×

zbMATH — the first resource for mathematics

Nonlinear dynamics, stability and control of the scan process in noncontacting atomic force microscopy. (English) Zbl 1178.74122
Summary: The nonlinear equations of motion for the scan process in noncontacting atomic force microscopy are consistently derived using the extended Hamilton’s principle. A modal dynamical system obtained from the continuum model reveals that scan control appears in the form of parametric excitation. The system is analyzed asymptotically and numerically to yield escape bounds limiting the noncontacting mode of operation. Approximate stability bounds are deduced from both a global Melnikov integral and a local Moon-Chirikov overlap criterion. The Melnikov-Holmes stability curve and the overlap criterion are found to be similar for small damping. However, for very small damping, typical of ultra-high vacuum conditions, where the Melnikov bound becomes trivial, the Moon-Chirikov criterion yields an improved stability threshold.

MSC:
74M05 Control, switches and devices (“smart materials”) in solid mechanics
74H55 Stability of dynamical problems in solid mechanics
74M25 Micromechanics of solids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Binning, G., Quate, C.F., Gerber, C.: Atomic force microscope. Phys. Rev. Lett. 56(9), 930–936 (1986) · doi:10.1103/PhysRevLett.56.930
[2] Giessible, F.J.: Advances in Atomic force microscopy. Rev. Mod. Phys. 75(3), 949–983 (2003) · doi:10.1103/RevModPhys.75.949
[3] Sarid, D.: Scanning Force Microscopy: With Applications to Electric, Magnetic, and Atomic Forces. Oxford University Press, New York (1991)
[4] Paulo, A.S., Garcia, R.: Unifying theory of tapping-mode atomic-force microscopy. Phys. Rev. B 66, 041406(R) (2002) · doi:10.1103/PhysRevB.66.041406
[5] Durig, U., Zuger, O., Stalder, A.: Interaction force detection in scanning probe microscopy: methods and applications. J. Appl. Phys. 72(5), 1778–1798 (1992) · doi:10.1063/1.352348
[6] Mertz, J., Marti, O., Mlynek, J.: Regulation of a microcantilever response by force feedback. Appl. Phys. Lett. 62(19), 2344–2346 (1993) · doi:10.1063/1.109413
[7] Ashhab, M., Salapaka, M.V., Dahleh, M., Mezic, I.: Dynamical analysis and control of microcantilevers. Automatica 35, 1663–1670 (1999) · Zbl 0941.93044 · doi:10.1016/S0005-1098(99)00077-1
[8] Perez, H., Zou, Q., Devasia, S.: Design and control of optimal scan trajectories: Scanning tunneling microscope example. ASME J. Dyn. Syst. Meas. Control 126, 187–197 (2004) · doi:10.1115/1.1636770
[9] Minne, S.C., Manalis, S.R., Quate, C.F.: Bringing Scanning Probe Microscopy Up to Speed. Kluwer, Boston (1999)
[10] Schitter, G., Fantner, G.E., Kindt, J.H., Thurner, P.J., Hansma, P.K.: On recent developments for high-speed atomic force microscopy. In: Proceedings of the 2005 IEEE/ASME, International Conference on Advanced Intelligent Mechatronics, Monterey, CA, July 2005
[11] Manalis, S.R., Minne, S.C., Quate, C.F.: Atomic force microscopy for high speed imaging using cantilevers with an integrated actuator and sensor. Appl. Phys. Lett. 68(6), 871–874 (1996) · doi:10.1063/1.116528
[12] Thompson, J.M.T.: Chaotic phenomena triggering the escape from a potential well. Proc. R. Soc. Lond. A 421, 195–225 (1989) · Zbl 0674.70035 · doi:10.1098/rspa.1989.0009
[13] Thompson, J.M.T.: Nonlinear Dynamics and Chaos. Wiley, New York (2002) · Zbl 1174.37300
[14] Erlandsson, R., Olsson, L.: Force interaction in low-amplitude ac-mode atomic force microscopy: cantilever simulation and comparison with data from Si(111)7\(\times\)7. Appl. Phys. A 66, S879–S883 (1998) · doi:10.1007/s003390051260
[15] Couturier, G., Nony, L., Boisgard, R., Aimé, J.P.: Stability of an oscillating tip in noncontact atomic force microscopy: theoretical and numerical investigations. J. Appl. Phys. 91(4), 2537–2543 (2002) · doi:10.1063/1.1428084
[16] Stark, R.W., Heckl, W.M.: Fourier transformed atomic force microscopy: tapping mode atomic force microscopy beyond the Hookian approximation. Surf. Sci. 457, 219–228 (2000) · doi:10.1016/S0039-6028(00)00378-2
[17] Rodriguez, T.R., Garcia, R.: Tip motion in amplitude modulation (tapping-mode) atomic-force microscopy: Comparison between continuous and point-mass models. Appl. Phys. Lett. 80(9), 1646–1648 (2002) · doi:10.1063/1.1456543
[18] Turner, J.A., Hirsekorn, S., Rabe, U., Arnold, W.: High-frequency response of atomic-force microscope cantilevers. Appl. Phys. Lett. 82(3), 966–979 (1997)
[19] Turner, J.A.: Non-linear vibrations of a beam with cantilever-Hertzian contact boundary conditions. J. Sound Vib. 275(1), 177–191 (2004) · doi:10.1016/S0022-460X(03)00791-0
[20] Wolf, K., Gottlieb, O.: Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer. J. Appl. Phys. 91(7), 4701–4709 (2002) · doi:10.1063/1.1458056
[21] Heim, L.O., Kappl, M., Butt, H.J.: Tilt of atomic force microscope cantilevers: Effect on spring constant and adhesion measurements. Langmuir 20, 2760–2764 (2004) · doi:10.1021/la036128m
[22] Cannara, R.J., Eglina, M., Carpick, R.W.: Lateral force calibration in atomic force microscopy: A new lateral force calibration method and general guidelines for optimization. Rev. Sci. Instrum. 77, 053701 (2006) · doi:10.1063/1.2198768
[23] Stiernstedt, J., Rutland, M.W., Attard, P.: A novel technique for the in situ calibration and measurement of friction with the atomic force microscope. Rev. Sci. Instrum. 76, 083710 (2005) · doi:10.1063/1.2006407
[24] Crespo da Silva, M.C., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams. I. Equation of motion. II. Forced motions. J. Struct. Mech. 6(4), 437–461 (1978)
[25] Israelachvili, J.: Intermolecular and Surface Forces. Academic Press, London (1992)
[26] Sarid, D., Ruskell, T.G., Workman, R.K., Chen, D.: Driven nonlinear atomic force microscopy cantilevers: from noncontact to tapping modes of operation. J. Vac. Sci. Technol. B 14(2), 864–867 (1996) · doi:10.1116/1.589163
[27] Nayfeh, A.H., Pai, P.F.: Non-linear non-planar parametric responses of an inextensional beam. Int. J. Non-Linear Mech. 24, 139–158 (1989) · Zbl 0673.73043 · doi:10.1016/0020-7462(89)90005-X
[28] Crespo da Silva, M.C.: Equations for nonlinear analysis of 3D motions of beams. Appl. Mech. Rev. 44(11), 51–59 (1991)
[29] Cannara, R.J., Brukman, M.J., Carpick, R.W.: Cantilever tilt compensation for variable-load atomic force microscopy. Rev. Sci. Instrum. 76, 053706 (2005) · doi:10.1063/1.1896624
[30] Meirovitch, L.: Fundamentals of Vibrations. McGraw-Hill, Boston (2001) · Zbl 0976.93008
[31] Stupnicka, W.S., Plaut, R.H., Hsieh, J.C.: Period doubling and chaos in unsymmetric structures under parametric excitation. J. Appl. Mech. 56(4), 947–952 (1989) · doi:10.1115/1.3176195
[32] Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983) · Zbl 0515.34001
[33] Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995) · Zbl 0848.34001
[34] Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
[35] Maris, J.P., Piednoir, A., Zambelli, T., Bouju, X., Gauthier, S.: Experimental investigation of resonance curves in dynamic force microscopy. Nanotechnology 14, 1036–1042 (2003) · doi:10.1088/0957-4484/14/9/319
[36] Albrecht, T.R., Grutter, P., Horne, D., Rugar, D.: Frequency modulation detection using high Q cantilevers for enhanced force microscope sensitivity. J. Appl. Phys. 69(2), 668–673 (1991) · doi:10.1063/1.347347
[37] Gottlieb, O., Champneys, A.R.: Global bifurcations of nonlinear thermoelastic microbeams subject to electrodynamic actuation. In: Rega, G., Vestroni, F. (eds.) Chaotic Dynamics and Control of Systems and Processes in Mechanics, pp. 117–126. Kluwer Academic, Dordrecht (2004)
[38] Yagasaki, K.: Chaotic dynamic of a quasi-periodically forced beam. Trans. J. Appl. Mech. 59, 161–167 (1992) · Zbl 0761.73067 · doi:10.1115/1.2899422
[39] Ashhab, M., Salapaka, M.V., Dahleh, M., Mezíc, I.: Melnikov-based dynamical analysis of microcantilevers in scanning probe microscopy. Nonlinear Dyn. 20(3), 197–220 (1999) · Zbl 0964.74041 · doi:10.1023/A:1008342408448
[40] Moon, F.C.: Chaotic and Fractal Dynamics. Wiley, New York (1992)
[41] Moon, F.C.: Experiments on chaotic motions of a forced nonlinear oscillator: Strange attractors. J. Appl. Mech. 47, 638–644 (1980) · doi:10.1115/1.3153746
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.