薄膜廣泛的應用於超大積體電路連接導線製程中，隨著元件的微小化、多層化、高密集度、高速運算、成本考量等的需求，又承受應力及電遷移效應等因素影響之下，如何提升電路運算速度是有待解決的課題，近年薄膜材料已逐漸導向銅取代鋁為主，對於奈米銅薄膜特性的研究尚未明朗，正是值得深入探究的課題，另則就傳統的薄膜機械性質仍以試誤方式進行反覆實驗，其步驟繁瑣費時不具經濟效益，而且拉伸過程的有效長度難以直接測量、切割薄膜時因內應力釋放造成褶皺、無法觀測材料內部暫態的破壞行為等尚存在技術的困難。就設計及製程的研發而言，薄膜材料機械性質是極為重要的考量因素，攸關整體結構的性能、可靠度及使用壽命，因此需要一種簡單、正確、快速的測試技術，所以本文將建構一套奈米銅薄膜拉伸理論及模擬系統，能有效掌握及預測薄膜機械性質參數，以探討應力行為、結構強度、差排演化的關係，如此可作為在設計研發及製程過程中，薄膜選材的重要參考依據，提升製程品質良率且縮短研發時程。
本文運用微觀的三維分子動力為理論基礎，以研究銅薄膜承受拉伸過程的應力行為與破壞機制，建構出一套完整的薄膜拉伸測試模型。首先銅薄膜原子遵循Morse勢能來描述，透過勢能對空間微分推導出作用力的分佈，運用Gear五階預測修正法來計算原子的運動軌跡，再使用截斷半徑配合Verlet鄰近表列進行演算，能有效率提升模擬速度，並且於平衡時需隨時監控系統原子的狀態，藉以判定系統是否趨於穩定後，即能執行結構拉伸的模擬，以取得機械性質及暫態行為進行分析後，更進一步，調控模擬系統的溫度、尺寸及應變率等因素，以研究結構參數對材料機械性質及破壞機制的影響，最後將模擬結果與文獻理論及實驗資料進行相互比較，以預估該理論分析與模擬系統的可行性及準確度，作為系統模型改善及修正的評估依據。
由本文所建構的奈米銅薄膜拉伸模擬，在不同系統溫度5K至1200K與應變率 下，若系統應變量為零時，發現存在的應力值不為零，此時預應力將提高了0.8 GPa，並討論系統溫度對應力曲線震盪的關係，此結果與相關文獻比對後，在不同材料亦有此現象發生，所以奈米尺度下尺寸及表面效應是不容忽略，此結果與巨觀薄膜材料有顯著不同的現象。又探討薄膜材料的降伏應力及應變、彈性模數及彈性能模數間呈現相依關係，分別由灰色理論進行擬合及預測後，將取得所述的四項關係式，所擬合及預測曲線的平均殘差為1~2 %，數據擬合的誤差極低，藉此能了解彈性能與材料韌性及延性的關係，最後就晶格旋轉、Lüders帶、多重滑移差排及能量演化等特性加以探討，如此充分改善以往僅能用理論及實驗探究物理現象，因此本文提供另一種研究奈米薄膜性質的測試方法，可作為薄膜在選材依據及設計準則，且有助未來在奈米機電系統中，作為薄膜強度的預估與設計參考，促使奈米薄膜運用於半導體產業上更為廣泛。

Thin film is widely applied to the process of connecting the conducting lines in ultra large integration circuit. With the demand of the component becoming slight, multi-layered, highly dense, high speed operating and the product cost, and the effects of stress-migration and electro-migration, how to promote operation speed is a pending topic to solve. In recent years, the filmic materials are gradually changed to copper to replace aluminum. Because the research result of the property of nano-scale copper thin film is not clear, it is an important topic worth exploring in depth. And in tradition, the experiments still used trial and error to test the film mechanical properties repeatedly, the steps took lots of time and did not have economic benefits, and the effective length was hard to measure directly in stretching, residual stress releasing caused folding when cutting film, it was unable to observe the transient destructive action inside the materials et cetera. For those reasons, there still exist technical difficulties. For research of designing and manufacturing, the mechanical property of its materials is an extremely important factor concerning the performance of the entire structure, reliability and working life. Therefore, we need a simple, correct and rapid testing technology. In this article, we will establish a stretching theory and simulating system about nano-scale copper thin film to evaluate and predict the parameter of thin film effectively for researching the relations of the stress, the strength of structure and the evolution of dislocation. In research of designing and manufacturing, it can be an important reference in choosing film material to increase product qualities and shorten researching time.
In this paper, submicroscopic method is employed to carry out three dimensional molecular dynamics theory and simulations of the mechanical properties of rectangular cross-section copper thin film. This proposed approach is utilized to elucidate the stress behavior, the deformation mechanism and the elastic-plastic fracture materials during the tension test of the nano-scale copper thin film. First, the copper atom should obey the Morse potential function, and we calculate it by using the potential to deduce spatial differentials. The Gear fifth order predictor-corrector method is adopted to calculate the positions of atoms, while the arithmetic method of local interactions by Velet's neighbor lists and cut-off are used to analyze the interactions among molecules in numerical calculation. That can promote the simulating speed efficiently and monitor the state of systemic atoms in equilibrium at any time. Then we can execute the simulation of structural tensile strength to get the mechanical properties and transient behavior in order to analyze. Furthermore, we can control the temperature, the scale and the strain-rate factors of simulating system to research how structural parameters affect the mechanical properties and failure mechanics. Finally, we compare the simulating results and references (or experimental datum) with each other to predict the practicability and accuracy of the theory and the simulating system to be the base of the model improvement and correction.
In this article, in the tensile test simulation of nano-scale copper thin film, we discovered in the state of different systemic temperature (5 K to 1200 K ) and strain-rate (2×108 sec-1). If the systemic strain value is zero, the existing stress is not zero. At this moment pre-stress will rise to 0.8 GPa. Then, we discuss the relationship between the systemic temperature and the stress curve quassation. Taking this result to compare with references, we can find this phenomenon also occurs in different materials. So in nano-scale, the size and the surface effect is not allowed to be neglected. This result shows different conspicuous phenomenon between nano-scale and large-scale. And then, we discuss the relations of the yield stress, the yield strain, the elasticity modulus and the resilience modulus. Then, gray theory is used to fit and predict the curve, we will discover the average residual error is 1 to 2 % between the fitting and predicting curves and we see the error is extremely small. By this cure, we can understand the relations of the esilience, the material toughness and the ductility. Finally, we discuss the crystal structure rotation, Lüders bands, the multiple glide dislocation and the energy behavior. The method can improve sufficiently the fact that the investigations of the physical phenomenon are only carried out by theories and experiments.
Therefore, this research proposed another method to test nano-scale copper thin film. This study of mechanical and fracture properties of thin film will be helpful to the strength prediction, the material choice and reference design of nano-electromechanical system. The method can also be used to provide a clear direction for further design studies.

1.Alpas, A.T., Embury, J.D., and Hardwick, D.A., “The Mechanical Properties of Laminated Microscale Composites of Al/Al2O3,” Journal of Materials Science, Vol. 25, pp. 1603-1609, 2004.
2.Arsenault, R.J., and Beeler, J.R., Computer Simulation in Material Science, ASM International, USA, 1988.
3.Ashurst, W.R., Yau, C., Carraro, C., Lee, C., Kluth, G.J., Howr, R.T., and Maboudian, R., “Alkene Based Monolayer Films as Anti-Stiction Coatings for Polysilicon MEMS,” Sensors and Actuators A: Physical, Vol. 91, pp. 239-248, 2001.
4.Aya, T., “Influence of Environmental Temperature on Yield Stress of Polymers,” JSME A, Vol. 40, pp. 343-348, 1997.
5.Badachhape, R.B., Margrave, J.L., and Brotzen, F.R., “Separation of Thin Aluminum Films from Silicon Substrates,” Thin Solid Films, Vol. 139, pp. L77-L78., 1986.
6.Baek, W.C., Ho, P.S., and Lee, J.G., “Stressmigration Studies on Dual Damascene Cu/Oxide and Cu/Low k Interconnects,” Materials Research Society, Vol. 741, pp. 249-255, 2004.
7.Beams, J.W., Structure and Properties of Thin Films, John Wiley and Sons, New Yark, 1959.
8.Becquart, C.S., Kim, D., Rifkin, J.A, and Clapp, P.C., “Fracture Properties of Metals and Alloys from Molecular Dynamics Simulation,” Materials Science and Engineering, Vol. 170, pp. 87-94, 1993.
9.Chandra, N., Namilae, S., and Shet, C., “Local Elastic Properties of Carbon Nanotubes in The Presence of Stone-Wales Defects,” Physical Review B, Vol. 69, pp. 94101-94112, 2004.
10.Chang, J., Cai, W., Bulatov, V.V., and Yip, S., “Molecular Dynamics Simulations of Motion of Edge and Screw Dislocations in A Metal,” Computational Materials Science, Vol. 23, pp. 111-115, 2002.
11.Chang, W.J., and Fang, T.H., “Influence of Temperature on Tensile and Fatigue Behavior of Nanoscale Copper Using Molecular Dynamics Simulation,” Journal of Physics and Chemistry of Solids, Vol. 64, pp. 1279-1283, 2003.
12.Chen, D.L., “Mixed Picture for Quantum System and The Effects of High Strain Rate on Mechanical Behavior of Gold Nanowires,” National Cheng Kung University, M. S., 2006.
13.Chen, D.L., Ju, C.C. Chen, T.C., and Fang, D.W., “Effects of Torsion on Mechanical Behavior of Nanostructure By Molecular Dynamics Analysis,” 18th CSME Conf., Vol. 3, pp. 1063-1070, 2001.
14.Chen, S., Ke, F., Minc, Z., and Bai, Y., “Atomistic Investigation of The Effects of Temperature and Surface Roughness on Diffusion Bonding Between Cu and Al,” Acta Materialia, Vol. 55, pp. 3169-3175, 2007.
15.Chowdhury, S., “Non-Contact AFM with A Nanoindentation Technique for Measuring The Mechanical Properties of Thin Films,” Nanotechnology, Vol. 15, pp. 1017-1022, 2004.
16.Cros, A., Aelfotoh, M.O., and Tu, K.N., “Formation, Oxidation, Electronic, and Electrical Properties of Copper Silicides,” Journal of Applied Physics, Vol.67, pp. 3328-3336, 1990.
17.Daly, S., Ravichandran, G., and Bhattacharya, K., “Stress-Induced Martensitic Phase Transformation in Thin Sheets of Nitinol,” Acta Materialia, Vol. 55, pp. 3593-3600, 2007.
18.Diamand, Y.S., Dubin, V., and Angyal, M., “Electroless Copper Deposition for ULSI,” Thin Solid Films, Vol. 262, pp. 93-103, 1995.
19.Evans, D.J., and Holian, B.L., “The Nose–Hoover Thermostat,” Journal of Chemical Physics, Vol. 83, pp. 4069-4074, 1985.
20.Fix, R., Gordon, R.G., and Hoffman, D.M., “Chemical Vapor Deposition of Vanadium, Niobium, and Tantalum Nitride Thin Films,” Chemistry of Materials, Vol. 5, pp.614-619, 1993.
21.Foiles, S.M., “Application of The Embedded-Atom Method to Liquid Transition Metals,” Physical Review B, Vol. 32, pp. 3409-3415, 1985.
22.Frenkel, D., Understanding Molecular Simulation from Algorithms to Applications, Academic Press, Burlington, 1996.
23.Fujii, T., and Akiniwa, Y., “Molecular Dynamics Analysis for Fracture Behaviour of Single Crystal Silicon Thin Film with Micro Notch,” Modelling and Simulation in Materials Science and Engineering, Vol. 14, pp. 73-83, 2006.
24.Fujisawa, S., Kikkawa, T., and Kizuka, T., “Direct Observation of Electromigration and Induced Stress in Cu Nanowire,” Japanese Journal of Applied Physics, Vol. 42, pp. 1433-1435, 2003.
25.Gall K., Horstemeyer, M.F., Schilfgaarde M.V., and Baskes, M.I., “Atomistic Simulations on The Tensile Debonding of An Aluminum-Silicon Interface,” Journal of The Mechanics and Physics of Solids, Vol. 48, pp.2183-2212, 2000.
26.Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equation, Prentice-Hall, Englewood Cliffs, NJ, USA, 1971, Chapter 9.
27.Gioia, G., and Ortiz, M., “Delamination of Compressed Thin Film. Advances in Applied Mechanics,” Vol. 33, pp. 120-192, 2001.
28.Girifalco, L.A., and Weizer, V.G., “Application of The Morse Potential Function to Cubic Metals,” Physical Review, Vol. 114, pp. 687-690, 1959.
29.Gotoh, Y., “Slip Patterns of Copper Whiskers Subjected to Tensile Deformation,” Physica Status Solidi A, Vol. 24, pp. 305-313, 1974.
30.Haile, J.M., Molecular Dynamics Simulation: Elementary Methods, John Wiley & Sons, New York, 1992.
31.Haque, M.A., and Saif, M.T.A., “In-Situ Tensile Testing of Nano-Scale Specimens in SEM and TEM,” Experimental Mechanics, Vol. 42, pp. 123-128, 2001.
32.Hsu, C.C., “A study of Defects on Micro/Nano Structures using Molecular Dynamics Simulation,” National Cheng Kung University, M. S., 2002.
33.Irving, J.H., and Kirkwood, J.G., “The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics,” Journal of Chemical Physics, Vol. 18, pp. 817-829, 1950.
34.Iwaki, T., “Molecular Dynamics Study on Stress-Strain in Very Thin Film: Size and Location of Region for Defining Stress and Strain,” JSME International Journal Series A, Vol. 39, pp. 346-353, 1996.
35.Jamting, A.K., Bell, J.M., and Swain, M.V., “Investigation of The Elastic Modulus of Thin Films Using Simple Biaxial Bending Techniques,” Thin Solid Films, Vol. 308-309, pp. 304-309, 1997.
36.Jeng, Y. R., and Tan, C.M., “Computer Simulation of Tension Experiments of A Thin Film Using An Atomic Model,” Physical Review B, Vol. 65, pp. 174107-174114, 2002.
37.Kang, J.W., and Hwang, H.J., “Mechanical Deformation Study of Copper Nanowire Using Atomistic Simulation,” Nanotechnology, Vol. 12, pp. 295-300, 2001.
38.Kelchner, C.L., Plimpton, S.J., and Hamilton, J.C., “Dislocation Nucleation and Defect Structure During Surface Indentation,” Physical Review B, Vol. 58, pp. 11085-11088, 1998.
39.Kirsten, W.Z., “Static and Dynamic Analysis of Failure Locations and Void Formation in Interconnects Due to Various Migration Mechanisms,” Materials Science in Semiconductor Processing, Vol. 6, pp. 85-92, 2003.
40.Koh, S.J.A., and Lee, H.P., “Molecular Dynamics Simulation of Size and Strain Rate Dependent Mechanical Response of FCC Metallic Nanowires,” Nanotechnology, Vol. 17, pp.3451-3467, 2006.
41.Komanduri, R., Chandrasekaran, N., and Raff, L.M. “Molecular Dynamics Simulation of Uniaxial Tension of Some Single Crystal Cubic Metals at Nanolevel,” International Journal of Mechanical Sciences, Vol. 43, pp. 2237-2260, 2001.
42.Landman, U., Luedtke, W.D., and Burnham, N.A., “Atomistic Mechanisms and Dynamics of Adhesion, Nanoindentation and Fracture,” Science, Vol. 248, pp. 454-461, 1990.
43.Leach, A.M., McDowell, M., and Gall, K., “Deformation of Top-Down and Bottom-Up Silver Nanowires,” Advanced Functional Materials, Vol. 17, pp. 43-53, 2006.
44.Liang, W., and Zhou, M., “Atomistic Simulations Reveal Shape Memory of FCC Metal Nanowires,” Physical Review B, Vol. 73, pp.115409-115411, 2006.
45.Lin, Z.C., and Huang, J.C., “A Study on A Rigid Body Boundary Layer Interface Force Model for Stress Calculation and Stress–Strain Behaviour of Nanoscale Uniaxial Tension,” Nanotechnology, Vol. 15, pp. 1509-1518, 2004.
46.Lutsko, J.F., “Stress and Elastic Constants in Anisotropic Solids: Molecular Dynamics Techniques,” Journal of Applied Physics, Vol. 64, pp. 1152-1154, 1988.
47.Miyazaki, N., and Shiozaki, Y., “Calculation of Mechanical Properties of Solids Using Molecular Dynamics Method,” JSME International Journal Series A, Vol. 39, p. 606, 1996.
48.Mizubayashi, H., Goto, K., and Ebisawa, T., “Anelasticity Study on Electromigration Effect in Cu Thin Films,” Materials Science and Engineering A, Vol. 442, pp. 342-346, 2006.
49.Nilsson, S.G., Borrise, X., and Montelius, L., “Size Effect on Young’s Modulus of Thin Chromium Cantilevers,” Applied Physics Letters, Vol. 85, pp. 3555-3557, 2004.
50.Ochoa, R., Swiler, T.P., and Simmons, J.H., “Molecular dynamics Studies of Brittle Failure in Silica: Effect of Thermal Vibrations,” Journal of Non-Crystalline Solids, Vol. 128, pp. 57-68, 1991.
51.Park, H.S., “Stress-Induced Martensitic Phase Transformation in Intermetallic Nickel Aluminum Nanowires,” Nano Letters, Vol. 6, pp. 958-962, 2006.
52.Park, H.S., and Zimmerman, J.A., “Stable Nanobridge Formation in <110> Gold Nanowires Under Tensile Deformation,” Scripta Materialia, Vol. 54, pp. 1127-1132, 2006.
53.Pharr, G.M., Oliver, W.C., and Brotzen, F.R., “On The Generality of The Relationship Among Contact Stiffness, Contact Area, and Elastic Modulus During Indentation,” Journal of Materials Research, Vol. 7, pp. 613-617, 1992.
54.Rahman, A., “Correclation in The Motion of Atoms in Liquid Argon,” Physical Review, Vol. 136, p. 405, 1964.
55.Rapaport, D., The Art of Molecular Dynamics Simulation, Cambridge University Press, London, 1997.
56.Schiotz, J., Rasmussen, T., and Jacobsen, K.W., ”Mechanical Deformation of Nanocrystalline Materials,” Philosophical Magazine Letters., Vol. 74, pp. 339-344, 1996.
57.Shen, M., Lin, W.H. Jian, W., and Liu, Z.L. “Copper Nanobelt Reorientation by Molecular Dynamics Simulation,” Chinese Physics Letters, Vol. 23, pp. 2721-2724, 2006.
58.Srolovitz, D., Maeda, K., Vitek,V. and Egami, T., “Structural Defects in Amorphous Solids Statistical Analysis of A Computer Model,” Philosophical Magazine A, Vol. 44, pp. 847-866, 1981.
59.Sun, Z.H., Wang, X.X., Soh, A.K., Wu, H.A., and Wang, Y., “Bending of Nanoscale Structures: Inconsistency Between Atomistic Simulation and Strain Gradient Elasticity Solution,” Computational Materials Science, Vol. 40, pp. 108-113, 2007.
60.Suresh, S., “Graded Materials for Resistance to Contact Deformation and Damage,” Science, Vol. 292, pp. 2447-2451, 2001.
61.Tabar, H.R., “Modelling The Nano-Scale Phenomena in Condensed Matter Physics Via Computer-Based Numerical Simulations,” Physics Reports, Vol. 325, pp. 239-310, 2000.
62.Tabata, O., Kawahata, K., and Sugiyama, S., “Mechanical Property Measurements of Thin Films Using Load Deflection of Composite Rectangular Membranes,” Sensors and Actuators, Vol. 20, pp. 135-141, 1989.
63.Tambe, N.S., “Scale Dependence of Micro/Nano-Friction and Adhesion of MEMS/NEMS Materials, Coatings and Lubricants,” Nanotechnology, Vol. 15, pp. 1561-1570, 2004.
64.Tokita, N., Hirabayashi, M., Azuma, C., and Dotera, T., “Voronoi Space Division of A Polymer: Topological Effects, Free Volume, and Surface End Segregation,” Journal of Chemical Physcs, Vol. 120, pp. 496-505, 2004.
65.Tong, C.J., Jiang, J.S., and Lin, M.T., “Using A New Microtensile System to Measure The Mechanical Properties of Sub-Micron Thick Thin Film materials,” 23th CSME Conf., Vol. 3, pp. 136-139, 2006.
66.Tsuchiya, T., Tabata, O., Sakata, J., and Taga, Y., “Specimen Size Effect on Tensile Strength of Surface-Micromachined Polycrystalline Silicon Thin Films,” Journal of Microelectromechanical Systems, Vol. 7, pp. 106-113, 1998.
67.Tsuchiya, T., Tabata, O., Sakata, J., and Taga, Y., “Specimen Size Effect on Tensile Strength of Surface Micromachined Polycrystalline Silicon Thin Films,” J. Microelectromech. Syst., pp. 106-112, 1998.
68.Vairagar, A.V., Mhaisalkara, S.G., and Krishnamoorthyb, A., “Effect of Surface Treatment on Electromigration in Sub-Micron Cu Damascene Interconnects,” Thin Solid Films, Vol. 462-463, pp. 325-329, 2004.
69.Wen, Y.H., “Size Effects on The Melting of Nickel Nanowires: A Molecular Dynamics Study,” Physica E, Vol. 25, pp. 47-54, 2004.
70.William, D.C., Materials Science and Engineering An Introduction 4/e, John Wiley & Sons, New York, 1998.
71.Wu, H.A., “Molecular Dynamics Study on Mechanics of Metal Nanowire,” Mechanics Research Communications, Vol. 33, pp. 9-16, 2006.
72.Xiang, Y., and Vlassak, J.J., “Bauschinger Effect in Thin Metal Films,” Scripta Materialia, Vol. 53, pp.177-182, 2005.
73.Zhou, L., Zhang, H., and Srolovitz, D.J., “A Size Effect in Grain Boundary Migration: A Molecular Dynamics Study of Bicrystal Thin Films,” Acta Materialia, Vol. 53, pp. 5273-5279, 2005.
74.Zimmerman, J.A., Kelchner, C.L., Klein, P.A., Hamilton, J.C., and Foiles, S.M., “Surface Step Effects on Nanoindentation,” Physical Review Letters, Vol. 87, pp. 165507-165511, 2001.