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A study of exchange interaction,magnetic anisotropies, and ion beam

induced effects in thin films ofCo2-based Heusler compounds

Oksana Gaier

Vom Fachbereich Physik der Technischen Universitat Kaiserslautern

zur Verleihung des akademischen GradesDoktor der Naturwissenschaften genehmigte Dissertation

Betreuer: Prof. Dr. B. HillebrandsZweitgutachter: Prof. Dr. C. Felser

Datum der wissenschaftlichen Aussprache: 15.07.2009

D 386

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Abstract

This thesis is devoted to the study of the exchange interaction and magneticanisotropies in thin films of Co2-based Heusler compounds with the chemical

composition Co2YZ. The strength of the exchange interaction plays an impor-tant role in the temperature dependence of the spin polarization of Heusler com-pounds, which is a crucial point for the performance of the devices incorporatingthese materials. Magnetic anisotropies strongly influence the magnetic domainconfiguration and the magnetization reversal processes, and additionally pro-vide insight into the spin-orbit interaction. Spin-orbit coupling is an importantparameter for applications of Heusler compounds in devices based on the spindegree of freedom.

The studies of exchange interaction and magnetic anisotropies in Heuslercompounds are carried out by Brillouin light scattering spectroscopy and mag-neto-optical Kerr effect magnetometry. Particular attention is given to the de-pendence of exchange and magnetic anisotropies on the chemical compositionand atomic ordering of the Heusler films. The influence of the chemical compo-sition on exchange and anisotropies is studied for compounds wherein Y=Mn,Feand Z=Al,Si, as well as the quaternary compound Co2Cr0.6Fe0.4Al. The influ-ence of the structural order is investigated for the Co2MnSi films exhibiting avarying degree of the L21 order. The gradual variation of the L21 order in theCo2MnSi films is achieved by annealing the samples at different temperaturesafter their deposition.

We observe that a larger ordering degree in the investigated Heusler com-pounds is related to a stronger exchange interaction, which is evidenced bylarger values of the exchange stiffness D found in the L21-ordered films com-pared to those with a less ordered B2 structure. Moreover, we observe a linearincrease of the exchange stiffnessDupon increasing number of valence electronsin the unit cell evoked by the change of composition (e.g. replacement of Mnby Fe). The highest D value is found for the Co2FeSi Heusler compound. Thisvalue is nearly as large as the highest D value reported up to this day whichis found for Fe1xCox alloys at a composition ofx 0.5. Furthermore, mostof the investigated films are found to exhibit a four-fold magneto-crystalline

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anisotropy, which becomes weaker with increasing ordering degree in the films.In L21-ordered films, the magneto-crystalline anisotropy is even found to be neg-

ligible, pointing to a vanishing spin-orbit coupling in the more ordered Heuslercompounds. This is a great advantage for possible applications based on thetransport of spins since the information carried by spins cannot be lost bycoupling to the orbital motion.

Given the strong dependence of the exchange interaction and magneticanisotropies on the structural order, this thesis also explores the structuralmodification of Heusler films using ion irradiation. The irradiation experimentsperformed in this thesis utilize both light 30 and 130 keV He+ and heavy 30 keVGa+ ions. Experiments with He+ are carried out for Co2MnSi films with aninitially predominant B2 order, with and without a simultaneous mild anneal.

Ga+ irradiation is performed for L21-ordered Co2FeSi films at room tempe-rature only. For the study of the influence of ion irradiation on the structural,magnetic and electronic properties of Heusler films various techniques are em-ployed including x-ray diffraction, Brillouin light scattering spectroscopy andmagneto-optical Kerr effect magnetometry. In some cases additional investiga-tions by means of superconducting quantum interference device magnetometry,photoemsission at high energies, and x-ray absorption and circular magneticdichroism are also carried out.

Our results show that the crystallinity of the Co2MnSi films remains largelyunaffected by the bombardment with He+ ions in the whole range of applied

fluences. Moreover, at particular fluences, we observe an improvement of mag-netic and electronic properites of the Co2MnSi layer towards the bulk materialafter the room temperature irradiation. However, no evidence of the transitionto the L21phase is found. For Ga

+ irradiation experiments on Co2FeSi films, wefind a strong destructive impact of the ion beam on the magnetic properties ofCo2FeSi. For larger fluences, we even observe a breakdown of the ferromagneticorder.

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Kurzfassung

Die vorliegende Arbeit beschaftigt sich mit der Untersuchung der Austausch-wechselwirkung und der magnetischen Anisotropien in dunnen Schichten aus

Co2-basierten Heusler-Verbindungen mit einer chemischen ZusammensetzungCo2YZ. Die Starke der Austauschwechselwirkung spielt eine wichtige Rolle furdas Verstehen der Temperaturabhangigkeit der Spinpolarisation der Heusler-Verbindungen. Dieses Verstandnis ist ein entscheidender Punkt fur die Ef-fizienz von Bauelementen, welche diese Materialien integrieren. Die magne-tischen Anisotropien beeinflussen im groen Mae die magnetische Mikrostruk-tur und den Ummagnetisierungsprozess, und stellen auerdem eine direkteVerbindung zu der Spin-Bahn-Wechselwirkung dar. Die Spin-Bahn-Kopplungist ein wichtiger Parameter fur technologische Anwendungen, welche auf demSpin-Freiheitsgrad basieren.

Die Untersuchungen der Austauschwechselwirkung und der magnetischenAnisotropien in dunnen Heusler-Schichten werden mittels Brillouin-Lichstreu-spektroskopie sowie magneto-optischer Kerr-Effekt-Magnetometrie durchge-fuhrt. Ein besonderes Augenmerk wird dabei auf die Abhangigkeit dieser mag-netischen Eigenschaften von der chemischen Zusammensetzung der Heusler-Schichten und von ihrer kristallographischen Ordnung gelegt. Der Einflu derchemischen Zusammensetzung auf die Austauschkonstanten und auf die mag-netischen Anisotropien wird an einer Serie von Verbindungen untersucht, inwelcher Y=Mn,Fe und Z=Al,Si sind, und welche auch die quaternare Verbin-dung Co2Cr0.6Fe0.4Al einschliet. Der Einflu der kristallographischen Ordnungwird an Co2MnSi-Schichten untersucht, deren L21-Ordnungsgrad graduell zu-

nimmt. Die Variation des Ordnungsgrades wird dabei durch die Variation derTemperatur erreicht, bei welcher die Schichten nach ihrer Deposition getempertwerden.

Es zeigt sich, dass ein hoherer Ordnungsgrad der Heusler-Verbindungen miteiner starkeren Austauschwechselwirkung korreliert. Die experimentell ermit-telte Spin-Steifigkeit D ist bei den L21-geordneten Verbindungen groer alsbei solchen, die eine weniger geordnete B2-Struktur aufweisen. Auerdem,wird eine lineare Zunahme von D mit zunehmender Anzahl der Valenzelek-

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tronen in der Einheitszelle beobachtet, wobei die Zunahme der Valenzelektro-nenzahl durch die Anderung der chemischen Zusammensetzung erzeugt wird

(z. B. Ersatz von Mangan durch Eisen). Die hochste Spin-Steifigkeit wird beiCo2FeSi beobachtet. Sie ist vergleichbar mit dem hochsten bis heute berichtetenD-Wert, welcher in Kobalt/Eisen-Legierungen beobachtet wurde. Daruberhinaus weisen die meisten untersuchten Heusler-Schichten eine vierfache mag-netokristalline Anisotropie auf, welche mit der sich verbessernden kristallogra-phischen Ordnung schwacher wird. In den Schichten mit der L21-Ordnung istsie sogar vernachlassigbar, was auf eine verschwindende Spin-Bahn-Kopplunghindeutet. Dies ist von einem groen Vorteil fur mogliche Anwendungen, die aufdem Transport von Spins basieren, da bei einer geringen Spin-Bahn-Kopplungdie Information, welche die Spins tragen, nicht verloren gehen kann.

In Anbetracht der starken Abhangigkeit der Austauschwechselwirkung undder magnetischen Anisotropien von der kristallographischen Ordnung, wirdim Rahmen der vorliegenden Arbeit zusatzlich die Modifikation der Heusler-Schichten durch Ionenbestrahlung untersucht. Die Bestrahlungsexperimentewerden mit 30 und 130 keV He+- sowie mit 30keV Ga+-Ionen durchgefuhrt.Dabei erfolgt die Bestrahlung mit He+-Ionen an Co2MnSi-Schichten, die nachihrer Herstellung grotenteils eine B2-Ordnung aufwiesen. Die Bestrahlungwird sowohl bei Raumtemperatur als auch bei erhohten Temperaturen durchge-fuhrt. Die Bestrahlung mit Ga+-Ionen erfolgt an L21-geordneten Co2FeSi -Schichten. Der Einflu der Ionenbestrahlung auf die strukturellen, magne-

tischen und elektronischen Eigenschaften der Heusler-Verbindungen wird mit-tels Rontgenbeugung, Brillouin-Lichtstreuspektroskopie sowie magneto-opti-scher Kerr-Effekt-Magnetometrie untersucht. In einigen Fallen werden auchPhotoemission bei hohen Energien, SQUID-Magnetometrie, Rontgenabsorptionund magnetischer Zirkulardichroismus verwendet.

Die Untersuchungen der mit He+-Ionen bestrahlten Co2MnSi -Schichtenzeigen, dass ihr kristalliner Zustand im gesamten Bereich der gewahlten Io-nenfluenzen nach der Bestrahlung grotenteils erhalten bleibt. Auerdem wirdbei bestimmten Fluenzen eine Verbesserung der magnetischen und elektro-nischen Eigenschaften beobachtet, sodass diese nach der Bestrahlung denen desVolumenmaterials nahe kommen. Jedoch wird kein Phasenubergang zur L21-Struktur beobachtet. Die Bestrahlung von Co2FeSi-Schichten mit Ga

+-Ionenhat dagegen einen destruktiven Einfluss auf die magnetischen Eigenschaftenvon Co2FeSi. Bei groeren Fluenzen wird sogar eine Zerstorung der ferromag-netischen Ordnung beobachtet.

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Contents

1 Introduction 1

2 Concepts in magnetism 52.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Exchange interactions . . . . . . . . . . . . . . . . . . . 52.1.2 Magnetic anisotropies . . . . . . . . . . . . . . . . . . . . 92.1.3 Magnetic domains . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Magneto-optical Kerr effect . . . . . . . . . . . . . . . . . . . . 142.2.1 Phenomenological description . . . . . . . . . . . . . . . 152.2.2 Microscopic origin . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Spin wave dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Landau-Lifshitz torque equation . . . . . . . . . . . . . . 192.3.2 Spin waves in single magnetic layers . . . . . . . . . . . . 202.3.3 Brillouin light scattering . . . . . . . . . . . . . . . . . . 22

3 Half-metallic Heusler compounds 253.1 Structural properties of Co2-based Heusler compounds . . . . . 263.2 Half-metallicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Origin of the band gap in Co2-based Heusler compounds 303.2.2 Effects destroying half metallicity . . . . . . . . . . . . . 31

3.3 Magnetic properties of Co2-based Heusler compounds . . . . . . 36

3.3.1 Magnetic moments and Slater-Pauling behavior . . . . . 363.3.2 Formation of local magnetic moments . . . . . . . . . . . 383.3.3 Exchange interaction between magnetic moments . . . . 393.3.4 Curie temperature . . . . . . . . . . . . . . . . . . . . . 403.3.5 Orbital magnetism . . . . . . . . . . . . . . . . . . . . . 41

4 Experimental methods 434.1 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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vi Contents

4.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.2 Application to Co2YZ Heusler compounds . . . . . . . . 45

4.1.3 X-ray diffractometer . . . . . . . . . . . . . . . . . . . . 47

4.2 Kerr effect techniques . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 MOKE magnetometry . . . . . . . . . . . . . . . . . . . 48

4.2.2 MOKE microscopy . . . . . . . . . . . . . . . . . . . . . 50

4.3 Brillouin light scattering spectroscopy . . . . . . . . . . . . . . . 51

4.3.1 Measurement geometries . . . . . . . . . . . . . . . . . . 51

4.3.2 Brillouin light scattering spectrometer . . . . . . . . . . 52

4.4 Ion irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.1 Ion-solid interactions . . . . . . . . . . . . . . . . . . . . 56

4.4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . 58

5 Experimental results I - Magnetic properties of Co2-based Heusler

films 59

5.1 Film preparation and pre-characterization . . . . . . . . . . . . 60

5.1.1 Overview of investigated samples . . . . . . . . . . . . . 60

5.1.2 XRD investigations of Co2MnSi films . . . . . . . . . . . 61

5.2 Determination of exchange constants . . . . . . . . . . . . . . . 64

5.2.1 Co2MnSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.2 Co2FeAl . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.3 Co2Cr0.6Fe0.4Al . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.4 Co2FeSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Magnetic anisotropies . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 Co2MnSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.2 Co2Cr0.6Fe0.4Al . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.3 Co2FeAl . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.4 Co2FeSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4 Magnetization reversal . . . . . . . . . . . . . . . . . . . . . . . 93

5.4.1 Magnetization reversal process . . . . . . . . . . . . . . . 93

5.4.2 Influence on coercivity . . . . . . . . . . . . . . . . . . . 96

5.5 Quadratic magneto-optical Kerr effect . . . . . . . . . . . . . . . 96

5.5.1 Co2FeSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5.2 Co2MnSi . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Contents vii

6 Experimental results II - Modification of Heusler films by ion

irradiation 1036.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.1.1 Overview of investigated samples . . . . . . . . . . . . . 1046.1.2 Simulations of the irradiation process . . . . . . . . . . . 106

6.2 He+ irradiation of Co2MnSi . . . . . . . . . . . . . . . . . . . . 1116.2.1 RT irradiation 30 keV He+ . . . . . . . . . . . . . . . . . 1126.2.2 RT irradiation with 130 keV He+ . . . . . . . . . . . . . 1226.2.3 Irradiation with 30 keV He+ above RT . . . . . . . . . . 1246.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3 Ga+ irradiation of Co2FeSi . . . . . . . . . . . . . . . . . . . . . 1296.3.1 MOKE investigations . . . . . . . . . . . . . . . . . . . . 129

6.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Summary and outlook 133

A Calculation of BLS intensity 137

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Chapter 1

Introduction

Heusler compounds, with the general composition X2YZ , which have beenknown since the beginning of the last century, are currently attracting great in-terest due to their possible use in the novel field of spin-dependent devices, alsoknown as spintronics or magnetoelectronics [1]. This interest in Heusler com-pounds stems from the half-metallic character of their spin-split band structure,i.e. metallic behavior for one spin component (majority spins), and insulatingbehavior for the other one (minority spins), as predicted by ab initio calcula-tions for many compounds of this material class [2,3]. As such, these materials

may exhibit a 100 % spin polarization at the Fermi level, which would makethem ideal candidates for spin polarizers or spin detectors, amongst other appli-cations.

In addition to half-metallicity, Heusler compounds exhibit several other fea-tures which make them suitable candidates for technological applications. Theirelectronic band structure, the width, and position of the insulating gap parti-cular can be tuned with respect to the Fermi level by changing composition [4].Consequently, the design of new materials with electronic and magnetic pro-perties adequate for a certain application is possible with this material class.Moreover, several Heusler compounds possess relatively high Curie tempera-tures [5, 6], which is a prerequisite for the stability of the performance of de-vices incorporating ferromagnetic materials. The highest Curie temperaturesare found for the so-called Co2-based Heusler compounds, which have the che-mical composition Co2YZ (with Y being a transition metal and Z an elementfrom the III-V groups) and crystallize in the L21 structure. This explains theparticular attention given to the Co2-based Heusler compounds.

An indispensable precondition for a successful implementation of Heuslercompounds into real devices, however, is a good understanding of their elec-tronic and magnetic properties. In the last decade, a lot of theoretical and

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2 Introduction

experimental work has been carried out to explain the appearance of the ferro-magnetic order in these materials and to elucidate the phenomena which might

be detrimental to the insulating gap in the minority spin channel. In parti-cular, it was found out that inelastic electron-magnon interactions, which areirrelevant at low temperatures but gain in importance at room temperature,can create states near the Fermi level in the minority gap, thus reducing theidealized 100 % spin polarization [79]. As such, the investigation of thermalmagnons in Heusler compounds is an important issue. In this context, a keyparameter is the exchange stiffness D which describes the energy of a magnon,and is related to the exchange constantA which expresses the energy of alignedspins in a magnetic material. The knowledge of these parameters also plays animportant role in micromagnetic simulations and the study of dynamic pheno-

mena. However, a systematic experimental investigation and comparison of theexchange stiffness in Heusler compounds is still lacking till today. In addition tothe inelastic electron-magnon interactions, atomic disorder in the crystal latticeis also found to have a detrimental effect on the spin polarization [10,11], since itcan create states in the insulating gap as well. The influence of various types ofdisorder which might occur in the L21Heusler structure on the electronic struc-ture of Heusler compounds is a topic of numerous theoretical and experimentalworks. However, the question of how the atomic disorder would influence theinelastic electron-magnon interactions has not yet been investigated.

Another magnetic property which is of equal importance for spintronics ap-

plications as the exchange interaction is the spin-orbit coupling. Since thisinteraction is responsible for the coupling of the spin to the orbital angularmomentum, it would have a strong impact on the depolarization of highly spinpolarized currents required in the spin-dependent devices. Therefore the in-vestigation of spin-orbit coupling for potential spintronics materials such asHeusler compounds is a crucial issue towards the optimization of the perfor-mance of devices based on these materials. A possible way to get insight intothe spin-orbit interaction inherent to these materials is a separate determina-tion of spin and orbital magnetic moments, which can be done by means ofx-ray magnetic circular dichroism for example. An easier way is the investi-gation of the magneto-crystalline anisotropies by magneto-optical techniques,an example of which is the magneto-optical Kerr effect magnetometry. Theanisotropy constants Kdetermined by such magneto-optical techniques are adirect measure for the strength of the spin-orbit interaction. There are fewreports dealing with the determination ofKvalues for the technologically rele-vant material class of Heusler compounds. However, a systematic experimentalinvestigation and comparison of anisotropy constants is not present for thesematerials.

The goal of this thesis is a systematic investigation of the exchange and

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Introduction 3

anisotropy constants for Co2-based Heusler compounds. For this purpose, stu-dies of thin films consisting of Co2-based Heusler compounds are carried out

by means of Brillouin light scattering spectroscopy and magneto-optical Kerreffect magnetometry. The investigations are performed with the particularaim of revealing the dependence of exchange and anisotropy constants on boththe chemical composition and atomic ordering of the Heusler films. Moreover,this thesis also explores the structural modification of Heusler films using ionirradiation. Usually, high-temperature annealing is applied to Heusler films toobtain the highly ordered L21 phase. This technique, however, often suffersfrom the drawback of interdiffusion between layers, and is not compatible withsemiconductors when envisaging devices consisting of Heusler-semiconductorhybrid structures. Irradiation with light ions such as He+ might provide an

alternative to the high-temperature annealing since it was shown to induce atransition from a disordered to a more ordered phase for the FePt(Pd) binaryalloys [12,13].

In the following, we first briefly introduce magnetic phenomena generallynecessary for the understanding of the performed experiments in Chapt. 2.Thereafter, a comprehensive overview of the crystallographic, electronic, andmagnetic properties of the Co2-based Heusler compounds will be given inChapt. 3, and the employed experimental methods will be described in Chapt. 4.Finally, studies of exchange and magnetic anisotropies of Co2-based Heuslercompounds will be presented in Chapt. 5, followed by the results obtained in

ion-irradiation experiments performed with Heusler films in Chapt. 6.

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4 Introduction

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Chapter 2

Concepts in magnetism

This Chapter gives an overview of magnetic phenomena which are importantfor the understanding of the present work. First of all, the exchange interac-tion and magnetic anisotropies are introduced, which form the core subject ofthis thesis. Thereafter, the macroscopic and microscopic behavior of a ferro-magnet in an external magnetic field is described. To lay out the fundamentalprinciples of the experimental techniques employed in this thesis introductorytheoretical descriptions of spin waves in thin layers and magneto-optical Kerreffect are given. It should be noted that it is not our goal to give a complete

mathematical description of these phenomena, and only those aspects will beaddressed in more detail which are necessary for the general understanding ofthe experiments performed within this work and for the discussion of the resultsin Chapts. 5 and 6. For more detailed descriptions, the reader is referred totextbooks such as Magnetism by J. Stohr and H. C. Siegmann [14].

2.1 Fundamentals

2.1.1 Exchange interactionsIn solids, the electronic orbitals of neighboring atoms overlap, which leads tothe correlation of electrons. This results in the interatomic exchange interactionthat makes the total energy of the crystal depend on the relative orientation ofspins localized on neighboring atoms. The exchange interaction is the largestmagnetic interaction in solids (1 eV) and is responsible for the existence ofparallel, i.e. ferromagnetic, and antiparallel, i.e. antiferromagnetic, spin align-ment. The exchange interaction might be mediated by different mechanisms

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6 Concepts in magnetism

depending on the material system under consideration. The most importantmechanisms are described below.

Direct exchange

The direct exchange arises from a direct overlap of electronic wave functions ofthe neighboring atoms and the Pauli exclusion principle, which requires differentsymmetry properties from the spatial and spin parts of the electronic wavefunction. In a two-spin system, the exchange energy is defined as the energydifference between the parallel and antiparallel spin configuration. For a many-electron system, the exchange energy is given by the expectation value of theHeisenberg Hamiltonian

Hex= 2i

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2.1 Fundamentals 7

a positive or negative sign, resulting in the parallel or antiparallel ground stateconfiguration of spins, respectively. The direct exchange is a short-range inter-

action. If the interatomic distance is too large (i.e. wave function overlap toosmall) the direct exchange coupling is not strong enough to overcome thermalexcitations, giving rise to paramagnetism.

Itinerant exchange

In metallic 3d systems, like Co, Fe or Ni, both the atomic moments and theexchange interaction are due to the delocalized, also called itinerant, electrons.It is thus not possible to describe the magnetic properties of metals in terms ofatomic or ionic moments coupled by the intersite exchange interaction, and one

has to rely on the band model of electrons. As a consequence of the Coulombrepulsion of electrons and their kinetic energy, the bands with opposite spinorientation are exchange split (Fig. 2.2) giving rise to a non-zero total mag-netic moment, and the appearance of ferromagnetism. This exchange splittingis commonly described using the Stoner model where the exchange splittingenergy is given by Eex = IM. Here, Iis the so-called Stoner exchange pa-rameter and M is the averaged atomic magnetization. Within this model, thestability of the spontaneous magnetic order is given by the Stoner criterion

I N(EF)> 1 , (2.2)

whereN(EF) is the density of states at the Fermi level. A high density of statesat the Fermi level and strong exchange splitting thus favor metallic ferromag-netism. The total magnetic moment is not fixed by atomic rules. Instead it

Density of states

En

ergy(eV)

4

2

0

-2

-4

-6

-8

4

2

0

-2

-4

-6

-8

Eex

s-electrons

d-electrons

Ni

EF

Figure 2.2: Spin resolved density of states of Ni adapted from Ref. [16]. Eexdenotes

the exchange splitting energy.

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depends on the unequal populations of electrons with different spin orientations,defined by the position of the Fermi level in the conduction band. Therefore,

the magnetic moments in 3d metals and their alloys exhibit quite a regulardependence on the number of valence electrons, which is known as the Slater-Pauling dependence [1719] and will be discussed in more detail in conjunctionwith the Heusler compounds in the next Chapter.

RKKY exchange

RKKY indirect exchange interaction (named after Rudermann, Kittel, Kasuyaand Yoshida) can play an important role when there is no direct overlap of thewave functions with unpaired electrons. In this case the interaction between

two magnetic moments at sites i and j is mediated by the polarization ofthe conduction electrons. Characteristic for this coupling mechanism is anoscillatory behavior of the exchange integral J which changes its sign as afunction of distance between the localized moments (Fig. 2.3). Amongst others,this mechanism explains the coupling of 4felectrons in rare earth materialsand is responsible for the oscillatory interlayer exchange coupling in multilayerGMR structures. In contrast to the direct exchange interaction, this type ofinteraction is long-range.

0 5 10 15 20 25-6

-4

-2

0

2

4

6

2k rF

J

(a.u.)

RKKY

-

-

-

Figure 2.3: RKKY exchange energy in dependence of the interatomic distance rmultiplied by the radius of the Fermi sphere kF (adapted from Ref. [15]).

Superexchange

Superexchange is another kind of indirect exchange interaction and is impor-tant in ionic solids such as the transition metal oxides and fluorides. The most

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2.1 Fundamentals 9

Mn MnO

Figure 2.4: Schematic illustration of the superexchange interaction between two Mn

atoms mediated by an oxygen atom.

prominent example is MnO. The interaction between the magnetic Mn atomsis mediated by the diamagnetic oxygen though the overlap of the metals 3dand oxygens 2p orbitals, and a partial delocalization of the involved electrons(Fig. 2.4). In the case of a parallel orientation of the magnetic moments locatedat the metal centers, no delocalization occurs, which makes the antiferromag-netic alignment energetically favorable. Generally, the size of the superexchangedepends on the magnitude of the magnetic moments on the metal atom, theorbital overlap between the metal, and the non-metallic element and the bondangle.

It should be noted that other exchange mechanisms exist apart from thosedescribed above. Biquadratic exchange [20, 21] and Dzyaloshinskii-Moriya in-teraction [22, 23] are just two examples of them. However, these mechanisms

will not be discussed here.

2.1.2 Magnetic anisotropies

In the absence of an external magnetic field, the magnetization Mof a magneticsolid usually tends to lie along one or several axes. Energy is required to displacethe magnetization from these preferential directions. The magnetic anisotropyis defined as the energy that is necessary to turnMinto any direction differentfrom the preferred axes. Magnetic anisotropies might be caused by differentmechanisms, and are generally described as different contributions Fani to the

free energy density of a magnetic system. For this, Fani is advantageouslyexpanded into a series of components i of the unit vector pointing into thedirection of magnetization:

Fani=i,j,k

Ki,j,ki1

j2

k3. (2.3)

The parametersKi,j,kin Eq. (2.3) are the so-called anisotropy constants, whichare accessible experimentally. The knowledge of these parameters is generally

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sufficient for the description of different anisotropy contributions.In thin films investigated within the framework of this thesis, magneto-

crystalline and shape anisotropy make a major contribution to the magneticanisotropy. In the following, these two contributions are described in moredetail. Depending on the magnetic system under consideration, various otheranisotropy contributions might become important. In thin films and multilayersstrain effects can give rise to the so-called magneto-elastic anisotropy. Inter-face anisotropy contributions might become relevant as well, depending on theinterface properties of the different layers. A comprehensive overview of thesecontributions and the description of their origins can be found for example inRef. [15].

Magneto-crystalline anisotropy

The origin of the magneto-crystalline anisotropy (MCA) lies in the spin-orbitinteraction which, in a simple semiclassical picture, is due to coupling of theorbital moment (resulting from the motion of the electron around the nucleus)and the spin angular momentum. The orbital motion represents a current loopthat gives rise to a magnetic field in the center of the loop. This field interactswith the spin angular momentum, coupling the spin and orbital moments. Ina quantum mechanical treatment, the spin-orbit interaction is described by theHamiltonian

HSO= e2

2m2ec2

1r

d(r)dr

s l= nl(r) s l . (2.4)

The spin and angular momentum (represented by the operators s and l inEq. (2.4), respectively) couple via the electrostatic potential of the nuclearcharges(r), which has the largest gradient d(r)/drfor small distancesrfromthe nucleus. The expectation value of nl(r) is called the spin-orbit couplingconstant, or spin-orbit parameter. Its value is of the order of 10-100 meV. Thespin-orbit interaction is thus considerably weaker than the exchange interaction(1 eV).

In single crystalline materials, the bonding is anisotropic, i.e. the overlapof the atomic wave functions depends on the crystallographic directions. Thisgives rise to an anisotropy of the orbital magnetic moment, which results indifferent values of the spin-orbit energy (Eq. (2.4)) associated to different crys-tallographic directions. The symmetry of the MCA is apparently that of thecrystal lattice. A quantitative treatment of the MCA by means ofab initiocalculations is still not satisfactorily possible. Therefore, a phenomenologi-cal description, in the form of a series expansion given in Eq. (2.3), is oftenused with i,j,k being the direction cosines defined with respect to particular

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2.1 Fundamentals 11

crystallographic axes. For the case of a cubic system, which is of a particu-lar importance for the present work, the number of coefficients in Eq. (2.3) is

substantially reduced and the MCA contribution is given by

F(cub)MCA =K0+ K1(

21

22+

22

23+

23

21) +K2

21

22

23 . (2.5)

The parameters K1 and K2 are the so-called first and second anisotropy con-stants for a cubic system. Usually, the interplay between these two anisotropyconstants determines the direction of hard and easy axes [15]. In case of a(001)-oriented film with an in-plane magnetization, a positive value ofK1 re-sults in an easy axes direction parallel to the100 crystallographic directions.The110 directions become the easy axes when K1 is negative.

Shape anisotropy

Besides the magneto-crystalline anisotropy described above the shape of thesample gives rise to magnetic anisotropy as well. Shape anisotropy is mediatedby the long-range dipolar interaction:

Edipdip= 120

i=j

1

r3ij

mi mj 3 (rij mi)(rij mj)

r2ij

. (2.6)

The summation is over all atomic magnetic dipoles mi and mj at a distance

rij and every pair of dipoles is only counted once.In a thin film with homogeneous magnetization, the dominant contribu-

tion to the dipolar energy arises from the demagnetizing field created by theuncompensated magnetic moments at the film surface. It is given by

E= 12

0 M2S sin

2 , (2.7)

where is the angle between the surface normal and the magnetization vectorMS. The dipolar anisotropy energy is thus minimized for an angle of 90

, i.e.for magnetic moments lying in the plane of the layer. The in-plane alignment

ofMS keeps the magnetic flux lines in the film plane, hence reducing the strayfield energy.

2.1.3 Magnetic domains

Since the beginning of the last century, it is a well known fact that the mag-netization is not uniform in a ferromagnetic material, but rather arranged intodomains with different orientation of magnetization. This domain configuration

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(a) (b)

Figure 2.5: 180 and 90 domain wall.

minimizes the stray field energy. Within each domain, the magnetic momentsare aligned parallel to each other and point into one of the preferential direc-

tions that are determined by the magnetic anisotropies in the absence of anapplied magnetic field. A large variety of domain patterns exists dependingon the specific properties of the ferromagnetic sample under investigation [24].Single crystalline ferromagnetic films with two easy axes in the sample plane,like those studied in this work, typically exhibit domain configuration, where ad-jacent domains are oriented at 90 or 180 with respect to one another (Fig. 2.5).

The change of the magnetization direction between the adjacent domainsdoes not occur abruptly but is characterized by a slight tilt of the microscopicmagnetic moments in the boundary regions. These boundary regions are severaltens of nanometers wide and called domain walls. Two types of domain walls aredistinguished, which are named after their discoverers Bloch and Neel. Thesetwo types of domain walls are schematically illustrated in Fig. 2.6. While inthe former the rotation of the magnetization vector occurs in a plane whichis parallel to the plane of the domain wall, in the latter the rotation takesplace in a plane perpendicular to it. Bloch walls are more common in bulk-likethick films, while Neel walls are often observed in thin films, where a surfacestray field is avoided by the rotation of the moments within the surface plane.The width of a domain wall is determined by the exchange and the anisotropyenergy [24].

(a) (b)

easy

axis

Figure 2.6: Rotation of magnetization in a (a) Bloch and (b) Neel wall [25]. The

local preferential direction of magnetization changes by 180 in both cases.

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2.1 Fundamentals 13

In an external magnetic field H0, which is assumed to be higher than acertain critical field, the system will try to align all the magnetic moments

parallel to the field direction in order to minimize the Zeeman energy

EZ= 0V

MH0d3r . (2.8)

In the above equation,0 is the vacuum permeability and Mis the magnetiza-tion distribution in a given volume V. The alignment of the magnetic momentscan occur through either the coherent rotation of magnetization or the growthof domains in which the magnetization lies in a favorable direction at the ex-pense of unfavorable ones by a motion of domain walls.

The response of a magnetic material to the external magnetic field is rep-

resented by a magnetization or hysteresis curve, an example of which is shownin Fig. 2.7. A hysteresis loop is characterized by the saturation magnetizationMS, the remanence Mr, the saturation field HS and the coercive field (or coer-civity)HC. TheHCstrongly depends on the details of the reversal mechanism.If a uniform rotation of the magnetization occurs, as is assumed in the Stoner-Wohlfarth model [26], HC is equal to the anisotropy field. In most systems,where the nucleation of domains plays an important role in the magnetizationreversal process (which is the case for the Heusler films studied in this thesis),the coercivity is usually smaller than the anisotropy field. Its value is deter-mined in addition to the anisotropy constants by the number of local defects.

For domain walls, defects act as pinning centers which increase the stabilityof the domain walls against the externally applied fields. This results in higher

Applied field, H

Magnetization

,M

HCHS

MS

MR

0

0

Figure 2.7: Definition of the main features of the hysteresis curve: saturation mag-

netization MS, remanence Mr, saturation field HS and coercive field or coercivity

HC.

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values ofHC.Experimentally, the magnetization reversal curves can be obtained by using

magneto-optical Kerr effect magnetometry. Since this technique was employedin this thesis, the underlying magneto-optical Kerr effect will be described inthe following. The experimental setup will be described in Chapt. 4.

2.2 Magneto-optical Kerr effect

When linearly polarized light is reflected from a magnetic film, its polarizationbecomes elliptic and the principal axis is rotated. This effect is known as themagneto-optical Kerr effect (MOKE) named after its discoverer John Kerr [27].

Accordingly, the rotation angle and ellipticity are referred to as Kerr angle and Kerr ellipticity . In general, these two quantities can be expressedin terms of complex amplitudes of the electric field vector. The quantities and are proportional to the magnetization of the film. Which component ofthe magnetization is probed in the experiment depends on the measurementgeometry.

Figure 2.8 illustrates three different Kerr effects, which are the (a) polar, (b)longitudinal and (c) transverse MOKE. For the polar effect, the magnetizationdirection is perpendicular to the film surface. For the longitudinal and trans-verse effect, the magnetization is lying in the film plane and is oriented either in

a parallel or perpendicular fashion with respect to the plane of light incidence.The polar MOKE is strongest at the perpendicular incidence. In addition tothe polar, linear and transverse MOKE which exhibit a linear dependence onthe respective components of magnetization (i.e. MP, ML and MT), there arealso higher order effects, for which the Kerr angle and ellipticity depend onproduct terms involving the polar, longitudinal and transverse magnetizationcomponents.

In a phenomenological picture, the different kinds of MOKE are described

polar longitudinal transverse

MP

ML

MT

Figure 2.8: Different magneto-optical Kerr effects that exist depending on the relative

orientation of the magnetization (red arrow), sample surface and the plane of the

incidence of light.

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2.2 Magneto-optical Kerr effect 15

using material tensors in combination with the Jones matrix formalism usu-ally introduced in textbooks on optics, or with a more generalized Yeh matrix

approach [28]. The microscopic origin of MOKE lies in the splitting of elec-tronic levels due to both exchange and spin-orbit interaction. In the followingtwo Subsections, both the phenomenological description and the microscopicorigin of MOKE are looked at in more detail.

2.2.1 Phenomenological description

Optical and magneto-optical properties of a magnetized crystal are described

by the permittivity tensor ij, which can be expanded into a series in thecomponents of the sample magnetization M:

ij =(0)ij + KijkMk+ Gijkl MkMl+ , (2.9)

where Mi are the components ofM. The terms (0)ij , Kijk and Gijkl denote

the components of the dielectric tensor and the linear and quadratic magneto-optical tensors, respectively [29]. The number of independent components ofthese tensors is reduced taking into account the crystal symmetry and Onsagers

relation:ij(M) =ji(M) . (2.10)

For cubic crystals, this results in only one free (complex) parameter in the

constant term (0)ij , another one in the linear term Kijk and three additional

parameters in the quadratic term Gijkl [30]:

(0)ij =d ij,

Kijk =K ,

Giiii=G11 ,

Giijj =G12, i =j ,G1212 =G1313 = G2323 =G44 ,

(2.11)

whereij is the Kronecker delta-function. Therefore, in the case of an in-planemagnetization, the off-diagonal elements ofij can be written as

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xy =yx =

2G44+ G2 (1 cos 4)

ML MT

G4

sin 4 (M2L M2T) ,

xz = zx = K ML ,yz = zy = K MT,

(2.12)

whereis the crystal orientation with respect to the plane of the light incidence.The termGis the so-called magneto-optical anisotropy parameter, and G =G11

G12

2G44. Analogous treatment of other crystal symmetries can be found

for example in Ref. [31,32].In the literature, the Kerr rotation and the Kerr ellipticity are often

regarded as the real and the imaginary part of a complex Kerr amplitude ,i.e. = i . can be given by the following expressions (up to the firstorder in the off-diagonal components of the permittivity tensor) [3336]

s= rpsrss

=Asyx+ Bs zx ,

p=rsprpp

= Apxy+ Bp xz.(2.13)

In the above equations, s and pstand for the polarization of the incident lightperpendicular and parallel to the plane of incidence, respectively. The termsrps, rss, rsp and rpp denote material specific reflection coefficients, whereas theterms with mixed indices give the fraction ofp-polarized light converted to s-polarized light after the reflection and vice versa. The weighting optical factorsAs/pandBs/pare even and odd functions of the angle of incidence, respectively,and can be expressed analytically in the limiting cases ofdFM /4 NFM ordFM /4 NFM. Here, dFM and NFM is the thickness and the refractivityindex of the ferromagnetic layer.

Substituting ij from Eq. (2.12) into Eq. (2.13) results in the following

expression for the complex Kerr amplitude [30,37]

s/p= As/p

2G44+G

2 (1 cos 4)

ML MT

As/p G4

sin 4 (M2L M2T) Bs/pK ML ,(2.14)

where + (-) is related to the Kerr s (p) effect. Equation (2.14) is a finalexpression of the Kerr effect in the case of an in-plane magnetized film with

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2.2 Magneto-optical Kerr effect 17

cubic symmetry. In addition to the linear longitudinal Kerr effect described bythe last term in Eq. (2.14), there are two separate quadratic contributions to

the total Kerr effect proportional to MLMT and M2L M2T. In contrast to thelinear effect, which does not depend on the crystal orientation , the quadraticeffect exhibits anisotropy, i.e. it oscillates as a function of. The magnitudeof the anisotropy of the quadratic signal is determined by the magneto-opticanisotropy parameter G.

2.2.2 Microscopic origin

As we have shown in the previous Subsection, the magneto-optical Kerr effect

is related to the off-diagonal permittivity tensor elements. The elements ofthe linear magneto-optical tensor can be expressed in terms of the microscopicelectronic structure using the Kubo formalism [38,39], according to which thedissipative part of the off-diagonal component of the conductivity tensor2 hasthe form (for >0)

Im[xy()] = e2

4 m2e

i,f

f(Ei) [1 f(Ef)]

[|i|p|f|2 |i|p+|f|2] (EfEi )) .(2.15)

In the above equation is the light frequency, the total volume, f(E) theFermi-Dirac function and p = pxpy are the linear momentum operators(e.g. px = i /x). The right hand side of Eq. (2.15) describes the absorp-tion of a photon by an electron through an electric dipole transition from anoccupied initial state|i to an unoccupied final state|f. The matrix elementsi|p+|f andi|p|f give the transition probabilities for left (+) and right() circularly polarized light, respectively. The Dirac-function expresses thecondition of energy conservation.

A non-vanishing Kerr effect thus requires different absorption probabilitiesfor + and light. This is only possible if the electronic levels involved in

the dipole transitions are split by both the exchange and spin-orbit coupling asschematically shown in Fig. 2.9 for the case ofp dtransitions. The selectionrules for these transitions are

s= 0 , l= 1 , m= 1 . (2.16)2The corresponding dispersive component Re[xy()] can be obtained using the Kramers-

Kronig relations [40]. The permittivity tensor is related to the conductivity tensor through

the expression ij =ij + (i/)ij .

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right circ.pol. lightm 1=

left circ.pol. lightm -1=

0 ex SO+ -

0 ex SO+ +

0 ex SO+-

0 ex SO- -

absorption

h-

(b)

d|21

|2-1

|10p p

|10

|21

|2-1

Fermilevel

d2ex

SO

m

1=

m

-1=

m

1=

m

-1=

0

spin-down e-

spin-up e-

(a)

2

SO

2

Figure 2.9: (a) Schematic representation of electric dipole transitions between mag-netically perturbedp and d states. For clarity reasons only transitions between the

levels with (l = 2,ml =1) and (l = 1,ml = 0) are shown. (b) Absorption spectrafor left and right circularly polarized light (adapted from Refs. [38,41]).

The allowed dipole transitions are represented by vertical arrows in Fig. 2.9(a).For clarity reasons only transitions between the levels with (l= 2, ml = 1) and(l = 1, ml = 0) are shown. When both the exchange and spin-orbit couplingare present, the absorption spectrum is different for left- and right-circularly

polarized light (Fig. 2.9(b)), which results in a non-zero Kerr effect. If eitherspin-orbit coupling or exchange is absent, the absorption spectrum becomesequal for both polarizations, leading to a vanishing Kerr effect.

2.3 Spin wave dynamics

The concept of spin waves, as dynamic eigen excitations of a magnetic system,was proposed by Bloch [42] in order to explain the temperature dependence ofthe magnetization of ferromagnetic materials [14,15]. A spin wave, or magnon,consists of a collection of spins that coherently precess about the magnetizationdirection (Fig. 2.10). In the case of small wavelengths, the spin-wave charac-teristics are mainly determined by the exchange interaction, whereas for spinwaves with large wavelengths the dipolar interaction dominates. Therefore twotypes of spin-wave modes are generally distinguished, the so-called dipolar andexchange modes. In the following an introduction to the properties of spin-waves in thin films is given, starting with the fundamental equation in the fieldof spin dynamics, the Landau-Lifshitz torque equation.

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2.3 Spin wave dynamics 19

M

Figure 2.10: Side view (top) and top view (bottom) of a spin wave in a one-

dimensional magnetic moment chain.

2.3.1 Landau-Lifshitz torque equation

The fundamental equation of motion in magnetism is the Landau-Lifshitz torqueequation [43]

1

dMdt

=MHeff, (2.17)which describes the precessional motion of the magnetization vector M in aneffective magnetic field Heff. The parameter = gB/ is the gyromagneticratio, where g is the spectroscopic splitting factor (also called Lande g-factor).The effective field Heffis the sum over all the external and internal magneticfields acting on the magnetization and is usually composed of the followingcontributions

Heff= H(t) + Hdem+ Hex+ Hani . (2.18)

H(t) includes an externally applied static field and fluctuating fields, whichoriginate from the precessional motion of spins and might be additionallycaused by an external microwave field. The term Hdem is the demagnetizingfield created by the uncompensated magnetic moments at the surface, whereasHex stands for the exchange field due to exchange interaction. The last termin Eq. (2.18) takes into account the internal fields associated with magneticanisotropies.

In order to obtain the dispersion relations of various spin-wave modes, theLandau-Lifshitz equation must be solved together with the Maxwell equationsin the so-called magnetostatic approximation

H= 0 , (2.19) (H+ 4M) = 0 . (2.20)

Since the fluctuations in M(t) and H(t) associated with spin waves are smallcompared to the static values, the magnetization and the field vectors are usu-ally split into time-independent static parts MS and H0 and dynamic partsm(t) and h(t), and the above set of equations is solved for the dynamic com-ponents only, assuming appropriate boundary conditions for surfaces and in-terfaces [44,45].

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2.3.2 Spin waves in single magnetic layers

As already mentioned in the introductory part of this Section, different spin-wave modes might appear in thin films. Their frequency can be largely domi-nated by either the exchange or the dipolar interaction.

Different types of dipolar spin wave modes are summarized in Fig. 2.11. Ifthe magnetizationMand the wave vector q lie both in the film plane and areperpendicular to each other a magnetostatic surface mode (MSSM) or Damon-Eshbach mode is excited. If both Mand q are collinear the so-called magneto-static backward volume modes (MSBVM) appears. The magnetostatic forwardvolume modes (MSFVM) are observed when M is oriented perpendicular to

the film surface. In the present work, MSSM geometry was exclusively used.Therefore, only this type of dipolar spin waves will be treated in more detail inthe following.

The Damon-Eshbach mode is characterized by the localization of the modeenergy (i.e. localization of the amplitude of the dynamic magnetization) closeto the film surface and an exponential decay of the precessional amplitudealong the film normal. Furthermore, it exhibits a nonreciprocal behavior, whichmeans that the propagation is possible for either positive or negative directionof the wave vector but not for both. In the magnetostatic limit, i.e. weakexchange contribution, and in the case of negligible anisotropies, the dispersion

Frequency(GHz)

q d||

magnetostatic backward

volume wave (MSBVW)

magnetostatic forwardvolume wave (MSFVW)

magnetostatic surfacewave (MSSW)

0 2 4 6 8 10

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0 q

q

qMS

MS

MS

Figure 2.11: Typology of the spin wave modes in a magnetic film as a function of the

directions of the magnetization MS and the in-plane wave vector q (adapted from

Ref. [46]).

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relation of the Damon-Eshbach mode is given by [46]:

(q)

2

DE

=H0(H0+ 4MS) + (2MS)2

1 e2qd , (2.21)whereq is the in-plane component of the wave vector qand H0 is the externalmagnetic field. For fixed experimental conditions (i.e.H0 andq are constant),the frequency of the Damon-Eshbach mode only depends on the saturationmagnetization. Therefore, trends observed forDEprovide valuable informationon the behavior ofMS.

Apart from the surface localized Damon-Eshbach mode, the so-called per-pendicular standing spin waves (PSSW) are accessible in the MSSW geometry

as well. These modes are of the exchange type and are formed by a superposi-tion of two spin waves propagating in opposite directions perpendicular to thefilm. The wave vectors of the PSSW modes are thus quantized,q= p/d, withp being the quantization number and d the film thickness. Their frequenciescan be approximately described by [46]:

(q)

2p

=

H+

2A

MSq2+

2A

MS

pd

2

H+ 2A

MS+ H

4MS/H

p/d 2

q2+

2A

MS p

d 2

+ 4MS ,(2.22)

where A is the exchange constant. The detection of PSSW modes thereforeallows for the determination of the exchange interaction in a given magneticsample. In the literature, the so-called exchange stiffness constant or spinstiffness D is frequently found along with the exchange constant A. Bothquantities are related through the expression [A5]:

A=DMS2gB

. (2.23)

Equations (2.21) and (2.22) provide a proper description of the spin-wavefrequencies as long as the condition qd 1 is fulfilled, and only in the fre-quency regions where no mode crossing occurs. For the spin waves investi-gated in the Heusler films in this thesis, q is typically 1.7 105 cm1, andd= 30 80 nm. Consequently,qd 1 is not longer valid. Therefore a nume-rical approach is necessary for the determination of the spin-wave frequencieswhich is based on a model described in detail in Refs. [44,45]. This numericalapproach also allows the determination ofMS and A values, as will be shownin Chapt. 5.

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2.3.3 Brillouin light scattering

Photons can interact with magnons, i.e. the elementary quanta of spin waves,via inelastic photon scattering which is known under the name of Brillouinlight scattering. Generally, two different scattering processes are distinguished,in which a magnon is created (Stokes process) or annihilated (anti-Stokes pro-cess), resulting in an energy shift of the scattered photon (Fig. 2.12). Bothphoton-magnon scattering processes satisfy the laws of energy and momentumconservation:

i= s , (2.24)ki= ks

k . (2.25)

In these expressions, i,s and ki,s denote the frequency and the wave vector ofthe incoming and the scattered photon, respectively. The parameters describingthe magnon are without any index. For light scattering from thin films theperpendicular component of the wave is not conserved anymore due to thebreak of translational symmetry. In this case, Eq. (2.25) is only valid for kwhich is the wave vector component parallel to the film plane.

Besides the one-magnon scattering processes schematically shown inFig. 2.12, higher order scattering processes exist, in which more than onemagnon are involved. Since the latter are not relevant for the present thesis,

they will not be considered further. In the literature, photon-magnon inter-actions are usually treated separately from the magneto-optical effects such

, k

i i, k

s=i-k =k-ks i

, k

i i, k

s=i+k =k ks i+

Stokes process anti-Stokes process

Figure 2.12: Schematic representation of photon-magnon scattering processes. A

magnon can be created (Stokes process) or annihilated (anti-Stokes process) resulting

in a gain or loss in energy and momentum of the scattered photon. i andki denote

frequency and wave vector of the incoming photon, s andks describe the scattered

photon and andk are assigned to the magnon.

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as Kerr or Faraday effect. However, they are closely related to each other.In the quantum mechanical picture, the magnon scattering and the classical

magneto-optical effects can be described by a common Hamiltonian [47, 48].But even without going into the details of the quantum mechanical treatment,the analogy between these effects can be shown as will be demonstrated in thefollowing.

Macroscopic description

In a classical treatment, the light scattering from a solid occurs due to theelectric polarization introduced in the solid by the external optical electric field.The relationship between the polarization Pand the electric field E is givenby

Pi= 0j

ijEj, (2.26)

whereij is the susceptibility tensor. The above expression can be rewritten asa function of the permittivity tensor introduced in Sect. 2.2.1 using the relationij =ij +ij. At this point, the common origin of the Brillouin light scatteringand the MOKE effect becomes evident. In the presence of spin waves, the off-diagonal permittivity tensor elements,i.e.the elements responsible for the lightscattering of magnons, have a similar form to Eq. (2.12), the only modification

being that the components of the static and dynamic magnetization vectorsMs and m(t) have to be used instead ofML and MT.

For crystals with cubic symmetry probed in the Damon-Eshbach geometry(Fig. 2.11), where the direction of the applied magnetic field and of the staticcomponent of magnetization is defined as thex-axis and the in-plane componentof the spin wave vector q lies along the y-axis, the microscopic spins precessabout thex-axis and the dynamic magnetization has only two components myand mz. In this case the off-diagonal components ofij related to spin wavescan be expressed as [49]

xy = yx =K mz+ 2G44Msmyxz = zx = Kmy+ 2G44Msmz,

(2.27)

where K and G44 are the components defined in conjunction with MOKE inSect. 2.2.1. It should be noted that in deriving Eq. (2.27) Ms was assumedto lie along one of the cubic crystalline axes and terms quadratic in mi wereneglected. The latter is justified since|m(t)| |Ms|.

Substituting Eq. (2.27) in Eq. (2.26) and assuming a linearly p-polarized

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incident wave (i.e. Ex = 0; Ey,Ez= 0) one obtains only one component ofP:

Px = (2G44MsEy KEz)my+ (KEy+ 2G44MsEz)my.

(2.28)

For thes-polarized incident wave having only the component Ex, the substitu-tion of Eq. (2.27) in Eq. (2.26) yields two components ofP:

Py = 2G44MsExmy KExmz,Pz =K Exmy+ 2G44MsExmz

(2.29)

These relations imply the well-known fact that an incident s-wave changes toa p-wave upon the interaction with a magnetic excitation and vice versa. Itis interesting to note that for the incoming photon the dynamic modulationof polarization expressed in Eq. (2.28) and (2.29) represents a phase gratingpropagating with the velocity of spin wave. Therefore, the photon-magnoninteraction can be understood in a classical picture as Bragg scattering froma moving grating resulting in the Doppler shift of the frequency of the Bragg-scattered light:

s=i k v i . (2.30)Vectors k and v in this expression denote the wave vector and the velocity ofthe spin wave.

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Chapter 3

Half-metallic Heusler

compounds

The interest in the intermetallic compounds with the general formula X2YZ,where X and Y are transition metals and Z an element from the III-V groups,emerged in 1903, when F. Heusler discovered Cu2MnAl to be ferromagnetic eventhough it consists only of non-ferromagnetic elements [50]. Since this discovery,numerous investigations were made to clarify the chemical and magnetic orde-

ring in these materials [5154] and different models were proposed to explain themicroscopic interactions leading to the appearance of ferromagnetism [5557].It was also found that some of the ternary intermetallic compounds with thechemical composition XYZ exhibit a ferromagnetic behavior as well. Todaythis class of materials is known as half- or semi-Heusler compounds, whereasthe X2YZ compounds are named full-Heusler compounds. Among the half-Heusler compounds, PtMnSb attracted particular attention due to its extremelylarge magneto-optical Kerr rotation [58], which made this compound a suitablematerial for the magneto-optical recording technology.

The discovery of the half-metallic property in NiMnSb and related com-pounds by de Groot et al. in 1983 [2] launched many numerical works devotedto the prediction of other half-metallic materials. Several other Heusler com-pounds turned out to posses the half-metallic property, as suggested from abinitio calculations. Among them are the Co2-based Heusler compounds, whichgained a particular interest and became most widely studied in this field dueto their high Curie temperatures. However, despite the huge achievements thathave been obtained in the last decade in magnetic tunnel junctions (MTJs)using Co2YZ compounds as electrodes [5964], an experimental proof of thehalf-metallicity in these materials is still missing. Different mechanisms that

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26 Half-metallic Heusler compounds

are thought to explain the suppression of the half-metallic state will be pre-sented in this Chapter. The main goal of this Chapter, however, is to provide

an overview of the crystallographic, electronic and magnetic properties of thefull-Heusler compounds where X=Co. A special attention will be given to thematerials studied within this work, which are Co2MnSi, Co2FeAl, Co2FeSi andCo2Cr0.6Fe0.4Al (CCFA in the following).

3.1 Structural properties of Co2-based Heusler

compounds

Heusler compounds with the general formula Co2YZ crystallize in the L21struc-ture which is shown in Fig. 3.1. The cubic unit cell consists of four interpene-trating fccsublattices, two of which are occupied by Co atoms, and the othertwo by the Y and Z atoms, respectively. The two Co sublattices are positionedat (0,0,0) and ( 1

2,12

,12

) in the unit cell, while those of Y and Z are at positions(14

,14

,14

) and (34

,34

,34

), respectively. The L21 structure represents the most or-dered phase of the Co2-based Heusler compounds. However, partial disordercan exist for a given chemical composition, in the form of interchange of atomsbetween different sublattices. If the ( 1

4,14

,14

) and ( 34

,34

,34

) sites are randomlyoccupied by either of Y or Z with a 50 % probability, the structure is referredto as the B2 structure. The interchange of atoms in Co and Y sublattices re-sults in the DO3 type of disorder. A random occupation of all four sublatticesleads to a structure which is called the A2 structure [6567].

Experimentally, the information about the structural properties, such as lat-

Co

Y

Z

Figure 3.1: L21 structure of Co2YZ Heusler alloys. The lattice consists of four

interpenetrating fcc sublattices, two of which are occupied by Co, the other two by

Y and Z atoms, respectively.

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3.1 Structural properties of Co2-based Heusler compounds 27

tice constants and the type of disorder present in the Heusler samples, can beobtained by x-ray and neutron scattering. The bulk lattice parameters of Co2-

based Heusler compounds, which are studied within this work or appear in thediscussions of experimental results in Chapts. 5 and 6, are listed in Table 3.1. Itis interesting to note that there is a systematically smaller lattice constant forSi containing compounds. Structural information concerning atomic disorderutilizing both x-ray and neutron scattering is provided by the analysis of rela-tive intensities of superstructure reflections, as will be discussed in more detailin Chapt. 4. Since the intensities of neutron diffraction peaks can be deter-mined in general with a much greater accuracy than those of the correspondingx-ray peaks [54], neutron scattering is a more favorable technique for the quan-titative determination of the degree and type of disorder. However, a neutron

source is not easily accessible. Thus, neutron scattering cannot be routinelyused for sample characterization. X-ray diffraction, which is easily accessible,suffers from the drawback of a very small difference in the atomic scatteringfactors between Co and second transition metal Y, to which the superstructurereflections are sensitive (Chapt. 4). Therefore, additional characterization byother techniques is usually necessary to eliminate possible ambiguities. In thiscontext, local experimental techniques such as x-ray absorption spectroscopy(XAS) in combination with circular magnetic dichroism (XMCD), Mossbauerspectroscopy and nuclear magnetic resonance spectroscopy (NMR), which aresensitive to the short-range order, are commonly used. In particular, NMR was

Compound NV (e/f.u.) Structure aexp (A) Ref.

CCFA 27.8 B2 5.737 [68]

Co2MnAl 28 B2 5.756 [5]

Co2MnSi 29 L21 5.654 [5]

Co2MnGe 29 L21 5.743 [5]

Co2FeAl 29 B2 5.730 [68]

Co2

FeGa 29 L21 5.741 [69]

Co2FeSi 30 L21 5.640 [70]

Table 3.1: Experimental bulk lattice parameters of selected Co2YZ Heusler com-

pounds. The listed compounds are sorted by the increasing number of valence elec-

trons NV (see also Sect. 3.3.1). Compounds studied within this thesis are marked

with (*).

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demonstrated to a be a very powerful method for the characterization of atomicdisorder [7173].

In order to obtain Heusler films with the well-ordered L21 structure, hightemperature annealing is usually required since the B2/L21 pase transition inthe bulk occurs at about 1000K in most of the Co2-based Heusler compounds[7476]. It is interesting to note, however, that the L21 phase is not alwaysthe most stable one from the thermodynamic point of view. Regarding theseries of materials studied in this thesis, CCFA preferably crystallizes in the B2structure [77] while L21 structure is more stable for Co2MnSi and Co2FeSi.

3.2 Half-metallicityThe concept of a half-metal was first introduced by de Groot et al. on thebasis of band structure calculations for the NiMnSb half-Heusler compound [2].This term is employed to describe material systems with an asymmetry of thespin-split band structure: while the majority spin band has a metallic behavior(i.e. a non-vanishing density of states (DOS) at the Fermi level), the minorityelectrons exhibit a semiconducting character (i.e.a band gap around the Fermilevel) (Fig. 3.2). The conduction electrons are thus 100% spin polarized.

Since the discovery of the half-metallicity in NiMnSb, many other materi-

als have been theoretically predicted to be half-metals, among them the Co2-based Heusler compounds. First theoretical evidence of the half-metallicityin Co2MnSi was given by Ishida et al. [78] and was later confirmed by variousauthors [7982]. Kubleret al. reported for the first time the existence of a bandgap for the minority electrons in the related Co2MnAl Heusler compound [83].In this compound, however, the Fermi energy falls into the upper tail of thevalence band and a non-vanishing spin-down DOS is found at the Fermi level(Fig. 3.2(b)). More recent calculations showed similar results for the electronicstructure of this material [80, 82, 84]. Nevertheless, Co2MnAl remains an in-teresting candidate for applications because its expected spin polarization isstill comparatively high. The half-metallic behavior of the Co2FeSi Heuslercompound has been reported by Wurmehl and collaborators [70] and Gercsiet al. [11]. In case of the Al containing counterpart, different results of bandstructure calculations can be found in the literature. While no half-metallicstate is found for Co2FeAl in Refs. [77,85], a clear band gap for minority spinsis shown in Refs. [11,82].

The differences in the reported electronic band structures mainly resultfrom different computational schemes used in ab initio calculations by diffe-rent authors. Most of the theoretical investigations are based on the density

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3.2 Half-metallicity 29

Co MnSi2

Co MnAl2

-6 -3 30

4

0

-4

4

0

-4 (b)

(a)

Energy (eV)

DOS

(states/eV)

Figure 3.2: Spin-resolved density of states (DOS) for (a) Co2MnSi and (b) Co2MnAl

[86]. The upper and lower panels in (a) and (b) show the DOS for the majority

(spin-up) and minority (spin-down) electrons, respectively, as indicated by the red

arrows.

functional theory (DFT), which expresses the ground state energy of an atomicsystem as a function of its electron density [87]. The main problem of DFTis that the exact potentials describing the exchange interaction of the elec-trons are not known and approximations are required. The basic assumptionunderlying these approximations is a delocalized or itinerant behavior of thevalence electrons. However, as will be discussed in Sect. 3.3.1, the Co2-basedHeusler compounds exhibit characteristics of systems with localized magneticmoments. Therefore, additional corrections such as electron-electron correla-tion potentials must be introduced in the calculations, which take into accounta partial localization of the valence electrons [82, 88]. This very rough insightinto the band structure calculations of the Heusler compounds makes evidentthat the prediction of the half-metallic state strongly depends on the detailsof the computational approach. In particular, the width and the position of

the minority band gap with respect to the Fermi level might vary dependingon the calculation scheme. It should be noted as well, that in most cases thehalf-metallic property is predicted for the ground state only, i.e. for T=0 K.

The electronic structure of Co2YZ Heusler compounds and their physicalproperties might be tuned by alloying with a fourth element, which opens anew way to design materials with desired characteristics [4,11,75,89,90]. In thequaternary alloys, the atoms in one of the sublattices are partially substitutedby the atoms of another element. CCFA, obtained from Co2CrAl by doping

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with Fe atoms, represents one of the most prominent examples in this classof Heusler compounds. In this case, a partial substitution of Cr resulted in a

large magnetoresistance effect of about 30% which was not observed for theinitial ternary compound [91]. Miura et al. investigated in detail the completesubstitutional series Co2Cr1xFexAl from the theoretical point of view. Similarto the case Co2MnAl, a non-vanishing DOS is expected in the minority bandof Co2Cr0.6Fe0.4Al according to these calculations, for both B2 and L21 orderedphases.

3.2.1 Origin of the band gap in Co2-based Heusler com-

pounds

Even though Heusler compounds are known for nearly a century, the mecha-nisms responsible for their particular electronic structure remained unexplainedfor a long time. In case of Co2-based compounds, it was just in the year 2002that Galanakis and collaborators proposed a mechanism leading to the appear-ance of the minority band gap which became generally accepted by the Heuslercommunity. As has been pointed out in their original work, the gap in theminority DOS of Co2-based Heusler compounds results from the hybridizationofd electrons taking place between the Co and X atoms as well as between theCo atoms sitting on second-neighbor positions [80]. In the following this will

be discussed in more detail.Figure 3.3 schematically shows the behavior ofd electrons for X=Mn. In afirst step the five d orbitals of each Co atom couple according to their symme-try group representations and form bonding (eg,t2g) and antibonding (eu,t1u)orbitals. In a second step the interaction between the hybridized Co states andthe d states of the Mn is considered. The doubly-degenerate Co eg orbitals hy-bridize with thee (dz2,dx2y2) orbitals of the Mn, creating a doubly-degeneratebonding and antibondingegstates, that are situated below and above the Fermilevel, respectively. The t2g Co orbitals couple with the t2 (dxy, dyx, dzx) statesof the Mn atom, giving rise to a low-lying triplet t2g state with a bonding char-acter and a triplet antibonding t2g state above the Fermi level. The remainingantibonding Co orbitals (eu and t1u) do not hybridize, since no states exist onthe Mn atom with the same symmetry group representation, and thus remainnon-bonding. The Fermi level falls between these two non-bonding Co orbitals.Therefore, a real minority gap exists in the Co2YZ Heusler compounds. Thesize of the gap is largely determined by the Co-Co interaction. It should benoted here, that in principle the same hybridization of electronic states mayoccur for the majority electrons as well, opening a band gap in the spin-upDOS. The exchange splitting, however, shifts the spin-up states of Mn to lower

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dxy,dyx,dyz

t2g

eg

,dx -y2 2dz2

dxy,dyx,dyz

,dx -y2 2dz2

eu

t1uCo Co

t2g

eg

eu

t1u

dxy,dyx,dyz

,dx -y2 2

dz2

eg

t2g

eg

t2g

Co-Co

Mn

EF

Figure 3.3: Schematic illustration of the origin of the minority band gap in Co2YZ

Heusler compounds proposed by Galanakis et al. [80].

energies where they form a common band with the Co states.

In the above discussion the Z atom has been completely disregarded. This isjustified by the fact that its low-lyingsandpstates do not directly contribute tothe formation of the minority band gap. However, since these states contributeto the total number of occupied and empty states, the Z atom is importantfor the position of the Fermi level within the minority band gap. In Fig. 3.2

for example, a clear shift of the Fermi level towards the conduction band isvisible when Al is substituted by Si in the Co2MnZ compound. Moreover, the sand pstates have been shown to play an important role for the distribution ofelectrons in the various symmetry distinguished states (t2g and eg) at Co andY sites [82]. This is of particular importance in the discussion of the specificmagnetic moments at the Co and Y sites as will be shown in Sect. 3.3.2.

3.2.2 Effects destroying half metallicity

We already mentioned before, that ab initio predictions of the half-metallicproperty for Co2-based Heusler compounds are generally made for the groundstate. Moreover, a bulk material is usually assumed in the theoretical inves-tigations, and coupling of the spin to the orbital moment is neglected. Half-metallicity has yet to be proven in thin films of Co2YZ Heusler compoundseven though high tunnel magnetoresistance (TMR) ratios corresponding to re-latively high spin polarization of the conducting electrons have been achievedat low temperatures in MTJs using these materials as electrodes [5964]. Thereason for the lack of the experimental evidence of the half-metallic state, i.e.

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100% spin polarization, are believed to be additional electronic states appear-ing in real thin film samples in the minority DOS and closing the band gap.

These additional states within the band gap might be introduced by variousmechanisms, which will be briefly described in the following. The mechanismsleading to the reduction of the theoretically predicted 100 % spin polarizationof Heusler compounds can partially be expected to play also an important rolefor the exchange interactions and magnetic anisotropies in these materials.

Disorder effects

The effect of structural disorder, such as antisites (replacement of one kind of

atom by another) and swaps (interexchange of atoms in different sublattices)on the half-metallic property of Co2-based Heusler compounds has been studiedtheoretically by various authors [10,11,77,84,90,92,93]. It has been found thatthe electronic structure of these materials can be significantly modified by thepresence of structural disorder, including a complete destruction of the half-metallic band gap. This detrimental effect arises from new minority electronicstates introduced at the Fermi level by the imperfections in the crystal lattice.The appearance of the new minority states can be understood as follows. Forthe hybridization of electronicdstates at Co and Y atoms leading to a gap in theminority DOS, the crystal field symmetry plays a crucial role (see Fig. 3.3 above

and related discussion). Swapping of the atoms or antisites induce changes inthe crystal field symmetry at a given Co or Y site, which results in additionalsplitting of the d orbitals. Hence, new states can appear in the region of theminority band gap.

The degree of the destructive impact of disorder varies substantially de-pending on the kind of disorder and the material system under consideration.For example, for Co2MnSi Picozzi and collaborators have shown that only inthe case of Co antisites (replacement of Mn by Co) is the half-metallic propertydestroyed due to the formation of a peak in the minority DOS at the Fermilevel [10]. In the case of Mn antisites (replacement of Co by Mn) as well asfor Co-Mn and Mn-Si swaps, the half-metallicity is preserved. For the samematerial system Galanakis and collaborators have demonstrated that a partialsubstitution of Mn by Si atoms (and vice versa) up to 20% does not destroythe half-metallic character of the Co2MnSi Heusler compound [86]. Similarresults have been reported for Co2FeSi by Gercsi et al. [11] and are shown inFig. 3.4. The half-metallicity is preserved when up to 25% of B2 disorder (dueto Fe-Si swaps) is introduced in the crystal structure, even though additionalelectronic states start to appear at the lower edge of the minority conductionband (Fig. 3.4(a)). The DO3 type of disorder broadens the gap, but 100% spin

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E-E (eV)F

DOS

(states/eV)

-6 -4 0-2-5 -3 -1 1 2

-40

-40

0

-40

20

-20

-20

0

20

0

20

-20

40

(a)

(b)

(c)

L21 B2 (25%)

DO (12.5%)3

A2 (12.5%)

Figure 3.4: Spin-resolved density of states (DOS) of Co2FeSi Heusler compound with

(a) B2, (b) DO3 and (c) A2 types of disorder taken from Ref. [11]. DOS of the L21

ordered phase is shown in grey.

polarization at the Fermi level is still conserved (Fig. 3.4(b)). In the presenceof the A2 type of disorder, however, new states appear at the Fermi level, thusclosing the half-metallic gap (Fig. 3.4(c)). In this context it is worth mention-ing that the DO3 type of disorder significantly increases the total energy of the

system, and thus is thought to be unlikely to happen [10,77].

Spin-orbit coupling

Even in an ideally prepared single crystal, electronic states in the half-metallicgap of the minority band may be introduced due to spin-orbit coupling. Thespin-orbit interaction couples the majority and minority spin channels to oneanother [94, 95]. This has the consequence that spin-flip scattering processescan take place, producing additional states in the gap of the minority band.Thus, 100% spin polarization cannot exist even in a hypothetically ideal crystalin the ground state. It should be noted, however, that the effect of spin-orbitinteraction on the spin polarization of Co2-based Heusler compounds is verysmall due to comparatively small orbital magnetic moments (see Sect. 3.3.5)and the spin polarization at the Fermi level can still be very high. Sargolzaeiand collaborators have demonstrated in their theoretical studies that in caseof Co2MnSi, the reduction of the spin polarization due to this effect is of only3% [96]. While the spin polarization is 100% without taking the spin-orbitcoupling into account, it amounts to 97% when the spin-orbit interaction is

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included in the band structure calculations. In case of Co2MnAl a decrease from97% to 95% is reported. At the edges of the band gap the effect of spin-orbit

interaction can become more pronounced since there exist spin-down states andtherefore the probability for the spin-flip scattering of the majority electronsis higher compared to the middle part of the band gap. The position of theFermi level within the half-metallic band gap will therefore largely determinethe impact of this effect on the spin polarization.

Temperature effects

At finite temperature there are mainly two effects giving rise to additional statesin the minority gap and thus destroying the half-metallic character. First,

thermal fluctuations of individual spins create a certain spin disorder and non-collinearity effects appear [97, 98]. In regions with short-range order, the localspin quantization axis is not necessarily parallel to the average moment di-rection. Moreover, spin axis of each atom can vary with respect to that of itsneighbors if the short-range order is not perfect. The latter is more significant inmulticomponent systems like Heusler compounds. Due to the non-collinearityof spins at T > 0, there always exists a partial projection of a spin-up wavefunction of each atom onto the spin-down states of its neighbors generatingnew states in the minority band gap. Note that in this case the non-vanishingDOS in the gap of the minority spin channel is symmetric with respect to the

Fermi energy, i.e. new states are generated for both electrons and holes. Thesecond source of depolarization are the so-called nonquasiparticle (NQP) stateswhich originate from a superposition of spin-up electron excitations and virtualmagnons [7, 9, 99]. In contrast to the non-collinearity effects, the NQP statesresult in an asymmetric band gap filling. At low temperatures these states areintroduced just above the Fermi energy. Note that if magnetic anisotropies aretaken into account, the cut-off of the NQP states is slightly above the Fermienergy. At elevated temperatures the distribution of the NQP states is broad-ened and crosses the Fermi energy. This results in the loss of the half-metallicproperty. Recently, experimental evidence of NQP states in the Co2MnSi Heus-ler compound has been given by MTJ spectroscopy measurements [100].

Apart from non-collinearity effects and the appearance of NQP states, othereffects leading to the finite temperature degradation of half-metallicity are dis-cussed in literature. For example, a change in hybridization strength betweenthe d states at elevated temperatures is proposed by Lezaic and collabora-tors [98], leading to a shift of the conduction and valence band edges in theminority DOS. This has the consequence that the Fermi energy is no longerlocated in the gap but falls into the minority conduction band and the half-metallicity is lost. Moreover, the defect densities will increase at T >0 leading

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to changes in the electronic structure described in the beginning of this Sub-section.

Surfaces and interfaces

At surfaces and interfaces the half-metallic property can be lost due to theintroduction of surface and interface states at the Fermi level in the mino-rity spin channel. Amongst the Co2-based Heusler compounds investigated inthis thesis, Co2MnSi is the one whose surface and interface properties havebeen most widely studied by ab initio calculations [78,101105]. For example,Galanakis has shown that the half-metallic character of this compound is lostat both Co- and MnSi-terminated surfaces [101]. A lower coordination num-ber of Co and Mn atoms at the surface reduces the hybridization between theCo minority d states and the Mn ones. For Co-terminated surfaces, the Cospin-down bands are shifted to higher energies with the effect that the Fermienergy falls in the middle of thed band. For MnSi-terminated surfaces the un-occupiedd-like Mn states shift to lower energies closing the minority band gap.Hashemifar and collaborators, however, have shown that the half-metallicityof the Co2MnSi Heusler compound can be preserved if the surface has theMnMn-termination [102].

The half-metallicity of the Co2MnSi/MgO and Co2MnSi/GaAs interfaceshave been theoretically studied as well [103105]. The former is of particularimportance for magnetic tunnel junctions, whereas the latter is considered forspin injection into semiconductors. Similar to the discussion of surfaces, in-terface state appear in the half-metallic gap because of the deviating atomicenvironment of Co and Mn atoms in the topmost layer. These new states atthe Fermi level can be filled by the electrons from the minority valence bandthrough inelastic scattering processes such as electron-phonon, electron-magnonand electron-electron scattering [106] and thus make the spin-down electronscontribute to the total conductivity of the interface. Therefore, the polarization

of the curren

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