為了縮小電子與光學元件的尺寸以及提高其效率，各元件上會鍍製各種不同之薄膜，而當薄膜鍍製於基材上後，由於薄膜與基材兩者材料之機械性質有所不同，不可避免的會產生相當之薄膜應力(Thin Film Stress)。元件之可靠度、穩定性與良率會直接受到薄膜應力之影響，因此精準估算薄膜應力至關重要。長期以來，1909年Stoney所推導之方程式常被用以計算薄膜應力。使用Stoney方程式時，僅需量測基材於鍍膜前後之表面曲率，再代入該方程式中即可求得薄膜應力。推導Stoney方程式時最重要之假設條件為所計算之薄膜應力必須為均勻且等雙軸向，然而鍍膜過程相當複雜，導致薄膜鍍製於基材後，薄膜應力幾乎不可能是均勻且等雙軸向的。因此，許多修正型之Stoney方程式被推導出來以計算非均勻薄膜應力，例如Riet所推導之修正型Stoney方程式(MSF-Riet)與Feng等人所推導之修正型Stoney方程式(MSF-Feng)。然而Stoney方程式、MSF-Riet與MSF-Feng係為理論推導所求得，它們的準確性皆尚未以實驗評估過。本論文利用加強曝光光彈理論(Enhanced Exposure Theory of Photoelasticity, EEToP)和菲佐(Fizeau)干涉儀分別量測PSM-1光彈材料鍍製不同厚度之二氧化矽(SiO2)薄膜的應力和曲率，且為了全域性地求得薄膜應力的分佈，本論文首次計算了基材各個方向之薄膜應力及其分量，再將由Stoney方程式與MSF-Riet計算之薄膜應力和以EEToP所求得之薄膜應力進行比較，評估了Stoney方程式與MSF-Riet之正確性和準確性。由比較的結果可得知，若僅在一個或兩個方向上使用單一曲率半徑將不足以適當地計算非均勻之薄膜應力，且會產生相當之偏差量。
為了量測試片之全域性曲率，本論文同時建立並研究了相干梯度感測(Coherent Gradient Sensing, CGS)技術，且利用一標準試片驗證了所建立之CGS量測系統之重複性與準確性。本論文亦利用MSF-Feng與CGS技術，成功求得基材於鍍膜後之不均勻的薄膜應力，並於實驗中探討試片內部由加工後產生之不同大小殘餘應力對鍍膜應力之影響，由實驗結果可知試片內之殘餘應力與薄膜應力間並無顯著之關聯性。而利用EEToP所計算之薄膜應力結果亦可評估MSF-Feng的準確性，由評估結果可知MSF-Feng與EEToP所計算之薄膜應力差異量相當大，故可知MSF-Feng尚未能完全準確計算非均勻之薄膜應力，仍有需要進一步推導新的修正型Stoney方程式。
本論文建立一實驗評估薄膜應力方程式準確性之方法，並同時建立一精確度與重複性高之CGS量測系統。預期透過本論文所建立之實驗評估方法與CGS量測系統，將可尋找出一可準確評估薄膜應力之方程式，如此即能嚴謹控管鍍膜產品品質並提升產品良率。

To reduce the size and/or enhance the effectiveness of electronic and optical components, a variety of thin films are coated on the components. Because of the difference of material properties of the substrate and thin film, certain level of thin film stress must exist in the thin film/substrate system. The reliability, stability, and yield rate of components are directly affected by the thin film stress. Therefore, accurate measurement of the thin film stress is essential. For over a century, the thin film stress has been determined by Stoney’s formula. The use of Stoney’s formula only requires the measurement of the radii of curvature of substrate before and after coating the thin film. However, in many practical applications, the assumptions of Stoney’s formula cannot be exactly followed, e.g. the state of stress must be uniform and equibiaxial. Hence, several modified Stoney’s formulas have been developed to determine the non-uniform thin film stress, e.g. modified Stoney’s formulas proposed by Riet (MSF-Riet) and Feng et al. (MSF-Feng). Nevertheless, Stoney’s formula, MSF-Riet and MSF-Feng have not been experimentally assessed. Therefore, in this dissertation, the stress and curvature of circular disks made of PSM-1 photoelastic material and coated with the silicon dioxide (SiO2) thin film of different thicknesses were measured by the enhanced exposure theory of photoelasticity (EEToP) and Fizeau interferometer, respectively. To thoroughly understand the distribution of thin film stress, for the first time, thin film stresses and their components were calculated in each direction of spcimen. After comparing thin film stresses determined by Stoney’s formula and MSF-Riet with EEToP, the correctness and accuracy of Stoney’s formula and MSF-Riet can be assessed. It was found that the use of a single radius of curvature in only one or two directions is not sufficiently appropriate to calculate the non-uniform thin film stress.
To measure the whole-field curvature of the specimen, coherent gradient sensing (CGS) technique was established and investigated in this dissertation. By using a standard specimen, the repeatability and accuracy of the CGS system were carefully checked. Besides, by adopting the MSF-Feng and CGS technique, the whole-field curvature of substrate can be measured to obtain non-uniform thin film stresses. The investigation of influence of residual stress produced by machining the specimen on the state of stress after coating was attempted. Because of the complexity of the coating process, no simple correlation between the residual stress and thin film stress after coating can be found. The accuracy of MSF-Feng was also evaluated by using the EEToP. An order of magnitude of difference of the results between MSF-Feng and EEToP was found. Therefore, it is clear that MSF-Feng is not sufficiently accurate to obtain the non-uniform thin film stress. Hence, further development of the new improved Stoney’s formula is still needed.
In this dissertation, an experimental method for evaluating the accuracy of thin film stress formula was established. In addition, the CGS measurement system with high accuracy and repeatability was established. It is hoped that the correct thin film stress formula will be developed by using the experimental evaluation method and CGS measurement system proposed in this dissertation so that the quality of the deposition could be rigorously controlled and the deposition yield rate could be improved.

CHAPTER 1. INTRODUCTION 1
1.1 Motivation for research 1
1.2 Literatures of Stoney’s formula 4
CHAPTER 2. THIN FILM GROWTH MECHANISMS 8
CHAPTER 3. THEORETICAL BACKGROUND 10
3.1 Thin Film Stress 10
3.1.1 Intrinsic Stress 11
3.1.2 Thermal stress 11
3.2 The biaxial stress state of thin film 15
3.3 The modified Stoney’s formulas 19
3.3.1 The modified Stoney’s formulas for silicon wafer 20
3.3.2 The modified Stoney’s formulas for anisotropic thin film stresses 20
3.3.3 The modified Stoney’s formula for flexible substrate 22
3.3.4 The modified Stoney’s formula for non-uniform curvatures of the substrate 23
3.4 Theory of Photoelasticity 25
3.5 Four-Step Phase Shifting Theory in Photoelasticity 27
3.6 Theory and methodology of coherent gradient sensing 30
3.7 Phase shifting technique of coherent gradient sensing 36
CHAPTER 4. THIN FILM STRESS MEASUREMENT TECHNIQUES 39
4.1 Cantilever method 39
4.2 Newton’s rings method 40
4.3 Optically levered laser technique 41
4.4 Laser interferometer 43
4.5 Coherent gradient sensing technique 44
4.6 X-ray diffraction technique 45
CHAPTER 5. EXPERIMENTAL SETUP AND TEST SPECIMENS 47
5.1 Test specimens 47
5.1.1 Test specimens of substrate 47
5.1.2 Thin film material 48
5.1.3 Concave spherical mirror 50
5.2 Experimental setup and equipment 50
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSIONS 57
6.1 Evaluation of correctness and accuracy of Stoney’s formula and MSF-Riet 58
6.1.1 Thin film stresses calculated by Stoney’s formula and MSF-Riet 58
6.1.1.1 Measurement of fringe pattern of PSM-1 disk by Fizeau interferometer 58
6.1.1.2 Positioning of PSM-1 disk in Fizeau interferometer 59
6.1.1.3 Measurement results of topography and radius of curvature of PSM-1 disks 60
6.1.1.4 The calculations of thin film stress by using Stoney’s formula 63
6.1.1.5 The calculations of thin film stress by using MSF-Riet 64
6.1.2 Measurement results of photoelasticity 66
6.1.2.1 Analysis of measurement results of photoelasticity 66
6.1.2.1.1 Methodology of enhanced exposure theory of photoelasticity (EEToP) 66
6.1.2.1.2 Measurement results of PSM-1 disks of EEToP 70
6.1.3 Evaluation of correlation of Stoney’s formula and MSF-Riet 71
6.1.3.1 Transformation of PSD of thin film of Stoney’s formula and MSF-Riet 71
6.1.3.2 Determination of equivalent thin film stress of EEToP 74
6.1.3.3 Comparisons between PSD-STONEY and PSD-ETFS-EEToP 75
6.1.3.4 Comparisons between PSD-Riet and PSD-ETFS-EEToP 76
6.2 Measurement accuracy of CGS technique 78
6.3 Full-field thin film stress measurement results 83
6.3.1 Measurement results of EEToP 83
6.3.2 Measurement results of CGS technique 86
6.3.3 Comparisons between whole-field thin film stress obtained by MSF-Feng and EEToP 89
CHAPTER 7. CONCLUSIONS 91
BIBLIOGRAPHY 96
ACKNOLEDGMENTS 110
Appendix A. LIST OF ABBREVIATIONS AND ACRONYMS 228
Appendix B. LIST OF SYMBOLS 230

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