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 Si 3 N 4 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. 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. Background - AFM Modes

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. Theory: Problem Statement Boundary Conditions :  Flexural Vibrations Clamped end: Free End:  Torsional Vibrations Clamped End: Free End: L a b 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/m 2 ) I - Area moment of inertia - ab 3 /12 (m 4 ) J - Polar moment of inertia - a 3 b/12 (m 4 ) G - Rigidity modulus (N/m 2 ) C T - Torsional Stiffness- ab 3 G/3 (Nm 2 )

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

9. Theory: Flexural Vibrations  A pplying 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. Shear Wave Transducer Sample Cantilever Excitation of Torsional Vibrations

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

18. Michelson Interferometer A R =a r *e i(  t-kz R ) A O =a o *e i(  t-k(z o -δ)) Laser Sample Detector B.S. Reference Mirror

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

20. Heterodyne Interferometry A R =a r *e i((  +  )t-kz R ) A O =a o *e i(  t-k(z o -δ)) Laser Sample Detector B.S. Reference Mirror Frequency Shifter

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

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

23. Mode Coupling  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 h b d m t - mass of the tip d - offset from the center of the beam b - thickness of the beam h - length of the tip

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

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

26. Mode Coupling Calculated Amplitude distribution based on mode coupling with H=0.025 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 Si 3 N 4 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. 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. Background - AFM Modes

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