## 1. Introduction

In modern fiber-optic communications and computer data communications, there is an increasing demand for highly integrated optical components and modules due to their potential advancements. However, as a modern, highly mature technology platform of integrated optical components, the silica-on-silicon (SOS) waveguide-based planar lightwave circuit (PLC) technology is promoting wide applications of the functional components formed by the micro/nanoscale waveguide structures in SiO

_{2} thin films [

1]. Among these functional components, optical power splitters, arrayed waveguide grating (AWG), variable optical attenuators (VOAs), and directional coupler (DC)-based optical switches and other passive components are playing desirable roles in increasing the data capacity and data delivery speed [

2,

3].

The well-known PLC technology was developed successfully based on the extensions and improvements of micro/nanoscale integrated microelectronic circuit technology with modern highly uniform processing technologies of thick silica layers, including the coating technology and the dry etching technology of thick uniform SiO

_{2} films [

4]. For the former, the plasma-enhanced chemical vapor deposition (PECVD) has been proliferated, and for the latter, the reactive ion etching (RIE) has been widely used for the ultra-deep etching of various thick films of semiconductors and insulators, and the inductively coupled plasma (ICP) is the upgraded tool of the plasma etching technique. The integrated optical technology of light signal channels is fabricated by a complicated yet successful procedure, including thin film deposition, diffusion, epitaxy, photolithography, etching, and annealing, etc. on a silicon substrate. Thus, these basic components are fabricated and interconnected by wires [

5]. In fact, in the silica waveguide components, the integrated circuit technology has strict requirements for thickness, refractive index, and residual stress of all the formed films. For the PLC technology, the residual stress of the optical device needs to be critically controlled; otherwise, the functional components of the light wave signal cannot be realized and even the high yield production of PLC chip-based products cannot be realized [

6,

7]. The commercial effect of PLC components requires high yields in both the R&D and mass-production processes, so the stable optical performance and product yield are two important issues to be addressed [

8].

Stress is a common phenomenon in the preparation process of the film. Wu and Lagally used scanning tunneling electron microscopy (STEM) to observe the coverage of Ge doping on Si wafers and improved the intrinsic surface stress anisotropy by increasing the coverage of Ge [

9]. Chen and Wolf analyzed the damage caused by the back grinding of the silicon wafer and measured the cross-section of the background wafer by Raman spectroscopy; it was found that the stress was diminished after 5 μm dry etching [

10]. Yao et al. studied a new method to improve the stress-induced distortion in Si substrates, which utilizes a micro-patterned silicon oxide layer on the back side of the substrate [

11]. Chen, Zhang, and Lin found the generation and evolution processes of the intrinsic stress in the coating process by PECVD, and further developed the systematic thermal cycling method and procedure to realize the stress relaxation [

12]. About the stress generation and evolution process during the polycrystalline film, Chason et al. proposed the dynamic contacting theory based on the island coalescence of adjacent-layer grains and gave the convincible contacting theoretical models for the growth-caused stress, one traditional intrinsic stress source of films [

13]. In our previous work, we found the structural stress of a multilayer coated thick SiO

_{2} film on a large wafer. The structural stress is originally formed by a deformation of the large multilayered wafer caused by the other two traditional stresses, the thermal stress and the growth-caused stress, but it is an independent stress and has a nonuniform distribution on the wafer from the center to the edge [

14]. This progress has successfully explained why there is always a big discrepancy between the measured residual stress and the theoretically calculated sum of the traditional thermal stress and growth-caused stress defined by Chason. As some practical establishments were made recently in solving the vital problems of SiO

_{2} films, including stress relaxation and high fabrication quality, the SiO

_{2} waveguide-based PLC devices have been attracting broad interest in research and development [

15,

16,

17]. In fact, the existence of film stress has an impact on the birefringence properties of the waveguides and further leads to the influence on the optical loss and polarization dependence of PLC devices [

3,

4]. Although the stress distribution on a large wafer is very important in promoting the PLC devices commercially adopted in the applications [

7,

8,

18,

19,

20], not much research focusing on the stress distribution on SiO

_{2} films has been reported so far.

The aims of this work were to study the stress distribution of thick SiO_{2} films on a large silicon wafer and the stress distribution dependence on the coating technique and conditions and find the intrinsic relationship between stress distribution and the technique conditions. Therefore, in this article, some numerical calculations were performed to investigate theoretically the thermal stress, growth stress, and structural stress. Moreover, 6-μm-thick SiO_{2} films were deposited on a 625-μm-thick 6-inch silicon wafer with PECVD technology and then the dependence of the stress distribution on gas flow rate and the doping dose were experimentally studied.

## 2. Theoretical Model

The silicon planar optical waveguide device chip is composed of a series of micro-scale waveguide channels that are generally formed by multilayer coatings, photolithographic operations, and annealing processes, so the stress components in the x-, y-, and z-directions in each film are different from one another [

7]. It is academically called non-equilibrium stress, and, thus, has different refractive index values for the transmission beams with different polarization states. Among them, if the beam is a transverse electromagnetic wave, the difference in refractive index between the vertically polarized light (S-polarization) and the horizontally polarized light (P-polarization) is a commonly known phenomenon of birefringence.

Figure 1 shows the vectors of stress fields imposing on the system of one buried rectangular waveguide channel fabricated with three layers of SiO

_{2} film where

Figure 1a presents a perspective view of the waveguide channel imposed by all the stress vectors and

Figure 1b is the cross-sectional view of a three-layer system, and all the stresses are from the aforementioned three stress sources.

In

Figure 1,

${\sigma}_{x}$,

${\sigma}_{y}$, and

${\sigma}_{z}$ are the components of the stress tensor in the x-, y-, and z- directions, respectively;

${\sigma}_{uc}$ represents the stress in the upper cladding layer; and

${\sigma}_{tt}$ is the thermal stress generated by the heater during heat conduction. Thus, the deposited film stress causes the entire birefringence of the film, which can define the refractive index change in the x- and y-directions caused by the film stress. Namely, for the transverse electric mode (TE-mode) and transverse magnetic mode (TM-mode), the stress will produce different changes in the film refractive index. The change values are defined by Equations (1) and (2) below [

21]:

where

${C}_{1}$ and

${C}_{2}$ are stress-optical constants (also called elastic constants), which are determined by Young’s modulus, Poisson’s ratio, and the stress tensor of the material.

${P}_{11}$ and

${P}_{12}$ are photoelastic tensor elements; then,

${C}_{1}$ and

${C}_{2}$ can be defined by Equations (3) and (4):

For silica glass materials, the two stress-optical constants are ${C}_{1}$ = 0.74 × 10^{−5} mm^{2}/kg and ${C}_{2}$ = 4.10 × 10^{−5} mm^{2}/kg.

In the SiO

_{2} optical waveguide devices, the waveguide structures are generally squared, and the thickness of the upper cladding layer and the pitch of the waveguide channels are much larger than the geometry of the waveguide core layer. Therefore, the orthogonal terms in the elements of the stress density tensor are equal, wherein the symmetric non-orthogonal terms are equal. In addition, the upper cladding layer and the core layer of the waveguide are formed by different deposition amounts but with the same deposition technique and the same procedure; therefore, they have almost the same Young’s modulus and Poisson’s ratio, that is,

${E}_{core}\approx {E}_{uc}=E$ and

${\nu}_{core}\approx {\nu}_{uc}=\nu $. The coefficient of thermal expansion (CTE) of the upper cladding layer, the core layer, and the substrate are

${\alpha}_{uc}$,

${\alpha}_{core}$, and

${\alpha}_{s}$, respectively. By referring to Huang’s theoretical model on the distributions and interactions of all the stress fields in the channel waveguide [

7], for a temperature difference

$\Delta T$, the thermal stresses in the x- and y-directions are defined by Equations (5) and (6), respectively:

Among them, determining the stress difference of the waveguide core layer between two directions is called biaxial stress, which can be obtained from the Equations (5) and (6):

At the same time, the stress in the z-direction is defined by Equation (8):

From the above relationship, the stress variation of the waveguide core layer in different directions can be obtained, and further results in the birefringence effect in the waveguide core layer. Therefore, generally, only Equation (7) is used for discussing the birefringence.

For the growth-caused stress of a SiO

_{2} film, Chason’s theoretical model is an effective combination of tensile stress and is expressed by [

15]:

with the definitions as

where

$\delta t$ is the time interval from the last sublayer to this grain-formed sublayer,

$\Delta \gamma $ is the energy difference between the surface and grain boundary,

$L$ is the grain size,

$\alpha $ is a constant depending on the deposition condition,

$\delta {\mu}_{s}$ is the increase in chemical potential of the surface because the film is not in equilibrium during deposition,

$\mathsf{\Omega}$ is the volume associated with adding an atom to the grain boundary,

$D$ is an effective diffusivity related to the rate of hopping from the surface into the triple junction,

$a$ is the normal size of the atom (same as the sublayer spacing),

$k$ is the Boltzmann constant,

$T$ is the absolute temperature,

$\overline{{E}_{i}}$ is the biaxial modulus of the i’th sublayer of film, and

${C}_{s}$ is a dimensionless concentration (fractional coverage) of mobile atoms on the surface that are free to make diffusive jumps into the triple junction.

For the structural stress of a SiO

_{2} film, our theoretical model is based on the interactions among all the layers and the interaction in any layer is dependent on the radius on the wafer defined by [

14]:

where

${E}_{f}$ is the elastic modulus of the finished film,

${E}_{j}$ is the elastic modulus of the j’th sublayer material,

${t}_{j}$ and

$t$ are the thickness of the j’th sublayer and the total thickness of the film configuration, respectively. At the condition of

${t}_{s}>>t$, we assume the individual sublayer to be unstrained with a relaxed length of

${d}_{i}$ after being deposited, the neutral thickness

${t}_{n}$ defined by:

where

${\nu}_{s}$ and

${\nu}_{i}$ are the Poisson ratios of the substrate and i’th sublayer of film, respectively. In Equation (15), 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

${\gamma}_{i}$ is a parameter for adjusting the relations between the strain and stress of all sublayers after the i’th sublayer is deposited, which is defined by

with the definition as that

${\beta}_{ij}$ is −1 for

$i>j$, 0 for

$i=j$, and 1 for

$i<j$, and the elastic modulus is also transferred to the effective modulus by the total widths of the i’th sublayer and bottom layer:

${w}_{i}$ and

${w}_{0}$ as

in Equations (13) and (14), 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 Equation (14) of the structural stress of film,

${\sigma}_{f,thry}$ is confirmed to be a function of

$r$.