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Semi-analytical model for thin film deformation analysis
Surface & Coatings Technology 231 (2013) 557–562
Contents lists available at ScienceDirect
Surface & Coatings Technology journal homepage: www.elsevier.com/locate/surfcoat
Semi-analytical model for thin film deformation analysis Chung-Jen Ou Department of Electrical Engineering, Hsiuping University of Science and Technology, No. 11, Gongye Rd., Dali District, Taichung 40128, Taiwan, ROC
a r t i c l e
i n f o
Available online 22 October 2012 Keywords: Thin film Thermal stress Elasticity Elastic deformation
a b s t r a c t Modelling of thermal deformation and the corresponding stress distribution are significant for the thin-film design. In order to evaluate the mechanical properties experienced by a thin film parameter on the rigid substrate, the author proposes a semi-analytical model to estimate the deformation of the thin film under various conditions. The physical interpretation and simplification of each term is obtained through the order-of-magnitude analysis. Analytical forms for the thin film deformation can be obtained, and the stress in the film can be easily determined using the proposed model through the constitutive law. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Several industrial devices encounter the problem of thermal stresses arising in thin layers with clamped edges, as in the case of thin films in microelectronics [1–3], thin layers of plastic or metal bonded to metal [4–6], thin glazes on ceramic tiles, Josephson junction devices [7], and so on. Several effective methods have been proposed for calculating thermal stress in a thin film on a substrate. Although the Stoney's equation and its extensions have been proposed for a curved substrate; [8–10] these equations are based on the existence of curvature R in the substrate surface. However, in many cases, the internal stress or deformation patterns of a thin film are required to be estimated for a flat substrate; in such cases the formulae, based on mechanics of material, that require the existence of R are not applicable. In order to overcome these problems, the author proposes a novel method based on the theory of elasticity and using the order-of-magnitude analysis. The following section discusses the theory of elastic deformation in Navier's form with regard to dilation, rotation and thermal gradient interpretations. Further in this section, appropriate dimensionless and the order of magnitude analysis will simplify the equations into simple second order differential equations. In the discussion section, several parameters and relationships reveal the properties of the proposed analytical solution, and a promising application of this method is revealed in the concluding section.
2. Theory 2.1. Governing equations Fig. 1(a, b) represents the thin film layer considered in this study. It is assumed to be an isotropic (not necessarily homogeneous), twodimensional, thin rectangular layer with length 2L and thickness δ. It E-mail address: [email protected]. 0257-8972/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.surfcoat.2012.10.028
is clear that δ is much smaller than L (δ ≪ L). The thin film layer is initially free of stress at a uniform temperature T0. Thermal stresses arise when the layer is heated to another uniform temperature T1. The temperature variation is represented as ΔT = T1 − T0. The mechanical properties of the thin film can be characterized by the coefficients of thermal expansion (CTE) α, Young's modulus E, and Poisson's ratio ν. On the basis of the theory of elasticity, Lame's form of thermal elastic stress can be expressed as follows: E ∂U 1 ν Þ EαΔT− EI σ XX ¼ ð − 1 þ ν ∂X 1−2ν ð1 þ ν Þð 1−2ν Þ E ∂U 1 E E ∂V ∂U Þ Þω ¼ σ XY ¼ ð þ ð Þð þ 1 þ ν ∂Y 2 1þν 2ð1 þ νÞ ∂X ∂Y
ð1Þ
where U and V are the displacements in the X and Y directions, respectively, and σij is the stress; the subscript ij indicates the tensor components. Among the geometric quantities, I ≡ ∂U / ∂X + ∂V / ∂Y is the dilation of the deformation and ω ≡ ∂V / ∂X − ∂U / ∂Y is the rotation within the thin film. Now, the stress component σXX is differentiated once with respect to X: ∂σ XX ∂X ! ∂ E ∂U 1 ν EαΔT− EI ¼ − 1 þ ν ∂X 1−2ν ð1 þ ν Þð1−2ν Þ ∂X 2 ! E ∂ U E ∂ ∂U Eν ∂I ¼ þ þ 2 2ð1 þ ν Þ ∂X 2ð1 þ ν Þ ∂X ∂X ð1 þ ν Þð1−2ν Þ ∂X ð2Þ !( Eα ∂ E ∂U ∂ Eν ⋅ ⋅ ½I þ − 1−2ν ∂X 1 þ ν ∂X ∂X ð1 þ ν Þð1−2νÞ ∂ Eα ΔT: − ∂X 1−2ν The first term, ∂ [(E/(1 + ν)) ∂ U/∂ X]/∂ X, can be divided into the summation of the three terms (with underline for clarifications) if
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C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
Fig. 1. Configuration and boundary conditions for thin film on rigid substrate.
there are no cracks and dislocations in U (which indicates that there exists a differentiable continuity for the thin-film displacement). Next, the first derivative of the stress component σXY is taken with respect to Y: 2 ∂σ XY E ∂ U E ∂ ∂V þ ¼ 2 2ð1 þ Þ ∂X ∂Y ∂Y ν Þ ∂Y 2ð1 þ ν ∂ E ∂V ∂U ⋅ þ : þ ∂Y 2ð1 þ ν Þ ∂X ∂Y
ð3Þ
Because of the equilibrium of force, Eqs. (2) and (3) can be combined into the stress equilibrium equation in X direction: ∂σ XX ∂σ XY þ ¼ 0: ∂X ∂Y
ð4Þ
It should be reiterated that material properties are considered as functions of spatial coordinates, which is a suitable consideration during the fabrication of the thin film. Therefore, the governing equation for displacement U is obtained as shown in Eq. (5).
1 ∂I 2 þ 2ν ∂ 2 ∇ Uþ α − ½ΔT þ F ¼ 0 1−2ν ∂X 1−2ν ∂X
ð5Þ
1 0 ∂ E ∂U ∂ Eν ⋅ ⋅I C þ 2ð1 þ ν Þ B ∂X B ∂X 1 þ ν ∂X ð1 þ νÞð1−2ν Þ C F≡ @ A: ∂ E ∂V ∂U ∂ Eα E ⋅ ΔT þ þ − ∂Y 2ð1 þ ν Þ ∂X ∂Y ∂X 1−2ν A similar equation can be used for governing the transformations X → Y, Y → X, U → V, V → U: 1 ∂I 2 þ 2ν ∂ 2 ∇ Vþ α − ½ΔT þ G ¼ 0 1−2ν ∂Y 1−2ν ∂Y
where 1 0 ∂ E ∂V ∂ Eν ⋅ ⋅ ½I C þ B 2ð1 þ νÞ B ∂Y 1 þ ν ∂Y ∂Y ð1 þ νÞð1−2νÞ C: G≡ @ ∂ A E ∂U ∂V ∂ Eα E þ ⋅ ΔT þ − ∂X 2ð1 þ νÞ ∂Y ∂X ∂Y 1−2ν
2.2. Dimensionless transformation Dimensionless parameters can be used to explore the important features of a thin-film structure. Let us consider the following dimensionless transformation:
x≡X=L; y≡Y=L; u≡U=U ref ; v≡V=U ref ; ΔT ¼ T ref Δθ; α ¼ α=αref ;
E ¼ E=Eref α ref LT ref ≡1 U ref
I¼
U ref I L
I ¼
∂u ∂v þ : ∂x ∂y
ð7Þ
Because the reference term for thermal expansion is the same in all the equations, we have two equations for the film displacement field (u, v):
where
Eqs. (5) and (6) are the fundamental governing equations for the present study, which can be referred to as a generalised Navier's model, since the leading terms of U and V are both dominated by the Laplacian operator ∇ 2. The associated boundary conditions for the thin film are shown in Fig. 1(b). Surfaces 1, 2, and 3 are under a free traction condition while surface 4 is under a rigid condition. Unlike the Stoney's equation, which requires the existence of a curvature R, this report deals with the estimation of stress in a thin film on a flat substrate.
"
# 2 2 ∂ ∂ 1 ∂I 2 þ 2ν ∂ α þ − uþ ½Δθ þ F ¼ 0 2 2 1−2ν ∂x 1−2ν ∂x ∂x ∂y
"
# ∂2 ∂2 1 ∂I 2 þ 2ν ∂ α ν þ þ − ½Δθ þ G ¼ 0 1−2ν ∂y 1−2ν ∂y ∂x2 ∂y2
ð6Þ where " 2ð1 þ νÞ ∂ E ∂u ∂ E ν ⋅ I ⋅ þ F ≡ E ∂x 1 þ ν ∂x ∂x ð1 þ ν Þð1−2ν Þ # ∂ E ∂v ∂u ∂ E α ⋅ Δθ þ þ − ∂y 2ð1 þ ν Þ ∂x ∂y ∂x 1−2ν
ð8Þ
C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
" 2ð1 þ ν Þ ∂ E ∂ν ∂ E ν ⋅ I ⋅ G ≡ þ E ∂y 1 þ ν ∂y ∂y ð1 þ ν Þð1−2νÞ # ∂ E ∂u ∂ν ∂ E α ⋅ Δθ : þ þ − ∂x 2ð1 þ ν Þ ∂y ∂x ∂y 1−2ν
559
2.4. General form and boundary conditions From Eq. (11), one can obtain the following equations: u ¼ ∬Pdydy þ yf ðxÞ þ kðxÞ
It should be noted that the advantage of this symmetry will be lost with the geometric arrangement of the thin film shown in Fig. 1. This is explained in the following section.
1 v¼− ∫I dy þ ∬Q dydy þ ygðxÞ þ hðxÞ 1−2ν
2.3. Order-of-magnitude analysis approach for thin-film deformation
where f(x), k(x), g(x), and h(x) are integrating functions, determined by the boundary conditions of the thin film. The corresponding first-order differential terms for u and v are
The analysis results of Eq. (8) can be determined using boundarylayer analysis techniques [11]. Due to the fact that δ indicates the thickness of the thin film, the dominant terms in Eq. (8) can be analysed by the following considerations: 2 2 ∂ u 1 ∂u 1 ∂ ν 1 ∂ ∂u ∂ν 1 ; ; : ; þ ∂y2 e δ2 ∂y e δ ∂y2 e δ ∂y ∂x ∂y e δ
ð9Þ
For the above dominant terms, the equations for displacement u and v for thin-film approximation are expressed as follows: "
"
# ! ! ∂2 v 2 þ 2ν 2ð1 þ ν Þ ∂ E ∂u ∂ ½ ⋅ − Δθ þ ðα 1−2ν E ∂y ∂x 2ð1 þ ν Þ ∂y ∂y2 ð10Þ ! 2ð1 þ ν Þ ∂ E α 1 ∂I Δθ þ ¼ 0: − E 1−2ν ∂y ∂y 1−2ν
As can be observed, in the case of deformation v, the differential of dilation (i.e. ∂I* / ∂y) contains an additional term. After using the order-of-magnitude analysis, there is no longer a symmetry property between the two deformation components. However, if we assume that the variations in the material properties of the thin film are not very significant, Eq. (10) can be further simplified to a more compact form as follows: "
"
# ∂2 u −P ¼ 0 ∂y2 2
#
∂u ¼ ∫Pdy þ f ðxÞ ∂y ∂v 1 d d dgðxÞ dhðxÞ ∫I dy þ ∬Q dydy þ y þ ¼− 1−2ν dx dx dx dx ∂x ð13Þ
It should be noted that within the coordinate system shown in Fig. 1, the function k(x) would be in the symmetrical form. From Eq. (13), one can easily obtain the stress within the film by the stress–strain relationship; [12] f(x), k(x), g(x), and h(x) are satisfied for the geometric boundary conditions of the thin film. In the following section, two typical cases of practical thin-film applications are investigated. The first case represents a thin-film condition without thermal variation along the y direction (Δθ(y) = 0). This case serves as a good approximation for film material with higher thermal conductivity since the rapid spreading of thermal energy can lead to the absence of thermal gradients. The second case assumes that there exists a thermal gradient along the y direction Δθ(y) = qy + r. This represents a film material with lower conductivity and exhibiting thermal resistivity in its coating layers. 3. Results and discussion
Case 1. ΔθðyÞ ¼ 0 and P ¼ Q ¼ 0:
∂ v 1 ∂I −Q ¼ 0 þ 1−2ν ∂y ∂y2
ð11Þ
where the definitions of P and Q are
∂u d df ðxÞ dkðxÞ þ ¼ ∬Pdydy þ y dx dx ∂x dx
∂v 1 I þ ∫Q dy þ g ðxÞ: ¼− 1−2ν ∂y
# 2 ∂ u 2 þ 2ν ∂ α − ½Δθ 1−2ν ∂x ∂y2 2ð1 þ ν Þ ∂ E ∂u 2ð1 þ ν Þ ∂ E α ⋅ Δθ − þ E E ∂y 2ð1 þ νÞ ∂y ∂x 1−2ν ¼0
ð12Þ
P¼
2 þ 2ν ∂ α ½Δθðx; yÞ 1−2ν ∂x
Q¼
2 þ 2ν ∂ α ½Δθðx; yÞ: 1−2ν ∂y
The first case represents a condition with no thermal gradients within the thin film. One can obtain a solution by assigning properties to the integrating functions f(x), k(x), g(x), and h(x). For achieving symmetry, the polynomials that generate f(x) and g(x) must be odd and even respectively. The superposition of these functions is naturally the solution for the thin film. This concept is quite similar to the Airy stress function method for elastic deformation. In Case 1, one can consider the approximation P → 0 and Q→ 0 and assume that f(x) =bx3, g(x) = ax2, and k(x) = h(x) = 0. Thus, the deformation field of the thin film can be expressed as 3
The advantage of these equations is that they are clearly secondorder differential equations in an iterative form. One can consider the dilation I* and the thermal terms P and Q as the driving sources for the thin film.
uðx; yÞ ¼ bx y 1 y 2 ∫ I dy þ ax y vðx; yÞ ¼ − 1−2ν 0
ð14Þ
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material configurations. One can integrate the total volume of the thin film to assess the compressibility of the film structure. Under such extreme condition of the thin film (i.e. incompressible condition), the following derivation is valid:
with the following derivatives: ∂u 1 d y ∂v 2 ∂u 3 ∂v ∫ I dy þ 2axy; ¼ 3bx y; ¼ bx ; ¼ − 0 1−2ν dx ∂x ∂y ∂x ∂y 1 2 I þ ax : ¼− 1−2ν
ð15Þ
δ
L
volume change ¼ ∫ ∫ I dxdy ¼ 0 −L
Note that dilation I* is included in the deformation v term. Although such an explicit term could be solved through numerical iterations, in this study, an analytical approach is employed by assuming that the configuration for the thin film is stable. The following relationship represents the equilibrium of the deformations:
I ¼
∂u ∂v 1 2 2 I þ ax : þ ¼ 3bx y− 1−2ν ∂x ∂y
ð16Þ
If the stability of the film is proved, then I* on the right hand side should be exactly equal to I* on the left hand side of Eq. (16). Based on this fact, the exact formula for I* become
I ¼
1−2ν 2 2 3bx y þ ax : 2−2ν
ð17Þ
Substituting this value of I* in Eq. (14), we can complete the final form for v:
−1 vðx; yÞ ¼ 2−2ν
3 2 2 2 2 bx y þ ax y þ ax y: 2
ð18Þ
Recall that Eq. (14) already gives the u(x,y) component. Restriction on coefficients a, b, and c is valid for volume expansions and
3 ¼ − bδ: 2
1−2ν 3 L δð2a þ 3bδÞ ¼ 0→a 6ð1−νÞ ð19Þ
The physical interpretation of the positive value of b is the expansion of the film at the edges; further, one can establish the incompressible criterion a = − 3bδ / 2 for the incompressible film. Fig. 2 shows the deformation of the thin film where δ = 0.1; L = 1; and b varies from 0 to 1.0, with a corresponding a value that satisfies the incompressible condition shown in Eq. (19). Note that these values are helpful for verifying the deformation pattern of the film. At the present moment the (thickness δ) / (length 2L) ratio is 5%, and the assumption of δ → 0 will be further consolidated in the next case by taking δ = 0.01. Case 2. ΔθðyÞ ¼ qy þ r and P ¼ 0; Q ≠0: The second case represents a condition where a thermal gradient term exists in the y direction. It is reasonable and practical to assume that the temperature difference Δθ is a linear function solely along the y direction. Hence, Δθ(x,y) is reduced to Δθ(y) = qy+ r with q and r having arbitrary real values. According to the definition in Eq. (11), it is clear that P = 0 and Q ≠ 0. As in Case 1, functions are selected as
Fig. 2. Deformation of thin film under an incompressible condition without a thermal term (δ = 0.1, L = 1, b = 0, 0.2, 0.4, 0.6, and 0.8, and a = −3bδ / 2).
C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
f(x) = bx 3, g(x) = ax2, and k(x) = h(x) = 0. Therefore, the deformation field u and v can be written as
Finally, substituting the solved dilation I* back in Eq. (20) leads to the definition of deformation v:
3
uðx; yÞ ¼ byx
y y y 1 2 ∫ I dy þ ∫ ∫ Q dy dy þ ax y: vðx; yÞ ¼ − 0 0 1−2ν 0
vðx; yÞ ¼ ð20Þ
We reiterate the definition of Q:
! 2 þ 2ν 2 þ 2ν Þα ðqy þ rÞ α ½ΔθðyÞ ¼ ð 1−2ν 1−2ν 2 þ 2ν 2 þ 2ν α q yþ α r ≡wy þ s: ¼ 1−2ν 1−2ν
∫ Qdy ¼ 0
ð21Þ
Coefficient s is a reference value and can be eliminated later. The physical interpretation of w is a combination of the thermal gradient and the thermal expansion capabilities of the film material. A positive value of w indicates that the difference between the internal temperatures of the thin film and the reference temperature increases along the y direction. Therefore, as in the case of Eqs. (15) and (16), dilation I* can be determined as follows:
I ¼
3 2 2 y2 2 2 2 bx y þ wy þ ax y þ w þ ax y: 2 2
1−2ν 2 2 3bx y þ 2wy þ ax : 2−2ν
ð22Þ
ð23Þ
The criterion for the volume change can provide the relationship between a, b, and w: 2
Integrating both sides leads to the definition of w: y
−1 2−2ν
3δ bL2 þ 2w 2 : 2aL þ 3δ bL þ 2w ≥0→a ≥− 2L2
2 þ 2ν ∂ α ½Δθðx; yÞ: Q¼ 1−2ν ∂y
561
ð24Þ
It is clear that if the thermal term w = 0, Eq. (24) will reduce to Eq. (19) with the equal sign for incompressible properties. At the same time, according to Eq. (21), q > 0 results in the requirement that w > 0. However, recent studies have shown that there exist solutions that allow q b 0 and still produce positive dilations of the thin film materials. Such solutions can be helpful for designing film structures for novel devices. Further, it should be noted that unlike the first case in which the characteristic length of the thin film is irrelevant, in the second case the criterion for the incompressible condition is related to the length 2L of the thin film. A series of thin-film deformations can be demonstrated under different parameters. To demonstrate the feasibility of the present model, this paper discusses the different deformation patterns, shown in Fig. 3. A scaling factor of 10 for the deformation is required to attain a better visualization of the deformation. Interestingly, the proposed method can also provide a possible solution that can potentially lead to the development of lens-type structures. This provides the prospect for a fabrication process for
Fig. 3. Deformation of thin film free from incompressible constraints with existing thermal term.
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C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
micro-lens structures using coating procedures and appropriate thermal gradients. The importance of the proposed semi-analytical solution is that the parameters can model the deformation of thin films. The figures show some parameters that represent various deformation configurations of the thin film and prove the feasibility of the proposed method.
Acknowledgement
4. Conclusions
References
Through the order of magnitude analysis approach, simple formulation consisting of thermal terms and related control parameters for analysing the deformation of coated thin films is proposed. This report discusses the physical interpretation of the governing equation, and the demonstrated cases reveal the robustness of the present method. Through this model the interaction between the gradient terms along the two orthogonal directions and various other parameters can be explored. Coating fabricators can provide first order optimization of the thin film parameters through this method. In the future, the catastrophic theory can be applied to understand the morphology of control space. Hence, fabricator can manipulate a more robust thin film with superior mechanical properties in the critical regions for novel opto-electric devices.
This work was sponsored by the National Science Council of the Republic of China (Taiwan) under the grants NSC100-2221-E-164-007 and NSC101-2221-E-164-014.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
J. Hu, L. Yang, M.W. Shin, Microelectron. J. 38 (2007) 157. D.S. Kim, B.C. Lee, Int. J. Solids Struct. 31 (1994) 1225. A. Krost, A. Dadgar, Mater. Sci. Eng. B93 (2002) 77. A.Y. Kuo, ASME J. Appl. Mech. Tech. Phys. 56 (1989) 585. M. Lee, I. Jasiuk, ASME J. Electron. Packag. 113 (1991) 173. S.L. Lee, C.R. Ou, ASME J. Heat Transfer 123 (2001) 205. J.C. Lambropoulos, S.M. Wan, J. Mater. Sci. Eng. A107 (1989) 169. Y. Zhang, Y. Zhaoa, J. Appl. Phys. 99 (2006) 053513. X. Feng, Y. Huang, Trans. ASME J. Appl. Mech. 74 (2007) 1276. Y.L. Shen, S. Suresh, I.A. Blech, J. Appl. Phys. 80 (1996) 1388. J.P. Holman, in: 6th ed., Heat Transfer, Chapter 6, 271, McGraw Hill, 1986. B.A. Boley, J.H. Weiner, B.A. Boley, J.H. Weiner, in: Dover edition, Theory of Thermal Stresses, Chapter 8, 246, 1988.
Contents lists available at ScienceDirect
Surface & Coatings Technology journal homepage: www.elsevier.com/locate/surfcoat
Semi-analytical model for thin film deformation analysis Chung-Jen Ou Department of Electrical Engineering, Hsiuping University of Science and Technology, No. 11, Gongye Rd., Dali District, Taichung 40128, Taiwan, ROC
a r t i c l e
i n f o
Available online 22 October 2012 Keywords: Thin film Thermal stress Elasticity Elastic deformation
a b s t r a c t Modelling of thermal deformation and the corresponding stress distribution are significant for the thin-film design. In order to evaluate the mechanical properties experienced by a thin film parameter on the rigid substrate, the author proposes a semi-analytical model to estimate the deformation of the thin film under various conditions. The physical interpretation and simplification of each term is obtained through the order-of-magnitude analysis. Analytical forms for the thin film deformation can be obtained, and the stress in the film can be easily determined using the proposed model through the constitutive law. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Several industrial devices encounter the problem of thermal stresses arising in thin layers with clamped edges, as in the case of thin films in microelectronics [1–3], thin layers of plastic or metal bonded to metal [4–6], thin glazes on ceramic tiles, Josephson junction devices [7], and so on. Several effective methods have been proposed for calculating thermal stress in a thin film on a substrate. Although the Stoney's equation and its extensions have been proposed for a curved substrate; [8–10] these equations are based on the existence of curvature R in the substrate surface. However, in many cases, the internal stress or deformation patterns of a thin film are required to be estimated for a flat substrate; in such cases the formulae, based on mechanics of material, that require the existence of R are not applicable. In order to overcome these problems, the author proposes a novel method based on the theory of elasticity and using the order-of-magnitude analysis. The following section discusses the theory of elastic deformation in Navier's form with regard to dilation, rotation and thermal gradient interpretations. Further in this section, appropriate dimensionless and the order of magnitude analysis will simplify the equations into simple second order differential equations. In the discussion section, several parameters and relationships reveal the properties of the proposed analytical solution, and a promising application of this method is revealed in the concluding section.
2. Theory 2.1. Governing equations Fig. 1(a, b) represents the thin film layer considered in this study. It is assumed to be an isotropic (not necessarily homogeneous), twodimensional, thin rectangular layer with length 2L and thickness δ. It E-mail address: [email protected]. 0257-8972/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.surfcoat.2012.10.028
is clear that δ is much smaller than L (δ ≪ L). The thin film layer is initially free of stress at a uniform temperature T0. Thermal stresses arise when the layer is heated to another uniform temperature T1. The temperature variation is represented as ΔT = T1 − T0. The mechanical properties of the thin film can be characterized by the coefficients of thermal expansion (CTE) α, Young's modulus E, and Poisson's ratio ν. On the basis of the theory of elasticity, Lame's form of thermal elastic stress can be expressed as follows: E ∂U 1 ν Þ EαΔT− EI σ XX ¼ ð − 1 þ ν ∂X 1−2ν ð1 þ ν Þð 1−2ν Þ E ∂U 1 E E ∂V ∂U Þ Þω ¼ σ XY ¼ ð þ ð Þð þ 1 þ ν ∂Y 2 1þν 2ð1 þ νÞ ∂X ∂Y
ð1Þ
where U and V are the displacements in the X and Y directions, respectively, and σij is the stress; the subscript ij indicates the tensor components. Among the geometric quantities, I ≡ ∂U / ∂X + ∂V / ∂Y is the dilation of the deformation and ω ≡ ∂V / ∂X − ∂U / ∂Y is the rotation within the thin film. Now, the stress component σXX is differentiated once with respect to X: ∂σ XX ∂X ! ∂ E ∂U 1 ν EαΔT− EI ¼ − 1 þ ν ∂X 1−2ν ð1 þ ν Þð1−2ν Þ ∂X 2 ! E ∂ U E ∂ ∂U Eν ∂I ¼ þ þ 2 2ð1 þ ν Þ ∂X 2ð1 þ ν Þ ∂X ∂X ð1 þ ν Þð1−2ν Þ ∂X ð2Þ !( Eα ∂ E ∂U ∂ Eν ⋅ ⋅ ½I þ − 1−2ν ∂X 1 þ ν ∂X ∂X ð1 þ ν Þð1−2νÞ ∂ Eα ΔT: − ∂X 1−2ν The first term, ∂ [(E/(1 + ν)) ∂ U/∂ X]/∂ X, can be divided into the summation of the three terms (with underline for clarifications) if
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C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
Fig. 1. Configuration and boundary conditions for thin film on rigid substrate.
there are no cracks and dislocations in U (which indicates that there exists a differentiable continuity for the thin-film displacement). Next, the first derivative of the stress component σXY is taken with respect to Y: 2 ∂σ XY E ∂ U E ∂ ∂V þ ¼ 2 2ð1 þ Þ ∂X ∂Y ∂Y ν Þ ∂Y 2ð1 þ ν ∂ E ∂V ∂U ⋅ þ : þ ∂Y 2ð1 þ ν Þ ∂X ∂Y
ð3Þ
Because of the equilibrium of force, Eqs. (2) and (3) can be combined into the stress equilibrium equation in X direction: ∂σ XX ∂σ XY þ ¼ 0: ∂X ∂Y
ð4Þ
It should be reiterated that material properties are considered as functions of spatial coordinates, which is a suitable consideration during the fabrication of the thin film. Therefore, the governing equation for displacement U is obtained as shown in Eq. (5).
1 ∂I 2 þ 2ν ∂ 2 ∇ Uþ α − ½ΔT þ F ¼ 0 1−2ν ∂X 1−2ν ∂X
ð5Þ
1 0 ∂ E ∂U ∂ Eν ⋅ ⋅I C þ 2ð1 þ ν Þ B ∂X B ∂X 1 þ ν ∂X ð1 þ νÞð1−2ν Þ C F≡ @ A: ∂ E ∂V ∂U ∂ Eα E ⋅ ΔT þ þ − ∂Y 2ð1 þ ν Þ ∂X ∂Y ∂X 1−2ν A similar equation can be used for governing the transformations X → Y, Y → X, U → V, V → U: 1 ∂I 2 þ 2ν ∂ 2 ∇ Vþ α − ½ΔT þ G ¼ 0 1−2ν ∂Y 1−2ν ∂Y
where 1 0 ∂ E ∂V ∂ Eν ⋅ ⋅ ½I C þ B 2ð1 þ νÞ B ∂Y 1 þ ν ∂Y ∂Y ð1 þ νÞð1−2νÞ C: G≡ @ ∂ A E ∂U ∂V ∂ Eα E þ ⋅ ΔT þ − ∂X 2ð1 þ νÞ ∂Y ∂X ∂Y 1−2ν
2.2. Dimensionless transformation Dimensionless parameters can be used to explore the important features of a thin-film structure. Let us consider the following dimensionless transformation:
x≡X=L; y≡Y=L; u≡U=U ref ; v≡V=U ref ; ΔT ¼ T ref Δθ; α ¼ α=αref ;
E ¼ E=Eref α ref LT ref ≡1 U ref
I¼
U ref I L
I ¼
∂u ∂v þ : ∂x ∂y
ð7Þ
Because the reference term for thermal expansion is the same in all the equations, we have two equations for the film displacement field (u, v):
where
Eqs. (5) and (6) are the fundamental governing equations for the present study, which can be referred to as a generalised Navier's model, since the leading terms of U and V are both dominated by the Laplacian operator ∇ 2. The associated boundary conditions for the thin film are shown in Fig. 1(b). Surfaces 1, 2, and 3 are under a free traction condition while surface 4 is under a rigid condition. Unlike the Stoney's equation, which requires the existence of a curvature R, this report deals with the estimation of stress in a thin film on a flat substrate.
"
# 2 2 ∂ ∂ 1 ∂I 2 þ 2ν ∂ α þ − uþ ½Δθ þ F ¼ 0 2 2 1−2ν ∂x 1−2ν ∂x ∂x ∂y
"
# ∂2 ∂2 1 ∂I 2 þ 2ν ∂ α ν þ þ − ½Δθ þ G ¼ 0 1−2ν ∂y 1−2ν ∂y ∂x2 ∂y2
ð6Þ where " 2ð1 þ νÞ ∂ E ∂u ∂ E ν ⋅ I ⋅ þ F ≡ E ∂x 1 þ ν ∂x ∂x ð1 þ ν Þð1−2ν Þ # ∂ E ∂v ∂u ∂ E α ⋅ Δθ þ þ − ∂y 2ð1 þ ν Þ ∂x ∂y ∂x 1−2ν
ð8Þ
C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
" 2ð1 þ ν Þ ∂ E ∂ν ∂ E ν ⋅ I ⋅ G ≡ þ E ∂y 1 þ ν ∂y ∂y ð1 þ ν Þð1−2νÞ # ∂ E ∂u ∂ν ∂ E α ⋅ Δθ : þ þ − ∂x 2ð1 þ ν Þ ∂y ∂x ∂y 1−2ν
559
2.4. General form and boundary conditions From Eq. (11), one can obtain the following equations: u ¼ ∬Pdydy þ yf ðxÞ þ kðxÞ
It should be noted that the advantage of this symmetry will be lost with the geometric arrangement of the thin film shown in Fig. 1. This is explained in the following section.
1 v¼− ∫I dy þ ∬Q dydy þ ygðxÞ þ hðxÞ 1−2ν
2.3. Order-of-magnitude analysis approach for thin-film deformation
where f(x), k(x), g(x), and h(x) are integrating functions, determined by the boundary conditions of the thin film. The corresponding first-order differential terms for u and v are
The analysis results of Eq. (8) can be determined using boundarylayer analysis techniques [11]. Due to the fact that δ indicates the thickness of the thin film, the dominant terms in Eq. (8) can be analysed by the following considerations: 2 2 ∂ u 1 ∂u 1 ∂ ν 1 ∂ ∂u ∂ν 1 ; ; : ; þ ∂y2 e δ2 ∂y e δ ∂y2 e δ ∂y ∂x ∂y e δ
ð9Þ
For the above dominant terms, the equations for displacement u and v for thin-film approximation are expressed as follows: "
"
# ! ! ∂2 v 2 þ 2ν 2ð1 þ ν Þ ∂ E ∂u ∂ ½ ⋅ − Δθ þ ðα 1−2ν E ∂y ∂x 2ð1 þ ν Þ ∂y ∂y2 ð10Þ ! 2ð1 þ ν Þ ∂ E α 1 ∂I Δθ þ ¼ 0: − E 1−2ν ∂y ∂y 1−2ν
As can be observed, in the case of deformation v, the differential of dilation (i.e. ∂I* / ∂y) contains an additional term. After using the order-of-magnitude analysis, there is no longer a symmetry property between the two deformation components. However, if we assume that the variations in the material properties of the thin film are not very significant, Eq. (10) can be further simplified to a more compact form as follows: "
"
# ∂2 u −P ¼ 0 ∂y2 2
#
∂u ¼ ∫Pdy þ f ðxÞ ∂y ∂v 1 d d dgðxÞ dhðxÞ ∫I dy þ ∬Q dydy þ y þ ¼− 1−2ν dx dx dx dx ∂x ð13Þ
It should be noted that within the coordinate system shown in Fig. 1, the function k(x) would be in the symmetrical form. From Eq. (13), one can easily obtain the stress within the film by the stress–strain relationship; [12] f(x), k(x), g(x), and h(x) are satisfied for the geometric boundary conditions of the thin film. In the following section, two typical cases of practical thin-film applications are investigated. The first case represents a thin-film condition without thermal variation along the y direction (Δθ(y) = 0). This case serves as a good approximation for film material with higher thermal conductivity since the rapid spreading of thermal energy can lead to the absence of thermal gradients. The second case assumes that there exists a thermal gradient along the y direction Δθ(y) = qy + r. This represents a film material with lower conductivity and exhibiting thermal resistivity in its coating layers. 3. Results and discussion
Case 1. ΔθðyÞ ¼ 0 and P ¼ Q ¼ 0:
∂ v 1 ∂I −Q ¼ 0 þ 1−2ν ∂y ∂y2
ð11Þ
where the definitions of P and Q are
∂u d df ðxÞ dkðxÞ þ ¼ ∬Pdydy þ y dx dx ∂x dx
∂v 1 I þ ∫Q dy þ g ðxÞ: ¼− 1−2ν ∂y
# 2 ∂ u 2 þ 2ν ∂ α − ½Δθ 1−2ν ∂x ∂y2 2ð1 þ ν Þ ∂ E ∂u 2ð1 þ ν Þ ∂ E α ⋅ Δθ − þ E E ∂y 2ð1 þ νÞ ∂y ∂x 1−2ν ¼0
ð12Þ
P¼
2 þ 2ν ∂ α ½Δθðx; yÞ 1−2ν ∂x
Q¼
2 þ 2ν ∂ α ½Δθðx; yÞ: 1−2ν ∂y
The first case represents a condition with no thermal gradients within the thin film. One can obtain a solution by assigning properties to the integrating functions f(x), k(x), g(x), and h(x). For achieving symmetry, the polynomials that generate f(x) and g(x) must be odd and even respectively. The superposition of these functions is naturally the solution for the thin film. This concept is quite similar to the Airy stress function method for elastic deformation. In Case 1, one can consider the approximation P → 0 and Q→ 0 and assume that f(x) =bx3, g(x) = ax2, and k(x) = h(x) = 0. Thus, the deformation field of the thin film can be expressed as 3
The advantage of these equations is that they are clearly secondorder differential equations in an iterative form. One can consider the dilation I* and the thermal terms P and Q as the driving sources for the thin film.
uðx; yÞ ¼ bx y 1 y 2 ∫ I dy þ ax y vðx; yÞ ¼ − 1−2ν 0
ð14Þ
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material configurations. One can integrate the total volume of the thin film to assess the compressibility of the film structure. Under such extreme condition of the thin film (i.e. incompressible condition), the following derivation is valid:
with the following derivatives: ∂u 1 d y ∂v 2 ∂u 3 ∂v ∫ I dy þ 2axy; ¼ 3bx y; ¼ bx ; ¼ − 0 1−2ν dx ∂x ∂y ∂x ∂y 1 2 I þ ax : ¼− 1−2ν
ð15Þ
δ
L
volume change ¼ ∫ ∫ I dxdy ¼ 0 −L
Note that dilation I* is included in the deformation v term. Although such an explicit term could be solved through numerical iterations, in this study, an analytical approach is employed by assuming that the configuration for the thin film is stable. The following relationship represents the equilibrium of the deformations:
I ¼
∂u ∂v 1 2 2 I þ ax : þ ¼ 3bx y− 1−2ν ∂x ∂y
ð16Þ
If the stability of the film is proved, then I* on the right hand side should be exactly equal to I* on the left hand side of Eq. (16). Based on this fact, the exact formula for I* become
I ¼
1−2ν 2 2 3bx y þ ax : 2−2ν
ð17Þ
Substituting this value of I* in Eq. (14), we can complete the final form for v:
−1 vðx; yÞ ¼ 2−2ν
3 2 2 2 2 bx y þ ax y þ ax y: 2
ð18Þ
Recall that Eq. (14) already gives the u(x,y) component. Restriction on coefficients a, b, and c is valid for volume expansions and
3 ¼ − bδ: 2
1−2ν 3 L δð2a þ 3bδÞ ¼ 0→a 6ð1−νÞ ð19Þ
The physical interpretation of the positive value of b is the expansion of the film at the edges; further, one can establish the incompressible criterion a = − 3bδ / 2 for the incompressible film. Fig. 2 shows the deformation of the thin film where δ = 0.1; L = 1; and b varies from 0 to 1.0, with a corresponding a value that satisfies the incompressible condition shown in Eq. (19). Note that these values are helpful for verifying the deformation pattern of the film. At the present moment the (thickness δ) / (length 2L) ratio is 5%, and the assumption of δ → 0 will be further consolidated in the next case by taking δ = 0.01. Case 2. ΔθðyÞ ¼ qy þ r and P ¼ 0; Q ≠0: The second case represents a condition where a thermal gradient term exists in the y direction. It is reasonable and practical to assume that the temperature difference Δθ is a linear function solely along the y direction. Hence, Δθ(x,y) is reduced to Δθ(y) = qy+ r with q and r having arbitrary real values. According to the definition in Eq. (11), it is clear that P = 0 and Q ≠ 0. As in Case 1, functions are selected as
Fig. 2. Deformation of thin film under an incompressible condition without a thermal term (δ = 0.1, L = 1, b = 0, 0.2, 0.4, 0.6, and 0.8, and a = −3bδ / 2).
C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
f(x) = bx 3, g(x) = ax2, and k(x) = h(x) = 0. Therefore, the deformation field u and v can be written as
Finally, substituting the solved dilation I* back in Eq. (20) leads to the definition of deformation v:
3
uðx; yÞ ¼ byx
y y y 1 2 ∫ I dy þ ∫ ∫ Q dy dy þ ax y: vðx; yÞ ¼ − 0 0 1−2ν 0
vðx; yÞ ¼ ð20Þ
We reiterate the definition of Q:
! 2 þ 2ν 2 þ 2ν Þα ðqy þ rÞ α ½ΔθðyÞ ¼ ð 1−2ν 1−2ν 2 þ 2ν 2 þ 2ν α q yþ α r ≡wy þ s: ¼ 1−2ν 1−2ν
∫ Qdy ¼ 0
ð21Þ
Coefficient s is a reference value and can be eliminated later. The physical interpretation of w is a combination of the thermal gradient and the thermal expansion capabilities of the film material. A positive value of w indicates that the difference between the internal temperatures of the thin film and the reference temperature increases along the y direction. Therefore, as in the case of Eqs. (15) and (16), dilation I* can be determined as follows:
I ¼
3 2 2 y2 2 2 2 bx y þ wy þ ax y þ w þ ax y: 2 2
1−2ν 2 2 3bx y þ 2wy þ ax : 2−2ν
ð22Þ
ð23Þ
The criterion for the volume change can provide the relationship between a, b, and w: 2
Integrating both sides leads to the definition of w: y
−1 2−2ν
3δ bL2 þ 2w 2 : 2aL þ 3δ bL þ 2w ≥0→a ≥− 2L2
2 þ 2ν ∂ α ½Δθðx; yÞ: Q¼ 1−2ν ∂y
561
ð24Þ
It is clear that if the thermal term w = 0, Eq. (24) will reduce to Eq. (19) with the equal sign for incompressible properties. At the same time, according to Eq. (21), q > 0 results in the requirement that w > 0. However, recent studies have shown that there exist solutions that allow q b 0 and still produce positive dilations of the thin film materials. Such solutions can be helpful for designing film structures for novel devices. Further, it should be noted that unlike the first case in which the characteristic length of the thin film is irrelevant, in the second case the criterion for the incompressible condition is related to the length 2L of the thin film. A series of thin-film deformations can be demonstrated under different parameters. To demonstrate the feasibility of the present model, this paper discusses the different deformation patterns, shown in Fig. 3. A scaling factor of 10 for the deformation is required to attain a better visualization of the deformation. Interestingly, the proposed method can also provide a possible solution that can potentially lead to the development of lens-type structures. This provides the prospect for a fabrication process for
Fig. 3. Deformation of thin film free from incompressible constraints with existing thermal term.
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C.-J. Ou / Surface & Coatings Technology 231 (2013) 557–562
micro-lens structures using coating procedures and appropriate thermal gradients. The importance of the proposed semi-analytical solution is that the parameters can model the deformation of thin films. The figures show some parameters that represent various deformation configurations of the thin film and prove the feasibility of the proposed method.
Acknowledgement
4. Conclusions
References
Through the order of magnitude analysis approach, simple formulation consisting of thermal terms and related control parameters for analysing the deformation of coated thin films is proposed. This report discusses the physical interpretation of the governing equation, and the demonstrated cases reveal the robustness of the present method. Through this model the interaction between the gradient terms along the two orthogonal directions and various other parameters can be explored. Coating fabricators can provide first order optimization of the thin film parameters through this method. In the future, the catastrophic theory can be applied to understand the morphology of control space. Hence, fabricator can manipulate a more robust thin film with superior mechanical properties in the critical regions for novel opto-electric devices.
This work was sponsored by the National Science Council of the Republic of China (Taiwan) under the grants NSC100-2221-E-164-007 and NSC101-2221-E-164-014.
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