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

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15.2.2 Lattice Structures and Structural Defects

Crystalline as well as amorphous (disordered) materials are regularly used in MST. One important

material in micromechanics of the latter type is glass. Another important material is silicon dioxide, SiO

which is used in crystalline form (quartz) as well as in amorphous form (for instance, low-temperature

oxide, LTO). Also silicon and most other relevant materials can be grown in both forms. From a

mechanical-strength viewpoint, an amorphous structure is sometimes preferred, due to the lack of active

slip planes for dislocation movement in such structures. In general, however, the strength performance

is more related to the distribution and geometry of microscopic ﬂaws in the material, especially surface

ﬂaws.

It would lead too far in the present context to deﬁne all crystalline lattice structures of interest in MST.

For this reason we will conﬁne ourselves to very brief descriptions of two important lattice types: the

diamond lattice type found in crystalline silicon and the zinc blend (ZnS) lattice type found in III-V

semiconductors.

The

diamond structure

is one of the simplest and most symmetric lattice types, and is found in Si and

Ge, for example. It consists of two face-centered cubic (fcc) lattices which are inserted into each other

in such a manner that they are shifted relative to each other by one quarter of a cube edge along all three

principal axes. Each atom is surrounded by four other atoms in a tetragonal conﬁguration.

The

zinc blende structure

is found in III-V compounds such as GaAs, InP, and InSb. It is identical with

the diamond structure apart from the fact that one of the two overlapping fcc lattices consists entirely

of the type III element (e.g., Ga) and the other entirely of the type V element (e.g., As). Every atom of

one kind is tetragonally surrounded by four atoms of the other kind, and crystallographic planes of any

chosen orientation are periodically arranged in parallel pairs consisting of one III-type and one V-type

atomic plane (in some orientations the parallel planes of a pair coincide).

Common crystal defects are

point defects

such as vacancies (one atom is missing), substitutionals (one

atom is replaced by an impurity atom), or interstitials (one atom is “squeezed in” between the ordinary

atoms). Other frequent crystal imperfections are

line defects

, such as dislocations, and more

complex

defects

, such as stacking faults or twins. All types of lattice defects affect the mechanical properties of a

crystal to a greater or lesser extent, but dislocations are the most detrimental of the lattice defects from

a mechanical-strength viewpoint due to their extremely high mobility (when a critical load limit,

the

yield limit,

has been exceeded).

Beyond the basic crystalline lattice structure (and the various types of lattice defects that may be

present in it), a number of

superstructures

can be of major importance to the mechanical behavior. The

grain structure of a polycrystalline material is one superstructure inﬂuencing the hardness and the yield

limit of the material, and precipitates of impurities, alloying substances, or intermediary phases are other

examples. The size and shape distribution of geometric ﬂaws, for instance, voids or cracks in the micron

or submicron range, is of crucial importance to the fracture strength of a brittle material. These super-

structures will be discussed in further detail in following sections.

Foreign atoms

of dopants, or contaminants such as oxygen, nitrogen, and carbon, commonly occur in

semiconductor materials, and are of great importance to their electronic properties (Hirsch, 1983;

TABLE 15.1

Melting Points (°C) of a Number

of Semiconductors and Other Materials

Si 1412 Nylon 137–150

Ge 937 Teﬂon 290

SiC 2537 Stainless steel 1400–1500

BN 4487 Al

2050

AlAs 1737 TiC 3100

GaAs 1238 HfC 3890

GaP 1467 SiO

1610

InP 1070 Glass ~700

InSb 536

Sumino, 1983a). At “normal” levels of doping or contamination in electronic components, the inﬂuence

of such impurities on the mechanical behavior is fairly limited, however. For extreme doping levels, some

inﬂuence on the plasticity behavior can be observed, especially at elevated temperatures, as will be

exempliﬁed later on.

15.3 Elasticity Properties

15.3.1 Isotropic Elasticity

For small deformations at room temperature most metals and ceramics (including conventional semi-

conductors) display a linear elastic behavior, i.e., they obey Hooke’s law, Equation 15.4, for the relation

between applied normal stress (

) and resulting normal strain (

). The corresponding relationship

between shear stress (

) and shear strain (

) is given by

(15.5)

where

is the

shear modulus

of the material. The Young’s modulus and the shear modulus are anisotropic

in crystalline materials. For ﬁne-grained polycrystalline materials, however, isotropic (averaged)

and

values are sometimes sufﬁcient.

When a linear-elastic material is subjected to a uniaxial strain (relative elongation)

= (

–

, its

cross-sectional dimension will diminish by a relative contraction

= (

–

. The ratio of these two

strains is a materials constant called the

Poisson’ s ratio

(15.6)

In isotropic media the elastic parameters are related by

(15.7)

The relative

volume change

caused by a uniaxial stress

is given by

(15.8)

where

is the

compressibility

(15.9)

The

bulk modulus

is deﬁned as the inverse value of the compressibility:

(15.10)

Multilayer structures consisting of different materials are frequent within micromechanics. For such

layered composites the Young’s moduli in the lateral and the transverse directions can be calculated from

(15.11)

τγ= G ,

νεε=

GE=+

()

[]

21 ν .

∆VV E K

=−

()

=12 3νσ σ ,

KE=−

()

31 2ν .

BKE== −

()

[]

1312ν .

EfE



∑

(15.12)

where E

||n

and E

⊥

are the Young’s moduli of the constituent materials in the two directions, and

are

the relative thickness fractions (= relative volume fractions) of the layers. Equations 15.11 and 15.12 are

applicable to stress-free multilayer structures built up of layers of individual thicknesses of ~100 nm or

more. In some superlattice structures the existence of a “supermodulus effect” has been suggested, i.e.,

the composite

values are supposed to radically deviate from the values predicted by conventional elastic

theories for multilayer structures or for homogeneous alloys. The existence of this effect is at present

under debate, and no physical model for it has been generally accepted as yet.

15.3.2 Anisotropic Elasticity

In single-crystalline materials the anisotropic elasticity is described by the elastic stiffness constants

(i,j = 1, 2, … 6) or, alternatively, by the elastic compliance constants

. These matrices are symmetric,

and in cubic crystals their number of elements is reduced by symmetry considerations to three indepen-

dent constants: C

, C

, and C

(or S

, S

, and S

). Table 15.2 gives typical room-temperature values

in gigapascals of these constants for a number of materials (Simmons and Wang, 1971).

The stiffness and compliance constants of cubic crystals are related by

(15.13)

(15.14)

(15.15)

Anisotropic values of E and ν can be calculated from these elastic constants. The Young’s modulus in the

crystallographic direction 〈lmn〉 is given by

(15.16)

TABLE 15.2 Values of Elastic Stiffness

Constants of a Number of Semiconductors

at 300 K (in units of GPa)

Si 165.78 63.94 79.62

Ge 129.11 48.58 67.04

GaAs 118.80 53.80 59.40

InP 102.20 57.60 46.00

InAs 83.29 45.26 39.59

Diamond 1076.4 125.2 577.4

The values are results published by different

workers, as compiled by Simmons et al.

(1971). The values for diamond were pub-

lished by van Enckevort (1994).

EfE

⊥⊥

∑

CC SS

11 12 11 12

−=−

()

−

CCSS

11 12 11 12

22+=+

()

−

44 44

−

12 2

11 11 12 44 1

ES S S S k=− −−

()

where:

(15.17)

and the directional cosines are normalized according to

(15.18)

For a longitudinal stress in the direction 〈lmn〉, resulting in a transverse strain in a perpendicular direction

〈ijk〉, the Poisson ratio is given by Brantley (1973):

(15.19)

where E is the Young’s modulus of the 〈lmn〉 direction given by Equation 15.16, and k

is given by

(15.20)

Orthonormality conditions, supplementing Equation 15.18, are

(15.21)

(15.22)

To illustrate the strong anisotropy of Young’s modulus and the Poisson ratio in crystalline semiconductors,

room-temperature values in various directions have been calculated and listed in Tables 15.3 and 15.4

for a few materials. The anisotropy of the Poisson ratio in a couple of semiconducting materials is

graphically illustrated by Figure 15.2. For the case of a hexagonal crystal structure, Thokala and

Chaudhuri (1995) calculated the Young’s modulus and the Poisson ratio for 6H–SiC, Al

, and AlN.

It is sometimes desirable to calculate the elastic properties of a randomly polycrystalline, but macro-

scopically isotropic, aggregate from the anisotropic single-crystal elastic constants. Theories for such

aggregate properties exist, and Simmons and Wang (1971) have tabulated so-called Voigt and Reuss

averages for a large number of crystalline materials. In polycrystalline thin ﬁlms the grain structure is

TABLE 15.3 Young’s Moduli E at 300 K

for Various Directions (in units of GPa)

Directions

<100> <110> <111> Poly

Si 130.2 169.2 187.9 163

Ge 102.5 137.5 155.2 132

GaAs 85.3 121.4 141.3 116

InP 60.7 93.4 113.9 89

InAs 51.4 79.3 96.7 76

Diamond 1050.3 1163.6 1207.0 1141

The polycrystalline results are mean values of

the Hashin and Shtrikman bounds, as calculated

by Simmons et al. (1971).

klm mn nl

222

()

lmn

222

1++=.

ν=− + − −

()

[]

SSSS kE

12 11 12 44 2

kil jm kn

222

()

ijk

222

1++= and

il jm kn++=0.

often strongly textured, and theories or expressions for elastic averaging over such morphologies also

exist (Brantley, 1973; Guckel et al., 1988; Maier-Schneider, 1995a). The resulting Young’s modulus of a

textured polycrystalline ﬁlm can deviate up to 10% from a nontextured polycrystalline ﬁlm.

15.4 Internal Stresses

The presence or nonpresence of internal stresses in a layered structure can be of great importance to the

mechanical behavior of the component, so for this reason some aspects of internal stresses will be

summarily discussed also in the present context. For instance, internal stresses can cause loss of adhesion

between the ﬁlm and the substrate and, consequently, lead to delamination failure of the composite.

They can have a beneﬁcial or detrimental effect on the fracture properties of the structure, by inhibiting

or promoting crack propagation in ﬁlm or substrate (Johansson et al., 1989). Furthermore, various

TABLE 15.4 Poisson Ratios ν at 300 K for Tension along [lmn]

and Contraction in the Perpendicular <ijk> Direction

System Si Ge GaAs InP InAs Diamond

[100]<010> 0.278 0.273 0.312 0.360 0.352 0.104

[100]<011> 0.278 0.273 0.312 0.360 0.352 0.104

[110]<001> 0.362 0.367 0.443 0.555 0.543 0.115

[110]<1

–

10> 0.062 0.026 0.021 0.015 0.001 0.008

[110]<1

–

11> 0.162 0.139 0.162 0.195 0.182 0.044

[111]<1

–

10> 0.180 0.157 0.188 0.238 0.222 0.045

[110]<1

–

12> 0.262 0.253 0.303 0.375 0.362 0.079

[111]<112

–

> 0.180 0.157 0.188 0.238 0.222 0.045

Poly 0.222 0.208 0.243 0.294 0.283 0.070

The poly values have been calculated by the method indicated in Table 15.3.

FIGURE 15.2 Illustration of the anisotropy of the Poisson ratio ν for a normalized (hypothetical) case of 100%

elastic straining along the [110] axis in Si and InAs. The outer contour illustrates a circular cross section of an

unstrained rod (a {110} plane), and the two inner contours illustrate cross sections in hypothetical states of 100%

elastic strain.

mechanisms of relaxation of internal stresses can have rather a drastic inﬂuence on the morphology of

ductile ﬁlms (Smith et al., 1991). Internal stresses in layered structures are of two fundamentally different

origins: thermal stresses caused by thermal mismatch between two adhering layers and intrinsic, or

microstructural, stresses generated during the deposition process.

15.4.1 Thermal Film Stress

One of the most important parameters in the generation of thermal stresses is the linear coefﬁcient of

thermal expansion (α), or, to be more precise, the difference in α for two adhering layers. The materials

parameter α is deﬁned as the relative elongation of a body per degree temperature rise:

(15.23)

which can be expressed as

(15.24)

where ε

therm

is the thermal strain and ∆T is the difference between the initial and the ﬁnal temperatures.

In cubic (isometric) single crystals, as well as in amorphous or polycrystalline materials, α is nearly

isotropic. In noncubic (anisometric) single crystals, α can be strongly anisotropic. In certain extreme

cases, e.g., Al

TiO

, LiAlSi

, and LiAlSiO

, α is positive in one direction and negative in a perpendicular

direction. This anisotropy can be utilized in micromechanical structures to control the spatial dimensions

by temperature variation. Some selected α values are found in Table 15.5.

A thermal stress is generated when the thermal expansion (or contraction) of one layer is prevented

by external forces of constraint, for instance, by adjacent layers with differing α values or differing

temperatures. In the case of a uniaxially clamped structure, the thermal stress caused by a temperature

difference ∆T can easily be calculated from Hooke’s law, Equation 15.4, and Equation 15.24:

(15.25)

For a thin ﬁlm on a thick substrate, we have biaxial stress conditions, and Hooke’s law is

(15.26)

TABLE 15.5 Linear Coefﬁcients of Thermal Expansion α

(in units of 10

–6

–1

)

Isometric

Anisometric

(⊥ Axis) ( Axis)

Si 2.4 SiO

(quartz) 13.7 7.5

GaAs 6.0 Graphite 1.0 27.0

AlAs 5.2 Al

8.3 9.0

SiC 4.5–5.0 Al

TiO

–2.6 11.5

Diamond 1.3 LiAlSi

6.5 –2.0

Glass 8.0 LiAlSiO

8.2 –17.6

Cu 16.2

α=

εα

therm

∆

σεα

therm therm

==EET∆ .

therm

−

∆∆,

where E

and ν

are the Young’s modulus and Poisson ratio of the ﬁlm material, and ∆α is the difference

in α between ﬁlm and substrate. It is apparent from Equation 15.26 that thermal stresses are minimized

by low E

values as well as by small differences in the expansion coefﬁcient and the temperature. The

latter difference is minimized by high thermal conductivities (k values) of the constituent materials.

In interfaces between differently oriented crystals, for instance, in grain boundaries, thermal stresses

can also be caused by anisotropy effects in E, α, and k. If the thermal stresses are not completely relaxed

upon cooling, which commonly occurs in thin-ﬁlm deposition, residual thermal stresses will be present

in the structure.

15.4.2 Intrinsic Film Stress

Internal stresses of intrinsic origin, on the other hand, are of a more complex physical nature and cannot

be expressed in terms of fundamental materials parameters. These nonthermal stresses are generated

during the ﬁlm growth process and strongly depend on which deposition technique is used and on

various process parameters. The magnitude of the intrinsic stresses can be very high, sometimes exceeding

the yield or fracture strengths of the corresponding bulk materials. Many theories to explain these stresses

have been suggested, and a summary is given in a review by Windischmann (1992). Intrinsic tensional

stresses have been attributed to grain boundary formation, to constrained shrinkage of disordered

material buried behind the advancing ﬁlm surface, and to attractive interatomic forces acting between

detached grains separated by a few atomic distances. Intrinsic compressive stresses, on the other hand,

have been attributed to impurity or working gas incorporation, increased defect density, and to recoil

implantation of ﬁlm atoms.

The total residual stress in a ﬁlm after deposition hence can be expressed as a sum of the thermal

residual and the intrinsic stresses:

(15.27)

where nonthermal stresses induced by lattice mismatch at the interface have been included among the

intrinsic stresses.

15.4.3 Substrate and Interface Stresses

Residual stresses in a ﬁlm will induce balancing stresses of opposite sign in the substrate. Using equilib-

rium relationships for forces and moments, the stress response induced in the substrate surface can be

expressed as

(15.28)

where t

and t

are the thicknesses of ﬁlm and substrate, respectively. Hence, in very thick substrates (t



), negligible stress response is induced. Equation 15.28 is derived for low t

ratios. If this ratio is larger

than 0.01, the error in σ

res

becomes noticeable (>5%).

All stresses in ﬁlm and substrate discussed so far are normal tensile or compressive stresses oriented

parallel to the interface. In a well-adhered ﬁlm–substrate composite no shear stresses are present in the

inner parts of the interface. Along the edges of a coated region, on the other hand, shear stresses may be

present for moment balance reasons. In thin, layered structures this effect is sometimes manifested by a

visible buckling of the edges. If nonbonded areas exist in the interior part of an interface, shear stresses

may be present along their boundaries and contribute in lowering the mechanical strength of the interface.

σσ σ

res therm intr

=+,

σσ

res res

=−4

15.5 Plasticity and Thermomechanical Properties

At room temperature, silicon, quartz, the III-V compounds, and many other materials used in MST

display a linear-elastic response to tensile stresses, all the way to brittle fracture. Hence, the plastic yield

limit is never reached, and plastic deformation or other effects based on dislocation slip will not occur

under tension at room temperature. At elevated temperature, however, or for high compressive loads at

room temperature, many of these materials may reach their yield limits before they reach the fracture

limit, in which case dislocation slip is activated and eventually they will deform plastically. Figure 15.3

illustrates the variation of the fracture limit σ

or the yield limit σ

of silicon as a function of temperature

(Yasutake et al., 1982a). In most applications plastic yield is undesirable and is, therefore, avoided by

ample dimensioning or by a “safe” materials selection. In some cases, however, the room temperature

yield limit is locally exceeded in high-compressive-stress ﬁelds which may be generated internally during

processing of multilayer structures or by, e.g., unintentional microscratches or microindentations. In yet

other cases dislocation slip may be activated by high operating temperatures and induce time-dependent

processes such as creep, aging, or fatigue.

15.5.1 Elastic–Ductile Response

The difference between elastic–brittle response and elastic–ductile response of a material is illustrated by

Figures 15.4a and b. The ﬁrst diagram illustrates the case when the fracture limit is lower than the critical

load limit for dislocation slip. The second diagram illustrates the opposite case; i.e., the dislocations (if

they exist from the start) are immobile until the critical resolved shear stress is reached in some slip

systems, where dislocation slip and eventually dislocation multiplication are initiated. The resulting plastic

deformation — contrary to elastic deformation — is irreversible upon unloading. The plastic curve

segment in Figure 15.4b is not as steep as the elastic curve segment, but still displays a positive slope

corresponding to a strain hardening effect. This effect is primarily due to the gradually increasing density

of dislocations, which tend to get entangled and obstruct further dislocation slip, hence demanding a

gradual increase of the applied stress in order to maintain the straining process.

FIGURE 15.3 Variation of fracture limit σ

and yield limit σ

of silicon as a function of temperature. The upper

curve is as-received CZ or FZ silicon (difference negligible), and the lower curve is CZ silicon annealed at 800°C for

100 h. For temperatures below 525

C (curves maxima) the curves illustrate the fracture limit σ

, above this transition

temperature they illustrate the plastic yield limit σ

. (From Yasutake, K. et al., 1982a.)

Much work has been devoted to the study of dislocations and plastic ﬂow in silicon and other

semiconductors. Essential parts of this work are surveyed in well-known articles by Alexander and Haasen

(1968) and by Hirsch (1985). In the plastic interval the dopant and impurity levels play a certain role

(Alexander and Haasen, 1968; Yonenaga and Sumino, 1978, 1984; Sumino et al., 1980, 1983, 1985; Sumino

and Imai, 1983; Imai and Sumino, 1983; Sumino, 1983b; Hirsch, 1985). At high levels of n-doping in Si,

the mobility of the dislocations is increased; i.e., the strain hardening effect is diminished. Impurities

such as oxygen or nitrogen atoms, on the other hand, tend to gather around the dislocations and hamper

their motion; so-called Cottrell atmospheres are formed around the dislocations. This means that higher

loads are required to “tear loose” the dislocations from these atmospheres, implying increased yield limit.

When the dislocations have been torn loose, they regain their high mobility, resulting in a marked yield

drop, see Figures 15.4b and 15.5.

The strain rate

ε during plastic deformation of semiconductors has been the subject of many studies

(Alexander and Haasen, 1968; Sumino, 1983a). Its dependency on temperature, effective resolved shear

stress, and dislocation velocity has been investigated in detail. Also the inﬂuence of doping levels,

impurities, and growth process (ﬂoat-zone growth, FZ, or Czochralski growth, CZ) have been studied,

and found to be considerable (Imai and Sumino, 1983; Sumino and Imai, 1983). Micromechanical

elements normally are designed to function below the yield limit, so the strain rate will only be discussed

here in connection with creep.

Microhardness is a complex materials parameter involving several properties of a more fundamental

nature, in particular plasticity properties. From a practical viewpoint the microhardness is a simple and

convenient measure of the susceptibility of a material to contact damage in the micron range (microin-

dentations, microscratches, etc.), and it plays a major role in most models describing tribological pro-

cesses. The microhardness can be measured by several methods, among which Vickers indentation and

Knoop indentation are most commonly used. In both methods a diamond stylus is pressed into the

surface of the body by a given load and at a given loading rate. Upon unloading, the size of the residual

FIGURE 15.4 (a) Linear elastic–brittle response.

(b) Elastic–ductile response.

indentation mark in the surface is measured and related to a characteristic hardness parameter (dimen-

sion: force/area). In the Vickers method the diamond stylus has the shape of a low proﬁle, square pyramid,

whereas the Knoop stylus has a rhombic pyramid shape.

In single-crystalline surfaces, the Vickers hardness (H

) is weakly anisotropic (Ericson et al., 1988).

Table 15.6 gives a few characteristic H

values for semiconductor surfaces of different crystallographic

orientations.

15.5.2 Time-Dependent Effects

Creep is a thermally activated deformation process which occurs under constant load (below the yield

limit) and during an extended period of time (Alexander and Haasen, 1968). Creep testing of brittle

materials is usually performed under compression. In Figure 15.6 two sets of typical creep curves (strain

vs. time) for Si under various loads and temperatures are displayed. In these curve sets, the points of

maximum creep rate

max

(i.e., the points of inﬂection) obey an exponential type of relationship (Reppich

et al., 1964):

(15.29)

where U is an activation energy and T is the temperature. This general creep behavior of Si is typical

also for Ge and many III-V compounds. Doping has a major inﬂuence on

max

. Doping of Si with As to

FIGURE 15.5 Resolved shear stress τ vs. shear strain γ

for tensile testing of single crystalline specimens at various

temperatures for (a) Si (Adapted from Yonenaga, I. and

Sumino, K., 1978, Phys. Status Solidi 50, 685.) and (b)

GaAs (Adapted from Sumino, K. et al., 1985, in Proc. 27th

Meeting, 145 Committee of JSPS, p. 91.)

exp ,

max

ετ=−

()

CUkT