Magnetic Ordering in Layered Magnets

ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2008

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 418

Magnetic Ordering in LayeredMagnets

MORENO MARCELLINI

ISSN 1651-6214ISBN 978-91-554-7147-7urn:nbn:se:uu:diva-8604

�� !"� !##$ �� %&�' (� �� ( �� ( ��) �� *�� +��)

��

,�� ,) !##$) ,�� -�� ,��) �� )�� "�$) $$ ��) ��) ./01 23$42�4''"43�"343)

�� ( �� 5*�� ( ��*�� 6�� *�� (�� *�� ( 7�89:##�; ��) /�� (�� <4�� (�� ((��)�� 4�� =�� ((�� :,-=+;�

/�� >�� .��(�� :/> .�; �� ?�� (�� :�1@;) �� 6�� *�� .�� :A�� 7��; �� @�� :0�� A��;)�� 4�� * � � �� ( �� *��

B4�� ( 7� �� *�� * �� ( �� *�� ?�� ((�� 5�� ( �� ?�� ) �� (�� 4�� ( �� )�� :.+C; �� 4��

��*�� (�� *�5� �� 7�89:##�; ��)�� .+C �� *�� ( �� *�� D& ��

�� 7�89:##�; �� *�� * �� ((�� )�� 4�� 7�89:##�; ��

�� ) �� *�� (� �� ((�� )�� (�� ( �� ( ��

�� 7�89:##�; �� *��5�� (�� ((�� ) �* 5�� ( �� ((�� ( �� )

�1@ �� * �� *�� 4�� )

� ��,�� (�� ,�� ?�� (��

�� !" #$%� �� &'(#)*) ��

E ,�� ,�� !##$

.//1 �F'�4F!�"

./01 23$42�4''"43�"343��&�&��&��&��4$F#" :��&88��)5�)��8��G��H��&�&��&��&��4$F#";

There is a K. in any life.Jonathan

List of Papers

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

I Martin Pärnaste, Moreno Marcellini, Erik Holmström, Nicolas Bock,

Jonas Fransson, Olle Eriksson and Björgvin Hjörvarsson (2007) Di-mensionality crossover in the induced magnetization of Pd layersJ. Phys.: Condens. Matter, 19:246213

II M. Pärnaste, M. Marcellini and B. Hjörvarsson (2005) Oscillatory ex-change coupling in the two dimensional limitJ. Phys.: Condens. Matter, 17:L477

III M. Marcellini, M. Pärnaste, B. Hjörvarsson, G. Nowak and H. Zabel(2008) The influence of interlayer exchange coupling on magneticorderign in Fe/V superlatticessubmitted to Journal of Magnetism and Magnetic Materials

IV M. Marcellini, M. Pärnaste, B. Hjörvarsson and M. Wolff (2008) Theinfluence of the distribution of the inherent ordering temperatureon the ordering in layered magnetssubmitted to Phys. Rev. B

V Björgvin Hjörvarsson, Moreno Marcellini, Martina Ahlberg and Max

Wolff (2008) The effect of boundaries on ordering in finite magnetssubmitted to Phys. Rev. LettersReprints were made with permission from the publishers.

I coauthored the following articles that are not included in this disserta-tion:

VI E. Holmstroöm, W. Olovsson, I. A. Abrikosov, A. M. N. Niklasson, B.Johansson, M. Gorgoi, O. Karis, S. Svensson, F. Schäfers, W. Braun, G.Öhrwall, G. Andersson, M. Marcellini, and W. Eberhardt (2006) Sam-ple Preserving Deep Interface Characterization TechniquePhys. Rev Letters, 97, 266106

VII M. Björck, Pärnaste, M. Marcellini, G. Andersson, B. Hjörvarsson(2007) The effect of strain and interfaces on the orbital moment in

Fe/V superlatticesJ. Magn. Magn. Mater. 313, 230-235

VIII K. O. Kvashnina, S. M. Butorin, A. Modin, I. Soroka, M. Marcellini,J.-H. Guo, L. Werme and J. Nordgren (2007) Changes in electronicstructure of copper films in aqueous solutionsJ. Phys.: Condens. Mater., 19, 26002

IX K. O. Kvashnina, S. M. Butorin, A. Modin, I. Soroka, M. Marcellini, J.

Nordgren, J.-H. Guo and L. Werme (2007) In situ X-ray absorptionstudy of copper films in ground water solutionsChemical Physics Letters, 447, 54-57

Comments to my contribution

The level of my contribution in any paper is reflected by the position of myname in the authors list. I actively partecipated in the planning of the experi-ment and in the manuscript preparation in the the following:

I Responsible for the samples preparation, partecipated in the MOKEmagnetic characterization and manuscript preparation.

II Responsible for the samples preparation, participated in the MOKE

magnetic characterization and manuscript preparation.

III Responsible for the samples preparation, data analysis and manuscriptpreparation, participated in the MOKE magnetic measurements.

IV Responsible for the whole project, participated in the MOKE measure-

ments.

V Responsible for sample preparation and data acquisition at ILL. Partic-ipated in the data analysis and manuscript preparation.

Minor contributions in the following:VI Responsible of samples preparation and participated in the data acqui-

sition.

VII Responsible of samples preparation and participated in the structuralcharacterization.

VIII Responsible of samples preparation.

IX Responsible of samples preparation.

5

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1 Growth of ultra thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Sputtering Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Molecular Beam Epitaxy (MBE) Process . . . . . . . . . . . . . . . . . 131.2.1 Molecular beam from effusion cells . . . . . . . . . . . . . . . . . 13

1.2.2 Electrons guns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Metallic superlattices/multilayers . . . . . . . . . . . . . . . . . . . . . . . 14

2 Structural characterisation of metallic superlattices . . . . . . . . . . . . . 19

2.1 Interaction between matter and X-ray . . . . . . . . . . . . . . . . . . . . 192.2 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 X-ray reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Growth of Fe/V(001) superlattices . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Magnetic characterisation of metallic superlattices . . . . . . . . . . . . . 394.1 Magneto optic Kerr effect (MOKE) . . . . . . . . . . . . . . . . . . . . . 394.2 Superconducting Quantum Interference Device (SQUID) . . . . . 434.3 Polarised neutron reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Order-disorder transitions and universality classes . . . . . . . . . . . . . 49

5.1 Magnetic order-disorder transitions . . . . . . . . . . . . . . . . . . . . . 495.2 1D universality class: Ising model . . . . . . . . . . . . . . . . . . . . . . 53

5.3 2D universality class: XY model . . . . . . . . . . . . . . . . . . . . . . . 545.4 3D universality class: Heisenberg model . . . . . . . . . . . . . . . . . 595.5 Extra universality class: Surface . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Oscillatory interlayer exchange coupling . . . . . . . . . . . . . . . . . . . . 616.1 The RKKY framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2 Exchange coupling in Fe/V(001) superlattices. . . . . . . . . . . . . . 637 Magnetic ordering in layered magnets.

Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Svensk Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Introduction

Magnetic heterostructures, with the shape of thin films, are present almost

any where: magnetic sensors in magnetic storage devices, magnetic cooling

devices, microattuators and motors, micro-pumps and many others.

However, none of them would be possible without the knowledge of thebase of magnetism when the size scales to nanometric scale. For such a reason,magnetic multilayers, hands made sandwiches of ferromagnetic (FM) and/ornot-magnetic (NM) and/or anti-ferromagnetic (AFM) materials, are powerfulstructures in order to address the basic investigation on magnetism.

The understanding of the magnetism at this scale cannot be carried out with-out the knowledge of the production techniques of such thin films (chapter 1)and their structural characterisation by using X-rays (chapter 2). The investi-gation of the optimisation of the growth of a specific system is described inchapter 3.The main goal of the scientific process is the investigation of the magnetic

properties. Three techniques of investigations are reviewed (chapter 4). Suchmeasurements show the existence of temperatures where the magnetisationvanishes. In the proximity of such ordering point, the magnetisation seems tofollow particular universality classes (chapter 5).In some case, two FM layer, spaced by a NM layer, can indirectly interact.

Such an interaction is described in chapter 6.The magnetic ordering in layered magnets has been addressed in five dif-

ferent cases. If one considers a single FM layer embedded in NM layers, themagnetic thickness does not correspond to the geometrical one as it has beenshown by a dimensional crossover in the magnetisation. The coupling betweentwo FM layers has been investigated as a function of the coupling strength,showing that in the weak coupling limit, there is no change in the universalityclass. In case of a multilayer, the long range interaction among the FM layersadds an extra degree of freedom changing the length scale and the magnetic di-mensionality. The effect of the coupling strength is different at the surface andin the interior. Different thicknesses at different positions are shown to play arole in the overall magnetisation of multilayers. Finally, it will be shown thatthe boundary conditions play a role in the determination of the universalityclass in a super-superlattice.

9

1. Growth of ultra thin films

Thin films are present in many devices used in every day life, one has just tothink to LCD screens, magnetic sensors, data storage devices, etc.

The production of ultra thin film has become feasible with the progress inthe area of growth technology and vacuum science. Firstly, the technologicalproblem of their production has been solved by improving vacuum technol-ogy. In order to exactly control the chemistry of the materials, ultra high vac-uums must be generated and at the present time, pressures of 10−10− 10−11Torr are easily generated by using a cascade of vacuum pumps.Secondly, many deposition techniques, which are able to exactly control the

physical dimensions of the thin films, have become available. Among all these

possible techniques, the most used techniques in metal thin films growth and

science are the:

1) molecular beam epitaxy or MBEand

2) magnetron sputtering.Both techniques give the best results in terms of controlling the thickness ofthe thin film, and in terms of controlling the chemistry of the materials.In this chapter the above introduced growth techniques will be described

as a first step of the scientific process, followed by general description of asuperlattice.

1.1 Sputtering Process

Sputtering is the dominating deposition process used for the synthesis ofmetallic thin films. Its main feature is the capability to evaporate manykinds of material from metals [1, 2, 3, 4] to either insulators [5, 6] orsemiconductors.

The goal of the sputtering process is to maintain a glow-discharge plasmain front of each material source, called the target. The plasma is caused bythe impact of energetic ions (usually Ar) accelerated through a potential drop,while the material to be etched or sputtered is kept at negative potentials. Theuse of Ar as opposed to other noble gases is due to Ar having the best ratiobetween economicity and great cross section1. But, in order to generally im-prove the growth of the thin films, other gases can be used together with the

1The cross section results to be proportional to the mass of the colliding particles and to the

mass of the collided one as (mAr/mmetal)4

11

Ar

Magnetic fieldlines

Target

Magnetron

N NS

Magnetic Poles

Ar+

Atoms

Figure 1.1: Sketch of sputtering process. Ar gas flows above the target. The Ar+

collides with the target and other atoms are ejected.

primary Ar [7]. Also the production, for example, of either nitrides or oxidescan be achieved by using reactive sputtering.

Briefly, the sputtering process is the following: Ar gas is introduced in thegrowth chamber at few mTorr. A dc or RF negative voltage is applied on thetarget. This voltage accelerates the electrons and ions present inside the cham-ber. The collisions among Ar atoms and electrons yield more ions because Aratoms are ionised by the interaction with the energetic electrons. The Ar+

ions are then accelerated towards the target and collide with the material. Inthe collision the energy of the ions is partially transferred to the atoms of thematerial and the etched atoms spread all around in random directions. Someof the sputtered atoms can reach the substrate and stay fixed there.In order to increase the sputtering rate, the targets sit on strong permanent

magnets: this device is called the magnetron. In such a way, the magnetic

field is coming out through the targets, see Fig. 1.1. Because of the presence

of a strong magnetic field, the secondary electrons, which are freed by the

scattering among Ar+ ions and material of the target, are effectively involved

in the sputtering process. The secondary electrons are thus confined above the

target and spin around the magnetic field lines. These energetic electrons cause

the creation of new ions in the collision with the Ar atoms. This increases thesputtering efficiency greatly.If the voltage drop is high enough a self-sustaining discharge glow is gen-

erated such that the process itself is self-sustaining.The growth rate can be mainly influenced in two ways:

• by applying different powers on the magnetrons• by applying different pressure during the process

12

One can assume that the average energy of the atoms is of the order of some

eV. The pressure allows the modulation of the energy of the molecular flux. In

first approximation, the mean free path of the atoms is inversely proportional

to the pressure. Thereby, at high pressure the mean free path is short and the

atoms are thermalized in a short time. On the other hand, at high pressure the

material flux from the target to the substrate is reduced.It must be remembered that no trace of Ar can be found on the final sample.

Ar gas does not chemically interact at all with the sputtered material. Gener-

ally, the only impurities that can be found inside the samples are oxygen and

carbon. The source of oxygen is the water that generally sticks on the wall of

the chamber and the carbon impurities degas from any organic material and

from the steel of the chamber’s walls. Other impurities come from the impu-

rities present in the Ar gas. The purity of the target generally is above 99.9%

and the purity of the Ar gas is better than 99.9999%. Therefore, the final purity

of the deposited material is above the 99.9%.

1.2 Molecular Beam Epitaxy (MBE) Process

The beauty of the molecular beam growth technique is the extreme cleanness

of the growth environment. The needed UHV condition of the growth cham-

ber (base pressure of≈ 10−10−10−11 Torr) allows the deposition of materialswith a high grade of purity. This method is especially suitable to growth semi-conductors and single crystal rare earths. [8, 9, 10] The deposition rate, that isthe amount of material that is deposited per unit of time, generally measuredin Å/s, is much smaller with respect to the deposition rate generally acceptedin sputtering growth. This very slow growth rate is reached without the helpof any external factor such as gases.

With respect to the sputtering where the kinetic energy of the molecular fluxcan be estimated on the order of tens of eV, the kinetic energy of the atomicflux is much lower than 1 eV. The only sources present in a MBE system arethermal sources as will be described later. Therefore, the only component ofenergy is the thermal energy. For example, the thermal energy of an atom at2000◦C is just≈ 0.2 eV. The small energy decreases the probability that atomsmove/hop randomly on the surface of the sample. In order to increase the mo-bility of the atoms on the surface, a much higher temperature of the substrateshould be used as opposed to the substrate temperature during the sputteringprocess. Two kinds of thermal sources exist in a MBE system: effusion cells(EC), and electrons guns. The latter are all-purpose sources and they are alsoused to evaporate high melting point materials.

1.2.1 Molecular beam from effusion cells

The effusion cells or Knudsen’s cells [11, 12] are the preferred sources usedto evaporate materials if the melting point temperature of the material is lowenough and the vapour pressure of the material is high enough. The latter is a

13

very important parameter since a molecular flux can already be established at

a temperature much lower than the melting point.The Knudsen’s cell is built of a heater and a crucible, protected by a shield

of molybdenum or tantalum, two refractory metals2. The crucible, which ismade up of a high melting point material, like Al2O3, Ta or W or pyrolytic

graphite, is filled with the material. Each material is evaporated from a suitably

chosen crucible since one should avoid the chemical reactions between the

two.The molecular flow is modulated by the modulation of the temperature of

the crucible. Effusion through the orifice of the cell gives a molecular beamwhich spreads in isotropic distribution over a sphere. The molecular flux canbe considered as constant as a function of time at fixed temperature.

1.2.2 Electrons guns

Electrons are emitted from a filament at high temperature by a thermioniceffect. In order to increase their energy, these are accelerated by an electricpotential of the order of some keV. Their trajectory is bent by strong perma-nent magnets and they are collimated by playing with electromagnetic sweepcoils whose purpose is to move the electrons beam onto the surface of thematerial. Using this method, the material is melted by the transfer of energyfrom the colliding electrons to the atoms of the material. In principle any kindof material can be melted down by this technique. The molecular flow is notstable in this kind of source and therefore the growth rate is tracked by usinga quartz crystal microbalance.

1.3 Metallic superlattices/multilayers

A regular repetition of layers of two or more different materials are calledmultilayers.In the special case when a long range structural coherence, much larger than

one bilayer thickness, is present along the growth direction, the more preciseterm superlattices [13] should be used.In order to briefly clarify how a superlattice is produced, the following ex-

ample is given: one should start with a single crystal substrate whose orien-

tation is defined by the indices (hkl). Substrates are generally oxides such asMgO or Al2O3 but also single crystal metal substrates can be used. By using

a deposition technique the metal A is deposited on the substrate. The deposi-tion is carried out by using a molecular flow from a source of the metalA. Thethickness of the metallic layer is then fixed at α mono-layer3 (ML) by the tim-ing of the opening and closing of the source. The metal B is deposited on topof the previous layerA in the same way. The thickness of this layer of material

2Refractory metal - A metal with a very high melting point such as tungsten, molybdenum,tantalum, niobium, chromium, vanadium and rhenium.3Mono-layer - Out-of-plane distance between two neighbouring atoms.

14

Mg

Fe/V

O

Figure 1.2: Distribution of the Fe/V(001) on the substrate. The bct (body centredtetragonally distorted) crystal structure of the superlattice is rotated of 45◦ with re-spect to the fcc (face centred cubic) structure of the substrate.

B is then fixed at β ML. If the combination AB is repeated in a predictablesequence as ABABA. . . , with the entire structure showing the same crystaldirection, such a film is called a superlattice with a selected crystal growthdirection (l′m′n′) which can be different with respect to the crystal orientationof the substrate.

In such a way two are the periodicities, given by Aα and Bβ with the de-fined superlattices lattice constant Λ which is given by the summation of the

2 thicknesses:

Λ = Aα +Bβ

Under these conditions the expression [AαBβ ]n correctly describes the super-lattice, where n represents the number of repetitions of the bilayers.One example of superlattice is the case of the epitaxial growth4 of

Fe/V(001) superlattices on MgO [14, 15]. It has been shown that this kind of

superlattices presents an epitaxial relationship5:

• Fe/V(001) ‖MgO(001)• Fe/V [110] ‖MgO[020].

In order to avoid any confusion, the term thin filmwill be used instead of eithermultilayer or superlattice in the rest of this work.

4Epitaxial Growth - The growth of one crystal on the surface of another crystal. The growthof the deposited crystal is oriented by the lattice structure of the original crystal.5Epitaxial Relationship - The orientation of the planes (hkl) and directions [uvw] in the filmrelative to those of the substrate.

15

Fe/V(001) superlattices on MgO are examples of hetero-epitaxial growth,

where the substrate and the thin film are two different materials. In this case

the thin film presents stresses and/or strains because lattice mismatch gener-

ally exists among substrate and materials. The strains are usually accommo-

dated elastically when the thin film is attached at the substrate.

The strains are defined as [16]

ε ≈ (as−a f )as

(1.1)

where as and a f are, respectively, the lattice constant of the substrate andthe lattice constant of the film. For example, one can calculate the strain ina Fe/V(001) superlattice attached on the MgO substrate: because of the rota-tion of the lattice plane of either Fe or V with respect to the lattice plane ofthe MgO, as = aMgO×

√2/2= 2.978 Å. The lattice constants of V = 3.03 Å

whereas the lattice constant of Fe = 2.868 Å. By applying Eq. 1.1, the follow-

ing results for ε are calculated:• ε =−1.73% for V on MgO• ε = 3.70% for Fe on MgO

• ε = 5.35% for Fe on V

It is clear the reason why all the Fe/V(001) superlattices studied in this workhad been grown by starting with V as first layer.Since the thickness of substrate is of some order of magnitude thicker than

the film (the thickness of the substrates is of the order of mm) in Eq. 1.1

one lets to accommodate the elastic stresses/strains into the film. A critical

thickness [16] exists below that a coherent thin film, without imperfection

can be grown. This critical thickness is a function of the elastic modulus and

biaxial strain ε of the materials.Three growth modes are defined depending on the thermodynamic equilib-

rium during the deposition: [17]

• Volmer - Weber equilibrium - The thermodynamic equilibrium is char-acterized by the existence of three dimensional crystals in contact with thesubstrate while the rest of the substrate is devoid of any condensed phase.That is, the growth of the thin film is caused by 3-dimension island coales-cence.

• Frank - van der Merwe equilibrium - The thermodynamic equilibrium

is characterized by the perfect cover of the substrate by the stacks of the

layers. This is the two dimensional crystal growth or layer by layer. Each

layer starts to grow on the top of the previous one [18].

• Stranski - Krastanov equilibrium -This is an intermediate equilibriumbetween the previous two. Starting from a bare substrate a limited numberof 2-dimensional layers can be formed until a 3-dimensional growth ofislands begins.

16

For our aims, the growth mode of the Fe/V(001) superlattices, which are

mainly studied in this work, has been described as Frank - van der Merwe

equilibrium [14].

17

2. Structural characterisation ofmetallic superlattices

A crystal structure is defined by the lattice,the lattice parameter d and sym-metry and the arrange of atoms with lattice unit. This lattice parameter, inthe special case of a cubic structure, represents the single length scale of thelattice. In a superlattice, one also introduces the lattice constant of the con-stituent bilayer Λ. Thereby, together with the d, Λ must also be added to thepossible length scales. In a thin film, one also adds the total thickness D as

an additional length scale. All of these length scales can be probed by X-ray

techniques such as X-ray diffraction for d and Λ and X-ray reflectivity for Λand D.In this chapter, the theory of X-ray interaction with matter will be intro-

duced. The focus will be on reflectivity and diffraction measurements. Exam-ples of the structural properties which can be extracted from those measure-ments will be shown.

2.1 Interaction between matter and X-ray

The energy spectrum or wavelength spectrum of electromagnetic waves ex-tends from those with long wavelength, radio frequencies, through the wholespectrum of infrared, visible light, ultraviolet, to those with ultrashort wave-length, X-rays and γ-rays.X-rays are used to investigate the structure and the magnetic characterisa-

tion of both hard and soft matter. The reasons for this are that X-rays canpenetrate the first Å of a specimen, so they can be diffracted by the crystaland the mathematical treatment of the diffraction can be applied. Also, theirenergy fit with energy which can excite the magnetic state of atoms.Presently, the main sources for X-rays radiation are X-rays tubes and syn-

chrotrons. In the first, an electron beam with an energy of tens of keV collides

with an anode. The electrons are quickly slowed down and the emitted ra-

diation is caused by the Bremsstrahlung effect. However, the X-ray can also

be absorbed by the electrons of the inner shell and these electrons leave the

atom. Thereby the spectrum is composed of a continuous spectrum together

with strong peaks. These peaks are caused by the decay of electrons from an

outer atomic shell to an empty inner atomic shell. It follows that the energies

of these peaks are quantised and element specific. These strong emission lines

can be selected by using a monochromator, see Fig. 2.1

19

Figure 2.1: Sketch of Bremsstrahlung emission from a generic anode. The highest

peaks represent the specific element emission due to the decay of the electrons from

the outer to the inner shells.

In synchrotrons X-rays are generated by placing undulator or bending mag-

nets in the way of the electron pulse. By forcing the electrons to follow a

curved path, they emit a continuous electromagnetic radiation known as syn-

chrotron radiation. This radiation is characterized by a very high intensity. The

specific energy can be selected by using a monochromator (generally more

than one). The brilliance of the source is at least 1010 times stronger than the

classical X-ray tube!X-rays can be modelled either as electromagnetic waves or as photons,

depending on the circumstances of the experiment. The relation between the

wavelength λ in Å and the energy of the photon ε in keV is the following:

λ =2π hc

ε=

Aε

where A = 12.398 Å keV−1.In scattering processes the radiation is described by using the wave-particle

duality and this description helps in understanding the X-ray reflectivity at theinterfaces and diffraction from a crystal.In brief, the electric field of the wave exerts a force on the electrons in the

matter which are accelerated and radiate the scattered wave. In these exper-iments, the process can be considered elastic if there is a negligible energytransfer to the sample. In order to describe the elastic process, a new variableQ, the scattering vector or wave vector transfer is introduced, as the follow-ing:

20

hQ = hk− hk′ (2.1)

where k and k′ are the initial and final momenta of the photon, with |k|= |k′|,see Fig. 2.2.

k k'

Q

Figure 2.2: Sketch of momenta which are involved in elastic scattering. Both in

diffraction and reflectivity the wavevector transfer Q is always perpendicular to the

plane of scattering.

So diffraction from a crystal and reflectivity from interfaces are examples

of elastic processes.In inelastic processes, the X-rays are conventionally described as photons.

In the scattering, the energy of the photon is partially transferred to the elec-

tron which recoils with its new momentum, see Fig. 2.3. The electron moment

is calculated by the Compton scattering equation:

q = k−k′ = k′kh

mec(1− cosψ)

where the constant hmec = λc is the Compton scattering length and ψ the scat-

tering angle.

For some element-specific energies, the X-ray can be absorbed by the atomand the excess energy is transferred to the electrons which leave the atom andthe atom is ionised. The general process is called photoelectron absorption andit is generally described by saying that the intensity of the beam exponentiallydecays by passing throughout the material, that is:

I(x) = I0 e−μx

where μ is the linear absorption coefficient which is element-specific.It follows that after the absorption and ejection of one electron by an atom,

the atom can decay to the ground state by fluorescence or the Auger effect. In

21

�(k, E)

(k', E')

particle

(q, �)

Figure 2.3: Sketch of momenta and energy which are involved in inelastic Comptonscattering.

the first case, one X-ray is emitted which is caused by the decay of one electronof the outer shell to the inner shell. In the second case, another electron isemitted from the outer shell while another electron of an inner shell decays onthe first hole. The technique of X-rays absorption is used in X-ray magneticcircular dichroism (XMCD) and photo-emission spectroscopy.

The basic idea of XMCD spectroscopy is that in magnetic materials thespin-up and spin-down electronic bands are split. The energy gap correspondsto some precise value ΔEgap = E(s, l) which is described by the Fermi’sgolden rule. It follows that using circular polarised light which is tuned onthat specific ΔE, just the electrons with either spin-up or spin-down can beexcited, that means, either absorption spectra or emission spectra can bemeasured, or the drain current from the sample can be collected.

2.2 X-ray diffraction

The wavelength of X-rays is of the order of 1 Å. This length is of the sameorder as the distance among atoms in a crystal. It follows that, if one thinksof X-rays as electromagnetic waves, X-rays can be diffracted by the regularstructure of a crystal.

The constructive interference pattern of the diffraction from a crystal withatomic distance among the layers d, is described by Bragg’s law

2d sinθ = nλ (2.2)

where n is an integer.Figure 2.5 shows how Bragg’s law can be derived. The necessary condition

for constructive interference is that the phase of the beams must coincide when

the incident angle equals the reflecting angle.

22

e-

e-

Figure 2.4: Sketch of the absorption of X-rays. The result is a decrement of the

intensity, emission of electrons and re-emission of X-rays.

The rays of the incident beam are always in phase and parallel up to thepoint at which the top beam strikes the top layer at atom A. The second beamcontinues to the next layer where it is scattered by atom C and it must travelthe extra distance AC + CB if the two beams are to continue travelling ad-

jacent and parallel. This extra distance must be an integral n multiple of thewavelength λ in order that the phases of the two beams are the same. By usingtrigonometry, it can be shown that the beams are in phase when Bragg’s law(Eq. 2.2) is fulfilled.On the other hand, Bragg’s law does not allow the calculation of the in-

tensity of the diffraction pattern. The scattering amplitude for each diffraction

condition can be derived as

Fcrystal(Q) = ∑r j

Fmolj (Q)eıQ·r j ∑

Rn

eıQ·Rn (2.3)

by taking into account the unit cell structure factor (first sum) and the lattice

sum (second sum). In this Eq. 2.3, r j is the position of the j’th molecule inthe unit cell, Rn specifies the origin of the unit cell, and Fmol

j is the molecular

form factor. The molecular form factor is defined as

Fmol(Q) = ∑ f j(Q)eıQ·r j (2.4)

where f j(Q) is the atomic form factor.

23

A B

C

�

d

�

Figure 2.5: Sketch of Bragg’s law (cut through a cubic crystal): two parallel beamsare in phase when the extra distance AC+CB is equal to n times the wavelength.

For a certain lattice, the lattice vector (�a1, �a2, �a3) represents the base and insuch a way a point Rn can be described by the relation

Rn = n1�a1+n2�a2+n3�a3.

Based on these definitions, a most convenient way to describe the plane ofdiffraction is by using the Miller indices (hkl). For example, for a given familyof scattering planes in a cubic symmetry, these are defined such that the plane

closest to the origin has the intercept (a1/h,a2/k,a3/l) on the base. Therefore,for a given scattering plane a dhkl lattice spacing is defined by

dhkl =a√

h2+ k2+ l2

The lattice sum in Eq. 2.3 represents the Fourier transform of the real space,

and generates the reciprocal space whose base vectors are (�a�1,�a

�2,�a

�3). The

reciprocal space is generated by applying the condition that

Q ·Rn = 2πn (2.5)

Thereby that condition is fulfilled if�ai ·�a�j = 2πδi j. The points in the reciprocal

lattice are defined by the vector

G = h�a�1+ k�a�

2+ l�a�3 (2.6)

Under this hypothesis, G satisfies the same condition of Eq. 2.5

24

20 30 40 50 60 70 80 90 100 110 120

2� (degrees)

Lo

g10 I

nte

nsit

y (

Arb

. u

nit

s)

(111)

(222)

(100)

(200)(400)

(300)

Figure 2.6: Diffraction measurements of a single of Pd95%Fe5%, whose the total thick-ness is 400 Å, which was grown on SrTiO3 (100) single crystal. The peaks indicated

by the squares are Bragg’s peaks of the alloy whereas the stars indicate the peaks from

the substrate.

G ·Rn = 2π(hn1+ kn1+ ln3) (2.7)

Because the terms in parenthesis of Eq. 2.7 are integer, the condition ofdiffraction is given by the equality

Q = R

which is the Laue condition for the observation of the X-ray diffraction.The regular structure of the lattice in the real space does not allow the

diffraction from all the Miller’s planes. This can be derived by calculating

the unit cell structure factor for different unit cells. For example, by consider-

ing a f cc (face cubic centred) structure and applying the first sum in Eq. 2.3, itfollows that if and only if (hkl) are all even or odd then the unit cell structurefactor is not zero. In case of a bcc (body cubic centred) structure the conditionfor scattering is fulfilled if and only if the h+ k + l is an even number.For example, in Fig. 2.6, Bragg’s peaks of SrTiO3 (100) substrate and an

alloy Pd95%Fe5% are shown. The structure of the alloy is fcc without any ro-tation. In such a way the peaks from the reflection from the plane (111) and(222) are clearly visible. The structure of SrTiO3 is a cubic peroskite but thedescription of this structure is will not be included in this thesis.

25

....

�{

D

d

dd

Figure 2.7: The different length scales in a superlattice.

The diffraction pattern from a superlattice has to include theΛ lengths scale,see Fig. 2.7 Because of the superlattices structure, which is defined by the

bilayer thickness and number of repetitions, the diffraction pattern is charac-

terized by the presence of additional peaks. These peaks are caused by the

additional periodicity arising from the chemical modulation. The positions of

these peaks are functions of the bilayer thickness Λ, and they are described bythe following formulation: [19]

2sinΘn

λ=1

d± n

Λ(2.8)

where d is the average lattice constant of the crystal,

d �Λ

α +β

where α and β are the number of monolayers of each material, for exampleas in [Aα /Bβ ]n. In Eq. 2.8, λ is the wavelength of the X-ray, n the label of thesuperlattices peak, and Θn is the position of the satellite labelled by n.In Fig. 2.8, the typical X-ray diffraction pattern from a superlattice, in such

a case [Fe3V11]20, is shown. Thus, from the positions of the satellites, theaverage bilayers thickness is determined.The high frequency fringes, which are called Laue’s fringes, are given by

the constructive interference between the total thickness and the d spacing.Any sample is generally capped with a protective layer of Pd or other oxideand this capping layer does not contribute to the interference because of thelack of epitaxial relation with the superlattice.

26

50 55 60 65 70 75

Lo

g10 I

nte

nsit

y (

Arb

. u

nit

s)

-1

-1

0

+1

+2

2� (degrees)Figure 2.8: X-rays diffraction scan of the superlattice [Fe3V11]20. The main diffrac-tion peak 0 is surrounded by two satellites per side. The out-of-plane correlation is of

the order of ≈330 Å.

26 28 30 32 34 36

Lo

g 10 I

nte

nsi

ty (

Arb

. U

nit

s)

� (degrees)

-1+1

0

Figure 2.9: Rocking curves around the peaks 0,−1, and+1 of the sample in Fig. 2.8.

27

An important value from Bragg’ scan is the full width at half maximum

(FWHM) which is related to the out-of-plane correlation. This is described by

Scherrer’s formula:

h =κλ

BcosθB(2.9)

with κ Scherrer’s constant [20], B the values of FWHM, λ the wavelength ofX-rays, θB is Bragg’s angle. Equation 2.9 is widely used in order to estimate

the crystal quality of the samples. However, one needs to know the instru-

mental resolution and the disorder of the crystal structure. If one considers the

samples in Fig. 2.6 and 2.8 and applies the previous equation, an out-of-plane

coherence of the crystal structure is calculated to be respectively 400 Å and

330 Å.

Additional information on the in-plane lateral correlation length [21] andmosaicity are determined by collecting the rocking curve, ω-scan or trans-verse scan around the main Bragg’s peaks. The sample is rotated on the plane

while the angle 2Θ is fixed. Strictly speaking, the Q is scanned transverselythrough Bragg’s reflection, see Fig. 2.10. Therefore, by means of the ω-scan,the x-y plane can be accessed. In Fig. 2.9, the rocking around the Bragg’s re-flection (200) in the same superlattice [Fe3V11]20 is shown together with the

rocking curve around the first satellites. More details can be found in Refer-

ences [22, 23, 24]. Strictly speaking, a sharp rocking curve is the evidence

Allowed

reflection

Qz

Qx

Q

Figure 2.10: Visualisation of transverse scan in the reciprocal space. The grey areaare not accessible because of geometrical reasons.

of high coherence of the lattice planes and crystal orientation because Bragg’scondition is fulfilled only at one narrow range. By fitting this shape with twoGaussians, one for the narrow peak and one for the diffuse scattering aroundthe narrow one, the lateral correlation length of the grain size can be estimated

28

by the width of the first one [21, 25, 26, 27]. On the other hand, the diffuse

scattering around the narrow peak is the measurement of the mosaicity of the

crystal. By integrating the area below the two Gaussians and by the ratio be-

tween those, one may estimate the largest component between mosaicity and

lateral smoothness.

2.3 X-ray reflectivity

In reflectivity measurements, the Q value is much closer to zero. Therefore

the length scales probed by the reflectivity, in the special case of multilayers,

are comparable with the thickness of bilayers Λ and the total thickness D,Fig. 2.7.In such measurements, X-rays can be described by using the optical the-

ory of electromagnetic waves. The refraction and/or reflection at the interface

between two media of an electromagnetic wave are described by Snell’s law:

�0

�i

�j

i

j�j

�i

Figure 2.11: Sketch of Snell’s law for optical wavelengths. The refractive index forX-ray is η < 1 so that Θ0 < Θi

ηi sinΘ = η j sinΘ j (2.10)

where ηi and η j are the refractive indices of the medium i and of the mediumj. The angle Θ0 is the angle between the normal to the surface and the inci-

dent beam whereas the angleΘi is the angle between the normal to the surfaceand the refracted beam, see Fig. 2.11. In this figure, two interfaces are shown:firstly, vacuum, whose refractive index is defined η0 = 1 and medium i; sec-ondly medium i and medium j.In the special case of interface vacuum-medium, the Eq. 2.10 becomes

29

sinΘ0 = ηi sinΘi

It can be shown that a critical angle Θc exists below which the light is totallyreflected: Θc is linked to the density of the material ρ .In the special case of X-rays the refractive index η is a complex unit. The

η fulfils the following relations:

η = 1−δ − ıβ (2.11)

where δ is a very small, of the order of few 10−5 to 10−6, β is the imaginarycomponent of the refractive index to account for the absorption effect. Bothare specific parameters for the medium.Because β and δ are very small quantities, the refractive index can be es-

timated as η ≈ 1− o(10−6). Taking into account Snell’s law (Eq. 2.10), anangle of total reflection is found to be Θc � 1◦.By using these principles, one can analyse the data in Fig. 2.12: the general

trend of the intensity beyond the critical angle is a function of I = I0(Q−4)(Q = 4π

λ sinΘ). The total reflection region extends until Qc ≈ 0.06 Å−1. Thiscritical angle can be used to determine the average composition in multilay-

ered structures [28].

By examining Fig. 2.13, one can see two strong and well defined peaks,namely Q1 and Q2. Those peaks reflect the out-of-plane chemical modulationof the sample (for a review see References [29, 30, 26, 31]).

Because of two different metals, that is two different electronic densities,the reflectivity index for X-rays changes at the interface, so (see Eq. 2.11)under the same conditions for total reflectivity, the beam reflects for particularangles, Eq. 2.10. The full and complete theory of reflectivity by X-rays hasbeen developed by Parrat by taking into account multiple reflections [32].The position of these peaks are correlated with the bilayers thickness (fre-

quency of the chemical modulation) Λ. On the other hand, the intensity andshape yield information on the roughness and interdiffusion at the interface. It

means that by analysing the shape of these peaks, the structure at these length

scales can be determined. The value of the length scale Λ is determined by thefollowing equation:

Λ = mλ2

1√sin2 θm− sin2 θc

(2.12)

where λ is the wavelength of X-rays, Λ is the bilayer thickness, θm is the

position of the peak, θc is the critical angle,m is the order of satellite. Equation2.12 may be used to calculate the experimental value of Λ if the values of θmand m are known.The fringes which are present both in Fig. 2.12 and on Fig. 2.13, are the

so-called Kiessig’s fringes [33]. The Eq. 2.12 may be approximated for small

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Q (Å-1)

Lo

g1

0 I

nte

nsit

y (

Arb

. u

nit

s)

Figure 2.12: Reflectivity measurement of a single layer of the alloy Pd95Fe5 whichwas grown on SrTiO3. The well defined and smooth total thickness is shown by the

presence of the Kiessig’s fringes.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Q (Å-1)

Lo

g1

0 I

nte

nsit

y (

Arb

. u

nit

s)

Q1

Q2

Figure 2.13: Reflectivity scan of [Fe3V7]41: the peaks Q1 and Q2 are the superlatticepeaks. The Kiessig’s fringes are well visible, meaning a well defined total thickness.

31

values of θ as θ 2j − θ 2c = j2(

λ2·ΛT

)(ΛT total thickness of the superlattice).

Each fringe of order j may be fitted with the previous expression.In order to estimate the roughness1 and interdiffusion2 the reflectivity scan

must be fitted to the exact solution as described by Parrat.

Commercial and open source software are available, however they slightlydiffer on how they model the reflectivity and the minimisation routines. Thesoftware used in most of the fits is GenX developed by M. Björk [35], basedon Parrat algorithm for the reflectivity calculation and the genetic algorithmfor the minimisation of the fitting parameters. Figure 2.14 shows a simulationusing GenX. The final parameters of the simulation are the values of the aver-age thicknesses of each layer, with their average roughness/interdiffusion. Inreflectivity experiments, it is not possible to discriminate between roughnessand interdiffusion, since both give the same components in the shape of thepeaks and shape of the whole scan. In order to estimate the roughness, themain peaks must be investigated by rocking curves as it has been previouslystated.

0 2 4 6 8 10 12 14

Lo

g10 I

nte

nsit

y (

Arb

. u

nit

s)

2� (degrees)

Data

Simulation

simulation

raw data

Figure 2.14: Simulation and real scan for the gradient sample GR- as described in

Paper IV.

1Roughness - Long range variation of the height of given layer, which may be transmittedthroughout the superlattices (correlated roughness) [34]2Interdiffusion - Atomic scale variations of the layers height, with possible interchange of thetwo materials [34].

32

3. Growth of Fe/V(001) superlattices

Nowadays superlattices of Fe/V(001) have been used as exemplary model instructural phase transitions [36, 37], magnetic order-disorder [38, 39, 40, 41,42, 43, 44, 34, 45, 46] and transport properties [39, 47, 45, 48]. One of thereasons for this success is the possibility of growing such a system in layerby layer mode as shown by Isberg et al. [14]. My contribution is given bythe application of this method on the new sputtering and MBE system called

Óðinn.In first approximation, it is not possible to apply a growth recipe, exper-

imentally found in one specific system, to another one. The first stage wasto find the new conditions for growing Fe/V(001) superlattices in that newsystem. A series of superlattices were grown at different temperatures, whichwere measured on the heater, and successively characterized by X-ray tech-niques. When the right growth temperature was determined, the second stepwas to try to see if different pressures and different magnetron powers hadsome effect on the overall sample qualities. By varying the pressure, at fixedmagnetron power, the growth rate, that is the amount of atoms which reachesthe substrate, experimentally measured in Å/s, proportionally changes. On theother hand, by changing the magnetron power at fixed pressure, the growthrate changes, too.

In summary, the criteria for the best growth conditions are those which givethese properties:

• small interdiffusion and roughness at the interfaces.• single crystal or maximisation of the out-of-plane and in-plane crystal co-herence

• small growth rate for Fe and V in order to become possible to preciselycontrol the thickness of both layers.

The [Fe11/V13]40 superlattice was designed in order to achieve these goals.The large thickness of the sample, together with the large thicknesses of the

single layers, allow for an easy structural characterisation of the samples. Theratio between the Fe and the V monolayer does not allow the presence ofdestructive interference in the X-ray pattern. After the preparation, each sam-ple was characterized by X-ray diffraction and reflectivity as well as rockingcurves through the main peaks, see Chap. 2. Figure 3.1 shows the reflectivitiesof four samples, which were grown at four different temperatures (measuredon the heater).

Because of the geometrical conditions between the heater and an externalwindow, it is not possible to directly measure the real temperature on the sub-

33

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Q (Å-1)

Lo

g 10 I

nte

nsi

ty (

Arb

. u

nit

s)700 °C

600 °C

575 °C

350 °C

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Figure 3.1: X-Ray reflectivity scan of four identical [Fe11/V13]40 superlattices whichwere grown at four different temperatures. The number of peaks is the marker of the

interdiffusion/roughness at the interfaces, see text.

strate by using a pyrometer which can be done in the other sputtering systems

which are present in our lab.One can see that in the reflectivity scan of the sample grown at 375 ◦C just

the peak Q1 is present whereas the samples grown at 575 ◦C, 700 ◦C and600 ◦C are characterized by more than one peak. (The meaning of these peaksis described in Chap. 2) In brief, one can say that the higher the index i ofFourier’s components, the sharpest the Fe/V interface.Therefore, if one only considers reflectivity, the best sample is that one

which is grown at 575 ◦C, followed by the samples at 600 ◦C, 700 ◦C and350 ◦C. The high frequency fringes (Kiessig’s fringes) on the reflectivity of thesample at 575 ◦C are the markers of the well-defined total thickness. Figure3.2 shows the X-ray diffraction pattern for the samples which are taken as

examples.

The description of this figure and the quantities which can be extracted,are fully described in Chap. 2. Briefly, as stated previously, for the case ofthe reflectivity scan, the number of satellites around the main peak, Q0, isthe marker of the well defined superlattice structure. The full width at halfmaximum (FWHM) of the main peak is correlated with the crystallinity ofthe film. The smaller this value the bigger the coherency. By applying theScherrer’s formulation 2.9, we can calculate the highest out-of-plane crystalcorrelation of about 600 Å for the samples at 600 ◦C and 575 ◦C. The othertwo samples are discarded because, in one hand the reflectivity of the sample

34

3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2

Q (Å-1)

Lo

g 10 I

nte

nsi

ty (

Arb

. U

nit

s)700°C

600°C

575°C

350°C

Q0

Q+1

Q+2

Q+3

Q-3

Q-2

Q-1

Figure 3.2:X-Ray diffraction pattern of four identical [Fe11/V13]40 which were grownat four different temperatures.

at 350 ◦C shows just one reflectivity peak with respect to the other three and,on the other hand, the width of the main peak Q0 of the sample at 700 ◦C ismuch broader than the others. One can list the samples, based on the diffrac-

tion scans, from the best to the worst, as 575 ◦C, 600 ◦C, 700 ◦C, and 350 ◦C.The direct estimate of the roughness at the Fe/V interface is given by mea-

suring the rocking curve around the peaks Qi of Fig. 3.3. The estimate of thewidth of the peak together with the estimate of the background plateau suggestthe amount of this defect and allows one to choose the sample which fulfilsthe previously displayed conditions.

After these considerations and examining Fig. 3.3, one can say that the sam-ples at 575 ◦C and at 600 ◦C do not show any difference whereas the sampleat 700 ◦C shows a much broader peak and larger plateau. It follows that theinterfaces Fe/V in the first two cases are rather smooth whereas a much higher

roughness defines the scattering profile of the sample at 700 ◦C [49]. Thiscan be as an indication that around this temperature the growth mode starts

to act like the Volmer - Weber mode. Indeed, the growth mode for the range

575-600 ◦C should be described by the Frank - van der Merwe equilibrium[14].

Bragg’s peak Q0 is also investigated by means of rocking curve. In such acase, the rocking curve displays the amount of crystallite and misalignmentsof the grains. In brief, because of the geometry of such a scan, misalignmentof the scattering plane will give a more wide rocking curve. In Fig. 3.4, therocking curves for three of the superlattices are displayed. By calculating the

35

–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

�� (degrees)

Lo

g 10 I

nte

nsi

ty (

Arb

. U

nit

s)700°C

600°C

575°C

Figure 3.3: Rocking curves on the peaks Q1 of the X ray reflectivity of Fig 3.1. Bothsamples at 575 ◦C and 600 ◦C show the same profile which points to a flat interface.Δω is measured with respect to the peak position.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Lo

g 10 I

nte

nsi

ty (

Arb

. U

nit

s)

700°C

600°C

575°C

�� (degrees)Figure 3.4: Rocking curves around the peaks Q1 of the X ray reflectivity of Fig 3.2.The sample at 700 ◦C presents the most sharp rocking curve with FWHM of 0.34◦.Δω is measured with respect to the peak position.

36

FWHM, the best crystallinity is given by the sample which was grown at

700 ◦C with a value of 0.34◦. On the other hand, by observing the Fig. 3.2,the quality and misalignments seem to be better for the samples at 575 ◦C and600 ◦C.At this point, by taking into account all the data and by keeping in mind

the conditions which have to be fulfilled, one can say that the best Fe/V(001)

superlattices are grown at a temperature of 575 ◦C. In such a case, the growthis 2-dimensional with small roughness at the interface; the superlattice is a

single crystal with small misalignments.The growth of superlattices at different pressures and powers did not show

any improvement of the overall quality of the superlattices.As it will be stated later on, the most critical parameter is the Fe/V interface

because the electronic scattering of the conduction electrons at the interface

plays a critical role on the oscillatory interlayer exchange coupling [50].

37

4. Magnetic characterisation ofmetallic superlattices

One way to magnetically characterise FM or AFM superstructures is by ex-ploiting the interaction between polarised light and magnetisation. Two of theeffects, the magneto optic Faraday effect in transmission and magneto opticKerr effect (MOKE) in reflection yield qualitative measurements of the mag-netisation. MOKE is widely becoming the master tool to investigate thin films[3, 2] and superstructures [51, 52] because of its versatility. On the other hand,MOKE can not be used to determine the absolute value of the magnetisation.SQUID (Superconducting Quantum Interference Device) is the appropriatetool to be used for the determination of the absolute value of the magnetisa-tion. If one considers a multilayer which is made of magnetic/non-magneticmaterial, then the absolute value of the magnetisation per atom is the aver-age magnetic moment. That means that MOKE and specially SQUID can notmeasure the magnetisation at different length scales. The technique which canbe used to discriminate the magnetic moment among different length scales ispolarised neutron reflectivity (PNR). It follows that only being able to handlethese techniques, one can achieve the complete knowledge of the magneticproperties of thin films.

4.1 Magneto optic Kerr effect (MOKE)

The magneto optical Kerr effect (MOKE) is described as the following: visi-ble and linearly polarised light, which is generated by a LASER, is reflectedfrom sample’s surface. Because of the interaction between the photons andthe density of states (DOS), the polarisation of the light changes as well as theellipticity [53]. Two ways exist to describe the effect. Firstly and macroscop-ically, the interaction light-magnetisation arises from the anti-symmetric, off-diagonal elements in the dielectric tensor. Secondly and microscopically, spin-orbit interaction gives different energy levels for different orbital states whichdifferently interact with the spin state of light. Thus, the optical response, suchas speed and absorption coefficient of light in the media, are different for leftand right polarised light. When the linear polarised light shines on the mag-netic sample, differences of the phase shifts of the left polarised part and theright polarised part make polarisation direction rotates. This effect gives Kerrrotation. The difference in absorption changes polarisation from linear to el-liptic. The effect gives Kerr ellipticity.

39

rps

rss= φs = φ ′s +φ

′′s (4.1)

rsp

rpp= φpp = φ ′p +φ

′′p (4.2)

Here, r is a reflection coefficient and p and s mean s−1 or p−polarisedlight2. The term φ ′ represents the Kerr rotation whereas the term φ ′′ representsthe Kerr ellipticity. For example, rps is the reflection coefficient from incoming

p−polarised light to outgoing s−polarised light. In magnetic material, bothKerr rotation and Kerr ellipticity exist. However, in metals, Kerr ellipticityusually dominates over Kerr rotation.The magnitude of the rotation of the polarisation is linearly proportional to

the net magnetisation M of the sample. In case of thin film and multilayers,it can be shown that the net rotation is a summation of the net rotation ofeach layer probed by the light [53]. It follows that is the total thickness of thethin film is smaller than the total probing depth (≈200 Å per metals) one canspeak of Surface MOKE. A sketch of MOKE setup is showed in Fig. 4.1.

sample

B

Laser

Polarizer

Analyzer

Detector

Faraday rotation

Helmhotz

coils

B

Figure 4.1: Main components and devices which are needed to build up a MOKE

instrument. In order to achieve high magnetic field, the Helmotz coils can be replaced

by a electromagnet.

This setup is like the Ac-MOKE setup “Homer”, where all the samples that

are presented in this work, had been characterized in their hysteresis loops and

susceptibility χ: the setup is described in details in References [42, 56].Depending on orientation between polarisation of the light and magnetisa-

tion, three geometries are possible [57]. Each different geometry can access

1s-polarised light - Light that is polarised perpendicular to the plane of incidence.2p-polarised light - Light that is polarised parallel to the plane of incidence.

40

to one of the three components of the magnetisation Mx, My, Mz, yielding a

qualitative value:

�

� �

��

�

� �

��

M

I R

Longitudinal geometry

Figure 4.2: The three different geometries for MOKE. M represents the direction of

the magnetisation that is probed.

• Polar geometry - The magnetization M points out-of-plane. Both polar-

ization and ellipticity change.

• Longitudinal geometry - The magnetization M lays on the plane. Theplane of incidence of the light is parallel to M. The plane of polarizationof the reflected beam changes as well as the ellipticity. The setup Homerbelongs to this class.

• Transverse geometry - Magnetization is normal to the plane of incidenceof the light but parallel to the sample surface. One measures both intensityvariation and phase shift of the reflected light.

The polar Kerr signal is usually an order of magnitude greater than the longitu-dinal Kerr effect, because of the great refraction index of metal. The boundary

41

conditions in each of the three geometries either allow or do not allow for a

Kerr rensponse for specific conditions and s− or p−polarization (page 728 ofRef. [58]).

Samples that are protected with some ten Å of non-magnetic layer can bemeasured with MOKE techniques because the Kerr rotation is only due to themagnetic part of the sample. But, one should keep in mind that the penetrationdepth of the light in the visible range is written as I = I0 exp(−t/λ ) where tis the optical path length and λ is the optical skin length. In case of metals,the value of λ lays between 100-200 Å. For example, all the samples whichwere measured by MOKE and that are presented in this work had been cappedwith 50-60 Å of Pd. Therefore, the common use of a metallic protective layerof some ten Å of non-magnetic material does not change the magneto-opticresponse. On the other hand, a protective layer of amorphous and transparentoxide such as alumina, will not change the penetration depth of the light in thesample.

T [K]80 100 120 140 160 180 200 220 240

Kerr

rota

tio

n (

Arb

. u

nit

)

Pd20Fe1Pd20

Figure 4.3: Remnant magnetization vs temperature for a single monolayer of Fe

embedded in Pd.

The longitudinal MOKE “Homer” setup allows the recording of hysteresisloops by slowly varying the temperature of the sample. In such a way, by ex-tracting the remnant magnetization from each loop, and because of the linearresponse of Kerr rotation as a function of the magnetization, one can measurethe change of the remnant magnetization as a function of the temperature,M vs T , Fig. 4.3. These data play an important role for the determination ofthe universality class of the magnetic order-disorder transition.

42

The rotation φ ′ may be positive or negative, depending on the reflectingmetal, that it the DOS, and depending on the energy of the incoming photons.

For example, at the wave length of the He-Ne laser, the φ ′ for Pd [54] is posi-tive whereas for Fe is negative [55]. Figure 4.4 shows two hysteresis loops as

an example of different Kerr rotations. Ac-susceptibility χac measurements

–8 –6 –4 –2 0 2 4 6 8

H [mT]

Kerr

Ro

tati

on

(A

rb.

Un

it) Fe/V

Fe/Pd

Figure 4.4: Hysteresis loops for two samples: the Kerr rotation for Fe/Pd is the oppo-site of Fe/V because Kerr rotation is dependent on the energy of the photons and on

the metal (See References [55, 54] and references therein).

are also possible in the “Homer” setup. The measurements of χac yields in-formation on the universality class of the magnetic order-disorder transitionand on the magnetic ordering temperature whose value is dependent on thefrequency on the strength of the exciting field [59, 60]. The χac measurementscan also be used in the determination of the order-disorder temperature in an-tiferromagnets [61]. This makes the χac an universal tool for the investigationof the magnetic ordering phenomena.

4.2 Superconducting Quantum Interference Device(SQUID)

The Superconducting Quantum Interference Device or SQUID has been a keyfactor in the investigation of magnetic materials and magnetic structures (for areview see Ref. [62]). Two are the keywords to understand this device: firstly,magnetic flux quantisation; secondly, Josephson effect [63].

43

The first one is a well known effect in superconductivity: as long as the

superconductor remains in superconducting state, the magnetic flux is trapped

within it. The magnetic flux is quantised Φ0 = hc2e , so that the change of the

flux can not be a continuous function.

x

x

Josephson

Junctions

Pickup

coil

SQUID loop

Feedback

signal

�I�V

��

Figure 4.5: Main components of a SQUID. The sample oscillates through the pick-upcoil.

The second effect accounts for the tunnelling of the superconducting cur-rent through a non-superconductor barrier without any voltage drop if the tun-nelling barrier is shorter than the coherence length (ξ ) of the Cooper pair andthe current is smaller than a critical one Ic. The tunnelling current, in presence

of a magnetic field, is generally written as:

I = I0sinπΦ/Φ0

πΦ/Φ0

where Φ is the total magnetic flux trapped in the junction. The effective

principle of a dc SQUID is really simple. The supercurrent in the SQUID

loop is above the Ic for the Josephson barrier. The magnetic sample oscil-

lates through a pickup coil and the generate current transferred to a input coil.

This is placed close to the SQUID loop and induces a magnetic flux Φ in theSQUID loop. A change in Φ induces a phase change in the superconducting

current. As Φ increases/decreases, the voltage at the Josephson barrier fol-lows the field. This voltage is used as a feedback voltage that nulls the fluxthat penetrates in the SQUID loop.Among all the uses of superconducting loops, magnetometry or suscep-

tometry are the main uses for the magnetic characterisation of materials. In

practice, the sample that one wants to measure, is magnetised inside a magnet

and oscillated through detection coils, whose shape and setup are depending

on the different system. The oscillations give rise to a change of flux that is

proportional to the magnetization M of the sample. This flux is continuously

measured by the SQUID loop. Thereby, the peak-to-peak signal, which is a

voltage change at the Josephson contacts, is proportional to twice the M.

44

0 50 100 150 200 250 300 350 4000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Temperature (K)

Magn

eti

c M

om

en

t (μ

B/

Fe)

NG

GR+

GR-

Figure 4.6: Squid data of gradient samples and non-gradient (Paper IV). If the sam-ple is known, one can measure the effective moment for magnetic moment carrier.

The effective measured values of the SQUID, which was used to carry out

our magnetometry measurements, are expressed in terms of emu3, that mea-sures the total moment of the sample. By knowing the composition and thenumber of carriers of magnetic moment, one can extrapolate the magneticmoment for carrier. Such measurements are, in any case, affected by errors inthe determination of the number of carrier. Such errors come from the non-accurate measurements of either the size of the sample or the weight of thesample in case of powder sample. Otherwise, the sensitivity of a commercialSQUID is on the order of 10−8 emu.

4.3 Polarised neutron reflectivity

A tool that probes the layer-by-layer magnetization at the most relevant lengthscales is neutrons reflectivity. Neutrons are widely used to investigate mag-netic structures in thin film [64], dynamic in thin films [65], superlattices[30, 66, 67] and superstructures [68].

The huge difference among neutrons and either photons or electrons is themuch weaker interaction with matter and the difficulty to produce those. Thisclaims for the access to big international facilities such as ILL.Neutrons have no charge but they have:

• a de Broglie wavelength which can be adjusted to few Å, λ = hmnv

3emu - electromagnetic unit is equivalent to 1 emu ≈ 1021μB.

45

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Q (Å-1)

Lo

g 10In

ten

sity

(A

rb. U

nit

s)X-Ray (Cu�1

)

Non-polarized Neutrons

Q1

Figure 4.7: Comparison between X-ray reflectivity and neutron reflectivity in the

case of a Fe/V(001) super-superlattice sample which was measured at ADAM beam

line at ILL.

• a momentum p = h/λ = hk

• magnetic dipole moment given by −γμnσ (σ Pauli spin operator)• the kinetic energy of a thermal or cold neutron is in the same range of theenergy of phonons or magnons.

It follows that the neutrons are subjected to nuclear force and magnetic field

B. Also, at certain energies, the neutrons can be captured by the nuclei givingrise to a nuclear reaction such as, for example, the nuclear fission of U235 of

Pu239.Because they exhibit the duality wave-particle, neutrons can be considered asa wave and the X-ray scattering theory which has been described in the pre-vious chapters, can be directly applied to the neutrons scattering. The onlyhypothesis are that the neutrons should be describe by a plane wave and theinteraction is weak, that is, the neutron beam is not fully reflected by the mat-ter. In order to access to the correct de Broglie wavelength (about some Å), theneutron beam should be “cold” or “thermal”, that means the average energyhas to be about hundreds of K which is about between 5 and 100 meV. Anexample of neutrons reflectivity is shown in Fig. 4.7.

Both neutrons and X-ray are diffracted in the same way, but neutrons allowthe identification of peaks that can be hidden in the large hump at small Qbecause of charge scattering. For example, in Fig. 4.7, the low Q < 0.2Å−1region, the X-ray reflectivity is dominated by a huge hump. This is due to

46

the huge scattering cross section of Pd, which caps the sample, in the X-ray.

On the other hand, the Pd is almost transparent for neutrons and this allows

the observation of two other peaks for Q < 0.2Å−1. Neutron reflectivity datacan be used as a complementary tool with respect to X-ray reflectivity to in-

vestigate the structure of a multilayer. Also, this becomes the master tool to

investigate multilayers whose layers have almost the same electron density.The interaction of neutrons with the atoms is generally written in terms of

Fermi’s pseudo-potential V±: [69]

V±(r) =2π h2

m(bn±bm)δ (r) (4.3)

where m denotes the mass of the neutron, bn the nuclear scattering length,

and bm the magnetic scattering length. The nuclear scattering length bn is de-pendent on the isotope and on the energy of the incoming neutrons and themagnetic scattering length bm is proportional to the magnetic moment.The superscripts highlight that the scattering potential is different for neu-

trons that are aligned parallel to the atomic moment (+) or are aligned anti-

parallel to the atomic moment (-).In a first approximation the neutrons refractive index n is equivalent to

n = 1− 1

2πρλ 2(bn±bm) (4.4)

where ρ is the number of nuclei per unit volume, and λ is the wavelength

of the incident neutrons [69]. Only the component of the magnetization on

the plane contributes to the refraction since in the most simple experiment the

sample is magnetised on the plane and it is perpendicular to the scattering

plane. By taking into account the Eq. 4.4 and the Eq. 4.3, it follows that the

refractive index is different for the neutrons (+) and neutrons (-). Like the X-

ray, neutrons are totally reflected by a flat surface if the angle of incidence is

smaller than the critical angle γc

γ±c ∼ λ (bn±bm)1/2

These different γ±c allow to select the neutrons with the selected spin by glanc-ing incidence reflection on multilayer mirror. The spin state is kept fixed by a

guiding field which is maintained along the path of the incident and scattered

beams. For example, the selected neutrons are (+) and they can be flipped to

(-) by a flipper which produces a radio frequency field perpendicular to the

guiding field.

Polarised neutrons scattering (PNR) allows the measurements of the mag-netic moment of the atoms at the most relevant length scales and it is self-calibrating since the spin-dependence of the reflectivity yields the total mag-netic moment of the layers [70].

47

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Q (Å-1)

Lo

g 10C

ou

nts

(A

rb. u

nit

s) R++

R--

NG

GR+

GR-

Figure 4.8: PNRmeasurements of three samples (see Paper IV). The different reflec-tivity from (+) and (-) are clearly visible as splitting between the scans. This splitting

is proportional at the magnetic moment probed at that length scale.

For example, in Fig. 4.8 the reflectivity measurements on three samples (seePaper IV) are shown. The (+) reflectivity is clearly different with respect to(-). This difference is called “spin-asymmetry”, S: [71]

S =R++−R−−

R++ +R−−

S is proportional to the amount of moment probed by neutrons at that lengthscale. In this specific case, the average bn for V takes on a negative valuewhereas the average bn for Fe is positive. By fitting these data by using thesoftware GenX, we can extract the magnetic moment per carrier. As opposedto SQUID, PNR can more precisely measure the magnetic moment since thetechnique is element selective and the measurements are not influenced at allby the substrate [72].

48

5. Order-disorder transitions anduniversality classes

Phase transitions phenomena occur in front of our eyes everyday: the boilingof water and the melting of ice are just two of such examples. Water undergoesa structural phase transition when it goes from the liquid to gaseous state, orwhen it goes from the solid to the liquid state. All kinds of phase transitionsoccurs in many other systems, and typically they involve some kind of changefrom an ordered state to a disordered one and viceversa. Examples includetransitions to superconducting, superfluid, and magnetic state [73, 74, 75, 76,77]. In this chapter we will focus on magnetic phase transitions a.k.a magneticorder-disorder transitions.

There are two milestones for the addressing of the critical phenomena.Firstly, the the scaling hypothesis by Widom [78, 79] and Kadanoff [80, 81].Secondly, the application of the renormalization group theory (RG) by Wil-son [82, 83, 84, 85, 86].

Within the framework of the RG, it is possible to quantitatively describe thecritical behaviour of, for example, liquids and ferromagnets, when they areclose to the critical point. In this sense, the theory of critical phenomena isuniversal and can be applied to any branch of physics in which phase transi-tions are relevant. In the rest of the chapter, order-disorder phenomena will bediscussed in connection with the experimental results both from literature andour works.

5.1 Magnetic order-disorder transitions

Order-disorder phenomena are defined by their own order parameters whosevanishing at some point in the phases space defines the critical point.In the special case of ferromagnetic order-disorder, the order parameter is

the spontaneous magnetisation M

M(T ) = limH→0+

M(H,T )

whereas in the antiferromagnetic case, the order parameter is the sub-latticemagnetisation M′ which is defined by the long range spin correlation: [74, 75]

limH→0+ |〈S0Sr〉|= (〈S0〉)2 ∝ (M′)2

49

In order to calculate the mean energy E, the Gibbs free energy G and the

Helmotz free energy F , the starting point is the definition of a Hamiltonian.The Hamiltonian of an ensemble of interacting spins defined in a lattice, H ,

is written as;

H = J ∑〈i, j〉

�Si · �S j− �H ∑m,n

�Szm,n

This is the Heisenberg classical Hamiltonian. The first term accounts for theinteraction among first-neighbour spins by the coupling constant J. The sec-ond term takes in to account the presence of a magnetic field H, which gener-ally is pointing on the z direction. Statistically, the partition function of such asystem is defined by

Z = ∑Si

e−H /kBT

where kB is the Boltzmann constant. The partition function is the only quan-

tity that is required to calculate all the thermodynamic functions of a system.

These include

• E = 1Z ∑i Eie

Ei/kBT , the Energy

• F = E−T S =− 1kBlogZ, Helmotz free energy

• G = F +∑i PiVi =− 1kB

∑i∂

∂Vi(Vi logZ), Gibbs free energy

All the other thermodynamic quantities can be calculated by knowing the re-

lation with one of the previous defined energies.A series of universality classes can be defined by taking in to account

a) spatial dimensionality D, depending on whether one is addressing ques-tions about dots (D = 0), chains (D = 1), planes (D = 2), volumes (D = 3).

b) the dimensionality d of the order parameter.

The universality classes which are defined by the dimensionality of d, arehistorically defined as

- Ising - one-dimensional spin.- XY - two-dimensional spin.

- Heisenberg - three-dimensional spin.The magnetic phase transitions described above, are knows as second orderphase transitions1 2, according to the classification given by Ehrenfest, or

1First-order phase transitions exhibit a discontinuity in the first derivative of the free energywith a thermodynamic variable2Second-order phase transitions show a discontinuity in a second derivative of the free energy.These include ferromagnetic phase transitions

50

continuous3, according to more recent classification [74]. The order-disorder

point is defined by the thermodynamic point where the second derivative of

the thermodynamic potential diverges.

Ferromagnetic (FM) order-disorder is characterized by the following con-ditions:

- long range order (LRO) in FM state

- short range order (SRO) in paramagnetic state

- critical point that defines the critical temperature Tc.

A way to understand such a system is by thinking to be at 0 K and to deal witha planar system of spins which can point in just two directions, for example,either up or down with respect to the plane. This ensemble shows a spon-taneous magnetisation M which implies long range order (LRO) among the

spins, all of them pointing in the same direction. Now we increase the temper-

ature up to the critical temperature Tc and the M vanishes. At and above this

temperature, the system switches from being fully ordered to a condition that

is dominated by short, local, range order (SRO).In particular, this means that the local fluctuations increase dramatically in

the vicinity of the critical temperature and destroy the LRO. For example, fluc-

tuations of the magnetisation may reach macroscopic magnitudes, as long as

the size of the system, and, correspondingly, the second-derivative of the free

energy becomes very large and may even tend to infinity for some wavelength.

Order-disorder phenomena are therefore focused on investigating the be-haviour of the order parameter and the behaviour of the other thermodynamicquantities close to the critical point. Here, one introduces the scaling hypoth-esis [78, 79, 80, 81]: in this framework it is assumed that any critical quantityscale as (Tc− T )λ for T → Tc. Therefore, the real task is to determine the

quantity λ , which is called the critical-point exponent.In first approximation, a good function f (x) that fulfils the scaling hypoth-

esis islim

x→0+f (x)∼ xλ

The critical exponent λ is therefore defined as

λ = limx→0+

log( f (x))logx

Away from the critical point, this approximation does not hold. It follows thathigher orders of correction are to be added [87], for example, the magnetisa-tion may be written as

M(T ) = B(Tc−T )λ [(1+bθ (Tc−T )θ +b1(Tc−T ) . . .],T < Tc

3Continuous phase transitions are described by a continuous first derivatives of the thermo-dynamic potential while the second derivatives are diverging or discontinuous. No latent heat

is associated with these transitions.

51

–1.0 –0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0.0 0.1

t

Magn

eti

zati

on

�=0.125

�=0.23

�=0.37

�=0.5

Figure 5.1: Different spin distributions (in the insets) and order parameter M(t). TheLRO at 0 K switches to SRO above t = 0. Inset: the full dots represent the 2 different

signs of the spins.

where λ and θ are universal exponents, whereas the other quantities are notuniversal. (see Ref. [88] and references there in)

A number of common critical-point exponents are defined, for example α ,β , γ , δ , ν , η . Each one describes the behaviour of a physical quantity whenthe system is coming close to the critical point [89]. For magnetic systems,

one defines the reduce temperature

t =T −Tc

Tc,

and the following expressions for the critical exponents can be written down:

- CV ∼ (t)−α , t → 0,H = 0,M = 0

- M ∼ (−t)β , t → 0−,H = 0

- χ ∼ (t)γ , t → 0,H = 0

- H ∼ |M|δ sgn(M), t = 0 critical isotherm

- ξ ∼ (t)−ν , t → 0, H = 0 correlation length

- Γ(r)∼ |r|−(d−2+η), t = 0, H = 0, M = 0 (pair correlation function)

Figure 5.1 shows an example of a spins system. At 0 K, all the spins point

out of plane (full circles) whereas a the t = 0, the net spontaneous magneti-sation is zero (same amount of full and empty circles) but the local order is

52

given by the local state of the spin at each point. The different β are related todifferent spin-dimensionalities.But, the application of the RG theory [84, 83, 82, 85, 86, 88], shows that

knowing only two of the exponents and the spatial dimensionality of the sys-

tem is sufficient to calculate all the others. This property of the critical expo-

nents gives rise to a series of scaling relations [90]. With these foundations,

one can analyse the universality classes of the ferromagnetic order-disorder

phenomena.

5.2 1D universality class: Ising model

The Hamiltonian of an Ising system is defined by

H = J ∑i, j

�Si · �S j (5.1)

with J the strength of the coupling. The �Si can have only 2 values, for exam-ple, either +1 or -1. It takes the name from Ising [91] who had studied thissystem for his PhD. He showed that a spins system that is described by theHamiltonian in the Eq. (5.1), could not show any long range order in the limitof one spatial dimension: the work of Ising was done on a spin ring. There-fore, such a system did not go through any phase transition. This result wastaken by the physics community to imply that such a system could not showany phase transition in any other order of spatial dimensionality.In 1944 Onsager [92] derived the exact partition function of the Ising model

in two dimensions. By knowing the partition function Z of a statistical system,F, G and CV can be calculated. Onsager’s solution allows the calculation of

the exponent α , which is related toCV :

CV (T ) = kB2

π

( 2JkBTc

)[− ln

(1− T

Tc

)+ ln

(kBTc

2J

)−

(1+

π4

)]

Thus, the heat capacity can be seen to diverge logarithmically as T → Tc,that is α = 0. The calculation and the mathematics of Onsager solution arecomplex, but his method was simplified by Kaufman [93] and it allowed tocalculate the short-range order in crystal lattice [94].The exact solution for the critical exponents β , which is related to the spon-

taneous magnetisation M, was found by Yang [95] in the 1951. An exampleof exact solution of such a spin system in a squared lattice is given by theEq. 5.2;

m =[1− (1− tanh2K)4

16tanh4K

] 18

[96]

53

K =J

kBT

which behaves as (T −Tc)18 per T → Tc.

By applying the scaling relations and the RG theory, the exact solutions for

the other critical exponents can be calculated and it follows that:

α = 0, β =1

8, γ =

7

4, δ = 15

For 3D Ising systems, the exact partition function, at the present time, has notbeen computed. A modern way to compute the critical exponents for such asystem is by using Monte Carlo calculations to simulate the behaviour of aset of a finite number of spins. The magnetisation M is thus seen to approach

the critical point with a power law decay whose exponent is in the order of

β = 0.3267(10) [97]. Other exponents that have been computed by MonteCarlo calculations include:

α = 0.110, β = 0.3267, γ = 1.237, δ = 4.786 [97]

Several uniaxial antiferromagnets and ferromagnets have a 2D Ising

behaviour [98, 99, 100, 101]. Also, non-magnetic systems such as absorbed

gases on surfaces (references in Ref. [77]) show the behaviour of an ideal

2D Ising order-disorder transition. The 3D Ising order-disorder transition

shows a magnetic behaviour that is much closer and indistinguishable

from the 3D-XY and 3D-Heisenberg because the approximated β values

fluctuate among β ≈ 0.31 and β ≈ 0.37 (such values depend on which kindof computational method has been used). An example of a real 3D-Isingsystem has been given by Li et al. [102]. The authors observed a dimensionalcross-over from a 2D-Ising to a 3D-Ising by measuring the exponent β .Assuming the spin dimensionality D does not change during the cross-over,

then the experimental value should belong to the 3D-Ising class.

5.3 2D universality class: XY model

In the XY model the dimensionality d of the spins increases to 2 and the spinscan freely rotate on the plane of the lattice. H is then written, in absence ofmagnetic field, as:

H =−J ∑<i j>

�S j ·�Si =−J ∑<i j>

cos(φi−φ j) (5.2)

In a such system made of infinite number of spins, it has been rigourously

shown that no long range order can exist [103, 104, 105] at any temperature

above 0 K. It follows that no ordinary phase transition is possible, specifically,

no magnetic long range order at any temperature above 0 K occurs.

54

However, if the size of the system is finite, then a spontaneous magnetisa-

tion M is possible above 0 K. This has been found in many real systems thatcan be described by the XY Hamiltonian, Eq. (5.2), such as those describedin References [106, 107, 108, 109].

0.0 0.5 1.0 1.5 2.0

Magn

eti

zati

on

2N=102N=1002N=10000

t

Figure 5.2: Magnetisation decays very slowly in terms of temperature t = kBT8πJ and

system size 2N, see Eq. 5.3.

The spontaneous magnetisation M, even if forbidden by the Mermin-Wagner theorem [103], is possible since the thermodynamic limit for M is

approached very slowly [110]. The magnetisation can be written in terms of

temperature and system size: [111]

m(N,T ) =( 1

2N

) kBT8πJ

(5.3)

where N is the number of spins and T is the temperature and J the couplingstrength among the spins. The magnetisation is thus the key-signature of finitesize 2D-XY behaviour, Fig. 5.2The exponent β has been calculated by Bramwell and Holdsworth [112]

β =3π2

128≈ 0.23

This critical exponent has been found in various systems [113, 114], whilst

this is not a critical exponent in the strict sense, in practice it treated as such

in experiment.

55

Figure 5.3: Bounded vortex and antivortex (Courtesy A. Taroni).

In the thermodynamic limit, the 2D-XY model exhibits some unique prop-erties. For instance, it undergoes a topological transition [115, 116, 117] withthe creation of bounded pairs of spin vortex and anti-vortex, Fig. 5.3The topological order is broken at the temperature where the pair of vortices

unbinds. Such a Kosterlitz-Thouless temperature TKT is given by

πJkBTKT

≈ 0.12

where J is the coupling strength among the spins. Above TKT free vortices

are possible. In this framework, the exponent η is found to depend on thetemperature:

η =kBT4πJ

. (5.4)

This is an unusual situation, because η is related to the correlation functionΓ(r) ∼ |r|−(d−2+η), and it normally only defined at Tc. Thus, Eq. 5.4 implies

that the 2D-XY model is critical over a range of temperatures. However, this

situation cannot persist up to arbitrarily high T , because the model must even-tually enter in a high T paramagnetic state, where the correlation decays ex-ponentially.

From an RG analysis [118], it is possible to show that at a temperature

Tc =πJkB

,

the continuum of critical point ends abruptly, with

η(TC) =1

4.

56

At this temperature, a number of other exponent may be determined, using the

scaling relations relations. For example

δ = 15

The exponent β for the 2D-XY model holds in a region

limT→T �

m(T )∼ (T −Tc)β [119]

where T � is a temperature close to TKT with the scaling given by

Tc(L)−TKT = 4[T �(L)−TKT ] =π2

c ln2L

That is, if one plots the lnm vs ln t, the result should be a straight line for allthe temperatures.

Figure 5.4: (From I) Remnant magnetisation versus reduced temperature for a seriesof δ -doped Pd/Fe layered magnets. The straight lines in connection with each datasetrepresent the slope in the fitted interval (between the two dashed lines), the exponent

β . The datasets are offset for clarity.

This results are shown in Fig. 5.4 from Papers I. The straight lines whoseslopes are β ≈ 0.23 fit very well on the experimental data set for the entireset of reduce temperatures while those whose slopes β are higher are fitted tothe line in a smaller region. The universal properties of a 2D-XY system aresummarised in Ref. [119].

Layered magnets with either no coupling or weak coupling among the mag-netic layers are able to display a 3D-XY behaviour. The magnetic excitations

57

Figure 5.5: Exponent β versus dFe for all samples in this study, as determined by

direct fitting and by double logarithmic plotting. The dashed horizontal lines represent

β of the 2D XY, 3D Ising, and 3D Heisenberg models. The solid line serves as a guideto the eye. Typical uncertainties in β and thickness are indicated.

are confined in the magnetic layers and can weakly interact with the othermagnons in the magnetic neighbour layers. The M in spin waves approxima-

tion for such weakly coupled 2D-XY layer is written by

M =

(J⊥J‖

) kBT8πJ⊥

[120]

where J⊥ is the coupling strength out-of-plane (or interlayer exchange cou-pling J′) and J‖ is the intralayer coupling strength.For the 3D-XY model there are just numerical solutions for the critical

exponents. The β exponent takes on a value ≈ 0.34 [121], being calculatedby Monte Carlo simulations. Many systems seems to share this universality

class [77, 112, 113].

Figure 5.5 is a compelling example of the crossover between 2D-XY to 3D-XY. In such a case, by increasing the Fe layer embedded in Pd, the magneticeffective thickness increases, giving rise to magnetic excitations out-of-planewith the in-plane magnetisation. The same cross-over has been observed byHo et al. [122].

58

5.4 3D universality class: Heisenberg model

The Heisenberg model is the most general one because it accounts for the

spins which can rotate on a sphere. The H is given by, in absence of anymagnetic field,

H = J ∑〈i, j〉

�Si · �S j

with the spins �Si free to rotate in the sphere: in practice, such a H describesa isotropic magnet with no disorder. No exact solution for the critical expo-nent for order parameter M exists as well as for the other critical exponents.Such exponents can be calculated by Monte Carlo simulations or high tem-perature expansion in the framework of renormalization group. In Ref. [77],a list of calculated exponents is reported, together with the experimental one:the average calculated β exponent is on the order of ≈ 0.37, which is similarto those one for the 3D-Ising and 3D-XY. One special case that can be affil-

iated with the Heisenberg model, is the case of spherical model [123], which

is the only solvable model in presence of an applied field. For D ≤ 2, thismodel does not show any phase transition, fulfilling the Mermin-Wagner the-orem, while at higher order of spatial dimensionality such a model displaysthe ferromagnetic transition at Tc. The critical exponent are function of D, forD < 4, thereafter they are independent, with β = 0.5. Such a model holds inany dimension and the solutions for the exponents are the same such as those

of the mean field model.

5.5 Extra universality class: Surface

The breaking of symmetry at the edge of a finite system can be seen as an

additional degree of freedom. If one considers a layered magnet, the out-of-

plane magnetic oscillations are confined in a shorter space, then just few of

them can be excited. The order-disorder at the surface becomes an extra uni-

versality class. This idea was developed by Binder who calculated the order-

disorder transition for an Ising system on the surface both exactly [124] and

by Monte Carlo calculation [125, 126, 127, 128, 129]. It is shown that the

magnetic surface is critical at a temperature T sur fc which can be either higher

or smaller than the Tc of the bulk. Such a T sur fc depends on the exchange cou-

pling J among the spins. The critical exponent β for a surface is equal to 1

and it collapse to the 3D-Ising β at some distance from the surface. Also sucha distance depends on the coupling J. There are some experimental exampleabout the exponent β , for example in References [130, 131, 132] much closerto 1.One way to explore the surface order-disorder is by using MOKE. Because

of the penetration of the light in metal is reduced to few 1-200 Å with ex-ponential decay, and because of the Kerr rotation is a summation of the Kerr

59

6 7 8 9 10 11 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

V Thickness (ML)

�

� direct fit

� log-log

Figure 5.6: Exponents β from Paper III. These seem to be closer to the theoreticalexponent for a surface magnetic transition than for a 3D-universality class.

rotation of the single layers, then the highest contribution to the Kerr rotation

is given by the surface. In Fig. 5.6, the exponents β for a series of Fe/V(001)superlattices are shown. None of them seems to belong to any of the univer-sality classes previously described. The exponents seem to be close to thosemeasured for Gd in Ref. [130, 132]. This highlights the importance of thesurface in the overall framework of the order-disorder magnetic transitions. Itfollows that the symmetry breaking of the surface generates another univer-sality class.

60

6. Oscillatory interlayer exchangecoupling

The oscillatory interlayer exchange coupling (IEC) between magnetic lay-ers plays an important role in modern technology. The discovery of a giantmagneto-resistence (GMR) [133, 134], caused by different ohmic resistancesfor spin-currents [135, 136, 137], allowed for the development of new deviceswhich are sensitive to magnetisation: a classical example is the hard-disksused for data storage.

The first evidence of IEC was found in rare earth multilayers [138, 139]while the first evidence in transition metals was found by Grünberg et al. [140]in Fe/Cr/Fe multilayers. In this latter case, antiferromagnetic interlayer inter-action, decaying regularly in strength with increasing thickness of Cr spacer,was observed. Such a ferro- or antiferro-magnetic interaction between twomagnetic layers separated by non-magnetic spacers is explained in the frame-work of the RKKY indirect exchange interaction (Rudermann-Kittel-Kasuya-Yosida) [141, 142, 143] modified by taking into account the discrete numberof layers.

6.1 The RKKY framework

The Rudermann-Kittel [141] interaction was previously introduced to explainthe indirect exchange coupling between nuclear magnetic moments by themeans of conduction electrons. Kasuya [142] successively discussed the ex-change interaction between conduction electrons and localised d-electronsand Yosida [143] addressed the problem of exchange interaction among s andd electrons. Such interaction describes an oscillating polarisation, rapidly de-creasing with the distance from the magnetic nucleus and concentrated close

to the magnetic impurity. These indirect interactions were proposed in order

to explain the broadening of hyperfine absorption in magnetic alloys and mag-

netic impurities in non-magnetic metals.

The first correct RKKY theory which addressed the IEC in multilayers wasdone in 1991 by Bruno et al. [144, 145]. The IEC was theoretically shown tobe oscillating as a function of discrete monolayer (ML) of the spacers. Thiswas in agreement with the original work of Grunberg et al. [140].

J′ = ∑α

Aα

D2sin(qαD+φα)[146] (6.1)

61

J'>0

m

mJ'<0

m

-m

Figure 6.1: On the left: the two magnetic layers are ferromagnetic coupled by a

positive IEC. On the right: the two magnetic layers are antiferromagnetic coupled by

a negative IEC. The thickness of the spacer changes the coupling sign and strength.

In Eq. 6.1, J′ is the IEC constant, Aα is the strength, φα a phase and D is thethickness of the spacer. The coupling decay is a function of D−2 as opposedto the the RKKY interactions which are dependent on D−3 [147].Figure 6.1 shows the classical building block used to understand the IEC:

one non-magnetic spacer is sandwiched between two ferromagnetic layers.The ferromagnetic layers induce a spin-polarisation inside the spacer and sucha polarisation extends and decays in the spacer. Eventually the polarisation canreach the second magnetic layer: thus, the two magnetic layers can indirectlyinteract. The spin-polarisation at the interface changes the electronic reflec-tivity indices in the spacer and these indexes are spin-dependent. It followsthat there are two different indexes for electrons with either spins parallel oranti-parallel to the magnetisation. In the spacer the electrons are reflected atboth interfaces: the interference between different waves gives a final wavefunction for the electrons in the spacer. This wave function gives a contribu-tion to the density of states which contributes to the total energy. This smallcontribution is the interlayer exchange coupling that is dependent on:

- the thickness of the spacer

- the wave function of the reflected electrons

- the spin-asymmetry of the reflectivity indexes at the interface of themagnetic-nonmagnetic layers [148, 149, 150]

The crystal orientation also plays a role in the determination of the strength

and sign of the IEC as pointed out by Stiles [151]. Because the electronic wave

functions are dependent on the crystal direction, the oscillatory IEC does the

62

same: reviews on the oscillatory IEC are found in References [150, 146, 152,

153, 154].The IEC has been found in many systems: for example Fe/Cr [140],

Co/Cu/Ni [155]; Fe/Pd and Fe/V(001) as discussed here.The effect of the IEC on the magnetic order-disorder of magnetic multi-

layers is given by the increase of the ordering temperature Tc in the case offerromagnetic coupling, while in the antiferromagnetic state the marker of thestrength is the coercive field [40].The IEC does depend on the temperature, as experimentally shown by Lind-

ner et al. [156]. This fact was later explained in terms of magnetic excitationsor spin-waves [157, 158] which dominate the strength of the IEC at tempera-

ture well above 0 K, eventually decaying to zero as a function of the tempera-

ture,

J′(T )≈ 1−α

(TTc

)3/2

Such behaviour is the same of magnetisation in proximity of 0 K. In particular,the larger the intralayer coupling strength, the weaker the decrease of J′ withT [159]. This highlights the importance of magnetic fluctuations on nano-magnetism [160].

Seeing that the IEC depends on the density of states at Fermi’s energy then itshould be possible to tune the strength and sign by changing these states. Thealloying of the non-magnetic spacer with H is one of the possibilities that hasbeen explored, such as in Fe/Nb superlattices [161], Ho/Y superlattices [162,163] and Fe/V(001) [41, 164, 165, 2].

Also, the roughness and interdiffusion at the interfaces between the spacersand the magnetic layers reduce the strength of the oscillations. In particular,the reduction is stronger for the short period oscillation without depending onthe thickness of the magnetic slabs. Interdiffusion plays the most relevant rolein the quenching of the periodicities [166].

6.2 Exchange coupling in Fe/V(001) superlattices.

The IEC in Fe/V(001) superlattices was theoretically investigated for the firsttime in 1993 [167]. This theoretical work, which was based on the trilayerFe2VxFe2, stated a double periodicity of 3 and 11 monolayers of V. The

strength and multi-periods in two different lattice orientations were, later,

shown to be different because Fermi’s surfaces are different for the two di-

rections of the lattice [168]. More recent theoretical work [169] shows an

oscillatory behaviour of the IEC rapidly decreasing as the V spacer thickness

increases, inset in Fig. 6.2. The IEC exhibits many local minimum values and

antiferro-magnetic couplings but, above 5 ML of V, the strength oscillates

around the zero value.

63

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16–0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

V thickness (ML)

J' (m

J/

m2)

Figure 6.2: Experimental IEC in Fe/V(001): the data are from, respectively, Vohl et.al [38] (squares) and Broddefalk et. al [39] (circles) Inset: Theoretical IEC betweentwo semi-infinite Fe slabs separated by n monolayers of V [169] (Courtesy from theauthor). The IEC is rapidly decreasing and oscillates around the 0 above 5 monolayers

of V.

Three antiferro-magnetic peaks in Fe/V(001) superlattices have been exper-imentally found at the thickness of 14, 21 and 28 monolayers of V [40, 34] asopposed to the theoretical ones. Another work [39] claims that antiferromag-netic range depends on the thickness of Fe with the maximum AFM couplingat 13 ML of V and 6 ML of Fe. The superlattices [Fe2/Vn]25 (5≤ n≤ 17) havebeen investigated by Eftimova et al. [46]. They stated that the AFM range isbetween 12 and 15 ML of V, with no coupling at 11 ML of V and maximumAFM strength at 13 ML of V.The IEC in Fe/V(001) superlattices has been also theoretically investigated

to define the roles of interdiffusion and roughness on the magnetic order-ing [50]. It is shown that the most important role is played by the roughnessthat quenches the short period oscillation in amplitude and changes the periodof oscillations. These effects can be seen by comparing the theoretical datawith the experimental ones. This is the main reason why the first experimentalAFM peak is at about 13 ML of V.The IEC can be tuned by introducing H in the spacer [41, 165, 163, 2]. The

addition of electrons in the spacer changes the density of states of the spacerand therefore the IEC can be adjusted at will.

64

7. Magnetic ordering in layeredmagnets.Summary of papers

The universality class of a series of layered magnets, namely PdFexPd 0.2 ≤x ≤ 1.6 ML, has been demostrated to be dependent on the effective magneticthickness of polarized Pd, Paper I.In the RKKY framework, the IEC (generally defined by J′) in Fe/V(001)

thin layered film is discussed, after having established the inherent magneticorder. The tuning of the IEC is shown by the change of the density of states.The IEC affects the ordering temperature while the universality class remainsthe same, Paper II.Evidences of non-universal class is shown in a series of Fe/V(001) super-

lattices highlighting the role of the magnetization at the surface, Paper IIIThe inherent ordering and the position of the magnetic layers, with J′ fixed,

is shown to play a role in magnetic order-disorder transitions, Paper IV.The effect of boundaries, such as the broken symmetry at the surface, brings

attention on the magnetic ordering at different length scales, Paper V.

7.1 Paper I: Dimensionality crossover in the inducedmagnetization of Pd layers.

Because of the proximity of Fe, Pd acquires a magnetic moment. Thus, themagnetic order-disorder transition can be investigated in sub-monolayers ofFe embedded in Pd. The magnetization, probed by MOKE, is due only toPd. Because all the imperfections at the interface are washed out by the largepolarization, extending until ≈ 10 ML of Pd, the effective magnetic thicknesscan be regarded as smooth and extending well above the sub-monolayer thick-

ness of Fe. A universality class cross-over from a 2D-XY to a 3D (Fig. 5.5) is

established at some thickness of Fe. This observation is explained in terms of

the creation of out-of-plane magnons whose ground state can be excited only

if the critical temperature of the 2D-XY slab is high enough.

65

Figure 7.1: From Paper II: Critical temperature vs √p for the bilayer sample[Fe3V14.4]2. Inset: Critical exponent β vs

√p. The two horizontal lines represent β

of the 3D Heisenberg- (top) and 2D XY- (bottom) models.

7.2 Paper II: Oscillatory exchange coupling in the twodimensional limit.

The inherent ordering of a single slab of Fe embedded in V and the orderingof two weakly coupled slabs of Fe have been established as belonging to the2D-XY universality class.Despite this, the magnetic ordering in layered magnets can be altered by

changing the density of states of the spacers by reversibly alloying them withH. In Fig. 7.1, the results of the measurements of the ordering temperatureTc for two weakly coupled Fe layers on H uptake are shown. Oscillatory be-

haviour in the Tc is the evidence of the change on the strength of the IEC whilethe overall universality class is not affected at all by the VHx alloy, see inset.

Furthermore, the H alloying allows the tuning of the magnetic moment ofboth Fe and V. In Fe/V(001) superlattices, the interface plays the role of tuningthe moments of Fe and V. The Fe moments are reduced while the V momentsare increased and antiparallel to the Fe atoms [170, 34, 171, 172, 173, 174].When H is alloyed with V, the Fe moment increases towards the bulk valuewhile the V moment turns back to zero, as has been experimentally [175] andtheoretically confirmed [176].

66

7 8 9 10 11120

140

160

180

200

220

240

260

V Thickness (ML)

TC

(K

)

J’

TC

Fitted Eq. 4

Figure 7.2: From Paper III: Tc versus V thickness in series of [Fe3Vx]20, 7≤ n≤ 11superlattices. The regular decay of the ordering temperature follows a logarithmic law,

with zero coupling for � 11 ML.

7.3 Paper III: The influence of interlayer exchangecoupling on magnetic orderign in Fe/V superlattices.

The interplay between magnetic ordering and IEC has been explored in a se-ries of superlattices, namely [Fe3Vx]20 with 7 ≤ x ≤ 11, in order to map outthe intensity of J′ and TC versus the thickness of V, from the strong coupledregime to uncoupled regime.

Results of the measurements of the Tc are shown in Fig. 7.2: it is clear thatthe change of Tc is not linear with respect to the spacer thickness. The single

slab of Fe can be considered as a 2D-XY magnet, and it follows that [112]

Tc = TKT +a

(lnL)2(7.1)

where TKT is the Thouless-Kosterlitz temperature and L is the size of the sys-tem. The Eq. 7.1, when IEC is introduced, can be written as

T (J′/J) = Tc,0+a

ln2(J′/J)

where Tc,0 is the value of a single, isolated Fe slab and J is the intra-layercoupling.

67

6 7 8 9 10 11 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

V Thickness (ML)

�

� direct fit

� log-log

Figure 7.3: From Paper III:β values for a set of [Fe3Vn]20, 7≤ n≤ 11, superlattices.These values are closer to those of a magnetic surface (page 59) than those of a 3D-XY system (page 54).

The spin-spatial dimensionality, which is defined by the β value of the mag-netization (page 52), of such a set of samples is unusual, being always thetypical values for an 3D-XY system (0.36).In the Fig. 7.3 these β values, determined in two different ways, are shown.

Such magnetic orderings are explained in terms of surface effect, (page 59).The magnetization is dominated by the surface contributions, whose β = 1,

because the magnetization is measured by MOKE (surface sensitive probe,

page 39).

7.4 Paper IV: The influence of the distribution of theinherent ordering temperature on the ordering in layeredmagnets.

A series of three samples was developed in order to address the influence of

the inherent magnetic ordering of individual magnetic layers on the overall

ordering and universality class.The IEC was fixed and two mirror symmetric samples with the Fe thickness

going from 2 to 3ML (and viceversa) were structurally and magnetically char-

acterized by SQUID, MOKE and PNR. As seen in Fig. 7.4, while the regular

superlattice shows a regular magnetic behaviour, with β ≈ 0.48, both of thegradient samples can not be included in any of the universality classes. This

68

100 150 200 250 300 350 4000.0

0.2

0.4

0.6

0.8

1.0

1.2

Temperature (K)

Ke

rr R

ota

tion

(a

rb.

un

its)

��

��

��

Figure 7.4: From Paper IV: Magnetic ordering in no magnetic field for a regularFe/V(001) superlattices compared with those of two gradient superlattices.

highlights the breakdown of the order-disorder modelling in the presence ofaltered inherent ordering of the magnetic layers.

The position in the superlattice of the weak link, that is the position of thethinnest Fe layer, plays a role in determining the overall ordering temperature.One can say that the magnetization starts to “melt” from this layer becausethe intra-layer coupling is smaller due to the missing coordination number.So, the sample GR- is seen to be melting from inside while the sample GR+is seen to be melting from outside.

7.5 Paper V: The effect of boundaries on ordering infinite magnets.

As stated in Paper III, the magnetic ordering in layered magnets depends onthe effects of the boundaries. In order to explore such effects, they should be

enhanced such that they are the main causes of the magnetic order-disorder

transition. Thus, a sample with enhanced boundaries has been developed and

investigated.

Figure 4.7 (not included in the paper) shows the the non-polarized neutronsreflectivity of the super-superlattice. Each individual peak is the Fourier com-ponent of the relative structural length scales of the sample. The peak Q7 isthe component relative to the bilayer thickness Λ1, while the peak Q1 is theconstructive interference from the length scale Λ3, given by the length of the

69

Figure 7.5: From Paper V: The effective exponents βi, corrensponding to the differ-

ent length scales probed by the experiments. Only the component Q1 resembles theexpected exponent for a 3D universality class.

magnetic stack and non-magnetic spacer. The Q2 is the Fourier component ofthe thickness of the magnetic stack Λ2. Such a sample has been magneti-cally characterized by means of PNR at ILL. The magnetic induction of eachFourier component can be directly mapped by the spin asymmetry, as definedby the difference in intensity of the two different neutrons spin channels. Bysuch measurements as function of the temperatures, both the Tc and the uni-versality class, given by the exponents β , can be determined.The results for the exponent βi, where the subscript indicated the Fourier

component, are shown in Fig. 7.5. It follows that only the first componentseems to belong to a 3D universality class, the dashed line, while the othercomponents are characterized by high β values, showing a magnetic meltingfrom the surface to the interior. This highlights the inherent importance ofboundaries on the order-disorder transitions. On the other hand, the overallorder-disorder transition point is well defined since all the ordering tempera-tures Tc,i collapse at the same value.

70

Svensk Sammanfattning

Människan har känt till magnetism sedan oräkneliga år. Redan de gamla grek-

erna kände till järnmineralen magnetits förmåga att dra till sig järn. Kom-

passen, som är en av de viktigaste uppfinningarna i mänsklighetens historia,

utnyttjar växelverkan mellan jordens magnetfält och en magnet. Den möjlig-

gör bland annat havssegling i stor skala. Även duvor är utrustade med några

korn av magnetiskt material i sina huvuden, som gör att de kan orientera sig

under sina resor. Dynamon utnyttjar växelverkan mellan ett magnetiskt fält

genererat av en magnet och en kopparspole för att skapa elektricitet.

Magneternas storlekar kan i variera i skala, alltifrån en enorm neutron-stjärna med sitt kolossala magnetfält till den enskilda järnatomen. I det förstafallet adderar ett stort antal partiklar sina magnetiska egenskaper för att ökamagnetismen. I det andra fallet fördelar sig elektronerna över bestämda banor,och varje elektron snurrar runt sig själv medurs eller moturs för att spara en-ergi.I likhet med duvorna kan man tänka sig att utnyttja en enda näve atomer för

att skapa en liten magnet. Vakuumteknologins framgångar börjar tillåta oss att

konstruera maskiner där det råder atmosfäriskt tryck på utsidan, medan trycket

på insidan är ungefär 1 000 000 000 000 gånger mindre! Genom förångning

av naturligt magnetiska grundämnen som järn, nickel och kobolt inuti dessa

maskiner, kan enskilda magneter förmås att växa till önskad storlek atom för

atom, lager för lager.Och deras tjocklek spelar roll!En tunn film kan idag ha tjocklek 4 Å, vilket är 10 000 000 gånger tunnare

än dagens mobiltelefoner, som kan ha en tjocklek på 4 millimeter. Med dessadimensioner är filmen betydligt mer magnetisk än mobiltelefonen. Och ävenom filmen inte är magnetisk vid rumstemperatur kan den vara det vid temper-aturer under nollstrecket, eller vid lägre temperaturer än de som kan uppmätasi Norrland, temperaturer då luften blir flytande.Tunnfilmsvetenskapen verkade tråkig fram till 1988 då 2007 års nobel-

pristagare Albert Fert och Peter Grünberg öppnade vägen till utvecklingen

av nya teknologiska prylar som Ipod, vilket är en hårddisk med utomordentlig

förmåga till datalagring, och magnetiska sensorer som exempelvis den elek-

troniska kompassen.

De två nobelpristagarna observerade att det elektriska motståndet mellantvå järnlager separerade av krom berodde på det magnetiska fält i vilket devar placerade, och att denna resistans blev enormt stort när två järnskikt rik-tade sina respektive magnetiseringar åt motsatta håll. I vissa fall bestämmer

71

sig faktiskt två tunna magnetiska filmer att ställa sig i motsatt riktning från

varandra.Den här effekten orsakas av Interlayer Exchange Coupling, utbyteskop-

pling mellan skikt, som studeras i den här avhandlingen, särskilt i det fall som

gäller för multilager av järn och vanadin. Det som man observerar är att mag-

netiseringen upphör vid temperaturer som varierar beroende på tjockleken hos

det vanadin som separerar järnlagren.

För att mäta magnetiseringen har man använt ljusets förmåga att identifieramagnetiseringens intensitet. Dessutom har dessa multilager undersökts medtekniker som utnyttjar växelverkan mellan magnetiska fält och supraledning.Båda teknikerna kan mäta medelmagnetiseringen och på ett ungefär atomer-nas enskilda magnetisering.

För att lära känna det exakta värdet av järnatomernas magnetisering harman använt kärnteknik, särskilt vad gäller växelverkan mellan neutroner (somtillsammans med protoner bygger upp atomkärnorna) och järnatomer. Det ärinte lätt att få tag i fria neutroner, därför har man utnyttjat tillgången till enstor facilitet som Institute Laue et Langvin (ILL), Laue och Langvin institutet,i Frankrike. Här produceras neutroner i en på ett adekvat sätt modifierad kärn-

reaktor som slösar lite energi. Samarbetet har gett optimala resultat som man

hoppas ska accepteras inom hela vetenskapssamhället.Vi återvänder till våra multilager, där tjockleken av det enskilda järnskik-

tet bestämmer temperaturen vid vilken magnetiseringen upphör. Detta bevisas

i två olika fall. I första fallet var järnet paketerat mellan två skikt av palla-

dium. Man har bevisat att ett enda skikt (och även mindre) av järnatomer

lyckas magnetisera palladiumatomer i hög grad. Med adekvata tekniker kan

man mäta magnetiseringen av endast palladiumatomerna.

I det andra fallet varierade tjockleken hos järn från lager till lager i ett mul-tilager av järn och vanadin. Positionen av de tunnaste järnlagren ger upphovtill två olika typer av magnetisering, beroende på om järnet finns i mitten ellerpå ytan av skiktet. Magnetiseringen av dessa tunna filmer har undersökts påalla skalor. Deras yta, deras volym och även varje enskild atom har studerats.

Vetenskapen om multilager är en exakt vetenskap: även den minimala skill-naden i tjocklek av en enda Ångström, mindre än ett lager atomer, förändrarmagnetiseringen. Därför är det viktigt att veta exakt vad man undersöker. Manmåste exempelvis veta hur tjock den tunna filmen är. Man skulle kunna tro attdet skulle vara mycket svårt att mäta de här olika tjocklekarna, men så ärinte fallet om man kan använda röntgenstrålning. Röntgenstrålningen reflek-teras genom tunna filmer på samma sätt som ljuset reflekteras i en spegel.Genom att noggrant mäta reflektionerna kan man även mäta atomernas täthetoch till och med avståndet mellan atomerna. Tack vare röntgenstrålning kanman dessutom mäta elektronernas och atomernas rotationsriktningar.Magnetisering upphör inte ögonblickligen vid vilken temperatur som helst,

utan den följer exakta, universella lagar som även kan förklara andra före-

teelser. I varje experiment har man försökt förstå vilken universell lag som

gav upphov till magnetiseringen. Vid ett tillfälle verkade det som om filmens

tjocklek gav en magnetiseringen som följde två universella lagar. I övriga fall

72

verkade inte magnetiseringen följa en förutbestämd universel lag, omman inte

introducerade teorin att magnetiseringen hade sin egen universela lag när dess

yta mättes.

Denna teori har bidragit till utvecklingen av en ny typ av multilager i vilketman försökt införa flera magnetiska ytor. Utforsknigen av denna tunna filmhar bevisat att olika magnetiska lagar kan uppmätas i samma tunna film.

73

“Dai Moreno, smettila di fare

l′ asse z e siediti con noi”

Da “Io, Chiara e Miriana”Acknowledgments

I will always appreciate the friendship of Martin (best wishes to him and his

family). Even if the culture, habit and way to think are at the opposite, I hope

we have got the best from each other.

I have enjoyed really much to work in this group with people from all overthe world: Hossein my office mate (we have learnt the best from each other,do you?), Panos for his young energy and the long trip to Trosa, Gabriella forbeing always available, Gunnar for his being is such way. Junaid because heunderstand me and I understand him. All the others, that I do not list (I cannot write an other thesis), for their roles in this group.Bjorgvin is special: he trusted me, brought me into the special world of

phase transitions and gave me the opportunity to learn how to be an au-

tonomous scientist and never said no to any of my ideas.

Adrian for the chats around the table, in the corridor, anywhere. I must say:your English is so perfect that sometime I do not understand.I will always say thanks to Sweden and Swedes for having trusted me. Now,

I do not think I can stay away from this country: I feel mine, even if I do not

understand the language.

Vi tacker Kristina för översättingen av min svenska sammanfattning ochTor, Gabriella och Martina för språkgranskningen.

Thanks Anne! for the revisioning of this work

The rest is just a list of people I met here and there: K. and G.

Laura, my sister for being always upset with me and my parents for having

told me “Go (but come back one day)”.

Donna Miriana, for the patience.Alessandra and Klara for their smiles.Federica, Chiara, Ezgi, Erika for being there.Then Gregor, Jailton, Moyseis&family, Paolino&family, Sergiu&family,

Paolone.Special thanks to: Giacomo e gli altri fregni, Michele, Gabriele&family,Nicola, Filippo for the unextinguishable support.

And Goffredo, perché siamo Campioni del Mondo.

And in the last, all the other lost friends.

Adrianooooooooooo!I am coming (perhaps. . . )

75

Game Over

76

Bibliography

[1] K. O. Kvashnina, S. M. Butorin, A. Modin, I. Soroka, M. Marcellini, J. Nord-

gren, J. H. Guo, and L. Werme. In situ x-ray absorption study of copper films

in ground water solutions. Chemical Physics Letters, 447(1-3):54–57, 2007.

[2] M. Pärnaste, M. Marcellini, and B. Hjörvarsson. Oscillatory exchange cou-

pling in the two-dimensional limit. Journal of Physics: Condensed Matter,17(44):L477–L483, 2005.

[3] Martin Pärnaste, Moreno Marcellini, Erik Holmström, Nicolas Bock, Jonas

Fransson, Olle Eriksson, and Björgvin Hjörvarsson. Dimensionality crossover

in the induced magnetization of pd layers. Journal of Physics: CondensedMatter, 19(24):246213 (7pp), 2007.

[4] E. Holmstrom, W. Olovsson, I. A. Abrikosov, A. M. N. Niklasson, B. Johans-

son, M. Gorgoi, O. Karis, S. Svensson, F. Schafers, W. Braun, G. Ohrwall,

G. Andersson, M. Marcellini, and W. Eberhardt. Sample preserving deep in-

terface characterization technique. Physical Review Letters, 97(26):266106,2006.

[5] A. Liebig, P. T. Korelis, H. Lidbaum, G. Andersson, K. Leifer, and B. Hjor-

varsson. Morphology of amorphous fe[sub 91]zr[sub 9]/al[sub 2]o[sub 3] mul-

tilayers: Dewetting and crystallization. Physical Review B (Condensed Matterand Materials Physics), 75(21):214202, 2007.

[6] Hossein Raanaei, Hans Lidbaum, Andreas Liebig, Klaus Leifer, and Björgvin

Hjörvarsson. Structural coherence and layer perfection in fe/mgo multilayers.

Journal of Physics: Condensed Matter, 20(5):055212 (5pp), 2008.

[7] A. Remhof, G. Nowak, A. Liebig, H. Zabel, and B. Hjörvarsson. Hydrogen

assisted growth of fe/v superlattices. Journal of Physics: Condensed Matter,18(35):L441–L445, 2006.

[8] J. Kwo, M. Hong, and S. Nakahara. Growth of rare-earth single crystals by

molecular beam epitaxy: The epitaxial relationship between hcp rare earth and

bcc niobium. Applied Physics Letters, 49(6):319–321, 1986.

[9] R.E. Camley, J. Kwo, M. Hong, and C. L. Chien. Magnetic properties of gd/dy

superlattices: experiments and theory. Phys. Rev. Lett., 64(22):2703, 1990.

[10] R. A. Cowley, D. F. McMorrow, A. Simpson, D. Jehan, P. Swaddling, R. C. C.

Ward, and M. R. Wells. Magnetic structures and interactions in ho/y, ho/lu, and

ho/er superlattices (invited). J. Appl. Phys., 76(10):6274–6277, 1994.

[11] M. Knudsen. Die gesetze der molekularströmung und der inneren reibungsströ-

mung der gase durch röhren. Annalen der Physik, 333(1):75–130, 1909.

77

[12] M. Knudsen. Die molekularströmung der gase durch offnungen und die effu-

sion. Annalen der Physik, 333(5):999–1016, 1909.

[13] Ivan K. Schuller, S. Kim, and C. Leighton. Magnetic superlattices and mul-

tilayers. Journal of Magnetism and Magnetic Materials, 200(1-3):571–582,1999.

[14] P Isberg, B Hjorvarsson, R Wappling, E. B. Svedberg, and L Hultman. Growth

of epitaxial fe/v (001) superlattice films. Vacuum, 48(5):483–489, 1997.

[15] P. Isberg, P. Granberg, E. B. Svedberg, B. Hjorvarsson, R. Wappling, and

P. Nordblad. Structure and magnetic properties of fe/v (110) superlattices.

Phys. Rev. B, 57(6):3531–3538, 1998.

[16] D.WilliamNix. Mechanical properties of thin film.Metallurgical TransactionsA, 20A:2217, 1989.

[17] A. Chamberod and J. Hillairet, editors. Metallic Multilayers. Trans Tech Pub-blications, 1989.

[18] F. C. Frank and J. H. van der Merwe. One-dimensional dislocations. ii. mis-

fitting monolayers and oriented overgrowth. Proceedings of the Royal Soci-ety of London. Series A, Mathematical and Physical Sciences (1934-1990),198(1053):216–225, 1949.

[19] Eric E. Fullerton, Ivan K. Schuller, H. Vanderstraeten, and Y. Bruynseraede.

Structural refinement of superlattices from x-ray diffraction. Phys. Rev. B,45(16):9292–9310, Apr 1992.

[20] J. I. Lanford and A. J. C. Wilson. Scherrer after sixty years: A survey and some

new results in the determination of crystallite size. J. Appl. Cryst., 11:102–113,1978.

[21] P. M. Reimer, H. Zabel, C. P. Flynn, and J. A. Dura. Extraordinary alignment

of nb films with sapphire and the effects of added hydrogen. Phys. Rev. B,45(19):11426–11429, May 1992.

[22] A R Wildes, R A Cowley, R C C Ward, M R Wells, C Jansen, L Wireen, and

J P Hill. The structure of epitaxially grown thin films: a study of niobium on

sapphire. Journal of Physics: Condensed Matter, 10(36):L631–L637, 1998.

[23] Bernd Wolfing, Katharina Theis-Brohl, Christoph Sutter, and Hartmut Zabel.

Afm and x-ray studies on the growth and quality of nb(110) on. Journal ofPhysics: Condensed Matter, 11(13):2669–2678, 1999.

[24] A. Gibaud, R. A. Cowley, D. McMorrow, R. C. C. Ward, and M. R. Wells.

High-resolution x-ray scattering study of the structure of nb thin film on al2o3.

Phys. Rev. B, 48:14463, 1993.

[25] P. Bödeker, A. Abromeit, K. Bröhl, P. Sonntag, N. Metoki, and H. Zabel.

Growth and xray characterization of co/cu (111) superlattices. Phys. Rev. B,47(4):2353, 1993.

78

[26] A. Stierle, A. Abromeit, N. Metoki, and H. Zabel. High resolution x-ray char-

acterization o co films on l2o3. J. Appl. Phys., 73:10, 1993.

[27] P F Fewster. X-ray diffraction from low-dimensional structures. Semiconduc-tor Science and Technology, 8(11):1915–1934, 1993.

[28] P. F. Miceli, D. A. Neumann, and H. Zabel. X-ray refractive index: A tool to

determine the average composition in multilayer structures. Applied PhysicsLetters, 48(1):24–26, 1986.

[29] J. Als-Nielsen and D. McMorrow. Elements of Modern X-Ray Physics. Wiley,2000.

[30] H. Zabel. X-ray and neutron reflectivity analysis of thin films and superlattices.

Applied Physics A: Materials Science & Processing, 58(3):159–168, 1994.

[31] C. Schiller, G.M. Martin, W.W. v.d. Hoogenhof, and J. Jorno. Fast and accurate

assesment of nanometer layers layers using grazing x-ray reflectometry. PhilipsJ. Res., 47:217–234, 1993.

[32] L. G. Parratt. Surface studies of solids by total reflection of x-rays. Phys. Rev.,95(2):359–369, Jul 1954.

[33] H. Kiessig. Interferenz von röntgenstrahlen an dünneb schichten. Ann. Physik,10:715–788, 1931.

[34] M. M. Schwickert, R. Coehoorn, M. A. Tomaz, E. Mayo, D. Lederman, W. L.

O’Brien, Tao Lin, and G. R. Harp. Magnetic moments, coupling, and interface

interdiffusion in fe/v(001) superlattices. Phys. Rev. B, 57(21):13681–13691,Jun 1998.

[35] M. Björck and G. Andersson. Genx: An extensible x-ray reflectivity refinement

software utilizing differential evolution. J. Appl. Cryst., 40:1174–1178, 2007.

[36] S. Olsson. Phase Transitions and Phase Formation of Hydrogen in Quasi-2DLattices. PhD thesis, Uppsala University, Teknisk-naturvetenskapliga veten-skapsområdet, Physics, Department of Physics, 2003.

[37] G. K. Palsson and B. Hjörvarsson. The limits of diffraction measurements.

Submitted to EPL, 2008.

[38] M. Vohl, J. A. Wolf, P. Grunberg, K. Sporl, D. Weller, and B. Zeper. Exchange

coupling of ferromagnetic layers across nonmagnetic interlayers. Journal ofMagnetism and Magnetic Materials, 93:403–406, 1991.

[39] A. Broddefalk, R. Mathieu, P. Nordblad, P. Blomqvist, R. Wäppling, J. Lu, and

E. Olsson. Interlayer exchange coupling and giant magnetoresistance in f e/v(001) superlattices. Phys. Rev. B, 65(21):214430, Jun 2002.

[40] G.R. Harp, M.M. Schwickert, M.A. Tomaz, Tao Lin, D. Lederman, E. Mayo,

and W.L. O’Brien. Competition between direct exchange and indirect

rkky coupling in fe/v(001) superlattices. Magnetics, IEEE Transactions on,34(4):864–866, 1998.

79

[41] B. Hjörvarsson, J. A. Dura, P. Isberg, T. Watanabe, T. J. Udovic, G. Andersson,

and C. F. Majkrzak. Reversible tuning of the magnetic exchange coupling in

fe/v (001) superlattices using hydrogen. Phys. Rev. Lett., 79(5):901–904, Aug1997.

[42] M. Pärnaste, M. van Kampen, R. Brucas, and B. Hjorvarsson. Temperature-

dependent magnetization and susceptibility of fe[sub n]/v[sub 7] superlattices.

Phys. Rev. B, 71(10):104426, 2005.

[43] Till Burkert, Peter Svedlindh, Gabriella Andersson, and Bjorgvin Hjörvarsson.

Magnetic ordering in a weakly coupled fe/v(001) superlattice. Phys. Rev. B,66(22):220402, 2002.

[44] P. Poulopoulos, P. Isberg, W. Platow, W. Wisny, M. Farle, B. Hjorvarsson, and

K. Baberschke. Magnetic anisotropy and exchange coupling in fenvm(0 0 1)

superlattices on mgo(0 0 1). Journal of Magnetism and Magnetic Materials,170(1-2):57–66, 1997.

[45] P. Granberg, P. Isberg, E. B. Svedberg, B. Hjorvarsson, P. Nordblad, and

R. Wappling. Antiferromagnetic coupling and giant magnetoresistance in

fe/v(0 0 1) superlattices. Journal of Magnetism and Magnetic Materials, 186(1-2):154–160, 1998.

[46] K. Eftimova, A. M. Blixt, B. Hjorvarsson, R. Laiho, J. Salminen, and J. Rait-

tila. Magnetic properties and coupling in fe(2 ml)/v(x ml) (x>5) superlattices.

Journal of Magnetism and Magnetic Materials, 246(1-2):54–61, 2002.

[47] P. Granberg, P. Nordblad, P. Isberg, B. Hjörvarsson, and R. Wäppling. Mag-

netic and transport properties of epitaxial fe/v(001) superlattice films. Phys.Rev. B, 54(2):1199–1204, Jul 1996.

[48] V. Meded, S. Olsson, P. Zahn, B. Hjövarsson, and S. Mirbt. Resistivity of

hydrogen-loaded fe/v and mo/v (100) superlattices: The role of vanadium ex-

pansion. Phys. Rev. B, 69(20):205409, May 2004.

[49] D. E. Savage, J. Kleiner, N. Schimke, Y.-H. Phang, T. Jankowski, J. Jacobs,

R. Kariotis, and M. G. Lagally. Determination of roughness correlations in

multilayer films for x-ray mirrors. Journal of Applied Physics, 69(3):1411–1424, 1991.

[50] E. Holmstrom, L. Nordstrom, L. Bergqvist, B. Skubic, B. Hjorvarsson, I. A.

Abrikosov, P. Svedlindh, and O. Eriksson. On the sharpness of the interfaces

in metallic multilayers. Proceedings of the National Academy of Sciences ofthe United States of America, 101(14):4742–4745, 2004.

[51] A.Westphalen, M.-S. Lee, A. Remhof, and H. Zabel. Invited article: Vector and

bragg magneto-optical kerr effect for the analysis of nanostructured magnetic

arrays. Review of Scientific Instruments, 78(12):121301, 2007.

[52] Andreas Westphalen, Till Schmitte, Kurt Westerholt, and Hartmut Zabel.

Bragg magneto-optical kerr effect measurements at co stripe arrays on fe(001).

Journal of Applied Physics, 97(7):073909, 2005.

80

[53] Z. Q. Qiu and S. D. Bader. Surface magneto-optic kerr. Review of ScientificInstruments, 71(3):1243–1255, 2000.

[54] A. N. Yaresko, L. Uba, and A. Ya. Perlov. Moke spectroscopy of palladium.

Phys. Rev. B., 58(12):7648, 1998.

[55] P. M. Oppener, T. Maurer, J. Sticht, and J. Kübler. Ab initio calculated

magneto-optical kerr effect of ferromagnetic metals: Fe and ni. Phys. Rev.B, 45(19):10924–10933, 1992.

[56] M Pärnaste. The Influence of Interlayer Exchange Coupling on Magnetic Or-dering in Fe-based Heterostructures. PhD thesis, Uppsala University - Dept.of Physic, 2007.

[57] A. K. Zvedin and V. A. Kotov. Modern magneto optics and magneto opticalmaterials. IOP Publishing Ltd, 1997.

[58] C. R. Brundle, C. A. j. Evans, and SWilson, editors. Encyclopedia of materialscharacterization. Material Characterization Series. Butterworth-HeinemannManning, 1992.

[59] Hans-Joachim Elmers, Jens Hauschild, and Ulrich Gradmann. Critical behav-

ior of the uniaxial ferromagnetic monolayer fe(110) on w(110). Phys. Rev. B,54(21):15224–15233, Dec 1996.

[60] U. Bovensiepen, C. Rudt, P. Poulopoulos, and K. Baberschke. Ac-

susceptibility of ni/w(1 1 0) ultrathin magnetic films: determination of the curie

temperature and critical behavior. Journal of Magnetism and Magnetic Mate-rials, 231(1):65–73, 2001.

[61] P. Poulopoulos, U. Bovensiepen, M. Farle, and K. Baberschke. Ac suscep-

tibility: a sensitive probe of interlayer coupling. Journal of Magnetism andMagnetic Materials, 212(1-2):17–22, 2000.

[62] R. L. Fagaly. Superconducting quantum interference device instruments and

applications. Review of Scientific Instruments, 77(10):101101, 2006.

[63] B. D. Josephson. Possible new effects in superconductive tunnelling. PhysicsLetters, 1(7):251–253, 1962.

[64] H. Fritzsche, Y. T. Liu, J. Hauschild, and H. Maletta. Magnetization of

uncovered and v-covered ultrathin fe(100) films on v(100). Phys. Rev. B,70(21):214406, 2004.

[65] A. Schreyer, T. Schmitte, R. Siebrecht, P. Bodeker, H. Zabel, S. H. Lee, R. W.

Erwin, C. F. Majkrzak, J. Kwo, and M. Hong. Neutron scattering on magnetic

thin films: Pushing the limits (invited). J. Appl. Phys., 87(9):5443–5448, 2000.

[66] Hartmut Zabel. Spin polarized neutron reflectivity of magnetic films and su-

perlattices. Physica B: Condensed Matter, 198(1-3):156–162, 1994.

[67] Hartmut Zabel, Ralf Siebrecht, and Andreas Schreyer. Neutron reflectometry

on magnetic thin films. Physica B: Condensed Matter, 276-278:17–21, 2000.

81

[68] H Zabel and K Theis-Brohl. Polarized neutron reflectivity and scattering

studies of magnetic heterostructures. Journal of Physics: Condensed Matter,15(5):S505–S517, 2003.

[69] G. L. Squires. Introduction to the theory of thermal neutron scattering. DoverPublications, Inc. Mineola (New York), 1996.

[70] J. A. C. Bland. Magnetic multilayers studied by polarised neutron reflection.

Physica B: Condensed Matter, 234-236:458–463, 1997.

[71] S. J. Blundell and J. A. C. Bland. Polarized neutron reflection as a probe of

magnetic films and multilayers. Phys. Rev. B, 46(6):3391–3400, Aug 1992.

[72] S. Hope, J. Lee, P. Rosenbusch, G. Lauhoff, J. A. C. Bland, A. Ercole, D. Buck-

nall, J. Penfold, H. J. Lauter, V. Lauter, and R. Cubitt. Thickness dependence

of the total magnetic moment per atom in the cu/ni/cu/si(001) system. Phys.Rev. B, 55(17):11422–11431, May 1997.

[73] L. P Kadanoff, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Pal-

ciauskas, M. Rayl, J. Swift, D. Aspnes, and J. Kane. Static phenomena near

critical points: Theory and experiments. Rev. Mod. Phys., 39(2):395, 1967.

[74] M E Fisher. The theory of equilibrium critical phenomena. Reports on Progressin Physics, 30(2):615–730, 1967.

[75] M E Fisher. The theory of equilibrium critical phenomena. Reports on Progressin Physics, 31(1):418, 1968.

[76] R.B. Stinchcombe. Polymers, Liquid Crystals, and Low-Dimensional Solids.Physics of Solids and Liquids. Plenum Press, New York and London, 1985.

[77] Andrea Pelissetto and Ettore Vicari. Critical phenomena and renormalization-

group theory. Physics Reports, 368(6):549–727, 2002.

[78] B. Widom. Equation of state in the neighborhood of the critical point. TheJournal of Chemical Physics, 43(11):3898–3905, 1965.

[79] B. Widom. Surface tension and molecular correlations near the critical point.

The Journal of Chemical Physics, 43(11):3892–3897, 1965.

[80] L. Kadanoff. Spin-spin correlations in the two-dimensional ising model. IlNuovo Cimento B (1965-1970), 44(2):276–305, 1966.

[81] L. P Kadanoff. Scaling laws for ising models near tc. Physics, 2:263, 1966.

[82] K. G. Wilson. Renormalization group and critical phenomena ii. Phys. Rev. B,4:3184, 1971.

[83] K. G. Wilson. Renormalization group and critical phenomena i. Phys. Rev. B,4:3174, 1971.

[84] K. G. Wilson. Renormalization group and strong interactions. Phys. Rev. D,3:1818–1846, 1971.

82

[85] K. G. Wilson and J. Kogut. The renormalization group and the expansion.

Phys. Rep., 12(2):75–199, 1974.

[86] K. G. Wilson. The renormalization group and critical phenomena. Rev. Mod.Phys., 55:583–600, 1983.

[87] Franz J. Wegner. Corrections to scaling laws. Phys. Rev. B, 5(11):4529–4536,Jun 1972.

[88] Michael E. Fisher. Renormalization group theory: Its basis and formulation in

statistical physics. Rev. Mod. Phys., 70(2):653–681, Apr 1998.

[89] H. E Stanley. Introduction to phase transition and critical phenomena. Inter-nation series of monographs on physics. Oxford science publications, 1971.

[90] Robert B. Griffiths. Ferromagnets and simple fluids near the critical

point: Some thermodynamic inequalities. The Journal of Chemical Physics,43(6):1958–1968, 1965.

[91] E. Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik,31:253, 1925.

[92] Lars Onsager. Crystal statistics. i. a two-dimensional model with an order-

disorder transition. Phys. Rev., 65(3-4):117–149, Feb 1944.

[93] Bruria Kaufman. Crystal statistics. ii. partition function evaluated by spinor

analysis. Phys. Rev., 76(8):1232–1243, Oct 1949.

[94] Bruria Kaufman and Lars Onsager. Crystal statistics. iii. short-range order in a

binary ising lattice. Phys. Rev., 76(8):1244–1252, Oct 1949.

[95] C. N. Yang. The spontaneus magnetization of a 2d ising model. Phys. Rev,85(5):808, 1951.

[96] T. D. Schultz, D. C. Mattis, and E. H Lieb. Two-dimensional ising model as a

soluble problem of many fermions. Rev. Mod. Phys., 36(3):856–871, Jul 1964.

[97] H. W. J. Blote, E. Luijten, and J. R. Heringa. Ising universality in three dimen-

sions: a monte carlo study. Journal of Physics A: Mathematical and General,28(22):6289–6313, 1995.

[98] C. Liu and S. D. Bader. 2-dimensional magnetic phase transition of ultrathin

iron films on pd(100). J. Appl. Phys., 67(9):5758, 1990.

[99] C. Rau, P. Mahavadi, and M. Lu. Magnetic order and critical behavior at sur-

faces of ultrathin fe(100)p(1 x 1) films on pd(100) substrates. J. Appl. Phys.,73(10):6757–6759, 1993.

[100] C. Rudt, P. Poulopoulos, J. Lindner, A. Scherz, H. Wende, K. Baberschke,

P. Blomquist, and R. Wappling. Absence of dimensional crossover in metallic

ferromagnetic superlattices. Phys. Rev. B, 65(22):220404, 2002.

[101] Ch. Wursch and D. Pescia. Test of scaling theory at a two-dimensional ising-

like transition using a monolayer of iron. Journal of Magnetism and MagneticMaterials, 177-181(Part 1):617–619, 1998.

83

[102] Yi Li and K. Baberschke. Dimensional crossover in ultrathin ni(111) films on

w(110). Phys. Rev. Lett., 68(8):1208–1211, Feb 1992.

[103] N. D. Mermin and H. Wagner. Absence of ferromagnetism or antiferromag-

netism in one- or two-dimensional isotropic heisenberg models. Phys. Rev.Lett., 17(22):1133–1136, Nov 1966.

[104] P. C. Hohenberg. Existence of long-range order in one and two dimensions.

Phys. Rev., 158:383–386, 1967.

[105] N. D: Mermin. Crystalline order in two dimensions. Phys. Rev., 176:250–254,1968.

[106] Kinshiro Hirakawa and Hideki Yoshizawa. Observation of the condensation

of magnons in the quasi-two-dimensional planar ferromagnet k2cuf4. J. Phys.Soc. Japan, 47(2):368–378, 1979.

[107] H. Zabel and S. M. Shapiro. Neutron scattering study of magnetic excitations

in intercalated cocl2. Phys. Rev. B, 36(13):7292–7295, Nov 1987.

[108] J Als-Nielsen, S T Bramwell, M THutchings, G JMcIntyre, and DVisser. Neu-

tron scattering investigation of the static critical properties of rb2crcl4. Journalof Physics: Condensed Matter, 5(42):7871–7892, 1993.

[109] D. G. Wiesler, H. Zabel, and S. M. Shapiro. Two dimensional xy type mag-

netism in intercalated graphite: an elastic and inelastic neutron scattering study.

Zeitschrift für Physik B Condensed Matter, 93(3):277–297, 1994.

[110] S. T. Bramwell and P. C. W. Holdsworth. Can the universal jump be observed

in two-dimensional xy magnets? J. Appl. Phys., 75(10):5955–5957, 1994.

[111] S. T. Bramwell and P. C. W. Holdsworth. Universality in two-dimensional

magnetic systems. J. Appl. Phys., 73(10):6096–6098, 1993.

[112] S T Bramwell and P CWHoldsworth. Magnetization and universal sub-critical

behaviour in two-dimensional xy magnets. Journal of Physics: CondensedMatter, 5(4):L53–L59, 1993.

[113] Andrea Taroni. Theoretical Investigations of Two-Dimensional Magnets. PhDthesis, University College London, 2007.

[114] Andrea Taroni, Steven T. Bramwell, and Peter C. W. Holdsworth. Universal

window for two dimensional critical exponents. 2008.

[115] J M Kosterlitz and D J Thouless. Long range order and metastability in two

dimensional solids and superfluids. (application of dislocation theory). Journalof Physics C: Solid State Physics, 5(11):L124–L126, 1972.

[116] J M Kosterlitz and D J Thouless. Ordering, metastability and phase transi-

tions in two-dimensional systems. Journal of Physics C: Solid State Physics,6(7):1181–1203, 1973.

[117] J M Kosterlitz. The critical properties of the two-dimensional xy model. Jour-nal of Physics C: Solid State Physics, 7(6):1046–1060, 1974.

84

[118] David R. Nelson and J. M. Kosterlitz. Universal jump in the superfluid den-

sity of two-dimensional superfluids. Phys. Rev. Lett., 39(19):1201–1205, Nov1977.

[119] S. T. Bramwell and P. C. W. Holdsworth. Magnetization: A characteristic of

the kosterlitz-thouless-berezinskii transition. Phys. Rev. B, 49(13):8811–8814,Apr 1994.

[120] Shinobu Hikami and Toshihiko Tsuneto. Phase transition of quasi-two dimen-

sional planar system. Progress of Theoretical Physics, 63(2):387–401, 1980.

[121] Kwangsik Nho and Efstratios Manousakis. Critical behavior of the planar mag-

net model in three dimensions. Phys. Rev. B, 59(17):11575–11578, May 1999.

[122] H. Y. Ho, Y. J. Chen, E. J. Hwang, S. K. Yu, and C. S. Shern. Depression

of curie temperature by surface structural phase transition. Applied PhysicsLetters, 90(14):142505, 2007.

[123] T. H. Berlin and M. Kac. The spherical model of a ferromagnet. Phys. Rev.,86(6):821–835, Jun 1952.

[124] K. Binder and P. C. Hohenberg. Phase transitions and static spin correlations

in ising models with free surfaces. Phys. Rev. B, 6(9):3461–3487, Nov 1972.

[125] K. Binder. Monte carlo computer experiments on critical phenomena and

metastable states. Advances in Physics, 23(6):917–939, 1974.

[126] K. Binder and P. C. Hohenberg. Surface effects on magnetic phase transitions.

Phys. Rev. B, 9(5):2194–2214, Mar 1974.

[127] K. Binder and D. P. Landau. Crossover scaling and critical behavior at the

"surface-bulk" multicritical point. Phys. Rev. Lett., 52(5):318–321, Jan 1984.

[128] K. Binder, D. P. Landau, and S. Wansleben. Wetting transitions near the bulk

critical point: Monte carlo simulations for the ising model. Phys. Rev. B,40(10):6971–6979, Oct 1989.

[129] D. P. Landau and K. Binder. Monte carlo study of surface phase transitions in

the three-dimensional ising model. Phys. Rev. B, 41(7):4633–4645, Mar 1990.

[130] S. Alvarado, M. Campagna, and H. Hopster. Surface magnetism of ni(100)

near the critical region by spin-polarized electron scattering. Phys. Rev. Lett.,48(1):51–54, Jan 1982.

[131] C. Rau and M. Robert. Surface magnetization of gd at the bulk curie tempera-

ture. Phys. Rev. Lett., 58(25):2714–2717, Jun 1987.

[132] C. S. Arnold and D. P. Pappas. Gd(0001): A semi-infinite three-dimensional

heisenberg ferromagnet with ordinary surface transition. Phys. Rev. Lett.,85(24):5202–5205, Dec 2000.

[133] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne,

G. Creuzet, A. Friederich, and J. Chazelas. Giant magnetoresistance of

(001)fe/(001)cr magnetic superlattices. Phys. Rev. Lett., 61(21):2472–2475,Nov 1988.

85

[134] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn. Enhanced magnetore-

sistance in layered magnetic structures with antiferromagnetic interlayer ex-

change. Phys. Rev. B, 39(7):4828–4830, Mar 1989.

[135] R. E. Camley and J. Barnas. Theory of giant magnetoresistance effects in

magnetic layered structures with antiferromagnetic coupling. Phys. Rev. Lett.,63(6):664–667, Aug 1989.

[136] J. Barnas, A. Fuss, R. E. Camley, P. Grünberg, and W. Zinn. Novel magnetore-

sistance effect in layered magnetic structures: Theory and experiment. Phys.Rev. B, 42(13):8110–8120, Nov 1990.

[137] Peter M. Levy, Shufeng Zhang, and Albert Fert. Electrical conductivity of mag-

netic multilayered structures. Phys. Rev. Lett., 65(13):1643–1646, Sep 1990.

[138] C. F. Majkrzak, J. W. Cable, J. Kwo, M. Hong, D. B. McWhan, Y. Yafet, J. V.

Waszczak, and C. Vettier. Observation of a magnetic antiphase domain struc-

ture with long-range order in a synthetic gd-y superlattice. Phys. Rev. Lett.,56(25):2700–2703, Jun 1986.

[139] M. B. Salamon, Shantanu Sinha, J. J. Rhyne, J. E. Cunningham, Ross W. Er-

win, Julie Borchers, and C. P. Flynn. Long-range incommensurate magnetic

order in a dy-y multilayer. Phys. Rev. Lett., 56(3):259–262, Jan 1986.

[140] P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers. Lay-

ered magnetic structures: Evidence for antiferromagnetic coupling of fe layers

across cr interlayers. Phys. Rev. Lett., 57(19):2442–2445, Nov 1986.

[141] M. A Rudermann and C Kittel. Indirect exchange coupling of nuclear magnetic

moments by conduction electrons. Phys. Rev., 96:99, 1954.

[142] Tadao Kasuya. A theory of metallic ferro- and antiferromagnetism on zener’s

model. Progress of Theoretical Physics, 16(1):45–57, 1956.

[143] K. Yosida. Magnetic properties of cu-mn alloys. Phys. Rev, 106:893, 1957.

[144] P. Bruno and C. Chappert. Oscillatory coupling between ferromagnetic layers

separated by a nonmagnetic metal spacer. Phys. Rev. Lett., 67(12):1602–1605,Sep 1991.

[145] P. Bruno and C. Chappert. Rudermann-kittel theory of oscillatory exchange

coupling. Phys. Rev. B, 46(1):261, 1992.

[146] P. Bruno. Interlayer exchange interactions in magnetic Multilayers. Mag-netism: Molecules to Materials III. Wiley-VCH Verlag, 2002.

[147] Y. Yafet. Ruderman-kittel-kasuya-yosida range function of a one-dimensional

free-electron gas. Phys. Rev. B, 36:3948, 1987.

[148] P. Bruno. Interlayer exchange coupling: a unified physical picture. Journal ofMagnetism and Magnetic Materials, 121(1-3):248–252, 1993.

[149] P. Bruno. Recent progress in the theory of interlayer exchange coupling (in-

vited). Journal of Applied Physics, 76(10):6972–6976, 1994.

86

[150] P. Bruno. Theory of interlayer magnetic coupling. Phys. Rev. B, 52(1):411,1995.

[151] M.D. Stiles. Exchange coupling in magnetic heterostructures. Phys. Rev. B,48(10):7238–7258, 1993.

[152] M. D. Stiles. Interlayer exchange coupling. Journal of Magnetism and Mag-netic Materials, 200(1-3):322–337, 1999.

[153] M.D. Stiles. Exchange Coupling in Magnetic Multilayers. Concepts Condens.Mat. Phys. EPG, 2002.

[154] M.D. Stiles. Interlayer Exchange Coupling. Ultrathin Magnetic Structures III.Springer-Verlag, 2004.

[155] A. Ney, F. Wilhelm, M. Farle, P. Poulopoulos, P. Srivastava, and K. Baber-

schke. Oscillations of the curie temperature and interlayer exchange coupling

in magnetic trilayers. Phys. Rev. B, 59(6):R3938–R3940, Feb 1999.

[156] J. Lindner, C. Rüdt, E. Kosubek, P. Poulopoulos, K. Baberschke, P. Blomquist,

R. Wäppling, and D. L. Mills. t3/2 dependence of the interlayer exchangecoupling in ferromagnetic multilayers. Phys. Rev. Lett., 88(16):167206, Apr2002.

[157] S. Schwieger, J. Kienert, K. Lenz, J. Lindner, K. Baberschke, and W. Nolt-

ing. Spin-wave excitations: The main source of the temperature dependence

of interlayer exchange coupling in nanostructures. Physical Review Letters,98(5):057205, 2007.

[158] S. Schwieger, J. Kienert, K. Lenz, J. Lindner, K. Baberschke, and W. Nolting.

Temperature dependence of interlayer exchange coupling. Journal of Mag-netism and Magnetic Materials, 310(2, Part 3):2301–2303, 2007.

[159] S. S. Kalarickal, X. Y. Xu, K. Lenz, W. Kuch, and K. Baberschke. Dominant

role of thermal magnon excitation in temperature dependence of interlayer ex-

change coupling: Experimental verification. Physical Review B (CondensedMatter and Materials Physics), 75(22):224429, 2007.

[160] K. Baberschke. Why are spin wave excitations all important in nanoscale mag-

netism? phys. stat. sol (b), 245(1):174–181, 2008.

[161] Ch. Rehm, F. Klose, D. Nagengast, H. Maletta, and A. Weidinger. Hydrogen-

induced switching of the magnetic coupling in an fe/nb multilayer. Physica B:Condensed Matter, 234-236:483–485, 1997.

[162] V Leiner, K Westerholt, B Hjorvarsson, and H Zabel. Tunability of the inter-

layer exchange coupling. Journal of Physics D: Applied Physics, 35(19):2377–2383, 2002.

[163] Vincent Leiner, Murat Ay, and Hartmut Zabel. Hydrogen and the mag-

netic interlayer exchange coupling: Variable magnetic interlayer correlation in

ho/y(00.1) superlattices. Physical Review B (Condensed Matter and MaterialsPhysics), 70(10):104429, 2004.

87

[164] D. Labergerie, C. Sutter, H. Zabel, and B. Hjorvarsson. Hydrogen induced

changes of the interlayer coupling in fe(3)/v(x) superlattices (x=11-16). Jour-nal of Magnetism and Magnetic Materials, 192(2):238–246, 1999.

[165] V. Leiner, K. Westerholt, A. M. Blixt, H. Zabel, and B. Hjörvarsson. Mag-

netic superlattices with variable interlayer exchange coupling: A new ap-

proach for the investigation of low-dimensional magnetism. Phys. Rev. Lett.,91(3):037202, Jul 2003.

[166] J. Kudrnovský, V. Drchal, I. Turek, M. Šob, and P. Weinberger. Interlayer

magnetic coupling: Effect of interface roughness. Phys. Rev. B, 53(9):5125–5128, Mar 1996.

[167] Mark van Schilfgaarde and Frank Herman. Simplified first principles approach

to exchange coupling in magnetic multilayers. Phys. Rev. Lett., 71(12):1923–1926, Sep 1993.

[168] Mark van Schilfgaarde, Frank Herman, Stuart S. P. Parkin, and Josef Ku-

drnovský. Theory of oscillatory exchange coupling in fe/(v,cr) and fe/(cr,mn).

Phys. Rev. Lett., 74(20):4063–4066, May 1995.

[169] Diana Iusan, M. Alouani, O. Bengone, and O. Eriksson. Effect of diffusion and

alloying on the magnetic and transport properties of fe/v/fe trilayers. PhysicalReview B (Condensed Matter and Materials Physics), 75(2):024412, 2007.

[170] M A Tomaz, W J Antel Jr, W L O’Brien, and G R Harp. Induced v moments in

fe/v(100), (211), and (110) superlattices studied using x-ray magnetic circular

dichroism. Journal of Physics: Condensed Matter, 9(11):L179–L184, 1997.

[171] M. Farle, A. N. Anisimov, K. Baberschke, J. Langer, and H. Maletta. Gyromag-

netic ratio and magnetization in fe/v superlattices. EPL (Europhysics Letters),49(5):658–664, 2000.

[172] A. Scherz, H. Wende, P. Poulopoulos, J. Lindner, K. Baberschke, P. Blomquist,

R. Wäppling, F. Wilhelm, and N. B. Brookes. Induced v and reduced fe mo-

ments at the interface of fe/v(001) superlattices. Phys. Rev. B, 64(18):180407,Oct 2001.

[173] O Eriksson, L Bergqvist, E Holmström, A Bergman, O LeBacq, S Frota-

Pessoa, B Hjörvarsson, and L Nordström. Magnetism of fe/v and fe/co multi-

layers. Journal of Physics: Condensed Matter, 15(5):S599–S615, 2003.

[174] M. Bjorck, M. Parnaste, M. Marcellini, G. Andersson, and B. Hjorvarsson.

The effect of strain and interfaces on the orbital moment in fe/v superlattices.

Journal of Magnetism and Magnetic Materials, 313(1):230–235, 2007.

[175] D. Labergerie, K. Westerholt, H. Zabel, and B. Hjorvarsson. Hydrogen in-

duced change of the atomic magnetic moments in fe/v-superlattices. Journalof Magnetism and Magnetic Materials, 225(3):373–380, 2001.

[176] V. Uzdin, D. Labergerie, K. Westerholt, H. Zabel, and B. Hjorvarsson. Evo-

lution of atomic magnetic moments in fe/v multilayers with hydrogen loading.

Journal of Magnetism and Magnetic Materials, 240(1-3):481–484, 2002.

88

Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 418

Editor: The Dean of the Faculty of Science and Technology

A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally through theseries Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)

Distribution: publications.uu.seurn:nbn:se:uu:diva-8604

ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2008

Magnetic Ordering in Layered Magnets

Documents

Transcript of Magnetic Ordering in Layered Magnets