Magnetic Force Microscopy is a development
of the technologies of Atomic Force Microscopy and Scanning
Tunneling Microscopy in order to image magnetization
patterns with high resolution and minimal sample preparation.
The technique relies on measuring the interaction with
the stray magnetic field emanating from the sample of
a sharp magnetic tip attached to a flexible cantilever
which is placed close to the surface (typically 10-500
nm).
The interaction is inferred by measuring
the cantilever deflection using either the tunnel effect,
optical interferometry or optical deflection. The image
is formed by scanning the tip laterally with respect
to the sample and recording the interaction strength
as a function of position.
A variety of different techniques
can be used to probe the interaction between tip and
sample. From those the non-contact ac-detection mode
MFM enjoys wide popularity in the literature. It operates
by monitoring the vertical dynamics of the cantilever
as it is scanned over the magnetised surface. A piezoelectric
bimorph is used to oscillate the flexible cantilever
that supports the probing tip transversely. This is
generally done by driving the cantilever at a fixed
frequency slightly higher than its mechanical resonance
and observing changes in the cantilever deflection amplitude. The
mechanical resonance frequency of the cantilever is
determined not only by its indigenous spring constant
but also by the vertical component of any force gradient
which it experiences since any such force which varies
with displacement appears in the cantilever equation
of motion as an additional effective spring. This, in
turn, shifts the resonance frequency and hence changes
the amplitude of the tip oscillation. Force gradients
can arise from several sources; magnetic force between
tip and sample magnetisations, interatomic forces between
tip and surface atoms and other forces such as electrostatic
interaction.
The MFM image is constructed by extracting
the magnetic force gradient component and suppressing
the interatomic force gradients. This suppression can
be done in several ways. The most straightforward is
to operate in the large separation limit of the interatomic
potential where the interatomic force is attractive
and the third derivative of the potential with respect
to tip height is small: thus the cantilever resonant
frequency is insensitive to topographical changes as
the tip is scanned.
This is applied in the tapping/lift
interleave technique developed by Digital Instruments.
A complete image acquisition is achieved by performing,
at evenly spaced positions along the so-called slow
translation axis, a set of two successive line scans
along the fast translation axis (perpendicular to the
slow translation axis). During the first of these two
scans, the tip flies very near to the sample surface
(10 nm at most). The tip-sample separation is permanently
adjusted by a feedback loop so as to maintain constant
the amplitude of the oscillation of the tip, the latter
being mainly subjected to short range Van der Waals
forces. This allows to generate a contour of constant
force gradient that defines the sample topography to
be stored. For the second of the two scans, the tip-sample
distance is increased to some value (typically ranging
from 50 to 200 nm in the case of the reported experiments)
chosen so that the lifted tip be then predominantly
subjected to the long range magnetic forces due to dipolar
interactions between the tip magnetisation and the stray
field emanating from the specimen. The feedback control
is turned off and the tip is driven along a trajectory
that mimics precisely the sample topography measured
during the first scan. The recorded signal then consists
of the variation of the magnetic force gradient at constant
height above the specimen surface, yielding a contrast
mapping of the magnetic stray field above the sample.
The disadvantage of a too large tip-sample
distance for the second (magnetic) scan is, that the
higher spatial frequencies in a magnetic field pattern
fall off rapidly with distance from the magnetization
distribution. As a result the resolution decreases with
which the magnetization distribution can be imaged.
A compromise has therefore to be found for the correct
tip-sample distance in each scan, to obtain good resolution
of the magnetic contrast (reduce distance) and at the
same time reduce any contributions from the interatomic
interactions (increase distance).
Because of the proportionality of
the recorded signal to the magnetic force derivative,
the technique employed provides a very good signal to
noise ratio. However, it concomitantly makes the recovering
of the local magnetic configuration within the sample
not straightforward. Indeed, modeling is most often
required to ensure a correct interpretation of the observed
MFM images. To evaluate theoretically the magnetic force
exerted on the tip and its gradient, various levels
of approximation are possible, depending on the degree
of complexity of the model used to describe the shape
and the magnetisation state of both the tip and the
sample. One simple but very instructive approximation
that we will use in the following is to assume that
the probing tip consists of a point dipole with effective
magnetic moment.
The MFM response is simply proportional
to the second derivative with respect to the z
coordinate of the vertical component of the stray field
produced by the sample at the tip location. It should
however be emphasized that this approach is only valid
under the assumption that the stray field from the sample
is not sufficient to alter the magnetisation in the
tip and that, conversely, the stray field from the tip
has no significant effect on the magnetization distribution
within the sample.
If, as is usual and was always the
case for the reported MFM experiments, the cantilever
is vibrated along the z axis perpendicular to
the x-y plane in which lies the flat substrate
supporting the sample and if the effective magnetic
moment of the tip is oriented along the z direction.
Thus, the MFM response is simply proportional to the
second derivative with respect to the z coordinate
of the vertical component of the stray field produced
by the sample at the tip location. It should however
be emphasized that this approach is only valid under
the assumption that the stray field from the sample
is not sufficient to alter the magnetisation in the
tip and that, conversely, the stray field from the tip
has no significant effect on the magnetization distribution
within the sample.
Magnetic Force Microscopy is a development
of the technologies of Atomic Force Microscopy and Scanning
Tunneling Microscopy in order to image magnetization
patterns with high resolution and minimal sample preparation.
The technique relies on measuring the interaction with
the stray magnetic field emanating from the sample of
a sharp magnetic tip attached to a flexible cantilever
which is placed close to the surface (typically 10-500
nm).
The interaction is inferred by measuring
the cantilever deflection using either the tunnel effect,
optical interferometry or optical deflection. The image
is formed by scanning the tip laterally with respect
to the sample and recording the interaction strength
as a function of position.
A variety of different techniques
can be used to probe the interaction between tip and
sample. From those the non-contact ac-detection mode
MFM enjoys wide popularity in the literature. It operates
by monitoring the vertical dynamics of the cantilever
as it is scanned over the magnetised surface. A piezoelectric
bimorph is used to oscillate the flexible cantilever
that supports the probing tip transversely. This is
generally done by driving the cantilever at a fixed
frequency slightly higher than its mechanical resonance
and observing changes in the cantilever deflection amplitude.
The MFM image is constructed by extracting
the magnetic force gradient component and suppressing
the interatomic force gradients. This suppression can
be done in several ways. The most straightforward is
to operate in the large separation limit of the interatomic
potential where the interatomic force is attractive
and the third derivative of the potential with respect
to tip height is small: thus the cantilever resonant
frequency is insensitive to topographical changes as
the tip is scanned.
This is applied in the tapping/lift
interleave technique developed by Digital Instruments.
A complete image acquisition is achieved by performing,
at evenly spaced positions along the so-called slow
translation axis, a set of two successive line scans
along the fast translation axis (perpendicular to the
slow translation axis). During the first of these two
scans, the tip flies very near to the sample surface
(10 nm at most). The tip-sample separation is permanently
adjusted by a feedback loop so as to maintain constant
the amplitude of the oscillation of the tip, the latter
being mainly subjected to short range Van der Waals
forces. This allows to generate a contour of constant
force gradient that defines the sample topography to
be stored. For the second of the two scans, the tip-sample
distance is increased to some value (typically ranging
from 50 to 200 nm in the case of the reported experiments)
chosen so that the lifted tip be then predominantly
subjected to the long range magnetic forces due to dipolar
interactions between the tip magnetisation and the stray
field emanating from the specimen. The feedback control
is turned off and the tip is driven along a trajectory
that mimics precisely the sample topography measured
during the first scan. The recorded signal then consists
of the variation of the magnetic force gradient at constant
height above the specimen surface, yielding a contrast
mapping of the magnetic stray field above the sample.
The disadvantage of a too large tip-sample
distance for the second (magnetic) scan is, that the
higher spatial frequencies in a magnetic field pattern
fall off rapidly with distance from the magnetization
distribution. As a result the resolution decreases with
which the magnetization distribution can be imaged.
A compromise has therefore to be found for the correct
tip-sample distance in each scan, to obtain good resolution
of the magnetic contrast (reduce distance) and at the
same time reduce any contributions from the interatomic
interactions (increase distance).
Because of the proportionality of
the recorded signal to the magnetic force derivative,
the technique employed provides a very good signal to
noise ratio. However, it concomitantly makes the recovering
of the local magnetic configuration within the sample
not straightforward. Indeed, modeling is most often
required to ensure a correct interpretation of the observed
MFM images. To evaluate theoretically the magnetic force
exerted on the tip and its gradient, various levels
of approximation are possible, depending on the degree
of complexity of the model used to describe the shape
and the magnetisation state of both the tip and the
sample. One simple but very instructive approximation
that we will use in the following is to assume that
the probing tip consists of a point dipole with effective
magnetic moment.
The MFM response is simply proportional
to the second derivative with respect to the z
coordinate of the vertical component of the stray field
produced by the sample at the tip location. It should
however be emphasized that this approach is only valid
under the assumption that the stray field from the sample
is not sufficient to alter the magnetisation in the
tip and that, conversely, the stray field from the tip
has no significant effect on the magnetization distribution
within the sample.
If, as is usual and was always the
case for the reported MFM experiments, the cantilever
is vibrated along the z axis perpendicular to
the x-y plane in which lies the flat substrate
supporting the sample and if the effective magnetic
moment of the tip is oriented along the z direction.
Thus, the MFM response is simply proportional to the
second derivative with respect to the z coordinate
of the vertical component of the stray field produced
by the sample at the tip location. It should however
be emphasized that this approach is only valid under
the assumption that the stray field from the sample
is not sufficient to alter the magnetisation in the
tip and that, conversely, the stray field from the tip
has no significant effect on the magnetization distribution
within the sample.
