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Nanoelectromechanical Systems Continuum Solid: Equations of Motion PASI 2006 - Bariloche, Argentina Lectures M.L. Roukes, Caltech NEMS: multiterminal mechanical devices introduction to nano- ELECTRICAL ELECTRICAL input output INPUT mechanical OUTPUT transducer mechanical transducer nanoelectromechanical systems SIGNAL mechanical SIGNAL stimulus system response mechanical perturbation ELECTRICAL control CONTROL transducer SIGNAL mechanical domain Michael Roukes -signal processing Kavli Nanoscience Institute Physics, Applied Physics, & -measurement / sensing Bioengineering -computation California Institute of Technology © 2006 Continuum Solid: Equations of Motion Isotropic solid with local displacement u(r,t) 2 ∂ u 2 ρ = ()()λ + μ ∇ ∇ ⋅u + μ∇ u μ(3λ + 2μ) ∂t 2 Young’s modulus E = Fundamentals: important concepts λ + μ λ, μ are the Lamé constants: λ – beam mechanics (review) λ = c μ = c = (c −c )/2 Poisson ratio ν = 12 44 11 12 2λ + 2μ – responsivity sensitivity –noise – dynamic range Boundary conditions define stress or strain: – nonlinear response Fixed boundary: u(rb ,t) = 0 for rb on boundary Stress-free boundary: T (rb ,t)⋅nˆ = 0 for rb on boundary, n ˆ unit normal Tii = (λ + 2μ)∇ ⋅u nˆ ⎛ ∂u ∂u j ⎞ T = μ⎜ i + ⎟ ij ⎜ ∂r ∂r ⎟ ⎝ j i ⎠ rb see Auld, Acoustic Waves and Fields in Solids (2nd ed.) 1990 © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © 2006 (c) Caltech, 2006 -all rights reserved- Page 1 of 28 1 PASI 2006 - Bariloche, Argentina Lectures M.L. Roukes, Caltech Flexural Beams: Equations of Motion Flexural Beams: Solutions Yn (x,t) = an (cosβn x − coshβn x)+ bn (sin βn x − sinh βn x) Doubly-clamped beam Cantilevered beam Doubly-clamped beam Y(x,t) an ≈ bn (a1 = 1.018b1,a2 = 0.9992b2 ,a3 = 1.000b3 …) Euler-Bernoulli theory: length >> width, thickness … βn L = 4.730, 7.853,10.996 x • Neutral axis displacement Y ωn = E 12ρ βnt • Width w, thickness t (along y) • Density ρ, modulus E = (3λ+2μ)/(λ+μ) Y(x,t) 1234 ∂2Y wt3 ∂4Y x ρwt + E = 0 ∂t 2 12 ∂x4 Boundaries: Clamped end: Y = 0, Y’ = 0 Free end: Y’’ = 0, Y’’’ = 0 See Timoshenko, Vibration Problems in Engineering (1974) © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors Flexural Beams: Solutions Flexural Beams: Damped Harmonic Motion undamped oscillator Yn (x,t) = an (cosβn x − coshβn x)+ bn (sin βn x − sinh βn x) amplitude Cantilevered beam time an ≈ −bn ()a1 = −1.362b1,a2 = −0.982b2,a3 = −1.008b3 … Y(x,t) damped decay βnL = 1.875,4.694,7.855… ωn = E 12ρ βnt exp(-ω t/2Q) x amplitude 0 1 2 3 frequency response ω0/Q Q resonance frequency: amplitude 1/2 ω0~ (Eeff / ρ) ω0 quality factor: frequency Q ~ 1/Δ © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors (c) Caltech, 2006 -all rights reserved- Page 2 of 28 2 PASI 2006 - Bariloche, Argentina Lectures M.L. Roukes, Caltech Single Mode Model: Damped SHO Single Mode Model: Damped SHO A damped, simple harmonic oscillator model describes the flexural motion of a A damped, simple harmonic oscillator model describes the flexural motion of a beam in the vicinity of the fundamental resonance with an accuracy to within beam in the vicinity of the fundamental resonance with an accuracy to within 1% for Q>10. 1% for Q>10. displacement applied force displacement applied force F ()ω F ()ω complex displacement complex displacement x()ω = 2 x()ω = 2 response function: []()KMi−−ω effω γ eff ω response function: []()KMi−−ω effω γ eff ω Q>>1 ⇒ resonant mechanical response comments: 106 10000 force constant Q =100 force constant 5 Q =1000 100 • this is a “dynamical version” of 10 =300 =100 1 4 Hooke’s law x = F /k 2 applied Parameters: 10 =30 0.01 Parameters: )| =10 ω 3 • the complex AC mechanical responsivity effective mass, M ( 10 0.0001 effective mass, M eff A 0.1 1 10 eff | 2 is defined as xF () ω =ℜ ()() ωω , dynamic stiffness (for point 10 dynamic stiffness (for point 1 loading at beam’s center), K 101 loading at beam’s center), K i.e. ℜ=()ω eff eff 2 0 10 []()KMi−−ω effω γ eff ω quality factor, Q 0.8 0.9 1.0 1.1 1.2 1.3 quality factor, Q Normalized Frequency, ω /ω © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © M.L. Roukes 2005 Ph/EE 118c, Ch227 0— Micro- and Nanoscale Sensors Single Mode Model: Damped SHO Single Mode Model: Cantilever DSHO A damped, simple harmonic oscillator model describes the flexural motion of a ω0 Eeff t beam in the vicinity of the fundamental resonance with an accuracy to within Resonance frequency (cantilever): = 0.159 2π ρ L2 1% for Q>10. 3 ⎛⎞t eff. force constant (cantilever): K = (0.2575) Ew F ()ω eff ⎜⎟ displacement x ()ω = ⎝⎠L 2 -1 response function: [()]()KM−−ω effωω iγ eff ω Dissipation: Q 2 Effective mass: Meff = keff / ω0 For the fundamental-mode response of a simple doubly-clamped beam, DC Response: ΔY = F/ Keff (constant force) effective mass: M eff = 0.735ltwρ N.B. For this simple treatment of beam 3 3 Resonant response: ΔY = Q F / Keff (at ω0) dynamic stiffness: κ eff = 32Et w / l elasticity, all of these factors apply only 2 to the specific mode considered. resonance frequency: ω0 = 2π (1.05) E / ρ (t / l ) ΔY eiωt Here, l × t × w are the beam’s dimensions, E is Young’s modulus and ρ is the mass density of the F/K QF/Keff beam. The above assumes the material is isotropic; for single-crystal devices anisotropy in the eff amplitude elastic constants yields resonance frequency dependence upon crystallographic orientation. iωt F e frequency ω © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors (c) Caltech, 2006 -all rights reserved- Page 3 of 28 3 PASI 2006 - Bariloche, Argentina Lectures M.L. Roukes, Caltech Fluctuation-Dissipation Theorem: Displacement and Force Noise Thermomechanical Noise x()ω = ℜ(ω)F(ω) ω0/Q Thermal force spectral density associated with finite Q: Q/k 1/k keff Response SF ()ν = 4kBTN TN = noise temperature ω0Q force spectral density 1/2 Displacement spectral density S (ω) for force density S (ω) (4kkBT/ω0Q) x F 2 ω 4kT Force Noise 0 BN SSxF()ωω=ℜ() = 2 ωω 2 QM eff ωω22−+0 ()0 ()Q displacement spectral density 1/2 (4ω Qk T) This yields thermal equilibrium of the average total energy: 0 B Displacement E = T +U = k T B N Frequency © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors The two key attributes for sensing: Displacement and Force Noise • responsivity x()ω = ℜ ()()ω F ω metric quantifying transduction (conversion between signal domains; generalization of “gain”) ω0/Q Q/k 1/k Response force spectral density • noise 1/2 imposes minimum detectable signal level; each element of a (4kkBT/ω0Q) system degrades the overall SNR (signal-to-noise ratio). Force Noise displacement spectral density Integrated energy 1 1/2 k T (4ω Qk T) 2 B 0 B Displacement Frequency © M.L. Roukes 2005 Ph/EE 118c, Ch227 — Micro- and Nanoscale Sensors © 2006 (c) Caltech, 2006 -all rights reserved- Page 4 of 28 4 PASI 2006 - Bariloche, Argentina Lectures M.L. Roukes, Caltech generalized measurement analysis: transduction generalized measurement analysis: responsivity SIGNAL ELECTRICAL SIGNAL ELECTRICAL DOMAIN, DOMAIN, DOMAIN, DOMAIN, “X ” “V ”or “I ” LOW NOISE “X ” “V ”or “I ” LOW NOISE input input SYSTEM ELECTRONIC SYSTEM ELECTRONIC transducer data transducer data UNDER MEASUREMENT UNDER MEASUREMENT (sensor) storage (sensor) storage STUDY signal electrical SYSTEM STUDY signal electrical SYSTEM input input X V, I Responsivity R = ∂∂VX/ a generalized representation of “gain” (here we’ve ignored I/O phase relationship and have assumed transducer linearity) Small signal analysis: δ VX=R ⋅δ © 2006 © 2006 example: magnetic force microscope NEMS attributes: power level for operation “ganged responsivities” fiber optic interferometer Signal Ceiling VBzkRm=∂/(1/) ∂ − ) [ int ] z microcantilever 1. miniature magnet dB ( provides coupling dynamic to magnetic sample ) range ∂B FmBm=⋅∇=() zz∂z m a “magnetization-to-force” Noise Floor transducer Input level ( Sample (e.g. magnetic dots) 2. mechanical force detector ( compliant mechanical element) 3. displacement readout x =−Fk/ ( e.g. fiber-optic interferometer ) a “force-to-displacement” VRx= int transducer a “position-to-voltage” transducer © 2006 © 2006 (c) Caltech, 2006 -all rights reserved- Page 5 of 28 5 PASI 2006 - Bariloche, Argentina Lectures M.L. Roukes, Caltech NEMS attributes: power level for operation “thermomechanical” noise floor • fundamental thermodynamic Signal Ceiling noise limit 4kTQB ) Sx = “displacement noise spectral density” kω0 dB units: [ (nm)2/Hz ] dynamic on resonance range Q>>1 ⇒ resonant mechanical response 106 10000 Q =100 5 Q =1000 100 Noise Floor 10 =300 Input level ( =100 1 is routinely determined by 4 2 10 =30 0.01 )| =10 ω 3 • thermomechanical noise ( 10 0.0001 A 0.1 1 10 | 102 but, in certain systems (at millikelvin 101 temperatures) is now approaching: 100 • quantum displacement fluctuations 0.8 0.9 1.0 1.1 1.2 1.3 Normalized Frequency, ω /ω (lecture 2) 0 © 2006 © 2006 “thermomechanical” noise floor NEMS attributes: power level for operation • fundamental thermodynamic noise limit 4kTQB Sx = nm-Scale “displacement noise spectral density” kω0 units: [ (nm)2/Hz ] Mechanical on resonance Mo Li, H.X Tang, M.L.
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