Fundamentals of Nanomechanical Resonators (eBook)

Silvan Schmid is Associate Professor of Micro and Nanotechnology at the Technical University of Denmark. His research focus is in micro- and nanomechanical resonators for sensor applications and fundamental research.Guillermo Villanueva is Professor at Ecole Polytechnique Federale de Lausanne (EPFL). His fields of expertise are MEMS/NEMS, sensors, oscillators, nonlinear and coupled dynamics, and fundamental noise processes.Michael Lee Roukes is Robert M. Abbey Professor of Physics, Applied Physics, and Biological Engineering at California Institute of Technology. He was founding Director of Caltech’s Kavli Nanoscience Institute from 2003-2006, and Co-Director from 2008-2013.

Preface 6
Contents 8
1 Resonance Frequency 10
1.1 Eigenmodes of Ideal Continuum Mechanical Structures 11
1.1.1 One-Dimensional Bending Vibrations 15
1.1.1.1 Free Bending Vibration of Beams 16
1.1.1.2 Free Bending Vibration of Beams Under Tensile Stress (Strings) 23
1.1.2 One-Dimensional Bulk Vibrations 26
1.1.3 Two-Dimensional Bending Vibrations 29
1.1.3.1 Free Bending Vibration of Plates 30
1.1.3.2 Free Bending Vibration of Plates Under Tensile Stress (Membranes) 34
1.1.4 Torsional Vibration of Thin Beams 36
1.2 Lumped-Element Model Resonator 38
1.2.1 Damped Linear Resonator 38
1.2.1.1 Free Undamped Vibration 39
1.2.1.2 Free Damped Vibration 40
1.2.1.3 Driven Damped Vibration 41
1.2.1.4 Quality Factor 44
1.2.1.5 Effective Parameters 47
1.2.1.6 Torsional Paddle Resonator 49
1.2.2 Coupled Linear Resonators 51
1.2.3 Damped Nonlinear Resonators 54
1.2.3.1 Sources of Nonlinearity 55
1.2.3.2 Solving the Nonlinear Equation of Motion 59
References 63
2 Quality Factor 66
2.1 Medium Interaction Losses 67
2.1.1 Liquid Damping 67
2.1.1.1 Resonator Immersed in Liquid 67
2.1.1.2 Liquid Inside the Resonator 68
2.1.2 Gas Damping 70
2.1.2.1 Fluidic Regime (Kn< 1)
2.1.2.2 Ballistic Regime (Kn> 1)
2.2 Clamping Loss 75
2.2.1 Cantilever Beams 76
2.2.2 Membranes 77
2.3 Intrinsic Damping 78
2.3.1 Intrinsic Damping Mechanisms 78
2.3.1.1 Friction Losses 79
2.3.1.2 Fundamental Losses 86
2.3.2 Damping Dilution in Strings and Membranes 90
2.3.2.1 Damping Dilution in Strings 91
2.3.2.2 Damping Dilution in Membranes 94
References 97
3 Responsivity 100
3.1 Frequency Response to Mass 101
3.1.1 Point Mass 102
3.1.1.1 Strings 104
3.1.1.2 Beams 106
3.1.2 Distributed Mass 108
3.2 Amplitude and Frequency Response to Force 110
3.2.1 Amplitude Response to a Force 110
3.2.1.1 Quasi-Static Force Sensing (??) 111
3.2.1.2 Resonant Force Sensing (?=?) 111
3.2.2 Frequency Response to a Force Gradient 111
3.2.2.1 Frequency Response to an Electrostatic Potential 112
3.3 Frequency Response to Ambient Temperature and Local Heating 114
3.3.1 Stress Released Resonators 115
3.3.2 Resonators Under Tensile Stress (Strings) 117
3.3.2.1 Ambient Temperature 117
3.3.2.2 Local Heating at String Center 120
References 122
4 Transduction 124
4.1 Electrodynamic (Actuation and Detection) 125
4.1.1 Lorentz Force on a Straight Wire 126
4.1.2 Electrodynamically Induced Voltage (Electromotive Force) 127
4.2 Electrostatic (Actuation and Detection) 128
4.2.1 Electrostatic Forces 129
4.2.1.1 Forces Between Electrodes 130
4.2.1.2 Dielectric Polarization Force 132
4.2.2 Capacitively Induced Current 135
4.2.3 Other Capacitive Detection Schemes 140
4.3 Thermoelastic (Actuation) 140
4.4 Piezoresistive (Detection) 141
4.5 Piezoelectric (Actuation and Detection) 143
4.5.1 Piezoelectric Actuation 145
4.5.2 Piezoelectric Detection 146
4.6 Optic (Actuation and Detection) 147
4.6.1 Optical Forces 147
4.6.2 Interferometric Detection 148
4.6.3 Beam Deflection Detection 150
4.6.3.1 Optical Leverage 150
4.6.3.2 End-Coupled Optical Waveguide 151
4.6.4 Plasmonic Detection 151
References 152
5 Measurement and Noise 157
5.1 Amplitude Noise 157
5.1.1 Fundamentals 158
5.1.1.1 Transduction Chain Noise Transfer 158
5.1.1.2 Noise Referred to Input (RTI) 158
5.1.2 Thermomechanical Fluctuations 159
5.1.2.1 Amplitude Calibration 162
5.1.3 Transduction Related Noise 163
5.1.3.1 Johnson–Nyquist Thermal Noise 163
5.1.3.2 Shot Noise 165
5.1.3.3 Hooge (1/f) ``Flicker'' Noise 166
5.1.3.4 Noise Equivalent Circuit 167
5.1.4 Amplifier Noise 167
5.1.4.1 Noise Figure and Noise Temperature 170
5.2 Frequency Noise 171
5.2.1 Phase-Locked Loop 171
5.2.2 Self-Sustained Oscillator 173
5.2.3 Allan Variance 176
References 178
Index 181