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研究生:林俊德
研究生(外文):Chun-Te Lin
論文名稱:使用原子-連體力學於奈米尺寸單晶IV-A族之機械性質研究
論文名稱(外文):Investigation of Nanoscale Single Crystal Group IV-A Mechanical Properties Using Atomistic-Continuum Mechanics (ACM)
指導教授:江國寧
指導教授(外文):Kuo-Ning Chiang
學位類別:博士
校院名稱:國立清華大學
系所名稱:動力機械工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2006
畢業學年度:95
語文別:英文
論文頁數:230
中文關鍵詞:原子-連體力學有限元素法單晶矽尺寸效應點缺陷勢位能
外文關鍵詞:atomistic-continuum mechanicsFinite element methodsingle crystal siliconsize effectpoint defectpotential function
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本論文將利用以有限元素法為基礎之原子連體力學,探討材料受單軸拉伸負載下,奈米尺寸單晶IV族元素於相異晶格面之機械性質,包含 (100), (110)及(111)之晶格面方向,亦進行模態分析反推材料機械性質。模擬結果顯示,奈米等級單晶矽之彈性模數與文獻中所發表的微米尺寸之楊氏係數實驗值相當。此外,本研究亦討論於不同應變條件下的單晶矽材料之機械性質,模擬結果得知,當應變介於0.1%至1% 之間時,其相對應之彈性模數均呈現定值。其次,原子連體力學模擬結果將受不同原子間勢位能而有所影響,本研究比較兩組常用於單晶矽之勢位能,包括Stillinger-Weber 與 Tersoff 勢位能。研究結果顯示,於(110)晶格面之彈性模數於此兩組勢位能下,具有相同的趨勢且其差異在5%以內。
再者,本論文使用連體力學之桁架分析方法,推導鑽石及類鑽石結構單晶格立方體的彈性模數之解析解,用以計算鑽石及類鑽石結構單晶矽之彈性模數。一旦得知材料晶格常數與原子間作用力,則可經由此解析解快速且精確的計算鑽石及類鑽石單晶結構立方體之彈性模數。同時,可利用此解析解驗證不同勢位能函數之參數的正確性與精確度。且該解析解可用於鑽石及類鑽石結構之化學元素,例如矽、碳, 鍺, 砷化鎵等半導體化合物。 此外,本論文亦利用解析解探討於大變形條件下的奈米等級單晶矽之機械性質。
除了利用單軸拉伸負載計算材料之彈性模數外,亦可利用模態分析反推材料之彈性模數。本論文使用原子連體力學所建立之原子模型,配合實際之原子質量,進行模態分析並透過結構動力學理論來反推材料之彈性模數。分析結果得知,其共振頻率與模態皆與由尤拉-伯努立樑原理所推導之懸臂樑解析解相符。進一步利用其第一共振頻率反推材料之彈性模數,其計算結果皆介於文獻上所記載的塊材之機械性質。
本論文亦利用原子連體力學配合Stillinger-Weber 勢位能函數,於單軸拉伸負載下,研究奈米尺寸單晶矽機械性質之尺寸效應。模擬結果顯示,奈米尺寸所相對應的材料機械性質具有高度之尺寸效應。此外,利用原子連體力學探討單晶矽於奈米尺度下之破壞現象。
最後,本論文使用原子連體力學,研究奈米尺寸下含點缺陷之單晶矽結構的機械性質。模擬結果顯示,單晶矽之彈性模數隨點缺陷的含量增加,呈線性遞減的趨勢。
綜合上述的分析結果,本論文所發展的原子連體力學與解析解,並不限制用於單晶矽材料,只要得知材料之原子間勢位能與原子排列即可利用此原子連體力學探討奈米尺寸之材料機械性質。此外原子連體力學所建立之原子模型不僅可用於單軸拉伸負載之靜態分析,亦可用於進行模態分析,其分析結果皆符合文獻上之塊材機械性質。
This dissertation uses atomistic-continuum mechanics (ACM) based on the finite element method (FEM) to investigate the elastic properties of the nanoscale single crystal group IV in different crystallography planes of (100), (110), and (111) under uniaxial tensile loading and modal analysis. The measured elastic modulus of SCS is in reasonable agreement with the results reported in the literature based on macro and micro scale experiments. The effect of strain through the variation of prescribed displacement on the measured mechanical properties was also investigated. It could be observed that the elastic modulus was nearly constant and relatively insensitive to the strain (0.1%~1%). Meanwhile, different interatomic potential functions were adopted to simulate and compare the mechanical properties, including the Stillinger-Weber and Tersoff potential. The ACM simulation results for both Stillinger-Weber and Tersoff potentials show a similar trend in the estimation of mechanical properties. The discrepancy of the elastic modulus between the Stillinger-Weber and Tersoff potential function on the (110) crystal plane were within 5%.
Furthermore, an analytic solution of single crystal silicon of diamond and diamond-like structure was derived to estimate the elastic (linear) behaviors of single lattice silicon based on the truss analysis of static mechanics. Through this analytical solution could reduce the complexity of the real bulk silicon an agreement on trends between the predicted elastic modulus and the corresponding bulk value is achieved. In addition, this analytic solution could apply to the other diamond and diamond-like structure, includes carbon, germanium and gallium arsenide. Moreover, the larger deformation was also considered to investigate the mechanical properties of the single crystal silicon.
In addition to the tensile test method, the modal analysis is also employed to estimate the elastic modulus of materials at nanoscale range. From simulation results showed that the natural frequency and mode shape are agree with the analytic solution base on the Euler-Bernoulli beam theory.
The size dependence of elastic properties of SCS at nanoscale was studied using ACM with Stillinger-Weber potential function. It was found that the mechanical properties are highly size dependent at nanoscale. Furthermore, fracture behavior of SCS at the nanoscale was also simulated using ACM. The simulation results reveal that the fracture phenomena prediction was accurate and appropriate using atomistic-continuum mechanics.
Finally, the defected model was constructed to investigate the elastic modulus of single crystal silicon under different crystalline plane using atomistic-continuum mechanics. The ACM simulation result reveals that the elastic modulus monotonously decreases with the concentration of vacancies defect increasing. This dissertation also investigates the size dependence elastic modulus with defect formation.
According to abovementioned simulation results, the use of an atomistic-continuum mechanics and analysis solution in the investigation of nanoscale materials was not limited to single crystal silicon, but it can also be applied to other nano-structured materials once the interatomic potential and the atomic structure of the material are always known. Therefore, atomic model is constructed using atomistic-continuum mechanics not only to apply in axis tensile loading test but also in the modal analysis. And the estimated of material properties under tensile loading and modal analysis was agreed with the bulk value of experiments in the literatures
ABSTRACT……………………………………………………………i
ABSTRACT (CHINESE)……………………………………………iii
ACKNOWLEDGEMENT …………………………………………………v
TABLE OF CONTENTS………………………………………………vi
LIST OF TABLES …………………………………………………ix
LIST OF FIGURES ………………………………………………xi


CHAPTER I. INTRODUCTION …………………1
1.1 Motivation of research ……………………… 1
1.2 Literature Survey ………………………………3
1.2.1 Review of elastic properties estimation by experiment..3
1.2.2 Review of elastic properties estimation by atomistic simulation……5
1.2.3 Review of elastic properties estimation by theoretical analysis……9
1.3 The objective and outline of research ………………10

CHAPTER II.THEORY…………………………………………………15
2.1 Geometrical characteristics of SCS in nature…………15
2.1.1 Covalent bond and other chemical bond.............15
2.1.2 Atomic structure of single crystal silicon…………17
2.2 Interatomic force and potential energy function……18
2.2.1 Attractive and repulsive forces ………………………20
2.2.2 Lennard-Jones potential …………………………………21
2.2.3 Morse potential ……………………………………………23
2.2.4 Brenner’s second generation empirical potential…24
2.2.5 Stillinger-Weber potential ……………………………25
2.2.6 Tersoff potential …………………………………………27
2.3 Numerical simulation strategy of many-particles system …30
2.3.1 Ab initio method (QM) ……………………………………30
2.3.2 Monte Carlo method (MC) …………………………………31
2.3.3 Molecular mechanics (MM) ………………………………32
2.3.4 Molecular dynamic method (MD) …………………………32
2.3.5 Atomistic-continuum mechanics (ACM) …………………36
2.4 Theory of vibration of cantilever beam ………………39
2.5 Crystallographic defect ……………………………………44
2.5.1 Point defect ………………………………………………44
2.5.2 Line defect …………………………………………………45
2.5.3 Area defect …………………………………………………45
2.5.4 Volume defect………………………………………………45
2.6 Statistical sampling method ………………………………46
2.6.1 Non-probability sampling procedure………………… 46
2.6.2 Probability sampling procedure………………………46

CHAPTER III. METHODOLOGY OF ATOMISTIC-CONTINUUM MECAHCNIS (ACM)…………………………………51
3.1 Basic assumptions …………………………………………51
3.2 Atomistic modeling of single crystal silicon for estimating the mechanical properties using static tensile loading ……………………………………54
3.3 Atomistic modeling of single crystal silicon for estimating the mechanical properties using modal analysis ………………………………………….55
3.4 Equivalent spring for describing bond angle motion of potential function……………56
3.5 Elastic modulus and Poisson’s ratio definition ………………………59

CHAPTER IV. ANALYTICAL MODEL OF DIAMOND AND DIAMOND-LIKE STRUCTURE …………………………………………………………61
4.1 Analytical solution of single lattice diamond and diamond-like structure ………………………………………61
4.1.1 (100) plane ………………………………………………67
4.1.2 (110) plane…………………………………… ………69
4.1.3 (111) plane………………………………………………71
4.2 Validation of the single lattice silicon ACM modeling through analytic model …………………………………………76
4.3 Elastic modulus estimation of others diamond and diamond-like structure: Carbon and Germanium, Gallium arsenide………………………………single lattice silicon ACM modeling through analytic model ………………………………77

CHAPTER V. NUMERICAL RESULT OF SINGLE CRYSTAL SILICON ACM MODEL USING STATIC TENSILE LOADING METHODsingle lattice silicon ACM modeling through analytic model ………………81
5.1 Validation of the single crystal silicon ACM model via experiments data …………………………………………………81
5.1.1 Elastic modulus estimation through ACM ……………82
5.1.2 Poisson’s ratio estimation through ACM …………83
5.1.3 Effect of prescribed displacement (strain) on the tensile response ……………………………………………………83
5.2 Comparative study of silicon empirical interatomic potentials …………………………………………………………84
5.3 Fracture phenomena of single crystal silicon under tensile loading ………………………………………………………85

CHAPTER VI. NUMERICAL RESULT OF SINGLE CRYSTAL SILICON USING MODAL ANALYSIS……………………………………………89
6.1 Elastic modulus estimation through first mode resonant frequency ……………………………………………………………89
6.2 Analog of continuum solid model …………………………………………………………………………92

CHAPTER VII. SIZE DEPENDENCE MECHANICAL PROPERTIES OF THE SINGLE CRYSTAL SILICON VIA ACM UNDER TENSILE LOADING……95
7.1 The boundary conditions and loading conditions …………………………………………………………………………95
7.2 Results and discussion ……………………………………95
7.2.1 Surface effect of thin film ……………………………96
7.2.2 Length effect………………………………………………98

CHAPTER VIII. MECHANICAL PROPERTIES OF DEFECTED MODEL OF THE SINGLE CRYSTAL SILICON VIA ACM ………………………101
8.1 Selection rule of the point defect ……………………102
8.2 Simple random sampling and random number generation…103
8.3 Results and discussions……………………………………103
8.3.1 The single crystal silicon ACM model with the concentration of defect under tensile loading……………103
8.3.2 Length dependent of the elastic properties of the single crystal silicon ACM model with the concentration of defect under tensile loading …………………………………107
8.3.3 Analysis of ultrathin beam ACM model under modal analysis………………………………………………………………111

CHAPTER IX. CONCLUSIONS AND FUTURE WORK……………………115
9.1 Conclusions …………………………………………………115
9.2 Recommendation for future study ………………………118

REFERENCES …………………………………………………………119
TABLE…………………………………………………………………131
FIGURES………………………………………………………………141
APPENDICES …………………………………………………………221
Appendix A: Local and global matrix assemble of single lattice cubic of single crystal silicon along (100), (110), and (111) crystallography plane………………………………221
Appendix B: Table of random number…………………………228
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