Download Quantifying electrostatic force contributions in

Supplementary material to:
Quantifying electrostatic force contributions in electrically biased nanoscale
interactions
C.Maragliano, A. Glia, M.Stefancich and M.Chiesa
Laboratory for Energy and NanoScience (LENS), Institute Center for Future Energy Systems (iFES),
Masdar Institute of Science and Technology, P.O. Box 54224, Abu Dhabi, UAE
S1. Derivation of the relation between electrostatic force derivative and phase of the oscillating cantilever
Since D-AFM allows measuring the phase shift with respect to the driving force, a possible method for
electrostatic force reconstruction is to find a relation between the phase shift and
. In EFM, such
relationship can be found by linearizing
as function of . Although the electrostatic term is, in general,
non-linear, for small oscillations ̅( ), one can approximate
( ( )) by means of a Taylor expansion
centered in . Assuming that the concavity of
( ( )) does not change considerably within the range of
̅( ) and when centered in , the tip-sample interaction term can be rewritten as:
( ( ))
( ( ))
( )
|
̅( )
( )
( ( ) )
(S.1)
where the Taylor expansion is truncated at the first order term. The combination of harmonic oscillator
equation and (1) leads to:
̅( )
( ( ))
̅( ) [
|
( )
]
̅( )
( )
(S.2)
Defining:
(S.3)
( ( ))
|
( )
(S.4)
(S.5)
with constant solution:
(
)
the time dependent response, assuming harmonicity, can be written as:
(S.6)
̅( )
(
)
(S.7)
The amplitude and phase response then follow in a useful form as:
( )
[(
( )
(
)⁄
)
(
⁄
(S.8)
)
(S.9)
) ]
(
Moreover, at the resonance frequency
:
( ( ))
|
(S.10)
( )
(S.11)
Thus, the phase shift
(
can finally be written as:
( ( ))
|
Defining
)
(
|
( )
)
( ( ))
( )
(S.12)
( )
and using the trigonometric relationship
( )
( )
, it follows
that:
( )
Furthermore if
( ( ))
(
|
( )
)
(S.13)
, by Taylor expansion:
( ( ))
|
(S.14)
( )
Equation (14) correlates a measurable quantity, i.e. the phase lag, with the tip-sample interaction force. By
setting
to be constant and greater than 10 nm and by plotting the phase lag as a function of
, we
obtain:
( ( ))
|
(
)
( )
where we neglected the contribution to
(
)
( ( )) due to short-range forces.
(S.15)
S2. Electrical tuning: setting the oscillation amplitude of the cantilever
In ac-EFM the oscillation amplitude of the cantilever at frequency ω is determined univocally by the
intensity of the electrostatic interaction between the tip and the sample. For a given tip-sample system,
this interaction depends on the amplitude of the alternating voltage
applied to the tip and on the tipsample separation . This means that, setting for example the voltage
to be constant, the value of
depends only on the tip-sample separation : it increases or decreases depending on whether the tip is
approached to or moved away from the sample. In this view it seems that, simply by recording the
oscillation amplitude of the cantilever as function of , it is possible to quantify the electrostatic
interaction between tip and sample. However, such conclusion is valid only if the electrostatic force does
not change considerably within an oscillation amplitude, as explained in eq (10). Given the shape of the
electrostatic force curve (proportional to the derivative of the tip-sample capacitance reported in Fig. 2),
this requisite is particularly critical at small tip-sample separation, where the electrostatic force curve
diverges. Thus, the electrical tuning, i.e. the procedure that allows setting and quantifying the oscillation
amplitude of the cantilever as function of frequency for a given voltage
and tip-sample separation ,
is of fundamental importance for the minimization of the error in the force reconstruction process. In
particular, the equilibrium tip-sample separation at which the electrical tuning is performed, for a given
tip-sample system and voltage
, is critical. This can be easily understood using a numerical example.
Let’s suppose to reconstruct the electrostatic force curve for a tip-sample separation ranging from 10 to
100 nm. Let’s apply a voltage
to the tip in order to have an oscillation amplitude of 20 Å at =100 nm.
Even though such amplitude might seem small enough to reconstruct the electrostatic force curve in the
all range, in the following we will show that this is not the case. Fig. S1 shows the error performed in the
reconstruction of the electrostatic force curve as well as the oscillation amplitude of the cantilever, both
as function of the tip-sample separation.
Fig. S1 (left) Error performed in reconstructing the force versus distance profile taking the oscillation amplitude as observable.
(right) amplitude versus tip-sample distance curve. In this case, the amplitude is set to be equal to 20 Å at 100 nm away from
the sample surface. Parameters used for this analysis: R=70 nm, θ=10 degrees, H=13 μm.
While the error at =100 nm is below 1%, at 20 nm away from the sample surface it increases up to 20%.
This is due to the fact that the oscillation amplitude has increased up to 3 nm and the electrostatic force
curve cannot be approximated with such amplitude in this region. This result demonstrates that setting
the oscillation amplitude at high tip-sample separation does not allow preserving from high error because
of the increasing trend of the amplitude. On the other hand, setting the oscillation amplitude at the
smallest tip-sample separation of interest (in this case 10 nm), we can determine directly the greatest
error performed in the electrostatic force curve reconstruction. The same conclusion can be derived in the
case in which the phase of the cantilever is taken as observable.
S3. Quantitative force versus distance measurements in amplitude modulation AFM: a novel force
inversion technique
We report the formalism proposed by Katan et al. [1] for extracting quantitative data from amplitude
modulation dynamic force-distance measurements. The method is basically based on the harmonic
oscillator model of the vibrating atomic force microscope cantilevers. It allows extracting both the
conservative and dissipative parts of the tip-sample interaction from a measurement of oscillation
amplitude and phase as a function of distance. The formalism is derived from Sader formula [2]. He
developed with his co-worker a theoretical framework to calculate the tip-sample interaction force from a
measured distance dependence of the frequency shift and dissipation signals in frequency modulation
atomic force microscopy (FM-AFM). The same theoretical framework has been extended to the realm of
amplitude modulation AFM (AM-AFM) by Katan et al and is here reported.
An AFM cantilever moving through a fluid (such as air) can be modeled as a point mass with an effective
mass , attached to a spring with spring constant . This system experiences a viscous drag with drag
coefficient
, where
is the resonance frequency of the cantilever and
is the quality factor of
the system. The equation of motion for the cantilever tip reads then:
( )
( )
( )
( ( ))
(
)
(1)
where
is the tip-sample interaction force [3] and the last term is the driving force with frequency
and a constant drive amplitude . The displacement ( ) can be expressed using a constant term and a
time dependent term ̅( ):
( )
̅( )
(2)
External forces acting on the cantilever are thus divided into the applied driving force
( ) and the
tip-sample interaction force
. Generally, we want to measure the latter.
can be divided into a
conservative force
and a dissipative force
( )
. A helpful discussion on the differences
between conservative and dissipative forces can be found in Sader et al. [2].
According to the formalism adopted by Katan et al., the two terms that compose
can be written as:
( )
( )
∫ ((
√ (
( )
)
( )
)
√ (
( )
∫ ((
( )
( )
)
)
)
( )
(3)
( )
( )
)
(4)
Where ( ) is the distance-dependent oscillation amplitude of the cantilever,
tip-sample approach,
is a function defined as:
is the distance of closest
( )
and
(
(
√ (
)
√ (
)
( ( )))
)
(5)
is another function defined as:
( )
( )(
In these functions,
as:
(
where
surface.
) ((
(
)
( ( ))
)
( ) is the distance-dependent phase of the oscillating cantilever and
)
)
(6)
is defined
(7)
is the amplitude of the oscillation of the cantilever when the tip is oscillating far from the
References
1. Katan, A.J., M.H.v. Es, and T.H. Oosterkamp, Quantitative force versus distance measurements in
amplitude modulation AFM: a novel force inversion technique. Nanotechnology, 2009. 20(16): p. 165703.
2. Sader, J.E., et al., Quantitative force measurements using frequency modulation atomic force
microscopy—theoretical foundations. Nanotechnology, 2005. 16(3): p. S94.
3. arc a . an
8): p. 197-301.
.
re Dynamic atomic force microscopy methods. Surface Science Reports, 2002. 47(6–