石英音叉振盪器自然頻率之分析與量測及線性外力對頻率之影響分析

Abstract

本文主要研究對象為 單端固定音叉式石英振盪器，分析石英音叉之特徵頻率及線性外力對石英音叉振盪器共振頻率的影響，並探討凡得瓦力造成共振頻率飄移量與量測初始距離之關係。本文針對異相振盪行為進行分析，中央音叉樑視為尤拉樑，質量塊則依據音叉樑異相振盪對質量塊所造成的力矩來建立翹曲形變的模型。耦合部分，假設其結構為彈性體，分別討論各自結構自由振動行為，再利用漢米頓定理推導出運動統御方程式及邊界條件，用以耦合機械性質連續關係；並討論石英音叉在幾何尺寸上對其特徵頻率的影響，分析尺寸對於共振頻率之敏感度，以此作製程誤差分析及光罩設計之參考並以此為音叉振盪器的設計依據，於頻率準確度與製程難度中取得平衡，以達到良率提升之目的。 此外，針對石英音叉振盪器分析在尖端受外力及凡得瓦力對共振頻率的影響，利用Lennard-Jones勢能模型推導出與距離有關的非線性力，並將此力藉由泰勒展開式擬線性化並將其代入所推導出的統御方程式及其邊界條件，並求得在不同距離時凡得瓦力對共振頻率的影響，建立此分析模型所推導出的凡得瓦力距離對頻率影響關係並應用於原子顯微鏡之量測精度與準確性的改良。

The thesis mainly studied about single-ended tuning fork quartz resonator, analysis the natural frequency of Quartz tuning fork (QTF) resonator, and the effect of resonant frequencies of the quartz tuning fork by linear external force, and discussion on the relation between the frequency shift caused by Van der Waals Force and the initial detected distance. Focusing on the analysis of the anti-phase mode, the study regards a central tuning fork as the Euler beam, and two ends of the proof masses built the warping model according to the moment caused by the anti-phase mode from the tuning fork to the proof masses. Assuming the coupling structure as an elastic body, and discussing the free vibration to each structure; using the Hamilton's principle to derive the governing equation and the boundary condition for coupling the continuous mechanical relationship. For understanding the impact of the process error to the resonant frequency and the convenience for designing, the study has analyzed the impact of the dimension to the sensitivity of the resonant frequency for being the references of the process error analysis and the design of the mask; in order to improve the yield, it must get the balance between the frequency accuracy and the difficulty during the process. Furthermore, for analyzing the effect of resonant frequencies of QTF by linear external force and Van der Waals force acting on the tip, this study using Lennard-Jones potential model to derive the nonlinear force related with distances and making the force linearization by Tayler expansion. Introducing the Van der Waals force into the governing equation and boundary conditions and analyzing the effect of resonant frequencies on different distances. This research make a relationship between the distance of force and frequency shift, this relationship can be use to improve the precision of atomic force microscope.