Bhushan B. Nanotribology and Nanomechanics: An Introduction

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23 Mechanical Properties of Micromachined Structures 1305

Young’s Modulus Measurements

Young’s modulus, E, is a material property critical to any structural device design.

It describes the elastic response of a material and relates stress, σ,andstrain,ε,by

σ = Eε. (23.5)

In bulk samples, E is often measured by loading a specimen under tension and

measuring displacement as a function of stress for a given length. While this is

far more diﬃcult for small structures, such as those fabricated from thin deposited

ﬁlms, it can be achieved with careful experimental techniques. Figure 23.7 shows a

schematic of one such measurement system [15]. The fringe detectors in the ﬁgure

detect the reﬂected laser signal from two gold lines deposited onto the polysilicon

specimen, which act as gauge markers. This enables the strain in the specimen dur-

ing loading to be monitored. Besides gold lines, Vickers indents placed in a nickel

specimen can also serve as gauge markers [16], or a speckle interferometry tech-

nique [17] can be used to determine strain in the specimen. Once the stress-versus-

strain behavior is measured, the slope of the curve is equal to E. By using a constant

load, such as a dead weight, and resistive heating, high-temperature creep can also

be investigated with this method [18].

In addition to the tensile test, Young’s modulus can be determined by other mea-

sures of stress–strain behavior. As seen in Fig. 23.8, a cantilever beam can be bent

by pushing on the free end with a nanoindenter [19]. The nanoindenter can moni-

tor the force applied and the displacement, and simple beam theory can convert the

displacement into strain in order to obtain E. A similar technique, shown schemati-

cally in Fig. 23.9 [20], involves pulling downwardon a cantilever beam by means of

an electrostatic force. An electrode is fabricated into the substrate beneath the can-

tilever beam, and a voltage is applied between the beam and the bottom electrode.

The force acting on the beam is equal to the electrostatic force corrected to include

the eﬀects of fringing ﬁelds acting on the sides of the beam, namely

F(x) =



g+ z(x)





0.65[g+ z(x)]



, (23.6)

Fringe

detector

Laser

Air

bearing

Piezoelectric

translator

Load

cell

Grips

Specimen Base

Fig. 23.7. Schematic of

a measurement system for

tensile loading of microma-

chined specimens [15]

1306 Harold Kahn

Identation

marker

Edge of beam

Indenter shaft

Diamond tip

point

Coil

Capacitance

Gage

Suspending

Magnet

Spring

Si Substrate

Fig. 23.8. Schematic of

a nanoindenter loading mech-

anism pushing on the end of

a cantilever beam [19]

Gap

V =0V

R =1/κ

Fig. 23.9. Schematic of a can-

tilever beam bending test us-

ing an electrostatic voltage

to pull the beam toward the

substrate [20]

where F(x) is the electrostatic force at x, ε

is the dielectric constant of air, g is

the gap between the beam and the bottom electrode, z(x) is the out-of-plane deﬂec-

tion of the beam, w is the beam width, and V is the applied voltage [20]. In this

work, the deﬂection of the beam as a function of position is measured using optical

interferometry. These measurements combine to give stress–strain behavior for the

cantilever beam. An extension of this technique uses doubly clamped beams instead

of cantilever beams. In this case, the deﬂection of the beam at a given electrostatic

force dependson the residual stress in the materialas well as Young’smodulus. This

method can therefore also be used to measure residual stresses in doubly clamped

beams.

Another device that can be fabricated from a thin ﬁlm and used to investigate

stress–strain behavior is a suspended membrane, as shown in Fig. 23.10 [21]. As

depicted in the schematic ﬁgure, the membrane is exposed to an elevated pressure

on one side, causing it to bulge in the opposite direction. The deﬂection of the mem-

braneis measuredby optical orother techniquesand relatedto the strain in themem-

brane. These membranes can be fabricated in any shape, typically square or circular.

Both analytical solutions and ﬁnite element analyses have been performed to relate

23 Mechanical Properties of Micromachined Structures 1307

Pressure (p)

Fig. 23.10. Schematic cross-

section of a microfabricated

membrane [21]

the deﬂection to the strain. Like the doubly clamped beams, both Young’s modulus

and residual stress play a role in the deﬂected shape. Both of these mechanicalprop-

erties can therefore be determined by the pressure-versus-deﬂectionperformance of

the membrane.

Another measurementbesides stress–strain behaviorthat can reveal the Young’s

modulus of a material is the determination of the natural resonance frequency. For

a cantilever, the resonance frequency, f

, for free undamped vibration is given by

4πl



3ρ



1/2

, (23.7)

where ρ, l and t are the density, length and thickness of the cantilever; λ

is the

eigenvalue, where i is an integer that describes the resonance mode number; for the

ﬁrstmode λ

= 1.875 [23].Giventhe geometryand density, measuring f

allows E to

be determined. The cantilever can be vibrated by a number of techniques, including

a laser, loudspeaker or piezoelectricshaker. The frequencythat producesthe highest

amplitude of vibration is the resonance frequency.

A micromachined device that uses an electrostatic comb drive and an AC signal

to generate the vibration of the structure is known as a lateral resonator [24]. One

example is shown in Fig. 23.11 [22]. When a voltage is applied across either set

of the interdigitated comb ﬁngers shown in Fig. 23.11, an electrostatic attraction is

generateddue to the increase in capacitance as the overlapbetween the comb ﬁngers

increases. The force, F, generated by the comb-driveis given by

F =

∂C

∂x

= nε

, (23.8)

where C is capacitance, x is the distance traveled by one comb-drive toward the

other, n is the number of pairs of comb ﬁngers in one drive, ε is the permittivity

of the ﬂuid between the ﬁngers, h is the height of the ﬁngers, g is the gap spacing

between the ﬁngers, and V is the applied voltage [24]. When an AC voltage at the

resonance frequency is applied across either of the two comb drives, the central

portion of the device will vibrate. In fact, since force depends on the square of the

voltagefor electrostatic actuation,for a time t, a dependentdrivevoltagev

(t)(given

(t) = V

+ v

sin(ωt) , (23.9)

1308 Harold Kahn

100 μm

Fig. 23.11. SEM of a polysi-

licon lateral resonator [22]

where V

is the DC bias and v

is the AC drive amplitude), the time-dependent

portion of the force will scale with

2ωV

cos(ωt)+ ωv

sin(2ωt) (23.10)

[24]. Therefore, if an AC drive signal is used with no DC bias, at resonance, the

frequency of the AC drive signal will be one half of the resonance frequency. For

this device, the resonance frequency f

will be

2π



sys



1/2

, (23.11)

where k

sys

is the spring constant of the support beams and M is the mass of the

portion of the device that vibrates. The spring constant is given by

sys

= 24EI/L

, (23.12)

I =

, (23.13)

where I is the moment of inertia of the beams, and L, h and w are the length, thick-

ness, and width of each beam. Therefore, by combining these equations and meas-

uring f

, it is possible to determine E.

One distinct advantage of the lateral resonator technique and the electrostati-

cally pulled cantilever techniquefor measuringYoung’s modulus is that they require

no external loading sources. Portions of the devices are electrically contacted, and

a voltage is applied. For the pure tension tests, as shown in Fig. 23.7, the specimen

must be attached to a loading system, which can be extremely diﬃcult for the very

small specimens discussed here, and any misalignment or eccentricity in the test

23 Mechanical Properties of Micromachined Structures 1309

could lead to unreliable results. However, the advantage of the externally loaded

technique is that there are no limitations on the type of materials that can be tested.

Conductivity is not a requirement,nor is any compatibilitywith electrical actuation.

Strength and Fracture Toughness Measurements

As one might expect, any of the techniques discussed in the previous section that

strain specimens in order to measure Young’s modulus can also be used to measure

fracture strength.The load is simply increased until the specimen breaks. As long as

either theload or the strainis measuredat fracture, and thegeometryof the specimen

is known, the maximum stress required for fracture σ

crit

can be determined, either

throughanalytical analysisor FEA. Depending on the geometryof the test, σ

crit

will

represent the tensile or bend strength of the material.

If the available force is limited, or if a localized fracture site is desired, stress

concentrations can be added to the specimens. These are typically notches micro-

machined into the edges of specimens. Focused ion beams have also been used to

carve stress concentrations into fracture specimens.

All of the external loading schemes, such as those shown in Figs. 23.7 and 23.8,

have been used to measure fracture strength. Also, the electrostatically loaded dou-

bly clamped beams can be pulled until they fracture. In this case, there is one com-

plication. The electrostatic force is inversely proportional to the distance between

the electrodes, and at a certain voltage, called the “pull-in voltage,” the attraction

between the beam and the substrate will become so great that the beam will imme-

diatelybe pulled intocontact with the bottomelectrode.As long as thefracture takes

place before the pull-in voltage is reached, the experiment will give valid results.

Other loading techniques have been used to generate fracture of microspeci-

mens. Figure 23.12 [25] shows one device designed to be pushed by the end of

a micromanipulatedneedle. The long beams that extendfrom the sides of the central

shuttle come into contact with anchored posts, and, at a critical degree of bending,

the beams will break oﬀ. Since the applied force cannot be measured in this tech-

nique, the experiment is continuously optically monitored during the test, and the

image of the beams just before fracture is analyzed to determine σ

crit

Another loading scheme that has been demonstrated for micromachined speci-

mens utilizes scratch drive actuators to load the specimens [26]. These types of ac-

Fig. 23.12. (a) SEM of a device for measuring bend strength of polysilicon beams; (b)image

of a test in process; (c) higher magniﬁcation view of one beam shortly before breaking [25]

1310 Harold Kahn

tuators work like inchworms, traveling across a substrate in discrete advances as an

electrostatic force is repeatedly applied between the actuator and the substrate. The

stepping motion can be made on the nanometer scale, depending on the frequency

of the applied voltage, and so it can be an acceptable approximation to continuous

loading. One advantage of this scheme is that very large forces can be generated

by relatively small devices. The exact forces generated cannot be measured, so (like

the techniquethat used micromanipulatedpushing)the testis continuouslyobserved

to determine the strain at fracture. Another advantage of this technique is that, like

the lateral resonatorand the electrically pulled cantilever,the loading takes place on-

chip, and therefore the diﬃculties associated with attaching and aligning an external

loading source are eliminated.

Another on-chip actuator used to load microspecimens is shown in Fig. 23.13,

along with three diﬀerent microspecimens [14]. Devices have been fabricated with

each of the three microspecimens integrated with the same electrostatic comb-drive

actuator. In all three cases, when a DC voltage is applied to the actuator, it moves

downward, as oriented in Fig. 23.13. This pulls down on the left end of each of

the three microspecimens, which are anchored on the right. The actuator contains

1486pairs of comb ﬁngers. The maximum voltage that can be applied is limited by

the breakdownvoltage of the medium in which the test takes place. In air, this limits

the voltage to less than 200V. As a result, given a ﬁnger height of 4 µmandagap

of 2µm, and using (23.8), the maximum force generated by this actuator is limited

to about 1 mN. Standard optical photolithography has a minimum feature size of

about 2µm. As a result, the electrostatic actuator cannot generate suﬃcient force to

perform a standard tensile test on MEMS structural materials such as polysilicon.

The microspecimens shown in Fig. 23.13 are therefore designed such that the stress

is ampliﬁed.

The specimen shown in Fig. 23.13b is designed to measure bend strength. It

contains a micromachined notch with a rootradius of 1µm. When the actuator pulls

downward on the left end of this specimen, the notch serves as a stress concentra-

tion, and when the stress at the notch root exceeds σ

crit

, the specimen fractures. The

specimen in Fig. 23.13c is designed to test tensile strength. When the left end of this

specimen is pulled downward, a tensile stress is generated in the upper thin hori-

zontal beam near the right end of the specimen. As the actuator continues to move

downward, the tensile stress in this beam will exceed the tensile strength, causing

fracture. Finally, the specimen in Fig. 23.13d is similar to that in Fig. 23.13b, except

that the notch is replaced by a sharp pre-crack that was produced by the Vickers

indent placed on the substrate near the specimen. When this specimen is loaded,

a stress intensity K is generated at the crack tip. When the stress intensity exceeds

a critical value K

, the crack propagates.K

is also referredto as the fracture tough-

ness.

The force generated by the electrostatic actuator can be calculated using (23.8).

However, (23.8) assumes a perfectly planar, two-dimensional device. In fact, when

actuated,the electric ﬁelds extend out of the plane of the device,and so (23.8) is just

an approximation. Instead, like many of the techniques discussed in this section, the

23 Mechanical Properties of Micromachined Structures 1311

500 μm

Microspecimen

100 μm

Fig. 23.13. (a)SEMofami-

cromachined device for con-

ducting strength tests; the

device consists of a large

comb-drive electrostatic ac-

tuator integrated with a mi-

crospecimen; (b)–(d)SEMs

of various microspecimens

for testing bend strength,

tensile strength and fracture

toughness, respectively [14]

test is continuously monitored, and the actuator displacement at the time of fracture

is recorded. Then FEA is used to determine the magnitude of the stress or stress

intensity seen by the specimen at the point of fracture.

In order to generate suﬃcient force to conduct tensile tests, a similar device

to that shown in Fig. 23.13 has been designed which uses an array of parallel plate

capacitors to providethe force, insteadof comb-drives[28]. In this way theavailable

force is increased but the maximum stroke is severely limited.

1312 Harold Kahn

Fatigue Measurements

A beneﬁt of the electrostatic actuator shown in Fig. 23.13 is that, besides mono-

tonic loading, it can generate cyclic loading. This allows the fatigue resistance of

materials to be studied. Simply by using an AC signal instead of a DC voltage, the

device can be driven at its resonant frequency. The amplitude of the resonance de-

pends on the magnitude of the AC signal. This amplitude can be increased until the

specimen breaks; this will investigate the low-cycle fatigue resistance. Otherwise,

the amplitudeof resonancecan be left constant at a level below that required for fast

fracture, and the device will resonate indeﬁnitely until the specimen breaks; this

will investigate high-cycle fatigue. It should be noted that the resonance frequency

of such a device is about 10kHz. Therefore, it is possible to stress a specimen for

over 10

cycles in less than a day. In addition to simple cyclic loading, a mean stress

can be superimposed on the cyclic load if a DC bias is added to the AC signal. In

this way, nonsymmetric cyclic loading (with a large tensile stress alternating with

a small compressive stress, or vice versa) can be studied.

Another device that can be used to investigate fatigue resistance in MEMS

materials is shown in Fig. 23.14 [27]. In this case, a large mass is attached to the

end of a notched cantilever beam. The mass contains two comb drives on opposite

ends. When an AC signal is applied to one comb drive, the device will resonate,

cyclically loading the notch. The comb drive on the opposite side is used as a cap-

acitive displacement sensor. This device contains many fewer comb ﬁngers than the

device shown in Fig. 23.13. As a result, it can apply cyclic loads by exploiting the

resonance frequency of the device, but it cannot supply suﬃcient force to achieve

monotonic loading.

Fatigueloadinghas also been studiedusing thesame externalloading techniques

shown in Fig. 23.7. In this case, the frequency of the cyclic load is considerably

100 μm 10 μm

Fig. 23.14. SEM of a device used to investigate fatigue; the image on the right is a higher

magniﬁcation view of the notch near the base ofthe moving part of the structure [27]: a) mass,

b) comb drive actuator, c) capacitive displacement sensor

23 Mechanical Properties of Micromachined Structures 1313

lower, since the resonance frequency of the device is not being utilized. This leads

to longer high-cycle testing times. Since the force is essentially unlimited, however,

this technique allows a variety of frequencies to be studied to determine their eﬀect

on the fatigue behavior.

Friction Measurements

Friction is another property that has been studied in micromachined structures. To

study friction, of course, two surfaces must be brought into contact with each other.

This is usually avoided at all costs for these devices because of the risk of stic-

tion. (Stiction is the term used when two surfaces that come into contact adhere so

strongly that they cannot be separated.) Even so, a few devices have been designed

that can investigate friction. One of these is shown schematically in Fig. 23.15 [29].

It consists of a movable structure with a comb-drive on one end and a cantilever

beam on the other. Beneath the cantilever, on the substrate, is a planar electrode.

The device is moved to one side using the comb-drive. Then a voltage is applied

between the cantilever beam and the substrate electrode. The voltage on the comb-

drive is then released. The device would normally return to its original position, to

relax the deﬂection in the truss suspensions, but the friction between the cantilever

and the substrate electrode holds it in place. The voltage to the substrate electrode is

slowly decreased until the device starts to slide. Given the electrostatic force gener-

ated bythe substrate electrodeand the stiﬀness of the truss suspensions,it is possible

to determine the static friction. For this device,bumps were fabricated on the bottom

of each cantilever beam. This limited the surface area that came into contact with

the substrate and so lowered the risk of stiction.

Anotherdevice designed to study friction is shown in Fig. 23.16 [30]. This tech-

nique uses a hinged cantilever. The portion near the free end acts as the friction

test structure, and the portion near the anchored end acts as the driver. The friction

Truss suspension

Position

marker

Dimple

Nitride

pad

=0 V

Nitride pad

Electrode Dimple

clamp

friction

= V

normal

suspension

friction

A–A B–B

Polysilicon

ground plane

a) b)

Fig. 23.15a,b. Schematics depicting a device used to study friction. (a) shows top and side

views of the device in its original position, and (b) shows views of the device after it has been

displaced using the comb-drive and clamped using the substrate electrode [29]

1314 Harold Kahn

200 μm

Slip

Fig. 23.16. Schematic cross-section and top-view optical micrograph of a hinged-cantilever

test structure for measuring friction in micromachined devices [30]

test structure is attracted to the substrate by means of electrostatic actuation, and

when a second electrostatic actuator pulls down the driver, the friction test structure

slips forwardby a length proportionalto the forces involved,including the frictional

force. This distance, however, has a maximum of 30nm, so all measurements must

be exceedingly accurate in order to investigate a range of forces. This test struc-

ture can be used to determine the friction coeﬃcients for surfaces with and without

lubricating coatings.

23.3 Measurements of Mechanical Properties

All of the techniques discussed in Sects. 23.1 and 23.2 have been used to measure

the mechanical properties of MEMS and NEMS materials. As a general rule, the

results from the various techniques have agreed well with each other, and the ar-

gument becomes which of the measurement techniques is easiest and most reliable

to perform. It is crucial to bear in mind, however, that certain properties (such as

strength) are process-dependent, and so the results taken at one laboratory will not

necessarily match those taken from another. This will be discussed in more detail in

Sect. 23.3.1.

23.3.1 Mechanical Properties of Polysilicon

In current MEMS technology, the most widely used structural material is polysili-

con deposited by low-pressure chemical vapor deposition (LPCVD). One reason for

the prevalenceof polysilicon is the large body of processing knowledge for this ma-

terial that has been developed by the integrated circuit community. Another reason,

of course, is that polysilicon possesses a number of qualities that are beneﬁcial to

MEMS devices, in particular high strength and Young’s modulus. Therefore, most

of the mechanical properties investigations performed on MEMS materials have fo-

cused on polysilicon.