1. Introduction

Almost all vacuum-deposited coatings are under stress [1]. Tensile and compressive stresses are two thin film stress states, consisting of intrinsic stress, thermal stress and surface tension. Residual stress has a complex impact on optical thin film devices, making the surface shape distortion, seriously impairing the optical wavefront. Excessive stress may even lead thin film devices to fail, which is very detrimental to optical system. Film stress research is crucial for increasing the precision and dependability of a devices. There is disagreement over the factors that contribute to the residual stress and the general trend in which each factor influences it [2].

Relevant studies on film-stress have been carried out both at home and abroad. Chun-Hway Hsueh used the plastic deformation approximation to theoretically explore the relationship between residual stresses and deformations of the film layers [3]. The effects of stress on the surface shape of an astronomical telescope mirror with an effective area of 2m² were simulated and investigated by BD Chalifoux et al [4]. HC Chen et al. investigated the impact of stress changes after annealing of TiO₂ films prepared by various techniques [5]. The impact of ion-beam-assistance on the stress of TiO₂/SiO₂ films was studied by Li Yu-Qiong et al [6]. Using the Kilcook model, Chen Tao highlighted the fact that the annealing effect during the film deposition process was the source of the intrinsic stress of the layer [7]. According to Fan Bin, the TiO₂/SiO₂ film layer on the BK7 substrate would change its stress state depending on the substrate's temperature, displaying the opposing stress scenarios [8]. Through the examination of infrared filters, Dong Mao-Jin et al. made the discovery of the thin-film edge stress effect [9]. Yang Li et al. pointed out that highly reflective dielectric multilayer films showed an increase in stress with the increase in film thickness [10]. Cheng Wang et al. investigated how heat treatment affects stress changes in films [11]. Kun Li et al. found that ZnS layers prepared with different evaporation rates possesses a compressive stress, but with the evaporation rate increased, the refractive index and the stresses of the film produced a change [12].The distribution of stress in a film needs to be further investigated since the state of residual stress in optical thin-film devices is intricate and has many affecting aspects, according to previous studies. This is because the layer's stress state is highly variable under different deposition techniques and parameters.

Titanium oxide film has a high refractive index, low absorption, and excellent physicochemical properties, which makes it a very important material in the visible/near-infrared region. EBE (electron beam evaporation) was used to deposit TiO₂ on H-K9L substrate with a diameter-to-thickness ratio of 10:1 and a thickness of t = 4 mm. The residual stress was calculated using the substrate deformation method. Based on the theory of thermal contraction effect and film stress from low-energy deposition methods, orthogonal experiments are used to investigate the effects of film thickness, annealing temperature, and annealing time on film stress from the optical properties, stress state, and crystalline structure of the film. This research has significant implications for the design of ultra-low surface-type precision thin film devices as well as low-stress optical thin film membrane systems.

2. Film stress

2.1 Theory of film stress distribution

Film stresses consist of thermal stresses, intrinsic stresses, and surface tension. Surface tension is small and generally negligible [1].

Thermal stresses is determined by the difference between the expansion coefficients of the film material and the substrate, and they can be described as follows for a single layer:

The intrinsic stress is closely related to the microstructure of the film, grain defects and processing parameters, etc. Hoffman et al. proposed a model that the intrinsic stress is related to the inter-grain elastic stresses generated during grain growth and merging, with an average value:

G.G. Stoney studied deeply the relationship between residual stress and deformation, and proposed a formula for calculating the residual stress of the layer [13]:

For a flat sample (Fig. 1), the vector height Power, radius of curvature R, and diameter D_s are satisfied:

 

Fig. 1. The relationship between Power, D_S, R and for a flat sample.

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Post-process to a film has a significant effect on stress. There have been many research and reports on the effect of heat-treatment on the film stress [14,–16]. However, since the heat-treatment process has many affecting factors and can not monitor the sample in a real-time, the effect of heat-treatment on optical properties, stress state and microstructure of the film is still in research. In conclusion, the influencing factors of film stress are complex, so previous reports vary greatly.

2.2 Methods for testing film stress

The primary methods for film stress measuring are: base deformation method, diffraction method, film vibration method, etc. Among them, the theoretical foundation of base deformation method is Stoney's formula, which is the most commonly used in testing film stress [1].

In Eq. (5), the Poisson's ratio E_S, the diameter D_S, and the thickness t_S are fixed once the substrate is determined. Young's modulus which varies very small with the temperature, is approximately constant. Therefore, the residual stress of a film can only be calculated if the film thickness t_f and the vector height difference ΔPower before and after coating have been established. The film thickness t_f can be calibrated by process experiment, and the vector height difference ΔPower before and after coating can be calculated by interferometer measurement.

3. Experiment designing

The TiO₂ film with different thicknesses were deposited by the same process on H-K9L material prepared by CDGM with diameter D_s = 40 mm and thickness t_s = 4 mm. The film thickness t_f was calibrated photometrically using LAMBDA900. The samples were heat-treated using the LAC atmosphere furnace made in Czech. The surface shape parameter is obtained by measuring the reflective wavefront using ZYGO interferometer. the ΔPower value is calculated using a self-developed program based on MATLAB software. By convention, a positive value of tensile stress shows a tendency for the surface to be concave inwards at the center, vice versa [17].

3.1 Film deposition process

Using a ZZSX-800M machine equipped with an INFICON IC5 quartz crystal controller, TiO₂ layers of different thicknesses were deposited using Ti₂O₃ material supplied by Umicore cop via EBE. The film deposition process parameters are: baking temperature 27°C, background vacuum 2.4 × 10⁻³ Pa, deposition rate 2A/s, process gas is O₂, and working vacuum is 3.4 × 10⁻² Pa (Fig. 2).

 

Fig. 2. The dispersion curve of TiO₂ film.

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3.2 Orthogonal experiment designing

It was noted in 1.1 that there are numerous factors influencing a film’s stress state, and that these factors interact with another one. The heat-treatment process, in particular, has a substantial effect on a film’s microstructure and alters film’s various properties. Using specification of the orthogonal table arrangement of multi-factors, the orthogonal experiment is a very efficient experimental design method to find the optimal level combinations [18], very suitable for this study.

It is essential to clarify the influencing factors and factor levels in orthogonal experiment. Process parameters are a necessary influencing factor to ensure the stability of the film’s optical characteristics among the factors discussed in 1.1. Considering the needs of the sample, such as weight reduction, mechanical strength, and mechanical accuracy, flat H-K9L with diameter-thickness ratio of 10:1 is chosen for the substrate. There are two optional categories of film thickness and annealing parameters remaining, so the physical thickness of the film, annealing temperature, and annealing time is selected in this research.

The film with optical thickness λ₀/4 is most commonly used in designing and preparing layer systems. According to the deposition parameters in 2.1, the physical thickness of TiO₂ film is maintained at 45.12nm∼142.96 nm when λ₀= 400nm∼1200 nm. Based on the actual layer system, the physical thickness factor levels of the layer were set as 30 nm, 80 nm, 130 nm, and 180 nm.

GJB 2485A-2019 requires optical films to be stable at a high temperature of 70 ± 2°C [19]. The annealing temperature is inevitably higher than 72°C if the layer satisfies the national military standard. According to the relevant literature, once the film anneals above 300°C, it will crystallize secondarily and grow the grain with the temperature rising, changing composition and optical properties of the film, increasing grain boundary stress, and even cracking the film, which will cause film failure [6]. To ensure the optical and other properties of the film, the annealing temperature should not exceed the deposition temperature of 270°C for TiO₂ films in this experiment. Therefore, the annealing temperature factor levels were set as 100°C, 150°C, 200°C, and 250°C;

At present there is no uniform standard for the level of the annealing time factor, taken as 0 min, 40 min, 80 min, 120 min.

So far, the number of orthogonal experiment factors and levels are determined, as shown in Table 1.

Table 1. Level table of orthogonal experimental factors

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3.3 Film microstructure testing

It is crucial to examine the microstructure and surface shape of the film under various annealing states because all the optical parameters, mechanical properties, and surface morphology are determined by them. AFM (atomic force microscope) is used to measure the film’s surface morphology.

4. Experimental results and analysis

4.1 Orthogonal experiment results

The experiment was carried out according to the orthogonal table for testing factors and levels shown in Table 1 in 2.2. The element’s Power values before and after deposited were measured, and ΔPower was computed. The residual stress was then calculate using Eq. (5). The experiment sequence, the ΔPower, and the related residual stress are shown in Table 2.

Table 2. Orthogonal experimental scheme and result^a

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4.2 Orthogonal experimental analysis

Based on the orthogonal experimental data in Table 2, the data was analyzed according to the method of assessing orthogonal experiment, and the results were shown in Table 3.

Table 3. Range analysis orthogonal experimental data

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The analysis of orthogonal experimental results shows that the film thickness is the main factor and the annealing time has the lowest influence affecting the residual stress of a film. According to the experiment factor level and experimental results analysis, the influence trend of each factor is drawn, as shown in Fig. 3.

 

Fig. 3. Range analysis of experimental factors and residual stress.

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As can be seen in Fig. 3(a): the residual stress decreases with the increase of the film thickness and reaches a minimum at about 80 nm, after which the residual stress increases. EBE is a low-energy deposition technique, the film stress state matches the thermal contraction effect model best. In the early stage of film deposition, there is a significant difference in the thermal expansion coefficient between the substrate and the film, thermal stress is relatively big as the main residual stress. With the film thickness increasing, the columnar structure with a certain degree of scalability can offset some of the residual stress, when the thickness of the film is about 80 nm, the residual stress of the sample reaches the minimum value. As the film deposition process continues, the film shows the intrinsic stress state of TiO₂ film, so the film layer stress increase.

As can be seen in Fig. 3(b), with the annealing temperature increasing, the residual stress of the film decreases gradually. Once the annealing temperature exceeds 200°C, the residual stress of the film increases rapidly. In the low-temperature annealing, the residual stress of the sample is released. When the annealing temperature exceeds 200°C, the TiO₂ film may slightly crystallize secondarily, and this tendency may not be noticed during the relevant measurement, but the anatase crystal phase gradually appeared. The two crystalline phases do not match with each other, as a result, the film stress increases quickly.

As can be seen in Fig. 3(c), during the annealing process, the residual stress of the film changes very little with annealing time, and when the annealing time is longer than 80 min, the residual stress of the film almost tends to stabilize. In the early stage of annealing, due to the difference between the glass substrate and the film in specific heat capacity and thermal expansion coefficient, there may be a small thermal gradient inside the sample, which causes the sample’s residual stress to increase slightly. As the annealing process continues, the sample reaches the thermal balance, and the effect of annealing time on residual stress is getting smaller and smaller, so after more than 80 min, the residual stress is almost unchanged.

4.3 Microstructure analysis

The surface morphology of the experimental samples 9^#, 10^#, 11^# and 12^# were measured using AFM and the results are shown in Fig. 4:

 

Fig. 4. Results of AFM testing

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It can be seen from the AFM measurement results of the samples under different annealing states that no regular crystalline mode appeared on the surface. Within the testing range, the peak-to-valley of the film’s surface profile ranged from 2.3 nm to 4.6 nm, the root-mean-square Sq deviation of the film’s surface profile is not greater than 0.7 nm, and the overall roughness of the film’s surface profile is small.

Surface roughness varies and tends to increase with residual stress in a manner similar to annealing temperature. The surface roughness initially diminishes, but it quickly increases if the annealing temperature exceeds 200°C. This phenomenon may be caused by the internal microstructure of the layer reorganizing during the initial stage of annealing, which reduces surface roughness. Since the secondary crystallization does not match the crystalline state of the original layer, the surface roughness increases when the annealing temperature is higher than 200°C.

It has been proved from another perspective that the post-treatment of low-temperature annealing is conducive to release residual stress in the sample. When the film is annealed under 200°C, the internal structure of the film is reorganized, but the film crystal structure does not change, so the residual stress and surface roughness decrease. When the annealing temperature is too high, it will make the film crystallize secondarily, due to the secondary crystallization and the original film’s crystalline state is not the same, it will make the internal stress and surface roughness increase.

5. Conclusion

EBE was used to deposit the TiO₂ film samples, which were then heat-treated and analyzed for spectra, surface morphology, and residual stresses. The effects of film layer thickness, annealing temperature, and annealing time on the film’s residual stress were investigated using the orthogonal experimental method based on the thermal contraction stress model of the film under low-energy deposition. Research indicate that the key factor influencing the TiO₂ film stress is film thickness, and that the annealing time has the least impact on the residual stress.