本論文通過模態分析探討了納米機械振盪器的運動性質.納米機械振盪器是自然的能量傳感器, 因為它們對施加的力（例如電力和光學力）非常敏感. 這使得它們適用於需要與弱信號交互的各種應用, 例如光學微波轉換器, 這是量子混合系統中必不可少的一個角色. 可用於研究納米機械振盪器的工具是模態分析, 它以固有頻率和模態形狀的形式確定其動態特性. 通過這種分析, 可以闡明系統的固有特性, 例如材料特性, 幾何形狀和邊界條件, 所有這些對於納米機械振盪器的利用都是至關重要的. 為了實現這一點, 需要一種驅動手段, 例如電動勢, 由於其可控性和局部性, 它允許對納米機械振盪器進行詳細探測. 此外, 空間檢測方法 (例如法布里-珀羅配置中機械振盪器的激光干涉測量法) 同樣重要. 在該實驗中, 由石墨和 NbSe_2 等二維材料製成的納米機械振盪器子類主要是因為它們的低質量和高靈活性, 而且還因為它們的電子和光學特性適合驅動和檢測手段. 通過觀察納米機械振盪器對電刺激變化的響應, 我們發現多層鼓面對電磁波的傳導能力比它們的超薄對應物提供的更多. 更重要的是, 我們能夠通過模態分析更深入地利用納米機械振盪器的空間方面. 我們不僅能夠通過實驗可視化納米機械模式, 我們還揭示了驅動力和共振模式的相互作用. 我們發現驅動力形狀與共振模式的投影決定了納米機械振盪在整個頻譜中的表現.

This dissertation explores the nature of motion of nanomechanical oscillators through modal analysis. Nanomechanical oscillators are natural transducers of energy because they respond very sensitively to applied forces such as electrical and optical forces. This makes them suitable for diverse applications that require interaction with weak signals, such as an optical microwave converter, a role that is essential in quantum hybrid systems. A tool that can be used to study nanomechanical oscillators is modal analysis, which determines its dynamic characteristics in forms of natural frequencies and modal shapes. Through this analysis, the inherent properties of the system such as material properties, geometry, and boundary conditions, all of which are critical for the utilization of nanomechanical oscillators, could be elucidated. To realize this, a driving means such as the electromotive force, which allows for the probing of nanomechanical oscillators in detail because of its controllability and locality, is necessary. Furthermore, a spatial detection method such as laser interferometry of mechanical oscillators in a Fabry-Perot configuration is equally crucial. For the experiment, a sub-class of nanomechanical oscillators made from two-dimensional materials, such as graphite and NbSe_2, were fabricated primarily because of their low mass and high flexibility, but also because their electronic and optical properties fit the driving and detection means. By observing the response of the nanomechanical oscillators with respect to the variations in electrical stimuli, we found that the transductive capability of multilayered drumheads for electromagnetic waves offer more than their ultrathin counterparts. More importantly we were able to make use of the spatiality aspect of the nanomechanical oscillators with more depth through modal analysis. Not only were we able to visualize the nanomechanical modes experimentally we also revealed the interaction of the driving force and the resonance modes. We discover that the projection of the shape of the driving force with the resonance modes determine how the nanomechanical oscillations behave across the frequency spectrum.

Abstract iii
Acknowledgements v
1 Nanomechanical oscillations 1
1.1 Nanomechanical oscillators in quantum systems . . . . . . . . . . . . . 1
1.2 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Mechanical tension and flexural rigidity . . . . . . . . . . . . . . . . . . 3
1.4 Electrical force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1 Time-dependent electrical force . . . . . . . . . . . . . . . . . . . 4
1.4.2 Electrical force influence to mechanical amplitude . . . . . . . . 5
1.5 Resonance modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.1 Normal mode expansion . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.2 Matrix form analysis of normal mode expansion . . . . . . . . . 7
2 Fabrication and detection of nanomechanical osillators 9
2.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Two-dimensional materials . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Deterministic transfer . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Optical detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Laser interferometry . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Electrically driven multilayered nanomechanical resonators 15
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Description and characterization of the device . . . . . . . . . . 15
3.2.2 Electrostatic tunability . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Optomechanical responsivity . . . . . . . . . . . . . . . . . . . . 19
3.3.2 Multilayered NMR model . . . . . . . . . . . . . . . . . . . . . . 20
3.3.3 Factors affecting tunability . . . . . . . . . . . . . . . . . . . . . 22
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Detailed simulations and analysis . . . . . . . . . . . . . . . . . . . . . . 26
3.5.1 Simulated results of mechanical frequency dependence . . . . . 26
3.5.2 Multiple interface approach . . . . . . . . . . . . . . . . . . . . . 28
4 Experimental modal analysis 30
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 Description and characterization of the device . . . . . . . . . . 31
4.2.2 Spatial response mapping . . . . . . . . . . . . . . . . . . . . . . 31
4.2.3 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Modal weights from experiment . . . . . . . . . . . . . . . . . . 34
Modal analysis without damping using FEM simulation . . . . 37
Modal weight dependence on driving frequency . . . . . . . . . 37
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.1 Effect of driving force on the modal shapes . . . . . . . . . . . . 39
4.3.2 Off-resonance modal analysis’ role in nanomechanical resonator
studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5.1 Experimental modal weight data . . . . . . . . . . . . . . . . . . 41
4.5.2 FEM simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.6 Detailed COMSOL simulations . . . . . . . . . . . . . . . . . . . . . . . 42
4.6.1 Modal weights for extended frequency range . . . . . . . . . . . 42
4.6.2 Modal weights dependency on eccentricity . . . . . . . . . . . . 45
4.6.3 Modal analysis using a beam geometry . . . . . . . . . . . . . . 46
4.6.4 Modal weights dependency on damping parameter . . . . . . . 47
5 Conclusion 48
Bibliography 49

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