Bhushan B. Nanotribology and Nanomechanics: An Introduction

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14 Mechanical Properties of Nanostructures 757

The SEM images show that below the critical loads the undoped Si(100) and

undopedpolysilicon ﬁlm were damagedby plowing,associated with the plasticﬂow

of the material and formation of debris on the sides of the scratch. For the SiC and

SiO

ﬁlms, in regionA, a plowingscratch track was found without anydebris on the

side of the scratch, which is probably responsible for the smoother curve and slower

increase in the coeﬃcient of friction before the critical load. After the critical load,

for the SiO

ﬁlm, delamination of the ﬁlm from the substrate occurred, followed by

cracking along the scratch track. For the SiC ﬁlm, only several small debris particles

were found without any cracks on the side of the scratch, which is responsible for

the small increase in the coeﬃcient of friction at the critical load. For the undoped

Si(100), cracks were found on the side of the scratch right from the critical load

and up, which is probably responsible for the big bursts in the friction proﬁle. For

the undoped polysilicon ﬁlm, no cracks were found on the side of the scratch at the

criticalload. Thismightresult fromgrain boundarieswhichcan stop thepropagation

of cracks. At the end of the scratch, some of the surface material was torn away and

cracks were found on the side of the scratch in the undoped Si(100). A couple of

small cracks were found in the undoped polysilicon and SiO

ﬁlms. No crack was

found in theSiC ﬁlm. Even at the end ofthe scratch, less debriswas found in theSiC

ﬁlm. A curly chip was found at the end of the scratch in both Ni

−

PandAuﬁlms.

This is a typical characteristic of ductile metal alloys. The Ni

−

P and Au ﬁlms were

damaged by plowing right from the beginning of the scratch with material pile-up

at the side of the scratch.

The scratch depth proﬁles obtained during and after the scratch on all samples

with respect to initial proﬁle, after the cylindrical curvature is removed, are plotted

in Fig. 14.9. Reduction in scratch depth is observed after scratching as compared to

that of during scratching. This reduction in scratch depth is attributed to an elastic

recovery after removal of the normal load. The scratch depth after scratching indi-

cates the ﬁnal depth which reﬂects the extent of permanent damage and plowing

of the tip into the sample surface, and is probably more relevant for visualizing the

damage that can occur in real applications. For the undoped Si(100), undoped poly-

silicon ﬁlm, and SiO

ﬁlm, there is a large scatter in the scratch depth data after the

critical loads, which is associated with the generation of cracks, material removal

and debris. The scratch depth proﬁle is smooth for the SiC ﬁlm. It is noted that the

SiC ﬁlm exhibits the lowestscratch depth among the samples examined.The scratch

depths of the undoped Si(100), undoped polysilicon ﬁlm and SiO

ﬁlm are compa-

rable. The Ni

−

P and Au ﬁlms exhibit much lager scratch depth than other samples.

The scratch depth of the Ni

−

P ﬁlm is smaller than that of the Au ﬁlm.

The critical loads estimated from friction proﬁles for all samples are compared

in Fig. 14.7 and Table 14.1. The SiC ﬁlm exhibits the highest critical load of about

14mN, as compared to other samples. The undoped Si(100) and undoped polysi-

licon ﬁlm show comparable critical load of about 11mN whereas the SiO

ﬁlm

shows a low critical load of about 9.5mN.TheNi

−

P and Au ﬁlms were damaged

by plowing right from the beginning of the scratch.

758 Bharat Bhushan

14.3.2 Bending Tests of Ceramic Nanobeams Using an AFM

Bending tests have been performed on Si and SiO

nanobeam arrays [62,84]. The

single-crystalsilicon bridgenanobeams werefabricated by bulk micromachiningin-

corporating enhanced-ﬁeld anodization using an AFM [61]. The Si nanobeams are

oriented along the [110] direction in the (001) plane. Subsequent thermal oxidation

of the beams results in formation of SiO

beams.The crosssectionof the nanobeams

is trapezoidal owing to the anisotropic wet etching process. SEM micrographs of Si

and SiO

nanobeam arrays and a schematic of the shape of a typical nanobeam are

shown in Fig. 14.10. The actual widths and thicknesses of nanobeams were meas-

ured using an AFM in tapping mode prior to tests using a standard Si tapping mode

tip (tip radius < 10nm). Surface roughness measurements of the nanobeam surfaces

in tapping mode yielded a σ of 0.7±0.2nm and peak-to-valley (P–V) distance of

4±1.2nmforSiandaσ of 0.8±0.3nm and a P–V distanceof 3.1±0.8nmforSiO

Prior to testing, the Si nanobeams were cleaned by immersing them in a “piranha

etch” solution (3:1 solution by volume of 98% sulphuric acid and 30% hydrogen

peroxide) for 10minutes to remove any organic contaminants.

Bending Strength

Figure 14.11 shows typical load displacement curves for Si and SiO

beams that

were bent to failure [62,84]. The upper width (w

) of the beams is indicated in the

ﬁgure. Also indicated in the ﬁgure are the elastic modulus values obtained from

the slope of the load displacement curve (14.5). All the beams tested showed linear

elastic behavior followed by abrupt failure, which is suggestive of brittle fracture.

Figure 14.12 shows the scatter in the values of elastic modulus obtained for both

Si and SiO

along with the average values (± standard deviation). The scatter in

the values may be due to diﬀerences in orientation of the beams with respect to the

trench and the loading point being a little oﬀ-center with respect to the beam span.

The average values are a little higher than the bulk values (169GPa for Si[110]

and 73 GPa for SiO

in Table 14.2). However the values of E obtained from (14.5)

have an error of about 20% due to the uncertainties in beam dimensions and spring

constant of the tip/cantilever (which aﬀects the measured load). Hence the elastic

modulusvalues on the nanoscale can be consideredto becomparable to bulk values.

5 μm

Si SiO

5 μm

Neutral axis

Fig. 14.10. (a) SEM micrographs of nanobeam arrays, and (b) a schematic of the shape of

a typical nanobeam. The trapezoidal cross-section is due to the anisotropic wet etching during

the fabrication [84]

14 Mechanical Properties of Nanostructures 759

100

2000

Load F

beam

(μN)

400 600 800 1000

Displacement, D

beam

(nm)

E <110> = 189 GPa

= 295 nm Si

2000

Load F

beam

(μN)

400 600 800 1000

Displacement, D

beam

(nm)

E = 83 GPa

= 250 nm SiO

Fig. 14.11. Typical load displacement curves of silicon and SiO

nanobeams. The curves are

linear until sudden failure, indicative of brittle fracture of the beams. The elastic modulus

(E) values calculated from the curves are shown. The dimensions of the Si beam were w

295 nm, w

= 484 nm and t = 255 nm, while those of the SiO

beam were w

= 250 nm,

= 560 nm and t = 425 nm [84]

220

190

160

130

100

200100

Elastic modulus E (GPa)

300 400 500 600

Beam upper width w

(nm)

[110] = 182 r 11 GPa

120

300200 400 500 600 700

Beam upper width w

(nm)

=85r 13 GPa

SiO

Elastic modulus E (GPa)

Fig. 14.12. Elastic modulus values measured for Si and SiO

. The average values are shown.

These are comparable to bulk values, which shows that elastic modulus shows no specimen

size dependence [62]

Table 14.2. Summary of measured parameters from quasi-static bending tests

Sample Elastic modulus Bending strength Fracture toughness

E (GPa) σ

(GPa) K

(MPa

√

Measured Bulk Measured Reported Estimated Reported Bulk

value (microscale) (microscale) value

Si 182±11 169

18±3 < 10

1.67±0.40.6–1.65

0.9

SiO

85±13 73

7.6±2 < 2

0.60±0.20.5–0.9

–

Si[110] [91]

[92]

[43,44,46–49,70,93,94]

[58]

[54–57]

[87]

Most of the beams when loaded quasi-statically at the center of the span broke

at the ends as shown in Fig. 14.13a, which is consistent with the fact that maximum

tensile stresses occur on the top surfaces near the ends. (See FEM stress distri-

bution results in Fig. 14.13b.) Figure 14.14 shows the values of bending strength

760 Bharat Bhushan

Si SiO

Maximum tensile

bending stress

–23 –17 –11 –4 2 9 15GPa

Maximum

compressive stress

5 μm5 μm

Fig. 14.13. (a)SEMmicro-

graphs of nanobeams that

failed during quasi-static

bending experiments. The

beams failed at or near the

ends, which is the location

of maximum tensile bending

stress [84], and (b) bending

stress distribution for silicon

nanobeam indicating that the

maximum tensile stresses oc-

cur on the top surfaces near

the ﬁxed ends

200100

Bending strength σ

(GPa)

300 400 500 600

Beam upper width w

(nm)

= 17.9 r 3 GPaSi

300200

Bending strength σ

(GPa)

400 500 600 700

Beam upper width w

(nm)

= 7.6 r 1.2 GPa

SiO

Fig. 14.14. Bending strength values obtained from bending experiments. Average values are

indicated. These values are much higher than values reported for microscale specimens, indi-

cating that bending strength shows a specimen size eﬀect [62]

obtained for diﬀerent beams. There appears to be no trend in bending strength with

the upper width (w

) of the beams. The large scatter is expected for the strength

of brittle materials, since they are dependent on pre-existing ﬂaw population in the

material and hence are statistical in nature. The Weibull distribution, a statistical

analysis, can be used to describe the scatter in the bending strength values. The

means of the Weibull distributions were found to be 17.9GPa and 7.6GPa forSi

and SiO

, respectively. Previously reported numbers of strengths range from 1–

6 GPa for silicon [44,46–49,51,54,70,93,94] and about 1 GPa for SiO

[58] mi-

croscale specimens. This clearly indicates that bending strength shows a specimen

14 Mechanical Properties of Nanostructures 761

2.5

2.0

1.5

1.0

0.5

0.0

0.50.0 1.0 1.5 2.0

Distance from notch tip r

(nm)

Estimated K

(MPa m

1/2

)

SiO

Fig. 14.15. Fracture toughness (K

) values of for increasing values of r corresponding to

distance between neighboring atoms in {111} planes of silicon (0.4 nm). Hence r values

between 0.4and1.6 nm are chosen. The K

values thus estimated are comparable to values

reported by others for both Si and SiO

[62]

size dependence. Strength of brittle materials is dependent on pre-existing ﬂaws in

the material. Since for nanoscale specimens, the volume is smaller than for micro

and macroscale specimens, the ﬂaw population will be smaller as well, resulting in

higher values of strength.

Fracture Toughness

Estimates of fracture toughness calculated using (14.10) for Si and SiO

are shown

in Fig. 14.15 [62]. The results show that the K

estimate for Si is about 1–

2MPa

√

m whereas for SiO

the estimate is about 0.5–0.9MPa

√

m.Thesevalues

are comparable to values reported by others on larger specimens for Si [54–57] and

SiO

[58].The high valuesobtained for Si could be due to the fact that the scratches,

despite being quite sharp, still have a ﬁnite radius of about 100 nm. The bulk value

for silicon is about 0.9MPa

√

m. Fracture toughness is considered to be a material

property and is believed to be independent of specimen size. The values obtained in

this study, given its limitations, appear to show that fracture toughness is compara-

ble, if not a little higher on the nanoscale.

Fatigue Strength

Fatigue strength measurements of Si nanobeams have been carried out by Sun-

dararajanand Bhushan[62] usingan AFM and Li and Bhushan[63] usinga nanoin-

denter. Various stress levels were applied to nanobeams by Sundararajan and

Bhushan [62]. The minimum stress was 3.5GPa for Si beams and 2.2GPaforSiO

beams. The frequency of applied load was 4.2Hz. In general, the fatigue life de-

creased with increasing mean stress as well as increasing stress amplitude. When

the stress amplitude was less than 15% of the bending strength, the fatigue life was

greater than 30,000 cycles for both Si and SiO

. However, the mean stress had to

be less than 30% of the bending strength for a life of greater than 30,000 for Si

whereas even at a mean stress of 43% of the bending strength, SiO

beams showed

762 Bharat Bhushan

Bending stress (GPa)

Number of cycles

> 30k cycles

Bending

strength

Bending stress (GPa)

Number of cycles

> 30k cycles

SiO

Bending

strength

Fig. 14.16. Fatigue test data showing applied bending stress as a function of number of cycles.

A single load–unload sequence is considered as 1 cycle. The bending strength data points are

therefore associated with 1/2 cycle, since failure occurs upon loading [62]

a life greater than 30,000. During fatigue, the beams broke under the loading point

or at the ends, when loaded at the center of the span. This was diﬀerent from the

quasi-static bending tests, where the beams broke at the ends almost every time.

This could be due to the fact that the stress levels under the load and at the ends

are not that diﬀerent and fatigue crack propagation could occur at either location.

Figure 14.16 shows a nanoscale S–N curve, with bending stress (S) as a function

of fatigue in cycles (N) with an apparent endurance life at lower stress. This study

clearly demonstrates that fatigue properties of nanoscale specimens can be studied.

SEM Observations of Fracture Surfaces

Figure 14.17 shows SEM images of the fracture surfaces of nanobeams broken dur-

ing quasi-static bending as well as fatigue [62]. In the quasi-static cases, the max-

imum tensile stresses occur on the top surface, so it is reasonable to assume that

fracture initiated at or near the top surface and propagated downward. The fracture

surfaces of the beams suggest a cleavage type of fracture. Silicon beam surfaces

show various ledges or facets, which is typical for crystalline brittle materials. Sili-

con usually fractures along the (111) plane due to this plane having the lowest sur-

face energy to overcome by a propagating crack. However, failure has also been

known to occur along the (110) planes in microscale specimens, despite the higher

energyrequired as comparedto the (111) planes [70]. The plane normal to the beam

direction in these samples is the (110) plane while (111) planes will be oriented at

◦

from the (110) plane. The presence of facets and irregularities on the silicon

surface in Fig. 14.17a suggest that it is a combination of these two types of frac-

tures that has occurred. Since the stress levels are very high for these specimens, it

is reasonable to assume that crack propagating forces will be high enough to result

in (110) type failures.

In contrast, the silicon fracture surfaces under fatigue, shown in Fig. 14.17b,ap-

pear very smooth without facets or irregularities. This is suggestive of low energy

fracture, i.e. of (111) type fracture. We do not see evidence of fatigue crack propa-

gation in the form of steps or striations on the fracture surface. We believe that for

14 Mechanical Properties of Nanostructures 763

Quasi-static bending

200 nm 200

SiO

200 nm

Fatigue

SiO

200 nm

Fatigue life = 2 k cycles

Mean stress =

8.0 GPa,

stress ampl. = 4.6 GPa

Fatigue life = 0.1 k cycles

Mean stress = 4.8 GPa

stress ampl. = 2.6 GPa

200 nm

200

SiO

200 nm

Fatigue life = 13.4 k cycles

Mean stress = 3.7 GPa

stress ampl. = 1.6 GPa

Fatigue life = 6 k cycles

Mean stress = 6.5 GPa

stress ampl. = 3.0 GPa

Fig. 14.17. SEM micrographs

of fracture surfaces of silicon

and SiO

beams subjected to

(a) quasi-static bending and

(b) fatigue [62]

the stress levels applied in these fatigue experiments, failure in silicon occurred via

cleavage associated with ‘static fatigue’ type of failures.

SiO

shows very smooth fracture surfaces for both quasi-static bending and fa-

tigue. This is in contrast to thehackled surfaceonemight expectfor the brittle failure

of an amorphous material on the macroscale. However, in larger scale fracture sur-

faces for such materials, the region near the crack initiation usually appears smooth

764 Bharat Bhushan

or mirror-like. Since the fracture surface here is so small and very near the crack

initiation site, it is not unreasonable to see such a smooth surface for SiO

on this

scale. There appears to be no diﬀerence between the fracture surfaces obtained by

quasi-static bending and fatigue for SiO

Summary of Mechanical Properties Measured Using Quasi-Static Bending Tests

Table 14.2 summarizes the various properties measured via quasi-static bending in

this study [62]. Also shown are bulk values of the parameters along with values re-

ported on larger scale specimens by other researchers. Elastic modulus and fracture

toughness values appear to be comparable to bulk values and show no dependence

on specimen size. However bending strength shows a clear specimen size depen-

dence with nanoscale numbers being twice as large as numbers reported for larger

scale specimens.

14.3.3 Bending Tests of Metallic Microbeams Using a Nanoindenter

Bending tests have been performed on Ni

−

P and Au microbeams [64]. The Ni

−

cantilever microbeams were fabricated by focused ion beam machining technique.

The dimensions were 10, ×12, ×50µm

. Notches with a depth of 3 µm and a tip

radius of 0.25µm were introduced in the microbeams to facilitate failure at a lower

load in the bending tests. The Au bridge microbeams were fabricated by electroplat-

ing technique.

Figure 14.18 shows the SEM images, load displacement curve and FEM stress

contour for the notched Ni–P cantilever microbeam that was bent to failure [64].

The distance between the loading position and the ﬁxed end is 40 µm. The 3 µm

Ni-P cantilever microbeam

Maximum

tensile

bending

stress

Compressive

Tensile

Load-displacement curve

Broken microbeam

FEM stress contour

at 20 mN

Load (mN)

46810

Displacement (μm)

20 μm

μm

– 3658 – 843 1970 4785 11599 21671MPa

20 μm

40 μm

10 μm

12 μm

Fig. 14.18. SEM micrographs of the new and broken beams, load displacement curve and

FEM stress contour for the notched Ni–P cantilever microbeam [64]

14 Mechanical Properties of Nanostructures 765

deep notch is 10µm from the ﬁxed end. The notched beam showed linear behavior

followed by abrupt failure. The FEM stress contour shows that there is higher stress

concentration at the notch tip. The maximum tensile stress σ

at the notch tip can

be analyzed by using Griﬃth fracture theory as follows [83],

≈ 2σ





, (14.11)

where σ

is the average applied tensile stress on the beam, c is the crack length,

and ρ is the crack tip radius. Therefore, elastic–plastic deformation will ﬁrst occur

locally at the end of the notch tip, followed by abrupt fracture failure after the σ

reaches the ultimate tensile strength of Ni

−

P, even though the rest of the beam is

still in the elastic regime. The SEM image of the fracture surface shows that the

fracture started right from the notch tip with plastic deformation characteristic. This

indicates that although local plastic deformation occurred at the notch tip area, the

whole beam failed catastrophically. The present study shows that FEM simulat-

ion can predict well the stress concentration, and helps in understanding the failure

mechanism of the notched beams.

Figure 14.19 shows the SEM images, load displacement curve and FEM stress

contour for the Au bridge microbeam that was deformed by the indenter [64]. The

recession gap between the beam and substrate is about 7 µm, which is not large

enough to break the beam at the load applied. From the load–displacement curve,

we note that the beamexperiencedelastic–plastic deformation.The FEM stress con-

tour shows that the maximum tensile stress is located at the ﬁxed ends whereas the

minimum compressive stress is located around the center of the beam. The SEM

– 956 – 657 – 358 – 60 240 539

MPa

838

Load-displacement curve

Load (mN)

468

Displacement (μm)

Au bridge microbeam

Deformed beam FEM stress contour at 3.2 mN

Compressive Tensile

Maximum tensile bending stress

200 μm

400 μm

Fig. 14.19. SEM micrographs

of the new and deformed

beams, load displacement

curve and FEM stress con-

tour for the Au bridge mi-

crobeam [64]

766 Bharat Bhushan

image shows that the beam has been permanently deformed.No crack was found on

the beam surface. The present study shows a possibility for mechanically forming

the Au ﬁlm into the shape as needed. This may help in designing/fabricating func-

tionally complex smart micro/nanodeviceswhich need conductorsfor power supply

and input/output signals.

14.3.4 Indentation and Scratch Tests of Polymeric Microbeams

Using a Nanoindenter

Studies have been conducted on two polymer microbeams made of PMMA and

PPMA with thickness of about 2 to 5µm [65]. PPMA was chosen due to its rela-

tively low glass transition temperature, allowing easy thermal processing of the ma-

terial during the microfabrication process. PMMA was chosen due to its wide use

in commercial biomedical applications. Table 14.3 summarizes published data on

these materials.

Hardness and Elastic Modulus

The hardness (H), elastic modulus (E) and creep of PPMA and PMMA beams were

measured at the supported region of the beams [65]. In Fig. 14.20, the indentation

location, where the H, E and creep were measured, is indicated by an arrow. Fig-

ure 14.21ashowsthe H and E of 2.9µm PPMA and 3.4 µmPMMAbeamsasafunc-

tion of contact depth [65]. The H and E were calculated by averaging the H and E

Bending

location

100 μm

20 μm

Indentation location

(H, E, creep)

Fig. 14.20. SEM images of

PPMA beams at low and high

magniﬁcations