Magnetic Order in Artificial Structures

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Livet är svårt men fjället är vårt. –Swedish proverb

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List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Magnetic structure and diffracted magneto-optics of patterned

amorphous multilayers

U. B. Arnalds, E. Th. Papaioannou, T. P. A. Hase, H. Raanaei, G. Andersson, T. R. Charlton, S. Langridge, and B. Hjörvarsson Physical Review B 82, 144434 (2010)

II X-ray resonant magnetic scattering from patterned multilayers

U. B. Arnalds, T. P. A. Hase, E. Th. Papaioannou, H. Raanaei, R. Abrudan, T. R. Charlton, S. Langridge, and B. Hjörvarsson

Submitted manuscript

III Melting artiﬁcial spin ice

V. Kapaklis, U. B. Arnalds, A. Harman-Clarke, E. Th. Papaioannou, M. Karimipour, P. Korelis, A. Taroni, P. C. W. Holdsworth, S. T. Bramwell, and B. Hjörvarsson

New Journal of Physics 14, 035009 (2012)

IV Photoemission electron microscopy of thermal ground state

ordering of artiﬁcial kagome spin ice building blocks

U. B. Arnalds, A. Farhan, V. Kapaklis, A. Weber, E. Th. Papaioannou, A. Balan, M. Ahlberg, R. Chopdekar, F. Nolting, B. Hjörvarsson, and L. J. Heyderman

Manuscript

V Element speciﬁc magnetization in Fe/Pd quantum well structures

T. P. A. Hase, M. S. Brewer, U. B. Arnalds, M. Ahlberg, V. Kapaklis, E. Th. Papaioannou, M. Björck, L. Bouchenoire, P. Thompson, D. Haskel, J. Lang, Y. Choi, C. Sánchez-Hanke, and B. Hjörvarsson

Manuscript

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Comments on my participation

The following is a brief description of my involvement in the publications: I Participated in the sample preparation. Performed the MOKE and diffracted

MOKE measurements and analysis, implemented the simulations and wrote most of the paper.

II Performed the XRMS measurements. Co-wrote the manuscript with T. P. A. Hase.

III Performed the patterning. Participated in MOKE measurements and analysis and contributed to the paper.

IV Performed the sample growth. Participated in PEEM imaging and anal-ysis. Wrote most of the manuscript.

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1. Introduction

Progress within the fields of thin film growth and lithographic techniques over the past decades has enabled the creation of artificial structures with length scales even down to nanometer dimensions [1, 2]. Through the introduction and tuning of artificial length scales in magnetic materials, by the control of material thickness, composition, and lateral extension, their properties can be influenced and even tailored to match a specific purpose. In such cases the lat-eral extension and design of the structures can for example be used to affect, or eliminate magnetic domain formation allowing the magnetic islands to be described as macroscopic magnetic moments. Using the shape of the mag-netic islands an anisotropy defined by the aspect ratio of the islands can be introduced creating preferred magnetization directions. In the case of arrays of magnetic islands their arrangement can be used to investigate the effect of dipolar interactions between elements or between layers in the case of mul-tilayered structures. Through such means it is therefore possible to create magnetic model systems of artificial structures, resembling atomistic systems, in which interaction energies, anisotropies, lattice arrangements and length scales can be tuned.

Recently, arrays of elongated magnetic nanoscale islands have become a topic of great interest with regard to the exploration of the role of geometric frustration in magnetic systems. Frustration arises when a system is incapable of simultaneous minimization of all competing pairwise interaction energies. Using nano-lithography patterning techniques artiﬁcial systems of nanomag-nets can be designed in which the interaction strengths and arrangement of the nanomagnetic elements can be controlled. Using magnetically sensitive imag-ing techniques the state of such systems can be determined allowimag-ing for the role of frustration to be directly investigated [3].

This thesis deals with the magnetic properties of artiﬁcially created mag-netic structures. Using the model systems described above, achievable in nano-patterned thin ﬁlm systems, the interplay of energy and length scales is explored. Selecting different characterization techniques, ranging from direct imaging methods to reciprocal space techniques, these attributes are investi-gated by exploring the state of the magnetic structures, extending from the atomic scale up to collective ordering phenomena of nano-magnetic elements.

Two main themes set the focus for study: laterally patterned Co68Fe24Zr8

/-Al2O3multilayer arrays of combined circular and ellipsoidal islands, and

pat-terned structures composed ofδ-doped Pd(Fe) thin ﬁlms. Both themes

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by lateral patterning. In both cases dipole interactions, between array elements as well as between individual layers in multilayer structures, play a vital role, modifying the magnetic response and inducing an ordering of the magnetizat-ion in the remanent state.

The two materials that comprise the focus of this work, Co68Fe24Zr8 and

δ-doped Pd(Fe), represent two distinct classes of materials. On the one hand,

Co68Fe24Zr8has a high Curie temperature and due to its amorphous nature can

be easily deposited into high quality multilayered structures onto pre-patterned

Si wafers. An interesting attribute of Co68Fe24Zr8 is that it can be imprinted

with a uniaxial anisotropy by the application of a magnetic ﬁeld during growth. This imprinted anisotropy can then be aligned to speciﬁc structural directions or lattice axes of patterned arrays. Attributes of arrays of combined multilay-ered elements and how they are discerned are discussed in papers I and II. The arrays span a large range of length scales, from their nanometer scale multi-layer periodicity to their lateral periodicity in the micrometer range. The at-tributes of these arrays are explored using different techniques, applicable for addressing the magnetization at different length scales, including magneto-optical techniques, micromagnetic simulations and x-ray resonant magnetic scattering.

On the other hand, δ-doped Pd(Fe) ﬁlms have a low Curie temperature

which can be tuned by controlling the thickness of the Fe layer embedded

in the Pd. As an example, δ-doped Pd(Fe) with an Fe layer thickness

cor-responding to 1 monolayer of Fe has a Curie temperature of 195 K, while for 0.2 monolayers the Curie temperature is 40 K. Fe impurities embedded in Pd polarize a sphere of about 1 nm radius resulting in a total magnetic

mo-ment of 9− 10 μBper Fe atom. For Fe layers this polarization extends up to

∼10 atomic distances (corresponding to ∼2 nm) from the Fe layer, creating

a smooth well defined magnetic layer. Many attributes of Pd(Fe) films have been described in previous papers by Pärnaste et al. [4] and by Papaioan-nou et al. [5]. In paper V element specific x-ray scattering measurements are used to explore the magnetization of the Fe layer and the polarized Pd layer independently. In papers III and IV the low Curie temperature and magnetic moment are utilized in the attempt to obtain a thermal ordering of arrays of magnetic moments illustrated in a thermal order-disorder transition of an ar-tificial square spin ice array, discussed in paper III, and thermal ground state ordering of finite arrays, discussed in paper IV.

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2. Magnetic properties

In this thesis the main interest is devoted to investigating the magnetic proper-ties and ordering of patterned arrays of thin ﬁlms and multilayered structures composed of ferromagnetic materials. The properties of ferromagnetic sys-tems are directly linked to their total energy to which several energy terms contribute, ranging from atomistic interactions up to energies which are man-ifested at a macroscopic scale. In this chapter a brief introduction is therefore given regarding the concepts related to the magnetic properties of thin ﬁlms and nanostructures, focusing on the ferromagnetic state and the energies as-sociated with a ferromagnetic body [6, 7, 8, 9]. For nanoscale ferromagnetic structures the dominant contributions to the total free energy are the exchange

energy Eex, the anisotropy energy Ea, the demagnetizing energy Ed, and the

Zeeman energy EZ, due to an external magnetic ﬁeld. Hence, the total energy,

Etot, can be described with

Etot=Eex+Ea+Ed+EZ. (2.1)

Other energy terms exist which contribute to the total energy such as magne-toelastic and magnetostrictive terms which are not discussed here. Taking into account neighboring elements, such as in the case of arrays of nanopatterned magnetic structures, dipolar interactions are furthermore introduced, affecting the total energy of the system.

2.1 Exchange energy

The exchange energy, Eex, lies at the heart of magnetic ordering, coupling

individual moments and promoting alignment of the spins in the case of ferro-magnetism. In the 3d transition metals Fe, Co, and Ni, a parallel alignment is preferred due to the exchange interaction. This induces a spontaneous align-ment of the magnetic moalign-ments and these materials are therefore called ferro-magnetic. In these materials the magnetism is carried by the 3d electrons. In the metallic state the 3d electrons are delocalized, or itinerant, thus forming a band structure. These bands are split into two sub-bands containing spin-up and spin-down electrons. The difference in the ﬁlling of these bands creates a spin imbalance resulting in a net magnetic moment [6]. As the exchange inter-action proceeds directly between neighboring atoms in the 3d ferromagnetic

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metals, it is known as direct exchange. For a many body system this exchange interaction can be expressed using the Heisenberg exchange Hamiltonian [9]

H =−2

∑

i> j

Ji jSi· Sj (2.2)

where J_{i j}is the exchange constant between spins S_i and S_j. If J_{i j} is positive

a parallel alignment of the spins is favored whereas a negative Ji j favors an

antiparallel alignment. Considering classical spins and taking only nearest neighbor interactions into account, with an exchange constant J, equation (2.2) can be written as

E =−JS2

∑

<i j>

cosϕi j (2.3)

whereϕ_{i j}is the angle between adjacent spins. Assumingϕ_{i j} is small for all

neighboring spins (as would be expected in a ferromagnetic material), cosϕi j

can be expanded to cosϕi j≈ 1−ϕi j2/2. The constant term does not depend on

the angle and can be neglected. Deﬁning the unit vectors of the moment, m,

which follow the spinsϕ we can write

|ϕi j| ≈ |mi− mj| ≈ |(ri j· ∇)|m (2.4)

where r_{i j}are distance vectors between spins i and j. Equation (2.3) can then

be written as

E =−JS2

∑

<i j>

[(ri j· ∇)m]2. (2.5)

Changing the summation to an integral over a magnetic body with a volume

V , the exchange energy can then be expressed in the continuum limit by Eex=A

V

[(∇m_x)2+ (∇m_y)2+ (∇m_z)2]dV (2.6)

where A is the exchange constant given by A = 2JS2_{z/a for a nearest neighbor}

distance a and z number of sites within each unit cell [9].

The magnetization of ferromagnetic materials is temperature dependent, taking its maximum value at zero and reducing as thermal excitations com-pete against the magnetic ordering. When the temperature reaches the Curie

temperature of the material, TC, the spontaneous magnetization is lost and the

material becomes paramagnetic. The nature of this transition and the value of

TCdepends on the material and is proportional to the strength of the exchange

interaction [9].

2.2 Magnetic anisotropy

Within a crystal lattice, coupling between the spin and orbital motion of elec-trons results in a magneto-crystalline anisotropy, dependent on the crystal

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symmetry, which gives rise to a preferred axis of the magnetization. This preferred axis is usually referred to as an easy axis, in contrast to the hard axis which represents an unfavorable axis of the magnetization. Depending on the crystal symmetry different cases of magneto-crystalline anisotropy occur, such as uniaxial and cubic. In the case of a uniaxial symmetry the anisotropy

energy density, ea, can be given by the expansion

euniax_a .=K1sin2(θ) + K2sin4(θ) + ... (2.7)

where K1, K2, ... are the anisotropy constants (in units of J/m3) andθ is the

angle between the anisotropy axis and the magnetization direction. Depending on the structure of a magnetic material additional anisotropy mechanisms can emerge. Symmetry breaking at surfaces and interfaces of materials can induce a preferred direction of the magnetization, referred to as surface anisotropy. Generally such effects tend to cause the moments to align perpendicularly to the interface. An imprinted anisotropy axis can be created by the application of an external magnetic ﬁeld. In papers I and II a ﬁeld induced imprinted

anisotropy in amorphous Co68Fe24Zr8 is used to deﬁne a uniaxial anisotropy

axis aligned to a lattice direction within a patterned array.

2.3 Dipolar interactions

The dipolar interaction energy of two magnetic dipoles depends on their

mu-tual alignment and separation. For two dipoles, μ₁ and μ₂, separated by a

distance vector r it can be given by

E = μ0 4πr3 (μ₁· μ₂)− 3 r2(μ1· r)(μ2· r) . (2.8)

Figure 2.1: Schematic

illustrat-ing the dipole-dipole interaction be-tween two magnetic dipoles, μ₁and μ2. In order to minimize their energy

the dipoles align in an antiferromag-netic arrangement.

At the atomic level equation (2.8) can be used to estimate the dipolar interaction en-ergy for two atomic moments. Assuming atomic distances, r = 1 Å and a moment of, μ = 1 μB, one obtains an energy of

the order of ∼10−23 J, corresponding to

a temperature of ∼1 K, well below the

Curie temperature observed for conven-tional magnetic materials (1043 K for Fe) [9].

Although the dipole ﬁeld of indepen-dent magnetic moments may seem small it is however manifested on a macroscopic

scale through the combined ﬁeld of all the moments in a magnetic body, gen-erating the demagnetizing ﬁeld and affecting domain formation. In the case

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of single domain structures the combined moment, and correspondingly, the energy of dipolar interactions between neighboring magnetic nanostructures can become large. In this case the interactions between elements can affect the magnetization as is discussed in papers I-IV.

2.4 Zeeman energy

The energy of a magnetic body in an external ﬁeld HA is called the Zeeman

energy and can be given by

EZ=−μ0

VM· HAdV. (2.9)

Where M(r) is the magnetization vector function. The Zeeman energy of a magnetic body is minimized when the magnetization is aligned parallel to the external ﬁeld.

2.5 Demagnetizing energy

The demagnetization energy, often referred to as the magnetostatic self energy, is the result of a magnetic body interacting with its own stray ﬁeld. The total demagnetization energy of a body is obtained by integration

Ed=−

1

2μ0

VM· HddV (2.10)

where Hd is the magnetic dipolar ﬁeld created by the magnetization

distri-bution. For uniformly magnetized bodies the demagnetization ﬁeld can be deﬁned with relation to the magnetization by

Hd=−NdM, (2.11)

where the demagnetizing factor, Nd, depends on the geometry of the magnetic

body. Due to the dependence on the geometry the demagnetizing energy gives rise to shape anisotropy. For a homogenous spherical body the components

of Nd are equal, i.e. Nx =Ny=Nz = 1/3. For elongated structures the Nd

component along the long axis is reduced, deﬁning a uniaxial preferred direc-tion of the magnetizadirec-tion, while the components along the short axes become larger. In the case of ultra thin ﬁlms, where the thickness is much smaller than

the lateral extension, the out-of-plane component of Nd nears a value of one

whereas the in-plane components are reduced to zero, resulting in a preference for the magnetization to lie within the plane. In this case the shape anisotropy

energy density is given by ed= 1₂μ0M2cos2ϑ where ϑ is the angle of the

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2.6 Magnetic domains

Figure 2.2: An artistic impression of the

ori-gin of domain formation. A single domain magnetic body has a large stray ﬁeld energy which is reduced by domain formation. Flux closure domains do not have any magnetic stray ﬁelds.

A uniformly magnetized body has a large stray ﬁeld energy. Through dividing the magnetizat-ion into domains this energy is reduced. The formation of mag-netic domains however requires the formation of domain walls, which cost energy, and the divi-sion into domains can therefore only continue while the reduction in the stray ﬁeld energy is greater than the energy required to form

the domain wall. In materials

which do not have a large uni-axial anisotropy, ﬂux closure do-mains can be formed in which the stray ﬁeld energy is zero, and in

the case of circular elements vortex states can become favored.

The width and energy of a domain wall depends on the magnitude of the exchange interaction, the anisotropy, and the dipole energy associated with the domain wall. In bulk materials Bloch walls are favored, in which the magnetization rotates in a plane parallel to the plane of the wall. In thin films Néel walls are favored, in which the magnetization rotation is confined in the plane as a dipolar energy cost would be associated with rotating the moments out of the plane of the film. An extensive account of magnetic domains and domain imaging methods has been given by Hubert and Schäfer [10].

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2.7 Single domain particles

Figure 2.3: The magnetization of a

sin-gle domain particle according to the Stoner-Wohlfarth model as a function of ﬁeld applied at 0◦, 45◦, and 90◦with respect to an easy axis deﬁned by a uniaxial anisotropy.

When the energy cost of form-ing domain walls is greater than the stray ﬁeld energy it can be energetically favorable for the magnetization of a magnetic body to be conﬁned to a single do-main. Changes in the magneti-zation therefore only occur by the rotation of the magnetization. The theory describing the behav-ior of single domain particles was developed by Stoner and Wohl-farth in 1948 [11]. The energy of a single domain particle with a magnetization M, volume V , and a uniaxial anisotropy K (which can be either due to shape or

crys-talline anisotropy) in an applied magnetic ﬁeld H is given by

E = KV sin2(θ) − MVH cos(φ − θ) (2.12)

whereφ and θ are the angles of the applied ﬁeld, H, and the magnetization, M,

respectively, with respect to the easy axis defined by the anisotropy. By min-imizing this energy the direction of the magnetization can be found for any direction or value of an applied magnetic field. Equation (2.12) can therefore be used to obtain magnetization loops for different angles of the applied field with respect to the easy axis, such as the ones shown in figure 2.3. The size limit at which a single domain state becomes favorable over a multidomain state depends on the particle geometry and its material properties [12]. For simple shapes, such as spherical particles, expressions for the size limit, can be obtained by comparing the energy cost of creating a domain wall and the energy gained through forming domains [8]. Generally the size limit scales

with∼√AK1/Ms2 where Ms is the saturation magnetization of the material.

Considering typical values of A, K1, and Msthe limit for spherical particles of

Co is for a diameter of 68 nm. Fe has a substantially lower anisotropy, K1, and

therefore a smaller critical diameter of 12 nm [8]. For patterned thin films the size limit for a single domain state will depend on the specific shape, thickness and aspect ratio of the structure and can even be affected by the fabrication process [12]. In papers III and IV elongated single domain islands (of size 250× 750 nm2 and 170× 470 nm2) ofδ-doped ultrathin films are observed, and in paper I layers of micrometer diameter, within multilayer structures, dis-play single domain states, in part induced by the stray field from neighboring layers.

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2.8 Thermal reversal

Figure 2.4: Schematic of the energy of a uniaxial

single domain particle as a function of the magn-etization angle. An energy barrier Erseparates the

two energy minima deﬁned by the anisotropy.

When the anisotropy en-ergy of a small single do-main magnetic particle is of the same order of magni-tude as the thermal energy

kBT , random thermal

ﬂuc-tuations of the

magnetizat-ion can occur [8]. In the

absence of an external ﬁeld, a particle with a uniaxial anisotropy K, due to mag-netocrystalline anisotropy or shape anisotropy, is given by the ﬁrst part of equation

(2.12), E = KV sin2(θ). The

energy will therefore have

two minima when the angle of the magnetization, θ, is aligned to the

anisotropy axis. Flipping the magnetization between these two energy

min-ima requires an activation energy given by Er=KV . When the thermal energy

kBT is comparable to Er the magnetization can be ﬂipped between the two

energy minima by thermal ﬂuctuations. The time scale of the reversal of the magnetization follows a Néel-Brown law

τ = τ0eEr/(kBT ), (2.13)

where the inverse attempt frequencyτ0takes a value of the order of∼10−10s

[6]. When the measurement time scale is much larger than the reversal time

scale, τm τ, the magnetization direction can ﬂip during the measurement

time and the magnetic response of the particle is superparamagnetic. When

τm is shorter thanτ the magnetization is blocked. The temperature at which

this transition occurs, i.e. whenτm≈ τ is called the blocking temperature. For

densely packed particles, interparticle interactions can not be neglected and

Erwill depend on the local environment and be modiﬁed by the dipolar ﬁelds

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3. Patterned multilayer structures

Lateral patterning of layered structures, composed of magnetic layers spaced with non-magnetic layers, offers the possibility to investigate the effect of dipolar interactions between layers at the edges of the structures. Using non-magnetic spacer layers, between the non-magnetic layers, coupling between the layers can be eliminated. Upon lateral patterning the layers can then only in-teract through their dipolar stray ﬁelds at the edges of the structures. As the thickness of layers can be of the order of nanometers, this interaction will be over a short length scale and can become a deﬁning parameter of the magnetic properties and remanent ordering.

The dipole interaction promotes an antiferromagnetic ordering and there-fore one expects, in the limit of each layer acting as a single domain and an absence of anisotropy, an antiferromagnetic arrangement of the magnetic lay-ers. Such an effect has been demonstrated by van. Kampen et al. for circular

Figure 3.1: Images illustrating the role of dipole interactions at the edges of laterally

patterned multilayers in which the magnetic layers are spaced with non-magnetic lay-ers. The arrows depict the magnetization of individual layers in the stack shown to the right for increasing magnetic field applied out of the image. At remanence an anti-ferromagnetic ordering occurs, for increasing field the outermost layers are the first to align to the field as they are less coupled to other layers. The images are reproduced with permission from van. Kampen et al. [14].

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multilayered pillars of permalloy and Al2O3 illustrating the possibility of

re-alizing artiﬁcial spin chains [14]. Through the combination of magnetization measurements and micromagnetic simulations they were able to show that the magnetization of each layer is single domain and that the dipolar coupling energy, scales with the diameter of the pillars, resulting in an antiferromagnet-ically ordered ground state at remanence (see ﬁgure 3.1).

An interesting consequence of dipole interactions between layers was ob-served by Choi et al. [15]. Using element selective x-ray resonant magnetic scattering they observed that due to the altered magnetic interactions in cir-cular multilayered structures, when two magnetic layers are spaced with a non-magnetic interlayer, a single domain state can become more favorable due to the stray ﬁeld from neighboring layers. When the number of dipole coupled layers is odd the interaction landscape can become quite different. As an antiferromagnetic ordering can not be achieved between all the layers this leads to frustration in the magnetization orientation of the layers. This effect is maximized when the total number of layers is three. If the magnetization is free to rotate in the plane the ground state of such structures is composed of a non-collinear helical arrangement of the magnetization of the layers [16].

In papers I and II the magnetic properties of multilayered structures

com-posed of magnetic layers of amorphous Co68Fe24Zr8and non magnetic Al2O3

are discussed and investigated using magneto-optical techniques, micromag-netic simulations and x-ray resonant magmicromag-netic scattering. In the following sec-tions their properties and the methodologies used for investigation are brieﬂy discussed.

Amorphous materials offer an additional choice to the conventional crys-talline magnetic materials. With respect to thin ﬁlm magnetism and multi-layered structures their attributes, such as the absence of atomic steps, can be highly advantageous as they can have a high degree of uniformity [17]. Generally amorphous materials are deﬁned by the absence of structural long range order. For 3d transition metals the amorphous nature can be achieved by alloying with glass forming elements such as B, P, C, or Si or with other tran-sition metals with substantially different radius. In the work presented here an amorphous phase of a Co and Fe alloy is achieved by the addition of 8 at.% of Zr, close to the limit of the Zr content needed in order to form the amorphous phase [18].

Amorphous CoZr alloys exhibit soft magnetic properties and a saturation

moment, Ms, comparable to elemental Co [19]. By substituting the Fe atoms

for Co in Co rich amorphous materials the saturation moment increases and the coercivity is reduced compared to amorphous CoZr [17, 20, 21, 22]. Through the application of an external ﬁeld during growth a uniaxial anisotropy axis can

be imprinted in Co68Fe24Zr8. Consequently, by creating multilayered

struc-tures where the field is applied in different directions, strucstruc-tures with layer specific anisotropies can be created as is discussed in reference [17]. In the work described in papers I and II growth under field is used in order to

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de-Figure 3.2: Schematic illustrating the patterning processes for the case of multilayers

grown on prepatterned substrates (left). The ﬁgures on the right show atomic force microscopy images of the patterned arrays used for the study and schematics illustrat-ing the layerillustrat-ing of the structures. The multilayered structures are composed of ten repetitions of 3 nm thick Co68Fe24Zr and 3 nm thick Al2O3layers. Both patterns have

a periodicity of d = 6μm along both principal directions. The circular islands have a

diameter of 6μm and the ellipses a major axis of 4.5 μm and a minor axis of 1.5 μm.

ﬁne a uniaxial imprinted anisotropy into multilayered structures aligned to a speciﬁc lattice direction of the patterned array.

3.1 Pattern fabrication

The multilayered ﬁlms investigated in this work were grown by magnetron

sputtering at room temperature. The Co68Fe24Zr8layers were grown from a

compound target maintaining a ﬁxed atomic ratio of the elements [17]. The

multilayer structures consisted of ten repetitions of 3 nm thick Co68Fe24Zr8

and 3 nm thick Al2O3starting and ending with an Al2O3layer.

The patterned multilayers were created utilizing pre-patterned Si substrates. Schematics depicting the patterning process are shown in figure 3.2. The patterns were written by electron beam lithography at the Micro and Nan-otechnology Centre at the Rutherford Appleton Laboratory, United Kingdom. A double layer poly(methyl methacrylate) (PMMA) based resist was used in which the underlying layer was more sensitive to the electron beam exposure. This produces an undercut in the patterned resist layer which reduces crown-ing at the edges of the patterned elements. As the properties of the patterned multilayers rely strongly on a well defined layering all the way to the edges of the elements such a patterning scheme is essential allowing the dipolar in-teractions at the edges of the structures to be clearly defined. AFM images depicting the two final patterns investigated, which set the focus for papers I

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Figure 3.3: Schematics illustrating the three different geometries that can be used to

investigate the Magneto-optical Kerr effect.

3.2 Magneto optical Kerr effect

The magneto-optical Kerr (MOKE) effect was discovered in 1877 by John Kerr [23]. It followed after the discovery of the Faraday effect, by Michael Faraday in 1845 [24]. The Faraday effect describes how the polarization vec-tor of linearly polarized light undergoes a rotation during transmission through a magnetic material. The magneto-optical Kerr effect is the analogous effect in reflection, in which the polarization vector undergoes a rotation upon reflec-tion from a surface. The magneto-optical Kerr effect can be used to provide a signal that is proportional to the magnetization of the sample, predominantly performed by illuminating a magnetic surface by a linearly polarized laser beam. By performing MOKE measurements as a magnetic field, H, is swept across a magnetic sample its magnetization as a function of field, M(H) can be recorded. More detailed descriptions regarding magneto-optics can be found in, e.g. references [25, 26, 27].

The Kerr effect is typically measured in one of three geometries depicted in ﬁgure 3.3 for different magnetization orientations with respect to the plane of the reﬂecting surface and the plane of incidence of the laser beam. In the lon-gitudinal geometry the magnetization is parallel to the sample surface and the plane of incidence of the light while in the polar MOKE geometry the magn-etization is perpendicular to the sample surface. In the transverse geometry the magnetization is parallel to the sample surface and perpendicular to the plane of incidence and the Kerr effect is not measured by the rotation of the polar-ization vector but by the change in intensity. Depending on the anisotropies involved in a magnetic system, whether there is a preference for the magneti-zation to point out of or in the plane of the sample the different geometries can therefore be used to investigate the different magnetization components.

The penetration depth of visible light in metals is of the order of 10 to 20 nm [28]. For thin ﬁlm studies the inﬂuence of the substrate on the observed mag-netic response is therefore limited as compared to bulk magnetization probes such as superconducting quantum interference device and vibrating sample magnetometry. In the case of multilayers a layer sensitivity occurs enhanc-ing the contribution of layers closer to the surface. In the case when different layers display contrasting responses a combined magnetization curve is

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ob-tained with the signal from each layer weighted by the attenuation of the beam [17]. When the magnetization of the top most layers do not correspond to the average magnetization the limited penetration depth can give rise to other artifacts, such as an inverted hysteresis, as is discussed in paper I where the dipolar stray ﬁeld from lower lying magnetic layers affect the response of the topmost layers.

In order to obtain a handle on the lateral magnetization, to observe do-main formation or the magnetization distribution of patterned samples, dif-ferent methods can be used such as the microscopy equivalent of MOKE, Kerr microscopy [10], magnetic force microscopy (MFM) [29], photoemis-sion electron microscopy (see discusphotoemis-sion in section 5.1), or by performing magnetization measurements at diffraction peaks as is discussed in the next section.

3.3 Diffracted MOKE

Lateral patterning of arrays opens up the possibility of using information con-tained in the light diffracted off the array, therefore allowing a characteriza-tion of the lateral magnetizacharacteriza-tion distribucharacteriza-tion through the magnetic form factor of the array [30]. As with the reﬂected beam in MOKE measurements the diffracted beams contain information on the magnetic state of the patterned samples. MOKE measurements performed at these diffracted beams are re-ferred to as diffracted MOKE (D-MOKE) [30] or Bragg MOKE [31]. While the information in the reﬂected beam corresponds to the average magnetization the magnetic information in each diffracted beam is proportional to the Fourier component of the magnetization for that diffraction order. Through magneti-zation measurements recorded at the diffracted beams the internal magnetizat-ion distributmagnetizat-ion can therefore be reconstructed. Diffracted MOKE measure-ments can therefore provide information on the magnetic domain structure and reversal mechanism within small particles and can even be used to distinguish the magnetic behavior of individual structures or parts of complex patterns as is discussed in paper I.

The combination of the Kerr effect and diffraction was ﬁrst described in the literature by Geoffroy et al. which reported on remarkably different

magn-etization loops recorded at diffraction peaks for SmCo4 square arrays with

a periodicity of 4 μm, compared to the standard MOKE curve [32]. Since

then, diffracted MOKE measurements have been performed on several dif-ferent systems such as circular dots [33], squares [34, 35], stripes [36, 37], and elliptical holes [38] in order to yield information regarding the internal magnetic structure and ﬁeld dependence of patterns. Furthermore, diffracted MOKE characterization can be used to determine and compare combined pat-terns. In the case of superlattice dot arrays, composed of two nonidentical elements per unit cell, diffracted MOKE measurements can be used to enable

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y Analyser −2nd −1st 1st _2nd 3rd 4th x z Laser 408 nm 51 mW Coils Iron Detector Sample Polarizer Diffracted beams Sample holder Reflected beam PEM φi

Figure 3.4: Schematic drawing illustrating a longitudinal MOKE setup and a picture

showing the distribution of diffraction peaks.

the switching of individual elements to be studied [39]. Paper I extends on a similar methodology employing different arrangements of elements to de-termine the switching of circular and ellipsoidal islands individually through MOKE measurements performed at the ﬁrst and second diffraction order.

Generally, a MOKE measurement performed on the specularly reﬂected beam is proportional to one component of the magnetization, such as m(x). As is discussed above, magnetization loops recorded at beams diffracted of a periodic array have been shown to be proportional to the magnetic form factor

f_nm of that component, where n is the diffraction order. The magneto-optical

signal recorded at a diffraction peak of order n is then given by the Fourier transform of the magnetic structure in one unit cell, S, by [30, 40, 33]

f_nm=

Sm(x) exp(inG· r)dS. (3.1)

For a periodic array the integration is carried out over the two dimensional unit cell, G is the reciprocal lattice vector, and r is the position vector within the cell. The periodicity of the patterns discussed in papers I and II is large compared to optical wavelengths allowing a large number of diffraction peaks to be observed as is shown in figure 3.4. Magnetization loops recorded for the two patterns (shown in figure 3.2) at the specularly reflected beam and at the first four diffraction peaks are shown in figure 3.5. The results reveal substantial differences indicating a strong impact of the Fourier component on the observed magnetization and the two distinct switching mechanism from the two elements of the pattern.

The combined patterns discussed in papers I and II can be considered as individual sublattices of the two elements comprising the patterns. For under-standing the observed magnetization of the different diffraction orders a sim-ple analysis can be made by considering the circular and ellipsoidal island as

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Figure 3.5: (a) Magnetization loops recorded at the specularly reﬂected and the ﬁrst

four diffraction beams from the patterns. Magnetization loops recorded at the spec-ularly reﬂected beam reveal the same behavior for the two patterns, whereas loops recored at the ﬁrst diffraction order show distinct differences owing to the lateral shift of the circular islands. The schematics show a standing-wave model illustrating the in-version of the circular island form factor between the two patterns at the 1st diffraction order.

non-interacting sublattices with independent magnetic form factors, f_n,cm and

f_n,em, where the integration in equation 3.1 is carried out for each sublattice.

The total magnetic form factor can then be given by

f_nm= f_n,cm +f_n,em. (3.2)

Comparing the two patterns,α and β, a shift of the circular island phase factor

by einπoccurs due to the lateral shift of the circular island sublattice by half the

periodicity between the two patterns along the [01] direction (defined by the major axis of the ellipse). An inverted form factor therefore occurs for the cir-cular island sublattices between the two arrays. This effect is clearly demon-strated in the magnetization loops recorded at the first diffraction order shown in figure 3.5 which show distinct responses. Direct comparison of the magn-etization loops can then be used to determine the individual magnmagn-etization of each sublattice. Taking the difference between the two loops the

magnetizat-ion of the ellipsoidal islands, Me, is canceled yielding the magnetization of the

circular islands, Mc. When the two curves are averaged the magnetization of

the circular islands, Mc, is canceled revealing the magnetization of the

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well to micromagnetic simulations illustrating that magnetic information car-ried by diffracted beams can be used to determine the magnetic structure of complex patterned multilayered arrays.

3.4 Micromagnetic simulations

Macroscopic measurements of individual elements, or a large number of sim-ilar structures, such as the recording of a magnetization loop for a nanopat-terned array, can yield a great deal of information regarding the global mag-netic behavior. Information regarding the internal magmag-netic structure can how-ever often be difﬁcult to glean from such measurements requiring direct mi-croscopy techniques or specialized methods such as diffracted MOKE mea-surements. Using micromagnetic simulations one can gain insight into the internal magnetic structure and behavior. They can therefore be used to in-vestigate complex structures, such as laterally patterned multilayers, in which underlying layers can affect the behavior while not being accessible using di-rect microscopy techniques.

Micromagnetic simulations rely on the theory of micromagnetism. Within micromagnetic theory the magnetization M(r) is considered as a continuous vector function of spatial coordinates. As is discussed in chapter 2 the to-tal free energy of a magnetic body is composed of the contribution from the exchange, anisotropy, demagnetization, and the external ﬁeld;

Etot=Eex+Ea+Ed+EZ. (3.3)

The effective ﬁeld, Heffacting upon the magnetization, M, can be determined

by differentiating the total energy with respect to the magnetization [41]

Heff=−

1

μ0V ∂Etot

∂M . (3.4)

The equilibrium magnetization conﬁguration can be found at the minima of the total energy, as pointed out by Brown [42], by solving

δEtot

δM = 0. (3.5)

In the equilibrium state this leads to the condition that the magnetization vec-tors are aligned parallel to the effective ﬁeld vector at all points. The equilib-rium magnetization conﬁguration of a magnetic structure can thus be mined through the minimization of the total energy or, alternatively, by

deter-mining a vanishing torque M× Heff exerted by the effective ﬁeld, Heff, using

the Landau-Lifshits equation [7]

dM

dt =−γM × Heff− γα Ms

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whereγ is the Landau-Lifshitz gyromagnetic ratio and α is a damping param-eter. The ﬁrst term in equation (3.6) describes the precession of M around the

magnetic ﬁeld direction Heff while the second term describes the relaxation

of the magnetization towards the direction of Heff due to a damping torque

scaled by the phenomenological damping constantα introduced in order for

the system to be able to reach an equilibrium state as opposed to a continuous

precession. The value of the damping parameterα depends on the material,

generally lying within in the range 0.005 to 0.2. When only the static magnetic conﬁguration of a magnetic system is sought and not its dynamics a higher

value ofα can be used (commonly α = 0.5). By this, a shorter equilibration

time can be obtained for the system while not noticeably inﬂuencing the end result.

When considering micromagnetic simulations the magnetic structure is dis-cretized into finite element cells. The exchange length defines the length at which the exchange energy is in equilibrium with the dominating anisotropy term, therefore defining the distance over which the magnetization is expected to change direction in the presence of anisotropies. It can therefore be used to determine the maximum cell size for the calculation in order to ensure that the magnetization is constant within each cell. When the shape anisotropy is dom-inant, the exchange length for a material with an exchange stiffness constant,

A, and a magnetization, Ms, is given by lex=

 A

Kd,

(3.7)

where Kd =μ0Ms2/2 is the stray ﬁeld energy constant.

Several programs have been developed to implement micromagnetic calcu-lations in which three dimensional spins can be simulated on a three dimen-sional grid. In paper I simulations regarding the field response of the circular and ellipsoidal islands were performed using the Object Oriented Micromag-netic Framework (OOMMF) [43]. As is discussed in paper I the magnetization loops obtained by the simulations were in good agreement with the measure-ments. Results of the micromagnetic simulations are shown in figure 3.6. Fig-ure 3.6(a) shows the integrated magnetization for each layer of a circular island as a function of applied field for the easy and hard axis directions defined by

the imprinted uniaxial anisotropy of the Co68Fe24Zr8 magnetic material. For

the easy axis the reversal of the magnetization occurs through individual layers switching one after the other through spin-flop transitions at different field val-ues due to the combined effect of the applied field, the imprinted anisotropy, and dipolar stray field of each layer affecting neighboring layers. For the hard axis the magnetization reversal occurs by the individual layers rotating out of the applied field direction towards the easy axis defined by the imprinted

uni-axial anisotropy, Ku. The magnetization distribution of the ﬁrst three layers of

the circular and ellipsoidal islands are shown in ﬁgure 3.6(b). The images il-lustrate how strongly the dipole interactions between layers at the edges of the

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(a)

(b)

Figure 3.6: Results of micromagnetic simulations for Co68Fe24Zr8/Al2O3

multilay-ered circular and ellipsoidal islands performed using OOMMF [43]. (a) The net mag-netic moment of individual layers depicted as arrows for different applied ﬁelds for the easy and hard axis directions. In both cases the magnetization of the individual layers arranges in an antiferromagnetic order at remanences aligned to the easy axis. (b) The magnetization for the top three layers in the circular and ellipsoidal islands for

an applied magnetic ﬁeld ofμ0H =−15 mT along the lateral direction. The arrows

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structures affect the magnetization distribution resulting in divergences at the edges within the layers. At high diffraction orders the corresponding length scale can become small highlighting these divergences introducing deviations in the magnetization loops such as the hump observed at the fourth diffraction order shown in ﬁgure 3.5.

3.5 X-ray resonant magnetic scattering

A limitation of magneto-optical measurements is the limited spatial resolution hindering investigations of the internal magnetic structure of nanomagnets. Using synchrotron radiation, with substantially shorter wavelengths, smaller length scales can be investigated. Combined with the possibility of using po-larized x-ray beams and tuning the photon energy close to a speciﬁc absorption edge, the magnetic properties of different materials can be studied [44, 45]. X-ray resonant magnetic scattering (XRMS) relies on the x-X-ray magnetic circular dichroism (XMCD) effect [6]. For 3d transition metals, investigations using

XMCD are mostly performed using soft x-ray synchrotron radiation at the L2

and L3absorption edges where electrons from the 2p core level states are

ex-cited by x-rays into empty 3d states above the Fermi level. The possibility for an incident x-ray to be absorbed depends on the polarization of the light.

Circularly polarized light carries an angular momentum±¯h depending on its

helicity (right or left circular polarization). Since XMCD involves the excita-tion of a core level electron and the subsequent absorpexcita-tion into an unoccupied band it can be described using a two step model. The ﬁrst step involves the

700 710 720 730 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Photon energy [eV]

Intensity [normalized] L 3 L 2 + − Difference

Figure 3.7: Representative XMCD energy scan of an Fe ﬁlm showing the L3and L2

absorption edges. The graph shows the recorded intensities for right (+) and left (-) circularly polarized x-ray photons and their difference. Data courtesy of V. Kapaklis.

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excitation of the core electron. The incident photon transfers its angular mo-mentum to the excited photoelectron. Since the 2p core level is split due to the spin-orbit coupling, part of the angular momentum can be transferred to the spin through the spin-orbit coupling. The excited photoelectrons are therefore spin-polarized with a polarization dependent on the incident helicity.

Further-more, since the 2p₁_/2 and 2p₃_/2 levels have opposite spin-orbit coupling the

spin-polarization will be opposite for excitations from these levels (for a ﬁxed helicity of the incident photon).

The second step involves the absorption of the spin polarized photoelec-tron into an unoccupied state. In the ferromagnetic materials the density of 3d states is different for electrons with a spin parallel or anti-parallel to the magnetization direction due to the exchange interaction. Due to this splitting the d-band will therefore act as a detector for the spin of the excited photoelec-trons. By taking the difference in the absorption for the two opposite helicities a signal proportional to the magnetization in the sample can therefore be ob-tained. Figure 3.7 shows an XMCD absorption spectra as a function of the incident photon energy. The ﬁgure shows the XMCD spectra for an Fe ﬁlm

recorded for energies around the L3and L2absorption edges showing how the

magnitude of the two absorption peaks depend on the helicity of the incident photons. By taking the difference in the intensities for the two helicities an intensity proportional to the magnetization can be obtained. Furthermore, by

analyzing the intensities of the magnetic contribution of the L3 and L2edges

the spin and orbital contributions to the magnetic moment can be obtained [6]. Measurements utilizing the XMCD effect generally rely on the secondary emission processes generated by the decay of the empty core level. As the empty core levels are ﬁlled, this can lead to the emission of ﬂuorescence radi-ation or secondary electrons emitted by Auger process’s and inelastic electron scattering.

As opposed to XMCD measurements which rely on the absorption of the incident photon, XRMS measurements rely on the detection of scattered pho-tons after they have interacted with a sample. Absorption and scattering of photons are connected through the optical theorem and the Kramers-Kronig relations [6, 46]. In addition to absorption measurements, the intensity of scat-tered x-rays with energies corresponding to the appropriate absorption edges can therefore be used to obtain element speciﬁc and magnetic information.

Turning to the theoretical background of resonant magnetic scattering, within the dipole approximation, the resonant electronic x-ray atomic scattering fac-tor of a magnetic atom can be written as [6, 47]

f (Q, E) = (ˆεf· ˆεi)F0(E) − i(ˆεf× ˆεi)· ˆmF1(E) (3.8)

where E is the energy of the incident x-ray and ˆm is the magnetization unit

vector. The scattering vector, Q, is deﬁned in terms of the unit polarization

vectors, ˆε_i and ˆε_f, describing the incident and scattered x-rays respectively.

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ampli-Figure 3.8: A schematic illustrating the scattering geometry for an XRMS

measure-ment. By selecting the sample and detector angles,ω and 2θ, the scattered intensity

can be recorded for different values of the in-plane and out-of-plane scattering com-ponents, Qxand Qz.

tude

F0(E) = f0+f(E) + i f(E) (3.9)

where f0 is the atomic scattering factor and f and f describe the real and

imaginary part of the resonant anomalous scattering factor contribution. The magnetic scattering amplitudes are contained in the second term of equation 3.8

F1(E) = f_m(E) + i f_m(E) (3.10)

and include the real, f_m, and imaginary, f_m, parts of the magnetic scattering

factor. The scattered intensity recorded using XRMS measurements is propor-tional to the square of the total structure factor and contains interference terms between the charge and magnetic amplitudes. This cross term can be accessed through the asymmetry ratio

R = I +_{− I}−

I+₊_I−. (3.11)

I+ and I− represent either the intensity recorded for the two helicities for a

ﬁxed magnetization or for a ﬁxed helicity and opposite orientations of the magnetization.

A schematic illustrating the scattering geometry for an XRMS setup is shown in ﬁgure 3.8. Scans probing the perpendicular sample structure, through

the out-of-plane momentum transfer vector, Qz, are performed by moving the

detector angle at twice the rate of the sample angle, thereby following the

spec-ularly reﬂected intensity with the detector. The sample and detector angles,ω

and 2θ, are given with respect to the incident x-ray beam (or alternatively with

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10−3 10−2 10−1 100 101

Detector [arb. units]

1st 2nd 3rd 4th H Pos. field, I+ Neg. field, I− 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 Q_z [nm−1] Asymmetry ratio Asymmetry ratio 770 775 780 0 1 2 3 Energy [eV]

Detector [arb. units]

Energy scan −1 0 1 1st Magnetization −1 0 1 2nd Field [mT] Detector [normalized] −1 0 1 3rd Field [mT] −30−20−10 0 10 20 30 −1 0 1 4th μ₀H [mT] Pattern α

Figure 3.9: Specularly reﬂected intensity for the multilayered sampleα (see ﬁgure

3.2), recorded for a ﬁxed helicity of the incident synchrotron radiation and for opposite saturation ﬁeld directions, discussed in further detail in paper II. The peaks correspond

to multilayer Bragg peaks occurring at Qz= 2π × n/Λ. Due to refraction effects the

position of the first Bragg peak is shifted for the two opposing saturation field values. An energy scan recorded at the first Bragg peak condition is shown in an inset as well as an indication of the energy value chosen, slightly below the maximum of the resonance. The insets on the right show magnetization loops obtained by recording the scattering intensity as a function of an applied field at the four Bragg peaks observed.

can be obtained from the angles through

Qz=

2π

λ [sin(2θ − ω) + sin(ω)] (3.12)

whereλ is the wavelength of the incident x-rays and the in-plane scattering

component, Q_x, can be obtained through

Qx=

2π

λ [cos(2θ − ω) − cos(ω)]. (3.13)

Layered structures, with a perpendicular periodicityΛ, such as the

multilay-ered structures discussed in papers I and II, give rise to Bragg peaks in the

Qzscan when Qz= 2π × n/Λ, where n is the order of the Bragg peak.

Fig-ure 3.9 shows the specularly reﬂected intensity (Qzscan) for the multilayered

sampleα shown in ﬁgure 3.2. Given the multilayer periodicity, Λ ≈ 6 nm,

Bragg peaks therefore occur at Q_z≈ n × 1.05 nm−1. The scattering

measure-ments along the Qzdirection presented in ﬁgure 3.9 were performed with an

x-ray energy of E = 777_{.0 keV (λ = 1.596 nm) corresponding to slightly}

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and therefore a larger penetration depth is achieved while still maintaining an element speciﬁc and magnetic contrast [48]. An in-plane periodicity, d,

can furthermore give rise to diffraction peaks at Qx= 2π × m/d, where m is

the diffraction order. As with diffracted MOKE, magnetization loops can be recorded at the diffraction peaks in order to obtain information on the magn-etization distribution.

In Paper II the individual magnetic responses of the circular and ellipsoidal islands are used to investigate how the observed magnetization depends on the specific scattering condition. The magnetization reversal of the circular and ellipsoidal islands is well understood from the micromagnetic simulations and the MOKE measurements. Within the XRMS measurements the dependence of the refractive index on the magnetic state of the layers and the convolution of the scattered intensity with the underlying structural and magnetic config-urations results in strong variations for different scattering conditions. This is highlighted in the observation of the intensity from the different elements measured in magnetization loops recorded at the specular Bragg conditions (shown in insets in figure 3.9) which scale approximately with the square of the surface area of the elements. The ellipsoidal islands therefore contribute a larger portion to the observed magnetization than observed in magnetization loops recorded using the magneto-optical Kerr effect, see figure 3.4.

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4. Artiﬁcial spin ice

Figure 4.1: Proton arrangement in

wa-ter ice. In the low energy conﬁgura-tion each oxygen atom has two hydrogen atoms near (forming the H2O molecule)

and two hydrogen atoms further away [49].

Frustration in physical systems is a uni-versal phenomenon which occurs when pairwise interactions can not all be sat-isﬁed at the same time [49]. The effects of frustration can be observed in nu-merous systems ranging from systems based on random interactions such as spin glasses [50] to geometrically or-dered systems. Perhaps the most com-mon frustrated system is water ice. In water ice each oxygen atom has four neighboring hydrogen atoms, two near

covalently bonded, forming the H2O

molecule, and two further away, hy-drogen bonded, belonging to neighbor-ing molecules. This condition of ar-ranging two protons near and two pro-tons far away from each oxygen atom is known as the ’ice rule’ [51] and can

be achieved in six different ways resulting in a degenerate ground state. This leads to a residual entropy at low temperatures, theoretically shown by Linus Pauling in 1935 [52] and revealed experimentally by Giauque and Stout in 1936 [53].

In magnetic materials frustration occurs when the magnetic moments are constrained so that the pairwise interaction energy of its constituents can not be minimized simultaneously. This is exempliﬁed by the archetypical frus-trated magnet of Ising spins on a triangular lattice shown in ﬁgure 4.2 [54, 55]. Neighboring magnetic moments tend to point in opposite directions which cre-ates an ambiguity for the third moment which is unable to point in the opposite direction of both its neighbors at the same time.

A magnetic analog to water ice are the pyrochlore materials holmium

ti-tanate Ho2Ti2O7 and dysprosium titanate Dy2Ti2O7. In these materials the

orientation of spins plays a similar role as the proton positions in water ice, and hence these materials are known as ’spin ice’ [56, 57, 49]. The lattice of the pyrochlore materials are built up of corner sharing tetrahedra occupied by

the magnetic rare earth ions (Dy3+or Ho3+). In these spin ices the ice rule of

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the effect of magnetic coupling and the strong anisotropy conﬁning the spins to an Ising like behavior [56, 58].

When the ice rule is adhered to, each tetrahedron is charge neutral, however thermal fluctuations can induce a spin to flip its direction creating two defects which can be viewed as a monopole-antimonopole pair [59, 60, 61]. By further reversing spins these defects can diffuse away from each other at an energy cost which is lower than the energy cost of the excitation [62, 63]. In order to maintain the validity of Maxwell’s equations Paul Dirac proposed that a monopole-antimonopole pair is connected by a flux string [64] which in spin ice corresponds to the chain of reversed spins. As with electrically charged particles in an electric field these magnetic charges can be manipulated by an external magnetic field and viewed in a context analogous to electricity and thereby referred to as ’magnetricity’ [62, 63].

Modern lithographic techniques have allowed the possibility to fabricate frustrated model systems using nanoscale magnetic thin film islands such as two dimensional analogues of the pyrochlore spin ices. Due to the similarity of such systems to the spin ice arrangements they are generally termed ’artificial spin ice’. In artificial spin ice each spin is composed of an elongated single do-main nanoscale magnetic island. Generally the islands have a width of 100 nm to 200 nm and an aspect ratio of 2 to 3. Due to shape anisotropy the major axis of the islands define an easy axis resulting in a preferred direction of the magn-etization and the islands can therefore be considered as Ising like macrospins. The advantage of artificial spin ice systems compared to the pyrochlore spin ice is that size and distance between islands can be tuned allowing for their interactions and the energy scales involved to be controlled. Compared to the pyrochlores the size of the nanomagnetic islands represents a major advan-tage as their magnetic configurations can be directly observed using magneti-cally sensitive imaging techniques such as magnetic force microscopy (MFM) [3, 65] and photoemission electron microscopy (PEEM) [66].

Figure 4.2: An example of geometrical frustration. Neighboring magnetic moments

tend to point in opposite directions which creates an ambiguity for the third moment which is unable to point in the opposite direction of both its neighbors at the same time.

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Figure 4.3: Schematics illustrating (a) square and (b) kagome artiﬁcial spin ice

ar-rays. In the square ice the ice rule is characterized by two moments pointing in and two moments pointing out of each vertex. For the kagome array the ice rule is char-acterized by two moments pointing in and one moment pointing out of each vertex, or vice versa.

Although lithographic techniques set no restrictions as to the design of ar-rays the focus for studies on artificial spin ice systems has been on square arrangements of Ising like dipole moments, figure 4.3(a), and on kagome ar-rangements, figure 4.3(b) (so named after the japanese word for a bamboo-woven basket pattern of interlaced triangles [67]).

Square ice is is composed of orthogonal chains of moments for which the intersection points deﬁne vertices of four moments. As each moment is as-sumed to be Ising like, able only to point along either of the directions deﬁned

by the major axis, each vertex therefore can have 24_{= 16 possible}

configura-tions, shown in figure 4.4. These configurations can be classified into 4 vertex types with different energies. The lowest energy state, ground state, is com-posed of the two degenerate type 1 vertices of staggered magnetic moments. Both the type 1 and type 2 vertices obey the ice rule of two moments pointing in and two moments pointing out resulting in charge neutrality. The higher en-ergy states in which there is a difference of the number of moments pointing in and out, types 3 and 4, can be described as magnetically charged.

A disadvantage of square artiﬁcial spin ice systems is that the interactions between the four islands at each vertex are not equal. This is demonstrated in the splitting of the charge neutral states into type 1 and type 2 vertices. This can be overcome in artiﬁcial kagome spin ice composed of hexagonal arrays of moments with three moments of equivalent interactions at each vertex, re-sulting in a highly frustrated system [68]. Here the ice rule is characterized by either two moments pointing in and one out of the vertex, or vice-versa [66, 69, 68].

The capability of tuning the energy scales in artiﬁcial spin ice systems and the capability of directly observing the state of such systems through

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mag-Figure 4.4: Schematics illustrating the 24_{possible vertex conﬁgurations of artiﬁcial}

square spin ice. The conﬁgurations are classiﬁed into four types depending on the vertex energy. In type 1 and type 2 vertices the ice rule of two moments pointing in and two moments pointing out is obeyed whereas type 3 and type 4 are charged vertices.

netic imaging techniques has spurred a number of investigations [3, 66, 70]. The first study of the role of frustration in artificial spin ice systems was re-ported by Wang et al. in 2006 [3]. Their study revealed short range ice like correlations and an absence of long range order. Further studies have reported direct observations of magnetic monopole defects and the flow of magnetic charge in artificial spin ice [71]. Emergent magnetic monopoles have been reported in the magnetization reversal of kagome spin ice [66, 72]. In this case the reversal proceeds through the nucleation of monopole-antimonopole pairs which dissociate along a 1-dimensional path of reversed magnetizations, defining a Dirac string.

Studies on spin ice systems have mostly focused on patterned structures composed of magnetic materials with a high magnetization and a high Curie temperature such as permalloy or cobalt. As the Curie temperature of these materials lies far above room temperature and the magnetic moments are large the energies associated with their reversal barriers and interaction are

equiva-lent to a temperature of the order of 104− 105K. These systems can therefore

be considered as quasi-static and the only way to manipulate their magnetic moments is through the application of magnetic ﬁelds [73, 74, 75, 76, 3]. Us-ing ac demagnetization protocols energy minimized states and short range or-der can be achieved which can be described by effective thermodynamics with a corresponding effective temperature [74, 77]. Obtaining long range ground state ordering has however not been realized using such demagnetizing proto-cols.

In order to achieve a thermal ground state ordering in artiﬁcial spin ice sys-tems the energy barrier for reversing the magnetization between the two low energy states deﬁned by the shape anisotropy must be thermally accessible. From equation (2.13), the available choices for achieving a thermal ground state ordering are thus, a higher temperature, a reduction in the energy bar-rier, or waiting for geological time scales. Considering the available materials selection and the possibilities for tuning the size and shape of elements in

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arti-ﬁcial spin ice systems a reduced barrier through the reduction in island volume and magnetic moment holds the greatest promise. The potential for creating an artiﬁcial spin ice system which can display thermal excitations therefore exists, allowing for thermal ground state ordering to occur as well as thermal dynamics of macrospins to be investigated.

The role of real thermodynamics in an artificial spin ice system was first reported by Morgan et al. in 2010 [65]. They observed long range ther-mal ground state ordering, and excitations above the ground state, indicative of thermal randomization occurring during the early stages of growth onto prepatterned substrates. In their method permalloy is grown onto prepatterned artificial square spin ice arrays. During growth a limited time window al-lows for thermal dynamics to take place and thermal ground state ordering and elementary excitations to occur. As the array islands become thicker the magnetization is locked in due to shape anisotropy and the final arrangement of the spins can be visualized directly using magnetic force microscopy.

Thermal order-disorder transitions of artiﬁcial spin ice systems have been studied theoretically indicating a temperature window in which a phase with separated monopoles may be exhibited [78]. Beyond magnetic artiﬁcial spin ice systems, thermal order-disorder transitions have been proposed for sys-tems composed of charged colloids, in which the interaction strengths can be adjusted [79].

Papers III and IV introduce the possibility of achieving thermal ordering in dipole interacting nanomagnetic systems by using a low Curie temperature

material. The material chosen is δ-doped Pd(Fe) thin ﬁlms for which the

Curie temperature and the magnetization can be tuned by the thickness of the Fe layer. By this the interaction energies and the shape anisotropy energy can be brought down to energy scales comparable to the experimental temperature. Paper III describes the investigation of a thermal order-disorder transition in an extended artiﬁcial square spin ice array observed in a global measurement. In paper IV the magnetization direction of individual islands composing ﬁnite arrays are directly imaged using photoemission electron microscopy, showing a high probability of thermal ground state ordering.

4.1 Material selection

In order to achieve thermal ground state ordering and thermal dynamics a re-duction in the total magnetic moment is needed while still maintaining a single domain nature of each magnetic element. A candidate for achieving such

ther-mal properties isδ-doped Pd(Fe) ﬁlms in which the magnetic moment, and

simultaneously the Curie temperature, can be tuned by the thickness of an Feδ-layer embedded in Pd. Compared to values reported by Mengotti et al. (1_{.1 × 10}−15Am2) [66] a reduction in the island magnetic moment by one