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國立中山大學機械與機電工程學系 博士論文 金屬薄膜於奈米壓痕之數值模擬與實驗研究 Numerical Simulation and Experimental Test of Nanoindentation Analysis on Metal Thin Film 研究生王中鼎 撰 指導教授錢志回 博士 中華民國 九十六 年 十 月 謝 誌 謝 誌 本論文承蒙指導教授 錢志回博士與 朱訓鵬博士悉心指導方得順利完成。兩位老師 的諄諄教誨與啟蒙，每遇困惑瓶頸，更且再三詳為析疑釋惑，初稿完成，復字斟句酌， 細心審閱修正，師恩浩瀚，謹申誠摯敬意。 承蒙口試委員 中正大學蕭庭郎教授、成功大學陳元方教授、中興大學黃敏睿教授 以及本系謝曉星教授、光灼華教授、任明華教授與楊旭光教授等於論文口試中，皆以豐 富之實務與學術經驗，費心審核愷切指正，使本文更臻完備，特誌由衷謝意。 修業期間，感謝本校材料系黃志青教授在材料領域的觀念指導；義守大學材料系簡 賸瑞教授在奈米壓痕方面的觀念交流與討論；日月光半導體製造公司賴逸少博士與楊秉 豐主任工程師在合作計劃上有形與無形的支持；金屬中心吳以德博士在研究及課業上的 協助；實驗應力實驗室及分子工程實驗室學弟妹們的幫忙；另外要特別感謝我的碩士指 導教授邵清安博士在研究觀念及做人處事上不斷地耐心指點與教誨；也要特別感謝分子 工程實驗室學弟李玟頡在分子動力學上的傾囊相助。 最後，謹以此論文獻予最敬愛的父母與妻子，尤其是我的妻子張淑女，在我正處於 軍旅生涯校級軍官的軍階與收入下，毅然而然地選擇退伍唸博班，在完全沒有經濟收入 及忙於課業與研究而疏於陪伴她的情況下，她不但要一肩擔負起家庭的經濟重擔，同時 還要肩負我的寶貝女兒的教養重責，由於她這樣無悔無怨的支持與鼓勵，方能使我在無 後顧之憂的情況下順利完成學業，點點滴滴感銘於心，在此深表由衷的感激。 王中鼎 謹誌 中華民國九十六年十月 于 西子灣 i Contents List of Tables...........iv List of Figures..v 摘要...vii Abstract.......ix Nomenclature......xi Chapter 1 Introduction..1 1.1 Motivation..1 1.2 Nanoindentation of Crystal Metal Orientation Surfaces........2 1.3 Nanoindentation of Multiscale Crystal Metals......5 1.4 Nanoindentation of Crystal Metal Semiconductors.......9 Chapter 2 Theory.........12 2.1 Governing Equation of Molecular Dynamics......12 2.2 Tight-Binding Potential....14 2.3 Initial Conditions........ 16 2.4 Periodic Boundary Condition......16 2.5 Rescaling ........18 2.6 Cell Link List Combined Verlet List........19 2.7 Leap-Frog .......20 2.8 Atomic Decomposition Algorithm.......21 2.9 Governing Equation of Finite Element ...22 2.10 Element Stiffness Matrix....26 ii 2.11 Numerical Computation................ 26 2.12 Direct Integration of Equations of Motion.....27 2.12.1 Derivation of General ulas.......28 2.12.2 Newmark’s β ............30 2.12.3 Average Acceleration ...32 2.13 Parallel Substructure .....35 Chapter 3 Simulation and Experiment Set Up of Nanoindentation....43 3.1 Simulation Set Up of Crystal Ni Metal Orientation Surfaces..............................43 3.2 Simulation Set Up of Multiscale Crystal Ni Metal..........................47 3.2.1 The Multiscale Model of Nanoindentation........ 47 3.2.2 MD/FE-HS Region.....49 3.2.3 Molecular Dynamics.......51 3.2.4 Finite Element ...............51 3.2.5 Nanoindentation Simulation...............54 3.3 Experiment Set Up of Crystal GaN Metal Semiconductor..................................55 Chapter 4 Results and Discussions.........69 4.1 The Nanoindentation Characteristics of Crystal Ni Metal Orientation Surfaces.69 4.1.1 Plastic Deation Characteristic during Nanoindentation......... 69 4.1.2 Pile-up Patterns after Nanoindentation.......74 4.1.3 Extracted Material Properties from Nanoindentation.....77 4.2 The Nanoindentation Accuracy of Multiscale Crystal Ni Metal..........82 4.3 The Nanoindentation Characteristics of Crystal GaN Metal Semiconductor..84 iii Chapter 5 Conclusion............100 5.1 Summary........100 5.2 Future Prospects.....101 Reference.........102 VITA............113 iv List of Tables Table 3-1 Mechanical properties of GaN thin films......60 Table 4-1 A comparison of the material properties of three nickel substrates...............90 v List of Figures Fig. 2.1 Cartesian frame.....38 Fig. 2.2 Lennard-Jones pairwise intermolecular potential............ 38 Fig. 2.3 Periodic boundary conditions.......... 39 Fig. 2.4 Cell link list ............ 39 Fig. 2.5 Verlet list ............. 40 Fig. 2.6 Cell link list combined verlet list ............ 40 Fig. 2.7 A sketch of atomic decomposition algorithm.............. 41 Fig. 2.8 A hexahedron element with 8 nodes................................ 41 Fig. 2.9 A tetrahedral element by using collapsing hexahedron element.............. 41 Fig. 2.10 A tetrahedral element with 6 nodes................................ 42 Fig. 2.11 Average acceleration ................................. 42 Fig. 2.12 A scheme for the parallel substructure in FEM........ 42 Fig. 3.1 Simulated nickel models ......... 61 Fig. 3.2 Multiscale Simulation models..... 62 Fig. 3.3 Close-up of the HS region and its surroundings in the nickel substrate...... 63 Fig. 3.4 The high perance computer.................. 64 Fig. 3.5 The experimental samples of GaN thin films.................. 65 Fig. 3.6 MTS NanoXP nanoindenter.......... 65 Fig. 3.7 Jeol JEM-2100F transmission electron microscopy............................ 66 Fig. 3.8 FEI Nova 220 focused ion beam......... 66 Fig. 3.9 Nanoindentation load–displacement curves of GaN thin films... 67 Fig. 3.10 The typical procedures for FIB milling sample preparation...... 68 vi Fig. 4.1 Indentation curves of the nickel substrates.............. 91 Fig. 4.2 Three dimension snapshots of nickel 100 substrate...... 92 Fig. 4.3 Three dimension snapshots of nickel 110 substrate...... 93 Fig. 4.4 Three dimension snapshots of nickel 111 substrate...... 94 Fig. 4.5 Pile-up patterns of the three nickel substrates............................. 95 Fig. 4.6 The cohesive energy curves of the three nickel substrates.. 96 Fig. 4.7 Indentation curves of both multiscale and full MD models.... 96 Fig. 4.8 Close-up snapshot for the deation morphology of multiscale model.. 97 Fig. 4.9 XTEM Bright-field images of specimen..... 98 Fig. 4.10 XTEM Bright-field images of GaN thin film........ 99 vii 摘要摘要 奈米壓痕檢測技術為目前量測奈米尺度材料之機械性質的最主要方法之一。隨 著近年來奈米科技的蓬勃發展，製程技術所能製造出的奈米元件尺寸愈來愈小，以 往巨觀的檢測技術已不敷使用，尤其當材料的尺寸到逹原子級的尺度時，其所需之 檢測精確度更是巨觀檢測技術所難以達到的，因此奈米壓痕檢測技術乃成為非常重 要的微觀檢測技術。本文即利用奈米壓痕之數值及實驗方法對金屬薄膜在探針的壓 痕作用下，對其材料特性進行定性及定量之分析。 本文研究方法第一部份先以分子動力學之平行運算對100、110及111三個定 向結晶面之鎳薄膜進行奈米壓痕之模擬分析，模擬結果顯示由探針作用所造成材料 塑性變形之應變能量乃是利用同質性孕核Homogeneous Nucleation的成形來儲 存，而利用{111}滑移面的錯位滑移Dislocation Sliding來釋放。三個定向結晶面之 壓痕曲線從曲線局部最高點到局部最低點的陡峭變化、壓痕表面的堆積隆起Pile-up 形貌之擴散程度，以及硬度Hardness與彈性模數Elastic Modulus之材料性質等， 均受材料{111}滑移面之滑移角的數目所影響；第二部份則利用分子動力學結合有限 元素法與平行運算之多尺度模擬對100定向結晶面之鎳薄膜進行奈米壓痕之模擬 分析，在與相同模擬條件之全分子動力學的模擬結果比較下，由壓痕曲線及壓痕變 形形貌之模擬結果驗証了本文對於分子動力學結合有限元素法所建立之多尺度模型 的正確性；第三部份則是利用奈米壓痕儀Nanoindenter結合聚焦離子束顯微鏡FIB 與穿透式電子顯微鏡TEM來對氮化鎵GaN薄膜因壓痕所引起的局部相變化機制 進行實驗分析。對三五族的氮化鎵材料而言，在壓痕負載的過程中，壓痕曲線會有 突然跳躍pop-in的現象產生，此現象乃為材料的錯位成核Dislocation Nucleation之 viii 變形機制，但在奈米壓痕過程中所導致的相變化機制則不存在於氮化鎵薄膜中。 關鍵詞關鍵詞 分子動力學，有限元素法，奈米壓痕，金屬薄膜，材料特性 ix Abstract Molecular dynamics MD simulations are applied to elucidate the anisotropic characteristics in the material responses for crystallographic nickel substrates with 100, 110 and 111 surface orientations during nanoindentation. The strain energy of the substrate rted by the tip is stored by the ation of the homogeneous nucleation, and is dissipated by the dislocation sliding of the {111} plane. The steep variations of the indentation curve from the local peak to the local minimums are affected by the numbers of slip angle of {111} sliding plane. The pile-up patterns of the three nickel substrates prove that the crystalline nickel materials demonstrate the pile-up phenomenon from nanoindentation on the nanoscale. The three crystallographic nickel substrates exhibit differing amounts of pile-up dislocation spreading at different crystallographic orientations. The effects of surface orientation in material properties of F.C.C. nickel material on the nanoscale are observable through the slip angle numbers of {111} sliding planes which influence hardness values, as well as the cohesive energy of different crystallographic surfaces that indicate Young’s modulus. Furthermore, the multiscale simulations are pered on the 100 monocrystal nickel substrate by using nanoindentation, compensating for MD limitation of a large specimen simulation without significant increase in the size of the problem. This study examines the accuracy of the coupling for the multiscale model by means of the indentation curve and the deation profile. Nanoindentation-induced mechanical deation in GaN thin films prepared by metal-organic chemical-vapor deposition MOCVD was investigated using the Berkovich diamond tip in combining with the cross-sectional transmission electron x microscopy XTEM. By using the focused ion beam FIB milling to accurately position the cross-section of the indented region, the XTEM results demonstrate that the major plastic deation was taking place through the propagation of dislocations. The present observations are in support of attributing the pop-ins appeared in the load-displacement curves to the massive dislocation activities occurring underneath the indenter during loading cycle. The absence of indentation-induced new phases might have been due to the stress relaxation via substrate and is also consistent with the fact that no discontinuity was found upon unloading. Key words Molecular dynamics, Finite element , Nanoindentation, Metal thin films, Material characteristics xi Nomenclature p A The contact area [C] Element damping matrix r E The reduced elastic modulus E Young’s modulus i F The interaction force of atom i H The Hamiltonian function d H The Hardness c h The contact depth [J] Jacobian matrix B k Boltzmann constant [K] Element stiffness matrix L The Lagrangian function i m The mass of atom i [M] Element mass matrix N Total atom numbers [N] The matrix of shape function i P The momentum of atom i max P The maximum load of the tip at the maximum depth Q r The vector of nodal displacement Q meanwhile, outside of the atomistic region, where strain fields are smoothly varying, continuum mechanics is used to describe the material deation behavior over larger length scales. Accordingly, multiscale s are a class of simulation s that have become useful and 7 important within the past decade [50]. Much of this is due to the fact that the governing physics and mechanics of deing media have been elucidated over the course of time. Another crucial factor at play is the recent explosion in computational power. As combining the two scale approaches, the multiscale leads straight towards a new revolution in computational mechanics. With regard to the literature review for the proposed research of multiscale in the past decades, a classic example is the pioneering work of Clementi and coworkers [51] in 1980s where they used quantum mechanics, atomistic dynamics, and fluid dynamics to predict the tidal circulation in Buzzard’s Bay Massachusetts. They first used high quality quantum mechanical s to uate the interaction of several water molecules. Since then, multiscale simulation has been applied to the various fields, including nanocomponent analysis [52-55], crack fracture analysis [51,53,55,56-61], wave propagation [57,62], nanoindentation [53,63-67], nanometric cutting [47], and stress distribution analysis [68-69]. Nowadays, there were many ologies to couple MD and FEM which had been successfully developed. Systematically, these ologies can be classified into several categories the bridging scale BSM [50,52,56-57,70-74], meshfree particle MPM [53-54], the quasicontinuum QCM [47,58,63-67,75-76], the coarse-grained molecular dynamics CGMD [77], the coupled atomistic dislocation dynamics CADD [78-80], the macroscopic atomistic ab initio dynamics MAAD [51,60], the coupling of length scales CLS [55,59], and the finite element and atomistic FEAt [61,81-82]. As the present article is not intended as an exhaustive review, the interested reader can refer to the multiple scale review papers of Curtin et al. [80], and Liu et al. [83]. 8 Among these previously mentioned literatures of multiscale research, a widely used approach is to implement the coupling through a so-called “handshake” region, which is essentially the interface between the MD and FEM regions. In this , the finite element mesh is graded down to the atomic lattice size in an overlap, or ‘‘handshake’’, region; the dynamics is governed by a total Hamiltonian function that combines the separate Hamiltonians of the different regions in an appropriate way [51,55,59-62,68-69,78-79,80-82,84]. In the handshake region, a certain weighted factor is pered between the MD description and the coarse grid description. These researches briefly include Dewald et al. [78] employed the CADD to quantify the error in dislocation driving forces for dislocations near the atomistic/continuum interface. Abraham [51] utilized the MAAD to simulate the crack growth of solid silicon slab corresponding to the stress wave propagation. Rudd et al. [55] pered the multiscale simulations of dynamical systems at finite temperature using the CLS with applications in fracture and Micro-Electro-Mechanical Systems MEMS. Izumi et al. [82] used the FEAt to analyze the shear-strain dominant field of silicon flat. Lidorikis et al. [69] adopted the CLS to study the stress distributions in the interface of silicon/silicon-nitride nanopixels. Liu [68] et al. established a full 3D model with a combination of MD and FE by the overlapping transition zone to study the stress distributions in an epitaxial island. In this paper, a newly coupled of the weighted function using the handshake region to combine MD and FE, proposed by Lidorikis et al. [69,84], was adopted in our hybrid FE/MD simulations, because this exhibits a simple characteristic and fits in with our demand for coupling the MD and FE regions. We take nanoindentation as an 9 investigation problem for assessing the perance of the coupled
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