Spin Structure of Domain Walls and Their Behaviour in Applied Fields and Currents

and Their Behaviour

in Applied Fields and Currents

DISSERTATION

ZUR ERLANGUNG DES AKADEMISCHEN GRADES DES DOKTORS DER NATURWISSENSCHAFTEN

AN DER UNIVERSITÄT KONSTANZ

MATHEMATISCH-NATURWISSENSCHAFTLICHE SEKTION FACHBEREICH PHYSIK

VORGELEGT VON DIRK BACKES

TAG DER MÜNDLICHEN PRÜFUNG: 8. FEBRUAR 2008 REFERENTEN:

PROF. DR. ULRICH RÜDIGER PROF. DR. JENS GOBRECHT

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5214/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-52141

In this thesis, the spin structure of domain walls in confined magnetic elements was determined and the behaviour of domain walls on the ap- plication of external magnetic fields and current pulse injection was ob- served. For this, magnetic elements of various shapes, materials, and substrates were prepared which were each dedicated to the purpose of the experiment. Different fabrication techniques were described, includ- ing several patterning processes and contacting of magnetic elements for current-pulse experiments.

The spin structure of transverse walls in constrictions down to 30 nm in permalloy wavy lines was measured using electron holography. It is known that the domain wall spin structure influences both the magne- toresistance and spin transfer torque effects. For the transverse walls, three different types were determined and the domain wall width was found to decrease faster than linearly with decreasing constriction width.

The magnetic fields needed to depin the domain walls from such constric- tions were measured using magnetoresistance measurements and were directly related to the domain wall spin structure.

While an approach was made to image the spin structure of patterned thin films of the halfmetal CrO2 coupled to a permalloy thin film us- ing photoemission electron microscopy (PEEM), a complicated coupling mechanism between the two layers was observed.

The spin structure of an array of crossed wires corresponding to an array of holes (antidots) in a cobalt thin film was investigated to under- stand the switching behaviour on the application of an external magnetic field. It was found that switching occurs by the nucleation and propaga-

tion of domain chains, and the moving chain ends can be pinned when they meet the ends of perpendicular domain chains or are blocked due to the formation of a 360^◦domain wall when they approach a perpendicular domain chain.

The current-induced domain wall motion in contacted permalloy wi- res on membrane samples was studied using electron holography. Effects due to the spin torque were separated from heating effects on the domain wall due to the current. A variety of effects such as domain wall transfor- mations, domain wall jumping between two pinning sites, or structural changes of the magnetic material crystallites due to the combined influ- ence of current pulses and heating was observed. A set of indicators was derived to distinguish current-induced domain wall motion due to the spin torque from the heating effects.

In dieser Arbeit wurde die Spinstruktur von Domänenwänden in mag- netischen Nanostrukturen und das Verhalten dieser Domänenwände un- ter dem Einfluß von magnetischen Feldern und elektrischen Strompulsen untersucht. Dazu waren magnetische Nanostrukturen nötig, die sich in Geometrie, Material, und verwendetem Substrat unterschieden, und die dem jeweiligen Zweck des Experiments angepaßt waren. Verschiedene Herstellungsmethoden wurden aufgeführt und einige Strukturierungs- prozesse vollständig beschrieben.

Die Spinstruktur von transversen Domänenwänden in Konstriktio- nen, die sich in Permalloy-Drähten befanden und die bis auf 30 nm ver- kleinert werden konnten, wurde mittels Elektronenholografie gemessen.

Es ist bekannt, daß die Spinstruktur der Domänenwand den Magneto- widerstand und auch den Spin-Torque-Effekt beeinflußt. Es wurden 3 verschiedene Typen transverser Wände gefunden, deren Domänenwand- breite schneller als linear mit der Konstriktionsbreite fiel. Durch Mag- netowiderstandsmessungen wurden die magnetischen Felder bestimmt, die nötig waren, um die Domänenwände von den Konstriktionen weg- zubewegen, und die Stärke dieses Feldes wurde mit der Spinstruktur in Verbindung gebracht.

Es wurde versucht, die Spinstruktur von strukturierten dünnen Fil- men aus dem Halbmetall CrO2 mit Photoemissionselektronenmikrosko- pie (PEEM) abzubilden. Bei diesem Versuch wurde ein komplizierter Kopplungsmechanismus zwischen dem CrO2-Film und einer darauf de- ponierten dünnen Schicht Permalloy beobachtet.

Die Spinstruktur von einem dünnen Cobalt-Film mit einem Lochmus-

ter (Antidot) wurde abgebildet, um das Umklappverhalten in einem ex- ternen magnetischen Feld zu verstehen. Dabei stellte sich heraus, daß das Umklappen durch Nukleation und Propagation von Domänenketten er- folgt. Diese sich bewegenden Domänenketten können gepinnt werden, wenn sie auf die Enden von senkrecht dazu laufende Domänenketten treffen, oder geblockt werden, indem sie mit senkrecht dazu laufenden Domänenketten eine 360^◦ Domänenwand bilden.

Permalloy-Drähte wurden auf Membranproben aufgebracht und kon- taktiert, um strom-induzierte Domänenwandpropagation mittels Elek- tronenholografie zu beobachten. Spin-Torque-Effekte wurden von Effek- ten aufgrund thermischer Anregung unterschieden. Eine Reihe von Ef- fekten wie Domänenwandtransformation, Domänenwandsprünge zwi- schen 2 Pinningzentren und sogar strukturelle Änderung der magneti- schen Kristallite können durch die thermische Anregung durch den Strom ausgelöst werden. Indikatoren wurden abgeleitet, damit zwischen strominduzierter Domänenwandpropagation, die auf den Spin-Torque- Effekt zurückzuführen sind, und Effekten aufgrund thermischer Anre- gung unterschieden werden kann.

Summary / Zusammenfassung i

List of Figures xi

List of Acronyms xii

Introduction 1

1 Theory 4

1.1 Microscopic Origins of Ferromagnetism . . . 5

1.1.1 Exchange Interaction . . . 5

1.1.2 Localized Model and Mean Field Approximation . 6 1.1.3 Band Model of Ferromagnetism . . . 6

1.2 Micromagnetic Systems . . . 9

1.2.1 Thermodynamics in Magnetism . . . 9

1.2.2 Energy Contributions . . . 10

1.2.3 Brown´s Equations and the Effective Field . . . 13

1.3 Magnetization Dynamics . . . 15

1.3.1 Landau-Lifshitz-Gilbert Equation . . . 15

1.3.2 Micromagnetic Simulations . . . 17

1.4 Spin Transfer Torque Model . . . 18

1.4.1 Spin Transfer Torque . . . 19

1.4.2 Landau-Lifshitz-Gilbert Equation with Spin Trans- fer Torque . . . 20

1.4.3 Adiabatic and Non-adiabatic Spin Torque . . . 21

1.4.4 Domain Wall Spin Structure Modifications . . . 24

1.5 Anisotropic Magnetoresistance (AMR) . . . 25

1.5.1 AMR Contribution to the Domain Wall Magnetore- sistance . . . 27

1.6 Halfmetallic Ferromagnets . . . 28

1.7 RKKY Coupling . . . 30

2 Fabrication 31 2.1 Patterning the Resist . . . 31

2.1.1 Electron Beam Writer . . . 32

2.1.2 Resist Technology . . . 35

2.1.3 Photolithography . . . 38

2.1.4 X-ray Interference Lithography . . . 39

2.2 Pattern Transfer Methods . . . 40

2.2.1 Deposition . . . 40

2.2.2 Properties of Films and Nanostructures . . . 44

2.2.3 Lift-off . . . 46

2.2.4 Etching . . . 48

2.3 Fabrication of Magnetic Structures and Devices . . . 49

2.3.1 Ferromagnetic Rings . . . 50

2.3.2 Wavy Lines with Constrictions . . . 50

2.3.3 Rings on Striplines . . . 52

2.3.4 Contacted Zigzag Lines . . . 53

2.3.5 Contacted Notched Rings with Antenna . . . 57

2.3.6 Patterning Techniques for Epitaxial Films . . . 61

3 Measurement Techniques 66 3.1 XMCD-PEEM . . . 67

3.1.1 X-ray Magnetic Circular Dichroism . . . 67

3.1.2 Photoemission Electron Microscopy (PEEM) . . . . 68

3.2 Lorentz Microscopy . . . 70

3.3 Electron Holography . . . 71

3.4 Kerr Microscopy . . . 74

3.5 Magnetoresistance Measurements . . . 76

4 Domain Walls in Confined Systems 78 4.1 Introduction to Domain Walls . . . 80

4.2 Transverse Domain Walls in Nanoconstrictions . . . 83

4.2.1 Introduction . . . 83

4.2.2 Domain Wall Types . . . 84

4.2.3 Domain Wall Type Distribution . . . 91

4.2.4 Domain Wall Width . . . 92

4.2.5 Comparison with Heisenberg Simulation . . . 94

4.2.6 Conclusion . . . 97

4.3 Pinning of Domain Walls . . . 98

4.3.1 Introduction . . . 98

4.3.2 Magnetoresistance Measurement . . . 99

4.3.3 Spin Structure of Domain Walls . . . 101

4.3.4 Simulations . . . 103

4.3.5 Conclusions . . . 105

4.4 Spin Structure of CrO2 . . . 106

4.4.1 Introduction . . . 106

4.4.2 Imaging of a Single Layer of CrO2 . . . 106

4.4.3 Indirect Imaging using a Py layer . . . 108

4.4.4 Conclusions . . . 113

5 Antidots 115 5.1 Introduction . . . 115

5.2 Experimental details . . . 116

5.3 Magnetization reversal . . . 118

5.4 Magnetic spin configurations . . . 124

5.5 Detailed reversal mechanism . . . 125

5.6 Size dependence of reversal . . . 129

5.7 Conclusions . . . 131

6 Effect of Current and Heating on Domain Walls 132

6.1 Introduction . . . 132

6.2 Temperature Effect on Spin Structure . . . 133

6.3 Current-induced Heating . . . 135

6.3.1 Transformation of the Spin Structure . . . 136

6.3.2 Domain Wall Motion due to Heating . . . 137

6.3.3 Vortex Annihilation . . . 138

6.3.4 Structural Changes by Heating . . . 139

6.4 Heat Conductance Improvement . . . 141

6.5 Conclusions . . . 143

7 Conclusions 145

Bibliography 149

Publication List 167

Acknowledgement 172

1.1 Sketch of the band model . . . 8

1.2 AMR contribution of a domain wall . . . 27

1.3 Spin-dependent density of states for a 3d-ferromagnet and for a halfmetallic ferromagnet . . . 28

2.1 Scheme of the column of the electron beam writer . . . 32

2.2 Scheme of Bézier curves . . . 35

2.3 Overview over exposure methods of photolithography . . 38

2.4 Scheme of x-ray interference lithography . . . 39

2.5 Scheme of the DC magnetron sputtering facility . . . 41

2.6 Scheme of a UHV system used for MBE growth . . . 43

2.7 Scheme of a CVD setup . . . 44

2.8 Sample holder for membrane lift-off . . . 46

2.9 Scheme of pattern writing and lift-off . . . 49

2.10 SEM images of a wavy line with notch forming a constriction 51 2.11 SEM images of a gold waveguide with permalloy elements 52 2.12 SEM images of contacted zig zag lines on silicon substrates 54 2.13 Contacted zig zag lines on membrane substrates . . . 56

2.14 Scheme of the evaporation of gold contacts on the edges of the membrane . . . 57

2.15 Scheme of a contacted notched ring with antenna . . . 58

2.16 Scheme of the overlay procedure using the GLOKOS routine 59 2.17 SEM image of a contacted notched ring with antenna . . . 61

2.18 Overview of dry etching techniques for patterning epitax- ial films . . . 62

2.19 SEM images of Fe3O4 rings and wires . . . 64

3.1 Illustration of the X-ray magnetic circular dichroism (XMCD) effect . . . 67

3.2 Scheme of a photoemission electron microscope (PEEM) . 68 3.3 Scheme of Lorentz microscopy techniques . . . 70

3.4 Scheme of electron holography . . . 72

3.5 Magnetic induction map of three-quarter rings . . . 74

3.6 Scheme of different MOKE effects . . . 75

3.7 Scheme of a Kerr microscope . . . 76

3.8 Scheme of a setup for pulse injection and resistance mea- surement . . . 77

4.1 Scheme of a Bloch and a Néel wall . . . 80

4.2 PEEM images and simulations of vortex and transverse walls 81 4.3 Scheme of a constriction formed by a notch . . . 84

4.4 Magnetic induction maps showing three transverse wall types . . . 85

4.5 Line profiles through a constriction . . . 87

4.6 Illustration of the fixed threshold method . . . 89

4.7 Shape reconstruction from the threshold positions . . . 90

4.8 Distribution of transverse wall types . . . 91

4.9 Dependence of the domain wall width on the ring width . 92 4.10 Dependence of domain wall angle and width on constric- tion width . . . 94

4.11 Heisenberg simulations of the spin structure in constrictions 96 4.12 SEM image of a wavy line with a notch . . . 99

4.13 Dependence of depinning fields on constriction width . . . 100

4.14 Direct imaging of spin structures around notches . . . 102

4.15 Energy potential around a notch forming a constriction . . 103

4.16 XAS and XMCD spectra of a CrO2film . . . 107

4.17 Remanent magnetization in a patterned CrO2/Py film af- ter saturation in easy axis direction . . . 108

4.18 Remanent magnetization after saturation in hard axis di-

rection . . . 109

4.19 Scheme of a CrO2wire . . . 111

4.20 An exeption for the observed remanent magnetization . . 112

5.1 Scheme of the antidot geometry . . . 116

5.2 Hysteresis loops . . . 117

5.3 XMCD images of domain chains . . . 118

5.4 Magnetization reversal via domain chains . . . 120

5.5 Magnetization reversal with perpendicular magnetic sen- sitivity direction . . . 121

5.6 Simulations of the magnetization reversal . . . 122

5.7 XMCD images showing the locations of ends of orthogonal domains . . . 123

5.8 Antidot configurations surrounding an antidot . . . 124

5.9 Snapshot of the development of spins . . . 126

5.10 Details of the micromagnetic simulations . . . 128

6.1 Spin structure during heating . . . 134

6.2 Multivortex walls . . . 136

6.3 Vortex annihilation . . . 139

6.4 Structural changes by heating . . . 140

6.5 Current-induced domain wall motion in a Py wire . . . 141

AMR Anisotropic magnetoresistance

CIDM Current-induced domain wall motion CVD Chemical vapor deposition

DOS Density of states

DWMR Domain wall magnetoresistance fcc face-centered cubic (crystal structure) GMR Giant magnetoresistance

hcp hexagonal close packed (crystal structure) LEED Low energy electron diffraction

LLG Landau-Lifshitz-Gilbert (equation) MBE Molecular beam epitaxy (system) MOKE Magnetic-optical Kerr effect

MR Magnetoresistance

MRAM Magnetic random access memory

PEEM Photoemission electron microscopy / microscope PMMA Polymethyl methacrylate

RHEED Reflection high energy electron diffraction SEM Scanning electron microscopy / microscope

SLS Swiss Light Source

TEM Transmission electron microscopy / microscope

UHV Ultra high vacuum

XAS X-ray absorption spectroscopy XMCD X-ray magnetic circular dichroism

XMCD-PEEM X-ray magnetic circular dichroism photoemission electron microscopy / microscope

Research into magnetic nanostructures became prominent when the tech- niques for the fabrication of sub-1µm magnetic elements became avail- able and the spin structure in the resulting magnetic nanostructures could be investigated using high-resolution measurement techniques. The in- fluence of the geometry on the spin structure in confined magnetic sys- tems gives rise to physical effects which have not been observed before.

In summary, the aims of this thesis are:

• Developing and optimizing patterning processes to provide mag- netic elements.

• Imaging of the spin structure in such magnetic elements with the focus on domain walls and their spin structure.

• Investigating the behaviour of magnetic domain walls in applied magnetic fields or currents.

To reach these aims, the following investigations are carried out:

The spin structure of domain walls and the influence of an applied magnetic field is studied (chapter 4). First the question is addressed of what happens with the spin structure of head-to-head domain walls if the dimensions are reduced. The theory predicts new domain wall types and a reduction of the domain wall width. For smaller walls a more efficient spin-torque and higher magnetoresistance effects are expected. Here, the spin structure of domain walls found in constrictions down to 30 nm is investigated and the domain wall types found are compared with the

theoretical predictions (section 4.2). It is determined whether the size re- duction of the constriction influences the width of the domain wall and the dependency of domain wall width on the constriction width is deter- mined.

In order to build devices the domain wall propagation either induced by fields or currents must be controlled. One possibility is to introduce artificial pinning sites, for example, creating notches which form a con- striction in a magnetic nanowire. By applying a magnetic field and mea- suring the position of the domain wall with the magnetoresistance effect, the strength of the field required to depin the domain wall from the con- striction is determined (section 4.3). This depinning field can be directly related to the spin structure of the domain wall and the energy landscape around the constriction can be determined.

There is currently a search for materials which give an improved spin- torque efficiency, i.e. faster domain wall movement and lower critical current densities. CrO2 is an example of a high spin-polarized material which is predicted to improve the spin-torque efficiency. An attempt is made here to image the spin structure of a patterned film of CrO2 (sec- tion 4.4). The goal is to find domain walls and to measure their spin structure which would allow to study the interaction with applied cur- rents.

In addition to constrictions, the influence of the existing spin struc- ture can serve as pinning sites for domain walls. To study such effects, the spin structures in an array of crossed ferromagnetic wires, which is equivalent to an array of holes (or antidot array) in a ferromagnetic film, are investigated (chapter 5). The influence of the holes, formed by the crossed wires, on the spin-configuration around the holes is determined and the switching of the magnetization of the antidot array on the appli- cation of an external magnetic field is investigated. It is found that the switching occurs by the nucleation and propagation of domain chains.

These moving domain chains ends can be pinned at the ends of perpen- dicular domain chains or the movement blocked due to the formation of a 360^◦domain wall when the chain ends approach a perpendicular domain

chain.

Finally, the behaviour of domain walls in permalloy nanowires is in- vestigated on the application of current-pulses. Here the goal is to sepa- rate the so-called spin torque effect due to spin-polarized currents from Ohmic heating due to the current pulse (chapter 6).

Theory

This chapter presents the theoretical background relevant for the physics and results presented in this thesis. In section 1.1, the microscopic origins of magnetism are briefly discussed, including the exchange interaction and localized as well as delocalized electron contributions to ferromag- netism. In section 1.2, these microscopic approaches are then comple- mented by a phenomenological view using thermodynamic potentials and macroscopic variables like magnetization and magnetic fields. This allows one to establish a straightforward relation to experiments, where only these macroscopic quantities are available. In section 1.3, the mi- cromagnetic simulations, which are frequently used today in magnetism research and which are based on this approach, are briefly described. In section 1.4, the spin transfer torque theory is discussed. This theory in- cludes the introduction of the interaction of current and magnetization into the micromagnetic description developed in the preceding section.

Different recent approaches are presented, in particular the possible roles of an adiabatic and a non-adiabatic spin-torque are described. In sec- tion 1.5, the anisotropic magnetoresistance effect in magnetic materials is treated. The material class of halfmetallic ferromagnets with the two prominent members Fe3O4and CrO2 is introduced in section 1.6 and the property of theoretically 100% spin polarization in these systems is dis- cussed. Finally, coupling by RKKY interchange is presented in section 1.7.

1.1 Microscopic Origins of Ferromagnetism

Two starting points exist for the microscopic description of ferromag- netism: In a localized model, the electrons responsible for the ferromag- netism are localized at an atom. In the band model, the relevant electrons are delocalized and can be described by the interaction with an effective field of the other electrons and atoms in the solid. Usually both descrip- tions will be necessary to fully understand a ferromagnetic system. How- ever, the band model is particularly relevant for the 3d-metals such as Fe, Co, Ni, and their alloys where delocalized 3d-electrons are responsible for the ferromagnetism, while in rare earth metals (with Sm, Eu, and Gd being prominent examples in magnetism) a localized electron theory is more suitable to describe the 4f-electrons, which cause the magnetic be- havior.

1.1.1 Exchange Interaction

The exchange interaction between single electrons can be regarded as the fundamental quantum mechanical effect that causes ferromagnetism. Ex- change favors a parallel alignment of neighboring spins. Due to the Pauli exclusion principle, which does not allow fermions like electrons to have the same quantum mechanical state (identical spin and location), the dis- tance between two electrons with the same spin is increased, which in turn reduces the Coulomb repulsion and therefore leads to a reduced en- ergy of the system even though the reduction of the Coulomb energy is related to an increase of the kinetic energy. The phenomenon can be de- scribed for two spins by an exchange Hamiltonian that takes the form

Hˆ =−2Jσˆ₁·σˆ₂, (1.1) where J is the exchange constant, i.e. the energy difference between parallel and antiparallel configuration, andσˆ_iare the Pauli spin matrices.

1.1.2 Localized Model and Mean Field Approximation

The Hamiltonian introduced above can be generalized to describe a spin lattice of localized electron spins:

Hˆ =−X

i,j

J_ijσˆ_i·σˆ_j. (1.2) Since this Heisenberg operator is non-linear, a solution of the problem can be often obtained only by linearization. The mean-field approxima- tion is such a linearization which reduces the problem to the interaction of one spin with a so-called mean field generated by all other spins [IL99].

The operator product in eqn. 1.2 is replaced by the product of the spin operator σˆ_i and the expectation value hˆσ_ii of the spin operator of the interacting neighbors. The according mean-field can be written as B_MF = _gµ¹

B

P

jJ_ijhˆσ_ii. Assuming an external magnetic field B₀, the Heisenberg Hamiltonian in the mean-field approximation is obtained in the form [IL99]

Hˆ =−gµ_BX

i

ˆ

σ_i·(B_MF+B₀). (1.3)

1.1.3 Band Model of Ferromagnetism

The band model assumes that each electron moves in an effective poten- tialV(r)created by the other electrons and ions of the crystal and that the eigenstates depend only on the spin. The eigenstates are solutions of the Schrödinger equation

− ~²

2m_eO²+V_σ(r)

φ_kσ(r)) =E_kσφ_kσ(r), (1.4) withm_e,kandσdenoting the electron mass, the wave vector, and the electron spin, respectively.

Taking into account the energy reduction due to the exchange interac- tion by renormalizing the one-electron energies, one obtains [IL99]

E↑(k) = E(k)−IN↑

N (1.5)

E↓(k) = E(k)−IN↓

N (1.6)

Here,E(k)are the energy values of a non-magnetic one-electron band structure,N↑ andN↓ are the numbers of electrons in the two spin states, andNthe total number of electrons. The so-called Stoner parameterIde- scribes the energy reduction due to the mentioned electron correlations.

By defining the normalized excess of spin up electrons as R = N_↑−N_↓

N , (1.7)

which is proportional to the magnetization, and by introducing E˜(k) =E(k)−IN_↑+N_↓

2N (1.8)

for convenience, we obtain

E_↑(k) = ˜E(k)−1

2IR (1.9)

E↓(k) = ˜E(k) + 1

2IR. (1.10)

The energy splitting depends on R or in other words on the relative occupation of the sub-bands for the two spin orientations. Since this oc- cupation is in turn given by the Fermi-Dirac distribution

f(E_kσ) = 1 exp

E_kσ−µ kBT

+ 1

, (1.11)

the self-consistency condition R= 1

N X

k

(fk↑(R)−fk↓(R)) (1.12) must be fulfilled. By inserting the above equations, this can be written as

Figure 1.1: (From [Har06]) Sketch of the band model. Non-ferromagnetic state, both subbands are equally occupied (left). Band splitting has occurred, the spin-up band is shifted, and therefore spin-down electrons have a higher density of states at the Fermi level.

R = 1 N

X

k





1 exp_E(k)−˜ 1

2IR−E_F kBT

+ 1

− 1

exp_E(k)+˜ 1 2IR−E_F kBT

+ 1



. (1.13) This equation can be resolved to R under certain conditions, under which a magnetic moment and thus ferromagnetism can exist. The term has to be expanded in powers of R. By restricting the treatment to T = 0K, which is the temperature where ferromagnetism should occur most probably, and by introducing the density of states per atom and spin orientationD(E˜ F) = _2N^V D(EF), the so-called Stoner criterion for the occurrence of ferromagnetism is finally obtained:

ID(E˜ F)>1 (1.14)

E_F is the Fermi energy. The detailed derivation of the Stoner criterion from eqn. 1.13 can be found in [IL99].

The spin-dependent density of states in a 3d-ferromagnet is schemat- ically shown in Fig. 1.1. The band splitting gives rise to the fact that elec- trons at the Fermi energy are spin-polarized, which means that a current flowing in such a material is spin-polarized, too. The so-called weak fer- romagnetism occurs, when the majority density of states is not fully oc-

cupied (e.g. Fe), the strong ferromagnetism with a fully occupied density of states for the majority electrons is found for example in Co or Ni. To determine the majority spin-direction of the magnetization the sum of the 3d-states below the Fermi energy have to be taken into account while for transport phenomena the spin-dependend density at the Fermi edge is important.

1.2 Micromagnetic Systems

Since the microscopic descriptions presented in section 1.1 do not provide parameters accessible to the experiment, a phenomenological approach with macroscopic variables such as the magnetization or an external field are often used to describe experimental findings and the behavior of mag- netic systems. Different energy contributions like exchange, anisotropy, dipolar coupling, and Zeeman energy have to be taken into account when calculating the magnetization configuration. This can be done analyti- cally in special cases using the Stoner-Wohlfarth model or by simulations as detailed in sections 1.2.3 and 1.3.2.

1.2.1 Thermodynamics in Magnetism

The macroscopic starting point for the description of a magnetic sys- tem are the thermodynamic potentials. The magnetic configuration will change under the external conditions in order to minimize its energy or more precisely to minimise the appropriate thermodynamic potential.

The external conditions include magnetic fields and temperature. They may also contain the influence of a current, which will not be considered in the following however, but discussed later in section 1.4.

The set of thermodynamic potentials suitable for description includes in general the internal energy, the enthalpy, the free energy, and the Gibbs free energy. Since each of these potentials remains constant if certain pa- rameters are kept constant, the appropriate choice of the potential de- pends on the experimental conditions. The Gibbs free energy [Ber98]

G(H, T) = U −T S−µ₀H·M (1.15) depending on the external field H and the temperature T is conve- nient in most situations. U denotes the internal energy andSthe entropy.

Since the magnetization M will be inhomogeneous over the sample, in most practical cases Mhas to be replaced by a vector fieldMˆ ≡M(r)so that the Gibbs free energy is replaced by the Landau free energy [Ber98]

G(M,ˆ H, T) =U(M)ˆ −T S−µ₀H·M.ˆ (1.16) This assumes already that the magnetizationMˆ can be regarded as a continuum vector function in the sample instead of taking into account single magnetic moments or spins. This so-called micromagnetic approx- imation is justified if the cell size is not larger than the exchange length.

1.2.2 Energy Contributions

1.2.2.1 Exchange Energy

According to eqn. 1.2, a misalignment of magnetic moments leads to an increase of energy. This exchange energy is therefore inevitably present in systems with inhomogeneous magnetization and it is intuitively clear that large gradients are related to a large energy penalty. A Taylor expan- sion of the exchange energy contribution as a function of the magnetiza- tion gradient in the lowest-order term yields

E_ex = A

|M|ˆ ² Z

(∇M)ˆ ²dV. (1.17)

The relation between the exchange stiffness A and the atomic scale parameterJij in eqn. 1.2 can be understood like this: By limiting the sum to nearest neighbors and interpreting the i as classical vectors one can find thatA∝kJ S²/a, whereJ is the nearest neighbor exchange constant, S the spin magnitude, a the lattice constant, and k a numerical factor depending on the lattice symmetry [Ber98].

1.2.2.2 Anisotropy Energies

Anisotropy energy contributions arise due to the fact that certain mag- netization directions in a system can be more favorable than others. The underlying symmetry breaking allows for classifying the anisotropies by their physical origin. The basic mechanism is the spin-orbit interaction with the prominent exception of the shape anisotropy, which is related to the stray field formation (see section 1.2.2.3).

Since the magnetic anisotropy results from the coupling between the orbitals of the crystalline structure and the spin moments via spin-orbit coupling, the magnetic anisotropy will reflect the symmetry of the crystal or exhibit a higher symmetry itself. In a cubic lattice with the crystal axes being the x, y, and z coordinates, the anisotropy energy density can be written as [Ber98]

ε_ani =K₀+K₁

sin²θsin²(2φ)

4 + cos²θ

sin²θ+K₂sin²(2φ)

16 sin²(2θ) +...

(1.18) in spherical coordinates or as

ε_ani =K₀+K₁ m²_xm²_y+m²_ym²_z+m²_zm²_x

+K₂ m²_xm²_ym²_z

+... (1.19) in a cartesian system with them_i being the components of the mag- netization direction m = M/|M| = (m_x, m_y, m_z). TheK_i, which can be positive or negative, define the easy and hard axes or planes of the sys- tem corresponding to minima and maxima of the anisotropy energy con- tribution. When a thin film is considered, in which only in-plane mag- netization directions are possible, this is reduced to a fourfold or biaxial in-plane anisotropy:

ε_ani=K₀+1

4K_biaxialsin²(2φ) +· · · (1.20) Also twofold (or uniaxial) anisotropies are observed. All anisotropy energy densities ε_ani can be directly transformed to anisotropy energies using

E_ani = Z

V

ε_ani(M(r))drˆ (1.21) Further magnetocrystalline anisotropy contributions exist, which do not play a significant role in this thesis and are therefore only briefly men- tioned for completeness:

• Anisotropy energy contributions can occur due to symmetry break- ing at the surface or interface of films. Due to the scaling with the inverse thickness they can dominate particularly in thin films and multilayers.

• Uniaxial anisotropy can be induced by application of an in-plane field during deposition.

• The magnetoelastic anisotropy arises in crystals were strain is pres- ent in the lattice.

• External stress gives rise to analog energy contributions like the in- ternal strain, but due to a different origin.

1.2.2.3 Stray Field Energy / Dipolar Coupling Energy

In bulk ferromagnets, the magnetic dipolar interaction is responsible for the existence of magnetic domains because the lowest energy state is achieved with magnetic flux closure configurations. The total energy con- tribution per unit volume can be written as [Ber98, JBdBdV96]

E_dip = 1 2µ₀

Z

H²_ddV =−1 2µ₀

Z

H_d·MdV, (1.22) where H_d is the magnetostatic or demagnetizing field of the sample.

Neglecting the discrete nature of matter the shape effect of the dipolar in- teraction in a ferromagnetic ellipsoid can be described via the anisotropic demagnetizing fieldH_d=−NMwith the shape-dependent demagnetiz- ing tensor N. For a thin film all tensor elements are zero except for the direction perpendicular to the layer for which N^⊥ = 1. Equation 1.22 gives in this case [JBdBdV96]

E_dip = 1

2µ₀M_s²cos²θ, (1.23) where it is assumed that the magnetization is uniform with a magni- tude equal to the saturation magnetizationM_s. θ denotes the angle with the film normal. It can be directly seen, that an in-plane configuration of the magnetization is energetically favorable.

1.2.2.4 Zeeman Energy

The Zeeman energy is the potential of a magnetic dipole moment in a magnetic field. For a homogeneous external fieldH₀, this energy contri- bution depends only on the average magnetization and not on the de- tailed spin structure of the system. The Zeeman energy can be written as

E_dip =−µ₀ Z

H₀·MdV. (1.24)

1.2.3 Brown´s Equations and the Effective Field

Using the energy terms discussed, the magnetization configuration for a given anisotropy and a given applied field can be calculated in principle.

Assuming a uniform magnetization, this can be done analytically using the so-called Stoner-Wohlfarth model. A detailed description how this can be performed is presented as an example for a system with a fourfold anisotropy in [Wer01, Klä03, TWK⁺06].

For an inhomogeneous magnetization, as encountered experimentally in this thesis, the calculations have to be done numerically (which practi- cally means computationally), in order to be able to compare experimen- tal results with the predictions of theory. Adding all energy contributions discussed above yields the following expression for the Landau free en- ergy:

G_L = (M,ˆ H) = Z

V

A

|M|ˆ ²

[∇M]ˆ ²+ε_ani(M)ˆ − 1

2µ₀H_d·Mˆ −µ₀H₀·Mˆ

! dV (1.25) Temperature effects are not included in this equation. In order to de- termine the (meta-)stable magnetization configurations for given param- eters such as the external field, a variational problem has to be solved.

The local minima of the Landau free energy have to be determined by varying the magnetization configuration of the system and fulfilling the conditions for the existence of a minimum _∂^∂_Mˆ G_L = 0 and _∂^∂_Mˆ²2G_L > 0. The difficulty lies in the fact that G_L is a functional of the entire vector fieldMˆ and thus one has to consider the infinite-dimensional functional space of all possible magnetization configurations.

Calculation of the variationδGLof the Landau free energy and taking into account the extremal condition δG_L = 0 yields a set of equations that must be fulfilled at equilibrium, known as Brown’s equations [Bro63, Ber98]:

Mˆ ×H_{ef f} = 0 (1.26)

Mˆ × ∂Mˆ

∂n = 0 (1.27)

with _∂n^∂ denoting the derivative normal to the surface. The first con- dition must be fulfilled for every point inside the sample, the second is a boundary condition for the surface. The effective fieldH_{ef f} is given by

H_{ef f} = 2

µ₀M_s² 5(A5M)− 1 µ₀

∂ε_ani

∂M +H_M +H₀. (1.28) It contains the applied fieldH0, the demagnetizing fieldHM, as well as the exchange and anisotropy contributions. The cross product in eqn.

1.26 shows that the torque on the magnetization due to the effective field has to become zero in equilibrium. The boundary condition 1.27 can be written in terms of the magnetization unit vector as m(r)× _∂n^∂ m(r) = 0,

which is equivalent to the simpler condition _∂n^∂ m(r) = 0. The mate- rial parameters like the exchange constant A, the saturation magnetiza- tionMs, all anisotropy constants determiningεani, as well as the external fields and the shape of the system have to be provided as input informa- tion. Solving Brown’s equations then yields the equilibrium magnetiza- tion distributionsMˆ.

However, the dynamical behavior of a system, e.g. the response to a change of the external field or the injection of a current, are not included so far, because Brown’s equations only describe the conditions to be ful- filled in the equilibrium state. The answer to this problem is given by the Landau-Lifshitz-Gilbert equation.

1.3 Magnetization Dynamics

The well established Landau-Lifshitz-Gilbert (LLG) equation describes the magnetization dynamics in a material without the influence of a cur- rent in form of a time-dependent differential equation. This basis is de- rived, before the micromagnetic simulations based on the LLG equation are discussed. The influence of a current is included into the description in frame of the spin transfer torque model, which is presented in the fol- lowing section.

1.3.1 Landau-Lifshitz-Gilbert Equation

From a quantum-mechanical starting point a classical desciption of the change of angular momentum can be derived as detailed in [Wie02] and [Hin02]. The torque exerted on a normalized magnetic moment S by a field is given by

∂S

∂t =−γ

µ_s(S×H⁰_{ef f}) (1.29) with the gyromagnetic ratio γ = gµ_B/~ of the electron (g = Landé factor, µ_B = Bohr magneton) and by normalizing the moment S = µ/µ_s.

The effective fieldH⁰_{ef f} is determined by the derivative of the Hamilton function H⁰_{ef f} = _∂S^∂ H. In the simplest case, where the Hamiltonian con- tains only the Zeeman term (H = µsS·B), the Larmor precession of the electron in an external magnetic field is directly obtained:

∂S

∂t =−γ(S×B). (1.30)

Already in 1935, Landau and Lifshitz [LL35] included a damping term into the description that takes into account the energy dissipation and the according relaxation of the magnetic moment towards the external field direction. This resulted in the equation

∂S(t)

∂t =−γ

µ_sS×H⁰_{ef f} − λ

µ_sS×(S×H⁰_{ef f}), (1.31) where isλthe damping constant introduced. However, investigations of the magnetization reversal time of a magnetic sphere as a function of the dampingλshowed that infinite damping unreasonably leads to zero reversal times (1/λ- dependence) [Kik56]. Gilbert [Gil55] derived the so- called Gilbert equation of motion

∂S(t)

∂t =−γ

µ_sS×H⁰_{ef f}

| {z }

precession

+α(S× ∂S(t)

∂t )

| {z }

damping

. (1.32)

As in the Landau-Lifshitz equation 1.31, the first term describes the precession and the second the damping. The Gilbert equation 1.32 can be transformed to the general form of the Landau-Lifshitz equation leading to the

∂S(t)

∂t =− γ (1 +α²)µs

S×H⁰_{ef f} − γα (1 +α²)µs

S×(S×H⁰_{ef f}), (1.33) which is usually referred to as the Landau-Lifshitz-Gilbert (LLG) equation of magnetization dynamics in the literature. Sometimes also the implicit form with time derivatives on both sides, the actual Gilbert equation 1.32, is referred to as LLG, which is correct in the sense that both equations can be transformed into each other as described.

It is important to discuss the units used in the equations above. In eqn. 1.29 the moment S is dimensionless with |S| = 1 and the effective field H⁰ef f is actually an effective energy. This way of description is fa- vorable for micromagnetic simulations and calculations. But one always has to take a very careful look at prefactors and the exact definition of variables introduced when comparing different publications and sources of information. When the effective field H⁰_{ef f} is given in correct units of magnetic fields (A/m) instead of energies ([H⁰_{ef f}] = J) and the magne- tization Mis introduced instead of the dimensionless magnetic moment S, eqn. 1.29 can be written as

∂M(t)

∂t =−µ₀γ(M×H_{ef f}). (1.34) This finally leads to an implicit LLG equation analogous to eqn. 1.32 of the form

∂M(t)

∂t =−µ0γ(M×Hef f) + α

M_s(M× ∂M(t)

∂t ), (1.35) whereM_sdenotes the saturation magnetization.

1.3.2 Micromagnetic Simulations

We now turn to the numerical treatment of the micromagnetic equations and the widely used solver named "Object Oriented Micromagnetic Framework" (OOMMF) [OOM, DP02]. The operating principle is the fol- lowing: First, the problem (sample shape) to be solved is subdivided into a two-dimensional grid of square cells with three-dimensional clas- sical spins situated at the center of each cell. In general, other methods also exist to discretize the problem. OOMMF also provides a true three- dimensional solver named OXS (OOMMF extensible solver), which is suitable for simulation of layered systems.

The cell size has to be smaller (at least not larger) than the relevant length scales on which the magnetization changes. This can be the ex- change length or the Bloch wall width

l_ex = s

2µ₀A

M_s² (1.36)

l_wall = rA

K, (1.37)

with Abeing the exchange constant and K being an anisotropy con- stant [HS98]. A cell size of 5 nm is a reasonable compromise between computation time and memory consumption on the one hand and a cor- rect modeling on the other hand if permalloy is chosen as material.

For each cell the effective fieldH_{ef f} as derived in section 1.2.3 must be calculated. The anisotropy and Zeeman energy terms are obtained assuming a constant magnetization in each cell. The exchange energy is then computed for the eight neighboring cells. The most time-consuming step is the calculation of the magnetostatic field energy, because it is nec- essary to sum over the energy contributions from all cells in the struc- ture. It is computed as the convolution of the magnetization against a kernel that describes the cell-to-cell magnetostatic interaction by us- ing fast Fourier transformation techniques, which reduces the number of operations to O(Nlog₂N) instead of O(N²), where N is the number of cells [DP02]. Then a numerical integration of the Landau-Lifshitz-Gilbert equation in the form given in eqn. 1.33 is performed. The procedure is it- erated yielding a step by step calculation of magnetization configurations with decreasing energy until the torque Mˆ ×H_{ef f} (see eqn. 1.26) of the effective field on the magnetization is below a chosen threshold value at each point of the system.

1.4 Spin Transfer Torque Model

The theoretical description of current-induced domain wall motion, or in other words the interaction of spin-polarized charge carriers with the magnetization of the material, is a complicated issue which is still the subject of much debate as it will be detailed in this section. Starting from the Landau-Lifshitz-Gilbert equation, the relevant approaches to include

interactions between current and magnetization into the description will be discussed. The main theoretical framework in this context is the spin transfer torque model.

1.4.1 Spin Transfer Torque

Relevant is the direct interaction between the spin of the charge carrier and the domain wall. This description was also pioneered by Berger [Ber84], who termed the phenomenon s-d exchange torque (between the localized 3d-electrons responsible for ferromagnetism and the delocal- ized 4s-electrons carrying the current), but today it is referred to as spin transfer torque or spin transfer effect. The s-d exchange potential is de- scribed by

V(x) = gµ_B

s·H_sd(x) + H_sd 2

, (1.38)

where s is the spin of the 4s-electron, H_sd(x) = 2J_sdhS(x)i/(gµ_B) is the exchange field with the s-d exchange integralJsd and the spin of the 3d-electrons, and the coordinatexnormal to the wall plane.

The first approach to integrate the spin transfer torque into the Landau-Lifshitz-Gilbert equation (eqn. 1.35) by adding additional terms was made by Slonczewski [Slo96]. He considered two ferromagnetic layers separated by a non-magnetic spacer with the macrospins S1,2 =

|S_1,2|ˆs_1,2. The torque exerted by a charge current I on a macrospin was found to be

∂S_1,2

∂t =Ig

eˆs1,2×(ˆs1׈s2) (1.39) with the electron chargeeand a prefactorg containing the spin polar- izationP of the current:

g =

−4 + 1 +P³

4P^3/2 (3 +ˆs₁·ˆs₂) ⁻¹

(1.40) First we see that the torque is proportional to the current and changes its sign when the current is reversed. Secondly, it depends on the spin

polarizationP, and the direction of the torque is expressed by the product ˆs_1,2 ×(ˆs₁ ׈s₂). When including this torque term into the LLG equation 1.35, we obtain a description of the system with a spin current.

1.4.2 Landau-Lifshitz-Gilbert Equation with Spin Trans- fer Torque

A key issue for the comparison of theoretical predictions and experimen- tal observations are micromagnetic simulations, which require the cor- rect addition of the torque term(s) to the Landau-Lifshitz-Gilbert equa- tion. Thiaville and co-workers [TNMV04, TNMS05] have suggested the following extension of the LLG equation, that incorporates a spin torque correction for local angular momentum transfer from current to magneti- zation:

∂M(t)

∂t =−γ₀M×H_{ef f} + α Ms

M× ∂M(t)

∂t −(u· 5)M

| {z }

extension

(1.41)

γ₀ =µ₀γ =µ₀gµ_B/~is the gyromagnetic ratio anduis the generalized velocity defined below in eqn. 1.47. They restrict themselves to adiabatic processes, i.e. a local equilibrium between the conduction electrons and the magnetization is assumed.

A reformulation of the description has been introduced by Li and Zhang [LZ04a]. In the low temperature regime the magnetization has a constant value and thus they obtain

τ_[LZ04a] =− 1

M_s²M×[M×(u· 5)M]. (1.42) This form of the spin torque is identical with that considered earlier by Bazaliy et al. [BJZ98] for metal-ferromagnet interfaces in the presence of a spin-polarized current

τ_[BJZ98] =− a Ms

M×[M×m] (1.43)

withmbeing the unit vector of the magnetization in the pinned layer.

This corresponds directly with the macrospin version of the torque sug- gested by Slonczewski [Slo96] (see eqn. 1.39). If the torqueτ_[LZ04a]is inte- grated across the multilayers investigated in [BJZ98], assuming that the free and the pinned layer are in single domain states, both torquesτ_[LZ04a]

and τ_[BJZ98] turn out to be equivalent. However, the velocity u depends on the spin polarization P of the bulk material, while in the multilayer situation the interfaces are important. Therefore a direct relation between uandais not obvious. The torque term [LZ04a] or the version suggested by Thiaville et al. [TNMV04] in eqn. 1.41 can be regarded as the continu- ous limit of eqn. 1.39.

In [LZ04a] it is shown that the spin transfer torque on a domain wall has many features in common with that at an interface as considered by Bazaliy et al. with the ratio τ_[LZ04a]/τ_[BJZ98] being given by the ratio t_F/W of the thickness of the ferromagnetic layer t_F to the width of the domain wallW, i.e. the torque is proportional to the volume of the mate- rial that experiences spin transfer effects. By implementing the modified LLG equation into micromagnetic code and applying the code to a 5 nm thick and 100 nm wide wire with a domain wall in the center, they find that the wall is moved by a current, but stops on a nanosecond timescale after a displacement in the sub-µm range.

Thiaville et al. [TNMV04] have also performed micromagnetic calcu- lations with their torque ansatz. They obtained current-induced domain wall motion above a critical current density.

1.4.3 Adiabatic and Non-adiabatic Spin Torque

Besides the adiabatic torque discussed so far [TNMV04, LZ04a, LZ04b]

also non-adiabatic contributions were considered [TNMS05, ZL04]. The adiabatic processes refer to the situation where the spins of the conduc- tion electrons can locally follow the magnetization, while in the case of non-adiabatic processes this is not possible and a mistracking between the conduction electron spins and the local magnetization occurs. Zhang

and Li [ZL04] include a non-adiabatic torque into the description.

This is deduced by calculating the response of the non-equilibrium conduction electron spins to a spatially and temporarily changing mag- netization and obtain four different torque contributions with two of them representing the adiabatic and the non-adiabatic processes. The other two torque contributions stem from the time dependent variation of the magnetization, are independent of the current density, and lead to corrections of the gyromagnetic ratio and the damping constant, respec- tively. In other words, they slightly affect only the two terms of the estab- lished Landau-Lifshitz-Gilbert equation 1.35 and are therefore of minor interest in this context. The two torques of interest result from spatial variations of the magnetization as present in a domain wall and can be identified as the adiabatic and the non-adiabatic torque, included as two additional terms:

∂M(t)

∂t = −γ₀M×H_{ef f} + α

M_sM× ∂M(t)

∂t (1.44)

− b_J M_s²M×

M×∂M

∂x

| {z }

adiabatic

− c_J

M_sM× ∂M

∂x

| {z }

non−adiabatic

,

where the constantsb_J andc_J are given by

b_J =j P µ_B

eM_s(1 +ξ²) and c_J =ξb_J. (1.45) ξis defined as the ratioτex/τsf of an exchange time (see [ZL04] for de- tails) and the spin-flip relaxation time. Its value is of the order of10⁻² for typical 3d-ferromagnets [ZL04]. The second non-adiabatic term is new and can be related to the mistracking of the conduction electron spins.

Also Thiaville and his coworkers [TNMS05] included a non-adiabatic term into the Landau-Lifshitz-Gilbert equation. They phenomenologi- cally introduced the additional term rather than Zhang and Li [ZL04], who physically derived it. The extended LLG equation then reads

∂M(t)

∂t =−γ₀M×H_{ef f}+ α

M_sM×∂M(t)

∂t −(u· 5)M

| {z }

adiabatic

− β

M_sM×[(u· 5)M]

| {z }

non−adiabatic

.

(1.46) The generalized velocity

u= gP µ_B

2eM_sj (1.47)

with j being the current density and P its spin polarization, points along the current direction. This phenomenological approach is then tested by micromagnetic simulations. The critical current density drops to zero, but values comparable with the experiment are only obtained when edge roughness is included. However, the domain wall velocities are still much too high compared with experiments [YON⁺04, KJA⁺05].

In order to compare the Thiaville approach [TNMS05] with the sug- gestion of Zhang and Li [ZL04] we can rewrite eqn. 1.46 in the form

∂M(t)

∂t = −γ₀M×H_{ef f} + α

M_sM×∂M(t)

∂t

− 1

M_s²M×(M×[(u⁰· 5)M])− β

M_sM×[(u⁰· 5)M].(1.48) It can be seen directly see that both approaches lead to identical ex- tended Landau-Lifshitz-Gilbert equations, but with slightly different ex- pressions for the generalized velocitiesu(eqn. 1.47) and u⁰, respectively, which are equal except a factor of1/(1 +ξ²)≈1if eqn. 1.44 and eqn. 1.48 are compared. The parameterβ , which was introduced phenomenologi- cally first, can be identified with

β = (λ_ex/λ_sf)², (1.49) whereλ_ex represents the exchange length andλ_sf the spin-flip length [TNMS05].

These equations provide different quantities for the experimentalist to check the validity of the theoretical descriptions. First of all,βor the ratio

ξ = c_J/b_J, depending on the theoretical description, determine the non- adiabaticity of the spin torque. The doping of 3d-ferromagnets like Py, Fe, and Co with rare earth metals might allow the modification of the spin- flip length and therefore the value of β. Different results in experiments due to the artificially modified ratio of adiabatic and non-adiabatic spin torque can reveal information on the roles and the strengths of the two torque contributions. Furthermore, changing the material will change the spin polarization P and the saturation magnetizationM_s and there- fore modify u. For example, halfmetallic ferromagnets with very high values of P like CrO2 (see section 1.6) and reducedM_s can be expected to show more pronounced spin torque effects compared to typical 3d- ferromagnets [LZ04b].

Tatara and coworkers very recently suggested a further modifica- tion of the Landau-Lifshitz equation [TTK⁺06]. In addition to the adi- abatic and the non-adiabatic terms discussed above, they introduce two other torquesτ_na and τ_pin due to non-adiabatic momentum transfer and pinning, respectively. Both non-adiabatic contributions are summed to β⁰ ≡ βP +βna, whereβ is due to spin relaxation, P is the spin polariza- tion of the current, and β_na a dimensionless wall resistivity due to high gradients in the domain wall magnetization [TTK⁺06]. Starting from the equations of motion for the domain wall [TK04], they identify a weak, an intermediate, and a strong pinning regime. The critical current den- sity is predicted to depend onβ⁰ in the weak pinning regime and on the damping parameter α in the strong pinning regime. However, existing experimental results of the critical current density in metals are not quan- titatively explained by these findings [TTK⁺06].

1.4.4 Domain Wall Spin Structure Modifications

In addition to domain wall motion induced by a current, changes of the domain wall spin structure have been predicted by theory. Thiaville et al. [TNMS05] observed periodic changes between vortex and transverse walls in the results of their micromagnetic simulations of a wall under

spin current in a Py wire. These predictions were qualitatively confirmed by experiments, where the transformation of a vortex wall to a trans- verse wall was directly observed with spin-SEM [KJA⁺05]. Results on current-induced domain wall spin structure changes were also obtained using XMCD-PEEM [KLH⁺06]. Also He, Li, and Zhang [HLZ06] found from their micromagnetic calculations that vortex domain walls tend to transform to transverse walls during the current-induced motion. After the transformation, the transverse walls stop and exhibit a higher critical current density due to stronger pinning, above which the wall velocity is equal to that of a vortex wall.

1.5 Anisotropic Magnetoresistance (AMR)

The general phenomenon that the resistance of a given magnetic sys- tem depends on a magnetic field and therefore its magnetization state is called the magnetoresistance (MR) effect. Such effects are of high im- portance for applications in data storage because stored information can be read out by a simple resistance measurement and easily be written by modifying the spin structure with e.g. an external magnetic field. Ap- plications in sensors are also very common today making use of the fact that a change of a magnetic field influences the magnetization of a sen- sor and therefore its readout signal. Many parameters of interest such as positions, angles, or velocities can be translated to a magnetic field change [GM04, CvdBR⁺97] and are thus available for measurement via a magnetoresistance effect. The number of reported magnetoresistance effects is large and still growing. The entirety of all effects is sometimes named XMR effect, where the "X" serves as a wildcard character.

The anisotropic magnetoresistance (AMR) was employed for the work of this thesis to detect the presence of DWs. It was observed for the first time by Thomson [Tho57] in 1857. Anisotropic scattering of the charge carriers caused by spin-orbit coupling leads to an anisotropy of the re- sistance. The resistivity tensor of a monodomain polycrystal with the

magnetization direction chosen arbitrarily along the z-axis can be written as [Mar05]

ρij =









ρ⊥ −ρH 0 ρ_H ρ⊥ 0 0 0 ρk









, (1.50)

where ρk and ρ⊥ are the longitudinal (along the current) and trans- verse (perpendicular to the current) components of the resistivity at zero field, respectively, andρH is the extraordinary Hall resistivity. Taking into account the relation

E_i =X

k

ρ_ikj_k (1.51)

between the electric field componentsE_iand the current components j_k, this corresponds to the following expression of the electric field

E =ρ⊥(B)j+ [ρk(B)−ρ⊥(B)][m·j]m+ρ_H(B)m×j (1.52) with the magnetic inductionB and the unit vector of the magnetiza- tion m. With the definition ρ = E·j/|j|² and θ being the angle between mandj

ρ= ρk+ρ⊥

2 + ρ_k−ρ_⊥

cos²θ− 1 2

. (1.53)

was obtained.

If a random orientation of magnetic domains in zero field is assumed, the mean value ofcos²θis equal to¹₂ in a film with in-plane magnetization so that the zero field resistivity is(ρk+ρ⊥)/2, while the saturation values are±(ρ_k−ρ⊥)/2for longitudinal and transverse fields, respectively. It can be seen that the effect does not directly depend on the applied field, but on the spin structure of the sample via the angle between magnetization and current.

Recently, efforts were made to calculate the AMR for Py quantita- tively. A semiclassical approach was used [RCDJDJ95] based on a two- current model, which treats currents of electrons with majority and mi-

Figure 1.2: (From [Büh05]) Schematics of the AMR contribution of a domain wall and its measurement. The current flow is indicated. (a) No wall is present and the resistance is high, (b) the presence of a domain wall (here a transverse wall) reduces the resistance.

nority spins independently and therefore assumes a weak spin-flip scat- tering. Ab initio calculations of the AMR for bulk Py alloys [KPSW03]

resulted in a very good agreement with experimental results [Smi51].

1.5.1 AMR Contribution to the Domain Wall Magnetore- sistance

The AMR effect can be specific to particular materials but also the sample spin structure and in particular domain wall spin structures can lead to a magnetoresistance signal. The term of "domain wall magnetoresistance"

(DWMR) is commonly used, but it is often unclear if the related effect is an intrinsic domain wall contribution - which would justify this term - or simply an extrinsic effect of the magnetization inside the wall giving rise to an AMR effect.

In the mesoscopic Py structures used in this work, the extrinsic AMR contribution of the domain wall is dominating [KVB⁺02, KVR⁺03]. In the ring geometry with electrical contacts widely used in this thesis, the cur- rent flows along the ring perimeter. If no domain walls are present, i.e.

the ring is in the so-called vortex state, the magnetization is aligned par- allel to a possible current flow everywhere in the ring (Fig. 1.2(a)). When a domain wall is present, it represents an area in which the magnetiza- tion has a component perpendicular to the current flow as depicted in Fig. 1.2(b). This leads to a negative resistance contribution of the domain

Figure 1.3: (From [Kön06])Schematics of spin-dependent density of states for (a) 3d transition metals and (b, c) for halfmetallic ferromagnets.

wall due to the AMR effect, which can be easily measured.

An intrinsic MR effect is ballistic magnetoresistance (BMR) which is the difference in resistance between parallel and antiparallel orientation of the magnetic moments at point contacts on a ballistic scale [GMZ99, TZMG99]. The dimensions of the point contacts have to be of the order of the mean free path of electrons in the system. There is much debate about the existence and the possible origins of the effect [HC03, EGE⁺04], which may also be due to a domain wall increase or decrease [TF97, RYZ⁺98].

1.6 Halfmetallic Ferromagnets

Halfmetallic ferromagnets are characterized by the combination of fer- romagnetic behavior and a particular electronic structure that leads to unusually high values of the spin polarization. In addition to Heusler al- loys [dGMVEB83], transition metal oxides such as Fe3O4[ZFP⁺05], CrO2

[Sch86], and manganites [vHWH⁺93] have been theoretically predicted to belong to the class of halfmetallic ferromagnets.

For CrO2, the spin-dependent density of states (DOS) exhibit a metal- lic character for one orientation the electrons due to occupied states at the Fermi level, while the second orientation exhibits a semiconducting be- havior with an energy gap at the Fermi level as visualized in Figs. 1.3(b).