Interactions Between Current and Domain Wall Spin Structures

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Interactions Between Current and Domain Wall Spin Structures

Universität Konstanz

DISSERTATION

ZUR ERLANGUNG DES AKADEMISCHEN GRADES DES DOKTORS DER NATURWISSENSCHAFTEN

AN DER UNIVERSITÄT KONSTANZ

MATHEMATISCH-NATURWISSENSCHAFTLICHE SEKTION FACHBEREICH PHYSIK

VORGELEGT VON MARKUS LAUFENBERG

TAG DER MÜNDLICHEN PRÜFUNG: 26. JULI 2006

REFERENTEN:

PROF. DR. ULRICH RÜDIGER PROF. DR. GÜNTER SCHATZ

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2006/1955/

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Summary

In this thesis, the interaction between current and domain wall spin structures in the 3d-ferromagnets Ni80Fe20(permalloy) and Co as well as in the halfmetallic ferromagnets Fe3O4 and CrO2 is investigated. Nanostructures of these materials with different shapes (rings, wires, rectangles) and different sizes and thicknesses are fabricated using molecular beam epitaxy, electron beam lithog- raphy, lift-off, as well as focused ion beam milling and various etching techniques. Samples designed for current injection and magnetoresistance measurements are electrically contacted with Au pads in an overlay process.

First, the spin structure of domain walls in these materials is studied using x-ray magnetic circular dichroism photoemission electron microscopy (XMCD- PEEM) with synchrotron light, because it is known that the wall spin structure influences the magnetoresistance and that spin transfer torque effects are closely related to the wall spin structure. Comprehensive domain wall type phase diagrams for 180^◦ head-to-head domain walls are obtained and explained using micromagnetic theory. Thermally induced wall type transformations are studied, the wall width is determined as a function of the element width, and the dipolar coupling between adjacent walls is investigated. The domain wall spin structures and their dependence on the magnetocrystalline anisotropy in Fe3O4

and CrO2structures are elucidated.

The current- and field-induced domain wall motion in NiFe structures is studied at variable temperatures using measurements of the anisotropic magnetoresistance contribution of the domain wall. The Joule heating of the current is determined and discriminated from intrinsic spin torque effects. The domain wall velocity and current-induced transformations of the domain wall spin structures are studied using XMCD-PEEM. In CrO2, the magnetoresistance contributions are probed, the Joule heating is measured, and current-assisted domain wall motion is studied at low temperatures.

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Zusammenfassung

In dieser Arbeit wird die Wechselwirkung zwischen Strom und der Spinstruk- tur von Domänenwänden in den 3d-Ferromagneten Ni80Fe20(Permalloy) und Co sowie in den halbmetallischen Ferromagneten Fe3O4 und CrO2 untersucht.

Nanostrukturen aus diesen Materialien (Ringe, Drähte, Rechtecke etc.) ver- schiedener Größe und Dicke werden unter Einsatz von Molekularstrahlepitaxie, Elektronenstrahl-Lithographie, Lift-off-Technik sowie fokussiertem Ionenstrahl- Milling und chemischen Ätzprozessen hergestellt. Proben, die für Stromin- jektion und Magnetowiderstandsmessungen bestimmt sind, werden in einem Overlay-Prozess mit Goldkontakten elektrisch kontaktiert.

Zunächst wird die Spinstruktur von Domänenwänden in diesen Materialien mit Hilfe von Photoemissionselektronenmikroskopie auf Basis von zirkularem Röntgendichroismus (XMCD-PEEM) mit Synchrotronstrahlung untersucht, weil bekannt ist, dass die Domänenwand-Spinstruktur den Magnetowider- stand beeinflusst und dass Spin-Transfer-Torque-Effekte in engem Bezug zur Spinstruktur stehen. Es ergeben sich Domänenwandtyp-Phasendiagramme für 180^◦ Kopf-an-Kopf-Domänenwände, die mit Hilfe der Theorie des Mikromag- netismus bestätigt werden. Thermisch induzierte Wandtyp-Transformationen werden untersucht, die Wandbreite als Funktion der Strukturbreite ermittelt und die dipolare Kopplung benachbarter Domänenwände abgebildet.

Die Domänenwand-Spinstrukturen und ihre Abhängigkeit von der magne- tokristallinen Anisotropie in Fe3O4und CrO2 Strukturen wird untersucht.

Strom- und feldinduzierte Domänenwandverschiebung in NiFe-Strukturen wird bei variablen Temperaturen über Messung des Domänenwandbeitrages zum anisotropen Magnetowiderstand studiert. Die Messung der Joulschen Erwärmung durch Strom erlaubt, thermische Effekte von intrinsischen Spin- Torque-Effekten zu separieren. Die Domänenwandgeschwindigkeit und strom- induzierte Transformationen der Domänenwand-Spinstruktur werden mit

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iii XMCD-PEEM bestimmt. In CrO2 werden die verschiedenen Magnetowider- standsbeiträge temperaturabhängig ermittelt, die Joulsche Erwärmung durch Strom gemessen und stromunterstützte Domänenwandverschiebung bei tiefen Temperaturen untersucht.

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List of Figures

1.1 Sketch of the band model of ferromagnetism . . . 8

1.2 Current distribution causing the hydromagnetic drag effect . . . . 18

1.3 Results of micromagnetic simulations including spin torque . . . 22

1.4 Spinwave spectrum modified by current . . . 29

1.5 Schematics of the intergrain tunneling magnetoresistance . . . 33

1.6 AMR contribution of a domain wall . . . 34

1.7 Spin-dependent density of states for 3d-ferromagnets and halfmetallic ferromagnets . . . 35

2.1 X-PEEM and LEEM imaging modes . . . 39

2.2 X-ray magnetic circular dichroism . . . 41

2.3 XAS and resulting XMCD spectra of Fe and Co . . . 42

2.4 Photographs of Elettra PEEM setup . . . 43

2.5 Schematic cross section and top view of the CryoVac cryostat . . . 45

2.6 Schematics of the cooling system . . . 46

2.7 Photograph and schematics of the vector field system . . . 47

2.8 Schematics and photographs of the cryostat sample holder . . . . 49

2.9 Measurement circuits used for resistance measurements . . . 50

2.10 Measurement circuit for pulse injection and consecutive resistance measurement . . . 51

2.11 Measurement circuit for investigation of Joule heating due to pulses 52 2.12 Schematics of the pulse injection setup for PEEM . . . 56

2.13 Standard Elmitec PEEM sample holder . . . 57

2.14 Elmitec PEEM sample holder modified for current injection . . . 58

2.15 PEEM sample holder for current injection (own design) . . . 58

2.16 Ray diagram of the TEM used for off-axis electron holography . . 60

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LIST OF FIGURES ix

2.17 Schematics of a scanning electron microscope . . . 61

3.1 Schematics of the MBE system used for growing NiFe and Co . . 65

3.2 Photographs of the MBE system used for growing Fe3O4 . . . 66

3.3 Schematic view of the CVD setup used for the preparation of CrO₂(100) films . . . 67

3.4 Standard sample design for XMCD-PEEM studies of rings . . . . 68

3.5 SEM images of NiFe ring structures for CIDM experiments . . . . 69

3.6 Schematics of the prepatterned substrate process . . . 71

3.7 XMCD-PEEM images of Fe3O4rings on prepatterned substrates . 71 3.8 Schematic representations of vortex and onion state . . . 75

4.1 Schematics of a cross-tie domain wall . . . 79

4.2 Zigzag domain walls in thin films . . . 80

4.3 Double vortex walls in thick ring elements . . . 81

4.4 PEEM images of rings with different geometries exhibiting trans- verse and vortex walls . . . 82

4.5 Domain wall type phase diagrams for NiFe and Co . . . 83

4.6 Schematics of the domain wall type energy landscape . . . 84

4.7 Ultrathin Co rings on prepatterned Si substrates . . . 86

4.8 Limiting cases of the structures described in the phase diagrams . 87 4.9 Spin structure transformations in NiFe rings due to heating . . . . 88

4.10 Images of a ring during a heating process showing the damage . 89 4.11 High-resolution PEEM images of domain walls . . . 91

4.12 Intensity profiles of domain walls . . . 92

4.13 Domain wall width as a function of the ring width . . . 93

4.14 PEEM images of interacting domain walls . . . 96

4.15 Domain wall types in rings as function of edge-to-edge spacing . 97 4.16 Stray field of a domain wall . . . 99

4.17 Distribution of energy barrier heights for vortex core nucleation . 100 4.18 PEEM images of nanostructures damaged by FIB patterning . . . 102

4.19 XAS and XMCD spectra of Fe3O4 . . . 103

4.20 XMCD-PEEM, MFM, and OOMF results of Fe3O4rings . . . 104

4.21 MFM and OOMMF results of rings magnetized along the hard axis 105 4.22 Fe3O4 zigzag lines with different widths . . . 106 4.23 MFM contrast formation in a ring magnetized along the easy axis 107

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LIST OF FIGURES x 4.24 MFM and OOMMF results of rings magnetized along the easy axis 108 4.25 SEM image of a Fe3O4prototype structure for CIDM experiments 109

4.26 PEEM images of a NiFe capped CrO2 wire . . . 110

4.27 PEEM images of a NiFe capped CrO2 film patterned with FIB . . 111

4.28 AFM and MFM images of domain patterns in CrO2 nanostructures 112 5.1 SEM image of a NiFe ring with simulation and MR measurement 117 5.2 Domain wall motion as function of current, field, and temperature 119 5.3 Joule heating due to current pulses . . . 121

5.4 Statistical distribution of the critical field . . . 125

5.5 SEM and PEEM images of sample with the initial magnetization . 127 5.6 Domain wall velocities of different wall types . . . 128

5.7 PEEM images of double vortex walls . . . 129

5.8 PEEM images of domain wall transformations after consecutive pulse injections . . . 130

5.9 Domain wall in a wide wire after consecutive pulse injections . . 134

5.10 Geometry dependent wall velocity as function of current density 135 5.11 Critical current density as function of the wire width . . . 137

5.12 Wall velocities as function of wire width and current density . . . 138

5.13 SEM images of a CrO2wire with constriction . . . 140

5.14 CrO2 hysteresis loops at 4.3 K . . . 141

5.15 Switching fields for the reversal of a CrO2wire . . . 142

5.16 Joule heating due to current in CrO2 . . . 143

5.17 Resistance as function of temperature for a CrO2wire . . . 143

5.18 Critical field for domain wall depinning in a CrO2 structure as function of temperature . . . 145

A.1 Conductance of NiFe wires as a function of thickness . . . 154

A.2 Comparison of NiFe and Au conductance in Au capped NiFe wires 156 B.1 Layout of theµ-BLS experimental setup . . . 159

B.2 Schematic view of a magnetic memory element . . . 160

B.3 Schematics of the sample fabrication process . . . 162

B.4 Spinwave spectra in the CoFe film surrounding the spin valve . . 163 B.5 Experimental and theoretical spinwave radiation patterns in CoFe 165

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List of Acronyms

AC Alternating current

AMR Anisotropic magnetoresistance

AFM Atomic force microscopy / microscope BLS Brillouin light scattering

BMR Ballistic magnetoresistance BNC British naval connector

CIDM Current-induced domain wall motion CIP Current in-plane

CPP Current perpendicular-to-plane CMR Colossal magnetoresistance CVD Chemical vapor deposition DC Direct current

DOS Density of states

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

GPIB General purpose interface bus

hcp hexagonal close packed (crystal structure) HV High voltage

ITMR Intergrain tunneling magnetoresistance LAN Local area network

LEED Low energy electron diffraction LEEM Low energy electron microscopy LHe Liquid helium

LIA Lock-in amplifier

LLG Landau-Lifshitz-Gilbert (equation)

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LIST OF ACRONYMS xii

LN2 Liquid nitrogen

MBE Molecular beam epitaxy (system)

MFM Magnetic force microscopy / microscope MOKE Magnetic-optical Kerr effect

MR Magnetoresistance

MRAM Magnetic random access memory µ-BLS Micro-focus Brillouin light scattering

PEEM Photoemission electron microscopy / microscope PMMA Polymethyl methacrylate

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

with polarization analysis

SLS Swiss Light Source

SQUID Superconducting quantum interference device STM Scanning tunneling microscopy / microscope TEM Transmission electron microscopy / microscope TMR Tunnel magnetoresistance

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

XMR Entirety of all magnetoresistance effects

X-PEEM X-ray photoemission electron microscopy / microscope XPS X-ray photoemission spectroscopy

XRD X-ray diffraction

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Introduction

Nanotechnology is a fast developing and growing field that covers parts of many disciplines in natural sciences and engineering. It deals with materials, structures, and processes on the nanometer scale. The impact of nanotechnology on society, for example in the fields of nanoelectronics or health care, are enormous and often unknown to many people being in contact with the achievements of this technology in their daily life [Sch04]. Nanomagnetism is the part of the nanotechnology that deals with magnetism on the nanometer scale and has a well known application in storage disc drives. Less known but as important for technological progress are sensors based on effects which stem from the field of nanomagnetism.

Disc drives store information by defining a certain direction of magnetization as a logical "1" and the opposite direction as a logical "0". Information is commonly written by a magnetic field that changes the magnetization direction of the bit. The readout process is realized by measuring the stray field with a field sensor. However, the permanent call for higher storage densities and faster data rates is pushing this technological approach to its fundamental limits. The limits in miniaturization of the bits as well as of the read-and-write head pose limits to the storage density. The time consuming mechanical positioning of the head and the rotation speed of the disc limit the data rates.

These limits can be partially overcome by magnetic random access memory (MRAM), which combines the advantages of the fast charge-based state- of-the-art DRAM with the non-volatility of magnetic information storage. Sin- gle storage cells are written by a magnetic field and read out by a simple resistance measurement. These storage cells consist of magnetic multilayer systems, so-called spin valves or tunnel junctions [Pri99, ERJ⁺02]. A new approach for writing information to such storage cells is the current-induced magnetization switching. Making use of the so-called spin transfer torque effect, magnetic

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Introduction 2 layers can be switched by a current flowing through the element. The design of a possible memory device is significantly simplified, because field generat- ing striplines are not needed any more. The spin transfer torque switching in magnetic multilayers has been studied intensively and was successfully demon- strated [MRK⁺99, HAN⁺04, CSK⁺04, Sun06].

Initiated by the research on spin torque effects in multilayers, also the current-induced manipulation of single domain walls in nanostructures has received rapidly increasing interest both due to fundamental interest in the physics involved and due to the application background. A prominent example for a domain wall based storage device is the recently proposed race- track memory [Par04]. The phenomenon of current-induced domain wall motion has been long known theoretically [Slo96, Ber84] as well as experimentally [FB85], but only recently controlled current-induced motion of single domain walls in magnetic nanostructures has been achieved (see for example [GBC⁺03, VAA⁺04, YON⁺04, RLK⁺05, KVB⁺05, KJA⁺05]). In particular, the observation of domain wall spin structure transformations due to current [KJA⁺05] points to the importance of comprehensive studies of domain wall spin structures. Furthermore, critical current densities for domain wall motion and wall velocities have been predicted [WNU04, HLZ06] and observed [NTM05, KJA⁺05] to depend on the wall spin structure. The ongoing discussion within theory about the correct model for the description of spin transfer torque effects (see for example [TK04, BM05, Bar06, TK06]) calls for in- depth studies of current-induced domain wall motion with a special focus on wall spin structure transformations in different geometries and materials. Since theory predominantly uses model systems at constant (mostly zero) temperature so far, the experiments have to be performed at constant temperatures to allow for direct comparison. Therefore, the inevitable Joule heating [YNT⁺05] has to be taken into account in order to separate thermal and intrinsic spin torque effects [YCM⁺06].

This thesis contributes from the experimental side to a deeper understanding of the domain wall spin structures in NiFe, Co, Fe3O4, and CrO2nanostructures and contains a comprehensive study of current- and field-induced domain wall motion at variable sample temperatures in NiFe. It is organized as follows:

In chapter 1, an introduction to the relevant theory is given with the main focus on the spin transfer torque theory, its development and present state, and

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Introduction 3 particular predictions of experimental results.

After that, the experimental techniques used are discussed in chapter 2. The X-ray magnetic circular dichroism photoemission electron microscopy (XMCD- PEEM) as the main magnetic imaging technique used is discussed including the developed setup for current pulse injection. Secondly, the bath cryostat setup for magnetoresistance measurements between 2 K and 300 K developed in the frame of this thesis is described in detail. Then the fabrication processes and the relevant general properties (magnetic and structural) of the investigated samples are briefly discussed in chapter 3.

Chapter 4 presents the results obtained on domain wall spin structures in NiFe, Co, Fe3O4, and CrO2, mainly by XMCD-PEEM. Comprehensive domain wall type phase diagrams for 180^◦ head-to-head domain walls in NiFe and Co are obtained and explained, thermally induced wall type transformations are studied, the wall width is determined as a function of the element width, and the dipolar coupling between adjacent walls is investigated. Furthermore, the domain wall spin structures and their dependence on the magnetocrystalline anisotropy in Fe3O4 and CrO2 structures are explored.

Chapter 5 contains the results on the interaction between domain walls and current. In particular, the current- and field-induced domain wall motion in NiFe structures is studied at constant temperatures using measurements of the anisotropic magnetoresistance contribution of a domain wall. The Joule heating due to the current is determined and the heating effects are discriminated from intrinsic spin torque effects. The wall velocity in current-induced domain wall motion and current-induced transformations of the domain wall spin structures are studied using XMCD-PEEM. Additionally, the magnetoresistance contributions in CrO2 are probed, the Joule heating is measured, and current-assisted domain wall motion is studied in CrO2at low temperatures.

Finally, the results are concluded and a brief outlook to possible future experiments is given.

Parts of this thesis have been published [LBB⁺06a, LBE⁺06, KLH⁺06, LKB⁺06, LBB⁺06b]. Reference will be made to this work where appropriate.

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Chapter 1

Theory

1.1 Overview

This chapter presents the theoretical background relevant for the physics and results presented in this thesis. First, the microscopic origins of magnetism are briefly discussed, including the exchange interaction and localized as well as delocalized electron contributions to ferromagnetism. These microscopic approaches are then complemented by a phenomenological view using thermodynamic potentials and macroscopic variables like magnetization and magnetic fields. This allows for establishing a straightforward relation to experiments, where only these macroscopic quantities are available. The micromagnetic simulations, which are frequently used nowadays in magnetism research and which base on this thermodynamic approach, are briefly described. Then the spin transfer torque theory is discussed in detail, since it is the main theoretical background for most of the results presented in this work. This theory includes 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. Magnetoresistance effects in magnetic materials, which constitute a complete family of effects, are treated in the next section with the focus on the anisotropic magnetoresistance, the intergrain tunneling magnetoresistance, and the domain wall contributions to the magnetoresistance. Finally, the material class of halfmetallic ferromagnets with the two prominent members Fe3O4 and CrO2 is introduced and the striking feature of theoretically 100% spin polarization in these systems is discussed.

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Theory 5

1.2 Microscopic Origins of Ferromagnetism

Two starting points exist for the microscopic description of ferromagnetism: In a localized model, the electrons responsible for the ferromagnetism 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 descriptions will be necessary to fully understand a ferromagnetic system. However, 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 behavior.

1.2.1 Exchange Interaction

The exchange interaction between single electrons can be regarded as the fundamental quantum mechanical effect that causes ferromagnetism. Exchange 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 distance between two electrons with the same spin is increased, which in turn reduces the Coulomb repulsion and therefore leads to a reduced energy of the system even though the reduction of the Coulomb energy is related to an increase of the kinetic energy.

The phenomenon can be described for two spins by an exchange Hamilto- nian that takes the form

Hˆ =−2Jˆσ₁·σˆ₂, (1.1) whereJis the exchange constant, i.e. the energy difference between parallel and antiparallel configuration, andσˆiare the Pauli spin matrices.

1.2.2 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_ijσ_i·σ_j, (1.2)

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Theory 6 Since this Heisenberg operator is non-linear, a solution of the problem can be often obtained only by linearization. The mean-field approximation 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 σ_i and the expectation valuehσ_ji of the spin operator of the interacting neighbors. The according mean-field can be written asB_MF = _gµ¹

B

P

jJ_ijhˆσ_ji. Assuming an external magnetic fieldB₀, the Heisenberg Hamiltonian in mean- field approximation is obtained in the form [IL99]

HˆMF=−gµ_BX

i

ˆσ_i·(B_MF+B₀). (1.3) 1.2.3 Band Model of Ferromagnetism

The band model assumes that each electron moves in an effective potentialV(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_e∇²+V_σ(r)

φkσ(r) =Ekσφkσ(r), (1.4) withme,kandσ denoting the electron mass, the wave vector, and the electron spin, respectively.

Taking into account the energy reduction due to the exchange interaction 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_↑ and N_↓ are the numbers of electrons in the two spin states, and N the total number of electrons. The so-called Stoner parameterI describes 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)

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Theory 7

for convenience, we obtain

E_↑(k) = E(k)˜ −1

2IR (1.9)

E_↓(k) = E(k) +˜ 1

2IR. (1.10)

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

f(Ekσ) = 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

R= 1 N

X

k

1 exp_E(_˜ _k₎₋¹

2IR−EF

kBT

+ 1 − 1 exp_E(_˜ _k₎₊¹

2IR−EF

kBT

+ 1

. (1.13)

This equation can be resolved toRunder 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= 0 K, 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(E_F), the so-called Stoner criterion for the occurrence of ferromagnetism is finally obtained:

ID˜(EF)>1 (1.14)

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

The spin-dependent density of states in a 3d-ferromagnet is schematically shown in Fig. 1.1. The band splitting gives rise to the fact that electrons 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 ferromagnetism occurs, when the majority density of states is not fully occupied (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 [Zel99].

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Theory 8

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.

1.3 Micromagnetic Systems

Since the microscopic descriptions presented in section 1.2 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 magnetic 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 analytically in special cases using the Stoner-Wohlfarth model or by simulations as detailed in sections 1.3.3 and 1.4.2.

1.3.1 Thermodynamics in Magnetism

The macroscopic starting point for the description of a magnetic system are the thermodynamic potentials. The magnetic configuration will change under the external conditions in order to minimize its energy or – more precisely – the according 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 1.5.

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 parameters are kept constant, the appropriate choice depends on the experimental conditions. The

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Theory 9

Gibbs free energy [Ber98]

G(H, T) =U−T S−µ₀H·M (1.15) depending on the external fieldHand the temperatureT is convenient in most situations. U denotes the internal energy andS the entropy. Since the magne- tizationM will be inhomogeneous over the sample in most practical cases, M has to be replaced by a vector field Mˆ ≡ M(r) so that the Gibbs free energy is replaced by the Landau free energy [Ber98]

G_L(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 approximation is justified if systems larger than interatomic distances are investigated. Its application is also limited to systems small enough so that an existing defect density does not do- minate the magnetic behavior, since such defects are difficult to include into the above description. Both conditions are fulfilled in the frame of this thesis.

1.3.2 Energy Contributions 1.3.2.1 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 expansion of the exchange energy contribution as a function of the magnetization gradient in the lowest-order term yields

E_ex= A

|Mˆ|² Z

∇Mˆ2

dV. (1.17)

The relation between the exchange stiffnessAand the atomic scale parameterJij

in eqn. 1.2 can be understood like this: By limiting the sum to nearest neighbors and interpreting theσ_ias classical vectors one can find thatA∝kJS²/a, where J is the nearest neighbor exchange constant,Sthe spin magnitude,athe lattice constant, andka numerical factor depending on the lattice symmetry [Ber98].

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Theory 10

1.3.2.2 Anisotropy Energies

Anisotropy energy contributions arise due to the fact that certain magnetization 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 1.3.2.3).

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 the m_i being the components of the magnetization direction m = M/|M| = (m_x, m_y, m_z). TheK_i, which can be positive or neg- ative, define the easy and hard axes or planes of the system corresponding to minima and maxima of the anisotropy energy contribution. When a thin film is considered, in which only in-plane magnetization 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 den- sitiesε_anican be directly transformed to anisotropy energies using

E_ani = Z

V

ε_ani M(r)ˆ

dr (1.21)

Further magnetocrystalline anisotropy contributions exist, which do not play a significant role in this thesis and are therefore only briefly mentioned for com- pleteness:

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

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Theory 11

• The magnetoelastic anisotropy arises in crystals were strain is present in the lattice. A typical situation for the occurrence is the epitaxial growth of multilayers with a lattice misfit that results in strain [COP⁺93, Lau02, FHG⁺06].

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

1.3.2.3 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 contribution 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. Neglect- ing the discrete nature of matter the shape effect of the dipolar interaction in a ferromagnetic ellipsoid can be described via the anisotropic demagnetizing field H_d =−NMwith the shape-dependent demagnetizing tensorN. For a thin film all tensor elements are zero except for the direction perpendicular to the layer for whichN^⊥= 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 magnitude equal to the saturation magnetizationM_s. θdenotes the angle with the film normal.

It can be directly seen, that an in-plane configuration of the magnetization is energetically favorable.

1.3.2.4 Zeeman Energy

The Zeeman energy is the potential of a magnetic dipole moment in a magnetic field. For a homogeneous external field H₀, this energy contribution depends only on the average magnetization and not on the detailed spin structure of the system. The Zeeman energy can be written as

E_dip=−µ₀ Z

H₀·MdV. (1.24)

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Theory 12

1.3.3 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 exemplarily 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 practically means computationally), in order to be able to compare experimental results with the predictions of theory. Adding all energy contributions discussed above yields the following expression for the Landau free energy¹:

G_L(M,ˆ H) = Z

V

A

|Mˆ|²

∇Mˆ2

+ε_ani(M)ˆ − 1

2µ₀H_d·Mˆ −µ₀H₀·Mˆ

dV (1.25) In order to determine the (meta-)stable magnetization configurations for given parameters 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 suffi- cient conditions for the existence of a minimum ^∂

∂MˆGL = 0 and ^∂²

∂Mˆ²GL > 0.

The difficulty lies in the fact that G_L is a functional of the entire vector field Mˆ and thus one has to consider the infinite-dimensional functional space of all possible magnetization configurations.

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

Mˆ ×H_eff = 0 (1.26)

Mˆ ×∂Mˆ

∂n = 0 (1.27)

with _∂n^∂ denoting the derivative in the outside direction normal to the surface.

The first condition must be fulfilled for every point inside the sample, the second is a boundary condition for the surface. The effective fieldH_eff is given by

H_eff= 2

µ₀M_s²∇(A∇M)− 1 µ₀

∂ε_ani

∂M +H_M +H₀. (1.28)

1Temperature effects are not included in this equation. They are also not included in the OOMMF code [OOM] used for micromagnetic simulations.

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Theory 13 It contains the applied fieldH₀, the demagnetizing fieldH_M, as well as the exchange and anisotropy contributions. The cross product in eqn. 1.26 shows, that thetorqueon 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 asm(r)× _∂n^∂ m(r) = 0, which is equivalent to the simpler condition _∂n^∂ m(r) = 0. The material parameters like the exchange constantA, the saturation magnetization M_s, all anisotropy constants determiningε_ani, as well as the external fields and the shape of the system have to be provided as input information. Solving Brown’s equations then yields the equilibrium magnetization distributionsM.ˆ

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 fulfilled in the equilibrium state. The answer to this problem is given by the Landau-Lifshitz-Gilbert equation.

1.4 Magnetization Dynamics

The well established Landau-Lifshitz-Gilbert (LLG) equation describes the magnetization dynamics in a material withoutthe influence of a current in form of a time dependent differential equation. This basis is derived, 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 following section.

1.4.1 Landau-Lifshitz-Gilbert Equation

The following description shows how the Landau-Lifshitz-Gilbert equation can be obtained from a quantum-mechanical starting point. It is based on the descriptions in [Wie02] and [Hin02].

The precession of an electron spin in an external magnetic field can be derived from the Heisenberg equation of motion

i~∂hˆs(t)i

∂t =

ˆs(t),Hˆ(ˆs(t))

, (1.29)

whereˆs(t)is the time dependent spin operator of the electron andHˆthe hamiltonian. The bracketsh. . .idenote the expectation value of an operator. The com-

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Theory 14 mutator

ˆs(t),Hˆ(ˆs(t))

can be approximated to first order in~to ˆs(t),Hˆ(ˆs(t))

=i~

ˆs(t)×∂Hˆˆs(t))

∂ˆs(t)

+O(~²) (1.30) by using the general commutation laws for angular momenta

sˆ_x,ˆs_y

=i~sˆ_zand cyclically and ˆ s_α,sˆ_α

= 0withα∈ {x, y, z}. (1.31) This results in the following equation of motion:

i~∂hˆs(t)i

∂t =i~

ˆs(t)×∂Hˆ(ˆs(t))

∂ˆs(t)

+O(~²) (1.32) The transition from quantum mechanics to classical mechanics in the limit~→0 can be performed using the Ehrenfest theorem [Ehr27], which identifies the spin s with the expectation value of the spin operatorˆs and the classical Hamilton functionHwith the HamiltonianHˆ. We obtain

∂s(t)

∂t =s(t)×∂H(s(t))

∂s(t) . (1.33)

The next step is to replace the spin s by the normalized magnetic moment S or the magnetization being the magnetic moment per volume unit, respectively.

Using s = µ/γ with the gyromagnetic ratio γ = gµ_B/~ of the electron (g = Landé factor, µB= Bohr magneton) and by normalizing the momentS =µ/µs, eqn. (1.33) transforms to

∂S

∂t =−γ

µ_s(S×H⁰_eff). (1.34)

The effective fieldH⁰_effis determined by the derivative of the Hamilton function H⁰_eff = _∂^∂_SH. In the simplest case, where the Hamiltonian contains only the Zeeman term (H=µ_sS·B), the Larmor precession of the electron in an external magnetic field is directly obtained:

∂S

∂t =−γ(S×B). (1.35)

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_eff− λ

µ_sS×(S×H⁰_eff), (1.36)

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Theory 15 where λis the damping constant introduced. However, investigations of the magnetization reversal time of a magnetic sphere as a function of the damping λshowed that infinite damping unreasonably leads to zero reversal times (1/λ- dependence) [Kik56]. Gilbert [Gil55] derived the so-called Gilbert equation of motion from the starting point of eqn. 1.33 using a Rayleigh dissipation function:

∂S(t)

∂t =−γ

µ_sS×H⁰_eff

| {z }

precession

+α

S×∂S(t)

∂t

| {z }

damping

(1.37)

As in the Landau-Lifshitz equation 1.36, the first term describes the precession and the second the damping. The Gilbert equation 1.37 can be transformed to the general form of the Landau-Lifshitz equation as detailed in the following.

Expanding eqn. 1.37 with "S×" on both sides and making use ofa×(b×c) = b(a·c)−c(a·b)leads to

S× ∂S(t)

∂t = −γ

µ_sS×(S×H⁰_eff) +αS

S· ∂S(t)

∂t

−α∂S(t)

∂t (S·S)

= −γ µs

S×(S×H⁰_eff)−α∂S(t)

∂t . (1.38)

Equating this result with eqn. 1.37 resolved toS× ^∂^S_∂t^(t) gives

∂S(t)

∂t =− γ

(1 +α²)µ_sS×H⁰_eff− γα

(1 +α²)µ_sS×(S×H⁰_eff), (1.39) 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 1.37, is referred to as LLG, which is correct in the sense that both equations can be transformed into each other as described. It should be mentioned, that both terms in eqn. 1.39 contain the damping constant α. Therefore, a separation between precession and damping contributions is not possible.

It is important to discuss the units used in the equations above. In eqn. 1.34 the momentSis dimensionless with[S] = 1and the effective fieldH⁰_eff is actu- ally an effective energy. This way of description is favorable 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 fieldH_eff is given in correct units of magnetic fields (A/m) instead of energies ([H⁰_eff] = J)

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Theory 16 and the magnetization Mis introduced instead of the dimensionless magnetic momentS, eqn. 1.34 can be written as

∂M(t)

∂t =−µ0γ(M×H_eff). (1.40) This finally leads to an implicit LLG equation analogous to eqn. 1.37 of the form

∂M(t)

∂t =−µ₀γ(M×H_eff) + α M_s

M×∂M(t)

∂t

, (1.41)

whereM_sdenotes the saturation magnetization.

1.4.2 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 following: First, the problem (sample shape) to be solved is subdivided into a two-dimensional grid of square cells with three-dimensional classical spins situated at the center of each cell.² 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 exchange length or the Bloch wall width

l_ex =

s2µ₀A

M_s² (1.42)

l_wall = rA

K, (1.43)

whereAare the exchange constant andKan anisotropy constant [HS98]. A cell size of 5 nm is a reasonable compromise between computation time and memory consumption on the one hand and a correct modeling on the other hand in most cases under investigation here.³ Smaller cell sizes were sometimes used and found not to influence the results of the simulations [KVB⁺04b].

2In 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. Furthermore, besides finite difference solvers like OOMMF, which dis- cretizes the problem in cubic cells with equidistant nodes, also finite element based algorithms exist, which subdivide the problem into polyhedral elements [CFK93]. The polyhedra can have a variable size and shape across the object to take into account small features, which require a refined discretization, as well as larger features, for which larger polyhedra help to save computation time [Her01]. A recent review can be found in [FS00].

3The particular cell sizes used are given together with the set of input parameters in the results chapter, where the individual simulations are discussed, as well as in table 3.1 on page 72.

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Theory 17 For each cell the effective fieldH_effas derived in section 1.3.3 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 necessary to sum over the energy contributions from all cells in the structure. It is computed as the convolution of the magnetization against a kernel that describes the cell-to-cell magnetostatic interaction by using fast Fourier transformation techniques, which reduces the number of operations toO(Nlog₂N)instead ofO(N²), whereN is the number of cells [DP02].

Then a numerical integration of the Landau-Lifshitz-Gilbert equation in the form given in eqn. 1.39 is performed using a second order predictor-corrector technique of the Adams type [DP02]. The procedure is iterated yielding a step by step calculation of magnetization configurations with decreasing energy until the torqueMˆ ×H_eff (see eqn. 1.26) of the effective field on the magnetization is below a chosen threshold value at each point of the system.

1.5 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.

1.5.1 Hydromagnetic Drag Force

During the last three decades, many efforts were made to describe current- induced effects on domain walls theoretically and different approaches were put forward. A comprehensive review of the development of the theory can be found in [Mar05]. In 1974 the concept of the hydromagnetic drag force was introduced [Ber74, Car74, Cha74]. Then the spin transfer torque theory was developed and different extensions of the Landau-Lifshitz-Gilbert equation for the

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Theory 18

Figure 1.2: (from [Ber78]) The non- uniform current distribution (a) in an uniaxial material with a wall can be decom- posed into a uniform distribution (b) and an eddy current loop (c) around the wall.

description of interactions between current and magnetization were suggested as detailed in section 1.5.2.

The basic principle of the hydromagnetic drag theory is the following: The tilted magnetization in the wall region gives rise to a Hall effect. The current path is therefore modified as visualized in Fig. 1.2(a) and can be regarded as a superposition of the undisturbed current flow (b) with an eddy current around the domain wall (c). This eddy current causes magnetic fields that exert forces on the domain wall so that the wall is displaced in the direction of the charge carrier drift.

The force is proportional to the cross-section of the wall and therefore to the film thickness, and so this hydrodynamic drag effect will be generally a significant issue only for film thicknesses larger than approximately 100 nm [Mar05].

Furthermore, the model of Berger [Ber74] was developed with the view on Néel or Bloch walls (see section 4.2) as present in bulk material of films. In 180^◦ head-to-head domain walls, an analogous geometrical argument as sketched in Fig. 1.2 for a Néel or Bloch wall shows that the hydromagnetic drag force is negligible in this geometry.

In the frame of this thesis, nanostructures are fabricated from thin film samples and mainly 180^◦ head-to-head domain walls are observed. Thus the hydromagnetic drag force does not play any significant role here.

1.5.2 s-d Exchange Force and Spin Transfer Torque

Particularly 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 force (between the localized 3d- electrons responsible for ferromagnetism and the delocalized 4s-electrons car- rying the current), but it is referred to as spin transfer torque or spin transfer

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Theory 19

effect nowadays. Starting from the s–d exchange potential V(x) =gµ_B

s·H_sd(x) +H_sd 2

, (1.44)

whereH_sd(x) =−2J_sdhS(x)i/(gµ_B)is the exchange field with the s–d exchange integralJ_sd and the coordinatexnormal to the wall plane, the force applied to the domain wall by the current is finally obtained as

F_x = 2M_s

µ_i (βv_e−v_w). (1.45)

Here,veare the drift velocity of the charge carriers,vwthe wall velocity,µithe intrinsic wall mobility, andβa dimensionless constant. The additive constant ^H₂^sd ensuringV(±∞) = 0consequently does not play any role in the expression of the force. The important underlying assumptions are the adiabatic approximation, which means that the angle between the spinsand the exchange fieldH_sd must be sufficiently small, as well as the classical treatment using Ohm’s law of the diffusive electron motion, which is justified if the electron wavelength is small compared to the wall width.

The first approach to integrate the spin transfer torque into the Landau- Lifshitz-Gilbert equation (eqn. 1.41) by adding additional terms was made by Slonczewski [Slo96]. He considered two ferromagnetic layers separated by a non-magnetic spacer with the macrospins S_1,2 = |S_1,2|ˆs_1,2. The torque exerted by a charge currentI on a macrospin was found to be

∂S_1,2

∂t =Ig

eˆs_1,2×(ˆs₁×ˆs₂) (1.46) with the electron charge eand a prefactorg containing the spin polarizationP of the current:

g=

−4 + 1 +P³

4P^3/2 (3 +ˆs₁·ˆs₂) −1

(1.47) First we see that the torque is proportional to the current and changes its sign when the current is reversed.⁴ Secondly, it depends on the spin polarizationP, and the direction of the torque is expressed by the productˆs_1,2×(ˆs₁×ˆs₂). When including this torque term into the LLG equation 1.41, we obtain a description of the system with a spin current.

4Viret and coworkers [VVOJ05] have suggested a current-induced pressure onto a tilted domain wall, which is quadratic in the current and therefore independent of its sign, but it is predicted to be dominating only at high current densities and in materials with large resistivity like magnetic semiconductors.

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Theory 20 Also the experimental research in the field of interactions between current and magnetization in this multilayer or so-called pillar geometry has attracted much interest in the last years. One of the first observations of current-induced switching in multilayers was reported by Myers et al. [MRK⁺99], current-driven microwave oscillations were observed by Kiselev et al. [KSK⁺03], and different temperature dependent experiments can be found in [MAS⁺02, TSR⁺04, KEG⁺04]. A time dependent study of current-driven magnetization dynamics is presented in [KES⁺05] and results on the magnetization reversal in systems with perpendicular anisotropy was recently published [MRK⁺06].

1.5.3 Landau-Lifshitz-Gilbert Equation with Spin Transfer Torque A key issue for the comparison of theoretical predictions and experimental observations are micromagnetic simulations, which require the correct addition of the torque term(s) to the Landau-Lifshitz-Gilbert equation. Thiaville and coworkers [TNMV04, TNMS05] have suggested the following extension of the LLG equation, that incorporates a spin torque correction for local angular mo- mentum transfer from current to magnetization:

∂M(t)

∂t =−γ₀M×H_eff+ α Ms

M×∂M(t)

∂t −(u· ∇)M

| {z }

extension

(1.48)

γ₀ = µ₀γ = µ₀gµ_B/~ is the gyromagnetic ratio and u is defined below in eqn. 1.54. They restrict themselves to adiabatic processes, i. e. a local equilibrium between the conduction electrons and the magnetization is assumed.

This description can be reformulated to obtain what Li and Zhang have introduced [LZ04a]. In the low temperature regime the magnetization has a constant value and thus they obtain

τ_[LZ04a] =− 1

M_s²M×[M×(u· ∇)M]. (1.49) This form of the spin torque is identical with what Bazaliy et al. [BJZ98]

have considered earlier for metal-ferromagnet interfaces in presence of a spin- polarized current

τ_[BJZ98]=− a

M_sM×[M×m] (1.50)

withmbeing the unit vector of the magnetization in the pinned layer. This cor- responds directly with the macrospin version of the torque suggested by Slon-

Interactions Between Current and Domain Wall Spin Structures