Manipulation of Magnetic Domain Walls and Vortices by Current Injection

(1)

Universität Konstanz

Fachbereich Physik

Manipulation of Magnetic Domain Walls and Vortices by Current Injection

Dissertation

ZUR ERLANGUNG DES AKADEMISCHEN GRADES:

DOKTOR DER NATURWISSENSCHAFTEN

vorgelegt von LUTZ HEYNE

2010

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-112081

URL: http://kops.ub.uni-konstanz.de/volltexte/2010/11208/

(2)

Dissertation der Universität Konstanz Tag der mündlichen Prüfung: 29.03.2010 Referent/in: Professor Dr. Mathias Kläui Referent/in: Professor Dr. Ulrich Nowak

(3)

Zusammenfassung

Die vorherrschenden Magnetisierungskonﬁgurationen in Mikrometer kleinen weich- magnetischen Drähten und Scheiben sind Domänenwände und Vortices. Diese Ar- beit beschäftigt sich mit der Manipulation solcher Konﬁgurationen mithilfe spin- polarisierter Ströme.

Für die Untersuchung der lithograﬁsch hergestellten Strukturen werden Rönt- gen-Photoemissions-Elektronen-Mikroskope verwendet. Unter Ausnutzung des zirkularen Dichroismus kann die Magnetisierung in der Probe hoch aufgelöst abge- bildet werden.

Die Experimente teilen sich in zwei Gruppen: Untersuchungen der strominduzierten Domänenwandbewegung in magnetischen Drähten und Untersuchungen zu strominduzierten Vortexkernverschiebungen in magnetischen Scheiben. Erstere beschäftigen sich unter anderen mit systematischen Studien zu den kritischen Stromdichten, die erforderlich sind, um Domänenwände zu verschieben. Des Weit- eren werden die strominduzierten Umwandlungen des Domänenwandtypes sowie die Abhängigkeit der Domänenwandgeschwindigkeit von der Stromdichte unter- sucht. Diese Studien beziehen sich auf das Material Permalloy (Ni80Fe20).

Die Ergebnisse erlauben Rückschlüsse bezüglisch des nicht-adiabatischen Spin- transfer Terms. Dieser Beitrag und der mit ihm assoziierte nicht-adiabatische Parameter β wird gegenwärtig kontrovers diskutiert. Die Untersuchungen zeigen unter anderem, dass der nicht-adiabatische Parameterβgrößer als die Dämpfungs- konstante α ist.

Die Studien zur strominduzierten Domänenwandverschiebung werden vervoll- ständigt durch Untersuchungen an anderen Materialien, wie zum Beispiel bei der Multilagenschicht Pt/CoFeB/Pt, in dem die Magnetisierung aus der Probenebene hinaus zeigt. In diesem System wird der dominante Einfluss des Oersted Feldes auf die magnetischen Domänenkonfiguration nachgewiesen. Dieser ermöglicht alter- nativ zu dem Spintransfer eine gezielte Manipulation der Spinkonfiguation über das Oersted Feld.

i

(4)

ii

Die Untersuchungen der strominduzierten Vortexkernverschiebung in magnetischen Scheiben basieren auf einer neuen Messmethode, die im Rahmen der Doktor- arbeit entwickelt wurde. Diese erlaubt es, während der Strominjektion Bilder der magnetischen Konﬁguration aufzunehmen. Unter Ausnutzung der speziellen Vortextopologie gelingt so erstmals eine Trennung der verschiedenen Eﬀekte, über die der Strom auf die Spinstruktur wirkt. Dadurch lassen sich die relativen Stärken des adiabatischen und nicht-adiabatischen Spin-Transfer Terms sowie des Oersted Feldes bestimmen. Insbesondere gelingt auf diese Weise eine verlässliche Bestim- mung des nicht-adiabatischen Parameters zu β = 0.15±0.08.

Die experimentellen Ergebnisse werden ergänzt durch analytische Rechnungen sowie mikromagnetische Simulationen, die einen besseren Vergleich von Theorie und experimentellen Ergebnissen gestatten und somit zu einem besseren Verständ- nis beitragen.

Die im Verlauf dieser Arbeit erhaltenen Ergebnisse zeigen deutlich die wichtige Rolle des nicht-adiabatischen Term für den Spintransfer zwischen injizierten Strom und lokaler Magnetisierung.

(5)

List of Figures

1.1 Band model of ferromagnetism . . . 7

1.2 Iso-energy surfaced for uniaxial and cubic anisotropy . . . 11

1.3 Magnetization inside a VC . . . 12

1.4 One- and two-dimensional domain walls . . . 13

1.5 Adiabatic spin-torque . . . 16

1.6 Velocity of a one dimensional DW . . . 24

2.1 Sample fabrication process . . . 28

2.2 Overlay process . . . 29

2.3 Sample resistance during sputtering . . . 30

2.4 AFM scan of permalloy wire . . . 31

3.1 Schematic setup of an undulator . . . 34

3.2 Schematic setup of the PEEM . . . 36

3.3 Photon absorption and electron emission process . . . 37

3.4 XAS spectra of permalloy . . . 39

3.5 UHV system at the SIM beamline (SLS) . . . 41

3.6 PEEM at the SIM beamline (SLS) . . . 42

3.7 Photograph of the modiﬁed ELMITEC sample holder . . . 43

3.8 Photograph and drawing of the self-designed sample holder . . . . 43

3.9 Schematic of the pulse injection setup . . . 44

3.10 Photograph of the microcontroller unit . . . 45

3.11 Transfer characteristics of PEEM HV cables . . . 46

3.12 Current pulse shape and Fourier frequency spectrum . . . 47

3.13 Schematic of the timing for the gating experiment . . . 48

3.14 Schematic of the MCP gating . . . 49

3.15 Pump-probe setup . . . 51

3.16 Hall sensor . . . 52 vi

(9)

LIST OF FIGURES vii

3.17 Magnetic ﬁeld of the PEEM objective lens . . . 53

3.18 PEEM compensation coil calibration . . . 53

4.1 SEM and XMCD image of Py wires . . . 57

4.2 Current-induced DW motion . . . 58

4.3 Distribution of the current-induced DW velocities in Py . . . 60

4.4 Critical current density as a function of 1/width . . . 61

4.5 Simulation of the current-induced DW transformation . . . 63

4.6 Simulation of current-induced DW transformations . . . 66

4.7 Current-induced DW transformations . . . 68

4.8 SEM and XMCD image of Py wire . . . 69

4.9 Current-induced DW velocity in a Py wire . . . 70

4.10 Schematic of the pump-probe setup . . . 72

4.11 Image sequence of subsequent current injections . . . 74

4.12 Current-induced DW velocity in Ho doped Py . . . 77

4.13 SEM and XMCD image of the CoFeB wire . . . 79

4.14 Current-induced DW motion in CoFeB . . . 80

4.15 Current-induced degrading of the CoFeB wires . . . 81

4.16 SEM image and MOKE signal of CoFeB/Pt multilayer . . . 84

4.17 Current injection into a CoFeB/Pt wire . . . 85

4.18 Calculated Oersted ﬁeld inside a wire . . . 86

5.1 Schematic spin-torque induced VC displacement . . . 91

5.2 VC displacement vs. magnetic ﬁeld . . . 94

5.3 VC displacement vs. β . . . 95

5.4 Simulation of D˜ as a function of the disk diameter . . . 96

5.5 VC radius for diﬀerent disk thicknesses . . . 97

5.6 VC trajectory for slowly increasing current density . . . 99

5.7 Current-induced VC displacement for diﬀerent cell sizes . . . 100

5.8 Schematic of a disk potential with additional pinning site . . . 101

5.9 Permanent VC displacement in permalloy disks . . . 103

5.10 Permanent VC displacement at large current densities . . . 104

5.11 Drift correction of the XMCD images . . . 105

5.12 Determination of the VC position . . . 105

5.13 Determination of the disk position . . . 106

5.14 Images of VC displacements in permalloy disk . . . 107

5.15 Simulation of the current ﬂow inside a disk . . . 108

(10)

viii LIST OF FIGURES

5.16 Current-induced image distortions . . . 109

5.17 VC displacements in permalloy disk . . . 110

6.1 Time line of the PhD thesis . . . 113

A.1 Schematic of the gating unit . . . 118

A.2 Schematic of the microcontroller control unit . . . 119

A.3 Schematic of the resistance measurement unit . . . 120

A.4 Schematic of the relay main unit . . . 121

A.5 Schematic of the relay hubs . . . 122

A.6 Schematic of the power supply . . . 123

A.7 Schematic of the sub unit used for magnetizing the sample . . . 124

B.1 Drawing of the sample holder ground plate . . . 125

B.2 Drawing of the sample holder ground plate . . . 126

B.3 Drawing of the sample holder housing . . . 127

B.4 Drawing of the ceramics . . . 128

B.5 Drawing of the sample holder cap . . . 129

B.6 Drawing of the washers for the sample holder . . . 130

B.7 Drawing of part 1 of the device to restrain the sample holder . . . 131

B.8 Drawing of part 2 of the device to restrain the sample holder . . . 132

(11)

List of Acronyms

AC Alternating current AFM Atomic force microscopy AMR Anisotropic magneto resistance at% atomic percent

DC Direct current DOS Density of states

DW Domain wall

FMR Ferromagnetic Resonance GMR Giant magneto resistance HF High frequency

HV High voltage

LLG Landau-Lifshitz-Gilbert MBE Molecular beam epitaxy MFM Magnetic force microscopy MOKE Magneto-optic Kerr eﬀect

PEEM Photo electron emission microscopy / microscope PMMA Polymethyl methacrylate

Py Permalloy (NiFe-alloy)

SEM Scanning electron microscopy / microscope SLS Swiss Light Source

TW Transverse wall UHV Ultra high vacuum

VC Vortex core

VW Vortex wall

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

X-PEEM X-ray photoemission electron microscopy / microscope

ix

(12)

(13)

Introduction

The last decades have taken us from computers that carried out calculations hardly any faster than humans and were as tall as whole buildings to miniature smart phones, that are many orders of magnitude smaller, faster and more versatile than their predecessors. This trend of miniaturization and faster devices is described by the famous Moore’s Law, which states that the density of integrated circuits roughly doubles every two years [Moo75, Lan06].

The progress that leads to smaller devices is facilitated by the growing ﬁeld of nanotechnology and the improvements made in the ﬁeld of nano-fabrication.

Nanomagnetism became an important branch in this area and is widely used for storage and sensor devices. For the future validity of Moore’s law, novel technolo- gies are required that allow for higher storage densities.

Controlled magnetization switching is a prerequisite to realizing ultra-high density magnetic memory devices. It was ﬁrst predicted by Berger that injected currents can be used to change the magnetization, for example to move magnetic domain walls in a wire [Ber84]. Therefore, some types of high density memories and logic devices are realized by using a magnetization switching mechanism based on domain wall motion [PHT08, CPRP07, IKH⁺08].

Experiments carried out on various systems such as ferromagnetic metallic and semiconducting wires show that domain walls are driven by a spin-polarized current above a certain threshold value [GBC⁺03, VAA⁺04, YON⁺04, KVB⁺05].

In metals, the motion appears to be induced by the combination of the spin-torque and an eﬀective force depending on the material and magnetization conﬁguration.

The switching speed for a memory device based on spin-torque and domain walls directly depends on the domain wall velocity, which therefore plays an important role. In addition to the domain wall speed, the threshold current density required to drive the motion is a crucial parameter for any application. Indeed current densities that are too high, result in excessive heating and may cause structural damage.

1

(14)

2 Introduction

A better theoretical understanding of the current driven domain wall motion is the key to designing structures for applications, where low current densities are needed. Most experimentally observed threshold current densities in metals are believed to arise from extrinsic pinning potentials due to defects [TKS08], but so far no quantitative argument about the origin of the threshold has been brought forward in detail. Besides inducing domain wall motion, the injected current can lead to domain wall transformation, which currently is only poorly understood and further complicates the comparison between experiment and theory.

Study of the interaction of spin-polarized currents with magnetic disks contain- ing a magnetic vortex core has only recently started [SOH⁺00, SNT⁺06, KNK⁺06, IKO06, CPS⁺07, YKN⁺07, GLK08]. The special topology of the vortex structure makes this structure a very promising candidate for a detailed study of the spin torque, since it provides the unique possibility to separate the diﬀerent spin torque terms. So far the resonant excitation of the vortex core using AC-currents was studied via transport measurements [Bed08, MTH⁺06, KNK⁺06] or by direct imaging [BMK⁺08, KFI⁺08] but an experimental study of the VC response to in-plane DC-currents is still missing.

In this work, results are presented on current-induced domain wall motion as well as on current-induced vortex core displacements in magnetic materials, that reveal novel insight into the physics involved. The thesis is organized as follows:

Chapter 1 gives a general introduction to the theory. Special emphasis is put on the micromagnetic description and the physical origin of the spin-torque terms.

Chapter 2 discusses the experimental techniques used during this thesis. It con- tains an explanation of the XMCD-PEEM experiments used for magnetic imaging and the relevant background such as the properties of synchrotron radiation and the circular dichroism eﬀect. The chapter is concluded by a description of the diﬀerent setups used for the current pulse injection and the related devices that were developed.

Chapter 3 details the sample properties and the special requirements for the samples studied with the techniques introduced in the previous chapter. The basics of the sample fabrication processes are explained.

Chapter 4 presents the results on current-induced domain wall motion in magnetic wires. The critical current densities, domain wall velocities and domain wall transformations are studied by direct high resolution imaging in pure and Holmium

(15)

Introduction 3

doped permalloy as well as in other materials. The experiments are complemented by corresponding micromagnetic simulations.

Chapter 5 deals with current-induced vortex core displacements in permalloy disks. A new measurement technique is introduced, that yields a direct and robust measure to determine the size of the non-adiabaticity constant β. The current- induced vortex core motion is also studied in detail by simulations and analytical calculations.

Chapter 6 summarizes the main results of this thesis. The chapter also includes a short overview of the conducted beamtimes and a recapitulation of the requirements for an eﬀective way of working at synchrotron sources. The chapter is rounded oﬀ by an outlook of promising further experiments.

Parts of the results presented in this thesis have been published in diﬀerent journals and a list of these publications is given on page 154. In the text, references are included wherever appropriate.

(16)

Chapter 1

Theory

1.1 Introduction

In the early 20th century Bohr and van Leeuwen proved independently that in a classical system the total magnetization must vanish in the absence of external fields [NR09]. Thus, ferromagnetism is a purely quantum mechanical effect and can only be understood in the framework of quantum theory, which also originates in the beginning of the 20th century. Up to the present day, the field of magnetism is as exciting as in the early days and still not all observed phenomena are understood thoroughly. In particular this applies to the interaction of the local magnetization with spin-polarized currents.

This chapter will provide the basic theoretical background relevant for this PhD thesis. The first section gives a short review of the microscopic origin of magnetism. The different contributions to the Landau free energy are also discussed. The Landau free energy defines the stable equilibrium configurations of the magnetic system via its local minima. Two static spin configurations are discussed that are for this work most important: vortex cores and domain walls. The Landau-Lifshitz-Gilbert equation of motion is introduced which describes how the system dynamically behaves. The sections deal with the interaction of the local magnetization with spin-polarized currents and the effect of current-induced domain wall (DW) motion.

More information on the theory of magnetism can be found for instance in [Aha00, NR09, Yos96].

4

(17)

1.2 Quantum Mechanical Origin of Magnetism 5

1.2 Quantum Mechanical Origin of Magnetism

For an alignment of the magnetic moments at zero field, an interaction between the magnetic moments is required. The first candidate for this interaction could be the magnetic dipolar interaction. However, a simple analysis reveals that the strength of this effect is orders of magnitudes too weak to explain the observed ferromagnetism with Curie temperatures more than 1000 K (1043 K for Fe and 1388 K for Co [SB00]). The exchange interaction introduced in the next section is suited to explain this long-range order.

1.2.1 Exchange Interaction

Two overlapping single electron wave functions ψ₁(r₁) and ψ₂(r₂) form a new state ψ(r1,r₂). The energyE of this coupled system depends on the relative spin orientations of the two electron spins (S₁ and S₂), although the Hamiltonian Hb of the system is not explicitly spin dependent.

E = Z Z

dr₁dr₂

ψ^∗(r₁,r₂)Hψ(rb ₁,r₂)

. (1.1)

The reason for this eﬀect is the Pauli exclusion principle. Electrons being fermions require a total antisymmetric wave function in respect to particle inversion. Gen- erally, wave functions can be split into spacial and spin parts. For fermions, a symmetric spin part therefore requires an antisymmetric spacial part and vice versa. In general, the two spacial wave functions have diﬀerent energies. Thus, the energy of the two-particle state can be written as:

E(S1,S₂) =E0−2J S₁·S₂, (1.2) withE₀being a constant term andJ the so-called exchange integral. The exchange integralJ can be evaluated by decomposing the two-particle stateψ(r₁,r₂)into a product of the single electron wave functions ψA and ψB and is deﬁned as:

J = Z Z

dr₁dr₂

ψ_A^∗(r₁)ψ^∗_B(r₂)Hψb _A(r₂)ψ_B(r₁)

. (1.3)

This integral is a manifestation of quantum eﬀects, since it arises from an inter- mixing of states and does not occur in a classical treatment of the problem. The

(18)

6 Theory

spin dependent exchange HamiltonianH_ex can now be written as:

H_ex=−2J S₁·S₂. (1.4)

Depending on the sign ofJ, either parallel alignment of the spins is favored (J >0) or an anti-parallel orientation is energetically favorable (J <0), which corresponds to ferromagnetic or anti-ferromagnetic coupling.

This result from a simple two-particle model motivates the deﬁnition of the Heisenberg Hamiltonian for a many particle system:

H_H =−X

ij

J_ijS_i·S_j, (1.5)

where J_ij is the exchange constant between the i^th and j^th electron with spin S_i and S_j. In general, J_ij depends on the distance and the arrangement of the two spins. However due to simplicity, often only next neighbors are considered.

The Heisenberg model is well suited to explain ferromagnetic and anti-ferromagnetic ordering in systems with localized magnetic moments, such as rare earth metals with their localized 4f-electrons like Gadolinium (ferromagnetic) or ionic solids like MnO (anti-ferromagnetic) or Fe₃O₄ (ferromagnetic).

1.2.2 Band model of itinerant ferromagnetism

In iron the magnetic moment per atom is 2.22µ_B[SB00]. This odd non-integral value cannot be explained on the basis of localized moments. Thus, to explain ferromagnetism in metals a diﬀerent approach must be taken.

In metals thes- orp-conduction electrons are delocalized and can move freely inside the metal. They can thereby mediate an exchange interaction between the more localized d-electrons, which can induce a magnetic ground state.

In a non-magnetic metal spin-up and spin-down electrons occur in equal numbers and no net magnetization remains as shown in Fig. 1.1(a). Shifting electrons from one spin band to the other, creates a non-zero magnetization M, that is directly proportional to the number of shifted electrons: M =µ_Bg(E_F)δE, with the density of states at the Fermi level g(EF) and δE the energy difference of the electrons that are shifted [See Fig. 1.1(b)]. The magnetization Minduces via the exchange interaction a mean field λMthat supports parallel alignment of the magnetic moments along this field with λ being a material dependent constant.

The magnetic moments aligned to the mean ﬁeld lower the potential energy E_pot

(19)

1.3 Quantum Mechanical Origin of Magnetism 7

Figure 1.1: (a) Band model sketch of a normal metal. (b) Electrons are shifted from one band to the other leading to a non-zero magnetization. (c) The two spin bands are shifted with respect to each other, resulting in equal Fermi levels as required at equilibrium.

by an amount of:

∆E_pot=−1

2µ₀MλM. (1.6)

The reduction of potential energy is accomplished by an increase of the kinetic en- ergyE_kin, since the energy of the shifted electrons is increased byδE [Fig. 1.1(b)].

Hence:

∆E_kin= 1

2g(E_F)δE², (1.7)

since 1/2g(E_F)δE electrons below the Fermi energy of one spin band are moved by δE above the Fermi energy to the other spin band.

If the total energy gain∆E = ∆Epot+ ∆E_kinis negative, the system can lower its energy by spontaneous spin ﬂip, the systems enters a ferromagnetic ground state.

∆E = ∆Epot+ ∆E_kin = 1

2δE²g(EF) 1−λµ0µ²_Bg(EF)

(1.8) With the so-called Stoner parameterU =λµ0µ²_B, the Stoner criterion for itinerant ferromagnetism reads:

U ·g(E_F)>1. (1.9)

The product of the Stoner parameter U and density of states at the Fermi level g(E_F) needs to be larger than unity to favor a ferromagnetic ground state. This for instance is the case for Co, Ni and Fe.

(20)

8 Theory

1.3 Micromagnetic Description and Landau Free En- ergy

The atomistic model discussed above is not applicable to larger systems, due to the large number of degrees of freedom that are of the same magnitude as the number of atoms in the system. To solve this problem, Brown derived a micromagnetic model, where the localized magnetic moments S_i are replaced by a continuous magnetization M(x)[Bro78].

The Landau free energy is a functional of the magnetization and deﬁnes the energy of the system and hence the stable magnetic conﬁgurations. Therefore it is important to know the various energy terms contributing to the this energy.

These are in detail:

1.3.1 Exchange Energy

As outlined above the exchange energy between two adjacent spins is given by E =−2JS₁S₂ (Eq. 1.4). In the continuum limit the sum over all spins is replaced by an integral over the sample volume. The scalar product can be rewritten using the angle θbetween two spins S_i,S_j:

SiSj =S²cos(θ)≈S²− 1

2S²θ² =S²−1

2(∇Sdx)². (1.10) The approximation holds for small angles θ. Replacing the spins by the local magnetization M and considering that constant terms are of no importance for the energy, the exchange energy can be written as:

E_ex= A M_s²

Z

dV(∇M(r))². (1.11)

The parameter A is the exchange stiﬀness and is directly proportional to the atomistic exchange integralJ. M_sis the saturation magnetization of the material.

1.3.2 Zeeman Energy

In an external magnetic ﬁelds H the magnetic moments M try to align to the ﬁeld to minimize the Zeeman energy. In the continuum limit the Zeeman energy is written as [Aha00]:

E_Z =−µ₀ Z

dVM(r)·H(r). (1.12)

(21)

1.3 Micromagnetic Description and Landau Free Energy 9

1.3.3 Stray Field Energy

The Zeeman energy arises from external fields, but the magnetic moments also interact with the dipolar field created by the neighboring moments. In the atomistic approach this stray field may be written as the sum over all dipolar fields of all other moments. In the continuum theory, the calculation of the stray field starts from the Maxwell equation:

∇B= 0. (1.13)

B can be decomposed using the relation: B=µo(Hs+M) (Hs is the stray ﬁeld without external ﬁelds). So equation 1.13 can be rewritten to:

∇H_s=−∇M. (1.14)

Considering that ∇ × H_s = 0 holds (H_s derives from dipolar ﬁelds) the last equation is transformed to a Poisson equation with the scalar potentialU and the magnetic charge density ρ:

∆U(r) =−ρ(r), (1.15)

with: H_s =−∇U, ρ(r) =−∇M. (1.16)

By solving this Poisson equation, the potential U and the stray field H_s are determined. The stray field acts as an external field on the magnetization and thus the stray field energy of the system is calculated by:

ES =−µ₀ 2

Z

V

dVH_s(r)M(r). (1.17)

To avoid counting each volume element twice, the factor 1/2 is included. The stray ﬁeld energy can be calculated alternatively by [Aha00]:

ES = µo

2 Z

dVHs(r)². (1.18)

It is important to note that the integral now extends over the total space. Equa- tion 1.18 implies that the stray field energy is always positive and the system therefore tries to minimize the stray field by confining it to the sample. The stray field energy depends strongly on the shape of the sample. For instance, in thin films the out-of-plane magnetization direction is energetically very unfavorable due to the higher stray field compared to the magnetization lying in-plane.

(22)

10 Theory

1.3.4 Magnetic Anisotropy Energy

In the simplest Heisenberg model the energy only depends on the relative orientation of the spins so it is isotropic. In real systems the spins are, for instance, linked to the crystallographic structure due to spin orbit coupling. Thus the energy might depend on the direction cosines α_i between the magnetization m = (α₁, α₂, α₃) and the crystallographic axes. By approximation this energy densityǫin a power series expansion of α_i, the anisotropy energy can be written as [Aha00]:

Eani = Z

V

dV ǫani(m(x)) (1.19)

ǫ_ani = ǫ₀+X

ij

b_ijα_iα_j+X

ijkl

b_ijklα_iα_jα_kα_l+. . . (1.20) Here it is already considered that under inversion of the magnetization the energy does not change. Therefore, E(M) = E(−M) and only even terms of α_i occur.

In crystallographic systems the symmetries impose some further constrains on the parameters b_ij.... For instance if only one preferred direction exists the so-called uniaxial anisotropy energy density ǫ^uniaxial_ani up to the 4th order reads:

ǫ^uniaxial_ani =K0+K1sin(θ)²+K2sin(θ)⁴+. . . (1.21) with θ being the angle between the magnetization and the z-axis. If K1 is negative (higher orders neglected) the energy is minimized for magnetic moments aligned along the z-axis which becomes the so-called easy axis. This is visualized in Fig. 1.2(a) where the anisotropy energy as a function of the magnetization direction is shown. IfK₁ is positive the z-axis is called the hard axis [see Fig. 1.2(b)].

In this case the magnetization direction favors the easy-plane, similar to the eﬀect of the shape anisotropy of thin magnetic ﬁlms.

For cubic symmetry the energy up to 6th order can be written as:

ǫ^cubic_ani =K₀+K₁(α²₁α²₂+α²₂α²₃+α²₃α²₁) +K₂α²₁α²₂α²₃+. . . (1.22) IfK₁is positive (again higher orders neglected) the magnetization favors alignment along one of the three main axis [see Fig. 1.2(d)]. If K₁ is negative, there are not 6 stable positions, but 12 [see Fig. 1.2(c)] and the stable positions are positioned between the x-y-z-axis.

(23)

1.4 Brown’s Equations of Static Equilibrium 11

Figure 1.2: Iso-energy surfaces of lowest order anisotropy. (a) and (b) are for the case of an uniaxial anisotropy. (c) and (d) show the case of an cubic anisotropy. The sign of the anisotropy constant K1is opposite for (a) and (b), and (c) and (d) respectively.

1.4 Brown’s Equations of Static Equilibrium

With the results of the last section, the Landau free energy reads:

E_tot=E_ex+E_Z+E_S+E_ani. (1.23) This energy will be minimized by the equilibrium magnetization M_eq. Thus the variation of Etot with respect to the direction cosines αi has to vanish:

δ_αiE_tot(M_eq) = 0, ∀α_i. (1.24) By solving these variational equations, Brown [Bro78] derived a set of equations that are used to determine the equilibrium magnetization:

M×H_{ef f} = 0, (1.25)

M×∂_nM = 0. (1.26)

The first equation has to be fulfilled for every point in the sample. The second equation is only valid at the boundary surface with ∂_n denoting the derivative in the direction of the surface. The effective field H_{ef f} originates from the Landau

(24)

12 Theory

free energy and is deﬁned by:

H_{ef f} =−∂H

∂M = 2A

M_s²∆M+H+H_s− 1 µ₀

∂ǫ_ani

∂M . (1.27)

In principle, by solving the above set of equations, the equilibrium magnetization M_eq can be calculated for every given system.

1.5 Static Magnetization Configurations

Two magnetic structures are important for this work, that are discussed below:

magnetic vortices and domain walls.

1.5.1 Magnetic Vortices

As previously mentioned, magnetic systems always try to minimize their stray ﬁeld.

This is achieved best by a flux closed magnetization, which effectively eliminates all stray fields [See Fig. 1.3(a)] . Nevertheless, in the center (vortex core) the fast changing magnetization direction would cause an explosion of the exchange energy.

This singularity is avoided by the tilting of the magnetization out-of-plane. The radius of the resulting VC δ_{V C} is of the order of the exchange lengthlex:

l_ex=

s2µ₀A

M_s² . (1.28)

For permalloy this exchange length is l_ex ≈5.7nm [HHK03]. The magnetization proﬁles for Bloch lines and VCs were calculated by Feldtkeller and Thomas [FT65]

as well as by Usov and Peschany [UP93]. Due to their small size it is diﬃ- cult to image VCs directly, and was achieved only recently by MFM imaging

Figure 1.3: (a) Simulation of the magnetization of a disk structure. (b) Enlarged disk center showing the VC. The arrows and the color code indicate the in-plane magnetization direction. (c) Out-of-plane magnetization component vs. distance from the VC center.

The blue line corresponds to the simulation and the red one to the analytical model.

(25)

1.5 Static Magnetization Configurations 13

in 2000 [SOH⁺00, RPS⁺00]. Using high-resolution spin-polarized scanning tun- neling microscopy it became also possible to resolve the internal structure of the VC [WWB⁺02]. Fig. 1.3(a) shows a micromagnetic simulation of a VC in a permalloy disk. The arrows and the color code indicate the in-plane magnetization direction. Fig. 1.3(b) displays the magniﬁed VC. The out-of-plane component of the simulated VC is shown as a function of the distance from the center in Fig. 1.3(c) by the blue line. The red curve corresponds to the analytical model by Usov and Peschany. Close to the VC center the agreement is good, but in the simulation the out-of-plane component oscillates slightly, which is not considered in the analytical model. This magnetization opposite to the VC magnetization is caused by the demagnetization ﬁeld created by the VC and is not considered in the analytical model.

The topology of a VC can be described by two numbers: its polarity p=±1, that is the direction of the out-of-plane component and its chirality c = ±1, that is the in-plane magnetization direction being clockwise or counter clockwise (counter clockwise in Fig. 1.3). Although small in size, the VC topology is of major importance for the dynamic behavior of the magnetic system (see also chapter 5).

1.5.2 Magnetic Domain Walls

Domain walls (DWs) are formed at the interface between two diﬀerent domains. In a simple one-dimensional model the moments are aligned along the x-direction with the moments in the domains being perpendicular to this axis (see Fig. 1.4). For such a spin chain two kinds of180^◦DWs exist: Néel walls, where the magnetization is rotating in the x-y plane [Née55] and Bloch walls with the magnetization rotating

Figure 1.4: Sketch of the magnetization direction inside a one-dimensional (a) Néel wall and a (b) Bloch wall. Simulations of a two-dimensional (c) transverse wall and a (d) vortex wall. The VC is indicated by the dot.

(26)

14 Theory

in the y-z plane [Blo32]. Fig. 1.4 (a) and (b) sketch the magnetization distribution inside a Néel and Bloch wall, respectively. The width of the DW is determined by the ratio of the exchange constant A and the eﬀective anisotropyK[HS98]:

δ_DW = rA

K. (1.29)

For Néel walls, the additional non-vanishing demagnetizing ﬁeld has to be considered, which increases the eﬀective anisotropy K and thus reduces the DW width.

In two-dimensional wires the situation is more complex and two different kinds of DWs are found: transverse walls (TW) and vortex walls (VW). Transverse walls [Fig. 1.4(c)] are similar to Néel walls; the magnetization is rotating only in one plane by180^◦. The triangular shape to the transverse wall is indicated by the two black lines. The internal structure of transverse walls also depends on the wire dimensions and not only symmetric ones as shown in Fig. 1.4(c) but also asym- metric ones exist [BSK⁺07].The topology of vortex walls is completely different [Fig. 1.4(d)]. Here the magnetization is curling around the center of the wall, the VC, where the magnetization points out-of-plane (see section 1.5.1) as indicated by the dot in the center of the DW in Fig. 1.4(d). The energy of a TW is domi- nated by its stray field, whereas for VWs the energy is governed by the exchange energy of the VC. In a wire with thickness t and width w, the stray field energy of a TW is proportional towt². The exchange energy of the VC in a VW in good approximation only depends on the wire thickness: t. Thus, if the product of t and w is smaller than a certain value, TWs are energetically favorable while for larger values of wt VWs prevail [MD97]. This theoretical dependence on the wire dimensions is in line with experimental results [KVB⁺04].

1.6 Magnetization Dynamics

A magnetic ﬁeld H exerts a torque on a magnetic moment M, that results in a gyroscopic motion of the moment around the ﬁeld:

dM

dt =−γ(M×H), (1.30)

with γ being the gyromagnetic ratio. This equation can easily be derived from the fundamental spin commutator relations and the Ehrenfest theorem for the expectation value of the spin direction [NR09].

(27)

1.7 Influence of spin-polarized Electrons on the Magnetization 15

This equation describes the undamped precession of the magnetic momentM around the field H. To consider energy dissipation, that will align the magnetization Malong the fieldHin finite time, Landau and Lifshitz [LL35] included an additional damping term, resulting in the equation:

dM

dt =−γ(M×H)−λM×(M×H), (1.31) with the damping constant λ. Gilbert used a slightly diﬀerent type of dissipation term [Gil55, Gil04], which is argued to be more correct in the case of large damping [Kik56, Mal87]. The Gilbert equation reads:

dM

dt =−γ(M×H) + α

M_sM×dM

dt . (1.32)

The Gilbert equation can be transformed to be similar to the Landau-Lifshitz (LL) equation (no time derivative on the right hand side). In this explicit form the equation is usually referred to as Landau-Lifshitz-Gilbert (LLG) equation. But also the actual Gilbert equation 1.32 is often referred to as LLG-equation.

It is worth noting that the LLG-equation 1.32 as well as the LL-equation 1.31 conserve the norm of the magnetization, since both damping terms are perpendicular to M. It can further be shown, that the two equations are identical when a renormalization of the time scale is done.

Replacing the fieldHby Brown’s effective fieldH_{ef f} =−_∂M^∂H yields the equation of motion of the magnetic system.

1.7 Influence of spin-polarized Electrons on the Mag- netization

It is well known that the sample resistance and thus the response to an injected current depends on the magnetization conﬁguration. This is caused by various mag- netoresistance eﬀects such as the anisotropic magneto resistance (AMR) [Tho57]

or the giant magneto resistance eﬀect (GMR) [BBF⁺88, BGSZ89], which will not be discussed here in detail.

But, we consider the reciprocal eﬀect: the inﬂuence of an injected current on the local magnetization also occurs. In 1984, Berger suggested that the spin- polarized conduction s-electrons can interact with the magnetic moments of the

(28)

16 Theory

more localized d-electrons and transport spin and angular momentum to the magnetic system [Ber84].

This so-called spin-torque interaction was integrated into the LLG-equation for a macro spin model [Ber96, Slo96, BJZ98]. In these early works, two coupled ferromagnetic layers with current ﬂowing from one layer to the other were studied.

The first layer polarizes the conduction electrons and if the second layer is not oriented parallel to the first one, the electron spin exerts a torque on the second layer magnetization trying to align it to the first layer.

Since the study of the spin-torques is a important issue of this work the following subsections will deal with this interaction in more detail.

1.7.1 Adiabatic Spin-Torque

If current is injected into a magnetic structure, the spins of the conduction elec- tronss(x)align to the local magnetizationM(x). It is assumed that this happens adiabatically, so that the electron spins adapt without delay to the varying local magnetization as indicated in Fig. 1.5. When the electrons pass a region where the magnetization direction changes, their spins follow this direction change and there will be a change of their spin direction (ds in Fig. 1.5) and hence of their angular momentum. Due to the conservation of angular momentum, this involves a transfer of angular momentum to the local magnetization that hence has to change its orientation in return (−dsin Fig. 1.5).

A currentIinjected into a wire (along the x-direction) with cross sectionAcar- ries during the timedta spin fraction ofds= ^~^IP_e dt. Here we also considered that the conduction electrons might not be fully spin polarized and the eﬀective spin po- larizationP ≤1is taken into account. When the local magnetizationMchanges, the electron spin will also change by the fractiondM/M_s=∇_xM(x)dx/M_s. Hence

Figure 1.5: The spin of the injected electrons (green arrows) adiabatically aligns to the local magnetization (blue arrows). When the electron spin schanges by dsthis angular momentum has to be transferred to the local magnetization.

(29)

the total amount of angular momentum transferred to the local magnetization is:

dS=−~∇_xMdx M_s

P Idt

e . (1.33)

Note the minus sign. It should also noted that only the perpendicular spin component can be transfered, since the saturation magnetization is constant. Hence, the above calculation is only valid in the limit of slow varying magnetizations. The angular momentum per volume (V =Adx) is directly proportional to the change of the magnetization:

dM = 1

2gµ_B· dS

~Adx. (1.34)

Considering that I/A equals the current density j and combining equations 1.33 and 1.34 ﬁnally yields:

dM =−gP µ_B 2eMs

j∇_xMdt.

This motivates the deﬁnition of

u= gP µ_B 2eMs

j, (1.35)

that has units of a velocity. When we allow arbitrary current directions the derivative of M in the direction of the current has to be used instead of ∇_xM. This

yields:

∂M

∂t

ast

=−(u· ∇)M. (1.36) This equation equals the result for the spin-torque obtained by Thiaville and coworkers [TNMV04, TNMS05]. If one assumes that the magnetization has a constant value (low temperature regime) this description is also consistent with results obtained by Li and Zhang [LZ04a]:

∂M

∂t

ast

=− 1

M_s²M×[M×(u· ∇)M]. (1.37) From these results one can expect that the spin-torque is largest in materials with low saturation magnetization M_s and high spin polarization P, since u will be maximized for a given current density.

(30)

18 Theory

1.7.2 Non-adiabatic Spin-Torque

The discussion above assumed that the injected electrons are always perfectly aligned with the local magnetization. However, for large magnetization gradients the injected conduction electrons might not be able to follow the local magnetization completely and will lag behind by a certain amount. This creates an additional torque on the magnetization, since the electron spin s in this case has a small component perpendicular toM.

The precession of this component induces similar to an external magnetic field a torque on the magnetization that is proportional to M×dM. For this reason the non-adiabatic term is also called field-like term. The torque strength is again proportional to the current density summarized by the parameter u. For an arbitrary current direction the derivative along the current direction has to be taken and one finally obtains:

∂M

∂t

nast

=βM×(u· ∇)M. (1.38) β is a measure for the mistracking of the electron spin, i. e for the degree of non-adiabaticity. For β = 0 the electron spin again perfectly follows the local magnetization and pure adiabatic spin-torque is present.

This derivation of the non-adiabatic spin-torque is quite simpliﬁed and the actual origin of the non-adiabaticity spin-torque term is still controversial and being discussed. Thiaville introduced this term phenomenologically to overcome discrepancies between experimental results and theory [TNMS05] (see also section 1.10).

Zhang and Li [ZL04] derived this term rigorously from theory by considering the exchange interaction between the local moments and the spin accumulation of the conduction electrons as spin-ﬂip scattering. They coupled the electrons of transport via an s-d Hamiltonian to the magnetic system. Considering also spin relaxation by spin-ﬂip scattering, they showed that the non-equilibrium spin density induced by the current and the variation of the local magnetization in turn creates an torque on the local magnetization. In this formulation, the non- adiabaticity is directly related to the spin-relaxation time τ_sf and the exchange energyJ_ex of the s-d Hamiltonian:

β_sf =τ_ex/τ_sf ≈10⁻², τ_ex=~/J_ex. (1.39)

(31)

In addition to the adiabatic and non-adiabatic spin-torque terms, they found two extra terms that depend on the time derivative of the magnetization. But these terms can be included as small corrections into the precession and the damping term. Later it was shown that not only spin relaxation due to spin-ﬂip scattering, but also due to spin-orbit interaction induces a β term [TE08].

Tatara and Kohno found a similar non-adiabatic term based on linear momentum transfer [TK04]. In this context the non-adiabaticity is closely related to the DW resistance ρ_DW[LZ97, TF97]:

β = γe∆²_DW

gµ_BP R0ρDW, (1.40)

with the DW width ∆_DW and the Hall resistance R₀.

In systems with fast varying magnetization, conduction electrons might be re- ﬂected at these high magnetization gradients [XZS06, WV04]. This corresponds to the true meaning of non-adiabatic transport and gives also rise to a non- adiabaticity constant β_na.

The eﬀective β therefore is a sum of the contribution due to spin-relaxation β_sf and non-adiabatic transport: β_na: β =β_sf +β_na. A more detailed discussion of the origin of β can be found in [TKS08, BTE08].

1.7.3 Discussion of the Spin-Torques

Integrating the two spin-torque terms into the LLG-equation 1.32 yields:

dM

dt =−γ(M×H_{ef f}) + α

M_sM×∂M

∂t −(u· ∇)M+βM×(u· ∇)M. (1.41) The two spin-torque contributions are always perpendicular to each other and thus deﬁne a basis in the plane orthogonal to the local magnetizationM. Therefore any torque acting on the magnetizationMcan be decomposed into these two terms (Since Ms is assumed to be constant only torques can act on the magnetization).

The remaining challenge is to determine the size of the two contributions.

The most hotly debated issue is the relation betweenβ and the damping constant α, since it has been predicted that α and β depend similarly on the band structure [GGSM09] or are even equal [BM05]. Xiao et al. [XZS06] cast doubt on the existence of the non-adiabaticity parameter caused by spin-ﬂip scattering, but ﬁnd an oscillating non-adiabatic torque for very high magnetization gradi-

(32)

20 Theory

ents. Kohno et al. [KTSS06] calculated the non-adiabaticity parameter microscop- ically and found that in general β 6= α. Garate et al. relate β as well as α to the band structure and intrinsic spin-orbit interactions [GGSM09]. From their work it follows that α and β depend similarly on temperature and disorder, but the value of the ratio β/α depends on the band structure and is in general different from unity. In contrast, Tserkovnyak et al. [TBBH05] as well as Barnes and Maekawa [BM05] claim that β should equal α. Controversy also exists concerning the type of damping, either Landau-Lifshitz [SSDZ07, SSDZ08, Sas09] or Gilbert [Smi08] damping, that has to be used. Whereas both formulations are identical in the absence of current, this is no longer the case when the spin transfer terms are included [SSDZ08, SSDZ07, SHK⁺09]. If a pure adiabatic spin-torque term is added to the LL-equation 1.31 with Landau-Lifshitz damping an additional termαM×(u·∇)Moccurs when transformed into the equivalent LLG-form (equation 1.32). Thus the adiabatic spin-torque includes a ﬁeld like term with β = α for Landau-Lifshitz type of damping. More general speaking: β_G =β_LL+α with β_Gbeing the non-adiabatic contribution in the LLG equation with Gilbert damping and β_LL for the Landau-Lifshitz damping. It was pointed out that the two equations are equivalent, if the adiabatic as well as the non-adiabatic spin-torque is considered and the choice of damping in this case is simply based on the ease of interpretation or convenience [SSDZ08].

However, to solve the discrepancies concerning the origin and the size of β, experiments addressing this topics are required. The measurement of β in real materials as well as its relation to α will give valuable information required to reﬁne the theory.

1.8 Thiele Formalism

Since an analytical treatment for realistic problems is not possible due to its com- plexity, either numerical solutions are required or appropriate approximations have to be done to simplify the problem. Under the assumption of steady-state motion Thiele [Thi72] derived an integrated equation for ﬁeld-induced DW motion. Keep- ing the internal magnetic structure constant, the systemM(x, t)at the timetcan

(33)

1.9 Thiele Formalism 21

be described by the displacementX(t) and the initial conﬁgurationM(x,0):

M(x, t) = M(x−X(t),0), (1.42) dX

dt = v, (1.43)

dM(x, t)

dt = −(v∇)M(x, t). (1.44) Considering only materials with spacially constant saturation magnetization and inserting 1.42 into the LLG equation 1.32, Thiele obtained a simpler equation of motion that is commonly referred to as the Thiele equation:

Fêx+G×v+αD¯ ·v= 0. (1.45) Fêx is the force acting on the system for instance due to stray fields and external fields. The vectorGis the so called gyrocoupling vector and theD¯ the dissipation tensor, that are defined by the following integrals over the sample volume:

G = −M_sµ₀ γ

Z

V

dV sin(θ_m) (∇θ_m× ∇φ_m), (1.46) D¯ = M_sµ₀

γ Z

V

dV ∇θ_m∇θ_m+ sin²(θ_m)∇φ_m∇φ_m

. (1.47)

The magnetization is written in spherical coordinates with the out-of-plane angle θ_mand the in-plane angleφ_m. The advantage of this formalism is that the internal structure of the detailed problem is condensed into the two integralsD¯ andG. So once these integrals are known the dynamic behavior of the system can easily be calculated.

Thiaville expanded the Thiele equation to include the inﬂuence of a spin- polarized current based on the expanded LLG-Equation 1.41 and derived the following equation [TNMS05]:

Fêx+G×(v−u) + ¯D·(αv−βu) = 0. (1.48) Equation 1.48 is well suited to study the combined influence of magnetic fields and spin-polarized currents on a magnetization system, for example for a DW in a wire or for a VC in a magnetic disk.

(34)

22 Theory

1.9 Micromagnetic Simulations

To allow for a comparison between experiment and theory, simulations are carried out using the LLG micromagnetic simulator [LLG]. The program is based on a finite difference version of the LLG-equation on a cubic grid. Starting from an initial configuration the Landau free energy and the effective field are calculated.

Here especially the calculation of the non-local demagnetizing field is time con- suming. This leads to a small change of the magnetization after a certain time dt and for this new configuration the new Landau free energy is calculated. This iteration will finally converge to the equilibrium position. Brown’s equation 1.25 provides an effective criterion, that is used to detect convergence [OOM]:

max(|M×H_{ef f}|)< ǫ, (1.49) withǫbeing the convergence threshold close to 0.

All simulations are done in two dimensions assuming a homogeneous magnetization in the z-direction - a valid assumption for thin structures. The discretization size in the x-y-plane is5nm×5nmunless noted otherwise. The LLG micromagnetic simulator also includes the two spin-torque terms and thus can be used to study the effect of spin-polarized currents on the magnetization. The material parameters used for permalloy are presented in Table 1.1. The damping constant is set to the above value when the dynamic behavior is studied. Otherwise, if only the final state is of interest, the damping is increased to speed up the simulation time. The non-adiabaticity parameter β is varied between 0 and 1 to study its influence on the magnetization dynamics.

Saturation magnetization Ms= 800·10³A/m Anisotropy energy K₁= 50J/m³ Exchange stiﬀness A= 13·10⁻¹²J/m damping constant α= 0.01

Table 1.1: Material parameters of permalloy used for the micromagnetic simulations

(35)

1.10 Domain Wall Dynamics 23

1.10 Domain Wall Dynamics

This sections deals with theoretical predictions and calculations concerning the interaction of spin-polarized currents with domain walls. This is done either by applying the Thiele formalism or by micromagnetic simulations. Employing the Thiele formalism has the advantage that analytical results can be obtained and the basic principles can be revealed. Since in this regime, one is restricted to more or less ﬁxed internal spin conﬁgurations and for instance DW transformations are neglected. Realistic problems therefore require full micromagnetic simulations.

The simplest approach is to consider a one-dimensional DW (see section 1.5.2), whose interaction with spin-polarized current can be studied analytically [LZ04b, TK04, TNMS05].

In the one-dimensional collective coordinate model the DW with its ﬁxed internal structure is described by its position X and its tilting angle φ. Employ- ing the Thiele formalism (section 1.8), the LLG equation (Eq. 1.32) is reduces to [JKL⁺08, BHBK09]:

−αX/∆ + ˙˙ φ = βu/∆, (1.50)

−X/∆˙ −αφ˙ = u/∆ +γ₀H_ksin(2φ)/2, (1.51) with the anisotropy field H_K = 2K/µ₀M_s. The DW width ∆ depends on the anisotropy K and varies with changing tilting angleφ. For simplicity it is considered to be constant in the following part. From the above set of equations it follow directly, that the DW motion is always in the direction of the electron flow (β >0, see Eq. 1.39) and does not depend on the DW type (head-to-head or tail-to-tail), in contrast to field-induced DW motion.

In the case of pure adiabatic transport (β = 0) the DW initially starts moving with velocity _(1+α¹ 2)u but stops just after a few nanosecond and when the current is switched oﬀ the DW even moves back to its original position [LZ04b]. So no continuous DW motion is observed and the adiabatic spin-torque only results in reversible DW displacement and in a ﬁnite rotation of the DW tilting angle φ.

However, this is only the case below the Walker breakdown [SW74]. For current densities above the threshold density u_W = γ₀∆H_K/2 the tilted angle φ cannot compensate the spin-torque: φstarts to oscillate and the DW moves continuously in the direction of the electron ﬂow [TK04]. For current densities much larger than this threshold, the DW velocityvagain approaches _(1+α¹ 2)u. This behavior cannot

(36)

24 Theory

explain the experimental observations. The critical current density (Walker breakdown current density) is about one order of magnitude larger than the densities required in the experiments [TNMS05].

The situation is diﬀerent, when the non-adiabatic term is included. In this case, no threshold current density exists and already small currents will displace the DW. The average DW velocityv for small currents scales asv=β/αu[ZL04].

Above the walker breakdown in the limit of large current densities, the averaged velocity v approaches ^1+βα_1+α2u. This behavior is summarized in Fig. 1.6 where the average DW velocity is shown as a function of the current density for various β values (Figure taken from Ref. [TNMS05]).

Tatara and Kohno studied the eﬀect of pinning on current-induced DW motion and found that it creates a critical current density also for a non-zero non- adiabaticity [TK04]. They ﬁnd an averaged DW velocity above the threshold that is proportional to q

j²−j_th². In the case of strong pinning, the critical current density j_th depends directly on the pinning potential strength. For weak pinning j_th is governed by the transverse anisotropy.

More realistic two dimensional models are hard to treat analytically due to their complex nature. Here micromagnetic simulations are the method of choice.

For transverse walls in perfect wires, results similar to the one-dimensional model were found by Thiaville and coworkers [TNMS05]. Introducing edge roughness in

Figure 1.6: (From [TNMS05]) Velocity of a one dimensional DW as a function of the current density for variousβ values (α=0.02).

(37)

1.10 Domain Wall Dynamics 25

their simulations, they found that the DW does not start moving below a certain critical current density, hence they conclude that extrinsic pinning governs the critical current densities. This is in agreement with previous studies on ﬁeld- induced DW motion, where it was also found that edge roughness has an strong inﬂuence on the DW mobility [NTM03].

It is further reported by He et al., that the critical current density decreases for higher non-adiabaticity [HZL05]. The same authors ﬁnd that the critical current density also depends on the DW structure [HLZ06]. In particular lower critical current densities were found for VWs compared to TWs. VWs can easily be pinned at defects close to the core position but are possibly less aﬀected by edge roughness [KMS95].

Due to the gyroscopic nature of the VC, the adiabatic spin-torque induces a VC motion perpendicular to the direction of the electron ﬂow. The direction of this movement is determined by the VC polarity [SO04]. Thus the VC in a vortex wall does not only move in the direction of the electron ﬂow but also towards one of the wire edges and will eventually be expelled from the wire forming a transverse wall [HLZ06, SLKL07]. This DW transformation process further complicates the analytical treatment of the DW motion. However, this transformation only occurs for the case for β 6=α, otherwise damping and precession term cancel the perpendicular spin-torque and the VW will keep its internal structure [HLZ06].

(38)

Chapter 2

Sample Fabrication and Characterization

2.1 Sample Requirements

The sample fabrication is a crucial step towards a successful experiment¹. The requirements for the samples can be split in two groups: scientiﬁc and experimental ones:

Scientific requirements

For the interaction of spin-polarized currents and the magnetization a large contribution of the adiabatic and non-adiabatic spin torque to the magnetization dynamics is desired. From equation 1.35 in section 1.7.1:

u= gP µ_B 2eMs

j, (2.1)

if follows that the materials used should have a small saturation magnetizationMs

and a spin polarization P close to one. In this work, primarily samples made of permalloy, an iron-nickel alloy (Ni80Fe20), are used withu[m/s]≈30·j[10¹²A/m²].

Other candidates that should be more promising are half metallic materials such as Fe₃O₄ and CrO₂ which have a high spin polarization and lower saturation magnetization. (for instance Fe₃O₄: M_s = 480 ·10³A/m and 90 % spin polarization [Har06]). However, up to the present date neither in Fe3O4 nor in CrO2

current-induced DW motion without external magnetic ﬁelds has been observed

1I am very grateful to all the people involved in the fabrication process.

26

(39)

2.2 Sample Requirements 27

even though the velocity uis about 4 times larger than in permalloy for identical current densities. One of the main problems is the high resistivity of these materials [BKF⁺07]. Permalloy is a good test candidate, due to its low coercivity, high susceptibility and high anisotropic magneto resistance. Furthermore, amorphous permalloy has a very low magneto-crystalline anisotropy. This makes permalloy very soft and the shape anisotropy is the dominating anisotropy type.

Experimental needs

Since the samples are intended to be used in an XMCD-PEEM (see chapter 3) certain points have to be considered:

• The material needs to have appropriate absorption edges in the energy range accessible by the synchrotron. The L₂ and L₃ edges of the 3d-metals Fe, Co, Ni range from 700 eV to about 850 eV, which is well in the range of the synchrotron and the PEEM beamline.

• On one side, the substrate has to be slightly conductive to avoid charging eﬀects in the electron microscope. On the other side, the substrate resistance has to be high compared to the resistance of the structures for current injection, otherwise the current will ﬂow though the substrate, shunting the structure.

• The sample resistance needs to be small enough to guarantee a suﬃcient current density, since the current injection setup is limited to a certain voltage.

• The sample size has to be smaller than 10x10 mm to ﬁt on the sample holder.

• The bond pads must be far away from the structures to make sure that they are hidden under the sample holder cap, since they could induce discharges in the PEEM.

• The sample has to be UHV compatible.

• The geometry of the structures should be adapted to the abilities of the spe- cific synchrotron source and end station. Some have a micro-focus, offering large fluxes on a limited field of view (for instance the BESSY end station) or allow for a large field of view (for instance the SLS end station). Some suffer from drift problems, making them unsuitable for long time experiments.

Manipulation of Magnetic Domain Walls and Vortices by Current Injection

Universität Konstanz

Fachbereich Physik