1. Simulation and Feedback control in Atomic Force Microscope
Michal Hrouzek, Alina Voda, Martin Stark, Joël Chevrier
Laboratoire d’Automatique de Grenoble, INP/UJF Grenoble
The European Synchrotron Radiation

2. 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

3. 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)

4. 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.

5. 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

6. 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.

7. 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

8. 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

9. 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

10. 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

11. 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

12. Sources of noise in AFM Electronic parts Mechanical 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

13. 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

14. 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

15. Cantilever model Second order differential equation model
Q liquid environment
air environment
vacuum environment
L mm length
w 10-50mm width
t 0.1-5mm thick
E GPa for silicon cantilevers
k N/m stiffness

16. Cantilever model Multimode model of the cantilever
E Modulus of elasticity
I Area moment of inertia
m Mass per unit length
L Cantilever length

17. Cantilever model Multimode model of the cantilever
E Modulus of elasticity
I Area moment of inertia
m Mass per unit length
L Cantilever length

18. Computer Simulation The cantilever properties used for multimode model
Computed properties of separated harmonic modes

19. Computer Simulation
Spectral analysis schema of the multi mode cantilever model

20. Computer Simulation - results
Thermal noise was the only excitation of the Cantilever.
(displayed spectra is an average over 100 FFT)

21. Measured spectra
non-contact silicon cantilever NSC12/50 (cantilever F)
(displayed spectra is an average over 1000 FFT)

22. 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)

23. 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

24. 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

25. Computer Simulation with controller
Driving with the PID regulator

26. 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

27. Computer Simulation with controller - results
Gain of the PID controller gain(t)

28. Computer Simulation with controller - results
Displacement of the driving bimorph zbimorph(t) = kbimorph zdrive(t)

29. Computer Simulation with controller - results
Displacement of the cantilever (red curve)
Displacement of the driving bimorph (blue curve)

30. 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

31. Interaction forces Properties Interaction forces 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, …

32. Interaction forces
Intensity of the van der Waals and repulsive forces.

33. 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 isn’t 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

34. Interaction forces model
Mathematical equations describing interaction forces between the tip and surface
Intensity of the interaction forces as a function of the separation distance.
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.

35. 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, Q=1
Approach curve with the cantilever, Q=100
This behavior has been observed at the experiment. (Martin Stark, Frederico Martins)

36. 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

37. 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 )

38. 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

39. 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

40. 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.

41. Acknowledgements
Frederico Martins
Mario Rodrigues
Raphaelle Dianoux

42. 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.