288
D. Passeri et al.
tip can be sufficient to polarize small ferromagnetic or superparamagnetic NPs [15],
the application of an external static magnetic field to polarize the NPs undoubtedly
leads to improved detection capabilities of MFM [16]. The intensity of the magnetic
field, and thus of the tip-sample interaction force, dramatically depends on the tipsample distance. Thus, the use of lift mode allows the tip to interact with the sample
at fixed distance z, which is constant on the scanned surface, and avoids the tip
to experience variations in the magnetic field caused not by the variation of sample
magnetic properties, but by the modulation of the tip-sample distance. Therefore, lift
mode effectively reduces the presence of anomalous contrast in magnetic images due
to changes in the tip-sample distance in case of uniform magnetic properties of the
sample [17]. Nevertheless, when the length scale of the features of the investigated
sample is comparable with that of the AFM tip, which typically occurs when NPs are
analyzed, even in lift mode variation in the local value of the tip-sample capacitance
C ts are observed [18]. If a nonzero voltage V ts is present between the tip and the
sample, the existence of a tip-sample capacitance which depends on the tip-sample
distance results in an electrostatic force acting on the tip which is given by [19]
F =
1
2
∂C ts
∂z
V
2
ts .
(12.3)
This additional force, which varies on the surface, affects the contrast in secondpass phase or frequency shift maps, resulting in the presence of topography-induced
electrostatic artifacts in magnetic images [20, 21]. In order to compensate such
spurious signals to obtain accurate quantitative nanomagnetic characterizations using
MFM, different approaches have been proposed. For instance, Jaafar et al. [21]
demonstrated the effectiveness of the combination of MFM with Kelvin probe force
microscopy (KPFM), in which KPFM is used to maintain V ts = 0 at each point during
the MFM imaging, thus nullify the tip-sample electrostatic force. Also, Angeloni et
al. [15, 22] proposed the MFM with controlled magnetization of the tip (CM-MFM),
in which the same surface is scanned twice, first in standard MFM and then with the
tip demagnetized in order to acquire the electrostatic contribution which is eventually
subtracted from the MFM image.
Other approach have been developed to enhance the nanocharacterization capabilities of MFM. In particular, while standard MFM is sensitive to static magnetic
fields, in recent years frequency modulated MFM (FM-MFM), also referred to as ac
field modulated MFM, has been developed to study the response of magnetic NPs to
oscillating magnetic fields [23]. In FM-MFM, during the second pass the cantilever
is set into oscillation at a frequency f c close to its free resonance frequency f 0 , which
is generally has high as a few hundreds of kilohertz. In addition to the static magnetic
field, applied by an external magnet and/or generated by the magnetic sample, an
oscillating magnetic field at frequency f m generally ranging from hundreds of hertz
to a few kilohertz is applied to the system. As a result, two sideband peaks at frequencies f c ± f m appear in the spectrum of the cantilever oscillation. If a hard magnetic
tip is used, the amplitude of the sideband peaks is proportional to the vertical gradient
of the vertical component of the ac magnetic field, i.e., to ∂ H
ac
z /∂z [24]. Conversely,
D. Passeri et al.
tip can be sufficient to polarize small ferromagnetic or superparamagnetic NPs [15],
the application of an external static magnetic field to polarize the NPs undoubtedly
leads to improved detection capabilities of MFM [16]. The intensity of the magnetic
field, and thus of the tip-sample interaction force, dramatically depends on the tipsample distance. Thus, the use of lift mode allows the tip to interact with the sample
at fixed distance z, which is constant on the scanned surface, and avoids the tip
to experience variations in the magnetic field caused not by the variation of sample
magnetic properties, but by the modulation of the tip-sample distance. Therefore, lift
mode effectively reduces the presence of anomalous contrast in magnetic images due
to changes in the tip-sample distance in case of uniform magnetic properties of the
sample [17]. Nevertheless, when the length scale of the features of the investigated
sample is comparable with that of the AFM tip, which typically occurs when NPs are
analyzed, even in lift mode variation in the local value of the tip-sample capacitance
C ts are observed [18]. If a nonzero voltage V ts is present between the tip and the
sample, the existence of a tip-sample capacitance which depends on the tip-sample
distance results in an electrostatic force acting on the tip which is given by [19]
F =
1
2
∂C ts
∂z
V
2
ts .
(12.3)
This additional force, which varies on the surface, affects the contrast in secondpass phase or frequency shift maps, resulting in the presence of topography-induced
electrostatic artifacts in magnetic images [20, 21]. In order to compensate such
spurious signals to obtain accurate quantitative nanomagnetic characterizations using
MFM, different approaches have been proposed. For instance, Jaafar et al. [21]
demonstrated the effectiveness of the combination of MFM with Kelvin probe force
microscopy (KPFM), in which KPFM is used to maintain V ts = 0 at each point during
the MFM imaging, thus nullify the tip-sample electrostatic force. Also, Angeloni et
al. [15, 22] proposed the MFM with controlled magnetization of the tip (CM-MFM),
in which the same surface is scanned twice, first in standard MFM and then with the
tip demagnetized in order to acquire the electrostatic contribution which is eventually
subtracted from the MFM image.
Other approach have been developed to enhance the nanocharacterization capabilities of MFM. In particular, while standard MFM is sensitive to static magnetic
fields, in recent years frequency modulated MFM (FM-MFM), also referred to as ac
field modulated MFM, has been developed to study the response of magnetic NPs to
oscillating magnetic fields [23]. In FM-MFM, during the second pass the cantilever
is set into oscillation at a frequency f c close to its free resonance frequency f 0 , which
is generally has high as a few hundreds of kilohertz. In addition to the static magnetic
field, applied by an external magnet and/or generated by the magnetic sample, an
oscillating magnetic field at frequency f m generally ranging from hundreds of hertz
to a few kilohertz is applied to the system. As a result, two sideband peaks at frequencies f c ± f m appear in the spectrum of the cantilever oscillation. If a hard magnetic
tip is used, the amplitude of the sideband peaks is proportional to the vertical gradient
of the vertical component of the ac magnetic field, i.e., to ∂ H
ac
z /∂z [24]. Conversely,