MODULATING MAGNETIC INTERACTIONS - IN METAMATERIALS AND AMORPHOUS ALLOYS - DIVA PORTAL
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Digital Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 2219
Modulating magnetic interactions
in metamaterials and amorphous alloys
NANNY STRANDQVIST
ACTA
UNIVERSITATIS
UPSALIENSIS ISSN 1651-6214
ISBN 978-91-513-1663-5
UPPSALA URN urn:nbn:se:uu:diva-488984
2022
Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 13 January 2023 at 09:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Sean Langridge (ISIS Neutron and Muon Source, Diffraction and Materials Division). Abstract Strandqvist, N. 2022. Modulating magnetic interactions. in metamaterials and amorphous alloys. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 2219. 74 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-1663-5. This thesis is focused on exploring and modulating magnetic interactions in metamaterials and amorphous alloys along one-, two-, and three-dimensions. First, thin films of alternating Fe and MgO are adapted to modulate magnetic interactions along one dimension. At the remanent state, the Fe layers exist in an antiferromagnetic order, achieved by interlayer exchange coupling originating from spin-polarized tunneling through the MgO layers. Altering the number of repeats can tune the strength of the coupling. This is attributed to the total extension of the samples and beyond-nearest-neighbor interactions. Similarly, decreasing the temperature results in an exponential increase of the coupling strength, accompanied by changes in the reversal character of the Fe layers and magnetic ground state. Next, magnetic modulations along two dimensions are investigated using lithographically patterned metamaterial consisting of arrays with mesospins - i.e., circular islands. Mesospins have degrees of freedom on two separate length scales, within and between the islands. Changing their size and lateral arrangement alters their behavior. The magnetic texture in small elements can be described as collinear with XY-like behavior, while larger islands result in magnetic vortices. Allowing the islands to interact by densely packing them in a square lattice alters the energy landscape. This is manifested by the interplay of intra- and inter-island interactions and leads to temperature-dependent transitions from a static to a dynamic state. The temperature dependence can be further altered by both element size and lattice orientation, leading to emergent behavior. The final part of this thesis explores the modulations of interactions in three dimensions through inherent disorder in magnetic amorphous alloys. The atomic distribution in amorphous alloys can be viewed as random. However, local composition at the nanometer scale is, in fact, homogeneous. Variations in the composition of amorphous CoAlZr alloys lead to changes in the local distribution of magnetic amorphous CoAlZr manifested by competing anisotropies. Finally, off-specular scattering performed on a magnetic amorphous FeZr alloy is used to investigate the compositional variations at the nanometer scale. Indeed, correlations are observed at low temperatures due to the sample relaxation. Keywords: Magnetic metamaterials, interlayer exchange coupling, superlattice, mesospins, magnetic nanostructures, emergence, amorphous alloys, CoAlZr, FeZr Nanny Strandqvist, Department of Physics and Astronomy, Materials Physics, 516, Uppsala University, SE-751 20 Uppsala, Sweden. © Nanny Strandqvist 2022 ISSN 1651-6214 ISBN 978-91-513-1663-5 URN urn:nbn:se:uu:diva-488984 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-488984)
Annars är man ingen människa utan bara en liten lort
- Astrid Lindgren
Contents
Abstract ...................................................................................... ii
1 Introduction ........................................................................... 1
2 One dimensional interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Coupling across insulating layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Fe/MgO(001) superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Field dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Two dimensional interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Inner texture of single mesospins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Interacting mesospins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Transition from collinear to vortex state . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Interplay of interactions and inner texture . . . . . . . . . . . . . . . . . . . . . . . 31
4 Three dimensional interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 The role of disorder and composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 The impact of compositional modulations and anisotropy . 38
4.3 Truncating three dimensional modulations . . . . . . . . . . . . . . . . . . . . . . . 40
5 Concluding thoughts ........................................................... 49
6 Populärvetenskaplig sammanfattning ............................... 51
7 Acknowledgment ................................................................. 53
8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.1 Magnetic characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.2 Scattering methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.3 Sample descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Bibliography ............................................................................. 67
|v
List of papers
This thesis is based on the following papers. Reprints were made with
permission from the publishers.
I The impact of number of repeats N on the interlayer
exchange in [Fe/MgO]N (001) superlattices
Tobias Warnatz, Fridrik Magnus, Nanny Strandqvist, Sarah Sanz,
Hasan Ali, Klaus Leifer, Alexei Vorobiev and Björgvin
Hjörvarsson
Scientific reports 11, 1942 (2021)
II Temperature-induced collapse of spin dimensionality in
magnetic metamaterials
Björn Erik Skovdal, Nanny Strandqvist, Henry Stopfel, Merlin
Pohlit, Tobias Warnatz, Samuel D. Slöetjes, Vassilios Kapaklis,
and Björgvin Hjörvarsson
Phys. Rev. B 104, 014434 (2021)
III Emergent anisotropy and textures in two dimensional
magnetic arrays
Nanny Strandqvist, Björn Erik Skovdal, Merlin Pohlit, Henry
Stopfel, Lisanne van Dijk, Vassilios Kapaklis, and Björgvin
Hjörvarsson
Phys. Rev. Materials 6, 105201 (2022)
IV Finding order in disorder: Magnetic coupling
distributions and competing anisotropies in an
amorphous metal alloy
Kristbjorg A. Thórarinsdóttir, Nanny Strandqvist, Vilborg V.
Sigurjónsdóttir, Einar. B. Thorsteinsson, Björgvin Hjörvarsson
and Fridrik Magnus
APL Mater. 10, 041103, (2022)
| viiOther publications not discussed in this thesis:
V Reversible exchange bias in epitaxial V2O3/Ni hybrid
magnetic heterostructures
Kristina Ignatova, Einar Baldur Thorsteinsson,
Nanny Strandqvist, Christina Vantaraki, Vassilos Kapaklis,
Anton Devishvili, Gunnar Karl Pálsson, Unnar B Arnalds
J. Phys.: Condens. Matter 34, 495001, (2022)
VI A Bibliometric Study on Swedish Neutron Users for the
Period 2006–2020
Hanna Barriga, Marité Cárdenas, Stephen Hall, Maja Hellsing,
Maths Karlsson, Adriano Pavan, Ru Peng, Nanny Strandqvist,
and Max Wolff
Neutron news 32, 28-33, (2021)
viii |Contribution statement
My contribution to each paper is briefly described below:
I Performed PNR experiment and analyzed the data. Discussed the
results and contributed to the manuscript.
II Performed MOKE and PEEM-XMCD experiments. Analyzed
MOKE data, discussed the results, and contributed to the
manuscript.
III Performed all experiments and micromagnetic simulations.
Analyzed the data and was the main responsible for writing the
manuscript.
IV Participated in sample design and fabrication. Analyzed data,
discussed the results, and contributed to the manuscript.
| ixChapter 1
Introduction
A flock of starlings can contain thousands upon thousands of individuals.
While filling the sky, it seems like they behave as a single mind, moving in
harmony, producing patterns by correlated motions, such as the example
illustrated in Fig. 1.1(a). Every individual starling exhibits a rather
ordinary behavior that is, in principle, not different from the behavior
of any other bird species. Nevertheless, when flying in a flock, starlings
are capable of creating mesmerizing patterns depending on intrinsic and
extrinsic signals. The underlying principle of swarming is the formation
of collective behavior and strong spatial coherence originating from a
short-range interaction between individuals. Swarming of starlings is,
therefore, one of many examples of emergent behavior in nature, where
the collective behavior is beyond the control of the individual parts.
In certain ways, a flock of starlings is an analogy to many-body in-
teractions in physical systems, including ferromagnetic materials that
spontaneously order [1–3]. The flock of birds chooses a unique direc-
tion, where each individual is viewed as having a velocity vector [1, 3].
The vector can be equated with a magnetic spin of an atom, which is
commonly depicted with an arrow having both magnitude and direc-
tion. If the atoms are located in close proximity at discrete positions in
a repetitive manner, they interact with each other with the same cou-
pling constant, commonly denoted as J . At temperatures T = 0, the
magnetization spontaneously orders and aligns in the same direction to
form a ferromagnetic phase (see schematic illustration in Fig. 1.1(b)).
When noise to the system is introduced, for example, by changing the
temperature, excitations can occur. As soon as the material heats up
T > 0, the spins begin to fluctuate, and the global magnetization de-
creases. The magnetization follows general rules governed by the equa-
tion M ∝ (1 − T /Tc )β . Here, the critical temperature Tc at which the
|1Introduction
a b
Magnetization
T=0
T > Tc
Tc
Temperature
Figure 1.1: (a) Illustration of bird swarming in a collective pattern emerging from a
global order, created with OpenAI. (b) Magnetization as a function of temperature for
a ferromagnetic material following the equation M ∝ (1 − T /Tc )β . At T = 0, all the
spins point in the same direction. As soon as the material heats up T > 0, excitations
are possible. At T > Tc , the spin orientations are random, the net magnetization is
lost, and the material is paramagnetic.
net magnetization is zero corresponds to the spins being randomly ori-
ented. β is the critical exponent, which is universal and does not depend
on the details of the physical system. The critical exponent is dictated
by both the spacial and spin dimensionality of the system. In the par-
ticular case illustrated in Fig. 1.1(b), β = 0.23 . Here, β belongs to the
universality class with a spatial dimensionality equal to 2 and spin di-
mensionality XY (2D-XY), meaning that the system extends in (finite)
two dimensions and the magnetic spins are allowed to rotate in the XY
plane [4].
In the same way, as for a ferromagnetic material, the behavioral rules
among a flock of starlings are guarded by a level of noise and environ-
mental perturbations. For instance, a predator might attack at one end
of the flock creating a ripple throughout the flock due to disturbed in-
teractions among individuals, which could be described as Tc . Thus, a
multiscale dependence will arise when the global order at a larger scale
is affected.
This brings us to the magnetic materials explored in this thesis. By
sample fabrication, we created artificial metamaterials and amorphous
alloys that were used to modulate interactions along different dimensions.
From our modulations, new phenomena arose due to collective behavior
at different length scales.
2 |Chapter 2
One dimensional interactions
Two ferromagnets separated by a nonmagnetic layer can couple with
each other. The interaction is called interlayer exchange coupling and
is one of the cornerstones of nanomagnetism. An epitome example of
interlayer exchange coupling is two iron layers separated by a magnesium
oxide insulator. The most commonly accepted model to describe the
interaction is based on quantum interference and confinement of one-
dimensional quantum potentials. The coupling between the Fe layers
is usually seen as a consequence of nearest-neighbor interactions due to
spin-polarized tunneling that leads to an antiferromagnetic order. In
the present chapter, the impact of the total extension of Fe/MgO(001)
superlattices and changes in magnetic properties caused by temperature
are studied with experimental methods.
2.1 Coupling across insulating layers
Interlayer exchange coupling (IEC) can be traced back to the 1980s. It
was predicted theoretically in 1986 [5], and later the same year, an anti-
ferromagnetic coupling was observed across metallic spacer layers in both
epitaxial Fe/Cr multilayers [6, 7] and rare earth multilayers [8, 9]. At
this point, the interest in exchange coupling grew, leading to many dis-
coveries. One of them was the oscillatory IEC observed when the thick-
ness of the metallic layer is altered [10]. These findings enable one to
tune the magnetic ordering from ferromagnetic (FM) - preferring parallel
configuration - to antiferromagnetic (AFM) having antiparallel order as
schematically illustrated in Fig. 2.1. The oscillatory behavior resembles
the one observed for Ruderman-Kittel-Kasuya-Yosida (RKKY) interac-
tions between magnetic impurities in a non-magnetic host [11] and is,
therefore, often termed RKKY-like interactions.
|3One dimensional interactions
The ability to control the oscillatory
behavior of IEC triggered the discovery
of giant magnetoresistance (GMR). GMR
Ferromagnetic
is an effect that is dependent on the spin-
Coupling, J
0 D dependent scattering of electrons. The re-
sistance displays a minimum when neigh-
Antiferromagnetic boring ferromagnetic layers are aligned
parallel (RFM ) and a maximum when the
layers are aligned antiparallel (RAFM ).
Figure 2.1: Schematic illustra- The ratio of the two gives relative
tion of the oscillatory behavior of
interlayer exchange coupling as a
magnetoresistance ΔR/R = [RAFM −
function of thickness D of a metal- R FM ]/RFM and is used as a measure of
lic spacer layer. the effect. When changing the magnetic
configuration from antiparallel to paral-
lel in the original Fe/Cr multilayers fab-
ricated in 1986, a ratio of roughly 50% was obtained at a 4.2 K, while 3%
was obtained at room temperature [12, 13]. The findings of the resistance
led to an explosion in research, advancement in growth techniques, and
eventually to new functionalities such as spin-valve sensors. It marked
the advent of the field of spintronics [14] and completely revolutionized
the magnetic recording industry by reducing the bit size and enhancing
the storage capacity. The discovery of GMR was therefore followed by
recognition through the award of the Nobel Prize in Physics 19 years
later [15].
In virtue of the GMR effect, the quest was to enhance the resistance.
The interest, therefore, partly turned towards magnetic multilayer stacks
based on FM layers separated by a thin insulating (I) layer, so-called
magnetic tunnel junctions (MTJs). The magnitude of magnetoresistance
in MTJs has - at least theoretically - no limit, and a massive surge has
been in this research field. The breakthrough came when the first ob-
servation of tunneling magnetoresistance (TMR) at room temperature
was achieved in amorphous aluminum oxide base systems [16, 17]. How-
ever, the amorphous structure turned out to be a problematic candidate,
and resistance of only 70% was obtained at room temperature, which is
lower than needed for most spintronic devices [18]. Instead, the interest
turned towards single-crystalline barriers, which finally led to the discov-
ery of giant TMR effects in Fe/MgO/Fe trilayers that showed resistance
of ∼200% [19, 20]. Even though Fe/MgO was a promising candidate,
it was not possible to smoothly implement them in the already exciting
devices, and instead, systems based on CoFeB were commercialized [21].
The progress on Fe/MgO systems has been tied to an improved under-
standing and control of the physical processes governing the coupling
mechanism. The heart of this chapter is, therefore, merely a work in-
4 |2.1 Coupling across insulating layers
spired by the fundamental nature of the IEC and the magnetic properties
in MTJs, especially Fe/MgO superlattices.
Quantum interference
The now widely accepted model for cou-
pling across an insulator is based on V0
Barrier (I)
a unified theory of quantum interfer-
VOne dimensional interactions
The above conceptual framework is
based on temperatures at T = 0. At finite
Ferromagnetic temperatures, the energy of the electrons
Coupling, J
which mediate the interlayer exchange
0 D coupling increase. Consequently, the elec-
trons become thermally activated and
Antiferromagnetic can populate excited electronics states,
which experience a lower tunneling bar-
rier. Thus, an increase in tunneling prob-
Figure 2.3: Schematic illustra- ability is expected [24]. Surprisingly few
tion of interlayer exchange cou- temperature-dependent studies have been
pling that decays exponentially as carried out on systems with insulator
a function of thickness D of an in-
sulating spacer layer.
spacers [30–34]. Even more, unexpect-
edly, are the dividing and opposite tem-
perature dependencies identified. Some
of the studies contradict the model predicted by quantum interference
[31, 33, 34], and a negative temperature coefficient has been observed
[35].
Coupling mechanisms and the role imperfections
In the previous section, we assume that the two ferromagnetic layers
and the insulating spacer are homogeneous and that the interfaces are
well-ordered, with perfect bonding leading to an indirect coupling. How-
ever, this is not always the case. As the electronic states determine
how effectively the electrons will transmit across the interface, it is also
reasonable to consider that any deviations in a sample will affect the
tunneling current as well as the magnetic behavior and strength of the
coupling. Thus, the interlayer exchange coupling mediated through an
insulator can be divided into two subcategories, namely, intrinsic and
extrinsic mechanisms.
To the subcategory of intrinsic mechanisms belongs the indirect cou-
pling described above. Nonetheless, the tunneling depends on the prop-
erties of both the ferromagnets and the insulating material itself [36].
The electrons which cross the barrier have different symmetry, and if the
symmetry of the Bloch state is conserved when crossing into the barrier,
the electrons will coherently tunnel. Coherent tunneling can be assumed
to occur in a barrier that is a perfect single-crystal. However, if the bar-
rier is amorphous, the lack of crystalline symmetry and disorder at the
interface is proposed to lead to a loss of coherence in transmission caused
by fluctuations in the insulator layer thickness. The loss of coherence,
in turn, can lead to a decrease in tunneling probabilities in comparison
to a perfect single-crystal [37]. Hence, the state of the interfaces and
6 |2.1 Coupling across insulating layers
a b c
FM
FM FM
I I
I
FM FM FM
d e f
FM FM
FM
I Defect/Impurity I I
FM FM FM
Figure 2.4: Schematic illustration of coupling induced by roughness due to cor-
related interfaces with (a) variation of the spacer resulting in an antiferromagnetic
alignment of adjacent layers, (b) constant spacer thickness (waviness) inducing a fer-
romagnetic order. (c) Coupling induced by uncorrelated roughness with variations of
the coupling strength leading to non-collinear magnetic order (90 ◦ ). The bottom of
the figure illustrates coupling induced (d) by magnetostatic coupling due to dipolar
fields arising at the edges, (e) defects and impurities in the spacer layer mediating
the coupling between the two FM layers, and (d) pinholes bridging a ferromagnetic
order through the spacer layer.
barrier material heavily influence the tunneling mechanism and coupling
strength.
An indirect coupling can be overshadowed by extrinsic mechanisms,
which then partially or entirely dictate the strength of the interlayer
exchange coupling and the magnetic order between adjacent layers. An
example of an extrinsic mechanism is correlated roughness, as illustrated
in Fig. 2.4(a-b). Here a periodic variation of the spacer layer thick-
ness where an antiferromagnetic coupling dominates, the alignment be-
tween adjacent layers will be antiferromagnetic [38]. While on the other
hand, having a correlated roughness with a constant spacer thickness1
is generally characterized by a ferromagnetic order. A roughness that is
non-correlated [see Fig. 2.4(c)] with fluctuating spacer thickness where
oscillations of the coupling strength lead to areas that are dominated
with ferromagnetic and other areas with antiferromagnetic coupling can
result in a non-collinear alignment (such as 90◦ ) of the magnetic layers
[40].
An additional extrinsic mechanism is magnetostatic coupling, which
refers to a coupling mediated by dipolar fields originating from the edges
of a sample as illustrated in Fig. 2.4(d). The dipolar field leads to a flux
closure between the FM layers, and if dominating, it leads to an anti-
ferromagnetic order. However, as the dipolar field decays exponentially
1
Often referred to as Orange-peel (Néel) coupling [39]
|7One dimensional interactions outside a film, one can neglect this mechanism for samples that exceed a lateral size of < 1 cm2 [41]. The last mechanisms described herein originate from deviations of a perfect insulator. Examples of such are localized impurity or defect states at the interface or in the insulating spacer layer, as illustrated in Fig. 2.4(e) [35]. Zhuravlev, et. al. has developed a model demonstrating that if the energy of these localized states matches the Fermi energy of the insulator, the tunneling exhibits a resonant character. Consequently, the antiferromagnetic coupling between the two ferromagnetic layers be- comes stronger, and the mechanism is often termed impurity-assisted coupling. In addition, the resonant character can lead to a thermal broadening and an increased strength of the antiferromagnetic coupling with decreasing temperature. Thus, the impurity-assisted coupling has been proposed as one possible reason behind the inconsistency of exper- imental observations for some ferromagnet/insulator systems [35]. The last mechanism described is pinholes, illustrated in Fig. 2.4(f). Pinholes lead to bridging the contact between the ferromagnetic layers through the insulating material. Depending on the extent and how ho- mogeneously they are distributed over the surface, these holes can result in either a ferromagnetic or non-collinear (90◦ ) magnetic order. Thus, having a thick insulator barrier, a coupling due to pinholes can generally be neglected [26, 38]. 2.2 Fe/MgO(001) superlattices The physical mechanism guarding the interlayer exchange coupling and the magnetic properties in Fe/MgO tunnel junctions has been and is still under investigation. Clearly, the crystalline quality affects tunneling con- ductance, and proper fabrication methods for trilayers and beyond are therefore essential to achieve. The structural quality has been optimized by alternating the growth of single crystalline α-Fe with body-centered cubic (bcc) structure and lattice constant a of 2.86 Å on top of MgO with a = 4.21 Å. Multiple repeats and epitaxial growth are then enabled by rotating the Fe lattice by 45◦ on top of the MgO, forsaking a slight lattice mismatch (aMgO /aFe ) of roughly 4% [47]. Meanwhile the optimization of a growth recipe for Fe/MgO structures, it was discovered how sensitive the interface quality is to obtain high tunneling conductance. For instance, it was found that MgO grown on Fe leads to the formation of a FeO due to the bonding between iron and oxygen. The formation of FeO might, in the end, affect the electronic structure at the boundary and the effectiveness of transmission across the interface [42–44]. Thus, due to difficulties characterizing the atomic structure, there is still limited information on the bonding and structure 8 |
2.2 Fe/MgO(001) superlattices
of the interface and to which extent and how detrimental the FeO is to
the magnetic properties and tunneling conductance.
Furthermore, the strength of the interlayer
exchange coupling is commonly elucidated, in-
N-1
ferring the total areal energy density of a mul-
tilayer system. The energy of a magnetic solid
M
N depends on the orientation of the magnetiza-
N J tion with respect to the crystal axes, known
M N+1 as magnetic anisotropy. Epitaxial Fe is sin-
dFe N+1 gle crystalline the magnetic properties are dic-
tated by the symmetry of the bct crystalline
structure. As a result, Fe has preferred mag-
N+2 netization directions, with an easy axis that
H is oriented along the (energetically fa-
vorable) and a hard axis along the
Figure 2.5: Schematic
view of a multilayer with in-
(energetically unfavorable). Assuming nearest
terlayer exchange coupling. neighbor interaction and a coherent in-plane
rotation accordingly for N number of layers,
the total energy can be elucidated by [45]:
N −1
1
E(N ) = − JN,N +1 cos(θN − θN +1 )
2 1
(2.1)
N
− Ms dFe H cos θN + Eanis
1
where J is the interlayer exchange coupling, Ms the saturation moment,
dFe the thickness of the magnetic layer layers, and H the externally
applied field as schematically displayed in Fig 2.5. The second term
in Eq. 2.1 is the Zeeman energy, while Eanis is the energy associated
with the magnetic anisotropy. Assuming that exchange and Zeeman
contributions dominate and that all layers are identical, the equation
can be reduced by minimizing the total energy to:
Hs Ms dFe 4J(1 − 1/N )
J(N ) = ⇒ Hs (N ) = (2.2)
4(1 − 1/N ) Ms dFe
here Hs is the saturation field, which is the field required to align the
layers parallel with an external field. Thus, the strength of the coupling
can, in this way, be determined by experimentally observed magnetiza-
tion obtained by hysteresis loops.
The remaining part of this chapter is dedicated to investigating the
magnetic properties in epitaxial [Fe(2.3nm)/MgO(1.7nm)]N (001) super-
lattices studied in Paper I (Ref. [46]). N indicates the number of bi-
|9One dimensional interactions
layer repetitions in the superlattice stack and has, in this chapter, been
varied between 2 and 10. The thickness of the Fe layers is kept the
same, and long-range dipolar interactions are therefore disturbed by the
outer boundaries of the thin film, which creates an out-of-plane magnetic
hard axis while maintaining the in-plane four-fold magnetocrystalline
anisotropy. The growth process for all of the samples mentioned herein
has been optimized, and a more detailed description of the fabrication
can be found in the thesis written by Tobias Warnatz [47]. The structural
quality of the samples can be seen as exceptionally high with smooth in-
terfaces, and related extrinsic mechanisms are therefore expected to be
negligible. In addition, the MgO layers investigated in this chapter are
1.7 nm, and pinholes are not expected to be present as this thickness is
above the critical thickness at which pinholes have been shown to impact
the magnetic properties [47]. The samples are all 1 cm2 , and stray field-
induced coupling, therefore, becomes insignificant. Consequently, the
most relevant interlayer exchange coupling discussed in the upcoming
sections can be viewed as only originating from intrinsic mechanisms.
2.3 Field dependence
This section is devoted to presenting and understanding the field de-
pendence and especially the relation to interlayer exchange coupling in
[Fe(2.3nm)/ MgO(1.7nm)]N (001) superlattices with 2 ≤ N ≤ 10. In or-
der to investigate the coupling between the Fe layers, in-plane magnetic
measurements were performed using Longitudinal Magneto-Optical Kerr
Microscopy (MOKE)2 .
Normalized hysteresis curves for [Fe/MgO]2 measured at room tem-
perature along the easy-axis (Fe[100]) and hard-axis (Fe[110]) is shown
in Fig. 2.6. Comparing the two hysteresis loops reveals a divergent
saturation field for the two directions, where the saturation field is the
field required to align the two Fe layers parallel with the external field,
indicated with Hs in the figure. The much smaller field, Hs = 2.4 mT,
which is obtained along the easy-axis, compared to Hs = 45 mT obtained
along the hard-axis, is a result of a strong magnetocrystalline anisotropy.
Furthermore, since MOKE is only sensitive to the magnetization parallel
to the scattering plane, the change in magnetization with the externally
applied field is proportional to the magnetization direction of the two
Fe layers. Using MOKE, therefore, makes it possible to elucidate the
reversal mechanism of the layers. For instance, along the hard axis, a
continuous decrease in magnetization can be observed when reducing the
field from Hs , which is a result of a coherent rotation of the Fe layers
towards the magnetic easy axis. On the other hand, a notable discrete
2
The experimental set-up and procedure is described in Appendix - Section 8.1
10 |2.3 Field dependence
1.0
a b
Normalized Kerr signal
Hs Hs
0.5
0.0
-0.5
Easy axis Hard axis
N=2 N=2
-1.0
-50 -25 0 25 50 -50 -25 0 25 50
0H (mT) 0H (mT)
Figure 2.6: Hysteresis loops measured along the (a) easy (Fe[100]) axis, and (b) hard
(Fe[110]) axis for [Fe(2.3nm)/ MgO(1.7nm)]2 (001) superlattice at room temperature.
Hs indicate the field required to align the two Fe layers parallel with the external field,
and the gray arrows and illustrations specify states at saturation and possible state
in the absence of an externally applied field. Adapted Paper I (Ref. [46]).
magnetic switching, i.e., digital hysteresis [46, 48], is visible for the easy
axis when reducing the field from saturation. This abrupt step is a sig-
nature of switching one individual Fe layer due to rapid nucleation and
motion of 90◦ domain walls across the entire sample [48–50].
Moreover, a remanent magnetization of roughly √ 0.50Ms is obtained
along the easy axis, while for the hard axis 1/ 2Ms of the magnetiza-
tion is retained after the field has been reduced to zero, with Ms being the
magnetization at saturation. These values are consistent with a 90◦ mag-
netic order of the two Fe layers. The 90◦ magnetic-layer configuration is
a metastable state as a result of the interplay between antiferromagnetic
coupling and the must stronger four-fold magnetocrystalline anisotropy
(approximately 20 times) [48, 51]. In addition, it is worth addressing
that each curve is an average of 30 field scans. Hence, the switching of
the layers is reproducible, resulting in the same behavior.
Increasing the number of bilayers to four (N = 4) has a profound
effect on the hysteresis, as shown in Fig. 2.7(a), for a field applied easy
(Fe[100]) axis. For instance, a negligible remanence is obtained at zero
field, recognized as a signature of the antiferromagnetic configuration.
Hence, the strength of the antiferromagnetic coupling is high enough in
comparison to the magnetocrystalline anisotropy to stabilize the antifer-
romagnetic order in the absence of an external field [48]. In addition,
three steps are visible between remanence and saturation. The step clos-
est to remanence (H1 ) is twice as large as the other two and reveals a
magnetization which is 0.5Ms . The change of magnetization is equal to
reversing two layers and arises from simultaneously switching the out-
ermost Fe layers. The outermost layers have one nearest neighbor, and
consequently, a lower field is required for the switching, an effect merely
arising from the difference in the number of interacting neighbors[46, 48].
Further increasing the field results in the switching of the remaining lay-
| 11One dimensional interactions
1.0
Normalized Kerr signal
a b c
Hs
0.5
H2
H1
0.0
-0.5
N=4 N=8 N=10
-1.0
-20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20
0H (mT) 0H (mT) 0H (mT)
Figure 2.7: Normalized hysteresis curves for [Fe/MgO]N (001) superlattices with (a)
N = 4, (b) N = 8, and (c) N = 10. The measurements are performed at room
temperature along the easy (Fe[100]) axis. The indicated letters in (a) represent the
switching field of the Fe layers, where H1 is the field required to switch the outermost
layers. The gray arrows indicate the magnetic alignment of the Fe layers at the
indicated external field. Adapted from Paper I (Ref. [46]).
ers. The saturation field (Hs ) represents the field required to align the
layers that are strongest coupled layers along the easy axis and can there-
fore be seen as a measure of the interlayer exchange coupling and can
be calculated by using Eq. 2.2. Similar behavior can be observed for
N = 8 and N = 10 as shown in 2.7(c-d). Here it becomes clear that
from now on, one has to distinguish between an interlayer exchange cou-
pling which results in a particular type of magnetic configuration, and
an antiferromagnetic coupling leading to a bilinear configuration where
adjacent layers have an antiparallel order.
MOKE is a fast technique that makes it possible to determine the
magnetization as a function of the external field and observe the layers’
switching. However, the obtained magnetization is a weighted average of
all layers in the superlattice stack, making interpretation of the switching
sequence of the layers difficult. Therefore, the switching sequence of
individual layers was determined with Polarized Neutron Reflectivity
(PNR) in combination with MOKE, as PNR results can be used to infer
the magnetic alignment of individual Fe layers qualitatively. The PNR
measurements were all performed at ILL at the beamline SuperADAM
[52], and the magnetic orientation of individual layers was determined
by fitting the collected data with GenX [53]. The results of the fitting
and measurement for all the samples can be found in Paper I [46].
An example of the switching behavior which has been determined by
PNR is illustrated in Fig. 2.8 for N = 8 bilayers3 . The numbers indicated
in the figure represent the external field used during PNR measurements.
Before the PNR measurements were performed, an external field of 500
mT was applied in the plane along the easy (Fe[100]) axis in order to
3
PNR data can be found in Paper I (Ref. [46]).
12 |2.3 Field dependence
Step 1 Step 2 Step 3 Remanence
1.00 8
Normlaized Kerr Signal N=8 1
Normlaized Kerr Signal
0.75 6
2
0.50 4
3
0.25 2
0.00
0 5 10 15
0H (mT) Field direction
Figure 2.8: (Left) Hysteresis loop measured along the easy axis for [Fe/MgO]8 with
numbers indicating the external fields used in PNR measurement. (Right) Schematic
illustration of the magnetic orientation of the Fe layers at external H1 , H2 and H3 ,
and remanence, determined form fitting the PNR data. Adapted from Paper I sup-
plementary information (Ref. [46]).
ensure that the sample was saturated, illustrated by Step 1 in Fig. 2.8.
Reducing the field from saturation (Step 1 → Step 2) and measuring at
an external field of 7.2 mT reveals a simultaneous switching of all odd
innermost layers. This confirms the results obtained from the hysteresis
loop where the normalized magnetization at 7.2 mT is ∼ 3/8Ms . Further
reducing the external field (Step 2 → Step 3) leads to the simultaneous
reversal of the innermost even layers. The fact that the outermost layers
are still pointing along the applied field at Step 1 confirms that these
are the weakest coupled layers, which is in harmony with only having
one nearest neighbor. At remanence, the layers should, therefore, have
an antiferromagnetic configuration in line with zero magnetization and
switching of the outermost layers.
Effect of total extension
Interlayer exchange coupling is often seen as a consequence of nearest-
neighbor interaction between adjacent ferromagnetic layers. However,
the switching sequence for the [Fe/MgO]N superlattices shown above is
difficult to rationalize solely based on a coupling mediated by nearest-
neighbor interaction. For example, to a first approximation, the inner-
most layers can be viewed as equal with respect to the field response.
Assuming that each layer only interacts with its nearest neighbors, all the
inner layers should simultaneously switch when reducing the field from
saturation. The hysteresis loops should therefore display two switching
fields. Thus, MOKE measurements, combined with PNR results, illus-
trate that this is not the case for any of the samples with N = 4, 8, and
10, as shown in Fig. 2.7 and Fig. 2.8.
| 13One dimensional interactions
4
3
Hs(N)/H1(N)
2
1
0
2 4 6 8 10 12
Number of Fe layers (N)
Figure 2.9: Normalized switching field for the outermost layers (H1 ) and the switch-
ing field required to align the layers parallel (Hs ) for samples with the number of
bilayer repetitions N . The dashed red lines correspond to the normalized switching
field for H1 , while the blue dashed line corresponds to the ratio Hs /H1 = 2 referring
to the normalized field required to switch the strongest coupled layers if coupling was
mediated by nearest neighbor interaction.
Furthermore, each step in the hysteresis loops along the easy axis is
proportional to the coupling strength of the layer(s) that is switching.
Since the outermost layers only have one nearest neighbor each, they
should experience half of the coupling compared to the innermost layers,
which have two. Hence, taking the ratio of the two switching fields should
result in Hs /H1 = 2. The ratio Hs /H1 is summarized in the Fig. 2.9 for
N = 2, 4, 8 and 10 repetitions. The figure illustrates that for N = 10,
the strength of the coupling of the outermost layers (Hs ≈ 16.9 mT) is
almost three times higher in comparison to the strength of the coupling
strength of the innermost layers (H1 ≈ 5.9 mT). The switching of the
outermost layers, and the saturation of the samples, can, therefore, not
be captured by nearest neighbor interactions when changing the number
of repeats. The changes can be argued to stem from two sources: beyond
nearest neighbor interaction and changes in the strength of the interlayer
exchange coupling between the layers due to the total extension of the
samples [46, 48].
2.4 Temperature dependence
The interlayer exchange coupling and resulting magnetic properties of
Fe/MgO(001) superlattices are guarded by a rather complex mecha-
nism. In order to get a complete grasp of the nature of the coupling
more reliably, temperature-dependent measurements were performed us-
ing MOKE. Fig. 2.10(a-c) shows hysteresis loops for the [Fe/MgO](001)
14 |2.4 Temperature dependence
1.0
a b c
Hs Hs
0.5 Hs
M/M0
0.0
-0.5
300 K 165 K 20 K
-1.0
-100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100
0H (mT) 0H (mT) 0H (mT)
Figure 2.10: Normalized hysteresis loops measured at (a) 300, (b) 165, and (c) 20
K along the easy (Fe[100]) easy axis for the Fe/MgO superlattice with N = 10. The
red arrow in the figures indicates the saturation field required to align all the layers
parallel with the external field Hs .
superlattice with N = 10 at T = 300, 165, and 20 K. The measurements
were performed along the easy (Fe[100]) axis and are normalized to sat-
uration magnetization Ms at T = 0. As shown in the figure, changing
the temperature remarkably affects the shape of the hysteresis curve. At
T = 300 K, digital hysteresis with clear steps is visible, and the field
required to align all the Fe layers parallel with the field (Hs ) is achieved
by applying a modest field of 16.9 mT. At 165 K, the loop is charac-
terized by a sudden change in magnetization close to remanence and
at ±60 mT. In between these steps, the magnetization changes linearly
with the field. The switching of the layers is no longer digital, bear-
ing a larger similarity to a coherent rotation of the layers. In addition,
the field required to align the layers parallel (Hs ) with the external field
has increased to ≈ 70 mT. Hence, the strength of the interlayer exchange
coupling between the Fe layers appears to increase with temperature. At
T = 20 K, the hysteresis loop is more S-shaped, and the discrete steps
have completely vanished. The interlayer exchange coupling between the
layers is at these temperatures, therefore, strong enough to overcome the
magnetically hard (Fe[110]) axis, and the field response is dominated by
coherent rotation. In addition, the coercivity has increased as well as
a clear remanence of ∼ 0.5Ms can be observed at these temperatures,
suggesting a change in the magnetic configuration of the Fe layers at zero
field.
Temperature dependence of the magnetic order
In order to establish the magnetic alignment of individual layers at dif-
ferent temperatures, Polarized neutron reflectivity (PNR) measurements
were performed. A schematic illustration of the experimental setup can
be found in Fig. 2.11(a). During each measurement, a guide field of
| 15One dimensional interactions
a b
Q1/2 Q1 NSF
100
295 K
External field
M|| Mtot 3
10
M|
Reflectivity
10 K
|
++
-+
SF
+ -
100
295 K
-- +- 2
10
4
10
10 K
0.00 0.05 0.10 0.15 0.20 0.25
Q (1/Å)
c d
295 K 10 K
Fe layer 295 K 10 K
1 100 70
2 -73 14
3 106 78
4 -72 5
5 100 77
6 -82 3
7 92 71
8 -83 -12
9 99 67
10 -90 2
Figure 2.11: (a) Schematic illustration of the experimental setup of a polarized
neutron reflection process with components of the magnetization perpendicular and
vertical to the initial polarization of the neutrons. (b) Polarized neutron reflectivity
measurements and fit performed with GenX [53] at 295 K and 10 K in an external field
close to remanence (1.5 mT and 20 mT). The data has been shifted (in intensity) for
clarity. The shaded grey areas represent the width of the widest Q1 and Q1/2 peak.
(c) Table for magnetization angles of individual Fe layers close to remanence obtained
by fitting PNR data, with every odd layer highlighted in bold. (d) Illustration of the
magnetic alignment of individual layers at 295 K and 10K close to remanence.
1.5 − 20 mT was used to maintain the neutron polarization parallel to
the in-plane axis of the samples, as indicated by the arrow in the figure.
Prior to each measurement, an external field of 500 mT was applied with
an electromagnet along the film plane, followed by reducing it to a value
close to remanence. To qualitatively infer the magnetic orientation, the
collected data for the non-spin-flip (NSF) R++ and R−− channels, as
well as the spin-flip (SF) R−+ channel, was fitted with GenX [53] fol-
lowing the procedure described in the Appendix - Section 8.2.
Fig. 2.11(b) shows PNR data, including fits obtained by GenX for
the NSF (R++ ) and the SF channel (R−+ ) collected at 295 and 10 K
in an external field close to remanence, 1.5 mT, and 20 mT, respec-
tively, for the sample with N = 10. At 295 K, the NSF channel shows
16 |2.4 Temperature dependence
a first-order Bragg peak Q1 at the scattering vector Q1 = 2π Λ = 0.172
Å−1 (gray shaded areas). Q1 originates from the structural periodic-
ity of the superlattice where Λ represents the thickness of the Fe/MgO
bilayer. The SF, being only sensitive to scattering with a magnetic ori-
gin, shows a well-defined Q1/2 peak. The Q1/2 peak corresponds to a
magnetic alignment in the transverse direction to the neutron polariza-
tion axis with twice the structural periodicity. That means the Fe layers
have an antiferromagnetic configuration, as schematically illustrated in
Fig. 2.11(d). The angles of each Fe layer were determined by fitting the
PNR curves, which revealed an angle of almost 180◦ between adjacent
layers as indicated in the table in Fig. 2.11(c). The layers are, therefore,
antiferromagnetically ordered at room temperature, consistent with the
interpretation of the MOKE data shown in Fig. 2.7(c). It is worth stress-
ing that a weak Q-half peak (Q1/2 ) can be observed in the NSF channel.
This peak can only arise due to magnet scattering, which suggests that
the sample exhibit a magnetic component along the direction of the ap-
plied field with twice the structural periodicity. This could be caused
by, for example, magnetic domains with antiferromagnetic alignment.
These domains may be randomly oriented, resulting in a projection that
is parallel and perpendicular to the polarization of incident neutrons.
At 10K, the PNR data shown in Fig. 2.11(b) reveal Q1 and Q1/2 peaks
in both the NSF and the SF channel. These peaks rise due to alternating
parallel and perpendicular components with respect to the external field
in every other Fe layer. Fitting the PNR data revealed that the magne-
tization in every even layer is oriented along the direction of the applied
field with an angle away from the polarization of incident neutrons, as
shown in Fig. 2.11(c). The remaining odd layers (highlighted in bold)
point along the transverse axis close to perpendicular to the initial polar-
ization of incident neutrons. Hence, a periodic alignment of the Fe layers
with an average of 70◦ is obtained at 10 K, as schematically illustrated
in Fig. 2.11(d). Thus, a transition from a magnetic structure with an
angle of π characterizing the magnetic order at room temperature to a
structure with an angle of >70◦ between the layers when decreasing the
temperature has been identified.
Temperature dependence of interlayer exchange coupling
The magnetic configuration at a given temperature can be obtained by
taking the ratio of remanent and saturation magnetization Mr /Ms ob-
tained from hysteresis loops. The ratio Mr /Ms is summarized in the top
of Fig. 2.12(a) for N = 10 bilayers. The temperature dependence reveals
two distinct regions, separated by a transition region. At temperatures
≥ 180 K, the Fe layers are aligned in an antiferromagnetic configuration,
defined by Mr /Ms = 0, as confirmed with MOKE and fitting of PNR
| 17One dimensional interactions
a b
0.6
N = 10 N = 2
0.4
Mr/Ms
102
90° AFM N = 4
0.2
N = 8
0.0 N = 10
Hs (mT)
103 N = 10
Hs (mT)
102 101
101
100 150 200 250 300
0 100 200 300 Temperature (K)
Temperature (K)
Figure 2.12: Temperature dependence of (a) the ratio Mr /Ms indicating a transition
of the magnetic configuration and the change in saturation field Hs for [Fe/MgO]10 .
Shaded areas indicate the temperature of the onset and end of the transition. (b)
Saturation field Hs as a function of temperature for N = 2, 4, 8, and 10. Shaded
regions indicate the temperature at which the increase of Hs is no longer exponential.
data. At 180K, the onset of a transition becomes apparent where Mr /Ms
gradually increases to a value close to Mr /Ms ≈ 0.5. This is consistent
with PNR data and indicates that Fe layers have a periodic alignment
of ≈ 70◦ between adjacent Fe layers at low temperatures.
The change in magnetic order is accompanied by a giant increase in
the field required to align all the Fe layers parallel with the external
field (Hs ), as shown in the bottom of Fig. 2.12(a) for N = 10. At
room temperature, Hs is roughly 16.9 mT and increases exponentially
til ≈ 180 K. At 180 K, Hs ≈ 50 mT and increases further to 940 mT at
10 K. Similar behavior were obtained for samples with N = 4 and 8 as
illustrated in Fig. 2.12(b), where the exponential increase has been fitted
as a guide for the eye. Here, it is clear that a change in the interlayer
exchange coupling for all three samples (N = 4, 8, and 10) takes place
at roughly 175 K, where the increase of Hs is no longer exponential with
decreasing temperature. For the sample with N = 2, on the other hand,
the change of the exponential increase takes place at roughly 125 K.
The increase of Hs with decreasing temperature indicates that the
interlayer exchange coupling between the layers is increasing. However,
the giant increase of the coupling contradicts the theoretical model based
on quantum interference developed by Bruno [28]. Nonetheless, it is in
line with the model based on impurities (impurity-assisted coupling) in
the tunneling barrier proposed by Zhuravlev [35]. Considering that the
increase of the interlayer exchange coupling also leads to a periodic align-
ment of roughly 70◦ between adjacent Fe layers as well as changes in the
reversal character (digital hysteresis loops to a coherent rotation of all the
layers), the results hint towards a change in the nature of the interlayer
18 |2.4 Temperature dependence
exchange coupling with temperature. Since the onset of the transition
shown in Fig. 2.12(b) takes place around the same temperature for the
N = 4, 8, and 10, it suggests that something intrinsic occurs in the MgO
barrier. Since the interface is a predominant factor determining the tun-
neling conductance, a FeO interface layer formed when MgO is grown on
top of Fe could be a possible explanation. However, it is difficult to tell
without further investigating the interface separating the Fe and MgO
layers.
| 19Chapter 3
Two dimensional interactions
The metamaterials studied in this chapter consist of flat (2D) circular
islands (75 - 450 nm) - mesospins - of ferromagnetic material. The
elements were patterned from 10 nm thick polycrystalline FePd film us-
ing lithography methods, and in this instance, the magnetocrystalline
anisotropy can be ignored. The lateral extent of the mesospin is much
larger than the thickness, and it is, therefore, more energetically favor-
able for the magnetization to lie in-plane. The circular shape allows for
inner degrees of freedom, and the magnetization within the mesospins
can either be described as a single-domain or textured state. If the ar-
rays are densely packed, the islands can interact due to external degrees
of freedom arising from the stray fields of the mesospins. This chapter
explores thermally active elements and their collective behavior arising
from a multiscale dependence due to an interplay of internal and external
degrees of freedom as a function of geometry and lateral extension.
3.1 Inner texture of single mesospins
The spontaneous magnetic order within a single ferromagnet with neg-
ligible magnetocrystalline anisotropy is ultimately determined by shape
and size. In a large ferromagnetic solid, the magnetization often breaks
into magnetic domains separated by domain walls. However, if the sam-
ple is reduced below the sub-micrometer size, the energy cost of domain
walls becomes too high in comparison to the cost of creating the stray
field, resulting in a single-domain state [4]. A flat, circularly shaped is-
land that is a few tens of nanometers thick and with lateral dimensions
on the order of hundreds of nm, the magnetization is conferred to lie
in the plane due to shape anisotropy. Magnetic texture in the form of
| 21Two dimensional interactions
Collinear O S C Vortex
m 1 0.97 ( 0) 0.85 ( 4) 0.66 ( 3) 0
Figure 3.1: Magnetic texture for collinear, O-, S-, C- and vortex state, depicting
uniform and nonuniform patterns. The average magnetization |m| has been nu-
merically calculated using Mumax3 [54] and is extracted along the horizontal axis of
mesospins with a diameter of 350 nm. |m| for islands with a diameter of 250 and
450 nm can be found in Paper IV. Data for |m| is adapted from Paper III (Ref.
[55].)
inner degrees of freedom within the island can then emerge, such as the
patterns illustrated in Fig. 3.1.
The simplest form of the internal structure is the collinear state which
can be treated as an indivisible building block without inner texture. The
magnetization points uniformly along one direction, and the net in-plane
magnetization | m | equals 1. When shape and crystalline anisotropy are
neglected, the mesospin behaves as a large spin. In other words, it acts
as a magnet with one north pole and one south pole, which is free to
rotate and can therefore act as a classically defined 2D-XY spin [56].
However, the circular shape allows for inner degrees of freedom, and the
magnetization can curve along the edge and form nonuniform patterns
such as O-, S-, and C-state [57–59]. These states can be viewed as
perturbations of the collinear state where | m | ranges from 0.97(0) for
the O-state to 0.66(3) for the C-state. Here, | m | has been determined
by using micromagnetic simulations performed by Mumax3 [54]. The
average magnetization has been extracted along the horizontal axis of
the mesospins indicated in Fig. 3.1 for islands with a diameter of 350
nm. The last listed state in Fig. 3.1 is the vortex state, which is utterly
different from the others. The vortex state is defined by the curling of
the magnetization creating total flux closure in the plane with a tiny out-
of-plane moment referred to as the vortex core [60–62]. Consequently,
the net magnetization in the plane (| m |) of the vortex texture can be
negligible.
The position of the vortex core can be used as a tool to relate the
collinear and vortex state as illustrated in the top of Fig. 3.2. The rela-
tion between the two states can be achieved by a bias from neighboring
islands [60, 63, 64], or by applying an external magnetic field [55, 65, 66].
When applying an external field along an in-plane direction, the vortex
core becomes displaced from the center. Pushing the vortex core out
from the mesospin, it is annihilated, and the magnetic texture resembles
a C-state. Further moving the vortex core leads to a continuous change
of the magnetic texture until it reaches the collinear state.
22 |3.1 Inner texture of single mesospins
2r
2.0
350 nm
1 (Normalized)
1.5
1.0
0.5
150 nm
E/Ev
0.0
75 nm
-0.5
0.0 1.0 2.0
Vortex core displacement (2r/D)
Figure 3.2: Energy landscape obtained by simulations performed in Mumax3 [54],
with energy barriers separating the collinear and vortex state for single islands with
D = 350, 150, and 75 nm. Each point is obtained numerically as the vortex core is
moved along a path resembling applying a magnetic field. The total energy E has
been normalized to the energy associated with the vortex state Ev . The illustration
at the top of the figure illustrates how the vortex core is moved. Adapted from Paper
II (Ref. [66]).
The energy associated with the transition from a vortex to a collinear
state through, for example, an external field can readily be calculated
by using Mumax3 [54]1 . Fig. 3.2 shows the energy landscape and an
activation barrier separation of the two states for a single mesospin with
diameters D = 350, 150, and 75 nm. Here the total energy has been
normalized to the energy associated with the vortex state Ev − 1. The
total energy has been defined as:
Etot = Es + Et (3.1)
where Es is magnetostatic energy being the cost of the stray field and Et
is the exchange energy rising from the cost of magnetic texture within
the mesospin. In the vortex state, E = Ev ≈ Et since the energy is
only dependent on the cost of texture and Es ≈ 0. As soon as the
vortex core is displaced from the center, a collinear magnetic component
arises with a corresponding stray field leading to an energy increase.
The energy reaches a maximum at (2r ≈ 0.9D) and can be seen as the
barrier separating the collinear at the vortex state. Moving the vortex
core further out, the magnetic texture changes, and the energy decrease
as it reaches the collinear state.
1
Details of the simulation procedure can be found in Appendix - Section 8.1
| 23Two dimensional interactions
The simulation results displayed in Fig. 3.2 re-
X-ray direction
0 3veal that the size of the islands determines the
3
m
a) ground state of the mesospins. The vortex tex-
D
ture is favored for large mesospins (D = 350 and
150 nm), while small islands (D = 75 nm) fa-
vor the collinear state. These calculations were
experimentally confirmed by photoemission elec-
3
tron microscopy (PEEM) imaging employing X-ray
b) Magnetic Circular Dichroism (XMCD) acquired at
SOLIEL, beamline HERMES [67], and Advanced
Light source (ALS) beamline 11.01 [68]2 . PEEM-
XMCD results are shown in Fig. 3.3 for mesospins
with a diameter of 350, 150, and 75 nm3 . For
the mesospins to be in the ground state when per-
c) forming the measurement, the samples were cooled
down from room temperature to roughly 100 K un-
der zero applied field. The mesospins are separated
with a distance of G = D+40 nm to ensure no inter-
action among the mesospins takes place. The con-
trast in the image shows the direction of the magne-
Figure 3.3: PEEM- tization with respect to the x-ray beam. White rep-
XMCD images of non- resents magnetization pointing to the right, while
interacting mesospins black corresponds to magnetization, which points
with diameter (a) 350 to the left. In between, there is a gray gradient that
(b) 150 and (c) 75 nm.
represents magnetization with a varying transverse
Adapted from Paper II
(Ref. [66]). component.
For the mesospins with D = 350 and 150 nm
found in Fig. 3.3(a-b), each island displays a white
and black pattern due to the continuous rotation of the in-plane magne-
tization. Depending on if the rotation is clockwise or anti-clockwise, the
black contrast is seen on the top of the islands, or vice versa. The checked
pattern among all the islands reveals a random rotation of the magnetic
texture within the islands for both samples, indicating negligible inter-
actions among the islands. The mesospins with D = 75 nm attain a
collinear state and can therefore be viewed as being two-dimensional
(XY-rotors) [56].
2
Experimental details and description regarding PEEM-XMCD can be found in Ap-
pendix - Section 8.1
3
Fabrication details in Paper II (Ref. [66]).
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