Domain Wall Dynamics in Magnetic Nanostructures

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Domain Wall Dynamics in Magnetic Nanostructures

DISSERTATION

ZUR ERLANGUNG DES AKADEMISCHEN GRADES DES DOKTORS DER NATURWISSENSCHAFTEN

AN DER UNIVERSITÄT KONSTANZ

MATHEMATISCH-NATURWISSENSCHAFTLICHE SEKTION FACHBEREICH PHYSIK

VORGELEGT VON DANIEL BEDAU

TAG DER MÜNDLICHEN PRÜFUNG: 20. FEBRUAR 2008 REFERENTEN:

PROF. DR. ULRICH RÜDIGER PROF. DR. WOLFGANG BELZIG

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5017/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-50177

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Zusammenfassung

In dieser Arbeit wird das dynamische Verhalten von magnetischen Domänen- wänden in lateral eingeschränkten magnetischen Nanostrukturen aus Permal- loy untersucht.

Für diese Messungen wird ein Hochfrequenzkryomagnetsystem benötigt, dessen Entwurf und Aufbau beschrieben werden. Das Kryomagnetsystem er- möglicht es, Proben bei Temperaturen von2 Kbis zu500 Kim Frequenzbereich von DC bis zu20 GHzzu untersuchen und dabei Magnetfelder von bis zu0.5 T unter beliebigen Winkeln in der Probenebene anzulegen.

Zunächst werden die statischen Eigenschaften der durch Kerben in den Na- nostrukturen erzeugten Domänenwand-Pinningpotentiale bestimmt. Aus der Winkelabhängigkeit des Depinningfeldes werden die Positionen von Vortex- wänden und von Transversalwänden in den Potentialmulden bestimmt und die Ergebnisse mit mikromagnetischen Simulationen verglichen.

Die kritischen Stromdichten und Feldstärken der strom- und feldinduzier- ten Domänenwandpropagation werden für Systeme mit und ohne Kerben bestimmt. Der Einfluß der Temperatur auf die kritischen Stromdichten und Feld- stärken wird untersucht und mit Voraussagen der Spintransfertheorie verglichen.

Es wird gezeigt, daß Domänenwände unter gewissen Voraussetzungen als Quasiteilchen beschrieben werden können, die sich in einer durch Konstriktio- nen und Defekte erzeugten Potentiallandschaft bewegen. Den Domänenwand- Quasiteilchen kann eine effektive träge Masse zugeordnet werden: eine gepinnte Domänenwand kann zu Schwingungen angeregt werden.

Ein neuartiger physikalischer Effekt, die homodyne Gleichrichtung an schwingenden Domänenwänden, wird vorgestellt. Die homodyne Gleichrich- tung ist ein empfindlicher Indikator der Schwingungsamplitude und ermöglicht eine genaue Vermessung der Domänenwandresonanzspektren in einem weiten Parameterbereich an Magnetfeldern, Temperaturen und Leistungen. Die so ge- wonnenen Eigenfrequenzen werden mit Werten verglichen, die aus Depinning- spektren erhalten wurden.

Die Gleichrichtungsspektren weisen eine charakteristische Linienform auf, die in mikromagnetischen Rechnungen reproduziert werden konnte. Aus der Linienform läßt sich die Polarität der Vortexkerne bestimmen und es wird gezeigt, daß sich die Vortexkerne bei resonanter Anregung mit sehr kleinen Ma- gnetfeldern schalten lassen.

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List of Figures

1.1 Domain wall velocity . . . 17

1.2 Photoemission spectrum . . . 21

1.3 XMCD contrast formation . . . 22

2.1 Experiment setup . . . 24

2.2 Vector magnet . . . 25

2.3 Circuit to decouple the sample from the power supply . . . 26

2.4 Design of the verticalfield coil . . . 27

2.5 Winding of the vertical coil . . . 27

2.6 Cryostat . . . 28

2.7 Flow diagram . . . 29

2.8 Dual line head . . . 30

2.9 Lock-in schematic 1 . . . 31

2.10 Lock-in schematic 2 . . . 31

2.11 Switching box . . . 33

2.12 Sample holder . . . 34

2.13 Sample in sample holder . . . 35

2.14 Switching box, detail . . . 36

2.15 Crosstalk of the sample holder . . . 37

2.16 Bandwidth measurement . . . 38

2.17 Sample carrier . . . 38

2.18 High-frequency test setup . . . 39

2.19 Time domain reflectogram . . . 40

2.20 Time domain reflectogram . . . 41

2.21 Microwave spectrum . . . 43

2.22 Power spectrum . . . 44

2.23 Low level microwave detection scheme . . . 45

2.24 RF choke . . . 45

2.25 MMIC amplifier . . . 46

2.26 PIN switch . . . 46

2.27 Bias tee . . . 46

2.28 Electronic switch (schematic) . . . 48

2.29 Electronic switch (photograph) . . . 49

2.30 Power distributor . . . 50

2.31 Destroyed sample . . . 51

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3.2 Contacting a microstrip . . . 62

3.3 Field and current distribution around the microstrip line . . . 63

3.4 Impedance of the microstrip . . . 63

3.5 Dispersion of the microstrip . . . 64

3.6 Transitions to planar waveguides. . . 65

3.7 Damping of the microstrip . . . 65

3.8 Field distribution around a coplanar waveguide . . . 66

3.9 Dispersion of the coplanar waveguide. . . 66

3.10 Damping coefficient for different planar waveguides. . . 67

3.11 Coplanar waveguide . . . 68

3.12 Impedance of coplanar waveguides . . . 68

3.13 Tapered transition . . . 69

3.14 Frequency sweep . . . 69

3.15 Power sweep . . . 70

3.16 Power calibration . . . 71

3.17 Configuration for power calibration . . . 72

3.18 Screenshot . . . 73

3.19 Magnetic ring with antenna . . . 74

3.20 Lift-off process . . . 77

3.21 Lift-off defect . . . 78

3.22 Sample after lift-off . . . 78

4.1 Two-point configuration . . . 80

4.2 Four-point configuration . . . 80

4.3 Hysteresis scheme . . . 81

4.4 Saturation scheme . . . 82

4.5 Rotation scheme . . . 82

4.6 Mode étoile scheme . . . 83

4.7 Mode étoile curve . . . 83

4.8 Field depinning . . . 84

4.9 Field depinning scheme . . . 85

4.10 Current depinning scheme . . . 85

4.11 Configuration for pulse injection . . . 86

4.12 Apply MW scheme . . . 87

4.13 Field depinning microwave scheme . . . 87

4.14 Configuration for frequency sweep . . . 88

4.15 Frequency sweep DC scheme . . . 89

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List of Figures ix

4.16 Configuration for frequency sweep DC . . . 89

4.17 Combined configuration . . . 90

4.18 Twofield frequency sweep scheme . . . 91

5.1 SEM images of rings . . . 93

5.2 Mode étoile . . . 95

5.3 Rotation scan . . . 96

5.4 Depinningfield as a function of the appliedfield direction . . . . 96

5.5 Hysteresis curve . . . 98

5.6 Angular dependence of the depinningfield . . . 99

5.7 MR hysteresis loop . . . 100

5.8 Permalloy ring and mode étoile curve . . . 103

5.9 Potential landscape and domain wall propagation diagram . . . . 104

5.10 Mode étoile and domain wall propagation diagram . . . 107

5.11 Criticalfield and critical current density . . . 108

5.12 SEM images and simulations . . . 112

5.13 Frequency Sweep DC Scheme . . . 113

5.14 Depinning spectrum transverse wall . . . 115

5.15 Depinning spectrum vortex wall . . . 116

5.16 Equivalent circuit of the sample . . . 117

5.17 Homodyne spectrum . . . 118

5.18 Field dependence of the frequency . . . 119

5.19 Dependence of the depinningfield on the current density . . . 120

5.20 Depinningfield . . . 121

5.21 Micromagnetic Simulation . . . 122

5.22 Homodyne detection, simulation . . . 122

5.23 Simulated DC signal . . . 123

5.24 SDOF model . . . 124

5.25 Normalized magnitude of the transfer function . . . 125

5.26 Phase angle of the transfer function . . . 125

5.27 Rectified voltage as calculated from the SDOF model . . . 126

5.28 MDOF model . . . 127

5.29 Inversion of the vortex core polarity . . . 128

5.30 Domain wall potential . . . 130

5.31 Curvature . . . 131

5.32 Field dependence of the peak frequency . . . 132

5.33 Asymmetric potential . . . 132

5.34 Rectification spectra . . . 133

5.35 Oscillation period . . . 134

5.36 Domain wall potential . . . 135

5.37 Micromagnetic simulation . . . 136

5.38 PEEM Sample . . . 137

5.39 Dynamic domain wall nucleation . . . 137

5.40 XMCD PEEM picture series . . . 138

5.41 Dynamic formation of domain walls . . . 140

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D.1 Block diagram . . . 162

D.2 Wiring plan . . . 163

D.3 Main controller . . . 164

D.4 Output stage driver . . . 165

D.5 Output stage . . . 166

D.6 NTC resistor breakage control . . . 167

D.7 Controller board, picture . . . 168

D.8 Controller board, schematic . . . 169

D.9 NTC driver . . . 170

D.10 Current sense . . . 171

D.11 Zero current detection . . . 172

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List of Acronyms

AC Alternating current

AMR Anisotropic magnetoresistance CIDM Current-induced domain wall motion DC Direct current

DW Domain wall

ESD Electrostatic discharge FEM Finite element method FMR Ferromagnetic resonance LIA Lock-in amplifier

LLB Landau-Lifshitz-Bloch equation

LLG Landau-Lifshitz-Gilbert equation, LLG micromagnetic simulator LN₂ Liquid nitrogen

MBE Molecular beam epitaxy MDOF Multiple degrees of freedom MFM Magnetic force microscopy MR Magnetoresistance

OOMMF Object oriented micromagnetic framework PCB Printed circuit board

PDE Partial differential equation

PEEM Photoemission electron microscope PMMA Polymethyl methacrylate

RKKY Ruderman-Kittel-Kasuya-Yosida SDOF Single degree of freedom

SEM Scanning electron microscope TDR Time-domain reflectometry TW Transverse wall

VW Vortex wall

XMCD X-ray magnetic circular dichroism

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Introduction

Recently interest in magnetic domain walls has been surging, fueled partly by a fundamental interest in the magnetic properties, the nanoscale spin structure and in particular by possible novel logic and memory applications based on domain walls [All02, Par97, AXC04, AXF⁺05, FCA⁺04].

As recently demonstrated, spin-transfer torque effects can be used to depin and displace a magnetic domain wall in a simple nanoscale single layer structure by injecting high current pulses [VAA⁺04, YON⁺04, KVB⁺05, KJA⁺05, HTB⁺06, HTR⁺07, SMYT04]. The effect has shown potential for achieving fast and reproducible switching in these simple single layer elements, in particular in the context of new memory and logic devices based on domain wall propagation [All02, Par97].

In contrast to field-induced domain wall motion, current-induced domain wall motion (CIDM) still lacks a thorough understanding. Most of the theoretical models describing the current interaction with wide domain walls are based on the adiabatic approximation [TK04a], but non-adiabatic corrections have been introduced [ZL04, ZLF02, TNMS05].

Logic devices with switching by field-induced [AXF⁺05], as well as by current-induced [Par97], domain wall motion [YON⁺04,KVB⁺05,VAA⁺04] have been suggested. In order to use current-induced domain wall motion in applications, the switching has to be simple and reproducible. To control the domain wall behaviour and in particular the switching, pinning centres as described below can be used to confine the wall propagation in field-induced switching [LDKRB01, FCA⁺04, HON⁺03] and also in current-induced switching [KVB⁺05, GLC⁺02]. Such pinning centres provide well-defined stable loca- tions for domain walls. Pinning centres can as well result from imperfections in the material [GBC⁺03, KVB⁺04b], but these are inherently hard to control.

Instead, in order to engineer pinning, artificially structured variations in the geometry of an element have been introduced. While protrusions in wires have been used [AXC04, HOKO05], constrictions have been shown to yield particularly well-defined pinning centres for domain walls [KVR⁺03, KVW⁺04, KER⁺05, GLC⁺02, ZP04, FCA⁺04]. In addition to applications, such as domain wall diodes in logic devices [AXC04], constrictions have also allowed the deter- mination of some more fundamental properties of domain walls, such as magnetoresistance effects associated with domain walls [KVR⁺03].

To probe the potential landscape around the pinning site, as well as the spin-

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by a domain wall. In [KER⁺05, KVR⁺03, KVW⁺04] the width and the depth of the potential have been determined. In this work we measure the angular dependency of the depinningfield to determine the equilibrium position of the domain wall with respect to the lateral constriction [BKR⁺07].

The magnetic moment of the electron is linked to an angular momentum, for this reason the spin configuration of a moving magnetic domain wall will be different from the spin configuration of a stationary domain wall. As the latter corresponds to the lowest energy state of the magnetization configuration, the first non-vanishing energy correction for a moving domain wall will be proportional to the square of the wall velocity [Dör48], and the domain wall will behave as if it had a positive effective mass and exhibit inertia.

This mass manifests itself in a quasiparticle behaviour of the domain walls.

Domain wall quasiparticles can be excited to oscillate in the pinning potential of a lateral constriction, the motion has been described in terms of a harmonic oscillator model [KDB⁺07]. Oscillations can be excited by AC magneticfields or by electric currents via the spin-transfer torque effect [TNMS05, LZ04a, ZL04].

Dynamic measurements have so far been limited to the investigation of the combined action offields and currents [HTB⁺06]. Only recentlyfirst dynamic measurements of pure current-induced domain wall motion have been reported [HTR⁺07]. Rather than using continuous currents, AC excitations were found to be more efficient, and have been reported to move domain walls at far lower current densities than in the case of stationary currents [SMYT04]. Excitations of domain walls using AC fields have been the subject of extensive research [WPS⁺06, NKP⁺06, BRG⁺05], as well as excitations of ferromagnetic materials without domain walls [YMKO07, YMO⁺07, YOS⁺07, GMW⁺07].

Saitoh and coworkers [SMYT04] probed the oscillation of a free domain wall in a wire by measuring the AC resistance under excitation. They did not use artificial pinning by a constriction, but generated a potential well by an external magneticfield. They found a peak in the AC resistance, which they attributed to a large-scale wall motion, though the actual extent of the wall motion was not directly determined.

In this work a novel physical effect is demonstrated, namely the homodyne rectification of microwave signals by a pinned magnetic domain wall resonating in the potential well generated by a geometrical shape variation [BKK⁺07]. The detected voltage is the DC component that results from the frequency modu- lation of the injected high-frequency current with the varying resistance of the

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Introduction 3 sample caused by the domain wall oscillation. Such a homodyne detection is well known from analogue or digital signal processing [Gec97], and is usually carried out using non-magnetic semiconductor devices [Feh86].

The resonance frequencies obtained by the homodyne detection method are independently verified by measuring the depinningfields of the domain wall under microwave excitation.

From these measurements we can thus completely characterize the potential landscape, since, in addition to the width and depth, the resonance frequency is a measure for the curvature of the attractive potential well. From the shape of the rectified signal we are able to determine the direction of the centre of the vortex domain wall, the vortex core. Since in our case we study the potential well generated by a geometric constriction, we have the externalfield as another degree of freedom, and wefind that the resonance frequency depends on the field, indicating that the curvature and thus the shape of the potential well can be tuned by applying a magneticfield.

This work is organized as follows:

Chapter 1 gives a short overview of the relevant theory. The emphasis is on the micromagnetic description of magnetism, which is presented after an introduction to the electronic origins of magnetism. We introduce the Landau-Lifshitz- Gilbert equation and discuss the adiabatic and non-adiabatic extensions to describe the interactions with electric currents.

The theoretical chapter closes with a short description of the x-ray magnetic circular dichroism, which allows to directly image spin structures with a very high time resolution down to15 ps, limited only by the pulse width of the syn- chrotrone.

Chapter 2 contains the technical details of the magnetotransport setup that was constructed as part of this work. We describe the vector magnet system, the cryostat, the high-frequency sample holder and the electronic equipment. In the second part of chapter 2, the dynamic PEEM that was used for the time-resolved imaging is discussed.

Chapter 3 describes the samples and the materials used. The layout of the structures is discussed with a special emphasis on the high-frequency aspects of the sample design. We present different approaches to couple high-frequency signals into our structure and describe the design principles of the waveguides used therefore. The chapter closes with a short discussion of the lift-off technique used to fabricate the samples characterized in this work.

Chapter 4 introduces the magnetotransport measurement techniques. We describe the individual measurements that are the building blocks of the experiments presented in this work. The experimental configurations are explained and the parameters are given for each experiment.

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In section 5.1.2 we study the influence of thermal activations on current and field-induced domain wall propagation. We measure the temperature dependence for bothfield- and current-induced domain wall propagation and show that, even though the thermal activation helps to overcome the pinning potential and therefore reduces the depinningfield, the intrinsic efficiency of the spin- transfer process is also reduced. This leads to an overall increase of the threshold current for higher temperatures.

In section 5.1.3 we present a novel physical effect, termed homodyne domain wall rectification, that stems from the mixing between an oscillatory AMR contribution and the exciting AC current. The homodyne rectification is used to study dynamic excitations of magnetic domain walls and the measured spectro are compared to those obtained from depinning experiments. Using micromagnetic simulations we were able to reproduce the rectifying action of the domain wall and identify the domain wall eigenmodes being excited.

In section 5.1.4 we further study the potential landscape around a domain wall. By varying the power level we are able to directly determine the shape of the pinning potential. In addition, a method is presented to totally determine the shape of the potential well for asymmetric potentials.

In the second part of chapter 5 the dynamic generation of magnetic domain walls is studied using XMCD-PEEM. Permalloy rings are fabricated on the central conductor of a coplanar waveguide. A short current pulse creates an Oerstedfield that saturates the sample. The relaxation of the saturated sample is studied and it is shown that the domain wall is formed in less than100 ps.

The results are summarized and an outlook is given on page 141.

Parts of this work have been published, a list of publications is given starting on page 195.

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Chapter 1

Theory

In this chapter we briefly discuss the electronic origins of magnetism. We present the Heisenberg and Ising spin models and the micromagnetic approximation, replacing the spin lattice with a continuousfield. We give the different energy contributions to the Landau free energy deriving Brown’s equations of magnetostatics. The Gilbert and Landau-Lifshitz-Gilbert equations are introduced with extensions to describe the interaction of the magnetization with electric currents.

The anisotropic magnetoresistance is briefly discussed. At the end of the chapter, a short introduction to the x-ray magnetic circular dichroism effect (XMCD) is given.

1.1 The Origin of Magnetism in Permalloy

A ferromagnet like Permalloy¹, exhibits spontaneous parallel alignment of the magnetic moments. To describe such spontaneous self-organization phenomena the interaction between the individual spins has to be understood.

Atfirst one could assume that the individual spins interact by dipolar coupling [MVH], which would favour an antiparallel orientation between single spins: Two magnetic dipoles with magnetic momentsµ_ithat are separated by a distancerhave a dipolar energy of:

E= µ₀ 4πr³

µ₁·µ₂− 3

r²(µ₁·r)(µ₂·r)

(1.1) For typical values (r = 0.1 nm,µ = 1µ_B ≈9.3×10⁻²⁴J/T) the dipolar energy isE =µ²_B/4πr³ ≈10⁻²³J. This energy is small compared to the thermal energy of the spins of4×10⁻²¹Jat room temperature. The dipolar coupling therefore could only play a role in the milli-Kelvin range, but cannot explain ferromag- netism, which exists up to about 1000^◦C [Blu01].

Instead of the dipolar coupling, the so called exchange interaction, which is a purely quantum mechanical phenomenon, is responsible for the magnetic

1Permalloy is an iron/nickel alloy, the material parameters of Permalloy are discussed in section 3.1.

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χ_S,T stands for the spin part of the wave function, which is either antisymmetric in case of the symmetric product state or symmetric for an antisymmetric product state.

As the degeneracy is2s+1for each state, the triplet will split into three levels in an externalfield. The names triplet and singlet are spectroscopic designation:

in an external magneticfield the triplet (s=1) state splits into three spectral lines while the singlet (s=0) energy remains unchanged.

The energies for the symmetric and antisymmetric product state are easily calculated:

E_S=< ψ_S |Hˆ |ψ_S >=

dr₁dr₂ψ_S^∗Hψˆ _S (1.4) E_T =< ψ_T |Hˆ |ψ_T >=

dr₁dr₂ψ^∗_THψˆ _T (1.5) We can rewrite the Hamiltonian of the two-particle system as a spin- dependent and a spin-independent part [Czy00, LL97a]:

Hˆ = 1

4(E_S+ 3E_T)−(E_S−E_T)S₁·S₂ (1.6) If we define the exchange constantJas the energy difference between both states

E_S−E_T = 2

dr₁dr₂ψ₁^∗(r₁)ψ^∗₂(r₂) ˆH ψ₁(r₂)ψ₂(r₁), (1.7) J = E_S−E_T

2 , (1.8)

we can rewrite the exchange Hamiltonian as

Hˆ_ex=−2J S₁·S₂. (1.9) This Hamiltonian couples twos = ¹₂ particles to a bosonic joint state with cu- mulative spins=0 or 1, corresponding to a singlet or a triplet. The definition of this Hamiltonian motivates the Heisenberg and Ising models, explained below, which describe the behaviour of coupled spins. From Eq. 1.7 it is obvious that Eq. 1.7 is just a convenient formulation for the additional electrostatic Coulomb interaction energy necessary to fulfil the Pauli principle.

The exchange constant can be easily calculated from the exchange integrals Eq. 1.7. Depending on the sign of the exchange constant, the two electrons will

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1.1. The Origin of Magnetism in Permalloy 7 couple either parallel or antiparallel. If the electrons are in closely overlapping atomic orbitals, the exchange constant is normally positive, favoring the triplet with its antisymmetric spatial state and confirming Hund’s rules. If the electrons are in the same atomic orbital the Pauli exclusion principle explicitly forbids the triplet state and a singlet is formed.

For molecular orbitals that are in general more extended the situation is less clear: Less Coulomb energy is gained from the antisymmetric state, therefore the singlet is often the lowest energy state. This is the case for the hydrogen molecule, where the ground state is a singlet with two antiparallel spins.

The existence of singlet ground states is typical for molecules, but not a general rule. In the majority of cases the exchange interaction is too low to force an alignment of the spins; therefore the molecule or crystal is nonmagnetic. Some transition metals, metal oxides and salts are magnetic, in these materials the exchange is strong enough to force an alignment of adjacent spins.

In rare earth metals the magnetism is dominated by the 4f electrons which do not overlap significantly with the orbitals of surrounding atoms. Without overlap the integral Eq. 1.8 and therefore the exchange Hamiltonian Eq. 1.9 are zero, it is obvious that direct exchange cannot describe the magnetic properties of these materials. Even though the direct exchange is zero, higher order interactions exist, which are mediated by other orbitals. In the magnetic oxides and salts the magnetism is mediated by nonmagnetic ions, these effects are called superexchange and double exchange.

Permalloy², like its constituents iron and nickel, is a 3d ferromagnet (the orbital momentum is quenched, therefore only spins are responsible for the fer- romagnetism). Even though there is some overlap between the 3d orbitals, the direct exchange cannot explain the strong ferromagnetic ordering in these metals. The main contribution stems from indirect exchange mediated by the 4s conduction electrons [Blu01].

Ferromagnets in which the exchange interaction is mediated by conduction electrons are calleditinerant ferromagnets. The fractional magnetic moment (e.g.

1.72µ_Bfor Co) of these material shows that the magnetism is not caused by individual magnetic ions. A technique to determine the distribution of the magnetic moments is neutron scattering [Moo66].

In the following section we will show, that under certain conditions spontaneous polarization of the conduction electrons will occur, leading to ferromag- netism.

Assume a nonmagnetic system with a density of statesD(E)and a dispersion relationE(k), that could be obtained by band structure calculations or photoemission spectroscopy.

If the system has a net magnetic polarizationM, the degeneracy between the two spin subbands is lifted. One subband is shifted up and the other band is shifted down by the same energy.

0Stoner introduced the phenomenological parameterI, which describes this energy shift as a function of the relative magnetic polarizationR, the latter being

2The properties of permalloy are discussed in section 3.1 on page 58.

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If we now assume a magnetizationM, the two subbands are shifted according to

E_↑(k) =E(k)− 1

2IR (1.12)

E_↓(k) =E(k) +1

2IR. (1.13)

The energy shift ¹₂IR is linked to the occupation numbers in Eq. 1.10 by the Fermi-Dirac distribution, which, substituted into Eq. 1.10, leads to the equation:

R= N_↑−N_↓ N = 1

N

k

⎛

⎝ 1 exp(^E(k)−_k¹²^IR−E^F

BT ) + 1

− 1

exp(^E(k)+_k¹²^IR−E^F

BT ) + 1

⎞

⎠. (1.14) This is an implicit equation for the relative polarizationR. If a solution for R= 0exists, the system will show a spontaneous magnetization [IL02].

In the low temperature limit this condition leads to the inequality

ID(E_F)/2>1. (1.15)

Eq. 1.15 is called the Stoner criterion. If the Stoner criterion is fulfilled, the system has a ferromagnetic ground state. Explicitly stated: the system will show spontaneous polarization if the product of the density of statesDat the Fermi level and the energy contribution from the exchange interaction are larger than one. The factor of ¹₂ stems from the fact thatDis the entire density of states for both subbands together.

For systems without a spontaneous polarization the susceptibility can be de- veloped in terms of the Stoner parameterI:

χ= χ₀

1−ID(E_F)+. . . . (1.16) From this equation we can expect a significant increase of the susceptibility for materials that nearly fulfil the Stoner criterion. For example platinum and pal- ladium show a strong paramagnetism, which is caused by the Stoner enhance- ment described above. In this case the paramagnetism does not obey Curie’s law, since the exchange energy contribution is larger then the thermal energy of the system.

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1.2. Ising and Heisenberg Model 9

1.2 Ising and Heisenberg Model

The definition of the exchange Hamiltonian in Eq. 1.9 as Hˆ = −2J S₁·S₂ is the foundation of a simple, yet fundamental class of models to describe magnetic behaviour.

We start from a lattice of single spins, corresponding to the magnetic moments, which might be localized at the lattice points of a crystal or might be randomly distributed, as in a spin glass [CC05]. The interaction between each of the magnetic moments is now modeled individually,Hˆ_ij =−2J_ijS_i·S_j is the contribution from the lattice points i and j to the exchange energy. The complete Hamiltonian thus has the form:

Hˆ =−

ij

J_ijS_i·S_j (1.17)

The exchange constantsJ_ijare obtained from Eq. 1.8 by substituting the integral Eq. 1.7, which needs to be computed only once for each type of pair. The proba- bility amplitudes in Eq. 1.7 can be computed by density functional theory or an approximation can be obtained from linear combinations of the atomic orbitals responsible for the bond. Often it is sufficient to take only the nearest neigh- bor interactions into account. This reduces considerably the number of terms in Eq. 1.17. Still, the model requiresO(n²)operations to compute< H >. Theˆ ground state is typically found by a meanfield simulation, i.e. the behaviour of each single spin is calculated individually in thefield created by all other spins, or by a Monte-Carlo simulations, often with simulated annealing. A complete simulation thus requires many steps and is computationally expensive.

The Heisenberg model allows for unity spins on a three dimensional Bloch sphere. A more restricted model is the Ising model, which limits the spins to one degree of freedom. The Ising model represents a system with a strong uniaxial anisotropy, where all spins are either parallel or antiparallel.

Heisenberg and Ising models can describe single spins and do not need an artificial discretization of the problem, in contrast to the continuum theory presented below. The whole reciprocal space is included in the simulation and the lower cut-off fork-vectors has a physical significance and is the same as for the real crystal. Therefore both models are able to accurately describe all lattice excitations, i.e. the entire spin-wave spectrum. In contrast, the continuum theory uses an artificial discretization, which unphysically restricts the lattice excitations of the system. The Heisenberg and Ising models are particularly suitable for localized spins, as in the case of 4f magnetism, but are less suitable for delo- calized magnetism. Itinerant systems can be simulated with these models, if the spins are localized and the interaction with the electron gas obeys an exchange like interaction, like in the case of the RKKY interaction.

The disadvantage of all atomistic lattice models is the computational complexity, which precludes the calculation of real world phenomena. For this reason, the continuum theory explained in the next section is more popular.

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M. For a constant temperature the sum of the different energy terms is called the Landau free energy (Eq. 1.27). The Landau free energy, like the Gibbs free energy, describes an isothermal isobaric thermodynamic system with the magnetization as an additional order parameter [Klä03]. The Landau free energy has minima at the thermodynamically stable magnetization configurations. Solving the variational problem offinding a minimal free energy for a magnetization configuration we obtain three coupled PDEs. These are called Brown’s equations [WFB78] and can be solved for the stable magnetization configuration.

In Brown’s approach, also called micromagnetism, the discretization can be chosen independently of the crystal lattice. This is the main advantage over Heisenberg or Ising-like spin models, which require simulation on the atomic scale, even though the characteristic length scales, domain width (Eq. 1.42) and exchange length (Eq. 1.41), are much larger: both are on the order of 10⁻⁹−10⁻⁷ m. The micromagnetic grid therefore can be chosen at least ten times coarser than for an atomistic Heisenberg or Ising model, thus allowing for a simulation of physically and technically relevant systems.

In the following sections the different energy contributions that lead to the formulation of the Landau free energy are formulated. Even though the energies are introduced phenomenologically, they are strongly linked with the material properties of the system and can be obtained independently. For lower temperatures the magnetizationM has a constant magnitude, therefore it is sometimes normalized with the saturation magnetization, m = M /M _s. A good introduc- tory overview about magnetostatics is given in [Dur68].

Exchange Energy In a ferromagnet the exchange constant is positive, therefore a parallel alignment of neighboring spins is favored. Magnetization gradients always lead to an increased energy, which is modeled by the following term:

E_exchange =A

dV(∇m) ² (1.18)

The parameterAis a stiffness which tries to align neighboring spins. Ais derived from a mechanical model; In [Ber98, WFB78] it is shown thatAis proportional to the exchange integralJ in the Hamiltonian Eq. 1.9. It is important to note that Eq. 1.18 is only thefirst non-vanishing term of an expansion in terms of

∇m. Correct results can only be obtained if ∇m is small, i.e. neighboring spins are nearly parallel.

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1.3. Micromagnetic Description 11 Zeeman Energy A single magnetic moment in an external magnetic field has the Zeeman energy E = −m ·H. Accordingly, the whole magnetization fieldm( r)leads to an energy contribution, which is called Zeeman energy:

E_zeeman =−µ₀M_s

dV H(r)·m( r) (1.19) Stray Field Energy The magnetization M(r) is the source of a magnetic field, called the demagnetizingfieldH_d:

∇·H_d=−∇·M (1.20)

The demagnetizingfield is responsible for an additional energy contribution, the dipolar energy. This is a Zeeman-like energy contribution which stems not from the externalfield, but from thefield created by the sample magnetization itself:

Edipole-dipole =−1 2µ₀M_s

dV H_d·m (1.21)

For an arbitrary given magnetizationM and sample shape the demagnetiz- ingfieldH_dcan be obtained by integrating over the volume and surface of the sample:

H_d(r) =

SampledV (r−r)∇·M( r)

|r−r |³ −

Surfacedn·M( r) (r−r)

|r−r |³ (1.22) A closed form solution exists only for simple geometrical shapes, for an ellipse it takes the form of a diagonal tensorN, from which thefield is easily calculated asH_d = −N M. In this exceptional case thefield inside the sample is uniform and always aligned with the magnetization. In a prolate ellipse the component of the demagnetizing tensor along the long axis has the smallest value. Eq. 1.21 has therefore a minimum if the magnetization is aligned parallel to the long axis.

The existence of preferred directions for the magnetization is called anisotropy.

The anisotropy induced by the dipolar coupling is called shape anisotropy.

If no closed form solution exists, Eq. 1.22 has to be integrated numerically on every lattice point. The computation of the dipolar coupling has thereforeO(n²) complexity and is the most computationally intensive step.

Eq. 1.22 is normally solved using Fourier techniques, reducing the complexity down toO(nlog(n)), but limiting solutions to equispaced discretizations.

Equispaced discretization are far from optimal, as the domain wall width

KA ≈ 100 nmenforces a discretization of the whole structure well below this scale.

Anisotropy Energies In anisotropic materials the energy depends on the local direction of the magnetization, i.e. there are preferred directions for the magnetization.

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For one preferred direction, i.e. a uniaxial anisotropy,takes the form:

_anisotropy(m( r)) =K₀+K₁cos(2α) +... (1.24) For a biaxial anisotropy, which favors alignment of the magnetization along the 0^◦and 90^◦directions the anisotropy has the form:

_anisotropy(m( r)) =K₀+K₁cos²(2α) +... (1.25) In both cases α is the angle between the magnetization and the magnetically easy axis

More complicated expressions are obtained for three-dimensional systems:

For a cubic system withα_i being the direction cosines between the magnetization and the crystal axes [Blü99] one obtains

_cubic(m( r) =K₀+K₁(α²₁α²₂+α²₂α²₃+α²₁α²₃)+K₂α²₁α²₂α²₃+K₃(α²₁α²₂+α²₂α²₃+α²₁α²₃)²+. . . . (1.26) Even though these energies look complicated, they are very easy and fast to compute, as they depend only on the local symmetry.

The Landau Free Energy The sum of the individual energy contributions introduced above is the Landau free energy:

G(M) =E_exchange+E_zeeman+Edipole-dipole+E_anisotropy. (1.27) This thermodynamic potential has minima at stable magnetization configurations. To obtain the stable configurations we have to solve the variational prob-

lem δG

δM = 0 (1.28)

for a minimum of the free energyG.

Differentiating Eq. 1.27 with respect to the magnetization leads to the definition of an effectivefield, which contains contributions from all energy terms.

Together with the minimal condition we obtain three coupled PDEs, which are known as Brown’s equations [WFB78]:

H_eff = 2

µ₀M_s²∇(A∇M)− 1

µ₀δ_anisotropy(m( r))δ M +H_d+H (1.29)

m×H_eff = 0 (1.30)

m×∂ m

∂n = 0 (1.31)

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1.4. Magnetization Dynamics 13

nis the surface normal. Unfortunately, Brown’s equations are not directly solv- able for any realistic problem, but the formulation ofH_effleads to an efficient iterative solution of the equation, with a convergence criterion derived from the second of Brown’s equations:

dV |m ×H_eff|≈0 (1.32)

Due to approximations of the exchange energy, the model is only valid if neighboring spins are nearly parallel, a condition normally fulfilled in ferromagnets due to the strong exchange coupling.

The model does not describe magnetization dynamics, but the definition of the effectivefield lends itself to a description of magnetization dynamics as the precession of the magnetization in the effectivefield, which is presented in the next section.

1.4 Magnetization Dynamics

Brown’s equations, presented in the previous section, only provide a static solution for the magnetization distribution. All interactions are included via the effectivefield, which is a variational derivative of the Landau free energy Eq. 1.27.

It is difficult to add further interactions, such as electric currents, since they would have to be included via the Landau free energy. In addition, the variational solution does not describe dynamic systems, and it is impossible to in- clude the spin-torque, since it is not a conservative force for which a potential could be formulated.

In the following sections we present the Landau-Lifshitz-Gilbert equation, an equation which describes magnetization dynamics. Furthermore, several extensions are proposed to describe interactions with spin-polarized electric currents.

1.4.1 The Landau-Lifshitz-Gilbert Equation

The Landau-Lifshitz-Gilbert equation uses the micromagnetic approximation of Brown’s equations, i.e. the magnetization is represented by a continuousfieldm rather than by an array of spins, but is valid for an atomic lattice, too. Instead of solving Eq. 1.27 for a minimum of the free energy, the Landau-Lifshitz-Gilbert equation models magnetization dynamics as a precession of the magnetization

min the effective magneticfieldH.

A single magnetic momentµ will experience a torque in an external mag- neticfieldH, since it has the properties of angular momentum. As the torque is always perpendicular to thefield and the magnetic moment, the moment will precess on a circular orbit with the Larmor frequencyω_L=eµ₀H/2m_e.

Accordingly, thefieldH will exert a similar torque∂ m/∂ton the magnetization. This torque is perpendicular to the magnetization and will cause a gyration

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Eq. 1.33 would predict an unphysical stationary gyration without any relaxation. Brown [WFB78] phenomenologically included a friction-like damping proportional to the velocity ^∂_∂t^m into Eq. 1.33:

∂ m

∂t =−γ_Gm ×H +α_Gm ×∂ m

∂t (1.34)

This equation is known as the Landau-Lifshitz-Gilbert equation [Gil55], α_G is called the damping coefficient. As α_G has been introduced by analogy, it has no direct physical relevance and includes different interactions from spin-wave excitations to losses due to spin-orbit coupling and the induction of eddy currents.α_Gcan be measured accurately by FMR spectroscopy [MMS05] or pulsed magnetometry [KSK02, DSK⁺02, ARSK00, SLCR99].

Rearranging the terms of the LLG equation Eq. 1.34 one obtains the Landau- Lifshitz equation, with modified coefficients:

∂ m

∂t =−γ_LLm ×H −α_LLm ×(m ×H) (1.35) The Landau-Lifshitz-Gilbert equation, which can be easily obtained from Eq. 1.35, is assumed to be physically more adequate, as the Landau-Lifshitz equation Eq. 1.35 predicts an infinitely fast variation of the magnetization in the limit of infinite damping [Kik56, Mal87].

The Landau-Lifshitz-Gilbert equation can be solved numerically without problems [SMB01,BRH93,GCJGE,GS05], but the calculation of the effectivefield H at each time step is computationally very expensive; about 90% of the simulation are spent calculating the effectivefield due to the long range interaction.

The discretization of the problem sets the lower cut-off frequency for magnonic excitations to an unphysical value. Temperature effects cannot be modeled directly, but corrections can be applied, as e.g. in the Landau-Lifshitz- Bloch equation [Gar97, CFNCG06].

1.4.2 Extensions to the Landau-Lifshitz-Gilbert Equation to Describe Interaction with Currents

The Landau-Lifshitz-Gilbert equation presented in Eq. 1.34 describes magnetization dynamics, but no interactions with spin-polarized currents or spin currents. Several approaches have been presented to add current interaction to the Landau-Lifshitz-Gilbert equation:

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1.4. Magnetization Dynamics 15 Adiabatic Interaction Li and Zhang [LZ04a] as well as Thiaville and Mil- tat [TNMV04, TNMS05] proposed similar extensions to the Landau–Lifshitz–

Gilbert equation to describe the interaction with an electrical current, the ex- tension only differ in their coefficients.

Both groups introduced a spin–transfer torque term τ as a function of the generalized velocity

u=jgP µ_B

2eM_s, (1.36)

which is proportional to the current densityj, and the spin polarizationP.

g= 2is the Landé g-factor of the spin angular momentum.

For the Li Zhang model [LZ04a], the spin-transfer torque is represented in the form

τ =− 1 M_s²

M ×(M ×(u·∇)) M), (1.37) which is similar to the spin-torque term proposed by Bazaliy [BJZ98] for trans- port across a metal–ferromagnet interface in a multilayer structure.

τ =− a

M_s²M ×(M ×M_pinned). (1.38)

M_pinnedis the magnetization direction of the pinned layer.

Thiaville and Miltat [TNMV04, TNMS05] proposed the following formulation of the spin-transfer torque:

τ =−(u·∇)M . (1.39)

For low temperatures the magnetization vectorM has a constant length, in this case the equation Eq. 1.37 proposed by Li and Zhang is equivalent to the formulation of Thiaville and Miltat in Eq. 1.39.

Li and Zhang included their spin torque term to the Landau-Lifshitz-Gilbert equation Eq. 1.34 to study current-driven domain wall dynamics [LZ04a,LZ04b].

It was shown that the term given in Eq. 1.37 does not permit a current-induced continuous domain wall movement below the walker breakdown limit [SW74].

During the current injection the domain walls are shifted only a few nm and then stop. If the current is switched off the domain walls return to their original position.

The model of Thiaville et al. predicts a critical current density for a stationary domain wall propagation, but this current density is much too high [TNMV04], and again lies above the Walker breakdown.

Both spin-torque terms cannot predict the experimentally observed continuous domain wall propagation for relatively low currents densities.

Nonadiabatic Interaction As the models presented above only predict continuous domain wall propagation [LZ04a, TNMV04] above the Walker breakdown, further additions to the Landau-Lifshitz-Gilbert equation were proposed.

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Zhang are given in Eq. 1.40, Zhang’s version uses a slightly reducedu, but oth- erwise the proposed extensions are identical:

˙

m=γ₀H ×m +α m×m˙ −(u·∇) m +β m×((u·∇) m) (1.40) Zhang et al. [ZLF02] identify the nonadiabaticity parameterβwith the squared ratio of exchange length and spin-flip length, β = (λ_exchange/λ_spin-flip)², but this cannot explain the observed temperature dependence of the threshold current [LBB⁺06b]. The spin-transfer torque due to this model increases with the gradient of the magnetization, a current will therefore have a stronger influence on thinner walls and on vortex cores with their large gradients around the vortex core.

The equation 1.40 has zero critical current for ideal wires, much lower than the current densities of2×10¹²A/m² observed. A higher critical current density has been explained by pinning at edge defects. Experimentally, domain wall velocities on the order of1 m/s were measured, while Eq. 1.40 predicts several 100 m/s. Again, this behaviour might explained by pinning of the domain walls, which is only overcome by thermal excitations to allow a stepwise movement.

Unfortunately, the Landau-Lifshitz-Gilbert equation Eq. 1.40 does not allow in- cluding thermal excitations if it is solved on a grid larger than the crystal lattice, since the coarse grid results in an unphysical low cut-off for spin wave excitations.

1.4.3 Predictions from the Landau-Lifshitz-Gilbert Equation

For applications such as the racetrack memory [Par97] we are interested in reasonably low switching currents. Typical measured current densities are 2×10¹²A/m² for Permalloy, much too high for any practical application. A memory device like the racetrack will only be feasible, if the domain walls can be shifted fast enough, but so far measured domain wall velocities of continuous propagations are on the order of1 m/s, which would be too low for a racetrack memory. Important predictions expected from micromagnetic simulations are therefore the critical current densities and domain wall velocities.

He and coworkers simulated current-induced depinning and found a de- crease of the current density for higher nonadiabaticity [HLZ05]. In the same article they identified the nonadiabatic interaction as responsible for the depinning, but showed that the adiabatic interaction initially shifts the domain wall

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1.4. Magnetization Dynamics 17

Figure 1.1: Average velocity computed for a transverse domain wall by micromagnetics in a 120×5 nm² cross-section wire as a function of the velocity u representing the spin-polarized current density, with the relative weightβ as a parameter. Open symbols denote vortex nucleation. The shaded area indicates the available experimental range foru. (a) Perfect wire and (b) wire with rough edges (mean grain sizeD= 10 nm). The dashed lines display afitted linear relation with a25 m/soffset. From Ref. [TNMS05].

far enough from the centre of the pinning potential, so that the nonadiabatic interaction only has to overcome a small part of the entire potential barrier. Fig. 1.1 shows the domain wall velocity as calculated by Thiaville for different values of the nonadiabaticity parameterβ. For perfect wires (a), even for very smallβthe threshold current is reduced to zero. For smaller values ofuthe domain wall ve- locityvincreases linearly withu,v= ^βu_α (filled dots). Above a certain threshold current vortices and antivortices are nucleated at the edges of the wire and the propagation velocity drops (open dots).

For wires with rough edges (b), the critical current density (proportional to the velocityu) is larger than zero. Above the threshold the domain wall prop- agates (filled dots), but for smaller values ofβ no propagation without vortex nucleation is possible (e.g.β = 0.01).

Tatara investigated domain walls excited by a high-frequency current [Tat05]

and found a reduction of the threshold current. Saitoh [SMYT04] used a resonant excitation to determine the domain wall mass, he reported a critical current density much lower than for the static case.

Thomas and coworkers [THJ⁺06] measured the depinning as a function of the width of the pulses and showed a reduction of the critical current density for a train of current pulses. He demonstrated a very sensitive dependence on the pulse length, which he linked to the precession of the system.

So far it is not evident, whether the lowering of the depinningfield for AC currents is because the total energy stored in the system is higher than the power available under stationary conditions, or by some other interaction: If the initial threshold current was on the same order of magnitude as the value for large- scale movements, we could not expect a reduction under resonant excitation, as the wall would never start to move. This argument favors a threshold current, which is very small or even zero and a domain wall, which is pinned solely by the lateral confinement. Saitoh measured a large-scale domain wall oscilla-

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wall transformations predicted by Thiaville [TNMS05] and observed by [Lau06]:

During a current injection the domain wall initially moves along the wire, but is transformed under the influence of the electric current. Typical transformations are the nucleation of vortices and the formation of multi-vortices, which have a much higher depinning current than single vortices. Vortex cores are driven out of the wire and annihilated at the edge, forming a transverse wall.

The transverse wall is pinned stronger than the vortex [HLZ06], due to its stray field, which is concentrated at the edge of the wire. In addition, the magnetization gradients in a transverse wall are smaller and therefore the spin-torque is weaker. If a vortex wall has been transformed to a transverse wall it will normally stay pinned. The transformations are current-induced and not due to heating effects [KLH⁺06].

In conclusion, we can state that, even though the edge roughness can be engineered to explain a measured current threshold, the models presented so far do not yield a reliable prediction of the critical current density. The velocities predicted are much higher than experimentally observed and, even though pinning and thermally assisted depinning sound like a reasonable explanation, thermal effects have not been included succesfully into micromagnetic simulations. On the other hand, domain wall movement, the motion of vortex cores, and domain wall transformations are described qualitatively by micromagnetic simulations [Lau06].

1.4.4 Micromagnetic Simulations

In this work the LLG micromagnetic simulator [Sch] and the Object Oriented Micromagnetic Framework (OOMMF) [DP02] were used. Both solve afinite- difference version of the LLG equation using an Adams-Bashford/Adams- Moulton implicit/explicit time integrator [DP02]. The stopping criterion is the same as for the iterative solution of Bloch’s equations given in Eq. 1.32.

The LLG micromagnetic simulator includes the complete Landau-Lifshitz- Gilbert equation with the two spin-torque terms (from Eq. 1.40) and was used to study current-induced domain wall motion. The parameters used in this work for Permalloy are given in the following table.

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1.5. Magnetoresistance Effects 19 Saturation magnetization M_s 800×10³ A/m

Anisotropy energy K₁ 0 —0.5×10² J/m³ Exchange stiffness A 13×10⁻¹²J/m Damping coefficient α 0.01

Nonadiabatic contribution β 0 — 1

Table 1.1: Material parameters used for micromagnetic simulations of Permalloy.

The Landau-Lifshitz-Gilbert equation is solved on a cubic grid by both solvers. The discretization has to befiner than the exchange length:

l=

2A

µ₀M_s² ≈5 nm, (1.41)

and the domain wall width:

l= A

K ≈100 nm (1.42)

Both length scales are much smaller than the typical sample size. As the magnetization is normally homogeneous away from the domain wall, FEM solvers like [HK98] use adaptive grids. The advantage of an adaptive grid incurs a large computational penalty because on a non-rectangular grid the demagnetizingfield cannot be calculated inO(nlog(n))time by a Fourier method. Finite element micromagnetic codes use an hierarchical multipole algorithm instead [LLOL06]. A freefinite element micromagnetic solver is available at [FFF⁺].

1.5 Magnetoresistance Effects

A Permalloy sample will show an electric resistance dependent on the angle between magnetization and current, this effect is called anisotropic magnetoresistance (AMR). The AMR is strong enough to be used in magnetoresis- tive read heads for harddisks [TKBK97] ; it wasfirst observed by Thomson in 1857 [Tho57]. For a typical sample the resistance will change up to about 1% for a total resistance of100 Ω.

Responsible for this resistance change is the spin orbit coupling, which causes different scattering amplitudes depending on the angle between the electronflow and the magnetization. The anisotropic electrical resistance can be represented by the diagonal resistance tensor (ρ)_ij, with the components tr(ρ) = (ρ_⊥, ρ_⊥, ρ).

The magnitude of ρ_⊥/ρ depends on the material, it is smaller than 1 for Permalloy and larger than 1 for iridium-doped Permalloy [MAK85]. The Hall resistance will give non-diagonal components of(ρ)_ij , effectively coupling the electricfields and currents along different axes by the Lorentz force, this contribution is proportional to the externalfieldB.

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If the electron spin is not aligned with the magnetization, nonadiabatic contribution from the last term of Eq. 1.40 have to be taken into account. This is only the case for very sharp gradients, e.g. around a vortex core, the small volume of the vortex core compared to the rest of the domain wall makes this effect negligible.

For arbitrary angles between current and magnetization the voltage for a given current path is given by:

dV j(r)M(r) cos²(θ), (1.44) As the current is itself dependent on the resistance, the equations have to be solved in a self-consistent manner, which is done e.g. in the micromagnetic simulator LLG [Sch].

Due to the high anisotropic magnetoresistance of Permalloy, magnetoresistance measurements are a very sensitive probe for the spin configurations in the samples, as described in section 4.1 on page 79.

1.6 X-ray Circular Magnetic Dichroism

In some materials left and right circularly polarized x-rays have different absorption coefficients. This effect is called magnetic circular dichroism and is used in the photoemission electron microscope described in section 2.9 to image spin structures.

Fig. 1.2 shows the photoemission spectrum of cobalt for circularly polarized x-rays of different helicity. The two absorption peaks, corresponding to the L₂ and L₃edges strongly depend on the helicity of the x-rays.

Fig. 1.3 shows a simplified model of the electronic structure of the 3d metals.

The degeneracy of the 2p level is lifted due to spin-orbit coupling, splitting the level into 2p_3/2 and 2p_1/2.

From the 2p levels electrons are excited into the partiallyfilled 3d levels by the x-rays, the L₂peak corresponds to the transition 2p_1/2→3d and the L₃peak corresponds to the transition 2p_3/2→3d. The background seen in Fig. 1.2 stems from excitations into s and p-like states. The photoelectrons measured in Fig. 1.2 are not the primary photoelectrons, but secondary Auger electrons. Another possibility is to measure not the photoelectrons but the intensity of the transmit- ted x-rays in an x-ray microscope [EFS⁺99]

Domain Wall Dynamics in Magnetic Nanostructures