Nanoelectromechanical Systems Continuum Solid: Equations of Motion

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.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    28 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us