Biomedical Engineering Reference
In-Depth Information
15.5
Noise in MFM
15.5.1
Thermal Noise
The force gradient detection in MFM is limited by the noise sources in the detec-
tion system. These include noise from the defl ection sensor, acoustic noise, and
thermal oscillations of the cantilever. For cantilevers with small spring constants,
noise from the defl ection sensor is usually negligible. Acoustic noise can often be
blocked out or fi ltered out from MFM images. The predominant limitation is the
rms amplitude arising from thermal oscillations, given by (in accordance with
equipartition theorem) [27]:
Ak Tk
T
=
(15.12)
B
where
T
is the temperature and
k
B
is Boltzmann's constant. However, when the
defl ection of the cantilever as measured by an optical beam defl ection system is
taken into account, the cantilever measures a change in inclination instead of a
vertical movement. Butt and Jaschke [31] showed that the mean “ virtual ” vertical
defl ection of a free cantilever, when measured using beam defl ection, is given by:
Ak T
=
43
k
(15.13)
T
B
which at a temperature of 22 °C amounts to:
(
)
A
=
0 074
.
nm
k
for diving board cantilever
(15.14)
T
(
)
and
A
=
0 056
.
nm
k
for V-shaped board cantilever
(15.15)
T
Thus, although V-shaped cantilevers are predicted to have a lower thermal oscil-
lation amplitude as compared to diving board cantilevers, the spring constant of
V-shaped cantilevers is generally much lower, resulting in higher values for
A
T
.
Further, it can be seen that for the static mode of MFM, where the use of V-shaped
cantilevers is more common, defl ection due to thermal oscillation can be much
larger than that due to magnetic interaction.
It is mainly in the dynamic or noncontact mode, where the cantilever is driven
at amplitudes much larger than
A
T
, that the changes in amplitude, and so on,
arising due to magnetic interactions can be ascertained. The force sensitivity in
MFM is thus defi ned as the minimum detectable force gradient despite the
thermal oscillations and other noise. The force sensitivity has been derived from
force spectral density (probability distribution of force noise over a range of fre-
quencies) as [29] :
kk TB
wQ
12
B
F
′
=
for AM
(15.16)
min
A
c
