**1. Introduction**

20 Will-be-set-by-IN-TECH

38 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

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Magnetic force microscopy (MFM) is a special mode of operation of the atomic force microscope (AFM). The technique employs a magnetic probe, which is brought close to a sample and interacts with the magnetic stray fields near the surface. The strength of the local magnetostatic interaction determines the vertical motion of the tip as it scans across the sample.

MFM was introduced shortly after the invention of the AFM (Martin & Wickramasinghe, 1987), and became popular as a technique that offers high imaging resolution without the need for special sample preparation or environmental conditions. Since the early 1990s, it has been widely used in the fundamental research of magnetic materials, as well as the development of magnetic recording components. MFM detects the quantity that is of particular interest for the magnetic recording process, namely the magnetic stray field produced by a magnetized medium or by a write head. The magnetic transition geometry and stray field configuration in longitudinal recording media is illustrated in Fig. 1 (Rugar et al., 1990). Nowadays, the main developments in MFM are focused on the quantitative analysis of data, improvement of resolution, and the application of external fields during measurements (Schwarz & Wiesendanger, 2008).

The interpretation of images acquired by MFM requires knowledge about the specific nearfield magnetostatic interaction between probe and sample. Therefore, this subject will be briefly discussed hereafter. Other topics to be considered are the properties of suitable probes, the achievable spatial resolution, and the inherent restrictions of the method. More detailed information can be found, e.g., in articles by Rugar et al., Porthun et al. and Hartmann. Valuable information can also be found in the works of Koch and Hendrych et al.

In the present chapter, we will also demonstrate some applications of the technique made by our research group in the study of magnetic vortices formation in sub-microsized structures, as well as further magnetic properties, of Si and Ge-based magnetic semiconductors thin films.

#### **2. Basics of magnetic contrast formation**

The operating principle of MFM is the same as in AFM. Both static and dynamic detection modes can be applied, but mainly the dynamic mode is considered here because it offers

Magnetic Force Microscopy: Basic Principles and Applications 41

The force derivative *F z* / can originate from a wide range of sources, including electrostatic probe-sample interactions, van der Waals forces, damping, or capillary forces (Porthun et al., 1998). However, MFM relies on those forces that arise from a long-range magnetostatic coupling between probe and sample. This coupling depends on the internal magnetic structure of the probe, which greatly complicates the mechanism of contrast

In general, a magnetized body, brought into the stray field of a sample, will have the

*E M H dV*

*F E M H dV*

The integration has to be carried out over the tip volume, or rather its magnetized part as illustrated in Fig. 2. Simplified models for the tip geometry and its magnetic structure are often used in order to make such calculations feasible. Another equivalent approach is to start the simulation with the tip stray field and to integrate over the sample volume (Porthun et al., 1998). According to Newton's third law, the force acting on the sample in the

<sup>0</sup> *tip sample tip*

0 is the vacuum permeability. The force acting on an MFM tip can thus be

<sup>0</sup> *tip sample ti <sup>p</sup>*

in the previous equation:

(2)

(3)

(1)

*F M H dV*

<sup>0</sup> *sample tip sample*

Fig. 2. Modelled MFM tip having a magnetic coating on a non-magnetic core. Parameters for

magnetic potential energy *E* (Porthun et al., 1998):

field of the tip is equal in magnitude to *F*

integration are indicated (Koch, 2005)

formation.

where

calculated by:

Fig. 1. Geometry of the magnetic stray field above a longitudinal magnetic medium (upper). Typical variation of the *Hx* and *Hz* components above the medium (lower) (Rugar et al., 1990)

better sensitivity. The cantilever (incorporating the tip) is excited to vibrate close to its resonance frequency, with a certain amplitude and a phase shift with respect to the drive signal. The deflection sensor of the microscope monitors the motion of the tip. Under the influence of a probe-sample interaction, the cantilever behaves as if it had a modified spring constant, *ccFz <sup>F</sup>* / , where *c* is the natural spring constant and *F z* / is the derivative of the interaction force relative to the perpendicular coordinate *z* . It is assumed that the cantilever is oriented parallel to the sample surface.

An attractive interaction with *F z* / 0 will effectively make the cantilever spring softer, so that its resonance frequency will decrease. A shift in resonance frequency will lead to a change of the oscillation amplitude of the probe and of its phase. All of these are measurable quantities that can be used to map the lateral variation of / *F z* . The most common detection method uses the amplitude signal and is referred to as amplitude modulation (AM). The cantilever is driven slightly away from resonance, where the slope of the amplitude-versus-frequency curve is high, in order to maximize the signal obtained from a given force derivative.

Measurement sensitivity, or the minimum detectable force derivative, has an inverse dependence on the *Q* value of the oscillating system (Hartmann, 1999). Therefore, a high *Q* value might seem advantageous, but this has the drawback that it increases the response time of the detection system. In situations where *Q* is necessarily high, for example when scanning in vacuum, a suitable alternative is the frequency modulation (FM) technique (Porthun et al., 1998; Hartmann, 1999). In this method the cantilever oscillates directly at its resonance frequency by using a feedback amplifier with amplitude control.

Fig. 1. Geometry of the magnetic stray field above a longitudinal magnetic medium (upper). Typical variation of the *Hx* and *Hz* components above the medium (lower) (Rugar et al.,

better sensitivity. The cantilever (incorporating the tip) is excited to vibrate close to its resonance frequency, with a certain amplitude and a phase shift with respect to the drive signal. The deflection sensor of the microscope monitors the motion of the tip. Under the influence of a probe-sample interaction, the cantilever behaves as if it had a modified spring constant, *ccFz <sup>F</sup>* / , where *c* is the natural spring constant and *F z* / is the derivative of the interaction force relative to the perpendicular coordinate *z* . It is assumed

An attractive interaction with *F z* / 0 will effectively make the cantilever spring softer, so that its resonance frequency will decrease. A shift in resonance frequency will lead to a change of the oscillation amplitude of the probe and of its phase. All of these are measurable quantities that can be used to map the lateral variation of / *F z* . The most common detection method uses the amplitude signal and is referred to as amplitude modulation (AM). The cantilever is driven slightly away from resonance, where the slope of the amplitude-versus-frequency curve is high, in order to maximize the signal obtained from a

Measurement sensitivity, or the minimum detectable force derivative, has an inverse dependence on the *Q* value of the oscillating system (Hartmann, 1999). Therefore, a high *Q* value might seem advantageous, but this has the drawback that it increases the response time of the detection system. In situations where *Q* is necessarily high, for example when scanning in vacuum, a suitable alternative is the frequency modulation (FM) technique (Porthun et al., 1998; Hartmann, 1999). In this method the cantilever oscillates directly at its

resonance frequency by using a feedback amplifier with amplitude control.

that the cantilever is oriented parallel to the sample surface.

1990)

given force derivative.

The force derivative *F z* / can originate from a wide range of sources, including electrostatic probe-sample interactions, van der Waals forces, damping, or capillary forces (Porthun et al., 1998). However, MFM relies on those forces that arise from a long-range magnetostatic coupling between probe and sample. This coupling depends on the internal magnetic structure of the probe, which greatly complicates the mechanism of contrast formation.

In general, a magnetized body, brought into the stray field of a sample, will have the magnetic potential energy *E* (Porthun et al., 1998):

$$E = -\mu\_0 \left[ \vec{M}\_{\text{tip}} \cdot \vec{H}\_{\text{sample}} dV\_{\text{tip}} \right] \tag{1}$$

where 0 is the vacuum permeability. The force acting on an MFM tip can thus be calculated by:

$$
\vec{F} = -\vec{\nabla}E = \mu\_0 \int \vec{\nabla} \left( \vec{M}\_{tip} \cdot \vec{H}\_{sample} \right) dV\_{tip} \tag{2}
$$

The integration has to be carried out over the tip volume, or rather its magnetized part as illustrated in Fig. 2. Simplified models for the tip geometry and its magnetic structure are often used in order to make such calculations feasible. Another equivalent approach is to start the simulation with the tip stray field and to integrate over the sample volume (Porthun et al., 1998). According to Newton's third law, the force acting on the sample in the field of the tip is equal in magnitude to *F* in the previous equation:

$$
\vec{F} = \mu\_0 \int \vec{\nabla} \left( \vec{M}\_{sample} \cdot \vec{H}\_{tip} \right) dV\_{sample} \tag{3}
$$

Fig. 2. Modelled MFM tip having a magnetic coating on a non-magnetic core. Parameters for integration are indicated (Koch, 2005)

The magnetostatic potential *<sup>s</sup> r* created by any ferromagnetic sample can be calculated from its magnetization vector field , *Ms <sup>r</sup>* (Hartmann, 1999):

$$\phi\_s(\vec{r}) = \frac{1}{4\pi} \left[ \int \frac{d^2 \vec{s}' \cdot \vec{M}\_s \left( \vec{r}' \right)}{\left| \vec{r} - \vec{r}' \right|} - \int d^3 \vec{r}' \frac{\vec{\nabla} \cdot \vec{M}\_s \left( \vec{r}' \right)}{\left| \vec{r} - \vec{r}' \right|} \right] \tag{4}$$

Magnetic Force Microscopy: Basic Principles and Applications 43

are the effective monopole and dipole moments of the probe.

The point-probe approximation yields satisfactory results in many cases of MFM contrast interpretation. However, a far more realistic approach can be achieved by considering the extended geometry of a probe. An example is the pseudodomain model (Hartmann, 1999), in which the unknown magnetization vector field near the probe apex, with its entire surface and volume charges, is modelled by a homogeneously magnetized prolate spheroid of suitable dimensions. The magnetic response of the probe outside this imaginary domain is neglected. This pseudodomain model allows interpretation of most results obtained by MFM on the basis of bulk probes. For probes with a different geometry, for example those where the magnetic region is confined to a thin layer, other appropriate models have been

Fig. 3 shows both the measured and calculated MFM response across a series of 5 µm longitudinal bits (Rugar et al., 1990). The signal was recorded as a constant force derivative contour. In this particular case, the tip was modelled as a uniformly magnetized truncated cone with a spherical cap, in agreement with the shape as observed by electron microscopy (Rugar et al., 1990). Note that for in-plane magnetized samples, interdomain boundaries are the only sources of magnetic stray field that can be externally detected by MFM. On the other hand, samples with perpendicular magnetic anisotropy produce extended surface charges that correspond to the upward and downward pointing domain magnetization. In this case the

near-surface stray field is directly related to the domain topology (Hartmann, 1999).

Fig. 3. (a) Contour of constant force derivative measured on a 5 µm bit sample. (b)

Corresponding model calculation of magnetic force derivative (adapted from Rugar et al.,

where *q* and *m*

1990)

developed (Rasa et al., 2002).

where , *s* is an outward normal vector from the sample surface. The first (two-dimensional) integral covers all surface charges created by magnetization components perpendicular to the surface, while the second (three-dimensional) integral contains the volume magnetic charges resulting from interior divergences of the magnetization vector field. The sample stray field is then given by *Hr r sample <sup>s</sup>* , which can be substituted in Equation (2) to calculate the interaction force *F* . In static mode the instrument detects the vertical component of the cantilever deflection, *F nF <sup>d</sup>* , where *<sup>n</sup>* is an outward unit normal from the cantilever surface. In the dynamic mode the compliance component or force derivative , *F r n nFr <sup>d</sup>* is detected (Hartmann, 1999).

A limitation in the use of MFM is that the magnetic configuration of the sensing probe is rarely known in detail. Although the general theory of contrast formation still holds, it is not possible to model the measured signal from first principles for an unknown domain structure of the magnetic probe. As a consequence, MFM can generally not be performed in a quantitative way, in the sense that a stray field would be detected in absolute units. Furthermore, because MFM is sensitive to the strength and polarity of near-surface stray fields produced by ferromagnetic samples, rather than to the magnetization itself, it is usually not straightforward to deduce the overall domain topology from an MFM image. The problem of reconstructing a concrete arrangement of inner and surface magnetic charges from the stray fields they produce is not solvable. MFM can, however, be used to compare the experimentally detected stray field variation of a micromagnetic object to that obtained from certain model calculations. This often enables to at least classify the magnetic object under investigation (Hartmann, 1999). Thus, even without detailed quantitative analysis, the qualitative information collected by the microscope can be very useful (Rugar et al., 1990).

#### **3. Modelling the MFM response**

If one wants to analyze the force derivative , *F r <sup>d</sup>* using Equations (2) and (4), then a model of the tip shape and magnetization must be constructed. Various levels of complexity are possible. Most models assume that both the tip and the sample are ideally hard magnetic materials, with a magnetization that is unaffected by the stray field from the other.

The simplest way to model a tip is with the point-probe approximation (Hartmann, 1999). The effective monopole and dipole moments of the probe are projected into a fictitious probe of infinitesimal size that is located a certain distance away from the sample surface. The unknown magnetic moments as well as the effective probe-sample separation are treated as free parameters to be fitted to experimental data. The force acting on the probe, which is immersed in the near-surface sample microfield, is given by (Hartmann, 1999):

$$
\vec{F} = \mu\_0 \left( q + \vec{m} \cdot \vec{\nabla} \right) \vec{H} \tag{5}
$$

2, , ,

integral covers all surface charges created by magnetization components perpendicular to the surface, while the second (three-dimensional) integral contains the volume magnetic charges resulting from interior divergences of the magnetization vector field. The sample

the cantilever surface. In the dynamic mode the compliance component or force derivative

A limitation in the use of MFM is that the magnetic configuration of the sensing probe is rarely known in detail. Although the general theory of contrast formation still holds, it is not possible to model the measured signal from first principles for an unknown domain structure of the magnetic probe. As a consequence, MFM can generally not be performed in a quantitative way, in the sense that a stray field would be detected in absolute units. Furthermore, because MFM is sensitive to the strength and polarity of near-surface stray fields produced by ferromagnetic samples, rather than to the magnetization itself, it is usually not straightforward to deduce the overall domain topology from an MFM image. The problem of reconstructing a concrete arrangement of inner and surface magnetic charges from the stray fields they produce is not solvable. MFM can, however, be used to compare the experimentally detected stray field variation of a micromagnetic object to that obtained from certain model calculations. This often enables to at least classify the magnetic object under investigation (Hartmann, 1999). Thus, even without detailed quantitative analysis, the qualitative information collected by the

, *F r <sup>d</sup>*

of the tip shape and magnetization must be constructed. Various levels of complexity are possible. Most models assume that both the tip and the sample are ideally hard magnetic

The simplest way to model a tip is with the point-probe approximation (Hartmann, 1999). The effective monopole and dipole moments of the probe are projected into a fictitious probe of infinitesimal size that is located a certain distance away from the sample surface. The unknown magnetic moments as well as the effective probe-sample separation are treated as free parameters to be fitted to experimental data. The force acting on the probe, which is immersed in the near-surface sample microfield, is given by (Hartmann, 1999):

> *F*

materials, with a magnetization that is unaffected by the stray field from the other.

*r d r*

, *Ms <sup>r</sup>* (Hartmann, 1999):

 

is an outward normal vector from the sample surface. The first (two-dimensional)

3 , , ,

*s s*

*ds M r M r*

*r r r r*

, where *<sup>n</sup>*

created by any ferromagnetic sample can be calculated

(4)

*<sup>s</sup>* , which can be substituted in Equation (2) to

. In static mode the instrument detects the vertical

is an outward unit normal from

using Equations (2) and (4), then a model

<sup>0</sup> *<sup>q</sup> m H* (5)

The magnetostatic potential

where , *s*

from its magnetization vector field

*s*

stray field is then given by *Hr r sample*

component of the cantilever deflection, *F nF <sup>d</sup>*

microscope can be very useful (Rugar et al., 1990).

If one wants to analyze the force derivative

**3. Modelling the MFM response** 

is detected (Hartmann, 1999).

calculate the interaction force *F*

, *F r n nFr <sup>d</sup>*

*<sup>s</sup> r*

> 1 4

where *q* and *m* are the effective monopole and dipole moments of the probe.

The point-probe approximation yields satisfactory results in many cases of MFM contrast interpretation. However, a far more realistic approach can be achieved by considering the extended geometry of a probe. An example is the pseudodomain model (Hartmann, 1999), in which the unknown magnetization vector field near the probe apex, with its entire surface and volume charges, is modelled by a homogeneously magnetized prolate spheroid of suitable dimensions. The magnetic response of the probe outside this imaginary domain is neglected. This pseudodomain model allows interpretation of most results obtained by MFM on the basis of bulk probes. For probes with a different geometry, for example those where the magnetic region is confined to a thin layer, other appropriate models have been developed (Rasa et al., 2002).

Fig. 3 shows both the measured and calculated MFM response across a series of 5 µm longitudinal bits (Rugar et al., 1990). The signal was recorded as a constant force derivative contour. In this particular case, the tip was modelled as a uniformly magnetized truncated cone with a spherical cap, in agreement with the shape as observed by electron microscopy (Rugar et al., 1990). Note that for in-plane magnetized samples, interdomain boundaries are the only sources of magnetic stray field that can be externally detected by MFM. On the other hand, samples with perpendicular magnetic anisotropy produce extended surface charges that correspond to the upward and downward pointing domain magnetization. In this case the near-surface stray field is directly related to the domain topology (Hartmann, 1999).

Fig. 3. (a) Contour of constant force derivative measured on a 5 µm bit sample. (b) Corresponding model calculation of magnetic force derivative (adapted from Rugar et al., 1990)

Magnetic Force Microscopy: Basic Principles and Applications 45

important due to the long-range nature of magnetic forces (Rugar et al., 1990). Originally, electrochemically etched wires of cobalt or nickel were used as cantilevers (Martin & Wickramasinghe, 1987). Thanks to the widespread use of AFM, cantilevers with integrated sharp tips are now fabricated in large numbers out of silicon-based materials. These tips can be coated with a thin layer of magnetic material for the purpose of MFM observations. A lot of effort has been spent on the optimization of magnetic tips in order to get quantitative information from MFM data (Rugar et al., 1990; Porthun et al., 1998; Hartmann, 1999). The problem is that in the coating of conventional tips, a pattern of magnetic domains will arrange, which reduces the effective magnetic moment of the tip. The exact domain structure is unknown and can even change during MFM operation. Nevertheless, some information on the magnetization state of selected probes has been acquired using electron

The spatial resolution in MFM imaging is related to the tip-sample distance, but also to the magnetized part of the tip that is actually exposed to the sample stray field. Thus in order to improve lateral resolution, it is beneficial to restrict the magnetically sensitive region to the smallest possible size. Ideally the effective volume of the probe would consist of a small single-domain ferromagnetic particle located at the probe apex. Socalled supertips have been developed based on this idea (Hartmann, 1999). However, there is a physical lower limit for the dimensions because an ultra-small particle becomes

The demand for a strong signal, produced by a small sensitive volume, indicates the need to maximize the magnetic moment in the tip. For this reason a single domain tip will give the best results and is also easier to describe theoretically. Materials with a high saturation magnetization should be used in order to limit the required volume. The well-defined magnetic state of a tip should be stable during scanning, and it should interfere as little as possible with the sample magnetization. A high switching field of the tip can be realized through the influence of shape anisotropy (Porthun et al., 1998; Hartmann, 1999), which forces the magnetization vector field near the probe apex to align with its axis of symmetry. Eventually, the smallest detail from which a sufficient signal-to-noise ratio can be gained is determined by the sensitivity of the deflection sensor, as well as the noise characteristics of

In the present work, we have employed etched silicon tips of the MESP type supplied by Bruker. These are standard probes for MFM, and have a pyramidal geometry (Fig. 5). The magnetic coating consists of ~ 10–150 nm of Co/Cr alloy (exact thickness and composition of the coatings are undisclosed). The cantilever has a length *L* of approximately 225 µm. As a result, the resonance frequency 0*f* is about 75 kHz. The coating has a coercivity of ~ 400 Oe and a magnetic moment of 110−13 emu. In order to ensure a predominant orientation of the magnetic vector field along the major probe axis, the thin film probes were magnetized (along the cantilever) prior to taking measurements. The Digital Instruments company offers a magnetizing device that possesses a permanent magnet. This apparatus ensures that the distance from the magnet to the tip is always the same in different magnetization procedures. Thus, taking into account that the magnetic field lines are dependent on the

holography (Rugar et al., 1990; Hartmann, 1999).

superparamagnetic.

the cantilever (Porthun et al., 1998).

distance, the reproducibility is then guaranteed.

Usually, the MFM response of a certain tip-sample configuration is calculated by an integration in the spatial domain, e.g., over the sample volume. Porthun et al. have proposed a different formalism, where the problem is approached in the frequency domain. This has the advantage that it shows some characteristics of the imaging process more clearly. To be specific, the sample magnetization distribution is split up into harmonics, each having a spatial wavelength and wavenumber 2 / *k* . The wavelength measures the length scale over which the magnetization vector goes through a complete rotation. Frequency components of the magnetic potential and the stray field are calculated separately. Then, the magnetic signal can be determined using Equation (2) for each of the stray field harmonics. For a specific (and simplified) tip-sample geometry (Fig. 4), the detected MFM signal is obtained by summing over all frequency components of the force derivative. The resulting signal, expressed in terms of sample magnetization and spatial frequency, forms a tip transfer function for the imaging process. An important observation is that the transfer function shows an exponential decay, exp*kz*<sup>0</sup> , with increasing tipsample distance 0*z* . It is thus crucial for high resolution to keep the tip-sample distance as small as possible. In addition, the dimensions (length, width, thickness) of a bar-type tip lead to specific decay rates both at high and low spatial frequencies. The latter illustrates that the finite size of a tip plays an important role in the imaging process. Therefore, a simple point-probe approximation is not sufficient to clarify how high and low spatial frequencies are attenuated. In the context of such a frequency domain description, the resolution can be defined as a minimum detectable wavelength which is determined by the noise limit of the detector system.

Fig. 4. One-dimensional model for the MFM measurement process (Porthun et al., 1998)

#### **4. Requirements for MFM tips**

The cantilever/tip assembly is obviously the critical element of a magnetic force microscope. Unlike in scanning tunnelling microscopy (STM) and repulsive-mode AFM, the tip shape is

Usually, the MFM response of a certain tip-sample configuration is calculated by an integration in the spatial domain, e.g., over the sample volume. Porthun et al. have proposed a different formalism, where the problem is approached in the frequency domain. This has the advantage that it shows some characteristics of the imaging process more clearly. To be specific, the sample magnetization distribution is split up into harmonics, each

and wavenumber 2 / *k*

Fig. 4. One-dimensional model for the MFM measurement process (Porthun et al., 1998)

The cantilever/tip assembly is obviously the critical element of a magnetic force microscope. Unlike in scanning tunnelling microscopy (STM) and repulsive-mode AFM, the tip shape is

length scale over which the magnetization vector goes through a complete rotation. Frequency components of the magnetic potential and the stray field are calculated separately. Then, the magnetic signal can be determined using Equation (2) for each of the stray field harmonics. For a specific (and simplified) tip-sample geometry (Fig. 4), the detected MFM signal is obtained by summing over all frequency components of the force derivative. The resulting signal, expressed in terms of sample magnetization and spatial frequency, forms a tip transfer function for the imaging process. An important observation is that the transfer function shows an exponential decay, exp*kz*<sup>0</sup> , with increasing tipsample distance 0*z* . It is thus crucial for high resolution to keep the tip-sample distance as small as possible. In addition, the dimensions (length, width, thickness) of a bar-type tip lead to specific decay rates both at high and low spatial frequencies. The latter illustrates that the finite size of a tip plays an important role in the imaging process. Therefore, a simple point-probe approximation is not sufficient to clarify how high and low spatial frequencies are attenuated. In the context of such a frequency domain description, the resolution can be defined as a minimum detectable wavelength which is determined by the

 

. The wavelength measures the

having a spatial wavelength

noise limit of the detector system.

**4. Requirements for MFM tips** 

important due to the long-range nature of magnetic forces (Rugar et al., 1990). Originally, electrochemically etched wires of cobalt or nickel were used as cantilevers (Martin & Wickramasinghe, 1987). Thanks to the widespread use of AFM, cantilevers with integrated sharp tips are now fabricated in large numbers out of silicon-based materials. These tips can be coated with a thin layer of magnetic material for the purpose of MFM observations. A lot of effort has been spent on the optimization of magnetic tips in order to get quantitative information from MFM data (Rugar et al., 1990; Porthun et al., 1998; Hartmann, 1999). The problem is that in the coating of conventional tips, a pattern of magnetic domains will arrange, which reduces the effective magnetic moment of the tip. The exact domain structure is unknown and can even change during MFM operation. Nevertheless, some information on the magnetization state of selected probes has been acquired using electron holography (Rugar et al., 1990; Hartmann, 1999).

The spatial resolution in MFM imaging is related to the tip-sample distance, but also to the magnetized part of the tip that is actually exposed to the sample stray field. Thus in order to improve lateral resolution, it is beneficial to restrict the magnetically sensitive region to the smallest possible size. Ideally the effective volume of the probe would consist of a small single-domain ferromagnetic particle located at the probe apex. Socalled supertips have been developed based on this idea (Hartmann, 1999). However, there is a physical lower limit for the dimensions because an ultra-small particle becomes superparamagnetic.

The demand for a strong signal, produced by a small sensitive volume, indicates the need to maximize the magnetic moment in the tip. For this reason a single domain tip will give the best results and is also easier to describe theoretically. Materials with a high saturation magnetization should be used in order to limit the required volume. The well-defined magnetic state of a tip should be stable during scanning, and it should interfere as little as possible with the sample magnetization. A high switching field of the tip can be realized through the influence of shape anisotropy (Porthun et al., 1998; Hartmann, 1999), which forces the magnetization vector field near the probe apex to align with its axis of symmetry. Eventually, the smallest detail from which a sufficient signal-to-noise ratio can be gained is determined by the sensitivity of the deflection sensor, as well as the noise characteristics of the cantilever (Porthun et al., 1998).

In the present work, we have employed etched silicon tips of the MESP type supplied by Bruker. These are standard probes for MFM, and have a pyramidal geometry (Fig. 5). The magnetic coating consists of ~ 10–150 nm of Co/Cr alloy (exact thickness and composition of the coatings are undisclosed). The cantilever has a length *L* of approximately 225 µm. As a result, the resonance frequency 0*f* is about 75 kHz. The coating has a coercivity of ~ 400 Oe and a magnetic moment of 110−13 emu. In order to ensure a predominant orientation of the magnetic vector field along the major probe axis, the thin film probes were magnetized (along the cantilever) prior to taking measurements. The Digital Instruments company offers a magnetizing device that possesses a permanent magnet. This apparatus ensures that the distance from the magnet to the tip is always the same in different magnetization procedures. Thus, taking into account that the magnetic field lines are dependent on the distance, the reproducibility is then guaranteed.

Magnetic Force Microscopy: Basic Principles and Applications 47

image. This height data is also used to move the tip at a constant local distance above the surface during the second (magnetic) scan line, during which the feedback is turned off.

In theory, topographic contributions should be eliminated in the second image.

Fig. 6. Outline of the lift mode principle. Magnetic information is recorded during the second pass (right panel). The constant height difference between the two scan lines is the

depends on the force derivative in the following manner (Porthun et al., 1998):

cantilever damping will be misinterpreted as change in resonance frequency.

0 / <sup>1</sup> *F z f f*

with 0*f* the free resonance frequency of the cantilever in the case of no tip sample interaction. In the amplitude detection, the cantilever is oscillated at a fixed frequency *ext* <sup>0</sup> *f f* , where in the case of / 0 *F z* the oscillation amplitude is already slightly below the maximum amplitude at 0*f* . When the resonance frequency changes this will result in a change in cantilever oscillation amplitude which can easily be detected. The disadvantage of this technique is that it is very slow for cantilevers with low damping and that a change in

It should be noted that an attractive interaction ( / 0 *F z* ) leads to a negative amplitude change (dark contrast in the image), while a repulsive interaction ( / 0 *F z* ) gives a

Finally, Fig. 7(b) shows a typical MFM image. In this case, the sample was a piece of metal evaporated tape: a standard sample that is used to check whether the microscope is correctly tuned to image magnetic materials (Koch, 2005). It is clear that no correlation exists between the topography data shown on the left, and the magnetic data on the right. Consequently,

*c*

(6)

Magnetic data can be recorded either as variations in amplitude, frequency, or phase of the cantilever oscillation. It is argued that phase detection and frequency modulation give the best results, with a higher signal-to-noise ratio (Porthun et al., 1998; Hartmann, 1999). However, these detection modes can require the addition of an electronics module to the microscope. In our MFM measurements we have used amplitude detection, which measures changes in the cantilever's amplitude of oscillation relative to the piezo drive. The signal

lift height *h* (adapted from Hendrych et al., 2007)

positive amplitude variation (bright contrast).

the separation of both contributions is successful.

Fig. 5. Scanning electron images of AFM probes like the ones used for MFM. The probes are coated with a magnetic thin film. Specifications are mentioned in the text (Bruker Corporation, 2011)
