system that re-adjusts the tip-sample distance. The required displacement of the z-piezo can be
recorded as topography signal.
In addition to the topography, EFM detects electrical forces, too. These forces are proportional to
the derivative of the capacitance
C
of the tip-sample arrangement with respect to the tip-sample
distance
z
and proportional to the potential difference
U
squared:
F
el
= −
1
2
d
C
d
z
U
2
.
(5)
A possible voltage dependency of
C
is neglected in this consideration.
The voltage
U
contains a direct voltage part
U
DC
and an alternating voltage part
U
AC
:
U
=
U
DC
U
AC
⋅
cos
ref
t
.
(6)
U
DC
consists of an additional applied bias voltage and, what is especially interesting from the
physical point of view, potential differences caused by different electronic work functions and
charges. For separating the impact of these electrical forces from other forces (e.g. van der Waals
forces),
U
has to be modulated at the frequency
ω
ref
. As a result of this, the measurable photo-
detector signal is modulated, too.
Inserting equation (6) in (5) and using power-reduction formulae of trigonometric functions one can
expect a force between tip and sample at the frequency
ω
ref
as well as at 2·
ω
ref
. Figure 7 confirms
this prediction: It shows the frequency spectrum of the cantilever oscillation with clear signals at the
mechanical excitation frequency
ω
r
and at the frequencies of the first and second harmonic of the
electrical excitation.
Now two additional lock-in amplifiers (see Figure 6) can be used
to analyze the photo-detector signal at the frequency
ω
ref
and 2·
ω
ref
simultaneously. The measured amplitudes of the signals are proportional to the strength of the
electrical forces.
Manual Anfatec PCI-Lockin Amplifier AMU5.0 – Rev. 1.01
Page 14 (34)
Figure 7: Frequency spectrum of the oscillating cantilever
