The development of a low temperature magnetic force microscope

MASTER

The development of a low temperature magnetic force microscope

Grijseels, S.C.M.

Award date:

2010

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microscope

S.C.M. Grijseels Master thesis

September 2009 - September 2010

Under supervision of:

M.Sc. J. Bocquel

Prof. dr. P.M. Koenraad Department of Applied Physics

Photonics & Semiconductor Nanophysics University of Technology Eindhoven

VAEE

‘Al zaai je liefde op duizend plekken, ze zal slechts op één ervan ontkiemen.’

and sensitivity. Extensive improvements and test measurements have been performed at room temperature, 77 K and 4.2 K, both with atomic force and magnetic force microscopy. The magnetic force microscopy res- olution and sensitivity have been investigated on hard disk drive samples and bit patterned media. On bit patterned media, we have shown that the spatial resolution was improved from ∼ 700 nm to ∼ 50 nm. Fur- thermore, we have been able to reduce the overall noise level of the setup by a factor 5. As a result of these improvements, the setup is now ready to be utilized for low temperature magnetic force microscopy on magnetic semiconductor nanostructures. In view of these prospects, we have investigated magnetic (Ga,Mn)As samples by means of magnetometry to gain insight in future challenges in conducting low temperature magnetic force measurements on (III,Mn)V semiconductor nanostructures.

1 Introduction 1

1.1 General Introduction . . . 1

1.2 Introduction to scanning probe microscopy . . . 2

1.2.1 Long–range attractive forces . . . 2

1.2.2 Contact and short–range repulsive forces . . . 5

1.3 Introduction to mass storage media . . . 5

1.3.1 Patterned media . . . 5

1.3.2 Exchange bias . . . 6

2 Fundamentals 9 2.1 Non–contact force microscopy . . . 9

2.2 Detection methods in dynamic force microscopy . . . 11

2.2.1 Amplitude modulation dynamic force microscopy . . . 11

2.2.2 Frequency modulation dynamic force microscopy . . . 13

2.2.3 Phase modulation dynamic force microscopy . . . 14

3 Theory of magnetic force microscopy 15 3.1 Magnetic recording theory . . . 15

3.1.1 Basics of magnetic contrast formation . . . 15

3.1.2 Contrast modeling . . . 17

3.1.3 Magnetic force detection . . . 20

3.2 Noise in magnetic force microscopy . . . 21

3.2.1 General discussion on noise . . . 21

3.2.2 Optical noise . . . 22

3.2.3 Thermal noise . . . 22

3.2.4 Electronic noise . . . 23

3.3 Limits of resolution . . . 24

4 Experimental setup 27 4.1 Introduction . . . 27

4.2 Mechanics . . . 27

4.2.1 Cantilever . . . 27

4.2.2 Piezostack . . . 30

4.2.3 Microscope stick . . . 33

4.2.4 Cryostat . . . 34

4.3 Optics . . . 34

4.3.1 Interferometer . . . 35

4.3.2 Fiber . . . 38

4.3.3 Laser . . . 40

4.3.4 Optical control . . . 40

4.4 Electronics . . . 41

4.4.1 The attocube setup . . . 42

4.4.2 The RHK setup . . . 42

4.4.3 Modes of operation . . . 43

4.5 Noise issues . . . 46

4.5.1 Mechanical noise . . . 46

4.5.2 Optical noise . . . 47

4.5.3 Electronic noise . . . 48

4.6 Modifications and alterations of the MFM . . . 48

5 MFM results 51 5.1 HDD results . . . 51

5.1.1 Conventional hard disk sample . . . 51

5.1.2 Magnetic tape . . . 54

5.1.3 New hard disk sample . . . 55

5.2 Investigation of resolution and sensitivity . . . 59

5.3 Bit patterned sample . . . 59

6 Outlook 63 6.1 Investigation of (Ga,Mn)As samples . . . 63

6.1.1 SQUID theory . . . 63

6.1.2 SQUID Results . . . 65

7 Conclusions 69 Acknowledgements 71 A Fourier analysis 73 A.1 Fourier transformation . . . 73

A.2 Transfer function theory . . . 73

B Contrast modeling 76 B.1 Thin film applications . . . 77

B.2 Vertical recording media . . . 79

B.3 Magneto–optical recording . . . 80

List of symbols 87

1.1 Units in the SI system and the cgs system . . . 7

4.1 Table of used cantilevers . . . 28

4.2 Table of used piezo elements . . . 31

4.3 Table of calibration results . . . 34

4.4 Table of output parameters for different modes of operation . . . 45

4.5 Table of modifications and improvements made to the MFM setup . . . 50

5.1 Table of the working parameters used for the HDD images . . . 51

5.2 Table of the working parameters used for the new HDD images . . . 57

5.3 Table of the working parameters used for the bit patterned images . . . 59

6.1 Table of (Ga,Mn)As samples . . . 64

6.2 Table of magnetization values for the four (Ga,Mn)As samples . . . 67

Introduction

In this introductory chapter, we will outline the main objectives that constitute this thesis. A brief introduction to scanning probe microscopy (SPM) is given and some of its the fundamental principles are discussed. Finally, from a SPM point of view, we will reflect upon some of the major challenges that the hard disk drive (HDD) industry is facing with respect to the formation and characterization of tiny magnetic particles and/or domains.

1.1 General Introduction

Magnetic force microscopy (MFM) [1] is an imaging technique based on atomic force microscopy (AFM) [2]

in which magnetic forces or magnetic force gradients between sample and a microscopic probe are measured to image the magnetic structure of a sample. Improving the sensitivity or more precisely the signal–to–noise ratio (S/N) of the experimental setup, constitutes a major objective in scanning probe microscopy. Noise in magnetic force microscopy was analyzed by several authors [3, 4, 5], both to reduce the noise in different applications, as well as to improve the sensitivity of the experimental setup.

In this thesis we will both explore the sensitivity limitations of the magnetic force microscope and gain full understanding of MFM in combination with phase–locked–loop (PLL) controlled frequency detection and excitation. Frequency modulation dynamic force microscopy (FM–DFM) is an operation mode in non contact atomic force microscopy (NC–AFM) that – when properly conducted – combines optimized sensitivity and resolution with fast and stable scanning mode operation. Nowadays, FM–DFM is a well established tool to image insulating surfaces with atomic resolution [6, 7]. Also in magnetic force microscopy, FM–DFM can be used to obtain high resolution MFM images [8]. However, up until now atomic resolution in magnetic force microscopy can only be achieved by using a detection method called magnetic exchange force microscopy (MExFM) [9]. This technique detects the short–range magnetic forces between tip and sample. In this thesis we will only focus on the long–range magnetic forces that exist between tip and sample. Furthermore, we will explore the possibilities of combining the high sensitivity setup with magnetic force microscopy at low temperatures (LT–MFM). The objective is to investigate (III,Mn)V semiconductor structures for its potential applications in spin–electronics (‘spintronics’), in which logic and memory operations may be integrated in a single device [10]. The key objective in spintronics research is to develop devices that allow for precise and simultaneous control of both charge and spin properties of the charge carriers. In particular (Ga,Mn)As, with a Curie temperature TC as high as 185 K [11], is one of the most promising candidates for such a device.

Besides the high TC, Mn has the highest magnetic moment for 3d metals (spin), when placed in bulk material (Fig. 1.1). Also, when Mn²⁺ions are located in III–V based diluted magnetic semiconductors (DMSs), they are a source of free holes (carriers). One of the interesting effects of these extra charge carriers is that they mediate interactions between magnetic moments localized on Mn ions and thereby lead to ferromagnetic ordering in III–V based DMSs.

With low temperature MFM, we want to investigate the origin and the nature of ferromagnetic order- ing within the GaMnAs layers [12]. The homogeneity of the Mn ions concentration in GaMnAs layers is subject to many investigations. Transport measurements [13] and characterization techniques like super- conducting quantum interference device (SQUID) [10] only show evidence of non–homogeneity of the Mn concentration. Several hypotheses about the nature of this non–homogeneity have been made. Among others, the presence of a ferromagnetic MnAs phase within the GaMnAs layers would explain the obser- vation of TC temperatures higher than the TC of GaMnAs. TC temperatures above room temperature

have been reported for MnAs [14]. Furthermore, it has been shown that under specific growth conditions, like post–growth annealing, MnAs precipitates are likely to be formed, causing an increase of the coerci- tivity HC [10]. We would like MFM to resolve such inclusions and their distribution within the layer but also to allow us to characterize areas where the MnAs ferromagnetic response may exists over much larger length scales [15]. With magnetic force microscopy, we expect to determine the factors leading to Mn inhomogeneities and see if these fluctuations themselves lead to the formation of magnetic domains.

Sc Ti V Cr Mn Fe Co Ni

0 2 3 4

1 5

monolayer adatoms impurity

Pd(001)

Magnetic moment [mB]

Figure 1.1: Comparison between the magnetic mo- ments per atom of 3d transition metals as adatoms on Pd(001), 3d monolayers on Pd(001), and 3d impurities in bulk Pd. The magnetic moment is expressed in units of µB = e~/2m, the Bohr magneton. (Adapted from [16])

A lot of work is being done to influence and control the magnetic properties of GaMnAs. For example, an ongoing work is the growth of strained GaMnAs layers on top of a quantum dot (QD) array. The strain (here induced by the QD array) is expected to lead to magnetic ordering within the GaMnAs layer. Recently, evidence of magnetic ordering by in- duced strain has been found in high resolution X–ray diffraction experiments [17]. However, imaging small magnetic domains of almost negligible magnitude, even at low and ultra low temperatures, has proven to be quite cumbersome. Therefore, we will first ex- plore the sensitivity of our setup by means of samples with small magnetic domains (∼ several nanometer) of considerable magnetization strengths. This can be done for instance by magnetic imaging of hard disk drives with high areal densities [bytes/in²]. More- over, investigation of samples processed by means of electron–beam (e−beam) lithography (§ 1.3.1), for example bit patterned magnetic recording (BPMR) media [18], is suitable to explore both the sensitivity and resolution limitations of the microscopic scheme.

1.2 Introduction to scanning probe microscopy

In scanning probe microscopy the interaction between a microscopic tip and a surface is measured as a function of the position of the tip relative to the sample. For this purpose, the tip is attached to a cantilever and the deflection of the cantilever is measured. In our microscope, this deflection is measured by means of an interferometric AFM system, which shall be explained in detail in chapter 4. Its principle is shown in Fig. 1.2.

The position of the sample is controlled by a piezoelectric actuator, which is a lead zirconate titanate^∗ (PZT) ceramic. Piezoelectric actuators convert electrical energy directly to mechanical energy and they make motion in the sub–nanometer range possible [20]. Scanning the material surface is done by actuating the sample in a raster pattern. For dynamic operation (chapter 2), the cantilever is excited by a bimorph piezoelectric plate called the dither piezo. When tip and sample are brought into close proximity, tip–sample interactions causes the cantilever to bend and as a result the distance d between tip and fiber changes, which results in a change in the interference spectrum. The forces existing between tip and sample can be classified according to the distance dependence of the underlying microscopic interaction potential φ^s_t(r, r^′). If the variations of the microscopic interaction potential φ^s_t(r, r^′) are small on the scale of interactomic distances, the integration can be carried out over a continuous tip and sample volume. These forces are short–range forces, for example chemical forces that vary over a distance of fractions of nanometers, contrary to long–range forces, which decay over several nanometers to tens of nanometers. The main long–range forces studied in this thesis result from van der Waals interactions, electrostatic interactions and magnetic interactions. Short–range forces, including chemical bonding forces (adhesion and capillary forces) and short–range electrostatic forces, will be only be discussed shortly in the introduction.

1.2.1 Long–range attractive forces

Long range attractive forces of the van der Waals type arise from instantaneous moments of otherwise nonpolar atoms and molecules, i.e. electromagnetic field fluctuations. Electromagnetic field fluctuations are universal,

∗Lead zirconate titanate (Pb[ZrxTi1-x] O3) is a ceramic perovskite material that shows a marked piezoelectric effect [19].

Feedback

& Detection System

Sample

PZT Scanner Laser

1 2 3

Cantilever

Laser output

Dither Piezo

Tip

50/50 coupler 4

d

(a) AFM without tip–sample interaction

Feedback

& Detection System

PZT Scanner Feedback

& Detection System

Laser

1 2 3

d

Laser output

Tip

50/50 coupler 4

Dither Piezo

Sample Cantilever

(b) AFM with tip–sample interaction

Figure 1.2: Schematic representation of the interferometric AFM system. The deflection of the cantilever is measured by means of a change in the optical path length d of the reflected laser light (chapter 4.3.1). Electronic and/or mechanical connections are indicated in black. Optical connections are indicated in green. The laser source, the fiber–end (ferrule) and the detection system are connected to a 50/50 coupler (chapter 4.3.3). Connection 3 in this case is not used. (a) The unbended cantilever without any tip–sample interaction. (b) The bended cantilever near the sample surface. In this case, the tip–sample interactions cause the cantilever to bend, hereby changing the length of the optical path.

which make van der Waals forces ever–present, independent of the chemical composition of the surfaces or the medium. Beside the spontaneous electric field fluctuations, which are the most important contribution in SPM studies, there are two other types of van der Waals forces. The first type arises from (permanent) electric dipole–

dipole interaction between molecules. In this case, both molecules tend to align to minimize their energy. The second type is the interaction between two molecules, with and without an electric dipole moment. The dipole moment of one molecule induces an electric field, which polarizes the other molecule (i.e. an induced dipole moment). Ever since the classical work of Lifshitz [21], it is commonly known to scientists that the microscopic approach, which is based on assuming simple pairwise additivity of the atomic or molecular contributions, does not rigorously characterize the collective macroscopic nature of the van der Waals interactions. In particular, the effects of multiple reflections of the intermolecular radiation field, and of a dielectric immersion medium surrounding probe and sample can only be characterized by the Lifshitz approach. This leads to a variety of phenomena relevant to long–range scanning force microscopy, see for example Hartmann [22]. In general, the van der Waals force can be described by the Lennard–Jones potential, as can be seen in Fig. 1.3a. At large enough distance the force is dominated by the −D/r⁷ term is attractive (non contact regime). At close distance, the C/r¹³ term dominates and the force is repulsive (contact regime). At intermediate distances, the force can both be repulsive and attractive, this is called the ‘semi-contact’ regime. When two spherical particles are brought in close proximity to each other, conservative interaction forces between these particles can be described by the Hamaker approximation [23]. Applying this approximation to SPM studies, where tip and sample are brought in close proximity, we write for the force F(r, t) between tip and surface

F(r) = −∇ 1 VsampleVtip

ˆ

Vsample

ˆ

Vtip

φ^s_t(r, r^′) d³r^′d³r. (1.1)

Here r = (x, y, z) is a coordinate system fixed to the sample and Vsample and Vtip denote the total sample and tip volumes. Considering only small longitudinal probe–sample separations, denoted by z (∼ several

r U

non contact region repulsive

attractive

‘semicontact’

region

contact region

(a) Lennard–Jones Potential for van der Waals forces

R a

D^z

z

(b) Micro–tip Figure 1.3: (a) The force exerted between tip and sample as a result of the Lennard–Jones potential U is given by FLJ(r) =_r^C₁₃

−

nanometer), we can neglect retardation effects of the radiation field between probe and sample. In this case, the nonretarded van der Waals force is given by a simple inverse power law of the type

Fn(z) = gnHn/zⁿ, (1.2)

where gn denotes a geometrical constant depending on the actual probe model and Hn is the nonretarded Hamaker constant, which reflects the dielectric contributions involved with the van der Waals force^†. As only surfaces covered uniformly with only one chemical species are studied (in the ideal case), the long–range van der Waals force does not vary locally (it does not lead to imaging contrast formation). Therefore, a simple model consisting of a uniform semi–infinite surface is used, in which the tip is described as a cone with a spherical cap (Fig. 1.3b). Then, the van der Waals force is given by [24]

FvdW(z) = −Hn

6

R

z²+ tan²α

z + Rα − Rα

z (z − Rα)



, (1.4)

where R is the tip’s radius, α is the angle of the tip’s cone and Rα= R(1 − sin α). If we neglect the canting of the cantilever^‡, Eq. 1.4 transforms to

FvdW(z) = −HnR

6z² . (1.5)

The van der Waals force for spherical tips is thus proportional to 1/z². For large distances, the force is attractive until the atoms of the tip and sample come so close together that the electron cloud begin to repel each other due to the Pauli principle^§. However, if the contact area between tip and sample involves tens to

†In SPM studies, the nonretarded Hamaker constant is given by

Hn= 3~

4π

∞

X

n=1

1 n³

∞

ˆ

0

d̟ ǫ1−ǫ3

ǫ1+ ǫ3

n

 ǫ2−ǫ3

ǫ2+ ǫ3

n

, (1.3)

with ǫj = ǫj(i̟) and ν = i̟ the imaginary frequency axis, and ǫ1, ǫ2 and ǫ3 denote the dielectric permittivity of the probe, sample and medium respectively [22].

‡The cantilever canting will be neglected throughout this thesis expect where necessary for the discussion of magnetic forces.

§For a modern approach to the calculation of van der Waals forces in SPM please study Hartmann [25] or for a more general description Israelachvili [26].

hundreds of atoms, the description of the effective repulsive force can be obtained without considering the Pauli repulsion and the ionic repulsion of the molecules.

Trapping of electrostatic charge in dielectric surfaces can also give rise to long–range electrostatic interaction forces. This will be the case in situations where an external electric field is applied between tip and sample and/or when an electrostatic potential difference exists between tip and sample. In the absence of an external field, if the separation of the tip and sample z is small in comparison to the tip’s radius R, the electrostatic force existing between tip and sample can be written as [27]

Fel = −πǫR z Φ^S_T
2

, (1.6)

with ǫ the electric permittivity and Φ^S_T the contact potential difference caused by the different work functions of tip and sample. The influence of electrostatic force is circumvented by proper grounding of both tip and sample.

1.2.2 Contact and short–range repulsive forces

The surface of two bodies are deformed when they are brought into mechanical contact. The deformation depends on the applied load and the properties of the material. Continuum elasticity theories describes the contact and adhesion between finite bodies under an external load. The problem of two elastic objects in contact, here the tip and the sample, consists in establishing and solving the relationship between the stress (σ) and strain (ε) tensors. This functional relationship is referred to as the constitutive equation and for isotropic materials it is deduced as

σij = ηεllδij + Gεij, (1.7)

where η is the Lamé coefficient and G the shear modulus (G = E/2(1 + υ), where υ is Poisson’s ratio)^¶. In ambient conditions a thin film of water is adsorbed on hydrophilic surfaces. At close proximity of tip and surface, a meniscus or liquid bridge may be formed between tip and sample. This meniscus implies an attractive or capillary force that shows a dependence with distance. When cooling down to liquid nitrogen or helium temperatures, the cantilever is placed in a quasivacuum environment (pressure ∼ 20 mbar) after being flushed with ⁴He gas. Therefore, we assume that capillary forces will be absent. In SPM studies, short–range forces are relevant in the contact regime, as for the magnetic force microscopy studies conducted in this thesis, only the presence of the long–range attractive van der Waals and magnetic forces are considered. The underlying theory relevant to magnetic SPM studies is presented in chapter 3.

1.3 Introduction to mass storage media

1.3.1 Patterned media

Nowadays, the HDD industry is at a critical technology crossroads. Recent studies forecast explosive growth of the digital universe from 281 × 10¹⁸(218 Exa) bytes in 2007 (about 45 Gb per person) to 1.8 × 10²¹(1.8 Zetta) bytes in 2011 [30]. Therefore it is of utmost importance to ensure a continuous increase in hard disk drive capacities to provide for the demand. The biggest lever for higher HDD capacities is to increase the areal density. In Fig. 1.4 the areal density trends of research targets as well as recent studies in HDD products are given as a function of time. As one can see, the conventional recording limit is set to 1 Tb/in² due to the so–called ‘superparamagnetic limit’. The key word in understanding this superparamagnetic limit is stability.

Most applications rely on the magnetic order of the nanoparticles being stable within time. However, with decreasing particle size, the magnetic anisotropy energy per particle responsible for holding the magnetic moment along certain directions becomes comparable to the thermal energy^k. When this happens, thermal fluctuations induce random flipping of the magnetic moment with time. As a result the nanoparticles lose their stable magnetic order and become superparamagnetic. For HDD applications, this imposes a trade–

off between signal–to–noise ratio (S/N) compared to the thermal stability of small grain media⁽¹⁾ and the

¶For SPM applications, various approximation models have been developed to describe the deformation of two bodies of which Johnson–Kendall–Roberts (JKR) [28] and Derjanguin–Muller–Toporov (DMT) [29] are the most famous ones.

kFor a sample with uniaxial anisotropy, which is the case for crystals with tetragonal and hexagonal symmetry, the magneto crystalline anisotropy energy is given by Ea= VK0+ K1sin²ϑ + K2sin⁴ϑ + ..., with Vsamplethe sample’s volume, Kiare the anisotropy coefficients and ϑ is the angle between the magnetic moment and the anisotropy axis [31].

Figure 1.4: Areal density and R&D targets. In the picture, LMR stands for longitudinal magnetic record- ing, whereas PMR stands for perpendicular magnetic recording. The latter is still under development or continuous. (Adapted from [18])

writability of a narrow track head⁽²⁾. This limits the ability to continue to scale conventional magnetic recording technology to higher areal densities. The advantages of patterning recording media were recognized as early as 1963 by Shew [32], who showed that discrete patterned tracks on a hard disk platter could reduce the crosstalk and noise problems associated with head positioning errors and allow increased tracking tolerances.

Although people thought that using patterned media could extend bit densities to 50 Gb/in²[33], more recent studies predict an areal density beyond 5 Tb/in² [18]. Up to now, the highest demonstrated areal density with a continuous perpendicular medium writing by conventional perpendicular magnetic recording (PMR) is (merely) 612 Gb/in². The major disadvantage of the BPMR is the need for write synchronization in which the write field must be timed to coincide with the locations of patterned media [34]. In this thesis, we will try to obtain magnetic images from a bit patterned sample provided by the Hitachi R&D Group [35]. This sample contains various grain sizes down to 20 nm, corresponding to an areal density of 530 Gb/in². This sample allows us to investigate the magnetic resolution of our setup.

1.3.2 Exchange bias

Hysteresis is well known in ferromagnetic (FM) materials. When an external magnetic field is applied to a ferromagnet, the atomic dipoles align themselves with the external field. Even when the external field is removed, part of the alignment will be retained: the material has become magnetized. The relationship between magnetic field strength H and magnetic flux density B(H) is not linear in such materials but has a sort of S−shaped loop as shown in Fig. 1.5a for grain–oriented electrical steel. For increasing levels of field strength, the magnetic induction curves up to a point where further increases in magnetic field strength will result in no further change in flux density. This condition is called magnetic saturation. If the magnetic field is now reduced linearly, the plotted relationship will follow a different curve back towards zero field strength at which point it will be offset from the original curve by an amount called the remanent flux density BR. In antiferromagnetic (AFM^∗∗) materials on the other hand, the magnetic moments of atoms or molecules align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. Therefore, the total magnetization vanishes. Ferromagnetic materials are characterized by the Curie or critical temperature TC, whereas antiferromagnetic materials are characterized by the Néel temperature TN. Above the critical temperature, the collective magnetism merges into paramagnetism with the corresponding characteristic behavior of the inverse susceptibility χ(T ) as function of temperature [31]. If an antiferromagnetic and a ferromagnetic material are stacked on top of each other, magnetic exchange coupling, or exchange bias, is induced at the interface between the FM and AFM system. At the interface, an extra source of

∗∗Throughout the remainder of this thesis the abbreviations FM and AFM will stand for frequency modulation and atomic force microscopy, respectively, unless indicated differently.

B_S B_R

H_S H_C

BH()

(a) B(H)−loops

m[10emu]–4

2

–2

–4 4

–800 –400 400 800

H [Oe]

T = 10 K HE

H_C 0

0

(b) AFM–FM system

Figure 1.5: (a) A family of B(H)−loops measured with a flux density that is sinusoidally modulated at 50 Hz with an amplitude varying from 0.3 T to 1.7 T. The material is conventional grain–oriented electrical steel. BR

denotes the remanent magnetic flux density, BSdenotes the saturation magnetic flux density, HS the saturation field and HCis the coercitivity. B(H)−loop is another name for hysteresis loop. (Adapted from [36]) (b) Hysteresis loop m(H), with m(H) the magnetic moment, of a FeFe2/Fe bilayer at T = 10 K after field cooling. The exchange bias HE and the coercitivity HC are indicated in the figure. (Adapted from [37]) In order to avoid any confusion about the units of magnetic quantities, the units in the SI system and the cgs system are listed in table 1.1.

anisotropy is introduced, leading to magnetic stability. This exchange coupling can be used to suppress the superparamagnetic limit [38]. Interface coupling due to exchange anisotropy is observed cooling the AFM–FM couple in the presence of a static magnetic field from a temperature above the Néel temperature, TN, but below the Curie temperature (TN < T < TC) to temperatures T < TN. The hysteresis loop of the AFM–

FM system at T < T_N after the field cool procedure, is shifted along the field axis generally in the opposite (‘negative’) direction to the cooling field, see Fig. 1.5b, i.e. the absolute value of coercive field for decreasing and increasing field is different. This loop shift is generally known as exchange bias, HE. The hysteresis loops also have an increased coercitivity HC, after the field cool procedure. Both these effects disappear at, or close to, the AFM Néel temperature confirming that it is the presence of the AFM material which causes this anisotropy. In last decade, much research has been conducted on ferromagnetic nanoparticles embedded in a paramagnetic matrix or in an antiferromagnetic matrix [38]. More recently however, also ferromagnetic semiconductors, in particular GaMnAs, where interlayer exchange coupling (IEC) between the GaMnAs – either antiferromagnetic or ferromagnetic – is expected to play a crucial role in controlling the spin configuration within structures. Unfortunately, the magnetic anisotropy, which is the most important physical quantity in determining the direction of the magnetization in a ferromagnetic, is found to depend on many parameters in GaMnAs, including temperature, strain and carrier density, which largely complicates the development of a practical spintronic application. However, the possibility of electrical control of magnetic

Table 1.1: Units in the SI system and the cgs^⋆ system (G: Gauss, Oe: Oerstedt, erg: energy, emu: electromagnetic unit)

Quantity Symbol SI unit cgs unit

magnetic induction B 10⁻⁴ T = 1 G

magnetic field H 10³/4π A/m = 1 Oe

magnetic moment m 10⁻³ J/T = 1 erg/G^⋆

magnetization M 10³ A/m = 1 Oe

magnetic susceptibility χ 4π = 1 emu/cm³

⋆cgs = centimetre–gramme–seconde.

⋆⋆1 erg = 10⁻⁷J.

anisotropy of a GaMnAs layer along with controlling IEC in GaMnAs multilayers by doping, suggests that such multilayers can potentially provide significant device advantages over metallic ferromagnetic multilayers [39].

As already suggested, a lot of additional research needs to be performed on GaMnAs layers, of which on of the most challenging will be to extend its Curie temperature to room temperature. Therefore, characterizing its magnetic properties by means of low temperature magnetic force microscopy could indeed proof to be a very usefool tool.

Fundamentals

2.1 Non–contact force microscopy

In this section, we will outline the most fundamental theories involved with atomic force microscopy as studied by Uwe Hartmann [40]. We will start with the simplest mode of operation of a non contact scanning force microscope, which consists in lifting the cantilever probe up to a certain distance from the sample surface to measure a long–range interaction in terms of a static force exerted on the probe. This mode of operation is in general referred to as the static or dc mode operation. When we monitor the deflection of the cantilever end resulting from the applied force, we define the deflection in one dimension as

∆z = Fz/k, (2.1)

z

y x

Figure 2.1: Three dimensional representation of the microcantilever–tip system.

where Fz is the force applied in the z direction. The static mode operation however, is not preferred in reality. Far more sensitive detection can be realized by utilizing the dynamic properties of the probe, e.g. by involving sinu- soidal excitation of the probe, also called the dynamic or ac mode operation. Excitation of the probe can be achieved by attaching the cantilever a bimorph piezoelec- tric plate, the dither piezo. In contrast to the detection of quasistatic forces, the response of the cantilever in the dy- namic mode is more complex. A thorough understanding of dynamic SPM operation requires to solve the equation of motion of the cantilever–tip ensemble under the influ- ence of tip–surface forces. A complete descriptive analysis is a formidable task that involves the solution of the equa- tion of motion of a three–dimensional object, a vibrating cantilever (Fig. 2.1). Some symmetry considerations allow to approximate the microcantilever by a one–dimensional object [41], then

EΥ∂⁴w(x, t)

∂x⁴ + µ∂²w(x, t)

∂t² = F (x, t), (2.2)

here w(x, t) is the transverse displacement of the cantilever, E is Young’s modulus, Υ is the moment of inertia, µ is the mass per unit of length of the cantilever, and F (x, t) is the force per unit of length exerted on the cantilever. Combined with the point–mass approximation^∗, the cantilever sensor may be represented by a simple harmonic oscillator consisting of a spring with a spring constant k and an effective mass m, yielding a resonance frequency of

ω0= rk

m. (2.3)

∗In the point–mass approximation the cantilever–tip ensemble is considered as a point–mass spring, see for example Rugar et al.[42]

If the cantilever is excited sinusoidally at its clamped end with a frequency ω and an amplitude A0= F0/m, the probe tip likewise oscillates sinusoidally with a certain amplitude Aδ, exhibiting a phase shift ∆ϕ with respect to the drive signal applied to the piezoelectric actuator. The deflection sensor of the force microscope monitors the motion of the probe tip, provided that its bandwidth is large enough. In the absence of tip–sample forces, the equation of motion in one dimension of the cantilever is given by

∂²z(t)

∂t² +ω0

Q

∂z(t)

∂t + ω²₀(z(t) − z⁰) = A0cos (ωt), (2.4) where z0 is the probe–sample distance at zero oscillation amplitude and z(t) is the momentum probe–sample separation. The quality factor Q of the spring itself stems from intrinsic properties of the cantilever, which are the lumped effective mass and the resonant frequency, and is determined by the damping factor γ in the following manner,

Q = mω0

γ . (2.5)

For the resonance drive frequency ωr, which takes into account the damping of the cantilever, but neglects the tip–sample interaction, we write

ωr= ω0

r 1 − 1

2Q². (2.6)

The influence of operation conditions, e.g. ambient air, liquid, ultrahigh vacuum (UHV) or low–temperature (LT), is reflected by the damping factor γ. The Q factor reflects the coupling to dissipative forces and can range from values below 100 for liquid or ambient gas conditions at an appropriate pressure, to more than 100, 000 which is sometimes obtained in UHV. Eq. 2.4 is the equation of a forced oscillator with damping. The general solution to this equation has a transient term and a steady state solution. Initially, both terms are prominent, after a time 2Q/ω0 however, the transient term is reduced by a factor 1/e [41], and the motion is dominated by the steady state solution

z(t) = z0+ A cos(ωt + α). (2.7)

The amplitude A of the probe oscillation is given by

A(ω) = A0ω²₀ q

(ω²− ω0²)²+ 4γ²ω²

. (2.8)

The phase shift between the probe oscillation and the excitation signal amounts to

∆ϕ = arctan

 2γω ω²− ω²0



. (2.9)

The above simplified formalism is based on the assumption that the oscillation amplitude A is sufficiently small in comparison with the length of the cantilever. The results derived so far however, describe only free cantilever oscillations, meaning that z0 is large enough, to prevent any influence of the sample on the probe oscillation. If z0 is now decreased such that a force Fts affects the motion of the cantilever then a term Fts/m has to be added to the right–hand side of Eq. 2.4, yielding

m∂²z

∂t² + γ∂z

∂t + k (z(t) − z0) = F0+ Fts. (2.10) In order to consider almost all interactions that could be relevant in SPM, one has to assume

F = F



z(t),∂z(t)

∂t



, (2.11)

which takes into account both the static and dynamic interactions. Since F covers probe–sample interactions of various types, in particular spatially nonlinear ones, the z(t) curves monitored by the deflection sensor and defined by Eq. 2.4 represent anharmonic oscillations, rather than the commonly know harmonic oscillations of the spring–type model. If however F (z(t)) can be substituted by a first–order Taylor series approximation, which is valid for Aδ ≪ |z⁰|, then the force microscope detects the compliance or vertical component of the

force gradient ∂F/∂z. On the basis of this approximation, the cantilever behaves under the influence of the probe–sample interactions as if it had a modified spring constant, given by

kef f= k −∂F

∂z, (2.12)

where k is the intrinsic spring constant, which has to be substituted into Eq. 2.3. As a result, an attractive probe–sample interaction, with ∂F/∂z > 0 will effectively soften the cantilever spring, while a repulsive interaction with ∂F/∂z < 0 will make it effectively stiffer. Converting Eq. 2.3 with the modified spring constant yields

ω = ω0

r 1 − 1

k

∂F

∂z. (2.13)

The minus sign comes from the fact that F is increasingly attractive as the tip approaches the surface; thus, a decrease of the resonance frequency is observed. Provided that ∂F/∂z ≪ k, or ∆ω ≪ ω⁰ – which is always the case in MFM and AFM applications – the shift in resonant frequency is given by

∆ω ≈ −1 2k

∂F

∂z. (2.14)

Summarizing, if we drive the lever at ω0, we can measure the amplitude eA0 due to this excitation and use Eq. 2.8 to calculate the new resonance fω0(t) =p

kef f/m, then

∂F

∂z = k 1 − 2a²+p

4Q²(a²− 1) + 1 2 (Q²− a²)

!

, (2.15)

where a = A0/ eA0 [43]. According to equations 2.8 and 2.9, a shift in the resonant frequency will result in a change of the probe oscillation amplitude Aδ and of the phase shift ∆ϕ between probe oscillation and driving signal. ∆ω, ∆A, and ∆ϕ are experimentally measurable quantities that can be used to map the spatial variation of ∂F/∂z. Additionally, phase and amplitude contain information about the damping coefficient γ.

The simple harmonic solution in Eq. 2.7 evidently shows that the dynamic mode of operation can be based on the employment of lock–in signal detection methods. The additional use of suitable feedback mechanisms opens up different variants of operation. These modes of operation will be discussed in the next sections.

2.2 Detection methods in dynamic force microscopy

2.2.1 Amplitude modulation dynamic force microscopy

The most commonly used detection method, generally referred to as slope detection or amplitude modulation (AM), involves driving the cantilever at a fixed frequency ω slightly off resonance. In amplitude modulation force gradients are measured by detecting changes in the cantilever amplitude A and/or phase ϕ. According to Eq. 2.13, a change in ∂F/∂z gives rise to a shift in the resonant frequency of ∆ω and, according to Eq. 2.8, to a corresponding shift ∆A in the amplitude of the cantilever vibration. Careful analysis [44] shows that the minimum detectable force gradient is given by

∂F

∂z



m,AM

=

s 2kkBT β

ω0Qhzosc² i, (2.16)

where hz²osci^1/2 is the root–mean–square amplitude of the driven cantilever vibration, kBBoltzmann’s constant, T the temperature of the system and β is the measurement bandwidth. High Q values are obtained in vacuum by reducing air damping (< 10⁻³mbar) or operating the cantilever at low temperatures by increasing the cantilever stiffness [45]. It might appear advantageous to maximize sensitivity by obtaining the highest possible Q. With slope detection, however, increasing the Q restricts the bandwidth β of the system. As previously discussed, if ∂F/∂z changes during scanning, the vibration amplitude settles to a new steady–state value after a sufficient length of time 2Q/ω0. Thus for a high Q cantilever (Q ∼ 50, 000) and a typical resonant frequency of 50 kHz, the maximum available bandwidth would be only 0.5 Hz, thereby restricting the dynamic range of the system. Therefore it is not useful to try to increase sensitivity by raising the Q to such high values.

Inevitably, for most vacuum applications (Q ∼ 100, 000), low temperature applications (Q ∼ 10, 000) and for

operation in higher mode excitation (Q ∼ 25, 000) [46], slope–detection is unsuitable^†. A possible solution to this problem is the application of Q controlled scanning force microscopy, which is discussed in the next paragraph, but for very high Q factors also this solution becomes inapplicable^‡. In this case, an alternative would be to switch to frequency modulation (FM) or phase modulation (PM) detection.

Q controlled dynamic force microscopy

When working in ambient or liquid conditions (UHV and/or low temperature), it might prove to be useful to make use of the Q control scheme, which allows the user to increase (decrease) the Q factor. During operation in conventional tapping mode, the function generator supplies not only the signal for the dither piezo; its signal also serves simultaneously as a reference for the lock–in amplifier. The setup therefore consist of an additional feedback circuit with an amplifier and a time shifter, which is also often referred to as the ‘phase’ shifter (also see chapter 4, Fig. 4.13). The signal of the displacement sensor is first fed into the amplifier and subsequently used to excite the dither piezo driving the cantilever in addition to the fixed frequency, constant amplitude signal from the function generator. Naturally the two driving mechanisms will be reflected in the corresponding equation of motion for the cantilever, which is given by

m∂²z(t)

∂t² + γ∂z(t)

∂t + k (z(t) − z⁰) + gkz(t − t⁰)

| {z }

Q control

= Adk cos (ωdt)

| {z }

External driving force

+Fts, (2.17)

in which g is the gain factor, t − t⁰ is the retarded time and Ad and ωd are the constant excitation amplitude and frequency respectively. The active feedback of the system is described by the retarded amplification of the signal, i.e. the tip position z measured at the retarded time t − t⁰and amplified by the gain factor g. The key feature in Q control is the positive feedback, which can be used to either increase or decrease the effective Q factor of the system. The basic idea of the feedback loop is to reduce the damping force acting on the cantilever by the surrounding system. Mathematically, the damping force [γ · ∂z(t)/∂t] has be compensated by the effective feedback term [gkz(t − t⁰)]. The time or phase shifter in the electronic setup controls the time delay between the sensor signal and the piezo excitation. This results in a phase shift between the cantilever oscillation and excitation. We describe this feature explicitly by a time shift t0. If we restrict ourselves to the steady–state solution only, i.e. t ≫ 0 and consider only a self–excitation of φ^′ = 90^◦, the self–excitation term becomes proportional to the cantilever velocity and allows us to estimate the effective Q, the amplitude and the phase shift from [48]

1 Qef f

= 1

Q− kg mωω0

, (2.18)

A(ω, g, φ^′) = Ad

q

(1 − (ω/ω0)²+ g cos φ^′)²+ ((1/Q0)(ω/ω0) − g sin φ^′)²

, (2.19)

tan φ^′= 1/Q0(ω/ω0) − g sin φ^′

1 − (ω/ω0)²+ g cos φ^′, (2.20)

where φ^′ is the phase shift between the self–excitation and the instantaneous deflection. If the Q control is switched off (g = 0), the equations reduce to the well–known resonance curves of an externally driven harmonic oscillator. As one can deduct from Eq. 2.19 φ^′= 90^◦is the optimum phase shift for the self–excitation because it does not modify the resonance frequency of the non–self–excited system, and at the same time it maximizes the effective Qef f, which in most cases, when the cantilever is driven nearby its resonance frequency, can also be defined as

Qef f = 1

1/Q0− g sin φ^′, (2.21)

which depends only on the gain factor g and the time (phase) shift t0. From this definition it is obvious that the gain factor must be limited to 0 ≤ g < 1/Q⁰, for larger values, i.e. g ≥ 1/Q⁰, the oscillation becomes unstable.

†The given values for quality factor the cantilever here may differ quite significantly for any given experiment, depending on size and shape of the cantilever itself and experimental conditions.

‡This might not be entirely true, see for example Kawai et al., who atomically resolved a Si(111)–7 × 7 surface by means of amplitude modulation dynamic force microscopy [47].

2.2.2 Frequency modulation dynamic force microscopy

j

(a) Frequency modulation detection

w /w_{2 0}

w /w_{1 0}

Q = 234 Q = 195

r b

DA₁ DA2

90

45 135

j[deg]

(b) Amplitude modulation detection

Figure 2.2: The cantilever amplitude A and phase ϕ as function of the excitation frequency ω as measured by the attocube system. (a) Frequency modulation detection. The phase difference ϕ between the excitation and cantilever response is maintained at 90^◦ to keep the cantilever oscillating at the resonance frequency. The red curve in this picture illustrates the response of the cantilever in the absence of a force gradient. The blue curve illustrates the frequency shift of the resonance peak in response to a force gradient. (b) Amplitude modulation detection in the absence of a force gradient for different Q factors. If the cantilever is driven at a frequency ω1

with Q = Qb, a change in Q results in a change in A of ∆A1. If the phase is maintained at 0.8 rad the frequency shifts from ω1to ω2and ∆A2is detected. Therefore, changes in the dissipation leads to errors in the force gradient measurement in the slope detection method. This problem is eliminated if the phase is properly set to 90^◦, because this phase occurs at the resonant frequency for all values of Q.

In order to obtain high–resolution images by increasing the quality factor of the cantilever, one has to use the frequency modulation (FM) scheme, developed by Albrecht et al. [44]. When used properly, not only the spatial resolution can be increased drastically, but also the signal–to–noise ratio (S/N) can be lowered significantly [8]. Contrary to amplitude modulation, the frequency modulation technique uses positive feedback to oscillate the cantilever at its resonant frequency. This is done by subsequently amplifying and phase shifting the cantilever detection signal by 90^◦. The resultant waveform is used as the excitation signal for the cantilever.

The amplification is adjusted by an automatic gain controller to keep the vibration amplitude constant. The dynamics of the cantilever is that of a self–driven oscillator, which in may aspects is different from the one generated by the constant excitation (both in frequency and amplitude) used in amplitude modulation. In frequency modulation detection, the spatial dependence of the frequency shift induced in the cantilever motion by the tip–sample interaction ∆f is used as the source of the contrast. During scanning the tip–sample distance is varied in order to achieve the set value for ∆f. Therefore, the topography in the images represents a map

of constant frequency shifts over the surface. The difference between AM and FM modulation is shown in Fig. 2.2. For frequency modulation detection Fig. 2.2a illustrates the cantilever response as a function of the excitation frequency. At resonance, there is a 90^◦ phase shift between the cantilever and response, such that if the phase shift is maintained at 90^◦, the cantilever will be excited at its resonant frequency. The effects of changes in Q are also eliminated by using the frequency modulation method. This can clearly been seen in Fig. 2.2b, if the phase shift is maintained at 0.8 rad, a change in Q results in a frequency change which is indistinguishable from a frequency shift due to a force gradient. The changes in Q are related to energy dissipation in tip–sample interaction and can be measured by a technique known as dissipation force microscopy. The minimum detectable force gradient in frequency modulation detection for thermally limited measurements (3.2.3) is given by Eq. 2.16 multiplied by a factor√

2 [44]

∂F

∂z



m,FM

= s

4kkBT β

ω0Qhzosc² i. (2.22)

Although the frequency modulation method resolves almost all of the problems of amplitude modulation detection, even further optimization of both stability and sensitivity is possible by operating with a technique that is called phase modulation detection.

2.2.3 Phase modulation dynamic force microscopy

In addition to the previous two detection methods, there is a third mode of operation. This technique is called phase modulation (PM) detection. In PM–AFM the cantilever is driven by an external oscillation signal with a fixed amplitude at or near its resonance frequency. The tip–sample interaction force Fts is detected as the change in phase. This mode of operation is comparable to that of the slope detection method (AM–AFM), but instead of the amplitude shift (Eq. 2.8), the phase shift (Eq. 2.9) is used as the feedback parameter. Operation in PM–AFM offers some advantages in comparison with AM– or FM–AFM. First, the capability to detect the tip–sample interaction force is more sensitive. The minimum detectable interaction force in PM–AFM is improved by one order compared with that in AM–AFM [49]. Second, the instability of cantilever dynamics is eliminated, allowing it to perform continuous and quantitative spectroscopy of the full range of the tip–sample force curve, in addition to topographic imaging [50]. Moreover, the collapse of the cantilever oscillation by strong contact between the tip and the surface is prevented, because the cantilever is always driven by the external oscillator signal through the oscillating control amplifier with automatic gain control (AGC). Finally, in PM–AFM the Q−control technique is applicable, which can be used effectively to increase the sensitivity of the detection. The minimum detectable compliance in PM–AFM for thermally limited measurements can be expressed as follows [51]

∂F

∂z



m,PM

=







1 hz_osc² i^1/2

q2kkBT β

Qω0 +^2k_Q²2^βψ_ds² for Qef f ≤^fβ⁰ 1

hz_osc² i^1/2

r

1 Qef f

2kkBT β

Q +^2k_Q²2^ω0ψ²_ds

for Qef f >^f_β⁰, (2.23) with Qef f as defined in Eq. 2.21, f0= ω0/2π the natural resonance frequency of the cantilever and ψds is the spectral detector noise density (also see chapter 3.2).

Summarizing, we can conclude that the different modes of operation can be described as

AM–AFM The cantilever is driven by an external oscillation signal with a fixed amplitude at or near its resonance frequency. If the detection signal amplitude is kept constant, this called the constant–

amplitude (CA) mode. If, on the other hand, the amplitude of the excitation is kept constant, the mode is called constant–excitation (CE) mode. The tip–sample interaction force F_ts is detected as the change in the cantilever (detector) oscillation amplitude.

FM-AFM The cantilever is always driven at the resonance frequency on the basis of a self–driven oscillator and its oscillation amplitude is kept at a constant amplitude by an automatic gain control circuit (AGC). This is the CE mode in frequency modulation detection. If the detection amplitude is kept constant, this is the CA mode in FM–AFM. Ftsis detected as the frequency shift of the cantilever (detector) frequency.

PM–AFM The cantilever is driven by an external signal at or near the resonance frequency in the CE or CA mode and Fts is detected as the phase shift.

Theory of magnetic force microscopy

3.1 Magnetic recording theory

3.1.1 Basics of magnetic contrast formation

If the probe and sample in a scanning force microscope exhibit a magnetostatic coupling, the magnetic force microscope is realized. The principle of magnetic force microscopy is very much like that of atomic force microscopy and has been described by many authors [52, 53]. Although some dare to assert that MFM is just an AFM with a magnetic tip, much smaller forces are measured in MFM. The most obvious difference here of course is the fact that every MFM is capable of AFM, whereas the other way around in general is not true.

To gain hindsight in the magnetic contrast formation, which is formed when a ferromagnetic tip and sample are brought in close approximation to each other, we shall adapt the point–dipole approximation as outlined by Uwe Hartmann [25, 40, 54].

The easiest way to address the problem of magnetic contrast model is to first consider the probe as a needle consisting of bulk material. In general, a sharp ferromagnetic needle naturally exhibits considerable magnetic shape anisotropy, which forces the magnetization vector field near the probe’s apex to predominantly align with the axis of symmetry of the probe. On the other hand, sufficiently far away from the apex region, where the probe’s cross–sectional area is almost constant, the more or less complex natural domain structure obtained in a ferromagnetic wire is established. This domain structure depends on the detailed material properties represented by the exchange, magnetocrystalline anisotropy, and magnetostriction energies^∗. Lattice defects, stresses, and the surface topology exhibit an additional influence on the domain structure. In magnetostatics, we can calculate the force on the magnetic tip by calculating of the free energy Ψ of the tip–sample system, this gradient of this energy then gives us the force vector F(r, t) = −∇Ψ = ∇ (m · B(r, t)), see for example Jackson [55]. In MFM, we are particularly interested in ∂Ψ/∂z.

There are two ways to calculate the magnetic potential; one can either consider the energy of the magnetic tip in the presence of the sample stray field, or the energy of the magnetic sample in the presence of the tip stray field. Although both are equally valid, it is preferred to consider the first case since the latter is much more complicated. We will start of by writing down the formula for the average macroscopic magnetization or magnetic moment density

M=X

i

Nihmⁱi, (3.1)

where Nithe average number per unit volume of molecules of type i and hmⁱi is the average molecular magnetic moment in a small volume at point r. In order to calculate the magnetic potential, we first have to make two assumption concerning probe’s geometry and its interaction with the sample. The first assumption is that the probe itself is modeled by a homogeneously magnetized prolate spheroid of suitable dimension, while the magnetic response of the probe outside this fictitious domain is completely neglected. The second assumption is that the dimensions and the magnitude of the homogeneous magnetization of the ellipsoidal domain are both completely rigid, i.e. independent of external stray fields produced by the sample. This forms the basis

∗Magnetostriction is a property of ferromagnetic materials that causes them to change their shape or dimensions during the process of magnetization. On a macroscopic level it subsequently consists of the migration of domain walls, followed by the rotation of the domains in response to the external magnetic fields.

(a) (b) ( ) c

net magnetization inhomog.

magnetization vector field

unknown

domain structure magnetically

ineffectual

homog.

magnetized single domain

point-dipole

micromagnetics magnetostatics

Figure 3.1: Three schematic representations of magnetic tips. (a) Realistic representation of a tip with ferro- magnetic coating. (b) Effective domain model used for contrast analysis in magnetic force microscopy. (c) Point–

probe approximation for modeling the tip. In this approximation the magnetic field sensed by the tip results from monopole and dipole expansion. In case the decay length of the stray field produced by the sample is smaller than the tip dimensions, the tip is considered as a magnetic monopole. For decay lengths larger than the tip’s dimensions, the monopole contribution can be neglected and only the dipole contribution needs to be taken into account.

of what we shall call the pseudodomain model, see Fig. 3.1b. With all of this in mind, for the magnetostatic potential created by any ferromagnetic sample we write

φs(r) = 1 4π

ˆ d²nˆs· M^S(r^′)

|r − r^′| − ˆ

d³r^′∇ · M^S(r^′)

|r − r^′|

!

, (3.2)

where φ_s is the magnetostatic potential, M^S(r^′) is the sample magnetization vector field, |R| = |r − r^′| and ˆ

ns is the outward unit normal vector from the sample surface. The first two–dimensional integral covers all surface charges created by magnetization components perpendicular to the bounding surface, whereas the latter three–dimensional integral contains the volume magnetic charges resulting from interior divergences of the magnetization vector field. The stray field is then given by H^S(r) = −∇φ^s(r). The magnetostatic free energy of a microprobe exposed to this stray field is

Ψ(r) = µ0

ˆ

d²nˆ_s· M^T(r^′) + ˆ

d³r^′∇r^′·h

φs(r^′)M^T(r^′)i

. (3.3)

The resulting force is then given by F(r^′) = −∇Ψ(r^′). According to the pseudodomain model, M^T(r^′), is divergence free and can be applied to any probe with an arbitrary magnetization M^T(r). In many cases of contrast interpretation, even further simplification of the probe’s magnetic behavior yields satisfactory results.

The effective monopole and dipole moments of the probe, resulting from a multipole expansion of Eq. 3.3, are projected into a fictitious probe of infinitesimal size that is located an appropriate distance away from the sample surface. The a priori unknown magnetic moments as well as the effective probe–sample separation are treated as free parameters to be fitted to the experimental data. This is known as the point–probe approximation(Fig. 3.1c). Depending on the decay length as produced by the sample stray field, the probe is modeled as a magnetic dipole (short decay lengths) or magnetic monopole^† (large decay lengths). The force acting on the probe, which is immersed into the near–surface sample microfield, is given by

†Of course, magnetic monopoles do not exist, but any arbitrary magnetization distribution can be described as a sum or integral over a collection of monopoles