A novel method for tapping atomic force microscopy

Abstract— In tapping mode atomic
force microscopy (TM-AFM), stiffness
range in which imaging can be done is
restricted due to lesser sensitivity at
higher stiffness values. In this work, a
controller for modifying the stiffness
and damping of a micro cantilever
sensor is proposed. Incorporating this
technique with conventional TM-AFM
can result higher stiffness range for
imaging.
Index Terms— Damping, sensitivity,
stiffness and TM-AFM.
I. INTRODUCTION
N March 1981 G. Binning, H. Rohrer, Ch.
Gerber and E. Weibel at the IBM Zurich
Research Laboratory observed vacuum
tunneling of electrons between a sharp tungsten
tip and a platinum sample. Combined with the
ability to scan the tip against the sample
surface, the scanning tunneling microscope
(STM) was born. Since then, this novel type of
microscopy has continuously broadened our
perception about atomic scale structures and
processes [1]. The development of the STM
technique has triggered the invention of a whole
family of scanning probe microscopes (SPM)
which make use of almost various kind of
interaction between a tip and a sample. This
class of SPM can provide information about
nanometer-scale properties of matter which is
often accessible by other experimental
techniques, thereby playing a key role in
nanotechnology.
Figure 1 Cantilever probe model
Tapping mode atomic force microscopy (TMAFM)
is one among various SPM methods.
TM-AFM uses a micro cantilever sensor,
oscillated at a fixed frequency near its
resonance. As shown in figure 1, this cantilever
probe is modeled as a spring mass system,
where interaction with a sample modifies the
stiffness and damping, thereby changing the
amplitude of cantilever oscillations. A feedback
controller is used to maintain the amplitude at a
given set point. Variation of amplitude with
respect to interaction stiffness, i.e. sensitivity,
becomes lesser at higher stiffness values as
shown in figure 2. This limits the stiffness range
in which imaging can be done.
Figure 2 Frequency responses of cantilever
probe
We propose a controller, designed using state
space analysis which can change stiffness and
damping of the micro cantilever sensor. As
resonance frequency is a function of stiffness,
by changing stiffness the resonance frequency
can be modified. Therefore, by reducing the
stiffness of micro cantilever by interaction
stiffness, high sensitivity can be retained.
Anil K M and Ashwin Lal
I
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II. DESIGN
1. Modeling Cantilever Probe in State Space
The cantilever probe shown in figure 1 can be
mathematically modeled as,

  
Here k, q and m are the equivalent stiffness,
damping and mass of the cantilever probe due
to interaction with sample. Applied voltage on
the cantilever is u and amplitude of oscillation
is x. This system can be represented in state
space with the following two equations [2, 3],
 


First equation is called state equation and the
second output equation. Here X is called state
vector, A the state matrix, B the input matrix, C
the output matrix, u the input and y the output.
Mathematically,

,   
      , 
 and
 .
2. Controller Design
Expanding the state equation will give,
   -  -

In this design, input in state equation i.e. ‘u’,
is modified adaptively to ‘u + BKX’, where B is
the input matrix, K the correction matrix and X
the state vector. The correction matrix K
consists of two terms one for stiffness
modification and the other for damping
modification. Mathematically,
 
Now, expanding the state equation will give,
  !  -   " 
 ! -  "

The above expression shows that stiffness is
now a function of Ks and damping a function of
Kd. The correction introduced in  for this
dependency is
#   # . This correction is realized using
the feedback loop shown in figure 3, where
there are two feedbacks correcting the 
concurrently.
Figure 3 Block diagram of the Controller
3. Realization of Controller
The controller shown in figure 3 is
implemented with the help of NI PXI-8106
Embedded Controller with GPIB card and NI
PXI-7831R FPGA card. Block level diagram of
the set up is shown in figure 4.
Figure 4 Block diagram of the set up
The experiments are done using a 32.7661
kHz quartz tuning fork as cantilever. The
tunneling current amplifier amplifies nanoampere
current through quartz tuning fork by a
factor of $ for acquisition using NI PXI
7831R. This FPGA card is programmed with a
controller which acquires controlled variable at
a sampling rate of 800kS/s using interleaved
sampling of four Analog Input channels. After
processing, it outputs manipulated variable to
the quarts tuning fork at 800kS/s. PXI 8106
Embedded Controller is used to control the
FPGA program, to modify parameters of the
controller implemented inside FPGA, to control
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SR844 RF Lock-In Amplifier, to acquire data
from SR844 RF Lock-In Amplifier, to plot
frequency response of the modified cantilever
probe and to synchronize all these operations.
Communication with SR844 RF Lock-In
Amplifier is done with the help of GPIB
interface as shown in figure 4.
The PXI 7831 FPGA VI and PXI 8106
Embedded Controller VI are shown in figure 5
and 6 respectively.
Figure 5 PXI 7831 FPGA VI
Figure 6 PXI 8106 Embedded Controller VI
III. RESULTS
Frequency responses of the quartz tuning fork
with and without feedback are shown in figure
7-11.
Figure 7 Natural frequency response
Figure 8 Frequency response with decreased
damping
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Figure 9 Frequency response with increased
damping
Figure 10 Frequency response with increased
stiffness
Figure 11 Frequency response with decreased
stiffness
The results show that stiffness as well as
damping of quartz tuning fork is modified
adaptively with the designed controller.
IV. CONCLUSION AND FUTURE WORK
The controller designed using state space
approach is able to change damping as well as
stiffness of a cantilever. The range in which
stiffness modification can be done is 1.2937e+7
times the mass of cantilever