Download Bragg-MOKE and Vector-MOKE investigations : magnetic reversal of

Bragg-MOKE and Vector-MOKE
Investigations:
Magnetic Reversal of Patterned
Microstripes
DISSERTATION
zur Erlangung des Grades eines Doktors der Naturwissenschaften
der Fakultät für Physik und Astronomie an der
Ruhr-Universität Bochum
vorgelegt von
Till Schmitte
Bochum 2002
Mit Genehmigung des Dekanats vom 07.11.2002 wurde die Dissertation in englischer
Sprache verfasst. Eine deutschsprachige Zusammenfassung befindet sich am Ende der
Arbeit.
Mit Genehmigung des Dekanats vom 11.11.2002 wurden Teile dieser Arbeit vorab
veröffentlicht. Eine Zusammenstellung befindet sich am Ende der Dissertation.
Dissertation eingereicht am 29.11.2002
Erstgutachter:
Prof. Dr. H. Zabel, Bochum
Zweitgutachter:
Prof. Dr. W. Kleemann, Duisburg
Disputation am
12.02.2003
Contents
I. Introduction
5
1. Introduction
7
2. Magnetism of thin films and thin film elements
2.1. Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Observation of domains and magnetic hysteresis in magnetic stripes
2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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II. Methods
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3. Magneto-optical Kerr effect of thin films and thin film grating structures 21
3.1. Theory of the Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1. Kerr effect - basics . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2. Electro-magnetic theory of the Kerr effect . . . . . . . . . . . . . 23
3.1.3. Second order contributions to the longitudinal Kerr effect . . . . . 25
3.2. Vector-MOKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3. Diffraction gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4. Bragg-MOKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1. Review of Bragg-MOKE literature . . . . . . . . . . . . . . . . . 32
3.4.2. Some simulations of Bragg-MOKE effects . . . . . . . . . . . . . . 39
3.4.3. Interference between stripe and substrate . . . . . . . . . . . . . . 44
3.5. MOKE setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.1. Standard setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.2. Measurement method . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.3. Extensions of the standard setup . . . . . . . . . . . . . . . . . . 54
4. Sample preparation
4.1. Thin film preparation . . . . . . .
4.1.1. Molecular beam epitaxy .
4.1.2. rf-Sputtering . . . . . . .
4.2. Lithography . . . . . . . . . . . .
4.2.1. Electron-beam lithography
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59
59
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60
60
61
1
Contents
4.2.2. Other Lithography techniques . .
4.2.3. Image transfer . . . . . . . . . . .
4.3. Imaging . . . . . . . . . . . . . . . . . .
4.3.1. Scanning electron microscopy . .
4.3.2. AFM and MFM . . . . . . . . . .
4.3.3. Microscopy and Kerr microscopy
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62
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III. Results and discussion
69
5. Anisotropy of Fe(001)
71
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2. Measurements and discussion . . . . . . . . . . . . . . . . . . . . . . . . 71
6. Fe-nanowires
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Sample preparation and experimental setup . . . . . . . . . . . . . . . .
6.3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1. Magnetic properties of the continuous Fe film . . . . . . . . . . .
6.3.2. Magnetic properties of the Fe nanowire array: longitudinal component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3. Magnetic properties of the Fe nanowire array: transverse component
6.4. Analysis and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
75
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77
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7. CoFe grating
7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Sample preparation . . . . . . . . . . . . . . . . . . . . .
7.3. Remagnetization process of the CoFe-grating . . . . . . .
7.3.1. Results from MOKE measurements . . . . . . . .
7.3.2. Results from Kerr-microscopy . . . . . . . . . . .
7.4. Bragg-MOKE measurements at the CoFe grating sample
7.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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9. Fe-gratings
9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2. Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
105
105
108
8. Ni-gratings
8.1. Introduction . . . . . . . . .
8.2. Experimental setup . . . . .
8.3. Results and Discussion . . .
8.3.1. Bragg-MOKE . . . .
8.3.2. MFM measurements
8.4. Summary and Conclusion .
2
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77
80
82
85
Contents
9.3.1. Single crystal film, sample A . . . . . . . . . . . . . . . . . . .
9.3.2. Polycrystalline Fe-gratings . . . . . . . . . . . . . . . . . . . .
9.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1. Saturation Bragg-MOKE signal . . . . . . . . . . . . . . . . .
9.4.2. Shape of Bragg-MOKE curves of the single crystalline sample
9.4.3. Shape of Bragg-MOKE curves of the polycrystalline sample .
9.5. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
10.Co gratings on a Fe-film
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . .
10.2. Experimental details . . . . . . . . . . . . . . . . . .
10.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1. Standard MOKE measurements . . . . . . . .
10.3.2. Bragg-MOKE measurements . . . . . . . . . .
10.3.3. Intensity measurements . . . . . . . . . . . . .
10.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1. Increasing Kerr effect in the spin valve region
10.4.2. Shape of Bragg-MOKE curves . . . . . . . . .
10.4.3. Bragg-MOKE amplitude . . . . . . . . . . . .
10.5. Summary and Conclusion . . . . . . . . . . . . . . .
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108
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126
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129
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131
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11.Further measurements
143
11.1. Diffuse Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.2. Fe grating with giant Kerr rotation . . . . . . . . . . . . . . . . . . . . . 144
12.Conclusions
147
Bibliography
153
Zusammenfassung
161
Publications
168
Acknowledgments
169
Lebenslauf
170
3
Contents
4
Part I.
Introduction
5
1. Introduction
Motivation
The understanding of the magnetization reversal process of artificially structured magnetic islands and wires is important both from a fundamental point of view and also for
potential magneto-electronic device applications [1] or mass storage devices. Of particular interest for the design of magnetic thin film devices such as read-heads and magnetic
random access memories (MRAM) is the magnetic domain structure within these microor nano-structured elements, their remanent magnetization, and the shape of their magnetic hysteresis loop. On the one hand, these parameters primarily depend on both
the shape and the aspect ratio of the magnetic elements, and on the other hand, they
depend on the intrinsic magnetic anisotropy constants of the magnetic material used
[2]. Particularly, if magnetic islands or wires are separated by only small distances,
long-range magnetic dipole interaction between the elements also has to be taken into
account [3].
Generally, the interest in this new materials raises new questions in the field of experimental techniques for the measurements of the micromagnetic properties.
Magnetic domain structures as well as the magnetization reversal process of nanostructured magnetic elements may be investigated by a number of experimental methods. On the one hand, magnetic domains of single magnetic elements may be imaged
in real space by various techniques such as Kerr-microscopy [4], Lorentz-microscopy [5],
scanning electron microscopy with polarization analysis (SEMPA) [6], X-ray magnetic
circular dicroism (XMCD) microscopy [7] or magnetic force microscopy (MFM) [8]. Hysteresis loops of magnetic elements are derived by evaluating the total size of magnetic
domains having a particular direction of the magnetization vector with respect to the
direction of the applied magnetic field. Also, the hysteresis loop of single domain magnetic elements may be measured with magnetic force microscopy, by using a calibrated
MFM-tip [9]. On the other hand, hysteresis loops of magnetic elements may as well
be measured via the magneto-optical Kerr effect (MOKE), superconducting quantum
interference device (SQUID) magnetometry or vibrating sample magnetometry (VSM)
for which the magnetization reversal process may as well be identified from the shape
of the corresponding hysteresis loop. However, resolution and accuracy of the latter
techniques ask for a large number of identical elements to be investigated in parallel,
such that the obtained hysteresis loop yields information upon the average magnetization reversal process of all elements, and not just upon the magnetization reversal of a
single element. Nevertheless, such techniques more easily allow for hysteresis loop measurements taken at various angles e.g. in orthogonal directions, from which the vector
of the magnetization can be reconstructed.
7
1. Introduction
In the present study two new techniques based on the magneto-optical Kerr effect
(MOKE) are explored and used to investigate the remagnetization process of arrays of
magnetic stripes or wires.
First, the MOKE can easily be operated as a vector-magnetometer. In addition to the
longitudinal MOKE geometry, also the perpendicular component of the magnetization
of nanowires is measured by applying an external magnetic field in a direction normal
to the plane of incidence. Both the longitudinal and the perpendicular field orientation
allows to derive a vector model for the magnetization process, following previous work
by Daboo et al. [10].
Second, a diffraction technique is introduced: Whereas the magneto-optical Kerr effect
is a well-established method for the investigation of thin film magnetism, the application
to samples with lateral structures of the order of the wavelength of the illuminating
laser light is new and challenging, promising to be a powerful technique. Here the
laterally structured sample acts as an optical grating leading to interference effects in
the reflected laser light. Principally scattering-techniques on periodic arrays of stripes
or dots can provide valuable information for the study of micromagnetism. When laser
light is reflected from these samples, Kerr hysteresis loops cannot only be measured in
specular reflection but also at diffraction spots of different order. This technique has
been named Bragg-MOKE. For instance, for ferromagnetic line gratings, the combination
of diffraction and the magneto-optical Kerr effect (MOKE) can yield information about
the mean lateral magnetization distribution [11]. The technique can also be used to
change the sign and the amplitude of the MOKE signal [12].
Whereas measurements of the Bragg-MOKE exist for the polar [12] and transverse
[11, 13, 14, 15] MOKE configuration, Bragg-MOKE hysteresis measurements in the
longitudinal geometry have not been published so far1 .
Aim of this thesis
This thesis has two main goals: One is to investigate the remagnetization process of
micro- and nano-patterned grating arrays and systematic studies of the remagnetization process will be presented. Geometrical factors as aspect ratio and angle between
stripes and magnetic field as well as material parameters are varied. Especially Fe is
in the focus of this thesis. This material can be prepared in polycrystalline and single
crystalline states each with different magnetic anisotropy and hence completely different
remagnetization processes. A variety of stripe arrays will be analyzed mainly using the
MOKE, and thus revealing integrated information of the magnetic properties.
The second goal is to to explore the potential of the magneto-optical techniques BraggMOKE and vector-MOKE. In particular Bragg-MOKE needs further investigations as
the observed effects are rather intriguing. Three main effects will be of interest:
• The influence of interference effects, e.g. between light reflected by the grating
structure and the surrounding substrate, may amplify the observed Kerr rotation.
• In diffraction geometries the incident angle is not identical to the reflecting angle.
1
8
There is one exception: In [16] the authors report on Kerr-spectra in longitudinal geometry but in a
different diffraction geometry (conical diffraction), no Bragg-MOKE hysteresis curves are discussed.
Therefore the question arises for off-specular Fresnel coefficients and how this will
influence the longitudinal Bragg-MOKE effect.
• During the remagnetization process domains will occur inside the magnetic stripes.
Correlated domain structures will influence the shape of the measured BraggMOKE curves.
This thesis will demonstrate each of these manifestations of the Bragg-MOKE effect and
qualitative explanations will be given.
The two main subjects of this thesis, namely remagnetization processes of magnetic
thin film elements and magneto-optics in diffraction geometry, are both discussed from
the experimentalist point of view: Different parameters of the systems under investigation are varied systematically and the effects are recorded. The explanations given follow
this phenomenological approach rather than an analytical or numerical description from
first principles.
Outline
The thesis on hand is organized in three parts. The first part gives an introduction to the
subject and a brief discussion of the domain structures observed in thin film elements.
The second part deals with the methods used here to investigate ferromagnetic gratings
in the nano- or micrometer scale. These are mainly the magneto-optical Kerr effect and
sample preparation techniques. The MOKE and the particular detection techniques
will be explained in detail and the actual state of research in the field of Bragg-MOKE
is discussed. The theoretical section of the second part also consists of a section in
which some basic models of the Bragg-MOKE effect are calculated analytically. The
experimental results are reported in the third part of this thesis. The chapters in this
part are organized following the different samples and series of samples prepared and
analyzed. In addition, the chapters are partial extensions of previously published work.
The third part ends with a conclusion and outlook.
9
1. Introduction
10
2. Magnetism of thin films and thin
film elements
The phenomenon of ferromagnetism in the 3d-metals (Fe, Co, Ni) is essentially due to a
quantum mechanical exchange energy, resulting from the Pauli principle and the almost
localized electrons of the 3d orbitals. The exchange energy leads to a spin asymmetry
in the 3d sub-band and thus a permanent moment of the metal.
In this chapter some basic facts about thin-film magnetism are summarized. The
origin and the theory of magnetism itself is not discussed any further, as many textbooks on this subject are available [17, 18, 19] and a decent discussion would be beyond
the scope of this thesis. In the first section ferromagnetic thin films are discussed in
a thermodynamic context and qualitative arguments for the existence of domains are
given. Subsequently, typical domain-structures of thin-film elements are reviewed and a
summary of relevant literature on magnetic stripes and grating structures is given.
2.1. Free energy
By definition, in thermodynamic equilibrium, any system will always be in a state of
minimum total free energy. The magnetic fields acting on the magnetic moments of
electrons create a local magnetization. In this field of micromagnetics the ferromagnetic
film is described by a vector-field m(~
~ r), where m is the reduced magnetization, m =
M/Ms . In a phenomenological approach several energy-terms are contributing to the
total free energy:
Exchange stiffness This term expresses the preference of a ferromagnet for a uniform
magnetization direction:
Z
Eex = A (grad m)
~ 2 dV,
(2.1)
where A is the exchange constant, a material parameter. This is the phenomenological
description of the quantum mechanical exchange energy. From this equation follows that
a infinite ferromagnet is in its energetic minimum if all magnetic moments are aligned
parallel. For non equilibrium cases (non-uniform magnetization distribution) the free
energy depends on the exchange constant A. Hard magnetic material (e.g. Co) has a
higher exchange constant than soft magnetic material (e.g. permalloy).
Crystalline anisotropy The energy of the ferromagnet depends on the relative orientation of the magnetization vector and the crystalline axes of the lattice. This is an
11
2. Magnetism of thin films and thin film elements
effect of the spin-orbit coupling and the crystal symmetry. Three types of crystalline
anisotropy can be distinguished: cubic, uniaxial and hexagonal. In this thesis mainly
Fe with a fourfold, cubic anisotropy is considered. The energy density of a magnetic
moment in polar coordinates is:
EK = (K1 + K2 sin2 θ) cos4 θ sin2 φ cos2 φ + K1 sin2 θ cos2 θ,
(2.2)
where φ and θ are the in-plane angle and out-of-plane angle, respectively. K1 is the
cubic anisotropy constant. In two dimensions and the case of a (001) oriented thin film
this reduces to
K1
EK =
sin2 (2φ).
(2.3)
4
The easy axes of Fe are aligned along the [100] directions of the crystal lattice.
Surface anisotropy Several reasons can lead to a twofold, uniaxial anisotropy. A
common example is the out-of-plane surface anisotropy leading to a strong easy axis
perpendicular to the surface of a thin film. This is found in several thin film systems,
such as Co/Pd. This effect is important for the technical implementation of magnetooptical storage devices, but will not be discussed in this thesis. Instead, many samples
display an in-plane uniaxial anisotropy due to steps at the surface or due to the artificial
structuring of the surface. In this cases a phenomenological expression is:
EU = KU sin2 (φ − φU ),
(2.4)
where KU is the uniaxial anisotropy constant and φU is the angle between the coordinate
axis and the easy axis of the uniaxial anisotropy.
Stray field energy The magnetized specimen produces a magnetic field itself, the stray
~ d . The systems tries to minimize the energy density of this field. The stray field
field H
energy is given by:
µ0 Z
~ d ∗ mdV.
H
~
(2.5)
Ed = −
2 sample
The stray field (also called demagnetizing field, the corresponding anisotropy is also
called shape anisotropy) depends on the shape of the specimen, a good approximation
for many situations is to assume a general ellipsoidal shape for the sample. Than the
demagnetizing field is
~ d = −N M
~S ,
H
(2.6)
with the symmetric demagnetizing tensor N . For the three axes of the ellipsoid, a, b, c,
the x component of N is [2]:
q
abc Z ∞ 2
Na =
[(a + η) (a2 + η)(b2 + η)(c2 + η)]−1 dη,
2 0
(2.7)
analogous expression are valid for the other directions. This expression can be evaluated
numerically to gain the demagnetizing tensor elements for an arbitrary ellipsoid.
12
2.1. Free energy
For the case of an infinitely extended plate the magnetization depends only on the
z-coordinate and Nc = 1. The stray field energy density is
µ0
(2.8)
Ed = Ms2 cos2 θ.
2
This model is a very good approximation for thin magnetic films. If no other anisotropy
favors out-of-plane magnetization the magnetization will remain in the film plane.
The stray field energy contribution takes the form of an uniaxial anisotropy with an
anisotropy constant KU,d = µ20 Ms2 . From the measurement of the polar (out-of-plane)
magnetic hysteresis of a thin film the anisotropy energy can be calculated by integrating
the hysteresis curve.
Another important geometry are small magnetic stripes with dimensions l, w, h,
length, width and height, respectively. It can be shown that for l w h the
demagnetizing factor Nw for a magnetization in the plane, but perpendicular to the
stripe is
h
Nw =
,
(2.9)
h+w
and therefore the stray field energy is
µ0 h
Ed,w =
M 2 sin2 φ,
(2.10)
2 h+w S
which is again of the form of a uniaxial (in-plane) anisotropy with the anisotropy constant
h
KU,w = µ20 Ms2 h+w
. In this case the easy axis of the anisotropy is in-plane and along the
stripes (φ = 0).
Zeeman energy The energy of the magnetic moment in an external field is given by:
EZ = −µ0 MS H cos(φ − φH ),
(2.11)
where H is the (homogenous) external field and φH the angle between the field and the
coordinate axis.
There are other contributions to the complete free energy function, like magnetoelastic and magnetostrictive contributions. These contributions are neglected throughout this thesis.
Sum of energies The complete energy function is the sum of all terms discussed above.
For the case of homogenously magnetized samples the exchange stiffness is always zero
and for the case of in-plane magnetized samples one finds:
K1
E(φ, H) = −µO MS H cos(φ − φH ) +
sin2 (2φ) + KU sin2 (φ − φU )
(2.12)
4
where the uniaxial anisotropy may be due to the shape anisotropy of magnetic elements,
to the (in-plane) surface anisotropy of a continuous film or a combination of both. In
this model the magnetization rotates from one direction into the other during remagnetization, discontinuities are not described. However, several practical cases can be
evaluated using the above formulas, as will be shown in the experimental sections.
If the exchange constant is small, such that the magnetization is not always homogenous, domains are formed which is discussed in the next section.
13
2. Magnetism of thin films and thin film elements
Figure 2.1.: Landau pattern of a square magnetic element.
2.2. Domains
The most simple situation is found if only the exchange energy and the stray field energy
is taken into account (these two contributions always exist). If the stray field energy
is dominating, as for soft magnetic material, magnetization patterns are formed that
prevent stray fields completely. An example is a magnetic disc. The magnetic moments
will form a closed circular structure, however, paying an energy-penalty by increasing
the exchange energy. These kind of structures have been observed in circular magnetic
dots [20, 2].
Domains in the more common sense are established if additionally magnetic anisotropy
is taken into account. Instead of smooth magnetization patterns domains with sharp
boundaries (domain walls) occur. Inside a domain the magnetization is homogenous and
parallel to an easy axis of the magnetic anisotropy. The occurrence and the shape of
domains are thus depending on the relative strengths of the three energy terms: exchange
energy, stray field energy and anisotropy. If an external field is applied it couples to the
system via the Zeeman energy.
Domains with discontinuous domain boundaries are also established in magnetic par-
Figure 2.2.: Equilibrium domain states of different permalloy elements, together with
the van-den-Berg construction of the domains, taken from [21]
14
2.2. Domains
Figure 2.3.: Construction of domain states in rectangular elements, see main text.
ticles like square or rectangular dots even without the existence of anisotropy axes. If
the magnetic pattern has sharp corners the demand for a zero stray field can only be fulfilled by forming lines of discontinuous magnetization distribution. Therefore magnetic
dots with square and rectangular shape display the typical Landau pattern, as depicted
in Fig. 2.1.
An extensive study of domains in thin film elements was performed by van Berg [22].
He invented a geometrical procedure to construct the equilibrium domain structure of
magnetic thin film elements. The algorithm is as follows:
• draw circles inscribed in the magnetic element which touch the edges at least at
two points.
• the centers of these circles form lines which correspond to the domain walls
• the magnetization is oriented perpendicular to the radius which runs to the point
of contact of the circle and the edge.
• if one circle touches the edges in more than two points the center of this circles
marks an intersection of domain walls.
As an example for this construction some measurements and schematic domain patterns
from [21] are reproduced in Fig. 2.2. In addition to the lowest energy pattern constructed
as explained above other higher energy patterns are also observed. If for instance a
magnetic rectangular element is divided into two virtual halves the domain pattern
can be constructed for the two halves separately using the van-den-Berg algorithm, see
Fig. 2.3. The resulting pattern is of higher energy but may be also stable depending
on the demagnetization process. Comparable domain states have been observed in [23].
If additional anisotropy energy is taken into account, the domain pattern may get even
more complex. An additional uniaxial anisotropy with the easy axis perpendicular to
the long side of the rectangular element in Fig. 2.3 would result in a stabilization of
the domains along the easy axis. The depicted state may than be the lowest energy
state. The most important result of van den Berg is that in most cases domains along
the edges (closure domains) will form in order to minimize the stray field. Closure
domains were experimentally observed in [24] for stripes of permalloy with an external
15
2. Magnetism of thin films and thin film elements
field perpendicular to the stripe axis: The internal region of the stripes is magnetized
along the field but depending on width and thickness more and more edge domains are
formed when the external field is reduced.
Two cases can be distinguished: first the domain state is dominated by the shape of the
specimen. This is the case for soft magnetic material. The domains can be constructed
with the van-den-Berg method. In this case the angle of the magnetization between two
domains is arbitrary reflecting the angle of the geometric shape of the element (e.g. 90◦
walls for square elements). Another case is a film or an element with anisotropy. In this
case the domains will be magnetized along an easy axis of the anisotropy. This leads
two 180◦ domain walls in uniaxial and 90◦ walls in fourfold anisotropy material. Wether
a material is dominated by the anisotropy or by the stray field is given by the parameter
Q = K/Kd , where K is a general anisotropy constant taking four- or two-fold crystal
anisotropy into account. For soft magnetic material Q 1.
2.3. Observation of domains and magnetic hysteresis in
magnetic stripes
Several measurements of the domain structure and the hysteresis of magnetic stripes
and wires can be found in the literature:
• Ebels et al. [25] have investigated Fe stripes on GaAs with the rather large width
of 15 µm. They found an induced uniaxial anisotropy due to edge effects and
a two step magnetization process with two different domain types, due to the
combination of four-fold anisotropy of Fe and the patterning.
• Shearwood et al. [3] report upon magneto-resistance and magnetization loops of
arrays of sub-micron sized Fe stripes. The stripes were held constant in shape
(0.5 µm width) but arrays with different separations between the elements where
produced. The result is that again an uniaxial anisotropy is induced and hints of
dipolar interactions depending on the separation were found.
• Hausmanns et al. [26, 27] show in combined work of experiment and simulation the
remagnetization behavior of Co nanowires (width: 150 to 4000 nm). They show
an increase of the coercive field with decreasing width of the wires proportional
to 1/w. In addition, the coercive field for different in-plane angles of the external
field was examined, showing a simple behavior consistent to a model where the
magnetization first rotates into the wire direction and afterwards switches by 180◦ .
• Because of the shape anisotropy 180◦ domain walls in small wires are expected
to be of the head-to-head type. This was theoretically confirmed by McMichael
et al. [28]. The head-to-head domain wall consist of additional intermediate domains with a magnetization perpendicular to the wire axis or forming vortex-like
structures.
• McCord et al [23] performed an intensive Kerr microscopy study on rectangular
permalloy elements. Very different domain structures were detected depending on
16
2.4. Conclusion
the magnetic history of the element. Typically closure domains with large internal
domains having perpendicular direction were observed.
• Mattheis et al. [24] also measured large edge domains aligned with the edge of the
stripes using Kerr microscopy for an external field direction perpendicular to the
wire axis.
2.4. Conclusion
The measurement and interpretation of domains in thin film elements is a very important
subject in the field of magneto-electronics and general research on magnetism. Therefore
there exist a large amount of publications on the subject. There are several approaches
to the problem:
• Today’s computer-power enables to calculate the domain structure of thin film
element. Several commercial and non-commercial programs are available. However, only in combination with the experiment the real domain structure can be
concluded. For special cases, like the spin structure inside of domain walls, the
numerical simulation is almost the only possibility to gain insight due to difficulties
observing very small magnetic structures.
• Measurements of integral physical properties like the hysteresis curves, transport
phenomena or dynamic properties gain important information on the magnetic system and can be used together with numerical simulations of the domain structure.
General features of the domain structure can be concluded. These kind of measurements provide important parameters of the complete system like remanence,
saturation magnetization, time constants or the magnetization vector.
• Direct observation of the domain structure using Kerr microscopy, MFM or other
techniques has the obvious advantage of directly imaging the domains, no simulation or assumptions are needed. However, every method has its specific limitations
such as resolution or problems with the contrast. In addition, it is often difficult
to obtain integral quantities such as remanence or coercive fields. Only a small
portion of a sample may be visualized and the overall behavior may not be detected.
The most comprehensive review of the subject is found in [29], where many methods,
the domain theory and a vast amount of examples are given.
The present thesis contributes to this field. The combination of diffraction and MOKE
will be shown to yield information about the domain structure and the hysteresis simultaneously.
17
2. Magnetism of thin films and thin film elements
18
Part II.
Methods
19
3. Magneto-optical Kerr effect of thin
films and thin film grating
structures
This chapter covers the theory and the experimental realization of Kerr effect measurements of thin films and ferromagnetic grating structures. The first sections explain the
theory of the Kerr effect, the fundamentals of vector-MOKE and the theory of diffraction
gratings.
The next section provides a review of the literature of the Bragg-MOKE effect. This
section closes with simulations of some of the Bragg-MOKE effects described in the
literature. This simulations are very important for a comparison of the experimental
results reported in the third part of this thesis.
In the last section of this chapter the experimental setup is introduced which was
used to measure the standard MOKE hysteresis curves, the vector-MOKE results and
the Bragg-MOKE curves.
3.1. Theory of the Kerr effect
3.1.1. Kerr effect - basics
In general the magneto-optical Kerr effect is the change of polarization and/or intensity
of a light beam reflected by a ferromagnetic surface. The measured quantity, e.g. the
rotation of the polarization, is a linear function of the magnetization of the ferromagnetic
material. A corresponding effect which has a quadratic dependence on the magnetization
is called Voigt or Cotton-Mouton effect. A special case of this will be discussed later in
Sec. 3.1.3. Another magneto-optical effect is the Faraday effect: the polarization of light
is rotated by transmitting light through dielectric material in the presence of a magnetic
field. In this case the rotation is proportional to the applied field. This effect is used in
the experimental setup (Sec. 3.5).
The simplest model of MOKE is to consider a Lorentz-Drude model of a metallic
film. The incident light wave causes the electrons in the metal to oscillate parallel to
the plane of polarization. In the absence of any magnetization the reflected light is
polarized in the same plane as the incident light, this is the regular component with an
amplitude RN . If a magnetization is thought to be acting on the oscillating electrons
like an internal magnetic field, the electrons exhibit a second motion due to the Lorentz
force. This second component is perpendicular to the direction of the magnetization and
21
3. Magneto-optical Kerr effect of thin films and thin film grating structures
Figure 3.1.: Geometry of the three magneto-optical Kerr effects, see main text.
perpendicular to the primary motion. The second component, RK , generates a secondary
amplitude of the reflected light which has to be superimposed onto the primary beam
[4].
In this framework one can understand easily the three general geometries of the
magneto-optical Kerr effect [4], which are displayed in Fig. 3.1:
a) In the polar geometry the magnetization is perpendicular to the reflecting surface.
A linear polarized wave generates a second component, which is strongest if the
angle of incidence is zero (perpendicular incidence, αi = 0). In addition, the effect
is independent of the direction of the polarization for αi = 0.
b) In the longitudinal configuration the magnetization is parallel to the reflecting
surface and parallel to the plane of incidence. The effect generates a polarization
rotation of the reflected beam in both cases, perpendicular (s-) and parallel (p-)
polarized light with respect to the plane of incidence. The sign of the Kerr rotation
in the two cases is different. The special case of perpendicular incidence generates
no Kerr rotation in either case, because RK points along the beam (s-polarization)
or RK is zero (p-polarization). Thus the measured Kerr effect increases with αi .
c) For the transverse configuration the magnetization is oriented perpendicular to
the plane of incidence and parallel to the surface. For p-polarized light this configuration causes a change of the amplitude of the reflected beam, but no Kerr
rotation.
The three cases can be combined to yield a formula of the Kerr effect for an arbitrary
magnetization and polarization of the electromagnetic wave. However in the present
work only the longitudinal configuration is used. The samples under investigation usually
exhibit no out-of-plane magnetization components, thus excluding the use of the polar
Kerr effect. The longitudinal case is preferred over the transverse case because of the
easier detection of polarization rotations than intensity shifts, as will be explained in
Sec. 3.5. The longitudinal measurements are performed using s-polarized light. A
polarization parallel to the plane of incidence would add an intensity modulation due to
the transverse Kerr effect.
The theory of the MOKE in the framework of a free electron gas as discussed above
has several shortcomings and the complexity of band structures of the ferromagnetic
materials demands a quantum-mechanical treatment. The magnetic moment of the
22
3.1. Theory of the Kerr effect
Figure 3.2.: Definition of the coordinate system used in the discussion of the Kerr effect.
ferromagnetic material is caused by the spin-asymmetry of the spin-up and spin-down
sub-bands of the 3d-band structure of Fe, Co or Ni. An incoming polarized light wave
interacts through its electric field with the electrons, and changes their orbital momentum. Because of the weak spin-orbit interaction this results in an interaction between
the electric field of the light wave and the magnetization. The incoming electromagnetic
wave can be split into left- and right-circular polarized eigenmodes, which have different
quantum-mechanical probabilities to excite spin-up or spin-down electrons of the 3dband near the Fermi-level. The exited electrons will emit electromagnetic waves, with
different circular polarization depending on their spin-state. Effectively this mechanism
results in matrix elements of a 2 × 2 reflection matrix Rc relating the Jones vector of the
incoming wave in a circular-polarization basis to the Jones vector of the emerging wave
(for a discussion of Jones vectors see Sec. 3.5 and [30]). The calculation of magnetooptical effects from first principles is a complicated task (see [31, 32, 33, 34]), however,
for the present investigations a theory in the framework of electro-magnetism is sufficient
and will be discussed in the next section. For an introduction to the theory of MOKE
see [35, 36].
3.1.2. Electro-magnetic theory of the Kerr effect
As discussed in the above section the reflection of a electromagnetic wave by a ferromagnetic surface can be split into the regular reflection, which is described by standard
Fresnel formulas and Fresnel reflection coefficients, and into a second component which
adds a small contribution polarized perpendicular to the regular component. The superposition of the two components leads to a rotation of the polarization axis. The
goal of the electromagnetic theory is to generalize the Fresnel formulas in order to yield
magneto-optical reflection coefficients.
In the following the situation depicted in Fig. 3.2 is assumed: A linear polarized wave
is incident on a ferromagnetic surface under the angle αi and reflected at αf = −αi . The
magnetic medium has the refractive index n1 . For the reflection at a magnetic medium
the use of a dielectric constant is not sufficient. In stead of this, in the dielectric law,
23
3. Magneto-optical Kerr effect of thin films and thin film grating structures
~ = E,
~ is a complex tensor, which can be written as [37, 38, 35]1 :
D


1
−iQmz iQmy

1
−iQmx 
= xx  iQmz
,
−Qmy iQmx
1
(3.1)
~ and the material
where the mi are the components of the magnetization vector M
constant Q is the magneto-optical constant, also called Voigts constant2 . In order to
take the absorption of the electromagnetic wave into account, the regular dielectric
constant xx is a complex number, the imaginary part corresponding to the absorption
coefficient. Equivalently, the refractive index n is complex. With the above dielectric
tensor the Maxwell equations have to be solved [38], which leads to a reflection matrix
R=
rpp rps
rsp rss
!
.
(3.2)
Eq. 3.2 relates the p- and s-polarized components of the incoming wave to the respective
components of the reflected wave. The coefficients rij are the ratio of the incident j
polarized electric field and reflected i polarized electric field. rss is the standard Fresnel
reflection coefficient, rpp is the standard Fresnel coefficient plus a term depending on
mx Q (transverse MOKE) and both the off-diagonal elements of R are functions of mz Q
and my Q (polar and longitudinal MOKE: see Fig. 3.2 for the definition of the coordinate
system). All coefficients are functions of the refractive index and the refraction angle
inside the ferromagnetic material. Explicit formulas can be found e.g. in [38].
The complex Kerr angles are defined as following:
rsp
p
ΘpK = θK
+ ipK =
,
(3.3)
rpp
rps
s
ΘsK = θK
+ isK =
.
(3.4)
rss
Here θK and K are the Kerr rotation and ellipticity, respectively, and the superscripts
denote whether the incoming light wave is polarized in the p- or s-state. In Ref. [38]
simplified formulas for different MOKE geometries are derived. As this thesis deals with
the longitudinal MOKE with s-polarized light, only this case is discussed. In general,
there are two situations which have to be distinguished. If the ferromagnetic film is
thick compared to the wavelength in the material, the MOKE signal is independent
of the thickness tF M . If the layer is thin, the MOKE signal is a function of tF M , for
ultrathin layers this is a linear function. For the laser wavelength used in the present
investigations (λ = 623.8 nm) bulk iron magneto-optical parameters are given in Tab 3.1
[39]. This leads to a wavelength inside of iron of ≈ 220 nm.
The longitudinal Kerr effect for the case of thick ferromagnetic films (i.e. the Kerr
effect is independent of the thickness) is derived from Eq. 3.3 and the formulas of rij for
mz = mx = 0 [38]:
cos αi tan α1 in0 n1 my Q
s
θK
=
,
(3.5)
cos(αi − α1 ) (n21 − n20 )
1
Different sign conventions are used in literature, the discussion in this thesis follows the proposed
scheme in [37]
2
~ is also called the gyromagnetic vector
The product QM
24
3.1. Theory of the Kerr effect
n
2.89 + 3.07i
Q
0.042 + 0.012i
α
β
1.00 0.13
Table 3.1.: Magneto-optical parameters of Fe. The refractive index and the Voigt constant were taken from [39]. The parameters for second order contributions,
α, β, were measured in [40] for a small angle of incidence (13◦ ) and for single
crystalline Fe(001)/GaAs films.
where n0 is the refractive index of the medium above the ferromagnetic film, for air
n0 = 1 is assumed. The angle of the refracted beam, α1 , has to be calculated using
Snell’s law: n0 sin αi = n1 sin α1 .
The interesting case of thin ferromagnetic films is difficult to solve as multiple reflections and interference have to be taken into account. However, a method described in
[41, 38] implements a matrix formalism which allows to calculate the MOKE signal for
complicated film structures and superlattices. A simplified formula for the case of a thin
ferromagnetic film on a non-magnetic substrate with the refractive index nsub (angle of
the refracted light in the substrate: αsub ) is given in [38] as
s
θK
=
4πn0 n1 nsub QdF M cos αi sin α1
.
λ(n0 cos αsub + nsub cos αi )(n0 cos αi − nsub cos αsub )
(3.6)
In more realistic situations also a layer on top of the ferromagnetic film, e.g. an oxide
layer, has to be taken into account.
In the present study magnetic films are considered with a typical thickness of dF M =
20 ... 50 nm, for which the approximation of a thick ferromagnetic film turned out to
be satisfactory. In Fig. 3.3 the Kerr rotation as given by Eq. 3.5 is plotted for the bulk
Fe parameters as a function of the incident angle. The function exhibits a maximum at
αi = 55◦ . For most experimental situations an incident angle of 45◦ is realistic for which
a Kerr rotation of θK = 0.068◦ can be expected. The ellipticity shows a maximum for
the same angle of incident, which is K = 1.85 · 10−3 rad.
3.1.3. Second order contributions to the longitudinal Kerr effect
The dielectric tensor in Eq. 3.1 is linear in the magnetization. As already mentioned,
there are cases where a second order contribution to are important. Especially single
crystalline Fe often exhibits strong second order effects. The second order effects are
~ and a second order term is added to the dielectric tensor in Eq. 3.1,
quadratic in M
which is given by [4]:
B1 m2x
B2 mx my B2 mx mz

B2 my mz 
B1 m2y
.
 B2 mx my
2
B2 mx mz B2 my mz
B1 mz


(3.7)
From this tensor expressions for the MOKE can be derived, as is shown in [42]. Experimental examples for the case of Fe can be found in [43, 40]. In Ref. [40] fits to MOKE
data in the longitudinal geometry show that the Kerr effect can be described effectively
by
θK ∝ my + αmy mx + βm2x .
(3.8)
25
3. Magneto-optical Kerr effect of thin films and thin film grating structures
0.08
0.07
Re(θsK) [°]
0.06
0.05
0.04
0.03
0.02
0.01
0
0
15
30
45
αi [°]
60
75
90
Figure 3.3.: Plot of the Kerr rotation as a function of the incident angle αi , as described
in Eq. 3.5 for bulk Fe and s-polarized light.
The two phenomenological parameters α and β were estimated in [40] for the case of
a single crystalline Fe(001) film on GaAs and are given in Tab. 3.1. As discussed in
[42] the second order contributions depend on the angle of incidence. The values in
Tab. 3.1 were taken at αi = 13◦ , for larger angles as were used in this thesis smaller
second order effects are expected. For the longitudinal configuration the magnetization
components mx and my in Eq. 3.8 can be identified with the two orthogonal magnetization components mt and ml along the transverse and longitudinal in-plane direction,
respectively. Therefore the second order effects lead to a contribution in the longitudinal MOKE of transverse magnetization components. If the re-magnetization process of
the sample under investigation is dominated by magnetization rotation processes, the
orthogonal magnetization component increases around zero field and will lead to strong
asymmetries in the measured hysteresis loop. Vice versa, if the re-magnetization process
involves only 180◦ domain wall movements, the magnetization in any domain is always
oriented parallel or antiparallel to the external field, thus no second order contributions
are detected. For Fe, tending to 90◦ domain walls, a combination of the two limiting
cases is expected.
3.2. Vector-MOKE
It is often advantageous to measure not only the component of the magnetization along
the applied field, but also the orthogonal magnetization component in order to reconstruct the magnetization vector from the measurement. The longitudinal MOKE can
be used as a vector-magnetometer in the following manner:
Magnetic hysteresis measurements were performed using a high resolution magneto-
26
3.2. Vector-MOKE
Figure 3.4.: Definition of the sample rotation χ and the angle φ of the magnetization
~ for the case of the longitudinal setup (a) and the perpendicular
vector M
setup (b). In order to measure the transverse magnetization component mt
the field and the sample are rotated by 90◦ , such that the angle χ is held
constant, but the magnetization component mt is in the scattering plane.
optical Kerr effect setup (MOKE) in the longitudinal configuration with s-polarized
light, which is able to measure the exact Kerr angle as a function of the applied magnetic
field. Details of the experimental setup can be found in Sec. 3.5. Here the magnetic
field lies in the scattering plane and the resulting Kerr angle is proportional to the
l
component of the magnetization vector along the field direction, θK
∝ ml , where ml is
~
~
the longitudinal component of M projected parallel to H. Additionally, the design of the
setup enables one to rotate the sample around its surface normal (angle χ), in order to
apply a magnetic field in different in-plane directions. This kind of measurement cannot
distinguish between a magnetization reversal via domain rotation and/or via domain
formation and wall motion. Therefore measurements were performed with the external
magnetic field oriented perpendicular to the scattering plane and the sample rotated
by 90◦ with respect to the scattering plane, keeping the rest of the setup constant. In
this perpendicular configuration MOKE detects the magnetization component parallel
t
to the scattering plane and perpendicular to the magnetic field, θK
∝ mt , as has been
shown by [10]. The geometry of the setup is sketched in Fig. 3.4. Both components, ml
~ sampled over the
and mt , yield the vector sum for the average magnetization vector M
region, which is illuminated by the laser spot. This area is ≈ 1mm2 . The magnetization
vector can be written as
~ =
M
ml
mt
!
= |M |
cos φ
sin φ
!
.
(3.9)
The proportionality constant between the Kerr angle θK and the two magnetization
components is a priori unknown. For the samples under investigation it was found that
in saturation the Kerr angle does not dependent on the sample rotation χ. Furthermore,
the angle of incidence of about 40◦ was kept constant for both set-ups. Therefore the
27
3. Magneto-optical Kerr effect of thin films and thin film grating structures
error - if at all - is tolerable by assuming the same proportionality constant for both
configurations. A source of error may be a contribution from the polar MOKE effect,
which would add a signal proportional to a magnetization component perpendicular to
the sample surface. In addition second order magneto-optical effects [42] (see Sec. 3.1.3)
could interfere with the following analysis. However, neglecting these potential problems,
one can write:
ml
cos φ
θl
=
= K
,
(3.10)
t
mt
sin φ
θK
from which follows the rotation angle of the magnetization vector:
θt
φ = arctan K
l
θK
!
.
(3.11)
Furthermore one can express |M |, normalized to the saturation magnetization:
l
|M |
θK
1
=
.
l,sat
sat
|M |
θK cos φ
(3.12)
Another possibility of yielding magnetic vector information from MOKE measurements is to use a combination of the transverse and longitudinal Kerr effect, as has been
shown by [44]. In the case of the transverse Kerr effect the magnetic information is
obtained from an intensity shift of the reflected light, which is proportional to the magnetization along the applied magnetic field perpendicular to the scattering plane. In this
geometry obviously a rotation of the polarization can be attributed to the longitudinal
Kerr effect which is then sensitive to the magnetization component perpendicular to the
magnetic field. Thus, by measuring both, the rotation and the intensity one can extract
information of two orthogonal magnetization components. Details of the procedure can
be found in [44]. The advantage here is that the magnetic field and the sample stay
in the same position and the two components can be measured simultaneously, as opposed to the geometry used in this work. The drawbacks are that the detection is more
complicated and the two signals yielded are not directly comparable concerning their
magnitude, because two different physical quantities are measured.
The results of vector-MOKE measurements provide important information which allow to distinguish between different magnetization reversals. Two limiting cases can
easily be separated (see Fig. 3.5):
~ |, is
• Coherent rotation (Fig. 3.5(a)): If the length of the magnetization vector,|M
constant during the reversal, the magnetization rotates from one direction into the
other.
• Domain formation (Fig. 3.5(b)): If only domains are formed, the angle of the magnetization stays always aligned with the external field but the magnitude changes.
In this case the transverse component is zero.
It is instructive to plot the transverse component and the angle φ as given by Eq. 3.11
and Eq. 3.12 as a function of the longitudinal magnetization component ml (the component parallel to the external field) For the case of coherent rotation the transverse
component is increased if the longitudinal component is decreases and and vice versa. If
no transverse component can be detected no rotation of the magnetization takes place
and the reversal is governed by domain processes.
28
3.3. Diffraction gratings
Figure 3.5.: Two limiting cases of magnetization reversal and the resulting vector-MOKE
measurements. (a) shows the case of coherent rotation. The reversal is
sketched and the transverse component, the angle and the magnitude of
the magnetization are plotted as a function of the magnetization along the
field. (b) depicts the case of domain formation and no rotation. The same
quantities are plotted as in (a).
3.3. Diffraction gratings
The most common diffraction experiments, and all experiments covered in this thesis,
are performed in the so-called Frauenhofer diffraction geometry. This means that the
source of light and the observer are at an infinite distance to the diffracting object. This
condition is easily satisfied by the use of a laser as light source. Thus a plane wave is
incident on the diffracting object, which is viewed as an object with a certain transfer
function, f (y). The complex function f (y) describes the reflection or transmission of the
amplitude of the electrical field vector and its absorption. In the following paragraphes
the scalar diffraction theory from one-dimensional grating structures is outlined. In
this context, scalar means that the diffraction is independent of the polarization of the
incident light.
Every point on the object acts as a new source of light, emitting a spherical wave, its
amplitude and phase given by the transmission function. In one dimension the resulting
diffraction pattern is the integral over the surface of the diffracting object multiplied
with a phase factor:
Z
ψ(k) = f (y)exp[−i(ky)]dy,
(3.13)
where k is a reciprocal space vector defined in this case via
k = k0 (sin αf − sin αi ).
(3.14)
The angles αi and αf define the directions of the incoming and diffracted beam, respectively. The wavenumber k0 is defined by k0 = 2π/λ, where λ is the wavelength of the
electromagnetic wave.
The simplest transfer function is that of a slit aperture of width a in one dimension:
fslit (y) = {
1 if |y| ≤ a/2
.
0 if |y| ≥ a/2
(3.15)
29
3. Magneto-optical Kerr effect of thin films and thin film grating structures
7
18
x 10
16
14
Intensity
12
10
8
6
4
2
0
−40
−30
−20
−10
0
αf [°]
10
20
30
40
Figure 3.6.: Calculated diffraction pattern for perpendicular incidence, a = 2.3 µm,
d = 5 µm, N = 8 and λ = 632 nm.
The resulting diffraction pattern is given by:
|ψ(k)|2 = a2
sin2 (ak/2)
.
(ak/2)2
(3.16)
Another important case is the diffraction from a finite array of diffracting objects. If
the transfer function is a regular spaced array of delta-functions the resulting intensity
pattern is:
sin2 (N dk/2)
|ψ(k)|2 =
,
(3.17)
sin2 (dk/2)
where N is the number of delta-functions contributing to the diffraction pattern and d is
the grating parameter. This function describes the well known intensity pattern from a
diffraction grating with major and minor intensity maxima. The intensity in the major
maxima is increasing with N , the number of minor intensity maxima between the major
maxima is (N − 2). For perpendicular incidence the major intensity maximum of order
n occurs if the Bragg-formula is satisfied:
d sin αf = nλ.
(3.18)
Considering the more general case of non-zero αi leads to the grating-equation:
d(sin αf − sin αi ) = nλ.
(3.19)
At this point it is important to note that the exact form of the grating equation depends
on the sign convention chosen. For Eq. 3.19 the angles in the first and third quadrant are
30
3.4. Bragg-MOKE
positive and angles in the second and fourth quadrant are negative (cartesian convention,
see [45]). Another source of confusion often occurs in comparison with x-ray scattering
techniques, where the angles are defined not relatively to the surface normal but with
respect to the surface. Therefore every sin function in the above formulae would be
converted to a cos for x-ray diffraction. On the other hand, the most common case
for x-ray scattering is diffraction from the lattice perpendicular to the surface, which
again leads to a rotation of the coordinate system of 90◦ . In total the form of the above
equation is the same for diffraction-gratings and Bragg-diffraction from crystals.
A more realistic case of a diffraction grating is to assume single slits with a transfer
function as given in Eq. 3.15, which are convoluted with the regular grating of deltafunctions as discussed above. From Fourier-theory it is known that the Fourier-transform
of a convolution of two functions is the product of the Fourier-transforms of the two
single functions (convolution-theorem). Therefore the intensity function of this case is
the product of Eq. 3.17 and Eq. 3.16, i.e. the intensity pattern shows maxima at the
same positions as for the delta-function array (Eq. 3.19 still holds), but the intensity
at the maxima display a modulation whose envelope is the intensity function of the
single slit. This model can be used in the most cases considered in this thesis. An
example of a diffraction pattern according to Eq. 3.17 assuming an envelope as given
in Eq. 3.16 is plotted in Fig. 3.6. Of course, an even more realistic intensity function
would be the Fourier-transform of the real transfer function, i.e. the modulation of the
(complex) reflection coefficient when viewed perpendicular to the stripes, also taking the
hight difference of the stripes and grooves into account. More details of the theory of
diffraction gratings can be found in [45] and elementary textbooks on optics (e.g. [46]).
It should be mentioned that the above theory of diffraction gratings is a scalar theory.
That means that during diffraction the two orthogonal polarization directions are not
coupled and the polarization state is conserved. Obviously this is inherently not the case
for the combination of diffraction and Kerr effect. Even non-magnetic, metallic gratings
couple the polarization directions, thus a vector-theory of diffraction is necessary [45].
A simple example is a grating consisting of thin metallic wires which has been used as
a polarizer. In this case the E-vector of the transmitted beam is aligned parallel to the
wires. The vector theory of diffraction is a broad subject in optics and several textbooks
and articles cover the matter, e.g. [47, 48, 49], and references therein. However, in this
thesis only the scalar theory is considered. Therefore results can not be fitted to models
exactly, but it will be shown that main features of the measurements can be discussed
in the framework of the scalar theory.
3.4. Bragg-MOKE
The term Bragg-MOKE stems from the used combination of the usual Kerr effect measurement and diffraction from a lateral structure, e.g. a diffraction grating. Instead of
analyzing the intensity or polarization rotation of the specular reflected beam, signals of
the diffracted beams are measured. This section first reviews the literature on the matter and than simulations of several effects playing an important role for Bragg-MOKE
are reported.
31
3. Magneto-optical Kerr effect of thin films and thin film grating structures
3.4.1. Review of Bragg-MOKE literature
First experiments
The first time the combination of Kerr effect and diffraction from grating structures was
mentioned in literature was 1993 by Geoffroy et al. [11]. In this article measurements
of the transverse Kerr effect from different diffracted intensities from a SmCo4 square
arrays with a grating parameter of 4 µm are reported. The loops measured at the diffraction spots did not simply reproduce the standard MOKE curve but showed remarkable
differences. The measurements are reproduced in Fig. 3.7. In addition, the magnitude
of the measured Kerr effect in saturation changed with the order of diffraction. In [11]
a simple explanation of the observed effects is offered in the framework of the scalar
diffraction theory as it is outlined in Sec. 3.3. In this case two main contributions have
to be taken into account:
• The diffracted light originates not only from the ferromagnetic grating but also
from the not ferromagnetic substrate. The phase shift, φh , between both contributions is depending on the height of the structure and the angle of diffraction and
incidence:
2πh
[1 + cos(αi + αf )],
(3.20)
φh =
λ cos αi
This contribution might change the amplitude of the measured signal.
• The magnetization distribution in the magnetic elements (i.e. the domains) give
rise to a weak modulation of the reflectivity in the elements. This magnetization
modulation contributes via Fourier transformation to the signal in the BraggMOKE hysteresis loops, but does not change the Kerr amplitude in saturation.
Especially the last point needs further explanation: As discussed in Sec. 3.1 the transverse Kerr effect measures the intensity shift as a function of the magnetization rather
than the polarization rotation as it is the case for the longitudinal geometry which is
discussed in this thesis. However, the magnetic contribution in the transverse Kerr signal is only a small contribution superimposed on the non-magnetic reflectivity signal.
The authors of [11] assume the same to be true for the Bragg-MOKE effect in transverse geometry. The measured intensity as a function of the magnetization at different
Bragg-spots is the product of the intensity due to the Fourier transform of the structural
grating and the Fourier transform of the magnetization distribution. When measured as
a function of the external field only the latter is changing, thus the Bragg-MOKE curve
is related to the change of the Fourier component with wave number n/d of the spatial
distribution of magnetization within the magnetic elements. The authors stress that the
coherent Bragg diffraction peaks carry information on the mean magnetization contribution in the patches. Following this assumptions the authors of [11] develop a model
assuming a simple two domain wall configuration in the square elements. The resulting
model calculation could qualitatively reproduce the measurements but no quantitative
correspondence was achieved. In particular the Kerr signal in saturation at different
Bragg-spots could not be explained.
Taking the article of Geoffroy et al. as a starting point several publications followed
both experimentally and theoretically. Again the two lines of investigations are visible:
32
3.4. Bragg-MOKE
Figure 3.7.: Transverse Bragg-MOKE measurements form a square dot array of hard
magnetic material with a grating parameter of 4 µm. The figure is taken
from Geoffroy et al. [11].
one is to understand the amplitude of the magneto-optical signal in saturation and
the other is to understand the shape of the Bragg-MOKE curves as resulting from the
domain states during the re-magnetization process.
Measurements and simulations of the saturation Bragg-MOKE amplitude in
transverse geometry
In the article of van Labeke et al. [13] (also see [50]) an experimental situation is constructed, which is essentially easier than the experiments discussed in [11]. The sample
under investigation was a one-dimensional grating structure, which was completely covered with a soft magnetic film, such that the grating and the grooves in between are
covered with ferromagnetic material (relief grating). The resulting grating was expected
to exhibit no domain structure related to the grating geometry. Therefore the measurements concentrated on the Bragg-MOKE amplitude in saturation as a function of
the incident angle rather than the shape of the hysteresis loops. Measurements were
carried out in transverse MOKE configuration and compared to calculations which used
33
3. Magneto-optical Kerr effect of thin films and thin film grating structures
the Rayleigh expansion and a perturbation method. The complex matrix formalism
calculations reproduced very well the measured Bragg-MOKE amplitudes without the
need to introduce phenomenological parameters (the optical constants were measured
on flat surfaces and used in the simulations). The variable used in this study was the
sum of the incident and diffracted angle of the laser beam for a given order of diffraction, Θ = αi + αf,n . The result is that the transverse Bragg-MOKE effect increases
for increasing Θ and for constant Θ the Bragg-MOKE amplitude increases with n. For
small values of Θ the curves can be approximated by a linear function. Another important observation was that the curves always pass through the origin, i.e. for Θ = 0 the
measured and calculated Kerr amplitude is zero. This means that if the diffracted beam
is directed along the incident beam (Littrow geometry, see [45]) no Bragg-MOKE curve
can be measured. Another important result from the calculation was, that in transverse
geometry the p- and s-polarized eigenmodes are not coupled, there is neither depolarization nor rotation of the polarization and the transverse Bragg-MOKE effect can only be
observed with p-polarized light, as already discussed for the standard transverse MOKE
in Sec. 3.1
The experiments were extended to inhomogeneous gratings (i.e. the grooves were non
magnetic) in [51, 52]. The same experiments were performed as before, but also the
relation between the transverse Bragg-MOKE amplitude as a function of n and αi and
the diffracted intensity without Kerr effect were investigated. It turned out that the
Kerr effect can be increased dramatically for n 6= 0 and increasing αi . The increase
was greatest for n = ±1, in this case a maximum combined with an abrupt change of
sign is observed for a special αi at which the diffracted intensity reaches a minimum.
The observed effects are explained with the interference of the light diffracted form the
ferromagnetic grating and the light diffracted by the non-ferromagnetic substrate-grating
formed by the grooves. Essentially the same effect takes place for the anti-reflection
coatings commonly used to enhance the Kerr effect by interference of reflected beams at
two surfaces. By varying the angle of incidence one can find a configuration in which the
light diffracted by the non-magnetic sub-grating compensates the Fresnel component of
the light diffracted by the ferromagnetic sub-grating. The total intensity is minimized
and consequently the relative change of intensity due to the transverse Kerr effect reaches
a maximum. The parameters for finding this maximum are the height of the stripes, the
angle of incidence and the optical constants of the material. This interference effect of
Bragg-MOKE amplitude amplification is superimposed to the effect due to the optical
properties of the ferromagnetic grating itself as discussed in [13] without non-magnetic
sub-grating.
Bragg-MOKE hysteresis loop simulations in transverse geometry
The article of Vial and Labeke [53] is an extension of the work of the same authors
[11, 13] in two respects: first, the model for the simulation of the transverse BraggMOKE effect, as it was presented in [13] (previous paragraph of this section), is further
extended to take into account inhomogeneous gratings and, second, the domain model
discussed in [11] is used together with the vectorial diffraction theory in order to model
Bragg-MOKE hysteresis loops. The general experimental and theoretical facts about the
Bragg-MOKE amplitude at different orders of magnitude and varying angle of incidence
34
3.4. Bragg-MOKE
from [13] are reproduced, but additionally Bragg-MOKE hysteresis loops are modelled,
with the same magnetic model as was used in [11].
The magnetic model used is based on the assumption that the magnetization direction inside magnetic domains is directed only parallel or anti-parallel to the external
field, which is in the transverse configuration perpendicular to the scattering plane and
along the stripes of the assumed grating structure. Hence, the magnetization is always
oriented along the easy axis of the stripe-induced uniaxial anisotropy. Two cases were
distinguished:
• one wall configuration: during re-magnetization only one 180◦ domain wall moves
from one side of the stripe to the other, the domain wall is oriented parallel to the
stripe. This configuration does not lead to a change of the shape of Bragg-MOKE
hysteresis loops compared to the standard MOKE curve.
• two wall configuration: in the middle of the stripe one domain with the opposite magnetization direction nucleates and the two domain walls propagate to the
sides of the stripe. The case of two domains nucleating at the edges and travelling inwards is equivalent. The simulation of this case leads to Bragg-MOKE
hysteresis curves similar to the one observed in [11], without the need of taking
phenomenological parameters into account.
As an example the simulations for a diffraction grating is reproduced in Fig. 3.8. Here
the transverse Bragg-MOKE signal is plotted as a function of the magnetization (and
not the external field), i.e. a Bragg-MOKE loop without change in shape as compared
to the specular loop would display a straight line (see also [11]). In this representation
the Bragg-MOKE loops only show the effect of the domains.
Transverse Bragg-MOKE from anti-dot arrays
Bragg-MOKE measurements performed at a different kind of sample were reported by
Vavassori et al. [14] and Guedes [15]. Instead of arrays of dots or stripes, antidots, i.e.
holes in a continuous thin film, are investigated. In the first article [14] a theory for the
Bragg-MOKE amplitude in saturation for the case of antidot arrays is developed. The
authors find simple expressions for the transverse MOKE signal at diffraction spots, valid
for p-polarized light. The theory described here is also a scalar diffraction theory taking
the form factor of magnetic discs into account. More interesting is the idea presented in
[14] to split the hole array into two sub-films with opposite magnetization direction in
order to model the hole array. In their description the hole array is represented by the
sum of a continuous film and a dot array with opposite magnetization direction. The
authors also report on small changes of the hysteresis loops taken at diffraction spots.
They relate the small changes to blade-like domains forming around the holes in the Fe
film. The small wiggles found around the coercive field of the Fe film are reported to be
independent of the order of diffraction and the direction of the external field. From this
it is concluded that the domains have the same periodicity as the hole array. The domain
state is also described by the addition of several virtual films. In order to illustrate this
a figure taken from [14] is reproduced in Fig. 3.9.
These experiments are extended to hole arrays with elliptical holes in [15]. In this
case the authors find a pronounced change of the slope of the Bragg-MOKE hysteresis
35
3. Magneto-optical Kerr effect of thin films and thin film grating structures
Figure 3.8.: Simulation of transverse Bragg-MOKE measurements from a diffractiongrating. The Kerr intensity is plotted as a function of the magnetization
for different order of diffraction p. The figure is taken from Vial and Labeke
[53].
loops if the elliptical holes are aligned with their long axis perpendicular to the field.
The shape of the Bragg-MOKE hysteresis loops also changes with order of diffraction.
This effect is attributed to domains forming around the coercive field which connect
holes in the 45◦ direction of the square lattice. The width of the domains is of the
order of the grating parameter. In [15] also a domain image is presented confirming
these assumptions. The Bragg-MOKE hysteresis loops are explained essentially with
arguments already given by Geoffroy [11]. A quantitative method is described, which
calculates the magnetic form factor of the domains. For standard MOKE the signal
is proportional to one component of the magnetization, e.g. my . The case of BraggMOKE can be described by replacing my with a magnetic form factor, f , which is the
Fourier-transform of the domain-structure in one unit cell, S, of the (anti)dot array:
f=
Z
S
my exp(ikr)dS,
(3.21)
where r is a position vector inside the area of the unit cell and k is the reciprocal
space vector corresponding to the diffraction order to be analyzed, i.e. k = 2πn/d.
The simulation presented in [15] for the assumed domains using Eq. 3.21 qualitatively
reproduce the measurements.
36
3.4. Bragg-MOKE
Figure 3.9.: Virtual addition of several domains, which was used by Vavassori et al. [14]
in order to construct a model of the observed wiggles in the Bragg-MOKE
curves.
Bragg-MOKE measurements and calculations in polar geometry
The article of Bardou et al. [54] reports measurements of the Bragg-MOKE effect in
polar MOKE geometry from dot arrays with perpendicular magnetic anisotropy. The
grating parameter ranges from 1.1 to 4 µm. The investigated set of samples consist
of densely packed arrays, with the dot diameter close to the grating parameter, and a
loosely packed array with the dot diameter equal to the half of the grating parameter.
The shape of the hysteresis curves is reported not to change at the diffraction spots as
compared to MOKE measurements at the specular reflected beam. However, interesting
features were found with respect to the saturation Kerr rotation:
• The Kerr rotation at the specular spot changes sign for the loosely packed dot
array as compared to the continuous film and the densely packed arrays.
• For a loosely packed array a change of sign of the Kerr rotation is reported for
the specular MOKE curve, the Bragg-MOKE curves exhibit the same sign of the
Kerr-rotation as the continuous film while the specular reflected beam changes
sign.
• The Bragg-MOKE Kerr rotation oscillates with a period of two as a function of
the order for a dot array with the dot diameter half of the grating parameter.
The qualitative explanation offered in [54] deals again with the phase difference between
the light reflected from the background substrate and the magnetic dots. For different dephasing during reflection and subsequent interference, a change of the measured
polarization rotation can be assumed.
The theoretical work of Suzuki et al. [12] tries to build the theoretical basis for the
observations in [54]. For the case of the polar Kerr effect, for large lattice parameters
compared with the wavelength and the magnetic dots being large enough that edge
effects can be ignored, the authors find very useful and concise relations for the BraggMOKE effect. The calculation first assumes a single unsupported dot of arbitrary shape,
for which the retarded potential is evaluated. From this the authors find expressions
for the polar Kerr-effect for diffracted light in non-specular directions, i.e. non-specular
complex reflection coefficients. The most important result is that in the case of the single
ferromagnetic island the polar Kerr effect is independent of the shape of the island. In a
37
3. Magneto-optical Kerr effect of thin films and thin film grating structures
second step the calculation is extended to the case of an array of magnetic unsupported
islands. In this case it is pointed out that nothing changes but the light obviously is
only diffracted in certain angles for which the obtained relations for the unsupported
islands remain valid. In the next step the substrate is taken into account and a phase
factor taking the hight difference into account enters the calculation. For this case only
the polar Kerr effect in specular reflection is explicitly discussed: For certain ratios
of the reflection coefficient of the non-magnetic substrate and the magnetic islands a
resonance-like amplification of the Kerr effect is expected as a function of the filling
factor of the array. Also a change of sign is predicted. This corresponds well to the
observations reported in [54].
What is left to do? - conclusion
As a conclusion from the work in the literature one finds that three effects contribute to
the subject:
• optical constants: The relaxation of the constrain αi = αf leads to off-specular
reflection coefficients, which have to be taken into account.
• interference effects: Interference between substrate and grating may add a significant contribution comparable to the effect of anti-reflection coatings (like ZnS)
commonly used in magneto-optics.
• domain structure: As the diffraction process is mathematically a Fourier transformation of the reflection coefficient distribution, which is depending on the domain
distribution, the Bragg-MOKE curves will depend on the actual magnetization
distribution. As this will change during the remagnetization process, the shape of
the Bragg-MOKE curve will be altered depending on the order of diffraction and
on the particular domain structure.
All Bragg-MOKE studies reported in the literature so far were carried out in the
transverse and in the polar geometry. The longitudinal geometry was not considered
although it is quite popular and has several advantages over the transverse Kerr effect:
• the longitudinal effect measures the absolute Kerr rotation and not the relative
intensity change. A more quantitative analysis may thus be possible.
• the longitudinal effect is often easier to measure.
• for two-fold symmetries (like diffraction gratings) the longitudinal effect enables
to measure with the field perpendicular to the stripes if the diffraction spots are
aligned in the plane of incidence. This geometry (hard axis) is magnetically more
interesting than the easy axis configuration of the transverse Bragg-MOKE measurements.
• the longitudinal Kerr effect can be used to construct a vector magnetometer.
Obviously, there are also disadvantages of using the longitudinal Kerr effect, e.g. the
additional second order effects may complicate the interpretation of the observed hysteresis loops and also a theory for the longitudinal Bragg-MOKE effect may be more
38
3.4. Bragg-MOKE
complicated. This was proved by Bangert et al. [55], who showed that magnetic gratings already change the polarization state of the diffracted light without taking the Kerr
effect into account.
What is also missing in the literature is a systematic study of the influence of different
geometries and materials of diffraction gratings on the observed Bragg-MOKE effects.
Several studies are meaningful:
• Change the stripe widths, but leave the grating parameter constant. In this situation the influence of different diffraction envelopes can be studies while the
angle of diffraction and thus the optical constants are constant for a given order
of diffraction. In addition, different stripe widths will change the influence of the
demagnetizing effects of the edges. This will lead to different domains structures
which can be analyzed.
• Change the material and leave the grating structure unchanged. Two cases can
be distinguished: a) only the material changes but magnetic anisotropies are constant. This may lead to a change of the optical parameters. b) change the magnetic
anisotropy, e.g. isotropic soft magnetic material versus more hard magnetic material with intrinsic magneto-crystalline anisotropy. This will enable to compare
different domain structures.
• Change the grating parameters and leave the relative stripe width constant. This
will change the angle of diffraction while the relative diffraction envelope of the
stripes is constant.
• Construct more complicated gratings, e.g. non-ferromagnetic gratings on ferromagnetic substrates, or ferromagnetic gratings on ferromagnetic substrates. This
will further demonstrate the influence of the different parameters on the BraggMOKE effects.
3.4.2. Some simulations of Bragg-MOKE effects
In this section some simulations of the effects which were described in the literature and
which were reviewed in the previous section is presented. The aim of this section is to
show the effect of simple cases, described with simple formulas, which will be used in
the discussion of the measurements of this thesis.
Fourier components of domains
As pointed out especially in [11] and [15], the magnetic signal obtained from hysteresis
measurements at diffraction spots represents the nth Fourier component of the magnetization distribution. Whereas Guedes [15] tries to model Bragg-MOKE hysteresis
curves using numerical integration methods, Geoffroy analytically calculates the Fourier
components for a simple, one-dimensional case. Both methods completely neglect the
complicated optics of the interaction, which was pointed out in [53]. However, general and qualitative agreement between model and measurements were found. Several
39
3. Magneto-optical Kerr effect of thin films and thin film grating structures
Figure 3.10.: Sketch of the re-magnetization models used for the simulation, see text. d
denotes the grating parameter i.e. the period of the structure, w the width
of the stripes and wd the width of the magnetic domain.
archetypical domain structures can be calculated analytically. All calculations assume
the magneto-optical signal to be given by the real part of the Fourier transformation :
Z
In = Re[
0
d
ikx
m(x)e
dx] =
Z
d
m(x) cos(kx)dx,
(3.22)
0
where d is the grating parameter (i.e. the volume of the unit cell in one dimension), the
wave vector is given by k = 2π
, and m(x) is the magnetization distribution. Note that
d
the magneto-optical signal as given in Eq. 3.22, is a function of the magnetization rather
than the external field, thus the effects of anisotropy and pinnig of domain walls are not
taken into account. These types of curves have been named diffraction hysteresis loops
(DHL) in [11, 53]. Experimentally DHL’s are obtained by plotting the Bragg-MOKE
curve of order n as a function of the normalized magnetization measured in specular
geometry. (Therefore the DHL for n = 0 is a linear function through the origin with
unit slope.)
Two 180◦ domain-wall model This is the model proposed in [11]. In its original form
it covers the case of magnetization reversal along the easy axis (external field parallel
to the stripes) and magnetic sensitivity direction also along the stripes, but can be
generalized to the case covered in this thesis, which is magnetization reversal along the
hard axis with the magnetic sensitivity direction also perpendicular to the stripes.
40
3.4. Bragg-MOKE
w=0.2*d
1
0.5
0
−0.5
−1
−1
n=1
n=2
n=3
0
In
In
0.5
w=0.33*d
1
n=1
n=2
n=3
−0.5
−0.5
0
m
0.5
−1
−1
1
−0.5
w=0.52*d
1
1
w=0.75*d
n=1
n=2
n=3
n=1
n=2
n=3
0.2
0
−0.5
−1
−1
0.5
0.4
0
In
In
0.5
0
m
−0.2
−0.5
0
m
0.5
1
−0.4
−1
−0.5
0
m
0.5
1
Figure 3.11.: Plots of Eq. 3.24 and Eq. 3.25: Model of Geoffroy [11] for the diffraction
hysteresis loop (DHL) of a system with two 180◦ domain walls. The solid
line corresponds to the ascending and the dashed line to the descending
branch of magnetization. The magneto-optical signal at the diffracted spot
of order n, In , is plotted as a function of the magnetization m of the sample.
Different cases for w and n are considered as indicated in the figure.
For saturation M = m(x)dx = ms ∗ w = 1, with the width of the stripe, w, and
ms = 1/w, the saturation magnetization density (see Fig. 3.10(a)). For the descending
branch (M is swept from positive to negative saturation), one domain with the reversed
magnetization direction nucleates in the middle of the stripe, Fig. 3.10(b) (two 180◦
domain walls). Therefore M = ms ∗ w − 2 ∗ ms ∗ wd , where wd is the actual width of the
)w
reversed domain. From this one can easily find wd = (1−M
. The domain pattern can
2
thus be viewed as a superposition of the saturated stripe with two times the domain of
negative magnetization. Therefore the DHL at order n is:
R
In = ms
"Z
w
cos(kx)dx − 2
0
#
w/2+wd /2
Z
cos(kx)dx ,
(3.23)
w/2−wd /2
which can be evaluated to:
d
πwn
cos
In =
πwn
d
"
πwn
πwn (1 − M )
sin
− 2 sin
d
d
2
For the descending branch the reversed domain is given by wd =
can be calculated to
d
πwn
cos
In =
πwn
d
"
!#
(1+M )w
2
πwn
πwn (1 + M )
− sin
+ 2 sin
d
d
2
.
(3.24)
and the DHL
!#
.
(3.25)
41
3. Magneto-optical Kerr effect of thin films and thin film grating structures
Some examples are plotted in Fig. 3.11. From Eq. 3.24 and Eq. 3.24 and Fig. 3.11 some
important conclusions can be drawn:
• Specular case: limn→0 In (M ) = M , as expected
• Saturation: For both the ascending and descending branch one finds: In (M ± 1) =
d
sin( 2πwn
)
2πwn
d
• For small πnw
the ascending and descending branch fall together, and for saturation
d
In = ±1. Thus, for small stripe widths relative to the lattice parameter and small
order of diffraction the DHL is identical to the specular case.
• An interesting case is w = d2 , for which one finds In (M = ±1) = 0 ∀n, see also
Fig. 3.11.
• Because of the symmetry of Eqs. 3.24 and 3.25, the two values of the ascending
and descending branch at M = 0 are equal with opposite sign and for most cases
(see Fig. 3.11) non-vanishing. As m = 0 corresponds to the coercive field, Hc ,
in the specular MOKE curve, it is clear that in general Hc of the Bragg-MOKE
curves is not identical to Hc of the specular measurement.
• All curves display a point-symmetry to the origin.
• The same results, only with exchanged signs, are obtained assuming not a central
reversed domain but two reversed domains at the edges of the stripes.
90◦ edge domains Another straightforward model is to assume domains formed at
the edges of the stripes with the magnetization direction along the stripe (two 90◦
domain walls). For the present geometry this leads to effective ”domains” at the edges
of the stripes with zero magnetization, as the magnetic sensitivity direction is oriented
perpendicular to the stripes, see Fig. 3.10(c). During the magnetization reversal the
central domain changes its width wd . Thus the reduced magnetization is M = wd /w. In
this model the ascending and descending magnetization branch is identical, no splitting
in the DHL occurs3 .
The magneto-optical signal is given by:
In (M ) =
1 Z w/2+wd /2
cos(kx)dx,
w w/2−wd /2
(3.26)
which leads to
d
πnw
πnw
In (M ) =
cos
sin
M .
(3.27)
πnw
d
d
Some examples are plotted in Fig. 3.12. The two extremal cases n → 0 and M = ±1 are
identical to those of the model with two 180◦ domain walls as discussed in the previous
paragraph. The DHL of this model always passes through the origin, In (0) = 0, and all
curves show point symmetry around the origin.
3
Therefore the term diffraction hysteresis loop is not exact and, in fact, in [11] the term DHL is only
used if the two branches of magnetization are distinct. However, for clarity in this thesis DHL
denotes the representation of the Bragg-MOKE curve as a function of the magnetization regardless
on wether splitting occurs or not.
42
3.4. Bragg-MOKE
Figure 3.12.: Plots of Eq. 3.27: Model with edge domains, see Fig. 3.10(c). The
magneto-optical signal at the diffracted spot of order n, In , is plotted as a
function of the magnetization m of the sample. Different cases for w and
n are considered as indicated in the figure.
w=0.2*d
2
0
−1
−2
−1
n=1
n=2
n=3
0.5
In
In
1
w=0.33*d
1
n=1
n=2
n=3
0
−0.5
−0.5
0
m
0.5
−1
−1
1
−0.5
w=0.52*d
0.5
1
w=0.75*d
0.1
0.5
n=1
n=2
n=3
0.05
0
In
In
0
m
n=1
n=2
n=3
0
−0.05
−0.1
−1
−0.5
0
m
0.5
1
−0.5
−1
−0.5
0
m
0.5
1
Figure 3.13.: Plots of Eq. 3.28: Remagnetization via coherent rotation or irregular
domain formation. The magneto-optical signal at the diffracted spot of
order n, In , is plotted as a function of the magnetization m of the sample.
Different cases for w and n are considered as indicated in the figure.
43
3. Magneto-optical Kerr effect of thin films and thin film grating structures
Coherent rotation For magnetic measurements along the hard axis a case without
the formation of domains is possible by the coherent rotation of the magnetization from
the hard axis in saturation into the easy axis in remanence (along the stripe axis).
Because only the magnetization component perpendicular to the stripes is recorded this
leads essentially to a change of the reduced magnetization without changing the domain
width, wd = w, as depicted in Fig. 3.10(d). The same model also covers the case of
irregular domains formed inside the stripe which leads upon averaging to a reduction of
the magnetization.
Because the argument of the cos function in Eq. 3.22 is independent on M , the resulting DHL’s are linear functions of M with zero axis intercept, given by:
d
2πnw
sin
M,
2πn
d
In (M ) =
(3.28)
some examples are plotted in Fig. 3.13. Note that for this case the shape of the BraggMOKE curve is not altered with respect to the specular MOKE curve. Only the Kerr
effect amplitude and sign may change.
One-domain wall configurations In addition to the above models, configurations with
only one domain wall can be considered. The first case is one 180◦ wall moving perpendicular to the wire axis, where the wall can start from both sides of the stripe. Assuming
equal probability of the reversed domain nucleating on either side leads to an average
magnetization profile with zero magnetization on each side of the stripe. Therefore this
model is identical to the model of two 90◦ walls nucleating on both sides and travelling
inwards as described in the above paragraphs.
The same considerations are true for one 90◦ wall. The averaging, however, does
not lead to zero magnetization at the edges. Therefore this case can be described by a
superposition of the model for two 90◦ walls and the coherent rotation model.
Both cases with one domain wall do not introduce new features to the observed DHL.
This situation changes if the assumption is lifted that the domains nucleate with equal
probability on each side. If one assumes one reversed domain starting always from the
same side of the stripe, the calculation leads to solutions in general similar to the ones
depicted in Fig. 3.11, i.e. the two branches are distinct and point symmetry is fulfilled.
However, the actual shape and oscillation periods are different. For the case of one 90◦
wall always starting at the same side, similar results are expected. For both cases it is
assumed that the single domain always starts on one side for the ascending branch and
on the other side for the descending branch. If this degeneracy is also lifted and the
domain always starts to nucleate at on given position, the calculation shows that the
two branches are distinct and additionally the point symmetry is lifted. This leads to
solutions where the end point of one complete magnetization cycle is not identical with
the starting point.
3.4.3. Interference between stripe and substrate
The ferromagnetic stripes do not form a grating in vacuum, but of course they are supported by the substrate and buffer layers. In this cases of reflection gratings the substrate
also forms a diffraction grating with the same lattice parameter as the ferromagnetic
44
3.4. Bragg-MOKE
1
6
(b) E =E =1; θ =0.01°
intensity
1
θB
[°]
K
0.5
0
(a) E1=E2=1; θK=0.01°
0.5
(d) E1=1; φ=8/10 π; θK=0.01°
intensity
θB
[°]
K
−3.1416 0 3.1416
φ [rad]
6
(c) E1=1; φ=8/10 π; θK=0.01°
0
−0.5
K
2
0
−3.1416 0 3.1416
φ [rad]
2
4
1
2
3
4
2
0
1
E2
2
3
E2
Figure 3.14.: Superposition of two polarized plane waves, as calculated in Eq. 3.29. In
B
the top row (a,b) the observed Kerr rotation, θK
, and the intensity is
plotted as a function of the phase shift , E1 = E2 = 2 and θK = 0.001◦
are held constant. In the bottom row (c,d) the observed Kerr rotation and
8
intensity is plotted as a function of E2 , E1 = 1, θK = 0.001◦ and φ = 10
π
are held constant.
stripes, but with a width of wsub = d − wF M . The substrate grating alone produces
diffraction spots at the same positions as the ferromagnetic grating, because the grating
parameter is equal. However, the substrate grating produces a pattern with altered
intensities and with an additional phase shift with respect to the ferromagnetic grating.
The resulting diffraction pattern is a coherent superposition of the two subgratings.
It is instructive to consider first two plain waves with arbitrary intensity, a phase shift
and, in addition, one of the waves exhibits a small rotation of the polarization direction.
Assuming that the main polarization direction is s-polarization the two waves are given
by:
!
!
Φ1,s
exp(iφ)
Φ1 =
= E1
,
Φ1,p
0
!
!
Φ2,s
cos θK
Φ2 =
= E2
.
Φ2,p
sin θK
(3.29)
(3.30)
The coherent superposition leads to the following observed Kerr-rotation and intensity:
<(sin θK )
,
<(exp(iφ) + cos θK )
I = |sin θK + exp(iφ) + cos θK |2 ,
B
θK
=
(3.31)
(3.32)
45
3. Magneto-optical Kerr effect of thin films and thin film grating structures
d=10; a=5.1
d=10; a=2.5
15
E1*0.5
E2
10
Intensity
Intensity
15
5
10
0
0
5
3
3
2.5
2.5
2
2
θB
/ θK
K
B
5
0
−5
θK / θK
E1*0.5
E2
1.5
−5
0
5
−5
0
n
5
1.5
1
1
0.5
0.5
−5
0
n
5
B
Figure 3.15.: Plots of θK
(Eq. 3.34) and intensity for d = 10 and a = 5.1 (left) and
a = 2.5 (right). The amplitude of the substrate-grating is diminished
(factor 0.5) and a phase shift is neglected.
which is plotted for some values in Fig. 3.14. It can be seen that for a phase shift
B
of φ = ±π the observed Kerr rotation, θK
, has a discontinuity (a). For the same
situation the intensity displays a minimum (c). For a given phase shift the observed
Kerr rotation also exhibits a discontinuity when varying the intensities. It can be shown
B
that a discontinuity of θK
is observed if
cos φ = −
E2
cos θK
E1
(3.33)
is fulfilled. More explicit, one finds for the observed Kerr rotation:
B
θK
=
E2 sin θK
.
E1 cos φ + E2 cos θK
(3.34)
B
A simple case is E1 = E2 = 1, if in addition θK and φ is assumed to be small θK
→ θ2K ;
π
B
if θK is small and φ = 2 one finds θK → θK .
The phase shift between the substrate and the grating is due to the path difference and
thus to the height of the stripes h. The phase difference is given in Eq. 3.20 [11]. It can
be seen that the phase difference increases with increasing αi and exhibits a maximum
for the case αi = −αf (Littrow-mounting, i.e. the diffracted beam is reflected back into
the incident beam).
As a rule of thumb one finds that h must be significantly larger than a tenth of the
wavelength in order to find large influence. This condition, however, is rarely fulfilled
as technical problems often impede very high aspect ratios. If the phase shift is smaller,
B
B
2
θK
depends only on the relative amplitudes: θK
≈ θK ( E1E+E
).
2
46
3.4. Bragg-MOKE
If the diffraction pattern of the substrate and the stripes are calculated separately,
the diffracted amplitudes E1 from the substrate and E2 from the ferromagnetic layer
may differ significantly for a given order of diffraction n. This leads to modulations in
the observed Kerr effect as a function of n. Using Eqs. 3.16, 3.14, 3.18, the diffraction
pattern for the stripes and the substrate-grating can be calculated. The result can be
inserted in Eq. 3.34. As example two situations are plotted in Fig. 3.15. For both cases
the phase shift is neglected and the amplitude of the substrate grating is multiplied by
1
in account for different reflection coefficients. The grating parameter is assumed to be
2
10. The left column displays the case for the stripe-width of 5.1 and the right column
for the stripe-width of 2.5. In the top panels the intensity for the two subgratings is
B
plotted separately and in the bottom panels the obtained values of θK
are plotted. It is
clearly seen that the observed Kerr rotation changes significantly and oscillates with a
period of 2 for a ≈ d/2 and with a period of 4 for a = d/4.
47
3. Magneto-optical Kerr effect of thin films and thin film grating structures
3.5. MOKE setup
All measurements in this thesis have been performed in the longitudinal MOKE configuration. As pointed out in Sec. 3.1 this means that the Kerr rotation angle θK has to be
detected in order to measure a magnetic hysteresis curve. The signal of the longitudinal
MOKE is up to first order proportional to the component of the magnetization lying
in the intersection of the scattering and the sample plane. In the following sections the
experimental realization of this concept will be discussed and an example will be given.
Furthermore extensions of the standard setup will be introduced which allow to measure
two orthogonal magnetization components (vector MOKE) and the setup for measuring
the Bragg-MOKE effect will be explained.
3.5.1. Standard setup
The most common application of the longitudinal MOKE is to measure hysteresis curves
of a thin magnetic film as a function of the relative in-plane orientation of the sample
to the external field. In this case the sample can be rotated around its surface normal
(angle χ) while the external field, H, is fixed in the scattering plane defined by the
incoming and outgoing light beams. From a sequence of hysteresis curves as a function
of χ one can deduce the in-plane anisotropy of the ferromagnetic film or the coupling
of ferromagnetic multilayers [56, 57, 58, 59]. This kind of anisotropy measurements
provide the basis for all subsequent investigations of the Bragg-MOKE effect and other
magnetic measurements. Therefore the measurement technique of the standard setup
will be discussed in detail in the following sections. The setup and the measuring
algorithm is based on the work of Th. Zeidler [60, 61]. More technical details can be
found in this references.
Mechanical alignment
Fig. 3.16 displays the geometry of the setup. A HeNe laser is used to shine polarized light
at an angle αi onto the sample, the sample environment with the magnet is depicted in
Fig. 3.17. The light is reflected under an angle αf = −αi and passes into the polarization
detection system, which is pictured in Fig. 3.18. The incident beam and the axis of the
detector system define the scattering plane. The sample is mounted between the poles of
an electromagnet (see Fig. 3.17), such that the field is in the scattering and the sample
plane. The sample can be rotated around its surface normal using a stepper motor. It is
important that during rotation the sample stays aligned perpendicular to the scattering
plane, because otherwise the reflected beam would not pass through the detector system
for any angle χ. Therefore the sample holder attached to the stepper motor can be
adjusted with a three-screw mechanism. The magnet provides a field of ≈ ±2600Oe for
a gap between the poles of 2.5 cm. As can be seen from Fig. 3.16 the accessible range
of αi is reduced for smaller gaps between the magnetic poles, because the magnetic coils
would disturb the optical path from the laser to the sample. For the gap of 2.5 cm
the maximum αi is 45◦ . The Kerr effect depends on αi , reaching a maximum of the
longitudinal MOKE effect typically around 55◦ (for Fe, see Sec. 3.1). From this follows
that higher external fields are only obtained at the cost of smaller Kerr signals.
48
3.5. MOKE setup
magnet coils
sample rotation c
sample
rotator
af
ai
s-polarizer
modulator
p-analyzer
HeNe laser
photodiode
Figure 3.16.: Schematic drawing of the standard setup to measure the longitudinal Kerr
effect.
Optical path
The light emitted from the HeNe laser is polarized perpendicular to the scattering plane
(s-polarized) in a Glan-Thompson prisma before it is reflected at the sample. The
Figure 3.17.: Photography of the sample environment of the MOKE apparatus.
49
3. Magneto-optical Kerr effect of thin films and thin film grating structures
Figure 3.18.: Photography of the detector setup.
reflected light passes through two Faraday setups before the polarization state is determined with an analyzer (crossed with respect to the polarizer: p-state) and a photodiode
acting as a detector. In the two Faraday setups (see Fig. 3.18) the light passes through
a glass-rod upon which an axial magnetic field is applied. The Faraday effect causes the
linear polarization of the light to rotate an angle θF , which is given by
θF = νlHa ,
(3.35)
where ν is the Verdet constant, l is the length of the glass-rod and Ha is the axial
magnetic field. Because the magnetic field is provided by a solenoid which is operated
with a current IF , the Faraday-rotation is essentially proportional to the current IF :
θF = kF IF ,
(3.36)
with kF being an experimental quantity which was determined to be kF = 0.7165
◦
A
.
One of the two Faraday setups is operated with an AC current at a frequency a little
smaller than 1kHz (the modulator) and the other Faraday-setup is operated with a DC
current (the rotator). The rotator is used to compensate the Kerr-rotation of the sample
and the modulator is used to modulate the polarization in order to use lock-in techniques
to detect the polarization state. This will be explained in the next section. Technical
details of the Faraday setup can be found in [60].
The high power (5 mW) of the HeNe laser is not necessarily needed for the standard
measurements, but it is helpful when the light diffracted from gratings or dot-arrays
is analyzed. The detector is a photodiode. A photomultiplier turned out to be less
useful as it is sensitive to the magnetic stray fields of the magnetic coils in the setup.
Measurements of very small Kerr rotations were improved by using the photodiode.
50
3.5. MOKE setup
3.5.2. Measurement method
The optical arrangement can be further analyzed by using the Jones-matrix method
[30]. The electrical field vector of an electromagnetic plane wave can be written as:
~ t) =
E(z,
|Ex | exp i(kz − ωt + δx )
|Ey | exp i(kz − ωt + δy )
!
Ex
Ey
=
!
,
(3.37)
where z is the propagation direction of the light wave and ω, k, δ are defined as usual.
As only the state of the polarization is of importance, the wave can be normalized and
the time and position dependent parts can be separated. The polarization of the wave
can thus be represented by a vector of two complex numbers:
|Ex | exp(iδx )
|Ey | exp(iδy )
~ =
E
!
.
(3.38)
This is the Jones vector of a plane wave. Linearly polarized light with an angle α to the
scattering plane can be expressed as
~ =
E
cos α
sin α
!
.
(3.39)
The general case of an elliptically polarized wave is given by
~ =
E
cos α cos − i sin α sin sin α cos + i cos α sin !
,
(3.40)
with being the ellipticity and α the tilting angle of the major axis of the polarization
ellipse.
Every optical element can be represented by a 2x2 matrix, T , expressing the change
of the polarization by passing light through this element:
~f = T E
~ i.
E
(3.41)
The easiest case is the Faraday rotator, which rotates the polarization for an angle α
and is thus represented by a simple rotation matrix:
TF araday =
cos α sin α
− sin α cos α
!
.
(3.42)
For the case of the Faraday modulator the rotation angle α is an oscillating function:
α = α0 sin ωt. The Jones matrix for every kind of optical element, like polarizers, λ/4plates and so forth, are tabulated in [30]. A sequence of optical elements is expressed
as the product of the representing Jones matrices. In this manner the polarization state
can be calculated for a complex setup. For the actual MOKE setup the product of Jones
matrices of the polarizer, the sample, the Faraday rotator, the Faraday modulator and
the analyzer has to be calculated. In general the sample is represented by the Jones
matrix:
!
r̃pp r̃ps
S=
,
(3.43)
r̃sp r̃ss
51
3. Magneto-optical Kerr effect of thin films and thin film grating structures
which is the magneto-optical Fresnel reflection matrix, where the complex numbers rij
express the ratio of the incident j polarized electric field and reflected i polarized electric
field. The measured Kerr rotation is given by:
ΘK = θK + iK =
r̃ps
r̃ss
(3.44)
From this considerations the intensity function of the setup can be calculated4 :
2
2
rps
rps
1 2
I = rss 1 + 2 − 1 − 2 cos(2δ0 sin ωt)+
2
rss
rss
rps
2
cos(φss − φps ) sin(2δ0 sin ωt)) ,
rss
"
!
(3.45)
where δ = δ0 cos ωt is the modulation amplitude of the Faraday modulator and the rotation of the Faraday rotator is assumed to be zero. In this equation the Kerr-rotation
cos(φss − φps ). The intensity function Eq. 3.45 is plotted
can be identified: θK = − rrps
ss
for a certain set of parameters in Fig. 3.19(a). Obviously two frequency components are
detected: the double fundamental frequency of the modulation and a smaller component
corresponding to the fundamental frequency. For zero Kerr rotation the fundamental
frequency vanishes completely. The panels (b) and (c) of Fig. 3.19 show the power spectrum and the phase of the Fourier transform of the intensity function in (a), calculated
using the FFT method.
In (b) the two peaks corresponding to the fundamental and the double frequency can
be seen. The phase information depicted in (c) is important to detect the sign of the
Kerr rotation. The difference of the two extremal points around the ground frequency
of the phase signal as a function of the Kerr rotation is plotted in Fig. 3.20. This kind
of signal can be obtained with a phase sensitive lock-in amplifier. The figure shows that
the signal S proportional to the Kerr-rotation. The same result can be obtained by an
expansion of Eq. 3.45 into a series of Bessel functions [62]:
"
1
rps
S=
1+
2
rss
2
!
rps 2
− J0 (2δ0 ) 1 −
− 4J1 (2δ0 )θK sin ωt−
rss
#
!
rps 2
2J2 (2δ0 ) 1 −
cos 2ωt + O(sin 3ωt) .
rss
(3.46)
This also proves: S ∝ θK .
It follows that two possible measuring techniques can be used: One can measure
directly the signal S. The advantage is that this is fast, the drawback is that S in Eq. 3.46
also depends on the intensity of the incident laser light. As a result the hysteresis has to
be measured several times and an average has to be taken in order to yield a sufficient
resolution, which will slow down the measurement, or, alternatively, one has to use
intensity stabilized lasers, which are more expensive. In this case the DC Faraday coil
can be omitted. For the direct measurement of S it is more difficult to extract the exact
value of the Kerr rotation, because the signal also depends on the other parameters
4
The calculation has been performed by W. Kleemann, Universität Duisburg, Germany [62].
52
3.5. MOKE setup
Intensity
0.0115
0.011
0.0105
0.01
0
0.002
0.004
0.006
0.008
0.01
2000
f [Hz]
4000
t[s]
0.2
(b)
phase(FFT)
power(FFT)
0.08
0.06
0.04
0.02
0
0
2000
f [Hz]
4000
(c)
0.1
0
−0.1
0
Figure 3.19.: (a) the intensity function Eq. 3.45 for f = ω/2π = 963Hz, δ0 = 2◦ ,
θK = −0.2◦ , θF = 0. (b) the power spectrum of the signal calculated with
FFT, and (c) the phaser of the FFT
0.15
0.1
phase−signal
0.05
0
−0.05
−0.1
−0.15
−0.2
−40
−20
0
Kerr−angle θK [°]
20
40
Figure 3.20.: Phase signal as it can be extracted from the Fourier-spectrum in Fig. 3.19
as a function of the Kerr-rotation.
53
3. Magneto-optical Kerr effect of thin films and thin film grating structures
(intensity, modulation amplitude) which may vary from experiment to experiment and
depend on material parameters of the sample (ellipticity, absorption).
The second modus is to compensate the Kerr rotation for each external field using
the DC Faraday rotator. In an ideal case S in Eq. 3.46 is zero if θK = −θF . Therefore
the signal is independent of the incident intensity and no stabilized lasers have to be
used. Also it is advantageous that the actual Kerr rotation is directly proportional to
the current through the Faraday coil and only has to be calibrated once. The drawback
here is the longer time needed for the compensation at each measuring point. However,
this is the technique used for all measurements in this thesis, a resolution of typically
10−4 ◦ has been achieved.
From Eq. 3.46 also follows, that the signal can be increased by increasing the amplitude of the Faraday modulation, which will also result in a higher resolution of the
measurement. For technical reasons the modulation amplitude is only ≈ 2◦ . Furthermore a high ellipticity of the light to be analyzed will result in a reduced signal and thus
reduced resolution.
Up to here only completely polarized light and ideal polarizers have been assumed.
If the polarizers are non ideal or the sample depolarizes a portion of the light beam, a
constant term has to be added to Eq. 3.46. In this case the compensation can not be
done to zero but only to a minimum of the lock-in signal. Of course this will also reduce
the accuracy of the measurement.
Data acquisition
The complete MOKE setup is computer controlled5 . The program controls the feedbackloop responsible for the compensation measurement, discussed in the previous section,
and controls the external field at the sample position. For each measuring point the
current IF and the external field H, measured in-situ using a calibrated Hall probe,
are written into a file. After one hysteresis measurement is performed, the Kerr rotation is calculated from eq. 3.36 using the calibrated proportionality constant kF . The
programm also controls the movement of the motor responsible for rotating the sample
(angle χ). The software provides an easy to use graphical user-interface as well as a
simple programming interface for specific extensions of the software, like measurements
of the intensity, angle dependent measurements and so forth.
3.5.3. Extensions of the standard setup
The standard MOKE setup described above has been modified in several ways in order
to meet the needs of the different measuring techniques applied.
Bragg-MOKE
The complete setup was mounted onto a goniometer in order to be able to chose the
angle of the incoming beam and the exit beam freely. A double rotary stage with two
independent axes was equipped with computer controlled step motors. The first stage
rotates the laser together with the polarizer and the second stage rotates the sample
5
The software was rewritten using the graphical programming language LabView
54
3.5. MOKE setup
Figure 3.21.: Geometry of the Bragg-MOKE setup. The s-polarized light in perpendicular incidence (αi = 0) is diffracted from the grating. The sample rotation
χ is selected, such that the magnetic field is perpendicular to the stripes.
The Kerr detector can be rotated perpendicular to the plane of incidence
by an angle αf .
environment consisting of the magnet and the sample holder. This setup is necessary
for the Bragg-MOKE measurements in order to measure at different diffracted beams.
The setup is schematically depicted in Fig. 3.21
For most of the Bragg-MOKE measurements the angle of incidence αi was chosen
0◦ , because this is the most symmetric case. Only in this case the exit angles of the
diffracted beams obey the relation αfn = −αf−n . Furthermore, for general symmetry
reasons hysteresis curves measured at an order n must be identical to the Bragg-MOKE
curve at order −n with only the sign changed, if αi = 0◦ . This can be seen with the
following argument:
1. For the geometry described above the incoming laser beam and the sample normal
form a common axis. The setup consisting of the sample, the magnet and the
detector shall be viewed as rotatable around this axis.
2. If the complete setup is rotated around this axis by 180◦ the detected MOKE
signal must be identical (including the sign). This rotation consists of the rotation
of the magnet and the detector.
3. If only the magnet is rotated 180◦ the sign of the measured MOKE signal is
55
3. Magneto-optical Kerr effect of thin films and thin film grating structures
reversed, because the field is reversed.
4. Therefore a rotation of the detector only (which is identical to a transformation
n → −n) must also lead to a change of the sign of the MOKE signal.
However, this highly symmetric configuration results in a smaller phase difference of the
waves diffracted by the grating structure and the waves diffracted from the substrate as
compared to a situation with non-zero incident angle and constant exit angle.
Vector-MOKE
A second magnet was constructed such that the direction of the magnetic field is perpendicular to the scattering plane. This geometry is used to measure the perpendicular
magnetization component as described in Sec. 3.2. In the experiment first the sample
is placed in the longitudinal configuration and the hysteresis curves are measured for
different in-plane rotation angles with increments of ∆χ. Then the setup is changed to
the perpendicular configuration and the sample rotation is repeated. A general source
of error in the setup is the definition of the angle χ: For both the measurement of
the longitudinal and transverse magnetization component the angle has to be redefined,
which adds an uncertainty of ±2◦ to the measurement of χ.
Vector-Bragg-MOKE
At this point it is important to note that combinations of Bragg-MOKE measurements
and Vector-MOKE measurements are drastically limited in the case of grating structures
(more generally: diffracting arrays with a two-fold symmetry):
1. The diffraction pattern of a grating structure lies in the plane of incidence only
if the stripes are oriented perpendicular to the plane of incidence (χ = 90◦ , hard
axis configuration).
2. For other χ the conical diffraction geometry [16] is established, in which the detector has to be tilted out of the plane of incidence. This is technically difficult
for the present setup, as the Faraday-rods and the analyzer are attached to the
detector.
3. Therefore, in the longitudinal geometry only the hard axis orientation is accessible
for Bragg-MOKE measurements. For the perpendicular configuration only the easy
axis orientation (field along the stripes) is accessible probing the perpendicular
magnetization component.
4. Therefore no Vector-Bragg-MOKE can be established with the present setup.
5. If the detector could be moved out of the plane of incidence, it remains to further
investigations if the results of Vector-Bragg-MOKE can be used for calculating a
vector-model of the remagnetization process. As in this case the scattering plane
is not identical to the plane of incidence, the constrains for the longitudinal Kerr
effect are not obeyed and a mixture of longitudinal and transverse Kerr effects can
be anticipated complicating the analysis of the results.
56
3.5. MOKE setup
Figure 3.22.: Example of the diffraction pattern of a magnetic grating. The figure displays a photography of the pattern as is can be observed on a screen.
However, these considerations are not true for square arrays of magnetic dots or comparable systems with a four-fold symmetry of the diffracting structure. In this case
equivalent diffraction spots occur in the plane of incidence as well as perpendicular to
it, therefore no rotation of the sample is necessary when changing from the standard
geometry to the perpendicular geometry.
Intensity-measurements
In addition to these modifications a simple photodiode can replace the complete polarization detection unit. This can be used to measure the integrated intensities of specular
or diffracted beams as a function of the involved angles. This information was useful
for clarifying the Bragg-MOKE effect as will be shown in the subsequent sections. As
an example the diffraction pattern of a diffraction grating is depicted in Fig. 3.22. The
photodiode is directed to the diffracted beams so that the complete visible intensity
around a given spot covers the photo-sensitive area of the detector.
57
3. Magneto-optical Kerr effect of thin films and thin film grating structures
58
4. Sample preparation
In this chapter the sample preparation including thin film preparation techniques and
lithography methods are explained. The resulting grating structures were analyzed using
some microscopy techniques which are also introduced.
4.1. Thin film preparation
The samples discussed in this thesis are laterally structured ferromagnetic thin films.
The samples were prepared by different physical vapor deposition techniques such as
molecular beam epitaxy or re-sputtering. The typical thickness of the ferromagnetic
films discussed here is in the range from 10 to 50 nm, thus no thin film effects like
perpendicular anisotropy or reduced critical temperatures have to be taken into account.
Apart from single films also more complex structures as magnetic multilayers, i.e. stacks
of different materials, are discussed. While the sequence of the layers deposited onto the
substrate is determined during the sample growth, the lateral structure of the sample is
achieved by means of electron beam lithography or other lithographic techniques.
In the following sections the different growth techniques are briefly presented and the
growth procedure of a single Fe layer is used to illustrate the two techniques.
4.1.1. Molecular beam epitaxy
Molecular beam epitaxy (MBE) uses directed beams of atoms or molecules which condense on a substrate. Ultra-high vacuum conditions ( 10−10 mbar) prevent contamination
of the growing film as well as corrosion of the heated source material. The material is
heated with a resistive heater (Knudsen cell) or the material is heated locally with a
high energy electron beam. The system has a variety of materials build in, and a shutter system enables to grow complicated multilayers. During the growth the substrate
temperature is typically enhanced in order to yield a single crystal film growth. The
growth can be monitored using the Reflection high energy electron diffraction (RHEED)
technique, where high energy electrons under grazing incidence are diffracted from the
surface of the sample. The resulting interference pattern gives information about the
quality of the samples [63]. A general review of MBE and related methods can be found
in [64, 56].
Using the MBE method single crystal Fe films were grown as follows: A sapphire
(Al2 O3 ) substrate in the (101̄2) (r-plane) orientation, previously annealed at 1050◦ C,
is first covered with a Nb buffer layer of ≈ 20 nm thickness. The growth temperature
is chosen to be 900◦ C. Afterwards the Nb buffer is annealed at 950◦ C. Subsequently a
59
4. Sample preparation
Cr buffer of about the same thickness is deposited at a substrate temperature of 450◦ C
and is annealed at 750◦ C. The resulting high quality Cr buffer is (001) orientated and
has about the same lattice parameter as the subsequent Fe film. The 20 nm Fe film
is deposited at 300◦ C and is not further annealed to prevent the intermixing of Fe and
Cr at the surface. The Fe film is covered with a 2 nm thin Cr cap layer. In this
configuration the Fe film grows along the (001) direction, has smooth surfaces and is
single crystalline in a single structural domain [65, 63] and exhibits a fourfold magnetocrystalline anisotropy [59]. For a detailed description of the growth of Fe(001) on this
system see [66, 58, 63, 65, 67]; a detailed description of the actual MBE machine used
can be found in [65].
4.1.2. rf-Sputtering
Another technique of fabricating thin films is the sputter technique, where a plasma
between the target material and the substrate is generated. The plasma ions, typically Ar
ions, ballistically extract target atoms from the cathode. Due to the special geometry of
the target, atoms are deposited onto the substrate. A special case is the sputter-process
for insulating material, where an rf driven plasma is used. A general introduction to
sputter deposition techniques can be found in [64]. The sputter process can be reversed,
i.e. it can be used to dry-etch material of the sample. In order to do this, the sample
to be etched is placed at a target position and material is removed from the sample
due to the Ar-ion bombardment. This process is important for the patterning of the
lateral structures as will be explained further below. More detailed information about
the actual sputter machine used can be found in [68, 69]
Some of the samples discussed in this work have been fabricated by sputtering Fe onto
an Al2 O3 (112̄0) (a-plane) substrate at room-temperature. This results in polycristalline
Fe films, basically without magnetic in-plane anisotropy.
4.2. Lithography
In order to produce any kind of lateral structures in the micrometer or even nanometer
scale a lithographic process is used. The main common feature of a great variety of
different lithographic techniques is the production of a mask made from organic material
and subsequently transferring this mask into the desired material. Essentially it is a three
step process:
• Production of the mask: an organic material is exposed to certain radiation, which
can be by visible light, UV-light, x-rays or electrons, changing the chemical properties of the material. In the developing process the exposed portions of the mask
are washed away and the not exposed parts of the film remain.
• thin film preparation: use MBE or sputter techniques to produce a thin film.
• image transfer: the mask has to be transferred into the thin film. There are two
possible ways to do so:
60
4.2. Lithography
Figure 4.1.: Two possible image transfer technologies: A) lift off technique; B) etching.
For further explanation see main text.
– lift-off: The mask is prepared on the substrate. Afterwards the thin film is
prepared onto the mask. In the lift-off process a reactive bath (e.g. aceton) is
used to wash out the remaining mask. The parts of the sample where material
was deposited on the mask are removed. The deposited material remains in
the ditches of the mask. This is illustrated in Fig. 4.1 A). It can be seen that
the lift-off technique essentially forms the negative image of the mask.
– etching: The thin film is prepared first on the substrate. Afterwards the mask
is fabricated on top. Different etching techniques can be used to transfer the
mask into the film. The illustration in Fig. 4.1 B) shows that this procedure
results in a positive image of the mask in the film.
For this thesis several techniques have been used to produce the grating structures. In
the following the mask preparation used is explained and afterwards the particular image
transfer is described.
4.2.1. Electron-beam lithography
Most of the grating samples were prepared using the electron-beam lithography technique. First the samples are coated with a double layer of resist (PMMA = Polymethylmethacrylate) of different molecular weight. The coating is done with a spin-coater: the
sample is rotated at high speed and a drop of PMMA is deposited on the sample (3000
61
4. Sample preparation
electron energy
beam size
beam current
dwell time
step size
write field
magnification
20 kV
100 nm
0.6 nA
0.006 s
0.05 µm
819.22 µm2
75x
Table 4.1.: Typical parameters of the electron-beam lithography setup which were used
to produce a PMMA grating mask for stripes of 5 µm grating parameter and
stripe widths from 1 to 4 µm
rpm for all samples in this thesis for 30 s). Afterwards the layer is tempered at 180◦ C
for 60 mins. This procedure is done twice to obtain two PMMA layers on top of each
other. The bottom layer has a lower molecular weight (200kg/mol) than the top layer
(950kg/mol), so that the bottom layer is more sensitive to the electron irradiation. The
electron irradiation cracks the long organic molecules of the resist and after exposure
the cracked, exposed, parts of the resist film are removed with a developing agent 1 . The
double layer technique results in a structure with a mushroom-like profile as depicted
in Fig. 4.2. This is especially useful for the lift-off techniques as the remover can more
easily solve the remaining PMMA than with a single layer technique. A general review
of electron beam lithography methods can be found in [70]
A commercial scanning electron microscope2 was equipped with a software3 that scans
the focused electron beam over the surface of a sample to form a previously designed
image. The desired image or pattern is designed with a integrated software of the
lithography system. The program is useful to design arrays of elements, like stripes, due
to the ability of the program to automatically multiply one element to form an array.
The software also controls the exposure time of the image. More information on the
particular implementation of electron beam lithography can be found in [71]. As an
example the lithography parameters of one grating structure are given in Tab. 4.1. The
smallest pattern produced with this system were stripes with a width of 0.5 µm.
4.2.2. Other Lithography techniques
Aside from the electron beam lithography technique other techniques exist, which are
used to pattern thin film samples. Particularly optical lithography is the standard tool
in the semiconductor industry. In this case an image of a previously prepared mask is
projected onto the resist layer using an optical setup. This technique is fast and the
mask can be used very often to produce the same pattern in the resist. For scientific use
this is a disadvantage, because the fast and easy procedure is payed by inflexibility in
the mask design. Principally the optical method is limited to the wavelength of the light
1
all chemical agents, the PMMA, the developer, and the remover are supplied by All Resist GmbH,
Berlin, Germany with the trade names AR-P-679-04, AR-600-56 and AR-600-70, respectively.
2
Philips SEM 515 located at the institute in Bochum
3
Elphy Quantum 1.3 of Raith GmbH, Dortmund, Germany
62
4.2. Lithography
used for the image. However, industrial solutions nowadays use UV light and are able
to produce structures smaller than 100 nm. In contrast, state of the art electron beam
facilities can provide structure sizes in the regime of 10 nm. The setups used for this
thesis, however, only provide a resolution of about 500 nm in the case of the electron
beam lithography and 1 µm in the case of optical lithography, which is sufficient as the
goal of this thesis is to perform diffraction experiments with visible light. The samples
prepared by optical lithography were provided by other teams, as will be indicated in
the chapters dealing with the experimental results.
An additional optical lithography method is the interference lithography technique. In
this case the resist is exposed by a laser interference pattern created by the interference of
a split laser beam (wavelength of 457.8 nm ) using a Michelson type interferometer setup
[72]. Samples patterned with this technique have stripes or dots with a lattice parameter
down to 150 nm. The advantage of this technique are the large areas (up to several
square centimeters), which can by patterned and the extremely well defined period of
the grating. The disadvantages are the inflexibility of the pattern and the typically
rounded profile of the developed patches. Samples prepared using this technique also
have been provided by a different team as will by indicated in respective subsequent
sections.
4.2.3. Image transfer
As already discussed above there are principally two different techniques: the lift-off and
the etching technique. Both procedures have been used for some of the samples under
investigation, but it turned out that a combination of both was especially suitable. The
procedure is shown in Fig. 4.2. First the thin film of Fe or other material is produced
by means of MBE or sputter techniques. Afterwards the grating mask is produced using
the double layer PMMA technique as described above. (step (b) in Fig. 4.2) In the next
step a Al2 O3 layer of about the same thickness as the Fe layer is sputtered on the mask.
Subsequently, the Al2 O3 film is patterned by means of the lift-off process. This results
in a Al2 O3 mask on top of the Fe film, which can be used as an etching mask. (step (d)
- (e) in Fig. 4.2). The etching process is done using the reversed sputter process. The
sputter-rate of Fe is approximately the same as for Al2 O3 , such that the sapphire mask
is completely removed when the grooves reach the bottom of the Fe layer.
If the PMMA mask is directly used as the etching mask one might run into problems,
because the thickness of the PMMA layer is much less homogeneous than the thickness
of sputtered Al2 O3 . In addition, the sputter rate of PMMA is approximately ten times
higher than the sputter rate of the metals used as thin films. Therefore it is difficult to
produce elements with a large aspect ratio, i.e. the ratio between height and width of
the elements. A disadvantage of the described combination of etching and lift-off is the
additional image transfer process compared to a simple lift-off or etching procedure. For
each image transfer, the image looses resolution and the edges become more rounded.
However, as will be shown further down, the resulting gratings are of excellent quality
with respect to edge sharpness and uniformity.
In practice, it turned out that the reversed sputter process used to etch the structure
into the thin film was difficult to control with respect to the etching rate. Therefore the
sample was dry-etched only for a short time and afterwards the thickness of the mag-
63
4. Sample preparation
Figure 4.2.: Patterning process used for fabricating array of stripes. (a) the (epitaxial)
Fe-film on the substrate is covered with two different photo-resists, (b) the
resist is exposed and developed, (c) a sapphire mask is deposited on the
resist, (d) via lift-off the resist is removed and the sapphire mask is transferred in the continuous film, (e) the resulting pattern is a negative image
of the exposed portions of the sample.
netic film was checked at an uncovered part of the sample. As it is difficult to measure
thicknesses from small portions of a film in the thickness-range of some nanometers, the
magneto-optical Kerr effect (see section 3.1) was used to monitor the relative thickness
as a function of the etching time. The Kerr signal for thin films is approximately a
linear function of the thickness [73] and, even more important, the goal of the grating
fabrication is to produce gratings with alternating magnetic properties, which are directly monitored with the use of the Kerr effect. As an example Fig. 4.3 shows the
dependence of the Kerr signal as a function of the etching time. The main disadvantage
of this procedure is that after each etching step the sample has to be transferred out of
the sputter machine and has to be adjusted in the MOKE setup. In addition, due to
the difficulties in controlling the etching rates, the procedure has to be repeated several
times.
4.3. Imaging
Imaging of the produced structures is important for the production process as well as
for the interpretation of the results of the different MOKE techniques used. The lateral
dimensions of the stripes are an important input to calculations e.g. the stray field
energy. In addition, the Bragg-MOKE technique will be shown to produce a Fourier
transformation of the domain state of the samples, which needs to be confronted with
the real space domain structure of the gratings. Therefore some domain imaging tools
(magnetic force microscopy and Kerr microscopy) were used. Information of the lateral structure is mainly obtained from scanning electron microscopy and atomic force
microscopy.
64
4.3. Imaging
0.07
0.06
Kerr signal θK [°]
0.05
0.04
0.03
0.02
0.01
0
5
10
15
20
25
30
35
40
total etching time [min]
45
50
55
Figure 4.3.: Kerr signal as a function of the etching time in the sputter machine. The
sample is a 20 nm thick single crystalline Fe film prepared by means of MBE
techniques on a Nb/Cr buffer.
4.3.1. Scanning electron microscopy
Scanning electron microscopy (SEM) is by far the most important and convenient instrument in the field of nano- and micro-lithography. It is used to structure the PMMA
mask and can than be used to analyze the mask and the resulting structure after developing, lift-of or etching. There is a large amount of literature about this standard
analysis tool, e.g. [74]. The actual SEM used is described in [71]. The instrument is
especially useful to study the resist (PMMA) masks, because there is usually a large
contrast between the organic material and the underlying metallic layer due to different
secondary electron yield. SEM was widely used to improve the imaging and developing
parameters of the lithography steps. There are, however, several shortcomings when
thin metallic structures on metallic underlayers should be imaged. The contrast is poor
and the height differences cannot be imaged. In addition, no absolute information of the
height of a structure can be obtained. Therefore, the atomic force microscopy technique
described below was used to study the completed grating samples. As the SEM was
used for the lithography as well as the imaging of the mask it defines the lateral length
scale of all structure discussed in this thesis. Measurements of calibration samples reveal
that this definition is accurate to better than 10% [75].
The SEM used was additionally equipped with a material analysis tool, the energy dispersive x-ray analysis (EDX). The energy of the x-ray spectrum caused by the absorbed
high energy electrons is analyzed. From this information the material composition of a
65
4. Sample preparation
sample can be gained. Interesting here was the possibility to analyze the sample locally
resolved, e.g. between or on the structures (with a resolution of ≈ 1 µm). The etching
processes were optimized using this information.
4.3.2. AFM and MFM
Atomic force microscopy (AFM) is a very versatile tool for analyzing nano- and microstructures. A tip attached to a cantilever is scanned using piezo-electric manipulators
over the sample and the tip to sample force is measured by the deflection of the cantilever. General information about AFM and the used instrument4 can be obtained from
[76]. The instrument is ideal for imaging single stripes or small portions of a grating,
the lateral resolution is ≈ 10 nm. AFM is the only possibility to gain absolute height
information of patterned media, the height resolution is ≈ 0.1 nm. The AFM is very
easy to use and fast to set-up, however, there are several shortcomings:
• one must be very careful with image abberations and artefacts
• the maximal field of sight is 30 × 30 µm2 , which makes it sometimes difficult to
find the interesting structures of the sample.
If the AFM is equipped with a magnetic tip and operated in the non-contact mode, it
can be used to image magnetic domains (magnetic force microscopy, MFM). The stray
field of the tip interacts with the stray field of the domains and domain walls. The MFM
mainly images the perpendicular components of the magnetic stray field in the vicinity
of the sample surface. Therefore it is especially suited to image films with perpendicular
magnetic anisotropy. The interpretation of domain images with in-plane magnetization
is difficult. Several setups at different laboratories were used, as will be indicated in the
following chapters.
4.3.3. Microscopy and Kerr microscopy
The most traditional way to image and analyze the lateral structures and gratings is
optical microscopy. It is by far the fastest and easiest method and important information
during the fabrication-process were gained. In addition, it is the best way to yield
overview images of large areas. The contrast for all systems under investigation is
very good, and special techniques like dark field images and transmitting illumination
can be used to further improve the contrast. Obviously, the resolution is limited and
therefore is is only a complementary method together with AFM and SEM. An optical
microscope can be modified that the sample is illuminated with polarized light and the
polarization of the reflected light is analyzed. In this way a Kerr-microscope can be
constructed, as explained in [4]. For several magnetic grids the domain structure was
imaged by Kerr microscopy in the longitudinal mode5 (see Sec. 3.1 and [4]). The weak
magneto-optical contrast was digitally enhanced by means of a background subtraction
technique [77]. The experimental setup has the option to apply in-plane magnetic fields
in any direction independently of the magneto-optical sensitivity direction. To visualize
4
5
Park Scientific Instruments, AutoProbe CP
the measurements were performed by J. McCord, IFW Dresden, Germany
66
4.3. Imaging
the magnetic domains within the narrow stripes, the highest possible optical resolution,
which is on the order of 0.3 µm for the given visible light illumination, was chosen. For
examples of Kerr microscopy studies at ferromagnetic elements see e.g. [23, 24].
67
4. Sample preparation
68
Part III.
Results and discussion
69
5. Anisotropy of Fe(001)
5.1. Introduction
Measurements of the anisotropy of a single crystal Fe/GaAs(001) film are presented1
as an example for the standard longitudinal MOKE geometry. This is a model system
for ferromagnetic/semiconducting heterostructures with promising potential technical
applications in the field of spin-electronics [1, 79] and has attracted much work in recent years [80, 81, 82, 83]. In this particular case a Fe film was prepared on a twodimensional electron-gas structure 2 [83, 78] which was intended to use for subsequent
magneto-electronic experiments. Here the results from the magneto-optical measurements are highlighted.
The measurements and the interpretation can serve as an example for the investigation
of Fe(001) on other substrate materials, which are more important for the microstructures discussed in the following chapters. The sample discussed here is a sample out of
a large set with different preparation conditions and compositions. Further details can
be found in [83, 78, 84]. However, all samples showed rather similar magnetic behavior.
5.2. Measurements and discussion
A thin Fe film was prepared on GaAs(001) substrate with a buried 2-dimensional electron
gas as described in [78]. The thickness of the Fe layer is tF e = 7 nm.
Ex-situ magnetic measurements at room temperature were carried out using the highresolution longitudinal magneto-optic Kerr effect (MOKE) set up as described in Sec. 3.5.
Magnetic hysteresis curves were measured with different in-plane rotational angles φH
between the in-plane applied field H and the in-plane crystallographic axes of the substrate (Fig. 5.1, inserts). A small coercive field, Hc , of 15 ± 5 Oe is measured for all
directions φH . The φH -dependence of the magnetic remanence (Fig. 5.1, full squares)
indicates the superposition of an in-plane 2-fold (uniaxial) magnetic anisotropy and an
in-plane 4-fold anisotropy, in agreement with earlier reports [80, 82, 83] about Fe films
on bulk GaAs(001). The uniaxial anisotropy has hard axes at φH = 90◦ and 270◦ , i.e.
along the [11̄0] direction of the substrate, and easy axes along [110]. The 4-fold hard
axes are observed at φH = 0◦ , 90◦ , 180◦ and 270◦ , i.e. along [1̄1̄0], [11̄0] etc., and easy
1
This section is based on a part of the article Magnetism and Interface Properties of Epitaxial Fe Films
on High-Mobility GaAs/Al0.35 Ga0.65 As(001) Two-Dimensional Electron Gas Heterostructures [78].
2
The GaAs substrate with a two-dimensional electron gas was prepared in the group of Wieck et al.,
Ruhr-Universität Bochum, Germany. The Fe layer was prepared by means of a MBE process in the
group of Keune et al., Gerhard-Mercator-Universität Duisburg, Germany.
71
5. Anisotropy of Fe(001)
φ
Figure 5.1.: MOKE results of the Fe/GaAs sample: The large figure displays the remanence as a function of the sample rotation with respect to the magnetic field
and the insets show typical hysteresis curves as example. The straight line
is the result of a simulation as is described in the main text.
directions along [01̄0], [100] etc., as expected for bulk bcc Fe. The origin of the 4-fold
anisotropy is the crystalline cubic anisotropy of bcc Fe, while the uniaxial anisotropy is
due to interface anisotropy [82].
In order to describe the behavior of the measured remanence (Fig. 5.1) a simple model
for the in-plan magnetization of a single-crystal thin film was used assuming a coherent
in-plane rotation of the magnetization vector. The total magnetic energy is given by
(see Sec. 2)
E(φ, H) = −µO MS H cos(φ − φH ) +
K1
sin2 (2φ) + KU sin2 (φ − φU )
4
(5.1)
The first, second and third term in Eq. 5.1 describe the Zeeman energy, the cubic
anisotropy energy, and the uniaxial anisotropy energy, respectively. φ, φH and φU are the
angles between the coordinate axis and the magnetization vector MS , the applied field H,
and the uniaxial easy axis, respectively, all oriented in the film plane. The magnetizationversus-field curve is a trajectory on the energy surface E(φ, H) starting at the maximum
72
5.2. Measurements and discussion
MS
K1
6
1.67 · 10 A/m 3.3 · 104 J/m3
KU
1.8 · 104 J/m3
φU
45◦
Table 5.1.: Magnetic parameters extracted from FMR and SQUID measurements [85] of
the Fe/GaAs film, which were used for the simulation of the MOKE results
(Fig. 5.1). The parameters are defined in the text, φU is the angle between
the hard axis of KU ([11̄0]) and the easy axis of K1 ([100]).
applied field (with MS and H aligned), and φ travelling through a local minimum on
the energy surface upon decreasing H. From the values φ(H = 0) on this trajectory the
remanence of the hysteresis loop can be calculated. (In the actual simulation a small
field value of H = 10 Oe prior to field reversal was chosen rather than H = 0). The
magnetization curves for different in-plane angles φH were simulated using this model.
From each simulation the angle φ at a small field (H = 10Oe) is recorded, and the lowfield magnetization, Ml , is calculated according to Ml (10Oe) = MS cos[φ(10Oe) − φH ].
The resulting function Ml = Ml (φH ), normalized to MS , is compared to the experimental
data.
The full-drawn line in Fig. 5.1 is the result of the simulation, where the magnetic
parameters as given in Tab. 5.1 were used (extracted from SQUID magnetometry and
ferromagnetic resonance (FMR) measurements on epitaxial Fe(7.7 nm)/GaAs(001) [86,
85]). The experimental and simulated data are in good agreement (Fig. 5.1). Note, that
the full-drawn line in Fig. 5.1 is not a least-squares fit to the experimental data. However,
the FMR parameters K1 and KU used here are in fair agreement with the corresponding
parameters of epitaxial Fe on bulk GaAs(001) obtained from Fig. 3 in Ref. [82] for tF e
= 53.7 ML (or 7.7 nm Fe): K1 = 3.7 · 104 J/m3 and KU = 1.6 · 104 J/m3 . Obviously, the
magnetic properties of the present epitaxial Fe(001) films on GaAs/Al0.35 Ga0.65 As(001)
heterostructures and of epitaxial Fe(001) films on bulk GaAs(001) (grown under similar
conditions [82]) are of similar quality.
At this point it should be stressed again that the Fe(001) films prepared by MBE techniques used in this thesis for preparing microstructures show very similar behavior with
respect to the coexistence of two- and fourfold in-plane anisotropy. Therefore the present
study can be regarded as a model for the system Fe on Cr(001)/Nb(001)/Al2 O3 (11̄02).
However, the observed two-fold anisotropy for the films used in this thesis is generally smaller, because the two-fold anisotropy decreases with increasing film-thickness.
The latter is a clear sign that the origin of the two-fold anisotropy in Fe on
Cr(001)/Nb(001)/Al2 O3 (11̄02) is a surface or interface effect. Furthermore the uniaxial anisotropy decreases with decreasing miscut of the substrate. The miscut of the
sapphire leades to steps at the surface which results in the observed uniaxial anisotropy
[59]. The magnetism and magnetic anisotropy of Fe on several substrates is the subject
of many publications, however, there is not much literature on Fe(001) on the particular
system Fe(001)/Cr(001)/Nb(001)/Al2 O3 (11̄02).
73
5. Anisotropy of Fe(001)
74
6. Fe-nanowires
6.1. Introduction
In this chapter1 the results of magneto-optical Kerr-effect (MOKE) measurements are
discussed, which were performed on a thin Fe film of 13 nm thickness, which has been
patterned into a periodic arrangement of nanowires by means of optical interference
lithography. The resulting array of nanowires consist of stripes having a width of 150 nm
and a periodicity of 300 nm. MOKE hysteresis loops are measured within magnetic fields
which are aligned in different directions, both parallel and perpendicular with respect
to the direction of the nanowires as well as for various angles in between. A particular
arrangement of the longitudinal Kerr effect measurement allows to identify both the
longitudinal and the transverse component of the magnetization of Fe nanowires (vectorMOKE, see Sec. 3.2). From this both the angle and the magnitude of the magnetization
~ is derived. For a non-parallel alignment of the nanowires with respect to the
vector M
direction of the external magnetic field, the hysteresis loops consist of a plateau region
with two coercive fields Hc1 and Hc2 , which is discussed as resulting from an anisotropic
pinning behavior of magnetic domains in a direction along and perpendicular to the
nanowires.
6.2. Sample preparation and experimental setup
The sample under investigation is a thin Fe film of 13 nm thickness which has been
transformed into a periodic nanowire array by an anisotropic plasma etching process after
film deposition. The Fe film was grown within a UHV-MBE system on a 1.7 cm × 1.7 cm
Al2 O3 (11̄02) (r-plane) substrate onto a 150 nm thick Nb buffer layer. In this case the
Nb layer has a (001) orientation as can be derived from the 3-dimensional epitaxial
relationship between niobium and sapphire [88] and as was revealed by x-ray scattering.
The Fe growth temperature was 120 ◦ C. Unlike the formation of epitaxial Fe(110) on
Nb(110) under the same growth conditions [89], Fe forms a polycrystalline layer on
Nb(001). The polycrystallinity of the Fe film is advantageous as it suppresses the the
intrinsic magnetic in-plane anisotropy, which is important for the present study as will be
discussed below. The Fe film was additionally covered with a 4 nm thick Nb film in order
to protect it against oxidation. Both the thickness and the roughness of the film were
analyzed by x-ray reflectometry. After film preparation, the sample was spin coated with
1
This chapter is based upon the article Magneto-optical study of the magnetization reversal process of
Fe nanowires, see Ref. [87]
75
z [Å]
6. Fe-nanowires
222
111
2
0
1
2
y [mm 1
]
0
]
x
m
[m
0
Figure 6.1.: Surface morphology of a periodic array of Fe nanowires on a Nb/sapphire
substrate imaged with atomic force microscopy. The 3-dimensional surface
graph covers a region of 2.5 × 2.5 µm2 .
a positive-photoresist. The resist was then exposed by a periodic line pattern created
by interference of a split laser beam (wavelength of 457.8 nm) by using a Michelson
type interferometer setup2 [90]. Subsequently, the resist was developed, resulting in a
periodically modulated resist mask on top of the Fe film with a cosine-squared thickness
modulation of 300 nm periodicity. The sample was then ion etched in a conventional rfsputtering system, using the modulated resist as an etching mask. During the sputtering
process, the etching rate of the resist is ≈ 10 times higher than that of the Fe film.
Therefore, the thickness of the resist mask was chosen such that it was initially about
10 times thicker than the metal film. As a result of the etching process, the periodically
structured resist mask is transferred into the underlying Fe film so that finally an array
of Fe wires on top of a Nb buffer was obtained. After lift-off of remains of the resist the
quality of the sample was checked by imaging its surface morphology with an atomic
force microscope (AFM). Fig. 6.1 shows a corresponding AFM image of 2.5 × 2.5 µm2
size. The measurement confirms (i) the regularity of the Fe nanowires having a width of
≈ 150 nm and a periodicity of 300 nm and (ii) that the wires are completely separated
from each other. Note, that the stripes have a sinusoidal shape, which is a consequence
of the lithography technique used.
The sample was measured by means of the standard longitudinal Kerr effect as described in Sec. 3.5, including rotation measurements in order to investigate the samples
magnetic anisotropy. In addition vector-MOKE measurements as described in Sec. 3.2
were carried out for different in-plane rotation angles of the sample. The measurement
geometry is sketched in Fig. 3.4.
2
The resist mask was prepared by S. Kirsch, Institut für Tieftemperaturphysik, Universität Duisburg,
Germany
76
6.3. Experimental results
6.3. Experimental results
6.3.1. Magnetic properties of the continuous Fe film
After film deposition and before plasma etching of the stripes reference MOKE measurements were carried out on the continuous Fe film. The upper panel of Fig. 6.2 presents
a typical MOKE hysteresis loop. The lower panel shows the Kerr rotation measured at
remanence normalized to the Kerr rotation in saturation as a function of the angle of
rotation χ about the surface normal of the Fe film. As can be readily seen from Fig.
6.2, the remanent magnetization amounts to about 88% of the saturation magnetization
and it is almost independent of the angle of rotation. Thus, the overall in-plane magnetic anisotropy is indeed negligible, as was purposely intended by choosing the above
mentioned growth conditions during film deposition. The coercive field of the continuous film is Hc = 250 Oe, as is typical for a poly-crystalline Fe film containing many
grain boundaries. Therefore, the Fe film is a good candidate for the investigation of an
induced magnetic anisotropy caused by a lithographic patterning process.
6.3.2. Magnetic properties of the Fe nanowire array: longitudinal
component
Fig. 6.3 shows three typical MOKE hysteresis loops taken for the array of Fe nanowires
with different in-plane angles χ as measured in the longitudinal configuration. The hysteresis loop in (a) has been recorded with the external magnetic field oriented parallel to
the wires. In this case, an almost squared hysteresis loop with a relatively large coercive
field Hc is found, which represents the typical behavior of a sample when magnetically
saturated along an easy axis of the magnetization. The coercive field Hc = 250 Oe indeed matches the value determined for the continuous film. However, the absolute values
of the Kerr rotation in saturation are reduced when compared to the unpatterned film.
This simply results from the fact that less material contributes to the measured signal
in the case of the patterned film. Note also, that the slope of the hysteresis loop at Hc
is less steep for the structured sample in comparison to the unpatterned film. This indicates that the magnetization reversal is dominated by a different switching mechanism.
Moreover, the patterning process obviously leads to an increase of the absolute value of
the remanence in the easy direction (parallel to the wires) suggesting a suppression of
domain formation.
Fig. 6.3(b) shows the corresponding hysteresis loop for the Fe nanowire sample after
the sample was rotated by 90◦ , i.e. when the Fe wires were oriented perpendicular to
both the external field and the plane of incidence. Here, a typical hard axis hysteresis
loop is obtained. The coercive field reduces to Hc = 50 Oe and the saturation field
increases to values beyond 1000 Oe, which is the maximum magnetic field value of the
present experimental setup.
Fig. 6.3(c) shows a MOKE hysteresis loop taken for an intermediate angle of rotation
χ = 45◦ . The hysteresis loop clearly exhibits a step-like behavior, when the direction of
the external field is reversed after saturation.
Some characteristic features of the hysteresis loops shown in Fig. 6.3 are presented
in Fig. 6.4 as a function of the angle of rotation χ. Fig. 6.4(a) shows the remanent
77
6. Fe-nanowires
0.04
θ [°]
0.02
K
0
−0.02
−0.04
−1500
−1000
−500
0
H [Oe]
500
1000
1500
0.8
0.7
θK
Rem
/ θK
Sat
0.9
0.6
0.5
0
45
90
χ [°]
135
180
Figure 6.2.: MOKE hysteresis loop of the unpatterned thin Fe film (upper panel). The
lower panel depicts the results of MOKE hysteresis loop measurements as
rem
a function of the angle of rotation of the unpatterned sample, where θK
sat
as measured at remanence is normalized to θK
as measured at saturation,
and plotted as a function of the angle of rotation χ, which is a measure of
the magnetic anisotropy.
rem
sat
Kerr signal θK
normalized to the Kerr signal at saturation θK
as a function of χ,
which yields information about the squareness of the hysteresis loops. According to
Fig. 6.4(a) the remanent Kerr signal is reduced significantly at certain angles χ without
reaching zero-values signifying the hard axis orientations (around 90◦ and 270◦ ). For
the corresponding angles χ along the easy axis orientations (0◦ and 180◦ ) the ratio
rem
sat
θK
/θK
measures almost unity. Similar behavior is obtained when the coercive field
Hc is plotted as a function of the angle of rotation χ in Fig. 6.4(b), from which two
pronounced minima of χ at around 90◦ and 270◦ are found. The small local maxima
within the minima at 90◦ and 270◦ reflect small changes of the anisotropy which are
also found in the continuous film at these angles (compare the lower panel of Fig. 6.2).
The hysteresis loops measured for intermediate angles of rotation χ exhibit steps, which
occur at characteristic magnetic fields Hc2 . Fig. 6.4(c) shows Hc2 as a function of the
angle of rotation χ. Hc2 exhibits maxima at the hard axes directions at 90◦ and 270◦ .
As mentioned above, hysteresis loops which are measured at intermediate angles of
rotation show certain steps at magnetic fields Hc2 . Some of those hysteresis loops are
78
6.3. Experimental results
Figure 6.3.: MOKE hysteresis loops as measured in the longitudinal configuration. (a)
the external magnetic field is oriented parallel to the nanowires, χ = 0◦ ; (b)
the magnetic field is oriented perpendicular to the nanowires, χ = 90◦ ; (c)
the nanowires are rotated by an angle of 45◦ with respect to the direction of
the external field. The insets depict correspondingly the orientation of the
stripes with respect to the directions of both the external magnetic field H
and the reflected laser beam.
displayed again in Fig. 6.5 on an enlarged scale, where only half of the hysteresis loops
as measured in longitudinal configuration are plotted. Obviously, Hc2 almost coincides
with Hc1 when measured along the easy axis (χ = 0◦ see Fig. 6.3(a)) and starts to
increase when the sample is rotated away from the easy axis (see Fig. 6.5). Finally, Hc2
l
fades out into the region of saturation when χ = 90◦ . Note that θK
values in between
Hc1 and Hc2 increase from almost zero to the saturation Kerr rotation which is measured
in the direction of the hard axis. Note also that even the hysteresis loop measured at
χ = 0◦ (Fig.6.3(a)) reveals a small step in the vicinity of the coercive field.
From the results of longitudinal MOKE measurements with the magnetic field direction within the plane of incidence one may indeed conclude that patterning of the thin
Fe film into an array of nanowires induces a strong uniaxial anisotropy, which results
79
6. Fe-nanowires
1
θrem
/ θsat
K
K
0.8
0.6
0.4
0.2
0
(a)
45
90
135
180
225
270
315
360
90
135
180
225
270
315
360
90
135
180
χ [°]
225
270
315
360
300
(b)
Hc1 [Oe]
200
100
0
0
45
600
(c)
Hc2 [Oe]
500
400
300
0
45
Figure 6.4.: Results from hysteresis loop measurements at different angles of rotation χ:
(a) Kerr rotation as measured at remanence normalized to the Kerr rotation
as measured at saturation, (b) coercive field Hc1 and (c) the position of steps
in the hysteresis loops measured at intermediate angles of rotation between
the hard axis and the easy axis (Hc2 ).
from the shape anisotropy of Fe nanowires. The saturation field measured with MOKE
along the hard axis is rather high, exceeding 1000 Oe.
6.3.3. Magnetic properties of the Fe nanowire array: transverse
component
As discussed above, also the transverse component of the magnetization vector was determined by Kerr effect measurements. Corresponding hysteresis loops for three directions
of the external magnetic field relative to the direction of the Fe nanowires are shown in
Fig. 6.6. The hysteresis loop reproduced in Fig. 6.6(a) was measured within a magnetic
field direction parallel to the Fe nanowires, i.e. along the easy axis of the magnetization.
Ideally, within this configuration the measured Kerr rotation should remain zero unless
80
6.3. Experimental results
0.03
χ=99°
χ=117°
χ=154°
0.025
↑ Hc2(99°)
θl [°]
0.02
K
↑ Hc2(117°)
0.015
0.01
↑ Hc2(154°)
0.005
0
−200
0
200
400
H [Oe]
600
800
1000
Figure 6.5.: Evolution of the characteristic magnetic field Hc2 as a function of the angle
of rotation χ. Different line shapes represent typical hysteresis loops as
taken in the longitudinal configuration for intermediate angles of rotation.
χ = 99◦ is close to the hard axis direction, i.e. when the external field is
applied perpendicular to the nanowires.
certain components of the magnetization are directed along the plane of incidence during
the magnetization reversal process. As can be seen from Fig. 6.6(a) the measured Kerr
rotation is indeed almost zero. A small signal is seen with a maximum value at zero
field. The origin of this behavior will be discussed below. The coercive field is the same
as measured for the easy direction in the longitudinal configuration. Fig. 6.6(b) shows
the hysteresis loop for a magnetic field pointing along the hard axis direction, i.e. in a
direction perpendicular to the wires and the plane of incidence. In this configuration
the measured Kerr rotation is larger than in the previous case and non-zero up to the
highest field values. From this behavior one can infer that the sample could not be
magnetically saturated in the hard axis direction, in agreement with the result shown
in Fig. 6.3 (b). Both measurements therefore represent minor loop measurements.
Fig. 6.6 (c) shows the transverse component of the Kerr rotation measured for an
angle of rotation of χ = 45◦ . Also here, the hysteresis loop exhibits steps similar to
the ones shown in Fig. 6.3(c). Note, that the coercive field, denoted here as Hc2 , is
almost identical with the switching field Hc2 as defined in the previous section for the
longitudinal magnetization component.
According to the MOKE hysteresis shown in Fig. 6.6(a), the transverse magnetization
component never reaches zero even at high magnetic fields, indicating that the sample is
never fully saturated in the present setup. This also corresponds to the hysteresis loop
measurements in the longitudinal configuration, for which always a small final slope in
the high-field region was found, even in the easy axis configuration (Fig. 6.3(a)). Reason
for this may be the presence of canted spins, which may result from the rough surface of
the sample. In addition, this effect may be pronounced within our measurement through
the high surface sensitivity of MOKE.
81
6. Fe-nanowires
Figure 6.6.: MOKE hysteresis loops with the magnetic field oriented perpendicular to
the plane of incidence, which probes the transverse component of the magnetization (a) the array of Fe nanowires is aligned parallel to the direction of
the external magnetic field (easy axis configuration); (b) the magnetic field
is applied perpendicular to the array of nanowires (hard axis configuration);
(c) the field is oriented at 45◦ to the Fe nanowires (intermediate configuration). The insets depict the orientation of the stripes with respect to both
the external magnetic field H and the reflected laser beam. Note that the
θK -scale is divided by two in comparison with what is shown in Fig. 6.3 for
the longitudinal configuration.
6.4. Analysis and discussion
The results of MOKE hysteresis loop measurements as a function of the sample rotation
and expressed in terms of the longitudinal and transverse component of the magnetization can be combined in order to calculate the angle and magnitude of the magnetization
vector by using equations 3.11 and 3.12. The corresponding results of such an analysis are displayed in Fig. 6.7, which provide an overview of the magnetization reversal
process of the Fe nanowires. Here, two ideal cases can easily be distinguished. If the
82
6.4. Analysis and discussion
→
←
360
φ [°]
|M| / |M|sat
270
180
90
1
(a) χ=0°
0.8
0.6
(b)
0.4
0.2
0
−1000
0
→
←
360
φ [°]
180
(c) χ=90°
0
1000
0
1000
0
H [Oe]
1000
1
|M| / |M|sat
270
90
0
−1000
1000
0.8
0.6
(d)
0.4
0.2
0
−1000
0
→
←
360
1
φ [°]
|M| / |M|sat
270
180
90
0
−1000
1000
(e) χ=45°
0.8
0.6
(f)
0.4
0.2
0
−1000
0
H [Oe]
1000
0
−1000
Figure 6.7.: Conversion of the results of MOKE measurements from Fig. 6.3and 6.6 by
using equations 3.11 and 3.12 into the angle of the magnetization vector
(left column) and the magnitude of the magnetization (right column). (a)
and (b) display the results for the external magnetic field directed along the
~
Fe nanowires (the easy axis); (c) and (d) show the results as obtained for H
perpendicular to the direction of the Fe nanowires (the hard axis); in (e) and
(f) the magnetic field is tilted by 45◦ with respect to the direction of the Fe
nanowires. The solid lines represent the measurements with the magnetic
field increased from negative to positive fields, the dashed lines correspond
to measurements with the magnetic field decreased from positive to negative
field values.
angle of the magnetization vector changes without changing the magnitude |M |, then
the magnetization reversal occurs via coherent rotation. On the other hand, if the angle
of the magnetization vector remains constant within a certain magnetic field range but
the magnitude |M | changes, then magnetic domains are formed.
Firstly, the magnetization reversal with the magnetic field oriented parallel to the Fe
nanowires (Fig. 6.7(a,b)) is discussed, i.e. when the magnetic field is directed along the
easy axis of the nanowires having a two-fold magnetic anisotropy. When the magnetic
field is swept from negative to positive values, the magnetization vector within the Fe
nanowires remains parallel to the direction of the external field (φ = 0◦ in Fig. 6.7(a)),
83
6. Fe-nanowires
until the magnetic field reaches a value which matches the coercive field for the easy
e.a.
~ suddenly flips from 0◦ to 180◦ and further on remains
axis Hc1
≈ +250 Oe. Then, M
parallel to the external magnetic field direction up to the highest magnetic field values.
Upon reduction of the magnetic field value and subsequent reversal of the magnetic field
direction the magnetization vector again rotates but in the opposite sense and reaches
e.a.
φ = 360◦ after passing the coercive field at Hc1
≈ −250 Oe. Both the starting point
~
(φ = 0◦ ) and the end point (φ = 360◦ ) are degenerate. The plot of the magnitude of M
sat
~
normalized to the saturation value M
in Fig. 6.7(b) shows a decrease of |M | over a
e.a.
relatively wide magnetic field range, reaching almost zero around ±Hc1
. This indicates
e.a.
that the magnetization reversal process around Hc1 is dominated by domain formation
within a wide magnetic field range. The shape of |M |(H) suggests that a number of
domains nucleate along the nanowires, rather than just one single domain wall travels
along each wire during the magnetization reversal process. A single domain would result
in a much more narrow range over which |M | switches.
The situation changes when the magnetic field is directed perpendicular to the
nanowires, i.e. along the hard axis direction, as shown in Fig. 6.7 (c and d). Here, the
magnitude of the magnetization |M | starts to decrease as soon as the magnetic field is reduced from its maximum value. At the same time the magnetization starts to change its
h.a.
direction. When approaching the coercive field for the hard axis orientation Hc1
= ±75
Oe, the magnitude of the magnetization drops in an even more rapid fashion, albeit it
also does not reach zero as in the previous case. Thus, for this case we assume that
a number of domains are formed along the wires immediately after leaving the saturation state, most likely as a result of large demagnetizing fields. Coherent rotation can
clearly be excluded to be the relevant magnetization reversal mechanism, however, some
rotational processes still remain to be active close to the coercivity. Around Hc1 the
angle of the magnetization changes rapidly (Fig. 6.7(c)). This indicates the switching
of domains which are oriented parallel to the field but perpendicular to the wire axis.
The hysteresis loops as measured in both the hard and easy directions also differ
significantly from each other in terms of the coercivity. Whereas Hc1 for the easy axis
loop more or less reproduces the value of the unpatterned film, its value is strongly
reduced for the hard axis loop. Obviously, patterning of the continuous Fe film does not
change the pinning behavior of domains along the wire axis. However, for the hard axis
orientation the domain walls parallel to the external field only have to move from one
side of the wires to the other side, which is probably aided by the rounded shape of the
wires, thus, leading to much smaller coercivities.
When the magnetic field is oriented 45◦ with respect to the Fe nanowires, we expect
to obtain a combination of the easy and hard axis behavior. Both the angle and the
magnitude of the magnetization vector are plotted for this case in Fig. 6.7 (e and
f). Starting at saturation within negative magnetic field values towards zero magnetic
~ changes from 0◦ to almost 45◦ , while the magnitude of M
~ only
field, the direction of M
slightly decreases. Within this first part of the hysteresis loop the situation is comparable
to what is found for the hard axis loop. Magnetic domains are nucleated along the
nanowires, which are oriented in only one direction, because the external magnetic field
has a component parallel to the wires. Close to Hc1 the remaining domains, which are
still oriented parallel to the external field, switch by 180◦ resulting in a decrease of the
84
6.5. Conclusions
~ . A further increase of the magnetic field leads to a meta-stable situation
magnitude of M
within the plateau region between Hc1 and Hc2 . Here, some of the domains appear to
be oriented parallel to the field and some domains are oriented parallel to the stripes.
~ is almost perpendicular to
This can be verified from the fact that the orientation of M
1
~
the stripes, and the magnitude of M is reduced to ≈ 2 of its saturation value. Only
after application of a magnetic field larger than Hc2 does the magnetization switch into a
direction parallel to the magnetic field direction and the magnitude regains its saturation
value. The domains which are oriented parallel to the nanowires switch their direction,
comparable to the case of the easy axis loop.
The above interpretation for the three main orientations of the Fe nanowires with
respect to the direction of the magnetic field (0◦ , 90◦ and 45◦ ) is consistent with measurements taken in various other in-plane directions. Fig. 6.4(c) displays Hc2 as a
function of the direction of the external field. Hc2 increases when approaching the hard
axes directions (90◦ and 270◦ ), and it is small otherwise. This can be explained by
the corresponding component of the external magnetic field that drives the switching
process. For instance, close to the hard axis orientation the external magnetic field component along the stripes becomes small, therefore, a higher external field is required to
switch the domains which are oriented parallel to the nanowires. This can be expressed
approximately through the relation: Hc2 = Hce.a. / cos χ.
The true domain structure of the nanowires is likely to be affected by the morphology
of the stripes. In particular, it might by affected by their waviness and by the fluctuations
of the wire-width and wire-height. Future magnetic force microscopy investigations
or spin-polarized scanning tunneling microscopy [91] will help to identify the domain
structure of the Fe nanowires. It should be mentioned that in contrast to the data
presented in [3], here no evidence for a dipolar interaction between the wires could be
discerned. This, however, most likely results from the fact that the separation of the
nanowires is of the same magnitude as compared to the wire-width.
6.5. Conclusions
With an analysis of magneto-optical hysteresis loop measurements taken for both longitudinal and transverse configurations the complex magnetization reversal behavior of
an array of Fe nanowires has been demonstrated. The Fe nanowires were fabricated by
patterning a continuous polycrystalline Fe film by using optical interference lithography.
Before patterning, essentially no in-plane magnetic anisotropy could be detected for the
continuous Fe film. The pattering process induces a uniaxial magnetic anisotropy with
the easy axis of magnetization directed parallel to the nanowires.
Due to the imposed shape anisotropy of the Fe nanowire array it is expected that
the magnetization reversal behavior is quite different for a magnetic field orientation
parallel and perpendicular to the nanowires. The magnetization reversal properties of
the nanowires were investigated by analyzing both the transverse and the longitudinal
components of the magneto-optical Kerr rotation. For a magnetic field orientation parallel to the nanowires an easy axis behavior of the magnetization reversal is observed
which is dominated by domain nucleation. For the magnetic field oriented perpendicular
to the nanowires the magnetization reversal is found to be also dominated by domain
85
6. Fe-nanowires
nucleation but with different pinning potentials for magnetic domains magnetized either
parallel or perpendicular to the wire-axis, leading to smaller coercivities for the latter
ones. For intermediate orientations of the magnetic field with respect to the direction
of nanowires (between hard and easy axis) a superposition of both magnetization reversal processes is found, resulting in a plateau-like region in the hysteresis loop due to a
metastable domain configuration. In summary, the analysis shows the potential usage of
a vector magnetometer in order to unveil the complex magnetization reversal processes
of nanoscaled magnetic wire-arrays.
Because of the small grating parameter no Bragg-MOKE experiments could be carried
out on this sample. The laser wavelength is too large. This problem does not exist for
radiation with smaller wavelengths like thermal neutrons or soft x-rays, both being also
very powerful magnetic probes. Measurements using polarized neutron reflectivity were
performed at this particular grating structure [92]. In this experiments hysteresis loops
were measured at the specular reflection and at a first order Bragg diffraction spot of
the lateral periodicity. The results of the neutron hysteresis loops agree well with the
presented MOKE measurements. However, for some orientations of the in-plane rotation
(angle χ) deviations of first order neutron hysteresis loop are detected compared with
zeroth order MOKE curve. Deviations between specular measurements and magnetic
measurements at Bragg-spots are generally expected for the case of Bragg-MOKE as
will be discussed in the following sections. Equivalent effects for neutron-scattering are
thus also probable. The results will be published [92].
86
7. CoFe grating
7.1. Introduction
In this chapter1 studies of the magnetization reversal of a laterally structured CoFe film
is reported using the magneto-optical Kerr effect in specular and diffraction geometry.
The lateral structure consists of 90 nm thick and 1.2 µm wide Co0.7 Fe0.3 stripes with
a grating period of 3 µm. The magnetization vector is measured by means of the
vector-MOKE technique for different orientations of the sample with respect to the field
directions. In addition, Kerr-microscopy was used for visualizing the domain state. Due
to the high aspect ratio of the individual stripes, the remagnetization process of the
stripe array is dominated by a single domain state over most of the field range. For the
easy axis direction a nucleation and domain wall movement is observed at the coercive
field. However, for all other orientations of the stripe array the magnetization reversal is
dominated by a coherent magnetization rotation up to the coercive field. For the hard
axis orientation the coherent rotation is complete. In addition, the measurements in
diffraction geometry (Bragg-MOKE) reveal small contributions from closure domains,
which were not detected using the standard methods.
7.2. Sample preparation
The basis of sample preparation for the present study are Co0.7 Fe0.3 thin polycrystalline
films grown by DC magnetron sputtering. A polycrystalline film is preferred to average the intrinsic magneto-crystalline anisotropy. The films have no further intentionally
induced anisotropy, such that their anisotropy is dominated by the shape anisotropy.
CoFe films with the quoted composition have been shown to exhibit the largest tunneling magnetoresistance [94] and have been introduced as electrodes in magnetic tunnel
junctions for magneto-electronic devices [95].
Here2 a 20 x 10 mm2 Al2 O3 (11̄02) substrate is used. The sample was spin coated with
Novolak photoresist, which was exposed by 442 nm light in a scanning laser lithography
setup and developed afterwards. A layer stack consisting of 5 nm Ta, 90 nm Co0.7 Fe0.3
and a 5 nm Ta protection layer was deposited onto the patterned photoresist. Finally,
the photoresist was removed via lift-off. The described procedure resulted in Co0.7 Fe0.3
stripes of 1.2 µm width and a grating parameter of d = 3 µm as can be seen from
1
This chapter is partially based on an excerpt of the article CoFe-stripes: magnetization reversal study
by polarized neutron scattering and magneto-optical Kerr effect [93].
2
The sample studied in this chapter was supplied by K. Rott and H. Brückel, Universität Bielefeld,
Germany
87
7. CoFe grating
Figure 7.1.: Surface topography of the array of Co0.7 Fe0.3 stripes obtained with an
atomic force microscope shown in a 3-dimensional surface view. The displayed area is 20 x 20 µm2 .
the AFM picture in Fig. 7.1. The grating structure covers the complete area of the
substrate.
7.3. Remagnetization process of the CoFe-grating
7.3.1. Results from MOKE measurements
The left row of Fig. 7.2 shows four typical longitudinal MOKE hysteresis loops taken from
the CoFe stripes with different in-plane angles χ. The hysteresis loop in (a) corresponds
to an external magnetic field oriented parallel to the stripes. In this case, an almost
square hysteresis loop is found, which represents the typical behavior of a sample when
magnetically saturated along an easy axis of the magnetization. The coercive field is
Hc = 140 Oe. The coercive field increases to Hc = 200 Oe and Hc = 320 Oe for
intermediate angles of rotation of 45◦ (c) and 63◦ (e), respectively. Fig. 7.2 (g) shows
the corresponding hysteresis loop for the CoFe stripe array oriented perpendicular to
both the external field and the plane of incidence. Here, a typical hard axis hysteresis
loop is obtained.
From the results of longitudinal MOKE measurements with the magnetic field direction within the plane of incidence follows that patterning of the thin CoFe film into
an array of stripes induces a strong uniaxial anisotropy, which results from the shape
anisotropy of CoFe stripes. The saturation field measured with MOKE along the hard
axis is rather high, exceeding 1000 Oe.
As discussed above, also the transverse component of the magnetization vector was
determined by Kerr effect measurements. Corresponding hysteresis loops for four directions of the external magnetic field relative to the direction of the CoFe stripes are
88
7.3. Remagnetization process of the CoFe-grating
θLK (long.)
0.02
θTK (trans.)
(a) χ=0°
(b)
(c) χ=45°
(d)
(e) χ=63°
(f)
(g) χ=90°
(h)
0
−0.02
0.02
0
θK [°]
−0.02
0.02
0
−0.02
0.02
0
−0.02
−3
−2
−1
0
1
2
3 −3 −2
H [kOe]
−1
0
1
2
3
Figure 7.2.: MOKE hysteresis loops measured in the longitudinal configuration (left figures) and in the transverse configuration (right figures). For both configurations in (a) and (b) the external magnetic field is oriented parallel to the
stripes (easy axis configuration); in (c) and (d) the stripes are rotated by 45◦
and in (e) and (f) by 63◦ with respect to the direction of the external field;
in (g) and (h) the magnetic field is oriented perpendicular to the stripes,
(hard axis configuration).
shown in the right column of Fig. 7.2. The hysteresis loop reproduced in Fig. 7.2(b) was
measured with a magnetic field parallel to the CoFe stripes, i.e. parallel to the easy axis.
Ideally, within this configuration the measured Kerr rotation should remain zero unless
components of the magnetization lie in the plane of incidence during the magnetization
reversal process. As can be seen from Fig. 7.2(b), the measured Kerr rotation is indeed
almost zero, thus no rotation of the magnetization occurs in this configuration. Fig. 7.2
(d) and (f) show the transverse component of the Kerr rotation measured for angles of
rotation of χ = 45◦ and χ = 63◦ , respectively. Fig. 7.2 (h) shows the hysteresis loop for
a magnetic field pointing along the hard axis direction, i.e. in a direction perpendicular
to the stripes and the plane of incidence. In these directions the measurements of the
transverse magnetization component always proves a rotation of the magnetization away
from the magnetic field.
From these measurements important conclusion concerning the remagnetization pro-
89
7. CoFe grating
1
|θK
Rem
Sat
/ θK |
0.8
0.6
0.4
0.2
0
−90
−45
0
45
90
χ [°]
135
180
Rem
Figure 7.3.: Kerr rotation in remanence, θK
, normalized to the saturation Kerr roSat
tation, θK , as a function of the sample rotation. The closed circles are
the measurements in the longitudinal configuration and the open squares
denote the measurements in the perpendicular configuration. The solid line
is a plot of the model according to Eq. 7.1 and the dashed line is a plot of
Eq. 7.2
cess of the CoFe stripes can be drawn. In general, two ideal cases can easily be distinguished: If the angle of the magnetization vector changes without changing the magnitude |M |, then the magnetization reversal occurs via coherent rotation. On the other
hand, if the angle of the magnetization vector remains constant within a certain magnetic
field range but the magnitude |M | changes, then magnetic domains are formed. The
hysteresis curves in Fig. 7.2 exhibit no discontinuity besides the jump at Hc , therefore
the magnetization process must be smooth, either dominated by rotation or by domain
nucleation. If only coherent rotation takes place, the magnetization in remanence must
be directed along the easy axis of the twofold anisotropy, i.e. along the stripes, without
Rem
forming any domains. In this case the longitudinal Kerr rotation in remanence, θK
must follow:
Rem,L
θK
= | cos χ|,
(7.1)
Sat
θK
and, correspondingly, the Kerr rotation angle in perpendicular configuration must follow:
Rem,T
θK
= | sin χ|.
Sat
θK
(7.2)
The measurements perfectly obey these rules as is depicted in Fig. 7.3. Therefore it is
90
7.3. Remagnetization process of the CoFe-grating
3000
2500
Htc [Oe]
2000
1500
1000
500
0
0
45
90
χ [°]
135
180
Figure 7.4.: Coercive field taken from the measurements in the perpendicular geometry
as a function of the sample rotation χ. The circles denote the measurement
and the solid line is a plot of Eq. 7.3.
clear that the magnetization process is governed by coherent rotation processes. Furthermore, the coercive field should also show such a simple behavior: Along the easy
axis the magnetization switches by 180◦ if the external field is equal to the pinning field
of the domains. If the sample is rotated away from the easy axis, the component of the
external field driving the switching process is reduced. Therefore one finds:
Hc (χ) =
Hce.a.
,
cos χ
(7.3)
which is plotted in Fig. 7.4. Clearly the plot confirms this notion. A similar behavior
was found in [26] for the remagnetization process of Co wires.
In summary: For fields parallel to the stripes the remagnetization process proceeds
by nucleation and domain wall motion within a very narrow region around the coercive
field, Hc . For field directions 0 < χ < 90◦ the remagnetization process is dominated
by a coherent rotation up to the coercive field, where some domains are formed. For
χ = 90◦ the remagnetization process appears to proceed entirely via a coherent rotation
of the magnetization vector and no switching takes place. In the hard axis direction the
magnetization vector describes a complete 360◦ rotation during the full magnetization
cycle without any discontinuity. For other directions a switching of the magnetization
of 180◦ is observed at Hc which can be viewed as a head-to-head domain wall movement
[28] through the stripe. Similar dependencies were found in [24], where the switching
process was found to be dominated by edge curling walls.
91
7. CoFe grating
Reason for this clear and straightforward remagnetization process is the high aspect
ratio of the stripes ( height
width = 0.075), leading to a strong shape anisotropy. The shape
anisotropy results in a two-fold anisotropy with an anisotropy energy density ES . From
a numerical integration of the magnetization curve in the hard axis direction follows
ES ≈ 730 Oe · Ms , where Ms is the saturation magnetization of the CoFe alloy. The
energy density of the shape anisotropy can be calculated according to Ref. [2], using
the relation Es = 2πN · Ms2 , where N is the tensor element of the demagnetizing tensor
in the appropriate direction. N can be calculated as an approximation for an ellipsoid
inscribed in the wire by using a formula given in [2]. Assuming Ms =1775 G for Co0.7 Fe0.3 ,
the theoretical shape anisotropy constant is 780 Oe · Ms , which is in good agreement
with our measurements. The value of the saturation magnetization Ms for Co0.7 Fe0.3 is
higher than the value for pure Fe and is given in Ref. [96] and references therein. More
recent experiments confirm this value [97].
7.3.2. Results from Kerr-microscopy
Figure 7.5.: Kerr microscopy pictures taken around Hc for different magnetic field to
sample alignment of χ = 0◦ (left, easy axis alignment) and χ = 73◦ (right).
The plane of incidence results in a top-down magneto-optical sensitivity axis
parallel to the stripes. The magnetization directions are indicated by arrows
in the χ = 0◦ map.
Kerr microscopy studies support the conclusions from the MOKE measurements. Images at different magnetic fields show a single domain state for a field below and above
Hc (not shown). Around Hc , domain nucleation and domain wall movement sets in.
The proposed mechanism of head to head domain walls through the stripes can be confirmed. From our observations we assume independent domain nucleation and domain
92
7.4. Bragg-MOKE measurements at the CoFe grating sample
wall movement for each single stripe. Possible existence of dipole-dipole interactions
between domains of adjacent stripes cannot be extracted from the Kerr microscopy observations in Fig. 7.5 which shows Kerr-microscopy pictures taken at Hc for two different
field angles χ of the sample rotation. In Fig. 7.5 (left) the easy axis case with no magnetization rotation for H 6= Hc is shown, and Fig. 7.5 (right) displays a picture taken
from an intermediate angle χ = 73◦ close to the hard axis case. In agreement with the
MOKE results, at Hc , similar Kerr microscopy pictures were found for different angles
χ of the sample rotation between χ = 0◦ and χ = 88◦ .
7.4. Bragg-MOKE measurements at the CoFe grating
sample
At the CoFe grating as described above Bragg-MOKE measurements were carried out
in the geometry discussed in Sec. 3.5. In particular, the angle of incidence was chosen
αi = 0◦ and the grating is oriented with the stripes perpendicular to the external field
(hard axis). In this geometry the first four orders of diffraction are accessible in the given
setup (see Sec. 3.5.3 and Fig. 3.21). The results of the Bragg-MOKE measurements are
reproduced in Fig. 7.6. For symmetry reasons (see Sec. 3.5.3) it is sufficient to discuss
only the measurements of positive order of diffraction which are shown in the figure.
The Bragg-MOKE curve at n = 2 is identical to the measurement of n = 1, therefore it
is omitted. The three Bragg-MOKE curves in Fig. 7.6 are normalized to the measured
Kerr-rotation in magnetic saturation. In order to compare the Bragg-MOKE amplitude
sat
(n), this value is plotted in Fig. 7.6(d) as a function of the order of
in saturation, θK
diffraction.
Sat
(n) is an increasing function of n. This effect is
As can be seen in Fig. 7.6(d), θK
a common feature observed in other measurements of the Bragg-MOKE effect as well
and will be discussed later in more detail (Sec. 9.4). At this point the discussion will
concentrate on the shape of the Bragg-MOKE curves independent on their amplitude.
At a first glance the three plotted Bragg-MOKE curves in Fig. 7.6(a-c) reproduce well
the hard axis MOKE loop depicted in Fig. 7.2(g). The loops are completely closed and
have a rather high saturation field. However, upon closer inspection, it is seen that the
saturation field differs for the different order of diffraction. In order to demonstrate this,
Fig. 7.7 shows the diffraction hysteresis loops (DHL) for the four orders of diffraction.
In this representation the Bragg-MOKE measurement is plotted as a function of the
MOKE curve in specular geometry which is assumed to be proportional to the averaged
magnetization, also see Sec. 3.4.1. Every deviation of the DHL from a straight line
indicates a difference in the Bragg-MOKE curve in the corresponding magnetization
regime. It is clearly seen that close to saturation magnetization (m/ms ≈ 1) the DHL of
order n = 1 and n = 2 display a higher Bragg-MOKE signal than expected (saturation
is approached at smaller fields than for the specular geometry). For n = 3 the opposite
is true. In a regime where the sample is already saturated on average the Bragg-MOKE
curve is still increasing. Only the DHL for n = 4 reproduces the ideal straight line with
unit slope.
This result is not expected as the anticipated remagnetization process from the vector-
93
7. CoFe grating
(b) n=3
1
1
0.5
0.5
θK / θSat
K
θK / θSat
K
(a) n=1
0
−0.5
−0.5
−1
−1
−4
−2
0
H [kOe]
(c) n=4
2
4
−4
−2
0
2
H [kOe]
B−MOKE amplitude
4
0.4
1
0.3
θsat
[°]
K
0.5
θK / θSat
K
0
0
0.2
−0.5
0.1
−1
−4
−2
0
H [kOe]
2
4
0
1
2
3
4
n
Figure 7.6.: Results of Bragg-MOKE measurements from the CoFe stripe array. (a), (b)
and (c) display the normalized Bragg-MOKE curves of order n = 1, 3 and
4, respectively. (d) shows the dependence of the saturation Bragg-MOKE
amplitude as a function of n.
MOKE and Kerr-microscopy studies above, together with the simulation of DHL’s in
Sec. 3.4.2 would lead to DHL which are linear functions for all n. For a straightforward
coherent rotation remagnetization process only the saturation Bragg-MOKE rotation
would be altered. DHL’s as measured from the present CoFe sample must be related
to a reversible formation of domains up to the highest field values measured. This was
demonstrated with the simulations in Sec. 3.4.2. However, compared to Fig. 3.12 the
deviation of the observed DHL’s from the ideal linear behavior is small. The conclusion drawn from the Bragg-MOKE measurements therefore is that edge-domains are
established which exist up to very high fields. These edge-domains effectively reduce
the magnetic width of the stripes. The measured DHL’s are therefore a superposition
of two model cases as discussed in Sec. 3.4.2, namely the edge-domain and the coherent
rotation models. Qualitatively the measured DHL agree with the simulations of these
two case in Sec. 3.4.2.
These domains probably reside at the edges of the stripes and are necessary for partly
compensating the magnetic flux at the sharp edges of the stripes. Examples of such
closure domains have been observed e.g. by McCord et al. [23]. As the magnetization
direction in these domains is alternating up and down the average magnetization of the
edge domains is zero.
94
7.5. Summary
(a) n=1
(b) n=2
1
B−MOKE signal
B−MOKE signal
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
m / ms
(c) n=3
−0.5
−0.5
0
0.5
m / ms
(d) n=4
1
−0.5
0
m / ms
1
1
B−MOKE signal
B−MOKE signal
0
−1
−1
1
1
0.5
0
−0.5
−1
−1
0.5
−0.5
0
m / ms
0.5
1
0.5
0
−0.5
−1
−1
0.5
Figure 7.7.: CoFe-stripe array: Conversion of the Bragg-MOKE curves to DHL’s.
7.5. Summary
In conclusion, we have studied the magnetization reversal behavior of CoFe stripes using Vector-MOKE, Bragg-MOKE and Kerr-microscopy. As was found from the VectorMOKE and Kerr-microscopy measurements, the remagnetization process of the present
sample presents a very simple case consisting from purely coherent rotation and a 180◦
domain wall at the coercive field. In addition, no interaction of the stripes where found.
However, the Bragg-MOKE measurement proofed that flux closure domains exist, which
were not detected using Kerr-microscopy at high magnetic fields. (Note that the Kerrmicroscopy images depicted in Fig. 7.5 were produced in small fields around Hc ). The
Bragg-MOKE techniques can give additional information which are difficult to obtain
with other methods. Certain orders of diffractions probe particular Fourier components
of the magnetization distribution and may be very sensitive to edge domains. Qualitatively the simulated archetypes of magnetization processes in Sec. 3.4.2 could be
Sat
reproduced, however, no qualitative agreement for the dependence of θK
(n) was found
using this model.
From this sample an extensive study with polarized neutron reflectometry (PNR) was
performed [93]. PNR at patterned magnetic films is a new and challenging task and
the straightforward remagnetization process of the CoFe stripes is an ideal sample for a
comparative study in order to evaluate the potential of PNR. The magnetization reversal
process was observed at the first order Bragg peak as function of the orientation of the
stripes with respect to the applied magnetic field and the scattering plane (the field was
95
7. CoFe grating
applied perpendicular to the scattering plane). From the different cross-sections, spin
asymmetry curves were calculated and compared to standard MOKE hysteresis loops.
The curves show a very good agreement with the observed MOKE hysteresis loops, from
which was concluded that PNR is a suitable method for studying patterned magnetic
samples.
Note, that here the standard specular MOKE curve is compared to the first order
diffracted PNR magnetization loop (Bragg-PNR). From the discussion above and the
simulations in Sec. 3.4.2 it is clear that also Bragg-PNR probes a certain Fourier component of the magnetization distribution. Therefore one would expect changes of the
Bragg-PNR curves qualitatively similar to those observed with Bragg-MOKE (Fig. 7.6),
which is not the case in [93]. However, due to the special physics of diffraction with
neutron beams [93, 98] it is possible to observe Bragg-PNR curves for in-plane sample
rotations from χ near to the easy axis up to χ near to the hard axis, but not exactly in
the hard axis configuration, which is the only possible configuration for Bragg-MOKE
(Sec. 3.5.3). Therefore Bragg-MOKE and Bragg-PNR display a certain lack of comparativeness. In addition Bragg-PNR curves of higher order than n = 1 could not be
observed.
96
8. Ni-gratings
8.1. Introduction
In this chapter1 measurements are presented of the Bragg-MOKE effect from Ni gratings
with rather large lattice parameter of 20 µm. Because of the large grating parameter
many interference spots are observable from which hysteresis loops can be taken.
The second idea of this system was to separate effects due to domains in the stripes
from pure optical phenomena. In order to do so two samples were prepared, the first one
consists of an Al-grating on top of a Ni film, and the second one of a simple Ni grating.
For the first case no correlated domains with the period of the grating are expected,
thus no influence of domains to the Bragg-MOKE curves should occur.
8.2. Experimental setup
The first optical grating (sample No. 1) has been prepared by depositing a Ni-film of
20 nm thickness on Si(111) and then preparing an array of Al-stripes with a lattice
parameter d = 20 µm, a thickness of 20 nm and a width of a single Al-stripe of wAl =
6 µm on top of the Ni-film. The Al-stripes are thick enough to prohibit the laser beam
to penetrate the Ni film below the stripes. The second sample (sample No. 2) is an
optical grating of Ni-stripes on Si(111) with a lattice parameter of d = 20 µm, a width
of a single Ni-stripe of wN i = 4 µm, and a thickness of 20 nm. Both gratings were
prepared by optical lithography and a lift-off process2 .
MOKE measurements with a HeNe laser as light source were carried out using a high
resolution Kerr angle setup in the longitudinal configuration as described in Sec. 3.5, also
see Fig. 8.1. The entire setup as depicted in Fig. 8.1 is mounted on a goniometer such
that the angle of incidence αi can be varied between 0◦ and 45◦ and the detector can
be moved from −45◦ to 45◦ . This is necessary in order to investigate the MOKE effect
at different orders of the diffracted light. The grating is oriented with the normal of
the surface in the scattering plane and the magnetic field perpendicular to the direction
of the stripes. The diffraction spots are numbered by the order of diffraction n, n = 0
denoting specular reflection. Positive (negative) n denotes an increasing (decreasing)
detector angle with respect to the surface normal. In addition to the measurements of
the Kerr angle we also measured the integrated intensity at the interference spots.
1
This chapter is based upon the article Magneto-optical Kerr Effects of Ferromagnetic Ni-gratings see
Ref. [99]
2
The samples were provided by K. Schädler and U. Kunze from Lehrstuhl für Werkstoffe der Elektrotechnik, Ruhr-Universität Bochum
97
8. Ni-gratings
Figure 8.1.: Schematic set-up of the Bragg-MOKE experiment with the Ni-grating on
Si between the pole faces of an electromagnet. The angle of incidence αi
is fixed, the grating can be rotated by the angle χ. The Kerr rotation is
measured at the different diffracted spot positions n.
Figure 8.2.: Representative examples of Bragg-MOKE hysteresis loops of sample No. 1
(Al stripes on Ni) measured at different orders of diffraction n.
98
8.3. Results and Discussion
8.3. Results and Discussion
8.3.1. Bragg-MOKE
First, the results obtained from the Al diffraction grating on Ni (sample No. 1)are
discussed. In Fig. 8.2 selected examples of hysteresis loops measured at different orders
of diffraction n are shown. The shape of the hysteresis loops is identical for all n, but
the amplitude of the Kerr signal changes strongly. In Fig. 8.3 the amplitude of the Kerr
signal at remanence and the intensity measured at different orders of diffraction n is
plotted. One clearly observes a periodic behavior with an oscillation period of ∆n = 3.
The Kerr intensity exhibits the same periodicity with a phase shift: the maximum of
the Kerr angle matches exactly the minimum in the intensity. The intensity distribution
in Fig. 8.3 results from a superposition of the diffraction pattern of the periodic lattice
with a lattice parameter of d = 20 µm and the diffraction pattern of a single stripe
with a width wAl ≈ 1/3d. The apparent periodicity in Fig. 8.3 is due to the Al-stripes
with a width of about 6 µm. The modulation from the diffraction due to the Ni-spacing
between the Al-stripes cannot be resolved, since its stripe width is approximately a
factor of 2 larger. The close correlation between the maxima of the Kerr angle and the
minima of the intensity can be attributed to the dominant contribution of the Ni stripes
rem
at these positions. In addition, the oscillation of θK
(n) is superimposed to a linear
rem
decrease of θK (n) from negative to positive n. The linear contribution changes sign for
n ≈ −18. As the measurements were carried out with αi ≈ 40◦ , the angle αf (n = −18)
approximately matches the position where the diffracted beam is directed back along
the surface normal. Vial and van Labeke [53] predicted a Kerr signal (in the transverse
Kerr geometry) of zero and a change of sign for the Littrow -mounting, in which the
diffracted beam is directed back into the incoming beam (αf = −αi ). This condition
would be fulfilled for the present case for n ≈ 36. Therefore the result in Fig. 8.3 is in
contradiction with the results in [53]. The linear contribution and the change in sign
will be subject to further investigations reported in the following chapters.
In Fig. 8.4 examples of hysteresis loops measured from the Ni grating on Si (sample
No. 2) are shown. One first should note that for this grating a strong enhancement of
the amplitude of the Kerr signal by a factor of up to 30 at certain diffraction spots is
observed. Contrary to sample No. 1 the shape of the hysteresis loops strongly depends
on the order of diffraction. Whereas at certain orders of diffraction (n = 1 shown in
Fig. 8.4) the hysteresis loop has a shape identical to that observed at specular reflection
n = 0, one observes anomalous hysteresis loops at other orders of diffraction (n = −3
and n = 9 in Fig. 8.4). Apparently these anomalous hysteresis loops contain a second
component of the Kerr signal with a high saturation field and a sign change when going
from +n to −n. The occurrence of the anomalous contribution in the hysteresis loops is
correlated with a rather low value of the Kerr signal for the normal hysteresis loop. It
is supposed that the anomalous contribution is caused by the magnetic microstructure
on the edges of the stripes, e.g. by closure domains with a magnetization component
perpendicular to the film plane. Similar anomalous contributions to the Bragg-MOKE
hysteresis loops in transverse geometry have been observed in [11, 53]. The authors show
by model calculations that the anomalous component can be attributed to the domain
pattern of the ferromagnetic stripes. In Fig. 8.5 the intensity and the Kerr amplitude
99
8. Ni-gratings
Figure 8.3.: Intensity of the diffracted light (upper panel) and Kerr signal amplitude at
remanence (lower panel) versus the order of diffraction for sample No. 1.
Figure 8.4.: Representative examples of Bragg-MOKE hysteresis loops of sample No 2
(Ni stripes on Si) measured at different orders of diffraction n.
100
8.3. Results and Discussion
Figure 8.5.: Intensity of the diffracted light (upper panel) and Kerr signal amplitude at
remanence (lower panel) versus the order of diffraction for sample No. 2.
for sample No. 2 is plotted for different orders of diffraction n. The periodicity in the
intensity distribution and the Kerr signal amplitude is ∆n = 5. This fits perfectly to the
width of the Ni-stripes wN i = 4 µm, which is 1/5 of the lattice parameter d = 20 µm.
Compared to the intensity pattern in Fig. 8.3, the intensity pattern in Fig. 8.5 appears
strongly smeared out. This may be caused by a superimposed damped modulation from
the diffraction of a single Si-stripe with a width of 16 µm. Contrary to the result in
Fig. 8.2 there is no phase shift between the intensity and the Kerr angle, i.e. the maxima
in the Kerr amplitude correlate with the maxima in the diffracted intensity. This is
plausible, since in this case the Ni-stripes are responsible for the intensity modulation.
8.3.2. MFM measurements
In order to further clarify the role of domains in this system force microscopy measurements have been carried out. Fig. 8.6(a) and (b)3 depict AFM and MFM measurements,
taken simultaneously from the identical portion of a Ni stripe of sample 2. From the
AFM measurement (a) the important information can be extracted, that the edges of
the Ni stripes show large ridges which stem from the lift-of process (white lines along
the stripe). These ridges are as high as 100 nm measured from the top of the stripes,
exceeding the stripe height by a factor of 10. Therefore it can be expected that a local
3
The measurements (a) and (b) were carried out from A. Carl and co-workers at the Institut für
Tieftemperaturphysik, Universität Duisburg, Germany
101
8. Ni-gratings
Figure 8.6.: (a) AFM image of a portion of one Ni stripe and (b) MFM image of the
same area taken from sample 2. (c) MFM image of a polycrystalline Ni/Si
film.
shape anisotropy will lead to strong magnetization components perpendicular to the
sample plane. The MFM image of the same stripe was taken in zero field. It displays
a magnetic contrast mainly at the edges of the stripes with alternating dark and light
regions. The center of the stripes show only small magnetic contrast. As the MFM tip
is mainly sensitive to perpendicular, out-of-plane, magnetization components it can be
concluded that edge domains with perpendicular magnetization triggered by the ridges
are formed. The MOKE signal in longitudinal configuration is proportional to the inplane and out-of-plane magnetization. Therefore the conclusion is reasonable that the
additional components in the diffraction MOKE curves in Fig. 8.4 are connected to the
out-of-plane magnetization of the edge domains.
Fig. 8.6(c) displays a MFM measurement of a small portion of a Ni film, which was
equivalently prepared as sample 2, but not structured. This measurement clearly shows
stripe domains which stem from the very complicated domain structure of Ni films. Al-
102
8.4. Summary and Conclusion
though the shape anisotropy leads to a overall in-plane magnetization of this film, the
fine domain structure shows out-of-plane magnetized stripe domains which are favored
because of an additional magnetostrictive anisotropy found in these films [100]. Hysteresis curves measured from this sample also shows a reduced remanence which can be
attributed to portions of the sample magnetized alternating perpendicular to the film.
More details of this measurements can be found in [76]. However, the size of the domains
is so small that many of the domains can be placed in the width of one stripe. Therefore
no diffraction signals at the position of the interference spots can be obtained which are
correlated to this small magnetic patterns. The stripe domains were not resolved in (b),
because the edge domains and the large structural hight differences cover the smaller
domains.
8.4. Summary and Conclusion
In the present chapter Bragg-MOKE measurements in longitudinal configuration were
demonstrated. Gratings with a relatively large lattice parameter of 20 µm were used
which enables one to observe the Bragg-MOKE effect up to higher orders of diffraction
than could be done before. It has been shown that it is possible to increase the amplitude
of the Kerr rotation by making use of diffraction from a ferromagnetic grating. In
principle, this effect is similar to the amplification of the Kerr signal of a ferromagnetic
homogeneous film using interference effects by coating the film with a dielectric cap
layer.
The Bragg-MOKE effect may be separated into optical effects only changing the
overall Kerr amplitude and effects of the magnetic substructure and it seems possible to
gain information about the magnetic domain pattern of the ferromagnetic stripes. Close
to the angle of extinction of the diffracted light from a single ferromagnetic stripe, the
Bragg-MOKE effect appears to be sensitive to the specific domain pattern at the edges
of the stripes.
The observed effects will be further analyzed and clarified in the subsequent chapters,
where measurements of gratings with constant grating parameter but changing stripe
width will proof the assumptions made above.
103
8. Ni-gratings
104
9. Fe-gratings
9.1. Introduction
The magneto-optical Kerr effect in longitudinal configuration is used to study hysteresis
loops of several Fe-gratings patterned by electron beam lithography. The Kerr effect is
not only detected in specular reflection but also off-specular at the intensity maxima
of the grating at different orders of diffraction n (Bragg-MOKE). The Kerr rotation
in saturation increases linearly with the order of diffraction n if the incoming beam is
at normal incidence, independent of the grating geometry. The shape of the hysteresis
curves changes for special orders of diffraction, indicating an enhanced sensitivity for
the formation of magnetic domains close to remanence: the nth order of diffraction is
sensitive to the nth order Fourier component of the magnetization distribution.
In this chapter1 two sets of gratings are under investigation. The first is made from
polycrystalline and the second set is made from single crystalline, (001) oriented Fe-film.
For both sets the grating parameter is constant at 5 µm and the stripe width is varied.
The magnetism and domain structure of the single crystalline gratings are analyzed in
different states of the patterning process showing the influence of connecting Fe between
the stripes.
9.2. Sample Preparation
The stripe arrays were prepared by means of electron beam lithography. In the first step
continuous thin film samples were prepared by molecular beam epitaxy (MBE) and by
rf-sputtering (see Sec. 4.1. The MBE sample of the present study (sample A) is a 20 nm
thick single crystalline Fe(001) film grown on a Cr(20 nm)/Nb(20 nm) buffer system on
Al2 O3 (11̄02) [58, 63, 65] (Fig. 9.1(a)). The base pressure during the MBE process was
10−10 mbar. The crystalline quality was checked in situ using RHEED and ex-situ by
x-ray diffraction techniques.
The sputtered sample (B) is a polycrystalline Fe film produced by rf sputtering at
a base pressure of 5 · 10−8 mbar. The Fe film was grown on Al2 O3 (112̄0) at room temperature (see Fig.9.1(b)). Again the structural properties were checked by ex-situ x-ray
diffraction and small angle x-ray reflectivity.
Before further processing the film, MOKE measurements of the samples were carried
out, in order to have hysteresis loops of the unpatterned film as a reference. The standard
MOKE measurements were carried out in longitudinal geometry at an angle of incidence
1
This section is an extension of the article Magneto-optical Kerr effect in the diffracted light of Fe
gratings[101]
105
9. Fe-gratings
Figure 9.1.: Sample design of the Fe-gratings. (a) shows sample A2 (single crystal
Fe(001)) in which four gratings with the stripes parallel to the in-plane
Fe(100) direction and one grating parallel to the Fe(110) direction were
etched. The situation after the completed etching procedure is depicted.
(b) displays the design of sample B (polycrystalline Fe) with four gratings.
wF e , t and d denote the stripe width, the thickness of the stripes and the
lattice parameter, respectively.
Figure 9.2.: Atomic force microscopy image of one grating of sample B with wF e =
2.1 µm
106
9.2. Sample Preparation
αi ≈ 45◦ with s-polarized light, i.e. with a polarization direction perpendicular to
the plane of incidence. The magnetic anisotropy was measured by rotating the sample
around its surface normal (angle χ in Fig. 3.21) and taking hysteresis curves for different
angles. Details of the MOKE setup can be found in Sec. 3.5.
From each of the thin films A and B several grating structures with different grating
geometries were produced. Therefore the film thickness and crystal quality is the same
for one set of gratings originating from one thin film. In a first step, the samples were
spin-coated with a double layer of PMMA. Using electron-beam lithography, the grating
structures were written onto the polymer film, which was subsequently patterned by
developing. Finally the grating structure was transferred into the metallic film using
ion-etching. The resulting grating structures were analyzed using scanning electron
microscopy and atomic force microscopy techniques. An example of the gratings thus
prepared is depicted in Fig. 9.2. The grating geometries used for the present study are
shown schematically in Fig. 9.1.
In more detail, the sputtered sample (B) was patterned using the double imagetransfer technique described in Sec. 4.1, i.e. after the mask fabrication via e-beam
lithography a Al2 O3 film of the same thickness as the Fe film was deposited and structured using the lift-of. Afterwards, the (negative) Al2 O3 image of the desired pattern is
etched into the Fe film. The Fe stripes therefore form a positive image of the stripes,
i.e. the surrounding, unstructured area of the sample is completely etched away.
In contrast, the single crystalline Fe sample were etched directly using the resist mask
for the etching process. Therefore the negative image is formed, leaving the partes of
the sample without grating untouched. Furthermore, the etching was performed in a
two step process, the first step results in grating structures where the gap between the
stripes is still covered with Fe connecting the adjacent stripes (this stage will be called
sample A1 ). The second etching step results in gratings which are completed in the
sense that no Fe connects the stripes (sample A2 )2 .
After the patterning process, Bragg-MOKE hysteresis curves were taken. The geometry of the Bragg-MOKE measurements for αi = 0 is sketched in Fig. 3.21, more details
of the Bragg-MOKE setup can be found in Sec. 3.5.3. The integrated intensities of the
diffraction spots were measured using a photo-diode instead of the Kerr-detector. In
addition, Kerr microscopy measurements were performed as described in Sec. 4.3.3.
2
Here an error in the publication [101] is corrected: in contrast to what is reported in [101], the
measurements in the article on the single crystalline Fe gratings are performed in the not completely
etched state (sample A1). Therefore some conclusions about the domain structure of this sample
are also wrong in [101]. This, however, does not change the results and conclusions of the article
with respect to the Bragg-MOKE effect, which is the main objective of [101]
107
9. Fe-gratings
0.05
K
θ [°]
(a)
0
−0.05
−400 −200
0
200
H [Oe]
400
0
200
H [Oe]
400
0.05
θK [°]
(b)
0
−0.05
−400 −200
Figure 9.3.: MOKE measurements of the single crystalline Fe film (sample A) in the
as prepared state. The two curves show MOKE measurements performed
along an easy axis (a) and a hard axis (b) of the fourfold magneto-crystalline
anisotropy. For the standard MOKE measurements αi = 45◦ .
9.3. Results
9.3.1. Single crystal film, sample A
Unstructured film
Longitudinal MOKE hysteresis curves of the unstructured Fe film are depicted in Fig. 9.3
for two orientations of the magnetic field along the magnetic easy (100) and hard (110)
axis. As expected, hysteresis curves along the easy axis are of square type, whereas the
hard axis hysteresis curve shows a reduced remanence. Measurements with the external
field applied in different in-plane directions, i.e. at different sample rotation angles
χ, reveal a fourfold symmetry, as expected for the Fe(001) film. Additionally, small
overshoots, i.e. asymmetries, of the MOKE hysteresis curve can be seen in Fig. 9.3, which
108
9.3. Results
Figure 9.4.: Kerr microscopy image of one of the single crystalline Fe-gratings before
they were completely etched through (sample A1). The contrast in this
image is mainly due to the magnetization direction of the domains.
can be attributed to second order magneto-optical effects [40, 43, 42], see Sec. 3.1.3. In
the case of the reversal process along the (110) axis the magnetization process combines
~ away from the hard axis (0◦ , along the applied
a rotation of the magnetization vector M
field) into the direction of the easy axis, at 45◦ from the applied field and by domain wall
~ has a nonzero transverse
movements. During the rotation the magnetization vector M
component, which results in the observed second order MOKE components. Both curves
show very small coercive fields, Hc ≈ 10 Oe, the observed behavior is consistent with
the coherent rotation model of the magnetization process [44].
Connected single crystalline Fe gratings, sample A1
In this section the measurements on the single crystalline Fe stripes (sample A1, see
Fig. 9.1(a)) are reported. As the Fe strips are connected to each other by the residual Fe
film in the gap, the magnetic domains in this situation are mostly independent on the
actual stripe structure, i.e. no edge domains or induced anisotropy are expected. The
situation was demonstrated by Kerr microscopy: In Fig. 9.4 domains which are larger
than the stripes can clearly be observed.
First measurements on the gratings from sample A1 with the direction of the stripes
parallel to the in-plane (100) axis are discussed. As the magneto-crystalline anisotropy
is fourfold, one easy axis lies parallel to the stripes, the other one perpendicular to
the stripes. The patterning of the gratings did not add a measurable twofold shape
109
9. Fe-gratings
0.1
n=1
n=2
n=3
K
θ [°]
0
−0.1
0.1
0.1
n=−2
0
n=−1
0
−0.1
−1
0
1
n=−3
−0.1
−1 0 1
−1
H [kOe]
0
1
Figure 9.5.: Bragg-MOKE hysteresis curves for one grating of sample A1, Fe-stripe
width is 3.9 µm.
0.1
wFe=3.9µm
w =2.4µm
Fe
w =2.0µm
Fe
w =1.3µm
Fe
sat
θK [°]
0.05
0
−0.05
−0.1
−3
−2
−1
0
n
1
2
3
Figure 9.6.: Bragg-MOKE angle in saturation as a function of the diffraction order for
sample A1
110
9.3. Results
θK [°]
0.1
0
(a) n=1
−0.1
−2
−1
0
1
H [kOe]
2
0
1
H [kOe]
2
θK [°]
0.1
0
(b) n=2
−0.1
−2
−1
Figure 9.7.: Bragg MOKE hysteresis loops with the grating parallel to the hard axis
of the magnetization for n = −1 (a) and n = −2 (b); wF e = 1.4 µm and
d = 5 µm.
anisotropy to the gratings of sample A1, which has been verified by rotating the gratings
around the surface normal and measuring sequences of MOKE hysteresis curves for
different angles χ. Thus in the geometry for the Bragg-MOKE measurements described
above the gratings are oriented perpendicular to the external field and with an easy axis
along the field. In this geometry, Bragg-MOKE measurements have been performed for
the first three positive and negative orders of diffraction of all four gratings. In Fig. 9.5
Bragg-MOKE hysteresis curves for the grating with a Fe line width of wF e = 3.9 µm are
presented as an example. It is clearly seen that the shape of the curves is independent of
the diffraction order n and that the shape of the hysteresis loops basically reflects the easy
axis curve from the unstructured film in Fig. 9.3. Only the coercive force is increased
sat
to Hc ≈ 45 Oe. In contrast, the absolute value of the Kerr rotation in saturation, θK
,
depends on n and changes sign when moving from positive to negative n. Fig. 9.6
111
9. Fe-gratings
log(I) [a.u.]
w =1.3µm
Fe
w =2.0µm
Fe
wFe=2.4µm
wFe=3.9µm
−5 −4 −3 −2 −1
0
n
1
2
3
4
5
Figure 9.8.: Integrated intensities for the gratings of sample A with the stripes parallel
to (100)
sat
sat
displays θK
(n) for all four gratings. The θK
(n)-curve is linear and independent of the
width of the Fe stripes wF e .
sat
No dependency of θK
on the diffracted intensities was found. Fig. 9.8 shows the
diffracted intensities as a function of n for perpendicular incidence. Although the intensity varies differently for different stripe widths, representing the diffraction envelope of
the single stripes, the measured Bragg-MOKE signal is not affected.
The situation changes if the stripes of the grating are along the in-plane (110) axis
of the Fe film, i.e. along the hard axis of the magnetization. Fig. 9.7 shows the
Bragg-MOKE hysteresis curves for n = 1, 2 and for wF e = 1.4 µm. The Bragg-MOKE
sat
rotation in saturation θK
(n) shows the same behavior as for the stripes along (100),
but the shape of the curves changes significantly. The overshoots are also seen in the
as prepared state of the sample A (see Fig. 9.3). However, the features are enhanced,
indicating that the contributions of second order magneto-optical effects have increased.
The contribution of second order effects decreases with increasing positive or negative
n (see Fig. 9.7).
Single crystalline Fe-gratings after etching, sample A2
After the remaining Fe connecting the stripes has been removed in an additional etching procedure, the magnetic properties of the single crystal Fe stripes were drastically
altered. Fig. 9.9 shows hysteresis curves for different stripe widths and different orientations of the field to the stripes. Only the gratings with the stripes parallel to the
112
9.3. Results
e.a.
h.a.
0.015
45°
2.4µm
0
Hc2
−0.015
θK [°]
0.015
2.0µm
0
H
−0.015
0.015
c2
1.7µm
0
H
−0.015
−0.5
c2
0
0.5−0.5
0
H [kOe]
0.5−0.5
0
0.5
Figure 9.9.: Standard MOKE measurements from the single crystalline Fe grating A2
in three different directions with respect to the external field. In the left
column measurements with the field perpendicular to the stripe axis are
depicted. The middle column represents the measurements with the field
along the stripes, and the right column with the field aligned 45◦ to the
stripes. The measurements were carried out at the arrays with wF e = 2.4,
2.0 and 1.7 µm (top to bottom row) as indicated in the figure. The definition
of Hc2 is also indicated in the hard axis plots.
in-plane easy axis of the magneto-crystalline anisotropy are under investigation, i.e. the
stripes are parallel to the Fe(100) direction.
It is clearly seen that in this situation the patterning induces an additional twofold anisotropy, which is strongest for the smallest stripes and surmounts the four-fold
crystalline anisotropy. The superposition of the two-fold anisotropy (with the hard axis
perpendicular to the stripes) and the fourfold crystalline anisotropy of Fe (with the easy
axes along and perpendicular to the stripes) has very interesting consequences. Below
a certain stripe width (the transition is between wF e = 3.9 µm and wF e = 2.4 µm) the
hysteresis exhibits a plateau region around zero field with almost zero magnetization
in the hard axis (with respect to the stripes). The system enters the plateau region at
a switching field Hc2 before the external field is reversed. At the field H = Hc2 the
condition:
Ek + Ez = Ek + EU
(9.1)
must be satisfied, with Ek , EU and Ez being the crystalline anisotropy energy density,
113
9. Fe-gratings
−3
4.5
x 10
Hc/(Ms/2−Hc)
4
3.5
3
2.5
2
4
4.5
5
5.5
6
1/w [1/m]
6.5
7
7.5
8
5
x 10
c2
( 1 ) for the gratings of sample A2 (see Eq. 9.2). From the
Figure 9.10.: Plot of 1 MHs −H
c2 w
2
linear fit to the data the height of the stripes can be determined, see text.
the anisotropy energy density and the Zeeman-energy, respectively. The Zeeman energy
at Hc2 is Ez = µ0 Ms Hc2 , and the uniaxial anisotropy is defined as EU = 12 µ0 N Ms2 . Here
h
the demagnetizing factor, N , is given by N = h+w
, where h is the thickness of the
Fe
stripes and wF e the width (see Sec. 7 and [2, 56]). This leads to (in SI units):
1
h
Ms
2 h + wF e
Hc2
1
⇐⇒ 1
=h .
w
Ms − Hc2
2
(9.2)
Hc2 =
Fig. 9.10 displays the data of Hc2 , plotted as
6
Hc2
1
Ms −Hc2
2
(9.3)
as a function of
1
.
w
Here the bulk
saturation magnetization of Fe (1.7 · 10 A/m) is chosen. From the linear fit to the data
follows h = 4.4 nm. Obviously, the error is quite large. However, AFM measurements
at the gratings confirm a height between the top of the stripes and the bottom of the
groove of ≈ 9 nm. This discrepancy may be explained by the presumably deeper grooves
and longer etching times than intended, i.e. during the etching the Fe-stripes are already
ablated and the grooves are etched into the underlying Cr buffer, thus the thickness of
Fe is smaller than the height of the stripes.
Bragg-MOKE measurements at the gratings from sample A2 have been performed,
some examples are plotted in Fig. 9.11. The measurements at the grating with wF e =
2.0 µm are representative for the measurements at the gratings for wF e = 2.4 µm and
114
9.3. Results
wFe = 2.0 µm
0.15
wFe = 3.7 µm
n=1
n=1
n=2
n=2
n=3
n=3
0
−0.15
K
θ [°]
0.15
0
−0.15
0.15
0
−0.15
−0.4
0
0.4
−0.4
H [kOe]
0
0.4
Figure 9.11.: Example Bragg-MOKE hysteresis curves of the single crystalline Fe grating, sample A2. The Fe stripe width and order of diffraction is given in
the figure.
wF e = 1.3 µm. However, the very interesting Bragg-spot n = 2 and wF e = 2.4 µm
could not be measured, as the diffracted intensity was too low. All these Bragg-MOKE
hysteresis curves show generally the same shape as the specular MOKE curves depicted
in Fig. 9.9. The only difference is that the plateau region around zero field is even
more pronounced and exhibits a smaller remanence compared to the specular MOKE
curves. In this case the diffraction at the gratings acts like a filter which excludes all
light reflected at the ferromagnetic surroundings of the gratings. Therefore the BraggMOKE curves show only the magnetic behavior of the gratings, whereas the signal at the
specular spot may be contaminated with signals from the neighborhood of the stripes.
The Bragg-MOKE curves at the stripes with wF e = 3.7 µm show for n = 1 and n = 2
no change of the shape with respect to the specular curve. However, the curve for n = 3
sat
displays an increase in the saturation field. The θK
(n) curve displays, as was observed
earlier, a monotonous behavior with increasing Kerr rotation for increasing n.
The domain pattern of the gratings on sample A2 was measured using Kerr microscpy3 . In all cases a demagnetized state is imaged, which was achieve by a decreasing, oscillating in-plane field. The results are shown in Fig. 9.12 for the gratings with
wF e = 2.4, 2.0 and 1.3 µm. For all images the magneto-optical sensitivity direction was
chosen to be along the stripes, i.e. black domains are magnetized parallel and white
domains antiparallel to the stripes, different shades of gray correspond to perpendicular
3
The measurements were performed together with J. McCord at the IFW Dresden, Germany
115
9. Fe-gratings
Figure 9.12.: Kerr microscopy images of the single crystal Fe-grating. All measurements
were performed in a demagnetized state with the magnetic sensitivity direction along the stripes. In the top row the field was oriented along the
strips during demagnetization (easy axis), in the bottom row the field was
oriented perpendicular to the stripes (hard axis). The Fe stripe width is
indicated in the figure. For this particular sample the gratings are surrounded by the unstructured Fe film. Therefore the domains of the Fe film
can be observed in some of the pictures, e.g. (a) and (c) in the bottom
part of the image.
magnetized or not-magnetized regions. In the first row the situation for a demagnetizing
field along the easy axis and in the bottom row along the hard axis is presented.
In the images (a) and (c) of Fig. 9.12 unpatterned surroundings of the gratings are
also visible. In these regions clearly the fourfold anisotropy can be detected, which
usually leads to 90◦ domain walls, i.e. in most cases neighboring domains exhibit a
transition from black or white into gray, but almost never from black to white (this
would correspond to a 180◦ domain wall). The direction of the walls is thus along the
hard axis, i.e. 45◦ to the stripes. In these two images it can be observed how the stripes
act as centers for the nucleation of domains in the unpatterned region.
Inside the stripes in the images (a), (c), (e) and (f) basically only black and white
contrasts are observed. The additional uniaxial anisotropy in this regions leads to locally
homogenous magnetized stripes with only 180◦ domain walls which are perpendicular
to the stripes. As 180◦ domain walls are unfavorable in the Fe system, the systems
tries to extend the domain wall width over a larger stripe-length, which can be observed
116
9.3. Results
particularly in image (a). In this case the uniaxial anisotropy is smallest and the regions
of opposite magnetization collinear to the stipes are separated by several small domains
displaying many 90◦ walls. For the cases of the smaller stripe widths the 180◦ wall does
not extend over a larger area because the uniaxial anisotropy favoring a 180◦ domain
wall is increased.
The situation changes slightly if the external field is directed perpendicular to the
stripes, as depicted in image (b) and (d). Because the external field essentially lessens
the uniaxial anisotropy, domains directed perpendicular to the strips are more favorable.
This can be observed in image (b). The regions of the stripes magnetized collinear
with its axis are separated by large areas with domains magnetized perpendicular to
the stripes of rhombic shape. These rhombic central domains (gray) are separated by
triangular edge-domains magnetized along the stripes (black or white). This pattern
fades out by approaching the homogenous magnetized regions giving the impression of
a helix-like structure.
For the case of the smallest stripes (images (e) and (f) in Fig. 9.12) the uniaxial
anisotropy is always big enough to force all domains in a collinear direction with respect
to the stripes.
9.3.2. Polycrystalline Fe-gratings
Next MOKE results obtained from the polycrystalline Fe-film in which five different
gratings have been etched (sample B, see Fig. 9.1(b)) are presented. In this case the
polycrystalline state and the larger thickness of the Fe-film leads to a different magnetization behavior.
Standard MOKE
Fig. 9.13 shows specular MOKE hysteresis curves obtained on sample B with different
gratings and with the magnetic field oriented along and perpendicular to the stripes.
The shape of the hysteresis loops in this case is dominated by the shape anisotropy
induced by the geometry of the gratings. The shape anisotropy is strongest for the case
of the narrowest stripes.
Bragg-MOKE
For these gratings measurements of the Bragg-MOKE effect in the same geometry as for
sat
sample A1 have been performed. The θK
(n) curve shows the same linear dependence
as seen for sample A1. If the interference spots from the gratings of sample B are in
the plane of incidence, which is the case for all measurements performed, the stripes are
perpendicular to the external field, i.e. the Bragg-MOKE hysteresis curves are obtained
in the hard axis configuration.
Selected examples of the measured Bragg-MOKE hysteresis curves are depicted in
Fig. 9.14. Because the measurement was performed in the hard axis direction, the
hysteresis curves have to be compared with the hard axis measurements in Fig. 9.13.
The rows of Fig. 9.14 display the Bragg-MOKE hysteresis curves for one stripe width
at different orders of diffraction n. The columns represent the same order of diffraction
117
9. Fe-gratings
Kerr signal [a.u.]
1
(a) 0.5 µm
(b) 2.1 µm
(c) 2.5 µm
(d) 3.7 µm
0
−1
1
0
−1
−2 −1
0
1
2
−2 −1
H [kOe]
0
1
2
Figure 9.13.: MOKE hysteresis loops measured along and perpendicular to the stripes
for the gratings with (a)wF e = 0.5 µm , (b) wF e = 2.1 µm, (c) wF e =
2.5 µm,(d) wF e = 3.3 µm. The hard axis (dashed line) is measured with
the external field perpendicular to the stripes and the easy axis (straight
line) with the field along the grating.
for different Fe-stripe widths wF e . Obviously the shape of the hysteresis curves do not
always represent the normal MOKE hysteresis curves as displayed in Fig. 9.13. This is
in strong contrast to the previous measurements of sample A.
In detail, for the case of wF e = 0.5 µm (first row in Fig. 9.14) essentially no change of
the shape of the Bragg-MOKE hysteresis curves as a function of n is observed. In the
case of wF e = 2.1 µm (second row in Fig. 9.14) no change for n = 1, positive overshoots
for n = 2 and a negative contribution for n = 3 are observed, which leads to an apparent
increase of the saturation field. In the third row the case of wF e = 2.5 µm is displayed,
where we find no change of the Bragg-MOKE hysteresis curves of n = 1 and n = 3, but
a strong negative contribution, which leads to sharp negative overshoots for n = 2. For
wF e = 3.3 µm (bottom row) almost no change for n = 1 and n = 2 is detected, and
again a negative contribution, which leads to an apparent increase of the saturation field
for n = 3.
For some of the Bragg-MOKE curves the DHL’s (see Sec. 3.4.2) were calculated.
The results are depicted in Fig. 9.15. In this representation the Bragg-MOKE signal
is plotted as a function of the normalized magnetization as obtained from the specular
measurements. This representation is independent on M as a function of H. The
normalized curves in Fig. 9.15 can be viewed as a superposition of a linear and a sine
function. The splitting between the two magnetization paths is small only for n = 2
and wF e = 2.5 µm (c) a significant splitting occurs. These results can be interpreted
118
9.3. Results
n=1
0.05
n=2
n=3
wFe=
0.5µ
0
−0.05
0.05 2.1µ
0
θK [°]
−0.05
0.05 2.5µ
0
−0.05
0.05 3.7µ
0
−0.05
−2
0
2 −2
0 2 −2
H [kOe]
0
2
Figure 9.14.: Bragg-MOKE hysteresis curves from the polycrystalline gratings (sample
B). Only the positive first three orders of diffraction are shown. Each row
in the figure represents the measurements of one grating with constant wF e ,
as indicated in the figure. wF e is increasing from the top to the bottom.
Each column displays measurements of a given constant order of diffraction
(n = 1...3 as indicated in the figure).
using the simple models developed in Sec. 3.4.2. The linear part thus stems from a
remagnetization process due to irregular domain formation or coherent rotation (i.e.
the magnetization component is reduced for decreasing fields, but no correlated effects
119
9. Fe-gratings
(a) w =2.1 n=1
(b) w =2.1 n=2
BMOKE−signal
Fe
Fe
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1
−0.5
0
0.5
1
−1
(c) w =2.5 n=2
BMOKE−signal
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
0.5
m / ms
0.5
1
Fe
1
−0.5
0
(d) w =3.7 n=3
Fe
−1
−0.5
1
−1
−0.5
0
0.5
m / ms
1
Figure 9.15.: Diffraction hysteresis loops, i.e. the Bragg-MOKE signal as a function of
the specular MOKE measurement in the h.a. axis orientation, of the Fegrating. The order of diffraction and Fe stripe width is indicated in the
figure.
occur) and the sine-like contribution can be attributed to edge-domains. If the domain
formation for the two magnetization paths is identical no splitting is expected. This
results will be discussed further in the subsequent sections.
Intensity
Now these results are compared with the measurements of the integrated intensity of
the diffraction spots depicted in Fig. 9.16. Here again oscillating intensities were found
which are due to the diffraction envelope of the single stripes. The intensity minima
are observed near the expected minima of the diffraction envelope of a single Fe-stripe.
From Figs. 9.14 and 9.16 it can be concluded that Bragg-MOKE hysteresis curves with
an anomalous shape coincide with a minimum in the intensity. For wF e = 0.5 µm no
intensity modulation is detected and all curves preserve the shape of the standard curve
measured in specular configuration. For wF e = 2.1 µm the anomalous Bragg-MOKE
curves are close to the intensity minimum at n = 2. In the case of wF e = 2.5 µm
a sharp intensity minimum exists at n = 2, which coincides with the Bragg-MOKE
curve with the strongest anomaly. The intensity pattern for wF e = 3.7 µm exhibits a
120
9.3. Results
w =0.5 µm
log(I) [a.u.]
Fe
wFe=2.1 µm
w =2.5 µm
Fe
wFe=3.7 µm
−6
−4
−2
0
n
2
4
6
Figure 9.16.: Intensity pattern of sample B. The specular reflected intensity is omitted.
minimum at n = 4, which is out of the range of the Bragg-MOKE setup, but already
the Bragg-MOKE curve for n = 3 is slightly anomalous.
Vector-MOKE at the polycrystalline Fe-grating samples
In order to further clarify the origin of the additional components measured in the
Bragg-MOKE curves of the polycrystalline sample (B) two different methods have been
used.
First, a virgin curve of the grating with wF e = 2.1 µm has been measured in BraggMOKE geometry (n = 2) identical to the result in Fig. 9.14 but after the sample was
saturated perpendicular to the stripes. In this case the stripes are first saturated in the
easy axis, than the sample is rotated into the hard axis direction and the Bragg-MOKE
curve is measured starting at zero field. The result is plotted in Fig. 9.17. Obviously
the additional contribution to the Bragg-MOKE signal is increased with this procedure
indicating that the additional component is somehow connected to magnetic domains
magnetized along the stripes, i.e. perpendicular to the field during the Bragg-MOKE
measurements.
Second, from the same sample and grating (B, 2.1 µm) a vector-MOKE measurement
was performed in specular geometry, which is presented in Fig. 9.18. The figure shows
the standard MOKE hysteresis curve in the hard axis as was already shown in Fig. 9.13.
In addition, a measurement is plotted after rotating the sample and the magnet, which is
proportional to the perpendicular magnetization component, MT , for a hysteresis along
the hard axis. It can be seen that this component is non-zero but small compared to the
121
9. Fe-gratings
0.06
θK [°]
0.03
0
−0.03
−0.06
−2000
−1500
−1000
−500
0
H [Oe]
500
1000
1500
2000
Figure 9.17.: Virgin curve of n = 2, wF e = 2.1 µm of the polycrystalline Fe stripe array
after saturation along the stripe axis.
longitudinal measurement. From this one can conclude that the remagnetization process
is mainly governed by domain processes, however, a small transverse magnetization
component exists and is not completely averaged out.
Kerr-microscopy
Kerr microscopy was used to image the domains at the coercive field of all gratings from
sample B. Fig. 9.19 displays some representative examples for hard axis magnetization.
In the top row the magneto-optical sensitivity direction is oriented parallel to the field
and perpendicular to the stripes. In the bottom row the the sensitivity direction is along
the stripes. In Fig. 9.19 measurements from the grating with wF e = 3.7µm (left column)
and wF e = 2.1µm (right column) are presented.
The measurements of the grating with the larger stripe-width shows that only few and
small edge domains with a magnetization along the stripes are formed (dark and light
contrast in (b)). Consistent with this observation, it is found in (a) that the magnetic
domains are mainly oriented perpendicular to the stripes. Moreover the domain pattern
in (a) seems to proof a certain correlation of the domains in adjacent stripes, i.e. the dark
or light domains are extended over more than one stripe. This result can be explained
by the combination of three effects:
• A small uniaxial anisotropy perpendicular to the strips. This was found with Kerr
microscopy at an unpatterned portion of the sample B. The anisotropy is very
small, so that the usual MOKE hysteresis measurements did not detect it.
122
9.3. Results
0.02
0.015
0.01
K
θ [°]
0.005
0
−0.005
−0.01
−0.015
−0.02
−2500 −2000 −1500 −1000 −500
0
500
H [Oe]
1000
1500
2000
2500
Figure 9.18.: Vector-MOKE of wF e = 2.1 µm of the polycrystalline Fe stripe array in the
h.a. axis orientation. The large curve is the longitudinal MOKE curve, the
smaller, gray curve is a plot of the measurement in the perpendicular configuration, i.e. is proportional to the transverse magnetization component
during reversal.
• Dipolar coupling between neighboring stripes. The coupling increases with decreasing distance. This effect is only observed for the broadest stripes.
• Vanishing shape anisotropy. The shape induced anisotropy with an easy axis along
the stripes is reduced with increasing stripe width.
Consistent to this the Kerr microscopy images of the grating with wF e = 2.1 µm
(Fig. 9.19(c,d)) do not show this effect. The image (c) displays no correlation and in
(d) strong edge domains are observed. In this case the dipolar coupling is reduced and
the shape anisotropy is dominant. The small uniaxial anisotropy perpendicular to the
stripes play no role for this stripe-width and separation.
In order to explore the domain pattern any further in Fig. 9.20(a) a magnification of
the Kerr image of the grid with wF e = 2.5 µm (sample B) in the demagnetized state
is displayed. Here, a very regular domain structure with closure domains at the stripe
edges is observed, as depicted schematically in Fig. 9.20(b). In the remanent state (not
shown) one essentially observes similar domains with one magnetization direction in the
interior of the stripes.
123
9. Fe-gratings
Figure 9.19.: Kerr microscopy images of two of the Fe gratings as indicated in the figure. All depicted measurements were performed at the coercive field, when
magnetized along the hard axis. The magneto-optical sensitivity direction
is parallel to the field in the top row and parallel to the stripes in the
bottom row.
9.4. Discussion
9.4.1. Saturation Bragg-MOKE signal
For an interpretation of the results first the dependence of the saturation Bragg-MOKE
signal as a function of the order of diffraction n is discussed, as shown in Fig. 9.6 (sample
sat
A1). This almost linear dependence of θK
(n) was observed not only for the samples
discussed in this chapter, but also for other ferromagnetic interference gratings with
different sample design and different materials. If the angle of incidence, αi , is chosen
sat
to be non-zero, the zero-point of θK
(n) shifts to the point at which the angle of the
diffracted spot crosses the surface normal (see Sec. 8 and [99]). This is a clear indication
that the sign of θK depends on the scattering geometry chosen.
sat
In order to explain the θK
(n) dependence in Fig. 9.6 qualitatively, a Lorentz-Drude
type model for the magneto-optical Kerr effect is assumed and the symmetry of the
124
9.4. Discussion
Figure 9.20.: (a) Kerr microscopic image in the demagnetized state of a grating from
sample B with the lattice parameter d = 5 µm and the stripe width wF e =
2.5 µm. The field direction during demagnetization was perpendicular
to the stripes. (b) Orientation of the magnetization within the domains
schematically
~
setup is considered: The incoming electromagnetic wave with the E-vector
parallel to
the stripes (see Fig. 3.21) induces a current oscillating perpendicular to the applied field.
~ of the ferromagnet cause a tilting
The Lorentz forces from the magnetic induction B
of this current out of the film-plane. The projection of the tilt angle in the direction
of the observer increases with the angle proportional to sin(αf ), thus explaining the
linear dependence at low angles in Fig. 9.6. This geometrical consideration is schematically depicted in Fig. 9.21. In other words, the observed dependence results from the
off-specular magneto-optical constants, which can in principle be measured with the
presented geometry, and which have been calculated in [12] for the case of the polar
sat
Kerr-effect. In addition, it was found that θK
(n) does not dependent on the stripe
width, wF e , i. e. is independent of the diffracted intensity. This result has also been
obtained from theoretical considerations in Ref. [12]. In contrast to this, in Sec. 8 an
sat
(n) was found, which was attributed to the diffraction envelope of a
oscillation of θK
single stripe. The main difference between the present diffraction gratings and those
studied in Sec. 8 is the smaller grating parameter of d = 5 µm compared to d = 20 µm
in Sec. 8.
In Ref. [12] the authors expect an important contribution from the interference between the metallic stripes and intermediate stripes on the substrate. The light diffracted
from the metallic stripes and from the substrate has a phase difference, which should
lead to interference phenomena. This effect has not been observed for our gratings,
125
9. Fe-gratings
Figure 9.21.: Illustration of the effect that leads to the increasing Kerr rotation with
increasing diffraction angle. The incoming beam is s-polarized at perpen~ out of the
dicular incidence. The Lorentz-force results in a tilting of E
sample plane. The projection into the direction of the observer increases
at higher order of diffraction.
because the height h = 20 . . . 50 nm of the stripes is small compared to the wavelength
λ = 632 nm of the illuminating laser. As has been discussed in Sec. 3.4.3, the contribution from interference phenomena depends on two factors: first, the phase shift between
the diffracted beams of grating and substrate, and, second, on the relative amplitudes.
The present result can be interpreted by assuming that the amplitude of the diffracted
waves from the substrate is always small compared to the amplitude diffracted from the
ferromagnetic grating. In this case the observed Kerr rotation (according to Eq. 3.34)
is mainly proportional to the Kerr effect of the ferromagnetic grating and essentially no
interference effects can be observed. The detected intensity pattern in Figs. 9.16 and 9.8
can easily be modelled by assuming a simple transmission grating as given in Eq. 3.16.
The substrate-grating is made of sapphire in the case of sample B which, as an insulator,
has a smaller reflectivity constant than the metallic stripes. The situation is different
in the case of sample A, where also the substrate-grating is of metallic material. One
possibility is that the etching procedures have created more roughness in the grooves
than on top of the stripes, and therefore have reduced the reflected intensity from in the
grooves.
9.4.2. Shape of Bragg-MOKE curves of the single crystalline
sample
To start with, the Bragg-MOKE curves for sample A1 for the case of the Fe-stripes
oriented along the in-plane Fe(110) direction (Fig. 9.7) are discussed. The standard
MOKE curves in this direction (Fig. 9.3(b)) clearly exhibit contributions from second
order MOKE effects. As discussed in [42, 40, 43], the Kerr angle in the longitudinal
126
9.4. Discussion
configuration can be expanded as (the polar Kerr effect is neglected)
θK = αML + βML MT + γMT2 ,
(9.4)
where ML and MT are the longitudinal and transverse magnetization components of the
~ . In the longitudinal MOKE geometry ML is the component
magnetization vector M
~ in
in the film plane along the external field and MT denotes the component of M
the film perpendicular to H. The coefficients β and γ depend on the ferromagnetic
material and the optical geometry. The occurrence of second order MOKE effects proves
that the magnetization rotates into the direction of the magnetic field. Details of the
magnetization process are very sensitive to the exact in-plane orientation (angle χ) of the
anisotropy axes relative to the external field [10]. This is the reason why the situation
of the virgin film (Fig. 9.3) is not exactly reproduced after patterning (Fig. 9.7). The
second term in Eq. 9.4 produces an asymmetric contribution to the MOKE hysteresis
curves, which obviously increases in the Bragg-MOKE effect as compared to the standard
MOKE measurement. The increasing second order effects can be explained with the
small angles involved in the Bragg-MOKE geometry at perpendicular incidence. Second
order effects increase in specular MOKE if αi = αf is small, for the case of perpendicular
incidence the linear longitudinal component completely vanishes and only second order
terms remain [42]. This geometry is comparable to the present case with αi = 0.
Unfortunately, a theory for second order effects which relaxes the constrain αi = αf is
not available at present.
Aside from the increasing second order effects, the Bragg-MOKE curves of sample A1
and A2 do change shape significantly as compared to the specular curves. As discussed
in Sec. 3.4.2 this must be connected to the fact that on average the magnetization distribution does not form domains inside the stripes. This can be seen from the different
Kerr microscopy images taken. The image of sample A1 (connected Fe stripes, Fig. 9.4)
shows domains which are larger than the Fe stripes, the magnetic structure is not correlated to the diffracting system. Therefore averaging over the stripes will just result in a
homogenous magnetization with changing amplitude as given by the specular hysteresis
curve. This case is equivalent to the coherent rotation model (of Sec. 3.4.2) and therefore
a Fourier decomposition can only result in changing amplitude but not changing shape
of the hysteresis.
The same argument holds for the Bragg-MOKE curves of sample A2. As was measured
by Kerr microscopy the magnetization in the stripes are not coupled and always directed
up or down parallel to the stripes. The regions with domains form to some extent broad
180◦ walls which, on average, have zero magnetization. The Fourier transform of this
case will again change only the amplitude but not the shape of the curves.
9.4.3. Shape of Bragg-MOKE curves of the polycrystalline sample
The situation is different for the Bragg-MOKE hysteresis curves of the polycrystalline
Fe-film (sample B). For this sample no macroscopic magnetocrystalline anisotropy exist
and the domain structure is essentially defined by the geometry of the stripes. The
stripes induce an uniaxial magnetic anisotropy, causing the formation of closure domains
at the edges of the stripes for a magnetization direction perpendicular to the stripes.
127
9. Fe-gratings
Kerr microscopy images of domains of some of the Fe grids (sample B) are reproduced
in Figs. 9.19 and 9.20.
For the magnetization reversal of this polycrystalline film one would not expect to observe contributions in the specular MOKE signal from the second order term in Eq. 9.4,
since the transverse component MT for a randomly oriented polycrystalline grain structure has as many positive as negative components. This is consistent with the fact that
no anomalous hysteresis loops in the specular MOKE measurements are found for any
of the polycrystalline Fe-gratings (see Fig. 9.13). Furthermore, the observed anomalous
hysteresis loops in Fig. 9.14 display a symmetry with respect to the origin, which is not
consistent with second order effects [14].
As has been discussed in Sec. 3.4.1 and 3.4.2 the anomalous shape of the BraggMOKE curves has been proven to be caused by the magnetic domain structure of the
patterned films. Following the derivation in Ref. [11, 15], the Bragg-MOKE hysteresis
curve for a diffraction spot of order n corresponds to the nth Fourier component of
the mean magnetization distribution within each stripe, i.e. is given by the magnetic
form factor of a single stripe, see Eq. 3.21. Knowing the magnetic domain structure as
a function of the applied field one can calculate the Bragg-MOKE hysteresis curve of
order n by numerical integration. In special cases the domain state can be described by
simple models which can be solved analytically, as has been shown in Sec. 3.4.2. The
domain structure of the Fe gratings B, as have been imaged in 9.19 and 9.20 can be
described by the edge-domain model of Sec. 3.4.2, i.e. a central domain aligned with
the field and edge domains of effectively zero magnetization. The simulations of this
model in Fig. 3.12 show that the diffraction hysteresis loops (DHL, see Sec. 3.4.1 for the
definition) are basically sinusoidal functions of the magnetization and are identical for
the ascending and descending branch. The strongest effect can be found for the case of
the width of the stripes being half the grating parameter. In this case the simulation is
qualitatively identical with the measurement, as depicted in Fig. 9.15.
A pronounced anomalous shape of the hysteresis curve is expected if the form factor
fm (n) vanishes in saturation and only the formation of domains gives a finite value for
fm . This is the case if
g wF e =
2π
n wF e = ±2π, ±4π, . . . .
d
(9.5)
For the series of Bragg-MOKE curves in Fig. 9.14 the reflection n = 2 and wF e = 2.5 µm
fulfills this condition and actually shows the most anomalous shape of the hysteresis loop.
The shape can also be explained regarding the calculations presented in Sec. 3.4.2.
The Bragg-MOKE curves were converted into DHL’s in Fig. 9.15. In this representation
they can be compared to the calculated curves in Sec. 3.4.2. Most of the curves can
be explained with a superposition of the edge-domain and the coherent rotation model.
The first of this models covers the case that the effective width of the stripes changes
during the remagnetization process and the latter that the magnetization changes homogenously its magnitude (coherent rotation or irregular domain formation). The calculations showed that in this cases sine like curves superimposed to linear curves are
expected, the sine like component being maximal at the situations given by Eq. 9.5.
The average domain distribution during remagnetization is therefore a combination of a
general decrease of the magnetization due to irregular domain formation combined with
128
9.5. Summary and Conclusions
the correlated formation of edge domains. For this cases a closed DHL is expected, i.e.
the domain formation is reversible, as is found in Fig. 9.15(a,b,d). This seems not to be
the case for the measurement of n = 2 and wF e = 2.5 µm, as depicted in Fig. 9.15(c). In
this case also a mechanism where the two magnetization paths are not identical have to
be taken into account. This was demonstrated in Sec. 3.4.2 with the two 180◦ domainwall model. However, an exact fit of the measured DHL to the calculated curve was not
possible, the model obviously gives only qualitative results.
This explanation is consistent with the measurements of the virgin curve after transverse saturation as was shown in Fig. 9.17. The area of the edge domains obviously
increases when magnetizing along the strips. Therefore the virgin curve displays an
enhanced anomalous contribution. Furthermore, the Vector-MOKE measurement in
Fig. 9.18 proves that for the hard axis magnetization process the transverse component
of the magnetization is small, i.e. the direction of the edge domains cancel each other
out.
From the Bragg-MOKE measurements no sign of dipolar coupling between the magnetic stripes was found. The coupling proved by Kerr-microscopy in the case of the
broadest stripes presumably only exist for very small fields. The observed coupling
in Fig. 9.19 leads to domains magnetized perpendicular to the stripes, the absence of
edge domains and the domains being extended over more than one stripe. In the onedimensional models discussed in Sec. 3.4.2 this would only lead to an averaged decrease
of the magnetization for small fields, as it is described by the coherent rotation model.
Dipolar coupling that would lead to any alternating magnetization in adjacent stripes
(i.e. the one-dimensional equivalent of a checker-board pattern) is not covered by the
models in Sec. 3.4.2, however, is also not observed by Kerr-microscopy for the present
gratings.
9.5. Summary and Conclusions
In this work the Bragg-MOKE effect in longitudinal geometry on three different types
of Fe-gratings (single crystalline connected, unconnected and polycrystalline) , keeping
the grating parameter constant, was investigated. The results can be separated into the
effects concerning the Bragg-MOKE amplitude in saturation and changes of the shape of
the hysteresis loop. Regarding the amplitude in saturation, a monotonous dependence
on the diffraction order is found. A qualitative explanation for this effect based on
geometrical considerations is given.
Concerning the shape of the hysteresis loops there are two different cases for the
observation of anomalous loops. If rotation processes are involved, as e.g. for the single
crystalline Fe-film with the field parallel to the hard magnetic axis, second order terms
contribute asymmetric components to the MOKE signal for specular reflection as well
as in higher order diffraction. For n 6= 0 the second order terms sometimes appear to
be enhanced compared to the specular case.
A different mechanism producing anomalous hysteresis loops which can be observed
only in higher order Bragg-MOKE is correlated with the domain patter formed during
the magnetization process. For certain orders of diffraction the domain formation can
drastically alter the shape of the hysteresis curve. Contrary to the case when second
129
9. Fe-gratings
order effects contribute to the hysteresis curve, the hysteresis loops in this case keep
inversion symmetry [14]. The anomalies in the hysteresis loops in diffraction of order
n represent the nth order Fourier component of the domain pattern and can get very
pronounced if the symmetry of the domain pattern matches with the wave vector of
the Fourier component. It has been shown that although the domain pattern is very
similar for comparable stripe width, the anomalies in the hysteresis loops depend very
sensitively on the ratio of the stripe width and the grating parameter.
130
10. Co gratings on a Fe-film
10.1. Introduction
In this chapter the magneto-optical Kerr effect in the diffracted light from Co gratings
with a grating parameter ranging from 5 to 15 µm on top of an Fe thin film is examined.
The Co gratings are decoupled from the Fe film with a Cr spacer layer. The system
forms a spin-valve structure as the coercive fields from Fe and Co are different. The
hysteresis loops measured at the diffracted spots reveal interesting amplification of the
Kerr signal in the field regime where the magnetization of Fe and Co is antiparallel.
In the field of magneto-electronics spin-valve structures play an important role. These
structures consist of two magnetic layers with different switching behavior so that in a
certain field region the magnetization inside the two layers is antiparallel leading to an
enhanced magneto-resistance. These systems can be realized with two methods: either
one layer is coupled to an antiferromagnet which provides a unidirectional pinnig and the
other layer is free. The coupled layer displays the exchange bias effect and has therefore a
modified switching field. The other possibility is to use two layers with different coercive
fields because of different materials used or different thicknesses prepared.
For the present study the second approach was used. An epitaxial Fe layer with a
small coercive field was deposited. On top a Co layer with a larger Hc was grown. The
two layers where completely decoupled by a thick Cr interlayer. Due to the surface
sensitivity of MOKE, magneto-optical measurements only probe the top layer of this
stack. Therefore the top Co and Cr layers were etched into different stripe arrays in
order to enable one to measure hysteresis curves of both layers simultaneously.
Furthermore, this setup enables to use the Bragg-MOKE effect as an additional tool
for the investigation of the system. The grating structures investigated in this chapter
always show a ratio of width to lattice parameter of 12 . In this situation the intensity of
the even order diffraction spots is strongly reduced and the signal is very sensitive to edge
domains as has been shown in Sec. 9.4. The prepared gratings have grating parameters
ranging from 5 to 15 µm. Increasing the width and grating parameter of the structures
should decrease potential dipolar coupling and should decrease the uniaxial anisotropy
induced in the Co stripes. As was shown in the previous sections, the amplitude of the
Kerr effect as a function of the diffraction order is a linear function. With the present
sample design it will be tested if this effect is indeed a function of the order of diffraction
or a function of the diffraction angle. Furthermore, in Sec. 8 interesting oscillations of
sat
θK
(n) were found for rather large grating parameter of 20 µm, an effect not reproduced
for the case of smaller Fe gratings in Sec. 9.4.
However, in order to separate the signals from the underlaying Fe layer and the Co
stripes it would also be a good idea to use asymmetric grating designs (width not half
131
10. Co gratings on a Fe-film
Figure 10.1.: Atomic force microscopy image of one of the Co on Fe gratings. The
imaged grating has a grating parameter of d = 5µm. The width of the Co
stripes is wCo = 2.8 µm and the hight of the structures is t ≈ 410 nm.
of grating parameter), because the diffraction envelope would lead to different weighting
of the two signals depending on the diffraction order under investigation, as has been
shown in Sec. 8.
10.2. Experimental details
Using the standard UHV-MBE process as explained in Sec. 4.1 a thin film sample on
Al2 O3 (11̄02) was prepared. On a Nb(001) (6 nm) / Cr(001) (27 nm) buffer system a
20 nm thin Fe(001) film was evaporated. The growth quality was checked in-situ using
RHEED. On top a 14 nm Cr spacer and a 27 nm Co layer was prepared, which is capped
with a thin 2 nm Cr film for oxidation protection. The sample quality was checked with
standard analysis techniques. The thin film stacking sequence is sketched in Fig. 10.2.
From the thin film sample several grating structures with different grating geometries
were fabricated in order to keep the structure of the gratings constant with respect to film
thickness and crystal quality. The gratings were produced with electron beam lithography and ion etching, as described in Sec. 4.1. The resulting grating morphology was
visualized using scanning electron microscopy and atomic force microscopy techniques,
a AFM image is presented in Fig. 10.1. The geometries of the gratings are depicted in
Fig. 10.2. Because of the inhomogeneity of the etching process, the difficulties in fitting
XRD data of such complex films and the errors of the AFM and EDX methods used,
the total inaccuracy of all thicknesses given is ≈ 10%. The lateral dimensions are better
defined, the error for wCo and d is < 1%. The width of the stripe, wCo , is approximately
half of the grating parameter, d, for all gratings. For technical reasons the deviation
from this ideal case is higher for the gratings with smaller grating parameters.
Before the etching process standard MOKE measurements of the samples were carried
out, in order to have a reference result of the unstructured film. After the patterning
132
10.3. Results
Figure 10.2.: Geometry of the five gratings prepared from the Fe/Cr/Co trilayer. The
Co strips and the Fe film in the grooves is still capped with a thin (≈2 nm)
Cr film, which is not shown in the drawing. The thicknesses t of the film
system is constant for all gratings. The grating parameters d and the stripe
widths wCo for the five gratings are given in the figure.
process Bragg-MOKE hysteresis curves were measured. The MOKE and Bragg-MOKE
technique was described in Sec. 3.5.3. For the present study the angle of incident was
zero (perpendicular incidence). As the grating parameter is different for the five gratings
under investigation the diffraction angle varies with the simple relation d sin αf = nλ,
where αf is the diffraction angle (see Fig. 3.21), n the order of diffraction, and λ =
632.2 nm is the Laser wavelength. This holds only if the angle of incidence αi = 0◦ .
Therefore an increasing number of diffraction spots was observed in the accessible range
of αf = ±40◦ of the setup for an increasing grating parameter.
10.3. Results
10.3.1. Standard MOKE measurements
The data obtained from the unstructured film is presented in Fig. 10.3(a). The MOKE
measurement is only sensitive to the Co film on top of the sample, because the Co film
and the Cr spacer are too thick to detect signals from the underlying Fe film. The
in-plane rotation angle, χ, for the measurement in Fig. 10.3(a) is chosen identical to
the Bragg-MOKE measurements, i.e. the hysteresis loop is directly comparable to the
Bragg-MOKE measurements of the structured sample. The orientation of χ is almost
parallel to an easy axis of the Co film as can be seen from the hysteresis loop with 100 %
remanence and a coercive field of HcCo = 340 Oe. Fig. 10.3(b) shows a hysteresis curve
of a Fe film on the same substrate buffer system, at identical growth conditions as the
sample described above, but without the Cr spacer and the Co layer. Therefore this
hysteresis represents the single Fe magnetization curve in the unstructured sample. The
coercive field is HcF e = 15 Oe< HcCo and the nucleation field, HNF e , of the Fe hysteresis
133
10. Co gratings on a Fe-film
0.05
(a)
0
−0.05
θK [°]
0.05
(b)
0
−0.05
0.05
(c)
0
−0.05
−1000
−500
0
H [Oe]
500
1000
Figure 10.3.: Standard (specular) MOKE measurements of (a) the unstructured film,
only the Co film contributed to the MOKE signal; (b) an Fe film prepared
identically as the Fe/Cr/Co sample, but without the Cr/Co layers on top;
(c) the Co grating with d = 7.5 µm.
loop has the same sign as the coercive field and is only slightly smaller. The measured Fe
hysteresis curve is not perfectly square, but after the magnetization switched, i.e. at a
little higher field than HcF e the hysteresis curve exhibits a small ”kink”. This behavior is
characteristic for a magnetic film with fourfold crystalline anisotropy and a superimposed
uniaxial anisotropy as is often found for Fe films on this particular substrate / buffer
system. The particular epitaxial relation of Nb(001) on Al2 03 (11̄02) leads to a tilting of
the (001) axis. The resulting steps of the Fe surface induce the uniaxial anisotropy. The
magnetization reversal does not consist of a simple 180◦ domain wall, but after switching
into a direction of a local minimum of the anisotropy energy the magnetization rotates
into the direction of the external field.
Because of the different coercive fields a spin valve behavior is expected in the combined sample. The standard MOKE hysteresis loop obtained from the reflected light of
134
10.3. Results
one of the Co gratings (d = 7.5 µm) is shown in Fig. 10.3(c). This combined hysteresis
loop displays steps which are characteristic for spin-valve systems. The coercive field of
the Fe film, HcF e , is preserved, the etching has not affected this property, in addition
the small ”kinks”of the pure Fe loop (b) are still visible in the Fe/Cr/Co grating measurement (c). In contrast, the coercive field of the Co stripes, HcCo,stripes , increased to
≈ 700 Oe after the patterning process (compare (a) and (c)). In addition, the Co stripes
width, wCo , has no effect on HcCo,stripes .
10.3.2. Bragg-MOKE measurements
Shape of Bragg-MOKE curves
Bragg-MOKE measurements on all observable diffraction spots from all gratings were
performed. The most obvious results is, that only two principally different shapes of
Bragg-MOKE hysteresis curves were detected. As an example normalized Bragg-MOKE
curves for the grating with d = 7.5 µm are presented in Fig. 10.4. Displayed are the
first six orders of diffraction for positive αf (positive n). It can be clearly seen that the
shape only differs for even and odd n. In order to point out the differences between
even and odd n, the case for n = 1 and 2 is displayed in an extra plot in Fig. 10.5.
For the odd orders of diffraction (Fig. 10.3(a,c,e)) the measured Kerr rotation remains
constant for decreasing field from saturation towards zero. At a small field just before
the external field is reversed, the Kerr rotation decreases into a steep dip around zero.
Note that this decrease cannot be identified with the nucleation field of Fe, because in
Fig. 10.3 HNF e was found to be positive, whereas in the Bragg-MOKE measurements the
first decrease of the Kerr rotation at negative external fields. After the dip the Kerr
rotation increases again and in the spin-valve region (between HcF e and HcCo ) a Kerr
rotation is observed which is essentially governed by the Co loop and not by the Fe
loop in contrast to the standard MOKE curve. In the standard MOKE measurement
(Fig. 10.3(c)) the Kerr rotation in the spin valve region has the same sign as the Kerr
rotation in saturation, whereas in the Bragg-MOKE measurements the Kerr rotation
in this region has the same sign as in remanence. Therefore the coercive field in the
standard MOKE measurement reproduces HcF e and in the Bragg-MOKE measurement
it reproduces HcCo of the patterned film.
The Bragg-MOKE hysteresis curves for even order of diffraction are reproduced in
Fig. 10.4(b,d,f). For these curves the coercive field is also that of the Co stripes and
the influence of the Fe film is smaller than in the standard MOKE curve. However, the
shape is different to the shape of the odd order diffraction loops. The dip around zero is
not observed and the Kerr rotation in the spin-valve regime exhibits a constant decrease
when the field is increased. In addition for n = 2 in Fig. 10.4(b) small kinks around
HcCo are detected. This feature exists only for n = 2, the higher even order of diffraction
exhibit only small steps at this position, see Fig. 10.4(d,f). For even order of diffraction
the apparent change in sign of TNF e is also observed as for the odd order loops, but due
to the absence of the ”dip” it is less pronounced.
135
10. Co gratings on a Fe-film
1
(a) n=1
(b) n=2
(c) n=3
(d) n=4
(e) n=5
(f) n=6
0
−1
θK / θK
sat
1
0
−1
1
0
−1
−1 −0.5
0
0.5
1
−1 −0.5
H [kOe]
0
0.5
1
Figure 10.4.: Normalized Bragg-MOKE hysteresis curves of the first six positive orders
of diffraction of the grating with wCo = 7.5 µm.
Bragg-MOKE amplitude
The amplitude of the Bragg-MOKE curves as a function of the diffraction angle are
displayed in Fig. 10.6(a-e) for all five gratings under investigation. An increase of the
measured Kerr rotation for increasing αf and a change of sign for αf = αi = 0 is
observed. The dependency is almost linear with a small superimposed oscillation, which
has a period of ∆n = 2. The superimposed oscillation is more pronounced for larger
grating parameters. For d = 5 (Fig. 10.6(a)) almost no oscillation is found, whereas the
oscillation is very strong for the largest grating parameters observed.
In Fig. 10.6(f) the result of a standard angle-dependent MOKE measurement of the
136
10.3. Results
1
n=1
n=2
θK / θsat
K
0.5
0
−0.5
−1
−2000
−1000
0
H [Oe]
1000
2000
Figure 10.5.: Normalized Bragg-MOKE hysteresis curves for n = 1, 2 of the grating with
wCo = 7.5 µm.
grating with d = 7.5 µm is shown. In this case the condition αi = αf is fulfilled (n = 0).
It is clearly seen that the measured Kerr rotation for the specular geometry is smaller
than in the Bragg-MOKE geometry for αi = 0 and n 6= 0. Assuming a linear function
∆θsat
sat
for θK
(αf ) one can extract a slope of ∆αKf ≈ 4 × 10−3 . For the specular case in Fig.
10.6(f) the slope is only ≈ 8 × 10−4 . Therefore the measured Kerr signal is increased
effectively by a factor of 5 by using the Bragg-MOKE geometry.
10.3.3. Intensity measurements
The result of measurements of the integrated intensity of the diffraction spots is shown
in Fig. 10.7 for the five gratings as a function of the diffraction angle. As the ratio
between the Co stripe width and the grating parameter is nearly 12 for all gratings, the
oscillation of ∆n = 2 is expected. However, as mentioned above, the gratings deviate
slightly from this ideal case. wCo is up to 12% larger than d/2 for the smallest grating
parameters. Therefore the oscillations in Fig. 10.7 are more and more pronounced when
increasing the grating parameter. Note that the oscillation of the integrated intensities
correspond to the oscillations of θK sat(n) in Fig. 10.6. The even orders of diffraction
show a strongly reduced intensity, which correspond to the minima of the oscillation in
sat
the θK
(n) functions. Furthermore the shape of the hysteresis curve changes for odd
and even n (see Fig. 10.4), i.e. changes with constructive or destructive interference
conditions.
137
10. Co gratings on a Fe-film
0.1
(a) d=5µm
(b) d=7.5µm
(c) d=10µm
(d) d=12µm
(e) d=15µm
(f) d=7.5µm,specular
0
−0.1
K
θsat [°]
0.1
0
−0.1
0.1
0
−0.1
−40 −20
0
20
40 −40 −20
αf [°]
0
20
40
Figure 10.6.: (a)-(e): saturation Bragg-MOKE rotation as a function of the diffraction
angle αf for the gratings as indicated in the figure. For all Bragg-MOKE
measurements the angle of incidence αi = 0. (f): saturation Kerr-rotation
for the grating with d = 7.5 µm as a function of the reflection angle αf = αi .
The scale in all subfigures is identical.
10.4. Discussion
There are basically three features which can be separated: first, the increased Kerr effect
in the spin valve region, second, the change of shape of the Fe curve (dip around zero)
and the change of shape of the Co curve (different slope around Hc for even order loops)
sat
and, third, the behavior of θK
(αf ).
138
10.4. Discussion
16
14
d=15 µm w
12
=7.5µm
Co
10
d=12 µm wCo=6.2µm
log(I) [a.u.]
8
6
d=10 µm wCo=5.2µm
4
2
d=8 µm wCo=4.2µm
0
−2
d=5 µm wCo=2.8µm
−4
−60
−40
−20
0
α [°]
20
40
60
f
Figure 10.7.: Integrated intensities of the diffraction spot of the five gratings. The grating parameter is indicated in the figure. The specular intensity (n = 0
is omitted.
10.4.1. Increasing Kerr effect in the spin valve region
The change of Kerr amplitude in the spin-valve region can be explained by the superFe
Co
position of the signal of the two subgratings. Assume θK
(H) and θK
(H) to the Kerr
loops of the Fe and the Co layer, respectively. Measurements of these curves were given
in Fig. 10.3(a) and (b). The resulting signal of the combined sample can viewed as the
m
mean value, θK
of the two signals:
m
θK
(H) =
Fe
Co
θK
(H) + θK
(H)
.
2
(10.1)
The result of this is plotted in Fig. 10.8(a). A better choice is to add the two curves
up coherently taking phase and amplitude differences into account. The reflected beam
from the Fe grating is assumed to be given by:
EsF e
EpF e
!
=
Fe
E0F e eiφ
Fe
cos θK
,
Fe
sin θK
!
(10.2)
where φ is a relative phase shift and E0 is the reflected amplitude. An equivalent
expression shall be valid for E Co . The combined signal is than:
EpF e + EpCo
θK = arctan
.
EsF e + EsCo
!
(10.3)
139
10. Co gratings on a Fe-film
0.05
(a)
0
−0.05
−1000
−500
0
500
1000
−500
0
500
1000
−500
0
H [Oe]
500
1000
0.05
θK [°]
(b)
0
−0.05
−1000
0.05
(c)
0
−0.05
−1000
Figure 10.8.: (a) median of the Fe and Co hysteresis as measured in Fig. 10.3(a) and (b);
(b) and (c) display the addition of the two curves according to Eq. 10.3
for the case φF e = φCo = 0, E0F e = 1 and E0Co = 5 in (b) and for the case
E0F e = E0Co = 1, φCo = 0 and φF e = 0.45π in (c).
This is plotted for the case φF e = φCo = 0, E0F e = 1 and E0Co = 5 in Fig. 10.8(b) and
for the case of E0F e = E0Co = 1, φCo = 0 and φF e = 0.45π in Fig. 10.8(c). It is clearly
seen that an enhanced Co amplitude or a large phase shift can result in the observed
effect. The large phase shift of nearly π2 cannot be due to the height difference of the two
gratings as was discussed in Sec. 3.4.3. Therefore it can be concluded that the enhanced
Kerr rotation in the spin valve regime is additionally due to an increasing amplitude
of the light diffracted by the Co grating compared to the Fe grating. This cannot be
attributed to the diffraction envelope of the two subgratings as in the present study the
width of the Fe and Co stripes are nearly equal. The increase in relative amplitude of
the Co grating can be explained:
• If the etching process is stopped too early the Fe film remains covered with a Cr
film, whereas the Co film may be already uncovered.
140
10.4. Discussion
• If the etching process continued too long the Fe film may be thinned out more
than the Co film.
• the etching process may roughen the surface of the Fe layer more than the Co
layer.
10.4.2. Shape of Bragg-MOKE curves
As already mentioned, the shape of the Bragg-MOKE curves is altered with respect
to the specular curves in two respects. First the dip of the loops around zero will be
discussed:
The dip around zero external field, mainly observed in the odd order loops in Fig. 10.4,
can be attributed to the Fe film because it is far from the Co coercive field in a region
where the Co stripes show full remanence. In Sec. 9.4 (Fig. 9.9 hysteresis loops of single
crystalline Fe stripes where shown. In that situation the additional uniaxial anisotropy
resulted in hysteresis curves with a sharp step just before reaching zero field, i.e. a
negative nucleation field HN . Assuming that the Fe film of the present study also
exhibits an induced uniaxial anisotropy it can be concluded that equivalent hysteresis
loops would be measured for the Fe film separately. However, originally it was intended
not to change the properties of the underlying Fe film. This obviously was not achieved.
An additional uniaxial anisotropy in the Fe film can be induced in two ways:
• Either, the Co stripes on top couple to the Fe film via their dipolar fields
• or, the etching process already modified the Fe film.
The uniaxial anisotropy is only observed at the diffracted intensities because in this case
only the parts of the sample having the grating period are filtered out. Contributions
of the surroundings or of not correlated domain formation are reduced.
The difference of shape of the curves of even and odd order of diffraction is mainly
observed around the coercive field of the Co stripes. In the odd order loops the behavior
is identical to the specular loops when taking the effects discussed above into account.
For the even order loops basically the slope of the Co hysteresis loops seems to be altered.
An additional component with a Kerr rotation opposite to the normal loop around Hc
would explain the observation. Such an additional component was found in Sec. 9.4 for
n = 2 and w = d/2 and was attributed to the existence of edge domains. As discussed in
Sec. 9.4 and Sec. 3.4.2 the Bragg-MOKE curve at diffraction order n is proportional to
the nth order Fourier component of the magnetization distribution of the stripes. Edge
domains lead to an effective reduction of the stripe width and can cause the observed
effects. In addition, the formation of closure domains at the edges is expected because
of the strong demagnetizing fields for an external field perpendicular to the edges, see
Sec. 2.
10.4.3. Bragg-MOKE amplitude
sat
(αf ) clearly consists of two contributions. An almost linear increase of
The curve of θK
sat
θK (αf ) was observed earlier, e.g. in Sec. 9.4. In was explained there using a straightforward picture and in the frame work of a Lorentz-Drude model of the Kerr effect,
141
10. Co gratings on a Fe-film
see Fig. 9.21. Superimposed to this an oscillating contribution is found in Fig. 10.6.
This can be explained by the interference effects of the underlaying Fe layer and the
Co grating, as it was discussed in Sec. 3.4.3. An example of this effect was plotted in
Fig. 3.15. It is clearly seen that a width to grating parameter ratio of 12 can lead to an
oscillation period of ∆n = 2. However, in the situation plotted in Fig. 3.15 the intensity
minima correspond to Kerr amplitude maxima, just opposite to the case observed in the
present study, see Fig. 10.6. This discrepancy may be solved if the actual situation of
two magnetic subgratings is taken into account in the simulations.
The oscillating contribution increases with increasing grating parameter. This is due
to the fact that the ideal case of w/d = 1/2 is approached for increasing grating parameters. In addition, an oscillation is better observable if the points are close to each other
and the number of points is much larger than the oscillation period, which also leads to
an apparent increase of the oscillating contribution with the grating parameter.
10.5. Summary and Conclusion
A spin valve like structure was under investigation. The first (Fe) layer was separated
by the second (Co) layer by a thick 20 nm Cr interlayer. The top Co layer and the Cr interlayer were patterned into different grating structures. In this study it has been shown
that Bragg-MOKE is a valuable tool for the investigation of such complex ferromagnetic
microstructures.
• The existence of edge domains was observed in the Co stripes.
• Several hints indicate that a uniaxial anisotropy is induced in the underlying Fe
layer.
In addition, the Bragg-MOKE amplitude is increased significantly which may help technically to analyze these kind of structures. However, in order to study further the
Bragg-MOKE effect it will be instructive to correlate the present measurements to Kerr
microscopy images. Furthermore the hysteresis curves of the two subsystems could be
measured separately using soft x-ray scattering techniques. This will assure the above
conclusions and also help in the understanding of the observed effects. The soft x-ray
techniques can also be used in a diffractional setup which may open up a new and
challenging area of research. Moreover, a decent theoretical treatment of the observed
optical effects is desirable.
142
11. Further measurements
This chapter groups two different experiments which are too short for discussing them
in separate chapters, but give additional important insight in the nature of the BraggMOKE effect that they make them worth mentioning.
11.1. Diffuse Kerr effect
In order to further clarify the dependence of the saturation Kerr rotation as a function
of the diffraction angle an experiment was designed which also allows to measure in the
off-specular condition but without actually fabricating a grating structure.
The sample prepared is rather simple: the unpolished backside of a standard sapphire
substrate was covered with a 30 nm thick Fe film using the sputter process as explained
in Sec. 4.1. The resulting film is extremely rough and no specular reflection can be
observed when the sample is mounted in the MOKE setup and subjected to the laser
0.07
0.06
0.05
K
θsat [°]
0.04
0.03
0.02
0.05
θK [°]
0.01
0
0
−0.05
−0.01
−0.02
−2000 −1000
−0.03
0
10
20
α [°]
30
0
H [Oe]
1000
40
2000
50
f
Figure 11.1.: Saturation Kerr rotation as a function of αf for a diffuse scattering Fe
film and perpendicular incidence. In the inset an example of the measured
hysteresis curves is plotted.
143
11. Further measurements
beam. Instead, a diffuse intensity distribution is observed in the complete solid angle
of 2π above the film plane. The angle of incidence of the laser was again chosen to
be perpendicular to the film plane. In this situation the diffuse scattered intensity in a
certain solid angle around three reflection angles was collected using a two lens setup and
focussed onto the Kerr-detector. The solid angle was comparatively small, the aperture
was 4 cm at a distance of 30 cm from the sample. The angle of reflection chosen were
32, 40 and 50◦ .
sat
The result is displayed in Fig. 11.1. The main panel shows the results of θK
(αf ). The
saturation Kerr rotation is the maximum value measured up to the field range of 2 kOe.
sat
The straight line is a linear fit to the data and the origin showing that θK
(αf ) indeed
has the same dependence as found from the grating experiments, thus fully justifying
the simple explanation of the experiment in terms of the Lorentz-Drude picture. The
inset in Fig. 11.1 shows the measured hysteresis at 50◦ . The measurement is of course
somewhat noisy, however, it proves that the MOKE hysteresis curves can be measured
also in the diffuse scattered light of rough surfaces.
11.2. Fe grating with giant Kerr rotation
In this section a set of samples produced from a 30 nm thick polycrystalline, sputtered
Fe film on sapphire (a-plane) is under investigation1 . The grating parameter is d =
4, 6, 8 µm and the Fe stripe width is wF e = 1.9, 2.7, 3.7 µm for the three gratings,
respectively. The detailed sample preparation technique is discussed in Sec. 4.1. Thus
wF e is ≈ 10% smaller than d/2, however, a strong ∆n = 2 oscillation in the intensity
spectrum is found. In this respect the three gratings are comparable to the set of samples
measured in Sec. 10 and the motivation for this study also follows the lines discussed
there.
The standard longitudinal MOKE curves show an increasing uniaxial anisotropy with
decreasing stripe width, but as wF e is generally larger than the Fe stripes discussed in
Sec. 9.4 the effect is less pronounced. Some examples of Bragg-MOKE hysteresis loops
(in the hard axis orientation) are depicted in Fig. 11.2. It is clearly seen that for n = 1
no anomalous hysteresis is detected. The same is true for other odd order of diffraction
loops. The loops for n = 2 show a different behavior. For d = 6 µm an additional
component is measured, which is identical to the measurements reported in Sec. 9.4.
However, this was only observed for d = 6 µ. For larger (d = 8) and smaller (d = 4) Fe
stripe widths this component was not observed.
The reason for the anomalous shape of the hysteresis cures for n = 2 is, as discussed
in the previous chapters, the existence of edge domains which lead to a variation of the
effective width of the stripes for the magneto-optical detection for the setup. The Fourier
transformation (Sec. 3.4.2) of this remagnetization behavior leads to sine-like additional
components in the diffraction hysteresis loop (DHL, see Sec. 3.4.2) which result in the
observed modification of the Bragg-MOKE hysteresis loop. For the broader Fe stripes
(d = 8 µm and wF e = 3.7 µm) the absence of this component means that edge domains
play a minor role for this grating. This is not surprising because the overall uniaxial
shape anisotropy is reduced and the relative region influenced by edge effects is reduced.
1
This section is based on parts of the article Magnetooptical Kerr effect of Fe-gratings [102]
144
11.2. Fe grating with giant Kerr rotation
n=−1
0.2
n=−2
8µm
0
−0.2
θK [°]
0.2
6µm
0
−0.2
0.2
4µm
0
−0.2
−0.5
0
0.5
−0.5
0
0.5
H [kOe]
Figure 11.2.: Bragg-MOKE hysteresis loops for n = −1 and n = −2 and the grating
parameter d = 4, 6 and 8 µm of the polycrystalline Fe grating.
However, the absence of the additional component for the smallest stripe width of this
set of gratings is surprising, but can be explained by the generally strongly reduced Kerr
signal for this particular Bragg-MOKE loop.
This brings the focus of the discussion to the Kerr signal as a function of the diffraction
order. The measurement is depicted in Fig. 11.3. Different to the measurements in
previous chapters the angle of incidence was non-zero, αi ≈ 40◦ . The Kerr signal in
Fig. 11.3 is plotted as a function of the diffraction angle, αf , for a better comparison of
the results of the three gratings with different d. The curve in Fig. 11.3 shows the up
to now expected behavior. The Kerr signal θK (αf ) consists of two components: first,
a linear dependence is observed which increases with increasing diffraction angle and
displays a change of sign for αf = 0. This behavior fits perfectly to the qualitative
145
11. Further measurements
d=8µm
d=6µm
d=4µm
0.5
0.4
θsat
[°]
K
0.3
0.2
0.1
0
−0.1
−60
−40
−20
0
20
α [°]
Figure 11.3.: Bragg-MOKE amplitude in saturation of the polycrystalline Fe gratings
as a function of the diffraction angle.
model of the Kerr amplitude in the framework of a Lorentz-Drude model as it was
developed in Sec. 9.4. In that case only perpendicular incidence was discussed but it is
clear that similar arguments for non-zero angle of incidence must also lead to a change
of sign of the Bragg-MOKE amplitude at αf = 0 (see Fig. 9.21). A similar dependence
was already depicted in Sec. 8 (Fig. 8.3). Second, an oscillation with the period ∆n = 2
is superimposed. The minima of the oscillating contribution coincide with the intensity
minima at the even order diffraction spots. The same behavior was already reported
from the Co on Fe gratings in Sec. 10. There are, however, two rather remarkable
exceptions: The Kerr loops for n = 2, d = 6 and 8 µm display a giant Kerr-signal
amplification.
This giant Kerr effect cannot be explained with the interference of light diffracted by
the substrate and the grating. The phase difference between the two components can be
calculated using Eq. 3.20. The result for the present geometry is φ ≈ π4 for n = −2 and
even smaller for n = 2. Following the calculation in Sec. 3.4.3, which correlate the phase
and intensity difference of the two components of the diffracted light to the observed Kerr
rotation, no such giant Kerr effect can be anticipated. Only for larger phase difference
> π2 an influence is predicted. All other attempts for the understanding of the observed
Bragg-MOKE effects discussed up to here also seem to fail to explain the observed giant
Kerr signal. Clearly, a decent theory of the longitudinal Bragg-MOKE effect, actually
calculating the vectorial amplitudes of the electromagnetic field, is necessary. However,
an amplification of the observed Kerr rotation of > 10 (compared the specular hysteresis
and the curve for n = −2) might be very useful for technical applications.
146
12. Conclusions
Understanding the remagnetization processes of artificially produced lateral magnetic
patterns, such as stripe or dot arrays, is of fundamental as well as of technical importance. Technical innovations such as magneto-electronics, MRAM’s, magnetic readheads and patterned magnetic media are not possible without fundamental research in
this subject.
There are two possibilities of analyzing the remagnetization process of magnetic thin
film elements. First, one can use lateral resolving, microscopic methods to image the
domains in single elements, and second, integrating methods such as MOKE or SQUID
can be used to gain information about remanence, coercive field and hysteresis. Regular
arrays may also be investigated using scattering methods, measuring magnetic properties
in reciprocal space.
The subject of this thesis was to explore the potential of new magneto-optical techniques and determine as example the magnetization reversal of patterned magnetic systems. The vector-MOKE technique gains additional integrated information about the
average magnetization vector. The Bragg-MOKE technique uses diffraction to gain further information about the lateral magnetization distribution in reciprocal space.
The systems under investigation were arrays of ferromagnetic stripes or wires with
grating parameters ranging from 300 nm to 20 µm and stripe widths from 10% to 75%
of the grating parameter. The grating structures were prepared from 10 - 20 nm thick
Fe, Ni or CoFe alloy films of single crystal or polycrystalline quality. The samples were
mainly prepared using electron beam lithography in combination with standard thin
film preparation techniques like MBE or sputtering.
The main tool of this thesis was the magneto-optical Kerr effect in the longitudinal
configuration. If grating structures are subjected to the laser beam of a MOKE setup
additional Bragg-reflections are observed which were used to gain further information
of the domain distribution and are used to enhance the Kerr-signal (Bragg-MOKE).
Furthermore, the MOKE technique can be used to gain vectorial information of the
magnetization process.
In the following the main results are briefly summarized and proposals for further
experiments are given.
Remagnetization process of ferromagnetic stripes
The measurements in the preceding chapters dealt with the remagnetization process and
the domain structure of ferromagnetic stripes and wires on the nano- and micrometer
scale. As the measurements were mainly done using integrating methods no detailed
analysis was possible, thus the main aim of this thesis was the exploration of the new
147
12. Conclusions
magneto-optical techniques rather than micromagnetic studies. However, several interesting magnetization processes were measured and could be described combining vectorMOKE, Bragg-MOKE and microscopy techniques such as MFM and Kerr-microscopy:
• The remagnetization pattern of Fe nanowires (Chap. 6) with a rounded shape and
a grating parameter of 300 nm consists of a complex domain nucleation process. A
plateau region in the hysteresis curve is most probable caused by different pinning
potentials of domains magnetized along or perpendicular to the wire axis.
• The remagnetization process of Ni stripes (width: 4 µm, Chap. 8)is almost not
affected by edge effects, due to the fine domain structure, one order of magnitude
smaller than the stripe width. Anomalies observed with the Bragg-MOKE technique are probably due to ridges at the edge of the stripes caused by the unperfect
production of the sample.
• The remagnetization process of CoFe stripes (width: 3 µm, see Chap. 7) is dominated by coherent rotation of the magnetization into the easy axis (along the
stripes) and a 180◦ domain wall switching. This behavior is due to the high shape
anisotropy of the CoFe stripes. In addition, the Bragg-MOKE technique provided
evidence for the existence of small edge domains magnetized along the wire edge.
• The remagnetization process of Fe stripes with different widths (ranging from 0.5
to 3.7 µm, see Chap. 9.4) showed very interesting features. The domain pattern depends on the relative strength of the fourfold crystalline anisotropy and the stripeinduced twofold anisotropy. For small separations of the stripes a coupling between
the magnetization of the individual stripes was observed with Kerr-microscopy.
If the fourfold anisotropy can be neglected the domains form a regular pattern
with strong edge domains observable with Bragg-MOKE. Stripes with crystalline
anisotropy showed a plateau region with zero magnetization.
Vector-MOKE
The vector-MOKE technique is an extension of the standard longitudinal Kerr effect.
An additional field applied perpendicular to the scattering plane allows for a measurement of the two orthogonal in-plane components of the magnetization and leads thus
to a measurement of the length and the angle of the magnetization vector. Therefore,
domain processes can be distinguished from rotational processes during the remagnetization. The technique was described in detail in this thesis and several examples of
measurements were given. In particular it was proven that the remagnetization process
of an array of Fe nanowires (Chap. 6) consists of a complex domain structure and that
an array of CoFe (Chap. 7) stripes changes its magnetization mainly by rotation of the
magnetization. In other cases the vector-MOKE techniques gave important information
for the interpretation of Bragg-MOKE measurements (Chap. 9.4).
Bragg-MOKE
In this thesis it was shown that the Bragg-MOKE technique is a valuable addition to
the MOKE technique for the examination of laterally patterned magnetic films. Mainly
148
Figure 12.1.: Illustration of the linear dependence of the Bragg-MOKE signal in rotation
as a function of the diffraction angle (all figures are taken from previous
sections of this thesis). From left to right: Because of the diffraction
geometry the exit angle is not simply the negative angle of incidence; this
off-specular geometry causes increasing Kerr rotations for increasing exit
angles; the effect was clearly observed for several grating structures.
qualitative explanations for the observed effects were given, which may help experimentalist to interpret their results. However, a more detailed theoretical analysis of the
longitudinal Bragg-MOKE effect, as it exists for other geometries [53] is certainly necessary. The deeper understanding of these effects also seems of technical interest since
it allows for a fast and effective characterization of the quality of patterned magnetic
media, which will play an important role in future magnetic data recording techniques.
In more detail, the Bragg-MOKE hysteresis curves show three effects:
• For measurements in diffraction geometry the constrain αi = αf is lifted, therefore
the usual Fresnel formulas describing the polarization cannot be valid. Qualitative
explanations in the framework of the Lorentz-Drude model of metals were developed in the preceding chapters. The saturation Kerr rotation of Bragg-MOKE
curves consists of an almost linear part which increases with increasing |αf | (increasing n) and is zero for αf = 0 (parallel to the surface normal). The effect is
depicted schematically in Fig. 12.1. This may be in contradiction with [53] were a
vanishing Kerr signal is found if the diffracted beam is antiparallel to the incident
beam (Littrow-mounting).
• The light diffracted by the spacing between the stripes will also contribute to the
total signal. The interference between light diffracted by the not ferromagnetic
substrate and the ferromagnetic stripes may lead to a modification of the observed
Kerr rotation much in the same way as anti-reflection coatings on thin films help to
increase the magneto-optical contrast in Kerr-microscopy. However, the thickness
of the stripes is small compared to the wavelength and thus the phase difference is
also small – no amplification of this kind was observed. Another related effect is
that the diffraction envelope of stripe and spacing will exhibit maxima or minima
149
12. Conclusions
Figure 12.2.: Illustration of the oscillating Bragg-MOKE signal in saturation as a function of the diffraction angle (all figures are taken from previous sections
of this thesis). From left to right: the sample morphology leads to phase
and amplitude differences in the light diffracted from the stripes and the
grooves; the diffraction envelope can be clearly observed by measuring the
intensity of the diffraction spots; the oscillating Kerr amplitude superimposed on a linear increase (see Fig. 12.1) reflects the oscillation period of
the intensity oscillations.
at different order of diffraction, thus the ferromagnetic signal may be enhanced
at certain order of diffraction. The effect of oscillating saturation Kerr signal
is schematically depicted in Fig 12.2. The understanding of the oscillating Kerr
signal in detail is a complicated task, in particular non-scalar diffraction theories
have to be used.
• The domain structure results in a non-uniform distribution of the magneto-optical
reflectivity constants at the surface of the stripes. Therefore, the nth order diffraction spots carries information of the nth order Fourier component of the magnetization distribution of the grating. If the individual stripes of the grating behave
more or less identical a magnetic form-factor can be defined which is the Fourier
transformation of the average magnetization distribution inside the stripes. In
this thesis several limiting cases in one dimension were calculated analytically
(Sec. 3.4.2) and compared with measured data. In Fig. 12.3 this effect is depicted
schematically. Although no exact fitting was possible, qualitative correspondence
was reached. In particular, if edge domains are formed, the effective width of the
stripes is reduced, which leads to a sinusoidal behavior of the Kerr signal as a
function of the magnetization. If only irregular domains are formed or coherent
rotation takes place, the effective width is constant but the magnitude of the magnetization inside the stripe is reduced. This case leads to linear functions of the
Kerr signal.
The use of diffraction for the magneto-optical measurements has also some technical
advantages:
• The Kerr signal in saturation can be increased by choosing high order of diffraction.
In addition, for special cases a resonance may occur, an enhancement of the Kerr
150
Figure 12.3.: Illustration of the effect of edge domains leading to anomalous BraggMOKE hysteresis shapes (all measurements were previously presented in
this thesis). From left to right: the Kerr-microscopy image reveals the existence of edge domains; a one dimensional model of one stripe is developed;
the Fourier-transformation of this model can be calculated analytically; the
calculation qualitatively corresponds to the Bragg-MOKE measurement in
a representation of the Kerr-rotation as a function of the magnetization.
rotation of a factor > 10 was reported (Sec. 11). However, the magneto-optic
figure of merit (product of Kerr rotation and intensity) may not be enhanced,
because the gain in Kerr rotation is payed by a strong decrease of the intensity at
high order of diffraction. This problem does not affect the measurements for the
present thesis, as the detection technique applied is only sensitive to the rotation
and is to a large extent independent of the intensity.
• Bragg-MOKE acts like a filter. Only signals from the grating can pass. Therefore,
gratings of only small total size, surrounded by ferromagnetic material can be
measured. Other integrating methods (like SQUID) have difficulties if the desired
signal is small and is accompanied by other ferromagnetic signals.
• It has been shown that also in the diffuse light of rough surfaces a MOKE signal
can be isolated (Sec. 11).
• The sensitivity to the magnetization at the edges of the stripes can be increased.
If measurements are carried out at reflexes with strongly reduced intensity (e.g.
n = 2 and w = d/2), the signal is very sensitive to changes of this ideal situation
(Chap. 9.4).
151
12. Conclusions
Outlook
The research field of nano- and microstructured magnetic elements is still growing triggered by the potential technical applications and the general interest in magnetic domains and patterns. The combination with the fascinating subject of diffraction makes
it even more challenging. This thesis proofed that the combination of diffraction and
magneto-optics is a valuable tool in this research field and that in itself it is not completely understood. This thesis helped to unravel some of the complicated effects which
were observed by Bragg-MOKE. However, many things remain to do, therefore this
thesis closes with a list of proposals for new experiments which should be done in the
future.
Two main directions of future work can be presented: first, just use the gathered
knowledge and investigate more complicated samples and magnetic structures and, second, try to further understand the Bragg-MOKE effect and develop methods to make
the Bragg-MOKE hysteresis curves predictable:
• Bragg-MOKE studies of the remagnetization process of dots or squares will provide
interesting results. The subject of coupling between adjacent elements has also
only been touched. Square lattices open up the possibility for further analysis
using diffraction techniques. In addition, a combination of vector-MOKE and
Bragg-MOKE is possible for arrays with a fourfold symmetry, i.e. the BraggMOKE signal can be detected with the applied field perpendicular and parallel to
the scattering plane with out the need to rotate the sample (see Sec. 3.5.3).
• The studies can be extended to different materials and a variety of other geometric
factors (e.g. height of the elements).
• Relief gratings and gratings of non-magnetic material on magnetic films have already been demonstrated, but may be further studied in order to measure offspecular magneto-optical constants independent of the domain structure.
• Innovative magnetic structures utilizing exchange bias or interlayer exchange coupling in order to modify the domain structure are of growing interest. BraggMOKE measurements are very sensitive to changes of the domain structure and
may thus help in this research subject.
Other MOKE methods like MOKE with a rotating field (ROTMOKE, see [103]) can be
combined with the Bragg-MOKE technique to gain additional information [104] and to
separate non-linear magneto-optical effects [103]. Last but not least, the Bragg-MOKE
studies presented here can be viewed as a preliminary study for measurements of inplane Bragg-spots of artificial magnetic structures using neutrons and soft x-rays. These
two probes have some advantages over the visible photons of the laser light, as larger
penetration depth (for neutrons), chemical sensitivity (for soft x-rays) and a generally
smaller wavelength enabling the investigation of even smaller patterns. Some general
physical ideas like the diffraction from the magnetization distribution are the same and
MOKE is some orders of magnitude cheaper.
Furthermore, methods have to be developed to model the Bragg-MOKE curves taking only the optic constants and the magnetization distribution into account. This has
152
already been done for the transverse Bragg-MOKE effect [53] and needs the expertise
of scientist in the field of diffractional optics. Another viewpoint is that the Fourier
spectrum of a given magnetization distribution can be calculated numerically, which
should at least predict the shape of measured Bragg-MOKE curve. Such an experiment
has recently been reported in [104]. The authors were able to model certain features of
a Bragg-MOKE measurement by numerically calculating the two-dimensional Fourier
transformation of a domain pattern obtained by using standard micromagnetic simulation tools. This is a very promising path for future investigations. In addition, one
could directly try to calculate the Fourier compounds using experimental data obtained
with Kerr microscopy.
Finally, it should be noted, that besides all technical and fundamental interest in
micro- and nanostructured magnetic media, it is also the general fascinating combination of classical Kerr-effect and diffraction which makes the subject of Bragg-MOKE
attractive to physicist of several disciplines. Because of this combination unexpected
data is produced and the shape of measured hysteresis loops astonished long-established
experts in magnetism. Only by careful analysis it is possible to separate the optics from
the magnetics. This thesis hopefully helped to develop physical transparent pictures for
some of the peculiarities of Bragg-MOKE.
153
12. Conclusions
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160
Zusammenfassung
Zusammenfassung
Einführung
Das Verständnis von Ummagnetisierungsvorgängen von künstlich strukturierten magnetischen Inseln, Partikeln oder Streifen, ist sowohl von fundamentalem Interesse, als
auch für die Anwendung von herausragender Bedeutung. Insbesondere die aktuelle Erforschung und Entwicklung der Magneto-Elektronik, wie z. B. MRAMs (magnetischen
Speicher mit wahlfreiem Zugriff), Leseköpfe für Festplatten und strukturierte magnetische Speichermedien, beruhen zu großen Teilen auf dem Verständnis von Ummagnetisierungsvorgängen und Domänenstrukturen.
Um die Ummagnetisierung experimentell zu bestimmen, sind zwei Zugänge möglich:
Zum einen kann man die Domänenstruktur ortsaufgelöst mit Methoden wie der KerrMikroskopie oder der magnetischen Kraft-Mikroskopie (MFM) bestimmen. Zum anderen
lassen sich magnetische Hysteresen mit integrierenden Methoden, wie dem magnetooptischen Kerr-Effekt (MOKE) oder SQUID (Supraleitende Quanteninterferenz Messung
zur Bestimmung der magnetischen Streufelder), messen. Hysteresen, Koerzitivfelder und
die remanente Magnetisierung können so sehr genau gemessen werden. Außerdem lassen
sich viele gleichartige magnetische Elemente gleichzeitig untersuchen.
In der vorliegenden Arbeit werden zwei neue magnetooptische Verfahren eingeführt
und auf Probensysteme aus magnetischen Streifen in der Größenordnung von Mikrometern angewandt.
• Die Vektor-MOKE Technik benutzt den longitudinalen Kerr-Effekt, um bei externen magnetischen Feldern in paralleler und senkrechter Orientierung zwei orthogonale Komponenten des Magnetisierungsvektors zu bestimmen. Aus diesen
Informationen lassen sich schließlich, mittels der Länge und Orientierung des Magnetisierungsvektors, Aussagen treffen, ob der Ummagnetisierungsvorgang aus einer reinen Rotation der Magnetisierung, reinen Domänenprozessen oder aus einer
Kombination von beiden besteht.
• Die Bragg-MOKE Technik nutzt Interferenzphänomene an regelmäßigen lateralen
Strukturen, um Aussagen über die gemittelte Domänenstruktur der magnetischen
Inseln zu erhalten. Dabei wird der longitudinale Kerr-Effekt an gebeugten Laserstrahlen der als Interferenzgitter wirkenden Probe detektiert. Insbesondere läßt
sich sehr einfach feststellen, inwiefern sich korrelierte Domänen, wie zum Beispiel
Randdomänen, ausbilden.
Die dabei untersuchten Probensysteme sind künstlich hergestellte magnetische Streifengitter. Die Breite der Streifen liegt zwischen 150 nm und 5 µm, die Schichtdicke ist
typisch 20 nm und der Gitterparameter wird von 300 nm bis zu 20 µm variiert. Die
Proben wurden mittels verschiedener lithographischer Methoden hergestellt.
Ziel dieser Arbeit ist insbesondere die Untersuchung des Bragg-MOKE Effekts selbst.
Dabei sind drei Punkte von Bedeutung:
• Die Beugungsgeometrie ist nicht mehr mit dem eigentlichen longitudinalen KerrEffekt identisch, da nun der Einfallswinkel nicht mehr gleich dem Ausfallswinkel
161
Zusammenfassung
ist. Daher spielen nicht-spekuläre Fresnel-Koeffizienten eine Rolle. Es stellt sich die
Frage, inwiefern sich ein Einfluss auf die gemessenen Kurven ergibt.
• Der Einfluss von Interferenzeffekten zwischen den magnetischen Streifen und
der dazwischen liegenden Substrat-Region kann zu zusätzlichen Phasenverschiebungen führen, wie es beispielsweise bei der Benutzung von Anti-ReflexionsBeschichtungen beobachtet wird.
• Aus grundsätzlichen Überlegungen folgt, dass die gemessene Bragg-MOKE Kurve
der Ordnung n die nte Komponente der Fourier-Transformation der Magnetisierungsverteilung des Gitters darstellt (magnetischer Formfaktor). Kann man dies
nutzen, um qualitative Informationen über die Domänenstruktur der Probe zu
erhalten, ohne ortsauflösende Methoden zu benutzen?
Die zentralen Aspekte dieser Arbeit liegen in der Untersuchung des Ummagnetisierungsvorgangs von magnetischen Gittern und der Untersuchung von optischen Effekten in der Bragg-MOKE Geometrie. Beide Aspekte werden vom Standpunkt der experimentellen Physik angegangen, verschiedene Parameter werden systematisch variiert
und daraus phänomenologische Beschreibungen hergeleitet. Analytische oder numerische
Herleitungen aus ersten Prinzipien stehen nicht im Vordergrund.
Magnetismus von magnetischen Nano- und Mikrostrukturen
Für den Magnetismus der untersuchten Streifengitter spielen zwei physikalische Phänomene eine herausragende Bedeutung:
Einerseits ist dies die magnetische Anisotropie, d.h. das Bestreben der Magnetisierungsverteilung innerhalb eines magnetischen Elements nicht nur gleichförmig sondern
auch entlang einer bestimmten Richtung ausgerichtet zu sein. Anisotropien werden entweder durch die Kristallstruktur oder durch die äußere Form vorgegeben. Beispielsweise
sind dünne Filme in der Regel in der Filmebene und magnetische Streifenstrukturen
entlang der Streifenrichtung magnetisiert (diese Richtungen bezeichnet man dann als
leichte Richtungen). Die Kristallstruktur von Fe führt dagegen bei einkristallinen Proben mit einer (001) Wachstumsrichtung zu einer vierzähligen Anisotropie mit leichten
Achsen entlang der (010) und (100) Richtung.
Andererseits bilden sich in vielen Fällen magnetische Domänen aus, d. h. die Magnetisierungsverteilung ist nicht mehr gleichförmig, sondern es existieren Bereiche mit unterschiedlichen Magnetisierungsrichtungen und diskontinuierlichen Grenzen dazwischen.
Domänen entstehen, um das Streufeld der Probe bzw. des magnetischen Elements zu
reduzieren. Die Domänenverteilung ist entweder durch die äußere Form (Richtung der
Magnetisierung entlang von Kanten) oder durch die oben erwähnten Anisotropien (Magnetisierung in einer Domäne entlang einer leichten Richtung) bestimmt.
Methoden
Probenpräparation Die Herstellung von lateral strukturierten Proben geschieht mit
Hilfe von lithographischen Methoden. Im interessanten Bereich von Strukturgrößen von
einigen hundert Nanometern bis hin zu einigen Mikrometern stehen mehrere Präparationsmethoden zur Verfügung. Allen Methoden gemein ist das Vorgehen in zwei Schritten:
162
Zusammenfassung
Erst wird ein organischer Film strukturiert (Maske), dann die resultierende Maske in
einen dünnen Film übertragen. Zur Herstellung der Maske wird der organische Film mit
Licht (optische Lithographie, Raster-Laser-Lithographie) oder mit Elektronen (RasterElektronen-Lithographie) belichtet und dann entwickelt. Die Struktur wird entweder
durch einen Ätzprozess, durch einen sog. lift-off -Prozess oder durch eine Kombination
beider in einen dünnen magnetischen Film übertragen. Der dünne Film (hier handelt es
sich um Schichtdicken von typischerweise 20 nm) wird mit Standartpräparationsmethoden wie Sputtern oder der Molekularstrahl-Epitaxie (MBE) hergestellt.
Auf diese Weise wurden regelmäßige Matrizen von magnetischen Streifen produziert,
die mit typischen Untersuchungsmethoden wie Raster-Elektronen-Mikroskopie, RasterKraft-Mikroskopie oder optischer Mikroskopie auf ihre Struktur untersucht werden konnten.
Magnetische Untersuchungsmethoden Zur magnetischen Charakterisierung der Mikrostrukturen wurde der longitudinale Kerr-Effekt benutzt. In seiner normalen Konfiguration wird linear polarisiertes Licht von der Probe reflektiert. Die Magnetisierung
der Probe führt daraufhin zu einer Änderung des Polarisationszustands des Laserlichts.
Diese Drehung der Polarisation wird mittels einer speziell entwickelten, hochauflösenden Analysetechnik als Funktion des äußeren Feldes detektiert. Daraus erhält man eine
Kurve, die proportional zur magnetischen Hysterese der Probe ist. Bei dieser Methode
wird die Magnetisierungskomponente entlang des Feldes bestimmt, wobei das Feld dazu in der Einfallsebene des Lichts und parallel zur Probenoberfläche orientiert ist. Die
Probe kann nun in verschiedene Richtungen (Streifenrichtung relativ zum Feld) gedreht
werden. Aus den jeweiligen Hysteresen lassen sich sehr detaillierte Aussagen über die
magnetische Anisotropie treffen. Wie bereits oben erwähnt, wird diese Methode im Rahmen dieser Arbeit erweitert, um sowohl Vektor-Informationen über die Magnetisierung,
als auch qualitative Aussagen über die Domänenstruktur zu erhalten.
Zur Messung von Bragg-MOKE Hysteresen wird die Probe so gedreht, dass die magnetischen Streifen senkrecht zur Einfallsebene des Lasers und senkrecht zum äußeren
Feld stehen. Dies entspricht im Allgemeinen der schweren Richtung der durch die Form
der Streifen induzierten Anisotropie. Die Streifen wirken wie ein Beugungsgitter und
erzeugen in der Einfallsebene ein Beugungsmuster. Der Detektor wird nun so verstellt,
dass das Licht eines Beugungsmaximums detektiert wird. Ansonsten wird die Hysterese
normal gemessen.
Ein anderer Weg wird mit ortsaufgelösten Messungen beschritten. Mit der KerrMikroskopie wird die Domänenstruktur eines kleinen Teils der Probe direkt sichtbar
gemacht. Dabei wird die oben erwähnte Polarisationsänderung im Mikroskop als Kontrast sichtbar. Eine zweite wichtige Methode ist die der Raster-Kraftmikroskopie (AFM)
verwandte Raster-Magnetokraft-Mikropskopie (MFM). Dabei wird eine nur Nanometer
große magnetische Spitze über die Probe bewegt und die Wechselwirkung mit den magnetischen Streufeldern der Domänen detektiert. Diese beiden abbildenden Methoden
wurden mit den aus MOKE Messungen erhaltenen Daten verglichen und erwiesen sich
bei der Interpretation der Bragg-MOKE Ergebnisse als sehr nützlich.
163
Zusammenfassung
Ergebnisse und Diskussion
Ummagnetisierung von ferromagnetischen Streifen Obwohl das Hauptaugenmerk
auf der Entwicklung der neuen magnetooptischen Methoden lag, wurden ferner interessante Ummagnetisierungsvorgänge an verschiedenen magnetischen Streifenstrukturen
gemessen:
• Der Ummagnetisierungsvorgang von Fe-Nanostreifen (Kap. 6) mit einem abgerundeten Querschnitt und einem Gitterparameter von 300 nm besteht aus einem komplexen Domänen-Nukleationsprozess. Ein Plateau-Bereich in der Hysterese deutet
auf unterschiedliche Pinnig-Potentiale für Domänen hin, welche entlang oder senkrecht zur Streifenrichtung magnetisiert sind.
• Der Ummagnetisierungsprozess von Ni-Streifen (Kap. 8) hingegen wird kaum von
Randeffekten beeinflusst, da eine sehr feine Domänenstruktur vorliegt, die eine
Größenordnung kleiner als die Streifenbreite ist. Anormale Bragg-MOKE Hysteresen konnten auf die Präparationsmethode, die zu starken Graten an den Rändern
führte, zurückgeführt werden.
• Reine Rotationsprozesse dominieren das magnetische Verhalten einer CoFeStreifen Probe (Kap. 7). Die Magnetisierung rotiert erst in die leichte Magnetisierungsrichtung (entlang der Streifen), bevor eine 180◦ -Domänenwand am Koerzitivfeld durch den Streifen läuft. Hier konnte mit der Bragg-MOKE Methode
nachgewiesen werden, dass sich zusätzlich sehr kleine Randdomänen ausbilden.
• Sehr intensiv wurden mehrer Streifengitter aus Fe untersucht (Kap. 9.4). Die Domänenstruktur hängt stark von der relativen Größe der vierzähligen Kristallanisotropie und der, durch die Form der Streifen gegebenen, zweizähligen Anisotropie
ab. Für kleine Abstände wurde zusätzlich mit Hilfe der Kerr-Mikroskopie eine
Kopplung zwischen den Streifen beobachtet. Falls die vierzählige Kristallanisotropie vernachlässigt werden kann, bilden die Domänen ein regelmäßiges Muster mit
starken Randdomänen. Dies wurde auch mit Bragg-MOKE nachgewiesen. Bei einer Dominanz der Kristallanisotropie wurde in den Hysteresen eine Plateau-Region
beobachtet.
Vektor-MOKE Die Vektor-MOKE Technik ist eine Weiterentwicklung des herkömmlichen longitudinalen Kerr-Effekts. Dabei wird ein äußeres Feld senkrecht zur Einfallsebene des Lasers angelegt und die Probe um 90◦ gedreht. Daraus kann dann der Winkel und
die Länge des Magnetisierungs-Vektors abgeleitet werden. Die Methode wurde im Detail erläutert (Kap. 3.2). Die Domänenstruktur insbesondere der CoFe-Streifen (Kap. 7)
und der Fe-Nanostreifen (Kap. 6) wurde mit Hilfe dieser Technik untersucht. Auch in
anderen Fällen lieferte Vektor-MOKE wertvolle Hinweise zur Interpretation, z.B. der
Bragg-MOKE Messungen (Kap. 9.4).
Bragg-MOKE In dieser Arbeit wurde gezeigt, dass die Bragg-MOKE Technik ein
wertvolles zusätzliches Werkzeug zur normalen Hysterese-Messung lateral strukturierter
magnetischer Filme darstellt. Es wurden hauptsächlich qualitative Erklärungen für die
164
Zusammenfassung
beobachteten Effekte diskutiert, theoretische Erklärungen stehen noch aus. Allerdings
sind bereits die qualitativen Erklärungen beachtenswert. Das tiefere Verständnis der beobachteten Bragg-MOKE Effekte ist auch von technischem Interesse, da Bragg-MOKE
zur schnellen Analyse der Qualität strukturierter magnetischer Speichermedien geeignet
scheint. Im Detail wurden drei Effekte beobachtet:
• Für den Fall der Messung in Beugungs-Geometrie ist die Bedingung Einfallswinkel=Ausfallswinkel aufgehoben. Das führt dazu, dass die gewöhnlichen FresnelKoeffizienten nicht benutzt werden können. Für diesen Fall wurde ein Modell im
Rahmen der Lorentz-Drude Theorie beschrieben (Kap. 9.4). Die Sättigungs-KerrRotation der Bragg-MOKE Hysteresen als Funktion des Beugungswinkels besteht
aus einem fast linearem Anteil, der mit zunehmendem Beugungswinkel zunimmt.
Dabei kommt es zu einem Vorzeichenwechsel, falls der gebeugte Strahl parallel zur
Proben-Normalen verläuft (0te Ordnung für senkrechten Einfall). Dieser Effekt ist
schematisch in Abb. 12.1 dargestellt.
• Das Licht, das von den Zwischenräumen gebeugt wird, trägt auch zum Gesamtsignal bei. Dabei kann es zu einer Verstärkung des Kerr-Signals kommen, wenn
der Phasenunterschied der vom Substrat und der von den Streifen gebeugten
Wellen annähernd π beträgt. Dieser Effekt ist der Verstärkung des Kerr-Effekts
durch Anti-Reflexions-Schichten äquivalent. Allerdings wurde dieser Effekt in Praxis nicht beobachtet, da die benutzten Schichtdicken relativ zur Laserwellenlänge
zu klein sind. Ein ähnlicher Effekt tritt auch aufgrund der unterschiedlichen Amplituden der beiden gebeugten Wellen auf. An bestimmten Beugungsordnungen kann
es dann zu einer Verstärkung oder Abschwächung des detektierten Bragg-MOKE
Signals kommen. Dieser Effekt ist mit der Intensität des gebeugten Lichtes korreliert und führt zu einer Oszillation des Kerr-Signals als Funktion der Beugungsordnung (siehe Kap. 8 und 10). Diese Oszillation ist der oben erwähnten, linearen
Abhängigkeit überlagert. Schematisch ist dieser Effekt in Abb. 12.2 dargestellt.
• Die Domänenstruktur führt zu einer nicht homogenen Verteilung der magnetooptischen Reflektionskoeffizienten innerhalb der Streifen. Daher enthält die BraggMOKE Hysterese nter Ordnung Informationen über die nte Fourier-Komponente
dieser Magnetisierungsverteilung. Falls sich die einzelnen Streifen alle ähnlich verhalten, ist es möglich einen magnetischen Formfaktor zu definieren, welcher die
Fourier-Transformierte der gemittelten Magnetisierungsverteilung darstellt. Verschiedene Spezialfälle für diesen magnetischen Formfaktor wurden in Kap. 3.4.2
analytisch berechnet. Die Ergebnisse lassen sich qualitativ mit den Messungen von
Bragg-MOKE Hysteresen vergleichen. Das Verfahren ist schematisch in Abb. 12.3
zusammengefasst. Insbesondere für den Fall von Randdomänen ergibt sich ein anschauliches Bild: Durch die Magnetisierung der Randdomänen entlang der Streifen tragen diese bei der Hysteresemessung in schwerer Richtung nicht mehr zum
magnetooptischen Signal bei. Effektiv wird also die Breite der Stege verändert.
Die Fourier-Transformation eines solchen Streifens variabler Breite ist eine Sinusfunktion. Falls nur unregelmäßige Domänen entstehen, verringert sich durch die
Mittelung nur die Größe der Magnetisierung, die effektive Stegbreite bleibt jedoch
konstant. In diesem Fall ist die Fouriertransformation eine lineare Funktion. Diese
165
Zusammenfassung
einfachen Fälle konnten in Kap. 7 und 9.4 an experimentellen Daten nachgewiesen
werden. In der Realität werden beide Fälle gleichzeitig vorkommen oder noch weit
kompliziertere Domänenmuster entstehen.
Die Kombination aus Interferenz, Beugung und dem magnetooptischen Kerr-Effekt
hat einige technische Vorteile:
• Mit Hilfe der Bragg-MOKE Technik kann es möglich sein den Kerr-Effekt erheblich
zu verstärken, indem man z.B. eine hohe Beugungsordnung zur Messung ausnutzt.
Eine Verstärkung von einem Faktor > 10 wurde in Kap. 11 nachgewiesen. Allerdings wird nicht unbedingt auch das Produkt von Intensität und Kerr-Rotation
vergrößert, da die Intensität bei hohen Ordnungen stark abnimmt. Messungen mit
der in Kap. 3.5 beschriebenen Detektionstechnik, sind davon weitgehend unabhängig.
• Bragg-MOKE wirkt wie ein Filter, so dass nur Signale der regelmäßigen Gitterstruktur detektiert werden. Daher ist es möglich Gitter von nur geringer Gesamtgröße magnetisch zu vermessen. Andere Methoden, wie SQUID, stossen hier u.U.
aufgrund der geringen Menge magnetischen Materials auf ihre Grenzen. Hinzukommt, dass bei Messungen der gesamten Probe auch nicht-magnetisches Material
zum Gesamtsignal beitragen kann.
• In der vorliegenden Arbeit wurde demonstriert, dass es möglich ist, auch von diffus streuenden Oberflächen eine magnetischen Hysterese mit MOKE zu messen
(Kap. 11).
• Die Sensitivität auf die Magnetisierung an den Rändern der Streifen kann erhöht
werden. Für bestimmte Beugungsordnungen mit stark reduzierter Intensität (z.B.
n = 2 und wF e = d/2) reagieren die Ergebnisse der Bragg-MOKE Untersuchung
sehr empfindlich auf leichte Veränderungen dieser idealen Situation. Kleine Veränderungen der optischen Eigenschaften können schon durch eine Änderung der
Magnetisierung an den Kanten hervorgerufen werden.
Ausblick Die Forschung an magnetischen Nano- und Mikrostrukturen ist wegen der
hohen technischen Relevanz nach wie vor ein sehr interessantes Gebiet. Die in dieser
Arbeit demonstrierten faszinierenden Effekte der Interferenz und Beugung stellen eine
weitere Herausforderung dar. Es wurde gezeigt, dass die Kombination aus Magnetooptik und Beugung ein wertvolles zusätzliches experimentelles Werkzeug bereitstellt,
allerdings sind die beobachteten Effekte noch nicht alle vollständig verstanden. Obwohl
die vorliegende Arbeit einen wesentlichen Beitrag zum Verständis geliefert hat, bleiben
wichtige Experimente durchzuführen.
Im Wesentlichen erscheinen zwei Richtungen für die zukünftige Forschung mit BraggMOKE sinnvoll: Zum einen kann man die bekannten Effekte ausnutzen und weitere,
auch kompliziertere, Probensysteme untersuchen. Zum anderen sollten Experimente zum
tieferen Verständnis des Bragg-MOKE Effekts selbst durchgeführt werden:
• Bragg-MOKE Untersuchungen des Ummagnetisierungsprozess an Gittern mit
kreisförmigen, rechteckigen oder quadratischen Elementen können interessante Ergebnisse liefern. Zum Beispiel wurden Effekte der magnetischen Kopplung zwischen
166
Zusammenfassung
den magnetischen Elementen bisher kaum behandelt. Auch bieten 2-dimensional
Gitter neue Möglichkeiten Interferenz und Beugungserscheinungen auszunutzen
und eine Kombination von Bragg-MOKE mit Vektor-MOKE bietet sich an.
• Die bisherigen Untersuchungen können auf weitere Materialklassen ausgedehnt
werden.
• Reliefartige Strukturen wurden bereits ansatzweise untersucht und bieten die Möglichkeit die magnetooptischen Konstanten unabhängig von der Domänenstruktur
zu bestimmen.
• Innovative magnetische Strukturen, die auf dem sog. exchange-bias Effekt oder
der Zwischenlagenaustauschkopplung basieren, zeigen neue und interessante Domänenstrukturen. Da Bragg-MOKE sehr sensitiv auf die Änderung der Domänenstruktur reagiert, können neue Erkenntnisse erwartet werden.
Nicht zuletzt sollte erwähnt werden, dass die vorgestellten Untersuchungen des BraggMOKE Effekts eine Vorstudie zu Messungen von magnetischen Phänomenen mit anderen
Strahlungsarten angesehen werden können. So können vergleichbare Experimente mit
weichen Röntgen-Photonen oder Neutronen durchgeführt werden. Diese beiden Sonden
haben verschiedene Vorteile gegenüber dem benutzten Laserlicht: so weisen Neutronen
eine größere Eindringtiefe auf und weiche Röntgenstrahlung kann benutzt werden, um
chemisch selektive Informationen zu erhalten. Außerdem haben beide Strahlungen erheblich kleinere Wellenlängen, so dass kleinere Strukturen untersucht werden können.
Desweiteren sollten Methoden entwickelt werden, wie die Bragg-MOKE Hysteresen
allein aus den optischen Konstanten und der Magnetisierungsverteilung hergeleitet werden können. Dazu wird man die Expertise von Wissenschaftlern aus dem Bereich der
Optik benötigen. Ein anderer Zugang ist über numerische Berechnungen der Fourierzerlegung von Magnetisierungsverteilungen möglich, welche die Form der Bragg-MOKE
Hysteresen richtig voraussagen sollten. Dazu können entweder mikromagnetische Modellrechnungen oder ortsaufgelöste Messungen des kompletten Ummagnetisierungsvorgangs
(z.B. mit einem Kerr-Mikroskop) als Ausgangspunkt gewählt werden.
Abschliessend sollte bemerkt werden, dass, neben dem technischem und fundamentalem Interesse an mikro- und nanostrukturierten magnetischen Materialien, es gerade
die Kombination aus klassischem Kerr-Effekt, Interferenz und Beugung ist, die die Faszination von Bragg-MOKE ausmacht. Deswegen kommt es zu ungewöhnlichen und unerwarteten Messergebnissen, die auch erfahrene Magnetismus-Experten staunen lassen.
Nur durch die sorgfältige Analyse der Daten kann eine Separation der optischen von den
magnetischen Effekten erreicht werden. Die vorliegende Arbeit konnte hoffentliche einen
Beitrag zur Entwicklung physikalisch transparenter Vorstellungen zur Interpratation von
Bragg-MOKE Messungen leisten.
167
Publications
List of publications which have resulted from
this work
• [78]: B. Roldan-Cuenya, M. Doi, W. Keune, S. Hoch, D. Reuter, A. Wieck, T.
Schmitte, H. Zabel Magnetism and Interface Properties of Epitaxial Fe Films on
High-Mobility GaAs/Al0.35Ga0.65As(001) Two-Dimensional Electron Gas Heterostructures in: Appl. Phys. Lett., accepted (2003).
• [87]: T. Schmitte, K. Theis-Bröhl, V. Leiner, H. Zabel, S. Kirsch, A. Carl Magnetooptical study of the magnetization reversal process of Fe nanowires in: J. Phys.:
Cond. Mat. 14, 7527 (2002).
• [93]: K. Theis-Bröhl, T. Schmitte, V. Leiner, H. Zabel, K. Rott, H. Brückl, J.
McCord CoFe-stripes: magnetization reversal study by polarized neutron scattering
and magneto-optical Kerr effect in: Phys. Rev. B, accepted (2003).
• [99]: T. Schmitte, T. Schemberg, K. Westerholt, H. Zabel, K. Schädler, U. Kunze
Magneto-optical Kerr effects of ferromagnetic Ni-gratings in: J. Appl. Phys. 87,
5630 (2000).
• [101]: T. Schmitte, K. Westerholt, H. Zabel Magneto-optical Kerr effect in the
diffracted light of Fe gratings in: J. Appl. Phys. 92, 4527 (2002).
• [102]: T. Schmitte, O. Schwöbken, S. Goek, K. Westerholt, H.Zabel Magnetooptical
Kerr effect of Fe-gratings in: J. Magn. Mag. Mat. 240, 24 (2002).
168
Acknowledgments
Acknowledgments
First, I would like to thank my advisor Prof. Dr. H. Zabel, who gave me the opportunity to work at his institute, for the valuable, open-minded discussions, his motivation
and constant support in every aspect of this thesis. His abilities to raise money, to put
science into intelligible (English) words and to teach all aspects of the theory of diffraction and scattering laid the basis for this thesis and also my personal progress in the
last years.
I owe many thanks to Prof. Dr. K Westerholt, for the discussions of new experimental
results and his valuable advice. In addition, I would like to thank him for the critical
reading of the manuscripts for the publications which have resulted from this work.
I would like to address my special thanks to all co-workers of external institutions
who have contributed to this successful work about magnetic microstructures. In particular, I like to mention: Prof. Dr. U. Kunze, Dr. S. F. Fischer, Th. Last and K.
Schädler from the Institut für Werkstoffe der Elektrotechnik, Ruhr-Universität Bochum
for valuable discussions and the preparation of several lithographic samples; Prof. W.
Keune, Dr. M. Doi and B. Roldan-Cuenya from the Laboratorium für Angewandte
Physik, Universität Duisburg for the preparation of Fe samples; Dr. H. Brückel and K.
Rott from the Institut für experimentelle Festkörperphysik, Universität Bielefeld for the
preparation of the CoFe sample; J. McCord from IFW Dresden for valuable discussions
and the Kerr-microscopy measurements; S. Kirsch and A. Carl from Laboratorium für
Tieftemperaturphysik, Universität Duisburg for the preparation of the Fe nanowires and
MFM measurements and R. Meckenstock and D. Spoddig from Institut für Festkörperspektroskopie, Universität Bochum for valuable discussions.
For the nice working atmosphere in the MOKE laboratory I would like to thank M.
Etzkorn, T. Schemberg, S. Gök, F. Radu and A. Westphalen.
The sample preparation is the basis of all this kind of research, for help with MBE,
sputtering, lithography and many other kinds of technical problems I would like to thank
O. Schwöbken, W. Oswald, P. Stauche, S. Erdt-Böhm and J. Podschwadek. Many things
would not have been possible without the aid of our mechanic and electronic workshops.
For help with any kind of computer problems I would like to thank M. Kneppe, M.
Hübener, S. Hachmann and H. Glowatzki.
For the proof-reading of this thesis I have to thank Dr. A. Remhof.
I would like to thank the complete group in Bochum for the nice and constructive
working-atmosphere, in particular I would like to mention Dr. R. Nötzel, Dr. J. Pflaum,
Dr. Ch. Sutter. Dr. D. Labergerie, Dr. G. Piaszinski, A. Bergmann, J. Grabis, Dr.
A. Schreyer, Dr. K. Theis-Bröhl for co-working and sometimes more sometimes less
important discussions during lunch, coffee or beer. Especially I would like to thank my
long-year friend and colleague Vincent Leiner and my room-mate Murat Ay.
Last but not least I have to thank my wife Stephanie and my child Jonas for their
support and, of course, my parents, relatives and friends are not forgotten!
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the
Sonderforschungsbereich SFB 491.
169
Lebenslauf
Lebenslauf
Name
Geburtsdatum
Geburtsort
Familienstand
Till Schmitte
30. Mai 1971
Herne
verheiratet, 1 Kind
1977 - 1981
1981 - 1990
Mai 1990
1990 - 1991
Grundschule in Bochum
Gymnasium Freiherr-vom-Stein Schule in Bochum
Abitur
Zivildienst
1991 - 1997
1993
1994 - 1995
Studium der Physik an der Ruhr-Universität in Bochum
Vordiplom
Austauschstudent an der University of Sussex, Brighton, UK
experimentelle Studienarbeit:
”Scanning electron microscopy with polarization analysis”,
1996 - 1997
Diplomarbeit
am Lehrstuhl für Experimentalphysik / Festkörperphyik ,
Ruhr-Universität Bochum (Prof. Dr. H. Zabel), Thema:
Magneto-optische Untersuchungen der Austauschkopplung an
nanostrukturierten epitaktischen Schichten
Dezember 1997 Diplom
170
1998 - 2002
Wissenschaftlicher Mitarbeiter
am Lehrstuhl für Experimentalphysik / Festkörperphysik,
Ruhr-Universität Bochum (Prof. Dr. H. Zabel)
seit 2003
Wissenschaftlicher Mitarbeiter
am Mannesmann Forschungsinstitut
Duisburg