Prezentace aplikace PowerPoint

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Simulation and Feedback control in Atomic Force
Microscope
Michal Hrouzek, Alina Voda, Martin Stark, Joël
Chevrier Laboratoire dAutomatique de Grenoble,
INP/UJF Grenoble The European Synchrotron
Radiation
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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Schema of Dynamic Force Microscopy
(excitation)

• Head of the AFM
• Driving loop
• Head positioning loop
• Stage with a sample
• x axis positioning loop
• y axis positioning loop

(set-value of Dw)
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Detection techniques in dynamic AFM

• Amplitude Modulation (AM)
• Original operation technique, Developed by Y.
  Matin J. Appl. Phys. 61 (10)
• Driver is exciting the cantilever with constant
  driving signal.
• Interaction forces affect the cantilever and
  lower the vibration amplitude.
• Change in amplitude depends directly on
  interaction force.
• Frequency Modulation (FM)
• Newer technique, Developed by T.R. Albrecht J.
  Appl. Phys. 69 (2)
• Driver with controller is exciting the cantilever
  to constant vibration amplitude.
• Interaction forces affect the cantilever and
  change the resonant frequency.
• Frequency shift depends directly on interaction
  force. Dw Df
• More sensitive compare to AM-technique
• Further would be treated only FM technique.

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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Feedback control in AFM

• Loop controlling position of AFM head in z axis
• Maintaining constant set-value of frequency shift
• Low frequency response (1 30 kHz)
• Cantilever driving loop (cantilever excitation)
• Exciting the cantilever and ensuring that contact
  with surface is not lost
• High frequency response (1kHz 1 MHz)
• Nonlinear behavior of the driver (due coupling
  with surface)
• Directly maintaining cantilever vibration
  amplitude
• Could be used for possible attenuation of thermal
  noise perturbation and influence of another
  noises.

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Position of the head and lever excitation
rzd(t), rzp(t) desired excitation and
set-value of frequency shift mz(t) measured
deflection of the cantilever ezd(t), ezp(t)
regulation errors of the bimorph and
head zdri(t), zpos(t) driving signal and head
positioning signal z(t) real deflection of the
cantilever
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Feedback control in AFM

• Loops controlling position of AFM stage in x-y
  axes
• Scanning
• Positioning of the stage with sample
• Simple movement in straight lines under the
  cantilever with tip
• Manipulation
• Particles manipulation at the surface
• Complex movement with many possible shapes
• Control of the applied force onto the particle is
  crucial

Scanning Manipulation
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Loops controlling stage position
rx,y(t) desired position piezo-electric
stack mx,y(t) measured (estimated) position of
the stage ex,y(t) regulation error ux,y(t)
driving signal yx,y(t) real position of the
stage
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Loops controlling stage position

• Accuracy problems with piezo-electric actuators
• Positioning nonlinearities (getting bigger with
  increasing speed)
• Usually are used piezo stacks with hysteresis 10
  - 15 of max. displacement
• (Harder stacks have smaller hysteresis but
  smaller displacement range)
• Drift due to creep
• (Could be reduced to curtain level by careful
  design of the stage)
• Measurement problems of stage position
• High level of noise of LVDT detectors get
  relevant at nano-meter resolution
• Observer based regulator can achieve better
  positioning resolution

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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Sources of noise in AFM

• Electronic parts
• Photo detector
• Optical path noise
• Thermal gradient
• Laser intensity noise
• Shot noise
• Laser mode noise
• Laser phase noise
• Electronic circuits
• Electrostatic noises
• Noise of amplifiers

• Mechanical parts
• Cantilever and tip
• Thermally induced cantilever noise
• Mechanical vibration
• Relaxation
• Air turbulences and acoustic waves
• Magnetic noises
• Electrostatic noises
• Chemical noises
• Bimorph
• Thermally induced noise
• Electrostatic noises

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Thermal noise

• Thermal noise is limiting AFM sensitivity
• Some noises could be lower by appropriate design
  and construction of AFM.

• The minimum detectable interaction force.
  (dynamic mode)

• k Spring constant, stiffness
• T Temperature
• kB Boltzmann constant
• b Measurement bandwidth
• A0 Vibration amplitude
• w0 Resonance frequency
• Q Quality factor

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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Cantilever model

• Second order differential equation model

• Q 0.1-1 liquid environment
• 1-1000 air environment
• 1000-100000 vacuum environment
• L 10-500mm length
• w 10-50mm width
• t 0.1-5mm thick
• E 130-180GPa for silicon cantilevers
• k 0.01-100N/m stiffness

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Cantilever model

• Multimode model of the cantilever

• E Modulus of elasticity
• I Area moment of inertia
• m Mass per unit length
• L Cantilever length

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Cantilever model

• Multimode model of the cantilever

• E Modulus of elasticity
• I Area moment of inertia
• m Mass per unit length
• L Cantilever length

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Computer Simulation

• The cantilever properties used for multimode model

• Computed properties of separated harmonic modes

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Computer Simulation

• Spectral analysis schema of the multi mode
  cantilever model

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Computer Simulation - results

• Thermal noise was the only excitation of the
  Cantilever.

(displayed spectra is an average over 100 FFT)
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Measured spectra

• non-contact silicon cantilever NSC12/50
  (cantilever F)

(displayed spectra is an average over 1000 FFT)
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Computer Simulation - results

• Time response of thermally driven cantilever
• -Time sequence 0 to 2ms -Time
  sequence 0 to 0.2ms (Zoom)

Zoom
Model initialization
Modeled thermal excitation (blue) (normal
distribution)
Position of the cantilever (red)
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Computer Simulation - results

• Time response of artificially driven cantilever
  at resonance
• blue curve exciting displacement
  zdriv(t)Z0sin(w0t)
• red curve displacement of the cantilever at its
  free end
• - Time sequence 0 to 10ms - Time sequence 0 to
  0.4ms (zoom)

Zoom
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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Computer Simulation with controller

• Driving with the PID regulator

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Computer Simulation with controller

• Driver with phase shift

• Band pass filter selects the frequencies of our
  interest (10kHz-100kHz)
• Amplitude detector gives numerical value of
  vibration amplitude A(t)
• Controller select the gain(t) that is
  multiplied by filtered cantilever position
• Low pass filter eliminates high frequency noise
• Gain amplification of the signal
• Phase shift optimal value is p/2

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Computer Simulation with
controller - results

• Gain of the PID controller gain(t)

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Computer Simulation with
controller - results

• Displacement of the driving bimorph zbimorph(t)
  kbimorph zdrive(t)

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Computer Simulation with
controller - results

• Displacement of the cantilever (red curve)
• Displacement of the driving bimorph (blue curve)

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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Interaction forces

• Properties
• They are non-linear
• Can be long-range or short-range, and attractive
  or repulsive.
• Forces are very sensitive to environmental
  conditions such as temperature, humidity, surface
  chemistry, and mechanical and electrical noises.
• Depend on the material, geometry and size of nano
  entities.
• Curtain forces are getting dominant in specific
  environments and some diminish.
• Interaction forces
• van der Waals (analogous to the gravity at the
  nano-scale)
• Casimir
• Thermal motion Exist for any material and
  depends only on temperature
• Capillary, Hydrogen and Covalent bonding, Steric,
  Hydropobolic, Double layer,

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Interaction forces

• Intensity of the van der Waals and repulsive
  forces.

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Interaction forces

• Approach curve of the cantilever.

1) Non-contact 2) Snap on point, spring
constant is smaller than attractive
force. zero) Equilibrium point, lever isnt
deflected in any direction. 3) Repulsive
interaction is dominant. 4) Maximum positive
deflection. 5) Capillarity holds the tip onto the
surface 6) snap off point, spring constant
overcomes the capillarity
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Interaction forces model

• Intensity of the interaction forces as a function
  of the separation distance.

• Mathematical equations describing interaction
  forces between the tip and surface

a0 intermolecular distance AH Hamaker
constant RS Sphere radius (end of the tip) E
Effective stiffness

• Numerical values used from reference S. I. Lee,
  Physical Review B, 66(115409), 2002.

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Interaction forces model - results

• Approach curve is simulated without any
  excitation, chip with the cantilever is slowly
  approaching the surface

Approach curve with the cantilever, Q1
Approach curve with the cantilever, Q100

• This behavior has been observed at the
  experiment. (Martin Stark, Frederico Martins)

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Interaction forces model - results

• Cantilever has been excited with constant driving
  signal zbimorph(t) Adrivesin(w0t)
• Harmonic outputs of separated modes has been
  recorded.
• Amplitude of first harmonic mode is decreasing
  with smaller separation distance.

Vibration amplitude
Separation distance
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Interaction forces model - results

• Amplitude of higher harmonic modes are increasing
  with smaller separation distance.
• Time sequences of one period are shown
• (second mode red curve, third mode green
  curve, fourth mode blue curve )

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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Outlines of the presentation

• AFM description
• Feedback control in AFM
• Sources of noise in AFM
• Cantilever model
• Driving loop with controller (model of PLL)
• Interaction forces
• Thermal noise in AFM
• Conclusion

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Conclusion

• Multimode cantilever model has been developed.
  Simulations have shown that the model is correct
  approximation of the dynamic system.
• Model of the interaction forces have been
  implemented into Matlab Simulink environment.
• Dynamic interaction between both models has been
  simulated and compared with measurements.
• Driving controller has been employed to control
  the excitation of the cantilever interacting with
  the surface.

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Acknowledgements

• Frederico Martins
• Mario Rodrigues
• Raphaelle Dianoux

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Future work

• Development of microscope stage controllers that
  are responsible for the sample positioning under
  the head with cantilever. This controller has to
  fulfill new requirements for speed and accuracy
  due to application of the AFM as a
  nano-manipulator.
• Thermal noise is second field of further work.
  Driving controller has to be redesigned to lower
  the nose signal ration to achieve better results
  in weak forces measurements.