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Effect of residual stresses on the strength and fracture energy of the brittle film: Multiple cracking analysis
Computational Materials Science 50 (2010) 246–252
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Effect of residual stresses on the strength and fracture energy of the brittle film: Multiple cracking analysis X.C. Zhang ⇑, C.J. Liu, F.Z. Xuan, Z.D. Wang, S.T. Tu Key Laboratory of Safety Science of Pressurized System, Ministry of Education, East China University of Science and Technology, Shanghai 200237, PR China
a r t i c l e
i n f o
Article history: Received 11 June 2010 Accepted 3 August 2010 Available online 15 September 2010 Keywords: Residual stress Film Fracture energy Interfacial shear strength Multiple cracking analysis
a b s t r a c t The aim of this paper was to address the effect of the residual stresses within the brittle film on the substrate on the film strength, fracture energy, and interfacial shear strength (IFSS). Special analyses were performed on the SiOx film/polyethulene terephthalate substrate systems. The residual stresses were evaluated by using the curvature method. The film strength, fracture energy, and IFSS were estimated on the basis of the multiple cracking analyses. In the multiple cracking analyses, the system was subjected to the combination of the residual stresses and the unidirectionally applied stress. Results showed that the relationship between the crack density in the film and the applied strain can be predicted by adopting the energy criterion on the basis of the knowledge on the residual stress distributions in the film segment. The film strength and fracture energy for the initiation of film cracking were almost proportional to the compressive residual stresses in the film. With increasing the compressive residual stresses within the film, the IFSS also increased. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The brittle films deposited on the relatively soft substrate materials have been widely used to improve the properties of the substrates under various conditions [1]. However, during the film deposition process, the residual stresses are inevitably generated due to the lattice mismatch and thermal mismatch between the film and substrate. The structural integrity and stability of the film/substrate systems are often influenced by the residual stresses in the film. For instance, the residual stresses can provide a driving force to form defects, which, in turn, degrade the device performance of film [2–5]. Hence, in the past decades, the residual stresses in the film/substrate structures have been of great interest to the material scientists and physicists. Different analytical models with closed-form solutions have been developed to predict the residual stresses in the film/substrate systems [6–10]. However, some of the existing models will lead to serious errors in predicting the residual stresses [11]. Recently, an analytical model has been developed for analysing the residual stress in multilayer and graded systems [12,13]. In the analytical model, the total strains in the individual layers are decomposed into an in-plane strain component and a bending strain component. The closed-form solutions for the residual stresses far away from the free edges of the system are obtained by balancing the in-plane forces and bending moments in the whole system. ⇑ Corresponding author. Tel.: +86 21 64253149; fax: +86 21 64253425. E-mail address: [email protected] (X.C. Zhang). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.08.010
The residual stresses in film/substrate system have an important influence on the strength and fracture toughness of the film. Different methods, such as indentation, have been developed to predict the relationship between them. For instance, Jungk et al. [14] investigated the fracture behavior and toughness of thin laser annealed ta-C films on silicon substrates using acoustic-sensing nanoindentation. Results showed that the crack length can be expressed as a function of both the indentation load and the residual stresses within the film. The tensile residual stresses may promote the crack propagation. Using the depth sensing nanoindentation, Bhowmick et al. [15] found the existence of the high compressive residual stresses within the columnar TiN film substantially improved the shear fracture strength of it. Using the microindentation, Xia et al. [16] evaluated the effect of the residual stresses on the fracture toughness of the thin film. Experimental results showed that the film fracture toughness decreased with increasing the tensile residual stress in the film, since the cracking of the film released the stored energy associated with the residual stress field. However, when the indentation method is used, the cracking propagation path can not be easily controlled owing to the micro-defects and residual stresses in the film. In such a case, it is difficult to quantitatively predict the effect of the residual stress on the strength or fracture toughness of film. Recently, the technique of multiple film cracking (MFC) has been used to evaluate the strength of the film [17–23]. By using this technique, the film/substrate system is subjected to a unidirectional strain. However, prior to loading, both the film and the substrate are subjected to residual stresses due to the mismatch strain.
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When the applied strain reaches a critical value, the cracking of the film perpendicular to the loading direction will be initiated. With increasing the applied strain, the number of the cracks increases and the roughly uniform spacing can be observed. The relationship between the crack density and the applied strain can be used to evaluate the strength and fracture energy of the film. However, despite the existence of a large amount of observational data on MFC of brittle films, the analytical treatment of the film-cracking problem has challenged theoreticians for about four decades [24]. But, the MFC can be analysed on the basis of the stress distribution prediction in the film segment with a certain cracking criterion [24,25]. The stress transfer between the film and the substrate during the loading and cracking processes in a planar geometry has been often analysed by adopting the concept of the shear-lag model initially developed by Cox [26]. Recently, Zhang et al. developed a rigorous model for predicting the stresses and MFC in a threedimension sense on the basis of a modified shear-lag model [27]. By comparing with the experimental results, the analytical model can accurately predict the relationship between the crack density of the film and the applied strain. The effects of the mismatch strain and film thickness on the crack density, the fracture energy, and the strength of the film can be reflected by using this model. The aim of this paper was to address the effect of the residual stresses within the film on the strength, the fracture energy and the interfacial shear strength (IFSS) of it on the basis of Zhang’s model [27]. First, the analytical model with the closed-form solutions for the residual stresses within the film/substrate structure was summarized. In this model, the relationship between the mismatch strain between the film and substrate and the residual stresses in the film can be seen. Then, Zhang’s model for MFC prediction was used to predict the effect of the residual stresses in the film on some fracture parameters of the film. Finally, the case of SiOx film/ polyethulene terephthalate (PET) substrate system was studied.
to represent the residual stress through the thickness of the film. The relationship between the average stress in the film, rr, and the curvature of the system can be expressed as [29]
2. Residual stress prediction
f1 ¼
During the film deposition process, a mismatch strain will be generated. The mismatch strain De can be characterized as ef ;0 es;0 , where es;0 and ef ;0 denote the strains resulting from the deformation in the substrate and the film, respectively. For the semiconductor films, such as GeSi and GaAs, the mismatch strain is mainly generated due to the lattice mismatch. For the sprayed coatings, the total mismatch strain is mainly composed of two parts, namely thermally mismatch strain due to the cooling of the whole system and intrinsically mismatch strain due to the shrinkage of the deposited particles. However, for the calculations of stresses, the origin and nature of the mismatch strains in the film/substrate system are not important [28]. Generally, when the thicknesses and elastic constants of substrate and film are given, the mismatch strain between the film and the substrate can be determined from the measured curvature. The linear relationship exists between the curvature of the film/substrate system and the mismatch strain, i.e. [29]
K¼
1 6Rgð1 þ gÞ De t s 1 þ 4Rg þ 6Rg2 þ 4Rg3 þ R2 g4
rr ¼ KE0f tf
1 þ Rg3 6Rg2 ð1 þ gÞ
ð2Þ
where E0f ¼ Ef =ð1 v f Þ is the biaxial modulus of the film, and t f denotes the film thickness. 3. MFC prediction Zhang’s model for MFC analyses were developed on the basis of the stress redistribution analyses of the film segment due to the free-edge effects. The closed-form solutions of the stress distribution in the film segment were obtained by a modified shear-lag model, in which the interfacial shear stress transfer was considered [27]. The stress redistribution can be expressed as a function of the mismatch strain, applied strain, width of the film segment (i.e., crack spacing), and the thicknesses and elastic properties of the film and substrate. Moreover, the stress gradient through the thickness of the film was considered. But, in the analytical modeling, the effective substrate thickness is used instead of the actual substrate thickness. The interfacial shear stress in the film segment, s0 , can be expressed as [27]
s0 ¼ E0f ðf1 ea þ f2 DeÞtf j
sinhðjxÞ coshðjlÞ
ð3Þ
where ea is the applied strain, l is the uniform half width of the film segments formed due to the multiple cracking, as shown in Fig. 1, and the parameter j with length dimension can be interpreted as the characteristic stress transfer distance, reflecting the efficiency of load transfer from substrate to film along the lateral direction [30]. The parameters f1, f2, and j can be expressed as
ð1 þ Rg3 Þ½ð1 v s Þð1 þ v f Þ þ Rgð1 v f v s Þ ð1 þ 3Rg2 þ 4Rg3 þ Rg þ R2 g4 Þð1 þ v f Þ
f2 ¼
1 j¼ tf
ð1 þ Rg3 Þ ð1 þ 3Rg þ 4Rg3 þ Rg þ R2 g4 Þ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3gð1 þ 3Rg2 þ 4Rg3 þ Rg þ R2 g4 Þ 2ð1 þ v s ÞRð1 þ Rg3 Þ
ð4aÞ
ð4bÞ
ð4cÞ
Generally, when the applied strain is relatively high, the multiple cracking in the film will be saturated. In such a case, the crack spacing in the film will not be further decreased and can be termed
ð1Þ
where K denotes the curvature, ts is the substrate thickness, De is the mismatch strain, R and g are respectively the biaxial modulus ratio and thickness ratio of the film and the substrate. In the biaxial-stress state, R ¼ Ef ð1 v s Þ=Es ð1 v f Þ, where E and v are respectively the Young’s modulus and Poisson’s ratio, the subscripts f and s denote the film and substrate, respectively. When the film/substrate thickness ratio is much <1.0, the residual stress gradient in the film can be ignored. In such a case, the average residual stress in the film at the centerline can be used
Fig. 1. Schematic showing (a) the film/substrate system subjected to applied strain and a unidirectionally applied strain and (b) multiple film cracking with a uniform crack spacing, 2l.
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as the saturated crack spacing, lsat. If the multiple cracking in the film is saturated and the value of lsat is given, the critical interfacial shear stress which was calculated from Eq. (3) can be used to characterize the interfacial shear strength (IFSS) [31]. In Zhang’s model, both the strength and energy criteria were adopted to predict the relationship between the crack density and the applied strain. The case of SiOx film/polyethulene terephthalate (PET) substrate system was used to illustrate the implementation of the analytical model. The relationship between the crack density and the applied strain predicted using the energy criterion agreed well with the measurements. By comparing with the energy criterion, the strength criterion predicted a higher crack density for the given film thickness and applied strain. The similar result was also obtained by Hsueh and Yanaka [25]. However, the strength criterion can be used to determine the fracture strength of the film. When the strength criterion was used to predict the MFC of the film, it was assumed that the cracking of the film segment would occur when the film center stress reached the film strength, rstr. In such a case, the film strength can be obtained by letting the crack spacing approach infinitely and the applied strain approach the critical strain, ec , at which the film cracking begins, i.e.,
rstr ¼ E0f ðf1 ec þ f2 DeÞ
Table 1 The data of the calculated mismatch strain, De, of the SiOx/PET system and the residual stress, rr , in the film, the measured critical applied strain, ec [20], for initial film cracking, and the saturated crack spacing, lsat . tf (nm)
ec (%)
De (%)
rr (MPa)
lsat (lm)
43 67 90 120 320
1.9565 1.5440 1.4282 1.0700 0.7301
0.7077 0.5959 0.6417 0.4060 0.3364
547.097 431.134 437.145 256.706 140.805
1.2138 1.2987 1.9048 2.0748 4.0580
It can be seen that C is proportional to e2c =j. In such a case, the normalized applied strain, ea =ec , is only related to the half width of the film segment and should be
ea ¼ ec
ð11Þ
The relationship between the fracture energy, C, and the fracture strength, rstr, can be determined by substituting Eq. (5) into Eq. (6), i.e.
ð5Þ
When the energy criterion was used, it was assumed that the cracking occurred when the strain energy used for the breaking of the film segment reached the energy required for the film cracking, 2tf C. By letting l ! 1, the fracture energy C can be expressed as
sffiffiffiffiffiffiffiffi 3 4f 6
C¼
3f 3 f4 2 f De r þ 5 rstr 2j 4E0f j str
ð12Þ
If f5 De=2j ! 0, rstr will be proportional to the square root of C. 4. Results and discussion
2
C¼
3ðf1 ec þ f2 DeÞ f3 f4 þ 2Deðf1 ec þ f2 DeÞf5 0 Ef 4j
ð6Þ
where
f3 ¼
ð1 þ v f Þð1 2v f v s þ v 2s Þ
ð1 v f v s Þ2 2 2 Rg ð1 þ gÞ f4 ¼ 1 þ 3 ð1 þ Rg3 Þ f5 ¼
2v s ð1 þ v s Þð1 v 2f Þ 1 v f v s Þ2
ð7aÞ ð7bÞ ð7cÞ
The applied strain can be expressed as a function of the half crack spacing l, i.e. [27],
ea ¼ De
f8 ð2f 2 f3 f4 f6 þ f5 f7 Þ 2f 1 f3 f4 f6
jl 1 sinhð2jlÞ þ 2jl tanhðjlÞ þ 2 4 coshð2jlÞ þ 1 1 sinhðjlÞ þ jl 2 coshðjlÞ þ 1
ð8Þ
f6 ¼ 2 tanh
f7 ¼ tanh
f8 ¼
jl 1 tanhðjlÞ 2 2
4.1. Summary of Yanaka et al.’s experiments [20]
ð9aÞ
ð9bÞ
Experiments of MFC have been performed on SiOx (x ’ 1:7) films with different thicknesses deposited on the PET substrates with a thickness of 12 lm by Yanaka et al. [20]. The films were deposited by using roll-to-roll vacuum evaporation. The variation of crack density of the film as a function of the applied strain
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ð2f 2 f3 f6 f4 þ f5 f7 Þ2 f3 f4 f6 4ðf22 f3 f6 f4 þ f2 f5 f7 Þ 3f 3 f4 ðf1 ec =De þ f2 Þ2 2P4 ðf1 ec =De þ f2 Þ
When the mismatch strain De is zero, the expression of fracture energy in Eq. (6) can be greatly simplified and becomes
C¼
Using the Eqs. (1)–(3), (4a)–(4c), (5), (6), (7a)–(7c), (8), (9a)– (9c), (10)–(12), the effect of the residual stresses in the film on the fracture strength, fracture energy and IFSS can be reflected if the parameters De, ec, and lsat have been known. In this paper, the case of SiOx film/PET substrate system is used to illustrate the influence of the residual stresses on the strength and fracture energy of the film. First, experiments of MFC for SiOx film/PET substrate system performed by Yanaka et al. [20] are summarized. Second, using fracture energy criterion, the predicted results of the crack densities for the SiOx/PET systems with different film thicknesses as functions of the applied strain are compared with the experimental data. The effect of the misfit strain on the crack density is also shown. Third, the effects of the film thickness and the residual stresses on the strength and fracture energy of the film are discussed. Finally, the effect of the residual stresses on the IFSS is investigated.
3f12 f3 f4 2 0 eE 4j c f
ð10Þ
ð9cÞ
was measured by using an in situ observation technology and then recorded in video cassette recorder. A constant displacement rate was applied unidirectionally to stretch the system. The detailed experimental procedure for determining the critical applied strain
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at which the film cracking begins can be seen in Ref. [20]. The measured data of critical applied strains, ec , for films with different thicknesses are listed in Table 1. The elastic moduli of the bare substrate and the film/substrate system were calculated from the initial slops of the stress–strain curves. The difference of measured elastic moduli was then used to estimate the elastic modulus of the film, Ef. However, the measured date of Ef showed high scattering for the systems with different film thicknesses. In such a case, an average value was adopted. The elastic moduli of the film and substrate thus obtained were [20]: Ef = 73 GPa, and Es = 6 GPa. It can be seen the elastic modulus of the film is close to that of the bulk fused silica [32]. The Poisson’s ratios of the film and the substrate are [19]: vf = 0.17, and vs = 0.35. Hence, for the SiOx/PET system, the material constant ratio R ffi 9.528. In the experiment of MFC, the film was subjected to the compressive residual stress before loading. However, the mismatch strain between the film and substrate, De, was not determined by Yanaka et al. [20]. But, the curvature of the film/substrate system during the film deposition was determined. Using Eqs. (1) and (2), the mismatch strains and the residual stresses in the film for individual systems can be determined and the results are also listed in Table. 1. Previous experimental results by Yanaka et al. [20] also showed that the crack spacing was generally varied around a mean value l. When the applied strain was given, the longest segments would break into halves with more or less half lengths, while shorter segments would remain unbroken. The shortest crack spacing is about a half of the longest crack spacing. Hence, the mean crack spacing (MCS), l, can be approximated as the average of the longest and the shortest crack spacings, such that [25]
l ¼ 1:5l
ð13Þ
The crack density can be defined as the form of the inverse of the average crack spacing, i.e., 1=l. In Yanaka et al.’s experiments, it can also be found that when the applied strain reaches the value of 6.40741%, the multiple cracking in the film are saturated. At this applied strain, the saturated crack spacing, lsat, for individual film/substrate systems can be determined and are also listed in Table 1. 4.2. Effective substrate thickness When the system is subjected to the applied loading, underneath the film segment, the substrate stress in the lateral direction (i.e., x-direction in Fig. 1) is partially transferred to the film and it becomes lower than the far-field stress. It has been argued that the thin film influences the stress field in the substrate only within a certain boundary zone in the neighborhood of the film [24,25]. Outside the boundary zone, the stress field in the substrate is not perturbed by the presence of the film. Hence, in the analytical modeling, the effective substrate thickness should be instead of the actual substrate thickness [33]. However, the effective substrate thickness is influenced by the materials constants of the substrate and the film and the width of the film [24]. Up to present, the analytical model for defining the effective substrate thickness has not been established. Hsueh and Wereszczak determined the relationship between the effective substrate thickness and the Young’s modulus ratio between the film and the substrate using finite element analysis (FEA) [24]. However, in their analyses, the perturbation of 1.85% of the substrate stress was used as the criterion. In this paper, the effective substrate is also estimated by FEA. However, the perturbation of 0% of the substrate stress was used as the criterion. A SiOx film segment on the PET substrate schematically shown in Fig. 2a is used in FEA. The geometry is symmetric and only half of the system is used in FEA modeling. The finite
Fig. 2. (a) Schematic showing the geometry used in FEA to examine perturbation of the substrate stress and (b) normalized substrate stress, rs =r0 , at the axial line through the thickness direction as functions of the normalized depth underneath the film, y=t f , with respect to different values of the half width of film.
element meshes near the film/substrate interface and in the region close to edges are refined to avoid the mesh size-effect on the magnitude of the stresses. The half width and thickness of substrate are, respectively, 100 and 500 times of the film thickness. While the right edge substrate is subjected to a uniform stress, r0, in the lateral direction. Underneath the film segment, the substrate stress, rs, is partially transferred to the film and then becomes lower than r0. The plane-strain condition is considered in FEA. The constraint in the lateral direction is imposed on the axial line to satisfy the symmetry condition, as shown in Fig. 2a. To examine the effect of the width of the film segment on the perturbation of the substrate stresses, the normalized substrate stresses, rs/r0, at the axial line through the thickness direction as functions of the depth underneath the film normalized by the film thickness, z/tf, with respect to different values for the half width of film segment are shown in Fig. 2b. It can be seen that, with increasing the width of the film segment, the effective substrate thickness increases. For instance, for the system with a half film width l=t f ¼ 20, the substrate stress is almost equal to the far-field stress at z=t f ¼ 73:4. However, for the system with a half film width of l=tf ¼ 80, the perturbation of the substrate stress can be ignored at z=tf P 90:6. When l=tf ¼ 50, the effective substrate thickness is 86.0tf. In such a case, the substrate stresses are respectively 0.2% higher and 0.4% lower than the far-field stress for the systems with half film widths of 20tf and 80tf. Hence, 86.0tf can be used as the effective substrate thickness for the SiOx/PET systems.
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4.3. Crack density prediction Using Eq. (8), the predicted MCSs for the SiOx/PET systems with different film thicknesses as functions of the applied strain are shown in Fig. 3. The predicted results are compared with the experimental data obtained by Yanaka et al. [20]. For a given film/substrate system, the MCS decreases with increasing the applied strain and the decreasing rate becomes smaller as the applied strain ea increases. The predicted relationship between the MCS and the applied strain agrees well with the measurements. However, when the film thickness is 43 nm, there is some discrepancy existing between the predicted results and the experimental results. Different reasons leads to this discrepancy. First, the present analytical modeling is performed on the basis of the linear elasticity assumption. However, the PET substrate is prone to be plastically deformed when the external loading is applied since its yield strain is about 2.2% [22]. The stress in film segment obtained from elasto-plastic solution is relatively lower than that obtained from elastic solution at a given ea [18]. If the substrate plasticity is considered, the predicted MCS will be increased. Second, although the three-dimensional stress analysis is performed in Zhang’s model, the mismatch strain, De, from the curvature was still evaluated on the basis of a two-dimensional case. In such
a case, the film stress may be highly evaluated, resulting in the highly predicted crack density and lowly predicted MCS. The normalized crack densities, t f =l, predicted using energy criterion as functions of the normalized applied strains, ea =ec , with respect to different film thicknesses are shown in Fig. 4. The measured data are also used for comparison. It can be seen that the predicted curves for the systems with the film thicknesses of 67 nm and 120 nm cannot be distinguished. From Table 1, it can be seen that when the film thicknesses are 67 nm and 120 nm, the ratio of the mismatch strain and the critical strain is about to be 0.38. The predicted curves for the systems with the film thicknesses of 90 nm and 320 nm cannot also be distinguished, as shown in Fig. 4. For these two systems, the ratio of De/ec is about 0.45. However, the normalized crack densities for these two systems are higher than those for the other three systems. Hence, if the ratio De/ec is controlled to be constant, the normalized crack density t f =l is almost not influenced by the film thickness. In such a case, the crack densities for the systems with different film thicknesses can be predicted when the crack density versus applied strain relationship has been measured at one film thickness. 4.4. Film strength and fracture energy The film strength, rstr, and fracture energy, C, for the film initial cracking respectively obtained from Eqs. (5) and (6) as functions of t1=2 are shown in Fig. 5a. With increasing the film thickness, both f the film strength and the fracture energy of the film decrease. It is
Fig. 3. The predicted crack spacings for the SiOx/PET systems with different film thicknesses as functions of the applied strain (symbols: measurements [20], lines: predictions).
Fig. 4. Normalized crack density, tf =l, predicted using energy criterion as functions of the normalized applied strains, ea =ec , with respect to different film thicknesses (symbols: measurements [20], lines: predictions).
Fig. 5. The film strength, rstr , and fracture energy, C, for the film initial cracking as functions of (a) square root of film thickness t1=2 and (b) compressive residual f stress in film, rr .
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interesting to find that both rstr and C are almost proportional to the inverse of the square root of the film thickness. For the film strength, the similar trend has been found by Yanaka et al. [20]. The predicted film strength decreases from 582.5 MPa to 160.3 GPa when the film thickness increases from 43 nm to 320 nm. The fracture energy of film ranges from 6.6 J/m2 to 4.7 J/ m2 when film thickness ranges from 43 nm to 320 nm. The reported strain energy release rates for the cracking amorphous and crystallized glasses are 8 J/m2 and 20 J/m2 [25], respectively, and the corresponding fracture energies are, 4 J/m2 and 10 J/m2, respectively. Hence, the predicted C for initial cracking of the film is in the order of that for the bulk glass. The film strength and fracture energy as functions of the residual stress in the film are shown in Fig. 5b. Except for the system with the film thickness of 90 nm, both the film strength and the fracture energy are almost proportional to the compressive residual stress in the film. Generally, with increasing the film thickness, the mismatch strain, De, between the film and substrate decreases due to the stress relaxation during the film deposition process, leading to the decrement of the residual stresses. However, for the system with a film thickness of 90 nm, the mismatch strain is 0.6417%, which is higher than the mismatch strains for the systems with film thicknesses of 67 nm and 120 nm, as shown in Table 1. Since the mismatch strain is predicted from the curvature of the system, there must be some error involving in the curvature measurement. Considering the error in the curvature measurement, it can be concluded that with increasing the compressive residual stress in the film, the strength and fracture energy of the film increases. pffiffiffiffi The variation of the square root of the fracture energy, C, for the film initial cracking along pffiffiffiffi the film strength, rstr, is shown in Fig. 6. It can be seen that C is almost proportional to rstr. With increasing the film strength, the fracture energy increases. Using fracture mechanics with small scale yielding, the similar result was also obtained by Hu and Evans [34]. They found that the energy release rate, Gc, of the film cracking through its thickness was proportional to the square of the film strength at which the first crack occurred. However, in the solution for Gc, a non-dimensional parameter which should be determined numerically existed. 4.5. Interfacial shear strength Using Eq. (3), the IFSSs for individual systems were calculated by setting l = lsat and the results are listed in Table 2. It should be noted that the IFSS is characterized by the maximum value of the interfacial shear stress when the multiple cracking is saturated. The predicted IFSS decreases from 98.79 MPa to 61.62 MPa when
Table 2 The data of the calculated IFSSs of the SiOx/PET systems with different thicknesses. tf (nm)
IFSS (MPa)
43 67 90 120 320
98.79071 80.92483 84.96647 77.88309 61.61682
Fig. 7. The variation of the interfacial shear strength of the SiOx/PET system along the compressive residual stress in the film.
the film thickness increases from 43 nm to 320 nm. The reported IFSS of the SiOx/PET system ranges from 20 MPa to 100 MPa [31]. Hence, the predicted data of IFSS agree well with the experimental results. However, from Table 2, it can be seen that the IFSS generally decreases with increasing the film thickness. This phenomenon is partially due to the variation of the residual stresses in the film. The IFSS as a function of the compressive residual stress in the film is shown in Fig. 7. It can be seen that with increasing the compressive residual stress in the SiOx film, the IFSS generally increases. It can also be conclude that the residual stress in the film is a key factor that controls the IFSS. Experimental results by Leterrier et al. [35] also showed that there is a linear relationship between the IFSS of the SiOx/PET system and the residual stress in the film. 5. Conclusions The effects of the residual stresses within the brittle films on the substrate on the film strength, fracture energy, and interfacial shear strength (IFSS) were investigated by using multiple film cracking analyses. Special analyses were performed on the SiOx film/polyethulene terephthalate substrate systems on the basis of the existing experimental data [20]. The conclusions are summarized as follows.
Fig. 6. The variation of the square root of the fracture energy, cracking along the film strength, rstr .
pffiffiffiffi C, for the film initial
(1) Both the film strength and the fracture energy for the initiation of film cracking are proportional to the compressive residual stresses in film. The film strength is almost proportional to the square root of the fracture energy. (2) The crack density in the film predicted by using the energy criterion agreed well with the experimental result. (3) The residual stress in the film is a key factor that controls the interfacial shear strength. With increasing the compressive residual stresses in the film, the interfacial shear strength of the film/substrate system increased.
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Acknowledgements The authors are grateful for the support by National Natural Science Foundations of China (50735001 and 10672058). The author X.C. Zhang is also grateful for the support by Shanghai Chenguang Planning Project (2008CG36) and Shanghai Rising-Star Program (08QA14023) and Ph.D. Programs Foundation of Ministry of Education of China (20090101120021). Reference [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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S. Bhowmick, V. Jayaram, S.K. Biswas, Acta Mater. 53 (2005) 2459–2468. Z.H. Xia, W.A. Curtin, B.W. Sheldon, Acta Mater. 52 (2004) 3507–3515. T.R. Watkins, D.J. Green, J. Amer. Ceram. Soc 76 (1993) 3066–3072. M. Yanaka, Y. Tsukahara, T. Okabe, N. Takeda, J. Appl. Phys. 90 (2001) 713–719. M. Yanaka, N. Nakaso, J. Mater. Sci. 33 (1998) 2111–2119. M. Yanaka, T. Miyamoto, Y. Tsukahara, N. Takeda, Compos. Interface 6 (1999) 409–422. Y. Leterrier, J. Andersons, Y. Pitton, J.-A.E. Månson, J. Polym. Sci. Part B: Polym. Phys. 35 (1997) 1463–1472. Y. Leterrier, Prog. Mater. Sci. 48 (2003) 1–55. L. Gan, J. Wang, R.M. Pilliar, Biomaterials 26 (2005) 189–196. C.H. Hsueh, A.A. Wereszczak, J. Appl. Phys. 96 (2004) 3501–3506. C.H. Hsueh, M. Yanaka, J. Mater. Sci. 38 (2003) 1809–1817. H.L. Cox, Brit. J. Appl. Phys. 3 (1952) 72–79. X.C. Zhang, F.Z. Xuan, Z.D. Wang, S.T. Tu, J. Appl. Phys. 104 (2008) 063520-1– 063520-8. S.C. Jain, H.E. Maes, K. Pinardi, IDe. Wolf, J. Appl. Phys. 79 (1996) 8145–8165. X.C. Zhang, B.S. Xu, H.D. Wang, Y.X. Wu, Y. Jiang, Surf. Coat. Technol. 201 (2007) 6715–6718. J. Andersons, Y. Leterrier, G. Tornare, P. Dumont, J.-A.E. Månson, Mech. Mater. 39 (2007) 834–844. Y. Leterrier, L. Boogh, J. Andersons, J.-A.E. Månson, J. Polym. Sci. Part B: Polym. Phys. 35 (1997) 1449–1461. Chronological Scientific Tables, National Astronomical Observatory, Maruzen, Tokyo, 1999, p. 453. (in Japanese). G.P. Cherepanov, J. Appl. Phys. 75 (1994) 844–849. M.S. Hu, A.G. EvansActa, Metall, Acta Mater. 27 (1989) 917–925. Y. Leterrier, Y. Wyser, J.-A.E. Månson, J. Adhesion. Sci. Technol. 15 (2001) 841– 865.
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Effect of residual stresses on the strength and fracture energy of the brittle film: Multiple cracking analysis X.C. Zhang ⇑, C.J. Liu, F.Z. Xuan, Z.D. Wang, S.T. Tu Key Laboratory of Safety Science of Pressurized System, Ministry of Education, East China University of Science and Technology, Shanghai 200237, PR China
a r t i c l e
i n f o
Article history: Received 11 June 2010 Accepted 3 August 2010 Available online 15 September 2010 Keywords: Residual stress Film Fracture energy Interfacial shear strength Multiple cracking analysis
a b s t r a c t The aim of this paper was to address the effect of the residual stresses within the brittle film on the substrate on the film strength, fracture energy, and interfacial shear strength (IFSS). Special analyses were performed on the SiOx film/polyethulene terephthalate substrate systems. The residual stresses were evaluated by using the curvature method. The film strength, fracture energy, and IFSS were estimated on the basis of the multiple cracking analyses. In the multiple cracking analyses, the system was subjected to the combination of the residual stresses and the unidirectionally applied stress. Results showed that the relationship between the crack density in the film and the applied strain can be predicted by adopting the energy criterion on the basis of the knowledge on the residual stress distributions in the film segment. The film strength and fracture energy for the initiation of film cracking were almost proportional to the compressive residual stresses in the film. With increasing the compressive residual stresses within the film, the IFSS also increased. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The brittle films deposited on the relatively soft substrate materials have been widely used to improve the properties of the substrates under various conditions [1]. However, during the film deposition process, the residual stresses are inevitably generated due to the lattice mismatch and thermal mismatch between the film and substrate. The structural integrity and stability of the film/substrate systems are often influenced by the residual stresses in the film. For instance, the residual stresses can provide a driving force to form defects, which, in turn, degrade the device performance of film [2–5]. Hence, in the past decades, the residual stresses in the film/substrate structures have been of great interest to the material scientists and physicists. Different analytical models with closed-form solutions have been developed to predict the residual stresses in the film/substrate systems [6–10]. However, some of the existing models will lead to serious errors in predicting the residual stresses [11]. Recently, an analytical model has been developed for analysing the residual stress in multilayer and graded systems [12,13]. In the analytical model, the total strains in the individual layers are decomposed into an in-plane strain component and a bending strain component. The closed-form solutions for the residual stresses far away from the free edges of the system are obtained by balancing the in-plane forces and bending moments in the whole system. ⇑ Corresponding author. Tel.: +86 21 64253149; fax: +86 21 64253425. E-mail address: [email protected] (X.C. Zhang). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.08.010
The residual stresses in film/substrate system have an important influence on the strength and fracture toughness of the film. Different methods, such as indentation, have been developed to predict the relationship between them. For instance, Jungk et al. [14] investigated the fracture behavior and toughness of thin laser annealed ta-C films on silicon substrates using acoustic-sensing nanoindentation. Results showed that the crack length can be expressed as a function of both the indentation load and the residual stresses within the film. The tensile residual stresses may promote the crack propagation. Using the depth sensing nanoindentation, Bhowmick et al. [15] found the existence of the high compressive residual stresses within the columnar TiN film substantially improved the shear fracture strength of it. Using the microindentation, Xia et al. [16] evaluated the effect of the residual stresses on the fracture toughness of the thin film. Experimental results showed that the film fracture toughness decreased with increasing the tensile residual stress in the film, since the cracking of the film released the stored energy associated with the residual stress field. However, when the indentation method is used, the cracking propagation path can not be easily controlled owing to the micro-defects and residual stresses in the film. In such a case, it is difficult to quantitatively predict the effect of the residual stress on the strength or fracture toughness of film. Recently, the technique of multiple film cracking (MFC) has been used to evaluate the strength of the film [17–23]. By using this technique, the film/substrate system is subjected to a unidirectional strain. However, prior to loading, both the film and the substrate are subjected to residual stresses due to the mismatch strain.
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When the applied strain reaches a critical value, the cracking of the film perpendicular to the loading direction will be initiated. With increasing the applied strain, the number of the cracks increases and the roughly uniform spacing can be observed. The relationship between the crack density and the applied strain can be used to evaluate the strength and fracture energy of the film. However, despite the existence of a large amount of observational data on MFC of brittle films, the analytical treatment of the film-cracking problem has challenged theoreticians for about four decades [24]. But, the MFC can be analysed on the basis of the stress distribution prediction in the film segment with a certain cracking criterion [24,25]. The stress transfer between the film and the substrate during the loading and cracking processes in a planar geometry has been often analysed by adopting the concept of the shear-lag model initially developed by Cox [26]. Recently, Zhang et al. developed a rigorous model for predicting the stresses and MFC in a threedimension sense on the basis of a modified shear-lag model [27]. By comparing with the experimental results, the analytical model can accurately predict the relationship between the crack density of the film and the applied strain. The effects of the mismatch strain and film thickness on the crack density, the fracture energy, and the strength of the film can be reflected by using this model. The aim of this paper was to address the effect of the residual stresses within the film on the strength, the fracture energy and the interfacial shear strength (IFSS) of it on the basis of Zhang’s model [27]. First, the analytical model with the closed-form solutions for the residual stresses within the film/substrate structure was summarized. In this model, the relationship between the mismatch strain between the film and substrate and the residual stresses in the film can be seen. Then, Zhang’s model for MFC prediction was used to predict the effect of the residual stresses in the film on some fracture parameters of the film. Finally, the case of SiOx film/ polyethulene terephthalate (PET) substrate system was studied.
to represent the residual stress through the thickness of the film. The relationship between the average stress in the film, rr, and the curvature of the system can be expressed as [29]
2. Residual stress prediction
f1 ¼
During the film deposition process, a mismatch strain will be generated. The mismatch strain De can be characterized as ef ;0 es;0 , where es;0 and ef ;0 denote the strains resulting from the deformation in the substrate and the film, respectively. For the semiconductor films, such as GeSi and GaAs, the mismatch strain is mainly generated due to the lattice mismatch. For the sprayed coatings, the total mismatch strain is mainly composed of two parts, namely thermally mismatch strain due to the cooling of the whole system and intrinsically mismatch strain due to the shrinkage of the deposited particles. However, for the calculations of stresses, the origin and nature of the mismatch strains in the film/substrate system are not important [28]. Generally, when the thicknesses and elastic constants of substrate and film are given, the mismatch strain between the film and the substrate can be determined from the measured curvature. The linear relationship exists between the curvature of the film/substrate system and the mismatch strain, i.e. [29]
K¼
1 6Rgð1 þ gÞ De t s 1 þ 4Rg þ 6Rg2 þ 4Rg3 þ R2 g4
rr ¼ KE0f tf
1 þ Rg3 6Rg2 ð1 þ gÞ
ð2Þ
where E0f ¼ Ef =ð1 v f Þ is the biaxial modulus of the film, and t f denotes the film thickness. 3. MFC prediction Zhang’s model for MFC analyses were developed on the basis of the stress redistribution analyses of the film segment due to the free-edge effects. The closed-form solutions of the stress distribution in the film segment were obtained by a modified shear-lag model, in which the interfacial shear stress transfer was considered [27]. The stress redistribution can be expressed as a function of the mismatch strain, applied strain, width of the film segment (i.e., crack spacing), and the thicknesses and elastic properties of the film and substrate. Moreover, the stress gradient through the thickness of the film was considered. But, in the analytical modeling, the effective substrate thickness is used instead of the actual substrate thickness. The interfacial shear stress in the film segment, s0 , can be expressed as [27]
s0 ¼ E0f ðf1 ea þ f2 DeÞtf j
sinhðjxÞ coshðjlÞ
ð3Þ
where ea is the applied strain, l is the uniform half width of the film segments formed due to the multiple cracking, as shown in Fig. 1, and the parameter j with length dimension can be interpreted as the characteristic stress transfer distance, reflecting the efficiency of load transfer from substrate to film along the lateral direction [30]. The parameters f1, f2, and j can be expressed as
ð1 þ Rg3 Þ½ð1 v s Þð1 þ v f Þ þ Rgð1 v f v s Þ ð1 þ 3Rg2 þ 4Rg3 þ Rg þ R2 g4 Þð1 þ v f Þ
f2 ¼
1 j¼ tf
ð1 þ Rg3 Þ ð1 þ 3Rg þ 4Rg3 þ Rg þ R2 g4 Þ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3gð1 þ 3Rg2 þ 4Rg3 þ Rg þ R2 g4 Þ 2ð1 þ v s ÞRð1 þ Rg3 Þ
ð4aÞ
ð4bÞ
ð4cÞ
Generally, when the applied strain is relatively high, the multiple cracking in the film will be saturated. In such a case, the crack spacing in the film will not be further decreased and can be termed
ð1Þ
where K denotes the curvature, ts is the substrate thickness, De is the mismatch strain, R and g are respectively the biaxial modulus ratio and thickness ratio of the film and the substrate. In the biaxial-stress state, R ¼ Ef ð1 v s Þ=Es ð1 v f Þ, where E and v are respectively the Young’s modulus and Poisson’s ratio, the subscripts f and s denote the film and substrate, respectively. When the film/substrate thickness ratio is much <1.0, the residual stress gradient in the film can be ignored. In such a case, the average residual stress in the film at the centerline can be used
Fig. 1. Schematic showing (a) the film/substrate system subjected to applied strain and a unidirectionally applied strain and (b) multiple film cracking with a uniform crack spacing, 2l.
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as the saturated crack spacing, lsat. If the multiple cracking in the film is saturated and the value of lsat is given, the critical interfacial shear stress which was calculated from Eq. (3) can be used to characterize the interfacial shear strength (IFSS) [31]. In Zhang’s model, both the strength and energy criteria were adopted to predict the relationship between the crack density and the applied strain. The case of SiOx film/polyethulene terephthalate (PET) substrate system was used to illustrate the implementation of the analytical model. The relationship between the crack density and the applied strain predicted using the energy criterion agreed well with the measurements. By comparing with the energy criterion, the strength criterion predicted a higher crack density for the given film thickness and applied strain. The similar result was also obtained by Hsueh and Yanaka [25]. However, the strength criterion can be used to determine the fracture strength of the film. When the strength criterion was used to predict the MFC of the film, it was assumed that the cracking of the film segment would occur when the film center stress reached the film strength, rstr. In such a case, the film strength can be obtained by letting the crack spacing approach infinitely and the applied strain approach the critical strain, ec , at which the film cracking begins, i.e.,
rstr ¼ E0f ðf1 ec þ f2 DeÞ
Table 1 The data of the calculated mismatch strain, De, of the SiOx/PET system and the residual stress, rr , in the film, the measured critical applied strain, ec [20], for initial film cracking, and the saturated crack spacing, lsat . tf (nm)
ec (%)
De (%)
rr (MPa)
lsat (lm)
43 67 90 120 320
1.9565 1.5440 1.4282 1.0700 0.7301
0.7077 0.5959 0.6417 0.4060 0.3364
547.097 431.134 437.145 256.706 140.805
1.2138 1.2987 1.9048 2.0748 4.0580
It can be seen that C is proportional to e2c =j. In such a case, the normalized applied strain, ea =ec , is only related to the half width of the film segment and should be
ea ¼ ec
ð11Þ
The relationship between the fracture energy, C, and the fracture strength, rstr, can be determined by substituting Eq. (5) into Eq. (6), i.e.
ð5Þ
When the energy criterion was used, it was assumed that the cracking occurred when the strain energy used for the breaking of the film segment reached the energy required for the film cracking, 2tf C. By letting l ! 1, the fracture energy C can be expressed as
sffiffiffiffiffiffiffiffi 3 4f 6
C¼
3f 3 f4 2 f De r þ 5 rstr 2j 4E0f j str
ð12Þ
If f5 De=2j ! 0, rstr will be proportional to the square root of C. 4. Results and discussion
2
C¼
3ðf1 ec þ f2 DeÞ f3 f4 þ 2Deðf1 ec þ f2 DeÞf5 0 Ef 4j
ð6Þ
where
f3 ¼
ð1 þ v f Þð1 2v f v s þ v 2s Þ
ð1 v f v s Þ2 2 2 Rg ð1 þ gÞ f4 ¼ 1 þ 3 ð1 þ Rg3 Þ f5 ¼
2v s ð1 þ v s Þð1 v 2f Þ 1 v f v s Þ2
ð7aÞ ð7bÞ ð7cÞ
The applied strain can be expressed as a function of the half crack spacing l, i.e. [27],
ea ¼ De
f8 ð2f 2 f3 f4 f6 þ f5 f7 Þ 2f 1 f3 f4 f6
jl 1 sinhð2jlÞ þ 2jl tanhðjlÞ þ 2 4 coshð2jlÞ þ 1 1 sinhðjlÞ þ jl 2 coshðjlÞ þ 1
ð8Þ
f6 ¼ 2 tanh
f7 ¼ tanh
f8 ¼
jl 1 tanhðjlÞ 2 2
4.1. Summary of Yanaka et al.’s experiments [20]
ð9aÞ
ð9bÞ
Experiments of MFC have been performed on SiOx (x ’ 1:7) films with different thicknesses deposited on the PET substrates with a thickness of 12 lm by Yanaka et al. [20]. The films were deposited by using roll-to-roll vacuum evaporation. The variation of crack density of the film as a function of the applied strain
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ð2f 2 f3 f6 f4 þ f5 f7 Þ2 f3 f4 f6 4ðf22 f3 f6 f4 þ f2 f5 f7 Þ 3f 3 f4 ðf1 ec =De þ f2 Þ2 2P4 ðf1 ec =De þ f2 Þ
When the mismatch strain De is zero, the expression of fracture energy in Eq. (6) can be greatly simplified and becomes
C¼
Using the Eqs. (1)–(3), (4a)–(4c), (5), (6), (7a)–(7c), (8), (9a)– (9c), (10)–(12), the effect of the residual stresses in the film on the fracture strength, fracture energy and IFSS can be reflected if the parameters De, ec, and lsat have been known. In this paper, the case of SiOx film/PET substrate system is used to illustrate the influence of the residual stresses on the strength and fracture energy of the film. First, experiments of MFC for SiOx film/PET substrate system performed by Yanaka et al. [20] are summarized. Second, using fracture energy criterion, the predicted results of the crack densities for the SiOx/PET systems with different film thicknesses as functions of the applied strain are compared with the experimental data. The effect of the misfit strain on the crack density is also shown. Third, the effects of the film thickness and the residual stresses on the strength and fracture energy of the film are discussed. Finally, the effect of the residual stresses on the IFSS is investigated.
3f12 f3 f4 2 0 eE 4j c f
ð10Þ
ð9cÞ
was measured by using an in situ observation technology and then recorded in video cassette recorder. A constant displacement rate was applied unidirectionally to stretch the system. The detailed experimental procedure for determining the critical applied strain
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at which the film cracking begins can be seen in Ref. [20]. The measured data of critical applied strains, ec , for films with different thicknesses are listed in Table 1. The elastic moduli of the bare substrate and the film/substrate system were calculated from the initial slops of the stress–strain curves. The difference of measured elastic moduli was then used to estimate the elastic modulus of the film, Ef. However, the measured date of Ef showed high scattering for the systems with different film thicknesses. In such a case, an average value was adopted. The elastic moduli of the film and substrate thus obtained were [20]: Ef = 73 GPa, and Es = 6 GPa. It can be seen the elastic modulus of the film is close to that of the bulk fused silica [32]. The Poisson’s ratios of the film and the substrate are [19]: vf = 0.17, and vs = 0.35. Hence, for the SiOx/PET system, the material constant ratio R ffi 9.528. In the experiment of MFC, the film was subjected to the compressive residual stress before loading. However, the mismatch strain between the film and substrate, De, was not determined by Yanaka et al. [20]. But, the curvature of the film/substrate system during the film deposition was determined. Using Eqs. (1) and (2), the mismatch strains and the residual stresses in the film for individual systems can be determined and the results are also listed in Table. 1. Previous experimental results by Yanaka et al. [20] also showed that the crack spacing was generally varied around a mean value l. When the applied strain was given, the longest segments would break into halves with more or less half lengths, while shorter segments would remain unbroken. The shortest crack spacing is about a half of the longest crack spacing. Hence, the mean crack spacing (MCS), l, can be approximated as the average of the longest and the shortest crack spacings, such that [25]
l ¼ 1:5l
ð13Þ
The crack density can be defined as the form of the inverse of the average crack spacing, i.e., 1=l. In Yanaka et al.’s experiments, it can also be found that when the applied strain reaches the value of 6.40741%, the multiple cracking in the film are saturated. At this applied strain, the saturated crack spacing, lsat, for individual film/substrate systems can be determined and are also listed in Table 1. 4.2. Effective substrate thickness When the system is subjected to the applied loading, underneath the film segment, the substrate stress in the lateral direction (i.e., x-direction in Fig. 1) is partially transferred to the film and it becomes lower than the far-field stress. It has been argued that the thin film influences the stress field in the substrate only within a certain boundary zone in the neighborhood of the film [24,25]. Outside the boundary zone, the stress field in the substrate is not perturbed by the presence of the film. Hence, in the analytical modeling, the effective substrate thickness should be instead of the actual substrate thickness [33]. However, the effective substrate thickness is influenced by the materials constants of the substrate and the film and the width of the film [24]. Up to present, the analytical model for defining the effective substrate thickness has not been established. Hsueh and Wereszczak determined the relationship between the effective substrate thickness and the Young’s modulus ratio between the film and the substrate using finite element analysis (FEA) [24]. However, in their analyses, the perturbation of 1.85% of the substrate stress was used as the criterion. In this paper, the effective substrate is also estimated by FEA. However, the perturbation of 0% of the substrate stress was used as the criterion. A SiOx film segment on the PET substrate schematically shown in Fig. 2a is used in FEA. The geometry is symmetric and only half of the system is used in FEA modeling. The finite
Fig. 2. (a) Schematic showing the geometry used in FEA to examine perturbation of the substrate stress and (b) normalized substrate stress, rs =r0 , at the axial line through the thickness direction as functions of the normalized depth underneath the film, y=t f , with respect to different values of the half width of film.
element meshes near the film/substrate interface and in the region close to edges are refined to avoid the mesh size-effect on the magnitude of the stresses. The half width and thickness of substrate are, respectively, 100 and 500 times of the film thickness. While the right edge substrate is subjected to a uniform stress, r0, in the lateral direction. Underneath the film segment, the substrate stress, rs, is partially transferred to the film and then becomes lower than r0. The plane-strain condition is considered in FEA. The constraint in the lateral direction is imposed on the axial line to satisfy the symmetry condition, as shown in Fig. 2a. To examine the effect of the width of the film segment on the perturbation of the substrate stresses, the normalized substrate stresses, rs/r0, at the axial line through the thickness direction as functions of the depth underneath the film normalized by the film thickness, z/tf, with respect to different values for the half width of film segment are shown in Fig. 2b. It can be seen that, with increasing the width of the film segment, the effective substrate thickness increases. For instance, for the system with a half film width l=t f ¼ 20, the substrate stress is almost equal to the far-field stress at z=t f ¼ 73:4. However, for the system with a half film width of l=tf ¼ 80, the perturbation of the substrate stress can be ignored at z=tf P 90:6. When l=tf ¼ 50, the effective substrate thickness is 86.0tf. In such a case, the substrate stresses are respectively 0.2% higher and 0.4% lower than the far-field stress for the systems with half film widths of 20tf and 80tf. Hence, 86.0tf can be used as the effective substrate thickness for the SiOx/PET systems.
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4.3. Crack density prediction Using Eq. (8), the predicted MCSs for the SiOx/PET systems with different film thicknesses as functions of the applied strain are shown in Fig. 3. The predicted results are compared with the experimental data obtained by Yanaka et al. [20]. For a given film/substrate system, the MCS decreases with increasing the applied strain and the decreasing rate becomes smaller as the applied strain ea increases. The predicted relationship between the MCS and the applied strain agrees well with the measurements. However, when the film thickness is 43 nm, there is some discrepancy existing between the predicted results and the experimental results. Different reasons leads to this discrepancy. First, the present analytical modeling is performed on the basis of the linear elasticity assumption. However, the PET substrate is prone to be plastically deformed when the external loading is applied since its yield strain is about 2.2% [22]. The stress in film segment obtained from elasto-plastic solution is relatively lower than that obtained from elastic solution at a given ea [18]. If the substrate plasticity is considered, the predicted MCS will be increased. Second, although the three-dimensional stress analysis is performed in Zhang’s model, the mismatch strain, De, from the curvature was still evaluated on the basis of a two-dimensional case. In such
a case, the film stress may be highly evaluated, resulting in the highly predicted crack density and lowly predicted MCS. The normalized crack densities, t f =l, predicted using energy criterion as functions of the normalized applied strains, ea =ec , with respect to different film thicknesses are shown in Fig. 4. The measured data are also used for comparison. It can be seen that the predicted curves for the systems with the film thicknesses of 67 nm and 120 nm cannot be distinguished. From Table 1, it can be seen that when the film thicknesses are 67 nm and 120 nm, the ratio of the mismatch strain and the critical strain is about to be 0.38. The predicted curves for the systems with the film thicknesses of 90 nm and 320 nm cannot also be distinguished, as shown in Fig. 4. For these two systems, the ratio of De/ec is about 0.45. However, the normalized crack densities for these two systems are higher than those for the other three systems. Hence, if the ratio De/ec is controlled to be constant, the normalized crack density t f =l is almost not influenced by the film thickness. In such a case, the crack densities for the systems with different film thicknesses can be predicted when the crack density versus applied strain relationship has been measured at one film thickness. 4.4. Film strength and fracture energy The film strength, rstr, and fracture energy, C, for the film initial cracking respectively obtained from Eqs. (5) and (6) as functions of t1=2 are shown in Fig. 5a. With increasing the film thickness, both f the film strength and the fracture energy of the film decrease. It is
Fig. 3. The predicted crack spacings for the SiOx/PET systems with different film thicknesses as functions of the applied strain (symbols: measurements [20], lines: predictions).
Fig. 4. Normalized crack density, tf =l, predicted using energy criterion as functions of the normalized applied strains, ea =ec , with respect to different film thicknesses (symbols: measurements [20], lines: predictions).
Fig. 5. The film strength, rstr , and fracture energy, C, for the film initial cracking as functions of (a) square root of film thickness t1=2 and (b) compressive residual f stress in film, rr .
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interesting to find that both rstr and C are almost proportional to the inverse of the square root of the film thickness. For the film strength, the similar trend has been found by Yanaka et al. [20]. The predicted film strength decreases from 582.5 MPa to 160.3 GPa when the film thickness increases from 43 nm to 320 nm. The fracture energy of film ranges from 6.6 J/m2 to 4.7 J/ m2 when film thickness ranges from 43 nm to 320 nm. The reported strain energy release rates for the cracking amorphous and crystallized glasses are 8 J/m2 and 20 J/m2 [25], respectively, and the corresponding fracture energies are, 4 J/m2 and 10 J/m2, respectively. Hence, the predicted C for initial cracking of the film is in the order of that for the bulk glass. The film strength and fracture energy as functions of the residual stress in the film are shown in Fig. 5b. Except for the system with the film thickness of 90 nm, both the film strength and the fracture energy are almost proportional to the compressive residual stress in the film. Generally, with increasing the film thickness, the mismatch strain, De, between the film and substrate decreases due to the stress relaxation during the film deposition process, leading to the decrement of the residual stresses. However, for the system with a film thickness of 90 nm, the mismatch strain is 0.6417%, which is higher than the mismatch strains for the systems with film thicknesses of 67 nm and 120 nm, as shown in Table 1. Since the mismatch strain is predicted from the curvature of the system, there must be some error involving in the curvature measurement. Considering the error in the curvature measurement, it can be concluded that with increasing the compressive residual stress in the film, the strength and fracture energy of the film increases. pffiffiffiffi The variation of the square root of the fracture energy, C, for the film initial cracking along pffiffiffiffi the film strength, rstr, is shown in Fig. 6. It can be seen that C is almost proportional to rstr. With increasing the film strength, the fracture energy increases. Using fracture mechanics with small scale yielding, the similar result was also obtained by Hu and Evans [34]. They found that the energy release rate, Gc, of the film cracking through its thickness was proportional to the square of the film strength at which the first crack occurred. However, in the solution for Gc, a non-dimensional parameter which should be determined numerically existed. 4.5. Interfacial shear strength Using Eq. (3), the IFSSs for individual systems were calculated by setting l = lsat and the results are listed in Table 2. It should be noted that the IFSS is characterized by the maximum value of the interfacial shear stress when the multiple cracking is saturated. The predicted IFSS decreases from 98.79 MPa to 61.62 MPa when
Table 2 The data of the calculated IFSSs of the SiOx/PET systems with different thicknesses. tf (nm)
IFSS (MPa)
43 67 90 120 320
98.79071 80.92483 84.96647 77.88309 61.61682
Fig. 7. The variation of the interfacial shear strength of the SiOx/PET system along the compressive residual stress in the film.
the film thickness increases from 43 nm to 320 nm. The reported IFSS of the SiOx/PET system ranges from 20 MPa to 100 MPa [31]. Hence, the predicted data of IFSS agree well with the experimental results. However, from Table 2, it can be seen that the IFSS generally decreases with increasing the film thickness. This phenomenon is partially due to the variation of the residual stresses in the film. The IFSS as a function of the compressive residual stress in the film is shown in Fig. 7. It can be seen that with increasing the compressive residual stress in the SiOx film, the IFSS generally increases. It can also be conclude that the residual stress in the film is a key factor that controls the IFSS. Experimental results by Leterrier et al. [35] also showed that there is a linear relationship between the IFSS of the SiOx/PET system and the residual stress in the film. 5. Conclusions The effects of the residual stresses within the brittle films on the substrate on the film strength, fracture energy, and interfacial shear strength (IFSS) were investigated by using multiple film cracking analyses. Special analyses were performed on the SiOx film/polyethulene terephthalate substrate systems on the basis of the existing experimental data [20]. The conclusions are summarized as follows.
Fig. 6. The variation of the square root of the fracture energy, cracking along the film strength, rstr .
pffiffiffiffi C, for the film initial
(1) Both the film strength and the fracture energy for the initiation of film cracking are proportional to the compressive residual stresses in film. The film strength is almost proportional to the square root of the fracture energy. (2) The crack density in the film predicted by using the energy criterion agreed well with the experimental result. (3) The residual stress in the film is a key factor that controls the interfacial shear strength. With increasing the compressive residual stresses in the film, the interfacial shear strength of the film/substrate system increased.
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Acknowledgements The authors are grateful for the support by National Natural Science Foundations of China (50735001 and 10672058). The author X.C. Zhang is also grateful for the support by Shanghai Chenguang Planning Project (2008CG36) and Shanghai Rising-Star Program (08QA14023) and Ph.D. Programs Foundation of Ministry of Education of China (20090101120021). Reference [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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