摘要
原子力顯微鏡藉由探針與試片間之作用力變化來量測非導體試片奈米等級之表面形貌，可於一般大氣環境或水溶液中進行量測，大幅增加其可應用之範圍。此外，原子力顯微鏡可藉由探針懸臂樑之變形量推求作用於奈米結構之作用力，因此可用於量測奈米結構之機械性質，成為奈米科技研究中極為重要之量測工具。
本研究首先針對原子力顯微鏡懸臂樑之彈簧常數與變形行為進行分析與實驗，探討原子力顯微鏡探針懸臂樑彈簧常數之準確度。分析方面，以有限元素套裝軟體ANSYS®分析表面金屬鍍層、懸臂樑幾何、各種複合作用力、懸臂樑下傾角度、材料性質、針尖質量與雷射光點聚焦位置等參數對探針懸臂樑彈簧常數之影響，並比較有限單元之分析結果與實驗量測數值、其他文獻公式與廠商提供資料間之差異，以驗證有限單元分析解之準確性。結果發現不同厚度之金屬反射鍍層對探針懸臂樑之扭轉與彎曲彈簧常數均有明顯之影響，當金屬鍍層厚度提升至200奈米時，NT-MDT CS12型之懸臂樑彎曲與扭轉彈簧常數分別上升了27.2%與26.9%。
探討材料常數時發現，懸臂樑之彎曲彈簧常數與楊氏係數間為正比關係，但彎曲彈簧常數隨波松比增加而呈現非線性遞增。此外，懸臂樑之側向彈簧常數隨波松比增加而逐漸減少。由於大多數探針懸臂樑採用非等向性材料-單晶矽製造，此非等向性材料性質之影響於研究中首先以CASTEP®軟體計算非等向性材料勁度矩陣後，再將該矩陣配合ANSYS®計算探針懸臂樑之各種彈簧常數與共振頻率。結果發現非等向性之材料性質對NanoWorld AT/CONT20及Microlever/Type C懸臂樑彈簧常數之影響可達23.89%與29.72%。文中並提出兩個有效率之公式來修正非等向性材料性質所產生之誤差。
探針幾何尺寸於製造過程所產生之誤差亦於研究中以有限單元法討論，厚度與寬度越小之懸臂樑具有較高之量測靈敏度，但相對更需要考慮幾何非線性所產之量測誤差。此外，探針針尖質量所造成之懸臂樑共振頻率偏移亦影響後續之彎曲彈簧常數計算與力量量測之準確性。文中亦探討不同設備所具有之不同探針懸臂樑下傾角度對其彈簧常數之影響。
於複合作用力與預應力影響之分析中可知，探針懸臂樑之扭轉角度與多方向之作用力間有耦合關係，須以有限單元分析求取其所產生之誤差。而由探針與試片間接觸作用力所產生之懸臂樑內預應力更嚴重影響懸臂樑之扭轉彈簧常數；1μN之接觸力可使NT-MDT CS12探針懸臂樑之扭轉彈簧常數增加30.9%。最後考慮量測狀況下之探針懸臂樑下傾角與受力情形，同時考慮波松比、下傾角、非線性材料性質與預應力之影響，求得更精確之懸臂樑彈簧常數與變形行為，以提高量測之準確度。
本研究針對各項可能影響碳管機械性質量測之參數進行有限單元分析與實驗。文中使用校正後之原子力顯微鏡進行實驗量測奈米碳管的機械性質，並配合有限單元分析探討碳管邊界條件之影響，以求得奈米碳管更準確之機械性質。

Abstract
Atomic force microscope (AFM) has many applications in science research under nano-scale or micro-scale. The mechanical properties of nanostructures can be measured from the deformation of cantilever of atomic force microscope. The accuracy of the mechanical properties depends on the precise calibration of the spring constant of AFM cantilevers. Thus, the spring constant of cantilever must be calibrated before measurement.
Many parameters such as coating layers, material properties, geometries, tilt angles, and landing forces, significantly affecting the spring constant of AFM were investigated in this study. The results obtained from the finite element method were compared with those determined from the existing equations and from experiment. The results calculated from our finite element model show good agreement with experimental data and with the ones from other published papers. The results show that both the bending and torsional spring constants of NT-MDT CS12 rectangular cantilevers increased with the increasing thickness of aluminum coating layer. The coating layers of AFM cantilevers should not be neglected, especially for thinner cantilevers.
The material properties directly affect the deformation characteristics of AFM cantilevers. The bending spring constant is proportional to the Young’s modulus and increases nonlinearly with Poisson’s ratio. However, the lateral spring constant decreases with increasing Poisson’s ratio. When the anisotropic material property of crystal silicon is considered, the commercial software CASTEP□ was adopted to obtain the anisotropic stiffness matrix of crystal silicon. Then, the anisotropic material matrix can be used in the finite element analysis. From the simulation results, the anisotropic material property significantly affects the spring constants of AFM cantilevers. Two equations were proposed to obtain the spring constants and the resonant frequencies of crystal silicon AFM cantilever with the axis located at different cantilever-crystal angles.
The geometry variations due to the fabrication tolerance were also studied by finite element method. The thinner and narrower cantilevers are more sensitive to the force measurement; however the geometry nonlinearity should be considered under large loadings to improve the accuracy of measurement. Moreover, the resonant frequency shift caused from the mass of image tip was studied to reduce the error of spring constant calculation.
During the force measurement process, the loading condition of the cantilever is very complex. During the cantilever landing process, the effects of combined loadings and pre-stress were analyzed by the finite element method. The landing force significantly affects the spring constant of AFM cantilevers, and should be considered. In order to precisely obtain the deformation behavior and spring constant of AFM cantilevers, the above mentioned parameters should be considered in the analysis.
Once the spring constants of AFM cantilevers were obtained, they were used to measure the mechanical properties of carbon nanotubes. The effects of loading positions and geometries on the mechanical properties measurement were also carried out by finite element method in this study.

References
1. G. Binnig, C. H. Rohrer, Ch. Gerber, and E. Weibel, “Tunneling Through a Controllable Vacuum Gap,” Applied Physics Letters, Vol. 40, No. 2, pp. 178-180, 1982.
2. G. Binnig, C. F. Quate, and Ch. Gerber, “Atmoic Force Microscope,” Physical Review Letters, Vol. 56, No. 9, pp. 930-933, 1986.
3. D. Sarid, P. Pax, L. Yi, S. Howells, M. Gallagher, T. Chen, V. Elings, and D. Bocek, “Improved Atomic Force Microscope using a Laser Diode Interferometer,” Review of Scientific Instruments, Vol. 63, No. 8, pp. 3905-3908, 1992.
4. J. E. Lennard-Jones, “Cohesion,” Proceedings of the Physical Society, Vol. 43, pp. 461-482, 1931.
5. J. B. D. Green, “Analytical Instrumentation Based on Force Measurements: Combinatorial Atomic Force Microscopy,” Analytica Chimica Acta, Vol. 496, pp. 267-277, 2003.
6. M. A. Poggi, E. D. Gadsby, L. A. Bottomley, W. P. King, E. Oroudjev, and H. Hasma, “Scanning Probe Microscopy,” Analytical Chemistry, Vol. 76, pp. 3429-3444, 2004.
7. R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and Ph. Avouris, “Single- and Multi-wall Carbon Nanotube Field-Effect Transistors,” Applied Physics Letters, Vol. 73, No. 17, pp. 2447-2449, 1998.
8. S. Elhadj, A. A. Chernov, and J. J. De Yoreo, “Solvent-Mediated Repair and Patterning of surfaces by AFM,” Nanotechnology, Vol. 19, pp. 105304, 2008.
9. T. W. Tombler, C. Zhou, L. Alexseyev, J. Kong, H. Dai, L. Liu, C. S. Jayanthi, M. Tang, and S. Y. Wu, “Reversible Electromechanical Characteristics of Carbon Nanotubes under Local-Probe Manipulation,” Nature, Vol. 405, pp. 405-769, 2000.
10. Y. Jin, K. Wang, W. Tan, P. Wu, Q. Wang, H. Huang, S. Huang, Z. Tang, and Q. Guo, “Monitoring Molecular Beacon/DNA Interactions using Atomic Force Microscopy,” Analytical Chemistry, Vol. 76, pp. 5721-5725, 2004.
11. H. Skulason and C. D. Frisbie, “Contact Mechanics Modeling of Pull-Off Measurements: Effect of Solvent, Probe Radius, and Chemical Binding Probability on the Detection of Single-Bond Rupture Forces by Atomic Force Microscopy,” Analytical Chemistry, Vol. 74, pp. 3096-3104, 2002.
12. F. Kühner, L. T. Costa, P. M. Bisch, S. Thalhammer, W. M. Hackl, and H. E. Gaub, “LexA-DNA Bond Strength by Single Molecule Force Spectroscopy,” Biophysical Journal, Vol. 87, pp. 2683–2690, 2004.
13. T. Boland, B. D. Ratner, “Direct Measurement of Hydrogen Bonding in DNA Nucleotide Bases by Atomic Force Microscopy,” Proceeding of National Academy of Sciences, Vol. 92, No. 12, pp. 5297-5310, 1995.
14. M. F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. F. Ruoff, “Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes under Tensile Load,” Science, Vol. 287, pp. 637-640, 2000.
15. M. F. Yu, B. S. Files, S. Arepalli, and R. S. Roff, “Tensile Loading of Ropes of Single Wall Carbon Nanotubes and Their Mechanical Properties,” Physical Review Letters, Vol. 84, pp. 5552-5555, 2000.
16. M. F. Yu, T. Kowalewski, and R. S. Ruoff, “Structure Analysis of Collapsed, and Twisted and Collapsed, Multiwalled Carbon Nanotubes by Atomic Force Microscopy,” Physical Review Letters, Vol. 86, No. 1, pp. 87-90, 2001.
17. M. F. Yu, M. J. Dyer, and R. S. Ruoff, “Structure and Mechanical Flexibility of Carbon Nanotube Ribbons: An Atomic-Force Microscopy Study,” Journal of Applied Physics, Vol. 89, No. 8, pp. 4554-4557, 2001.
18. M. F. Yu, B. I. Yakobson, and R. S. Ruoff, “Controlled Sliding and Pullout of Nested Shells in Individual Multiwalled Carbon Nanotubes,” Journal of Physical Chemistry B, Vol. 104, pp. 8764-8767, 2000.
19. J. P. Salvetat, G. Andrew, D. Briggs, J. M. Bonard, R. R. Bacsa, A. J. Kulik, T. Stockli, N. A. Burnham, and L. Forro, “Elastic and Shear Moduli of Single-Walled Carbon Nanotube Ropes,” Physical Review Letters, Vol. 82, No. 5, pp. 944-947, 1999.
20. E. W. Wong, P. E. Sheehan, C. M. Lieber, “Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes,” Science, Vol. 277, pp. 1971-1975, 1997.
21. S. Cuenot, C. Fretigny, S. D. Champagne, and B. Nysten, “Measurement of Elastic Modulus of Nanotubes by Resonant Contact Atomic Force Microscopy,” Journal of Applied Physics, Vol. 93, No. 9, pp. 5650-5655, 2003.
22. H. Ni and X. Li, “Young’s Modulus of ZnO Nanobelts Measured using Atomic Force Microscopy and Nanoindentation Techniques,” Nanotechnology, Vol. 17, pp. 3591-3597, 2006.
23. X. Chen, S. Zhang, D. A. Dikin, W. Ding, R. S. Ruoff, L. Pan, and Y. Nakayama, “Mechanics of a Carbon Nanocoil," NanoLetters, Vol. 3, No. 9, pp. 1299-1304, 2003.
24. A. Volodin, M. Ahlskog, E. Seynaeve, C. V. Haesendonck, A. Fonseca, and J. B. Nagy, “Imaging the Elastic Properties of Coiled Carbon Nanotubes with Atomic Force Microscopy,” Physical Review Letters, Vol. 84, No. 15, pp. 3342-3345, 2000.
25. T. Hayashida, L. Pan, Y. Nakayama, “Mechanical and Electrical Properties of Carbon Tubule Nanocoils,” Physica B, Vol. 323, pp. 352-353, 2002.
26. S. Motojima, M. Kawaguchi, K. Nozaki, and H. Iwanaga, “Growth of Regularly Coiled Carbon Filaments by Ni Catalyzed Pyrolysis of Acetylene, and Their Morphology and Extension Characteristics,” Applied Physics Letters, Vol. 56, No. 4, pp. 321-323, 1990.
27. D. Dietzel, M. Faucher, A. Iaia, J. P. Aimé, S. Marsaudon, A. M. Bonnot, V. Bouchiat, and G. Couturier, “Analysis of Mechanical Properties of Single Wall Carbon Nanotubes Fixed at a Tip Apex by Atomic Force Microscopy,” Nanotechnology, Vol. 16, pp. S73-S78, 2005.
28. J. Stiernstedt, M. W. Rutland, and P. Attard, “A Novel Technique for the in-situ Calibration and Measurement of Friction with the Atomic Force Microscope,” Review of Scientific Instruments, Vol. 76, pp. 083710, 2005.
29. S. Breakspear, J. R. Smith, T. G. Nevell, and J. Tsibouklis, “Friction Coefficient Mapping using the Atomic Force Microscope,” Surface and Interface Analysis, Vol. 36, pp. 1330-1334, 2004.
30. M. Reinstädtler, U. Rabe, V. Scherer, U. Hartmann, A. Goldade, B. Bhushan, and W. Arnold, “On the Nanoscale Measurement of Friction using Atomic-Force Microscope Cantilever Torsional Resonances,” Applied Physics Letters, Vol. 82, No. 16, pp. 2604-2606, 2003.
31. D. F. Arias, D. M. Marulanda, A. M. Baena, and A. Devia, ” Determination of Friction Coefficient on ZrN and TiN using Lateral Force Microscopy (LFM),” Wear, Vol. 261, pp. 1232-1236, 2006.
32. Y. D. Yan, S. Dong, and T. Sun, “3D Force Components Measurement in AFM Scratching Tests.” Ultramicroscopy, Vol. 105, pp. 62-71, 2005.
33. R. J. Emerson IV and T. A. Camesano, “On the Importance of Precise Calibration Techniques for an Atomic Force Microscope,” Ultramicroscopy, Vol. 106, pp. 413-422, 2006.
34. N. A. Burnham, X. Chen, C. S. Hodges, G. A. Matei, E. J. Thoreson, C. J. Roberts, M. C. Davies and S. J. B. Tendler, “Comparison of Calibration Methods for Atomic-Force Microscopy Cantilevers,” Nanotechnology, Vol. 14, pp. 1-6, 2004.
35. R. G. Cain, M. G. Reitsma, S. Biggs, and N. W. Page, “Quantitative Comparison of Three Calibration Techniques for the Lateral Force Microscope,” Review of Scientific Instruments, Vol. 72, No. 8, pp. 304-3312, 2001.
36. J. E. Sader and L. White, “Theoretical Analysis of the Static Deflection of Plates for Atomic Force Microscope Applications,” Journal of Applied Physics, Vol. 74, No. 1, pp. 1-9, 1993.
37. J. E. Sader, “Susceptibility of Atomic Force Microscope Cantilevers to Lateral Forces,” Review of Scientific Instruments, Vol. 74, No. 4, pp. 2438-2443, 2003.
38. J. E. Sader and C. P. Green, “In-plane Deformation of Cantilever Plates with Applications to Lateral Force Microscopy,” Review of Scientific Instruments, Vol. 75, No. 4, pp. 878-883, 2004.
39. C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and Torsional Spring Constants of Atomic Force Microscope Cantilevers,” Review of Scientific Instruments, Vol. 75, No. 6, pp. 1988-1996, 2004.
40. J. M. Neumeister and W. A. Ducker, “Lateral, Normal, and Longitudinal Spring Constants of Atomic Force Microscopy Cantilevers,“ Review of Scientific Instruments, Vol. 65, No. 8, pp. 2527-2531, 1994.
41. R. J. Warmack, X. Y. Zheng, T. Thundat, and D. P. Allison, “Friction Effects in the Deflection of Atomic Force Microscope Cantilevers,” Review of Scientific Instruments, Vol. 65, No. 2, pp. 394-399, 1994.
42. M. A. Poggi, A. W. McFarland, J. S. Colton, and. L. A. Bottomley, “A Method for Calculating the Spring Constant of Atomic Force Microscopy Cantilevers with a Nonrectangular Cross Section,” Analytical Chemistry, Vol. 77, pp. 1192-1195, 2005.
43. L. O. Heim, M. Kappl, and H. J. Butt, “Tilt of Atomic Force Microscope Cantilevers: Effect on Spring Constant and Adhesion Measurements,“ Langmuir, Vol. 20, pp. 2760-2764, 2004.
44. R. Lévy and M. Maaloum, “Measuring the Spring Constant of Atomic Force Microscope Cantilevers: Thermal Fluctuations and other Methods,” Nanotechnology, Vol. 13 pp. 33-37, 2002.
45. R. W. Stark, T. Drobek, and W. M. Heckl, “Thermomechanical Noise of a Free V-shaped Cantilever for Atomic-Force Microscopy,” Ultramicroscopy, Vol. 86, pp. 207-215, 2001.
46. H. J. Butt and M. Jaschke, “Calculation of Thermal Noise in Atomic Force Microscopy,” Nanotechnology, Vol. 6, pp. 1-7, 1995.
47. J. L. Hutter and J. Bechfoefer, “Calibration of Atomic-Force Microscope Tips,” Review of Scientific Instruments, Vol. 64, No. 7, pp. 1868-1873, 1993.
48. J. P. Cleveland, S. Manne, D. Bocek, and P. K. Hansma, “A Nondestructive Method for Determining the Spring Constant of Cantilevers for Scanning Force Microscopy,” Review of Scientific Instruments, Vol. 64, No. 2, pp. 403-405, 1993.
49. C. T. Gibson, B. L. Weeks, J. R. I. Lee, C. Abell, and T. Rayment, “A Nondestructive Technique for Determining the Spring Constant of Atomic Force Microscope Cantilevers,” Review of Scientific Instruments, Vol. 72, No. 5, pp. 2340-2343, 2001.
50. C. T. Gibson, B. L. Weeks, C. Abell, T. Rayment, and S. Myhra, “Calibration of AFM Cantilever Spring Constants,” Ultramicroscopy, Vol. 97, pp. 113-118, 2003.
51. J. E. Sader, I. Larson, P. Mulvaney, and. L. R. White, “Method for the Calibration of Atomic Force Microscope Cantilevers,” Review of Scientific Instruments, Vol. 66, No. 7, pp. 3789-3798, 1995.
52. J. E. Sader, J. W. M. Chon, and P. Mulvaney, “Calibration of Rectangular Atomic Force Microscope Cantilevers,” Review of Scientific Instruments, Vol. 70, No. 10, pp. 3967-3969, 1999.
53. J. E. Sader, “Frequency Response of Cantilever Beams Immersed in Viscous Fluids with Applications to the Atomic Force Microscope,” Journal pf Applied Physics, Vol. 84, No. 1, pp. 64-76, 1998.
54. J. W. M. Chon, P. Mulvaney, and J. E. Sader, “Experimental Validation of Theoretical Models for the Frequency Response of Atomic Force Microscope Cantilever Beams Immersed in Fluids,” Journal pf Applied Physics, Vol. 87, No. 8, pp. 3978-3988, 2000.
55. G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, and A. Huang, “Resonant Response of Scanning Force Microscopy Cantilevers,” Review of Scientific Instruments, Vol. 65, No. 8, pp. 2532-2537, 1994.
56. J. D. Holbery and V. L. Eden, “A Comparison of Scanning Microscopy Cantilever Force Constants Determined using a Nanoindentation Testing Apparatus,” Journal of Micromechanics and Microengineering, Vol. 10, pp. 85-92, 2000.
57. A. Torii, M. Sasaki, K. Hane, and S. Okuma, ” A Method for Determining the Spring Constant of Cantilevers for Atomic Force Microscopy,” Measurement Science and Technology, Vol. 7, pp. 179-184, 1996.
58. C. T. Gibson, G. S. Watson, and S. Myhra, “Determination of the Spring Constants of Probes for Force Microscopy/Spectroscopy,” Nannotechnology, Vol. 7, pp. 259-262, 1996.
59. P. J. Cumpson and J. Hedley, “Accurate Analytical Measurements in the Atomic Force Microscope: A Microfabricated Spring Constant Standard Potentially Traceable to the SI,” Nanotechnology, Vol. 14, pp. 1279-1288, 2003.
60. P. J. Cumpson, P. Zhdan, and J. Hedley, “Calibration of AFM Cantilever Stiffness: A Microfabricated Array of Reflective Springs,” Ultramicroscopy, Vol. 100, pp. 241-251, 2004.
61. P. J. Cumpson, J. Hedey, and C. A. Clifford, “Microelectromechanical Device for Lateral Force Calibration in the Atomic Force Microscope: Lateral Electrical Nanobalance,” Journal of Vacuum Science and Technology B, Vol. 23, pp. 1992-1997, 2005.
62. J. H. Fabian, L. Scandella, H. Fuhrmann, R. Berger, T. Mezzacasa, Ch. Musil, J. Gobrecht, and E. Meyer, “Finite Element Calculations and Fabrication of Cantilever Sensors for Nanoscale Detection,” Ultramicroscopy, Vol. 82, pp. 69-77, 2000.
63. M. Müller, T. Schimmel, P. Häußler, H. Fettig, O. Müller, and A. Albers, “Finite Element Analysis of V-shaped Cantilevers for Atomic Force Microscopy Under Normal and Lateral Force Loads,“ Surface and Interface Analysis, Vol. 38, pp. 1090-1095, 2006.
64. J. L. Hazel and V. V. Tsukruk, “Spring Constants of Composite Ceramic/Gold Cantilevers for Scanning Probe Microscopy,” Thin Solid Film, Vol. 339, pp. 249-257, 1999.
65. C. A. Clifford and M. P. Seah, “The Determination of Atomic Force Microscope Cantilever Spring Constants via Dimensional Methods for Nanomechanical Analysis,” Nanotechnology, Vol. 16, pp. 1666-1680, 2005.
66. S. Jeon, Y. Braiman, and T. Thundat, “Torsional Spring Constant Obtained for an Atomic Force Microscope Cantilever,” Applied Physics Letters, Vol. 84, No. 10, pp. 1795.1797, 2004.
67. A. Feiler, P. Attard, and I. Larson, “Calibration of the Torsional Spring Constant and the Lateral Photodiode Response of Frictional Force Microscopes,” Review of Scientific Instruments, Vol. 71, No. 7, pp. 2746-2750, 2000.
68. M. Varenberg, I. Etsion, and G. Halperin, “An Improved Wedge Calibration Method for Lateral Force in Atomic Force Microscopy,” Review of Scientific Instruments, Vol. 74, No. 7, pp. 3362-3367, 2003.
69. S. Xie, W. Li, Z. Pan, B. Chang, and L. Sun, “Mechanical and Physical Properties on Carbon Nanotube,” Journal of Physics and Chemistry of Solids, Vol. 61, pp. 1153-1158, 2000.
70. B. G. Demczyk, Y. M. Wang, J. Cumings, M. Hetman, W. Han, A. Zettl, and R. O. Ritchie, “Direct Mechanical Measurement of the Tensile Strength and Elastic Modulus of Multiwalled Carbon Nanotubes,” Materials Science and Engineering A, Vol. 334, pp. 173-178, 2002.
71. X. Li, H. Gao, C. J. Murphy, and K. K. Caswell, “Nanoidentation of Silver Nanowires,” Nano Letters, Vol. 3, No. 11, pp. 1495-1498, 2003.
72. S. Cuenot, S. D. Champagne, and B. Nysten, “Elastic Modulus of Polypyrrole Nanotubes,” Physical Review Letters, Vol. 85, No. 8, pp. 1690-1693, 2000.
73. D. Qian, G. J. Wanger, W. K. Liu, M. F. Yu, and R. S. Ruoff, “Mechanics of Carbon Nanotubes,“ Applied Mechanics Reviews, Vol. 55, No. 6, pp. 495-533, 2002.
74. O. Lourie, and H. D. Wagner, “Transmission Electron Microscopy Observations of Fracture of Single-Wall Carbon Nanotubes under Axial Tension,” Applied Physics Letters, Vol. 73, No. 24, pp. 3527-3529, 1998.
75. O. Lourie, D. M. Cox, and H. D. Wagner, “Buckling and Collapse of Embedded Carbon Nanotubes,” Physical Review Letters, Vol. 81, No. 8, pp. 1638-1641, 1998.
76. H. E. Troiani, M. M. Yoshida, G. A. C. Bragado, M. A. L. Marques, A. Rubio, J. A. Ascencio, and M. J. Yacaman, “Direct Observation of the Mechanical Properties of Single-Walled Carbon Nanotubes and Their Junctions at the Atomic Level,” Nano Letters, Vol. 3, No. 6, pp. 751-755, 2003.
77. D. A. Walters, L. M. Ericson, M. J. Casavant, J. Liu, D. T. Colbert, K. A. Smith, and R. E. Smalley, “Elastic Strain of Freely Suspended Single-Wall Carbon Nanotube Ropes,” Applied Physics Letters, Vol. 74, No. 25, pp. 3803-3805, 1999.
78. P. A. Williams, S. J. Papadakis, A. M. Patel, M. R. Falvo, S. Washburn, and R. Superfine, “Torsional Response and Stiffening of Individual Multiwalled Carbon Nanotubes,” Physical Review Letters, Vol. 89, No. 25, pp. 255502, 2002.
79. P. A. Williams, S. J. Papadakis, A. M. Patel, M. R. Falvo, S, Washburn, and R. Superfine, “Fabrication of Nanometer-Scale Mechanical Devices Incorporating Individual Multiwalled Carbon Nanotubes as Torsional Springs,” Applied Physics Letters, Vol. 82, No. 5, pp. 805-807, 2003.
80. A. M. Fennimore, T. D. Yuzvinsky, W. Q. Han M. S. Fuhrer, J. Cumings, and A. Zettl, “Rotational Actuators Based on Carbon Nanotubes,” Nature, Vol. 424, pp. 408-410, 2003.
81. J. Bernholc, C. Brabec, M. B. Nardelli, A. Maiti, C. Roland, and B. I. Yakobson, “Theory of growth andmechanical properties of nanotubes,” Applied Physics A, Vol. 67, pp. 39-46, 1998.
82. T. Belytschko, S. P. Xiao, G. C. Schatz and R. Ruoff, “Atomistic Simulations of Nanotube Fracture,” Physical Review B, Vol. 65, pp. 235430, 2002.
83. M. M. Franco, H. Kuzmany, and F. Zerbetto, “Mechanical Interactions in All-Carbon Peapods,” Journal of Physics Chemistry B, Vol. 107, pp. 6986-6990, 2003.
84. N. Yao and V. Lordi, “Young’s Modulus of Single-Walled Carbon Nanotubes,” Journal of Applied Physics, Vol. 84, No. 4, pp. 1939-1943, 1998.
85. C. H. Kiang, M. Endo, P. M. Ajayan, G. Dresselhaus, and M. S. Dresselhaus, “Size Effects in Carbon Nanotubes,“ Physical Review Letters, Vol. 81, No. 9, pp. 1869-1872, 1998.
86. D. Srivastava, M. Menon, and K. Cho, “Nanoplasticity of Single-Wall Carbon Nanotubes under Uniaxial Compression,” Physical Review Letters, Vol. 83, No. 15, pp. 2973-2976, 1999.
87. T. Xiao and Liao, “Non-linear Elastic Response of Fullerene Balls under Uniform and Axial Deformations,” Nanotechnology, Vol. 14, pp. 1197-1202, 2003.
88. T. Xiao, X. Xu, and K. Liao, “Characterization of Nonlinear Elasticity and Elastic Instability in Single-Walled Carbon Nanotubes,” Journal of Applied Physics, Vol. 95, No. 12, pp. 8145-8148, 2004.
89. B. I. Yakobson, C. J. Brabec, and J. Bernholc, “Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response,” Physical Review Letters, Vol. 76, No. 14, pp. 2511-2514, 1996.
90. M. R. Falvo, G. J. Clary, R. M. Taylor II, V. Chi, F. P. B. Jr, S. Washburn, and R. Superfine, “Bending and Buckling of Carbon Nanotubes under Large Strain,” Nature, Vol. 389. pp. 582-584, 1997.
91. Y. Hirai, S. Nishimaki, H. Mori, Y. Kimoto, and Y. Tanaka, ”Molecular Dynamics Studies on Mechanical Properties of Carbon Nano Tubes with Pinhole Defects,” Proceeding of Microprocesses and Nanotechnology Conference, pp. 128-129, Nov. 6-8, 2002.
92. C. Li and T. W. Chou, “Elastic Properties of Single-Walled Carbon Nanotubes in Transverse Directions,” Physical Review B, Vol. 69, pp. 073401, 2004.
93. V. Lordi and N. Yao, “Radial Compression and Controlled Cutting of Carbon Nanotubes,” Journal of Chemical Physics, Vol. 109, No. 6, pp. 2509-2512, 1998.
94. S. Iijima, C. Brabec, A. Maiti, and J. Bernholc, “Structural Flecibility of Carbon Nanotubes,” Journal of Chemical Physics, Vol. 104, No. 5, pp. 2089-2092, 1996.
95. J. F. Despres, E. Daguerre, and K. Lafdi, “Flexibility of Graphene Layers in Carbon Nanotubes,” Carbon, Vol. 33, No. 1, pp. 87-92, 1995.
96. W. Shen, B. Jiang, B. S.. Han, and S. S. Xie, “Investigation of the Radial Compression of Carbon Nanotubes with a Scanning Probe Microscope,“ Physical Review Letters, Vol. 84, No. 16, pp. 3634-3637, 2000.
97. T. Hertel, R. Martel, and P. Avouris, “Manipulation of Individual Carbon Nanotubes and Their Interaction with Surfaces,” Journal of Physics Chemistry B, Vol. 102, pp. 910-915, 1998.
98. W. H. Knechtel, G. S. Düsberg, W. J. Blau, E. Hernández ,and A. Rubio, “Reversible Bending of Carbon Nanotubes using a Transmission Electron Microscope,“ Applied Physics Letters, Vol. 73, No. 14, pp. 1961-1963, 1998.
99. V. H. Crespi, N. G. Chopra, M. L. Cohen, A. Zettl, and S. G. Louie, “Anisotropic Electron-Beam Damage and the Collapse of Carbon Nanotubes,” Physical Review B, Vol. 54, No. 8, pp. 5927-5931, 1996.
100. G. L. Hwang and K. C. Hwang, “Breakage, Fusion, and Healing of Carbon Nanotubes,” Nano Letters, Vol. 1, No. 8, pp. 435-438, 2001.
101. L. M. Peng, Z. L. Zhang, Z. Q. Xue, Q. D. Wu, Z. N. Gu, and D. G. Pettifor, “Stability of Carbon Nanotubes: How Small Can They Be?,” Physical Review Letters, Vol. 85, No. 15, pp. 3249-3252, 2000.
102. L. J. Hall, V. R. Coluci, D. S. Galvão, M. E. Kozlov, M. Zhang, S. O. Dantas, R. H. Baughman, “Sign Change of Poisson’s Ratio for Carbon Nanotube Sheets,” Science, Vol. 320, pp. 504-507, 2008
103. ANSYS Element Reference 000855. English Edition. SAS IP, Inc.
104. M.C. Lin and M.K. Yeh, "Buckling and Postbuckling of Spherical Shells with Apical Cutout under Ring Load," AIAA Journal, Vol. 33, No. 9, pp. 1728-1733, 1995.
105. R. S. Gates and J. R. Pratt, “Prototype Cantilevers for SI-traceable Nanonewton Force calibration,” Measurement Science and Technology, Vol. 17, pp. 2852-2860, 2006.
106. C. T. Gibson, D. A. Smith, and C. J. Roberts, “Calibration of Silicon Atomic Force Microscope Cantilevers,” Nanotechnology, Vol. 16, pp. 234-238, 2005.
107. J. F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford, U.K., 1960
108. R. S. Gates and J. R. Pratt, “Prototype Cantilevers for SI-Traceable Nanonewton Force Calibration,” Material Science and Technology, Vol. 17. pp. 2852-2860, 2006.
109. J. J. Wortman and R. A. Evans, “Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon and Germanium,” Journal of Applied Physics, Vol. 38, No. 1, pp. 153-156, 1965.
110. J. J. Hall, “Electronic Effects in the Elastic Constants of n-Type Silicon,” Physical Review, Vol. 161, No. 3, pp. 756-761, 1967.
111. H. J. Mcskinmin and P. Anderatch Jr., “Measurement of Third-Order Moduli of Silicon and Germanium,” Journal of Applied Physics, Vol. 35, No. 11, pp. 3312-3319, 1964.
112. CASTEP is a commercial implementation of the CAmbridge Serial Total Energy Package into of the Cerius2 code of Molecular Simulations Inc. described in V. Milman, B. Winkler, J. A. White, C. J. Pickard, M. C. Payne, E. V. Akhmatskaya, and R. H. Nobes, “Electronic Structure, Properties, and Phase Stability of Inorganic Crystals: A Pseudopotential Plane-Wave Study,” International Journal of Quantum Chemistry, Vol. 77, pp. 895-910, 2000.
113. CASTEP User Guide, English Edition, 2004, Accelrys, Inc.
114. N. H. Tai, M. K. Yeh, and J. H. Liu, “Enhancement of the Mechanical Propertries of Carbon Nanotube/Phenolic Composites using a Carbon Nanotube Network as the Reinforcement,” Carbon, Vol. 42, pp. 2735-2777, 2004.
115. http://www.nanoworld.com/print/General_Description.html (2008)
116. http://www.nanosensors.com/faq.html (2008)
117. J. Cho, E. C. Park, and E. Yoon, “AFM probe Tips using Heavy Boron-Doped Silicon Cantilevers Realized in a <100> Bulk Silicon Wafer,” Microprocesses and Nanotechnology Conference, pp. 230-231, Tokyo, Japan, July 2000.