Investigation for the structural stress of SiO2 thin films and its distribution on the large-wafer created by plasma enhanced chemical vapor deposition

For a multilayered configuration of SiO2 film created by plasma enhanced chemical vapor deposition (PECVD), the thermal stress and growth-caused stress are two intrinsic stresses. In this work, based on the interactions of all the layers of film, a nonlinearly distributed structural stress over a large substrate is found. The numerical simulations for the nonlinear distribution of the structural stress and the uniform distributions of the two intrinsic stresses are carried out. As a result, the tensile structural stress decreases by ∼4x105MPa from center to edge of a 6” silicon wafer and the compressive growth-caused stress increases by ∼5x105MPa corresponding to the growth-rate increase of 40nm/s, which matches a ∼120MPa distribution of residual compressive stress obtained with in-situ measurements of film samples. In simulations, it is also discovered that the initial curvature of substrate has an impressive influence on the later grown film.For a multilayered configuration of SiO2 film created by plasma enhanced chemical vapor deposition (PECVD), the thermal stress and growth-caused stress are two intrinsic stresses. In this work, based on the interactions of all the layers of film, a nonlinearly distributed structural stress over a large substrate is found. The numerical simulations for the nonlinear distribution of the structural stress and the uniform distributions of the two intrinsic stresses are carried out. As a result, the tensile structural stress decreases by ∼4x105MPa from center to edge of a 6” silicon wafer and the compressive growth-caused stress increases by ∼5x105MPa corresponding to the growth-rate increase of 40nm/s, which matches a ∼120MPa distribution of residual compressive stress obtained with in-situ measurements of film samples. In simulations, it is also discovered that the initial curvature of substrate has an impressive influence on the later grown film.

For a multilayered configuration of SiO 2 film created by plasma enhanced chemical vapor deposition (PECVD), the thermal stress and growth-caused stress are two intrinsic stresses. In this work, based on the interactions of all the layers of film, a nonlinearly distributed structural stress over a large substrate is found. The numerical simulations for the nonlinear distribution of the structural stress and the uniform distributions of the two intrinsic stresses are carried out. As a result, the tensile structural stress decreases by ∼4x10 5 MPa from center to edge of a 6" silicon wafer and the compressive growth-caused stress increases by ∼5x10 5 MPa corresponding to the growth-rate increase of 40nm/s, which matches a ∼120MPa distribution of residual compressive stress obtained with in-situ measurements of film samples. In simulations, it is also discovered that the initial curvature of substrate has an impressive influence on the later grown film. © 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5045516

I. INTRODUCTION
As a mature technical platform, silicon dioxide (SiO 2 ) waveguide based planar lightwave circuit (PLC) technology is driving the wide adoptions and applications of functional components. 1,2 In research and development of the PLC components, the uniformity and repeatability of refractive index and absorption of a light wave in dielectric thin films are of the important interests. [3][4][5][6][7][8][9] In these works, Townsend et al. first performed research on the strain conservation of a multilayered configuration of film and gave a method to modeling stress mechanism; 3 Hsueh started research with the elastic deformation of thin-film to model the asymmetric stress at two orthogonal directions, and then considered the bending caused strains of thin-film; 4 Zhao et al. developed the strain compatibility and the force conservative condition between any two layers in the successive depositions to control the birefringence in optical waveguides; 5 Peralta et al. investigated the depositing conditions of SiO 2 films to compress the birefringence of waveguides; 6 Huang and Yan performed the systematic studies of all possible stresses around waveguide channels; 7 and Heise and Narevich investigated the birefringence on the polarization dependent loss (PDL) of silica-waveguide Mach-Zehnder interferometer (MZI) typed optical switches; 8 Zheng et al. improved the uniformity of SiO 2 waveguides by introducing a silica ridge in a high vacuum gap, 9 and we ever investigated the performance dependences of the silicawaveguide MZI-typed optical switches on the physical parameters determined by manufacturing and then demonstrated a method to controlling the manufacturing quality in our previous work. 10 a ) sundg@cust.edu.cn We know, when evaluating the maturity of optical waveguide devices and techniques, in addition to the central performance specifications, there is also another factor presenting both the physical property and application potential of PLC components -the stress distribution around waveguides because it plays an important role in reducing the R&D time and yielding a high success-rate, resulting in the broad applications and proliferation of the PLC based optical products in industry. 9,10 So, in this work, the dependence of the structural stress on the position of wafer is taken as a key cutting point for studying the stress distribution of SiO 2 films.
Research on the physical properties of SiO 2 films produced by plasma-enhanced chemical vapor deposition (PECVD) was started around the begin of 1980s, and the some significant achievements have led to the common conclusion that the depositing conditions of machine play a dominant role in determining the stress properties. [11][12][13][14][15][16] In 1986 Kobeta and Irene first measured the stress distribution of the PECVD-grown SiO 2 films, 11 then in 1991 Windischenmann et al. proposed the flame hydrolysis deposition (FHD) to tailor the over cladding, 12 further in 1996 Carlotti et al. compared the stress properties among the different PECVD depositing conditions, 13 and Chen et al. investigated the thermal process effects on the stress properties. 14 Of the forming mechanisms of film stresses, Hertz first proposed an early contact theory of isotropic elastic spheres of growing dielectric films in 1985 for the case having no frictional interaction. 15 Then, around the begin of this century, Freund and Chason extended this contact theory to the case of elastic solids. 16 In this theory, the deposition of thin film is started with an array of hemispherical grains, then an island having an interaction of two forces to coalesce a thin film, in which the interaction of two forces leads to a stress. In 2012, Chason et al. proposed a kinetic theory to improve their contact theory of the growth-caused stress of films. 17 Then, the tensile stress gradually relaxes and the compressive stress is generated in a triple junction formed by the surface of the last deposited sublayer and the two grain boundaries on this surface. 18,19 In summary, all the above studies are attributed to the two intrinsic stresses that are created during the deposition process and the temperature change process between the inside and outside of vapor chamber, however, these two sources cannot cover all the stresses. So, this work is devoted to investigating another stress source, the structural stress.
In this work, we comprehensively investigate the interactions among multiple layers and substrate to model the structural stress distribution of SiO 2 film over a large substrate in Sec. II. Further, in Sec. III, we carry out the numerical simulations for the distribution of structural stress on the large Si wafer. In Sec. IV, we measure the stress distributions of several SiO 2 thin films deposited on a 6" Si wafer, then discuss the discrepancies between the experimental results and the numerical simulation values in amplitude and further prove a potential of growth-caused stress. Thereby, in Sec. V, we simulate the dependence of growth-caused stress on growth-rate and further analyze the balancing process of these stresses. Finally, in Sec. VI, we give the conclusions.

II. MODELS FOR ALL THE STRESSES OF THIN FILMS
A. Theoretical models for defining the structural stress and its distribution Figure 1 schematically shows the dual forming processes of structural stress and growth-caused stress of a multilayered SiO 2 film deposited on a thick substrate: Fig. 1(a) is a film shape where the structural stress is caused by the interactions among all the layers and substrate and Fig. 1(b) is a schematic forming process of the growth-caused stress with the film evolution from the isolate grains to a continuous film in a triple junction between two grains and a surface of the last sublayer.
In a multilayered configuration of thin film as shown in Fig. 1, as the growth-caused stress is formed during deposition, the interactions among all the layers and substrate also generate a new stress -the structural stress. Of difference from the growth-caused stress is that the structural stress is mainly dependent on the curvature of multilayered film configuration and related to r, consequently changing the distribution of the final residual stress of the film, while the growth-caused stress depends on the deposition conditions such as the substrate temperature, the gas flow/pressure, and the rf power. In terms of Townsend's theoretical model, we define the structural stress as 3 Where E f is the elastic modulus of the finished film, E j is the elastic modulus of the jth sublayer material, t j and t are the thickness of the jth sublayer and the total thickness of the film configuration, respectively. At the condition of t s >> t, we assume the individual sublayers to be unstrained with relaxed length of d i after the ith sublayer is deposited, if ν s and ν i are Poisson ratios of the substrate and the ith sublayer of film, respectively, the corresponding thickness at this situation is defined as the neutral thickness t n defined by: In Eq. (2), the curvature K f should be variable with radius r, so it is theoretically determined by the multilayered configuration with the elastic modulus of each sublayer as Where γ i is a parameter for adjusting the relations between the strain and stress of all sublayers after the ith sublayer is deposited, which is defined by With the definition as that β ij is -1 for i > j, 0 for i = j, and 1 for i < j. In this model of defining the stress, the elastic modulus is also transferred to the effective modulus by the total widths of the ith sublayer and bottom layer: w i and w 0 as In Eqs. (3) and (4), both d i and E * i are the functions of radius r, so the key parameters t n and K f are certainly the functions of r. Therefore, from the definition model Eq. (1) of the structural stress of film, σ f ,thry is confirmed to be a function of r.

B. Discussions for the two intrinsic stresses
As the first intrinsic stress of films, the thermal stress has a common equation as 20 Where ∆α = α f − α s , α s and α f are the coefficient of thermal extension (CTEs) of substrate and film, respectively, and ∆T is the temperature difference between the depositing and room conditions. To control the thermal stress of thin films, most of research works are basically focused on two methods: (i) lowering the temperature difference ∆T and (ii) relaxing both the biaxial modulus of film E f and the CTE difference ∆α with an appropriate annealing process.
As the second intrinsic stress of a thin film, the growth-caused stress has also been attracting much research and typically the prestigious establishment is the kinetic model that combines the contact theory of films and the equilibrium of surface energy in film growing process. In terms of the Chason's kinetic model, after the deposition of the ith sublayer of film, the growth-caused stress is an effective combination of tensile stress and compressive stress defined by: With the definitions as and Where δt is the time interval from the last sublayer to this grain-formed sublayer, ∆γ the energy difference between surface and grain boundary, L the grain size, α a constant depending on the deposition condition, δ µ s the increase in chemical potential of the surface because the film is not in equilibrium during deposition, Ω the volume associated with adding an atom to the grain boundary, D an effective diffusivity related to the rate of hopping from the surface into the triple junction, a the normal size of the atom (same as the sublayer spacing), k the Boltzmann constant, T the absolute temperature, E i the biaxial modulus of the ith sublayer of film, and C s a dimensionless concentration (fractional coverage) of mobile atoms on the surface that are free to make diffusive jumps into the triple junction. Finally, the average growth-caused stress of films can be calculated by: In accordance with our comparing studies of the contact theory and kinetic evolution theory developed by Freund and Chason, 16,17 we set a further understanding as that the growth-caused stress is based on a new conservation of the stress caused forces of each deposited grain, whereas the thermal stress is from a new conservation of the stress caused forces of whole wafer. We know, the deposition parameters used in the set of equations (7a)-(7e), ∆γ, L, δt and α are dependent of the depositing conditions: the temperature, mixed ratio of gases, pressure and radio frequency (rf) power, but they are independent of position r. To the contrary, β is dependent on r because the effective elastic modulus E * i defined by Eq. (5) is the function of r, and further is τ too. Accordingly, in our simulations, if the structural stress of a film is much lower than the two intrinsic stresses, the effect of the film curvature on E * i can be ignorable, otherwise a modulation effect onto the elastic parameters of film needs to be introduced into a new formula as

A. Simulations for the structural stress distribution
At the outset a single layer film is set, then for the structural stress defined by Eqs. (1)-(3), the physical parameters related to the elastic stress such as the Young modulus and CTE are not only dependent on the measured position (namely, the distance of measured position from the center, r), but also affected by the other layers and substrate. In accordance with Eq. (3), the starting layer -substrate curvature K s has an impact upon the final curvature value, so it undoubtedly has an impact upon the structural stress. For a multilayered configuration of films shown in Fig. 1, the 3.0 and 6.0µm thick SiO 2 films are produced by PECVD technique on a 6" diameter wafer having a thickness of 625µm, then by setting three K s values: 1.0, 10.0 and 100.0m −1 , the dependences of relative curvature K f /K s of film on r are numerically simulated as depicted in Figs. 2(a) and 2(b), respectively.
Note from Fig. 2 that when the distance r is from 5mm to 65mm, the K f /K s of both either 3.0µm or 6.0µm film almost stays on the constants for the first two K s values: 1.0 and 10.0m −1 , while it is a nonlinear increase for the K s value of 100.0m −1 . Thus, we conclude that the curvature of substrate has a paramount important influence on the structural stress distribution, the much lower K s value is preferred to compress the curvature of film. So, by selecting K s = 1.0m −1 , with Eqs. significant finding from Fig. 3 is that the structural stresses are always positive for these two film thickness values, namely, indicating the tensile stresses that slowly decreases with the film thickness. We also notice from Fig. 3(a) that, for a 3.0µm thick film, the structural stress nonlinearly decreases by 4 × 10 5 MPa when r is from 5mm to 70mm. in contrast to the 3.0µm thick film, for a 6.0µm thick film, the structural stress also nonlinearly increases by 4 × 10 5 MPa, too. In order to comprehensively investigate the dependences of the film structural stress on film thickness and measured position on the wafer, we select a wide range of film thickness: 3-18µm to simulate the structural stress distribution and property of the film with these two parameters, consequently obtain a two-dimensional (2D) distribution of the structural stress as shown in Fig. 4. Note that the structural stress implicitly expresses the distribution trend of the SiO 2 film structural stress that quickly nonlinearly decreases with the measured position r and slowly nonlinearly decreases with the film thickness t.
In terms of the above derivations, simulations and discussions, we can conclude that the structural stress is an extra-stress apart from the two intrinsic stresses: the thermal stress and the growth-caused stress, and finally influences the total residual stress distribution on the whole wafer. In addition, we can notice in Eq. (6) that the CTE difference ∆α only depends on material, so the thermal stress is independent of the position of substrate, to the contrary, the growth-caused stress defined by Eq. (9) should have a variable distribution on the large substrate.

B. Simulations for the thermal stress and analysis
As analyzed in Sec. II, the interactions among all the sublayers and substrate can cause a nonlinear distribution of structural stress of 3-18µm SiO 2 thin film on a 6" Si wafer, but the absolute stress distribution from the center to the edge of wafer can reach the level of 4 × 10 5 MPa, namely, the absolute change of structural stress almost stay at the same level. Compared with the usual values of intrinsic thermal stress, the positive values of structural stress σ str are relatively very high. Then, we numerically calculate the distribution of the new biaxial elastic modulus E * f of 6µm thick SiO 2 thin film as depicted in Fig. 5. Note that the variation of biaxial elastic modulus caused by the multilayer interactions is too small to take into account, namely, it is only (E * f /E f − 1) ≈ 0.0013%. Thus, in the intrinsic thermal stress calculation, this effect is ignorable.
To study the effect of structural stress on the intrinsic thermal stress of thin films produced by PECVD, it is very necessary to simulate the influences of the new E * f on the thermal stress. In Eq. (6), for ∆α = α f − α s , a typical value is around 2.9 × 10 −6 (1/K) and ∆T that is used for depositing the films  is around 400 • C, then we obtain the numerical results for the ∆α range from −4.0 × 10 −6 K −1 to −2.0 × 10 −6 K −1 as shown in Fig. 6. Note that the amplitude of intrinsic thermal stress is not only uniformly distributed on a large Si wafer, but its amplitude is much lower than the corresponding numerical simulation results shown in Fig. 3. In addition, the theoretical model for defining the thermal stress of thin films Eq. (6) does not include the thickness of film, which immediately gives rise to a conclusion as that this thermal stress model is based on an average stress value of the film. For the illustrative case, at the condition of ∆α = 4.0 × 10 −6 (1/K) and ∆T = 400 o C in Fig. 6, the thermal stress value is only -120MPa. However, the structural stress is an extra-stress imposed on the sum of two intrinsic stresses: the growth-caused stress and the thermal stress. Finally, the sum of three stresses leads to the steady state stress σ ss . By investigating the values and properties of some physical and chemical parameters that are used in the set of equations from (7a) to (7e), we find that the growth-caused stress has its very complex dependences on the growth process and parameters. As mentioned above, the curvature effect of the whole film has little effect on the biaxial elastic modulus E f of film, so σ thm does not depend on the r of wafer. 17,21,22 The growth-caused stress σ grw and the structural stress σ str are intimately linked during the deposition, then the resulting residual stress further determines the central parameter α f of the thermal stress σ thm . During the deposition of the ith sublayer of film, its σ grw is best fitting to a kinetic competition model of different mechanism that not only depends on the growth rate, the diffusivity and the grain size, but it is also related to the absolute temperature T, then produces the direct effects on such the central physical parameters of the deposited film as E f and α f , which further affect σ str with the curvature K f and σ thm with the variations of both E f and α f . Consequently, the final steady state stress σ ss (i.e., the residual stress of film) is the algebraic sum of these three stresses as σ ss = σ str + σ thm + σ grw (10) All the above numerical simulations are performed by the Matlab Programs.

IV. EXPERIMENTAL MEASUREMENTS AND DISCUSSION
The nine points for stress distribution measurements are schematically shown in Fig. 7. In fact, as labeled with dashed line in Fig. 7, the nine points stand for nine circles having different radii with an equal adjacent radius difference of 15mm from the center to the edge of wafer, namely, point 5 is at the center, points 4 and 6 are at the radius of 15mm, points 3 and 7 at 30mm, points 2 and 8 at 45mm, and points 1 and 9 at 60mm, the significant outside circle. Then we measure the birefringence values on the nine selected positions on wafer by using the Metricon 2010 Prism Coupler and further calculate the stress distribution with the theoretical model relating to the birefringence and the stress of film. 23 As a practical work, at the depositing temperature of 400 • C we create three samples of 6µm thick SiO 2 thin films with respect to three different SiH 4 flux values: 217, 250 and 265sccm and then create the other two samples of 3µm thick SiO 2 thin films with respect to two different BPSG fluxes.
At the wavelength of 1539nm we measure the birefringence values of all the samples of SiO 2 thin films first, and then obtain the corresponding residual stress values of three 6µm thick SiO 2 films and two 3µm thick SiO 2 films as shown in Figs. 8(a) and 8(b), respectively. Note from Fig. 8(a) that all the stress values are minus, which stand for the compressive stresses of SiO 2 films, in which one SiH 4 flux value, 217sccm, obviously shows a compressive stress increase of more than 120MPa from the center to edge of wafer. When the SiH 4 flux increase to 250sccm, this variation gradually becomes weak, but the amplitude of the compressive stress is uniformly higher than the case of 217sccm. Note from Fig. 8(b) that two different BPSG fluxes have the same compressive stress increases of more than 120MPa from the center to the edge of wafer. Thereby, it is concluded from Fig. 8 that a trend of the final stress distribution from the center to the edge of wafer can reach more than 120MPa.
Obviously, the experimental results of compressive stress shown in Fig. 8   in Fig. 3 and the thermal stress shown in Fig. 6. In terms of the obvious differences between the experimental results and the numerical calculation results of (σ str + σ thm ), we can conclude, as predicted above, σ grw of SiO 2 films is not only the compressive stress at the order of 10 5 MPa, but it also has the similar distribution with σ str under the same deposition conditions.

V. ANALYSIS FOR THE DIFFERENCES BETWEEN THE NUMERICAL VALUES AND EXPERIMENTAL RESULTS
As mentioned above, the numerical simulations show that, when r is from 5mm to 70mm, the structural stress has the changes of about 4 × 10 5 MPa for both the 3.0 and 6.0µm SiO 2 films, the thermal stress is at the level of −120MPa and uniformly distributes on wafer, while the measured stress variance from the center to the edge of wafer is only about −100MPa. Consequently, it is concluded that only the growth-caused stress can form such big stress differences. With Eqs. (7a) -(7e) we obtain the simulation results of the growth-rate dependence of growth-caused stress with respect to three normalized atom sizes: 0.25, 0.50 and 0.75µm as shown in Fig. 9. Note that the growth-caused stress is −8.0 × 10 5 ∼ −2.0×10 5 MPa when the growth rate is from 10nm/s to 50nm/s, which can exactly balance the numerical values of the structural stress, leading to a conclusion that at the lower growth rate (e.g. <20nm/s), the lower the normalized size of atoms leads to the higher compressive  growth-caused stress. Accordingly, of the parameters ∆γ, L, α, δ µ s and D, some can be used to manipulate the distribution of σ grw .

VI. CONCLUSIONS
The conclusions in this work are that the nonlinear distribution of a structural stress over the wafer and the unbalance of the algebraic sum of the structural stress and the thermal stress with the experimental measurements of the final residual stress gives a convincible way to estimating the growth-caused stress.