# Design and Modeling of a High-Speed Scanner for Atomic Force Microscopy

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3 III. MODELING OF THE SCANNER In this section a simplified mathematical model of the scanner is derived that explains some dynamics of the system along the vertical positioning axis. The device is a complicated mechanical system with many oscillatory modes. We start by developing a simple mathematical model that captures only one mode. The scanner system is modeled as two masses supported by a spring with damping. A schematic diagram is shown in Fig. 3. When operating the Z-piezo we are interested in the transfer function from the elongation to the top of the piezo stack m. We have G G x x ( m m s cs k, s cs k s s (. s s m () This transfer function is characterized by three parameters: The resonance frequency k /( m m ) of the total mass oscillating against the spring. The relative damping c. k( m m ) The parameter = m / (m +m ) which is determined by the ratio of the masses. Fig. 3. Model of the high-speed scanner in the vertical direction. Let m and m be the masses, k the effective spring constant, and c the damping coefficient. Furthermore, let the positions of the center of the masses be x and x. The elongation of the piezo stack is = x - x. Let the reaction force created by elongation of the piezo stack be F. A momentum balance gives the following model for the system. The parameters can be determined from a step response. The parameter can be obtained from the initial and final values of the step, and the frequency and relative damping from the oscillation period and decay. Examples of step responses for different parameters are shown in Fig. 4. d x m F dt d x dx m c kx F dt dt Taking the Laplace transform gives and we get L( X ( X X ( m s X ( m s L( X ( X ( ms m m s cs k L(. m s m s cs k cs k (, m s cs k Fig. 4. Step responses for the model () with =. and =.7 (solid blue),.8 (dashed red) and.9 (dotted blue) (upper curve. The lower curves show the step responses when a time constant of the power amplifier of A =.7/ has been added. More accurate modeling can be obtained from the frequency response. The Bode plots of the transfer function are shown in Fig Authorized licensed use limited to: IEEE Xplore. Downloaded on October 6, 8 at 8:44 from IEEE Xplore. Restrictions apply.

4 output. The input of the power amplifier for the Z piezo is regarded as the system input. The input steps are generated by a function generator (HP 33A, Palo Alto, CA). Fig. 6 shows a measured step response at an input step height of 8mV (blue signal) that is amplified by the power amplifier to.v (red signal). This voltage is applied to the piezo and represents a nominal step height of 34 nm. The cantilever deflection is measured by the optical lever setup of the AFM using a custom made high-bandwidth deflection electronics (green signal) and the electronics of the commercial AFM (purple signal), respectively. The measured step response (green signal) is in good agreement with the simulation in Fig. 4. Fig. 5. Bode plot of the model () with =. and =.7 (solid blue),.8 (dashed red) and.9 (dotted blue). We have i ), G x ( i where for small the peak of the frequency response occurs approximately at and we have G x. ( i ) 4 For small the dip in the frequency response occurs approximately at and we have / ( i ) i. i G x For =.9 and =. the expressions give a peak of.69 and a dip of.38, and the dip occurs at /. 5. IV. EXPERIMENTAL DATA The model derived in Section III is verified by experimental data presented in this section. Step responses of the scanner in the Z-direction are measured. A piece of a silicon wafer is glued on top of the piezo as a standard specimen. The Z-position of the sample is sensed by the AFM cantilever operated in contact mode without scanning. For this experiment the photo diode signal displaying the cantilever deflection represents the system Fig. 6. Measured step response of the high-speed AFM system in Z- direction. System input: input of the power amplifier; system output: AFM deflection signal. (A) Input step signal (blue, 5mV/div), (B) output of the power amplifier (red, 5mV/div), (C) highbandwidth cantilever deflection (green, mv/div), (D) low-bandwidth cantilever deflection (purple, V/div); time scale: 5 μs/div. Amplitude versus frequency data of the amplifier and scanner system in the vertical direction are recorded with a network analyzer (4395A, Agilent, Palo Alto, CA). The input to the system is again the input of the power amplifier. The system output is the vibration of the top of the sample on the Z piezo measured by a scanning laser vibrometer (OFV-3, Polytec, Waldbronn, Germany). The sinusoidal frequency sweep with constant excitation amplitude is also generated by the network analyzer. Using the vibrometer we recorded velocity data of the piezo top during the frequency sweep. The velocity data are converted into position data by dividing gain by frequency and subtracting 9 from the phase, according to an integration of a sinusoidal signal. The nominal time-delay of the vibrometer of.9 microseconds is also removed from the phase data. Fig. 7 shows a typical bode plot that has been recorded. The most dominant frequency peak occurs at about 4 khz and is in good agreement with the simulation shown in Fig Authorized licensed use limited to: IEEE Xplore. Downloaded on October 6, 8 at 8:44 from IEEE Xplore. Restrictions apply.

6 deflection is done with an analogue proportional-integral feedback controller. Images are recorded in contact mode, where the topography image (represented by the feedback signal) and the deflection image (representing the control error) are displayed. VI. CONCLUSION In this paper we present the design and a simplified mathematical model of a high-speed scanner for an atomic force microscope. The model obtained from first principles gives a good basis for implementing a model-based controller for better control performance and an observer for more accurate conversion of the control action into the topography signal [6][7][9]. Experimental step responses and frequency spectra verify the mathematical model and enable to obtain quantitative values of some unknown parameters of the mathematical model. Performance results demonstrating the high-speed imaging capability of the new AFM system are shown. ACKNOWLEDGMENT The authors would like to thank Dr. Johannes Kindt for fruitful discussions. [] D. Croft, G. Shed, S. Devasia, Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy applications, ASME Journal of Dynamic Systems, Measurement, and Control 3, p () [3] Q. Zou, K.K. Leang, E. Sadoun, M.J. Reed, S. Devasia, Control issues in high-speed AFM for biological applications: collagen imaging example, Asian Journal of Control 6(), p [4] G. Schitter, A. Stemmer, Identification and open-loop tracking control of a piezoelectric tube scanner for high-speed scanning probe microscopy, IEEE Transactions on Control Systems Technology (3), p (4) [5] A. Daniela, S. Salapaka, M.V. Salapaka, M. Dahleh, Piezoelectric scanners for atomic force microscopes: design of lateral sensors, identification and control, Proc. of the 999 American Control Conference, San Diego, CA, p.53-7 (999) [6] S. Salapaka, A. Sebastian, J.P. Cleveland, M.V. Salapaka, high bandwidth nano-positioner: a robust control approach Review of Scientific Instruments 73(9), p.33-4 () [7] G. Schitter, A. Stemmer, F. Allgower, Robust two-degree-of-freedom control of an atomic force microscope, Asian Journal of Control 6(), p (4) [8] K. El Rifai, O.M. El Rifai, K. Youcef-Toumi, On Dual Actuation in Atomic Force Microscopes, Proc. of the 4 American Control Conference, Boston, MA, p. (4) [9] S. Salapaka, T. De, A. Sebastian, Sample-profile estimate for fast atomic force micrsocopy, Applied Physics Letters 87, p.53 (5) [] G. Schitter, G.E. Fantner, J.H. Kindt, P.J. Thurner, P.K. Hansma, On recent developments for high speed atomic force microscopy, Proc. of the Conference on Advanced Intelligent Mechatronics 5, Monerey, CA, p.6-4 (5) [] G.E. Fantner, P. Hegarty, J.H. Kindt, G. Schitter, G.A.G Citade, P.K. Hansma, Data acquisition system for high-speed atomic force microscopy, Review of Scientific Instruments 76, p.68 (5) REFERENCES [] G. Binnig, C.F. Quate, and C. Gerber, Atomic force microscope, Physical Review Letters 56(9), p (986) [] D. Sarid, Scanning Force Microscopy, New York: Oxford University Press, 994 [3] P.K. Hansma, V.B. Elings, O. Marti, C.E. Bracker, Scanning tunneling microscopy and atomic force microscopy: application to biology and technology, Science 4(4876), p.9-6 (988) [4] A.L. Weisenhorn, J.E. Mac Dougall, S.A.C. Gould, S.D. Cox, W.S. Wise, J. Massie, P. Maivald, V.B. Elings, G.D. Stucky, P.K. Hansma, Imaging and manipulating molecules on a zeolite surface with an atomic force microscope, Science 47(4948), p.33-3 (99) [5] Q. Zhong, D. Inniss, K. Kjoller, V.B. Elings, Fractured polymer/silica fiber surface studied by tapping mode atomic force micrsocpoy, Surface Science Letters 9, p.l688-9 (993) [6] G. Schitter, P. Menold, H.F. Knapp, F. Allgower, A. Stemmer, High performance feedback for fast scanning atomic force microscopes, Review of Scientific Instruments 7(8), p.33-7 () [7] G. Binnig, D.P.E. Smith, Single-tube three dimensional scanner for scanning tunneling microscopy, Review of Scientific Instruments 57(8), p (986) [8] H.J. Mamin, H. Birk, P. Wimmer, D. Rugar, High speed scanning tunneling microscopy: principles and applications, Journal of Applied Physics 75(), p.6-8 (994) [9] S.R. Manalis, S.C. Minne, C.F. Quate, Atomic force microscopy for high speed imaging using cantilevers with an integrated actuator and sensor, Applied Physics Letters 68(6), p.87-3 (996) [] T. Ando, N. Kodera, E. Takai, D. Maruyama, K. Saito, A. Toda, A high-speed atomic force microscope for studying biological macromolecules, Proceedings of the National Academy of Sciences USA 98(), p () [] J.H. Kindt, G.E. Fantner, J.A. Cutroni, P.K. Hansma, Rigid design of fast scanning probe microscopes using finite element analysis, Ultramicroscopy (3-4), p (4) 57 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 6, 8 at 8:44 from IEEE Xplore. Restrictions apply.

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