測量電磁場引起的現象一直是一個熱門的研究主題，這樣的研究提高了對於奈米尺度元件中微弱作用力的分辨率。這些研究中通常需要具有奈米級橫截面的機械諧振器做為光機換能器，以提高對於機械振盪行為的分辨率和靈敏度。這些奈米機械諧振器 (Nanomechanical Resonators, NMR)的靈敏度超過了電子同類設備。

本論文研究了施加在奈米機械諧振器上的電動、光熱、和布朗效應。電動效應源於對 NMR 中的電容施加 AC 和 DC 電壓，導致電容的受迫振盪。光熱效應源於NMR對於入射電磁波的吸收，導致NMR因受限熱膨脹而發生頻率偏移。NMR振動行為因熱激發產生的起伏則與布朗效應有關。

上述效應的研究由於不易觀察，具有極高挑戰性。此乃因為NMR鼓面振動之位移會隨著鼓面縮小而下降，導致讀取電路之訊雜比劣化。為了解決此問題，我們採用次微米級厚度的凡德瓦(van der Waals)材料，如二硒化鈮(NbSe2$) )和石墨(graphite)，並將其製備成懸浮的圓形鼓面。這樣的微機械鼓面連同基板形成法布里-伯羅干涉儀結構，其干涉效果提供了NMR振盪行為在光學或微波頻段的檢測機制。

NMR的振幅經過校準可用於測量其電力驅動效應。 根據多層干涉法 (MIA)，光波在多層結構中的干涉可用於確定NMR與任意基板間的空氣間隙與NMR鼓面厚度。利用這些參數，量測信號將被轉換 成NMR振盪行為之資訊，包括數百皮米的振幅和鼓面上數百皮牛頓的力。

NMR鼓面共振頻率對探測光入射功率的靈敏依賴可用來量化NMR因光吸收引起的光熱效應。所測得的靈敏度變化與理論模型吻合良好，該模型考慮了對於法布里-伯羅腔所誘導的吸收與靜態位移的調變。此外，經由上述機制所測得的NbSe2$多層材料之熱導率為14.5 W m$^{−1}$ K$^{−1}$，與文獻結果非常吻合。根據模型所預測的光熱響應對NMR厚度和材料的相依性，採用低張力、低熱導率的薄鼓面將可實現可調靈敏度的輻射熱計。

最後，超導“法布里-伯羅”微波共振腔可用於解析懸浮石墨諧振器因布朗運動引發的振盪模式。所測得的各模式之頻率間隔顯示，懸浮石墨諧振器振盪在mK溫度下表現得像固定邊界平板。此外，諧振器基模的邊帶功率隨微波探測訊號功率呈線性增加。此線性度有助於確定單光子極限下 約110 mHz 的光機械耦合強度。

對於量測學、輻射熱測量、及量子計算相關應用，本論文研究成果及發現將有助於優化可調性奈米諧振器的開發。

Measuring electromagnetic field-induced phenomena has been a subject of ongoing research that led to improved resolution of weak forces, which is associated with nanoscale phenomena. These endeavors require a vibrating transducer whose resolution and sensitivity improve by reducing one of its cross-section dimensions to nanoscale dimensions. Named nanomechanical resonators (NMRs), these nanoscale devices make force meters with sensitivities surpassing their electronic counterparts.

This dissertation investigates electromotive, photothermal, and Brownian effects exerted on the nanomechanical resonator. The electromotive effect originates from the application of AC and DC
voltages to the NMR, setting the vibrating capacitor to oscillatory motion. The photothermal effect in NMRs emerges from the incident absorption of an electromagnetic wave on the NMR surface, causing the NMR frequency to shift due to constrained thermal expansion. Brownian effect pertains to collective vibrational fluctuations of the NMR originating from thermal excitations.

These phenomena are challenging to resolve as the NMR displacement scales with shrinking lateral dimensions, reducing the signal-to-noise ratio measured by the readout circuitry. Circumventing this issue requires circular drumheads made from freestanding layered van der Waals materials such as NbSe$_{2}$ and graphite with submicron thicknesses. Furthermore, Fabry-Pérot interferometric readouts and microwave cavities are used to detect the vibrational displacement of NMRs.

The calibrated vibrational amplitudes of the NMR quantify the electromotive actuation effects. Wave interferences of light from multiple layers at optical wavelengths, termed the ”Multilayer Interference Approach (MIA),” are exploited to determine spacer gap heights and resonator thicknesses on arbitrary substrates. These parameters converted the resolved signals to hundreds of picometers in vibrational amplitude and hundreds of piconewtons of forces for our drumhead NMRs.

The photothermal effect of the NMR due to optical absorption is then quantified using the sensitivity of the resonant frequency shift of electromotive-driven drumheads with the incident power of the probe laser beam. The measured sensitivity variations of the NMR drumheads agree well with a proposed model that considers modulation of Fabry-Pérot cavity-induced absorption with static displacement. Furthermore, a thermal conductivity of multilayered NbSe$_{2}$ is determined to be 14.5 W m$^{−1}$ K$^{−1}$, which is in good agreement with literature results. The model predicts the thickness, and material dependencies of the photothermal responsivity. Thinner drumheads with low tension and small thermal conductivity values enable bolometers with displacement-tunable sensitivity.

Finally, the modes of a suspended graphite resonator enabled by Brownian motion are resolved using a superconducting microwave cavity, a microwave analog of Fabry-Pérot cavities. The measured resonant frequency spacings between the resolved modes suggest the device behaves like a clamped plate at millikelvin temperatures. Furthermore, the mechanical sideband power of the fundamental mode increases linearly with the incident microwave probe power. The linearity helped determine an optomechanical coupling strength of 110 mHz in the single-photon limit.

The findings regarding the investigated effects would hopefully translate to an optimized design for tunable NMR-based optoelectromechanical systems for metrology, bolometric, and possible quantum computer applications.

Abstract ZH i
Abstract EN ii
Preface iii
Acknowledgement iv
1 Introduction 1
1.1 Micro-and Nano-electromechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation with van der Waals Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Content of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Nanomechanics, Electromechanics, and Optomechanics 5
2.1 Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Equation of Motion of a Driven Damped Mechanical Oscillator . . . . . . . . . . . 5
2.1.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Electromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Electromotive Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Static Displacements of Tensioned Plates due to Electrostatic Force . . . . . . . . 9
2.2.3 Vibrational Amplitude due to Electromotive Force . . . . . . . . . . . . . . . . . . 11
2.3 Optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Reflectance from Fabry-P´erot Cavities derived from Multilayer Interference Approach 14
2.3.2 Amplitude-Modulated Sideband Voltages from Microwave Cavities . . . . . . . . . 17
3 Device Fabrication and Measurement 21
3.1 vdW Nanomechanical Resonator Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Selection of vdW Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Micromechanical Exfoliation with Polydimethylsiloxane Disks . . . . . . . . . . . . 23
3.1.3 Photolithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.4 Procedure for Stamp Transfer Process . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Optical Interferometry Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Microwave Interferometry Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Vibrational Amplitude Calibration of Nanomechanical Resonators Based on van der
Waals Materials by Fabry-P´erot Interferometry 30
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.2 Setbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Measurement and Characterization Technique . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Determining NMR Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Key Merits of MIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Photothermal Responsivity of Nanomechanical Resonators Based on van der Waals
Materials 40
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Design and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.1 Brief Description Regarding the Device Structure, Actuation, and Detection of
Niobium Diselenide NMRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Understanding Photothermal Responsivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3.1 Effect of Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3.2 Description of the Photothermal Effect . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3.3 Using the Multilayer Interference Approach to Calculate the Fabry-P´erot Absorbance 44
5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4.1 Measured Driven Mechanical Responses . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4.2 Modelling of the Photothermal Effect on Tensioned Drum Plates . . . . . . . . . . 45
5.4.3 Variations of f0 on the Laser Spot Position . . . . . . . . . . . . . . . . . . . . . . 48
5.4.4 Influence of Increasing Static Displacement on the Photothermal Responsivities of
NbSe2 Drumheads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4.5 Influence of Drum Thickness on the Profile representing the Photothermal Responsivities
of NbSe2 Drumheads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4.6 Benchmarking Simulated Results with those of NMR Designs Communicated Elsewhere
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6 Microwave Interferometry of Graphite Mechanical Resonators 53
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 Device and Cavity Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 Power Spectral Density of Nanomechanical Modes . . . . . . . . . . . . . . . . . . . . . . 56
6.4 Extracting Single-Photon Optomechanical Coupling Strength . . . . . . . . . . . . . . . . 58
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7 Conclusion, and Outlook 61

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