Current methods to study atomic force microscope (AFM) cantilever dynamics use model simplification or are based on the non-trivial solution of the equation of motion. As an alternative method, transfer function analysis gives a more complete description of system dynamics. In this work a transfer function study of two different AFM configurations, the point force and base driven cantilever, is presented. Exact analytical expressions of the infinite dimensional transfer function are derived for cantilever deflection and slope along the cantilever. Frequency response and transfer function infinite product expansion are obtained for the case where system outputs are set at the free end of the cantilever. The frequency response reflects the full complexity of cantilever dynamics affected by the presence of an infinite number of poles and zeros. An analytical expression for all the zeros and poles of the system is obtained. From the frequency response and pole-zero investigations it is shown how cantilever actuation and output measurement affect AFM operation and cantilever dynamics modelling. Transfer function analysis of AFM cantilevers opens the possibility of model based AFM operation to increase imaging and manipulation performance.

Volume Subject Area:
Design Engineering
Topics:
Atomic force microscopy, Cantilevers, Transfer functions, Dynamics (Mechanics), Frequency response, Poles (Building), Deflection, Equations of motion, Imaging, Modeling, System dynamics
1.
Binnig
G.
,
Quate
C. F.
,
Gerber
C.
,
1986
, “
Atomic force microscopy
,”
Physical Review Letters
,
56
(
9
), pp.
930
–
933
.
2.
Colton R.J., Engel A., Frommer J.E., Gaub H.E., Gewirth A., Guckenberger R., Rabe J., Heckl W.M., Parkinson B., 1998, Procedures in scanning probe microscopies, John Wiley and Sons, Chichester.
3.
Wouters
D.
,
Schubert
U. S.
,
2004
, “
Nanolithography and Nanochemistry: Probe-Related Patterning Techniques and Chemical Modification for Nanometer-Sized Devices
,”
Angewandte Chemie International Edition
,
43
(
19
), pp.
2480
–
2495
.
4.
Rubio-Sierra
F. J.
,
Heckl
W. M.
,
Stark
R. W.
,
2005
, “
Nanomanipulation by Atomic Force Microscopy
,”
Advanced Engineering Materials
,
7
(
4
), pp.
193
–
196
.
5.
Burnham
N. A.
,
Gremaud
G.
,
Kulik
A. J.
,
Gallo
P. J.
,
Oulevey
F.
,
1996
, “
Materials’ properties measurements: Choosing the optimal scanning probe microscope configuration
,”
Journal of Vacuum Science and Technology B
,
14
, pp.
1308
–
1312
.
6.
Meirovitch L., 1967, Analytical Methods in Vibrations, Macmillan, New York.
7.
Stark
R. W.
,
Schitter
G.
,
Stark
M.
,
Guckenberger
R.
,
Stemmer
A.
,
2004
, “
State space model of freely vibrating and surface-coupled cantilever dynamics in atomic force microscopy
,”
Physical Review B
,
69
(
8
),
085412
085412
.
8.
Dareing
D. W
,
Thundat
T.
,
Sangmin
J.
,
Nicholson
M.
,
2005
, “
Modal analysis of microcantilever sensors with environmental damping
,”
Journal of Applied Physics
,
97
(
8
),
084902
084902
.
9.
Junkins J. L., Kim Y., 1993, Introduction to Dynamics and Control of Flexible Structures, AIAA, Washington, D.C.
10.
Goodson
R. E.
,
1970
, “
Distributed System Simulation Using Infinite Product Expansions
,”
Simulation
,
15
(
6
), pp.
255
–
263
.
11.
Spector
V. A.
,
Flashner
H.
,
1990
, “
Modeling and Design Implications of Noncollocated Control in Flexible Systems
,”
ASME Journal of Dynamic Systems, Measurement and Control
,
112
(
2
), pp.
186
–
193
.
12.
Florin
E. L.
,
Radmacher
M.
,
Fleck
B.
,
Gaub
H. E.
,
1994
,
Review of Scientific Instruments
,
65
(
3
), pp.
639
–
643
.
13.
Rugar
D.
,
Mamin
H. J.
,
Erlandsson
R.
,
Stern
J. E.
,
Terris
B. D.
,
1988
, “
Force microscope using a fiber-optic displacement sensor
,”
Review of Scientific Instruments
,
59
(
11
), pp.
2337
–
2340
.
14.
Meyer
G.
,
Amer
N. M.
,
1990
, “
Optical-beam-deflection atomic force microscopy: The NaCl (001) surface
,”
Applied Physics Letters
,
56
(
21
), pp.
2100
–
2101
.
15.
Simmons G., 1991, Differential Equations with Applications and Historical Notes, McGraw-Hill, New York.
16.
Franklin G.F., Powell J.D., Emami-Naeini A., 1986, Feedback Control of Dynamic Systems, Addison-Wesley, Reading, MA.
17.
Garcia
R.
,
Perez
R.
,
2002
, “
Dynamic atomic force microscopy methods
,”
Surface Science Reports
,
47
(
6–8
), pp.
197
–
301
.
This content is only available via PDF.
Copyright © 2005
 by ASME
You do not currently have access to this content.