Bhushan B. Nanotribology and Nanomechanics: An Introduction

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1294 Bharat Bhushan

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Mechanical Properties of Micromachined Structures

Harold Kahn

Summary. To be able to accurately design structures and make reliability predictions in any

ﬁeld, it is ﬁrst necessary to know the mechanical properties of the materials that make up the

structural components. The devices encountered in the ﬁelds of microelectromechanical sys-

tems (MEMS) and nanoelectromechanical systems (NEMS), are necessarily very small, and

so the processing techniques and the microstructures of the materials used in these devices

may diﬀer signiﬁcantly from bulk structures. Also, the surface-area-to-volume ratios in such

structures are much higher than in bulk samples, and so surface properties become much more

important. In short, it cannot be assumed that the mechanical properties measured for a bulk

specimen of a material will apply when the same material is used in MEMS and NEMS. This

chapter will review the techniques that have been used to determine the mechanical properties

of micromachined structures, especially residual stress, strength and Young’s modulus. The

experimental measurements that have been performed will then be summarized, in particular

the values obtained for polycrystalline silicon (polysilicon).

23.1 Measuring Mechanical Properties of Films on Substrates

In order to accurately determine the mechanical properties of very small structures,

it is necessary to test specimens made from the same materials, processed in the

same way, and of the same approximate size. Not surprisingly it is often diﬃcult

to handle specimens this small. One solution is to test the properties of ﬁlms that

remain on substrates. Micro- and nanomachined structures are typically fabricated

from ﬁlms that are initially deposited onto a substrate, are subsequently patterned

and etched into the appropriate shapes, and are then ﬁnally released from the sub-

strate. If the testing is performed on the continuous ﬁlm, before patterning and re-

lease, the substrate can be used as an eﬀective “handle” for the specimen (in this

case, the ﬁlm). Of course, since the ﬁlms are attached to the substrate, the types of

tests possible are severely limited.

1298 Harold Kahn

23.1.1 Residual Stress Measurements

One common measurementeasily performedon ﬁlms attachedto substrates is resid-

ual ﬁlm stress. The curvature of the substrate is measured before and after ﬁlm

deposition. Curvature can be measured in a number of ways. The most common

technique is to scan a laser across the surface (or scan the substrate beneath the

laser) and detect the angle of the reﬂected signal. Alternatively, proﬁlometry, opti-

cal interferometry or even atomic force microscopy can be used. As expected, tools

that map a surface or perform multiple linear scans can give more accurate readings

than tools that measure only a single scan.

Assuming that the ﬁlm is thin compared to the substrate, the average residual

stress in the ﬁlm, σ

, is given by the Stoney equation,

(1−ν

)



−



, (23.1)

where the subscripts f and s refer to the ﬁlm and substrate, respectively; t is thick-

ness, E is Young’s modulus, ν is Poisson’s ratio, and R is the radius of curvature

before (R

)andafter(R

) ﬁlm deposition [2]. For a typical (100)-oriented silicon

substrate, E/(1−ν) (also known as the biaxial modulus) is equal to 180.5GPa,in-

dependent of in-plane rotation [3]. This investigation can be performed on the as-

deposited ﬁlm or after any subsequent annealing step, provided no changes occur to

the substrate.

This measurement will reveal the average residual stress of the ﬁlm. Typically,

however, the residual stresses of deposited ﬁlms will vary throughout the thick-

ness of the ﬁlm. One way to detect this, using substrate curvature techniques, is to

etch away a fraction of the ﬁlm and repeat the curvature measurement. This can

be iterated any number of times to obtain a residual stress proﬁle for the ﬁlm [4].

Alternatively, toolshavebeen designedthat can measure thesubstratecurvaturedur-

Biaxial stress (MPa)

Temperature (°C)

500

300

200

100

100 200 300 400

–100

–200

Heating

Yield stress

Tension

Compression

Cooling

Stress-Temperature plot

for annealed aluminium film

on silicon (thickness = 590nm)

–EΔQ

1–ν

Fig. 23.1. Typical results for

residual stress as a function of

temperature for an aluminum

ﬁlm on a silicon substrate [1].

The stresses were determined

by measuring the curvature

of the substrate before and

after ﬁlm deposition, using

the reﬂected signal of a laser

scanned across the substrate

surface

23 Mechanical Properties of Micromachined Structures 1299

ing the deposition process itself, in order to obtain information on how the stresses

evolve [5].

An additional feature of some of these tools is the ability to heat the substrates

while performing the stress measurement. An example of the results obtained in

such an experiment is shown in Fig. 23.1 [1], for an aluminum ﬁlm on a silicon

substrate. The slope of the heating curve gives the diﬀerence in thermal expansion

between the ﬁlm and the substrate. When the heating curve changes slope and be-

comes nearly horizontal, the yield strength of the ﬁlm has been reached.

23.1.2 Mechanical Measurements Using Nanoindentation

Aside from residual stress, it is diﬃcult to measure the mechanical properties of

ﬁlms attached to substrates without the measurementbeing aﬀected by the presence

of the substrate. Recent developments in nanoindentation equipment have allowed

this technique to be used in some cases. With specially designed tools, indentation

can be performed using very low loads. If the ﬁlms being investigated are thick and

rigid enough, measurements can be made that are not inﬂuenced by the presence

of the substrate. Of course, this can be veriﬁed by depositing the same ﬁlm onto

diﬀerent substrates. By continuously monitoring the displacement as well as the

load during indentation, a variety of properties can be measured, includinghardness

and Young’s modulus [6]. This area is covered in more detail in a separate chapter.

For brittle materials, cracks can be generated by indentation, and strength infor-

mation can be gathered. But the exact stress ﬁelds created during the indentation

process are not known exactly, and therefore quantitative values for strength are dif-

ﬁcult to determine. Anisotropic etching of single-crystal silicon has been performed

to create 30-µm-tall structures that were then indented to examine fracture tough-

ness [7], but this is not possible with most materials.

23.2 Micromachined Structures

for Measuring Mechanical Properties

Certainly the most direct way to measure the mechanical properties of small struc-

tures is to fabricate structures that would be conducive to such tests. Fabrication

techniques are suﬃciently advanced that virtually any design can be realized, at

least in two dimensions. Two basic types of devices are used for mechanical prop-

erty testing: “passive” structures and “active” structures.

23.2.1 Passive Structures

As mentioned previously, the main diﬃculty encountered when testing very small

specimens is handling. One way to circumvent this problem is to use passive struc-

tures. These structures are designed to act as soon as they are released from the

substrate and to provide whatever information they are designed to supply without

1300 Harold Kahn

further manipulation. For all of these passive structures, the forces acting on them

come from the internalresidual stresses of the structural material. For deviceson the

micron scale or smaller, gravitationalforces can be neglected, and therefore internal

stresses are the only source of actuation force.

Stress Measurements

Since internal residual stresses act upon the passive devices when they are released,

it is naturalto design a devicethat can beused to measureresidual stresses.One such

device, a rotating microstrain gage, is shown in Fig. 23.2. There are many diﬀerent

microstrain gauge designs, but all operate via the same principle. In Fig. 23.2a, the

large pads, labeled A, will remain anchored to the substrate when the rest of the de-

vice is released. Upon release, the device will expand or contract in order to relieve

its internal residual stresses. A structure under tension will contract, and a struc-

ture under compression will expand. For the structure in Fig. 23.2, compressive

stress will cause the legs to lengthen. Since the two opposing legs are not attached

to the central beam at the same point – they are oﬀset, they will cause the central

beam to rotate when they expand. The device in Fig. 23.2 contains two indepen-

dent gauges that point to one another. At the ends of the two central beams are two

parts of a Vernier scale. By observing this scale, one can measure the rotation of the

beams.

If the connections between the legs and the central beams were simple pin con-

nections, the strain, ε, of the legs (the fraction of expansion or contraction) could be

600 μm

500 μm30 μm

20 μm

Leg

Indicator

beam

Fig. 23.2. (a) Microstrain gauge fabricated from polysilicon; (b) shows a close-up of the

Vernier scale before release, and (c) shows the same area after release

23 Mechanical Properties of Micromachined Structures 1301

determined simply by the measured rotation and the geometry of the device, namely

ε =

beam

oﬀset

central

leg

, (23.2)

where d

beam

is the lateral deﬂection of the end of one central beam, d

oﬀset

is the dis-

tance between the connections of the opposing legs, L

central

is the length of the cen-

tral beam (measured to the center point between the leg connections), and L

leg

is the

length of the leg. However, since the entire device was fabricated from a single po-

lysilicon ﬁlm, this cannot be the case; some bending must occur at the connections.

As a result, to get an accurate determinationof the strain relieved upon release,ﬁnite

element analysis (FEA) of the structure must be performed.This is a commonsitua-

tion for microdevices. FEA is a powerfultool for determining the displacements and

stresses of nonideal geometries. One drawback is that the Young’s modulus of the

material must be known in order to do the FEA as well as to convert the measured

strain into a stress value. But Young’s moduli are known for many micromachined

materials or they can be measured using other techniques.

Other devices besides rotating strain gauges have been designed that can meas-

ure residual stresses. One of the simplest is a doubly clamped beam, a long, narrow

beam of constant width and thickness that is anchored to the substrate at both ends.

If the beam contains a tensile stress it will remain straight, but if the beam contains

a compressivestress it will buckle if its length exceeds a critical value, l

, according

to the Euler buckling criterion [8],

= −





, (23.3)

where ε

is the residual strain in the beam and h is the width or thickness of the

beam, whicheveris less. To determine the residualstrain,a series of doubly clamped

beams of varying lengths are fabricated. In this way, the critical length for buckling,

, can be deduced after release. One problem with this technique is that during the

release process, any turbulence in the solution will lead to enhanced buckling of the

beams, and a low value for l

will be obtained.

For ﬁlms with tensile stresses, a similar analysis can be performed using ring-

and-beam structures, also called Guckel rings after their inventor, Henry Guckel.

A schematic of this design is shown in Fig. 23.3 [8]. Tensile stress in the outer

ring will cause it to contract. This will lead to compressive stress in the central

beam, even though the material was originally tensile before release. The amount of

compression in the central beam can be determined analytically from the geometry

of the device and the residual strain of the material. Again, by changing the length

of the central beam it is possible to determine l

, and then the residual strain can be

deduced.

Stress Gradient Measurements

For structures fabricated from thin deposited ﬁlms, the stress gradient can be just as

important as the stress itself. Figure 23.4 shows a portion of a silicon microactuator.

1302 Harold Kahn

Fig. 23.3. Schematic showing

(a) top view and (b) side view

of Guckel ring structures [8].

The dashed lines in (a) indi-

cate the anchors

The device is designed to be completely planar; however, stress gradients in the ﬁlm

cause the structures to bend. This ﬁgure illustrates the importance of characterizing

and controlling stress gradients, and it also demonstrates that stress gradients are

most easily measured for a simple cantilever beam. By measuring the end deﬂection

δ of a cantilever beam of length l and thickness t, the stress gradient, dσ/ dt is

determined by [9]

dσ

2δ

1−ν

. (23.4)

The magnitude of the end deﬂection can be measured by microscopy, optical

interferometry, or any other technique.

Another useful structure for measuring stress gradients is a spiral. For this struc-

ture, the end of the spiral not anchored to the substrate will move out-of-plane. The

diameter of the spiral will also contract, and the free end of the spiral will rotate

when released [10].

50 μm

Fig. 23.4. Scanning elec-

tron micrograph (SEM) of

a portion of a silicon mi-

croactuator. Residual stress

gradients in the silicon cause

the structure to bend

23 Mechanical Properties of Micromachined Structures 1303

Strength and Fracture Toughness Measurements

As mentioned above, if a doubly clamped beam contains a tensile stress, it will re-

main taut when released because it cannot relieve any of its stress by contracting.

This tensile stress can be thought of as a tensile load being applied at the ends of

the beam. If this tensile load exceeds the tensile strength of the material, the beam

will break. Since the tensile stress can be measured, as discussed in the Sect. 23.2.1,

this technique can be used to gather information on the strength of materials. Fig-

ure 23.5 shows two diﬀerent beam designs that have been used to measure strength.

The device shown in Fig. 23.5a was fabricated from a tensile polysilicon ﬁlm [11].

Diﬀerent beams were designedwith varying lengths of the wider regions (markedl

in the ﬁgure). In this manner,the load applied to the narrow center beam was varied,

even though the entire ﬁlm contained a uniform residualtensile stress. For l

greater

than a critical value, the narrow center beam fractured, giving a measurement for

the tensile strength of polysilicon.

The design shown in Fig. 23.5b was fabricated from a tensile Si

ﬁlm [12].

As seen in the ﬁgure, a stress concentration was included in the beam, to ensure the

fracture strength would be exceeded. In this case, a notch was etched into one side

of the beam. Since the stress concentration is not symmetric with regard to the beam

axis, this results in a large bending moment at that position, and the test measures

the bend strength of the material. Again, like the beams shown in Fig. 23.5a, the

geometry of various beams fabricated from the same ﬁlm were varied, to vary the

maximum stress seen at the notch. By seeing which beams fracture at the stress

concentration after release, the strength can be determined.

The fracture toughness of a material can be determined with a similar tech-

nique, but an atomically sharp pre-crack is used instead of a stress concentration.

Sharp pre-cracks can be introducedinto micromachined structures before release by

adding a Vickers indent onto the substrate, near the device; the radial crack formed

by the indentwill propagateinto the overlyingstructure[14]. Accordingly,the beam

Top view

= 50 μm

2 μm

Anchor Narrow

center

beam

Tensile stress

= 20 –1200 μm I

= 20 μm

Implantation window

Fig. 23.5a,b. Schematic de-

signs of doubly clamped

beams with stress concen-

trations used for measuring

strength. (a) was fabricated

from polysilicon [11] and

(b) was fabricated from

[12]

1304 Harold Kahn

10 μm

Fig. 23.6. (a)SEMof

a 500 µm-long polysilicon

beam with a Vickers indent

placed near its center; (b)

higher magniﬁcation SEM of

theareaneartheindentshow-

ing the pre-crack traveling

from the substrate into the

beam [13]

with a sharp pre-crack, shown in Fig. 23.6, was fabricated using polysilicon [13].

Due to the stochastic nature of indentation, the initial pre-crack length varies from

beam to beam. Because of this, even though the geometry of the beam remains iden-

tical, the stress intensity, K, at the pre-crack tip will vary. Upon release, only those

pre-cracks whose K exceeds the fracture toughness of the material, K

, will prop-

agate, and in this way upper and lower bounds for K

can be determined for the

material.

For all of the beams discussed in this section, ﬁnite element analysis is required

to determine the stress concentrations and stress intensities. Even though approx-

imate analytical solutions may exist for these designs, the actual fabricated struc-

ture will not have idealized geometries. For example, corners will never be per-

fectly sharp, and cracks will never be perfectly straight. This reinforces the idea

that FEA is a powerful tool when determining mechanical properties of very small

structures.

23.2.2 Active Structures

As discussed above, it is very convenient to design structures that act upon release

to provide informationon the mechanical properties of the structural materials. This

is not always possible, however. For example, those passive devices just discussed

rely on residual stresses to create the changes (rotation or fracture) that occur upon

release, but many materials do not contain high residual stresses as-deposited,or the

processing scheme of the device precludes the generation of residual stresses. Also,

some mechanical properties, such as fatigue resistance, require motion before they

can be studied. Active devices are therefore used. These are acted upon by a force

(the source of this force can be integrated into the device itself or can be external

to the device) in order to create a change, and the mechanical properties are studied

via the response to the force.