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Ericson, E.; et. al “Mechanical Properties of Materials in Microstructure...” Handbook of Micro/Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press LLC, 1999 © 1999 by CRC Press LLC 15 Mechanical Properties of Materials in Microstructure Technology Fredric Ericson and Jan-Åke Schweitz 15.1 Introduction 15.2 Cohesion and Crystal Structures System Energy and Interatomic Binding • Lattice Structures and Structural Defects 15.3 Elasticity Properties Isotropic Elasticity • Anisotropic Elasticity 15.4 Internal Stresses Thermal Film Stress • Intrinsic Film Stress • Substrate and Interface Stresses 15.5 Plasticity and Thermomechanical Properties Elastic–Ductile Response • Time-Dependent Effects 15.6 Fracture Properties Fracture Limit and Fracture Toughness • Some Fracture Data • Fracture Initiation • Weibull Statistics • Fatigue 15.7 Adhesive Properties and Influence of Coatings Adhesion • Influence of Coatings 15.8 Testing General Test Structures and Testing Methods • Elasticity Testing by Static Techniques • Elasticity Testing by Dynamic Techniques • Testing of Other Properties 15.9 Modeling and Error Analysis Single-Layer Beam • Two-Layer Beam • Resonant Beam • Micro vs. Bulk Results 15.10 Summary and Conclusions References 15.1 Introduction Mechanical properties are of critical importance to any material that is used for transmission of forces or moments, or just for sustaining loads. The gradual introduction of microcomponents in practical © 1999 by CRC Press LLC applications within microstructure technology (MST) has instigated an increasing demand for insight into the fundamental factors that determine the mechanical integrity of such elements, for instance, their long-term reliability or how to choose proper safety limits in design and use. The mechanical integrity of a microsystem is not only of importance in mechanical applications. It is not unusual that mechanical (or thermomechanical) integrity is a prerequisite for reliable performance of microsystems with primarily nonmechanical functions, e.g., electric, optic, or thermal. Do we have enough knowledge about mechanical properties to determine the long-time reliability of a micromechanical component or to choose proper safety limits? In general, the answer to this question is negative. Are the properties of bulk materials applicable to microsystems? Again, we do not know for certain in every case; we cannot even be absolutely sure that bulk data on the fundamental elastic constants are valid for a micromachined element. Furthermore, the materials used in semiconductor technology are usually well characterized from an electronic viewpoint, but from a mechanical viewpoint they are in many cases more or less uncharted. In some cases we do not even have access to bulk data. This leads to the conclusion that much more work is required on the systematic exploration of the mechanical properties of microsized elements, as well as on the influence of the manufacturing processes on these properties. This chapter aims to define some basic concepts concerning mechanical properties, and to relate these concepts to experimental procedures and to some practial design aspects. For many properties we give numerical examples, if they exist, mostly concerning silicon and related materials. Sometimes comparisons of these materials with other types of materials are made. Silicon-based micromechanics is predominant today. In the future, mechanically high-performing materials like SiC may be frequently used in micromachined structures in high-temperature applications, for instance. For this reason, a number of thermomechanical phenomena are briefly defined and discussed in this chapter. 15.2 Cohesion and Crystal Structures 15.2.1 System Energy and Interatomic Binding Mechanical properties such as elasticity, plasticity, fracture strength, adhesion, internal stresses, etc., depend on the fundamental mechanisms of cohesion between atoms. Basically, the atoms in a solid material are held together by electrostatic attraction between charges of opposite signs. Magnetic forces are of minor importance to the cohesion. The potential energy of interatomic binding consists of terms of classical electrostatic interaction as well as terms of electrostatic quantum interaction (exchange effects). At equilibrium, the attractive potential energy Uo is balanced by the repulsive kinetic energy To in a state of minimum system energy Eo : Eo = U o + To , (15.1) see Figure 15.1. Neglecting boundary effects, the balance between potential and kinetic energy at equilibrium is given by the virial theorem: 2To + U o = 0 , (15.2) where To is positive (repulsive) and Uo is negative (attractive). In Figure 15.1 the parameter a represents some measure that is proportional to the average interatomic distance, for instance, the lattice parameter. If the state of equilibrium is shifted by external forces, compressive or tensile, the value of a will decrease or increase, and the system will move along the curve of Figure 15.1 away from the state of minimum system energy. The virial theorem is then modified into © 1999 by CRC Press LLC FIGURE 15.1 Crystal system energy E vs. lattice spacing parameter a. 2T + U = −a ∂E , ∂a (15.3) where F = – ∂E/∂a is the force striving to restore equilibrium at minimum energy. For small deviations from equilibrium, E is a harmonic function of a in most materials, and the restoring force F is a linear function of the change in a. This is one way of expressing Hooke’s law of elasticity: σ = Eε , (15.4) where σ is the applied stress (force per area unit), ε is the resulting strain ∆a/ao, and E is Young’s modulus of the material. Hence, Young’s modulus (= linear modulus of elasticity) is proportional to the second derivative of the system energy E with respect to the lattice parameter a at equilibrium (a = ao ). Depending on the electron structure of the constituent atoms, the mechanism of cohesion can vary from weak dipole interaction (van der Waals interaction) to strong covalent binding. In silicon the latter type is predominant, but some materials of interest for micromachining also exhibit ionic binding or metallic binding. In III-V semiconductors, for instance, the cohesion is of a mixed covalent and ionic nature, and in tungsten metallic binding is predominant. Ionic binding is found in chemical compounds such as common salt, NaCl. One or more electrons are transferred from one type of atom to the other, whereby electrostatic attraction between ions of opposite electric charge occurs. Metallic binding occurs in metallic elements or compounds. In this case the loosely bound valence electrons are disconnected from the atoms, and form a quasi-free electron gas, the conduction electron gas. These electrons are at liberty to move between the ion cores and to a large extent also through them. The high mobility of these electrons gives rise to the good electric conductivity found in most metals. Hence, the positive ions are immersed in a sea of negative conduction electrons, which act as a fluid cement holding the ions together. Covalent binding is found in some elemental solids (e.g., silicon) as well as in very complicated molecular structures (e.g., polymers). The cohesive mechanism is complex, but in a very simplified picture it can be described in terms of negative-valence electrons preferring to locate themselves in the regions between a positive ion and its nearest neighbors, by which an electrostatic coupling occurs. This type of binding is usually strong and “directional”; i.e., the interatomic bonds are formed at specific angles, resulting in well-defined molecular or crystalline structures. The strength of the interatomic bonds is decisive for the stiffness and the brittleness of the crystal. Strong and directional covalent bonds give silicon high stiffness and strength. In GaAs the bonds are of a mixed covalent and ionic type, making this material less stiff and more fragile than Si. Also the melting points (Table 15.1) are affected by the bond strength. © 1999 by CRC Press LLC TABLE 15.1 Melting Points (°C) of a Number of Semiconductors and Other Materials Si Ge SiC BN AlAs GaAs GaP InP InSb 1412 937 2537 4487 1737 1238 1467 1070 536 Nylon Teflon Stainless steel Al2O3 TiC HfC SiO2 Glass 137–150 290 1400–1500 2050 3100 3890 1610 ~700 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, SiO2 , 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 superstructures 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; © 1999 by CRC Press LLC 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 semiconductors) display a linear elastic behavior, i.e., they obey Hooke’s law, Equation 15.4, for the relation between applied normal stress (σ) and resulting normal strain (ε). The corresponding relationship between shear stress (τ) and shear strain (γ) is given by τ = Gγ , (15.5) where G 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) E and G values are sometimes sufficient. When a linear-elastic material is subjected to a uniaxial strain (relative elongation) ε = (L – Lo)/Lo , its cross-sectional dimension will diminish by a relative contraction εc = (do – d)/do . The ratio of these two strains is a materials constant called the Poisson’ s ratio: ν = εc ε . (15.6) In isotropic media the elastic parameters are related by [ ( )] G = E 2 1+ ν . (15.7) The relative volume change caused by a uniaxial stress σ is given by ( ) ∆V Vo = 1 − 2ν σ E = Kσ 3 , (15.8) where K is the compressibility: ( ) K = 3 1 − 2ν E . (15.9) The bulk modulus is defined as the inverse value of the compressibility: [ ( )] B = 1 K = E 3 1 − 2ν . (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 N E = ∑f E n n n =1 © 1999 by CRC Press LLC , (15.11) TABLE 15.2 Values of Elastic Stiffness Constants of a Number of Semiconductors at 300 K (in units of GPa) Si Ge GaAs InP InAs Diamond C11 C12 C44 165.78 129.11 118.80 102.20 83.29 1076.4 63.94 48.58 53.80 57.60 45.26 125.2 79.62 67.04 59.40 46.00 39.59 577.4 The values are results published by different workers, as compiled by Simmons et al. (1971). The values for diamond were published by van Enckevort (1994). N 1 E⊥ = ∑f n E ⊥n , (15.12) n =1 where E||n and E⊥n are the Young’s moduli of the constituent materials in the two directions, and fn 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 E 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 Cij (i,j = 1, 2, … 6) or, alternatively, by the elastic compliance constants Sij . These matrices are symmetric, and in cubic crystals their number of elements is reduced by symmetry considerations to three independent constants: C11 , C12 , and C44 (or S11, S12 , and S44). 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 ( C11 − C12 = S11 − S12 ( ) −1 C11 + 2C12 = S11 + 2S12 ) , −1 (15.13) , C44 = S44−1 . (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 ( ) 1 E = S11 − 2 S11 − S12 − S44 2 k1 , © 1999 by CRC Press LLC (15.16) TABLE 15.3 Young’s Moduli E at 300 K for Various Directions (in units of GPa) Directions Si Ge GaAs InP InAs Diamond <100> <110> <111> Poly 130.2 102.5 85.3 60.7 51.4 1050.3 169.2 137.5 121.4 93.4 79.3 1163.6 187.9 155.2 141.3 113.9 96.7 1207.0 163 132 116 89 76 1141 The polycrystalline results are mean values of the Hashin and Shtrikman bounds, as calculated by Simmons et al. (1971). where: ( ) ( ) ( ) 2 2 2 k1 = lm + mn + nl , (15.17) and the directional cosines are normalized according to l 2 + m2 + n2 = 1. (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): [ ( ) ] ν = − S12 + S11 − S12 − S44 2 k2 E , (15.19) where E is the Young’s modulus of the 〈lmn〉 direction given by Equation 15.16, and k2 is given by () ( ) ( ) 2 2 2 k2 = il + jm + kn . (15.20) Orthonormality conditions, supplementing Equation 15.18, are i 2 + j 2 + k 2 = 1 and (15.21) il + jm + kn = 0. (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, Al2O3, and AlN. It is sometimes desirable to calculate the elastic properties of a randomly polycrystalline, but macroscopically 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 © 1999 by CRC Press LLC TABLE 15.4 Poisson Ratios ν at 300 K for Tension along [lmn] and Contraction in the Perpendicular Direction System [100]<010> [100]<011> [110]<001> – [110]<110> – [110]<111> – [111]<110> – [110]<112> – [111]<112> Poly Si Ge GaAs InP InAs Diamond 0.278 0.278 0.362 0.062 0.162 0.180 0.262 0.180 0.222 0.273 0.273 0.367 0.026 0.139 0.157 0.253 0.157 0.208 0.312 0.312 0.443 0.021 0.162 0.188 0.303 0.188 0.243 0.360 0.360 0.555 0.015 0.195 0.238 0.375 0.238 0.294 0.352 0.352 0.543 0.001 0.182 0.222 0.362 0.222 0.283 0.104 0.104 0.115 0.008 0.044 0.045 0.079 0.045 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. 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 © 1999 by CRC Press LLC TABLE 15.5 Linear Coefficients of Thermal Expansion α (in units of 10–6 K–1) Anisometric Isometric Si GaAs AlAs SiC Diamond Glass Cu 2.4 6.0 5.2 4.5–5.0 1.3 8.0 16.2 (⊥ Axis) ( Axis) 13.7 1.0 8.3 –2.6 6.5 8.2 7.5 27.0 9.0 11.5 –2.0 –17.6 SiO2 (quartz) Graphite Al2O3 Al2TiO5 LiAlSi2O6 LiAlSiO4 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: α= 1 dL , Lo dT (15.23) ∆L = α∆T , Lo (15.24) which can be expressed as ε therm = where εtherm 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., Al2TiO5, LiAlSi2O6, and LiAlSiO4, 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: σ therm = Eε therm = Eα ∆T . (15.25) For a thin film on a thick substrate, we have biaxial stress conditions, and Hooke’s law is σ therm = © 1999 by CRC Press LLC Ef 1− νf ∆α ∆T , (15.26)