Nonlinear dynamics in Atomic Force

8/7/2019 Nonlinear dynamics in Atomic Force

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Arvind Raman, Associate Professor

Mechanical EngineeringBirck Nanotechnology Center

Students: Shuiqing Hu, M. Moreno-Moreno (UAM, Spain), S. Ruetzel(Darmstadt, Germany)

Collaborators: Steve Howell (Sandia), Scott Crittenden (ARL)Soo-Il Lee (U of Seoul, Korea), Ron Reifenberger (Physics, Purdue)

J. Gomez-Herrero (UAM, Spain)

Nonlinear dynamics in

Atomic Force M icroscopy

Nonlinear dynamics inNonlinear dynamics in

Atomic Force M icroscopyAtomic Force M icroscopy


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OutlineOutline

Nonlinearities everywhere

Whither chaos?

Parametric resonance- medieval physics in theAtomic Force Microscope


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BifurcationsBifurcations

40 42 44 46 48 500

100

200

300

400

TipAmplitude(nm)

40 42 44 46 48 50-3

-2

-1

0

Excitation Frequency (kHz)

Ph

ase(rad)

43 44 45

150

170

190

43 44-1.2

-0.5

EG&G 7260Lock-In

Deflection

Detector

GPIB

NanoTecController

Piezotube

ADC

Z

PiezoControl

Oscillator

Sample

Bimorph

Non-Contact

Signal

Laser

A. Si tip / HOPG sample z=90 nm, frequency sweep

NanotecTMAFMLee, Raman et al, Phys Rev B (2002),

Ultramicroscopy (2003)Also Khle et al, Garca et al, Stark, et al

Aim et al, L. Wang*)B. If experiment is repeated much closer to sample the tipsuddenly sticks to the sample in certain frequency ranges


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Computed responseComputed response

40 42 44 46 48 500

100

200

300

TipAm

plitude(nm)

40 42 44 46 48 50-3

-2

-1

0

Excitation Frequency (kHz)

Phase(rad)

44.3 44.7178

183

43.4 43.7172

176

180

43.3 43.7-1.2

-1.1

SN3

SN1

SN2 SN4

SN3

SN4SN1

SN2

Further from sample, 4 bifurcations (SN1 - SN4) Closer to sample 2 SNs & 2 PDs near fundamental resonance Identifying bifurcation really needs knowledge of unstable branch

Z=90nm Z=2nm

AUTO computation

Lee, Raman et al, Phys Rev B (2002)

0 20 40 60 800

1

2

3

Tipam

plitude(nm)

0 20 40 60 80-3

-2

-1

0

Excitation Freq (kHz)

Phase(rad)

PD's

PDPD

PD PD PD's

SN

SN

SN

SN

Ruetzel, Raman et al, Proc. Roy. Soc. (2003)Lee, Raman et al, Ultramicroscopy (2003)


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Analyt ical approachesAnalyt ical approaches

Analytical analysis for nonlinear dynamics in AM-AFM:

Chen et al. (1995), Whangbo et al. (1998) -harmonic

approximation method Nony et al. (1999) a variational approachbased on

the principle of least action

Wang (1998) - Krylov-Bogoliubov-Mitroposky (KBM)asymptotic approximation, Sebastian & Salapaka,Hlscher & Schwarz, Belikov, Magonov

Yagasaki (2004) - averaging methodand anextended version of the subharmonic Melnikov

Others for chaotic oscillations


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The averaging methodThe averaging method

From Sanders and Verhurlst


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Internal resonance* based AFMInternal resonance* based AFM Experiments performed using

47kHz microcantilever on wild

and

mutant bacteriorhodopsinmembrane

2nd bending mode freq ~7*1st

1st

and 2nd mode couple onlyin the presence of nonlinearinteraction forces (local van

derWaals and electrostatic forces)

0 2 4 6 8 10 12

-60

-40

-20

0

20

40

c)

H3H2

B2/H7

H1

PSD

(dB)

Frequency (kHz)

On mica (50 % setpoint)

Probing attractive forces at the nanoscale using higher harmonic dynamic force microscopy,Crittenden, Raman, Reifenberger (PRB, 2005)

* Nayfeh and Mook (1979)


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Internal resonance based AFMInternal resonance based AFM3500 nm x 3500 nm scans

proteins Lipid deposits

Topography Second harmonic image

Seventh harmonic image

Clear distinction between lipids and proteins

Presence of internal resonance critical in themethod The method shows promise for themeasurement of

local attractive forces of soft biomolecules Can be extended to electrostatic forceProbing attractive forces at the nanoscale using higher harmonic dynamic force microscopy,Crittenden, Raman, Reifenberger (PRB 2005)


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OutlineOutline


Whither chaos?



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Internal resonance* based AFMInternal resonance* based AFM Experiments performed using

47kHz microcantilever on wild

and

mutant bacteriorhodopsinmembrane

2nd bending mode freq ~7*1st

1st

and 2nd mode couple onlyin the presence of nonlinearinteraction forces (local van

derWaals and electrostatic forces)

0 2 4 6 8 10 12

-60

-40

-20

0

20

40

c)

H3H2

B2/H7

H1

PSD

(dB)

Frequency (kHz)

On mica (50 % setpoint)

Probing attractive forces at the nanoscale using higher harmonic dynamic force microscopy,Crittenden, Raman, Reifenberger (PRB, 2005)

* Nayfeh and Mook (1979)


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f0 2f0 3f0

Realistic Case

3f0/2f0/2

Ideal Case

f0

Dynamic AFM uses the 1st harmonic amplitude(A) ofcantilever oscillation as control feedback

Subharmonics and chaoscan introduce metrology errors

Urban myth? (Burnham et al1995, Salapaka et al2001)

Homoclinic chaos

Grazing bifurcations (non-smooth, discontinuous interaction forces)

Strong forcing

MotivationMotivation

Frequency

Amplitud

e

A

Z0 A


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Experiments performedExperiments performed

Cantilever set Force Modulation Ultrasharp NSC12-C TESP

Resonance Frequency (kHz) 66.32.1 130.21.8 310.15.9

Stiffness (N/m) 1.50.1 2.90.2 32.00.4Quality factor 1428 13011 1485

Under nitrogen, 1st harmonic Amplitude equals 80nm

Under nitrogen, 1st

harmonic Amplitude equals 140nm

In air, 1st harmonic Amplitude equals 80nm

Experimental condition

(=0.980,=0, and

=1.020)In air, 1st harmonic Amplitude equals 140nm

Summary of the properties and experimental conditions for 3 sets ofmicrocantilevers (the stiffness, resonance frequency and qualityfactors are ascertained from experimental data and are averagedover the 10 microcantilevers within each set).

More than 30 microcantilevers are operated under 3 drivefrequency, with various operating conditions and each set ofexperiment is repeated 3-6 times . Results are consistent.

S. Hu and A. Raman, Phys. Rev. Lett (2006)


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0 2 4 6 8 10 12 14 160

0.05

0.1

0.15

LyapunovExpo

nent

0 2 4 6 8 10 12 14 160

10

20

30

Drive Amplitude (nm)

NoiseLimit(%)

Lyapunov Exponent

Noise Limit

Attactive Regime Attractive-Repulsive Regime

Period-1 Chaotic Period-2 Chaotic

Response at resonanceResponse at resonance

0 100 200 300 400 500 600 700

0

0.2

0.4

0.6

0.8

1

-50

0

50

100

DriveAmplitude(Volt)

Tip Deflection Power Spectral Density

Frequency (kHz)PowerSpectralDensity(dB/Hz)

Attractive

Repulsive

Setpoint amplitude constant, setpoint ratio increased

Grazing bifurcations similar to those in impact oscillators

under reasonable operating conditions

S. Hu and A. Raman, Phys. Rev. Lett (2006) Also Stark et al. Nanotechnology (2006)

O l


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OutlineOutline


Whither chaos?


C d l d S i d C l


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J. San Martn-O Botafumeiro-parametric pumping in the middle ages, Am. J. Phys.

Censer

The pumping friars

Catedral de Santiago de Compostela

Parametric excitation-theoryParametric excitation-theory


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22 200 02

cos( ) ( ) 0,d d

G t t G

dt Q dt

+ + + =


For G=0, homogeneous solution A e(-0t/2Q)cos(0t+) decaysexponentially

For G>Gthreshold, a homogenous solution A cos(/2 t+) growswhen n~20 n=1,2,3..

00

G

0 /2Gthreshold=0/Q

Growingperiodic

oscillations

Decaying

quasi-periodicoscillations

A

G>Gthreshold

20


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22 200 02

cos( ) ( ) 0,d d

G t t G

dt Q dt

+ + + =

Parametric excitation theoryParametric excitation theory

For G=0, homogeneous solution A e(-0t/2Q)cos(0t+) decaysexponentially

For G>Gthreshold, a homogenous solution A cos(/2 t+) growswhen n~20 n=1,2,3..

00

G

0 /2Gthreshold=0/Q

Growingperiodic

oscillations

Decaying

quasi-periodicoscillations

A

G>Gthreshold

20


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parametrically?

parametrically?

Rugar and Gruetter, PRL, 67(6), 1991

Dougherty et al., Meas. Sci. Tech., 7, 1733, 1996

K. L. Turner et al., APL (2006)

Others:

M. Stark et al., PRB (2005)Patil and Dharmadhikari,Appl. Surf. Sci. (2003)


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ExcitationExcitation

282500 283000 283500 284000 284500 285000

0

10

20

30

40

50

60

70

Am

plitude(nm)

Driving Frequency (Hz)

Conventional

Excitation

Parametric

Excitation

Non-Lorentzian, extremely sharp, controllable widthresonance peak

60 Hz

()conventional(/2)parametric

Why does this circuit work?Why does this circuit work?


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Why does this circuit work?Why does this circuit work?

[ ] 0( ) ( ) ( ) ( ) ( , ) ( , ) ( , , )cantilever fluidic tip samplemx t k x t z t cx t F x x F x x F x x d + + = + +&& & & & &

[ ] ( )( ) ( ) ( ) cosz t G x t z t t =

z(t)

x(t)

2( )cos( )1 ( ) ~ ( ) cos( ) ( )

1 cos( )

Gx t t If G z t Gx t t O G

G t


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The Instabi lity TongueThe Instabili ty Tongue

The instability tongues can be computed in Gparameter space.Assuming , and small damping , then

Experimental results are from Moreno-Moreno et al. APL (200286.3 286.4 286.5 286.6 286.7 286.8 286.9

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

Experimental Data

Theoretical prediction

Self-Sustained

Parametric

Oscillation

Half of excitation frequency (kHz)

G/Gth

If , then the equation of motionbecomes the linear damped Mathieu's equation.

Potential ApplicationsPotential Applications


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Potential ApplicationsPotential Applications

DNA on mica Topography

High sensitivity attractive mode and tapping mode

imaging, dissipation maps Dynamic AFM in liquids

Mass sensing using microcantilevers

Silicon grating

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Nonlinear dynamics in Atomic Force