Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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optical interferometric technique the relative displacement between the gold lines could be measured

yielding both the axial and the transverse strain. By combining the measured strain with the applied

stress, both the Young’s modulus and the Poisson ratio for polysilicon were determined. The measured

data agree well with the calculated bulk values given in Table 15.3 and 15.4 in Section 4.3.2.

Young’s modulus was also determined by Ogawa et al. (1996) from tensile tests of thin Al and Ti ﬁlms,

where the elongation was determined by measuring the relative displacement of two gauge marks on the

specimens by a double-ﬁeld-of-view light microscope combined with image analysis.

Other static techniques are concerned with deﬂection of membranes, bridges (doubly supported

beams), or released structures of more complex geometries. In the membrane technique a thin diaphragm,

circular or square, of the material to be tested is supported by a surrounding rigid frame. A uniform

pressure is applied to one side of the membrane, and the bulging of the membrane is detected. This

technique has been applied to membranes of polyimide (Allen et al., 1987), polysilicon (Maier-Schneider

et al., 1995), Au and Al (Paviot et al., 1995) and SiC (Tong and Mehregany, 1992; Yamaguchi et al., 1995).

Polysilicon membranes environmentally exposed to HF solutions during a period of time have been

investigated in a similar way (Walker et al., 1990), and composite SiN

/poly-Si membranes have been

used to determine the Young’s moduli of nitride ﬁlms on polysilicon substrate (Tabata et al., 1989).

Measurements on thin SiN

single-layer membranes by applying a point load with a Nanoindenter have

also been performed (Hong et al., 1990).

Young’s modulus measurement by static deﬂection of a single-crystalline Si bridge has been carried

out (Najaﬁ and Suzuki, 1989). In this case the doubly supported beam was deﬂected electrostatically by

a capacitive drive electrode at the middle of the beam (Figure 15.9). In another experimental setup, a

low-stress SiN

bridge was deﬂected by a stylus-type surface proﬁler (Tai and Muller, 1990). A related

test conﬁguration is the so-called bridge-slider, a long beam ﬁxed at one end and released at the other.

The released end is enclosed in a housing which allows movement only in the length direction of the

beam. When the movable end is slid toward the ﬁxed end, the beam bulges out-of-plane due to the

compression, and Young’s modulus can be evaluated from this deformation. Such experiments have been

reported for polysilicon (Tai and Muller, 1990).

Other test geometries, such as ring-and-beam structures

(Guckel et al., 1988) or T-shape structures

(Allen et al., 1987) have also been suggested for static measurement of the elastic properties of released

micromachined structures.

15.8.3 Elasticity Testing by Dynamic Techniques

Elastic data have been extracted from dynamic experiments with cantilever beam or bridge structures,

and lately also with comb-drive structures suspended in elastic springs. Petersen and Guarnieri, 1979,

pioneers in the ﬁeld of dynamic microtesting, measured the transverse mechanical resonant frequencies

of cantilever beams, micromachined in a number of thin insulating ﬁlms deposited by various methods.

The beam vibration was electrostatically excited, and the resonant frequency was measured by detection

of the movement of a reﬂected laser beam. By using a similar technique, elastic modulus measurements

have been carried out on annealed polysilicon cantilever beams

(Putty et al., 1989) and on boron-doped,

FIGURE 15.9 A principle sketch showing a doubly supported bridge structure for Young’s modulus measurements.

(Najaﬁ, K. and Suzuki, K., 1989, in Proc. IEEE Micro Electro Mechanical Systems, Salt Lake City, UT, February 20–22,

p. 96. With permission.)

single-crystalline, 〈110〉-oriented silicon beams

(Ding et al., 1990). Dynamic modulus measurement using

microbridges (doubly supported beams) have been made on polysilicon

(Guckel et al., 1988) and on

boron-doped 〈100〉-oriented single-crystalline silicon beams

(Zhang et al., 1991) Also, laterally moving

comb-drive structures have been used to extract E values for polysilicon

(Hirano et al., 1991; Pratt et al.,

1991; Biebl et al., 1995) and electroless plated Ni ﬁlms (Roy et al., 1995).

Determinations of the linear-elastic temperature coefﬁcients for single-crystalline silicon were made

by Bourgeois et al. (1995) with the aid of three different resonating structures.

15.8.4 Testing of Other Properties

Micromechanical testing of the fracture strength is performed by static loading of microbeams (cantilevers

or bridges), but occasional strength tests on membranes have also been performed (Walker et al., 1990).

Resonant experiments for micromechanical strength testing have not been reported so far, but a resonant

micromechanical device for fatigue testing has been described (Connally and Brown, 1991). Strength

tests have been reported for single-layer, single-crystalline beams of silicon

(Johansson et al., 1988a;

Ericson and Schweitz, 1990; Ding et al., 1990) and GaAs

(Hjort et al., 1994), as well as on released single-

layer beams of polysilicon

(Tai and Muller, 1988), silicon oxide

(Weihs et al., 1988; Nix, 1989), polyimide

(Allen et al., 1987; Mehregany et al., 1987), and gold

(Weihs et al., 1988). The fracture behavior of two-

layer beams has been studied for SiO

/Si

(Johansson et al., 1989; Ericson and Schweitz, 1990) TiN/Si

(Johansson et al., 1989) Ti/Si (Johansson et al., 1989; Ljungcrantz et al., 1993) and for Al/Si (Johansson

et al., 1989).

Besides the beam-bending method, uniaxial tensile testing methods have recently been published.

Greek et al. (1997) performed tensile tests on surface-micromachined polysilicon beams using a micro-

manipulator in an SEM. The free end of the beam has the form of a ring that is gripped by the testing

probe and moved by piezoelectric actuators. Other methods for gripping of the beam during a tensile

test have been used by Tsuchiya et al. (1997) who utilized electrostatic forces built up by an applied

voltage, and Read and Marshall (1996) who simply glued the test probe to the specimen. Gripping the

test beam with a probe in a ring seems to be the best method since the rounded surfaces interact to align

the test structure in the direction of the motion.

In principle, it is possible to study the plasticity behavior of thin elements micromachined into suitable

test geometries by the same means as the elasticity or fracture behavior is studied. Elasticity and fracture,

however, are the two extremes of deformation, whereas plasticity is a deviation from linear stress–strain

behavior between these extremes. Hence, high-resolution measuring equipment is required to detect, for

example, the onset of plastic ﬂow (the yield limit) and the strain-hardening behavior of the material

beyond that limit. Such measurements have been reported for free-standing gold cantilevers

(Weihs et al.,

1988, 1989), and the technique has been described in review articles by Nix (1989) and Vinci and Bravman

(1991). The plastic yield limit of a gold ﬁlm deposited by electron beam evaporation was determined to

be 340 MPa, and a clear strain hardening effect (increased yield limit) was detected by repeated

load–unload cycling.

Numerous methods for internal stress measurement exist, including X-ray diffraction techniques,

techniques for measuring curvature, techniques for load deﬂection of membranes using buckling effects,

and techniques utilizing surface-micromachined indicator structures giving an output depending on the

internal stress state.

The most commonly used method is the wafer curvature technique, where the curvature is obtained

by optical or mechanical surface-proﬁling measurements. The micromechanical beam curvature tech-

nique has been used by several authors, e.g., Johansson et al. (1989) and Ljungcrantz et al. (1993), who

studied sputter-deposited coatings of Al, Ti, and TiN on Si cantilever beams. The beam deﬂections in

these cases were measured optically and gave the biaxial stress state in the ﬁlm materials. One general

comment on a fundamental difference between the commonly used method of X-ray diffraction (XRD)

for stress analysis, on the one hand, and the micromechanical beam curvature technique, on the other,

should be emphasized in the present context. In the XRD method the lattice dilation (elastic strain) is

measured, and knowledge of the elastic constants is required to evaluate the elastic stress. Often such

information is missing for thin ﬁlms, in which cases the two-layer beam curvature method is an attractive

alternative. In this method the strain is evaluated from the observed curvature, and the stress from

considerations of the equilibrium of forces and moments, without the use of any materials parameters

for the ﬁlm. Hence, the internal stress and strain can be obtained for a completely unknown thin-ﬁlm

material on a known substrate material.

To determine the internal stress state in a free-standing thin ﬁlm other methods have to be applied

because the stress state in the ﬁlm changes when it is freed from the substrate. Bulge testing is one of

these methods where a uniform pressure is applied to a membrane and the measured deﬂection not only

yields the Young’s modulus but also the internal ﬁlm stress. Tabata et al. (1989) made measurements on

polysilicon and silicon nitride, Maier-Schneider et al. (1995) on polysilicon, and Paviot et al. (1995)

investigated Au and Al thin ﬁlms.

Different kinds of free-standing micromachined structures have also been utilized to derive the stress

state more locally than the other techniques. van Drieënhuizen et al. (1993) compared different structures,

for instance, a so-called diamond structure capable of measuring both compressive and tensile stress.

Also a rotating structure, where an internal strain is converted to a rotation of an indicator arm, was

presented and realized in polysilicon (Figure 15.10). Rotating structures of different designs have also

been employed by other authors (Lin et al., 1993; Ericson et al., 1997; Zhang et al., 1997) to study thin

ﬁlms and thick ﬁlms of polysilicon and thin ﬁlms of silicon nitride.

If the stress in the ﬁlm material not only is of a biaxial nature but also varies through the thickness

of the ﬁlm, it could be disastrous for micromechanical components. To measure these internal stress

gradients, Fan et al. (1990b) presented spiral microstructures freed from the substrate which bent into

different shapes depending on the sign of the gradient. Also cantilever beams etched out of the ﬁlm

material bending upward or downward have been used in the same sense as the spirals (Ericson et al.,

1997).

FIGURE 15.10 A principle sketch of a rotating structure for measurement of the internal stress (van Drieënhuizen

et al., 1993; Lin et al., 1993; Ericson et al., 1997; Zhang et al., 1997). Internal strain in the two actuator beams is

converted to a rotation of the indicator arm when the whole structure (except for the ﬁxed plates) is released from

the substrate.

Several types of adhesion tests have been devised, for example, the “Scotch tape” test, the scratch test,

the linear peel test, and the inﬂated membrane test

(Valli, 1986; Mittal, 1978). The latter test has been

performed on micromachined structures

(Senturia et al., 1987), but in general little work on adhesion

testing has been done within micromechanics. This is probably because in most practical cases a simple

scratch test is sufﬁcient to determine if a coating is prone to ﬂake off. If the coating ﬂakes off, it is a good

idea to try to modify the surface pretreatment and the deposition process until it does not. Some type

of proof testing might be necessary. But usually it is not necessary, or even possible, to evaluate such a

test in terms of a well-deﬁned adhesive strength parameter. Normally, a simple ranking in terms of some

externally applied load, for instance, a critical pressure in an inﬂated membrane test or a critical load in

a scratch test, is quite sufﬁcient.

15.9 Modeling and Error Analysis

The adequacy of an evaluation model is just as important as the accuracy and reproducibility of the

experimental procedure. A theoretical model describing a real physical process will always be idealized

to some extent. For instance, simplifying assumptions are usually made regarding boundary conditions

and deformation states. Care must be taken to deﬁne the geometric structure properly, to use the correct

constitutive relationships between stress and strain, and to introduce a realistic overall deformation

behavior into the model, etc. Such care has not always been taken in the past, resulting in contradictory

results from, e.g., different elastic property measurements on micromachined structures.

Naturally, the choice of evaluation model is governed by the choice of experimental method, the

geometric test conﬁguration, the testing conditions, etc. The existing models are too numerous to allow

a detailed analysis of their merits and limitations. As one example, however, bending of a micromachined

cantilever beam will be discussed, and some basic relationships given. These relationships will be used

for comparisons with other types of tests, and taken as starting points for some fundamental consider-

ations concerning the error analysis.

15.9.1 Single-Layer Beam

The maximum stress σ

, suffered by a cantilever beam of length L which is deﬂected by a perpendicular

force F applied to its free end is

(15.41)

where I is the moment of inertia of the beam cross section and c is the distance between the neutral layer

(the zero-stress plane during bending) and the upper beam surface. For a rectangular beam cross section,

we have

(15.42)

(15.43)

It is important to note that no material properties (e.g., Young’s modulus) appear in the stress evaluation

model of Equation 15.41. The single experimental input variable that enters this expression is the applied

load F; all other parameters are geometric constants. Not even the deﬂection of the beam, caused by F,

need be known. From an error analysis viewpoint, one observes that the evaluated stress depends linearly

(or inverse linearly) on all input parameters, except on the beam thickness which enters as t

–2

. Conse-

quently, a relative thickness error will be doubled in the evaluated stress, whereas all other relative input

FL I c=

()

Iwt=

12,

ct= 2.

errors add up in the stress error without ampliﬁcation. This is an important observation, since it is not

trivial to make an accurate measurement of the thickness of a micromachined element.

From a beam-bending experiment, the Young’s modulus can be determined from the following expression:

(15.44)

where δ is the deﬂection of the beam at the loading point. F and δ must be independently measured in

the experiment, and both values enter Equation 15.44 to the ﬁrst power. Two geometric parameters, L

and t (via I), enter the expression to the third power. For this reason this evaluation model puts high

demands on the numerical accuracy of the geometric parameters. In micromachined structures these

demands are not always easy to meet, especially not concerning the thickness. For this reason, bending

experiments on cantilever beams are not well suited for determining E. Pure tensile tests are better since

in this conﬁguration the thickness enters the E expression to the ﬁrst power.

The above discussion is centered on the problem of the thickness error, which in practice is one of the

most important sources of errors. There are, however, several other such sources which should be taken

into consideration in the evaluation of a beam-bending experiment; for instance, erroneous input data or

errors due to oversimplifying assumptions in the deformation model. These errors are systematically

reviewed by Schweitz (1992), and suitable corrections are suggested. In general, the evaluation of an

experiment beneﬁts greatly from numerical simulations (FEA), as complements to the analytical modeling.

15.9.2 Two-Layer Beam

Analytical models for evaluating bending experiments with two-layer beams also exist (Johansson et al.,

1989). In these models the elastic properties of the substrate material usually must be known, in order

to evaluate the properties of the deposited ﬁlm material. There exists one comparative test method,

however, which does not require explicit knowledge of the substrate properties (Schweitz, 1991). Accord-

ing to this method, one beam of the substrate material (or one set of beams) is ﬁrst tested in an uncoated

condition, and force-vs.-deﬂection curves are registered. Next the beam is coated by, e.g., sputtering or

evaporation techniques, and then tested again. The difference between the F(δ) curves of the coated and

the uncoated beams, respectively, is caused by the ﬁlm (Figure 15.11). For any given deﬂection δ

the

ﬁlm stress can be expressed in terms of the increase in force (∆F) required to deﬂect the coated beam to

the same δ

value as the uncoated beam:

(15.45)

FIGURE 15.11 Schematic illustration of force vs. deﬂection curves

for uncoated and coated beams.

EFL I L=−

()

()( )

13νδδ, ,

fs f

wt t t

tt=

()

2 ∆

, .



is the maximum ﬁlm stress, and w is the beam width. In this method the stress can be determined

without using constitutive relationships for the two materials, and hence without previous knowledge of

the elasticity or plasticity parameters of any of the materials. From Equations 15.41 and 15.45 and Hooke’s

law, the Young’s modulus of the ﬁlm can be evaluated. Assuming that t

 t

, and E

not  E

, a ﬁrst

approximation is (Schweitz, 1992)

(15.46)

where F

is the force required to give the uncoated beam the designated deﬂection δ

15.9.3 Resonant Beam

In resonant experiments with single-layer cantilever beams, the resonant frequency ω is related to the

beam geometry and its material properties by

(15.47)

where ρ is the mass density of the beam material. E can be evaluated using this relationship. In this

evaluation model, E is proportional to t

–2

(static beam bending: t

–3

) and to L

(static beam bending: L

Hence, the dynamic model is less sensitive to thickness errors than the model of static deﬂection, but

still more sensitive to the same error than the model of static in-plane tension.

15.9.4 Micro vs. Bulk Results

It has been pointed out that different measurements of the Young’s modulus on micromachined structures

yield contradictory results, and that “micro” results sometimes disagree with generally accepted bulk

values (Schweitz, 1992). In some cases, when the E values measured on micromachined silicon structures

have been found to differ drastically (up to 30%) from the bulk value, the inﬂuence of the dopants has

been suggested to be the cause. However, according to alloying theory, not even the most extreme doping

levels in silicon would affect the E value to more than a few percent (Hall, 1967; Kim, 1981). Dislocation

networks generated by the doping process have also been suggested as possible causes. But dislocations

do not in general affect the E value much, not even in high density, except under very special circumstances

which are unlikely to prevail in doped silicon at room temperature. The high surface-to-volume ratio of

thin microelements has also been proposed as an explanation. But the inﬂuence of surface effects on the

elastic properties becomes negligible for thicknesses exceeding a few nanometers. Grain size or texture

may have some effect (see Section 15.4.3.2) for narrow structures, but this effect still is too small and

too scarce to explain the alleged difference between micro and bulk results.

In fact, to date no convincing physical explanation to the observed discrepancies between the accepted

bulk E value, on the one hand, and recently measured “micro” values on the other, has been put forward.

But many of the reported micromechanical experiments appear to have been evaluated by means of

oversimpliﬁed theoretical models. This, in combination with uncertain input values, at present seems to

be the most probable cause of the observed scatter in the “micro” values, as well as of the discrepancy

between some “micro” results and bulk values.

15.10 Summary and Conclusions

This chapter is not primarily intended for readers who are specialists in materials science, but rather for

workers who are electronics engineers or other types of specialists within the multidisciplinary ﬁeld of

≈

























∆













0 162

MST. For this reason, some basics of crystal cohesion, elasticity, plasticity, thermomechanics, and fracture

have been reviewed, along with a few mechanical systems properties of particular interest within micro-

mechanics: internal stresses and adhesion.

Mechanical testing of micromachined structures and systems is of paramount signiﬁcance to the

development of microdevices with high demands on mechanical integrity and long-term reliability. In

this context the accuracy and reproducibility of the testing procedure, as well as the adequacy of the

evaluation method, is of central importance. The most serious problem today is perhaps the poor

reproducibility of test results. The experimental techniques used are not consistent, and other workers

ﬁnd it difﬁcult to verify results. The best way to bring this problem under control probably is to introduce

a limited number of well-deﬁned, standardized test procedures that can be performed and evaluated by

independent researchers. This could be an important task for, e.g., an alert, international research

organization in the ﬁeld of microstructure technology.

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