Bhushan B. Nanotribology and Nanomechanics: An Introduction

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14 Mechanical Properties of Nanostructures 747

where k is the stiﬀness of the tip/cantilever. In this manner, a load displacement

curve for each nanobeam can be obtained.

The photodetector sensitivity of the cantilever needs to be calibrated to obtain

tip

in nm. For this calibration, the tip is pushed against a smooth diamond sample

by moving the Z-piezo over a known distance. For the hard diamond material, the

actual deﬂection of the tip can be assumed to be the same as the Z-piezo travel

piezo

), and the photodetector sensitivity (S) for the cantilever setup is determined

S = D

piezo

/dV

AFM

nm/V . (14.4)

In the measurements, D

tip

is given as dV

AFM

×S.

Since a sharp tip would result in an undesirable large local indentation, Sun-

dararajan and Bhushan [62] used a diamond tip which was a worn (blunt) diamond

tip. Indentation experiments using this tip on a silicon substrate yielded a residual

depth of less than 8nm at a maximum load of 120µN, which is negligible com-

pared to displacements of the beams (several hundred nm). Hence we can assume

that negligible local indentation or damage is created during the bending process

of the beams and that the displacement calculated from (14.2) is from the beam

structure.

Elastic Modulus and Bending Strength

Elasticmodulus and bendingstrength(fracture stress) of the beams can be estimated

by equations based on the assumption that the beams follow linear elastic theory of

an isotropic material. This is probably valid since the beams have high length-to-

width /w and length-to-thickness /t ratios and also since the length direction is

along the principal stress direction during the test. For a ﬁxed elastic beam loaded

at the center of the span, the elastic modulus is expressed as

E =



192I

m , (14.5)

where  is the beam length, I is the area moment of inertia for the beam cross-

section and m is the slope of the load-displacement curve during bending [85]. The

area moment of inertia for a beam with trapezoidal cross section, is calculated from

the following equation

I =

+ w

(

+ w

)

, (14.6)

where w

and w

are the upper and lower widths, respectively, and t is the thickness

of the beam. According to linear elastic theory, for a centrally loaded beam, the

moment diagramis shown in Fig. 14.2. The maximummoments are generated at the

ends (negative moment) and under the loading point (positive moment) as shown

in Fig. 14.2. The bending stresses generated in the beam are proportional to the

moments and are compressive or tensile about the neutral axis (line of zero stress).

748 Bharat Bhushan

Beam

Neutral

axis

Positive

Bending

moment

Fig. 14.2. A schematic of the

bending moments generated

in the beam during a quasi-

static bending experiment,

with the load at the center

of the span. The maximum

moments occur under the

load and at the ﬁxed ends.

Due to the trapezoidal cross

section, the maximum tensile

bending stresses occur at the

top surfaces at the ﬁxed ends

The maximum tensile stress (σ

, which is the bending strength or fracture stress) is

produced on the top surface at both the ends and is given by [85]

max

e

, (14.7)

where F

max

is the applied load at failure and e

is the distance of the top surface

from the neutral plane of the beam cross-section and is given by [85]

(

+ 2w

)

(

+ w

)

. (14.8)

Although the moment value at the center of the beam is the same as at the ends, the

tensile stresses at the center (generated on the bottom surface) are less than those

generated at the ends (per (14.7)) because the distance from the neutral axis to the

bottom surface is less than e

. This is because of the trapezoidal cross section of the

beam, which results in the neutral axis being closer to the bottom surface than the

top (Fig. 14.2).

In the preceding analysis, the beams were assumed to haveﬁxed ends. However,

in the nanobeams used by Sundararajan and Bhushan [62], the underside of the

beams was pinned over some distance on either side of the span. Hence a ﬁnite

element model of the beams was created to see if the diﬀerence in the boundary

conditions aﬀected the stresses and displacements of the beams. It was found that

the diﬀerence in the stresses was less than 1%. This indicates that the boundary

conditions near the ends of the actual beams are not that diﬀerent from that of ﬁxed

ends. Therefore the bending strength values can be calculated from (14.7).

Fracture Toughness

Fracture toughness is another important parameter for brittle materials such as sili-

con. In the case of the nanobeamarrays, these are not best suited for fracture tough-

ness measurements because they do not possess regions of uniform stress during

14 Mechanical Properties of Nanostructures 749

1500

750

250

500

nm nm

250

Scratch

–50

250 5000

b)a)

Beam

Substrate

Diamond tip

Load Stress

concentration

Fracture

load

Maximum tensile

bending stress

– 25 – 16 – 11 – 4 3 10 17

GPa

Fig. 14.3. (a) Schematic of technique to generate a defect (crack) of known dimensions in

order to estimate fracture toughness. A diamond tip is used to generate a scratch across the

width of the beam. When the beam is loaded as shown, a stress concentration is formed at

the bottom of the scratch. The fracture load is then used to evaluate the stresses using FEM.

(b) AFM 3-D image and 2-D proﬁle of a typical scratch. (c) Finite element model results

verifying that the maximum bending stress occurs at the bottom of the scratch [62]

bending. Sundararajan and Bhushan [62] developed a methodology and its steps

are outlined schematically in Fig. 14.3a. First, a crack of known geometry is intro-

duced in the region of maximum tensile bending stress, i.e. on the top surface near

the ends of the beam. This is achieved by generating a scratch at high normal load

across the width (w

) of the beam using a sharp diamond tip (radius < 100nm).

A typical scratch thus generated is shown in Fig. 14.3b. By bending the beam as

shown, a stress concentration will be formed under the scratch. This will lead to

failure of the beam under the scratch once a critical load (fracture load) is attained.

The fracture load and relevant dimensions of the scratch are input into the FEM

model, which is used to generate the fracture stress plots. Figure 14.3c shows an

FEM simulation of one such experiment, which reveals that the maximum stress

does occur under the scratch.

If we assume that the scratch tip acts as a crack tip, a bending stress will tend

to open the crack in Mode I. In this case, the stress ﬁeld around the crack tip can

be described by the stress intensity parameter K

(for Mode I) for linear elastic

materials [86]. In particular the stresses corresponding to the bending stresses are

described by

σ =

√

2πr

cos







1+ sin





sin



3θ



(14.9)

for every point p(r,θ) around the crack tip as shown in Fig. 14.4.If we substitute the

fracture stress (σ

) into the left hand side of (14.9), then the K

value can be sub-

stituted by its critical value, which is the fracture toughness K

.Now,thefracture

stress can be determined for the point (r = 0,θ = 0), i.e. right under the crack tip as

explained above.However, we cannot substitute r = 0 in (14.9). The alternativeis to

substitute a value for r, which is as close to zero as possible. For silicon, a reason-

750 Bharat Bhushan

Bending

stress

Crack

p(r, θ)

Fig. 14.4. Schematic of crack tip and coordinate systems used in (14.9) to describe a stress

ﬁeld around the crack tip in terms of the stress intensity parameter, K

[62]

able number is the distance between neighboringatoms in the (111) plane, the plane

along which silicon exhibits the lowest fracture energy. This value was calculated

from silicon unit cell dimensions of 0.5431nm [87] to be 0.4 nm (half of the face

diagonal). This assumes that Si displays no plastic zone around the crack tip, which

is reasonable since in tension, silicon is not known to display much plastic defor-

mation at room temperature. Sundararajan and Bhushan [62] used values r = 0.4

to 1.6 nm (i.e. distances up to 4 times the distance between the nearest neighbor-

ing atoms) to estimate the fracture toughness for both Si and SiO

according to the

following equation

= σ

√

2πrr= 0.4to1.6nm. (14.10)

Fatigue Strength

In addition to the properties mentioned so far that can be evaluated from quasi-static

bending tests, the fatigue propertiesof nanostructuresare also of interest. This is es-

pecially true for MEMS/NEMS involving vibrating structures such as oscillators

and comb drives [88] and hinges in digital micromirror devices [89]. To study the

fatigue propertiesof the nanobeams,Sundararajanand Bhushan[62] appliedmono-

tonic cyclic stresses using an AFM, Fig. 14.5a. Similar to the bending test, the dia-

mond tip is ﬁrst positioned at the center of the beam span. In order to ensure that the

tip is always in contact with the beam (as opposed to impacting it), the piezo is ﬁrst

extended by a distance D

, which ensures a minimum stress on the beam. After this

extension, a cyclic displacement of amplitude D

is applied continuously until fail-

ure of the beam occurs. This results in the application of a cyclic load to the beam.

The maximum frequency of the cyclic load that could be attained using the AFM by

Sundararajanand Bhushan [62] was 4.2 Hz. The vertical deﬂection signal of the tip

is monitored throughout the experiment. The signal follows the pattern of the piezo

input up to failure, which is indicated by a sudden drop in the signal. During initial

runs, piezo drift was observed that caused the piezo to gradually move away from

the beam (i.e. to retract), resulting in a continuous decrease in the applied normal

load. In order to compensate for this, the piezo is given a ﬁnite extension of 75nm

every 300seconds as shown in Fig. 14.5a. This results in keeping the applied loads

fairly constant. The normal load variation (calculated from the vertical deﬂection

signal) from a fatigue test is shown in Fig. 14.5b. The cyclic stress amplitudes (cor-

14 Mechanical Properties of Nanostructures 751

Beam

Piezo

Cyclic load

Frequency = 4.2 Hz

Compensate for piezo drift

75 nm every 300 s

Time

Piezo extension

125

100

Normal load (μN)

0 200 400 600

Time (s)

Piezo drift

compensation

Failure

Fig. 14.5. (a) Schematic

showing the details of the

technique to study fatigue

behavior of the nanobeams.

The diamond tip is located

at the middle of the span

and a cyclic load at 4.2Hz

is applied to the beam by

forcing the piezo to move in

the pattern shown. An ex-

tension is made every 300 s

to compensate for the piezo

drift to ensure that the load on

the beam is kept fairly con-

stant. (b) Data from a fatigue

experiment on a nanobeam

until failure. The normal load

is computed from the raw

vertical deﬂection signal.

The compensations for piezo

drift keep the load fairly con-

stant [62]

responding to D

) and fatigue lives are recorded for every sample tested. Values for

are set such that minimum stress levels are about 20% of the bending strengths.

14.2.3 Bending Tests of Micro/Nanostructures Using a Nanoindenter

Quasi-static bending tests of micro/nanostructuresare also carried out using a nano-

indenter(Fig. 14.6) [63–65].The advantageof nanoindenteris that loads up to about

400mN, higher than that in AFM (up to about 100µN), can be used for structures

requiringhigh loadsfor experiments.To avoid theindenter tip pushinginto thespec-

imen, a blunt tip is used in the bending and fatigue tests. For ceramic and metallic

752 Bharat Bhushan

Capacitive

displacement gauge

Heating

stage

Load application coil/magnet

Indenter

support springs

Indenter

support springs

Lateral force probe

Lateral motion stage

Load frame

Indenter column

Indenter tip

Beam

sample

Fig. 14.6. Schematic of mi-

cro/nanoscale bending test

using a nanoindenter

beam samples, Li et al. [64] used a diamond conical indenter with a radius of 1µm

and an included angle of 60

◦

. For polymer beam samples, Wei et al. [65] reported

that the diamond tip penetrated the polymer beams easily and caused considerable

plastic deformation during the bending test, which led to signiﬁcant errors in the

measurements. To avoid this issue, the diamond tip was dip-coated with PMMA

(about 1–2 µm thick) by dipping the tip in the 2% PMMA (wt/wt)solution forabout

5 seconds. Load position used was at the center of the span for the bridge beams and

at 10µmoﬀ from the free end of cantilever beams. An optical microscope with

a magniﬁcation of 1500 × or an in-situ AFM is used to locate the loading position.

Then the specimen is moved under the indenter location with a resolution of about

200 nm in longitudinal direction and less than 100nm in lateral direction.

Using the analysis presented earlier, elastic modulus and bending strength of

the beams can be obtained from the load displacement curves [64,65]. For fatigue

tests, an oscillating load is applied and contact stiﬀness is measured during the tests.

A signiﬁcant drop in the contact stiﬀness during the test is a measure of number of

cycles to failure [63].

14.3 Experimental Results and Discussion

14.3.1 Indentation and Scratch Tests of Various Ceramic and Metals

Using a Micro/Nanoindenter

Studies have been conducted on ﬁve diﬀerent materials: undoped single-crystal

Si(100), undoped polysilicon ﬁlm, SiO

ﬁlm, SiC ﬁlm, electroless deposited Ni–

14 Mechanical Properties of Nanostructures 753

11.5wt% P amorphousﬁlm, and electroplatedAu ﬁlm [64,75,76].A 3µm thickpo-

lysilicon ﬁlm was deposited by a low pressure chemical vapor deposition

(LPCVD) process on an Si(100) substrate. The 1µmthickSiO

ﬁlm was de-

posited by a plasma enhanced chemical vapor deposition (PECVD) process on an

Si(111) substrate. A 3µm thick 3C-SiC ﬁlm was epitaxially grown using an atmo-

spheric pressure chemical vapor deposition (APCVD) process on Si(100) substrate.

A12µmthickNi

−

P ﬁlm was electroless plated on a 0.8mmthickAl−4.5wt%Mg

alloy substrate. A 3µm thick Au ﬁlm was electroplated on an Si(100) substrate.

Hardness and Elastic Modulus

Hardness and elastic modulus measurements are made using a nanoindenter [64].

The hardness and elastic modulus values of various materials at a peak indentation

depth of 50nm are summarized in Fig. 14.7 and Table 14.1. The SiC ﬁlm exhibits

the highest hardness of about 25GPa an elastic modulus of about 395GPa among

the samples examined, followed by the undoped Si(100), undoped polysilicon ﬁlm,

SiO

ﬁlm, Ni

−

P ﬁlm, and Au ﬁlm. The hardness and elastic modulus data of the

undoped Si(100) and undoped polysilicon ﬁlm are comparable. For the metal al-

loy ﬁlms, the Ni

−

P ﬁlm exhibits higher hardness and elastic modulus than the Au

ﬁlm.

Fracture Toughness

The optical images of Vickers indentations made using a microindenter,at a normal

load of 0.5N held for 15 s on the undoped Si(100), undoped polysilicon ﬁlm, and

SiC ﬁlm are shown in Fig. 14.8 [76]. The SiC ﬁlm exhibits the smallest indenta-

tion mark, followed by the undoped polysilicon ﬁlm and undoped Si(100). These

Vickers indentation depths are smaller than one-third of the ﬁlm thickness. Thus,

the inﬂuence of substrate on the fracture toughness of the ﬁlms can be ignored. In

addition to the indentation marks, radial cracks are observed, emanating from the

indentation corners. The SiC ﬁlm shows the longest radial crack length, followed

by the undoped Si(100) and undoped polysilicon ﬁlm. The radial cracks for the un-

doped Si(100) are straight whereas those for the SiC, undoped polysilicon ﬁlm are

Table 14.1. Hardness, elastic modulus, fracture toughness and critical load results of the bulk

single-crystal Si(100) and thin ﬁlms of undoped polysilicon, SiO

,SiC,Ni

−

PandAu

Samples Hardness Elastic modulus Fracture toughness Critical load

(GPa) (GPa) (MPam

1/2

)(mN)

Undoped Si(100) 12 165 0.75 11

Undoped polysilicon ﬁlm 12 167 1.11 11

SiO

ﬁlm 9.5 144 0.58 (Bulk) 9.5

SiC ﬁlm 24.5 395 0.78 14

−

Pﬁlm 6.5 130 0.4 (Plowing)

Au ﬁlm 4 72 0.4 (Plowing)

754 Bharat Bhushan

not straight but go in a zigzag manner. The fracture toughness (K

) is calculated

using (14.1).

The fracture toughness values of all samples are summarized in Fig. 14.7 and

Table 14.1. The SiO

ﬁlm used in this study is about 1 µm thick, which is not thick

enough for fracture toughness measurement. The fracture toughness value of bulk

silica are listed instead for a reference. The Ni

−

P and Au ﬁlms exhibit very high

fracture toughness values that cannot be measured by indentation methods. For

other samples, the undoped polysilicon ﬁlm has the highest value, followed by the

Critical load (mN)

Plowing

(MPa m

1/2

)

Bulk

500

400

300

200

100

Elastic modulus (GPa)

Hardness (GPa)

Undoped

Si (100)

Undoped

poly-

silicon

film

SiO

film

SiC

film

Ni-P

film

Au-

film

Fig. 14.7. Bar chart summa-

rizing the hardness, elastic

modulus, fracture toughness,

and critical load (from scratch

tests) results of the bulk un-

doped single-crystal Si(100)

and thin ﬁlms of undoped

polysilicon, SiO

,SiC,Ni

−

andAu[64]

14 Mechanical Properties of Nanostructures 755

Undoped Si (100) Undoped polysilicon film

5 μm

SiC film

Fig. 14.8. Optical images of Vickers indentations made at a normal load of 0.5Nheldfor15s

on the undoped Si(100), undoped polysilicon ﬁlm, and SiC ﬁlm [76]

undoped Si(100), SiC ﬁlm, and SiO

ﬁlm. For the undoped polysilicon ﬁlm, the

grain boundaries can stop the radial cracks and change the propagation directions

of the radial cracks, making the propagation of these cracks more diﬃcult. Values

of fracture toughness for the undoped Si(100) and SiC ﬁlm are comparable. Since

the undoped Si(100) and SiC ﬁlm are single crystal, no grain boundaries are present

to stop the radial cracks and change the propagation directions of the radial cracks.

This is why the SiC ﬁlm shows a lower fracture toughness value than the bulk poly-

crystal SiC materials of 3.6MPam

1/2

[90].

Scratch Resistance

Scratch resistance of various materials have been studied using a nanoindenter by

Li et al. [64]. Figure 14.9 compares the coeﬃcient of friction and scratch depth pro-

ﬁles as a function of increasing normal load and optical images of three regions

over scratches: at the beginning of the scratch (indicated by A on the friction pro-

ﬁle), at the point of initiation of damage at which the coeﬃcientof friction increases

to a high value or increase abruptly (indicated by B on the friction proﬁle), and to-

wards the end of the scratch (indicated by C on the friction proﬁle) for all samples.

Note that the ramp loads for the Ni

−

P and Au range are from 0.2 to 5 mN whereas

the ramp loads for other samples are from 0.2 to 20mN. All samples exhibit a con-

tinuous increase in the coeﬃcient of friction with increasing normal load from the

beginning of the scratch. The continuous increase in the coeﬃcient of friction dur-

ing scratching is attributed to the increasing plowing of the sample by the tip with

increasing normal load, as shown in the SEM images in Fig. 14.9. The abrupt in-

crease in the coeﬃcient of friction is associated with catastrophic failure as well

as signiﬁcant plowing of the tip into the sample. Before the critical load, the coeﬃ-

cient of friction ofthe undoped polysilicon,SiC and SiO

ﬁlms increased at a slower

rate, and was smoother than that of the other samples. The undoped Si(100)exhibits

some bursts in the friction proﬁles before the critical load. At the critical load, the

SiC and undoped polysilicon ﬁlms exhibit a small increase in the coeﬃcient of fric-

tion whereas the undoped Si(100) and undoped polysilicon ﬁlm exhibit a sudden

increase in the coeﬃcient of friction. The Ni

−

P and Au ﬁlms show a continuous

increase in the coeﬃcient of friction, indicating the behavior of a ductile metal. The

756 Bharat Bhushan

Coefficient of friction Scratch depth (nm)

Before scratch

During scratch

After scratch

Friction

1.2

0.8

0.4

0.0

50 101520

500

– 500

– 1000

– 1500

Undoped polysilicon film

1.2

0.8

0.4

0.0

50 101520

500

– 500

– 1000

– 1500

SiO

film

1.2

0.8

0.4

0.0

50 101520

500

– 500

– 1000

– 1500

SiC film

1.2

0.8

0.4

0.0

10234

1000

– 1000

–3000

Ni-P film

1.2

0.8

0.4

0.0

1023

1000

– 1000

–3000

Au film

1.2

0.8

0.4

0.0

50 101520

500

– 500

– 1000

– 1500

Undoped Si(100)

Region A Region B Region C

Normal load (mN)

2 μm

Fig. 14.9. Coeﬃcient of friction and scratch depth proﬁles as a function of increasing nor-

mal load and optical images of three regions over scratches: at the beginning of the scratch

(indicated by A on the friction proﬁle), at the point of initiation of damage at which the coef-

ﬁcient of friction increases to a high value or increase abruptly (indicated by B on the friction

proﬁle), and towards the end of the scratch (indicated by C on the friction proﬁle) for all

samples [64]

bursts in the friction proﬁle might result from the plastic deformation and material

pile-up in front of the scratch tip. The Au ﬁlm exhibits a higher coeﬃcient of fric-

tionthantheNi

−

P ﬁlm. This is because the Au ﬁlm has lower hardness and elastic

modulus values than the Ni

−

Pﬁlm.