 lm,m,rh, andrth

were 2.00, 0.46GPa, and 0.82GPa, respectively. In (a) to (c), the grey bandsindicate the overlap of the strength distributions.



the basis of the surface morphology of the sample, DRIEscallops or pits, as indicated by the sample web width, nearideal (8.2–7.5 lm) or reduced (less than 7.5 lm). Thestrength distributions of the groups overlapped as shown inFigs. 10(a) and 10(b), and fits to the two groups wereperformed separately. The solid lines in Fig. 10(c) indicatethe best-fit distributions, described by m, rh, and rth valuesof 4.576 1.18, (0.906 0.26) GPa, and (1.326 0.24) GPa,respectively, for the high strength group, and values of2.006 0.12, (0.466 0.02) GPa, and (0.826 0.02) GPa forthe low strength group. The uncertainties are standarddeviations of the fitted values. The R2 values for these fitswere.0.99, consistent with the agreement between the dataand the fitted lines in Fig. 10(c). The overlap in the strengthdistributions was in the range (1.4–1.9) GPa as indicated bythe grey bands in Fig. 10.

C. Fractography and surface topography

The expected fracture cleavage planes for Si tested intension along a ,110. direction are {111} and {110},24

giving rise to expected fracture planes for the web of thetheta specimens as illustrated in the schematic diagram ofFig. 11(a). FESEM images of web fracture surfaces areshown in Figs. 11(b) and 11(c) for arch theta samples thatcontained DRIE scallop etch features [Fig. 11(b)] andpitting etch features [Fig. 11(c)] on the web surfaces. Inboth cases, features on the fracture surfaces, includingchanges in fracture plane, cleavage step hackle, and arc-ribs surrounding a cathedral mirror,64,65 enabled the fractureorigin to be identified. In the two examples shown, fractureinitiated on a {110} plane perpendicular to the web longaxis, and imposed tension, and after some small propagationdistance deflected onto the smaller fracture energy {111}planes. Step hackle was predominantly restricted to these{111} planes, and arc-ribs and associated cathedral mirrorswere predominantly restricted to the initial {110} planes.The hackle, arc-ribs, and mirrors all radiated away froma single location that indicated the fracture origin, similar toSi {110} fracture observed previously.65 Also in both cases,it appeared that as soon as the propagating cracks formedarc-ribs, the cracks deflected from the {110} to {111}planes. The arrows in the magnified images of Figs. 11(b)and 11(c) indicate the fracture origin and the clear mirrorcenter on the fracture surface. In the two examples, thefracture-inducing flaw is not obvious, although the websurfaces are clearly not flat at the fracture origin. Thesefeatures were observed on the majority of the nearly 20fracture surfaces examined, with one each of a complete{110} and a {111}-to-{110} failure mode also observed.Observations of fractured samples did not indicate initialfracture in any region other than the web segment.

Measurements of fracture mirrors were performed onmagnified images of fracture surfaces: The mirror radiuswas taken as half the separation of the first discerniblearc-ribs delineating the mirror at or near the web surface.

FIG. 11. (a) Schematic diagram of the {111} and {110} fracture planesthat occur in the specimen web segment. (b,c) FESEM images of websegment fracture surfaces for the scallop and pit samples, respectively. Inboth examples fracture initiated on {110} fracture planes and changed to{111} fracture planes once the arc-ribs around the fracture mirror weregenerated. The mirror location and corresponding origin location isindicated by the arrow; in both (b) and (c), fracture initiated at a sidewalletch feature. Cleavage step hackle radiates from the fracture mirror andorigin. The size of the fracture mirror in the scallop-etch example in (b) isclearly smaller than the mirror in the pit-etch example in (c).



Mirror radii, R, varied from just over 200 to 1100 nm. Com-bining the mirror radii with the strength values for all samplesexamined and using Eq. (4) generated a mean and standarddeviationmirror constant ofA5 (1.26 0.4)MPam1/2, whichcompares with the constant of (1.61 6 0.14) MPa m1/2

determined previously65 for Si {110} fracture on a {110}tensile surface on micro- rather than nano-scale mirrors.

Figure 12 showsAFM images and topographic line scansof a DRIE scallop surface [Fig. 12(a)] and a pittedsurface [Fig. 12(b)]. The images are similar to the FESEMpresented in Fig. 5. The peak-to-valley topography of theDRIE scallop surface along the loading direction parallel tothe scallops, and presumably perpendicular to a fracture-initiatingflawand the observed{110} initial fracture surface,was approximately 35 nm.The peak-to-valley topography ofthe pitted surface along the loading direction was approxi-mately 250 nm, although the valley base level was muchmore irregular than that observed for the scalloped surface togive surface features approximately 150 nm in scale. Thestrength value ranges for the two strength distributions notedin Section IV.B and delineated in Fig. 10(c) are 0.8–1.9 and1.4–2.7 GPa. Using these strength values in Eq. (3) providesthe corresponding critical flaw size ranges of 200–35 and65–18 nm, respectively. As a consequence of the over-lapping strength distributions, the estimated flaw size rangesalso overlap. The pairs of dashed horizontal lines in the linescan graphs of Fig. 12 are separated by 35 [Fig. 12(a)] and150 nm [Fig. 12(b)] indicating that the upper bounds of thecalculated critical flaw sizes are comparable to the extremesof the surface topography for both surface types.

Figure 13(a) shows a schematic diagram of the failure-causing flaw as described by Eq. (3), a sharp edge crack oflength cs on a planar surface. Consideration of the surfaceroughness suggests a better representation of the flaw is thatof a sharp crack of length cf at the bottom of semiellipt-ical notch of depth a, as shown in Fig. 13(b). Fracture

mechanics analysis69 shows that the relation betweenstrength and size for a semielliptical notch with a crackflaw is Eq. (3) with cs5 a + cf. Setting a as the characteristictopography dimensions above and using the estimated flawsizes in this relation suggests crack lengths, cf, in the rangeof a few nanometers to a few tens of nanometers.

V. DISCUSSION AND CONCLUSIONS

The microfabricated miniaturized theta specimenspresented here, the original Durelli geometry and thenew arch geometry, along with associated additionaldesign features and calibration and testing protocols,provide a simple and effective method for measuringthe tensile strength of brittle materials at the microscale.Microfabrication methods allow many samples to be

FIG. 12. AFM images and line-scans of sidewall etch surfaces with (a) scallops and (b) pits. The surface features along the length of the scallops andpits are similar to the maximum flaw sizes calculated via linear elastic fracture mechanics.

FIG. 13. Schematic diagrams of (a) a sharp crack on a planar surfaceand (b) a short sharp crack at the root of a semielliptical notch that ismore representative of failure-causing flaws on rough etched surfaces.



formed at once, and the ease of use of the combined samplegeometry and testing protocol allows many measurementsto be performed in a time-effective manner, thus enablingstatistically relevant numbers of strength measurements tobe obtained; hundreds of samples were formed on a singlewafer and a single test time was about 5 min. Much of theease of use and measurement efficacy arises from twocritical design features: (1) Forming the samples in a thindevice layer on top of a thick handle wafer allowed thickbars incorporating thin samples to be handled easily. Thisenabled sample mounting in the IIT systemwith negligibleloading hysteresis and very small system loading compli-ance. Direct checking of the measured load–displacementresponses to verify the accuracy of the compliance andstress and strain analyses was thus a simple matter. Ease ofsample handling also enabled straightforward recovery ofbroken samples for post-failure analysis. (2) The incorpo-ration of a specimen hat and the use of a spherical IITloading probe maximized the uniformity of stress in theweb and minimized secondary, non-web stresses. This ledto maximized testing yield with all samples appearing tofail in the intended web segment. The arch theta specimenwas designed to have smaller regions of secondary stress,but the hat appears to have mitigated secondary stresseffects for both Durelli and arch designs. Additional design,testing, and analysis features, such as separating the samplesby blocks on the bars (Figs. 3 and 5), using a sample breakdetection routine, and incorporating variations in webwidthinto the stress and strain analysis (Fig. 4), also increasedtesting yield and measurement accuracy and precision.

The measurements on the etched Si samples demon-strated that the theta specimen can be used to achievea materials science and engineering goal of establishingprocessing–structure–properties relations, albeit here an un-intended demonstration. Variations in the etching process ledto two different surface structures that in turn led to twodifferent strength properties (Figs. 5, 10, and 12). Surface-roughnessmeasurements suggest that the strength-controllingflaw sizes in the samples were comparable to the surfaceroughness, which appeared in two forms: regular scallopsabout 35 nm in scale and irregular pitting about 150 nm inscale. The two flaw size populations gave rise to two strengthpopulations, a stronger one associatedwith the smaller rough-ness front-side device-layer DRIE scallops and a weakerone associated with the unintended larger roughness pitting.The processing step leading to the pitting was probably theback-side handle-waferDRIE, step (c) in Fig. 3. The pitsweremost likely formed when the SiO2 isolation layer separatingthe device and handle layers was fractured or puncturedduring steps (b) or (c), leading to backside DRIE gasespassing through the SiO2 and re-etching the front-sidesurfaces during step (c). An example of this is shown inFig. 14, which is a FESEM image of a (non-web) samplefragment. The fragment smooth face at the front of the imagewas the location of the SiO2 layer, and regular front-face

DRIE scallops can be observed on an isolatedmesa at the rearof the fragment. The left face is clearly etched with a pittedsurface to leave the mesa and leading to a thinner sample,consistent with the observation that “rough” webs were alsothinner. The formation of this unintended rough surface wasnot related to specimen geometry, affecting Durelli and archdesigns approximately equally. As a consequence, as bothdesigns had approximately the same stressed area, thestrengths of both designs were the same, consistent with theimplication fromFig. 1 that if processing and stressed area arethe same then similar strengths should be observed.

Other test specimens that have enabled statisticallymeaningful numbers of small-scale tensile strength meas-urements have often used more complex geometries micro-fabricated from multilayer polycrystalline material; theseinclude pull tabs54,70,71 and chains of links72 for whichspecialized mechanical loading systems were required andon-chip tensile bars for which electrical connection wasrequired for thermal actuation of the tests.73,74 Simplergeometries microfabricated from single-layer SOI materialhave also been demonstrated: In one case, large numbers ofsamples in the form of beams were tested, using loading byan AFM and an indentation system45; in another, a tensiledog-bone geometry was demonstrated,56 although the spec-imen does not lend itself well to testing large numbers ofsamples and a specialized mechanical loading system isrequired. In both SOI cases, as here, good agreementbetween the measured small-scale Young’s modulus andthe bulk valuewas found, and the strength of the sampleswascontrolled by flaw sizes linked to the surface roughness.Although the theta specimen is unlikely to reach thethroughput of the multilayer specimens (thousands of speci-mens at a rate of one per minute72), the advantages of thetheta geometry are that the samples are formed from a single

FIG. 14. FESEM image of an inhomogeneously etched surface ofa sample fragment. The flat surface at the front of the image was initiallycovered with an oxide separating the specimen device layer and the handlewafer (not shown). The re-etched pitted surface is apparent on the left of thefragment, leaving a small mesa of device layer DRIE surface remaining.



layer, which need not be silicon (copper for example56), thegeometry is simple, and specialized equipment beyond an IITsystem is not required. In addition, samples need not neces-sarily be formed by lithography and microfabrication, butby focused ion-beam75 or laser machining,76 for example,leading to even greater flexibility in selection of materials.

An additional, technologically important goal ofmaterialsscience and engineering is to establish the linkage ofmaterialproperties (set by processing and structure) to the perfor-mance of components formed from the material. The thetaspecimen can be used to great effect to predict the manu-facturing yield and operational reliability ofMEMS compo-nents. Strength distributions measured with theta specimenscan be used to predict the strength distributions of othercomponents fabricated by the same process; the thetastrength distribution is used as input to extrapolate to thescale of the components (in themanner of the dashed lines inFig. 1). If the loading spectrum and reliability of the compo-nents are known or assumed, the component strength distri-bution can thence be used to predict the lifetime spectrum ofthe device. Alternatively, theta specimens could be incor-porated into witness chips on MEMS wafers to ensure thatfabrication processes remain within specification or toenable lifetime assurance on a wafer-by-wafer basis.

A critical enabling element of the theta specimen pro-tocol in enhancing yield and reliability is the ability to testa statistically relevant number of samples such that a lowerbound or threshold value of a strength distribution,rth, canbe determined with sufficient precision to be useful indesign. If a precise strength threshold can be determined,the manufacturing and reliability design processes changefrom the stochastic process of minimizing the probabilitythat a component or system will fail to the deterministicprocess of controlling the component geometry or manu-facturing or operating environments such that a componentis never exposed to stresses above the threshold strengthand thus never fails. Here the relative precision of thethreshold strength was a few percent, suggesting thatdesigning components to within a safety factor of 2 oreven less is feasible.

ACKNOWLEDGMENTS

The authors thank George D. Quinn of the NationalInstitute of Standards and Technology (NIST) for assistanceand guidance with fractography and Weibull statistics.Researchperformed inpart at theNISTCenter forNanoscaleScience and Technology.

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Documents

INVITED FEATURE PAPER Deformation and fracture …...INVITED FEATURE PAPER Deformation and fracture of single-crystal silicon theta-like specimens Michael S. Gaither,a) Frank W. DelRio,