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Stress mechanisms of SiO2 and Nb2O5 thin films sputtered on flexible substrates investigated by finite element method
Surface & Coatings Technology 344 (2018) 449–457
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Stress mechanisms of SiO2 and Nb2O5 thin films sputtered on flexible substrates investigated by finite element method
T
⁎
Hsi-Chao Chena,b, , Chen-Yu Huanga, Po-Wei Chengb a b
Department of Electronic Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan Graduate School of Optoelectronics Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan
A R T I C LE I N FO
A B S T R A C T
Keywords: Stress mechanism Sputtering Flexible substrate Numerical analysis Finite element method (FEM)
The stress mechanisms of SiO2 and Nb2O5 thin films sputtering on BK-7 glass, PET, and PC flexible substrates were investigated by the numerical analysis of the finite element method (FEM). Residual stress is a combination of thermal and intrinsic stresses during the sputtering process. The thermal stress results from the difference in thermal expansion coefficients and the substrate temperature between the film and substrate. The sputtering process is complicated; hence, the intrinsic stress was simulated by an equivalent-room-temperature (ERT) technique of FEM and a polynomial fitting curve, which could reduce the error ratio to less than 2%. The experimental verification of residual stresses has been conducted using the self-made Twyman-Green interferometer and shadow moiré interferometer with Stoney and modified Stoney formulas for glass and flexible substrates, respectively. However, in examining the stress mechanism of the BK-7 hard substrate, it was found that the degree of intrinsic stress was directly proportional to the D factor (Unit cell ratio) of SiO2 and Nb2O5 thin films, and that the error ratio was about 3%; while the stress mechanisms of PET and PC flexible substrates show that the residual stresses were directly proportional to the D factor, and the error ratio was about 6%.
1. Introduction Thin films are usually in a state of elastic mechanical stress during and after deposition, thus, the stress behavior of film is very important in all applications of thin films with respect to durability, stability, and usability. The first investigation regarding mechanical stress in thin film was conducted in 1858 on electro-deposited antimony film [1], and a more quantitative method was used to investigate the mechanical stress of thin film in 1909 by Stoney [2]. Thereafter, a series of researches on the measurements and explanations of the origin and quantitative calculation of mechanical stress, as based on a grain boundary interaction model for crystalline films, have been published [3–6]. It is well known that oxide films prepared by magnetron sputtering usually have a high packing density, and therefore, show excellent stability under the changing conditions of relative humidity. While these sputtered films are generally amorphous or low polycrystalline at the substrate temperature of room temperature, the signified crystalline directions are random, and the crystalline will be arrayed with the increased the substrate temperature [7,8]. Then, the mechanical stress or named residual stress is a consequence of the high film density, as caused by the higher energy of the condensing and bombarding atoms and molecules [9,10]. Then, thermal stress could be created by the variations in
⁎
thermal expansion coefficients and temperature between the film and substrate, and the nucleation growth of thin film could result in intrinsic stress during the deposition process. Due to the complexity of the deposition process and the plasma environment during sputtering deposition, it is difficult to analyze intrinsic stress. Intrinsic stress results from the target material, plasma power, chamber pressure, and structure organization to influence the atom dislocation, growth defects, film adhesion, etc., which also impact the amount of residual stress. In the metal film, residual stress induces the phenomenon of hillock and whisker of atomic displacement, causing film peeling due to tensile stress [11]. However, in the dielectric film of metal-oxidation, the residual stress could result in the shrinking of the film, which cracks the film due to compressive stress [12]. As is widely known, BK-7 glass, PET, and PC flexible plates are the commonly-used substrates, and SiO2 and Nb2O5 are the most widelyused low and high refractive index oxide thin films, respectively, for optical interference coatings. These films are hard and chemically resistant, transparent in the visible range, and have stability refractive indices, which demonstrate excellent mechanical and environmental stability [13]. These oxide thin films created by deposition on different substrates using magnetron sputtering find wide applications in microelectronics, optics, semiconductor multi-layer, superconducting
Corresponding author at: Department of Electronic Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan. E-mail address: [email protected] (H.-C. Chen).
https://doi.org/10.1016/j.surfcoat.2018.03.051 Received 25 September 2017; Received in revised form 15 March 2018; Accepted 16 March 2018 Available online 16 March 2018 0257-8972/ © 2018 Elsevier B.V. All rights reserved.
Surface & Coatings Technology 344 (2018) 449–457
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Fig. 1. A diagram of stress distribution for flexible substrates.
systems, data storage systems, organic light emission displays, and flexible thin film solar cells [14–19]. Simultaneously, the vigorous developments of computing technology have allowed more researchers to use numerical analysis to simulate stress [20–23]. Finite element analysis (FEM) is a very powerful technique used to model thin-film structures and implement processing with static and dynamic analyses [24–26]. The purpose of this research is to use FEM to analyze the residual stress in oxide thin film (SiO2 and Nb2O5) using different substrates. Thermal stress can be simulated by examining the differences in temperature and thermal expansion coefficients between the substrate and films during the sputtering process. Next, the intrinsic stress is derived using the equivalent-room-temperature (ERT) technique [35] and a polynomial fitting curve, which can reduce the error ratio. A D factor (Unit cell) was used to determine the relationship degree of residual or intrinsic stresses of these oxide thin films. The stress mechanism of the BK-7 hard substrate is intrinsic stress, which is directly proportional to the D factor of SiO2 and Nb2O5 thin films. On the contrary, the stress mechanisms of PET and PC flexible substrates show the residual stresses to be directly proportional to the D factor of SiO2 and Nb2O5 thin films. These phenomena were verified by the residual stress measurements with Twyman-green and shadow moiré interferometers.
Ef Ef εtf0 = αf (Tr − T0 ) 1 − νf 1 − νf Es Es = εts0 = αs (Tr − T0) 1 − νs 1 − νs
σtf0 = σts0
(3)
where Ef and Es are the elastic modules for film and substrate, and vf and vs are the Poisson ratios for film and substrate, respectively. The strain for film and substrate are then expressed as:
εtf = αf (Tr − T0) +
f (1 − νf ) t f Ef
f (1 − νs ) εts = αs (Tr − T0) − ts Es
(4)
where tf and ts are the film thicknesses for film and substrate, respectively, and f is the internal reacting stress related to the thickness of thin film, as resulted from the temperature variant. The strains of the film and substrate are in the same positions due to a continuous boundary condition, thus, the internal reacting stress f is expressed, as follows:
(αs − αf )(Tr − T0 )
f=
1 − νf Ef t f
+
1 − νs Es ts
(5)
Then, the thermal stress can be expressed as: 2. Theoretical formula and experimental measurement
σtf =
Ef
2.1. Residual and thermal stresses
σtf =
(6)
ts
Ef 1 − νf
(αs − αf )(Tr − T0)
(7)
The residual stress is a combination of thermal and intrinsic stresses without external force during the sputtering process. The residual stress can be expressed, as follows, for the glass substrate using the Stoney equation [2]:
(1)
During the sputtering process, an Ar+ plasma sputters the target to achieve metal oxidation at high temperatures. Once the deposition process ends, the temperature can fall to room temperature, with thermal stress resulting primarily from the concentrates of the thin film and substrate of different thermal expansion coefficients. The original thermal strain can be expressed as:
σf =
Es ts2 ⎛ 1 Es ts2 f 1 ⎞ (k − k 0) = − = tf 6t f ⎝ R R0 ⎠ 6t f ⎜
⎟
(8)
Fig. 1 shows the stress distribution for flexible substrates during the sputtering process. Residual stress derivation for the flexible substrate must consider the thin film strain, thus, the total strain energy density U (r,z) can be expressed as [27]:
εtf0 = αf (Tr − T0 ) εts0 = αs (Tr − T0)
Es
If (1 − νf)/Eftf > > (1 − νs)/Ests the thermal stress can be expressed as:
Residual stress can be separated into thermal stress, intrinsic stress, and extrinsic stress. As residual stress is always a combination of thermal and intrinsic stresses without external force during the sputtering process, the residual stress can be expressed as:
σresidual = σthermal + σintrinsic
(αs − αf )(Tr − T0 ) f = 1 − νf 1 − νs t f tf +
(2)
where Tr is the deposition (or substrate) temperature, T0 is the original temperature, and αf and αs are the thermal expansion coefficients of the film and substrate, respectively. The original thermal stress for the film and substrate are then expressed as:
t
U (r , z ) =
450
t
for − 2s < z < 2s ⎧ Es (ε0 − kz )2 2 ⎨ Ef (ε0 − kz + εm ) for ts < z < ts + t f 2 2 ⎩
(9)
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Fig. 2. The substrate shape and boundary condition for (a) BK-7 glass, (b) PET and PC flexible substrate.
Total potential energy V(ε0, k) for the thin film is expressed as:
V (ε0 , k ) = 2π
R
ts +t 2 f t − s 2
∫0 ∫
U (r , z ) rdzdr
expansion, nucleation shrinkage, and uniformity of boundary, during the deposition processing. ANSYS software is used for finite element analysis, where it is necessary to model residual stress as a material property for the condition of a thin-film structure. However, it is assumed that the residual stress is uniformly distributed through the entire thin film. In FEM analysis, the PLANE82 cell is used for 2-D solid modeling. PLANE82 is a classical structural element of a two-dimensional entity, and consists of the quadrangle and two-order elements. The element assembly can be used as both a plane element (plane stress or plane strain) and as an axis-symmetric element (z-axis) [32,33], as shown in Fig. 2. In the simulation process, we assumed the substrate was flat before sputtering. After sputtering, the film and substrate were bending with a reference center of point O, as shown in Fig. 3. As any point on the film surface is the same distance from the O point, it is also known as the circle radius of R = 1/k for the Stoney equation. When the 1/k increased with the increase in bending, the O point was closer to the substrate. ΔOAB is an isosceles triangle, thus, perpendicular line OC could bisect this to obtain a right-angled triangle OCA. The ΔBDA is similar to ΔOCA with the same vertex angle A and right angle 90o. Therefore:
(10)
In the energy balance of the flexible substrate, ∂V/∂ε0 = 0 and ∂V/ ∂R = 0, the modified Stoney residual stress for flexible stress can be expressed as [28,29]:
σf = Ef εm =
ν=
νs + νf 2
Y f∗
(Ys ts2 − Yf t f2) + 4Ys Yf ts t f (ts + t f )2 6(1 +
; Ys =
ν∗) Ys Yf ts t f
(ts + t f )
1+
Y ∗f t f
(k − k 0 )
Y s∗ ts
Ef Ef Es Es ; Yf = ; Y s∗ = ; Y f∗ = 1 − νs 1 − νf 1 − νs2 1 − νf2 (11)
Intrinsic stress mainly comes from structural defects and deposition process, such as the interface stress, special layout of micropores or dislocation, lattice mismatch of film and substrate, phase change of thin film, permeation of impurities from external material or gas, adsorption and deaquation of water vapor, and structural damages caused by sputtering or other energies. Seeking a full understanding of the reasons for intrinsic stress is a complex process. There are many models to describe the reasons for intrinsic stress [30], including the defect model, surface tension model, grain boundary model [31], and the peening model [3].
BA AD BD = = OA AC OC
(12)
BA, AD and AC are the known numbers in the simulation results, thus, the bending radius (R) of OA could be calculated by Eq. 12. From the post-treatment of FEM, all positions of elements and nodes regarding the radius of the bending after sputtering could be obtained. However, as the k0 is zero at R0 = ∞ before sputtering and k is equal to
2.2. Simulation model of the finite element method Residual stress is caused by non-uniform processing conditions and material properties, such as temperature, coefficient of thermal
Fig. 3. Residual stress simulated by the bending curve with FEM.
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while the PET and PC flexible substrates are the same square size of 40 × 50 mm and a thickness of 0.125 and 0.178 mm, respectively. The thin film thicknesses of all oxide thin films are about 300 nm. Intrinsic stress is complex and results from both material properties and deposition defects, which are not easily described. However, as the vacuum sputtering process can reduce the effects of environment and impurities, the intrinsic stresses result mainly from material properties, such as the elasticity module, material constants, etc. For this reason, an equivalent-room-temperature (ERT) technique is used to simulate intrinsic stress by the superposition of the thermal and intrinsic stresses [35]. The concept is that the intrinsic strain is resulted only from the material and the boundary condition of thin film. In other words, we allowed the strain resulted from the film to reach the size of intrinsic strain to simulate the load of the intrinsic stress. As the sputtering process is in a vacuum and has low impurity, the load came from the temperature variation for the intrinsic stress. Since intrinsic stress occurs prior to thermal stress, the ERT temperature is set as TERT and the deposition temperature (or substrate temperature) is TD. Then, the formula equation for the intrinsic stress can be expressed as:
Table 1 Material and geometric properties of different oxide thin films and substrates [34].
E (GPa) ν α (10−6/K) Area (mm2) Thickness (μm)
SiO2
Nb2O5
BK7
PET
PC
74.5 0.164 0.55 – 0.3
60.0 0.2 5.8 – 0.3
81.0 0.208 7.25 Π*12.72 1500
2.65 0.405 56.67 40*50 125
2.35 0.37 67.75 40*50 178
1/R after sputtering, we could calculate these residual stresses from the Stoney or modulated Stoney equations for hard and flexible substrates, respectively. We assume the materials are homogeneous and isotropic for all substrates and oxide thin films. In the FEM model, PLANE82 is used for 2D quadrilateral quadratic structural element, and the thermal equation is shown, as follows:
∂qy ⎞ ∂q ∂T − ⎛⎜ x + ⎟ = pcp ∂ ∂ x y ∂t ⎠ ⎝
(13)
σ=
where qx and qy are the displacements of two degrees of freeform, ρ is the density (kg/m3), and Cp is the specific heat (J/kg K). The relationship between temperature and element is expressed as:
∑ Ni (x,y) Ti (t ) i=1
+
1 − νs t f Es ts
(16)
Fig. 4 shows the simulation flow of ERT technology. First, experimental data are used as a reference residual stress. Secondly, room temperature (or ambient temperature) is set as TR and deposition temperature (TD) for the temperature of the substrate, and we then looked for the TERT temperature. Thirdly, a thin film cooled from TD to TR could simulate the thermal stress using FEM. Lastly, a thin film cooled from TERT to TD could simulate the intrinsic stress, thus, realizing that the error ratio of intrinsic stress must be less than 1%. There are many material and boundary properties that can affect the growth of intrinsic stress. However, these thin films are deposited by sputtering within a high vacuum environment and dense package density, thus, the intrinsic stress effect results mainly from the material microstructure [36]. Especially, the flexible substrate could increase the effect of oxidation's material structure. Based on this concept, the same substrate is used to analyze the different oxidation thin films of SiO2 and Nb2O5. The simulation process shows the relation of the unit cell, which can be expressed as:
(14)
where Ni denotes the coordinate values of the ith node in the Nth element. In the deposition process the conversion equation for thermal stress is dependent on thermal conduction (Kc) and thermal convection (Kh).
[C ]{Tm + 1} = {Rh }Δt − [K c ] + [Kh ]Δt {Tm} + [C ]{Tm}
1 − νf Ef
n
T (x , y, t ) =
−αf (TD − TERT )
(15)
where [C] is the material constant and {Rh} is the thermal convection of the rigid body. The simulation model assumes that all oxide thin films (SiO2 and Nb2O5) are sputtered on various substrates (BK-7, PET, and PC). Table 1 shows the material and geometric properties of different oxide thin films and substrates. E is the elastic modulus, ν is the Poisson's ratio, and α is the coefficient of thermal expansion (CTE). The BK-7 substrate is a circle with a 1 in. (25.4 mm) diameter and thickness of 1.5 mm;
⎧ σ1 = ⎪ ⎪ ⎨σ = ⎪ 2 ⎪ ⎩
−αf 1 (TD − TERT 1 ) 1 − νf 1 Ef 1
+
1 − νs t f 1 Es ts
−αf 2 (TD − TERT2 ) 1 − νf 2 1 − νs t f 2 Ef 2
+
Es
∝ Unit cellf 1 × Deposition Environment ∝ Unit cellf 2 × Deposition Environment
ts
(17)
TERT2 temperature could be obtained by the idea deposition
Fig. 5. The simulation flow of stress estimate by D factor.
Fig. 4. The simulation flow of ERT technology.
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2.3. Residual stress measured by Twyman-green interferometer and shadow moiré interferometer
Table 2 Material constant for the monoclinic SiO2 and Nb2O5 thin film. Parameters
SiO2(monoclinic)
Nb2O5(hexagonal)
a (Å) b (Å) c (Å) α(Å) β(Å) γ(Å) Unit cell(Å3)
4.2166 4.0206 7.6423 – 119.667 – 112.58
3.607 – 3.925 – – – 44.22
The bending curvature of residual stress in BK-7 glass is very small and not easily observed by the human eye, while the bending curvatures of PET and PC flexible substrates are larger and easily seen. The bending curvatures are measured using both a self-made TwymanGreen interferometer and a shadow moiré interferometer for BK-7 and flexible substrates, respectively. The bending radius of BK-7 glass is determined using a phase-shifting Twyman-Green interferometer, which measures the variation in the fringes caused by the deposition of film on the substrate, as shown in Fig. 6. The plane wavefront is divided in amplitude by a beam splitter, and the reflected and transmitted beams travel to a reference plate and a test plate. After being reflected by both the reference plate and the test plate, the interference pattern can be seen on a monitor through a CCD camera. Five phase shifting fringe patterns are obtained by moving the reference plate to five equally-spaced positions of λ/8 with a computer-controlled piezoelectric transducer (PZT) translation device. The phase of the fringe is then calculated from the digitized intensities at each point in the interferograms, using Hariharan's algorithm [39]. In the calculation, the stress in the coating is assumed to be isotropically and homogeneously distributed, thus, the deformation of the substrate is much smaller than the substrate's thickness. The systematic measurement tolerance is ± 25 MPa [40]. The residual stresses of the PET and PC substrates are measured by the phase-shifting interferometry applied in the double beam shadow moiré interferometer. This technique also uses an automatic measurement system to catch the interferograms, and then, calculates the residual stress of the flexible substrate. As the results of the shadow moiré interferometer are symmetrical, this measurement system is found to be stable and highly precise. White light is used as the light source, as it can prevent speckle noise. The beam splitter divides one beam into two beams, which are then passed through a reference grating at an incident angle of 45°, and project two deformation shadow gratings onto the
environment: 1 − νf 2
TERT 2 = TD −
αf 1 (TD − TERT 1 )
Ef 2
αf 2
1 − νf 1 Ef 1 1 − νf 2
= TD −
αf 1 (TD − TERT 1 )
Ef 2
αf 2
1 − νf 1 Ef 1
+
1 − νs t f 2 Es ts
+
⎜ 1 − νs t f 1 Unit Es ts ⎝
+
1 − νs t f 2 Es ts
+
1 − νs t f 1 Es ts
⎛ Unit cellf 2 ⎞ ⎟ cellf 1 ⎠ (D factor ) (18)
where the D factor is the degree relationship of the unit cell ratio for two different oxide thin films, with the simulation process as shown in Fig. 5. The sputtering process without heating the substrate is always polycrystalline in all directions, which is to close the first crystalline type. When the Nb2O5 thin film is sputtered at a low temperature, the crystalline turns from the non-crystal to the first crystal type δ-Nb2O5 or TT-Nb2O5 [37]. In the other method, pressure and temperature are two dominant factors used to form three crystalline types of tetragonal, monoclinic, and cubic SiO2 film growth. As the deposition pressure of 1 × 10−3 Torr is at medium vacuum, the crystalline form of SiO2 is monoclinic [38]. Table 2 shows the material constant for the monoclinic SiO2 and hexagonal Nb2O5 thin film.
Fig. 6. Schematic drawing of a phase shifting Twyman-Green interferometer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 7. Schematic representation of a phase shift shadow moiré interferometer.
Fig. 9. The simulated intrinsic stress of the SiO2 and Nb2O5 thin films sputtered on BK-7 glass with FEM by D factor.
Fig. 8. Residual, thermal, and intrinsic stresses of SiO2 and Nb2O5 thin film sputtering on different substrates simulated by FEM.
Table 3 The analysis of stress ratio for SiO2 to Nb2O5 thin films sputtering on different substrates. Sub.
BK7 PET PC
Unit cell ratio
2.55 2.55 2.55
thermal stress. Fig. 8 exhibits these simulated stresses for SiO2 and Nb2O5 thin films sputtered on different substrates. The results exhibit that these residual stresses of oxide thin films are compressive stresses for general phenomena. While these thermal stresses are also compressive stresses, the thermal stress of a flexible substrate is larger than that of a glass substrate, resulting in the large thermal expansion coefficient of the flexible substrate. Since the intrinsic stresses of these flexible substrates are tensile stresses, the residual stress size of the SiO2 thin film is about twice that of the Nb2O5 thin film, and the stress size of a PC substrate is also twice that of the PET substrate. In the same flexible substrate, the thermal stress size of the SiO2 thin film is 1.3 times that of the Nb2O5 thin film. However, the maximum residual stress is found in the BK7/SiO2 thin film, while the minimum is found in the PET/Nb2O5 thin film. We infer that the unit cell of the SiO2 thin film is larger than that of the Nb2O5 thin film, thus, the residual stress of the SiO2 thin film is always larger than the Nb2O5 thin film.
Stress ratio Residual stress
Thermal stress
Intrinsic stress
3.02 2.43 2.40
5.73 1.31 1.30
2.75 0.85 0.24
object's surface, as illustrated in Fig. 7. The systematic measurement tolerance is ± 9.4 MPa with a 1.26% error [41,42]. 3. Results and discussions 3.1. Stress simulation by FEM These SiO2 and Nb2O5 thin films are deposited on BK-7 glass, PET, and PC flexible substrates by magnetron sputtering, and residual stresses are simulated with the bending curve after deposition by FEM. Thermal stresses are also simulated by FEM with different temperatures and thermal expansion coefficients between the film and substrate. Intrinsic stresses are derived through subtraction of residual stress and
3.2. Stress mechanism of BK-7 glass, PET and PC flexible substrate with D factor These residual stresses are measured by Twyman-Green and Shadow moiré interferometers for glass and flexible substrates, respectively. The 454
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to the D factor, and the ratio size is about 2.43 and 2.40 for PET and PC flexible substrates, respectively, which is close to 2.25. We suggest that the oxidation film is a highly elastic module (> 50 GPa), while the PET and PC flexible substrates are of low elasticity (< 5 GPa), thus, the deposition defects result from the thermal expansion and boundary continuing to induce the dominant residual stress. However, the intrinsic stresses, as simulated by FEM with the ratio of the D factor for the BK-7 glass substrate, are about −3.25% and 3.44% errors for SiO2 and Nb2O5, respectively, as shown in Fig. 9. Simultaneously, the residual stress simulated by FEM with the ratio of the D factor for PET flexible substrates is about 4.7% and − 4.49% errors for SiO2 and Nb2O5, respectively. In the same way, the residual stress simulated by FEM with the ratio of D factor for PC flexible substrates is about 6.26% and − 5.89% errors for SiO2 and Nb2O5, respectively. These errors of simulated stresses are all close to 6% by FEM with the ERT technique, as shown in Fig. 10. 3.3. A polynomial fitting curve for the simulation precision of intrinsic stress Fig. 10. The simulated residual stress of the SiO2 and Nb2O5 thin films sputtered on PET and PC flexible substrates with FEM by D factor.
Intrinsic stress is a complex phenomenon and difficult to simulate. In order to promote the simulated precision of the FEM for intrinsic stress with an ERT technique, a polynomial fitting curve is used to fix the ERT value for the SiO2 and Nb2O5 thin films deposited with different substrates, as shown in Fig. 11. These related equations could be determined using polynomial regression analysis to reduce the simulated convergence time and increase the correct super-position values. There are seven ERT values that needed to be fit: -6005 K, -3880 K, -1755 K, 400 K, 2495 K, 4620 K, 6745 K and 400 K under the free-force situation. The norms of the residuals of the fitting values are required to be below 0.01, and all the fitting equations are shown in Table 4. The fitting curve equations used to simulate the intrinsic stress by FEM with ERT technology could reduce the error to 0.01%, 0.02%, and 0.01% for SiO2 thin film sputtering on BK-7, PET, and PC substrates, respectively. Simultaneously, the intrinsic stress, as simulated by the fitting curve, meant that the errors were reduced to 0.07%, 1.32%, and 0.10% for the Nb2O5 thin film sputtering on BK-7, PET, and PC substrates, respectively. However, the FEM simulated errors of the intrinsic stress of the SiO2 and Nb2O5 thin films deposited on a flexible substrate could be
thermal stresses could be simultaneously measured by the difference of the temperature and thermal expansion coefficients between the film and substrate, and then, the intrinsic stress is calculated by subtracting residual and thermal stresses for SiO2 and Nb2O5 thin films sputtered with different substrates. The D factor of the unit cell ratio for SiO2 to Nb2O5 thin films is 2.25, with a ratio of 112.58 to 44.22 Å3, as listed in Table 2. All stress ratios of the SiO2 to Nb2O5 thin films sputtered on different substrates are shown in Table 3 for residual, thermal, and intrinsic stresses, respectively. However, the stress mechanism of the BK-7 glass indicates that the intrinsic stress is directly proportional to the D factor (unit cell ratio), and the ratio size is about 2.75, which is close to 2.25. Apparently, the BK-7 glass and oxide thin film comprise a highly elastic module, thus, the deposition defect comes mainly from molecule accumulation and boundary disorder to induce the dominator of intrinsic stress. On the contrary, the stress mechanism of the PET and PC flexible substrates show that the residual stresses are in direct ratio
Fig. 11. A polynomial fitting curve for intrinsic stress simulation with SiO2 and Nb2O5 thin films sputtered on different substrates.
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Acknowledgement
Table 4 A polynomial fitting equation and error ratio of intrinsic stress for SiO2 and Nb2O5 thin films sputtered on different substrates with FEM. Thin film
A polynomial fitting equation
Error (%)
BK7/SiO2 PET/SiO2
σ = (0.054017)X + (−21.606) σ = (−1.2471e-11)X3 + (2.4385e-8)X2 + (0.041085) X + (−16.436) σ = (−4.2582e-12)X3 + (1.3049e-8)X2 + (0.041073) X + (−16.43) σ = (4.663e-9)X2 + (0.45966)X + (−183.87) σ = (4.9611e-22)X6 + (6.5559e-17)X5 + (−1.6641e13)X4 +(−6.8136e-9)X3 + (9.0913e-6)X2 + (0.35074) X + (−141.31) σ = (8.4685e-23)X6 + (1.4601e-17)X5 + (−3.9403e14)X4 +(−2.7315e-9)X3 + (3.9263e-6)X2 + (0.3565) X + (−143.05)
0.01% 0.02%
PC/SiO2 BK7/Nb2O5 PET/Nb2O5
PC/Nb2O5
The authors would like to thank the Ministry of Science and Technology of Taiwan for financially supporting this research under contract No. MOST 105-2221-E-224-029- and MOST 106-2813-C-224002-E. References
0.01% 0.07% 1.32%
[1] G. Gore, On the Properties of Electro-deposited Antimony, 1st ed., Trans. Roy. Soc, London, 1858. [2] G.G. Stoney, The tension of metallic films deposited by electrolysis, Proc. Roy. Soc. A 82 (1909) 172–175. [3] F.M. D'Heurle, Aluminum films deposited by rf sputtering, Metall. Trans. A. 1 (1970) 725–732. [4] D.W. Hoffman, J.A. Thornton, Effects of substrate orientation and rotation on internal stresses in sputtered metal films, J. Vac. Sci. Technol. 16 (2) (1979) 134–137. [5] H.K. Puller, J.M. Maser, The origin of mechanical stress in vacuum deposited MgF2 and ZnS films, Thin Solid Films 59 (1979) 65–76. [6] K.H. Muller, Stress and microstructure of sputter-deposited thin films: molecular dynamics investigations, J. Appl. Phys. 62 (1987) 1796–1799. [7] F.M. D'Heurle, J.M. Harper, Note on the origin of intrinsic stresses in films deposited via evaporation and sputtering, Thin Solid Films 171 (1989) 81–88. [8] H.C. Chen, C.C. Lee, C.C. Jaing, M.H. Shiao, C.J. Lu, F.S. Shieu, Effects of temperature on columnar microstructure and recrystallization of TiO2 film produced by ion-assisted deposition, Appl. Opt. 45 (9) (2006) 1979–1984. [9] J.Y. Robic, H. Leplan, Y. Paulean, B. Rafin, Residual stress in silicon dioxide thin films produced by ion-assisted deposition, Thin Solid Films 389 (1996) 34–39. [10] H.C. Chen, K.S. Lee, C.C. Lee, Annealing dependence of residual stress and optical properties of TiO2 thin film deposited by different deposition methods, Appl. Opt. 47 (13) (2008) 284–287. [11] M. Ohring, Materials Science of Thin Films, 2nd ed., Academic Press, 2001. [12] S. Ben Amor, G. Baud, J.P. Besse, M. Jacquet, Elaboration and characterization of titania coatings, Thin Solid Films 293 (1997) 163–169. [13] H.A. Macleod, Characteristics of thin-film dielectric material, Thin-Film Optical Filters, 3rd ed., Institute of Physics Publishing Press, London, 2001, pp. 621–627. [14] M. Grätzel, Ultrafast colour displays, Nature 409 (2001) 575–576. [15] D.R. Cairns, G.P. Crawford, Electromechanical properties of transparent conducting substrates for flexible electronic displays, Proc. IEEE 93 (8) (2005) 1451–1458. [16] T. Minami, Transparent conducting oxide semiconductors for transparent electrodes, Semicond. Sci. Technol. 20 (2005) S35–S44. [17] G.K. Mor, O.K. Varghese, M. Paulose, K. Shankar, C.A. Grimes, A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications, Sol. Energy Mater. Sol. Cells 90 (2006) 2011–2075. [18] A. Subrahmanyam, C.S. Kumar, K.M. Karuppasamy, Anote on fast protonic solid state electrochromic device: NiOx/Ta2O5/WO3−x, Sol. Energy Mater. Sol. Cells 91 (2007) 62–66. [19] X.F. Zhu, B. Zheng, J. Gao, G.P. Zhang, Evaluation of the crack-initiation strain of a Cu-Ni multilayer on a flexible substrate, Scr. Mater. 60 (2009) 178–181. [20] F. Pourahmadi, P. Barth, K. Petersen, Modeling of thermal and mechanical stresses in silicon microstructures, Sensors Actuators A Phys. 23 (1–3) (1990) 850–855. [21] A. Steegen, K. Maex, Silicide-induced stress in Si: origin and consequences for MOS technologies, Mater. Sci. Eng. R 38 (1) (2002) 1–53. [22] J.A. Martins, L.P. Cardoso, J.A. Fraymann, S.T. Button, Analyses of residual stresses on stamped valves by X-ray diffraction and finite elements method, J. Mater. Process. Technol. 179 (2006) 30–35. [23] R. Roos, G. Kress, M. Barbezat, P. Ermanni, Enhanced model for interlaminar normal stress in singly curved laminates, Compos. Struct. 80 (2007) 327–333. [24] A.O. Cifuentes, I.A. Shareef, Some modeling issues on the finite element computation of thermal stresses in metal lines, ASME J. Electron. Packag. 115 (1993) 392–403. [25] M.K. Lin, A.O. Abatan, C.A. Rogers, Application of commercial finite element codes for the analysis of induced strain-actuated structures, J. Intell. Mater. Syst. Struct. 5 (1994) 869–875. [26] M. Toparli, F. Sen, O. Culha, E. Celik, Thermal stress analysis of HVOF sprayed WC–Co/NiAl multilayer coatings on stainless steel substrate using finite element methods, J. Mater. Process. Technol. 190 (2007) 26–32. [27] L.B. Freund, J.A. Floro, E. Chason, Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations, Appl. Phys. Lett. 74 (14) (1999) 3–5. [28] Y. Zhang, Y.P. Zhao, Applicability range of Stoney's formula and modified formulas for a film/substrate bilayer, J. Appl. Physiol. 99 (5) (2006) (053513-1~7). [29] K.S. Lee, C.J. Tang, H.C. Chen, C.C. Lee, Measurement of stress in aluminum film coated on a flexible substrate by the shadow moiré method, Appl. Opt. 47 (13) (2008) 315–318. [30] R.W. Hoffman, H.S. Story, Stress annealing in copper films, J. Appl. Physiol. 27 (2) (1956) (193–193). [31] B.A. Movchan, A.V. Demchishin, Study of the structures and properties of thick vacuum condensates of nickel, titanium, tungsten, aluminum oxide and zirconium
0.10%
Fig. 12. The intrinsic stresses for SiO2 and Nb2O5 thin films sputtered on different substrates by a polynomial fitting.
reduced to below 2%. The intrinsic stresses of SiO2 and Nb2O5 thin films sputtered on different substrates by a polynomial fitting are shown in Fig. 12. 4. Conclusions The stress mechanism of SiO2 and Nb2O5 thin films sputtered on BK7 glass, PET, and PC flexible substrates were investigated by FEM. The residual stress was a combination of thermal stress and intrinsic stress, and without external force used during the sputtering process. The thermal and residual stresses could be calculated by numerical simulation of FEM. The residual stress size of SiO2 films was about twice that of Nb2O5 film; the stress size of PC substrate also was twice that of the PET substrate. In the same flexible substrate, the thermal stress of SiO2 was 1.3 times that of Nb2O5. However, the maximum residual stress was in the BK7/SiO2 thin film and the minimum was in the PET/Nb2O5 thin film. In the BK-7 glass substrate, the ratio of intrinsic stress was close to the D factor of SiO2 and Nb2O5 thin film, and the error was about 3%. While in the PET and PC flexible substrates, the ratio of residual stress was close to the ratio of the D factor of the SiO2 and Nb2O5 thin films, and the error was about 6%. The fitting curved equations were used to simulate the intrinsic stress by FEM with ERT technology, which could potentially reduce the error to less than 2%.
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chemical behavior, Catal. Today 8 (1) (1990) 27–36. [38] M.T. Dove, M.S. Craig, D.A. Keen, W.G. Marshall, S.A.T. Trachenko, M.G. Tucker, Crystal structure of the high-pressure monoclinic phase-II of cristobalite, SiO2, Mineral. Mag. 64 (3) (2000) 569–576. [39] P. Hariharan, B.F. Oreb, T. Eiju, Digital phase-shifting interferometry: a simple error- compensating phase calculation algorithm, Appl. Opt. 26 (1987) 2504–2506. [40] C.L. Tien, C.C. Lee, C.C. Jiang, The measurement of thin film stress using phase shifting interferometry, J. Mod. Opt. 47 (2000) 839–849. [41] K.T. Huang, H.C. Chen, Automatic measurement and stress analysis of ITO/PET flexible substrate by shadow moiré interferometer with phase-shifting interferometry, J. Disp. Technol. 10 (7) (2014) 609–614. [42] H.C. Chen, P.W. Cheng, K.T. Huang, Biaxial stress and optoelectronic properties of Al-doped ZnO thin films deposited on flexible substrates by radio frequency magnetron sputtering, Appl. Opt. 56 (4) (2016) c163–c167.
dioxide, Phys. Metals Metallogr. 28 (1969) 83–90. [32] T.R. Chandrupatla, A.D. Belegundu, Introduction to Finite Elements in Engineering, 3rd ed., Prentice Hall, 2002. [33] Y. Li, J.Q. Xu, L.D. Fu, Fundamental solution for bonded materials with a free surface parallel to the interface. Part I: solution of concentrated forces acting at the inside of the material with a free surface, Int. J. Solids Struct. 41 (2004) 7075–7089. [34] T.C. Chen, G.B. Lee, Determination of stress-optical and thermal-optical coefficients of Nb2O5 thin film material, J. Appl. Physiol. 101 (4) (2007) (063506-1~6). [35] M.N.G. Nejhad, C.L. Pan, H.W. Feng, Intrinsic strain modeling and residual stress analysis for thin-film processing of layered structures, J. Electron. Packag. 125 (1) (2003) 4–17. [36] H. Windischmann, An intrinsic stress scaling law for polycrystalline thin films prepared by ion beam sputtering, J. Appl. Physiol. 62 (5) (1987) 1800–1807. [37] E.I. Ko, J.G. Weissman, Structures of niobium pentoxide and their implications on
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Stress mechanisms of SiO2 and Nb2O5 thin films sputtered on flexible substrates investigated by finite element method
T
⁎
Hsi-Chao Chena,b, , Chen-Yu Huanga, Po-Wei Chengb a b
Department of Electronic Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan Graduate School of Optoelectronics Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan
A R T I C LE I N FO
A B S T R A C T
Keywords: Stress mechanism Sputtering Flexible substrate Numerical analysis Finite element method (FEM)
The stress mechanisms of SiO2 and Nb2O5 thin films sputtering on BK-7 glass, PET, and PC flexible substrates were investigated by the numerical analysis of the finite element method (FEM). Residual stress is a combination of thermal and intrinsic stresses during the sputtering process. The thermal stress results from the difference in thermal expansion coefficients and the substrate temperature between the film and substrate. The sputtering process is complicated; hence, the intrinsic stress was simulated by an equivalent-room-temperature (ERT) technique of FEM and a polynomial fitting curve, which could reduce the error ratio to less than 2%. The experimental verification of residual stresses has been conducted using the self-made Twyman-Green interferometer and shadow moiré interferometer with Stoney and modified Stoney formulas for glass and flexible substrates, respectively. However, in examining the stress mechanism of the BK-7 hard substrate, it was found that the degree of intrinsic stress was directly proportional to the D factor (Unit cell ratio) of SiO2 and Nb2O5 thin films, and that the error ratio was about 3%; while the stress mechanisms of PET and PC flexible substrates show that the residual stresses were directly proportional to the D factor, and the error ratio was about 6%.
1. Introduction Thin films are usually in a state of elastic mechanical stress during and after deposition, thus, the stress behavior of film is very important in all applications of thin films with respect to durability, stability, and usability. The first investigation regarding mechanical stress in thin film was conducted in 1858 on electro-deposited antimony film [1], and a more quantitative method was used to investigate the mechanical stress of thin film in 1909 by Stoney [2]. Thereafter, a series of researches on the measurements and explanations of the origin and quantitative calculation of mechanical stress, as based on a grain boundary interaction model for crystalline films, have been published [3–6]. It is well known that oxide films prepared by magnetron sputtering usually have a high packing density, and therefore, show excellent stability under the changing conditions of relative humidity. While these sputtered films are generally amorphous or low polycrystalline at the substrate temperature of room temperature, the signified crystalline directions are random, and the crystalline will be arrayed with the increased the substrate temperature [7,8]. Then, the mechanical stress or named residual stress is a consequence of the high film density, as caused by the higher energy of the condensing and bombarding atoms and molecules [9,10]. Then, thermal stress could be created by the variations in
⁎
thermal expansion coefficients and temperature between the film and substrate, and the nucleation growth of thin film could result in intrinsic stress during the deposition process. Due to the complexity of the deposition process and the plasma environment during sputtering deposition, it is difficult to analyze intrinsic stress. Intrinsic stress results from the target material, plasma power, chamber pressure, and structure organization to influence the atom dislocation, growth defects, film adhesion, etc., which also impact the amount of residual stress. In the metal film, residual stress induces the phenomenon of hillock and whisker of atomic displacement, causing film peeling due to tensile stress [11]. However, in the dielectric film of metal-oxidation, the residual stress could result in the shrinking of the film, which cracks the film due to compressive stress [12]. As is widely known, BK-7 glass, PET, and PC flexible plates are the commonly-used substrates, and SiO2 and Nb2O5 are the most widelyused low and high refractive index oxide thin films, respectively, for optical interference coatings. These films are hard and chemically resistant, transparent in the visible range, and have stability refractive indices, which demonstrate excellent mechanical and environmental stability [13]. These oxide thin films created by deposition on different substrates using magnetron sputtering find wide applications in microelectronics, optics, semiconductor multi-layer, superconducting
Corresponding author at: Department of Electronic Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan. E-mail address: [email protected] (H.-C. Chen).
https://doi.org/10.1016/j.surfcoat.2018.03.051 Received 25 September 2017; Received in revised form 15 March 2018; Accepted 16 March 2018 Available online 16 March 2018 0257-8972/ © 2018 Elsevier B.V. All rights reserved.
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Fig. 1. A diagram of stress distribution for flexible substrates.
systems, data storage systems, organic light emission displays, and flexible thin film solar cells [14–19]. Simultaneously, the vigorous developments of computing technology have allowed more researchers to use numerical analysis to simulate stress [20–23]. Finite element analysis (FEM) is a very powerful technique used to model thin-film structures and implement processing with static and dynamic analyses [24–26]. The purpose of this research is to use FEM to analyze the residual stress in oxide thin film (SiO2 and Nb2O5) using different substrates. Thermal stress can be simulated by examining the differences in temperature and thermal expansion coefficients between the substrate and films during the sputtering process. Next, the intrinsic stress is derived using the equivalent-room-temperature (ERT) technique [35] and a polynomial fitting curve, which can reduce the error ratio. A D factor (Unit cell) was used to determine the relationship degree of residual or intrinsic stresses of these oxide thin films. The stress mechanism of the BK-7 hard substrate is intrinsic stress, which is directly proportional to the D factor of SiO2 and Nb2O5 thin films. On the contrary, the stress mechanisms of PET and PC flexible substrates show the residual stresses to be directly proportional to the D factor of SiO2 and Nb2O5 thin films. These phenomena were verified by the residual stress measurements with Twyman-green and shadow moiré interferometers.
Ef Ef εtf0 = αf (Tr − T0 ) 1 − νf 1 − νf Es Es = εts0 = αs (Tr − T0) 1 − νs 1 − νs
σtf0 = σts0
(3)
where Ef and Es are the elastic modules for film and substrate, and vf and vs are the Poisson ratios for film and substrate, respectively. The strain for film and substrate are then expressed as:
εtf = αf (Tr − T0) +
f (1 − νf ) t f Ef
f (1 − νs ) εts = αs (Tr − T0) − ts Es
(4)
where tf and ts are the film thicknesses for film and substrate, respectively, and f is the internal reacting stress related to the thickness of thin film, as resulted from the temperature variant. The strains of the film and substrate are in the same positions due to a continuous boundary condition, thus, the internal reacting stress f is expressed, as follows:
(αs − αf )(Tr − T0 )
f=
1 − νf Ef t f
+
1 − νs Es ts
(5)
Then, the thermal stress can be expressed as: 2. Theoretical formula and experimental measurement
σtf =
Ef
2.1. Residual and thermal stresses
σtf =
(6)
ts
Ef 1 − νf
(αs − αf )(Tr − T0)
(7)
The residual stress is a combination of thermal and intrinsic stresses without external force during the sputtering process. The residual stress can be expressed, as follows, for the glass substrate using the Stoney equation [2]:
(1)
During the sputtering process, an Ar+ plasma sputters the target to achieve metal oxidation at high temperatures. Once the deposition process ends, the temperature can fall to room temperature, with thermal stress resulting primarily from the concentrates of the thin film and substrate of different thermal expansion coefficients. The original thermal strain can be expressed as:
σf =
Es ts2 ⎛ 1 Es ts2 f 1 ⎞ (k − k 0) = − = tf 6t f ⎝ R R0 ⎠ 6t f ⎜
⎟
(8)
Fig. 1 shows the stress distribution for flexible substrates during the sputtering process. Residual stress derivation for the flexible substrate must consider the thin film strain, thus, the total strain energy density U (r,z) can be expressed as [27]:
εtf0 = αf (Tr − T0 ) εts0 = αs (Tr − T0)
Es
If (1 − νf)/Eftf > > (1 − νs)/Ests the thermal stress can be expressed as:
Residual stress can be separated into thermal stress, intrinsic stress, and extrinsic stress. As residual stress is always a combination of thermal and intrinsic stresses without external force during the sputtering process, the residual stress can be expressed as:
σresidual = σthermal + σintrinsic
(αs − αf )(Tr − T0 ) f = 1 − νf 1 − νs t f tf +
(2)
where Tr is the deposition (or substrate) temperature, T0 is the original temperature, and αf and αs are the thermal expansion coefficients of the film and substrate, respectively. The original thermal stress for the film and substrate are then expressed as:
t
U (r , z ) =
450
t
for − 2s < z < 2s ⎧ Es (ε0 − kz )2 2 ⎨ Ef (ε0 − kz + εm ) for ts < z < ts + t f 2 2 ⎩
(9)
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Fig. 2. The substrate shape and boundary condition for (a) BK-7 glass, (b) PET and PC flexible substrate.
Total potential energy V(ε0, k) for the thin film is expressed as:
V (ε0 , k ) = 2π
R
ts +t 2 f t − s 2
∫0 ∫
U (r , z ) rdzdr
expansion, nucleation shrinkage, and uniformity of boundary, during the deposition processing. ANSYS software is used for finite element analysis, where it is necessary to model residual stress as a material property for the condition of a thin-film structure. However, it is assumed that the residual stress is uniformly distributed through the entire thin film. In FEM analysis, the PLANE82 cell is used for 2-D solid modeling. PLANE82 is a classical structural element of a two-dimensional entity, and consists of the quadrangle and two-order elements. The element assembly can be used as both a plane element (plane stress or plane strain) and as an axis-symmetric element (z-axis) [32,33], as shown in Fig. 2. In the simulation process, we assumed the substrate was flat before sputtering. After sputtering, the film and substrate were bending with a reference center of point O, as shown in Fig. 3. As any point on the film surface is the same distance from the O point, it is also known as the circle radius of R = 1/k for the Stoney equation. When the 1/k increased with the increase in bending, the O point was closer to the substrate. ΔOAB is an isosceles triangle, thus, perpendicular line OC could bisect this to obtain a right-angled triangle OCA. The ΔBDA is similar to ΔOCA with the same vertex angle A and right angle 90o. Therefore:
(10)
In the energy balance of the flexible substrate, ∂V/∂ε0 = 0 and ∂V/ ∂R = 0, the modified Stoney residual stress for flexible stress can be expressed as [28,29]:
σf = Ef εm =
ν=
νs + νf 2
Y f∗
(Ys ts2 − Yf t f2) + 4Ys Yf ts t f (ts + t f )2 6(1 +
; Ys =
ν∗) Ys Yf ts t f
(ts + t f )
1+
Y ∗f t f
(k − k 0 )
Y s∗ ts
Ef Ef Es Es ; Yf = ; Y s∗ = ; Y f∗ = 1 − νs 1 − νf 1 − νs2 1 − νf2 (11)
Intrinsic stress mainly comes from structural defects and deposition process, such as the interface stress, special layout of micropores or dislocation, lattice mismatch of film and substrate, phase change of thin film, permeation of impurities from external material or gas, adsorption and deaquation of water vapor, and structural damages caused by sputtering or other energies. Seeking a full understanding of the reasons for intrinsic stress is a complex process. There are many models to describe the reasons for intrinsic stress [30], including the defect model, surface tension model, grain boundary model [31], and the peening model [3].
BA AD BD = = OA AC OC
(12)
BA, AD and AC are the known numbers in the simulation results, thus, the bending radius (R) of OA could be calculated by Eq. 12. From the post-treatment of FEM, all positions of elements and nodes regarding the radius of the bending after sputtering could be obtained. However, as the k0 is zero at R0 = ∞ before sputtering and k is equal to
2.2. Simulation model of the finite element method Residual stress is caused by non-uniform processing conditions and material properties, such as temperature, coefficient of thermal
Fig. 3. Residual stress simulated by the bending curve with FEM.
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while the PET and PC flexible substrates are the same square size of 40 × 50 mm and a thickness of 0.125 and 0.178 mm, respectively. The thin film thicknesses of all oxide thin films are about 300 nm. Intrinsic stress is complex and results from both material properties and deposition defects, which are not easily described. However, as the vacuum sputtering process can reduce the effects of environment and impurities, the intrinsic stresses result mainly from material properties, such as the elasticity module, material constants, etc. For this reason, an equivalent-room-temperature (ERT) technique is used to simulate intrinsic stress by the superposition of the thermal and intrinsic stresses [35]. The concept is that the intrinsic strain is resulted only from the material and the boundary condition of thin film. In other words, we allowed the strain resulted from the film to reach the size of intrinsic strain to simulate the load of the intrinsic stress. As the sputtering process is in a vacuum and has low impurity, the load came from the temperature variation for the intrinsic stress. Since intrinsic stress occurs prior to thermal stress, the ERT temperature is set as TERT and the deposition temperature (or substrate temperature) is TD. Then, the formula equation for the intrinsic stress can be expressed as:
Table 1 Material and geometric properties of different oxide thin films and substrates [34].
E (GPa) ν α (10−6/K) Area (mm2) Thickness (μm)
SiO2
Nb2O5
BK7
PET
PC
74.5 0.164 0.55 – 0.3
60.0 0.2 5.8 – 0.3
81.0 0.208 7.25 Π*12.72 1500
2.65 0.405 56.67 40*50 125
2.35 0.37 67.75 40*50 178
1/R after sputtering, we could calculate these residual stresses from the Stoney or modulated Stoney equations for hard and flexible substrates, respectively. We assume the materials are homogeneous and isotropic for all substrates and oxide thin films. In the FEM model, PLANE82 is used for 2D quadrilateral quadratic structural element, and the thermal equation is shown, as follows:
∂qy ⎞ ∂q ∂T − ⎛⎜ x + ⎟ = pcp ∂ ∂ x y ∂t ⎠ ⎝
(13)
σ=
where qx and qy are the displacements of two degrees of freeform, ρ is the density (kg/m3), and Cp is the specific heat (J/kg K). The relationship between temperature and element is expressed as:
∑ Ni (x,y) Ti (t ) i=1
+
1 − νs t f Es ts
(16)
Fig. 4 shows the simulation flow of ERT technology. First, experimental data are used as a reference residual stress. Secondly, room temperature (or ambient temperature) is set as TR and deposition temperature (TD) for the temperature of the substrate, and we then looked for the TERT temperature. Thirdly, a thin film cooled from TD to TR could simulate the thermal stress using FEM. Lastly, a thin film cooled from TERT to TD could simulate the intrinsic stress, thus, realizing that the error ratio of intrinsic stress must be less than 1%. There are many material and boundary properties that can affect the growth of intrinsic stress. However, these thin films are deposited by sputtering within a high vacuum environment and dense package density, thus, the intrinsic stress effect results mainly from the material microstructure [36]. Especially, the flexible substrate could increase the effect of oxidation's material structure. Based on this concept, the same substrate is used to analyze the different oxidation thin films of SiO2 and Nb2O5. The simulation process shows the relation of the unit cell, which can be expressed as:
(14)
where Ni denotes the coordinate values of the ith node in the Nth element. In the deposition process the conversion equation for thermal stress is dependent on thermal conduction (Kc) and thermal convection (Kh).
[C ]{Tm + 1} = {Rh }Δt − [K c ] + [Kh ]Δt {Tm} + [C ]{Tm}
1 − νf Ef
n
T (x , y, t ) =
−αf (TD − TERT )
(15)
where [C] is the material constant and {Rh} is the thermal convection of the rigid body. The simulation model assumes that all oxide thin films (SiO2 and Nb2O5) are sputtered on various substrates (BK-7, PET, and PC). Table 1 shows the material and geometric properties of different oxide thin films and substrates. E is the elastic modulus, ν is the Poisson's ratio, and α is the coefficient of thermal expansion (CTE). The BK-7 substrate is a circle with a 1 in. (25.4 mm) diameter and thickness of 1.5 mm;
⎧ σ1 = ⎪ ⎪ ⎨σ = ⎪ 2 ⎪ ⎩
−αf 1 (TD − TERT 1 ) 1 − νf 1 Ef 1
+
1 − νs t f 1 Es ts
−αf 2 (TD − TERT2 ) 1 − νf 2 1 − νs t f 2 Ef 2
+
Es
∝ Unit cellf 1 × Deposition Environment ∝ Unit cellf 2 × Deposition Environment
ts
(17)
TERT2 temperature could be obtained by the idea deposition
Fig. 5. The simulation flow of stress estimate by D factor.
Fig. 4. The simulation flow of ERT technology.
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2.3. Residual stress measured by Twyman-green interferometer and shadow moiré interferometer
Table 2 Material constant for the monoclinic SiO2 and Nb2O5 thin film. Parameters
SiO2(monoclinic)
Nb2O5(hexagonal)
a (Å) b (Å) c (Å) α(Å) β(Å) γ(Å) Unit cell(Å3)
4.2166 4.0206 7.6423 – 119.667 – 112.58
3.607 – 3.925 – – – 44.22
The bending curvature of residual stress in BK-7 glass is very small and not easily observed by the human eye, while the bending curvatures of PET and PC flexible substrates are larger and easily seen. The bending curvatures are measured using both a self-made TwymanGreen interferometer and a shadow moiré interferometer for BK-7 and flexible substrates, respectively. The bending radius of BK-7 glass is determined using a phase-shifting Twyman-Green interferometer, which measures the variation in the fringes caused by the deposition of film on the substrate, as shown in Fig. 6. The plane wavefront is divided in amplitude by a beam splitter, and the reflected and transmitted beams travel to a reference plate and a test plate. After being reflected by both the reference plate and the test plate, the interference pattern can be seen on a monitor through a CCD camera. Five phase shifting fringe patterns are obtained by moving the reference plate to five equally-spaced positions of λ/8 with a computer-controlled piezoelectric transducer (PZT) translation device. The phase of the fringe is then calculated from the digitized intensities at each point in the interferograms, using Hariharan's algorithm [39]. In the calculation, the stress in the coating is assumed to be isotropically and homogeneously distributed, thus, the deformation of the substrate is much smaller than the substrate's thickness. The systematic measurement tolerance is ± 25 MPa [40]. The residual stresses of the PET and PC substrates are measured by the phase-shifting interferometry applied in the double beam shadow moiré interferometer. This technique also uses an automatic measurement system to catch the interferograms, and then, calculates the residual stress of the flexible substrate. As the results of the shadow moiré interferometer are symmetrical, this measurement system is found to be stable and highly precise. White light is used as the light source, as it can prevent speckle noise. The beam splitter divides one beam into two beams, which are then passed through a reference grating at an incident angle of 45°, and project two deformation shadow gratings onto the
environment: 1 − νf 2
TERT 2 = TD −
αf 1 (TD − TERT 1 )
Ef 2
αf 2
1 − νf 1 Ef 1 1 − νf 2
= TD −
αf 1 (TD − TERT 1 )
Ef 2
αf 2
1 − νf 1 Ef 1
+
1 − νs t f 2 Es ts
+
⎜ 1 − νs t f 1 Unit Es ts ⎝
+
1 − νs t f 2 Es ts
+
1 − νs t f 1 Es ts
⎛ Unit cellf 2 ⎞ ⎟ cellf 1 ⎠ (D factor ) (18)
where the D factor is the degree relationship of the unit cell ratio for two different oxide thin films, with the simulation process as shown in Fig. 5. The sputtering process without heating the substrate is always polycrystalline in all directions, which is to close the first crystalline type. When the Nb2O5 thin film is sputtered at a low temperature, the crystalline turns from the non-crystal to the first crystal type δ-Nb2O5 or TT-Nb2O5 [37]. In the other method, pressure and temperature are two dominant factors used to form three crystalline types of tetragonal, monoclinic, and cubic SiO2 film growth. As the deposition pressure of 1 × 10−3 Torr is at medium vacuum, the crystalline form of SiO2 is monoclinic [38]. Table 2 shows the material constant for the monoclinic SiO2 and hexagonal Nb2O5 thin film.
Fig. 6. Schematic drawing of a phase shifting Twyman-Green interferometer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 7. Schematic representation of a phase shift shadow moiré interferometer.
Fig. 9. The simulated intrinsic stress of the SiO2 and Nb2O5 thin films sputtered on BK-7 glass with FEM by D factor.
Fig. 8. Residual, thermal, and intrinsic stresses of SiO2 and Nb2O5 thin film sputtering on different substrates simulated by FEM.
Table 3 The analysis of stress ratio for SiO2 to Nb2O5 thin films sputtering on different substrates. Sub.
BK7 PET PC
Unit cell ratio
2.55 2.55 2.55
thermal stress. Fig. 8 exhibits these simulated stresses for SiO2 and Nb2O5 thin films sputtered on different substrates. The results exhibit that these residual stresses of oxide thin films are compressive stresses for general phenomena. While these thermal stresses are also compressive stresses, the thermal stress of a flexible substrate is larger than that of a glass substrate, resulting in the large thermal expansion coefficient of the flexible substrate. Since the intrinsic stresses of these flexible substrates are tensile stresses, the residual stress size of the SiO2 thin film is about twice that of the Nb2O5 thin film, and the stress size of a PC substrate is also twice that of the PET substrate. In the same flexible substrate, the thermal stress size of the SiO2 thin film is 1.3 times that of the Nb2O5 thin film. However, the maximum residual stress is found in the BK7/SiO2 thin film, while the minimum is found in the PET/Nb2O5 thin film. We infer that the unit cell of the SiO2 thin film is larger than that of the Nb2O5 thin film, thus, the residual stress of the SiO2 thin film is always larger than the Nb2O5 thin film.
Stress ratio Residual stress
Thermal stress
Intrinsic stress
3.02 2.43 2.40
5.73 1.31 1.30
2.75 0.85 0.24
object's surface, as illustrated in Fig. 7. The systematic measurement tolerance is ± 9.4 MPa with a 1.26% error [41,42]. 3. Results and discussions 3.1. Stress simulation by FEM These SiO2 and Nb2O5 thin films are deposited on BK-7 glass, PET, and PC flexible substrates by magnetron sputtering, and residual stresses are simulated with the bending curve after deposition by FEM. Thermal stresses are also simulated by FEM with different temperatures and thermal expansion coefficients between the film and substrate. Intrinsic stresses are derived through subtraction of residual stress and
3.2. Stress mechanism of BK-7 glass, PET and PC flexible substrate with D factor These residual stresses are measured by Twyman-Green and Shadow moiré interferometers for glass and flexible substrates, respectively. The 454
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to the D factor, and the ratio size is about 2.43 and 2.40 for PET and PC flexible substrates, respectively, which is close to 2.25. We suggest that the oxidation film is a highly elastic module (> 50 GPa), while the PET and PC flexible substrates are of low elasticity (< 5 GPa), thus, the deposition defects result from the thermal expansion and boundary continuing to induce the dominant residual stress. However, the intrinsic stresses, as simulated by FEM with the ratio of the D factor for the BK-7 glass substrate, are about −3.25% and 3.44% errors for SiO2 and Nb2O5, respectively, as shown in Fig. 9. Simultaneously, the residual stress simulated by FEM with the ratio of the D factor for PET flexible substrates is about 4.7% and − 4.49% errors for SiO2 and Nb2O5, respectively. In the same way, the residual stress simulated by FEM with the ratio of D factor for PC flexible substrates is about 6.26% and − 5.89% errors for SiO2 and Nb2O5, respectively. These errors of simulated stresses are all close to 6% by FEM with the ERT technique, as shown in Fig. 10. 3.3. A polynomial fitting curve for the simulation precision of intrinsic stress Fig. 10. The simulated residual stress of the SiO2 and Nb2O5 thin films sputtered on PET and PC flexible substrates with FEM by D factor.
Intrinsic stress is a complex phenomenon and difficult to simulate. In order to promote the simulated precision of the FEM for intrinsic stress with an ERT technique, a polynomial fitting curve is used to fix the ERT value for the SiO2 and Nb2O5 thin films deposited with different substrates, as shown in Fig. 11. These related equations could be determined using polynomial regression analysis to reduce the simulated convergence time and increase the correct super-position values. There are seven ERT values that needed to be fit: -6005 K, -3880 K, -1755 K, 400 K, 2495 K, 4620 K, 6745 K and 400 K under the free-force situation. The norms of the residuals of the fitting values are required to be below 0.01, and all the fitting equations are shown in Table 4. The fitting curve equations used to simulate the intrinsic stress by FEM with ERT technology could reduce the error to 0.01%, 0.02%, and 0.01% for SiO2 thin film sputtering on BK-7, PET, and PC substrates, respectively. Simultaneously, the intrinsic stress, as simulated by the fitting curve, meant that the errors were reduced to 0.07%, 1.32%, and 0.10% for the Nb2O5 thin film sputtering on BK-7, PET, and PC substrates, respectively. However, the FEM simulated errors of the intrinsic stress of the SiO2 and Nb2O5 thin films deposited on a flexible substrate could be
thermal stresses could be simultaneously measured by the difference of the temperature and thermal expansion coefficients between the film and substrate, and then, the intrinsic stress is calculated by subtracting residual and thermal stresses for SiO2 and Nb2O5 thin films sputtered with different substrates. The D factor of the unit cell ratio for SiO2 to Nb2O5 thin films is 2.25, with a ratio of 112.58 to 44.22 Å3, as listed in Table 2. All stress ratios of the SiO2 to Nb2O5 thin films sputtered on different substrates are shown in Table 3 for residual, thermal, and intrinsic stresses, respectively. However, the stress mechanism of the BK-7 glass indicates that the intrinsic stress is directly proportional to the D factor (unit cell ratio), and the ratio size is about 2.75, which is close to 2.25. Apparently, the BK-7 glass and oxide thin film comprise a highly elastic module, thus, the deposition defect comes mainly from molecule accumulation and boundary disorder to induce the dominator of intrinsic stress. On the contrary, the stress mechanism of the PET and PC flexible substrates show that the residual stresses are in direct ratio
Fig. 11. A polynomial fitting curve for intrinsic stress simulation with SiO2 and Nb2O5 thin films sputtered on different substrates.
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Acknowledgement
Table 4 A polynomial fitting equation and error ratio of intrinsic stress for SiO2 and Nb2O5 thin films sputtered on different substrates with FEM. Thin film
A polynomial fitting equation
Error (%)
BK7/SiO2 PET/SiO2
σ = (0.054017)X + (−21.606) σ = (−1.2471e-11)X3 + (2.4385e-8)X2 + (0.041085) X + (−16.436) σ = (−4.2582e-12)X3 + (1.3049e-8)X2 + (0.041073) X + (−16.43) σ = (4.663e-9)X2 + (0.45966)X + (−183.87) σ = (4.9611e-22)X6 + (6.5559e-17)X5 + (−1.6641e13)X4 +(−6.8136e-9)X3 + (9.0913e-6)X2 + (0.35074) X + (−141.31) σ = (8.4685e-23)X6 + (1.4601e-17)X5 + (−3.9403e14)X4 +(−2.7315e-9)X3 + (3.9263e-6)X2 + (0.3565) X + (−143.05)
0.01% 0.02%
PC/SiO2 BK7/Nb2O5 PET/Nb2O5
PC/Nb2O5
The authors would like to thank the Ministry of Science and Technology of Taiwan for financially supporting this research under contract No. MOST 105-2221-E-224-029- and MOST 106-2813-C-224002-E. References
0.01% 0.07% 1.32%
[1] G. Gore, On the Properties of Electro-deposited Antimony, 1st ed., Trans. Roy. Soc, London, 1858. [2] G.G. Stoney, The tension of metallic films deposited by electrolysis, Proc. Roy. Soc. A 82 (1909) 172–175. [3] F.M. D'Heurle, Aluminum films deposited by rf sputtering, Metall. Trans. A. 1 (1970) 725–732. [4] D.W. Hoffman, J.A. Thornton, Effects of substrate orientation and rotation on internal stresses in sputtered metal films, J. Vac. Sci. Technol. 16 (2) (1979) 134–137. [5] H.K. Puller, J.M. Maser, The origin of mechanical stress in vacuum deposited MgF2 and ZnS films, Thin Solid Films 59 (1979) 65–76. [6] K.H. Muller, Stress and microstructure of sputter-deposited thin films: molecular dynamics investigations, J. Appl. Phys. 62 (1987) 1796–1799. [7] F.M. D'Heurle, J.M. Harper, Note on the origin of intrinsic stresses in films deposited via evaporation and sputtering, Thin Solid Films 171 (1989) 81–88. [8] H.C. Chen, C.C. Lee, C.C. Jaing, M.H. Shiao, C.J. Lu, F.S. Shieu, Effects of temperature on columnar microstructure and recrystallization of TiO2 film produced by ion-assisted deposition, Appl. Opt. 45 (9) (2006) 1979–1984. [9] J.Y. Robic, H. Leplan, Y. Paulean, B. Rafin, Residual stress in silicon dioxide thin films produced by ion-assisted deposition, Thin Solid Films 389 (1996) 34–39. [10] H.C. Chen, K.S. Lee, C.C. Lee, Annealing dependence of residual stress and optical properties of TiO2 thin film deposited by different deposition methods, Appl. Opt. 47 (13) (2008) 284–287. [11] M. Ohring, Materials Science of Thin Films, 2nd ed., Academic Press, 2001. [12] S. Ben Amor, G. Baud, J.P. Besse, M. Jacquet, Elaboration and characterization of titania coatings, Thin Solid Films 293 (1997) 163–169. [13] H.A. Macleod, Characteristics of thin-film dielectric material, Thin-Film Optical Filters, 3rd ed., Institute of Physics Publishing Press, London, 2001, pp. 621–627. [14] M. Grätzel, Ultrafast colour displays, Nature 409 (2001) 575–576. [15] D.R. Cairns, G.P. Crawford, Electromechanical properties of transparent conducting substrates for flexible electronic displays, Proc. IEEE 93 (8) (2005) 1451–1458. [16] T. Minami, Transparent conducting oxide semiconductors for transparent electrodes, Semicond. Sci. Technol. 20 (2005) S35–S44. [17] G.K. Mor, O.K. Varghese, M. Paulose, K. Shankar, C.A. Grimes, A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications, Sol. Energy Mater. Sol. Cells 90 (2006) 2011–2075. [18] A. Subrahmanyam, C.S. Kumar, K.M. Karuppasamy, Anote on fast protonic solid state electrochromic device: NiOx/Ta2O5/WO3−x, Sol. Energy Mater. Sol. Cells 91 (2007) 62–66. [19] X.F. Zhu, B. Zheng, J. Gao, G.P. Zhang, Evaluation of the crack-initiation strain of a Cu-Ni multilayer on a flexible substrate, Scr. Mater. 60 (2009) 178–181. [20] F. Pourahmadi, P. Barth, K. Petersen, Modeling of thermal and mechanical stresses in silicon microstructures, Sensors Actuators A Phys. 23 (1–3) (1990) 850–855. [21] A. Steegen, K. Maex, Silicide-induced stress in Si: origin and consequences for MOS technologies, Mater. Sci. Eng. R 38 (1) (2002) 1–53. [22] J.A. Martins, L.P. Cardoso, J.A. Fraymann, S.T. Button, Analyses of residual stresses on stamped valves by X-ray diffraction and finite elements method, J. Mater. Process. Technol. 179 (2006) 30–35. [23] R. Roos, G. Kress, M. Barbezat, P. Ermanni, Enhanced model for interlaminar normal stress in singly curved laminates, Compos. Struct. 80 (2007) 327–333. [24] A.O. Cifuentes, I.A. Shareef, Some modeling issues on the finite element computation of thermal stresses in metal lines, ASME J. Electron. Packag. 115 (1993) 392–403. [25] M.K. Lin, A.O. Abatan, C.A. Rogers, Application of commercial finite element codes for the analysis of induced strain-actuated structures, J. Intell. Mater. Syst. Struct. 5 (1994) 869–875. [26] M. Toparli, F. Sen, O. Culha, E. Celik, Thermal stress analysis of HVOF sprayed WC–Co/NiAl multilayer coatings on stainless steel substrate using finite element methods, J. Mater. Process. Technol. 190 (2007) 26–32. [27] L.B. Freund, J.A. Floro, E. Chason, Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations, Appl. Phys. Lett. 74 (14) (1999) 3–5. [28] Y. Zhang, Y.P. Zhao, Applicability range of Stoney's formula and modified formulas for a film/substrate bilayer, J. Appl. Physiol. 99 (5) (2006) (053513-1~7). [29] K.S. Lee, C.J. Tang, H.C. Chen, C.C. Lee, Measurement of stress in aluminum film coated on a flexible substrate by the shadow moiré method, Appl. Opt. 47 (13) (2008) 315–318. [30] R.W. Hoffman, H.S. Story, Stress annealing in copper films, J. Appl. Physiol. 27 (2) (1956) (193–193). [31] B.A. Movchan, A.V. Demchishin, Study of the structures and properties of thick vacuum condensates of nickel, titanium, tungsten, aluminum oxide and zirconium
0.10%
Fig. 12. The intrinsic stresses for SiO2 and Nb2O5 thin films sputtered on different substrates by a polynomial fitting.
reduced to below 2%. The intrinsic stresses of SiO2 and Nb2O5 thin films sputtered on different substrates by a polynomial fitting are shown in Fig. 12. 4. Conclusions The stress mechanism of SiO2 and Nb2O5 thin films sputtered on BK7 glass, PET, and PC flexible substrates were investigated by FEM. The residual stress was a combination of thermal stress and intrinsic stress, and without external force used during the sputtering process. The thermal and residual stresses could be calculated by numerical simulation of FEM. The residual stress size of SiO2 films was about twice that of Nb2O5 film; the stress size of PC substrate also was twice that of the PET substrate. In the same flexible substrate, the thermal stress of SiO2 was 1.3 times that of Nb2O5. However, the maximum residual stress was in the BK7/SiO2 thin film and the minimum was in the PET/Nb2O5 thin film. In the BK-7 glass substrate, the ratio of intrinsic stress was close to the D factor of SiO2 and Nb2O5 thin film, and the error was about 3%. While in the PET and PC flexible substrates, the ratio of residual stress was close to the ratio of the D factor of the SiO2 and Nb2O5 thin films, and the error was about 6%. The fitting curved equations were used to simulate the intrinsic stress by FEM with ERT technology, which could potentially reduce the error to less than 2%.
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chemical behavior, Catal. Today 8 (1) (1990) 27–36. [38] M.T. Dove, M.S. Craig, D.A. Keen, W.G. Marshall, S.A.T. Trachenko, M.G. Tucker, Crystal structure of the high-pressure monoclinic phase-II of cristobalite, SiO2, Mineral. Mag. 64 (3) (2000) 569–576. [39] P. Hariharan, B.F. Oreb, T. Eiju, Digital phase-shifting interferometry: a simple error- compensating phase calculation algorithm, Appl. Opt. 26 (1987) 2504–2506. [40] C.L. Tien, C.C. Lee, C.C. Jiang, The measurement of thin film stress using phase shifting interferometry, J. Mod. Opt. 47 (2000) 839–849. [41] K.T. Huang, H.C. Chen, Automatic measurement and stress analysis of ITO/PET flexible substrate by shadow moiré interferometer with phase-shifting interferometry, J. Disp. Technol. 10 (7) (2014) 609–614. [42] H.C. Chen, P.W. Cheng, K.T. Huang, Biaxial stress and optoelectronic properties of Al-doped ZnO thin films deposited on flexible substrates by radio frequency magnetron sputtering, Appl. Opt. 56 (4) (2016) c163–c167.
dioxide, Phys. Metals Metallogr. 28 (1969) 83–90. [32] T.R. Chandrupatla, A.D. Belegundu, Introduction to Finite Elements in Engineering, 3rd ed., Prentice Hall, 2002. [33] Y. Li, J.Q. Xu, L.D. Fu, Fundamental solution for bonded materials with a free surface parallel to the interface. Part I: solution of concentrated forces acting at the inside of the material with a free surface, Int. J. Solids Struct. 41 (2004) 7075–7089. [34] T.C. Chen, G.B. Lee, Determination of stress-optical and thermal-optical coefficients of Nb2O5 thin film material, J. Appl. Physiol. 101 (4) (2007) (063506-1~6). [35] M.N.G. Nejhad, C.L. Pan, H.W. Feng, Intrinsic strain modeling and residual stress analysis for thin-film processing of layered structures, J. Electron. Packag. 125 (1) (2003) 4–17. [36] H. Windischmann, An intrinsic stress scaling law for polycrystalline thin films prepared by ion beam sputtering, J. Appl. Physiol. 62 (5) (1987) 1800–1807. [37] E.I. Ko, J.G. Weissman, Structures of niobium pentoxide and their implications on
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