Fundamentals of Nanomechanical Resonators

Description

Now in an updated second edition, this classroom-tested textbook introduces and summarizes the latest models and skills required to design and optimize nanomechanical resonators, taking a top-down approach that uses macroscopic formulas to model the devices. The authors cover the electrical and mechanical aspects of nanoelectromechanical system (NEMS) devices in six expanded and revised chapters on lumped-element model resonators, continuum mechanical resonators, damping, transduction, responsivity, and measurements and noise. The applied approach found in this book is appropriate for engineering students and researchers working with micro and nanomechanical resonators. Preface Contents List of Symbols 1 Lumped-Element Model Resonators 1.1 Damped Linear Resonator 1.1.1 Free Undamped Vibration 1.1.2 Free Damped Vibration 1.1.3 Driven Damped Vibration 1.1.3.1 Quality Factor 1.1.3.2 Transient Response 1.2 Coupled Linear Resonators 1.2.1 Coupled Undamped Resonators 1.2.1.1 Avoided Crossing 1.2.2 Driven Coupled Resonators 1.3 Damped Nonlinear Resonators 1.3.1 Sources of Nonlinearity 1.3.1.1 Material Nonlinearity 1.3.1.2 Geometric Nonlinearity 1.3.1.3 Actuation Nonlinearity 1.3.1.4 Detection Nonlinearity 1.3.1.5 Nonlinear Damping 1.3.1.6 Other Types of Nonlinearity 1.4 Parametric Amplification References 2 Continuum Mechanical Resonators 2.1 Rayleigh-Ritz Method 2.2 One-Dimensional Bending Vibrations 2.2.1 Free Bending Vibration of Beams 2.2.1.1 Cantilevers 2.2.1.2 Bridges 2.2.2 Free Bending Vibration of Beams Under Tensile Stress (Strings) 2.3 Two-Dimensional Bending Vibrations 2.3.1 Free Bending Vibration of Plates 2.3.1.1 Rectangular Plates 2.3.1.2 Circular Plates 2.3.2 Free Bending Vibration of Membranes 2.3.2.1 Rectangular Membranes 2.3.2.2 Circular Membranes 2.4 One-Dimensional Bulk Vibrations 2.5 Torsional Vibration of Thin Beams 2.5.1 Torsional Paddle Resonator 2.6 Effective Parameters 2.6.1 Energy Approach 2.6.2 Galerkin's Method 2.7 Geometric Nonlinearity 2.7.1 Geometric Nonlinearity of Doubly Clamped Beams and Strings 2.7.2 Geometric Nonlinearity of Rectangular Plates or Membranes 2.7.3 Nonlinear Mode Coupling References 3 Damping 3.1 Medium Interaction Losses 3.1.1 Liquid Damping 3.1.1.1 Resonator Immersed in Liquid 3.1.1.2 Liquid Inside the Resonator 3.1.2 Gas Damping 3.1.2.1 Fluidic Regime (Kn1) 3.2 Clamping Loss 3.2.1 Cantilever Beams 3.2.2 Membranes 3.2.3 Strategies Against Clamping Losses 3.3 Intrinsic Loss 3.3.1 Friction Losses 3.3.1.1 Time-Temperature Equivalence 3.3.1.2 Surface Friction 3.3.2 Fundamental Losses 3.3.2.1 Thermoelastic Damping 3.3.2.2 Phonon-Phonon Interaction Loss (Akhiezer Damping) 3.3.3 Dissipation Dilution 3.3.3.1 Dissipation Dilution in Strings 3.3.3.2 Dissipation Dilution in Membranes 3.3.3.3 Soft Clamping References 4 Transduction 4.1 Electrodynamic Actuation and Detection 4.1.1 Lorentz Force on a Wire 4.1.2 Electrodynamically Induced Voltage (Electromotive Force) 4.2 Electrostatic Actuation and Detection 4.2.1 Electrostatic Forces 4.2.1.1 Forces Between Electrodes 4.2.1.2 Dielectric Polarization Force 4.2.1.3 Electrostatic Actuation Nonlinearity 4.2.2 Capacitively Induced Current 4.2.2.1 Transimpedance Amplifier 4.2.2.2 LC Filter 4.2.3 Other Capacitive Detection Schemes 4.3 Thermoelastic Actuation 4.4 Piezoresistive Detection 4.5 Piezoelectric Actuation and Detection 4.5.1 Piezoelectric Actuation 4.5.2 Piezoelectric Detection 4.6 Optic Actuation and Detection 4.6.1 Optical Forces 4.6.2 Interferometric Detection 4.6.2.1 Fabry-Pérot Cavity 4.6.2.2 Mach-Zehnder Interferometer 4.6.2.3 Michelson Interferometer 4.6.2.4 Laser-Doppler Vibrometer 4.6.3 Beam Deflection Detection 4.6.3.1 Optical Leverage 4.6.3.2 End-Coupled Optical Waveguide 4.6.4 Scattering Optomechanical Transduction References 5 Measurements and Noise 5.1 Amplitude Noise 5.1.1 Fundamentals 5.1.1.1 Transduction Chain Noise Transfer 5.1.1.2 Noise Referred to Input (RTI) 5.1.2 Thermomechanical Fluctuations 5.1.2.1 Amplitude Calibration and Thermometry 5.1.3 Transduction-Related Noise 5.1.3.1 Johnson-Nyquist Thermal Noise 5.1.3.2 Shot Noise 5.1.3.3 Hooge (1/f) ``Flicker'' Noise 5.1.3.4 Noise Equivalent Circuit 5.1.3.5 Amplifier Noise 5.1.3.6 Noise Figure and Noise Temperature 5.2 Noise in Resonance Frequency Measurements 5.2.1 Allan Variance 5.2.2 Frequency Noise from Thermomechanical and Transduction Noise 5.2.2.1 Open-Loop Scheme (OLS) 5.2.2.2 Phase-Locked Loop (PLL) 5.2.2.3 Self-Sustained Oscillator (SSO) 5.2.3 Frequency Noise from Thermal Fluctuations References 6 Responsivity and Sensitivity 6.1 Response to Change of Mass (Rm) 6.1.1 Response to Point Mass 6.1.1.1 Response of Strings to Point Mass 6.1.1.2 Response of Beams to Point Mass 6.1.2 Response to Distributed Mass 6.2 Response to Change of Effective Spring Constant 6.2.1 Response to Force Gradient (RF') 6.2.1.1 Electrostatic Force Gradients in Parallel Plate Capacitors 6.2.2 Response to Temperature (RT) 6.2.2.1 Response to Temperature of Stressed Resonators 6.2.2.2 Response to Temperature of Unstressed Resonators 6.2.3 Response to Local Heating (RP) 6.2.3.1 Local Heating at String Center 6.2.3.2 Even Heating of Drumhead 6.2.3.3 Local Heating of Drumhead Center References Index