Michael Reinstaedtler , Ute Rabe , Volker Scherer , Joseph A.Turner , Walter Arnold

1. Imaging of flexural and torsional resonance modes of atomic force microscopy cantilevers using optical interferometry Michael Reinstaedtler , Ute Rabe , Volker Scherer , Joseph A.Turner , Walter Arnold Surface Science 532-535(2003) 1152-1158 Date : 13th October 2005 Presenter : Ashwin Kumar

2. Background - Operation of the AFM A sharp tip is scanned over the sample surface the tip is maintained at a constant force (to obtain height information), or height (to obtain force information) above the sample surface Tips are typically made from Si3N4 or Si, and extended down from the end of a cantilever An optical detection system is used, in which a diode laser is focussed on the back of a reflective cantilever As the tip moves up and down with the contour of the surface, the laser beam is deflected off the attached cantilever into a dual element photodiode AFM Schematic

3. Background - AFM Modes Contact Mode the tip scans the sample in close contact with the surface The force on the tip is repulsive with a mean value of 10 -9 N the deflection of the cantilever is sensed and compared in a DC feedback amplifier to some desired value of deflection Non-Contact Mode (used when tip contact might alter the sample surface) In this mode the tip hovers 50 - 150 Angstrom above the sample surface Attractive Van der Waals forces acting between the tip and the sample are detected topographic images are constructed by scanning the tip above the surface

4. Background - AFM Modes Tapping Mode:(sample surfaces that are easily damaged ) The cantilever assembly is oscillated at or near the cantilever's resonant frequency the cantilever is oscillated with a high amplitude when the tip is not in contact with the surface The oscillating tip is then moved toward the surface until it begins to lightly touch, or tap the surface. During scanning, the vertically oscillating tip alternately contacts the surface and lifts off The reduction in oscillation amplitude is used to identify and measure surface features.

5. Motivation Earlier Work involved determination of contact stiffness and localized elastic modulus measurement of the surface The vibrational spectrum of the cantilever is used to discern local elastic data. It becomes imperative to understand the vibrational spectra completely to perform the above mentioned measurements The free vibrational response would help to characterize the cantilever or the probe Moreover, since the boundary conditions are also changed during the contact mode resonance, Free vibrational response and imaging the mode shape would help as a tool for calibration or standard. * Ultrasonics 38(2000) 430-437 * Journal of Applied Physics, 82(1997) 966 * Review of Scientific Instruments 67(1996) 3281

6. In a Nutshell Excite and Detect the torsional vibrations of the AFM cantilevers. Examine the features of the torsional vibration spectrum Image the flexural and torsional resonance modes Use a model based approach to explain the spurious modes in the spectrum

7. b a L Theory: Problem Statement L - length of the beam (m) a - width of the beam (m) b - thickness of the beam (m) E - Elastic Modulus of the beam (N/m2) I - Area moment of inertia - ab3/12 (m4) J - Polar moment of inertia - a3b/12 (m4) G - Rigidity modulus (N/m2) CT - Torsional Stiffness- ab3G/3 (Nm2) Boundary Conditions : Flexural Vibrations Clamped end: Free End: Torsional Vibrations Clamped End: Free End:

8. Theory: Flexural Vibrations Equation of motion for the bending modes The general solution of the form The dispersion relation:

9. Theory: Flexural Vibrations Applying the Boundary Conditions: The Characteristic Equation - Bending-mode eigenfrequencies: Amplitude Distribution:

10. Theory: Torsional Vibrations Equation of motion for the torsional modes The general solution of the form Applying the boundary conditions: * Jerry H. Ginsberg , Mechanical and Structural Vibrations,2001

11. Experimental Setup

12. Longitudinal Vs Shear Wave Propogation

13. Excitation of Torsional Vibrations Cantilever Sample Shear Wave Transducer

14. Beam Deflection Setup Spatial variations of reflected beam are detected Transverse vibrations cause vertical movement of the spot Torsional vibrations cause horizontal movement of the spot If the light beam moves up or down, If the light beam moves right or left * Handbook of Nano-Technology,Springer,2003

15. Experimental Results Optical Micrograph of the cantilever

16. Interferometric Measuring System Spot Size : 2-5 microns Step Size : 2 microns

17. Optical Detection Of Vibration of the Beam  IncidentBeam A= a*ei(ωt-k(z-2δ)) Reflected Beam • Phase Information is lost during Intensity or Power Measurements • Interferometric systems are used to convert phase change into intensity variations

18. Michelson Interferometer Reference Mirror Sample • AR=ar*ei(t-kzR) • AO=ao*ei(t-k(zo-δ)) Laser B.S. Detector

19. Output Intensity Vs Optical Path Length Region of Best Sensitivity

20. Heterodyne Interferometry Reference Mirror Frequency Shifter • AR=ar*ei((+)t-kzR) • AO=ao*ei(t-k(zo-δ)) Laser B.S. Sample Detector

21. Phase locked loop demodulator Mixer LPF2 O/p Detector Input VCO LPF1

22. Amplitude and Phase distribution - Measured Amplitude and Phase distribution - Calculated

23. d b h Mode Coupling mt - mass of the tip d - offset from the center of the beam b - thickness of the beam h - length of the tip Asymmetrical shape of the modes - Geometrical asymmetries - Tip not aligned with the center of the beam - Tip is in force interaction with the sample surface

24. Mode Coupling Coupling Description: - Equation of motion: Boundary Conditions at the free end:(x = L)

25. Mode Coupling From Previous Results: In Case of free oscillations: Coupling Parameter H :

26. Calculated Amplitude distribution based on mode coupling with H=0.025 Mode Coupling Resonance Mode at 265 Khz - The mode does not fit into the mode coupling analysis - Most likely occurs due to nonlinear coupling into flexural motion

27. Conclusion Verification of standard flexural and torsional modes in the vibration spectrum by imaging the mode shapes and comparing them with the model based expected pattern Mode Coupling due to geometrical and mass asymmetries account for a number of resonances Large strain values leads to non-linear mixing of modes

28. Beam Deflection Setup Spatial variations of reflected beam are detected Transverse vibrations cause vertical movement of the spot Torsional vibrations cause horizontal movement of the spot If the light beam moves up or down, If the light beam moves right or left

29. Background - Operation of the AFM A sharp tip is scanned over the sample surface the tip is maintained at a constant force (to obtain height information), or height (to obtain force information) above the sample surface Tips are typically made from Si3N4 or Si, and extended down from the end of a cantilever An optical detection system is used, in which a diode laser is focussed on the back of a reflective cantilever As the tip moves up and down with the contour of the surface, the laser beam is deflected off the attached cantilever into a dual element photodiode AFM Schematic

30. Background - AFM Modes Contact Mode the tip scans the sample in close contact with the surface The force on the tip is repulsive with a mean value of 10 -9 N the deflection of the cantilever is sensed and compared in a DC feedback amplifier to some desired value of deflection Non-Contact Mode (used when tip contact might alter the sample surface) In this mode the tip hovers 50 - 150 Angstrom above the sample surface Attractive Van der Waals forces acting between the tip and the sample are detected topographic images are constructed by scanning the tip above the surface

31. Background - AFM Modes Tapping Mode:(sample surfaces that are easily damaged ) The cantilever assembly is oscillated at or near the cantilever's resonant frequency the cantilever is oscillated with a high amplitude when the tip is not in contact with the surface The oscillating tip is then moved toward the surface until it begins to lightly touch, or tap the surface. During scanning, the vertically oscillating tip alternately contacts the surface and lifts off The reduction in oscillation amplitude is used to identify and measure surface features.

32. Motivation Earlier Work involved determination of contact stiffness and localized elastic modulus measurement of the surface The vibrational spectrum of the cantilever is used to discern local elastic data. It becomes imperative to understand the vibrational spectra completely to perform the above mentioned measurements The free vibrational response would help to characterize the cantilever or the probe Moreover, since the boundary conditions are also changed during the contact mode resonance, Free vibrational response and imaging the mode shape would help as a tool for calibration or standard. * Ultrasonics 38(2000) 430-437 * Journal of Applied Physics, 82(1997) 966 * Review of Scientific Instruments 67(1996) 3281