Noise, AFMs, and Nanomechanical Biosensors

Transcription

1 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Noise, AFMs, and Nanomechanical Biosensors with Mark Paul (Virginia Tech), and the Caltech BioNEMS Collaboration Support: DARPA

2 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Outline Motivation: MEMS and NEMS BioNEMS: Fluctuations in the linear regime

3 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, [From M. R. Roukes, Caltech]

4 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Single crystal silicon [From Craighead, Science 290, 1532 (2000)]

5 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Diamond Film [From Sekaric et al., Appl. Phys. Lett. 81, 4445 (2002)]

6 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Array of µ-scale oscillators [From Buks and Roukes J. MEMS. 11, 802 (2002)]

7 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Self-Oscillations [Zalalutdinov et al., Appl. Phys. Lett. 79, 695 (2001)]

8 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, MicroElectroMechanical Systems and NEMS Tiny mechanical oscillators: driven, dissipative nonequilibrium nonlinear collective (arrays) noisy (potentially) quantum Goals Apply knowledge from statistical mechanics, nonlinear dynamics, pattern formation etc. to technologically important questions Investigate stochastic and nonlinear dynamics, and pattern formation in new regimes

9 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, BioNEMS - Single BioMolecule Detector/Probe

10 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, BioNEMS Prototype (Arlett et. al, Nobel Symposium 131, August 2005)

11 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Example Design Parameters b Wall Localized stress L 1 L Localized stress h w y x z Dimensions: L = 3µ, w = 100nm, t = 30nm, L 1 = 0.6µ, b = 33nm Material: ρ = 2230Kg/m 3, E = N/m 2 Results: Spring constant K = 8.7mN/m; vacuum frequency ν 0 6MHz

12 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Atomic Force Microscopy (AFM)

13 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Noise in micro-cantilevers Thermal fluctuations (Brownian motion) important for: BioNEMS: detection scheme AFM: calibration Goals: Correct formulation of fluctuations for analytic calculations Practical scheme for numerical calculations of realistic geometries

14 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Previous approach (Sader 1998) Model molecular collisions with cantilever as white noise force uniformly distributed along cantilever Calculate modal response x n (ω) for periodic driving force F(ω) (resonance curves) interesting frequency dependent mass loading and damping from coupling to fluid Calculate fluctuation of tip displacement as sum of mode responses for constant F(ω) 2

15 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Problems This approaches is formally incorrect and hard to implement for realistic geometries and strong damping: Noise force is not white Noise force is not uniformly distributed along surface Mode fluctuations are not in general independent Difficult to calculate coupled elastic-fluid modes, and many needed for strong damping

16 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Fluid Dynamics Issues with ν the kinematic viscosity η/ρ. u t + u u = p + ν 2 u, u = 0 Fluid dynamics is (relatively) easy if we can neglect the inertial terms. For typical BioNEMS/AFM: u u = O(u 2 ) is negligible because of tiny oscillation amplitudes Important parameter is the Strouhal number S = ωw2 4ν 1.6 ω frequency 2π 1 MHz w width 1µ ν kinematic viscosity 10 6 m 2 s 1 Low Reynolds number flow: linear but can t take S = 0

17 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Simple Picture (Sader) Potential flow Diffusing vorticity S -1/2

18 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Stokes Theory Viscous force on sphere of radius a moving with speed v is F/v = 6πρνa Viscous force per unit length of cylinder of radius a is given by γ = F/v = πρν S Im Ɣ(S) with Ɣ(S) = 1 + 4iK 1( i is) isk0 ( i is) Effective mass per unit length from fluid M = πa 2 ρ Re Ɣ(S) Q Re Ɣ(S) Im Ɣ(S) (Other parameter T = π 4 ρ ρ s w t = mass of cylinder of fluid mass of cantilever 2)

19 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, γ/πρν Q Q, γ/πρν S For small S: SƔ(S) 4i 1 2 log ( ) 4 S CE + i π 4

20 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, New approach: fluctuation-dissipation theorem (Paul and MCC, 2004) Equilibrium fluctuations can be related to the decay of a prepared initial condition (near equilibrium) thermodynamics: Onsager regression hypothesis statistical mechanics: fluctuation-dissipation theorem, linear response theory, Kubo formalism

21 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, New approach: fluctuation-dissipation theorem (Paul and MCC, 2004) Equilibrium fluctuations can be related to the decay of a prepared initial condition (near equilibrium) thermodynamics: Onsager regression hypothesis statistical mechanics: fluctuation-dissipation theorem, linear response theory, Kubo formalism Consider Hamiltonian H = H 0 F(t)A H 0 A(r 1...r N, p 1...p N ) F(t) unperturbed Hamiltonian system observable (small) time dependent force

22 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, F(t) equilibrium under H 0 + H ρ=ρ H (r N,p N ) H=H 0 + H H=H 0 t

23 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, F(t) F 0 H=H 0 + H H=H 0 <B(t)> t δb(t)δa(0) e = k B T B(t) F 0 t

24 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (e.g. see Introduction to Modern Statistical Mechanics by Chandler)

25 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (e.g. see Introduction to Modern Statistical Mechanics by Chandler) To calculate the change in a measurement B(t) due to the application of a small field F(t)that gives a perturbation to the Hamiltonian H = F(t)A.

26 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (e.g. see Introduction to Modern Statistical Mechanics by Chandler) To calculate the change in a measurement B(t) due to the application of a small field F(t)that gives a perturbation to the Hamiltonian H = F(t)A. The time dependence is given by the evolution of r N (t), p N (t) according to Hamilton s equations.

27 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (e.g. see Introduction to Modern Statistical Mechanics by Chandler) To calculate the change in a measurement B(t) due to the application of a small field F(t)that gives a perturbation to the Hamiltonian H = F(t)A. The time dependence is given by the evolution of r N (t), p N (t) according to Hamilton s equations. We can calculate averages in terms of the known distribution ρ ( r N, p N ) at t = 0: ( ) ( ) B(t) = dr N dp N ρ r N, p N B r N (t), p N (t) where r N (t) is the phase space coordinate that evolves from the value r N at t = 0.

28 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (e.g. see Introduction to Modern Statistical Mechanics by Chandler) To calculate the change in a measurement B(t) due to the application of a small field F(t)that gives a perturbation to the Hamiltonian H = F(t)A. The time dependence is given by the evolution of r N (t), p N (t) according to Hamilton s equations. We can calculate averages in terms of the known distribution ρ ( r N, p N ) at t = 0: ( ) ( ) B(t) = dr N dp N ρ r N, p N B r N (t), p N (t) where r N (t) is the phase space coordinate that evolves from the value r N at t = 0. We could equivalently follow the time evolution of ρ through Liouville s equation and instead evaluate ( ) B(t) = dr N dp N ρ r N, p N,t B (r ) N, p N but the first form is more convenient.

29 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (cont.) Consider the special case of the step force F(t)of magnitude F 0 :

30 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (cont.) Consider the special case of the step force F(t)of magnitude F 0 : For t 0 the distribution is the equilibrium one for the perturbed Hamiltonian H ( r N, p N ) = H 0 + H ρ(r N, p N ) = e β(h 0+ H ) dr N dp N e β(h 0+ H ) so that writing Tr dr N dp N. B (0) = Tre β(h 0+ H ) B ( r N, p N ) Tre β(h 0+ H )

31 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (cont.) Consider the special case of the step force F(t)of magnitude F 0 : For t 0 the distribution is the equilibrium one for the perturbed Hamiltonian H ( r N, p N ) = H 0 + H ρ(r N, p N ) = e β(h 0+ H ) dr N dp N e β(h 0+ H ) so that writing Tr dr N dp N. B (0) = Tre β(h 0+ H ) B ( r N, p N ) Tre β(h 0+ H ) For t 0 we let r N (t), p N (t) for each member of the ensemble evolve under the Hamiltonian, now H 0, from its value r N, p N at t = 0, so that B(t) = Tre β(h 0+ H ) B ( r N (t), p N (t) ) Tre β(h. 0+ H ) Note that the integral is over r N, p N r N (0), p N (0), and H = H ( r N, p N ) etc.

32 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (cont.) It is now a simple matter to expand the exponentials to first order in H (F 0 small!) B(t) Tre βh 0(1 β H)B ( r N (t), p N (t) ) Tre βh 0(1 β H) to give B(t) = B 0 β [ H B(t) 0 B 0 H 0 ] + O ( H ) 2 where <> 0 denotes the average over the ensemble for an unperturbed system i.e. using ρ 0 = e βh 0/Tre βh 0.

33 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Derivation (cont.) It is now a simple matter to expand the exponentials to first order in H (F 0 small!) B(t) Tre βh 0(1 β H)B ( r N (t), p N (t) ) Tre βh 0(1 β H) to give B(t) = B 0 β [ H B(t) 0 B 0 H 0 ] + O ( H ) 2 where <> 0 denotes the average over the ensemble for an unperturbed system i.e. using ρ 0 = e βh 0/Tre βh 0. Finally writing δb(t) = B(t) B 0 etc., and noticing that putting in the form of H H = F 0 A(r N, p N ) = F 0 A (0) gives for the change in the measurement B(t) = B(t) B 0 B(t) = βf 0 δa (0) δb(t) 0.

34 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Application to single cantilever Assume observable is tip displacement X(t) Apply small step force of strength F 0 to tip Calculate or simulate deterministic decay of X(t) for t>0. Then C XX (t) = δx(t)δx(0) e = k B T X(t) F 0 Fourier transform of C XX (t) gives power spectrum of X fluctuations G X (ω)

35 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Advantages Correct! Essentially no approximations in formulation assume X(t) given by deterministic calculation also in implementation assume continuum description Incorporates full elastic-fluid coupling non-white, spatially dependent noise no assumption on independence of mode fluctuations complex geometries Single numerical calculation over decay time gives complete power spectrum Can be modified for other measurement protocols by appropriate choice of conjugate force AFM: deflection of light (angle near tip) BioNEMS: curvature near pivot (piezoresistivity)

36 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Single cantilever b Wall Localized stress L 1 L Localized stress h w y x z Dimensions:L = 3µ, W = 100nm, L 1 = 0.6µ, b = 33nm Material: ρ = 2230Kg/m 3, E = N/m 2

37 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Device schematic

38 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Adjacent cantilevers Correlation of Brownian fluctuations δx 2 (t)δx 1 (0) e = k B T X 2(t) F 1

39 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Device schematic

40 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Results: single cantilever 3d Elastic-fluid code from CFD Research Corporation 1µs force sensitivity: K G X (ν) 1MHz 7pN

41 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Results: adjacent cantilevers Autocorrelation of the noise for Cantilever 1 x 1 (0)x 1 (t) (nm 2 ) x 1 (0)x 2 (t) (nm 2 ) F t(µs) s=5h F 1 s=h Step force initial condition t=0 t Crosscorrelation of the noise for Cantilever t(µs)

42 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Comparison with AFM experiments 1.2 x µ 20.11µ 0.573µ Asylum Research AFM (Clarke et al., 2005) Dashed line: calculations from fluctuation-dissipation approach Dotted line: calculations from Sader (1998) approach

43 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Wall effects 1.2 x Amplitude Frequency x 10 4

44 Noise, AFMs, and Nanomechanical Biosensors: Lancaster University, November, Conclusions I ve described one aspect of theoretically modelling micron and submicron scale oscillators Linear fluctuations in solution [Paul and MCC, Phys. Rev. Lett. 92, (2004)] Other areas of interest: Nonlinear collective effects of parametrically driven high-q arrays [Lifshitz and MCC, Phys. Rev. B67, (2003)] Analysis of a QND scheme to measure the discrete levels in quantum harmonic oscillator [Santamore, Doherty, and MCC, Phys. Rev. B70, (2004)] Synchronization due to nonlinear frequency pulling and reactive coupling [MCC, Zumdieck, Lifshitz, and Rogers, Phys. Rev. Lett. 93, (2004)] Noise induced transitions between driven (nonequilibrium) states Single nonlinear oscillator [cf. Aldridge and Cleland, Phys. Rev. Lett. 94, (2005) ] Collective states in arrays of oscillators