Spin Transport in Ferromagnetic and

Antiferromagnetic Textures

Dissertation by

Collins Ashu Akosa

In Partial Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

King Abdullah University of Science and Technology, Thuwal,

Kingdom of Saudi Arabia

©November 2016

Collins Ashu Akosa

All Rights Reserved

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The dissertation of Collins Ashu Akosa is approved by the examination committee

Committee Chairperson: Prof. Aurelien Manchon

Committee Member: Prof. Husam Niman Alshareef

Committee Member: Prof. Tao Wu

Committee Member: Prof. Jurgen Kosel

Committee Member: Prof. Dieter Suess

3

ABSTRACT

Spin Transport in Ferromagnetic and Antiferromagnetic

Textures

Collins Ashu Akosa

In this dissertation, we provide an accurate description of spin transport in mag-

netic textures and in particular, we investigate in detail, the nature of spin torque

and magnetic damping in such systems. Indeed, as will be further discussed in this

thesis, the current-driven velocity of magnetic textures is related to the ratio be-

tween the so-called non-adiabatic torque and magnetic damping. Uncovering the

physics underlying these phenomena can lead to the optimal design of magnetic sys-

tems with improved efficiency. We identified three interesting classes of systems

which have attracted enormous research interest (i) Magnetic textures in systems

with broken inversion symmetry: We investigate the nature of magnetic damping

in non-centrosymmetric ferromagnets. Based on phenomenological and microscopic

derivations, we show that the magnetic damping becomes chiral, i.e. depends on

the chirality of the magnetic texture. (ii) Ferromagnetic domain walls, skyrmions

and vortices: We address the physics of spin transport in sharp disordered magnetic

domain walls and vortex cores. We demonstrate that upon spin-independent scatter-

ing, the non-adiabatic torque can be significantly enhanced. Such an enhancement

is large for vortex cores compared to transverse domain walls. We also show that

the topological spin currents flowing in these structures dramatically enhances the

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non-adiabaticity, an effect unique to non-trivial topological textures (iii) Antiferro-

magnetic skyrmions: We extend this study to antiferromagnetic skyrmions and show

that such an enhanced topological torque also exist in these systems. Even more

interestingly, while such a non-adiabatic torque influences the undesirable transverse

velocity of ferromagnetic skyrmions, in antiferromagnetic skyrmions, the topological

non-adiabatic torque directly determines the longitudinal velocity. As a consequence,

scaling down the antiferromagnetic skyrmion results in a much more efficient spin

torque.

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DEDICATION

To my parents Mr. & Mrs. Ashu Bernard,

who have taught me much more than

spintronics and given me the most

important gift of all, love.

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ACKNOWLEDGEMENTS

This dissertation is an accumulation of countless hours of hard work - not just by me,

but also others who over the last five years have helped me a great deal and whose

valuable guidance on both my career and life could never be overemphasized.

First, I would like to express my sincere gratitude to my dissertation advisor Prof.

Aurelien Manchon for giving me the opportunity to work in his group at KAUST,

and for introducing me to this fascinating field of Spintronics. I am thankful for

the amount of time and energy that he spent in helping me think through prob-

lems, improve my writing and presentations. Aurelien provided invaluable insight

and feedback on my work, and gave me countless opportunities to participate in

schools, workshops and conferences on spintronics. It was a great pleasure to conduct

research under his direction and I hope that this is only the starting point of future

collaborations in spintronics.

I would also like to thank Profs. Mathias Klaui, Kyung-Jin Lee, Ioan Mihai Miron,

Gilles Gaudin, Oleg A. Tretiakov, Dr. Andre Bisig, and Papa Ndiaye Birame, with

whom it has been a great pleasure to collaborate during the course of my thesis.

Special thanks also goes to my committee members Profs. Husam Niman Alsha-

reef, Tao Wu, Jurgen Kosel, and Dieter Suess, for their interest in my work, and the

time and effort they made to read and give feedback on my thesis.

Last but not the least, I would like to thank all those unforgettable faces engraved

in my mind, who never hesitated to stand beside me through all the progresses,

confusions, difficulties and happiness during the course of my thesis at KAUST.

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TABLE OF CONTENTS

Copyright 1

Examination Committee Approval 2

Abstract 3

Dedication 5

Acknowledgements 6

List of Figures 11

List of Tables 16

1 Introduction 17

1.1 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 Non-collinear magnetic textures . . . . . . . . . . . . . . . . . . . . . 21

1.3 Antiferromagnetic textures . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Objectives and contributions . . . . . . . . . . . . . . . . . . . . . . . 25

2 Background 27

2.1 Magnetic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Basics of micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Dynamical equation . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Micromagnetic energies . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Current-induced torques . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.1 Spin transfer torque . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.2 Spin-orbit torques . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Non-collinear magnetic textures . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 One-dimensional magnetic textures . . . . . . . . . . . . . . . 45

2.4.2 Two-dimensional magnetic textures . . . . . . . . . . . . . . . 56

8

2.4.3 Antiferromagnetic textures . . . . . . . . . . . . . . . . . . . . 61

2.5 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5.1 The s-d model . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5.2 Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . 68

2.5.3 Spin diffusion model . . . . . . . . . . . . . . . . . . . . . . . 75

3 Phenomenology of chiral damping 77

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3 Microscopic origin of the chiral damping . . . . . . . . . . . . . . . . 80

3.4 Domain wall motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Role of spin diffusion in current-induced domain wall motion 92

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.1 Tight-binding model for disordered ferromagnets . . . . . . . . 96

4.2.2 Drift-diffusion model . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Current-induced domain wall motion . . . . . . . . . . . . . . . . . . 106

4.3.1 Transverse wall . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3.2 Vortex structure . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Intrinsic non-adiabatic topological torque 115

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2 Emergent electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.1 Premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.2 Topological spin torque . . . . . . . . . . . . . . . . . . . . . . 121

5.2.3 Non-adiabaticity parameter . . . . . . . . . . . . . . . . . . . 123

5.2.4 Damping parameter . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3.1 Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . 125

5.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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6 Topological transport in antiferromagnetic skyrmions 133

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Emergent electrodynamics for antiferromagnetic textures . . . . . . . 135

6.2.1 Topological spin Hall effect . . . . . . . . . . . . . . . . . . . . 138

6.2.2 Topological torque . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.4 Current-induced dynamics of antiferromagnetic skyrmion . . . . . . . 144

6.5 Discussions and conclusions . . . . . . . . . . . . . . . . . . . . . . . 146

7 Conclusions and perspectives 147

References 150

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11

LIST OF FIGURES

1.1 (Color online). Schematic diagram of STT-MRAM architecture for in-

plane (a) and the more efficient out-of-plane (b) magnetization. Fig-

ures extracted from [1] . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 (Color online). Schematic diagram of the racetrack memory in ferro-

magnetic nanowires. Figure extracted from [2] . . . . . . . . . . . . . 21

1.3 (Color online). Illustration of the spin structure of a domain wall (a)

and vortex (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 (Color online). Illustration of the spin structure of a ferromagnetic (a)

and antiferromagnetic (b) skyrmion. Figures extracted from [3, 4] . . 24

2.1 (Color online). Semi-classical schematic description of an electron. . . 28

2.2 (Color online). A uniformly magnetized sample is characterized by the

three magnetic vectors B, H, and M, as shown. The magnetic field

inside the FM is called the demagnetizing field Hd while that outside

is called the stray field Hs. . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 (Color online). Shows the dynamics of the magnetisation vector about

an effective field governed by a) precession on a constant energy contour

and b) damping effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 (Color online). Shows a uniformly magnetized sample with magneti-

zation (red). Finite system sizes results to accumulation of magnetic

charges at the surfaces producing stray fields (gray) outside the sam-

ple. Occurrence of stray fields results in demagnetizing (black) fields

inside the sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 (Color online) Illustration of spin transfer torque mechanism. (a)

Transport through a spin valve made up of thick ferromagnetic layer

FM1, nonmagnetic spacer NM and a thin ferromagnetic layer FM2. (b)

Interface of NM/FM2: the absorption of the transverse spin component

in FM2 results to spin transfer torque on the magnetization of the free

layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12

2.6 (Color online) A schematic illustration of a magnetic structure showing

how the magnetostatic energy of the system can be reduced by dividing

the structure into domains. . . . . . . . . . . . . . . . . . . . . . . . . 45

2.7 (Color online) Magnetisation texture in (a) Bloch and (b) Neel domain

wall, showing the higher magnetostatic energy due to surface (external)

magnetic charges for Bloch walls in thin films. . . . . . . . . . . . . . 48

2.8 (Color online). Thickness dependence of domain wall energy for Bloch

and Neel wall for Co81Ir19. Figure extracted from [5]. . . . . . . . . . 48

2.9 (Color online). Illustration of domain wall dynamics under the com-

bined action of current and field. . . . . . . . . . . . . . . . . . . . . 55

2.10 Illustration of the spin-dependent band structure of the s-d model.

The s band has a parabolic dispersion relation with a spin-split energy

Jsd. The d band on its part is narrower with a significant difference in

the number of up and down states at the Fermi energy. . . . . . . . . 67

2.11 (Color online) . Schematic diagram of four terminal system made up of

a central scattering region (blue boxed) attached to four leads L, T, R,

B at chemical potentials µL, µT , µR and µB respectively. The scattering

region contains an isolated magnetic skyrmion while the four leads

are ideal ferromagnet/antiferromagnet to insure smooth magnetisation

variation from the leads to the scattering region. In this set-up, a

charge bias is added to lead-L and lead-R while the induced transverse

charge or/and spin current is probed in lead-T and lead-B. . . . . . . 69

3.1 (Color online) Illustration of AHE induced (a) and s-d DMI mediated

(b) chiral damping. In addition to the constant isotropic exchange field

in the direction of the local moments (red) about which the itinerant

electrons (black) precess on a constant energy contour, in the presence

of s-d DMI, the itinerant electrons feels an additional spatially varying

field (blue) which turns to align its spin perpendicular to the local mo-

ment. This field which varies in both magnitude and direction results

in a damped precession which depends on the chirality of the wall. . . 81

13

3.2 (Color online) (a,b) Domain wall velocity as a function of perpendic-

ular magnetic field for different chiral damping strengths, at Hx =0

mT and Hx =25 mT, respectively; (c) corresponding effective damp-

ing. (d) Domain wall velocity as a function of in-plane magnetic field

for different chiral damping strengths, and for Hz =5 mT; (e,f) cor-

responding azimuthal angle and effective damping, respectively. The

parameters are α= 0.5 , HWB = 12.5 mT. In all simulations, we start

with a Bloch configuration (ϕ = π/2) and allow the system to relax to

steady state condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.3 (Color online) Normalized domain wall velocity as a function of in-

plane magnetic field Hx in the presence of a driving field Hz = 20mT.

In-plane magnetic field pushes the domain from a Bloch to a Neel

configuration and hence increases the azimuthal angle ϕ thereby mod-

ifying the damping of the wall. A kink in the velocity is observed at

Hx=−Hk, when the domain wall saturates in the Neel configuration

when the damping becomes independent on the in-plane field Hx. . . 90

4.1 (Color online) Sketch of the one-dimensional Bloch wall implemented

on a two-dimensional square lattice as used in the tight-binding model

described in the main text. . . . . . . . . . . . . . . . . . . . . . . . 97

4.2 (Color online) Spatial distribution of the adiabatic and non-adiabatic

torque for different current spin polarization. (a) P = -1.0, (b) P =

-0.56, and (c) P = -0.36. The adiabatic torque remains essentially

constant, while the non-adiabatic torque shows strong depends on the

current spin polarization. . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 (Color online) (a) Non-adiabaticity parameter as a function of the do-

main wall width in the absence (red) and presence of spin-conserving

impurity scattering (black) of strength Vimp = 0.1 eV; (b) Non-adiabaticity

parameter as a function of spin-conserving impurity scattering strength

for a domain wall width of 8a. In these calculations, we set the polar-

ization to P =-0.56. In the absence of disorder, the non-adiabaticity

is governed by spin mistracking while it is dominated by de-phasing

when introducing disorder in the structure. The fitted parameters are

β0 = 2.92, β0 = 8.9 × 10−3, β∗0 = 4.27 × 10−6, β1 = 2.87 × 10−4,

β2 = 5.24× 10−2, λL = 0.97, λL = 0.64 and λ∗L = 25. . . . . . . . . . 100

14

4.4 (Color online) Spatial profile of the y- (a,c) and z-components (b,d) of

the magnetization of a typical Bloch wall (a,b) and vortex core (c,d). 105

4.5 (Color online) Normalized velocity of a one dimensional Bloch wall,

below v<x (a) and above Walker breakdown v>x (b). Numerical and an-

alytic results are shown with dotted and solid lines respectively. The

parameters used are α = 0.005 and β0 = 0.01, χ = 0.2. (c) Spin-

mistracking non-adiabaticity as a function of the transverse spin diffu-

sion length λex for decreasing domain wall widths, calculated based on

Ref. [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.6 (Color online) Normalized transverse vy (a) and longitudinal vx (b)

velocities of the vortex core. (c) Polar angle of the vortex core as a

function of the core radius. Numerical and analytic results are shown

with dotted and solid lines respectively. The parameters used are R =

150 nm, α = 0.005, β0 = 0.01 and χ = 0.2. . . . . . . . . . . . . . . . 108

5.1 Schematic diagram of tight-binding system made up of a central scat-

tering region (isolated magnetic skyrmion/vortex) attached to two fer-

romagnetic leads (red boxed) L and R at chemical potentials µL and µR

respectively. To ensure smooth magnetization variation from the leads

to the scattering region, we consider an optimal system size compared

to the radius (R r0). . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2 Two-dimensional profile of torque components showing contributions

from different spin current sources. The adiabatic torque is dominated

by the contribution from the longitudinal spin current (a) compared

to the transverse spin current (b). The converse is true for the non-

adiabatic torque which is dominated by contribution from the trans-

verse spin current (d) compared to the longitudinal spin current (c). . 128

5.3 One-dimensional profile of torque components showing contribution

from different spin current sources and the spin density. The adiabatic

torque (a) is dominated by the contribution from Jxs while the non-

adiabatic torque (b) is dominated by contribution from Jys . Results

shows a very good match between the torque calculated from the spin

density (S) and that calculated from the spin current Js = Jxs + Jys . . 129

15

5.4 Effective adiabatic and non-adiabatic torque dependence on the skyrmion

radius. The adiabatic torque (a) is almost non-dependent on the radius

of the sykrmion while the non-adiabatic torque (b) shows substantial

dependence on the skyrmion radiusr0. Sfit represents fit of our numer-

ical data to analytical result. . . . . . . . . . . . . . . . . . . . . . . . 130

6.1 (Color online). Schematic diagram of tight-binding system made up

of a central scattering region containing isolated antiferromagnetic

skyrmion attached to two antiferromagnetic leads (red boxed) L and

R at chemical potentials µL and µR respectively. . . . . . . . . . . . . 140

6.2 (Color online). Topological properties of ferromagnetic and antifer-

romagnetic skyrmions. Although THE vanishes in antiferromagnetic

skyrmions (a), TSHE (b) can be gigantic very close to the antiferro-

magnetic band gap in clean systems. . . . . . . . . . . . . . . . . . . 141

6.3 (Color online). Two-dimensional profile of the transverse topological

spin current (a,c) and the corresponding non-adiabatic torque (b,d).

The transverse spin current and corresponding torque (c, d), are gigan-

tic close to the antiferromagnetic band gap compared to their values

far away (a,b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.4 (Color online). Robustness of TSHE in ferromagnetic and antiferro-

magnetic skyrmions realistic systems. Unlike ferromagnetic skyrmions

(a), not only is TSHE in antiferromagnetic skyrmions (b) robust, but

it increases with the strength of spin-independent impurity scattering.

In the presence of spin-conserving scattering, the THSE in both fer-

romagnetic and antiferromagnetic skyrmions increases with a decrease

in the skyrmion size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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LIST OF TABLES

4.1 Summary of measurements of non-adiabaticity for different materials.

The table entries shows values of β(β/α) for vortex walls (VW), in-

plane transverse walls (TW) and out-of-plane Bloch walls (BW). . . . 94

17

Chapter 1

Introduction

The sharing, storing and processing of information constitutes some of the greatest

driving forces behind human technological development. Information processing has

witness a substantial evolution from oral story telling in the ancient times, through

written languages in the 15th century, analog recording in the early 20th century, to

the present digital recording technologies in which information is encoded in memory

bits and transistors as ”0 ” and ”1 ”. In accordance with Moore’s prediction of an

exponential increase of the number of transistors per integrated semiconductor circuit

due to technological developments [7], we have witnessed doubling of the processing

power of computers roughly every 18 months [8] due to the ever-increasing, more effi-

cient manufacturing technologies. However, there is strong indication of an inevitable

halt to this trend because as electronics components become very small, quantum ef-

fects from single electron become important to consider. Indeed, the thermal noise

in nanoscale electronics components resulting from the coulombic interactions be-

tween electrons at room temperature can become very destructive and dissipative.

Therefore there has been growing interest in developing information processing tech-

nologies that do not heavily rely on the electric current, central to which entails the

understanding of the fundamental properties of an electron.

Electronic transport and magnetism are at the backbone of today’s information

technology. Even though magnetism has recorded tremendous success in high-density

18

storage such as hard disks, much needs to be done to fully integrate magnetic storage

into today’s electronic circuits. Such integration requires a proper understanding of

various microscopic mechanisms involved in the conversion of electrical current into

magnetic information and vice versa. Amongst these mechanisms is application of

the Faraday’s law that uses the magnetic field-induced electromotive force (electrical

current) to read information stored in magnetic bits, while the reciprocal effect uses

the Oersted field generated by electrical current to write information in magnetic

bits. Another mechanism which turns out to be more efficient than the Faraday

induction incorporated in magnetic tapes and hard disks makes use of anisotropic

magnetoresistance (AMR) effect [9] in which the relative angle between the local

magnetization and electric current influences the electrical resistance of the sample.

In the late 80s, the discovery of the giant magnetoresistance (GMR) effect [10, 11],

and eventually the tuneling magnetoresistance (TMR) [12] effect revealed that the

electrical resistance of a multilayer device hugely depends on the relative direction

of the magnetization of its layers. This discovery is considered the birth of the field

of spintronics-an emerging technology although still in its infancy stage as far as

commercialization is concerned.

1.1 Spintronics

Unlike in conventional semiconductor-based microelectronics in which information is

carried by the charge of an electron, the central technological motivation behind spin-

tronics is to carry information through the spin of electrons to create new functional-

ities and devices. Over two decades since its introduction into the magnetic recording

media industry, sensors based on magnetoresistance effects with ever-increasing data

densities have dominated conventional hard drives. However, the extensive interest

in spintronics research is largely due to the fact that its technological applications

19

are hardly limited to conventional magnetic recording media. As a matter fact, spin-

tronics devices based on Magnetic Tunnel Junctions (MTJs) [13, 14] are considered

one of the most innovative and promising candidates for the next generation memory

and logic chips as they promises high-speed, non-volatility, and strong Complemen-

tary metal-oxide-semiconductor (CMOS) compatibility. Their mode of operation in

the data storage industry for Magnetic random access memory (MRAM), directly

employs magnetoresistance effects to encode binary data or information in the mag-

netization of small multilayer device made up of a magnetically hard (fixed) and soft

(free) layer as the resistance of their relative orientational states. MRAM design has

attracted significant research interest as it provides substantial improvement over con-

ventional memories as it assures: non-volatility, reduced power consumption, orders

of magnitude increase read and write times with no write-cycle wear limitations.

This technology however uses the Oersted field generated by current traversing

the wires which suffers two major setbacks: scaling and non-locality. With respect to

scaling, the current density and heating increases with decrease in size of the cells.

With respect to non-locality, for very high storage densities, the long-range nature

of the Oersted field generated by current may also affect neighboring cells leading to

read-write errors and thus limit operational cell size to over 150nm. Therefore there

have been growing need to exploit technologies that do not rely solely on magnetic

field to control the magnetic state of devices.

The theoretical prediction [15, 16] and early experimental verification [17, 18, 19,

20] that current traversing a magnetic sample may alter its magnetic configuration

through the transfer of angular momentum, paved the way for a surge in research

efforts for potential application in devices such as the next generation GMR read

heads. These devices characterized by small sizes and high current densities, high-

density MRAM and current tunable high-frequency oscillators. These high-density

memories are called spin transfer torque magnetic random access memories (STT-

20

(a) (b)

Figure 1.1: (Color online). Schematic diagram of STT-MRAM architecture for in-plane (a) and the more efficient out-of-plane (b) magnetization. Figures extractedfrom [1]

MRAM) with operational principle very similar to the MRAM, the major difference

being that the orientation of the free ferromagnetic layer is changed by passing a

current through the cells. In such a setup, the current flows only through a distinct

memory cell and does not influence neighboring cells as it may be the case for the

writing fields of field-driven MRAMs. STT-MRAM possesses the huge advantage of

operating with much lower power consumption compared to field driven MRAM, and

shows great promises as future memory technology with prototype already success-

fully demonstrated with commercialization underway [21, 22]. However, since the

STT-MRAMs are based on a trilayer setup as shown in Figure 1.1, the need for a

polarizer has some limitations as far as technological implementation is concerned.

Notably, the generation of stray fields as well as experiencing spin transfer torque

from the free layer that can perturb the sensing. These issues have led researchers to

conceive alternative ways of generating spin torque using charge current without the

need for an external polarizer. In this regard, there has been enormous research effort

aimed at exploiting the spin-orbit coupling, which exploits the connection between

spin and orbital angular momenta, to generate spin currents and densities from which

a spin-orbit torque can be exerted on a ferromagnet [23, 24].

21

Figure 1.2: (Color online). Schematic diagram of the racetrack memory in ferromag-netic nanowires. Figure extracted from [2]

1.2 Non-collinear magnetic textures

An exciting direction exploits the use of non-collinear magnetic textures as candi-

date for the next generation high-density memories. This possibility relies on recent

improvement in lithographical patterning techniques coupled to the advancement in

computing power which has made it possible to investigate individual domain walls,

skyrmions or vortices and perform micro magnetic simulations on the static and dy-

namics of these magnetic structures. Indeed, Stuart Parkin proposed a novel memory

device called Racetrack memory [2] which is a chip-based magnetic memory in which

bits of information are stored as magnetic domains in magnetic nanowires as shown

in Figure 1.2. In such set-up, the spin transfer torque (STT) is used to push several

magnetic domains through a long magnetic nanowire. In particular, small magnetic

domains are injected at the bottom of a U-shaped wire and pushed up into one of

the ends of the wire by a current through STT. Therefore by using nanosecond long

current pulses, successive bits can be written and pushed up the nanowire and read

22

by pushing back with reversed current pulses towards any reading device such as

TMR sensor at the chip surface. The racetrack memory thus merges of low-cost,

nonvolatility of magnetic storage and high speed of solid state memory [25].

When spin-polarized electrons flow through magnetic textures such as domain

walls, vortices and skyrmions, their spin is reoriented. Therefore, just as in the case

for GMR, spin dependent effects are expected to be central to understanding elec-

trons scattering in such magnetic textures [26]. Indeed, over the past two decades,

there have been extensive scientific investigations on magnetic domain walls, vor-

tices and skyrmions. These studies have been central to (i) understanding many

observed magnetic properties such as magnetization reversal and coherent rotation

processes in ferromagnets, and (ii) an emerging pathway to magnetic memories, sen-

sors and logic through current-induced manipulation of non-collinear magnetic tex-

tures in nanowires.

Domain walls are boundaries in ferromagnets that separate regions with distinct

magnetization directions (magnetic domains) and in which the magnetic profile varies

continuously as shown in Figure 1.3 (a). These domain walls are formed as a result of

the competition between the magnetic exchange on the one hand and magnetocrys-

talline anisotropy and the magnetostatic interactions on the other hand.

Vortices on their part, are flux-closure, two-dimensional topological soliton states

that are created during reversal in soft magnetic materials such as Permalloy. They

are formed as a consequence of competition between different magnetic interactions

notably the dipole-dipole and the exchange interactions. The creation of vortex struc-

ture in these materials ensures the absence of both edge and surface magnetic charges

with small and localised stray field around the core interior as shown in Figure 1.3(b).

The reduction of the long range stray fields reduces the magnetostatic interaction be-

tween neighboring vortex and thus making such magnetic systems strong candidate

for high-density magnetic storage devices.

23

(a)

(b)

Figure 1.3: (Color online). Illustration of the spin structure of a domain wall (a) andvortex (b).

Similar to vortices, another class of two-dimensional topological soliton states

which has generated a lot of research interest are magnetic skyrmions. Magnetic

skyrmions as shown in 1.4 (a) have been theoretically predicted to exist in magnetic

materials with antisymmetric exchange interactions [27] and experimentally observed

[28, 29]. Isolated skyrmions are strong candidates for digital bits encoding [30] with

potential applications in novel electronic and magnetic devices such as the racetrack

memory [2, 31, 32, 33, 34, 35, 36] and the skyrmion logic gates [37].

1.3 Antiferromagnetic textures

Over the past years, antiferromagnets have played a very important but passive role

in magnetic storage devices in which they have been used primarily as pinning layers.

Recently, emerging technologies have been proposed with antiferromagnetic materi-

als playing an active role such as, governing the transport properties of the devices

[38, 39, 40, 41, 42, 43]. This is principally due to the fact that antiferromagnets

(i) can easily be integrated with ferromagnetic components, (ii) no stray fields and

(iii) are characterized by ultrafast response (∼ THz range, while ferromagnets’ re-

sponse is in the ∼ GHz range). Therefore, the control of antiferromagnetic textures

24

(a) (b)

Figure 1.4: (Color online). Illustration of the spin structure of a ferromagnetic (a)and antiferromagnetic (b) skyrmion. Figures extracted from [3, 4]

by electric current bears appealing promises. Fundamental to the understanding of

current-induced motion of antiferromagnetic textures is the detailed knowledge of

the influence on dynamics of inhomogeneous magnetic textures by current. Indeed,

while magnetic textures such as ferromagnetic skyrmions show extensive potential

for application in memories and logic devices, a potential bottle neck remains the

detrimental effect of the skyrmion Hall effect which drives skyrmion towards the lat-

eral edge of the nanostructure [36, 44, 45, 46]. And since this undesirable transverse

motion increases with driving current, an increase in current can eventually destroy

the skyrmion. Interestingly, it has been shown that this skyrmion Hall effect is totally

suppressed in antiferromagnetic skyrmion [see Figure 1.4 (b)] [36, 46] and therefore

provides a promising alternative to ferromagnetic skyrmion as a memory device.

25

1.4 Motivation

The prospect of storing and transferring information through the use of electric cur-

rent to control the position of a magnetic textures has attracted a lot of enthusiasm

due to its enormous potential application in magnetic memory and logic devices

[47, 48, 49, 2, 50, 51]. A detail understanding of the physics of spin transport phe-

nomena is essential to enhance the performance of present spintronics devices, as well

as in designing new devices for future applications. Since a single nanowire can con-

tain multiple domain walls, vortices or skyrmions at arbitrary positions, an individual

magnetic element is expected to encode a much greater amount of data. Even though

at the first thought, the magnetostatic interactions between domain walls, vortices or

skyrmions might limit the data density that can be achieved in this way, the prospect

of using antiferromagnetic textures with vanishing magnetostatic fields to overcome

this limitation is quite appealing and thus requires thorough investigation.

1.5 Objectives and contributions

The objective of this thesis is to provide an accurate description of spin transport in

magnetic textures and in particular to investigate in detail, the nature of spin torque

and magnetic damping in such systems. Indeed, as will be further discussed in this

thesis, the current-driven velocity of magnetic textures is related to the ratio between

the so-called non-adiabatic torque and magnetic damping. Uncovering the physics

underlying these phenomena can lead to the optimal design of magnetic systems

with improved efficiency. We identified three interesting classes of systems which

have attracted enormous research interest lately: (i) magnetic textures in systems

with broken inversion symmetry (ii) ferromagnetic skyrmions and vortices and (iii)

antiferromagnetic skyrmions.

After reviewing theoretical background of spin transport in Chapter 2, Chapter 3

26

presents an investigation on the nature of magnetic damping in non-centrosymmetric

ferromagnets. Based on phenomenological and microscopic derivations, we show that

the magnetic damping becomes chiral, i.e. depends on the chirality of the magnetic

texture. In Chapter 4, we address the physics of spin transport in sharp disordered

magnetic domain walls and vortex cores. We demonstrate that upon spin-independent

scattering, the non-adiabatic torque can be significantly enhanced. Such an enhance-

ment is large for vortex cores compared to transverse domain walls. In Chapter 5, we

concentrate our attention on the topological transport in non-trivial textures such as

vortices and, more importantly skyrmions. We show that the topological spin currents

flowing in these structures dramatically enhance the non-adiabaticity, an effect unique

to non-trivial topological textures. Finally, in Chapter 6 we extend this study to an-

tiferromagnetic skyrmions and show that such an enhanced topological torque also

exists in these systems. Even more interestingly, while such a non-adiabatic torque

influences the transverse velocity of ferromagnetic skyrmions, in antiferromagnetic

skyrmions, the topological non-adiabatic torque directly determines the longitudinal

velocity. As a consequence, scaling down the antiferromagnetic skyrmion results in

a much more efficient spin torque. In Chapter 7, we present our conclusions and

perspectives.

27

Chapter 2

Background

2.1 Magnetic materials

Electrons are commonly visualized as point-like charged particles orbiting round the

atomic nuclei in as planets in the solar system. The reality is of course pretty far from

this unphysical picture as the electron in addition to its charge has an intrinsic fun-

damental property called the spin. Although the spin of the electron has a relativistic

origin [52], it is insightful to apply a quasi-classical representation and consider the

electron as a rigid sphere rotating around its own axis, with the spin axis being the

axis of rotation as illustrated in Figure 2.1. Indeed, according to classical electrody-

namics [53], the magnetic dipole moment created by a rotating charge −e is given

by

µm = − eg

2mL, (2.1)

where m is the rotating mass, L is the angular momentum and g is called the Lande

g-factor. When applied to the intrinsic spin angular momentum of the electron Se,

the magnetic dipole moment of the electron’s spin is given as

µe = −γSe, (2.2)

28

Figure 2.1: (Color online). Semi-classical schematic description of an electron.

where γ = geµB/~ is the gyromatic ratio, ~ is the reduced Planck’s constant, µB is

the Bohr magneton and ge ≈ 2 is the electron g-factor.

All matter consists of finite number of electrons which by virtue of Pauli exclusion

principle [54] minimizes their energy by pairing up spin-up and spin-down electrons.

The result of which is in an overall vanishing total intrinsic spin of electron cloud for

most materials except for a class of materials called magnetic materials with unpaired

electrons in their valence shell. In magnetic materials, neighbouring spins couple with

each other through the exchange interaction. Broadly speaking, in centrosymmetric

magnetic materials, there are two dominant (symmetric) exchange interactions: ferro-

magnetic and antiferromagnetic coupling. Ferromagnetic coupling favors the parallel

alignment of the spin axis of an electron with its neighbour’s while antiferromagnetic

coupling favors an anti-parallel alignment. However, in magnetic materials with in-

version asymmetry, an exchange interaction that is antisymmetric upon exchange of

neighboring spins, called Dzyaloshinskii-Moriya interaction (DMI) [55, 56, 57] favours

the perpendicular alignment of spin axes [see Section 2.2.2 for more details], and can

result to some exotic magnetic ordering with defined chirality such as skyrmion, chi-

29

ral domain walls and helical spin textures. The interplay between ferromagnetic or

antiferromagnetic interactions and DMI in addition to other interactions in magnetic

materials leads to very exotic and interesting magnetic ordering in real materials

[58, 59, 60]

2.2 Basics of micromagnetism

In the 20th century, in an effort to bridge the gap between Maxwell’s macroscopic

description of electromagnetism and the quantum mechanical theories of magnetic

dipoles, the concept of micromagnetism was put forward Brown [61]. Micromag-

netism can therefore be thought of as a continuum theory of magnetic moments which

underlies the description of magnetic microstructure. As an illustration, consider a

ferromagnetic sample of volume Ω and small volume dVr represented by the position

vector r ∈ Ω such that dVr is large enough to contain a huge number N of elementary

magnetic moments µj , j ∈ (1...N) but small enough so that the magnetic moment

of the volume can be considered uniform. Then the magnetization M(r, t) is defined

as magnetic moment per unit volume or

M(r, t) =

∑Nj=1µj

dVr. (2.3)

In the micromagnetic framework, the small volumes Vr that can vary in spatial scale

from nanometer to micrometer due to short- and long-range interactions between

magnetic moments are called magnetic domains. These magnetic domains form the

microstructures that links the basic physical properties to the macroscopic properties

of magnetic materials. Magnetic materials minimize their total energy through the

formation magnetic domains especially in confined samples in which the domains

reduce the dipolar energy contribution. In ferromagnets, the magnetic induction or

30

Figure 2.2: (Color online). A uniformly magnetized sample is characterized by thethree magnetic vectors B, H, and M, as shown. The magnetic field inside the FM iscalled the demagnetizing field Hd while that outside is called the stray field Hs.

flux density B, magnetization M and magnetic field H are related by

B = µ0(H + M) (SI units) (2.4)

where µ0 = 4π × 10−7TA−1m is the permeability of free space which is a measure

of the ration B/H in vacuum. To better understand the nature of these vectors, we

consider a uniformly magnetized sample as shown in Figure 2.2. The magnetic field

inside a magnetic material is called the demagnetization field-since its orientation is

opposite to that of the magnetization thus tends to destroy it. Outside the magnetic

material, the magnetic field is usually called stray field-since it loops around in space.

In an effort to find the magnetization distribution that minimizes the total mag-

netic energy associated with the different magnetic interactions, Landau and Lifshitz

[62] used a variational principle that leads to a set of non-linear and non-local dif-

ferential equations, otherwise known as micromagnetic equations. In the following

section, we present a quantum mechanical derivation of this equation.

31

2.2.1 Dynamical equation

Our starting point is the quantum mechanical Heisenberg equation of a single spin

S. Using Ehrenfest theorem, the dynamics of the j component of S is given by

dSjdt

=i

~[H, Sj] =

i

~

(∑k

∂H∂Sk

[Sk, Sj] +O(~2)

), (2.5)

where H is the Hamiltonian of the system. Using the commutation relation of spin-12

particles

[Sk, Sj] = −i~∑l

εjklSl, (2.6)

we have

dSjdt

=∑k,l

∂H∂Sk

εjklSl +O(~) (2.7)

or equivalently

dS

dt= −S× ∂H

∂S+O(~). (2.8)

Since the mean distance between spins is small compared to the characteristics length

scale over which the magnetization changes, the semi-classical approximation can be

invoked by replacing S by the magnetization vector M and the limit ~→ 0 to have

dM

dt= −γM×Heff (2.9)

where γ = gµB/~ is the gyromagnetic ratio and the effective field

Heff =1

γ

⟨∂H∂S

⟩= − δE

µ0δM(2.10)

where E is the energy of the system. Eq. (2.9) describes magnetization precession

around the effective field on a constant energy contour as shown in Figure 2.3(a).

32

Figure 2.3: (Color online). Shows the dynamics of the magnetisation vector about aneffective field governed by a) precession on a constant energy contour and b) dampingeffects.

This is the case because taking the time rate of change of the total energy yields

dE

dt=

∫dV

δE

δM· dMdt

=

∫dV (−µ0Heff) · (−γM×Heff) = 0. (2.11)

However, since dissipative processes are omnipresent at the macroscopic level in dy-

namical processes, different approaches have been used to incorporate magnetic damp-

ing. In 1955, Gilbert introduced a viscous force term proportional to the derivative

of the magnetization to obtain the Landau-Lifshitz-Gilbert (LLG) equation

dM

dt= −γM×Heff +

α

Ms

M× dM

dt, (2.12)

where α is a phenomenological material-dependent constant with values ranging from

0.001 to 0.1. This term is justified as Eq. (2.9) could be derived from a Lagrangian

formulation in which the generalized coordinates are the components of magnetization

vector. Another form of the damping can be obtained from Eq. (2.12) through- taking

the vector product with M on both side of Eq. (2.12)

M× dM

dt= −γM× (M×Heff)− αMs

dM

dt(2.13)

33

and substituting Eq. (2.13) into Eq. (2.12), to obtain [62]

dM

dt= −γ′M×Heff −

λLL

M2s

M× (M×Heff) (2.14)

where γ′ = γ/(1 + α2) is a renormalized gyromagnetic ratio and λLL = γ′αMs. Eq.

(2.14) was introduced by Landau and Lifshitz to account for dissipation.

It is straight forward to see that in dynamical systems energy is dissipated, that

is dEdt< 0, as depicted in Figure 2.3(b). To see that, first we re-cast Eq. (2.13) into

Heff =1

γM2s

M× dM

dt+

1

M2s

(M ·Heff)M +α

γMs

dM

dt. (2.15)

and then using Eq. (2.15), we have

dE

dt=

∫dV

δE

δM· dMdt

=

∫dV (−µ0Heff) · dM

dt= − µ0α

γMs

∫dV

(dM

dt

)2

< 0, (2.16)

Eq. (2.12) describes the dynamics of a magnetic system as such to fully understand

such dynamics, knowledge of the effective field associated to the different contribution

of the micromagnetic energy [c.f. Eq. (2.10)] is required. In the following section, we

present a brief overview of these interactions and associated micromagnetic energies

from which the effective fields are calculated based on the variational principle.

2.2.2 Micromagnetic energies

Zeeman energy

In magnetic materials, the magnetic field energy has two contribution, the external

field energy and the demagnetizing field energy. The first contribution also called

the Zeeman energy represents the interaction energy of the magnetization M with an

34

external applied magnetic field Hext given as

EZ = −µ0

∫dVM ·Hext (2.17)

Exchange energy

The exchange energy is a purely quantum mechanical short-range Coulomb interac-

tion between electrons with overlapping orbitals. The atomic moments are determined

by the intra-atomic exchange while the interatomic exchange between neighbouring

magnetic atoms is responsible for magnetic ordering in magnetic materials. The most

general expression for the exchange energy in a system with spins Si at sites i in a

lattice can be written as

Eex = −1

2

∑i 6=j

Si · JijSj, (2.18)

where Jij are generalized exchange tensors whose strength decreases rapidly with

increasing distance between spins. The exchange tensor can be decomposed into

Jij = JijI + J sij + J a

ij, (2.19)

where I is an identity tensor, Jij = 13 Tr (Jij), J s

ij = 12 (Jij + Jji) and J a

ij = 12 (Jij − Jji)

are isotropic exchange, traceless symmetric anisotropic and anti-symmetric exchange

tensors respectively.

The magnetic state of magnetic materials is predominantly determined by the

isotropic exchange because it is the largest exchange energy 0.1-10 eV (10 eV in room

temperature antiferromagnets, 1eV in transition metal ferromagnet and 100meV in

GaMnAs), while the anisotropic exchange energy can only attain values of up to 5%

of the isotropic energy [63].

35

Isotropic exchange energy A generic property of ferromagnetic materials is the

appearance of a spontaneous magnetization M below the Curie temperature Tc due to

spontaneous symmetry breaking associated with energy of the order of kBTc/atom ≈

0.1 eV/atom. Heisenberg explained ferromagnetism through the isotropic exchange

interaction which is a consequence of non-relativistic quantum mechanics and the

Pauli exclusion principle: the effective coulomb repulsion for electrons pairs with an-

tiparallel spins is stronger than for parallel spins, this prevents electrons with parallel

spins to occupy the same orbital state. The isotropic exchange energy is given as

Eisoex = −1

2

∑i 6=j

JijSi · Sj. (2.20)

If we limit the exchange interaction only to nearest neighbours such that Jij = J

which is indeed a good approximation especially for 3d metals where this interaction

is short range, then exchange in the continuum limit reduces to

Eisoex =

A

M2s

∫dV

[(∂M

∂x

)2

+

(∂M

∂y

)2

+

(∂M

∂z

)2]

(2.21)

where A = JnS2/a is the exchange stiffness (constant), n is the number of atoms

(spins) per unit cell and a is the lattice constant. The typical length scale over

which the exchange energy dominates the magnetic body and below which the body

is expected to behave as a single coherent domain is called the exchange length given

by lex =√

2A/µ0Ms. From the exchange energy and using Eq. (2.10) the effective

field due to the exchange interaction is given as

Hexeff =

2A

µ0M2s

∇2M. (2.22)

Anisotropic exchange energy Similar to the isotropic exchange, the anisotropic

symmetric exchange energy is unchanged upon the interchange of two interacting

36

spins. However it is dependent on the relative direction of the spins and the lattice.

The total anisotropic exchange energy is given by

Eaniex = −1

2

∑i 6=j

Si · J sijSj. (2.23)

This type of exchange can result from any microscopic exchange mechanism ranging

from RKKY exchange, super-exchange to double exchange depending on the chemical

composition, geometry and size of the system [64].

Antisymmetric exchange energy: DMI In 1958, using symmetry arguments,

Dzyaloshinskii predicted the existence of an antisymmetric exchange from the com-

bination of reduced symmetry and spin-orbit coupling [56]. Two years later, Moriya

proposed a microscopic mechanism that explains the existence of such exchange in

systems with spin-orbit coupling [55, 57]. Indeed, the antisymmetric component of

Eq. (2.18) can be re-cast into

Eaex = −1

2

∑i 6=j

Si · J aijSj = −1

2

∑i 6=j

Dij · (Si × Sj), (2.24)

where

Dxij =

1

2(J yz

ij − Jzyij ) , Dy

ij =1

2(J zx

ij − J xzij ) , Dz

ij =1

2(J xy

ij − Jyxij ). (2.25)

An interesting playground for DMI which has attracted a lot of research interest

are perpendicularly magnetized ultra-thin films embedded between dissimilar materi-

als such that structural inversion symmetry is broken along the film normal (z-axis).

In such case the DMI arises as a result spin-orbit coupling in the electronic struc-

ture induced by inversion-asymmetric crystal fields near the surface [65]. The DMI

is characterized by the Dzyaloshinskii vector along z× u, where u is the direction of

37

magnetization gradient [66, 65, 67, 68]. In the continuum limit the DMI energy is

given as

EDMI =D

M2s

∫dVM · (∇z ×M), (2.26)

where ∇z = z×∇ and D is the DM energy density that characterizes the strength

of the DMI. From the energy, the effective field due to the DMI as is given as

HDMIeff = − D

µ0M2s

∇z ×M. (2.27)

Anisotropy energy

In ferromagnetic materials, there is usually a preferred direction with respect to the

shape of the material or/and the crystalline axes on which the magnetization lies.

This property is called magnetic anisotropy whose degree and strength is measured

by the anisotropy energy with typical values of the order 10−6 to 10−3eV per atom.

The magnetic anisotropy thus breaks the rotational invariance with respect to the spin

quantization axis and is a small relativistic correction to the Hamiltonian. Magnetic

anisotropies range from magnetocrystalline, shape, surface, to stress anisotropies [69].

Our focus in this section is on magnetocrystalline, shape and surface anisotropies

which play a central role in magnetic DW structures and dynamics in thin films.

Magnetocrystalline anisotropy energy arises from the asymmetry in the crys-

tals due to spin-orbit coupling between the orbitals of the crystalline structure and

the spin moments given by the Hamiltonian

HSOC = 2e

(1

2mc

)2

s · (E× p), (2.28)

where s and p are the spin and momentum operators respectively. This coupling

is largest in the neighbourhood of the nucleus and in particular, for 3d metals the

38

electrostatic potential can be considered as spherically symmetric. In which case, the

electric field can be written as

E = −dΦ

dr

r

r, (2.29)

this gives a spin-orbit Hamiltonian of the form

HSO = ξs · l, (2.30)

where l = r× p is the angular momentum operators and ξ is the spin-orbit constant

given as

ξ(r) = −2e

r

dΦ

dr

(1

2mc

)2

. (2.31)

The determination of the directional dependence of the magnetocrystalline energy

is done through the expansion of the directional cosines of the magnetization vector

in such a way that the underlined symmetries of the crystal are respected. The

uniaxial anisotropy which is dominant in tetragonal, hexagonal and rhombohedral

structures is the simplest kind of magneto crystalline anisotropy. In systems with

uniaxial anisotropy, the anisotropy energy is given by

EU = −∫dV

(K1

M2s

(u ·M)2 +K2

M4s

(u ·M)4

)(2.32)

where u defines the direction of the easy axis. The corresponding effective field due

to uniaxial anisotropy is given as

HUeff =

2K1

µ0M2s

(u ·M) +4K2

µ0M4s

(u ·M)3 (2.33)

where K1 and K2 are constant anisotropy energy densities whose strength are used to

classify magnets as either hard magnetic materials as in Co, FePt with typical values

K1 ∼ 106 Jm−3 or soft magnetic materials as in Fe with typical values K1 ∼ 104

39

Jm−3.

Dipolar energy

As the size of the magnetic sample increases, the exchange interaction weakens and

starts competing with an opposing effect: the dipolar interaction. This contribution of

the magnetic energy also called shape anisotropy or demagnetizing energy represents

the magnetic field generated by the magnetic body itself and can be dominant in

finite polycrystalline samples. This interaction can be understood as follows: at large

distances the atomic moments can be considered as dipoles which unlike exchange

interaction, are long-range and favour an anti-aligment of the moments. This energy

term arises as a direct consequence of Maxwell equation ∇ · B = 0, which implies

that a magnetic (demagnetizing) field is generated in the presence of any divergence

of the magnetization

∇ ·H = −∇ ·M. (2.34)

Given that each magnetic moment represents a magnetic dipole, this amounts to a

contribution of total magnetic energy inside the sample given as

Edem = −1

2µ0

∫dVHd ·M, (2.35)

the factor of 12

arising because it is a self-energy contribution. We note that ∇ ·M is

not defined at the surface as illustrated in Figure 2.4 and therefore magnetic surface

charges are given by

σ = M · n, (2.36)

where n is a unit surface normal. Magnetic materials tends to minimize these

magnetic charges by aligning their magnetization parallel to the surfaces which even-

tually leads to a magnetization pattern with flux-closure and therefore will favor multi

domain formation. The calculation of Hd is very complicated and for an arbitrary

40

Figure 2.4: (Color online). Shows a uniformly magnetized sample with magnetization(red). Finite system sizes results to accumulation of magnetic charges at the surfacesproducing stray fields (gray) outside the sample. Occurrence of stray fields results indemagnetizing (black) fields inside the sample.

sample shape is is given by

Hd =

∫V

(r− r′)∇ ·M(r′)

|r− r′|2dV −

∫∂V

(r− r′)σ(r′)

|r− r′|2dS (2.37)

For a macrosopic description, the demagnetizing field and magnetization are related

through

Hd = −N ·M (2.38)

where N is the demagnetizing tensors whose components depends on the aspect ratio

of the sample.

2.3 Current-induced torques

2.3.1 Spin transfer torque

In 1996, Slonczewski [16] and Berger [15] independently predicted that when a charge

current passes through two ferromagnets separated by a nonmagnetic layer, spin an-

41

Figure 2.5: (Color online) Illustration of spin transfer torque mechanism. (a) Trans-port through a spin valve made up of thick ferromagnetic layer FM1, nonmagneticspacer NM and a thin ferromagnetic layer FM2. (b) Interface of NM/FM2: the ab-sorption of the transverse spin component in FM2 results to spin transfer torque onthe magnetization of the free layer.

gular momentum can be transferred from one ferromagnet to the other. This angular

momentum transfer is capable of inducing magnetic excitations and eventually mag-

netization switching in one of the layers. This effect called spin transfer torque has

been experimentally verified [17, 18, 19, 20]. The original experimental device con-

sisted of two ferromagnetic cobalt layers - a thick (10 nm) called the fixed layer and

a thin (2.5 nm) called the free layer separated by a copper spacer (6 nm) layer. An

illustration of the spin transfer mechanism is shown in Figure 2.5. As electrons pass

through the fixed layer, their spin precesses and eventually aligns along the direction

of the fixed layers magnetization P and therefore the current becomes spin polarized

along P. This spin polarized current (spin current) passes through the nonmagnetic

spacer and eventually hits the free layer. If P and M are collinear, the spin of the

propagating electrons remains unchanged and therefore no magnetization dynamics

is observed. However, when P and M are not collinear, the moving electrons’ spin di-

rection is re-oriented to align along the local magnetization direction of the free layer

in order to minimize the energy of the system. During this process, the spin com-

ponent of the moving electrons parallel to M (longitudinal spin component) remains

unchanged while its component transverse to M disappears at the NM/FM2 as shown

42

in Figure 2.5(b). The conservation of angular momentum implies that the transverse

spin component is transferred to the local magnetization. As a result, a spin transfer

torque is exerted on the magnetization of the free layer. For sufficiently large spin

current, it is possible to observe magnetic excitations and eventually magnetization

switching of the free layer [70].

The concept of spin transfer torque can readily be extended to magnetic textures

that are characterized with an inherent non-collinear magnetization profile. Indeed,

Berger in the 1970s [71] predicted that when a spin current traverses a magnetic

texture, the spin polarization direction of the conduction electrons becomes aligned

with that of the local magnetization through the action of the exchange interaction.

Furthermore, since the exchange interaction conserves the total spin, the angular

momentum transferred to the local magnetization is equivalent to a torque on the

magnetization which results in a displacement of the magnetic texture in the direction

of electron flow. The corresponding torque is called the adiabatic spin transfer torque

given as

Tad = −bJ(u ·∇)M, (2.39)

where u is the direction of applied current which we consider throughout this thesis

to be along x. bJ = PgµBJe/2eMs is the magnitude of spin drift velocity which

represents the maximum attainable speed of the magnetic texture, P is the current

polarization, g is the Lande g-factor, µB is the Bohr Magneton, e is the electron

charge, and Ms is the saturation magnetization. The picture of only adiabatic spin

transfer mechanism resulted to a major discrepancy with experimental results. In-

deed, the intrinsic current thresholds calculated for steady propagation of a domain

wall in a perfect wire were found to be more than 10 times larger than experimental

values [72, 73]! Ever since, there has been tremendous effort to fully understand the

spin transfer mechanisms capable of explaining the experimental findings. In this

regards, it became clear that there was a need to consider deviation from a perfect

43

adiabaticity. This puzzle was partly resolved through the works of Zhang and Li [74]

and Thiaville et. al. [75] who introduced a corresponding non-adiabatic spin transfer

torque characterized by a dimension-less parameter β called the non-adiabatcity and

given as

Tna =βbJMs

M× (u ·∇)M. (2.40)

This torque contribution is expected to be smaller than its adiabatic counterpart

but it turns out to significantly alter the dynamics of the magnetic textures and

in particular it dictates the critical current density and the terminal velocity of

the texture. Although there is consensus on the existence of such non-adiabatic

torque, there is no consensus on its physical origin and amplitude. The nature of

non-adiabatic torque has been the subject of extensive experimental and theoretical

studies [76, 77, 78, 79, 80, 81, 2, 82, 83, 84, 84, 85, 86, 87, 88, 89, 90, 91, 92]

2.3.2 Spin-orbit torques

Unlike the spin transfer torque mechanism that requires non-collinear magnetization

as discussed in Section 2.3.1, current induced spin-orbit torques transfer angular mo-

mentum between the orbital and spin degree of freedom through spin-orbit coupling.

This mechanism that requires only a single ferromagnet turns out to be a very effi-

cient way to achieve magnetization switching, and has attracted enormous research

interest in a wide range of systems ranging from ultrathin ferromagnetic heterostruc-

tures [23, 24, 93, 94] to diluted magnetic semiconductors [95, 96, 97]. In a magnetic

bilayer grown along the z-direction and where the current is injected in-plane along

the x-direction, experimental and theoretical investigations show that current induced

spin-orbit torques adopt the general form [98, 99, 100, 101, 102, 103, 104]

T = T ||m× (y ×m) + T⊥y ×m, (2.41)

44

where the torque components are usually called the in-plane (T ||) and out-of-plane

(T⊥) torques.

2.4 Non-collinear magnetic textures

The equilibrium structure of domains which make up magnetic materials is the stable

magnetization configuration arising from the competition between the macroscopic

long-range antiferromagnetic dipole-dipole interaction and the microscopic short-

range ferromagnetic exchange interaction. As an illustration suppose there exists

a uniformly magnetized sample as shown Figure 2.6 (a). This will result in the ac-

cumulation of surface magnetic charges and as a result, a large magnetostatic energy

associated with it. If this single domain is divided into two sub-domains as shown in

Figure 2.6 (b), the magnetostatic energy is reduced by a factor of two and as a rule

of thumb, if a single domain is divided into N sub-domains the magnetostatic energy

reduces by a factor of 1/N . Indeed, it is even possible to have a closure structure as

shown in Figure 2.6 (c) for which the magneto static energy is zero which could be the

case only in materials without uniaxial anisotropy. However, the division of domains

into sub-domains does not continue indefinitely since magnetostatic energy is not the

only energy term involved. As a matter of fact the introduction of a domain is ac-

companied by the apparition of domain walls that cost energy, as the latter competes

with the exchange energy. Things might even get more complicated in materials with

broken inversion symmetry and strong spin-orbit coupling characterized with DMI.

In such scenario, the competition between the exchange, Dzyaloshinskii-Moriya and

dipolar interactions can give rise to exotic magnetic textures [27].

45

Figure 2.6: (Color online) A schematic illustration of a magnetic structure showinghow the magnetostatic energy of the system can be reduced by dividing the structureinto domains.

2.4.1 One-dimensional magnetic textures

The boundaries of domains in magnetic materials are characterized by a regions in

which the magnetization varies continuously called domain walls. The structure

of these domain walls is a result of the competition between the exchange inter-

action, which prefers to keep neighbouring spins aligned in parallel, and the magnetic

anisotropy, which prefers to keep spins aligned along one of the preferred axis of easy

magnetization.

Statics

For an infinite system with uniaxial anisotropy, magnetostatic effects are minimal,

and therefore the structure of the domain wall is solely determined by the exchange

and anisotropy energies. In such scenario, starting from the energy per unit area

E =

∫ +∞

−∞

[A

(dθ

dx

)2

+K1 sin2 θ

]dx, (2.42)

and since the integrand in Eq. (2.42) is positive (for A, K1 > 0 ) it follows that the

solution of minimal energy per unit area is a monotonic function θ(x). In particular,

46

applying the boundary conditions: θ(x) monotonically decreases to 0 as x → −∞

and monotonically increases to π as x → +∞. The application of the variational

derivative to Eq. (2.42) yields the solution

θ(x) = arccos

[tanh

(−√K1

Ax

)]. (2.43)

Bulk materials: Bloch walls In bulk materials, the magnetostatic energy con-

tribution can be neglected, therefore the direction of the magnetization is dictated

by the magnetocrystalline anisotropy of the material. The competition between the

exchange and anisotropy usually results in a Bloch domain wall separating two anti-

parallel magnetic domains involving a 180 rotation of the magnetization over N

spins. The exchange energy of two spins is given by

E = −2JS1 · S2 = −2JS2 cos θ, (2.44)

where θ is the angle between the two spins. In the case of a strong exchange, this

represents a small and equal spin rotation angle of θ = π/Na0 and a resulting exchange

spin rotation energy density of

Eex = JS2 π2

Na20

, (2.45)

where a0 is the lattice constant. On the other hand, the anisotropy energy density

contribution over the N spins is given by

Ea = K1

N∑i=1

sin2 θi ≈ a0NK1. (2.46)

47

The total energy density of the Bloch wall is therefore given as

EBW = Eex + Ea =JS2π2

a20

1

N+ a0K1N. (2.47)

The number of spins N0 that minimizes the total energy is given as

N0 =π

a0

√JS2/a

K1

=π

a0

√A

K1

. (2.48)

This defines the domain wall width given as

λBWdw = a0N0 = π

√A

K1

(2.49)

and the associated minimum energy density

EBW = 2π√AK1. (2.50)

The total energy density from Eq. (2.47) shows that - while the exchange energy

can be minimized by the distribution of the rotation over a large number of spins

(smoothly varying textures) the anisotropy on its part favours small rotations (abrupt

textures).

Thin films: Neel walls In the above analysis of the Bloch domain walls in bulk

material, we ignored the magnetostatic energy contribution. However, in thin films,

the domain walls induces accumulation of magnetic charges on the surface that results

in a significant magnetostatic energy density contribution given as −µ0M · Hd =

12µ0M

2s cos θ, where Hd is the demagnetizing field and θ is the angle between the Hd

and M. This contribution can dominate over the anisotropy energy and eventually

leads in a Neel domain wall characterized by magnetization rotation within the plane

of the film as shown in Figure 2.7. Following similar analysis as in the case of bulk

48

Figure 2.7: (Color online) Magnetisation texture in (a) Bloch and (b) Neel domainwall, showing the higher magnetostatic energy due to surface (external) magneticcharges for Bloch walls in thin films.

material we obtain the associated domain wall width and minimum energy density

given as

λNWdw = π

√A

K1

(1 +

2K1

µ0M2s

)(2.51)

and

ENW = 2

√AK1

(1 +

µ0M2s

2K1

)(2.52)

respectively. As shown in Figure 2.8, depending on width of the film we expect either

Figure 2.8: (Color online). Thickness dependence of domain wall energy for Blochand Neel wall for Co81Ir19. Figure extracted from [5].

the Bloch or Neel to be favorable. However, we note that this analysis is based on the

49

assumption of constant magnetization rotation and does not take into account other

effects such as surface anisotropies which become dominant in ultra thin films.

Dynamics

Magnetic systems are characterized by equilibrium and metastable magnetic states

that correspond to global and local minima respectively of the total energy of the

system. The dynamical process by which the magnetization goes from one of such

state to another under the influence of a driving force - be it an external magnetic

field, temperature or a charge current - is called the magnetization dynamics. The

characteristic time involved in such process strongly depends on the type of excitation,

material parameters and the spatial dimensions of the magnetic system. As it turns

out, relatively simple models have been successfully used to describe the dynamics of

magnetic textures even with its non-uniform nature. This stems from the fact that the

magnetic texture non-uniformity (the domain wall width, skyrmion or vortex radius)

is fixed (and considered constant) by energies (anisotropy , exchange DMI) which

are much larger than the Zeeman energy from the applied field. In what follows, we

discuss different contributions to the motion of non-collinear magnetic textures. In

doing so, we make a very general presentation of field and current driven dynamics of

non-collinear textures to incorporate both bulk magnetic textures and thin films tri-

layers with broken inversion symmetry along the z direction, which are good playing

grounds for DMI, and/or spin-orbit torques.

To fully understand the interplay between field and current driven dynamics of

non-collinear magnetic textures, we consider a general equation of motion governing

the dynamics of any magnetic texture given by the extended LLG equation

dM

dt= −γM×Heff +

α

Ms

M× dM

dt+ T, (2.53)

50

where Heff is the effective field as discussed in Section 2.2.2, which in the case of thin

films with broken inversion asymmetry is given as

Heff =2A

µ0Ms

∇2m− D

µ0Ms

∇z ×m + (Hz + [Hu −NzMs]mz) z

+ (Hx −NxMsmx)x + (Hy −NyMsmy)y, (2.54)

A is the exchange stiffness, Hu = 2Kuµ0Ms

is the uniaxial anisotropy field, Hi and Ni

(i = x, y, z) are respectively the external applied magnetic field’s components and

the demagnetizing factors satisfying Nx + Ny + Nz = 1 with values that depends on

the geometry of the system and D is the strength of DMI. In such systems, the spin

torque acting on the magnetization M due to charge current has different sources as

discussed in Section 2.3 and is given as

T = − bJ

Ms

(u ·∇)M +βbJ

Ms

M× [(u ·∇)M] + γH ||

Ms

M× (y ×M). (2.55)

On right hand side (RHS) of Eq. (2.55), the first and second terms are the adiabatic

and non-adiabatic (with non-adiabaticity parameter β) torque respectively that the

itinerant electrons exerts and the last term is the damping-like torque contribution

due to spin-orbit torques [68, 98, 99, 100, 101, 102, 103, 104]

In the context of magnetic domain walls, the position of a domain wall can be

manipulated by the action of applied magnetic field, spin polarized current and even

through magnetostatic interactions with other domain walls [105, 106]. Impurities

and/or disorders which are omnipresent in real materials however suppress the action

of these driving forces and thus can influence the nature of the dynamics of the

magnetization. The current- and field-driven dynamics of magnetic domain walls

are usually parsed into several regimes, depending on the relative magnitude of the

applied force compared with disorder strength. For weak applied force, the wall moves

in the creep regime, where thermal activation plays a crucial role in the de-pinning

51

process, while for larger force it moves in the flow regime, essentially independent on

temperature. When the external force exceeds the intrinsic pinning of the domain

wall, the wall structure can undergo structural deformation, resulting in a turbulent

regime [107, 108].

The first means by which magnetic domain walls can be manipulated is through

the application of an external magnetic field that is usually included in the effective

field Heff given by Eq. (2.10) in the equation of motion of the magnetization given by

LLG equation Eq. (2.12). In confined nano-structures which is the focus of this thesis,

this external magnetic field can result in domain wall nucleation or motion. In the case

of motion, the domain wall moves in such a way that the Zeeman energy is minimized:

the magnetic domain anti-parallel to the external field shrinks while the one parallel

to the applied field grows in size. Therefore a change in field direction results in

a change in direction of the domain wall motion. In perpendicularly magnetized

ultrathin multilayers, the disorder is so strong that the domain wall is more likely

to move in the creep regime and large forces are needed to drive the wall into flow

and turbulent regimes [109, 110, 111, 112]. In the creep regime, the disorder is very

important and its magnitude is comparable to the driving force. The driving force f

has to overcome random energy barrier EB introduced by the disorders which results

in magnetization dynamics that follows a power law EB ∝ f−µ, where µ > 0 is a

creep universal exponent. Studies on the field-driven domain wall motion in metallic

ferromagnets in the creep regime, reveals that the velocity of the domain wall is given

by [109]

v = v0 exp(−κ0H

−µz /kBT

), (2.56)

where v0 and κ0 are normalization constants, kB is the Boltzmann constant, T is the

temperature, and Hz is the strength of the magnetic field. Here, the creep exponent

µ = 14, is in agreement with theoretical prediction in two-dimensional systems [113].

The second means by which magnetic domain walls can be manipulated is through

52

the transfer of angular momentum from the itinerant electrons flowing as a result of

charge current to the local magnetization. Unlike the field-driven domain wall motion,

there is hardly any consensus on the creep exponent µ for the current-driven domain

wall motion in which the reported effective energy barrier scales as EB ∝ J−µ, where

J is the current density. Some studies [110] have reported µ = 13

pointing to the notion

that current-driven and field-driven domain wall motion are quantitatively different.

Indeed, although the non-adiabatic component of current-induced spin transfer torque

is qualitatively very similar to the magnetic field as we shall see later in this thesis,

the adiabatic component of the spin transfer torque presents different properties that

could account for this difference in the creep exponent.

On the other hand, when the driving forces are sufficiently strong compared to

the disorders as is the case with some experiments studies [79, 114, 115, 78, 76, 112],

the domain wall motion is described in the context of the flow regime through the

extended LLG equation Eq. (2.53). To fully understand this process, we consider one-

dimensional perpendicularly magnetized Bloch/Neel wall with magnetization varying

along x which is going to be considered as the direction of current flow, i.e. u = x.

The magnetization direction can be written in spherical coordinates as m(x, t) =

(cos Φ sin θ, sin Φ sin θ, cos θ) with azimuth and polar angles given respectively as Φ =

Φ(t) and θ = 2 arctan(

exp[s (x−X(t))

λdw

]). X(t) represents the position of the domain

wall centre at time t, λdw is the domain wall width, s = ±1 defines the chirality of

the domain wall such that ↑↓ for s = +1 and ↓↑ for s = −1.

To make our analysis tractable, we first define some notation and important rela-

tions.

dm

dt=

∂θ

∂X

dX

dt

∂m

∂θ+ sin θ

dΦ

dt

∂m

∂Φ= −s sin θ

λdw

dX

dt

∂m

∂θ+ sin θ

dΦ

dt

∂m

∂Φ(2.57a)

(u ·∇)m = ∇xm =s sin θ

λdw

∂m

∂θ(2.57b)

53

where the unit vectors m,∂m∂θ

, and ∂m∂Φ

are such that

m · ∂m

∂θ= m · ∂m

∂Φ=∂m

∂θ· ∂m

∂Φ= 0 (2.58a)

m× ∂m

∂θ=∂m

∂Φ,∂m

∂θ× ∂m

∂Φ= m ,

∂m

∂Φ×m =

∂m

∂θ. (2.58b)

The dynamics of the domain wall amounts to finding the solutions X(t) and Φ(t) of

Eq. (2.53) under the action of the effective field given by Eq. (2.54) and the torque

given by Eq. (2.55). This can be achieved in two major steps

• Multiply Eq. (2.53) by m and project the resulting equation on the unit vectors

∂m∂θ

and ∂m∂Φ

• Integrate the resulting equations over space to obtain a coupled non-linear dif-

ferential equations that describe the dynamics of the domain wall centre X(t)

and the tilt angle Φ(t).

This procedure without going into much details of the analytics, yields the coupled

equations

dX

dτ= sα

(Hz −

sβbJ

γλdw

+π

2H|| cos Φ

)− sHxy

d sin 2Φ (2.59)

− bJ

γλdw

+sπ

2(Hx + sHDMI) sin Φ− sπ

2Hy cos Φ

dΦ

dτ=

(Hz −

sβbJ

γλdw

+π

2H || cos Φ

)+ αHxy

d sin 2Φ (2.60)

+αsbJ

γλdw

− απ

2(Hx + sHDMI) sin Φ +

απ

2Hy cos Φ.

In the expression, τ = γt/(1+α2), Hxyd = 1

2 (Nx−Ny)Ms is the in-plane demagnetizing

field, and HDMI = D2µ0Msλdw

is the DMI field. Before we proceed, let us make a couple

of interesting remarks on Eqs. (2.60) and (2.59).

54

• The DMI acts as an effective in-plane magnetic field that depends on the chi-

rality of the structure.

• The damping-like torque influences the dynamics of Neel walls only and is gen-

erally more efficient than the spin transfer torque.

• The non-adiabatic torque and the z-component of the applied magnetic field

have a qualitatively similar effect as they both move the domain wall. However,

current-driven dynamics differ slightly from field driven dynamics because in

addition to the non-adiabatic torque, the adiabatic torque involved with current-

driven dynamics distorts the domain wall structure.

The application of a weak magnetic field leads to domain wall velocity that increases

linearly with the magnetic field/current density until a critical field is reached - the

Walker breakdown field/current density HW/JWB [116]. This motion can be pictured

as solution of Eqs (2.59) and (2.60) for which ∂τΦ = 0 (domain wall structure remains

rigid) i.e. the domain wall remains in a Neel or a Bloch wall state and moves with a

steady-state velocity given as

dX

dt= s

λdw

αγ

(Hz −

sβbJ

γλdw

+π

2H || cos Φ

). (2.61)

Above the Walker breakdown field/current density, due to stronger precession

of the magnetization about the effective field, the domain wall structure starts to

undergo periodic structural changes and eventually oscillates from one structure to

another this leads to an overall decrease of velocity as a function of the applied

field/current density. The onset of this process (dΦdτ6= 0) defines the Walker breakdown

field (HWB) and current density (JWB), which, in the absence of damping-like torque

55

Figure 2.9: (Color online). Illustration of domain wall dynamics under the combinedaction of current and field.

and DMI, read

H±WB = ±(α|Hxy

d | ∓ sα− βγλdw

bJ

)(2.62a)

and

J±WB = ±(eγMs

µBPsλdw

α− β[α|Hxy

d | ∓Hz]

). (2.62b)

The process of structural change is periodic, therefore the new domain structure

eventually starts to accelerate again until the next structural transformation oc-

curs [See Figure 2.9]. It is worth noticing here that the existence of a finite Walker

breakdown field or current density is determined by the in-plane demagnetizing field

Hxyd = 1

2 (Nx − Ny)Ms and therefore for cylindrical nano-wires for which Nx = Ny,

Hxyd = 0, we expect no such breakdown resulting in massless domain wall motion

[117]. More interestingly, in antiferromagnetic domain walls the absence of stray

fields will also result in no Walker breaker field/current density [118, 119, 120].

56

2.4.2 Two-dimensional magnetic textures

Statics

Vortex The vortex configuration is widespread in nature, and can be found in vari-

ety of systems ranging from the order parameter in type II superconductors, density

of matter in blackholes to two-dimensional ferromagnets. All these systems are char-

acterized by a physical parameter that curls around a central singularity called the

vortex core. The very first micromagnetic studies on the structure of magnetic vor-

tices was carried out by Feldtkeller et. al. [121]. This paved the way to further stud-

ies notably by Kosterlitz et. al [122], who could stabilize vortex states under certain

conditions using an XY model that describes the interaction of classical magnetic mo-

ments confined in a plane. They interpreted these states as topological defects whose

structure cannot be reduced to a uniform or continuous state by any finite transfor-

mation. In magnetically soft materials, the exchange length lex =√A/2µ0M2

s is only

a few nanometers. Therefore, since the exchange length defines the typical length

scale below which magnetic bodies are expected to behave as a single uniform do-

main, different magnetic textures can be stabilized depending on the dimensions and

shape of the sample. These structures are direct consequence of competition between

different magnetic interactions notably, the dipole-dipole and the exchange interac-

tions. Indeed, studies with submicron disk-shaped (magnetic nano-dots) Permalloy

(Ni80Fe20) reveal the stabilization a vortex state that ensures the absence of both edge

surface and volume magnetic charges [123].

It turns out that magnetic nanodots with vortex as their ground state show poten-

tial application as high density magnetic storage, nonvolatile MRAM and STT-RAM

[]. Indeed, its conserved quantities can be associated with bit of information such as

the chirality-sense of in-plane magnetization rotation and the polarity-sense of magne-

tization of the vortex core. In addition to its important role in magnetization reversal

57

[124, 125], different techniques such as time-resolved Kerr miscrocopy [126], Brillouin

light scattering by spin waves [127], Micro-SQUID [128], and X-ray imaging [129] have

all been used with great success to observe high frequency dynamical properties of the

vortex state. Owing to its central role in spintronics, the spin-torque effect has been

widely been studied in non-linear magnetic structures such as vortex. These studies

ranges from the excitation of circular vortex motion by an AC [130, 131, 132] or a DC

[133] spin-polarized current, to the theoretical prediction[134, 135] and experimental

observation [81] of the control of vortex polarity using a spin-polarized current.

Skyrmions In the 1960s, British physicist Tony Skyrme proposed skyrmions as a

topological solution of the 3D non-linear sigma model to account for the stability of

hadrons in particle physics [136]. Skyrme’s hypothesis stemmed from the fact that in

field theory, particles are defined as wave-excitation of the field that cannot be changed

by any deformation of the field configuration or in mathematical terms, characterized

by a topological integer that remains unchanged (topologically protected). In the

context of magnetism, a magnetic skyrmion is the smallest possible perturbation

to a uniform magnet that gives rise to an emergent electrodynamics that cannot be

described simply using the Maxwell’s equations. [137, 138] Skyrmions are very stable,

very small (few nanometers in size), and can be efficiently manipulated with spin-

polarised current with minimal dissipation making them a very good candidate to

store and process information with unprecedentedly high densities and low energy

consumption [59, 30]. The small size of magnetic skyrmions had make it very difficult

to be observe in experiments. Recently, using a high spatial resolution magnetically

sensitive scanning tunnelling microscopy technique, Romming et. al. [139] were able

to characterise the internal structure of magnetic skyrmions in a metal film.

There are severals means of creating magnetic skyrmions including from a notch

[31], from spin currents in nanodisk [140, 141], using ultrashort optical laser pulse

58

[142] and from domain wall pair [143, 37]. Skyrmions have been observed in the B20

crystal lattice of MnSi [28] and Fe0.5Co0.5Si [29] characterized with broken inversion

symmetry and strong spin-orbit coupling. In these materials the competition between

the Heisenberg exchange that favours the parallel alignment of neighbouring spins

and Dzyaloshinkii-Moriya interaction that favours the perpendicular alignment of

neighbouring spins results in the stabilization of a skyrmionic texture with definite

chirality. Skyrmions have also been observed in multi-layer thin films structure of a

heavy metal with strong spin-orbit coupling and a ferromagnet in which interfacial

inversion symmetry provides the necessary DM interaction to stabilize a skyrmionic

texture, as is the case of palladium-iron (PdFe) bilayer grown on a (111) iridium

surface [139].

These two-dimensional topological textures are characterized by three topological

integers :

• Vorticity (q = ±1): also called the winding number that measures the total

angle through which m rotate in a circular contour around the core [144] hence

directly related to the topology of the magnetic moment curling in the XY

plane.

• Polarity (p = ±1): defines the relative orientation of the magnetic moments

inside the vortex/skyrmion core with respect to the plane of the ferromagnet.

• Chirality (c = ±1): gives the sense of rotation of m in a circular contour around

the vortex/skyrmion that can rotate clockwise (c = +1) or counterclockwise

(c = −1).

Through out this thesis, these magnetic textures are given in spherical coordinates by

m = (cos Φ sin θ, sin Φ sin θ, cos θ), where Φ and θ are the azimuth and polar angles

respectively. We adopted the profile of skyrmions and vortices such their azimuth

59

angle are both given by

Φ(x, y) = qArg(x+ iy) + cπ

2, (2.63)

with the difference between these textures arises from the polar anlge. For an isolated

magnetic skyrmion the polar angle is given [145, 146]

cos θ = pr2

0 − r2

r20 + r2

, (2.64)

while for an isolated vortex it is given as

cos θ =

pr20−r2

r20+r2 , r ≤ r0

0, r0 < r ≤ R(2.65)

where r0 is the radius of the skyrmion/vortex core.

Dynamics

Magnetic vortices and skyrmions are characterized by very rich magnetization dy-

namics, and in this section, we present a very general description of the dynamics of

these topologically non-trivial textures. We consider the dynamics of a ferromagnetic

nano-disk containing an isolated magnetic vortex/skyrmion on top of a heavy metal

with strong spin-orbit coupling such that there is broken inversion symmetry at the

interface along the z direction. As mentioned above, this condition presents a good

play ground to study the influence of both spin Hall effect and DMI on the dynamics

of these textures. Our consideration are based on the Thiele’s formalism of general-

ized forces acting on rigid magnetic structures [147], which amounts to assuming that

the derivative of the magnetization vector m satisfies

m = −(v ·∇)m = −vx∂xm− vy∂ym, (2.66)

60

v is the velocity of the structure’s centre. Similar to the dynamics of one-dimensional

textures described in Section 2.4.1, the equation of motion governing their dynamics

is given by Eq. (2.53) under the action of the effective field and spin torque given

respectively by Eqs. (2.54) and (2.55). This can be achieved as follows

• Multiply Eq. (2.53) by m×

• Project the resulting equation on two mutually orthogonal vectors ∂xm and

∂ym

• Integrate the result over the volume of the magnetic strip Ω to a balance equa-

tion of the forces acting on the structure from which the velocity components

can be extracted.

This procedure in the absence of external magnetic field yields

∫Ω

[G× (v − bJx) + D · (αv + βbJx)] dΩ = 0, (2.67)

where

G = sin θ [∇θ ×∇Φ] , (2.68)

and

Dij = ∂iθ∂jθ + sin2 θ∂iΦ∂jΦ. (2.69)

After some algebra the longitudinal and transverse velocities are given as

vx = −(1 + αβ)

1 + α2bJ (2.70a)

vy = −pq (β − α)

1 + α2bJ (2.70b)

61

for an isolated magnetic skyrmion and

vx = −(1 + C2αβ)

1 + C2α2bJ (2.71a)

vy = −pqC (β − α)

1 + C2α2bJ , (2.71b)

for an isolated vortex structure, where C = 1 + 0.5 ln(R/r0).

2.4.3 Antiferromagnetic textures

In ferromagnets, the formation of domain walls, vortices and to a lesser extend

skyrmions is due to competition between the macroscopic dipole-dipole interactions

and the microscopic exchange interaction as discussed in Sections 2.4.1 and 2.4.2.

Due to the absence of stray fields in homogeneous antiferromagnetic materials be-

cause of zero equilibrium magnetization, it is not obvious to expect similar magnetic

textures for antiferromagnets. Furthermore, even if antiferromagnets do exhibit simi-

lar magnetic textures, other mechanisms aside from competition of the exchange and

dipole-dipole interactions should be responsible for their existence. Indeed, antiferro-

magnetic domain walls do exist in crystals and there are prospect of antiferromagnetic

vortices and skyrmions [4, 148, 149]. However, other defects such as grain boundaries,

dislocations and crystallographic twins have been suggested to be responsible for the

stabilization of these textures [150]. It has also been suggested that growing an

antiferromagnetic layer on top of a ferromagnetic layer forces the spins of the anti-

ferromagnet to be align along the easy axis of the ferromagnet due to exchange bias

effect [151], and therefore resulting in antiferromagnetic domain wall. However, over

the past decades, researchers encountered enormous difficulties to observe antiferro-

magnetic textures largely due to the then available experimental techniques which

relied mainly on neutron diffraction [152] and thus were unable to sense the antiferro-

magnetic order parameter. This information is readily available now through experi-

62

mental techniques based on X-ray magnetic linear dichroism (XMLD) [153, 154, 155]

which is able to detect the direction of antiferromagnetic order parameter from the

absorption spectrum. The observed antiferromagnetic domain walls vary from few

lattice constants in monolayer antiferromagnetic Fe [58] and Cr [156] to about 100nm

in NiO antiferromagnetic insulators [157].

Two methods for antiferromagnetic skyrmion creation have been proposed recently

based on micromagnetic lattice simulations: (i) the conversion of pair of antiferromag-

netic domain walls in nano-wires junction and (ii) injection of vertical spin-polarised

current to a nano-disk with an antiferromagnetic ground state [4].

Dynamics of antiferromagnetic textures

In this section, we present the theoretical scheme to study the dynamics of antiferro-

magnetic textures [158, 159, 160, 161]. Our derivation is based on a staggered G-type

magnetic textures. Let mi be the local magnetic moment at site i, then any local

non-equilibrium spin density generated regardless of its source (i.e. applied current,

external field or even magnetization dynamics) can in general be decomposed into

Si ≈ Si0mi + Sina∂xm

i + Siadmi × ∂xmi, (2.72)

where S0i (Siad, S

ina). Since this local non-equilibrium spin density is not totally

aligned along the local magnetization mi, this will set the local moments into preces-

sion which results in a local torque on the local moments given as

Ti ∼ Si ×mi = −γJ iad∂xmi + γJ inam

i × ∂xmi, (2.73)

where γ is the gyromatic ratio. The first term on the right hand side of Eq. (2.73) is

the adiabatic contribution to the torque while the second term is the non-adiabatic

torque contribution. For a crystal lattice in which each unit cell contains two equiv-

63

alent magnetic sites a and b, we can define the total magnetization and the anti-

ferromagnetic Neel order parameter as m(r, t) = ma(r, t) + mb(r, t) and l(r, t) =

ma(r, t)−mb(r, t) respectively. In the absence of magnetic fields and at equilibrium,

m = 0 while l is finite and homogeneous.

The dynamics of this system can then easily be modelled using the coupled ex-

tended LLG equation in the presence of spin transfer torque given as

∂tma = −γma ×Ha

eff + αama × ∂tma + Ta (2.74a)

∂tmb = −γmb ×Hb

eff + αbmb × ∂tmb + Tb, (2.74b)

where αi is the Gilbert damping constant for the i-sublattice, Hieff is the effective field

felt by the i−sublattice and Ti is the spin torque exerted on the i−sublattice given by

Eq. (2.73). To proceed, we re-write ma and mb in terms of the total magnetization

m and n(r, t) = l(r, t)/|l(r, t)|, i.e m = ma + mb and n = ma −mb such that the

sub-lattice torques are given as

Ta = −γJaad∂xn+γJanan×∂xn−γJaad∂xm+γJanam×∂xn+γJanan×∂xm+γJanam×∂xm

(2.75a)

Tb = γJ bad∂xn+γJ bnan×∂xn−γJ bad∂xm−γJ bnam×∂xn−γJ bnan×∂xm+γJ bnam×∂xm.

(2.75b)

Using Eq. (6.14), we obtain the equation of motion for m and n corresponding to

Eq. (6.13) as

∂tm =[γfn − αn∂tn− γJ+

na∂xn− γJ−adn× ∂xn]× n + τ nl

m (2.76a)

∂tn =[γfm − αm∂tm− γJ−na∂xn− γJ+

adn× ∂xn]× n + τ nl

n , (2.76b)

where αm and αn are damping constant corresponding to m and n respectively,

64

J±ad = 12(Jaad ± J bad), J±na = 1

4(Jana ± J bna). The torques terms τ nl

m and τ nln on the right

hand side of Eq. (2.76) given as

τ nlm = γfm×m−γJ+

ad∂xm+γJ−nam×∂xn+γJ−nan×∂xm+γJ+nam×∂xm−αm+∂tm×m

(2.77)

and

τ nln = γfn×m−γJ−ad∂xm+γJ−nam×∂xm+γJ+

nam×∂xn+γJ+nan×∂xm−αn+∂tn×m

(2.78)

respectively are non-linear and will be neglected throughout our analysis. In Eq.

(2.76), the effective fields fn(m) are derived from the free energy with leading order

[53]

F =

∫dr

[a

2m2 +

A

2(∂2xn + ∂2

yn + ∂2zn)−H ·m

](2.79)

as

fm =1

2(Heff

a + Heffb ) = −δmF = −am + n× (H× n) (2.80a)

fn =1

2(Heff

a −Heffb ) = −δnF = An× (∇2n× n)− n× (m×H). (2.80b)

Using simple vectorial calculus, we can solve for the total magnetization m from Eqs.

(2.76a) and (6.16a) in terms of the n as

m =1

a

[n×H +

1

γ∂tn− αmfn + (J+

ad + αmJ+na)∂xn− (J−na − αmJ−ad)n× ∂xn

]× n,

(2.81)

where γ = γ/(1 + αmαn). The substitution of Eq. (2.81) into Eq. (2.76b) yields the

equation of motion of the Neel order parameter for antiferomagnetic textures which

up to linear order in m, n, H, Jad and Jna is given as [159]

1

γ∂2t n = a

[γfn − αn∂tn− γJ+

na∂xn]− n× ∂tH + fnl, (2.82)

65

where

fnl = −aγJ−adn× ∂xn + αm∂tfn − (∂tJ+ad + αm∂tJ

+na)∂xn + (∂tJ

−na − αm∂tJ−ad)n× ∂xn.

(2.83)

We neglect fnl for the rest of this thesis.

Eq. (6.17) is a second order differential for n, and shows that the dynamics of

the Neel order parameter is fundamentally different from the dynamics of the mag-

netization in ferromagnetic textures described by the first order differential equation

given by Eq. (2.53). If we consider n to depend on the coordinate of the centre of

the texture i.e. n(r, t) = n(r−X0(t)), where X0(t) = X(t)x + Y (t)y, Eq. (6.17) can

be transformed using

∂tn = −∂tX∂xn− ∂tY ∂yn and ∂2t n = −∂2

tX∂xn− ∂2t Y ∂yn +O([∂tX]2) +O([∂tY ]2)

(2.84)

into

1

aγγ

(∂2tX∂xn + ∂2

t Y ∂yn)

+αn

γ(∂tX∂xn + ∂tY ∂yn) = J+

na∂xn +1

aγn× ∂tH− fn.

(2.85)

And finally, the projection of Eq. (2.85) on two orthogonal vectors ∂xn and ∂yn and

integrating over the volume Ω, to obtain

M(∂2t X0 + aγαn∂tX0

)= F , (2.86)

where M is a tensor of the effective mass of the texture with components given as

Mij =1

aγγ

∫Ω

dΩ∂in · ∂jn, (2.87)

66

and the force acting on the texture F is given by

Fi =

∫Ω

dΩ(J+na∂xn +

1

aγn× ∂tH− fn) · ∂in. (2.88)

From Eq. (2.86), the dynamics of a rigid antiferromagnetic texture can be considered

as an inertial quasi particle with an effective mass that obeys Newton’s second law.

2.5 Theoretical models

In this thesis, we have investigated the nature of spin transport in magnetic textures.

In this section, we present the theoretical framework in which these models have been

developed.

2.5.1 The s-d model

Even though the electronic structures of transition metal ferromagnets are very com-

plicated, characterized by hybridization and spin-dependent band structures from

electrons in the s, p, and d bands, there exist elegant transport theories that capture

the basic physical mechanisms involved. Proposed originally by Mott [162], the s-d

model is now omnipresent in spin transport theory and in particular domain walls

transport models.

In the s-d representation, the electronic structure is made up of two bands of s

and d type. In this picture, the exact form of the d -band is irrelevant and is char-

acterized by electrons with large effective mass and in principle considered localized

with negligible contribution to the transport of the material. The d -electrons fills up

the up and down sub-band with respect to some quantization axis that defines the

majority and minority spin direction, and thereby the net magnetization represented

by a classical vector M(r) of the material. On the other hand, the s band described

by a simple parabolic dispersion characterized by electrons with low effective mass

67

Figure 2.10: Illustration of the spin-dependent band structure of the s-d model. Thes band has a parabolic dispersion relation with a spin-split energy Jsd. The d bandon its part is narrower with a significant difference in the number of up and downstates at the Fermi energy.

are considered itinerant. In general, the s-electrons are treated quantum mechani-

cally, while the d -electrons are only accounted for through their overall magnetization

vector. Therefore, the s-electrons are represented by the spin operator s = ~2σ and

are responsible for the transport of the material.

Due to the isotropic exchange interaction, the itinerant s-electrons couple to the

localized d -electrons represented by the s-d Hamiltonian given as

Hsd = − Jsd

2Ms

M(r) · σ, (2.89)

Jsd is the spin-split energy gap between up and down s sub-bands. Figure 2.10 is a

schematic illustration of the band structure for the s-d model.

68

2.5.2 Tight-binding model

The tight-binding model is a standard technique developed by Bloch in 1928 [54] and

often used to obtain a fast and fairly good electronic structure calculation of various

quantum systems. The popularity of this method stems from its efficiency in large

simulations and ability to provide first level understanding of electronic behavior of

physical systems. In the tight-binding approach, the wave functions are written as

linear combination of localized independent atomic orbitals with fixed energy associ-

ated to each state. In such a representation, an appropriate form of the Hamiltonian

is assumed which is readily constructed and diagonalized. The Hamiltonian basically

comprises the on-site and overlap matrix elements which represent the on-site energy

and the strength of the interaction between electrons from one atom and those they

are bound to (usually first few neighbouring atoms). The beauty of such a method is

that only the symmetry and not the exact form of the orbitals is relevant. The values

of the orbital overlap and onsite energies are obtained from experiment or fits from

first principle simulations [163].

In this section, we present the numerical method used in the calculation of non-

equilibrium quantities based on the KWANT code [164] such as charge and spin

densities/currents in as well as spin-resolve conductance for non-collinear magnetic

textures. To numerically calculate these quantities within the Landauer-Buttiker

formalism, we consider a finite scattering region attached to periodic semi-infinite

leads that acts as wave guides that lead plane waves into and out of the scattering

region. The Hamiltonian of such a system can be written as

H =∑ij

Hijc†icj, (2.90)

where Hij are elements of an infinite Hermitian matrix, c†i (cj) are the fermionic cre-

ation (annihilation) operators, i and j represent the degrees of freedom of the system

69

Figure 2.11: (Color online) . Schematic diagram of four terminal system made up ofa central scattering region (blue boxed) attached to four leads L, T, R, B at chemicalpotentials µL, µT , µR and µB respectively. The scattering region contains an isolatedmagnetic skyrmion while the four leads are ideal ferromagnet/antiferromagnet toinsure smooth magnetisation variation from the leads to the scattering region. Inthis set-up, a charge bias is added to lead-L and lead-R while the induced transversecharge or/and spin current is probed in lead-T and lead-B.

and take the form i = r, α, where r is the lattice coordinates, and α represents internal

degrees of freedom which can include for example spin, atomic orbitals, and Nambu

electron-hole degree of freedom for superconductivity. Therefore, the tight-binding

Hamiltonian in Eq. (2.90) can be expressed in first quantization as the sum of the

Hamiltonian fragments

Hrr′ =∑αα′

Hrαr′α′ |rα〉〈r′α′| (2.91)

that couple site r and r’. The scattering problem is solved within the wave function

formulation that is mathematically equivalent to the non-equilibrium Green’s function

formalism by virtue of Fisher-Lee relation [165].

70

Topological Hall angle

Here, we present the scheme used in Chapters 5 and 6 for the calculation of the ter-

minal voltages and their corresponding terminal currents which is essential for quan-

tifying the Hall effect associated with the topology of the magnetic texture otherwise

called Topological Hall Effect (THE). To achieve this, we consider a four-terminal

system as shown in Figure 2.11 in which a longitudinal charge bias is added to leads

L and R and the induced transverse charge or/and spin current is probed using leads

T and B. The terminal charge current of the m-lead, where m ∈ (L, T,R,B) is

calculated as

Iem =e2

2π~∑

n 6=m,σ,σ′

(T σ

′σnm Vm − T σσ

′

mn Vn

)(2.92)

where T σ′σ

nm is the probability of electron from the m-lead with spin σ to be transmitted

to the n-lead with spin σ′ which is explicitly given as

IeL =e2

2π~∑σ,σ′

([T σ

′σTL + T σ

′σRL + T σ

′σBL ]VL − T σσ

′

LT VT − T σσ′

LR VR − T σσ′

LB VB

)(2.93a)

IeT =e2

2π~∑σ,σ′

(−T σσ′

TL VL + [T σ′σ

LT + T σ′σ

RT + T σ′σ

BT ]VT − T σσ′

TR VR − T σσ′

TB VB

)(2.93b)

IeR =e2

2π~∑σ,σ′

(−T σσ′

RL VL − T σσ′

RT VT + [T σ′σ

LR + T σ′σ

TR + T σ′σ

BR ]VR − T σσ′

RB VB

)(2.93c)

IeB =e2

2π~∑σ,σ′

(−T σσ′

BL VL − T σσ′

BT VT − T σσ′

BR VR + [T σ′σ

LB + T σ′σ

TB + T σ′σ

RB ]VB

)(2.93d)

or equivalently in matrix representation as

IeL

IeT

IeR

IeB

=

e2

2π~T

VL

VT

VR

VB

, (2.94)

71

where

T =∑σ,σ′

[Tσ′σ

TL + Tσ′σ

RL + Tσ′σ

BL ] −Tσσ′

LT −Tσσ′

LR −Tσσ′

LB

−Tσσ′

TL [Tσ′σ

LT + Tσ′σ

RT + Tσ′σ

BT ] −Tσσ′

TR −Tσσ′

TB

−Tσσ′

RL −Tσσ′

RT [Tσ′σ

LR + Tσ′σ

TR + Tσ′σ

BR ] −Tσσ′

RB

−Tσσ′

BL −Tσσ′

BT −Tσσ′

BR [Tσ′σ

LB + Tσ′σ

TB + Tσ′σ

RB ]

,

(2.95)

The linear system described by Eq. 2.94 is not independent and therefore the asso-

ciated matrix T is not invertible. We however observe that since these equations are

dependent, we can without loss of generality set one of the voltage VB = 0 to obtain

a reduced invertible system given as

IeL

IeT

IeR

=e2

2π~T R

VL

VT

VR

, (2.96)

where

T R =∑σ,σ′

[T σ

′σTL + T σ

′σRL + T σ

′σBL ] −T σσ′

LT −T σσ′LR

−T σσ′TL [T σ

′σLT + T σ

′σRT + T σ

′σBT ] −T σσ′

TR

−T σσ′RL −T σσ′

RT [T σ′σ

LR + T σ′σ

TR + T σ′σ

BR ]

,

(2.97)

or equivalently VL

VT

VR

=2π~e2R

I

0

−I

, (2.98)

where R = T R−1, IL = −IR = I and IT = 0. From which we straight-forwardly

obtain the expression for the terminal voltages as

VL =R11 −R13

R11 +R33 −R13 −R31

µL − µRe

(2.99a)

72

VT =R21 −R23

R11 +R33 −R13 −R31

µL − µRe

(2.99b)

VR =R31 −R33

R11 +R33 −R13 −R31

µL − µRe

(2.99c)

VB = 0, (2.99d)

and their corresponding terminal currents as

IeL =e2

2π~[T11VL + T12VT + T13VR] (2.100a)

IeT =e2

2π~[T21VL + T22VT + T23VR] (2.100b)

IeR =e2

2π~[T31VL + T32VT + T33VR]. (2.100c)

Finally, we calculate the Topological Hall angle θTH as

θTH = tan−1

(VT − VBVL − VR

). (2.101)

Topological spin Hall angle

The calculation of the Topological spin Hall angle is a little more tricky. First no-

tice that in the calculation of the Topological Hall angle presented in the previous

section, we sum over of all possible spin states in Eq. 2.94 and the equations there

after. What this means is that we do not need to keep track of spin state of the in-

jected and measured current. However, it is crucial to keep record of the spin state of

the injected and measured current to correctly calculate the spin current. Therefore

spin resolved transmission coefficients are empirical in this case. We achieved this

by defining separate sub-lattices for up and down spin species such that while the

texture mixes/couple both sub-lattices, the leads which are pure ferromagnetic/anti-

ferromagnetic remain de-coupled. This way we are able to calculate the spin-resolved

73

transmission coefficients and hence the spin currents. First we define the quantities

T inmn = T ↑↑mn + T ↑↓mn − T ↓↓mn − T ↓↑mn (2.102a)

T outmn = T ↑↑mn + T ↓↑mn − T ↓↓mn − T ↑↓mn (2.102b)

From which the terminal spin currents are defined as [166, 167]

Ism =e

4π

∑n6=m

(T outnmVm − T in

mnVn)

(2.103)

i.e.

IsL =e

4π

([T out

TL + T outRL + T out

BL ]VL − T inLTVT − T in

LRVR − T inLBVB

)(2.104a)

IsT =e

4π

(−T in

TLVL + [T outLT + T out

RT + T outBT ]VT − T in

TRVR − T inTBVB

)(2.104b)

IsR =e

4π

(−T in

RLVL − T inRTVT + [T out

TR + T outLR + T out

BR ]VR − T inRBVB

)(2.104c)

IsB =e

4π

(−T in

BLVL − T inBTVT − T in

BRVR + [T outTB + T out

RB + T outLB ]VB

). (2.104d)

We then proceed to calculate the terminal voltages and spin current using the same

scheme used above for Topological Hall angles which entails a simple replacement of

the matrix T

Ts =

[T outTL + T out

RL + T outBL ] −T in

LT −T inLR −T in

LB

−T inTL [T out

LT + T outRT + T out

BT ] −T inTR −T in

TB

−T inRL −T in

RT [T outLR + T out

TR + T outBR ] −T in

RB

−T inBL −T in

BT −T inBR [T out

LB + T outTB + T out

RB ]

,

(2.105)

74

and follow the same scheme to calculate the terminal spin currents. From the terminal

voltages, spin and charge currents, we can calculated Topological spin Hall angle as

θTSH =2e

~

(IsT − IsBIeL − IeR

)(2.106)

Local spin/charge density and current

In this section, we present the scheme used in Chapters 5 and 6 for the calculation of

local spin density and spin current density. The local spin density Sαn at site n and

local spin current density Jαn−1 between site n − 1 and n at a particular transport

energy for electrons from the α-lead can be calculated from their respective operators

Sαn =~2

∑ν

(Ψα↑∗nν ,Ψ

α↓∗nν)σ(Ψα↑

nν ,Ψα↓nν)

T (2.107)

and

Jαn−1 =1

2i

∑ν

tnν,n−1ν(Ψα↑∗nν ,Ψ

α↓∗nν)σ(Ψn−1ν ,

α↑Ψα↓n−1ν)

T + c.c, (2.108)

where Ψασnν are the ν-th propagating mode with spin-σ originating from the α-lead at

site n. The quantum mechanical average is calculated by integrating over the small

energy window εF − eV2 and εF + eV

2 . Such an integration is important to smear out

the spatial oscillations due to quantum interferences [167]. Hence, the spin density

at site n reads

〈Sn〉 =∑α

∫dε

2πfαS

αn, (2.109)

where fα is the Fermi-Dirac function for lead-α. A similar formula applies for 〈Jn−1〉.

The corresponding charge density and current density can be obtained from Eqs.

(5.15) and (5.16) by replacing ~2σ by eI where I is the 2× 2 identity matrix.

75

2.5.3 Spin diffusion model

In real materials, impurities are omnipresent and in the limit of strong disorder, the

transport of electrons through magnetic textures can be described using the drift-

diffusion model which strictly speaking is valid for texture non-uniformities (domain

wall widths or radius of skyrmion/vortex cores) small compared to the electron mean

free path. This model forms the basis of our studies in Chapters 3 and 4. Here,

we present an overview of this model for magnetic textures. Let us consider the

Hamiltonian

H =p2

2m+ Jsdm · σ, (2.110)

where Jsd is the s-d exchange interaction between the local moments and the itinerant

electrons. The application of Enhrefest theorem to Eq. (2.110) yields

∂ts =∂〈σ〉∂t

=1

i~〈[σ,H]〉 = − 1

τex

s×m. (2.111)

From Eq. (3.8), the semi-classical Bloch equation for the itinerant spin density s is

given as

∂ts + ∇ · J +1

τex

s×m = −Γre, (2.112)

where J = −D∇⊗ s, is the spin current density with D = v2F τ/3 being the diffusion

constant while Γre captures effects due to the spin relaxation and dephasing. The

spin and spin density can be written in the form s = nsm + δs, where ns (δs) is

(non-)equilibrium spin density. This representation corresponds to writting the (non-

)equilibrium spin current density as J0 = −bJu ⊗m (δJ = −D∇ ⊗ δs) where u is

the direction of current flow and bJ = µBPG0E/e. This transform Eq. (3.9) into

−D∇2δs +1

τex

δs×m +1

τϕm× (δs×m) +

1

τsf

δs = bJ(u ·∇)m, (2.113)

76

where the term on the RHS of Eq. (3.10) is the driving source for the generated

non-equilibrium spin density. In this case, the source is the apply current flowing

through the system. In Chapter 3 we show that spin pumping resulting from time

varying magnetic textures can also act as non-equilibrium spin density source. In

writing down Eq. (3.10) we have used the relaxation time approximation with

Γre =1

τϕm× (δs×m) +

1

τsf

δs. (2.114)

Eq. (3.10) is the stationary-state drift-diffusion equation for itinerant electrons flow-

ing in a magnetic texture. The solution of the non-equilibrium spin density from this

equation induces a spin transfer torque on the local magnetization which will be the

topic of Chapters 3 and 4.

77

Chapter 3

Phenomenology of chiral damping

A phenomenology of magnetic chiral damping is proposed in the context of magnetic

materials lacking inversion symmetry. We show that the magnetic damping tensor

acquires a component linear in magnetization gradient in the form of Lifshitz invari-

ants. We propose different microscopic mechanisms that can produce such a damping

in ferromagnetic metals, among which local spin pumping in the presence of anoma-

lous Hall effect and an effective ”s-d” Dzyaloshinskii-Moriya antisymmetric exchange.

The implication of this chiral damping in terms of domain wall motion is investigated

in the flow and creep regimes.

3.1 Introduction

Understanding energy relaxation processes of fast dissipative systems at the nanoscale

is of paramount importance for the smart design and operation of ultrafast nano

devices. In this respect, magnetic heterostructures have drawn increasing enthusi-

asm in the past ten years with the sub-picosecond optical control of magnetic order

[168, 169] and the promises of ultrafast domain wall motion in asymmetric metallic

multilayers [98, 100]. In fact, the observation of ultrahigh current-driven velocity

in ultra-thin multilayers has come as a surprise and triggered intense investigations

on the physics emerging from symmetry breaking in strong spin-orbit coupled mag-

78

nets [170, 171, 172, 173]. These studies have unravelled the essential role played by

Dzyaloshinskii-Moriya interaction [56, 55] (DMI), an antisymmetric exchange inter-

action that emerges in magnets lacking spatial inversion symmetry. This interaction

forces neighboring spin to align perpendicular to each other and competes with the fer-

romagnetic exchange resulting in distorted textures such as spin spirals or skyrmions,

as observed in bulk inversion asymmetric magnets [59] as well as at the interface of

transition metals [174, 175, 176, 177, 178]. In perpendicularly magnetized domain

walls, this interaction favors Neel over Bloch configuration [179, 180, 181, 182], a key

ingredient explaining most of the thought-provoking observations reported to date

[68]. The dynamics of spin waves can also be modified by DMI, which distorts the

energy dispersion [183] and results in a relaxation that depends on the propagation

direction [184, 185].

A crucial aspect of fast dynamical processes is the nature of the energy relaxation.

In the hydrodynamic limit of magnetic systems, this dissipation is written in the form

of a non-local tensor [see Eq. (3.1)] whose complex physics is associated with a wide

variety of mechanisms such as many-magnon scattering [186] and itinerant electron

spin relaxation [187, 188, 189]. Since the energy relaxation rate of spin waves depends

on their wave vector (the higher the spin wave energy, the stronger its dissipation),

the magnetic damping of smooth magnetic textures (i.e. in the long wavelength limit

q) depends on the inverse of the exchange length, q ∼ 1/λdw [190]. In inversion sym-

metric systems, this results in a correction to the magnetic damping of the order of

1/λ2dw [191, 192, 193, 194, 195]. However, in magnetic systems lacking inversion sym-

metry such as the systems in which DMI is observed, the energy dissipation becomes

chiral: a component linear in the magnetization gradient emerges thereby fulfilling

Neumann’s principle stating that ”any physical properties of a system possesses the

symmetry of that system”.

In the present work, we explore the nature of the magnetic damping in non-

79

centrosymmetric magnets and propose different physical mechanisms resulting in the

emergence of a chiral damping. This chiral damping significantly impacts the motion

of magnetic domain walls in both flow and creep regimes, opening appealing avenues

to solve recent puzzling observations [196, 197].

3.2 Symmetry considerations

The equation of motion governing the dynamics of continuous magnetic textures is

given by Landau-Lifschitz-Gilbert (LLG) equation

∂tm = −γm×Heff + m(r)×∫dr′α(r, r′)∂tm(r′), (3.1)

where m(r, t) = M(r, t)/Ms is a unit vector in the direction of the magnetization

M(r, t), γ is the gyromagnetic ratio, Heff is the effective field incorporating the ex-

ternal applied, anisotropy, exchange, DMI and demagnetizing fields, and α(r, r′) is

the magnetic damping expressed as a non-local second-rank tensor. In the limit of

smooth textures, the tensorial components of the damping is a function of the mag-

netization direction and of its spatial gradients, αij = αij(m,∇m). Performing an

expansion up to the first order in magnetization gradient yields [198]

αij = αij0 +∑lm

Kijlmmlmm +

∑klm

Lijklmmk∂lmm. (3.2)

The first term is the isotropic damping, the second term amounts for the magnetocrys-

talline anisotropy and the third term is the chiral damping. Only terms bilinear in

magnetization direction mi, i.e. even under time reversal symmetry, are retained in

the expansion. Spatial inversion symmetry breaking imposes the third term of Eq.

(3.2) to reduce to a sum of Lifshitz antisymmetric invariants, ∝ mk∂lmm −mm∂lmk

(i.e. Lijklm = −Lijmlk). In the case of a cubic three-dimensional system with bulk spa-

80

tial inversion symmetry breaking (e.g. B20 or ZnS crystals), all three crystallographic

directions (x , y and z) are equivalent, and the chiral damping adopts the general

form

αij = αij0 + αij3dλdwm · (∇×m), (3.3)

where λdw defines the characteristic exchange length. In a two-dimensional system

with interfacial symmetry breaking along z, i.e. invariant under C∞z rotation sym-

metry, the damping takes the form

αij = αij0 + αijz λdwm · (∇z ×m), (3.4)

where ∇z = z×∇.

As dictated by Neumann’s principle, the chiral damping possesses the same sym-

metry as DMI for these systems (e.g. see Refs. [59, 68]). In addition, Onsager

reciprocity imposes αij = αji, and hence, one can construct a chiral damping up to

linear order in magnetization gradient based on symmetry arguments. These consid-

erations suggest that such a damping is present in non-centrosymmetric (ferro, ferri

and antiferro)magnets in general, not limited to metals. However, this phenomenol-

ogy does not provide information regarding the strength of the chiral damping itself,

the relative values of the off-diagonal tensor elements (∼ αi 6=j) compared to the diag-

onal ones (∼ αii) nor does it indicate the underlying physical mechanisms responsible

for it. Let us now turn our attention towards the microscopic origin of such chiral

damping.

3.3 Microscopic origin of the chiral damping

We focus our attention on magnetic metals where the damping is driven by the

spin relaxation of itinerant electrons [187, 188, 189]. In ferromagnets with interfacial

81

Figure 3.1: (Color online) Illustration of AHE induced (a) and s-d DMI mediated (b)chiral damping. In addition to the constant isotropic exchange field in the directionof the local moments (red) about which the itinerant electrons (black) precess on aconstant energy contour, in the presence of s-d DMI, the itinerant electrons feels anadditional spatially varying field (blue) which turns to align its spin perpendicular tothe local moment. This field which varies in both magnitude and direction results ina damped precession which depends on the chirality of the wall.

Rashba spin-orbit coupling, it has been shown that the magnetic damping adopts

the form of a tensor linear in both magnetization gradient and Rashba strength [199,

145, 200]. Besides this effect, we here propose two additional mechanisms that can

contribute to the chiral damping.

The first mechanism as illustrated in Fig. 3.1 (a) arises from the interplay between

spin motive force and anomalous Hall effect. Indeed, it has been shown recently that

time-dependent spin textures generate a local spin current Jsi = gµB~G0

4e2∂tm × ∂im

[201], where G0 is the electrical conductivity. This local spin current which flows

along ei direction and polarized along ∂tm×∂im has been shown to induce magnetic

dissipation at the second order of spatial gradient [74]. Interestingly, in the presence

of anomalous Hall effect in the ferromagnet, this primary spin current Jsi is converted

into a secondary spin current Jsj = θHP [Jsi · (ei × ej)]ei × ej, where θH is the spin

82

Hall angle and P is the spin polarization in the ferromagnet. This secondary spin

current which flows along ej and polarized along ei × ej can be injected into an

adjacent spin sink with strong spin relaxation, thereby inducing a dissipative torque

on the magnetization which is very similar to the spin pumping mechanism [202].

Considering a one-dimensional domain wall along x deposited on a heavy metal with

an interface normal to z, we obtain a damping torque on the form [see Section ??]

τ = AθHgµB~PG0

4e2Msd[(∂tm× ∂im) · y]m× (y ×m), (3.5)

A being a renormalization factor arising from the spin current backflow (A → 1 for

strong spin relaxation [202]), Ms the saturation magnetization and d the thickness of

the ferromagnet. Assuming G0 = 2 × 107 S·m−1, P = 0.7, θH = 5 %, d = 1 nm and

Ms = 800 emu/cc, we find a chiral damping of the order of 0.015 for a domain wall

thickness λdw = 1 nm. This damping torque is proportional to ∼ sin 2ϕ, ϕ being the

azimuthal angle of the magnetization, and vanishes when the wall is either in Bloch

(ϕ = π/2) or Neel configuration (ϕ = 0).

The second mechanism is related to DMI. Indeed, as demonstrated in Ref. [203],

for transition metal ferromagnets, both localized (pd hybridized) and delocalized elec-

trons (spd hybridized) contribute to DMI and therefore one can parse the total spin

moment Si into localized (d-dominated) and delocalized (s-dominated) contributions

Si = Sdi + ssi and the resulting DMI between sites i and j can be phenomenologically

rewritten up to the linear order in ss as Dij ·Si×Sj = Dddij ·Sdi ×Sdj +Dsd

ij ·Sdi × ssj . The

first term only involves orbital overlap between localized states while the second term

describes the chiral exchange between the local and itinerant spins. In the continuous

limit, the Hamiltonian of the itinerant electrons can be written as

H =p2

2m+ Jsdm · σ +

αR

~(z× p) · σ +

αD

~[(z× p)×m] · σ, (3.6)

83

where σ is the vector of Pauli spin matrices, Jex, αR and αD = Dsda (a is the

lattice parameter) are the strength of the exchange, Rashba spin-orbit and s-d DM

interactions respectively. The electron spin within the magnetic texture therefore

moves in a trajectory defined by the drift velocity

v = ∂pHsd =p

m− αR

~z× σ − αD

~z× (m× σ). (3.7)

The last term in Eq. (3.7) can also be interpreted as the source of magnetic relaxation

[204, 205] which as it turns out, introduces a chirality in the magnetic dissipation

which is mediated however in this case by itinerant electrons. In other words, the

itinerant electrons experiences an additional spatially varying effective field of the

form ∼ (z× p)×m as illustrated in Fig. 3.1(b).

Following Section 2.5.3, we apply of Enhrefest theorem to 3.6 to obtain

∂ts =∂〈σ〉∂t

=1

i~〈[σ,H]〉 = − 1

τex

s×m + bR∇z × s + bD(∇z ×m)× s (3.8)

where ∇z = z×∇, τex = ~/2Jsd is the precession time, bR = 2αR/~ and bD = 2αD/~

are the typical velocities associated with Rashba and DM interactions. From Eq.

(3.8), the semi-classical Bloch equation for the itinerant spin density s is given as

∂ts + ∇ · J +1

τex

s×m− bR∇z × s− bD(∇z ×m)× s = −Γre, (3.9)

where J = −D∇⊗ s, is the spin current density with D = v2F τ/3 being the diffusion

constant while Γre captures effects due to the spin relaxation and dephasing. The

spin and spin density can be written in the form s = nsm + δs, where ns (δs) is

(non-)equilibrium spin density. This representation corresponds to writting the (non-

)equilibrium spin current density as J0 = −bJu ⊗m (δJ = −D∇ ⊗ δs) where u is

84

the direction of current flow and bJ = µBPG0E/e. This transform Eq. (3.9) into

−D∇2δs+1

τex

δs×m+1

τϕm×(δs×m)+

1

τsf

δs−bR∇z×δs−bD(∇z×m)×δs = fs, (3.10)

where fs is the driving source for the generated non-equilibrium spin density which

we identify as coming from two different sources - the flowing current [bJ(u ·∇)m]

and spin pumping as a result of non-steady state magnetization [−ns∂tm]. Our focus

on this study is on magnetization dissipation therefore we neglect effect due to apply

current.

To obtain a good approximation for an analytic solution for the non-equilibrium

spin density δs from Eq. (3.10), we consider smooth magnetic texture such that we

can use gradient expansion method in order of the gradient of the magnetization.

First ,we re-cast Eq. (3.10) into

1

τex

δs×m +1

τϕm× (δs×m) +

1

τsf

δs + gc = −ns∂tm, (3.11)

where gc represents higher order corrections given as

gc = −bR∇z × δs− bD(∇z ×m)× δs−D∇2δs. (3.12)

At the lowest order, we set gc = 0, and simple vectorial calculus yields

−τ||τex

m× (δs×m) + δs×m = nsτ||m× ∂tm, (3.13a)

δs×m +τex

τ||m× (δs×m) = −nsτex∂tm, (3.13b)

where 1τ||

= 1τϕ

+ 1τsf

from which the zero-th order solution for δs

δs = −nsτex[ξ∂tm + m× ∂tm] (3.14)

85

Higher order solution is obtained by the substitution of δs given by Eq. (3.14) into

Eq. (3.12) and then repeating the scheme above but this time with gc 6= 0. This

scheme yields at the second order in magnetization gradient

δs = −nsξ (1 + 2bDτex[m · (∇z ×m)]) ∂tm (3.15)

+ ns(1 + (1− ξ2)bDτex[m · (∇z ×m)]

)m× ∂tm

− nsξbRτex[∇z × ∂tm]×m + nsbRτex[∇z × (m× ∂tm)]×m

− nsξbRτexξm× ([∇z × ∂tm]×m) + nsbRτexξm× ([∇z × (m× ∂tm)]×m).

From which we computed the corresponding torque τ generated by a precessing mag-

netization ∼ ∂tm as

τ/ns ≈ (1 + ξχ− βm×) [−∂tm (3.16)

+λR[∇z × (m× ∂tm + ξ∂tm)]⊥

+ λD[(∇z ×m)× (m× ∂tm + ξ∂tm)]⊥] .

In this expression, β = τex/τsf , χ = τex/τϕ, ξ = χ + β, ns = ns/(1 + ξ2) and the

subscript ⊥ indicates that the torque is defined perpendicular to the magnetization

m. The first term has been derived previously [74], the second term (λD = τexbD)

arises from the s-d DMI exchange, and the third term (λR = τexbR) arises from

Rashba spin-orbit coupling [199, 145, 200]. The total torque τ contributes both to the

renormalization of the gyromatic ratio (terms that preserve time-reversal symmetry)

and to dissipation (terms that break time-reversal symmetry). This form is more

general than the expansion given in Eq. (3.2). Assuming αR ≈ 10−11 eV·m [98, 100],

αD ≈ Dsda = 10−11 eV·m [203] and τex ≈ 10−14 s, we obtain λR = λD = 0.3 nm.

Considering that ns = 0.1 and τϕ ≈ τsf ≈ 10−13 s, the chiral damping is of the order

86

of nsλR/λdw ≈0.03 and nsξλD/λdw ≈ 0.003 for λdw =1 nm. Notice that the chiral

damping arising from s-d DMI (∝ Dsd) does not necessarily scale with the DMI itself

(∝ Ddd) as both effects:-chiral damping and asymmetric exchange, involve different

electron orbitals.

3.4 Domain wall motion

To illustrate the effect of this chiral damping on the dynamics of magnetic textures,

we first derive the equation of motion of a one-dimensional perpendicular domain wall,

commonly observed in heavy metal/ferromagnet asymmetric multilayers [98, 100, 170,

171, 172, 173]. The magnetization is defined m = (cosϕ sin θ, sinϕ sin θ, cos θ), where

ϕ = ϕ(t) is the azimuthal angle and θ(x) = 2 tan−1 (exp[s(x−X)/λdw]), X being

the domain wall centre, and s = ±1 defining the domain wall chirality (↑↓ or ↓↑,

respectively). The magnetic domain wall is submitted to a magnetic anisotropy field

Hk = Hk sin θ sinϕy which favors a Bloch configuration and an applied magnetic field

H = Hxx which favors a Neel configuration. The damping torque is given by Eq.

(3.16) and the rigid domain wall dynamics is described by the coupled equations

∂τϕ = Hz + sπ

2(α− sµ cosϕ)(Hx sinϕ− Hk

2sin 2ϕ),

s∂τX

λdw

= (α− sν cosϕ)Hz −π

2(Hx sinϕ− Hk

2sin 2ϕ), (3.17)

where we defined τ = γt, µ = (λD − ξλS)(πns/4λdw), and ν = (βλD + λS)(πns/4λdw)

(α, ν, µ 1). In Eq. (3.17), we neglected renormalization of the gyromagnetic ratio

and only considered the dissipative components. The damping due to s-d DMI and

Rashba spin-orbit coupling both produce a contribution ∝ s cosϕ that depends on

the domain wall chirality s and on the azimuthal angle ϕ. Notice that the DM field

does not explicitly enters these equations since it can be simply modeled by a chiral

in-plane field ∼ sHx [68].

87

Let us now investigate the influence of this damping on the field-driven motion of a

domain wall submitted to both perpendicular (Hz) and in-plane (Hx) magnetic fields.

In this example, we chose µ = ν = αc for simplicity. Figure 4.1(a) shows the steady

state velocity of the domain wall as a function of Hz for difference chiral damping

strengths and Hx = 0. These velocity curves display the usual Walker breakdown

around Hz ≈ 12 − 19 mT. Below Walker breakdown and in the absence of in-plane

field Hx, the domain wall azimuthal angle ϕ obeys Hz = sπ4Hk(α− sαc cosϕ) sin 2ϕ,

which produces a kink around Hz ≈ sαcHWB/α where HWB = απHk/4, which is

associated with a jump in the effective damping, αeff = α− αc cosϕ [see Fig. 4.1(c)].

For a large in-plane field Hx, a Neel configuration (ϕ ≈ 0, π) is stabilized and the

damping becomes slightly independent on Hx. Hence, the kink disappears as shown

in Fig. 4.1(b). More interestingly, when the domain wall is tuned from Bloch to

Neel configuration by an in-plane field [see Figs. 4.1(d)-(f)], the effective damping

is strongly modified [Fig. 4.1(f)] and domain wall velocity becomes asymmetric as

reported on Fig. 4.1(d). Notice that the kink is still observable at small negative

in-plane field.

We now turn our attention towards the creep regime, which is of most importance

for field-driven domain wall motion in ultrathin disordered multilayers [197, 196]. In

this case, the creep law predicts [113]

vcreep = v0 exp(−(Tc/T )(Hc/Hz)

1/4)

(3.18)

where Tc is the critical temperature (close to the de-pinning temperature in fact),

T is the sample temperature, Hz is the applied field and Hc is the critical field

needed to overcome the disorder pinning potential. This expression assumes Hz

Hc, and Tc T . The exponent gathers only terms accounting for the disordered

energy landscape of the system and does not include any dissipative contributions

88

−50 0 500

0.5

1

1.5

Φ/π

µ0H

z = 5 mT

(e)

αc = 0 α

c = 0.25α α

c = 0.5α α

c = 0.75α

−50 0 50

0.2

0.4

0.6

0.8

µ0H

x(mT)

αeff

µ0H

z = 5 mT

(f)

−50 0 50

10

20

30

Ve

loc

ity

(m

/s)

µ0H

z = 5 mT

(d)

−40 −20 0 20 40

0.4

0.6

0.8

µ0H

z(mT)

αeff

(c)

µ0H

x = 0 mT

−40 −20 0 20 40

−20

0

20

Ve

loc

ity

(m

/s)

(a)

µ0H

x = 0 mT

−40 −20 0 20 40

−50

0

50

Ve

loc

ity

(m

/s)

(b)

µ0H

x = 25 mT

Figure 3.2: (Color online) (a,b) Domain wall velocity as a function of perpendicularmagnetic field for different chiral damping strengths, at Hx =0 mT and Hx =25mT, respectively; (c) corresponding effective damping. (d) Domain wall velocity asa function of in-plane magnetic field for different chiral damping strengths, and forHz =5 mT; (e,f) corresponding azimuthal angle and effective damping, respectively.The parameters are α= 0.5 , HWB = 12.5 mT. In all simulations, we start with a Blochconfiguration (ϕ = π/2) and allow the system to relax to steady state condition.

in principle. In contrast, the coefficient v0 depends on both the energy landscape as

well as on the viscosity of the elastic wall. In the limit of an over-damped membrane

(α∂tm ∂tm), v0 ≈ γλdwHc/α [113]. It is mostly probable that even for strongly

disordered ferromagnets, the creeping domain wall is not in the over-damped regime,

although very large damping (up to α ≈ 0.5) have been reported in Pt/Co/AlOx

and Pt/Co/Pt systems [98, 100]. However, since the theoretical description of the

creep motion of intermediate damped systems is not fully understood, we propose to

investigate the impact of chiral damping in this limit.

To evaluate the impact of the chiral damping, we follow the procedure proposed

89

by Je et al. [206] and rewrite the creep law

vcreep(H) =η0σdw(Hx)

β

αeff(Hx)exp

(−χ0

σ1/4dw (Hx)

σ1/4dw (0)

H− 1

4z

). (3.19)

Here, σdw is the (chiral) energy of the domain wall and β is a coefficient that describes

the scaling law between the critical force Hc and the domain wall energy and η0 and

χ0 are normalization factors that can be chosen to fit experimental data [206, 196].

The domain wall energy reads

σdw = σ0 + πλdwµ0Ms

[1

2Hk cosϕ−Hx

]cosϕ, (3.20)

where the first term σ0 accounts for the ϕ-independent contribution to the magnetic

energy, the second term is the magnetic anisotropy field Hk favoring Bloch configu-

ration and the last term Hx is the longitudinal field favoring Neel configuration. The

energy minimization ∂ϕσdw = 0 gives [206]

σdw

σ0

=

1− (Hx)2/HdwHk : |Hx| ≤ Hk

1 + (Hk − 2|Hx|)/Hdw : |Hx| > Hk

(3.21)

where Hdw = π2λdwµ0Ms/σ0. The magnetic damping αeff is written in the simplest

form αeff = α + sαc cosϕ. To investigate the impact of chiral damping on the creep

motion, we chose Hdw = 1T , and Hk = 50mT . The normalized velocity v(Hx)/v(0)

[v(0) = η0σdw(0)β

αeff(0)] of the domain wall as a function of the in-plane field and for

different strengths of chiral damping αc is represented on Fig. 3.3. For |Hx| < Hk,

the velocity varies smoothly while the domain wall changes from one Neel chirality

to another. For |Hx| > Hk, the domain wall saturated in the Neel configuration and

the velocity follows the exponential law given above. These results agree with Ref.

[196]. Including DMI in the calculation (i.e. Hx → Hx + sHDMI) only results in a

90

Figure 3.3: (Color online) Normalized domain wall velocity as a function of in-planemagnetic field Hx in the presence of a driving field Hz = 20mT. In-plane magneticfield pushes the domain from a Bloch to a Neel configuration and hence increases theazimuthal angle ϕ thereby modifying the damping of the wall. A kink in the velocityis observed at Hx=−Hk, when the domain wall saturates in the Neel configurationwhen the damping becomes independent on the in-plane field Hx.

horizontal shift of the velocity curve in Fig. 3.3.

3.5 Conclusion

The symmetry principles discussed in this Letter are general and ensure the existence

of such a chiral damping in any magnetic structures (ferromagnets, antiferromag-

nets, chiral magnets, but also metals and insulators etc.) presenting spatial inversion

symmetry breaking. The physical mechanisms responsible for this chiral damping

can be spin-orbit coupling but also dipolar coupling or magnetic frustrations as in

the case of DMI. However, because different orbital characters are involved, chiral

damping and DMI do not necessarily scale with each other. The recent observa-

tion of chiral signature in the viscosity of creeping magnetic domain walls [196] as

well as asymmetric damping of spin waves [184] are clear evidence of such a chiral

91

damping in non-centrosymmetric magnetic systems. Finally, chiral damping is not

limited to magnetic systems and is expected to have significant impact on the trans-

port properties of systems whose dynamics is dominated by viscosity such as vortex

motion in type-II superconductors, crystal growth, or percolation in porous media

[207, 208, 209].

92

Chapter 4

Role of spin diffusion in

current-induced domain wall

motion

Current-induced spin transfer torque and magnetization dynamics in the presence of

spin diffusion in disordered magnetic textures is studied theoretically. We demon-

strate using tight-binding calculations that weak, spin-conserving impurity scattering

dramatically enhances the non-adiabaticity. To further explore this mechanism, a

phenomenological drift-diffusion model for incoherent spin transport is investigated.

We show that incoherent spin diffusion indeed produces an additional spatially de-

pendent torque of the form ∼∇2[m×(u·∇)m]+ξ∇2[(u·∇)m], where m is the local

magnetization direction, u is the direction of injected current and ξ is a parameter

characterizing the spin dynamics (precession, de-phasing and spin-flip). This torque,

which scales as the inverse square of the domain wall width, only weakly enhances

the longitudinal velocity of a transverse domain wall but significantly enhances the

transverse velocity of vortex walls. The spatial dependent spin transfer torque uncov-

ered in this study is expected to have significant impact on the current-driven motion

of abrupt two-dimensional textures such as vortices, skyrmions and merons.

93

4.1 Introduction

The control of the magnetic state of nanoscale heterostructures [16, 210] such as

magnetic domain walls [76, 77, 78, 79] and vortex cores [80, 81] by a spin polarized

charge current is attracting increasing interest as a promising mechanism for innova-

tive memory devices [2]. Identifying the nature of the torque exerted by the injected

current on the domain wall itself has constituted a stimulating challenge resulting in

the observation of unique dynamical behaviors [76, 77, 78, 79, 80, 81, 2, 82, 83, 84]

and raising seminal questions concerning the transport of itinerant spins in inhomo-

geneous magnetic textures [74, 72, 88, 211, 6, 212, 191]. The most widely accepted

form of the spin transfer torque exerted by a charge current on a smoothly varying

magnetic texture m(r, t) is [74, 72]

T = −bJ(u ·∇)m + βbJm× (u ·∇)m (4.1)

where bJ is the adiabatic spin torque, β describes the so-called non-adiabaticity of

the spin torque and u is the direction of current injection. The non-adiabaticity

β is generated by different mechanisms such as spin relaxation [74, 211], spin-orbit

coupling [213] and magnetic texture-induced spin mistracking [88, 6, 211]. In addition

to non-adiabaticity, it has been found that the Gilbert damping α can also be affected

(enhanced) by the spin texture [191]. From the viewpoint of domain wall dynamics,

the longitudinal (in the direction of current injection) velocity for transverse walls and

the transverse (perpendicular to the direction of current injection) velocity for vortex

walls is controlled by the ratio of β to α [72]. Therefore, experimental efforts have been

expended in accurately determining β and α for a wide range of magnetic materials

and domain wall widths [76, 77, 78, 79, 80, 81, 2, 82, 83, 84]. Experiments have

shown that this ratio depends on the domain wall structure (Bloch, Neel or vortex)

and that vortex domain walls exhibit a much larger non-adiabaticity (β ≈ 8α to 10α)

94

[84, 85, 86, 87] compared to transverse domain walls (β ≈ α) as summarized in Table

4.1. The authors attribute these large non-adiabaticities to the narrow character of

domain walls in vortex structures (≈10 nm). As a matter of fact, Tatara et al. [88],

Xiao et al. [6] and Wessely et al. [89] demonstrated that in abrupt domain walls

the itinerant spin cannot adiabatically follow the local spin texture, resulting in an

enhancement of the non-adiabaticity. Burrowes et al. [90] on the other hand, tested a

very sharp transverse wall of about 1 nm using FePt nano-wires and found that such

a narrow domain wall does not cause a significant increase in the non-adiabaticity.

Table 4.1: Summary of measurements of non-adiabaticity for different materials. Thetable entries shows values of β(β/α) for vortex walls (VW), in-plane transverse walls(TW) and out-of-plane Bloch walls (BW).

Material β(β/α)

Py (VW) 0.018(2)[80, 81],0.04(8)[84]0.073(9)[85],0.15(10)[86],0.15(9.2)[87]

Py (TW) 0.01(1.2)[85],0.02(1)[77],0.13(10)[214]FePt (BW) 0.06(1)[90]CoNi (BW) 0.022(1)[90]Pt/Co (BW) 0.35(2)[215]

Theoretical investigations [211, 216] have also shown a damped oscillatory behav-

ior of the non-adiabatic torque when increasing the domain wall width which may

account for the non-adiabaticity enhancement for abrupt domain walls. However,

these models are applied to transverse walls only and do not readily explain the

observed differences between transverse and vortex walls.

The experimental observations [84, 85, 86, 90] indicate that the nature of the

non-adiabatic spin torque exerted on abrupt magnetization patterns could be related

to their dimensionality. Indeed, whereas a magnetic transverse wall varies along one

direction only (∂xm 6= 0, ∂ym = 0), the magnetization in a vortex structure varies

along two directions (∂xm 6= 0, ∂ym 6= 0). Therefore, one approach to explain the

95

different non-adiabaticities of abrupt transverse domain walls and vortex structures is

to consider a mechanism that couples both x and y directions. It has been shown that

a transverse spin current caused by anomalous Hall effect increases the transverse ve-

locity of vortex cores while leaving the transverse domain walls essentially unchanged

[217, 92]. However, its contribution is of the order of the damping constant α and

thus is insufficient to explain the results observed in Refs. [84, 85, 86]. An alternate

mechanism is the diffusion of spin accumulation along the domain wall. This effect,

neglected in the original theory of Zhang and Li [74], has recently been proposed

to be responsible for an enhancement of the non-adiabaticity of sharp domain walls

[217, 92].

In this Article, we demonstrate both numerically and analytically that scatter-

ing against spin-independent disorder dramatically increases the non-adiabatic spin

torque and can significantly affect the motion of abrupt domain walls. The physics

of non-adiabaticity in magnetic domain walls involves two types of mechanisms: (i)

the coherent precession of the spin around the local magnetization and (ii) spin de-

phasing and relaxation induced by disorder. In the absence of disorder, the spin

precession around a smoothly varying magnetization results in a vanishingly small

non-adiabaticity [6]. When a small amount of spin-independent disorder is intro-

duced, the non-adiabaticity increases due to the enhancement of the spin de-phasing

which we interpret in terms of an effective Elliott-Yafet spin-flip scattering enabled

by the abrupt magnetic texture. As a consequence, the resulting non-adiabaticity

increases when the domain wall width decreases and when increasing the strength

and the amount of disorder. These features are demonstrated in Section 4.2.1 using a

tight-binding model (weak disorder regime) and Section 4.2.2 using a drift-diffusion

model (strong disorder regime). In Section 4.3, we show that the resulting non-

adiabatic torque significantly affects the current-driven motion of Bloch walls and

vortex cores and may result in an effective increase of the transverse velocity of vor-

96

tex walls by an order of magnitude. A discussion of the relevance of our study to

previous studies is given in Section 4.4 and Section 4.5 concludes this Article.

4.2 Theoretical model

4.2.1 Tight-binding model for disordered ferromagnets

In this section, we present a framework to calculate the local spin transfer torque for

both ballistic and diffusive transport in magnetic textures. In particular, we consider

a single band tight-binding model on a two-dimensional square lattice as implemented

in Kwant software [164]. Our system can be described by the Hamiltonian

H =∑i

εic+i ci +

∑<ij>

tc+i cj + Jsd

∑i

c+i mi · σci

where εi is the onsite energy, t is the hoping parameter, the sum < ij > is over

nearest neighbors, mi is a unit vector in the direction of the local moment at site

i, Jsd is the strength of the exchange energy that couples the local moment to the

itinerant electrons with spin represented by the Pauli matrices σ. c+i = (c↑i , c

↓i )

+ is

the spinor form of the usual fermionic creation operator. Without loss of generality,

we consider current along the x-axis and choose the z-axis as the quantization axis.

Our calculations are performed on a one-dimensional Bloch wall with spin texture as

illustrated in Fig. 4.1 in spherical coordinates given by θ(x) = 2 tan−1(exp[x/λdw])

and φ = π/2, where λdw is the domain wall width.

To calculate the transport properties of interest in our system, we first calculate

the non-equilibrium spin density λdwsi from the local spin density si for both electrons

coming from the left (L) and right (R), as λdwsi = sLi − sRi [218]. The local spin

torque Ti is defined as the torque exerted by the non-equilibrium spin density on the

97

Figure 4.1: (Color online) Sketch of the one-dimensional Bloch wall implemented ona two-dimensional square lattice as used in the tight-binding model described in themain text.

local moment at site i and reads

Ti = Jsdδsi ×mi (4.2)

To extract the different torque components, the torque is rewritten in the form Ti =

T iad∇xmi+Tinadmi×∇xmi where T iad(nad) is the local adiabatic (non-adiabatic) torque

component, from which the local non-adiabaticity parameter ( defined as the ratio of

the local non-adiabatic and adiabatic torques, i.e. βi = T inad/Tiad) is extracted. In all

our calculations, we use the parameters: system size 401×31a2, where a is the lattice

constant, t = 1.0 eV, Jsd = 0.8 eV and transport energy EF= 7.0 eV.

Fig. 4.2 shows ballistic calculations of the spatial distribution of the adiabatic

and non-adiabatic spin transfer torques for different current spin polarizations of the

ferromagnet for a domain wall width of 8a. In our calculations, we tune the current

spin polarization P = (G↑ − G↓)/(G↑ + G↓), G↑(↓) being the conductance for spin

up (down) channel at the Fermi level, by shifting the onsite energy of the system.

The non-adiabatic torque which strongly depends on the current spin polarization

has been normalized by the non-adiabaticity at the center of the wall (βmax) and

shows an oscillating behavior which we attribute to the precession of the propagating

98

-0.5

0

0.5

1

-0.5

0

0.5

1

Norm

aliz

ed t

orq

ue

com

ponen

ts

150 200 250Lattice site

-0.5

0

0.5

1

Adiabatic torque (Non-adiabatic torque)/βmax

(a)

(b)

(c)

P = -1.0

P = -0.56

P = -0.36

βmax

= -1.08 x 10-7

βmax

= -8.66 x 10-5

βmax

= -7.4 x 10-2

Figure 4.2: (Color online) Spatial distribution of the adiabatic and non-adiabatictorque for different current spin polarization. (a) P = -1.0, (b) P = -0.56, and (c) P= -0.36. The adiabatic torque remains essentially constant, while the non-adiabatictorque shows strong depends on the current spin polarization.

states around the local moments in the form ei(k↑−k↓)·x, where k↑(↓) is the wave vector

in the transport direction [219, 220, 221]. As shown in Fig. 4.3 (a), the average

non-adiabaticity defined as the average over space of the local non-adiabaticity, β =∑i βi/L with L being the length of the system, is related to the domain wall width

by β = β0 exp (−λdw/λL), where λL depends on the current spin polarization and

controls the length scale for which the adiabatic approximation holds [6].

We model spin-conserving impurity scattering through the addition of a ran-

dom potential Vi to the constant onsite energy ε0 as εi = ε0 + Vi such that Vi ∈

[−0.5Vimp, 0.5Vimp]. To calculate the transport properties of our system, we calcu-

late the statistical average over a thousand different disorder configurations. For the

range of impurities considered, the mean free path is much larger than the domain

wall width, which corresponds to realistic experimental situations. Figs. 4.3 (a) and

99

4.3 (b) shows that in the presence of spin-conserving impurity scattering, the non-

adiabaticity is significantly increased as compared to the ballistic case and follows the

empirical law β = β0 exp (−λdw/λL[1− λdw/λ∗L]), where λ∗L and λL depend on both

the impurity strength and current spin polarization.

This result shows that (i) spin mis-tracking is the dominant mechanism in disorder-

free systems and is only significant in weak ferromagnets and abrupt spin textures

which is consistent with Xiao et al. [6]. (ii) In the presence of spin-conserving impurity

scattering, the non-adiabaticity is enhanced by two to three orders of magnitude even

in the absence of explicit spin-flip scattering. Such an enhancement can be attributed

to the onset of spin relaxation due to the combined effect of momentum scattering and

magnetic textures. Indeed, a magnetic texture induces a mixing of the spin states in

the rotating frame of the magnetization. This spin mixing is for instance the physical

ground of ballistic domain wall resistance [222]. In other words, a magnetic texture

mixes the spin projections as spin-orbit coupling does. The introduction of spin-

conserving disorder relaxes the linear momentum, and results in an Elliott-Yafet-type

spin relaxation [223, 224]. Therefore, disorder-induced momentum relaxation coupled

with domain wall-induced spin mixing results in the onset of spin relaxation. Since

this spin relaxation has been recently shown to enhance the domain wall resistance

[225], it is reasonable that it also induces an enhancement of the non-adiabaticity [see

[88], Eq. (10)]. This demonstrates the seminal role played by disorder and momentum

scattering on spin transfer torque in abrupt magnetic textures.

To further explore this mechanism, we address the nature of the non-adiabaticity

in the diffusive regime in which disorder is treated incoherently. In this regime,

the amount and strength of the disorder must be such that the mean free path is

smaller than the system size. However, modeling such a system using a quantum

mechanical tight-binding model requires a large system size as well as averaging over

a large number of disorder configurations and is therefore computationally demanding.

100

6 8 10 12Domain wall width (a)

10-8

10-6

10-4

10-2

No

n-a

dia

bat

icit

y (

β)

0 0.05 0.1 0.15 0.2Disorder strength (eV)

0

0.5

1

1.5

2

No

n-a

dia

bat

icit

y (

β)

~ β0e

-∆/λL

~ β0

* + β

1V

imp+ β

2V

imp

2

(a)

disorder

disorder-free

(b)x 10

-3

∆ = 8a~ β0

~ e

-∆/λL

~ (1 - ∆/λ

L* )

Figure 4.3: (Color online) (a) Non-adiabaticity parameter as a function of the domainwall width in the absence (red) and presence of spin-conserving impurity scattering(black) of strength Vimp = 0.1 eV; (b) Non-adiabaticity parameter as a function ofspin-conserving impurity scattering strength for a domain wall width of 8a. In thesecalculations, we set the polarization to P =-0.56. In the absence of disorder, thenon-adiabaticity is governed by spin mistracking while it is dominated by de-phasingwhen introducing disorder in the structure. The fitted parameters are β0 = 2.92,β0 = 8.9 × 10−3, β∗0 = 4.27 × 10−6, β1 = 2.87 × 10−4, β2 = 5.24 × 10−2, λL = 0.97,λL = 0.64 and λ∗L = 25.

Therefore, the non-adiabaticity in this regime is modeled following a semi-classical

drift-diffusion approach.

4.2.2 Drift-diffusion model

We now investigate the role of spin diffusion within a drift-diffusion model which

in principle applies in the limit of strong disorder and strictly speaking is valid for

domain walls width larger than the electron mean free path. The itinerant electrons

evolving in magnetic textures are described by the one-electron Hamiltonian

H =p2

2m+ Jsdσ ·m(r, t), (4.3)

where m = cosφ sin θex + sinφ sin θey + cos θez is the time- and position- dependent

magnetization direction (|m| = 1), p is the momentum, σ is the Pauli matrix and

101

Jsd is the exchange coupling energy. The itinerant electron spin operator satisfies the

generalized spin continuity equation

∂tσ + ∇ · J =1

i~[σ,H]− Γre(σ,m), (4.4)

where J is the spin current operator, Γre represents spin relaxation and de-phasing

due to scattering against spin-orbit coupled, magnetic and non-magnetic impurities.

The drift-diffusion equation for spin dynamics has been derived using quantum kinet-

ics [226] and generalized random matrix theory [227] from which the spin accumula-

tion s = 〈σ〉 in the diffusive regime satisfies the equation

∂ts = −∇ · J − 1

τexs×m− Γre(s,m), (4.5)

where τex = ~/2Jsd is the spin precession time (≈ 10−15 s to 10−14 s), J = 〈J 〉 is the

spin current density tensor such that J = −D∇⊗s, where D is the diffusion constant

and Γre = 〈Γre〉. In systems such as disordered ferromagnetic metallic nano-wires, the

local momentum scattering tends to counteract the influence of the external electric

field resulting in an additional diffusion term in both charge and spin currents [227].

The dynamics of the magnetization and that of the itinerant spins are considered

to be decoupled since the dynamics of conduction electrons is fast compared to that of

magnetization. As a result, the spin density can be written in the form s = nsm+δs,

where ns (δs) is the (non-) equilibrium spin density (ns ≈ 10−2Ms). Similarly, the

spin current density takes the form J = J0 + δJ . J0 = −bJu ⊗m is the adiabatic

spin current density, where bJ = µBPG0E/e, P is the spin polarization, µB is the

Bohr magneton, G0 is the electrical conductivity in the absence of magnetic texture,

u is the direction of injected current. δJ = −D∇δs is the non-adiabatic spin current

density which arises from carrier diffusion. Using the relaxation time approximation,

102

Γre is given by

Γre(s,m) =1

τsfλdws +

1

τφm× (λdws×m), (4.6)

where τsf is the phenomenological spin-flip relaxation time (≈ 10−13 s to 10−12 s) and

τφ is the spin dephasing time.

Note that Eq. (4.6) differs from Ref. [74] by the presence of a dephasing term

(∝ 1/τφ) which arises from the destructive interference of non-equilibrium precessing

spins with different wave vector directions. Microscopic investigations of spin dephas-

ing at the interface between a normal metal and a strong ferromagnet using realistic

Fermi surfaces have shown that this effect destroys the transverse component of itin-

erant spins within a few monolayers [219, 220, 221]. In bulk disordered ferromagnets,

recent derivations using quantum Boltzmann equation [104] or generalized random

matrix theory [228] have demonstrated that the interplay between spin precession and

impurity scattering introduces a spin dephasing. The presence of spin dephasing as

we will show below, has an important implication in terms of spin torque definition.

In particular, the torque T is defined as the amount of spin current absorbed by the

local magnetization less the spin density lost to the environment. Therefore from Eq.

(4.5), we obtain

T = −∇ · J − 1

τsfδs =

1

τexδs×m +

1

τφm× (λdws×m). (4.7)

In the following, we use this definition to express the spin torque up to the third

order in spatial gradient of the magnetization.

103

Spin transfer torque from drift-diffusion

In the framework of the above representation, the drift-diffusion equation for the

non-equilibrium spin density is given by

−D∇2δs +1

τexδs×m +

1

τφm× (δs×m) +

1

τsfδs

= bJ∇xm− ∂tδs− ns∂tm +Dns∇2m. (4.8)

The right hand side (RHS) of Eq. (4.8) acts as a source of itinerant spin dynamics

within the spin texture for current applied along the x -dirextion. We solve Eq. (4.8)

for the steady state (∂tδs = ∂tm = 0) solution of the non-equilibrium spin density

from which the torque can be calculated using Eq. (4.7).

In systems with wide and smoothly varying spin textures such that the domain

wall width (radius of vortex core)W and the spin diffusion length λsf =√Dτsf satisfies

W 2 λ2sf , the first term in the left hand side (LHS) of Eq. (4.8) can be neglected

and in this case, the itinerant non-equilibrium spin density reduces to [74, 75]

δs ≈ bJ τex [ξ∇xm + m×∇xm] , (4.9)

where ξ = β0 + χ, β0 = τex/τsf , χ = τex/τφ and τex = τex/(1 + ξ2).

However, for abrupt spin textures or vortex structure for which W 2 ∼ λ2sf , the

first term in the LHS of equation Eq. (4.8) can no longer be neglected. Before we

present the complete analytical results, we first attempt to give some insight of the

impact of the diffusion term [first term in LHS of Eq. (4.8)] on the spin torque.

Let us start with a more general consideration of vector representation in space.

Since the unit vectors in the direction of (m, ∇xm, m × ∇xm) form a basis in 3

dimensions, any vector in space can be written as linear combination of these basis

vectors. Therefore, without loss of generality, we can write the first term in the LHS

104

of Eq. (4.8) in the form

−D∇2δs =DW 2 [Gδs + Ba∇xm + Bnm×∇xm] , (4.10)

where G, Ba and Bn are spatially- and spin texture-dependent scalars. Inserting Eq.

(4.10) into Eq. (4.8) we obtain

1

τexδs×m +

1

τφm× (δs×m) +

1

τsf

(1 + G λ2

sf

W 2

)δs

=(bJ − Ba DW 2

)∇xm− Bn DW 2 m×∇xm. (4.11)

We have made no assumption in arriving at Eq. (4.11) which is a general steady

state solution of Eq. (4.8). Moreover, its form depicts a very simple but instructive

aspect of abrupt spin textures compared to wide and smoothly varying textures or

homogeneous ferromagnets.

First, in the LHS of Eq. (4.11), the third term represents a spatially dependent

renormalisation of the spin diffusion length represented by an effective spin diffusion

length λeff given by

1

λ2eff

=1

λ2sf

(1 + G(r)

λ2sf

W 2

). (4.12)

This renormalisation which is due to spin diffusion is as expected strongest at the

domain wall centre and reduces outwardly as represented by the coefficient G as in

the case of Bloch wall and vortex structure [see Section 4.3].

Second, the renormalisation of the spin diffusion length (the effective spin diffusion

length is reduced within the domain wall for abrupt textures), represents an effec-

tive enhancement of the non-adiabatiticity of the texture represented at first order

approximation as

βeff = β0

(1 + G(r)

λ2sf

W 2

). (4.13)

The complete picture of the effect of spin diffusion can be obtained after solving for

105

Figure 4.4: (Color online) Spatial profile of the y- (a,c) and z-components (b,d) ofthe magnetization of a typical Bloch wall (a,b) and vortex core (c,d).

the non-equilibrium spin density from Eq. (4.11) and using Eq. (4.7) to obtain the

torque. Without going into much details of the analytics, the spin torque is given by

T = (ηbJ − ηdiff)∇xm− (βbJ + βdiff)m×∇xm, (4.14)

where η = (1 +χξ)/(1 + ξ2) and β = β0/(1 + ξ2) are the adiabatic and non-adiabatic

torque in the absence of spin diffusion respectively [this represents the spin torque

obtained from the spin density given by equation Eq. (4.9)]. The contribution of

spin diffusion to the adiabatic and non-adiabatic torque are ηdiff and βdiff respectively

given as

ηdiff =DW 2 [ηBa + βBn] and βdiff =

DW 2 [ηBn − βBa]. (4.15)

Notice that spin diffusion reduces (enhances) the adiabatic (non-adiabatic) torque

therefore enhances the non-adiabaticity parameter.

106

5 10 15 20 251

1.5

2

Domain wall width (nm)

No

rma

lize

d v

elo

city (a)

λ ex

(nm)

0.3

0.5

0.8

1.0

0.5 (Numerical)

5 10 15 20 25

0.99

0.995

1

Domain wall width (nm)

No

rma

lize

d v

elo

city (b)

λ ex

(nm)

0.3

0.5

0.8

1.0

0.5 (Numerical)

0.5 1 1.5 2 2.5

0.1

0.2

0.3

λex

(nm)

No

n−

ad

iab

aticity

(c)

W=1nm

3nm 5nm 7nm

9nm

Figure 4.5: (Color online) Normalized velocity of a one dimensional Bloch wall, belowv<x (a) and above Walker breakdown v>x (b). Numerical and analytic results areshown with dotted and solid lines respectively. The parameters used are α = 0.005and β0 = 0.01, χ = 0.2. (c) Spin-mistracking non-adiabaticity as a function of thetransverse spin diffusion length λex for decreasing domain wall widths, calculatedbased on Ref. [6].

Therefore, it appears clear that the standard torque can be significantly modified

by effects due to spin diffusion which only becomes dominant over ballistic spin mis-

tracking for abrupt spin textures or vortex structures. In particular, for long spin

precession lengths (in the case of weak ferromagnets) or abrupt spin textures (strong

anisotropy or vortex structure), there is an enhancement of non-adiabaticity as the

spin de-phasing is not strong enough to compensate the mis-tracking due to the spin

precession.

In the following section we combine fully numerical and analytical treatment of the

domain wall velocity for a Bloch wall and a vortex structure to uncover the importance

of corrections due to spin diffusion. Our Numerical treatment entails (i) solving

Eq. (4.8) using Comsol Multiphysics software, (ii) calculate the spin torque and its

components using the torque definition in Eq. (4.7), (iii) calculate the velocities of

the domain wall using Thiele’s formalism for spin textures [147].

4.3 Current-induced domain wall motion

We study the influence of the spin diffusion on two typical magnetization patterns:

a Bloch wall and an vortex structure with magnetization profile as shown in Fig.

107

4.4. In both cases, the magnetization dynamics is governed by the Landau-Lifshitz-

Gilbert-Slonczewski (LLGS) equation

∂tm = −γm×Heff + αm× ∂tm + T, (4.16)

where γ is the gyromatic ratio, α is the Gilbert damping constant, T is the spin

transfer torque. Heff is the effective field given by

Heff = 2A∇2m + [Hz + (Hk +Hdz) cos θ]ez

+Hdx cosφ sinφex +Hdy sinφ sin θey, (4.17)

where A is the exchange constant, Hz is the external applied field along the z-axis,

Hk is the uniaxial anisotropy field and Hdi(i = x, y, z) are the demagnetizing field

along the i axis whose relative strength depends on the dimensions of the nanowire

and satisfies the relation∑

iHdi = −4πMs, where Ms is the saturation magnetization

of the ferromagnet.

4.3.1 Transverse wall

We focus on the case of perpendicularly magnetized systems, as the case of in-

plane magnetized systems produces similar results. In a perpendicularly magnetized

nanowire, a one dimensional Bloch wall is described by θ(x) = 2 tan−1(e(x−X)/λdw

), φ =

φ(t), where X and λdw are the wall centre and width respectively. Figs. 4.4(a) and

4.4(b) shows the y- and z-components respectively of the magnetization of the wall.

To calculate the geometric scalars G , Ba ,Bn and B, we proceed by gradient expansion

method which entails starting with the zeroth order solution of Eq. (4.9) for λdws to

108

5 10 15 20 251

5

10

15

Domain wall width (nm)

No

rma

lize

d v

elo

city (a)

λ ex

(nm)

0.3

0.5

0.8

1.0

0.5 (Numerical)

5 10 15 20 25

0.94

0.96

0.98

1

Domain wall width (nm)

No

rma

lize

d v

elo

city (b)

λ ex

(nm)

0.3

0.5

0.8

1.0

0.5 (Numerical)

5 10 15 20 250

0.1

0.2

Domain wall width (nm)

Po

lar

an

gle

(ra

d)

(c) λ ex

(nm)

0.3

0.5

0.8

1.0

0.5 (Numerical)

Figure 4.6: (Color online) Normalized transverse vy (a) and longitudinal vx (b) ve-locities of the vortex core. (c) Polar angle of the vortex core as a function of thecore radius. Numerical and analytic results are shown with dotted and solid linesrespectively. The parameters used are R = 150 nm, α = 0.005, β0 = 0.01 andχ = 0.2.

get

−D∇2δs =Dλ2

dw

[ξ(3 sin2 θ − 1)∇xm + cos 2θm×∇xm

].

(4.18)

Next, Eq. (4.18) is rewritten in the form of Eq. (4.10) and comparing coefficients of

the basis vectors with Eq. (4.10) we obtain G = ξ(3 sin2 θ − 1) , Ba = bJ τex(1− ξ)G ,

Bn = bJ τex(G + cos 2θ) and B = 0.

The dynamics of the domain wall is given by the coupled differential equations

∂τX =λdw

sin θ[αγHθ + γHφ + T · eθ − αT · eφ], (4.19)

∂τφ =1

sin θ[αγHφ − γHθ + T · eφ + αT · eθ], (4.20)

where τ = t/(1+α2), eφ = cosφey−sinφex , eθ = cosφ cos θex+sinφ cos θey−sin θez,

Hθ,φ = Heff · eθ,φ and T is the total spin transfer torque calculated from Eq. (4.14).

In the absence of external applied field, at the lowest order, we obtain the average

velocity of the domain wall below the Walker breakdown (∂τφ = 0) as

v<x ≈ − βα

(1 +

1

3β∗[1− 2ξ]

λ2sf

λ2dw

)bJ , (4.21)

109

where β∗ = η − βξ.

Above the Walker breakdown (∂τφ 6= 0), the average velocity of the domain wall

is given by

v>x ≈ −bJ1 + α2

(η + αβ − 1

3β[η∗ − αβ∗[1− 2ξ] + 2ξη]

λ2sf

λ2dw

),

(4.22)

where η∗ = ηξ + β.

The additional non-adiabaticity given by the second term in the RHS of Eq.

(4.21) is governed by the ratio λ2sf/λ

2dw, and since the velocity of the domain wall

is governed by the non-adiabatic torque below the Walker breakdown, spin diffusion

only significantly affects the longitudinal wall velocity below the Walker breakdown

by enhancing the effective non-adiabaticity [cf. Eq. (5.12)].

Figs. 4.5(a) and 4.5(b) display analytical and numerical results of the velocities of

a Bloch wall as a function of the domain wall width λdw. The velocities are normalized

to the case without spin diffusion (wide and smoothly varying texture). Below the

Walker breakdown [Fig. 4.5(a)], the velocity is moderately affected by the domain

wall width. For example, when λdw = 5 nm, the normalized velocity increases by

about a factor of 2 for λex = 0.8 nm (equivalent to NiFe with Jsd ≈ 0.5 eV, see also

Ref. [229]). Above the Walker breakdown [Fig. 4.5(b)], the velocity is simply not

affected by the domain wall width (less than 1%). As can be seen from the numerical

calculations for λex = 0.5nm, the numerical calculations are in very good agreement

with our analytical predictions.

To assess the importance of the spin diffusion mechanism compared to the ballistic

spin mis-tracking mechanism, we numerically calculated the non-adiabaticity caused

by the ballistic spin mistracking based on Ref. [6]. The non-adiabaticity parameter

evaluated numerically is defined as the ratio between the non-adiabatic torque and the

110

adiabatic torque at the center of the wall. From Fig. 4.5(c), we see that the estimated

contribution of the ballistic spin mis-tracking to the non-adiabaticity remains very

limited and since most of the ferromagnetic materials used in experiments are strong

ferromagnets (λex < 0.8 nm), this torque has a sizable influence for abrupt domain

walls only.

4.3.2 Vortex structure

In a two-dimensional magnetic stripe, a vortex structure can be described by θ(x, y) =

2 tan−1(r/r0) for r =√x2 + y2 ≤ r0, θ = π/2 for r0 < r ≤ R, and φ = Arg(x, y) +

π/2, where r0 (R) is the inner (outer) radius of the vortex core. Figs. 4.4(c) and

4.4(d) show the y- and z-components of the magnetization. As in the case for the

Bloch wall, we calculate the geometric scalars G , Ba and Bn by gradient expansion

method to obtain

−D∇2δs =Dr20

2r20

r2 sin2 θ [ξ∇xm + m×∇xm] , (4.23)

and the geometric scalars G =2r2

0

r2 ξ sin2 θ , Ba = bJ τex(1 − ξ)G and Bn = bJ τex(1 −

ξ)G/ξ.

To extract the velocities, we use Thiele’s description in which Eq. (4.16) can be

expressed in the form of the sum of forces exerted on the wall [147]. By multiplying

Eq. (4.16) on the left by m× and projecting the obtained equation on −∂im one

obtains

−∫

Ω

[∂tm · (α∂im + m× ∂im)] (4.24)

=

∫Ω

[∂im · γHeff + T · (m× ∂im)] ,

for i = x, y. The integral∫

ΩdΩ runs over the volume Ω of the magnetic stripe.

111

In the absence of external applied field, after some algebra, the longitudinal and

transverse velocities of the vortex core are given by

vx = − bJ1 + α2

[η + αβ −Rβ (η∗ − αβ∗) λ2

sf

r20

], (4.25)

vy =bJ

1 + α2

[β − αη +Rβ (β∗ + αη∗)

λ2sf

r20

], (4.26)

where R is a geometric factor given by R = [17/3− r20/R

2]/[1 + ln(R/r0)].

Figs. 4.6(a) and 4.6(b) display the transverse and longitudinal velocities of the

vortex core as a function of the core radius r0. While the longitudinal velocity is not

significantly affected, the transverse velocity is dramatically enhanced in the presence

of spin diffusion. For r0 = 5 nm, the transverse velocity can be increased by a factor of

10 for a transverse spin diffusion length λex = 0.8 nm. This key result shows that spin

diffusion cannot be neglected for vortex spin textures, in contrast to previous belief

where only much more abrupt spin structures were considered to lead to an enhanced

β. This indicates that the traditional way to extract β must be reconsidered and

that a more in-depth analysis of the velocities needs to be performed. This can be

phenomenologically understood as the re-normalization of the effective spin diffusion

length as itinerant electrons move across the vortex core represented by

λeff = λsf/

√1 +

8ξr40

(r2+r20)2

λ2sf

r20

. (4.27)

This shows that as itinerant electrons flow across the vortex core, their effective spin

diffusion length reduces by a factor of√

1 + 8ξλ2sf/r

20 which represents an equivalent

enhancement of the non-adibaticity parameter. As can be seen from the numerical

calculations for λex = 0.5nm, the numerical calculations are in very good agree-

ment with our analytical predictions. It should be noted however that the analytical

treatment does not include higher order terms which are taken into account in the

112

numerical calculations, resulting in the slight deviation at very short core radius.

Another way to extract the non-adiabaticity parameter is to estimate the polar

angle tan−1(vy/vx) acquired by the vortex core after current injection [86]. As shown

in Fig. 4.6(c), the polar angle is dramatically enhanced at small core sizes due to the

transverse spin diffusion. Therefore, for vortex cores, the transverse spin diffusion

provides a sizable contribution to the non-adiabaticity.

4.4 Discussion

Up to now, only spin-flip and ballistic mis-tracking were known to be responsible

for the non-adiabatic torque. In Section 4.2.1, we show that in a disordered system

even in the absence of explicit spin-flip, spin de-phasing induced by weak disorder

dramatically enhances the non-adiabaticity parameter. This is interpreted in terms of

an Elliott-Yafet-type relaxation where the magnetic texture plays the role of spin-orbit

coupling. Increasing the amount of disorder (or reducing the width of the domain wall)

results in an increase of the non-adiabaticity. In the case of strong disorder, the system

is dragged towards the diffusive regime (i.e. the mean free path becomes smaller than

the system size). However, as mentioned in Section 4.2.1, modeling such a regime

using a quantum mechanical tight-binding model is computationally demanding and

we adopt a semi-classical drift-diffusion approach to explore this regime. Analytical

expressions of the non-adiabaticity are derived in the diffusive limit and we explicitly

show that the spin diffusion length is enhanced by the conjunction of spin diffusion

and sharp magnetic texture [see Eq. (4.12)], resulting in a larger non-adiabaticity

parameter [see Eq. (5.12)], which is consistent with the results from the tight-binding

model. Our calculations [see Figs. 4.6(b) and 4.6(c)] are consistent with the large

non-adiabaticities measured in Refs. [84, 85, 86] for vortex cores in NiFe. In these

structures the transverse spin diffusion length is about λex ≈ 0.8 nm [219, 220, 221,

113

229] and the radius r0 ≈ 5 nm, which yields effective velocities about 10 times larger

than in the absence of spin diffusion. Although the length scale for which our model is

applicable lies within the borderline regime of validity of the drift-diffusion formalism,

our calculations provide a consistent explanation of the enhancement of the non-

adiabaticity parameter and hence domain wall velocity between the mis-tracking-

dominated (clean) and spin diffusion-dominated regime (strong disorder). Finally, in

magnetic wires with strong perpendicular magnetic anisotropy, such as Co/Ni, FePt

or Co/Pt [90, 215], the domain wall width is quite small (∼ 5 nm) and therefore one

would expect the non-adiabaticity to be quite large. Nonetheless, these materials are

strong ferromagnets and therefore the mis-tracking between the itinerant spin and the

magnetic texture is quite weak, resulting in a weak correction to the non-adiabaticity

(β ∼ α).

4.5 Conclusion

The role of diffusion of spin accumulation in current-induced domain wall motion has

been studied theoretically using the drift diffusion model. In particular, we consider

an abrupt Bloch wall and a vortex structure and investigate how spin diffusion affects

the non-adiabaticity parameter and hence the velocity of the wall. Our results show

that while spin diffusion only has a moderate effect on the longitudinal velocity of a

Bloch wall, it significantly enhances the transverse velocity of the vortex core up to

an order of magnitude. We show that for abrupt spin textures, the diffusion of spin

accumulation produces a supplementary spatially dependent torque which depends on

the topology of the texture and is proportional to the square of the ratio of the effective

spin diffusion length and the domain wall width. Our results uncover the significant

difference between a transverse wall and a vortex core: in the latter, the abrupt texture

results in a stronger re-normalization (reduction) of the effective spin diffusion length

114

resulting in a stronger enhancement of the non-adiabaticity parameter.

115

Chapter 5

Intrinsic non-adiabatic topological

torque

We propose that topological spin currents flowing in topologically non-trivial magnetic

textures, such as magnetic skyrmions and vortices, produce an intrinsic non-adiabatic

torque of the form Tt ∼ [(∂xm×∂ym)·m]∂ym. We show that this torque, which is ab-

sent in one-dimensional domain walls and/or non-topological textures, is responsible

for the enhanced non-adiabaticity parameter observed in magnetic vortices compared

to one-dimensional textures. The impact of this torque on the motion of magnetic

skyrmions is expected to be crucial, especially to determine their robustness against

defects and pinning centers.

5.1 Introduction

The search for the efficient electrical manipulation of magnetic textures has recently

received a major boost with the observation of ultrafast current-driven domain wall

motion in non-centrosymmetric transition metal ferromagnets [115, 100, 170, 173],

and very low de-pinning current threshold of bulk magnetic skyrmions [230, 231, 232].

The latter are topological magnetic defects [59] that present some similarities with

the more traditional magnetic vortices [233, 84, 85, 86]. In both cases, the magnetic

116

topology induces a Lorentz force on the flowing electrons, resulting in topological

Hall effect [234, 59] (see also Ref. [235]). Both classes of magnetic textures also

experience a Hall effect when driven by an electric flow, an effect sometimes referred

to as skyrmion Hall effect [59]. In fact, it is important to emphasize that the current-

driven characteristics of skyrmions are mostly similar to that of magnetic vortices.

For instance, the universal current-velocity relation [236, 31], as well as the colossal

spin transfer torque effect at the edge [237] have already been predicted in the case of

magnetic vortices [233]. Notwithstanding, skyrmions display some striking differences

with respect to magnetic vortices. As a matter of fact, vortex walls are non-local

objects composed of a vortex core and two transverse walls [233, 84, 85, 86], expanding

over several exchange lengths in the wire. In contrast, skyrmions are localized objects

with a limited expansion from a few tens to hundred nanometers [59]. This localization

has a dramatic impact on the manner skyrmions behave in the presence of pinning

centers (defects, notches etc.). When driven by a current, they can deform their

texture, and thereby avoid pinning [236, 31, 237, 32, 238, 239]. In contrast, magnetic

vortices are much more sensitive to defects and get easily pinned.

Since the robustness of skyrmions with respect to defects might hold the key for

efficient ultra-dense data storage [30], it is crucial to develop a precise understanding

of the nature of the torque exerted on these objects. Recently, Iwasaki et. al. have

shown that such a robustness is partly due to the presence of a large non-adiabatic

torque [236]. Indeed, in magnetic textures the spin transfer torque can be generally

expressed as

T = −bJ(u · ∇)M +βbJ

Ms

M× (u · ∇)M, (5.1)

where u is the direction of current injection, m = M/Ms is a unit vector in the

direction of the magnetization M and Ms is the saturation magnetization. The first

term in Eq. (5.1) is the adiabatic torque while the second term is the non-adiabatic

torque [74, 75]. It is well known that while the non-adiabatic parameter in transverse

117

walls is quite small β ≈ α (α being the magnetic damping of the homogeneous

magnet), it is much larger in magnetic vortices, β ≈ 8 − 10α [84, 85, 86, 92]. To

the best of our knowledge, there is currently no available estimation of the non-

adiabaticity parameter in skyrmions, but one can reasonably speculate that it should

be of the same order of magnitude as that of magnetic vortices. The recent avalanche

of experimental observations of magnetic skyrmions in transition metal multilayers

might soon shed light on this question [177, 240, 241, 242, 178].

That being said, the nature of non-adiabaticity in skyrmions and vortex walls has

been scarcely addressed. In transverse domain walls, two major physical mechanisms

have been uncovered: spin relaxation [74] and spin mistracking [88]. Spin relaxation

produces a non-adiabatic torque β ∼ τex/τsf , where τex is the precession time of

the spin about the local magnetization, while τsf is the spin relaxation time. Spin

mistracking is the quantum mis-alignement of the flowing electron spin with respect

to the local magnetic texture and provides a non-adiabaticity parameter β ∼ e−ξλdw

that exponentially decreases with the domain wall width λdw. It is therefore limited

to extremely (atomically) sharp domain walls [212, 6]. In Chapter 4, we also showed

that spin diffusion enhances the non-adiabatic torque when the domain wall width is

of the order of the spin relaxation length [91]. However, none of these effects explains

the large difference in non-adiabaticity between transverse walls on the one hand

and skyrmions and vortices on the other hand. In a recent work, we proposed that

the topological currents induced by the topological Hall effect can enhance the non-

adiabatic parameter in magnetic vortices [92]. Such an intimate connection between

topological Hall effect and non-adiabaticity has also been pointed out by Jonietz et

al. [230], but to the best of our knowledge, no theoretical work addresses this issue

thoroughly, and all micromagnetic simulations on skyrmions dynamics so far assume

a constant β parameter [236, 31, 237, 32].

As an in-depth follow-up of Ref. [92], the present work investigates the nature

118

of the non-adiabaticity in skyrmions analytically and numerically, and demonstrates

that the texture-induced emergent magnetic field inherent in these structures, induces

a non-local non-adiabatic torque on the magnetic texture. We provide an explicit ex-

pression of the torque and highlight its connection with the spin and charge topological

Hall effect.

5.2 Emergent electrodynamics

5.2.1 Premises

It is well-known that when conduction electrons flow in a smooth and slow magnetic

texture, m(r, t), their spin adiabatically changes orientation so that these electrons

acquire a Berry phase [243, 244, 245, 246]. This geometrical phase is attributed to

an emergent electromagnetic field (Eem,Bem) determined by the magnetic texture

gradients [247, 248, 201, 249, 212, 250, 251]. Indeed, a time-dependent magnetic

texture induces local charge and spin currents through the action of the so-called spin

electromotive force [247, 201, 191, 252, 253, 254, 255]. For the sake of completeness,

we derive below this emergent electromagnetic field. Let us considering the simplest

Hamiltonian of an s-d metal in the presence of a smooth magnetic texture given as

H =p2

2m+ Jsdσ ·m(r, t). (5.2)

The Schrodinger equation corresponding to Eq. (5.2) can be re-written in the rotating

frame of the magnetization, using the unitary transformation U = e−iθ2σ·n where

n = z×m/|z×m| to obtain

H =(p− eA)2

2m+ Jsdσz + eV , (5.3)

119

where the vector and scalar potentials are given respectively as A = − ~2eσ · (m ×

∂im)ei and V = ~2eσ · (m × ∂tm). As a consequence, the spin-polarized carriers feel

an emerging electromagnetic field on the form [255, 248, 201, 249, 212, 250, 251, 252,

253, 191, 254]

Esem =

s~2e

[m · (∂tm× ∂im)]ei, (5.4a)

Bsem = −s~

2e[m · (∂xm× ∂ym)]z. (5.4b)

The electric field is proportional to the first derivative in time and space and therefore

a moving magnetic texture induces a charge current [201, 249, 212, 250, 251] and a

self-damping [191, 254]. The effect of the magnetic field has so far been overlooked

as it requires a second order derivative in space, and is generally considered small.

Interestingly, this emergent magnetic field turns out to be crucial to understand the

spin transport involved in these textures. Indeed,it creates a local ordinary Hall

current such that the spin-dependent local charge current driven by the external

electric field reads [92]

jse = σs0E + σs0Esem +

σsHBH

E×Bsem +

σsHBH

Esem ×Bs

em, (5.5)

where σs0 and σsH are respectively the spin-dependent longitudinal and ordinary Hall

conductivities, E is the external electric field, and BH = |Bsem|. Inspecting Eq. (5.5),

we note that there are two sources of charge or spin currents (i) through the external

electric field E and (ii) through the emergent electric field driven by the time-variation

of the magnetic texture, Esem. Therefore, our calculation is able to capture the physics

of the motion of the itinerant electrons [Topological (spin) Hall effect] or the magnetic

texture itself [skyrmion (vortex) Hall effect]. Let us now assume a rigid magnetic

structure for which the time derivative of the magnetization is such that ∂tM =

−(v · ∇)M, where v = vxx + vyy is the velocity of the magnetic structure, and,

120

without loss of generality, an electric field applied along x such that E = Ex. We

can obtain the expressions for the local spin current tensor, Js = M⊗ (j↑e − j↓e), and

charge current vector, je = j↑e + j↓e, from Eq. (5.5). Explicitly, we obtain

J s =[bJ + λ2

E

(vy − λ2

HvxNxy)Nxy

]M⊗ x (5.6a)

−[PH

P0

λ2HbJ + λ2

E

(vx + λ2

HvyNxy)]NxyM⊗ y,

je = σ0

[E +

~P0

2e

(vy −

PH

P0

λ2HvxNxy

)Nxy

]x (5.6b)

−σ0

[λ2

HE +~P0

2e

(vx +

PH

P0

λ2HvyNxy

)]Nxyy,

where bJ = ~P0σ0E/2|e|Ms, σa = σ↑a + σ↓a, Pa = (σ↑a − σ↓a)/σa, (a = 0, H), λ2E =

~2σ0/4e2Ms, λ

2H = ~σH/2|e|σ0BH, and Nµν(r) = m · (∂µm×∂νm) where ν, µ ∈ (x, y).

Finally, ⊗ is the direct product between spin space and real space. Eqs. (5.6a) and

(5.6b) constitute a central result in this work. They contain information about the

motion of itinerant electrons as they traverse a smooth magnetic texture. Indeed, in

its explicit form, one is able to explain the subtle difference between non-adiabatic

transport in one- and two-dimensional textures. In particular, in addition to the usual

constant adiabatic spin current moving along the direction of the applied current

(∼ x) common in one-dimensional textures, the presence of a non-zero topological

charge, Nxy 6= 0 (as it is the case for magnetic skyrmions and vortices), leads to a

texture-induced emergent magnetic field that induces an additional spatially varying

spin current along both the longitudinal (∼ x) and transverse (∼ y) directions to the

electric field E. This longitudinal spin current is responsible for (i) topological spin

and charge Hall effects already observed in topological textures such as skyrmions

[248, 234, 59] (see also Ref. [235]) and as we propose, (ii) enhanced non-adiabaticity

already observed in vortices [92]. To clarify a potential mis-conception, we note

121

here that magnetization variation in more than one direction is a necessary but not

sufficient condition to observe these effects. The sufficient condition is a non-zero

topological charge (Nxy 6= 0), which is the case for magnetic textures such as vortices

and skyrmions.

The topological charge and spin Hall effects arising from the magnetic texture

induced-emergent electromagnetic field given by Eq. (5.4) can be quantified by the

charge and spin Hall angles defined respectively as θTH = tan−1(∫

jyed2r/∫jxe d

2r)

and θTSH = tan−1(

2e~

∫J ys d

2r/∫jxe d

2r)

to obtain

θTH ∼ −Qλ2H and θTSH ∼ QPHλ

2H, (5.7)

where Q is the topological number defined as Q = 14π

∫Nxyd2r. From Eq. (5.7),

one can straightforwardly deduce that θTSH ≈ −PHθTH which, although very simple,

turns out to be far reaching as it captures most of the important physics in static

magnetic textures as confirmed by our numerical calculations in the following section.

5.2.2 Topological spin torque

In the previous section, we reminded the basics of the topologically-driven spin and

charge currents in magnetic textures. Let us now investigate the impact of this spin

current on the dynamics of the magnetic texture itself. By virtue of the spin transfer

mechanism, this spin current exerts a torque on the local magnetization, Tt = −∇·Js,

which explicitly reads

Tt = −[bJ + λ2

E

(vy − λ2

HvxNxy)Nxy

]∂xM (5.8)

+

[PH

P0

λ2HbJ + λ2

E

(vx + λ2

HvyNxy)]Nxy∂yM.

122

A remarkable consequence of Eq. (5.8) is that, since ∂ym ∼ m × ∂xm in magnetic

vortices and skyrmions, the finite topological charge (Nxy 6= 0) induces an intrinsic

topological non-adiabatic spin transfer torque. This non-adiabatic torque is intrinsic

as it does not rely on impurities or defects (in contrast with the non-adiabaticity

studied in Refs. [74, 91]), and topological since its origin is associated to the topology

of the magnetic texture.

To quantify these effects, we study the dynamics of an isolated magnetic skyrmion

and vortex under the action of the torque given in Eq. (5.8) in the context of Thiele

formalism of generalized forces acting on a rigid magnetic structure [147]. The equa-

tion of motion governing the dynamics of these structures is given by the extended

LLG equation

∂tM = −γM×Heff +α

Ms

M× ∂tM + T (5.9)

where the torque T is given as

T = Tt +βbJ

Ms

M× ∂xM (5.10)

with β being a spatially constant non-adiabatic parameter arising from, e.g., spin

relaxation [74]. To make our analysis simple without missing any interesting physics,

we adopt the magnetization profile of an isolated skyrmion and vortex core in spherical

coordinates as m = (sin θ cos Φ, sin θ sin Φ, cos θ), where the polar angle θ is defined

for an isolated skyrmion as cos θ = p(r20 − r2)/(r2

0 + r2), and for an isolated vortex as

cos θ = p(r20 − r2)/(r2

0 + r2) for r ≤ r0, and θ = π/2 for r > r0. p = ±1 defines the

skyrmion (vortex core) polarity, r0 defines the radius of the skyrmion (vortex) core.

For both textures, the azimutal angle is defined as Φ = qArg(x + iy) + cπ/2, where

q = ±1 is the topological charge and the c = ±1 defines the in-plane curling direction

otherwise called the chirality. For these magnetic profiles, the topological number

123

Q = 14π

∫Nxyd2r is such that Q = 1

2pq for an isolated vortex core and Q = (1−S)pq

for an isolated skyrmion, where S = r20/(r

20 + R2) → 0 for R r0. Using these

profiles, we obtain the analytical expressions of the velocity components as

vx = −ηeff + αeffβeff

η2eff + α2

eff

bJ , (5.11a)

vy = pqηeffβeff − αeff

η2eff + α2

eff

bJ , (5.11b)

where the effective paremeters ηeff , βeff and αeff depend on the magnetic texture.

For the sake of completeness only, we consider the effect of the re-normalization

of the gyromagnetic ratio represented by ηeff which is of order of unity and equals

1− 16S4

5

λ2E

r20

λ2H

r20

and 1− 315

λ2E

r20

λ2H

r20

for an isolated skyrmion and vortex respectively, where

Sk =∑i=k

i=0(S)i. However, this is not the focus of this study as this effect is very small

and can be neglected.

5.2.3 Non-adiabaticity parameter

The dynamics given by Eqs. (5.11a) and (5.11b) correctly describes the motion of

both an isolated skyrmion and vortex core. The dynamics of these two structures is

very similar, the major difference being contained in the effective parameters ηeff , βeff

and αeff . The first effective parameter of interest in this study is βeff , which provides

a direct connection between emergent field-induced topological Hall effect and non-

adiabaticity. Indeed, while there has been much discussion about the mechanisms re-

sponsible for non-adiabatic spin transfer in the literature [74, 88], the proposed mech-

anisms although very successful in describing non-adiabatic effects in one-dimensional

domain walls, have failed to address the large non-adiabaticity measured in vortex

walls. Here, we show that emergent-field induced topological torque gives rise to an

124

additional non-adiabatic torque and a resulting effective non-adiabaticity parameter

given as

βeff =

β + 4S2

3PH

P0

λ2H

r20, for skyrmion

βC + 73PH

P0

λ2H

r20, for vortex core,

(5.12)

where the geometric factor C = 1 + ln√R/r0. Eq. (5.12) reveals that associated

with these textures is an intrinsic non-adiabaticity parameter which results in an

overall enhancement of their one-dimensional and/or non-topological counterpart, β.

This enhancement of the non-adiabaticity parameter is a direct consequence of the

topology-induced transverse spin current J ys , as shown in Eq. (5.6a).

5.2.4 Damping parameter

In the context of magnetic textures dynamics, it is practically impossible to discuss

the non-adiabaticity parameter βeff without a mention of the damping parameter αeff

as both parameters govern the motion of these structures, see Eq. (5.11b). Different

mechanisms for magnetic dissipation have been proposed [186, 187, 188, 189, 256,

191, 254]. Our calculation reveals that a non-steady state magnetization induces an

emergent electric field that results in an intrinsic damping solely due to the topological

nature of these textures and thus an overall enhancement of the damping given as

αeff =

α + 4S2

3

λ2E

r20, for skyrmion

αC + 73

λ2E

r20, for vortex,

(5.13)

which is consistent with Ref. [254].

These result are far reaching as they provide a very transparent mechanism which

explains the subtle difference in the measured non-adiabaticity in vortices compared

one-dimensional domain walls [92]. However, as we have shown, two dimensional

magnetization variation is a necessary but far from a sufficient condition for this

125

effect, with the sufficient condition being a non-zero topological charge (Nxy 6= 0) as

discussed in preceding section.

5.3 Numerical results

5.3.1 Tight-binding model

The derivation proposed in the previous section relied on the adiabatic transport

of conduction electrons in a smooth and slowly varying magnetic textures. In other

words, the spin of the conduction electrons remains aligned on the local magnetization

direction and no spin mis-tracking is considered [88]. In principle, these formulas do

not hold when the magnetic texture becomes sharp (i.e. when the skyrmion size is of

the order of the spin precession length). In this section, we verify our analytical results

numerically using a tight-binding model to test the validity of the adiabatic model

discussed in the previous section. The local spin/charge densities and currents as well

as the corresponding spin transfer torque are computed numerically using the non-

equilibrium wave function formalism [164]. The system is composed of a scattering

region containing an isolated skyrmion or vortex core, attached to two ferromagnetic

leads as shown in Figure 5.1. We model this system as a two dimensional square

lattice with lattice constant a0, described by the Hamiltonian

H =∑µ

εµc+µ cµ − t

∑<ij>

c+µ cj − Jsd

∑µ

c+µmµ · σcµ, (5.14)

where εµ is the onsite energy, t is the hopping parameter, the sum < ij > is restricted

to nearest neighbours, mµ is a unit vector along the local moment at site i coupled

by exchange energy of strength Jsd to the itinerant electrons with spin represented by

the Pauli matrices σ. The label i and j represent the lattice site and c+µ = (c↑µ, c

↓µ)+

is the usual fermionic creation operator in the spinor form. The z-axis is chosen

126

Figure 5.1: Schematic diagram of tight-binding system made up of a central scatteringregion (isolated magnetic skyrmion/vortex) attached to two ferromagnetic leads (redboxed) L and R at chemical potentials µL and µR respectively. To ensure smoothmagnetization variation from the leads to the scattering region, we consider an optimalsystem size compared to the radius (R r0).

as the quantization axis while the x-axis is the direction of current flow. For all

our numerical calculations except stated otherwise, we used the parameters: hopping

constant t = 1, exchange energy Jsd = 2t/3, onsite energy εµ = 0, transport energy

εF = −4.8Jsd, bias eV = 0.1Jsd, and a large system size of 401 × 401a20 to ensure

smooth magnetization variation from system to leads to avoid unphysical oscillations

of torques close to the leads. The local spin density Sαn at site n and local spin current

density Jαn−1 between site n− 1 and n at a particular transport energy for electrons

from the α-lead can be calculated from their respective operators

Sαn =~2

∑ν

Ψα↑∗nν

Ψα↓∗nν

T

σ

Ψα↑nν

Ψα↓nν

(5.15)

127

and

Jαn−1 =1

2i

∑ν

tnν,n−1ν

Ψα↑∗nν

Ψα↓∗nν

T

σ

Ψα↑n−1ν

Ψα↓n−1ν

+ c.c, (5.16)

where Ψασnν are ν propagating mode of the spin-σ wave functions from the α-lead at

site n. The quantum mechanical average is calculated by integrating over the small

energy window εF − eV2 and εF + eV

2 as

〈Sn〉 =∑α

∫dε

2πfαS

αn, (5.17)

where fα is the Fermi-Dirac function for lead-α. A similar formula apply for 〈Jn−1〉.

The corresponding charge density and current density can be obtained from Eqs.

(5.15) and (5.16) by replacing ~2σ by eI where I is the 2× 2 identity matrix.

5.3.2 Results and discussion

Our objective in this section is to ascertain the source of the different contributions to

the adiabatic and non-adiabatic torques to uncover their respective microscopic origin

and establish a direct correspondence with our theoretical predictions in Section 5.2.

The most natural and reliable method for calculating the torque Tn at site n on the

local moment mn is by using the local spin densities as

Tn =2Jsd

~〈Sn〉 ×mn, (5.18)

since this is a conserved quantity, and therefore well defined. However, this method

gives little or no information about the microscopic origin of the torque. Therefore,

we also calculate the torque from the local spin current as

Tn = 〈Jn〉 − 〈Jn−1〉. (5.19)

128

Figure 5.2: Two-dimensional profile of torque components showing contributions fromdifferent spin current sources. The adiabatic torque is dominated by the contributionfrom the longitudinal spin current (a) compared to the transverse spin current (b).The converse is true for the non-adiabatic torque which is dominated by contributionfrom the transverse spin current (d) compared to the longitudinal spin current (c).

This way, we are able to unambiguously separate the different contributions to the

torque arising from the spin current along y and x, quantify its origin, and compare

our numerical results with our analytical predictions. A caveat to this method how-

ever is that the spin current is a non-conserved quantity especially in systems with

spin-orbit coupling or sharp magnetic textures (i.e. sizable spin mistracking). This

notwithstanding, since our considerations are based on smooth magnetic textures we

can argue that the spin current is well defined and to make sure of this, we used

both definitions of the local torque and ensured that the calculated torque using both

methods are the same. We deduce the local adiabatic T adn and non-adiabatic T na

n

129

Figure 5.3: One-dimensional profile of torque components showing contribution fromdifferent spin current sources and the spin density. The adiabatic torque (a) is domi-nated by the contribution from Jxs while the non-adiabatic torque (b) is dominated bycontribution from Jys . Results shows a very good match between the torque calculatedfrom the spin density (S) and that calculated from the spin current Js = Jxs + Jys .

torque contribution from the calculated local torque Tn by recasting the local torque

in the form Tn = T adn ∂xmn−T na

n mn×∂xmn. Throughout this study except otherwise

stated, the torques are reported in units of 2Jsd/~.

Our numerical results for the two-dimensional profile of spin transfer torque com-

ponents [calculated using the spin current definition in Eq. (5.19)] for an isolated

skyrmion of core radius 10a0 is shown in Figure 5.2. As shown in Figures 5.2 (a) and

130

Figure 5.4: Effective adiabatic and non-adiabatic torque dependence on the skyrmionradius. The adiabatic torque (a) is almost non-dependent on the radius of thesykrmion while the non-adiabatic torque (b) shows substantial dependence on theskyrmion radiusr0. Sfit represents fit of our numerical data to analytical result.

(b), the adiabatic torque is dominated by the contribution of the longitudinal spin

current (Jxs ), which is at least an order of magnitude larger than the contribution

from the transverse spin current (Jys ). The non-adiabatic torque [Figures 5.2 (c) and

(d)], is largely dominated by contribution of the transverse spin current (Jys ). These

results confirm the analysis based on the analytical derivations of the previous section,

Eqs. (5.6a) and (5.8).

Figure 5.3 displays the one-dimensional profile of the torque components obtained

by summing over the transverse direction for a skyrmion radius r0 = 10a0. Con-

sistently with Figure 5.2, these results show that the non-adiabatic torque is domi-

nated by the texture-induced transverse spin current while its adiabatic counterpart

is dominated by the longitudinal spin current. In addition, we also performed these

calculations using Eq. (5.18) (open circles). The results obtained by this method

overlap with the results obtained using Eq. (5.19). This indicates that for skyrmions

with radius r0 ≥ 10a0 the spin mistracking is negligible.

Finally, we investigated the scaling laws governing the different contributions of

131

the topological torque components with respect to the skyrmion radius. To achieve

this, we calculated the normalized adiabatic and non-adiabatic torque components by

integrating the projections of the local torque on m×∂ym and m×∂xm respectively

and normalize accordingly i.e.

Tad =

[∫T · (m× ∂ym)d2r

]/

∫MsNxyd

2r (5.20a)

Tna =

[∫T · (m× ∂xm)d2r

]/

∫pqMsNxyd

2r. (5.20b)

As shown in Figures 5.4(a) and (b), while the adiabatic torque is almost independent

on the skyrmion radius, the non-adiabatic torque shows a substantial dependence

on the skyrmion radius r0 which is in accordance with our analytical results. As a

matter of fact, a simple fit of our numerical data to our analytical result given by Eq.

(5.8), yields an effective non-adiabatic parameter βeff = 3.5β for a skyrmion radius of

10a0, where β is the constant non-adiabaticity parameter obtained in the limit of very

large skyrmion radius. For all the range of skyrmion sizes investigated, we also find

that the adiabatic assumption exploited to derive the torque expression, Eq. (5.8), is

valid. In other words, it confirms that the large non-adiabatic torque in topologically

non-trivial magnetic textures can not be explained by spin mistracking.

5.4 Conclusion

We investigated the nature of adiabatic and non-adiabatic spin transfer torque in

topologically non-trivial magnetic textures, such as skyrmions and magnetic vortex

cores. We showed that the topological spin current flowing through such textures

induce an intrinsic topological non-adiabatic torque, Tt ∼ [(∂xm × ∂ym) ·m]∂ym.

Our numerical calculations confirm the physics highlighted by our analytical deriva-

tions and confirm that spin transport is mostly adiabatic up to a very good accuracy

132

in these structures, thereby ruling out spin mis-tracking. Besides providing a rea-

sonable explanation for the enhanced non-adiabaticity in skyrmions and magnetic

vortices, our theory opens interesting perspectives for the investigation of current-

driven skyrmion dynamics. As a matter of fact, it has been recently proposed that

the peculiar robustness of magnetic skyrmions against defects is related to the pres-

ence of non-adiabaticity [236]. Therefore, understanding the role of this topological

non-adiabatic torque when magnetic skyrmions interact with defects is of crucial

importance to control current-driven skyrmion motion and achieve fast velocities

[230, 231, 232, 177, 178].

133

Chapter 6

Topological transport in

antiferromagnetic skyrmions

6.1 Introduction

As the spintronics community continues to search for high-efficiency, high-density and

low power consuming spintronic devices, new materials other than the conventional

ferromagnets are constantly being explored. Spin-orbit torques have been successfully

used to achieve efficient, ultrafast and reliable magnetization switching in magnetic

hetero-structures made of a bilayer of a ferromagnet on top of a material with strong

spin orbit coupling in the presence of a constant in-plane magnetic field applied

collinear to current. This effect has been widely studied in materials ranging from

heavy metals Pt [257, 258, 259], Ta [260, 261, 262, 263, 264], W [265], Hf [266],

W/Hf [267], Ir-doped Cu [268] to topological insulator (Bi, Sb)Te [269]. However,

the requirement of simultaneously applying a constant magnetic field and current in

order to achieve switching poses a technological challenge leading to the quest for

new material systems.

Recently, spin-orbit transport research has been extended to antiferromagnets

which present the advantage of (i) absence of overall magnetization, and (ii) ultrafast

dynamics. In particular, it has been shown that spin-orbit torques drive antiferro-

134

magnetic domain walls much faster than their ferromagnetic counterpart [120, 119].

Furthermore, large anomalous Hall effect (AHE) have been predicted in non-collinear

antiferromagnet Mn3Ir [270]. The origin of this AHE is attributed to an interplay

between broken time-reversal symmetry and spin-orbit coupling, similar to spin Hall

effect (SHE). Indeed, experimental measurements shows that metallic antiferromag-

nets with strong spin-orbit coupling can induce magnetization switching on an adja-

cent ferromagnetic layer [271, 272]. Finally, inverse spin Hall effect has been observed

in AFM/FM systems [273, 261]. All these observations are promising indications of

the existence of a direct SHE in antiferromagnets capable of inducing magnetization

switching.

Interestingly, it has also been shown recently that magnetic skyrmions, that usu-

ally require a critical current density of 4 to 5 orders of magnitude smaller than

domain walls, can be stabilized in non-centrosymmetric B20 coumponds with strong

Dzyaloshinskii-Moriya interaction [236, 31, 237, 32, 238, 239]. This property in ad-

dition to less sensitivity to defects [239, 274, 275], enhanced non-adiabatic torque

[276] and substantial SHE [277, 235] makes magnetic materials with skyrmionic tex-

tures good candidate for technological applications, including the bilayer structures

discussed above. Although this possibility looks very appealing, such application is

limited by - (i) small charge-to-spin current conversion, (ii) enhanced non-adiabatic

torque that directly influences the undesirable transverse motion of skyrmions. In-

deed, this transverse motion of skyrmions known as skyrmion Hall effect (SkHE)

is a consequence of a non-zero magnus force inherent in ferromagnetic skyrmions.

In a typical current-driven dynamics of ferromagnetic skyrmions, SkHE drives the

skyrmions towards the lateral edge of the nanostructure so that it can eventually get

destroyed for large current densities. It turns out that antiferromagnetic skyrmions

are characterized by a vanishing magnus force ( i.e., the absence of SkHE) [46, 4],

which makes these entities particularly interesting to explore.

135

In this work, we conduct an extensive study of spin-dependent transport in antifer-

romagnetic skyrmions and explore the possibilities of efficient magnetization manipu-

lation in skyrmionic textures with antiferromagnetic order. We show that transverse

topological spin current flowing in antiferromagnetic skyrmions also results in an

enhancement of the non-adiabatic torque which directly influences the longitudinal

motion of the skyrmion. Furthermore, this non-adiabatic torque increases when de-

creasing the skyrmion size, and therefore downscaling results in a much higher torque

efficiency. Our calculations show that for clean systems, a significant TSHE only

occurs very close to the antiferromagnetic band gap which indicates that enormous

material engineering is required to maximize this effect for very clean materials. Nev-

ertherless, the possibility of impurity-induced enhancement of this effect over a wider

energy range is quite appealing since impurities are omnipresent in real materials.

6.2 Emergent electrodynamics for antiferromagnetic

textures

We consider a two-dimensional skyrmionic system with a G-type antiferromagnetic

order on a square lattice with lattice constant a0. In the low energy limit, the

Hamiltonian of the system can be described in the coupled sublattice-spin space

(|A〉, |B〉)⊗ (| ⇑〉, | ⇓〉) given as

H = γkτx ⊗ I2 + Jsdτz ⊗ n · σ, (6.1)

where In is an n × n identity matrix, Jsd is the exchange energy, n = l/|l| is a unit

vector in the direction of the Neel vector l. γk = −2t(cos kxa0 + cos kya0), τ and σ

are the Pauli matrices of the sublattice [(|A〉, |B〉)] and spin [(| ⇑〉, | ⇓〉)] subspaces

respectively. The eigenvalues and eigenstates corresponding to Eq. (6.1) are given

136

respectively as

εs(k) = s√γ2k + J2

sd (6.2)

and

Ψσs =

1√2

[√1 + sσPk|A〉+ s

√1− sσPk|B〉

]⊗ |σ〉, (6.3)

where s = ±1 represents the bands above (s = +1) and below (s = −1) the band gap

and Pk = Jsd/√γ2k + J2

sd is the polarization of the local (sublattice-resolved) density

of states. We follow Ref. [276], by writing the Schrodinger equation corresponding to

Eq. (6.1) in the rotating frame of n, using the unitary transformation U = e−iθ2σ·nu

where nu = z× n/|z× n| to obtain

H =

([p− eA]2

2m− E0

)τx ⊗ I2 + Jsdτz ⊗ σz + eVI4. (6.4)

The vector and scalar potentials in Eq. (6.4) are given respectively as A = − h2e

[ση ·

(n × ∂in)]ei and V = h2e

[ση · (n × ∂tn)]z, where the spin operator is defined as

ση = 12(I2 + ητz)⊗ σ, with η = +1 for the A and η = −1 and for the B sublattices.

A direct consequence of these potentials is that the spin-dependent carriers feel an

emerging electromagnetic (EEM) field for the η-sublattice [46], that can be written

in the basis of the eigenstates in Eq. (6.3) as

Eη,σe,s = σ(1 + sησPk)

~2eNt,iei (6.5a)

Bη,σe,s = −σ(1 + sησPk)

~2eNx,yz, (6.5b)

where Nν,µ = (∂νn × ∂µn) · n [ν, µ ∈ (t, x, y)], σ = +1 for spin-up and σ = −1 for

spin-down. Under the action of an external electric field E which we assume along x

i.e. E = Ex, the resulting spin-dependent local carrier current density is given as

jη,σe,s = ση,σ0,s E + ση,σ0,s Eη,σe,s +

ση,σHBηs

E×Bη,σe,s +

ση,σHBηs

Eη,σe,s ×Bη,σ

e,s , (6.6)

137

where Bηs is a constant with dimension of magnetic induction, ση0,s and σηH,s are the

normal and Hall conductivity respectively. We neglect the fourth term on the right

hand side of Eq. (6.6) which is a higher order correction that simply renormalizes the

gyromatic ratio [276].

In what follows, we consider a rigid magnetic texture such that ∂tn = −(v ·

∇)n, where v is the velocity of the structure. To keep our analysis tractable we

make the following simplifications. (i) We set s = 1, therefore we consider only

the top-band in the rest of this paper. (ii) We define two length scales associated

with the emergent magnetic and electric field as ληH =√~σηH/2eσ

η0B

η and ληE =√~2ση0/4e

2Ms respectively. (iii) We consider the sublattices to be strongly coupled

such that even though there is local polarisation of the density of states (Pk), no spin

current flows in the system i.e ση,+ν = ση,−ν = 12σην = 1

4σν , for ν ∈ (0, H). However,

in the presence of spin-conserving impurity scattering which is omnipresent in real

materials, the last equality does not necessarily hold i.e. σaν 6= σbν . Furthermore, since

this type of impurity scattering conserve the spin, the spin degenerency inherent in

antiferromagnets is not affected. This makes diffusive transport in antiferromagnets

textures much richer than their ferromagnetic counterpart. The local charge and spin

current densities are calculated using Eq. (6.6) as

jηcc = jη,+e + jη,−e (6.7a)

and

Jηsc =~2e〈jη,σe ⊗ σ〉 (6.7b)

respectively, where the average 〈...〉 is calculated with respect to the eigenstates given

138

by Eq. (6.3). We obtain explicitly

jηcc = ση0

(E + ηPk

~2eNxyvy

)x (6.8a)

+ ηPkση0

(ληH

2E − ~2evx

)Nxyy

Jηsc =(bηJ + (1 + P 2

k )ληE2vyNxy

)l⊗ x (6.8b)

− (1 + P 2k )

(η

PkbηJλ

ηH

2 + ληE2vx

)Nxyl⊗ y,

where l = Msn and bηJ = ηPk~ση0E/2eMs. Eq. (6.8a) reveals that in saturated

antiferromagnetic textures, there is no topological (skyrmion) Hall effect [jcc = j+cc +

j−cc = σ0Ex] consistent with Barker et. al. [46].

6.2.1 Topological spin Hall effect

In this section, we present detail analysis of the consequences of the spin current given

by Eq. (6.8b). First, we discuss steady state conditions (∂tn = 0) since we are more

interested in the non-adiabaticity parameter. We however acknowledge the fact that

non-steady state conditions (i.e. when the skyrmion is moving) induce spin pumping

effects that will result in a re-normalization of the damping parameter [276] which

is out of the scope of this present study. At steady state therefore, the spin current

density reads

Jηsc = bηJ l⊗ x− ηPkληH2bηJNxyl⊗ y, (6.9)

where Pk = (1 + P 2k )/Pk. Before we proceed, a couple of quick comments drawn

from Eq. (6.9) are in order. (i) The longitudinal spin current is staggered while the

transverse (topological) spin current is not. A direct consequence is that antiferro-

magnetic skyrmions and by extension any topologically non-trivial antiferromagnetic

texture e.g antiferromagnetic vortices, experiences a finite TSHE. (ii) The strength

139

of the TSHE is determined by the polarisation of the local density of states Pk and

the Hall conductivity σηH . Since P 2k takes values between 0 and 1, it can at most

double this effect. Therefore any significant change in TSHE in these materials can

be attributed properties of the Hall conductivity.

6.2.2 Topological torque

The flow of spin current in magnetic materials represents the flow of angular mo-

mentum that can be transferred to the local magnetization through the spin transfer

mechanism as discussed in Section 2.3. The torque corresponding to the topological

spin current in Eq. (6.9) obtained by definition as τ ηt = −∇ · Jηsc/Ms is given as

τ ηt = −bηJ∂xn + ηPkληH

2bηJNxy∂yn. (6.10)

At this point we have almost everything we need to study the current induced

motion of antiferromagnetic skyrmions. To proceed, we need to define the mag-

netic profile of an antiferromagnetic skyrmion. In this regard, we consider the pro-

file defined in spherical coordinates as mi = (−1)ix+iy(cos Φ sin θ, sin Φ sin θ, cos θ)

where the azimutal and polar angles are given by Φ(x, y) = qArg(x + iy) + cπ2

and

cos θ = p(r20−r2)/(r2

0 +r2) respectively. This describes an antiferromagnetic skyrmion

with topological charge q, polarisation p and chirality c. Using this texture, the topo-

logical torque in Eq. (6.10) can be re-written in the form

τ ηt = −bηJ ∂xn + ηpqPkληH

2bηJ Nxyn× ∂xn. (6.11)

For the sake of completeness we incorporate the non-adiabatic effects arising from, for

example, spin relaxation [74] with constant non-adiabaticity parameter β such that

140

τ η = τ ηt + ηβbηJn× ∂xn with the total torque at η given as

τ η = −bηJ∂xn + ηbηJ

(β + pqPkλ

ηH

2Nxy

)n× ∂xn. (6.12)

From Eq. (6.12), it appears clear that just as in ferromagnetic skyrmions, the trans-

verse topological spin current flowing in antiferromagnetic skyrmions directly en-

hances the non-adiabatic torque. In the following section we study the impact of

this intrinsic topological torque on the current-driven motion of antiferromagnetic

skyrmions.

Figure 6.1: (Color online). Schematic diagram of tight-binding system made up of acentral scattering region containing isolated antiferromagnetic skyrmion attached totwo antiferromagnetic leads (red boxed) L and R at chemical potentials µL and µRrespectively.

6.3 Numerical results

In this section, we present a numerical verification of our analytical model discussed

in the previous section. To do this, we follow the method outlined in Chapter 5 by

considering a system made up of a scattering region containing an isolated antiferro-

141

-6 -4 -2 0 2 4 6

Transport energy (Jsd

)

-0.015

-0.01

-0.005

0

0.005

0.01

θT

H

FS

GS

-6 -4 -2 0 2 4 6

Transport energy (Jsd

)

-0.4

-0.2

0

0.2

0.4

θT

SH

20×FS

GS

(a) (b)

Ea

Eb

Figure 6.2: (Color online). Topological properties of ferromagnetic and antiferromag-netic skyrmions. Although THE vanishes in antiferromagnetic skyrmions (a), TSHE(b) can be gigantic very close to the antiferromagnetic band gap in clean systems.

magnetic skyrmion and two attached antiferromagnetic leads as shown in Fig. 6.1.

This system is modelled as a two dimensional square lattice described by the Hamil-

tonian given by Eq. (5.14). We calculate the local spin/charge densities and currents

as well as the corresponding spin transfer torque using the non-equilibrium wave func-

tion formalism [164] as outlined in Section 2.5.2. The calculation of the charge and

spin Hall effect can be achieved by adding a top and bottom antiferromagnetic leads

to the system as described Sections 2.5.2. Furthermore, to investigate the effect of

disorder, we model spin-conserving impurity scattering by the randomization of the

onsite energy εµ = ε0 + Vµ, such that Vµ ∈ [−0.5Vimp, 0.5Vimp], where Vimp is the

impurity strength. We consider a system size of 4× 101× 101a20, Jsd = 2t/3, ε0 = 0

and t = 1.

We first benchmark our system by calculating θTH and θTSHE which quantify the

THE and TSHE respectively, as a function of transport energy for a clean system and

compare with the case of an equivalent system containing a ferromagnetic skyrmion.

As shown in Figure 6.4 (a), THE vanishes in antiferromagnetic skyrmions as a result

of effective cancellation of the magnus from each sublattice [46] as predicted by Eq.

(6.8a). Figure 6.4 (b) shows that the THSE is finite and becomes gigantic very close

142

Figure 6.3: (Color online). Two-dimensional profile of the transverse topological spincurrent (a,c) and the corresponding non-adiabatic torque (b,d). The transverse spincurrent and corresponding torque (c, d), are gigantic close to the antiferromagneticband gap compared to their values far away (a,b).

to the antiferromagnetic band gap in a clean system. This enhancement very close

to the band gap in the clean limit - can be interpreted in terms of an increase in the

intrinsic texture-induced effective spin-orbit coupling in addition to strong spin-flip

scattering occurring very close to the antiferromagnetic band gap. However since

this strong enhancement only occurs in a small energy window close to the band

gab, enormous material engineering is required to take advantage of this effect for

application.

We identify two regions of interest in the band structure represented by green

rectangles in Figure 6.4 (b) to investigate the nature of the TSHE. Figure 6.3 shows the

profile of the transverse spin current and its corresponding spin torque. In accordance

with Eqs. (6.9) and (6.12), this shows that the non-adiabatic torque is governed by

143

0 1 2

Impurity strength (Jsd

)

-0.01

-0.008

-0.006

-0.004

-0.002

0

θT

SH

Ferromagnetic skyrmion

r0 = 8a

0

r0 = 16a

0

r0 = 24a

0

0 1 2

Impurity strength (Jsd

)

-0.05

0

0.05

0.1

0.15

θT

SH

Antiferromagnetic skyrmion

r0 = 8a

0

r0 = 16a

0

r0 = 24a

0

(a) (b)

E = Ea

E = Ea

Figure 6.4: (Color online). Robustness of TSHE in ferromagnetic and antiferromag-netic skyrmions realistic systems. Unlike ferromagnetic skyrmions (a), not only isTSHE in antiferromagnetic skyrmions (b) robust, but it increases with the strengthof spin-independent impurity scattering. In the presence of spin-conserving scatter-ing, the THSE in both ferromagnetic and antiferromagnetic skyrmions increases witha decrease in the skyrmion size.

the transverse topological spin current. As shown, the spin current/torque becomes

gigantic very close (c/d) to the band gap.

Finally, since impurities are omnipresent in real materials, we investigate the effect

of spin-conserving impurity scattering on the THSE in antiferromagnetic skyrmions

and compare with their ferromagnetic counterpart. Figure 6.4 (b) shows that TSHE

in antiferromagnetic skyrmions survives spin-conserving impurity, and more inter-

estingly, increases wth the strength of the impurities. Furthermore, this impurity-

enhanced THSE increases when decreasing the skyrmion size. We interpret this en-

hancement as a result of impurity-induced asymmetry in the sublattice density of

states that gives rise to different scattering rates. Next, we compare these results

with an equivalent ferromagnetic skyrmion. As shown in Figure 6.4 (a), unlike in an-

tiferromagnetic skyrmions, ferromagnetic skyrmions do not survive strong impurity

scattering.

144

6.4 Current-induced dynamics of antiferromagnetic

skyrmion

To investigate the impact of the topological torque given by Eq. (6.12) on the dy-

namics of an antiferromagnetic skyrmion, we follow the standard theoretical scheme

to study the dynamics of antiferromagnetic textures on a crystal lattice. This en-

tails considering each unit cell to contain two equivalent magnetic sites a and b,

such that the total magnetization and antiferromagnetic Neel order parameter can

be written in terms of the sublattice magnetization m(r, t) = ma(r, t) + mb(r, t) and

l(r, t) = ma(r, t)−mb(r, t) respectively [158, 159, 160, 161]. This allows the dynam-

ics of the system to be modelled using the coupled extended Landau-Lifshitz-Gilbert

equation in the presence of spin transfer torque given as

∂tmη = −γmη ×Hη

eff + αηmη × ∂tmη + τ η, (6.13)

where αη, Hηeff and τ η are the Gilbert damping constant, effective field and the spin

torque on the η-sublattice respectively. The local torque in Eq. (6.13) can then be

written in terms of m and n = l/|l| as

τ η = ηγJad∂xn− γJnan× ∂xn, (6.14)

where η = +1 for a-sublattice and η = −1 for b-sublattice. The effective fields in

Eq. (6.13) are derived from the free energy with leading order [53] in the absence of

external magnetic field given by

F =

∫dr

[a

2m2 +

A

2(∂2xn + ∂2

yn)

](6.15)

145

as

fm =1

2(Heff

a + Heffb ) = −δmF = −am (6.16a)

fn =1

2(Heff

a −Heffb ) = −δnF = A∇2n. (6.16b)

In Eq. (6.16), a and A are the homogeneous and inhomogeneous exchange constant.

Putting all these together the equation of motion of the Neel order parameter that

describes the dynamics of antiferromagnetic textures reads [159]

1

γ∂2t n = a

[γfn − αn∂tn + γJ+

na∂xn]. (6.17)

Comparing Eqs. (6.12) and (6.14) we obtain

J+na =

1

γbJ

(β + pqNxyPkλ2

H

)(6.18)

where J+na = 1

4(J+

na + J−na).

After the substitution of Eq. (6.18) into Eq. (2.86), the terminal velocity of the

skyrmion is given as

vx = ∂tX = − bJαn

(β + Pk

4λ2H

3r20

)and vy = ∂tY = 0. (6.19)

This shows that the motion of an antiferromagnetic skyrmion is different from that

of a ferromagnetic skyrmion in that: (i) there is no transverse motion (vy = 0) in

antiferromagnetic skyrmion as a result of an effective cancellation of the magnus forces

acting on the structure (ii) the enhanced non-adiabatic torque directly influences the

longitudinal motion of antiferromagnetic skyrmions. The latter strikingly contrasts

with the case of ferromagnetic skyrmions, where the non-adiabatic torque generates

the undesirable transverse motion of the skyrmions.

146

6.5 Discussions and conclusions

It has been shown that the interaction of ferromagnetic skyrmions with defects that

are omnipresent in real materials is very weak resulting in their low sensitivity to

these defects [274, 275]. However, (i) this interaction increases with decreasing the

skyrmion size [275]; (ii) a decrease in skyrmion size increases the non-adiabatic torque

[276] that drives the undesirable transverse motion of skyrmions. Therefore high-

density memory applications using ferromagnetic skyrmions are limited by higher

de-pinning current densities and enhanced undesired transverse motion as a result

of downscaling the size of skyrmions. In contrast, antiferromagnetic skyrmions are

driven by the non-adiabatic torque while exhibiting no transverse motion. Moreover,

this non-adiabatic torque, being topological in nature, increases when the skyrmion

size decreases. As a result, the current-driven motion increases when the skyrmion

becomes smaller. This unique property could be a great advantage to compensate

the increasing pinning.

147

Chapter 7

Conclusions and perspectives

In this dissertation, we have provided an accurate description of spin transport in

magnetic textures. In particular, we investigated in detail - the nature of spin torque

and magnetic damping in such magnetic textures. In Chapter 3, we investigated the

nature of magnetic damping in non-centrosymmetric ferromagnets. Based on phe-

nomenological and microscopic derivations, we showed that the magnetic damping

becomes chiral, i.e. depends on the chirality of the magnetic texture. In Chapter 4,

we discussed the physics of spin transport in sharp disordered magnetic domain walls

and vortex cores. We demonstrated that spin-independent scattering significantly

enhances the non-adiabatic torque especially in vortex cores compared to transverse

domain walls. The remaining part of the thesis was devoted to topologically non-

trivial magnetic textures such as vortices and skyrmions. In Chapter 5, we showed

that the topological spin currents flowing in these structures dramatically enhances

the non-adiabaticity, an effect unique to non-trivial topological textures. Finally,

in Chapter 6 we extended this study to antiferromagnetic skyrmions and show that

such an enhanced topological torque also exist in these systems. Even more inter-

estingly, while such a non-adiabatic torque influences the transverse velocity of fer-

romagnetic skyrmions, in antiferromagnetic skyrmions, the topological non-adiabatic

torque directly determines the longitudinal velocity. As a consequence, scaling down

the antiferromagnetic skyrmion results in a much more efficient spin torque.

148

These studies open up exciting future directions. First of all, the use of magnetic

materials in magnetic storage media relies on the representation of bits of information

as two stable states up and down stored, for example as, ”1 ” and ”0 ”. Convention-

ally, to switch the magnetization orientation from one state to another, energy is

usually supplied to the system that drives the system away from its stable state. To

eventually return to a stable state, the system has to remove this surplus energy (and

angular momentum associated with it) by dissipation. Therefore magnetic damping

places a limit to the speed at which information can be processed (read and write) in

magnetic media. This damping is usually taken as a constant (Gilbert damping) in

both bulk magnetic materials and thin layers of the ferromagnetic materials. There

are recent indications that magnetic damping should in principle be non-local and

by virtue of Neumann’s principle possesses the symmetries of the underlying system

[278, 279, 91, 145]. Indeed, first principle calculations show an enhanced damping due

to strong interfacial dissipation especially in systems with strong spin-orbit coupling

[280]. Therefore, there is a need to develop micromagnetic models and experimen-

tal techniques that will incorporate and measure this non-local damping. This will

pave the way to estimating the chiral contribution to magnetic damping discussed in

Chapter 3 for non-centrosymmetric magnets.

Another direction of interest addresses the nature of the non-adiabatic torque for

future micromagnetic models. We showed in Chapter 4 for sharp magnetic textures,

and in Chapters 5 and 6 for topologically non-trivial magnetic textures, that the non-

adiabatic torque acquires a non-local contribution which can actually be the dominant

contribution to non-adiabaticity. Therefore developing micromagnetic models that

fully take into account these non-local torques will be crucial to fully understand the

dynamics of topologically non-trivial textures such as skyrmions and vortices.

Finally, antiferromagnetic skyrmions have been shown to experience no skyrmion

Hall effect in addition to the absence of stray fields which makes them very promis-

149

ing for high density memory devices. We showed in Chapters 5 and 6 that the

non-adiabatic torque which drives the motion of these textures is dominated by a

topological contribution that scales as the inverse square of the vortex/skyrmion ra-

dius. This property makes it very appealing to properly investigate the creep motion

of these topologically non-trivial textures using micromagnetic simulations.

150

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