原子力顯微鏡探針懸臂樑彈簧常數及奈米碳管機械性質量測

Spring Constant of Atomic Force Microscope Cantilevers and Mechanical Property Measurement of Carbon Nanotubes

陳伯毅(Bo-Yi Chen)

指導教授 ： 葉孟考 ； 戴念華

國立清華大學/工學院/動力機械工程學系/博士(2008年)

https://doi.org/10.6843/NTHU.2008.00278

摘要

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。

關鍵字

原子力顯微鏡 ； 彈簧常數 ； 奈米碳管 ； 機械性質量測

並列摘要

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.

並列關鍵字

Atomic force microscope ； spring constants ； carbon nanotubes ； mechanical properties measurement

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