Copyright

by

Alvaro Sebastian Nunez

2006

The Dissertation Committee for Alvaro Sebastian Nunez

certifies that this is the approved version of the following dissertation:

Interaction between Collective Coordinates and

Quasiparticles in Spintronic Devices

Committee:

Allan H. MacDonald, Supervisor

Jim Erskine

Jim Chelikowsky

C.-K.(Ken) Shih

Brian Korgel

Interaction between Collective Coordinates and

Quasiparticles in Spintronic Devices

by

Alvaro Sebastian Nunez, Bs. Sc. Physics

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

August 2006

Con todo mi amor a Viviana y Penelope,

mis dos chiquititas:

Si no fuera porque tus ojos tienen color de luna

de dıa con arcilla, con trabajo, con fuego,

y aprisionada tienes la agilidad del aire,

si no fuera porque eres una semana de ambar, si no fuera por que eres

el momento amarillo

en que el otono sube por las enredaderas

y eres aun el pan que la luna fragante

elabora paseando su harina por el cielo oh, bien amada, yo no te

amarıa!

En tu abrazo yo abrazo lo que existe,

la arena, el tiempo, el arbol de la lluvia, y todo vive para que yo viva:

sin ir tan lejos puedo verlo todo:

veo en tu vida todo lo viviente.

Pablo Neruda (soneto VIII)

Acknowledgments

It is a great pleasure to thank my advisor Allan MacDonald for all the help and

inspiration he gave for this work to be done. His trust in me, his patience with my

lack of experience or knowledge (or usually both) and, more generally, his positive

attitude to handle difficulties and complexities, have made a big imprint on me, that

I hope will last for a long time. I would like to thanks Seagate Technology Inc. for

financial support for this work.

I would also like to thank my undergraduate professors Fernando Lund, Romualdo

Tabensky and Nelson Zamorano, for their support and help.

Together with Allan, his big group of students, post-docs and staff, made of my stay

in Austin, not only the most fulfilling years of my intellectual life, but also years

that I’ll remember with joy and love.

Specials thanks to Joaquin Fernandez-Rossier and to Enrico Rossi, both coauthors

of some of the work here presented. Besides their great friendship, they were always

willing to exchange ideas and to teach me lots of things.

Some other work presented here was done in collaboration with Rembert Duine,

who also helped me in the painful painful process of proofreading the first drafts of

this work.

The help of Becky Drake was also very important. Thank you very much for all the

help and patience!

This work also was possible thanks to discussions and the collaboration of many peo-

v

ple, among them Paul Haney, Ramin Abolfath, Mathias Braun and Anton Burkov.

Of course all the imperfections that, with certainty, still populate this work are my

entire responsibility.

Finalmente no puedo dejar pasar esta oportunidad para agradecer el amor, la pa-

ciencia y todas las alegrias que me han dado, a mi pequena familia: Viviana y

Penelope, Fogata de amor y guıa, Razon de vivir mi vida.

Alvaro Sebastian Nunez

The University of Texas at Austin

August 2006

vi

Interaction between Collective Coordinates and

Quasiparticles in Spintronic Devices

Publication No.

Alvaro Sebastian Nunez, Ph.D.

The University of Texas at Austin, 2006

Supervisor: Allan H. MacDonald

In this dissertation several aspects of the interaction of collective and quasi-particle

degrees of freedom are studied. This is done in the context of spin dependent

transport effects with applications for spintronics devices.

In ferromagnetic metals the effects of quasi-particle currents on spin textures, either

domain wall structures or spin waves, are discussed. In nano-magnetic heterostruc-

tures, the effects acquire the form of spin transfer torques. The microscopic origin

of these effects, as discussed in this work, relies on the relation between exchange

fields and spin densities. The presence of the current modifies the spin density. In

consequence the exchange fields are also affected by the current. It is these modifi-

cations on the exchange fields that are able to alter the dynamics of the collective

fields.

It is shown how this rather abstract picture of spin transfer reduces to the usual

description, that can be found in the extensive literature on the subject, based on

a bookkeeping argument and on spin conservation. The most important feature of

this picture, as discussed in the text, is that it allows for generalizations of the spin

vii

transfer effects to systems were the spin conservation arguments fail or are of little

use. We discuss applications of this view to spin transfer torques on systems with

spin-orbit interaction and for systems with antiferromagnetic elements.

In the latter case, a preliminary model study of spin dependent transport in anti-

ferromagnets is reported, it has revealed that i) giant magnetoresistive effects are

possible, and ii) nanostructures containing antiferromagnetic elements will exhibit

current-induced magnetization dynamics. In particular it turns out that, contrary

to the ferromagnetic case, the spin transfer torques act throughout the entire free

antiferromagnet to cooperatively switch it, a result of the special symmetries of the

antiferromagnetic state. This implies that the critical current for inducing collective

magnetization dynamics is likely to be lower in antiferromagnetic metal nanostruc-

tures than in ferromagnetic spin valves.

viii

Table of Contents

Acknowledgments v

Abstract vii

List of Figures xiii

Chapter 1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Plan of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2 Basic Elements of Spintronics 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Giant-magneto resistive effects . . . . . . . . . . . . . . . . . . . . . 10

2.3 Spin transfer effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Semiconductor Spintronics . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 3 Non-equilibrium Formalism for Transport in Mesoscopic

Systems 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Landauer-Butikker Formalism . . . . . . . . . . . . . . . . . . . . . . 22

3.3 NEGF formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Basic Considerations in Non-Equilibrium Field Theory. . . . 27

ix

3.3.2 Basic properties of the Non-Equilibrium Green’s functions . . 31

3.3.3 Field Equations and Perturbations in Keldysh Space . . . . . 35

3.3.4 Application: Tunneling current . . . . . . . . . . . . . . . . . 42

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 4 Current-induced dynamics in a Ferromagnet 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Dynamics of a Ferromagnet: Landau-Lifshitz equation . . . . . . . . 51

4.2.1 Microscopic Description of low energy modes . . . . . . . . . 51

4.2.2 Spin-wave Doppler shift as a Spin-Torque Effect . . . . . . . 59

4.2.3 Spin wave description . . . . . . . . . . . . . . . . . . . . . . 60

4.2.4 Enhanced Spin-Wave Damping at finite Current . . . . . . . 66

4.3 Current induced Domain wall dynamics . . . . . . . . . . . . . . . . 71

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Numerical Solution of the Landau-Lifshitz equation in the

presence of a current . . . . . . . . . . . . . . . . . . . . . . 74

4.3.3 Hamiltonian form of Landau-Lifshitz equation . . . . . . . . 77

4.3.4 Bloch Domain Wall . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.5 Motion of a Rigid Domain Wall Driven by an External Mag-

netic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.6 Motion of a Rigid Domain Wall Driven by an Current . . . . 87

4.3.7 Beyond the rigid approximation: Modification of the shape of

the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 5 Theory of Spin Transfer Phenomena in Magnetic Metals

and Semiconductors 93

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Basic Phenomenology of Spin transfer effects . . . . . . . . . . . . . 95

x

5.3 Microscopic Theory of Spin Transfer . . . . . . . . . . . . . . . . . . 99

5.3.1 Quasiparticle Spin Dynamics . . . . . . . . . . . . . . . . . . 102

5.3.2 Collective Magnetization Dynamics: . . . . . . . . . . . . . . 102

5.3.3 Spin-Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Toy-Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.1 Effect of spin-orbit interaction . . . . . . . . . . . . . . . . . 106

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Chapter 6 Antiferromagnetic Spintronics 114

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Scattering in Single Q Antiferromagnets . . . . . . . . . . . . . . . . 116

6.3 Antiferromagnetic giant magnetoresistance . . . . . . . . . . . . . . . 120

6.3.1 Elementary Local Spin Model . . . . . . . . . . . . . . . . . . 121

6.4 Tight-Binding Non-equilibrium Calculation . . . . . . . . . . . . . . 125

6.4.1 Transmission through oscillating 1D exchange fields . . . . . 128

6.4.2 Spin Filter Effect suppression . . . . . . . . . . . . . . . . . . 128

6.5 Current-driven switching of an antiferromagnet . . . . . . . . . . . . 129

6.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 135

Chapter 7 Conclusions and Outlook 138

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Appendix A Basic calculations 142

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.2 Pauli Spin Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.3 Discrete Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . 144

A.3.1 Recursive Green’s Function Algorithm . . . . . . . . . . . . . 146

A.4 Manipulations in Keldysh Space . . . . . . . . . . . . . . . . . . . . 148

xi

A.4.1 Keldysh Rotations . . . . . . . . . . . . . . . . . . . . . . . . 148

A.4.2 Lehmann Spectral Representation . . . . . . . . . . . . . . . 149

Appendix B Spin Transfer torques in piece-wise constant ferromag-

nets 152

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B.2 Spin current conservation . . . . . . . . . . . . . . . . . . . . . . . . 153

B.3 Spin filter effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

B.4 Spin transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Appendix C Some Scattering Matrix Properties in magnetic systems162

C.1 The general properties of the FM scattering matrix . . . . . . . . . . 162

C.2 Composed Transmission of an AFM and FM hybrid . . . . . . . . . 165

C.3 Outline of a proof of the periodicity of the transverse spin density . 166

Bibliography 171

Vita 186

xii

List of Figures

1.1 Increases in areal density and shipped capacity of magnetic storage over time. 6

2.1 Schematic band diagrams the spin transport in a parallel (a) and antiparallel

(b) configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Types of magnetoresistance. . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Illustration of the spin transfer torque in a spin valve consisting of a

pinned and free ferromagnetic layer. The torque on the spin angular

momentum of the electrons, indicated by the dotted arrow, has to be

accompanied by a reaction torque on the magnetization of the free

ferromagnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 A schematics representation of a Datta-Das transistor. The channel, in

between the p-doped InGaAs and the insulating layer InAlAs (in red in the

left panel), is exposed to the effects of the gate modifying the strength of

the spin-orbit interaction. This makes the electronic spins, coming from

the ferromagnetic source, precess allowing it’s entrance to the drain if they

reach it with the right spin orientation (right panel top) and blocking the

transport if they reach the drain misaligned (right panel bottom). . . . . . 16

2.5 Some remarkable spintronics effects that have been found in DMS. . . . . 17

3.1 The canonical problem to be solved. . . . . . . . . . . . . . . . . . . . . 21

xiii

3.2 Contour Ct is a closed-time contour. . . . . . . . . . . . . . . . . . . . . 30

3.3 Keldysh Contour Ct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 The diagram representation of the free particle Green’s function. The double

line without indexes will denote the matrix in Keldysh space. . . . . . . . 36

3.5 The basic model to describe a system coupled to electrodes. A potential

difference between the electrodes will create a current across the system. . 44

4.1 Cartoon of the torques driving the magnetization dynamics, (a) the

usual ferromagnetic precession is driven by a torque of the form ~Heff×~M, and (b) a dissipation torque driving the magnetization toward its

equilibrium position. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Current modified spin-wave spectrum . . . . . . . . . . . . . . . . . 65

4.3 Two mechanisms of current-induced magnetic domain wall motion.

The dashed-dotted line illustrates the electron transferring its spin

angular momentum to the domain wall, leading to motion. The dot-

ted line illustrates momentum transfer: the electron scatters off the

domain wall and gives the domain wall a momentum kick. . . . . . 72

4.4 Exact solution of the Landau-Lifshitz equations for the parameters

indicated. The different plots are: (top-left panel) A 3D represen-

tation of the Ωz component. The horizontal axis is the space label

in units of the domain wall width. The axis entering the plane of

the page is the time axis in units of 1/(αωuniaxial). The third dimen-

sion is the dimensionless z-component of the magnetization vector.

(rest of panels) A 2D representation of the different coordinates of

the director vector. Here Q is infinity (no in-plane anisotropy) and

the domain wall responds as a straight line with velocity X = J1+α2 .

As the domain moves the components in the hard plane precess. . . 78

xiv

4.5 Exact solution of the Landau-Lifshitz equations for the parameters

indicated. The different plots are: (top-left panel) A 3D represen-

tation of the Ωz component. The horizontal axis is the space label

in units of the domain wall width. The axis entering the plane of

the page is the time axis in units of 1/(αωany). The third dimension

is the dimensionless z-component of the magnetization vector. (rest

of panels) A 2D representation of the different coordinates of the di-

rector vector. Here Q is finite but still large enough as to allow the

domain wall motion. For a finite value of Q, domain wall moves but

there are some oscillations on top of the straight line motion. As the

domain moves the components in the hard plane precess. . . . . . . 79

4.6 Exact solution of the Landau-Lifshitz equations for the parameters

indicated. The different plots are: (top-left panel) A 3D representa-

tion of the Ωz component. The horizontal axis is the space label in

units of the domain wall width. The axis entering the plane of the

page is the time axis in units of 1/(αωany). The third dimension is

the dimensionless z-component of the magnetization vector. (rest of

panels) A 2D representation of the different coordinates of the direc-

tor vector. Q is even smaller approaching the critical situation and

the wiggles become stronger. . . . . . . . . . . . . . . . . . . . . . . 80

xv

4.7 Exact solution of the Landau-Lifshitz equations for the parameters

indicated. The different plots are: (top-left panel) A 3D representa-

tion of the Ωz component. The horizontal axis is the space label in

units of the domain wall width. The axis entering the plane of the

page is the time axis in units of 1/(αωany). The third dimension is

the dimensionless z-component of the magnetization vector. (rest of

panels) A 2D representation of the different coordinates of the direc-

tor vector. Q is small enough as to stop the motion of the domain

wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.8 The definition of the polar angles used as independent fields in the

theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.9 Left panel: Cartoon of a Bloch Domain wall of width λ. Right panel:

plot of the Mz and My components of the magnetization along the

domain, and the energy density. Mx is zero to avoid magnetostatic

torques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.10 Average velocity 〈X〉 for the domain wall as a function of Q and

the driving field h. The color code represents the relative value of

〈X〉/(h/α), we see that is constant, equal to 1, below the Walker

limit represented by the dashed line. Beyond that limit the system

acquires an oscillatory behavior characterized by zero average velocity. 86

4.11 Left panel: average velocity 〈X〉 as function of the anisotropy param-

eter Q and the current J . Below the critical current Jcr(Q) described

by the dashed line we have a fixed point at zero velocity, and above

that current non-zero velocities appear. Right panel: 〈X〉/Jcr(Q)

as a function of J/Jcr(Q) for several values of Q. Above the criti-

cal current all the curves collapse into the dashed line described by

〈X〉 =√J2 − J2

cr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xvi

4.12 Left panel: average velocity 〈X〉 as function of the anisotropy param-

eter Q and the current J . Below the critical current Jcr(Q) described

by the dashed line we have a fixed point at zero velocity, and above

that current non-zero velocities appear. Right panel: 〈X〉/Jcr(Q)

as a function of J/Jcr(Q) for several values of Q. Above the criti-

cal current all the curves collapse into the dashed line described by

〈X〉 =√J2 − J2

cr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1 Spin transfer mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2 (a) Cartoon of a point contact between two ferromagnets that display the

spin transfer effect. The current goes from one magnet through the point

contact to the other magnet where it creates a spin transfer torque that

drives the second magnet out of its equilibrium position. (c) Differential

resistance as a function of current[1]. As the current is increased to a certain

critical value, the parallel configuration (of low resistance) becomes unstable

and the free magnet is switch to be antiparallel to the pinned magnet. The

jump in resistance is the GMR effect, and is identical to the jump measured

independently by switching the free magnet with an applied magnetic field. 98

5.3 Spin transfer dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Spin Transfer in orbital representation . . . . . . . . . . . . . . . . . . . 101

5.5 Toy model described in the text, a 2DEG with ferromagnetic regions. . . . 107

5.6 Right movers Fermi Surface in a Rashba System. . . . . . . . . . . . . . 108

5.7 Transport spin density per unit current in the case without spin-orbit inter-

action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.1 (a) Effective resistance arrays that represents a parallel configuration in a

conventional GMR device. (b) same for antiparallel.

122

xvii

6.2 (a) Effective resistance arrays that represents a parallel configuration in

a AFM-GMR device. (b) same for antiparallel. No GMR effect can be

observed from the classical system. . . . . . . . . . . . . . . . . . . . . 122

6.3 (a) Scattering process for right-going incoming electrons. (b) same for left-

goers. Both processes are included in the S matrix.

123

6.4 The model heterostructure for which we perform our calculations. . 125

6.5 Landauer-Buttiker conductance as a function of the angle θ between

the magnetization orientations Ωi on opposite sides of the paramag-

netic spacer layer. There is a sizable giant magnetoresistance effect,

with larger conductance at smaller θ and weak dependence on layer

thicknesses. These results were obtained for ∆/t = 1 and ǫi = 0. . . 126

6.6 The Transmission coefficient of an oscillating exchange field. . . . . . . . . 129

6.7 Local spin-transfer torques in the down-stream antiferromagnet. The

in-plane spin transfer is staggered and therefore effective in driving

coherent order parameter dynamics. The out-of-plane spin-transfer

component is ineffective because it is not staggered. These results

were obtained for ∆/t = 1, ǫi = 0, θ = π/2, N = 50, and M = 50. . . 132

6.8 Total spin transfer torque action on the downstream antiferromagnet,

as a function of θ. We used the parameters ∆/t = 1 and ǫi = 0. . . . 132

6.9 Derivative of the total spin transfer torque per unit current, Mg(θ =

π), acting on the downstream antiferromagnet with respect to the

angle θ at θ = π as a function of M . We used the parameters ∆/t = 1

and ǫi = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xviii

A.1 The tunnelling part of the Hamiltonian dresses the propagation on one side

(blue) with events of tunnelling to the other side (red). That can be repre-

sented by a self energy that in this simple case equals the amplitude of two

tunnelling events from one side to the other and then back. . . . . . . . . 145

A.2 This cartoon represents a generic system whose Green’s function is going to

be calculated using the recursive Green’s function algorithm. Note that the

system can have any shape, with varying width and can even have holes. . 147

B.1 Many channel interference leading to spin-transfer . . . . . . . . . . . . . 158

xix

Chapter 1

Introduction

Condensed matter physics studies the collective behavior of systems with a macro-

scopic number of constituents. The physical behavior of a system with many con-

stituents is not a simple agglomerate of the behavior of each of its parts. The

microscopic physical laws are supplemented by certain “organizational principles”

that only acquire meaning as the number of constituents is increased, and acquire

their full power as the thermodynamic limit is approached. Those principles are

quite different in their nature than the laws of the constituent elements. Their real-

ity is said to be emergent, in the sense of not being explicitly contained in the laws

that rule the world at a microscopic level. The collective behavior of a macroscopic

number of particles often does not even qualitatively reflects the details the of mi-

croscopic rules that define their dynamics. A remarkable example of this is the fact

that in the low energy limit, a great variety of systems can be described by field

theories defined on a continuum of degrees of freedom, where the discrete nature of

the underlying system appears only as convenient ultraviolet cutoffs that regularize

the high energy behavior of the fields.

The qualitative difference between the microscopic and macroscopic descrip-

tions of the same system is one of the most profound features of the whole body of

1

knowledge associated with condensed matter.

Two foundational examples of this view of nature were given by Landau in

his seminal works on the nature of phase transitions and of Fermi liquids. Those

two examples are the cornerstones of the whole field of condensed matter physics

and of the notion of emergence. The phases of matter, according to the first theory,

are sharply distinct from each other as a consequence of their symmetries. In this

way, a transition to a more ordered state is associated with a loss (or breaking) of

a symmetry. Examples of this are abundant, and in this work we are going to be

mostly concerned with the breaking of the spin-rotational symmetry that is at the

heart of the magnetic states of matter. In solid state systems the most widespread

state of matter, by far is the Fermi Liquid. A Fermi liquid is described by the usual

Schrodinger equation of a many body system. Even though this equation does not

respect single particle momentum as a good quantum number, the collective be-

havior of the system forces, in the thermodynamic limit, the system to ensure the

existence of a Fermi surface, in single particle momentum space, as being a pre-

ferred “reference” momentum locus, from where the excitations of the system are

defined. The condensation of the abundant electronic degrees of freedom into the

much smaller number of degrees of freedom associated with these ordered states is

the basic organizing principle that rules the low energy behavior of a macroscopic

number of electrons under normal circumstances. It is the basis of the theory of

metals, semiconductors, metallic ferromagnets, metallic antiferromagnets, supercon-

ductors, etc. The whole system of many particles can be accurately described by

an account of the order parameter and quasi-particle excitations around the Fermi-

surface. These two are well defined and distinct modes of excitations that we have

available in a solid. The collective coordinate (order parameter) fluctuations are

accurately described by bosonic fields with a well behaved low energy limit. In

particular the symmetry restoring -Goldstone- modes have a gapless spectrum as a

2

consequence of the symmetry of the microscopic Hamiltonian. On the other hand

the quasi-particle excitations can be described by fermionic fields that are well de-

fined only in the immediate vicinity of the fermi-surface. Though they look like

electronic excitations, they are, actually, also collective excitations involving the ex-

citation of many electrons at a time. Quasi-particles as a consequence are distinct

from electrons, they have a different mass (in some cases it can even reach thou-

sands of times the electron mass) and a finite lifetime. Because these excitations

correspond to so different kinds of disturbances they are uncoupled at low energies.

In equilibrium then, they correspond to two disconnected parts of the spectrum of

the system. Any coupling between them, since they have zero energy, will lead to an

immediate reconstruction of the ground state. The present work studies the basic

interactions that arise between collective coordinates and quasi-particle degrees of

freedom in magnetic systems when they are out of equilibrium. This coupling is

interesting in several senses as is discussed in the next section.

1.1 Introduction

Magnetism abounds with dichotomies: It was known to the

ancients and yet is the focus of exciting new research; its man-

ifestations are apparent to every schoolchild yet its origins are

rooted deep in quantum mechanics and relativity; its applica-

tions underlie huge industries yet its understanding -even in

iron- is still incomplete [2].

In a ferromagnet, one particular direction of space is chosen to be the preferred

orientation of the electronic spins. This spontaneous broken symmetry allows the

existence of low energy excitations corresponding, in a magnet, to collective modes

involving a change of the orientation of a macroscopic number of electronic spins.

These low-energy excitations are known as spin-waves. It is in this feature on which

3

most of the effects discussed in this work rely upon. The symmetry breaking is,

ultimately, due to Coulomb interactions among the electrons and to their fermionic

character. The interplay of both gives rise to what is known as the “exchange in-

teraction” [3]. The exchange interaction is an effective spin-dependent interaction

arising from Coulomb repulsion in a fermionic system. The electrostatic energy of

a symmetric spin state is reduced due to the antisymmetry of the spatial part of

the many body wave-function with symmetric spin part. The exchange interac-

tion is dominant over other terms describing the dynamics of the electrons and its

strength is the reason that the low energy excitations correspond to macroscopic

spin reorientations. If we were to characterize the exchange interaction grossly by

a single energy scale it would turn out to be in the range of 0.7 eV-1.0 eV. This

corresponds to the parameter I in the stoner model (to be discussed later on). It

can also be related to the coefficient U in the interaction term of the Hubbard

Model. The value corresponds to a suitable parametrization of the ab initio results

of LSDA calculations[4]. The specific orientation of the direction of a magnet is

associated with smaller energies, related to relativistic corrections (dipole-dipole in-

teractions giving rise to what is known as magnetostatic anisotropy and spin orbit

interactions associates with the crystalline anisotropy). The final configuration of

the macroscopic spin density field is then associated with those smaller terms. It

is easy to manipulate the magnetization orientation with weak magnetic fields or

with currents, as we will discuss. This provides an inexpensive knob to manipulate

the direction of a macroscopic number of spins. Many technological applications (in

particular non-volatile storage of information technologies) have depended on this

effects for decades. They have been subjected to detailed, extensive studies, and

a robust theory of those effects has being built on the basis of phenomenological

considerations since the early works of Landau and Lifshitz [5] and Brown [6]. With

the advent of ab initio methods those results were given a sound microscopic ba-

4

sis. The topic that concerns us in this work is the study of how this remarkably

successful picture is modified by the introduction of non-equilibrium effects. The in-

teraction between the macroscopic spin density field and the charged quasi-particle

excitations responsible for electric currents remained largely unexplored for a long

time until the development of new techniques of nanofabrication made it possible

to create samples where those effects were relevant. Since then such effects have

received increasing attention from the community. Such attention arises mainly

from two complementary sources. The first one is that the effects associated with

the interplay between magnetization and quasi-particles have provided interesting

phenomena like giant magnetoresistance effects and spin transfer torques. Giant

Magnetoresistance (GMR) effects are associated with the change of the electric re-

sistance that can be attained by manipulating the relative orientation of magnets

in heterostructures of nanoscale dimension [7]. Spin Transfer Torques (ST) are as-

sociated with the exchange between the quasi-particle and magnetization angular

momentum [8, 9]. These two remarkable effects are of importance in technological

applications, the first one led the revolution in hard-disk technologies of the late 90’s

(see figure (1.1)). The second one may be of similar relevance, but both the science

and technology are still in an evolutionary state. The other reason is that this in-

terplay is the first example of the vast field of spintronics1, a multidisciplinary effort

that focuses on the manipulation of electronic spin degree of freedom as a source of

control, flexibility and efficiency in electronic applications [10, 11, 12].

The interplay between magnetization and current, is a manifestation of a gen-

eral situation involving non-equilibrium collective physics. Indeed, circumstances in

which the non-equilibrium quasi-particles affect and are affected by the collective

coordinate (order parameter) is a quite general situation. In particular the physics

of Josephson Junctions and of Andreev reflection are manifestation of such an inter-

1This is a popular short form of spin electronics also known as magneto electronics.

5

Figure 1.1: Increases in areal density and shipped capacity of magnetic storage over time.The inset plot shows the total capacity of hard drives shipped per year; in 2002 that shippedcapacity was 10 EB (1 EB 1018B ) worth of data. Figure taken from [13]

play taking place in superconductor systems[14, 15]. Nanomechanical applications

also profit from such an interplay. In these systems, phonons are excited and ma-

nipulated by currents, Current-induced forces can be used to manipulate molecules

and nanocontacts [16]. Similar situations also are present in Quantum Hall Bilayer

systems[17], and in Quantum Hall ferromagnets.

1.2 Plan of work

In Chap. ( 2) we introduce some basic terminology and phenomenology that will

help to set the stage for the calculations that are going to follow. The basic intro-

duction to spintronics presented in that chapter has no intent other than to provide

some sense of completeness to this work and is not supposed to be a review of

the subject. Excellent reviews are available [18, 19, 20, 12]. In Chap. ( 3) we

6

present some formal results concerning the formalisms appropriate to describe theo-

retically the non-equilibrium situation. These are basically the Landauer-Buttikker,

and the Non-equilibrium Green’s Functions formalisms. With the aid of the NEGF

formalism we describe the basic model problems that will be used to describe the

non-equilibrium state in nano-electronics, namely a mesoscopic system connected

to leads. In Chap.( 4) the basic features of spin dynamics in a ferromagnet are

discussed. The description is used to argue for modifications to the Landau-Lifshtiz

equation for the magnetization density when currents flow through a magnet. Those

modifications have two physical effects (a) a shift in the spin waves dispersion re-

lation, (b) a collective motion of spin textures such as domain walls. Numerical

examples are discussed. The following chapter (Chap. ( 5)) deals with similar

contributions to the spin dynamics, this time in the case of a spin valve. These

effects, known as spin transfer effects are under study since they might provide a

key element in the writing process of magnetic storage technologies. Finally, in the

main chapter of this work (Chap. ( 6)) we discussed the possibility of implementing

a phenomenology similar to the spin transfer effects in systems that have antifer-

romagnetic elements. The analysis is carried on by means of generic symmetry

arguments and also by direct calculations using the formalism described in Chap.

( 3).

7

Chapter 2

Basic Elements of Spintronics

2.1 Introduction

Broadly speaking spintronics is a concept relating several collective efforts, ranging

from basic sciences to purely technological applications, which have in common the

study of processes that manipulate and probe the electronic spin degree-of-freedom.

An example of this are the strong and robust magneto-transport effects that occur

in metallic ferromagnets (anisotropic, tunnel, and giant magneto-resistance, for ex-

ample) resulting from the sensitivity of magnetization orientation to external fields,

combined with the strong magnetization-orientation dependence of the spin poten-

tials felt by the current-carrying quasi-particles. This fundamentally interesting class

of effects has been exploited in information storage technology for some time, and

new variations continue to be discovered and explored. Conventional electronics,

as opposed to spin-electronics, has as its main focus the control, manipulation and

detection of the electronic charge. This paradigm has been of great importance for

the interplay between science and technology. The rich phenomenological tapestry

that has been formed by the conjunction of several, subtle and physically fundamen-

tal effects in semiconductor systems stands as a major success of late 20th century

8

science. The appropriate manipulation of electronic spin is expected to provide an

even richer scenario. It is believed to have room for improving current technologies

and for the development of radically new ones [11, 12]. Just like in charge-electronic

based technologies, the implementation of a spintronic device requires that we reach

understanding of several aspects of its dynamics. The behavior of the spin degree

of freedom is non-trivial and highly non-classical. This fact stands as both a major

obstacle to the fulfillment of spintronic operations and also as a major source of new

and rich phenomenology. This new phenomena is to be exploited in the search of

new and more efficient applications. The field of spintronics is developing up into

several subfields that study the behavior of spins under different regimes:

• In magnets the spins are bound to behave collectively (at macroscopic numbers

at a time). This made possible to manipulate them with rather weak external

fields. In this way it has been possible to use them to create devices that are

ultra sensible to small magnetic fields [7]. This effect has become the de facto

standard used in present day hard disk technology1, [21, 22]. Similar effects

take place also for metallic antiferromagnets [23, 24].

• Besides the effect of collective exchange fields in ferromagnets and antiferro-

magnets, spin dynamics is affected in a complicated way by the presence of

spin-orbit interaction. This coupling corresponds to relativistic corrections to

the simple Pauli Hamiltonian [25]. It is characterized, in the solid-state set-

ting [26] by the presence of a momentum dependent spin splitting. Physically

this implies that different quasi-particles will have their spins precessing at

different rates and around different axis, defined by their momenta. Its role is

of great importance in determining the actual behavior of spins. Again it has

the dual nature. First as an obstacle for the application of naive ideas. Then

as powerful tool that provides us with a crank that, when properly mastered,

1See http://www.almaden.ibm.com/st/magnetism/ms/

9

can be used to reach operational capabilities in the manipulation of the spins

[27, 28].

• The effect of spin-orbit interaction acquires its full complexity in the presence

of disorder. Here, we have two natural complications. The microscopic nature

of the disorder also comes accompanied by a disorder-related (or extrinsic, as

it has become common practice to call it) spin-orbit interaction. On the other

hand the lack of momentum conservation, and therefore of precession axis due

the band (intrinsic) spin orbit coupling leads to spin dephasing.

In this chapter we review the basic features, phenomenology and principles that are

the background of the following chapters. We present a description of spintronics

effects in ferromagnetic metals, in spin valves and domain wall configurations. Later

we discuss briefly the main concepts behind spintronics in semiconductors.

2.2 Giant-magneto resistive effects

The birth-date for the field of spintronics is usually set at the discovery, in 1988, of

giant magnetoresistance [7]. In a magnetic super-lattice of (001)Fe/(001)Cr a change

in the resistance of the sample was observed, as big as 50% at 4.2K, when a magnetic

field was applied. However, as it is usually the case in science, it is appropriate to

regard this discovery as the culmination of a series of interesting investigations.

Indeed, the nature of spin transport in metals has been under study at least since

the early work of Mott [29], where the notion that currents in a ferromagnet are

spin polarized was first introduced. It is easy to obtain an estimate of the spin

polarization of the current flowing through a ferromagnet, by just thinking in terms

of the Drude theory of transport. The difference of spin-up and spin-down currents

is given by just the difference in densities, and we get the following expression for

10

Figure 2.1: Schematic band diagrams for spin transport between ferromagnets with parallel(a) and antiparallel (b) configurations (taken from [31]). The continuous line representtunneling in the up-channel and the dashed line refers to tunneling of the spin down channel.In the parallel case we have conductivities for each channel proportional to N 1

↑N 2

↑ and

N 1

↓N 2

↓ , respectively. In the antiparallel case they are proportional to N 1

↑N 2

↓ and N 1

↓N 2

↑

respectively.

the current polarization:

P ≡ J↑(EF) − J↓(EF)

J↑(EF) + J↓(EF)=

N↑(EF) −N↓(EF)

N↑(EF) + N↓(EF). (2.1)

The first experimental signature of spin dependent transport came only after 30

years, when in a series of remarkable experiments Tedrow and Meservey [30], using

superconductor/ferromagnet tunnel junctions (e.g. Al/Al2O3/Fe), were able to

directly verify Mott’s ideas and give a measurement of the current polarization

P. Their results indicated a spin polarization ranging from 10% to 45%. It was

Julliere[32], using ideas similar to the ones of Tedrow and Meservey, who created

the first spin valve using two ferromagnets separated by a tunneling junction. The

essence of the effect can be understood in a very simple way using the standard

theory for tunneling across barriers[33]. The conductance σ, of a tunnel junction

is, according to Fermi’s Golden Rule estimations, proportional to the tunnel rate

across the barrier and to the densities of states at each side of the junction:

σ = 4πe2NLNR|T|2. (2.2)

11

Assuming that the tunneling rate is spin independent and that we can regard the

transport of different spin channels as parallel transport, we easily obtain Julliere’s

formula, relating the fractional change in resistance between parallel and antiparallel

configurations and the spin-polarization at each side of the junction:

R↑↑ −R↑↓

R↑↑=

2P1P2

1 − P1P2. (2.3)

From Eq. (2.3) a TMR ratio ranging between 2% and 50% is obtained depending

on the material2. The effect was related to the spin-dependent transmission in the

super-lattice. Since then, much progress has been made both in understanding and

improving this effect especially through advances in the consistent fabrication of

layered structures that are relatively free of pinholes and have relatively weak inter-

diffusion across interfaces. As mentioned above this effect was at the center of the

“hard-disk revolution” of the late nineties, when the storage capacity of hard-disks

exploded in a matter of few years by at least three orders of magnitude. Indeed, it is

on this physical effect that most of the read-heads of hard-disks are currently based.

The idea behind the effect is simple. Since the electrons feel the large exchange

fields their transport properties will be affected by the relative orientations of the

layers in a super-lattice. On the other hand, the relative orientation of the layers can

be manipulated easily by an external magnetic field. We have a way to change the

potential profiles that the electron has to travel across by orders of the 0.5-1.0 eV,

by just applying “small” magnetic fields, where small means that the direct energy

splitting for the different spin species is at most of the order of meV. Note that this

remarkable situation (“meV causes” having “eV effects”) is a direct consequence of

2It must be emphasized that the argument leading to this figures is incomplete. The TMR de-pendence on the polarizations of the ferromagnets is complicated by several effects, most deviationscome from the spin dependence of the tunneling rates, which can be expected since in the vicinityof the tunnel barrier the states in the metals are modified. This modification in the states alsoinduced a non-trivial change in the polarization right at the interface with the subsequent changein the transport properties. Similar arguments indicate that the TMR must depend on the tunnelbarrier width. Nevertheless experiments with Fe/Al2O3/Fe tunneling junctions [34] have reportedTMR ranging from 30% (at 4.2 K) to 18% (at room temperature). TMR’s as high as 50% at roomtemperature have been found by several groups.

12

Figure 2.2: Types of magnetoresistance. (a) AMR results from bulk spin-polarized scat-tering within a ferromagnetic metal. (b) Colossal magnetoresistance (CMR) results frominteractions predominantly between adjacent atoms in certain crystalline perovskites. (c)GMR results from interfacial spin-polarized scattering between ferromagnets separated byconducting spacers in a heterogeneous magnetic material. (d) Tunneling magnetoresistance(TMR) in magnetic tunnel junctions results from spin filtering as spin-polarized electronstunnel across an insulating barrier from one ferromagnet to another. (e) Anomalous MRfrom domain wall effects has been observed in single-crystal ferromagnetic Fe whiskers andpatterned magnetic wires. (f) Ballistic MR (BMR) is another type of domain wall effectin the limit of very narrow constrictions where the conductance may be quantized. Figuretaken from [13].

the situation described at the beginning, large single particle excitation energies but

small collective excitation energies.

2.3 Spin transfer effects

Spin transfer torques correspond to the reciprocal action of the currents onto the

magnets. The idea is to consider a magnetic heterostructure like the one described

in fig.(5.2a) (a spin valve). If a current flows across the system, it has been shown

13

Figure 2.3: Illustration of the spin transfer torque in a spin valve consisting of apinned and free ferromagnetic layer. The torque on the spin angular momentum ofthe electrons, indicated by the dotted arrow, has to be accompanied by a reactiontorque on the magnetization of the free ferromagnet.

that the magnetic configuration can be altered in response to the exchange fields

created by the non-equilibrium quasiparticles. These sort of effects were predicted

to take place in nano-magnetic heterostructures in the seminal, independent, works

of Berger and Slonczewski [8, 9]. The effects have been demonstrated in several

experiments using magnetic nano-pilars[1, 35, 36], multilayers [37], magnetic point

contacts [38, 39, 40, 41], and even epitaxially grown diluted magnetic semiconduc-

tors.

The pinned ferromagnet polarizes electrons entering the device from the left.

The free ferromagnet changes the direction of the spin angular momentum. The

electrons align with the direction of magnetization of the free ferromagnet. This

change in angular momentum, i.e., torque, is indicated by the dotted arrow in Fig.

(2.3). Because of conservation of total spin, it has to be accompanied by a reaction

torque on the free ferromagnet. An electron entering the free ferromagnet will align

its spin with the local magnetization on a microscopic length scale. The basic length-

14

scale associated with this elastic spin decay is given by the destructive interference

of a great number of channels with different phases corresponds to the scale of the

transverse-spin coherence length. In metallic ferromagnets this turns out to be in

the same length scale of the Fermi wavelength. Hence, the spin transfer torque only

acts on the first few atomic layers of the free ferromagnet. It is the stiffness of the

ferromagnetic state that forces responses of the entire free ferromagnet only. In

competition with the above theoretical picture that appeals to spin conservation,

there is a theoretical picture which assumes that spin flips of the accumulated spins

at the interface between the spacer and the free ferromagnet emit spin waves that

become coherent and lead to reversal[38]. The latter picture is more successful in

describing the temperature dependence of the critical current for reversal, the former

appears to describe the dynamics of the system very well [35, 42, 43]. The physics

of the spin transfer torque and the debate between these two pictures is still an

open issue of both experimental and theoretical ferromagnetic metal spintronics.

The basic source of discrepancy is the drastic differences between the two pictures

used to start the analysis, one dealing with ballistic electrons precessing around the

magnetization, the other with different spin species diffusing around the sample. As

mentioned before, I believe that the research proposed in this project will also add

to the understanding of the physics of spin transfer torques in ferromagnetic metals.

2.4 Semiconductor Spintronics

Paramagnetic semiconductor spintronics is the subfield of spintronics where the

principal effects are associated with the interplay between intrinsic and extrinsic

spin-orbit spin splitting. Spin orbit coupling in paramagnetic semiconductors has

become an interesting tool for the manipulation of spins. The whole field of spin-

tronics acquired its impressive momentum, when it was suggested that tunable spin

orbit strength could be used to control the orientation of electronic spins. The

15

Figure 2.4: A schematics representation of a Datta-Das transistor. The channel, inbetween the p-doped InGaAs and the insulating layer InAlAs (in red in the left panel),is exposed to the effects of the gate modifying the strength of the spin-orbit interaction.This makes the electronic spins, coming from the ferromagnetic source, precess allowing it’sentrance to the drain if they reach it with the right spin orientation (right panel top) andblocking the transport if they reach the drain misaligned (right panel bottom).

Datta-Das transistor [44] irrupted in the field as a paradigmatic example of what

possibilities a proper control of the electronic spin could provide. Here a field-effect

is used to tune the spin-orbit interaction of a channel connecting two ferromagnetic

leads. The conductivity of the system will depend on the matching of the spins as

the leave the source with their spins polarized along one orientation (fixed by the

ferromagnetic moment of the source) and precess until they reach the drain. At the

same time, it also became an example of the difficulties that the researcher would

face along the way of fulfilling effective spin control. Problems are at the spin injec-

tion process, uncontrollable spin dynamics associated with spin decoherence arising

from the very presence of spin-orbit interaction, disorder effects, etc.

The problem of spin dynamics in semiconductors is quite interesting. The re-

markable sensitivity of semiconductor properties, initially regarded as a problematic

featurehas become the basis of most of our advanced technological applications. The

very same sensitivity has counterparts in the spin dynamics. The extreme sensibil-

ity of the behavior of spins to detailed features of the semiconductor has presented

serious obstacles to implement a spin-dependent semiconductor device[45]. It is

however reasonable to suppose that with advances in the manipulation techniques,

spintronics applications will soon be a reality.

16

Figure 2.5: Some remarkable spintronics effects that have been found in DMS. (a) Showshow magnetism can be turned on or off [46], in a field effect device by just adjusting thegate voltage. Left Panel: When holes are depleted from the (Ga:Mn)As layer it becomesparamagnetic. Right Panel: When the gate is adjusted to increment the hole populationthe system becomes ferromagnetic. (b) The use of a DMS in a LED lead the the generationof polarized light [47].

Diluted Magnetic Semiconductors

Diluted Magnetic Semiconductors are ternary alloys created by doping suitable non-

magnetic semiconductors with magnetic atoms. As mentioned already the basic

properties of a semiconductor can be affected quite strongly with a rather discrete

amount of doping. In the case of a magnetic semiconductors, extremely diluted

distributions of dopants magnetic atoms, are capable to change the behavior of the

sample, from non-magnetic to ferromagnetic. The basic properties of the paramag-

netic host lattice are retained, and therefore all the myriad of possibilities that are

associated with semiconductor physics. [27]

Spin Hall Effect

The anomalous Hall effect corresponds to the appearance of a Hall signal in a metal-

lic ferromagnetic sample that doesn’t arise from the Lorentz force due to an external

magnetic field. The magnitude of this potential difference is related to the compo-

17

nent of the magnetization along the axis perpendicular to the plane in which the

transport measurements are made. This effect, of course, refers to an intrinsic

”anomalous” contribution to this potential drop, rather than to the trivial ”nor-

mal” Hall Effect associated with the magnetostatic fields that are associated with

the magnetization. The basic physics associated with this effect has remained in

debate. This is despite the fact that the Anomalous Hall effect was discovered long

ago [48]. This confusion in the community has its roots on the fact that there

are several possible sources for the effect. The experimental outcome depends on

a detailed balance of competing factors. It is universally agreed that the effect

requires spin orbit interactions. The basic mechanisms that are believed to con-

tribute to the anomalous Hall signal are divided into extrinsic and intrinsic ones.

The intrinsic mechanisms [49] are associated with spin-orbit coupling in the ballistic

bands. Their nature is somewhat independent of impurities and defects. The ex-

trinsic mechanisms are associated with spin-orbit coupling directly in the impurity

potential. A flurry of recent theoretical work was motivated by the discovery of the

Berry phase [50, 51]. The quantum Hall conductance was interpreted in terms of

the Berry curvature in Ref. [52]. The quantization of the conductance was given a

profound geometrical meaning associated with the Gauss-Bonnet theorem. Ref. [53]

re-opened the theoretical investigation of the Karplus-Luttinger theory. The new

Berry-phase perspective provides a better understanding of the anomalous velocity

term. These developments have supported comparison with experiment in a robust

manner [54]. The influence of these new theoretical efforts extends well beyond the

original AHE problem.

The spin Hall Effect is a novel effect involving coupling between charge and spin

transport. It was predicted [55, 56] that, in a system with spin-orbit interactions,

the creation of a charge current has as a consequence the appearance of spin cur-

rents propagating along the transverse directions. A remarkable feature of this phe-

18

nomenon is that such a transverse response does not require breaking time-reversal

symmetry by means of a magnetic field or ferromagnetism. These predictions mo-

tivated a series of experimental studies involving semiconductor heterostructures.

The experiments show [57, 58] that, indeed as expected, the presence of a current

induced spin accumulation along the transverse region at the boundaries. The im-

pact of these discoveries was remarkable [59]. The development of this topic has not

been exempt from polemic. Just like in the case of anomalous Hall Effect, the source

of confusion is the role played by impurities. A, still ongoing, intense debate con-

cerning the nature and importance of spin currents and spin dynamics in SO-coupled

system was opened by these works. Spin orbit interactions break the independent

conservation of spin and orbital angular momentum. Since it is not a conserved

quantity, the spin accumulation is not necessarily related to a spin current. These

complications make the interpretation of the experiments rather obscure.

19

Chapter 3

Non-equilibrium Formalism for

Transport in Mesoscopic

Systems

In this chapter we introduce two basic formalisms that are going to be used in

the remaining chapters of this work. Although none of this work is original, the

description of this formalisms is necessary to make the thesis self-contained.

3.1 Introduction

With the development of experimental techniques to handle mesoscopic systems it

has become possible to develop concrete tunable implementations of the canoni-

cal idealizations of the quantum world: double-slit experiments, Aharonov-Bohm

like interferometers, tunneling, etc. These non-local effects presented a challenge

to the usual theories of transport. The solution of this challenge required a drastic

conceptual departure. The theoretical problem that arises is that when electrons

are flowing through a mesoscopic system it is hard to give an accurate theoretical

20

representation of them in terms of few parameters, such as electrical or thermal

conductivity. This is in contrast with the usual description for macroscopic systems

in a Drude-like picture, where the transport description is achieved by studying

the motion of classical particles. A quantum mechanical description, on the other

hand, involves a study in terms of wave-functions, and the complexities of such a

representation can be easily imagined. Not only are the wave functions extended

objects (actually the non-locality of the quantum world is implemented into quan-

tum mechanics by precisely this feature) but they also depend strongly on several

experimentally uncontrolled features of the system, such as defects and impurities.

For concreteness let’s talk about a “canonical” problem, consisting of a meso-

scopic system connected by separated channels to a set of independent reservoirs.

The non-equilibrium features of this system are encoded by the fact that the inde-

pendent reservoirs can have different chemical potentials. When connected to each

other through the system, the reservoirs will try to reach equilibrium by means of

the interchange of electrons. Hence a current will flow across the mesoscopic system.

Figure 3.1: The canonical problem to be solved. A mesoscopic sample is connected toindependent reservoirs, with different chemical potentials, µi. Each reservoir attempts torestore the system to equilibrium by interchanging electrons. A current flows through thesystem as a consequence.

There are two complementary ways to handle the problem. On one side

we have the Landauer-Butikker formalism, which formally reduces the problem to

21

a scattering problem, with the system as a scatterer and the leads, as many as

these might be, as scattering channels connected to independent reservoirs. In this

picture, the non-local quantum mechanical effects are build in the scattering matrix

of the system.

A different approach to non-equilibrium is to consider the electrodes and the

system to be isolated from each other, initially and separately in local equilibrium.

Then turn the connection on. A real time description of this problem can be achieved

using non-equilibrium Green’s functions (NEGF) techniques. We note here that the

distinction between the two pictures above, that can be confusing since we can

anyway use Green’s functions to calculate the scattering matrices. It could be

thought that the two formalisms are the same. It can be said that the Green’s

functions are of more generality than the Landauer-Butikker formalism, at least in

their usual implementations. The NEGF formalism can in principle handle time

dependent scatterers (such as magnons or phonons) while the Landauer-Butikker

picture can not. On the other hand the Landauer-Butikker formalism provides a

simple and robust description of transport, allowing some generic calculations and

studies of for example, the quantization of the conductance, Onsager reciprocity

symmetries and noise spectrum properties.

3.2 Landauer-Butikker Formalism

The basis of the Landauer-Butikker formalism is simple. Once the main step is

done, the rest follows from the simple rules of quantum mechanics. Moreover the

step to be taken is a conceptual one, not a mathematical manipulation.

The usual framework to describe transport in a macroscopic sample, let’s

say a piece of copper is (one intensive parameter) the conductivity σij. This single

parameter describes, for an infinitesimal volume element of the material, Ohm’s law.

This means that the current density at that element is given by ~ji = σij~Ej, where

22

~E is the net electric field at the volume element. To calculate the net current flow

of the system as a response to an external potential difference, i.e. to evaluate its

conductance, one merely needs to calculate the net electric fields in every element

and use Ohm’s law to calculate the current densities, from where the conductance

follows. The entire mathematical framework of solid state theory was then oriented

(at least in all transport calculations) to calculate the conductivity tensor, and its

dependence on the system properties, such as impurity concentration and external

magnetic fields. The problem in performing this sort of calculations in a mesoscopic

system is quite simple: Ohm’s law is not an appropriate description of the physics

the electrons at the mesoscopic scale. A quantity such as the conductivity simply

doesn’t exist for a small system. Only when the system size is big enough to allow

a semiclassical description, can we describe the transport in terms of an intensive

quantity. A few copper atoms do not have a well defined conductivity , whereas a

big sample of metallic copper does. For a problem involving a few copper atoms, we

need to solve the full Schrodinger equation in order to obtain a quantification of the

currents in the system. The conductivity is an emergent property1. The Landauer-

Butikker formalism starts by realizing that the correct description of transport must

be given in terms of the quantum mechanical wave-functions instead. After this step

is taken the arguments follows smoothly from a simple 1-D quantum mechanical

problem.

When an electronic wave-packet incoming from the left faces a potential barrier it

is well known that can either be reflected by the barrier or be transmitted, with

probabilities R and T respectively. The idea is that when the system is connected

to the right and left to reservoirs with different chemical potentials there will be

1As the system size increases the solution of the quantum mechanical problem becomes more andmore involved, to make intractable after the number of atoms reaches a modest amount (∼ 103).We should note however that by saying emergent we do not refer only to the quantitative value ofthe “corrections” to the semiclassical picture becoming small. The difference is qualitative ratherthan quantitative: the very concept of conductivity is ill defined for small system. This qualitativedifference is the main content conveyed by the word “emergent”.

23

a net difference between electrons coming from one side and the other. Let’s say

that the left reservoir’s chemical potential µL is bigger than the right one µR. The

left electrode will have a window of energy, from µR to µL of occupied states that

are not compensated by states from the right. Let’s call this window the transport

window T . The electrons from the left electrode will enter the system and cross it,

giving rise to the current:

I =∑

k∈T

eT(Ek)vk, (3.1)

where the sum extends over all the states within the quantum window, the velocity

of the wave packet is vk ≡ 1~∂Ek/∂k, and the transmission coefficient to get across

the scatterer for a state with a given energy Ek is T(Ek). By changing variables to

energies the velocity of the wave-packets cancel out with the density of states and

we obtain (2 is for spin degeneracy):

I =2e

~

∫ µL

µR

dET(E). (3.2)

For small bias potentials, we have, calling eδV = µL−µR, which transforms chemical

potentials into actual bias potentials; I = 2e2

~T(EF )δV . This gives the usual relation

for the conductance in terms of the transmission coefficient:

G =2e2

~T(EF ). (3.3)

There is a finite intrinsic resistance (defined as the inverse of the conductance

as in the usual Ohm’s law) associated with transport across a quantum element.

From charge conservation, the best transmission that can be obtained is 1, and this

limit is not associated with limitations to our abilities to build perfect systems, it is

a limit on the perfect system itself. It comes from the basic principles of quantum

mechanics. This seems at odds with our usual understanding that resistance (as

actually its very name implies) is a measure of something in the materials that op-

poses the flow of charge. Even the perfect system with nothing in it but a channel

24

where the electron moves freely has a finite resistance. This is known as the quan-

tized contact resistance. A system exhibiting a resistance limited by the number of

transport channels is known as a quantum point contact.

This raises a natural question concerning dissipation. The resistance asso-

ciated with Eq. (3.3) can be related with dissipation or heating of the sample.

However in the derivation we considered only elastic processes that are incapable

of rendering such a dissipation. The question is: where “are” the dissipative pro-

cesses? The answer is hidden in our naive assumption concerning the electrodes.

By saying that the electrodes are Fermi-seas with a given chemical potential, we

are assuming that they are in equilibrium. As the electrons leave one electrode and

enter the other, quasi-particles are created on each side that are out of equilibrium.

Dissipative processes then enter into play to relax those excitations and keep the

electrodes in equilibrium. All these physical processes have a time scale faster that

the electronic transport itself. The dissipation is due to those processes. Assum-

ing that the energy dissipation on the reservoirs is efficient the Landauer argument

counts the degree to which the system is out of equilibrium, which depends only on

the channel properties.

With this physical picture in mind it is now straightforward to extend the

treatment to many leads. If we have a set of leads connected to the system, each

with its independent chemical potential as in figure (3.1) the flow of electrons from

lead α to lead β is determined by 2e2

~Tαβ(µα − µβ). Denoting the transmission

probability from lead i to lead j by Tij, the current and potential at the lead i by

Ii and Vi, respectively, the relation Ii = e2

h

∑j Tij(Vi − Vj), holds. This relation

is the fundamental result of the Landauer-Butikker formalism. We note here that

we haven’t said a word on how to determine the transmission coefficients, leaving

them as parameters describing the system. To find the transmission coefficients it is

necessary in principle to solve the whole quantum mechanical problem of scattering

25

by the system, a problem that is in general untractable. There are important prop-

erties that follow not from the detailed values of the transmission coefficients, but

rather from relations among them. Such relations can be found based on general

arguments, such as symmetries, that apply to the general scattering matrix.

We will find several techniques that will allow us to find the coefficients.

Among them the Green’s function formalism is the most general and versatile one

and will be discussed in the next section.

3.3 NEGF formalism

In this section we introduce the basic theoretical formalism that will guide the major

developments described in this thesis, the non-equilibrium Green’s functions formal-

ism. This formalism was developed during the 60’s by several people, among them

Schwinger[60], Keldysh[61], and Kadanoff and Baym [62]. Those approaches dif-

fered in their mathematical methods but they convey, essentially, the same physical

content. The Kadanoff-Baym formalism established differential equations for var-

ious Greens functions like those described below. The Kadanoff-Baym differential

equations were written in a semiclassical terminology (basically the one of wave

packet widths, momenta and positions2) and then it was possible to derive semi-

classical approximations and corrections to them. They derived the now famous

Quantum Boltzmann Equation, that allowed them to calculate transport properties

of several systems. The Keldysh formalism is the very same non-equilibrium Green’s

functions formalism but an integral representation. Since it does not restrict the

2This is called the Wigner representation. In the density-matrix (or Green’s function) we canalways perform the change of variables:

ρ(x1, x2) ≡ ρ(X,x), (3.4)

where X = (x1 + x2)/2 and x = x1 − x2. For homogenous systems it is clear that ρ(X,k) ∼Rdx eikxρ(X,x) and we can build the semiclassical approximation regarding the system as a set of

wave-packets with position X and momentum k.

26

relevant variables to be describable by wave-packets, it is particularly useful in han-

dling non-equilibrium situations in mesoscopic systems. It is this formalism that we

are going to derive and illustrate in this chapter.

3.3.1 Basic Considerations in Non-Equilibrium Field Theory.

The field theoretical description of a condensed matter system is usually written

down [33, 63, 64] by writing a perturbation series, in terms of certain small param-

eter, in terms of the small parameters appropriate to the problem, for the Green’s

function describing the system. The treatment, despite is overwhelming power and

versatility, is restricted to handle (quasi-) equilibrium situations in which the system

remains (quasi-) stationary and it is well described by a time-independent Hamilto-

nian. A path integral representation for those expansions can also be worked out in

terms of path integrals over coherent states [65, 66].

To study non-equilibrium situations an extension of the formalism is needed.

Such an extension can be done from several perspectives. The Non-equilibrium

Green’s Function formalism that we are going to discuss next is based on the closed

time contour formalism [60] and was developed mostly by Keldysh [61]. Here we are

going to express the main results using a coherent state path integral. The main

results are the ones described in [67].

Time loops and Keldysh Green’s functions

At this point we are going to treat a quite generic quantum system behaving under

the dynamics described by a time dependent Hamiltonian:

H = H0 + Hint, (3.5)

in thermodynamical equilibrium with a thermal bath with temperature T = 1β .

The state of this system is fully described, in the grand canonical ensemble, by the

27

density-matrix[68, 69]:

ρ(H) ≡ exp (−βH− µN )

Tr exp (−βH− µN ) (3.6)

The averages of physical quantities are determined by:

〈O〉 = Tr ρ(H)O . (3.7)

On the other hand if a non-equilibrium situation is induced by applying a time-

dependent potential:

H(t) = H + δH(t), (3.8)

we need to explicitly calculate the time evolution of the operators:

〈O(t)〉 = Trρ(H)OH(t)

. (3.9)

where OH(t) correspond to the Heisenberg representation of the time-evolution of

O. If we denote O0(t) as the Heisenberg operators for the Hamiltonian H0 we can

easily find the expression:

O(t) = U†(t, t0)O0(t0)U(t, t0). (3.10)

In the expression above U corresponds to the time evolution operator.

U(t, t0) = T exp

−i

∫ t

t0

H(t)

. (3.11)

The evaluation of that kind of expression is a complex issue. However we have to our

advantage the fact that in equilibrium theory we are faced with a similar expression.

The idea is to develop a theory similar to the equilibrium one. The only difference is

that the explicit time dependence of δH is, in the equilibrium situation, completely

artificial. In that treatment the “interaction” Hamiltonian is modeled to be “turned

on” and then “turned off” adiabatically.

In equilibrium field-theory the system is supposed to be in an intermediate

step of an infinitely long cycle starting (at t = −∞) and ending (at t = ∞) in a non-

interacting state. The adiabaticity of such an artificial time-dependence ensures, for

28

systems with non-degenerate ground states, that the states at the extremes of the

cycle differ by nothing but a Berry phase factor [70, 71]. Such a phase is usually

written down in terms of the scattering matrix S(−∞,∞).

|Ψ(t = −∞)〉 = S(−∞,∞) |Ψ(t = ∞)〉 . (3.12)

The latter relation allow us to calculate averages of T -ordered products in a simpler

manner. The averages being taken at the ground state |Ψ0〉 ≡ |Ψ(t = −∞)〉, are

simplified into purely time ordered structures as follows3:

〈Ψ(t = −∞)|T O|Ψ(t = −∞)〉 ≡ 〈Ψ(t = −∞)|T O|Ψ(t = ∞)〉〈Ψ0|S(−∞,∞)|Ψ0〉

. (3.13)

We see how the ket evaluated at t = ∞ simply goes along the stream of time

orderings in the operator. It is convenient to have an average of a completely time-

ordered expression; since in that situation we can use the Wick theorem to reduce

it to pair-wise averages of time ordered creation and destruction operators, i.e. the

Green’s functions of the system : G12 = 〈T c(1)c†(2)〉. This is the basis of the

diagrammatic expressions upon which the technique relies in order to evaluate a

sum of an infinite sub-series of terms in the perturbation theory [33, 64, 65, 5].

The clear distinction from the equilibrium case is that the non-equilibrium

time evolution depends explicitly on time in a non-adiabatic manner. This means

that the state at t → ∞ not always easily related to the state at t → −∞. In

[60, 61, 72, 73] a formalism is developed to treat the out of equilibrium case. This

technique is based on the use of time-loops to give a solution to this problem. The

technique uses a “path” in time that instead of going to t = ∞ it goes back to

t = −∞. The price to pay for this is that we now face integrals, orderings and

evolutions on a time-loop which makes things harder.

3The denominator of the following expression, known as the vacuum polarization bubble, playsa very important role in the formalism, it can be shown to be responsible for the cancelation of theunconnected diagrams.

29

Figure 3.2: Contour Ct is a closed-time contour [60]. The contour has two branches, onegoing from the past to the future and another in the opposite direction. The contour orderingoperator reorders its arguments in such a way that operators acting at times previous inthe contour are located to the left. The reordering is performed in accordance with the(anti-)commutation relations obeyed by the operators.

The evolution of an operator when the Hamiltonian depends explicitly on

time is given by:

O(t) = U†(t, t0)O0 U(t, t0). (3.14)

With this a simpler expression for O(t) is given by:

O(t) = TCt

exp

(−i∫

Ct

dτδH(τ)

)O0(t)

, (3.15)

where Ct is the time contour represented in figure 3.2. All the terms in the expression

above have well known path-integral representations [65, 66]. If we consider now

the operator O to be a combination of products of single particle operators (charge

density, charge current, spin density, spin currents etc.), we can write everything in

terms of a single path-integral generating functional Z, over Grassmann variables

fields:

Z[J(x, t), J(x, t)] =

∫DΨ(x, t)DΨ(x, t) exp

iSK

[Ψ(x, t), Ψ(x, t)

], (3.16)

where,

SK

[Ψ(x, t), Ψ(x, t)

]=

∫

C∞

dt

∫dxΨ(x, t)

(i∂

∂t+

∇2

2m− V(x)

)Ψ(x, t)

+

∫

C∞

dt

∫dx(Ψ(x, t)J(x, t) + Ψ(x, t)J(x, t)

). (3.17)

We have absorbed the initial density matrix in a path integral over imaginary

30

Figure 3.3: Contour Ct. The contour has three branches, one going along the complex planethat configures the equilibrium situation at t0, another going from the past to the futureand another in the opposite direction. The contour ordering operator reorder its argumentsin such a way that operators acting at times previous in the contour are located to the left.

time as is customary. This point should be carefully examined when we include

interactions. For example, in describing a broken symmetry state it is clear that

a Hubbard-Stratonovic transformation must now include auxiliary fields defined all

along the Closed Time Contour. We extend the notion of Green’s function defined

normally by ordering under the time axis to a “Non-equilibrium” Green’s Function

defined by ordering under the contour Ct.

3.3.2 Basic properties of the Non-Equilibrium Green’s functions

Naturally the range of the temporal variable can also be extended to cover the whole

contour. Given two instants t1 and t1′ we have four possibilities for locating them

on the contour Ct, that gives rise to four different Green’s functions. With that the

time dependent 1-body Green’s function, can be written as:

G(1, 1′) = Tr[ρ(t0)TC∞(Ψ(x, t)Ψ(x′, t′))

]

=1

i2δ2

δJ(x, t)δJ(x′, t′)Z[J, J

]∣∣∣∣J,J=0

. (3.18)

In the same way we can define the n-body Green’s functions by:

G(1 · · · n, 1′ · · · n′) =1

i2n

δ2n

δJ(1) · · · δJ(n)δJ(1′) · · · δJ(n′)Z[J, J

]∣∣∣∣J,J=0

(3.19)

31

Now we are going to focus on the 1-body Green’s function. This will help us to

develop further the notation and to find some useful relations that we are going to

extend also for the multi-body Green’s functions. The four cases for the 1-body

non-equilibrium Green’s functions are:

• t1 and t2 are in the lower branch of the CTP.

• t1 and t2 are in the upper branch of the CTP.

• t1 in the lower and t2 in the upper branch of the CTP.

• t1 in the upper and t2 in the lower branch of the CTP.

The respective non-equilibrium Green’s function become:

iG−−12 = 〈T Ψ1Ψ

†2〉

=

〈Ψ1Ψ†2〉 t1 > t2

∓〈Ψ†2Ψ1〉 t1 < t2,

(3.20)

iG++12 = 〈T Ψ1Ψ

†2〉

=

∓〈Ψ†2Ψ1〉 t1 > t2

〈Ψ1Ψ†2〉 t1 < t2,

(3.21)

iG−+12 = 〈Ψ1Ψ

†2〉, (3.22)

iG+−12 = ∓〈Ψ†

2Ψ1〉. (3.23)

Clearly, they correspond to the different values that the contour-ordered Green’s

function can have depending on the position of its arguments on the contour (the

superscript ± means that the corresponding time variable is in the lower(upper)

part of the path). The notation at this point starts to vary in the literature. The

following notations are commonly used for these four Greens functions:

G

G++ G+−

G−+ G−−

≡

GF G+

G− GF

≡

GT G<

G> GT

(3.24)

32

Here the F stands for the “Feynmann causal propagator” and the T for “time

ordered propagator”. The “lesser” greens function G< will be the most important

element of the theory, from its very definition we can see how it is closely related

to 1-body operator expectation values. These quantities are not independent. A

relation that emerges from their definitions is:

G−− +G++ = G−+ +G+−. (3.25)

We are going to take advantage of this relation by reducing the problem to three

independent variables by using the above relation:

GR(1, 1′) = G−−(1, 1′) −G−+(1, 1′)

= G+−(1, 1′) −G++(1, 1′)

GA = G−−(1, 1′) −G+−(1, 1′)

= G−+(1, 1′) −G++(1, 1′)

GK = G+−(1, 1′) +G−+(1, 1′)

= G−−(1, 1′) +G++(1, 1′) (3.26)

where GR and GA functions correspond to the usual retarded and advanced Green’s

functions respectively. The transformation between these two kinds of Green’s func-

tions was first implemented by Keldysh and it is called a Keldysh-rotation. In the

appendix (A.4.1) we show how they can be cast in terms of rotations in Keldysh

space. The inverse relation can be written as:

G =1

2GR

1 −1

1 −1

+

1

2GA

1 1

−1 −1

+

1

2GK

1 1

1 1

, (3.27)

or using the spinors in Keldysh space ξ =

1

1

and η =

1

−1

we can write:

Gµν =1

2GRξµην +

1

2GAηµξν +

1

2GKξµξν , (3.28)

33

The relations between these two representations (called in Ref. [67] single time and

physical representations) can be made more systematic and generalized to n-body

Green’s functions. This process is expressed in a simpler way by regarding the

transformation between the different representations of Green’s functions as simple

transformations in the sources rather than in the fields themselves.

Keldysh Rotations at the Generating Functional Level

The source term can be separated as:

∫

C∞

dxdtJ(x, t)Ψ†(x, t) ≡∫ ∞

−∞dtdx

(J+(x, t)Ψ†

+x, t) − J−(x, t)Ψ†−(x, t)

), (3.29)

where the sub-indexes are indicative of the place in the CTP that the operators

are taken, and the minus sign separating the two contributions reflects the differ-

ent directions on integrations along the path. This separation allow us to write

symbolically:

Gµν = − δ2Z

δJµδJν(3.30)

It is convenient to introduce a different parametrization of the sources, in terms of

the difference and sum of upper and lower branches:

J∆(t) ≡ J+(t) − J−(t) = ηµJµ, (3.31)

Jc(t) ≡ 1

2(J+(t) + J−(t)) =

1

2ξµJµ, (3.32)

and the source term becomes:

∫J∆ Ψ†

c + Jc Ψ†∆, (3.33)

with the corresponding fields:

Ψ†∆(t) ≡ Ψ†

+(t) − Ψ†−(t) = ηµΨ†

µ, (3.34)

Ψ†c(t) ≡ 1

2

(Ψ†

+(t) + Ψ†−(t)

)=

1

2ξµΨ†

µ. (3.35)

34

Again, we can invert these relations, using:

Jµ = Ψ†cξµ +

1

2J∆ηµ, (3.36)

Ψ†µ = Ψ†

cξµ +1

2Ψ†

∆ηµ, (3.37)

with the corresponding changes in the derivatives:

δ

δJµ=

1

2ξµ

δ

δJc+ ηµ

δ

δJ∆, (3.38)

δ

δJc= ξµ

δ

δJµ, (3.39)

δ

δJ∆=

1

2ηµ

δ

δJµ. (3.40)

Equipped with these relations we can prove that all the Green’s functions are just

functional derivatives of the same generating functional with respect to different

variables:

GA = − δ2Z

δJ∆δJc, (3.41)

GR = − δ2Z

δJcδJ∆, (3.42)

GK = − δ2Z

δJcδJc, (3.43)

3.3.3 Field Equations and Perturbations in Keldysh Space

Basic Perturbation Expansion in Keldysh Space

The main merit of the Non-Equilibrium Green’s functions formalism is that it ex-

presses a generic time-dependent behavior in the form of a time-ordered expectation

value (see fig.(3.3)). This allow us to implement a non-equilibrium version of the

Wick theorem and to write down a series expansion just as in the usual case. The

catch however is the emergence of quite cumbersome expressions due to the different

combinations of time-branches that might appear. To illustrate the perturbation se-

ries that appears in the Keldysh formalism, we are going to take the simple example

of free fermions under the influence of a external potential V (~x, t).

35

Figure 3.4: The diagram representation of the free particle Green’s function. The doubleline without indexes will denote the matrix in Keldysh space.

To first order we obtain:

G(1)(1, 1′) =

∫dx2

∫

CK

dτ2 G(0)CK

(1, 2)V (2)G(0)CK

(2, 1′). (3.44)

By doing the separation:∫CK

dτ →∫∞−∞ dt−

∫∞−∞ dt we can write the term in terms

of standard functions. The different combinations of positions of 1 and 1′ give rise

to four corrections: G

(1)++(1, 1′) G

(1)+−(1, 1′)

G(1)−+(1, 1′) G

(1)−−(1, 1′)

=

∫dt2

G

(0)++(1, 2) G

(0)+−(1, 2)

G(0)−+(1, 2) G

(0)−−(1, 2)

× (3.45)

×

V (2) 0

0 −V (2)

G

(0)++(2, 1′) G

(0)+−(2, 1′)

G(0)−+(2, 1′) G

(0)−−(2, 1′)

or, in the more compact notation in Keldysh space:

G(1)(1, 1′) = G(0)(1, 2)V (2)G(0)(2, 1′). (3.46)

We see how standard calculations can become quite intricate by the presence of the

four entries in the Green’s function. Now, the advantage of the generating functional

approach is that we can reduce the effort, by the use of formal field-equations.

Functional Field Equations

Let us define the generating functional for the connected correlators:

W[J, J] = −i log Z[J, J], (3.47)

36

The proof that this expression does indeed generate the connected correlators is

cumbersome but it reduces ultimately to a combinatorial problem. It is displayed in

full detail in the treatise of Zinn-Justin [74]. The formal treatment goes as follows.

Let the 1-point correlators be defined by:

δW

δJ(1)≡ Ψc(1), (3.48)

δW

δJ(1)≡ Ψc(1),

they can of course be regarded as functions of the sources J and J , or vice-versa,

the sources be regarded as functions of the fields. We perform the usual Legendre

transformation into the vertex generating functional:

Γ[Ψc, Ψc] = W[J, J] −∫d1(Ψc(1)J(1) + Ψc(1)J(1)

), (3.49)

with the consequence:

δΓ

δΨ(1)= −J(1) (3.50)

δΓ

δΨ(1)= −J(1) (3.51)

this implies, by taking the derivative with respect to J(2), and using equations (3.48)

and (3.49), the following identity:

∫δ2W

δJ(2)δJ(3)

δ2Γ

δΨc(3)δΨc(1)d3 = −δJ(1)

δJ(2). (3.52)

Finally, identifying the first element inside the integral as the connected 1-body

Green’s function, we conclude:

∫G(1, 2)Γ(2, 3)d2 = δ(1 − 3), (3.53)

and similarly (by taking the derivative with respect to J) we obtain:

∫Γ(1, 2)G(2, 3)d2 = δ(1 − 3), (3.54)

37

both expressions are known as Dyson Field Equations. Γ is known as the 1-particle

irreducible (1PI) vertex function,

Γ(1, 2) ≡ δ2Γ

δΨc(1)δΨc(2). (3.55)

The above might seem extremely formal, however, what we have achieved is very

important and practical. We have found field equations for the correlation functions

that are generally valid. Now, we just need to find approximations to the 1PI vertex

and proceed to calculate the Green’s functions. Note, that to write this expression

as a relation of matrices on Keldysh-space, we need to keep track of the (-) that

follows the integrals on the negative branch. This implies that the correct form of

the Dyson equation in terms of Keldysh-matrices is4:

∫d3 Γ(1, 3)σ3G(3, 2) = σ3δ(1 − 2) (3.56)

It should also be emphasized that the same procedure could have been followed

using the “mixed” sources J∆ and Jc from the previous section in order to define

a “physical” time Dyson equation. Again, that equation involves less functional

dependencies, but is further away from the observables. The Dyson equation in

such a representation is:

∫d3 Γ(1, 3)σ1G(3, 2) = σ1δ(1 − 2). (3.57)

This is the same equation that could have been reached from performing a Keldysh-

rotation on both the Green’s function and the vertex. Note that the usual Keldysh-

space representation of the generating functional also holds:

Γ =

0 ΓA

ΓR ΓK

. (3.58)

4here the bold-face in Γ is a reference to its matrix character, and it not the vertex generatingfunctional

38

with the inverse relation:

Γµν =1

2ΓRξµην +

1

2ΓAηµξν +

1

2ΓKξµξν , (3.59)

just like the one for the Green’s functions.

Equilibrium Green’s and vertex functions

The calculation of the equilibrium Green’s functions for a free fermion system is

straightforward [33, 63]. The basic results are:

g<(ω) = i nF(ω)A(ω); (3.60)

g>(ω) = −i [1 − nF(ω)] A(ω); (3.61)

gt(ω) = [1 − nF(ω)] gR(ω) + nF(ω)gA(ω); (3.62)

gt(ω) = − [1 − nF(ω)] gA(ω) − nF(ω)gR(ω); (3.63)

where gR/A(ω) = (ω ± iη − H0)−1. The Green’s function in Keldysh space is:

G0 =

gt g<

g> gt

(3.64)

Assuming that Γ(0) satisfies the Dyson equation with G0 we obtain:

Γ(0) =

γt γ<

γ> γt

. (3.65)

It becomes simple to treat this equation in the “physical time” representation:

G0 =

0 gA

gR gK

; (3.66)

Γ(0) =

0 γA

γR γK

, (3.67)

39

from which, using eq.(3.57), we obtain5

γA =(gA)−1

, (3.68)

γR =(gR)−1

, (3.69)

γK = −γRgKγA ≡ 0, (3.70)

directly. Going back to the single time representation:

Γ(0) =

γt γ<

γ> γt

=

12

(γR + γA

)−1

2

(γR − γA

)

12

(γR − γA

)−1

2

(γR + γA

)

=

(ω − H0) 0

0 − (ω − H0)

(3.71)

Perturbation Series and Quantum Boltzmann Equation

The standard notation for the 1PI vertex function is:

Γ(1, 2) = Γ(0)(1, 2) − Σ(1, 2), (3.72)

where Γ(0) is the 1PI vertex function of the free fermion system, and Σ, the self

energy, stands for the corrections (in a non-perturbative expression, to all orders)

due to deviations form the ideal system. The detailed properties of the Keldysh-

space self energy depend on the precise form of the perturbating mechanism (either

for e-e interactions, electron-phonon coupling, external fields, etc). The details

for the case of a system connected to leads are going to be described in the next

section. However, regardless of the specific form of the self energy we have that as

a consequence of the Dyson equation it can always be written as:

Σ =

ΣT Σ<

Σ> ΣT

, (3.73)

5The product γRgKγA can be related to the difference between the inverse of the advanced andretarded greens functions, i.e. ∝ the infinitesimal η.

40

whose elements satisfy the usual constraint on Keldysh-space:

ΣT + ΣT = Σ< + Σ>, (3.74)

leading to the same physical representation:

Σ =

0 ΣA

ΣR ΣK

, (3.75)

Equation (3.57) reads: 0

(gA)−1

− ΣA(gR)−1

− ΣR −ΣK

σ1

0 GA

GR GK

= σ1, (3.76)

whose components imply that:

GA =

((gA)−1

− ΣA)−1

, (3.77)

GR =

((gR)−1

− ΣR)−1

, (3.78)

GK = GRΣKGA. (3.79)

The, by now usual, rotation in Keldysh space (eq. (3.28)) leads us to:

G< =1

2

(GR −GA −GRΣKGA

). (3.80)

This expression is sufficient for the problems studied in this thesis. We can, however,

represent all the Green’s functions using a somewhat more standard notation by

expressing everything in terms of the single-time self energies. In the end we obtain

after simple manipulations:

GR = gR(1 + ΣRGR

), (3.81)

GA = gA(1 + ΣAGA

), (3.82)

G≶ =(1 +GRΣR

)g≶(1 +GAΣA

)+GRΣ≶GA, (3.83)

Gt =(1 +GRΣR

)gt

(1 +GAΣA

)+GRΣtG

A, (3.84)

Gt =(1 +GRΣR

)gt

(1 +GAΣA

)+GRΣtG

A, (3.85)

41

where (·)≶ stands for either (·)< or (·)>. Equation (3.83) is the most important

relation in non-equilibrium field theory. It is the starting point for the derivation

of the Boltzmann equation. We must keep in mind that all these equations are

empty statements relating the different green’s functions and they convey no phys-

ical information concerning the details of the system. They do however, indeed

have some general information about statistics). In order to use them to their full

power we must supplement them with “constitutive relations” specifying further

the self-energies associated with the processes we want to take into account. In the

next section we are going to evaluate exactly (in a non-perturbative sense) the self

energies associated with the connection of a system to leads. Other examples are

impurity scattering and phonon scattering, where the self energy contribution can

be evaluated using perturbative methods (those are similar, though, to the standard

equilibrium methods).

3.3.4 Application: Tunneling current

Physical Considerations

As an example we are going to consider transport across a tunneling junction[75, 76].

This calculation is very important for the discussions that appear in the following

chapters. Consider the system described in figure (3.5). An insulator (I) connecting

two metallic leads (M). If a bias potential is induced between the two metallic leads,

a current will travel across the insulator. We need to calculate the tunneling trans-

mission probability and the current bias characteristic using the ideas described in

the section above. Some physical considerations are in order to help us better un-

derstand the results to be obtained. The issue of dissipation is perhaps most crucial

in understanding the physics of ballistic transport. It is important to emphasize

that the dissipation in a ballistic junction always takes place in the leads. What is

remarkable is that the resistance is determined by transmission coefficients in the

42

junction itself, that have nothing to do with the processes that will cause the actual

dissipation. The reason for the independence of the resistance of a junction and the

dissipative mechanisms in the leads can be clearly understood as follows. Consider

the system in fig.(3.5) an assume that at t → −∞, the system is absolutely decou-

pled from the leads, which in turn are in equilibrium with independent reservoirs

whose difference in chemical potential correspond to the bias difference. We then

turn on the connection of the reservoirs to the system. The electrons will then flow

across the system in a futile attempt to restore the thermodynamical equilibrium

between the leads. When the reservoir collecting electrons receives electrons from

the emitter, they will be creating a non-equilibrium distribution. We assume that

here is where the dissipation of the leads enters to attempt a restoration of the

thermodynamic equilibrium. The idea is simple then, the dissipation mechanisms

act only to restore the equilibrium and then it is natural that the dissipation is just

proportional to the amount to which the equilibrium is disturbed, this is the net cur-

rent, determined only at the junction. In order to calculate the ballistic current in a

tunnel junction we need then to evaluate the elastic mechanisms associated with the

scattering at the junction, but we also need to model the inelastic mechanisms that

drive the leads toward equilibrium. These are mostly phonons and electron-electron

interactions. In rigor this effects might be captured relying in some Caldeira-Leggett

like model [77, 78]. In other chapters we are going to describe some mechanism to

restore equilibrium in the leads.

Note that in the case of an interacting system, the non-equilibrium changes in density

will cause changes in the mean-field potentials, leading to a modification of the

junction scattering profiles. It is natural that such changes are going to be confined

to the proximity of the contacts. Those effects can be accounted for in this formalism

by simple extending the region to be regarded as system in non-equilibrium to

include the part of the leads modified by the contacts.

43

Model Hamiltonian

The system consists on two electrodes (L and R) described by an equilibrium distri-

bution different for each one, and a junction in contact with both. The electrodes

and the junction will be described as simple non-interacting electron systems.

Figure 3.5: The basic model to describe a system coupled to electrodes. A potentialdifference between the electrodes will create a current across the system.

H = HJ + HE + HC (3.86)

where HJ is the Hamiltonian of the junction, HE the Hamiltonian of the leads and

HC is the coupling between the junction and the electrodes. The Hamiltonian of

the system can be written as:

H =∑

〈ri,rj〉,σ

trirj r†ri,σ

rrj,σ +∑

〈li,lj〉,σ

tlilj l†li,σllj,σ +∑

〈i,j〉,σ

tij d†i,σdj,σ

+∑

ri∈∂R

j∈∂R,σ

trij

(r†ri,σdj,σ + d

†j,σrrj,σ

)+∑

li∈∂L

j∈∂L,σ

tlij

(l†li,σ

dj,σ + d†j,σllj,σ

)(3.87)

The notation is self-explanatory, rrj,σ, llj,σ and dj,σ annihilate fermions at positions

rj, lj and j respectively located in the right lead, the left lead, and the system. trirj ,

tlilj and tij are the hopping parameters in the right lead, left lead, and system. 〈·〉corresponds in this equation to a sum over nearest neighbors, while ∂L is the region

of the left lead in contact with the system the region ∂L (and similarly for R) in the

system.

Generating functional

Since we are interested in the properties of the system (example: density and current

going through it) we attach to the path integral current sources at sites in the

44

junction only. The generating function is:

Z[Ji(t), Ji(t)] =

∫D2lliD2rriD2di exp iSK [rri(t), lli(t), di(t), rri(t), lli(t), di(t)] ,

(3.88)

where the action can be separated in several pieces:

SK = SLEADS + SSYSTEM + SCOUPLING + SSOURCES (3.89)

SLEADS =

∫

C∞

dt

∑

ri

rri(t)(i∂tδrirj + trirj

)rrj(t) +

∑

li

lli(t)(i∂tδlilj + tlilj

)llj(t)

(3.90)

SSYSTEM =

∫

C∞

dt∑

i

di(t) (i∂tδij + tij) dj(t) (3.91)

SCOUPLING =

∫

C∞

dt∑

ri∈∂R

j∈∂R,σ

trij

(rri,σ(t)dj,σ(t) + dj,σ(t)rrj,σ(t)

)

+

∫

C∞

dt∑

li∈∂L

j∈∂L,σ

tlij

(lli,σ(t)dj,σ(t) + dj,σ(t)llj,σ(t)

)(3.92)

SSOURCES =

∫

C∞

dt∑

i

di(t)Ji(t) + Ji(t)di(t) (3.93)

Since all the terms are quadratic in the coherent states we can perform the integrals

explicitly, starting from integrating out the leads. Here we need to assume that each

lead is described by the equilibrium conditions, although the chemical potentials in

different leads are allowed to be different.

Seff =

∫

C∞

dt

∫

C∞

dt′ di(t)((i∂tδij + tij) δ(t− t′) − Σij(t, t

′))dj(t

′)

+

∫

C∞

dt∑

i

di(t)Ji(t) + Ji(t)di(t). (3.94)

Finally, we can integrate the fermions definitively and so we are left with a path

integral involving the currents only:

Seff =

∫

C∞

dt

∫

C∞

dt′Ji(t)Gij(t, t′)Jj(t

′) (3.95)

45

Fisher-Lee Formula

We are now ready to obtain the final expression that characterizes the transport

through the system. Finally, we are going to extend the result to a slightly more

general family of systems. To evaluate the Green’s functions we can evaluate the

derivatives with respect to the sources in the effective action Eq.(3.95). However

we have already done the algebraic aspects of such calculations leading to equation

(3.83). In the present context, since we are only interested in the lesser-green’s

function, we write:

G< = GRΣ<GA (3.96)

where we can read directly the lesser self energy from equation (3.95). It is basically

equal to the lesser green’s functions of the leads at the contact points times the

hopping parameter at the contact.

Now, the retarded/advanced self energy is:

ΣR/A = ΣR/A

R + ΣR/A

L (3.97)

separated in contributions from each lead. For each lead we have the self energy

contribution:

ΣR/A

R,L,L′ =∑

L1,L2≤0

tL,L1gR/A

R,L1,L2tL2,L′ (3.98)

ΣR/A

L,L,L′ =∑

L1,L2≥N

tL,L1gR/A

L,L1,L2tL2,L′ (3.99)

The imaginary part of the self energy is:

ΓR/L = i(ΣRR/L − ΣA

R/L

)(3.100)

It follows that,

Σ< = i (nLΓL + nRΓR) . (3.101)

This expression defines all the equilibrium and non-equilibrium properties of the

system. Before writing down the final solution, let us generalize the results a bit,

46

in order to make the applications in the rest of this work more straightforward,

we will use the notation of [79]. We consider a system with several bands in more

than one dimension. We must define, fermion operators with labels indicating the

extra degrees of freedom, ΨλkL, where λ is a band index, k a wave-vector index

on the transverse directions, and L the lattice index. The non-equilibrium Green’s

functions are now,

G<λL,λ′L′(k; t, t′) = i

⟨Ψ†

λ′kL′(t′) Ψ

λkL(t)⟩

(3.102)

G>λL,λ′L′(k; t, t′) = −i

⟨Ψ

λkL(t) Ψ†

λ′kL′(t′)⟩

(3.103)

and so on. The observables of interest are, the electron density:

NL = − 2i

Aδ

∑

k

∫dE

2π

∑

λ

G<λL,λL(k;E) (3.104)

and the current density:

JL = i2e

A

∫dE

2π

∑

k λλ′

L1,L2

tλL1;λ′L2G<λ′L2,λL1

(k;E) − tλL2;λ′L1G<λ′L1,λL2

(k;E)(3.105)

Since for this system all the derivations apply, we can also find the following relation:

J =2e

A

∑

k

∫dE

2πTr(ΓRG

RΓLGA)(nR − nL) , (3.106)

known as the generalized Fisher-Lee relation.

3.4 Conclusions

The journey we have made in this chapter of the thesis has been through an intricate

morass of sometimes confusing formalism. Nevertheless, what has been described so

far is important and will be used recurrently in the subsequent chapters. A moment

of pause is in order before we undertake the problem of using these tools in the

magnetic state. This will be done right at the start of next chapter (after introducing

47

some additional formalities related to the inclusion of the order parameter field).

Basically we have started with a quite generic problem, namely a generic field theory

driven out of equilibrium by some arbitrary disturbance. The generality of such a

situation called for a strictly formal manipulation in terms of time dependent density

matrices. Those considerations led us to the notion of contour time path integral,

that is useful to bypass some problems arising due to the time-dependence of the

disturbance. The time-path allowed us to define “time-ordered” correlators (ordered

in the sense of the time-path). The latter turn out to be the non-equilibrium Green’s

functions and several of their properties were discussed. In particular, a set of field

equations was derived giving rise to the Dyson-equation for the non-equilibrium

Green’s functions. The general formalism was illustrated by applying it to the case of

a system connected to leads. The Fisher-Lee relation was proved, bringing together

the NEGF and Landau-Buttiker approaches to to describe quantum transport.

48

Chapter 4

Current-induced dynamics in a

Ferromagnet

The contents of this chapter are partially based on the article: J. Fernandez-Rossier,

M. Braun, A. S. Nunez, A. H. MacDonald, Influence of a Uniform Current on

Collective Magnetization Dynamics in a Ferromagnetic Metal, Phys.

Rev. B 69, 174412 (2004), cond-mat/0311522.

4.1 Introduction

The strong and robust magnetotransport effects that occur in metallic ferromagnets

(anisotropic, tunnel, and giant magnetoresistance for example [80, 81]) result from

the sensitivity of magnetization orientation to external fields, combined with the

strong spin and magnetization-orientation dependent potentials felt by the current-

carrying quasiparticles. This fundamentally interesting class of effects has been ex-

ploited in information storage technology for some time, and new variations continue

to be discovered and explored . Attention has turned more recently to a distinct class

of phenomena in which the relationship between quasiparticle and collective proper-

49

ties is inverted, effects in which control of the quasiparticle state is used to manipu-

late collective properties rather than vice-versa. Of particular importance is the the-

oretical prediction [8, 9] of current induced magnetization switching and related spin

transfer effects in ferromagnetic multilayers. The conditions necessary to achieve

observable effects have been experimentally realized and the predictions of theory

largely confirmed by a number of groups [38, 82, 83, 84, 85, 1, 35, 37, 86, 87, 88]

over the past several years.

Current-induced switching is expected [8, 9, 89, 90, 91, 92, 93, 94, 95, 96,

97, 98, 99, 100, 101] to occur in magnetically inhomogeneous systems containing

two or more weakly coupled magnetic layers. The work presented in this paper was

motivated by a theoretical prediction of Bazaily, Jones, and Zhang (hereafter BJZ),

who argued that the energy functional of a uniform bulk half-metallic ferromagnet

contains a term linear in the current of the quasiparticles [101], i.e. that collective

magnetic properties can be influenced by currents even in a homogeneous bulk

ferromagnetic metal. The current-induced term in the energy functional identified by

BJZ implies an additional contribution to the Landau-Lifshitz equations of motion

and, in a quantum theory, to a change proportional to ~q · ~j in the magnon energy

ǫ(~q). (Here ~q is the magnon or spin-wave wavevector and ~j is the current density in

the ferromagnet.) The BJZ theory predicts that a sufficiently large current density

will appreciably soften spin waves at finite wavevectors and eventually lead to an

instability of a uniform ferromagnet. The current densities necessary to produce an

instability were estimated by BJZ to be of order 108 A cm−2, roughly the same scale

as the current densities at which spin-transfer phenomena are realized, apparently

suggesting to some that these two phenomena are deeply related.

In this chapter we establish that modification of spin-wave dynamics by a

current is a generic feature of all uniform bulk metallic ferromagnets, not restricted

to the half-metallic case considered by BJZ. We find that, in the general case, the

50

extra term in the spin wave spectrum

δǫ(~q) ∝ ~q · ~J (4.1)

where ~J is the spin current, i.e., the current carried by the majority carriers minus

the current carried by the minority carriers 1. In the half metallic case ~J = ~j, recov-

ering the result of Reference [101]. For reasons that will become apparent later, we

refer to the extra term in the spin wave dispersion as the spin wave Doppler shift,

although this terminology ignores the role of underlying lattice as we shall explain.

We also study the effect of a uniform current on spin-wave damping. The usual

Gilbert damping law γ =∝ ǫ(~q = 0), has an additional contribution proportional to

the spin-current density. In our picture, a uniform current modifies collective mag-

netization dynamics because it alters the distribution of quasiparticles in momentum

space. The spin-transfer mechanism that operates in inhomogeneous ferromagnets

[8, 9], on the hand, is based on a current mediated transfer of the quasiparticle

spin-distribution between magnetic layers that are separated in real space.

4.2 Dynamics of a Ferromagnet: Landau-Lifshitz equa-

tion

4.2.1 Microscopic Description of low energy modes

So far in this work we have invoked repeatedly the notion of low energy modes

associated with the broken symmetry in a ferromagnet. In this section we are going

to state the main aspects of the physics associated with those modes, and use them

to argue in favor a phenomenological model that describes them, the Landau-Lifshitz

equation[5, 102, 103, 104]. It is not the intention of the author to give a complete

description of the status of the immense field of ferromagnetism but just to describe

1More precisely, ~J ≡ e~N

P~k

∂ǫ(~k)

∂~k

hn↑

~k− n↓

~k

i, where N is the number of sites in the lattice, and

↑ and ↓ are defined in the axis of the average magnetization.

51

the basic issues associated with the dynamics of ferromagnetic metals. It will be

easier to start from a toy-model calculation that illustrate the main aspects of the

manipulations that we want to describe.

Model Hamiltonian

The following discussion is based in the one giving in [105] to derive the effective low-

energy Lagrangian of an antiferromagnet (the nonlinear σ-model. Similar arguments

are applied to the ferromagnetic case in [106].

Let us start from the Hubbard model on a 3D lattice. The Hamiltonian is:

H = −∑

r,r′,σ

c†rσ(tr,r′ + µ

)cr′σ + U

∑

r

c†r↑cr′↑c

†r↓cr′↓ (4.2)

Here, c†rσ is an electron creation operator at site r and spin σ, tr,r′ correspond to

the hopping parameters that we take to account for nearest neighbor hopping only.

U is the on-site repulsion energy. The sum over r is taken over the cubic lattice.

The equilibrium state of the system described by eq.(4.2) can be represented by a

path-integral over imaginary time with effective action[65, 107, 108, 109]:

S =

∫ β

0dτ∑

rr′

Ψr

(∂τ − µ− tr,r′

)Ψr′

+ U∫ β

0dτ∑

r

ψr↑ψr↑ψr↓ψr↓ (4.3)

where we have defined the spinor:

Ψr =

ψr↑

ψr↓

, (4.4)

In order to describe the condensed phase we can express the interacting part of the

action in the following decomposed expression[105]:

ψr↑ψr↑ψr↓ψr↓ =1

4

(ΨrΨr

)2 − 1

4

(Ψr σ ·Ωr Ψr

)2, (4.5)

52

where an integral over the unit vector Ωr must be done at every place and instant

in time so to ensure spin rotation invariance.2 In this manner we can write:

Z =

∫D2ΨrD [∆c,∆s,Ω] exp

(∫ β

0dτ∑

rr′

Ψr

(∂τ − µ− tr,r′

)Ψr′

−∫ β

0dτ∑

r

1

U(∆2

cr + ∆2sr

)− Ψr (i∆cr + ∆sr σ · Ωr ) Ψr

)(4.6)

Mean-Field equations

The mean-field equations for the system are obtained by looking for saddle-point

conditions over the fields ∆c, ∆s and Ω. The isotropy of the spin problem ensures

that any direction of the field Ω is equivalent with any other and then we choose the

mean-field solution pointing along an arbitrary axis hereafter label as the z-axis.3

For the other fields we obtain:

−i∆c = −U2〈ΨrΨr〉 (4.7)

∆s =U2〈Ψr σ3 Ψr〉. (4.8)

Given these solutions, the effective action for the electrons is a non-interacting one:

Seff =

∫ β

0dτ∑

rr′

Ψr

((∂τ + i∆c + ∆s σ3 − µ) δr,r′ − tr,r′

)Ψr′ . (4.9)

From this expression, the right hand side of equations (4.7) and (4.8) can be eval-

uated, given values for ∆c and ∆s. The resulting equations can be solved self-

consistently to obtain unique values for

2This method has the advantage of keeping the symmetry unbroken in the Lagrangian. Thesymmetry is only broken when the saddle point equations are solved. Different methods commonin the literature break the symmetry already at the Lagrangian level[107]. The above complicatesthe expressions for the low-energy effective Lagrangian[105].

3This is the crucial step in describing the broken symmetry state. Detailed analysis of this issueis given, for example in [65, 110, 111, 112].

53

Spin Fluctuations

We now study the dynamics of fluctuations in the orientation of the order parameter.

We are looking for the effective action for the low energy modes, this action should

involve the dynamics of the orientation of the magnetization at long wave lengths and

low frequencies. At every site we can decompose the orientation into perpendicular

components. mr is a unit vector that represents the low frequency-long wavelength

part of the excitation, while Lr is a high frequency local fluctuating mode, that we

are going to assume to be small. The magnetization is:

Ωr(τ) = mr

√1 − L2

r + Lr (4.10)

If we fix the magnitude of the exchange fields we get the following expression for

the free energy:

Z =

∫D2ΨrD [∆c,∆s,Ω] exp

(∫ β

0dτ∑

rr′

Ψr

(∂τ − µ+ i∆cr − tr,r′

)Ψr′

+ ∆s

∫ β

0dτ∑

r

Ψr (σ ·Ωr(τ) ) Ψr

)(4.11)

in order to be able to take advantage of the slow (low frequency-long wavelength)

dynamics we can rotate the spin basis for the electrons at every instant and location.

We write:

Ψr = RrΦr (4.12)

where Rr is a SU(2)/U(1)4 matrix that aligns the local value of m with any given

direction (let’s say the z-axis.)

σ ·mr = Rr σ3 R†r (4.13)

4Here, the SU(2) stands for spin rotations and the U(1) correspond to consider rotations aroundthe axis of the local magnetization as the identity. The special unitary group, denoted SU(N), is thegroup of unitary matrices of range NxN with unit determinant. SU(N) is a subgroup of the unitarygroup U(N), including all NxN unitary matrices. The notation A/B stands for the quotient groupbetween the groups A and B.

54

Again, following the notation of [105], we use the identity:

ΦrR†rRr+eµ

Φr = Φr exp (∂µ − iAµr ) Φr. (4.14)

With this transformation the action becomes

Srotated =

∫ β

0dτ∑

r

Φr

(∂τ − A0

r − (i∆c + µ) − 2t∑

i

cos(−i∂i − Ai

r

)

− ∆s

(σ3

√1 − l2r + lr · σ

))Φr (4.15)

where we have defined:

A0r = −R†

r ∂τ Rr (4.16)

Air = iR†

r ∂i Rr, (4.17)

i.e. the SU(2) gauge fields, which are spin-operators. The condition of low energy

modes is imposed by regarding the magnitude of those fields as small and expanding

the action to quadratic order in the fields and in the fluctuations lr. The expansion

leads to an action that can be written as the sum of a number of terms:

SBerryPhase = −∫ β

0dτ∑

µνr

jνµrA

νµr (4.18)

SExchange =t

2

∫ β

0dτ∑

µνr

Aνµr

2Φr cos (−i∂µ)Φr + c.c. (4.19)

Sl = −∆s

∫ β

0dτ∑

νr

lνrjν0r (4.20)

Sl2 =∆s

2

∫ β

0dτ∑

r

l2rj30r (4.21)

The spin-density and spin-current are defined as:

jν0r = ΦrσνΦr, (4.22)

jνµr = tΦr sin (−i∂µ)σνΦr + c.c., (4.23)

55

By integrating out the electronic degrees of freedom we obtain an effective action:

Seff [mr,Lr] = 〈SBerryPhase〉 + 〈SExchange〉

+ 〈Sl〉 + 〈Sl2〉 −1

2〈S2

BerryPhase〉 −1

2〈S2

l 〉 − 〈SBerryPhaseSl〉(4.24)

Let’s neglect the fluctuations around the long-wave-length excitations, that is to

say, let’s make lr = 0. We have:

Seff [mr] = 〈SBerryPhase〉 + 〈SExchange〉 −1

2〈S2

BerryPhase〉 (4.25)

where we have explicitly:

〈SBerryPhase〉 = −∫ β

0dτ∑

r

〈jνµr〉Aν

µr (4.26)

〈SExchange〉 =

∫ β

0dτ∑

µνr

1

4〈K〉 Aν

µr2 (4.27)

〈S2BerryPhase〉 =

∫ β

0dτ

∫ β

0dτ ′∑

rr′

〈jνµr j

ν′

µ′r′〉AνµrA

ν′

µ′r′ (4.28)

To evaluate the single particle averages is straightforward. In equilibrium, the elec-

tronic system can support no charge currents and it is therefore only the charge

density that enter into the calculation. The spin densities are also easy to take into

account, an easy symmetry analysis shows that the only allowed spin density is along

the z-axis. Since the averages are taken with respect to a homogenous system with

no spin-orbit interactions, the spin currents must vanish together with the charge

currents, and therefore we obtain:

〈jνµr〉 = δ3,νδ0,µM (4.29)

The kinetic energy expectation value 〈K〉 can also be directly evaluated in terms of

the density of electrons. The main problems in dealing with this action are hidden

in the current-current (and spin current-spin current) correlation functions. This

term is explicitly non local in time and therefore contains the physical mechanisms

56

responsible for damping. In the model system we have addressed there is no room

for such mechanisms. The inclusion of those processes requires dealing with mag-

netic disorder, spin orbit coupling or magnetostriction effects. We are not going to

include them at this stage and leave that discussion for future work on the matter.

The treatment of damping that we are going to take is based on the phenomeno-

logical Landau-Lifshitz equations. The square of the non-abelian gauge field Aνµr

2

is proportional to the square of the gradient:

∑

ν

Aνµr

2 = (∂µΩ)2 (4.30)

which together with the usual expressions for the dynamical equations of a spin

system lead us to:∂Ω

∂t= ρsΩ ×∇2Ω + αΩ × ∂Ω

∂t(4.31)

where we have introduced the spin damping term as a phenomenological constant

α. The inclusion of the fluctuations of the order parameter l will lead to a renormal-

ization of the effective Hamiltonian. These effects are related to magnon-magnon

vertex corrections. The basic features of magnetism are captured in this equation.

The existence of long lived long wave-length excitations follows directly in the form

of the dispersion relation ω = ρsk2 that follows from the equation for the magnetiza-

tion orientation dynamics.. The inclusion of external fields and magnetic anisotropy

can be accomplished by adding precession terms to this equation.

∂Ω

∂t= ρsΩ ×∇2Ω + Ω× Heff + αΩ × ∂Ω

∂t(4.32)

A pictorial representation of the different terms is shown in Fig.( 4.1)

here are other ways to derive Landau-Lifshitz-Gilbert equations, some per-

haps simpler and more direct. One advantage of the derivation presented above is

that we can immediately understand the influence of a transport current on long-

wavelength magnetization dynamics. When a current is driven across a ferromagnet,

57

Figure 4.1: Cartoon of the torques driving the magnetization dynamics, (a) theusual ferromagnetic precession is driven by a torque of the form ~Heff × ~M, and (b)a dissipation torque driving the magnetization toward its equilibrium position.

a spin current will be associated with it, equal to the electron current times the po-

larization.The expectation value for the current gets modified to account for the

spin current

〈jνµr〉 = δ3,νδ0,µM + δ3,νJ s

µ (4.33)

creating a term in the action that corresponds to a space-dependent Berry-phase.

This additional term in the action of the magnetization implies a modification of

the final Landau-Lifshtiz equation:

∂Ω

∂t= J

siΩ× (Ω×∇iΩ) + ρsΩ ×∇2Ω + Ω× Heff + αΩ × ∂Ω

∂t(4.34)

The relation Ω× (Ω×∇iΩ) ∼ ∇iΩ, holds since the fixed length of the unit vector

Ω, allow us to give a direct interpretation of this new term. The left hand side and

the new term can be collected together in the equation in order to write it as a

convective derivative. In absence of damping (i.e. α = 0) we can absorb the new

term by a suitable Galilean boost. This tells us that the basic effect of the new

term is to push the magnetic texture along with the spin current drift. This effect

58

correspond then to a spin-wave doppler shift. Of course the damping processes will

modify this simple picture, but they will not change the basic effect of a overall

momentum dependent shift in frequency.

4.2.2 Spin-wave Doppler shift as a Spin-Torque Effect

In this section we explain how the influence of an uniform current on magnetization

dynamics can be understood as a special case of a spin-torque effect[8, 9]. The latter

takes place when a spin current coming from a magnet spin polarized along ~M1 enters

in a magnet spin polarized along ~M2. In this circumstance there is an imbalance

between the incoming and the outgoing transverse component (with respect to ~M2)

of the spin currents in magnet 2. Because of spin conservation (resulting from

the rotational invariance of the system), the imbalance in the spin flux across the

boundaries of magnet 2 must be compensated by a change of the magnetization of

that magnet, which is described by a new term in the Landau Lifshitz equation

[8, 9]. The microscopic origin of the spin current imbalance can be understood as

a destructive interference effect, originating in the fact that the steady state spin

current is a sum over stationary states with a broad distribution in momentum space

[8, 9]. Alternatively, it is possible to understand the spin current flux imbalance as a

destructive interference in the time domain. At a given instant of time, the outgoing

current-carrying quasiparticles spent differing amounts of time in magnet 2. The

average over that distribution results in a vanishing transverse spin component in

the outgoing flux.

The above argument, connecting spin flux imbalance and spin-torque, applies

to a system in which the inhomogeneous magnetization is described by a piecewise

constant function. It is our contention that the spin wave Doppler shift can be

understood by applying the same argument to the case of smoothly varying mag-

netization. We consider again a magnet with charge current ~j, and spin current

59

~J . We assume that current flows in the x direction and, importantly, that the spin

current is locally parallel to the magnetization orientation ~J (x) = jsΩ(x). It can

be shown that this is the case in a wide range of situations.

The spin density reads ~S(x) = S0Ω(x) where S0 is the average spin polariza-

tion. We focus on the slab centered at x and bounded by x− dx and x+ dx. Spins

are injected into the slab at the rate jsΩ(x− dx) and leave at the rate jsΩ(x+ dx).

The resulting spin current imbalance is 2dxjs∂xΩ. Therefore, there must be a spin

transfer to the local magnetization:

d~S(x)

dt

∣∣∣∣∣ST

= js∂xΩ (4.35)

Now, using |Ω|2 = 1 at every point of the space we obtain:

d~S(x)

dt

∣∣∣∣∣ST

= jsΩ(x) × (∂xΩ(x) × Ω(x)) (4.36)

which is exactly the same result obtained in [101]. Including this term in the Lan-

dau Lifshitz equation and solving for small perturbations around the homogeneous

ground state (spin waves) results into the spin wave Doppler shift discussed in previ-

ous sections. In conclusion, this argument demonstrates that the spin-wave Doppler

shift and spin transfer torques are different limits of the same physical phenom-

ena, the transfer of angular momentum from the quasiparticles to the collective

magnetization whenever the latter is not spatially uniform.

4.2.3 Spin wave description

Spin waves without current

We are interested in the dynamics of the collective coordinate, so that the static

solution obtained by solving the mean field approximation is insufficient. To describe

the elementary collective excitations we need to study small amplitude dynamic

60

fluctuations of the collective coordinate around the static solution:

~∆i(τ) ≃ ~∆cl + δ~∆i(τ) (4.37)

We introduce Eq.( 4.37) into the effective action (Eq.( 4.28)) and neglect terms of

order[δ~∆i(τ)

]3and higher. The resulting actionScl(~∆

cl)+SSW, where the first term

is the classical approximation to the effective action and the fluctuation correction

is:

SSW =1

2βN∑

Q,a,b

δ∆a(Q)Kab(Q)δ∆b(−Q) (4.38)

where Q is a shorthand for ~q, iνn, and a, b stand for Cartesian coordinates. Note

that the bosonic fields, δ~∆(Q) are dimensionless and the Kernel K has dimensions

of inverse energy. This action defines a field theory for the spin fluctuations. The

equilibrium Matsubara Green function, Dab(~q, iνn) , is given [3, 65] by the inverse of

the spin fluctuation Kernel, Kab(Q). Analytical expressions for Kab(Q) are readily

evaluated for the case of parabolic bands and are appealed to below. We obtain the

retarded spin fluctuation propagator by analytical continuation of the Matsubara

propagator: Dretab (~q, ω) = Dab(~q, iνn → ω + i0+) The imaginary part of the retarded

propagator summarizes the spectrum and the damping of the spin fluctuations most

directly.

The theory defined by Eq.( 4.38) includes two types of spin fluctuations

which are very different: i) longitudinal fluctuations (parallel to n), or amplitude

modes and ii) transverse fluctuations (perpendicular to n), or spin waves. The

amplitude modes involve a change in the magnitude of the local spin splitting, ∆,

and are either over damped or appear at energies above the continuum of spin-

diagonal particle-hole excitations. In contrast, the spin waves are gapless in the

limit ~q = 0, in agreement with the Goldstone theorem, and are often weakly damped

even in realistic situations where magnetic anisotropy induces a non-zero gap. Note

that the amplitude modes decouple from the spin wave modes for small amplitude

61

fluctuations. For x = n, we can write

Kab(Q) =

K|| 0 0

0 Kyy Kyz

0 Kzy Kzz

(4.39)

Since the low energy dynamics of a metallic ferromagnet is governed by transverse

spin fluctuations, we do not discuss longitudinal fluctuations further. After analytic

continuation, we obtain the following result for the inverse of the retarded transverse

spin fluctuation Green function (Dret)−1, which is diagonal when we rotate from y, z

to +z ± iy chiral representations. The diagonal elements are then

Dret± (~q, ω) =

4U

3

1

1 + 23UΓ(±~q,±ω)

(4.40)

where Γ(~q, ω) is the Lindhard function evaluated with the spin-split mean-field

bands:

Γ(~q, ω) =1

N∑

~k

n↑~k− n↓~k+~q

ǫ↑~k− ǫ↓~k+~q

+ ω + i0+(4.41)

where nσ~k

is shorthand for the Fermi-Dirac occupation function nF

[ǫσ~k

]for the quasi-

particle occupation numbers. Eqs. ( 4.40) and ( 4.41) make it clear that the spin

wave spectrum is a functional of the occupation function nF for the quasi-particles

in the spin-split bands. The influence of a current on the spin-wave spectrum will

enter our theory through non-equilibrium values of these occupation numbers.

In the case of parabolic bands (still without current), the Taylor expansion

of the Lindhardt function in the low-energy low-frequency limit gives the following

result for the spin wave propagator:

Dret± (~q, ω) =

4U∆

3

1

ω ± ρq2(4.42)

where ρ is the spin stiffness which is easily computed analytically in this case.

The poles of Eq.( 4.42) give the well known result for the spin wave dispersion,

62

ω = ±ρq2. Several remarks are in order: i) In real systems, spin-orbit interactions

lift spin rotational invariance, resulting in a gap for the q = 0 spin waves. The size

of the gap is typically of order of 1 µeV. ii) The interplay between disorder and spin

orbit interactions, absent in the above model, gives rise to a broadening of the spin

wave spectrum, even at small frequency and momentum. In Section V we address

this issue and discuss how damping is changed in the presence of a current.

Spin waves with current

In the previous subsection we derived the spin wave spectrum of a metallic ferromag-

net in thermal equilibrium. Equations (4.40) and (4.41) establish a clear connection

between spin waves and quasiparticle distributions. In order to address the same

problem in the presence of a current, a non-equilibrium formalism is needed. By

taking advantage of the formulation discussed above in which collective excitations

interact with fermion particle-hole excitations we are able to appeal to established

results for harmonic oscillators weakly coupled to a bath. In the equilibrium case,

the fact that the low-energy Hamiltonian for magnetization-orientation fluctuations

is that of a harmonic oscillator follows by expanding the fluctuation action to lead-

ing order in ω to show that y and z direction fluctuations are canonically conjugate.

In our model magnons are coupled to a bath of spin-flip particle-hole excitations.

Following system-bath weak coupling master equation analyses[113] we find that the

collective dynamics in the presence of a non-equilibrium current-carrying quasipar-

ticle system differs from the equilibrium one simply by replacing Fermi occupation

numbers by the non-equilibrium occupation numbers of the current-carrying state.

The following term therefore appears in the Taylor expansion of the Lindhardt func-

tion Γ:∂Γ

∂qi

∣∣∣∣q=ω=0

=1

N∆2

∑

~k

∂ǫ(~k)

∂ki

[n↑~k

− n↓~k

](4.43)

63

Since this expression uses the easy direction x as the spin-quantization axis, the x

(spin) component of the spin current is:

~J ≡ e

~N∑

~k

∂ǫ(~k)

∂~k

[n↑~k

− n↓~k

](4.44)

so that∂Γ

∂qi

∣∣∣∣q=ω=0

=~

e∆2Ji (4.45)

The quantity Ji, the component of the spin current polarized along the magneti-

zation direction n = x and flowing along the i axis, is the difference between the

current carried by majority and minority carriers. In equilibrium there is no cur-

rent and no linear term occurs in the wavevector Taylor series expansion, leading to

quadratic magnon dispersion as obtained in Eq.( 4.42). When (charge) current flows

through the ferromagnet, the difference in carrier density and mobility between ma-

jority and minority bands inevitably gives rise to a nonzero spin current [114]. We

therefore obtain the following spectrum for spin waves in the presence of a current:

ω = ρq2 − 2U

3∆

~

e~q · ~J (4.46)

This equation is the central result of this thesis. Notice that it is in precise agreement

with the single-mode-approximation expression since ∆ = 2U3 (n↑−n↓); in that case,

however, the explicit expression was derived for the case of free-particle parabolic

bands only. Eq.( 4.46) states that the spin wave spectrum of metallic ferromagnet

driven by a current is modified in proportion to the resulting spin current.

In the half metallic case, when the density of minority carriers is zero, the

spin current is equal to the total current and we recover the result of BJZ [101]. In

that limit ∆ = 2U3 n and ρ ≃ ~2

2m , leading to

ω =~

2

2mq2 − ~

en~q ·~j =

~2

2mq2 − ~~q · ~vD (4.47)

where we have expressed the current as ~j = en~vD with ~vD the drift velocity, gener-

alizing the half-metallic simple Doppler shift result to non-parabolic bands.

64

0 2 4 6 8 10q (µm

−1)

0

1

2

3

4

ω(q

) (µ

eV)

j=0j=5 10

8 Acm

−2

j=1.1 109 A cm

−2

Figure 4.2: Current modified spin-wave spectrum

Spin wave instability

Eqs. ( 4.46) and ( 4.47), taken at face value, predict that the energy of a spin

waves is negative and therefore that the uniform ferromagnetic state is destabilized

by an arbitrarily small current. If this were really true, it would presumably be a

rather obvious and well known experimental fact. It is not true because spin waves

in real ferromagnetic materials have a gap due to both spin-orbit interactions and

magnetostatic energy. Inserting by hand this (ferromagnetic resonance) gap, the

spin wave dispersion reads:

ω = ω0 + ρq2 − 2U

3∆

~

e~q · ~J (4.48)

so that it takes a critical spin current to close the spin wave gap. In Fig.(2) we

plot the current driven spin wave spectrum assuming ω0 = 1µeV , the electronic

density of iron (n = 1.17 1023 cm−3) and a Doppler shift given by q vD. The

critical current so estimated is ∼ 1.1 109 A cm−2 for a typical system. This critical

current could be much lower, perhaps by several orders of magnitude, in metallic

ferromagnets in which material parameters have been tuned to minimize the spin-

wave gap. Experimental searches for current-driven anomalies in permalloy thin

films, for example, could prove to be fruitful.

65

Spin wave action with current

In the small ω and small ~q limit, the spin waves are independent and their action

is equivalent to that of an ensemble of non interacting harmonic oscillators,indexed

with the label ~q. The Matsubara action for a single oscillator mode is the frequency

sum of

[p~q, x~q

]

12M~q

−iω2iω2

K~q

2

p~q

x~q

(4.49)

where the diagonal terms are the Hamiltonian part of the action and the off-diagonal

term can be interpreted as a Berry phase. For the spin waves, the analog of p and

x are, modulo some constants, δ∆y, δ∆z . In this representation, the low ω and low

~q spin wave action reads:

χ−1⊥ (ω, ~q) =

ρ~q · ~q −iω

iω ρ~q · ~q

+

2U

3∆

~

e~J · ~q

0 −i

i 0

(4.50)

This representation makes it clear that the spin wave Doppler shift appears as a

modification of the term which couples the canonically conjugate variables, δ∆y and

δ∆z, i.e., the spin wave Doppler shift modifies the Berry phase. When expressed in

this way, the spin-wave Doppler shift is partly analogous to the change in superfluid

velocity in a superfluid that carries a finite mass current, and the stability limit we

have discussed is partly analogous to the Landau criterion for the critical velocity of

a superfluid. These analogies are closer in the case of ideal easy-plane ferromagnets,

which like superfluids have collective modes with linear dispersion instead of a having

a gap.

4.2.4 Enhanced Spin-Wave Damping at finite Current

In the previous sections we have shown how the dispersion of spin waves in a metallic

ferromagnet is affected by current flow, and we have obtained results compatible with

66

those of BJZ [101]. In this section we address a problem which, to our knowledge,

has remained unexplored so far: how does the current flow affects the lifetime of the

spin waves.

A ferromagnetic resonance (FMR) experiment probes the dynamics of the

coherent or ~q = 0 spin wave mode. The signal linewidth is inversely proportional

to the coherent mode lifetime, the time that it takes for a transverse magnetic fluc-

tuation to relax back to zero. Spin waves have a finite lifetime because they are

coupled to each other and to other degrees of freedom, including phonons and elec-

tronic quasiparticles. In ferromagnetic metals, the quasiparticles are an important

part of the dissipative environment for the spin waves [115, 116, 117, 118]. and we

can therefore expect that quasiparticle current flow affects the spin wave lifetime to

some degree. In order to discuss this effect, it is useful to first develop the theory

of quasiparticle spin-wave damping in equilibrium.

Damping at zero current

The elementary excitation energies for the ferromagnetic phase of the Hubbard

model, are specified by the locations of poles in Eq.( 4.40). The damping rate is

proportional to the imaginary part of the transverse fluctuation propagator. Ac-

cording to Eq.( 4.40), the damping of a spin wave with frequency ω and momentum

~q, γ(~q, ω) = −2Im [Γ(ω, ~q)] is given by:

γ(~q, ω) =2π

N∑

~k

[n↑~k

− n↓~k+~q

]δ[ǫ↑~k

− ǫ↓~k+~q+ ω

](4.51)

In the absence of disorder, this quantity is nonzero when |~q| is comparable to

kF↑ − kF↓ or when ω ≃ ∆, the band spin-splitting. Either disorder, which breaks

translational symmetry leading to violations of momentum conservation selection

rules, or spin-orbit interactions, which cause all quasiparticles to have mixed spin

character, will lead to a finite electronic damping rate at characteristic collective

67

motion frequencies. Because this damping is extrinsic, however, its numerical value

is usually difficult to estimate. It is often not known whether coupling to electronic

quasiparticles, phonons, or other degrees of freedom dominates the damping.

Formally generalizing Eq.( 4.51) to the case with disorder and spin orbit

interactions leads to

γ(ω) ∝∑

~k,~k′,ν,ν′

Sν,ν′(~k,~k′)(nν

~k− nν′

~k′

)δ[ǫν~k − ǫν

′

~k′ + ω]

(4.52)

where Sν,ν′(~k,~k′) ≡ |〈~k, ν|S(−)|~k′, ν ′〉|2 is a matrix element between disorder broad-

ened initial and final quasiparticle states, labeled by momentum ~k and band index

ν (but not Bloch states0. Averaging out the extrinsic dependence on wavevector

labels by letting Sν,ν′(~k,~k′) → Sν,ν′ we obtain

γ(ω) = n2∑

ν,ν′

Sν,ν′

∫dǫ

∫dǫ′Nν(ǫ)Nν′(ǫ′) ×

×(n(ǫ) − n(ǫ′)

)δ[ǫ− ǫ′ + ω

](4.53)

where Nν(ǫ) is the density of states of the band ν. For ω of the order of the

ferromagnetic resonance frequency, we can expand Eq. (4.53) to lowest order in ω:

γ(ω) ≃ ω

n2

∑

ν,ν′

Sν,ν′Nν(ǫF )Nν′(ǫF )

(4.54)

This result can be considered a microscopic justification of the Gilbert damping

law, which states that the damping rate is linearly proportional to the resonance

frequency and vanishes at ω = 0. The proportionality between frequency and damp-

ing rate follows from phase space considerations: the higher the spin wave frequency

ω, the larger the number of quasiparticle spin flip processes compatible with energy

conservation.

68

Damping at finite current

We analyze how a current modifies quasiparticle damping, we again appeal to the

picture of magnons as harmonic oscillators coupled to a bath of particle-hole ex-

citations and borrow results from master equation results for oscillators weakly

coupled to a bath. For magnetization in the ‘↑’ direction, magnon creation is ac-

companied by quasiparticle-spin raising and magnon annihilation is accompanied

by quasiparticle-spin lowering. It turns out that only the difference between the

rate of quasiparticle up-to-down and quasiparticle down-to-up transitions enters the

equation that describes the magnetization evolution. This transition rate difference

leads to the same combination of quasiparticle occupation numbers as in Eq.( 4.54),

except that the occupation numbers characterize the current-carrying state and are

not Fermi factors. For metals we can use the standard approximate form[119] for

the quasiparticle distribution function in a current carrying state:

gν~k

= nν~k− e ~E · ~vν(~k)τν(ǫ

ν~k)

[− ∂n

∂ǫ

∣∣∣∣ǫ=ǫν

~k

](4.55)

Because of the independent sums over ~k and ~k′ in Eq.( 4.52), and because it is a

simple difference of Fermi factors that enters the damping expression, we conclude

that the quasiparticle damping correction will vanish to leading order in the spin-

dependent drift velocities vσD. We reach this conclusion even though the phase space

for spin-flip quasiparticle transitions at the spin-wave energy is altered by a factor

∼ 1 when ǫF × vD

vF∼ ǫ0, where ǫF is a characteristic quasiparticle energy scale,

i.e. the up-to-down and down-to-up transition rates change significantly when this

condition is met, but not their difference. To obtain a crude estimate for the current

at which this condition is satisfied we use the following data[119] for iron: n ≈ 1.7

1023, Fermi velocity ∼ 1.98 108 cm s−1. The drift velocity corresponding to a

current density of 10βA cm−2 is vd = jen ≃ 10β−4 cm s−1. The typical energy of

a long-wavelength magnon is ∼ 10−6 eV. Therefore, current densities of the order

69

of 106 A cm−2 and larger will substantially change the coupling of spin-waves to

their quasiparticle environment. Although this change will influence the spin-wave

density-matrix, magnetization fluctuation damping itself will not be altered by this

mechanism until much stronger currents are reached.

Two magnon damping

In the previous subsections we have calculated the damping of the lowest energy

spin wave due to its coupling to the reservoir of quasiparticles. In this section we

study damping of the coherent rotation mode (~q = 0 spin wave) due to its coupling

to finite ~q spin waves. This mechanism is known as two magnon scattering and is

efficient when the coherent rotation mode is degenerate with finite ~q spin waves [120],

a circumstance that sometimes arises due to magnetostatic interactions. The main

point we wish to raise here is that because of the spin-wave Doppler shift, precisely

this situation arises when the ferromagnet is driven by a current. As in the previous

subsection, we assume that some type of disorder lifts momentum conservation. The

effective Hamiltonian for the spin waves reads:

H = ω0b†0b0 +

∑

~q 6=0

ω(~q)b†~qb~q + b†0∑

~q 6=0

g~q√Vb~q + h.c. (4.56)

where b~q is the annihilation operator for the spin wave with momentum ~q and g~q is

some unspecified matrix element accounting for disorder induced elastic scattering

of the spin waves. Equation (4.56) is the well Hamiltonian known for a damped

harmonic oscillator can be solved exactly The damping rate for the ~q = 0 spin wave

reads:

γ( ~J ) =2π

~

∫d~q

(2π)3|g~q|2δ(ω0 − ω~q) (4.57)

Now we use ω0 − ω~q = ρq2 − a~q · ~J . After a straightforward calculation we obtain:

γ( ~J ) =g2

4π

a| ~J |ρ2

(4.58)

70

where we have approximated g~q ≃ g. Hence, in the presence of elastic spin wave

scattering, renormalization of the spin wave spectrum due to the current will enhance

the damping of the lowest spin wave mode. Unlike the Gilbert model, the damping

rate given by equation (4.58) is independent of ω0, implying that the dimensionless

Gilbert damping coefficient would decline with external field if this mechanism were

dominant.

4.3 Current induced Domain wall dynamics

4.3.1 Introduction

The way in which a current influences magnetization can cause domain wall motion

was first suggested in the pioneering works of Luc Berger [121]. A domain wall

separates two domains with different directions of magnetization, an example of

which is shown in Fig.( 4.3 ). Luc Berger’s treatment was of a deep intuitive nature.

It dealt with quasi-classical arguments concerning the behavior of electrons in a space

dependent exchange field. More detailed quantum treatments of this problem have

shown that the basic conclusions of those semiclassical arguments are correct [122,

123]. The basic ideas have been demonstrated experimentally using ferromagnetic

metal nanowires [124]. The physics behind current-driven domain wall motion comes

from two complementary effects. The first is a spin transfer effect similar to that

in spin valves. In a first approximation the spin of the electron will be aligned with

the local magnetization. An electron traveling through a domain wall, illustrated by

the dashed-dotted line in Fig. 3, will therefore change its spin angular momentum.

Because of conservation of total spin this change in spin angular momentum has to

be transferred to the local magnetization and leads to domain wall motion. The

system responds collectively with an overall shift of the domain wall. The second

mechanism is called momentum transfer, and results from the nonzero resistance

71

Figure 4.3: Two mechanisms of current-induced magnetic domain wall motion. Thedashed-dotted line illustrates the electron transferring its spin angular momentum tothe domain wall, leading to motion. The dotted line illustrates momentum transfer:the electron scatters off the domain wall and gives the domain wall a momentumkick.

72

of a domain wall due to the scattering of conduction electrons off the domain wall

[122]. An electron being reflected off the domain wall, illustrated by the dotted line

in Fig. 3, gives the domain wall a momentum kick that also leads to domain wall

motion. The relative importance of these two mechanisms depends on the width of

the domain: for narrow domain walls momentum transfer dominates, whereas for

wide walls spin transfer is believed to be more important. Possible applications of the

manipulations of domain walls with current are information storage and alternatives

to current electronic logic circuits. [125].

Displacements of domain walls under the influence of external magnetic fields

[126, 127] are dominated by the damping constant. This is naturally expected

since the basic nature of the domain wall motion process can be understood as a

relaxation process. Domain wall motion leads to the growth of the energetically

favorable domain with the external field. The relaxation process is stopped either

when (a) the energetically favorable domain completely absorbs the unfavorable

one, or (b) when a potential barrier is created that overcomes the gains in energy.

This potential barrier arises from magnetostatic effects. Similar situations arise in

the case of current induced motion. It turns out that the physics of current-driven

ferromagnetic domains is closely related to the fact that a current changes the energy

of spin wave excitations in a ferromagnet [101, 128, 100, 129]. Like the theory of

spin transfer, the theory of current-driven domain wall motion is still under debate.

The controversy here is whether or not the domain wall is intrinsically pinned, i.e.,

whether or not the critical current for moving the domain wall is zero in the absence

of extrinsic pinning [122, 123]. The basic physical issue under debate is precisely

related to the main topic of this proposal: the behavior of the spin of the electron

moving around through a non-trivial spin potential.

73

4.3.2 Numerical Solution of the Landau-Lifshitz equation in the

presence of a current

In this section we are going to solve directly the differential equations describing

the dynamics of the magnetization order parameter at long wave-lengths. These

equations are the Landau-Liftshitz equations discussed earlier in this chapter. They

are based on the knowledge of the energy associated with an arbitrary magnetic

configuration. In the previous section we argued for the general form of these equa-

tions and gave some generic physical meaning for the different terms that appear

in the expressions for the magnetic energy. The energy has in general several con-

tributions, but it can be mainly regarded as the contribution of four terms. The

Laplacian term that appeared in our derivation of an effective action for a ferro-

magnet. In the magnetism literature it is known as the exchange energy and the

coefficient that characterizes the size of this term is accurately known for different

materials by comparison with experiment. Its physical origin is the strong Coulomb

repulsion between electrons that can be reduced by arranging the spatial parts of the

wave functions to be antisymmetrical with respect to electronic permutations. In

the extreme case, of a fully polarized electron system, the many body wave function

has a node in any point (in the many-body coordinate space) where any two elec-

trons share the same location. This reduces the interaction energy matrix elements.

The fermionic nature of the electrons requires that under these circumstances the

spin part of the many body wave function must be symmetrical and therefore the

electrons have a net energy gain by polarizing their spins to become aligned. This

energy must compete with the loss of kinetic energy associated with deviations from

the spin independent Fermi sea. In a ferromagnet these two contributions balance

each other at a non-zero value for the magnetization density. The exchange energy

also imposes a penalty for having spatial modulation of the magnetization. In addi-

tion to the exhange energy, every ferromagnet has a “band” or ‘magnetocrystalline’

74

energy. Here the word band stands for the effects associated with spin-orbit inter-

action effects in the bands that would like to correlate the magnetization direction

with the underlying atomic lattice. Finally, the contribution that is conceptually

simplest but the most difficult to estimate quantitatively, the “shape” anisotropy

energy. The term in the energy functional responsible for shape anisotropy arises

from the continuum limit of dipole-dipole interactions between individual spin mag-

netic dipoles. The way in which it appears in the LL equations corresponds to

a mean-field treatment of these interactions. Interactions between the magnetic

dipoles carried by electrons are almost always ignored in condensed matter physics

and are important in ferromagnets only because many moments are aligned. This

is a long range interaction[130] and is quite complicated to evaluate for a given

(arbitrary) magnetization configuration. In what follows we assume, for simplicity,

that taking into account the main magnetostatic effect (namely the energy penalty

for magnetic poles) suffices for an account to the main features of this contribution.

The general form of the Landau-Lifshitz equations for a system in equilibrium is:

∂M

∂t= −γM × δE

δM+ αM × ∂M

∂t(4.59)

Where E is given by:

E = Edemag + Euniaxial + Eexchange (4.60)

Edemag =

∫dx(2πM2

x

), (4.61)

Euniaxial = −∫

dxK

M20

(M2

z

), (4.62)

Eexchange =

∫dx

A

M20

|∇M |2 . (4.63)

The competition between the uniaxial anisotropy and the exchange contributions

to the energy set the domain wall width λ =√

KA . Also the uniaxial energy sets a

scale for simple precession around the easy axis. The frequency associated with this

75

scale is KM0

. These scales can be used to define units of time and length and the

equations of motion become:

∂Ω

∂t= −Ω× δE

δΩ+ αΩ × ∂Ω

∂t, (4.64)

where E =∫ǫdx:

ǫ = |∇Ω|2 − Ω2z +

1

QΩ2x (4.65)

Q measures the hard plane anisotropy. For Q → ∞ there is full isotropy within

the plane. For finite Q there is a energy penalty along the x-axis. We can look

numerically for a solution containing a domain wall. First we confine our problem

to one space and one time dimensions. We assume that there is full homogeneity

along the other spacial dimensions. We solve the differential equation from above

with boundary conditions having opposite orientations at opposite boundaries. This

can be done easily. If we take the edges to be really far away from the domain wall,

this means at a distance much greater than the domain wall width, we recover a

solution indistinguishable from the soliton-like solution in infinite space (see below).

The presence of the current enters in the dynamics, according to last section, in the

form of a spin-transfer torque. The addition of this term to the dynamics should be

in the form of (∂Ω

∂t

)

BJZ

= JΩ × (Ω ×∇xΩ) . (4.66)

Where J correspond to the spin current in the units defined above. This term

can be easily included in the numerical solutions. The results are illustrated and

discussed in the panels that follow. The behavior is fully characterized by the

dimensionless parameters J , Q, and α. A realistic treatment of the different energy

contributions to the Landau-Lifshitz equation would involve an exact evaluation of

the magnetostatic energy term. Such calculation and the solution of the resulting

micromagnetic Landau-Lifshitz equations is quite demanding from the numerical

point of view. The calculations we illustrate here due to the simplicity of our energy

76

functional necessarily left out several effects associated with the deformation of the

domain, such as the formation of vortex structures or others.5

4.3.3 Hamiltonian form of Landau-Lifshitz equation

As we have already discussed, the Landau-Lifshitz equation describes the time evo-

lution of the magnetization field in a ferromagnet:

∂ ~M

∂t= − γ

M~M × δE

δ ~M+

α

M~M × ∂ ~M

∂t(4.67)

the first term of the right hand side describes the standard precession of a spin

under the influence of an effective magnetic field. The second term accounts for

the relaxation mechanism that tend to make the magnetization pointing along the

magnetic field.

Since the energy of the system is Etotal =∫dV E, its rate of change is:

Etotal =

∫dV

δE

δ ~M· ∂

~M

∂t= −αM

γ

∫ (∂ ~M

∂t

)2

dV, (4.68)

where in the last expression we have used the LL equation. Since the last term is

clearly negative we have that the presence of α describe energy being diverted out

of the magnetic system, usually toward the lattice. Since the field dynamics we are

describing conserve the norm of ~M, we can always describe the fields with polar

angles. Let ~M = (cosφ sin θ, sinφ sin θ, cos θ). We can then write the LL equations

as:

sin θ∂θ

∂t= − γ

M

δE

δφ− α sin2 θ

∂φ

∂t(4.69)

sin θ∂φ

∂t=

γ

M

δE

δθ+ α

∂θ

∂t(4.70)

5At this point it seems adequate to mention the existence of a packed set of numerical routinesOOMMF [131] that handle the difficulties of magnetostatic effects in some restricted geometries.

77

J = 0.4 Q = α = 0.1

Ωx

Ωy

Ωz

Ωz

Figure 4.4: Exact solution of the Landau-Lifshitz equations for the parametersindicated. The different plots are: (top-left panel) A 3D representation of the Ωz

component. The horizontal axis is the space label in units of the domain wall width.The axis entering the plane of the page is the time axis in units of 1/(αωuniaxial).The third dimension is the dimensionless z-component of the magnetization vector.(rest of panels) A 2D representation of the different coordinates of the directorvector. Here Q is infinity (no in-plane anisotropy) and the domain wall responds asa straight line with velocity X = J

1+α2 . As the domain moves the components inthe hard plane precess.

78

J = 0.4 Q = 1/ 0.3 α = 0.1

Ωx

Ωy

Ωz

Ωz

Figure 4.5: Exact solution of the Landau-Lifshitz equations for the parametersindicated. The different plots are: (top-left panel) A 3D representation of the Ωz

component. The horizontal axis is the space label in units of the domain wall width.The axis entering the plane of the page is the time axis in units of 1/(αωany). Thethird dimension is the dimensionless z-component of the magnetization vector. (restof panels) A 2D representation of the different coordinates of the director vector.Here Q is finite but still large enough as to allow the domain wall motion. Fora finite value of Q, domain wall moves but there are some oscillations on top ofthe straight line motion. As the domain moves the components in the hard planeprecess.

79

J = 0.4 Q = 1/ 0.5 α = 0.1

Ωx

Ωy

Ωz

Ωz

Figure 4.6: Exact solution of the Landau-Lifshitz equations for the parametersindicated. The different plots are: (top-left panel) A 3D representation of the Ωz

component. The horizontal axis is the space label in units of the domain wall width.The axis entering the plane of the page is the time axis in units of 1/(αωany). Thethird dimension is the dimensionless z-component of the magnetization vector. (restof panels) A 2D representation of the different coordinates of the director vector. Qis even smaller approaching the critical situation and the wiggles become stronger.

80

J = 0.4 Q = 1.0 α = 0.1

Ωx

Ωy

Ωz

Ωz

Figure 4.7: Exact solution of the Landau-Lifshitz equations for the parametersindicated. The different plots are: (top-left panel) A 3D representation of the Ωz

component. The horizontal axis is the space label in units of the domain wall width.The axis entering the plane of the page is the time axis in units of 1/(αωany). Thethird dimension is the dimensionless z-component of the magnetization vector. (restof panels) A 2D representation of the different coordinates of the director vector. Qis small enough as to stop the motion of the domain wall.

81

Figure 4.8: The definition of the polar angles used as independent fields in thetheory.

By changing variables to Φ = cos θ (the projection of the magnetization along

the easy axis) we obtain, using sin θθ = −∂tΦ, and (δθE)/ sin θ = −δΦE:

Φ =γ

M

δE

δφ+ α

√1 − Φ2φ (4.71)

φ = − γ

M

δE

δΦ− α√

1 − Φ2Φ (4.72)

where the Hamiltonian structure is evident when the damping terms are neglected.

. The action is:

S =

∫dtd3x

(φΦ − γ

ME(Φ, φ)

)(4.73)

and the dissipative function[132] is:

R = −1

2

∫dtd3x E =

α

2

∫dtd3x

(Φ2

1 − Φ2+ (1 − Φ2)φ2

)(4.74)

The equations of motion can be written as:

δSδΦ

=δRδΦ

, andδSδφ

=δRδφ

(4.75)

82

4.3.4 Bloch Domain Wall

We now focus on discussion of a Bloch domain wall. The exchange energy is:

Eexchange =A

M2

((∇Mx)2 + (∇My)

2 + (∇Mz)2), (4.76)

or in terms of the canonical coordinates,

Eexchange = A

((∇Φ)2

1 − Φ2+ (1 − Φ2)(∇φ)2

). (4.77)

The anisotropy energy is approximate by the uniaxial term,

Euniaxial = Ku sin2 θ = Ku(1 − Φ2), (4.78)

and assuming that the magnetic structure varies only along the x direction, we get

for the demagnetizing field:

Edemag = 2πM2 sin2 θ cos2 φ = 2πM2(1 − Φ2) cos2 φ (4.79)

using λ =√A/Ku as the unit of length we obtain for the total energy per unit

transverse area;

Etotal =√AKu

∫dx

((∇Φ)2

1 − Φ2+ (1 − Φ2)(∇φ)2 + (1 − Φ2) +

1

Q(1 − Φ2) cos2 φ

)

(4.80)

A static field is a stationary point of such a functional. The minimization condition

on the variable φ is;

∇2φ− 1

2Qsin(2φ) = −∇ log(1 − Φ2) (4.81)

and for the variable Φ

∇2Φ = −Φ(∇Φ)2

1 − Φ2− Φ(1 − Φ2)(1 + (∇φ)2 +

1

Qcos2 φ) (4.82)

A Bloch domain wall consists of a soliton solution of those equations of the form:

φ = ±π2, (4.83)

Φ = ± tanh(x) (4.84)

83

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

-1

1

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2

x/ λ

M z

x/ λ

M y

E

Figure 4.9: Left panel: Cartoon of a Bloch Domain wall of width λ. Right panel:plot of the Mz and My components of the magnetization along the domain, and theenergy density. Mx is zero to avoid magnetostatic torques.

It is easy to evaluate the energy of such a domain it turns out to be equal to 4√AKu

above the energy of the system per unit transverse area without a domain wall. The

stationary domain wall is a consequence of subtle compensation at each point in

space and the avoidance of magnetostatic torques.

The addition of “external” torques, will upset that detailed compensation

giving rise in some cases to domain wall motion. In these notes we are going to de-

scribe two mechanisms of domain wall propagation, one driven by external magnetic

fields [127, 133, 134] and another driven by currents [121, 122].

4.3.5 Motion of a Rigid Domain Wall Driven by an External Mag-

netic Field

The first simple study we can make of domain wall motion is a rigid one driven by

and external magnetic field along the −z direction. The energy functional becomes:

E(Φ, φ) =

((∇Φ)2

1 − Φ2+ (1 − Φ2)(∇φ)2 + (1 − Φ2) +

1

Q(1 − Φ2) cos2 φ

)+ hΦ (4.85)

and the action is;

S =

∫dtdx

(Φφ−E(Φ, φ)

), (4.86)

84

We look for a solution that corresponds to rigid undeformed motion of the domain

wall.. Rigid means that the magnetic structure is just drifting rigidly. Let X be the

center of the domain wall. Associated with that motion there is a precession that

changes φ from π/2 by an amount that in the present approximation we regard as

constant in space (but changing with time). Then we can write,

Φ(x, t) = Φ0(x−X(t), 0) (4.87)

φ(x, t) =π

2+ p(t) (4.88)

where Φ0(x) = tanh(x). The action can be calculated now in terms of the new

”‘collective”’ coordinates X and p.

Φ = −X∂xΦ0(x−X) (4.89)

φ = p (4.90)

We obtain:

Seff =

∫dt

(−2Xp+

2

Qcos2(p) + h

∫dxΦ0(x−X)

)(4.91)

The dissipation function is:

R = α(X2 + p2

)(4.92)

Then the equations of motion are:

δSδX

=δRδX

−→ p+ h = αX (4.93)

δSδp

=δRδp

−→ −X +1

Qsin(2p) = αp (4.94)

from here we look for a fixed point of the system which is given by X = h/α,

sin(2p) = Qh/α. Note that the solution is blocked at h > α/Q. This constitutes

the Walker limit. Beyond that limit we can solve the equation:

p =h

1 + α2+

α

1 + α2

1

Qsin(2p) −→

∫dp

Λ + Γ sin(2p)= t (4.95)

85

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

00.005

0.01

0.015

0.0

1.0

0.5Q

h

Figure 4.10: Average velocity 〈X〉 for the domain wall as a function of Q and thedriving field h. The color code represents the relative value of 〈X〉/(h/α), we seethat is constant, equal to 1, below the Walker limit represented by the dashed line.Beyond that limit the system acquires an oscillatory behavior characterized by zeroaverage velocity.

86

Below the WL the integral is:

t =1

2√

Γ2 − Λ2log

(Λ tan p+ Γ −

√Γ2 − Λ2

Λ tan p+ Γ −√

Γ2 − Λ2

)(4.96)

which implies:

(tan p+

Γ

Λ

)tanh(

√Γ2 − Λ2 t) +

√Γ2

Λ2− 1 = 0 (4.97)

which after the transient (as t −→ ∞) implies again, X = h/α. Above the WL we

have:

tan p = −Γ

Λ+

√1 − Γ2

Λ2tan(

√Λ2 − Γ2 t) (4.98)

which clearly indicates some sort of oscillations that appear on the motion of the

domain wall.

4.3.6 Motion of a Rigid Domain Wall Driven by an Current

The effects of a current in a ferromagnet have been the subject of many interesting

theoretical and experimental studies, especially in recent years. The main effects of

interest here are ones associated with spin transfer phenomena [8, 9]. Those effects

acquire a very interesting form in the case of a smoothly varying magnetization

profile [101, 128]. In such a case the torque exerted by the non-equilibrium current-

carrying quasiparticles modifies the LL equation in the following way:

∂ ~M

∂t= − γ

M~M × δE

δ ~M+

α

M~M × ∂ ~M

∂t+ Jk

~M × (~M × ∂k~M). (4.99)

Using the fact that the norm of ~M is preserved we can write the above as:

∂ ~M

∂t= − γ

M~M × δE

δ ~M+

α

M~M × ∂ ~M

∂t+(~J · ∇

)~M (4.100)

Note that those equations can be cast in the same Hamiltonian form as in the

absence of current, by writing the action as:

S =

∫dtd3x

(φΦ + ~J · φ∇Φ − γ

ME(Φ, φ)

)(4.101)

87

In the case of a Bloch domain wall the action is:

S =

∫dtdx

(Φφ+ J φ∂xΦ

−(

(∇Φ)2

1 − Φ2+ (1 − Φ2)(∇φ)2 + (1 − Φ2) +

1

Q(1 − Φ2) cos2 φ

)),(4.102)

and within the rigid approximation becomes (notice that the current is coupled to

p whereas the field was coupled to X.):

Seff = 2

∫dt

(pX + J p− 1

Qsin2 p

)(4.103)

Then the equations of motion are:

δSδX

=δRδX

−→ −p = αX (4.104)

δSδp

=δRδp

−→ X + J − 1

Qsin(2p) = αp (4.105)

The equations then have a fixed point at X = 0 and p = 0 and sin(2p) = QJ , as

long as |QJ | < 1 [122]. Note that the existence of solutions for the rigid domain

wall, in no way means that those are good descriptions of the system. However

a close look at figures (4.4, 4.5, 4.6, 4.7), shows that at least for a wide range of

parameters the rigid wall approximation seems quite reasonable. In what follows we

are going to focus on effects that appear beyond this approximation.

4.3.7 Beyond the rigid approximation: Modification of the shape

of the wall

The presence in the action of terms linear in the spin waves coordinates show that our

starting point is not a stationary value of the action. A stationary value of the action

will be a much more adequate starting point. Going back to the action, we now use

it to calculate the best ansatz for a moving domain wall. Let Φ(x, t) = Φ(x −X)

and φ(x, t) = π/2+ p, where now we don’t know the field Φ(x). Then the dynamics

will be specified by minimizing:

88

0 1 2 3 4 5

1

2

3

4

5

00.005

0.01

0.015

Q

J

1.0

0.5

0.0 0 1 2 3 4 50

1

2

3

4

5

J/J cr

<X>.

Jcr

Figure 4.11: Left panel: average velocity 〈X〉 as function of the anisotropy parameterQ and the current J . Below the critical current Jcr(Q) described by the dashed linewe have a fixed point at zero velocity, and above that current non-zero velocitiesappear. Right panel: 〈X〉/Jcr(Q) as a function of J/Jcr(Q) for several values of Q.Above the critical current all the curves collapse into the dashed line described by〈X〉 =

√J2 − J2

cr.

89

S =

∫dtdx

(−(pX −J p)∂xΦ −

((∇Φ)2

1 − Φ2+ (1 − Φ2) +

1

Q(1 − Φ2) sin2 p

)),

(4.106)

To minimize that action may seem a complicated problem, but it is very

simple [127, 133, 134]. We only need to rescale the spatial dimensions by making

Φ(x) = Φ(x/Σ) where Σ2 = (1 + 1Q sin2 p) > 1 we get:

S =

∫dtdx′

(−(pX − J p)∂xΦ − Σ

((∇Φ)2

1 − Φ2+ (1 − Φ2)

)), (4.107)

The solution is then a scaled domain wall Φ(x) = tanh(Σx), moving with collectives

coordinates minimizing the action:

Seff =

∫dt(−pX − J p− 2Σ(p)

)(4.108)

and the dissipation function becomes:

R = α

∫dt

(ΣX2 +

p2

Σ

)(4.109)

The equations of motion are then,

δSδX

=δRδX

−→ −p = αΣ X (4.110)

δSδp

=δRδp

−→ X + J − 2dΣ

dp= α

p

Σ(4.111)

The static point condition X = p = 0 implies, again, a relation between p

and J given by:

J =1

Σ

dΣ2

dp(4.112)

and therefore we have that the critical current is given by:

Jcr = Max

1

Σ

dΣ2

dp

(4.113)

with the asymptotic behavior

Jcr −→

2√Q as Q→ 0

1Q as Q→ ∞

(4.114)

90

0 p4

p2

0.25

0.5

0.75

1

1 2 3 4 5

1

2

3

4

5

00.005

0.01

0.015

0.0J

0.0

0.5

1.0

Q

0 1 2 3 4 50

1

2

3

4

5

Figure 4.12: Left panel: average velocity 〈X〉 as function of the anisotropy parameterQ and the current J . Below the critical current Jcr(Q) described by the dashed linewe have a fixed point at zero velocity, and above that current non-zero velocitiesappear. Right panel: 〈X〉/Jcr(Q) as a function of J/Jcr(Q) for several values of Q.Above the critical current all the curves collapse into the dashed line described by〈X〉 =

√J2 − J2

cr.

91

The new dynamics described by this equations can be easily related to the

dynamics described by the Tatara-Berger set of equations in the case of big Q.

In general however, they are different. For small Q big discrepancies between the

critical currents are expected.

92

Chapter 5

Theory of Spin Transfer

Phenomena in Magnetic Metals

and Semiconductors

The contents of this chapter are partially based on the article: Alvaro S. Nunez and

Allan H. MacDonald, Spin Transfer Without Spin Conservation, “The Pro-

ceedings of the 8th International Symposium on Foundations of Quantum Mechanics

in the Light of New Technology” to be published by World Scientific Publishing Co.,

cond-mat/0403710.

5.1 Introduction

In recent years fundamental aspects of magnetism that are obscured in bulk ma-

terials have been cleanly identified and systematically studied in magnetic nanos-

tructures. These new phenomena, including giant-magneto resistance[7], inter-layer

coupling[135] and spin transfer, have collectively weaved a rich phenomenological

tapestry that has already enabled several new technological applications[81] and

93

promises more in the future. The transfer[8, 9] of magnetization from quasiparti-

cles to collective degrees of freedom in transition metal ferromagnets has received

attention recently because of experimental [38, 82, 83, 84, 85, 1, 35, 37, 86, 87, 88]

and theoretical[89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 136] progress that has

motivated basic science interest in this many-electron phenomenon, and because of

the possibility that the effect might prove to be a useful way to write magnetic

information. The key theoretical ideas that underly this effect were proposed some

time ago[8, 9] and rest heavily on bookkeeping which follows the flow of spin-angular

momentum through the system. Recent advances in nanomagnetism have made it

possible to compare these ideas with experimental observations and explore them

more fully.

The non-equilibrium state of a current-carrying many-electron system can

be quite complicated. It is remarkable then, that in the case of a nanomagnet

with a spin-valve geometry, the main effect of a transport bias voltage is to in-

troduce a non-equilibrium torque that acts on the magnetic condensate and has an

extremely simple form. The influence of a bias voltage on order parameter dynamics

is an example of a type of non-equilibrium physics that appears likely to arise with

greater frequently as nanoscale transport is explored in more and more contexts.

For example, it has recently been argued[137] on the basis of mean-field-theory con-

siderations, that the magnetic transition temperatures and other thermodynamic

magnetic properties of a magnetically isolated film can also be altered by transport

bias voltages. These two examples motivate a more general and formal examination

of the equilibrium statistical mechanics of collective variables in conductors with a

bias voltage than is attempted here.

In this article we focus our attention on a theory of spin transfer that does

not rest on an appeal to conservation of total spin, focusing instead on its origin

in the change in the exchange-field experienced by quasiparticles in the presence of

94

non-zero transport currents. Our discussion of spin-transfer sees the effect of a spe-

cific example of a larger class of phenomena that occur in any interacting electron

system that can be described by a time-dependent mean-field theory. Our approach

can assess whether or not the current-driven magnetization dynamics in a particular

geometry will be coherent, and can predict the efficacy of spin-transfer when spin-

orbit interactions are present. Although we can employ any time-dependent mean

field theory of a metallic ferromagnet we expect that for magnetic metals most ap-

plications would be in the framework of ab initio spin-density-functional theory[138]

(SDFT) which is accurate for many of these systems.

Our approach makes it clear that closely related phenomena occur in any

physical system with interactions between quasiparticles and collective coordinates,

even systems without broken symmetries. We briefly discuss applications to[17]

semiconductor electron-electron bilayers and antiferromagnetic metal nanoparticle

circuits.

5.2 Basic Phenomenology of Spin transfer effects

Spin transfer torques correspond to the reciprocal action of the currents on the mag-

netization. The idea is to consider a magnetic heterostructure like the one described

in fig.(5.2a) (a spin valve). If a current flows across the system, it has been shown

that the magnetic configuration can be altered in response to the exchange fields

created by the non-equilibrium quasiparticles. These sort of effects were predicted

to take place in nano-magnetic heterostructures in the seminal, independent, works

of Berger and Slonczewski [8, 9]. The effects have been demonstrated in several

experiments using magnetic nano-pilars[1, 35, 36], multilayers [37], magnetic point

contacts [38, 39, 40, 41], and even epitaxially grown diluted magnetic semiconduc-

tors. The consequences of non-equilibrium excitations that appear in response to

a current, can be quite complicated. It is remarkable then, that in the case of a

95

spin-valve the non-equilibrium torques acquire an extremely simple form. This form

can be easily obtained by appealing to conservation laws. We consider a magnetic

spin valve geometry, with two nano-magnets. Let’s call the two magnetizations Ω1

and Ω2. The net torque can only depend on the two magnetizations, and is clear

that the most general form the torque can have is:

Γ2ST = γout(Ω1,Ω2)Ω1 × Ω2 + γin(Ω1,Ω2)Ω2 × (Ω1 × Ω2) . (5.1)

The subscribts in the coefficients γin and γout refer to the direction of the torque

relative to the common plane of Ω1 and Ω2. For sufficiently weak currents these

torques will be linear in current, since such a dependence is allowed by symmetry.

Indeed, as will become apparent from the following discussion, even very strong

currents remain in the limit in which γin and γout are proportional to current.

These torques must be added to the equation of motion for the dynamics of the

second magnet. We assume that the first magnet is pinned.1. The equation of

motion of the ferromagnet is usually written as:

dΩ2

dt= −Ω2 ×

δF2

δΩ2+ αΩ2 ×

(Ω2 ×

δF2

δΩ2

)+ Γ2

ST. (5.3)

The essence of this effect is that while the first “out-of-plane” torque is basically

a change in the free energy (the free energy is locally corrected by an amount

δF = γoutΩ1 · Ω2), the effects of the “in-plane” torque acts as an energy pump or

drain. If we could tune its sign by a proper choice of parameters we would have

a source of negative effective damping. The net damping (the sum of the intrinsic

damping and the spin transfer torque) can be canceled and even make negative,

rendering unstable an otherwise perfectly stable geometric arrangement of the mag-

netization. The basic questions are then, “Is there an in plane torque?” and “How

1This is in spite the fact that we can also write a similar expression for the torque in this magnet:

Γ1ST = eγoutΩ2 ×Ω1 + eγinΩ1 × (Ω2 × Ω1) , (5.2)

This pinning can be implemented in several ways. In some works the pinned ferromagnet is justmake large enough. Another strategy uses a more complex hetero-structure including an anti-ferromagnetic layer that shifts up the coercitivity by means of the exchange-bias effect[139].

96

Figure 5.1: Illustration of the spin transfer torque in a spin valve consisting of a pinnedand free ferromagnetic layer. Because of conservation of total spin angular momentum, thetorque on the spin angular momentum of the electrons, indicated by the dotted arrow, hasto be accompanied by a reaction torque on the magnetization of the free ferromagnet.

big is it?”. In other words what is the physics describing γin. To understand the cur-

rent induced behavior of the magnetization it is convenient to consider the behavior

of the spin-polarization of current that crosses a single ferromagnetic layer. Let us

choose the spin quantization axis to be aligned with the local magnetization direc-

tion. Because of the exchange potential in the ferromagnet, electrons with different

spin polarizations experience quite different spin polarizations as they pass through

the device, giving rise to different transmission coefficients. As a consequence, the

current leaving the ferromagnet will become polarized even if the incoming one is

completely unpolarized (an explicit example of this simple and quite generic effect,

often called spin filter effect, is described in section B.3).

The influence of the first magnet on the unpolarized incoming current allows us

to regard the current going in to the second magnet as a current spin polarized

along the magnetization of the first magnet Ω1. This polarized spin current is in

97

Figure 5.2: (a) Cartoon of a point contact between two ferromagnets that display the spintransfer effect. The current goes from one magnet through the point contact to the othermagnet where it creates a spin transfer torque that drives the second magnet out of itsequilibrium position. (c) Differential resistance as a function of current[1]. As the currentis increased to a certain critical value, the parallel configuration (of low resistance) becomesunstable and the free magnet is switch to be antiparallel to the pinned magnet. The jumpin resistance is the GMR effect, and is identical to the jump measured independently byswitching the free magnet with an applied magnetic field.

turn filtered by the second ferromagnet. The second magnet then converts a spin

polarized current along Ω1 into a spin current polarized along Ω2. Because of the

overall conservation of spin angular momentum, the torque exerted by the collec-

tive exchange field on the quasiparticles must be accompanied by a reaction torque

exerted by the quasiparticles on the exchange field. (We will discuss later how to

view this effect from a more microscopic point of view.) We see that this difference

in spin current must be precisely the amount of torque exerted on the ferromagnet

by the non-equilibrium quasiparticles (see section B.2). Since the reorientation is

within a plane, these effect gives rise to an in-plane torque. We have that γin ∼ g P I

where P is the polarization of the spin current, g is a factor that account for some

features that have been swept under the rug in this argument mostly due to the

non-local nature of the electron transport and the role of the interfaces (more on

this will be discussed later in this work). After the instability point is reached the

subsequent dynamics of the magnet can be quite complex. Basically there are three

98

regimes that are predicted from equation (5.3). These dynamical regimes are all

observed in experiments [35]. One is a full switching of the magnetization. The

idea is that another stable state, whose stability is left unaffected or enhanced by

the spin transfer torque, is reached. Afterwards the magnetization is forced to stay

there. It is usually the case, for the geometries studied in experiments, that this

second configuration points in the direction opposite to the original configuration.

The spin current that destabilizes the first configuration actually stabilizes the sec-

ond direction. This current-induced magnetization-switching has been studied with

the technological potential of providing a possible set-up for an MRAM (magnetic

random access memory2). Basically the reading can be done with the standard

GMR (also present in these samples and actually used as a probe for the relative

orientations of the magnets) and the current induced spin switching can be used to

write. The other dynamical regimes are essentially described by oscillatory behavior,

either periodic or chaotic. Those are going to be described later on.

5.3 Microscopic Theory of Spin Transfer

Our microscopic picture of spin-transfer is summarized schematically in Fig.(5.4).

In spin density functional theory, SDFT, order in a metallic ferromagnet is charac-

terized by excess occupation of majority-spin orbitals, at a band energy cost smaller

than the exchange-correlation energy gain. (Adopting the common terminology

of magnetism, we refer to the spin-independent and spin-dependent parts of the

exchange-correlation fields of SDFT below as scalar and exchange potentials.) In

the ordered state, majority and minority spin quasiparticles are brought into equi-

librium by an exchange field that is approximately proportional to the magnetiza-

tion magnitude and points in the majority-spin direction. The spin-orientation of

the singly occupied majority-spin orbitals is the collective-coordinate, the magne-

2For a review of the basic requirements of an MRAM see [140, 141]

99

Figure 5.3: Numerical solution of the Landau-Lifshitz dynamics under the effects of a spintransfer torque. The red line is described by the magnetization vector as it flips from thenorth to the south pole in response to the current. The change in precession sense at theequator correspond to the change in the effective exchange field. This field is proportionalto the z component, and therefore change its sign at the equator.

100

Figure 5.4: Left panel: Ground state of a metallic ferromagnet. The low-energy collectivedegree of freedom is the spin-orientation of singly occupied orbitals. Right panel: Quasi-particles experience a strong exchange field ~∆ that brings majority and minority spins intoequilibrium. Because this field is parallel to the magnetization it does not produce a torque.In an inhomogeneous ferromagnet, the spin orientation of the transport orbitals in a win-dow of width eV at the Fermi energy can differ from the magnetization orientation. Thespin-transfer torque is produced by the transport-orbital contribution to the exchange field.

tization orientation, that plays the lead role in most magnetic phenomena. The

non-equilibrium current-carrying state of a ferromagnetic metal thin film can then

be described using a scattering or non-equilibrium Greens function formulation of

transport theory[142] and as explained in Chapter 3. The current is due to elec-

trons in a narrow transport window with width eV centered on the Fermi energy,

and can be evaluated by solving the quasiparticle Schroedinger equation for elec-

trons incident from the high-potential-energy side of the film. The spin-transfer

effect occurs when the spin-polarization of these transport electrons is not parallel

to the magnetization, producing a transport induced exchange field around which

the magnetization precesses. We expand on this picture below and illustrate its

utility by applying it to a toy-model two-dimensional ferromagnet with Rashba[143]

spin-orbit interactions.

101

5.3.1 Quasiparticle Spin Dynamics

We start by considering single-particle Hamiltonians of the form

H =p2

2m+ V (~r) − 1

2~∆(~r) · ~τ , (5.4)

where V (~r) and ~∆(~r) are arbitrary scalar and exchange potentials and ~τ is the Pauli

spin-matrix vector. In the local-spin-density approximation[138] (LSDA) of SDFT,

~∆(~r) = ∆0(n(~r),m(~r))m(~r) where m is a unit vector, ~m = mm is the total spin-

density at ~r obtained in equilibrium by summing over all occupied orbitals, and the

magnitude of the exchange field (∆0(n,m)) is the quasiparticle spin-splitting of a

polarized uniform electron gas. The spin-density contribution from a single orbital

Ψα is ~sα(~r) = Ψ†α(~r) ~τ Ψα(~r)/2. The time-dependent quasiparticle Schroedinger

equation therefore implies that

dsα,j(~r)

dt= ∇iJ

iα,j(~r) +

1

~

[~∆ × ~sα(~r)

]j

(5.5)

where the spin current tensor for orbital α is defined by,

J iα,j(~r) =

1

2mIm(Ψ†

α(~r)τj∇iΨα(~r)). (5.6)

This equation exhibits the separate contributions to individual quasiparticle spin dy-

namics from convective spin flow, the source of the conservative term, and precession

around the exchange field ~∆. Both sides of Eq.(5.5) vanish when the quasiparticle

spinor solves a time-independent Schroedinger equation.

5.3.2 Collective Magnetization Dynamics:

The time-dependence of the total magnetization is obtained by summing Eq.(5.5)

over all occupied orbitals.

dmj(~r)

dt=∑

α

∇iJiα,j(~r) +

1

~

[~∆ × ~m(~r)

]j

(5.7)

102

where J iα,j is the contribution to the spin-current from orbital α. The main point

we wish to make here is that (in the LSDA) ~∆ is proportional to ~m at each point in

space-time so that (at least in the absence of transport currents) the second term on

the right vanishes. The collective magnetization dynamics[144] is driven not by the

large effective fields seen by the quasiparticles, but by external and demagnetization

fields and spin-orbit coupling effects that have been neglected to this point in the

discussion, and by the divergence of the collective spin-current[145] in the first term.

A complete description of magnetization dynamics would require that the neglected

terms be included, and that damping due to magnetophonon and other couplings be

recognized. In practice, thin film magnetization dynamics can usually be successfully

described using a partially phenomenological micromagnetic theory approach[146]

in which the long-wavelength limit of the microscopic physics is represented by a

small number of material parameters that specify magnetic anisotropy, stiffness,

and damping. We adopt that pragmatic approach here, replacing the microscopic

Eq.(5.7) by the phenomenological Landau-Liftshitz equation

∂m

∂t= m × ~Heff + α m × ∂m

∂t, (5.8)

where α is the damping parameter,

~Heff(~r) ≡ δEMM[m]

δm(~r)(5.9)

is the effective field that drives the long-wavelength collective dynamics of an elec-

trically isolated sample, and EMM[m] is the micromagnetic energy functional.

5.3.3 Spin-Transfer

When current flows through a ferromagnet, the transport orbitals are few in number

and make a negligibly small contribution to the magnitude of the magnetization.

In an inhomogeneous magnetic system, however, they can make an important con-

tribution to the exchange field ~∆ as we now explain. The slow dynamics of the

103

collective magnetization can be ignored in the transport theory, appealing to an

adiabatic approximation. Our approach to spin-transfer is based on a scattering

theory formulation[142] in which properties of interest can be expressed in terms of

scattering solutions of the time-independent Schroedinger equation defined by the

instantaneous value of ~∆. Transport electrons will in general make a contribution to

the spin-density that is small but perpendicular to the magnetization 3. We define

this transport contribution to the spin-density as ~mtr. Because it is perpendicu-

lar to the magnetization, its contribution to the exchange-field experienced by all

quasiparticles

δ~HST = ∆0(n,m)~mtr

m(5.10)

produces a spin-torque that can be comparable to that produced by ~Heff. It follows

that the influence of a transport current on magnetization dynamics is captured by

replacing ~Heff in Eq.(5.8) by ~Heff + δ~HST. This proposal is the central idea of our

paper. We note here that the above doesn’t correspond, by means of Eq.(5.9), to

just a correction in EMM[m]. The net correction δ~HST to the effective dynamics

under non-equilibrium configurations can be separated into two contributions. One

“conservative” part which can be written as a corresponding correction to EMM[m],

and one “non-conservative” part that pump (or drain) energy to (from) the system.

A remarkable feature of the spin-valve geometry is that this non-conservative part

is the dominant part (an example of this is given in Fig. (5.7a)). In this way the

main effect of the spin-transfer torques in the dynamics is to create an “effective

damping”4. Although the behavior of this term makes it compete, in the dynamical

equations of the magnet, with the damping it is important to note that its origins

are not in a disorganized reservoir but in a coherent precession of the electrons in

3The transport orbitals will also, in general, contribute to the total spin-density component inthe direction of the magnetization. This effect alters the exchange field along the magnetizationdirection and does not produce a spin torque.

4Note that, this “effective damping”, can compensate the intrinsic damping[8, 9], signaling theinstability that precedes the switching, in current induced switching experiments.

104

the transport window.

The separation we have made here between transport orbitals and condensate

orbitals is reminiscent of the separation between conduction electrons and local

moments that is often made in models of magnetic systems. In diluted-magnetic-

semiconcutor ferromagnets, for example, these models often have a quantitative[19]

validity. In transition metal ferromagnets so-called s − d models of this type has

some qualitatively validity, but since the transition metal bands cross the Fermi level,

cannot be justified systematically. In the s − d model description of spin-transfer,

the magnetic condensate is associated entirely with the local moments and transport

with the conduction electrons. The mean-field exchange interaction between local

moments and transport electrons that carry current through a region with a non-

collinear magnetization produces a torque through the mechanism described above.

In our formulation of spin-transfer torque theory, the separation between transport

orbitals and the condensate order parameter is based only on the existence of a

transport energy window near the Fermi energy.

Our proposal can be related to the common approach in which spin-transfer

is computed from spin current fluxes. In the absence of spin-orbit coupling, summing

over all transport orbitals and applying Eq.(5.5) implies a relationship between the

transport magnetization and the transport spin currents:

[~∆(~r) × ~mtr(~r)

]j

= −~∇iJtr,ij (~r) (5.11)

where J tr,ij is the spin-current tensor summed over all transport orbitals. Note that

the net spin current flux through any small volume is always perpendicular to the

magnetization. It follows from Eq.(5.11) that

δ~HST(~r) =∇i

~J tr,i(~r) × m

m. (5.12)

When Eq.(5.12) is inserted in Eq.(5.8) it implies a contribution to the local rate

of spin-density change in any small volume proportional to the net flux of spin

105

current into that volume; in other words it implies that the bookkeeping theory

of spin-transfer applies locally, a property that can be traced in this instance to

the local spin-density-approximation (LSDA) of SDFT. The local approximation for

exchange interactions has its greatest validity when the magnetization varies slowly

on an atomic length scale, in long-wavelength spin waves or in typical domain walls

for example. This observation helps explain why a simple spin-transfer argument[?]

is able to account for the influence of a current on spin-waves in a homogeneous

ferromagnet[101] and on the propagation of a domain wall[122]. When spin-orbit

interactions are present, Eq.(5.12) is no longer valid.

Eqs.(5.8) and (5.10) provide explicit expressions for the effective magnetic

fields that drive magnetization precession at each point in space and time. Using

these equations it is possible to explore the consequences of spatial variation in spin-

transfer torque magnitude and direction, and of spin-orbit interactions. These have

a dominant importance in ferromagnetic semiconductors[147], where spin transfer

effects have been successfully demonstrated[148].

5.4 Toy-Model Calculations

In this section we implement the program described above for two examples one

involving a tunneling Hamiltonian between a ferromagnetic system and two leads,

one being a magnet whose magnetic moment is misaligned with the moment of the

system. The other case we handle is the case of a ferromagnetic 2DEG with Rashba

spin-orbit interaction. We study the behavior of the spin transfer efficiency as the

spin-orbit is tuned.

5.4.1 Effect of spin-orbit interaction

We illustrate our theory by evaluating ~mtr(~r) for a toy model containing a ferro-

magnetic two-dimensional electron system with Rashba spin-orbit interactions. The

106

Figure 5.5: Toy model described in the text, a 2DEG with ferromagnetic regions. In ourcalculations we apply periodic boundary conditions in the transverse y direction. A spin-transfer torque is present when the two magnetization directions are not aligned. The insetshows the Fermi surfaces of the two ferromagnets in which are identical in the absence ofspin-orbit coupling and and indicates the transverse channel ky range over which one ofthe two Schroedinger equation solutions is an evanescent spinor. The Schroedinger equationsolutions for electrons incident from x→ −∞ can be solved by elementary but tedious calcu-lations in which the spinors and their derivatives are required satisfy appropriate continuityconditions at the interfaces.

107

Figure 5.6: Right movers Fermi Surface in a Rashba System. The dashed lines are de-scribing the behavior of the imaginary part of the wave-vector of the state with the Fermienergy, of course, for the usual states depicted in the inset of Fig.(5.5) the imaginary part iszero, and the dashed lines are in the ky axis. In the region between the red (minority states)and the black (majority states) lines the imaginary part grows, indicating either a decaying(evanescent) or increasing behavior of the wave function as it moves to the right or left.In the case of finite spin-orbit coupling the wave vectors can be negative. This of coursedoes not affect their “Right mover” status, since in the presence of spin-orbit interactionthe velocity operator is modified by an amount that just cancels this shift.

108

model system, illustrated in Fig.(5.5), is intended to capture key features of the

spin-transfer effect. We take the width of the pinned magnet to infinity, neglect

the paramagnetic spacer that is required in practice to eliminate exchange coupling

between the two magnets, and assume for simplicity that there is no band offset

between the two ferromagnets and that the two exchange fields are equal in mag-

nitude. Current flows from the pinned magnet, through the free magnet, into a

paramagnetic metal that functions as a load. The spin-orbit interaction is assumed

to be confined to the free magnet region. 5 The above simplifications allow us to

write the Hamiltonian of the system as:

H =p2

2m− ~∆(x) · ~s+ λ(x), ~p × z · ~s,

where ~∆ correspond to the local exchange vector, and λ is non-zero only on the

region with spin-orbit coupling. For this model we evaluated ~mtr(~r) in a current-

carrying system using the Landauer-Buttiker approach[142]. In the linear response

regime, this requires that the Schroedinger equation be solved at the Fermi energy

for all transverse channels for electrons incident from the left.

It is helpful at this point to make contact with the usual description of spin-

transfer. In its simplest version, spin-transfer theory assumes complete transfer,

i.e. that the incoming current is spin-aligned in the fixed magnet direction and the

outgoing current is spin-aligned in the free magnet direction. To the extent that the

complete transfer assumption is valid, the torque is in the plane defined by the two

magnetization orientations, which we refer to as the transfer plane. Microscopically

[89, 90, 91, 92, 93, 94, 95, 96, 97, 100, 98, 99, 136] the component of the outgoing

current perpendicular to the transfer plane is expected to be very small because of

interference between precessing magnetizations in different channels.

It follows from Eq.(5.10) that the spatially averaged spin orientation of the

5To ensure Hermiticity we write HSO = (λ, ~p × z) · ~s, where the symbol ·, · denotes theoperator anticonmutator .

109

transport electrons should be approximately perpendicular to the transfer plane. It

can be verified that this is indeed true by directly evaluating ~mtr(~r). This simple

intuitive argument is not exact, however. In particular, the incoming spin current

is not necessarily polarized along the pinned magnet magnetization, because of in-

terference between incident and reflected quasiparticle waves that complicates the

spin-transfer torques and also because it fails to account for electrons that are de-

scribed by spinors with evanescent components. (See the inset of Fig.(5.5)). In

any microscopic calculation these effects and others conspire to produce a relatively

small component of the torque that is perpendicular to the transfer plane, and

correspondingly to a component of ~mtr(~r) that is in the transfer plane.

In Fig.(5.7(a)) we plot values of ~mtr(~r) per unit current averaged over the free

magnet space as a function of the angle between the two magnetization orientations,

in the case without spin-orbit interaction. We have taken the free magnet orientation

be the z direction and the pinned magnet to be in the z − x plane with polar angle

θ.

When spin-orbit interactions are included, the strength of the spin-transfer

torque must be evaluated using the transport spin densities. The bookkeeping argu-

ment, based on total spin conservation, is no longer valid. The quasiparticle spins

not only are no longer conserved due to momentum-dependent effective magnetic

fields that represent spin-orbit coupling. As we see in Fig.(5.7), the spin-transfer

effect is not only reduced in magnitude but its dependence on θ no longer approx-

imates the simple complete transfer expression. A measure of how the effect is

destroyed by the spin-orbit interaction is given by the magnitude of the spin trans-

fer efficiency g, defined as the value of the in plane torque per unit current at the

optimum geometry, θ = π/2. In Fig.(5.7(c)] we show the efficiency as a function

of the spin orbit interaction strength. We see that when the spin-orbit interaction

strength is comparable to the exchange spin splitting the effect is strongly reduced

110

S y

S x

S y

(b)(a)

(c)

θ θ

Figure 5.7: (a) Transport spin density per unit current in the case without spin-orbit inter-action. mtr

y is the component perpendicular to the transfer plane (“non-conservative”)whilemtr

x is the smaller component in the transfer plane (“conservative”) that is contributed byevanescent spinors. Both components are normalized to the maximum mtr

y which occurs forθ = π/2. (b)Non-equilibrium spin density per unit current perpendicular to the transferplane for different spin-orbit interaction strengths. It follows that from these results that thespin-transfer torque is reduced in efficiently and altered in angle dependence by spin-orbitinteractions. (c)Spin transfer efficiency, g, normalized by the ST efficiency in the absenceof spin-orbit coupling, as a function of the spin-orbit strength, for several widths of the freemagnet. The spin-transfer effect becomes weak when the spin-orbit splitting is comparablewith the exchange splitting.

111

except for the case of extremely thin layers.

5.5 Discussion

We have presented a formalism that allow us to evaluate the interplay between

transport currents and magnetization dynamics in very general circumstances. This

formalism can address open issues in magneto-transport theory including the pos-

sible importance of incoherent nanomagnet magnetization dynamics in metal spin-

transfer phenomena, and the influence of the spin-orbit interactions on spin-transfer

in diluted magnetic semiconductor ferromagnets. Our theory of spin-transfer is

formulated in terms of the change in the effective Hamiltonian that describes all

quasiparticles, even ones well away from the Fermi energy, when a conductor is

placed in a non-equilibrium state by connecting it to two reservoirs with different

chemical potentials. From this point of view, related phenomena occur in nearly any

electronic systems, although they will not always lead to experimental effects that

are as interesting and experimentally robust as they ones that occur in ferromagnetic

metals.

The approach to electron-electron interaction related non-linear transport

effects explained in this chapter has recently been applied[17] to quantum Hall bi-

layers and to circuits that contain antiferromagnetic metals. In the case of quantum

Hall bilayers, the collective coordinates of interest are the interlayer phase and pop-

ulation differences, which play the same role as azimuthal and polar angles of the

magnetization in a ferromagnet, pseudospins rather than spins. Bilayer quantum

Hall systems have spontaneous interlayer phase coherence (pseudospin ferromag-

netism) and pseudospin transfer torques have been invoked to explain the sudden

drop in interlayer conductance with bias voltage seen in experiment[149]. We an-

ticipate that similar transport effects can occur even in systems that do not have

interlayer phase coherence, notably in bilayer electron systems in the absence of an

112

external magnetic field. In antiferromagnetic metals circuits, it has recently been

predicted[17] that large spin-transfer torques appear because of quasiparticle scat-

tering properties related to combined spatial and spin symmetries. In this case

the spin-torques cannot be related to conservation of total spin. Other examples

include quantum wells with tilted magnetic fields, in which the Hartree-potential

that defines the two-dimensional transport channel is itelf altered by a bias voltage.

In all these effects, the quasiparticle band structure can no longer can be regarded

as fixed for a given system. Instead changes in quasiparticle band structure, and

non-equilibrium changes in the quasiparticle Hamiltonian density matrix appear as

interdependent responses to circuit bias voltages.

113

Chapter 6

Antiferromagnetic Spintronics

The contents of this chapter are partially based on the article: Alvaro S. Nunez,

Rembert Duine, and A.H. MacDonald, Antiferromagnetic Metals Spintron-

ics, Physical Review B, 73, 214426 (2006), cond-mat/0510797.

6.1 Introduction

Spintronics in ferromagnetic metals[11] is based on one hand on the dependence of

resistance on magnetic microstructure [7], and on the other hand on the ability to

alter magnetic microstructures with transport currents [8, 9, 38, 39, 38, 82, 83, 84,

85, 1, 35, 37, 86, 87, 88]. These effects are often largest and most robust in circuits

containing ferromagnetic nanoparticles that have a spatial extent smaller than a

domain wall width and therefore largely coherent magnetization dynamics. In this

chapter we point out that similar effects occur in circuits containing antiferromag-

netic metals. The systems that we have in mind are antiferromagnetic transition

metals similar to Cr[150] and its alloys[151] or the rock salt structure intermetallics

[152] used as exchange bias materials which are well described by time-dependent

mean-field-theory in its density-functional theory[138] setting.

114

Our proposal that currents can alter the micromagnetic state of an antifer-

romagnet may seem surprising since spin-torque effects in ferromagnets [89, 90, 91,

92, 93, 94, 95, 96, 97, 98, 99, 100, 136] are usually discussed in terms of conser-

vation of total spin, a quantity that is not related to the staggered moment order

parameter of an antiferromagnet. Our arguments are based on a microscopic pic-

ture of spin-torques[153] in which they are viewed as a consequence of changes in the

exchange-correlation effective magnetic fields experienced by all quasiparticles in the

transport steady state. A spin torque that drives the staggered-moment orientation

n must also be staggered, and will be produced[153] by the exchange potential due

to an unstaggered transport electron spin-density in the plane perpendicular to n.

The required alteration in torque is produced by the alternating moment orienta-

tions in the antiferromagnet rather than the transport electron exchange field. As

we now explain the transverse spin-densities necessary for a staggered torque occur

generically in circuits containing antiferromagnetic elements.

The key observations behind our theory concern the scattering properties of a

single channel containing non-collinear antiferromagnetic elements with a staggered

exchange field that varies periodically along the channel and is commensurate with

an underlying lattice that has inversion symmetry. For an antiferromagnetic ele-

ment that is invariant under simultaneous spatial and staggered moment inversion

it follows from standard one-dimensional scattering theory [154] considerations that

transmission through an individual antiferromagnetic element is spin-independent,

and that the spin-dependent reflection amplitude from the antiferromagnet or any

period thereof has the form r = rs1 + rt n · ~τ , where n is the order parameter ori-

entation and ~τ are the Pauli spin matrices; rs and rt are proportional to sums and

differences of reflection amplitudes for incident spins oriented along and opposite to

the staggered moment. The reflection amplitude for a spinors incident from opposite

sides differ by changing the sign of n and the transmission amplitudes are identical.

115

It then follows from composition rules for transmission and reflection amplitudes in

a compound circuit containing paramagnetic source and drain electrodes and two

antiferromagnetic elements with staggered moment orientations n1 and n2 separated

by a paramagnetic spacer (see Fig. 6.4) that the transport electron spin-density in

the n1 × n2 direction is periodic in the antiferromagnets. (We define the direction

of ni to be the direction of the local moment opposite the spacer.) The spin-torques

that appear in this type of circuit therefore act through the entire volume of each

antiferromagnet.

A proof of this property will be presented in the appendix Sec. (C.3). Here

we illustrate the potential consequences of this property by using non-equilibrium

Greens function techniques to evaluate antiferromagnetic giant magnetoresistance

(AGMR) effects and layer-dependent spin-torques in model two-dimensional circuits

containing paramagnetic and antiferromagnetic elements. We focus on the most fa-

vorable case in which the antiferromagnet has a single Q spin-density-wave state

with Q in the current direction. In the following we first explain the model sys-

tem that we study and the non-equilibrium Greens function calculation that we

use to evaluate magnetoresistance and spin-torque effects. We conclude that under

favorable circumstances, both effects can be as large as the ones that occur in ferro-

magnets. We then estimate typical critical current for switching an antiferromagnet.

Finally, we discuss some of the challenges that stand in the way of realizing these

effects experimentally.

6.2 Scattering in Single Q Antiferromagnets

In this section we find the limitations placed by symmetry on the single-channel

quasiparticle scattering matrix of a one-dimensional antiferromagnet. In an an-

tiferromagnet the quasiparticles satisfy a Schroedinger equation with an exchange

Zeeman field with oscillatory spatial dependence in the direction of the order param-

116

eter of the antiferromagnet. We assume that a single period of the spin-density-wave

is invariant under the combined effects of time reversal and spatial inversion. (Note

that time reversal includes a spin flip in the present spin-12 case.) This assumption is

valid for a spin-density wave that is commensurate with an underlying lattice that

has inversion symmetry. The generalization from one-dimension to two or three

dimensions is trivial for a single-Q spin-density wave state with the wavevector Q

oriented along the current direction. An antiferromagnet circuit element composed

of any integer number of spin-density-wave periods is also invariant under this sym-

metry operation.

We first define some notation conventions. We denote the asymptotic wave

functions traveling to the right (x→ ∞) and to the left (x→ −∞) by

Ψ−∞(x) = |−∞R〉 eikx + |−∞L〉 e−ikx ; (6.1)

Ψ∞(x) = |∞R〉 eikx + |∞L〉 e−ikx , (6.2)

where |∞R〉 , · · · and |∞L〉 , · · · are the spinor coefficients of the right and left goers,

respectively. The scattering matrix expresses the outgoing spinors in terms of the

incoming spinors:

|−∞L〉

|∞R〉

= S

|−∞R〉

|∞L〉

with S in turn expressed in terms

of 2×2 transmission and reflection matrices S =

r t′

t r′

. We choose the direction

of the Zeeman field in the antiferromagent, n, to be the spin quantization axis.

Invariance under simultaneous rotation of n and quasiparticle spins allows us to

write each transmission and reflection matrix in the scattering matrix as a sum of

a triplet and a singlet parts

S = Ss + Stn · τ . (6.3)

Now, the operation space inversion-time reversal symmetry transform the

117

wave functions into:

Ψ−∞(x) = iσy |−∞∗L〉 eikx + iσy |−∞∗

R〉 e−ikx ; (6.4)

Ψ∞(x) = iσy |∞∗L〉 eikx + iσy |∞∗

R〉 e−ikx , (6.5)

Because the system is invariant under the space inversion-time reversal symmetry

operation, the components of this transformed scattering wave functions must be re-

lated by the same scattering matrix. This condition imposes the following symmetry

constraint on S:

S† =

0 σy

σy 0

S∗

0 σy

σy 0

. (6.6)

By rewriting this constraint explicitly in terms of the reflection and transmission

matrices we obtain

r′s − r′tτz = rs + rtτz ; (6.7)

ts − ttτz = ts + ttτz ; (6.8)

t′s − t′tτz = t′s + t′tτz . (6.9)

It follows that tt = t′t = 0 and that r′t = −rt. The most general form of S allowed

by this symmetry operation is

S =

rs + rt n · τ t′s

ts rs − rt n · τ

. (6.10)

However the parameter space is further constrained by unitarity. This allows us to

write

rs = ieiν sin Θ cos Φ ;

rt = eiν sin Θ sinΦ ;

t′s = ei(ν−ξ) cos Θ ;

ts = ei(ν+ξ) cos Θ , (6.11)

118

where ξ and ν are phases that so far are independent parameters, and Θ and Φ

are the polar coordinates of a sphere of radius unity. This is the most general form

for spin-dependent scattering by a integer number of periods of a one-dimensional

spin-density-wave. In terms of the rotation matrix QΦ = exp (iΦ n · τ ), we obtain:

S = eiν

sin Θ QΦ cos Θ e−iξ1

cos Θ eiξ1 sin Θ Q−Φ

. (6.12)

In this form, we can easily conclude that transmitted electrons will preserve their

spins orientations, while reflected electrons will emerge from the system with their

spin orientations rotated around the order parameter in opposite senses depending

on their direction of incidence. This is to be contrasted with the case of a ferro-

magnetic scatterer. In that case, both the transmitted and reflected electrons are

rotated, besides, the rotations are independent of the direction of incidence. As

a direct consequence of this elementary, but general, considerations we reach the

conclusion that single antiferromagnetic layers cannot act as spin filters, in other

words, the spin polarization of a current will be conserved as it crosses an isolated

antiferromagnetic element. We emphasize that while the transmission coefficients of

an antiferromagnet are spin-singlets, the reflection coefficient are still non-trivial, in-

deed for an incoming unpolarized current, while the transmitted current will be still

unpolarized, the reflected current will be spin-polarized along the order parameter

direction. This fact is the main property that is behind the further developments

to be described below.

We now briefly discuss the consequences of this result for circuits with non-

collinear antiferromagnetic elements. In an array for multiple non-collinear anti-

ferromagnets, each one will fail to induce spin polarization, however the multiple

reflection process at each interface will lead to a non-trivial spin-current configu-

ration. Most importantly, for two antiferromagnets with respective staggered mo-

ment orientations n1 and n2 separated by an arbitrary paramagnetic spacer we are

119

able to prove that the the out-of-plane spin density, i.e., the spin density in the

n⊥ ≡ n1 × n2/|n1 × n2| direction is periodic with the lattice in the paramagnetic

part of the system, and periodic with the same period as the spin density wave in the

antiferromagnets. These spin-densities will produce a contribution to the exchange

correlation field that is out of the plane of either antiferromagnet; the average out-

of-plane spin-density will produce a staggered field that will drive spatially coherent

precession of the antiferromagnetic order parameter and can lead to order parameter

reorientation. Because the spin-density is periodic in each antiferromagnet, it will

not decay away from the interface in either antiferromagnet and will therefore lead to

spin transfer torques that act throughout the entire volumes of the antiferromagnet

elements. As we discuss later, this surprising property could potentially lead to low

critical currents for induced order-parameter dynamics. A proof of this property is

outlined in the Appendix. In the next sections we illustrate its consequences for spin

dependent transport by performing non-equilibrium Greens function calculations on

tight-binding model antiferromagnets.

6.3 Antiferromagnetic giant magnetoresistance

The results of the previous section provide a simple way to calculate the dependence

of the resistance of a circuit containing antiferromagnetic elements on the relative

orientation of the order parameters, an effect that we refer to as antiferromagnetic

giant magnetoresistance (AGMR).For simplicity, we consider two identical antifer-

romagnets with scattering matrices given by Eqs. (6.10) and (6.11), with different

order parameter orientations denoted by n1 and n2. Note that throughout this pa-

per we define n1 and n2 to be the direction of the moments opposite the spacer.

We denote the distance between the antiferromagnetic layers by L. As discussed

in the Appendix we calculate the scattering matrix of the compound system using

120

standard composition rules [155, 154]. The result is given by Eq. (C.38):

|∞R〉 = t2Kt1 |−∞R〉 , (6.13)

where the multiple reflection kernel is defined by K = (1− r′1r2)−1. The net trans-

mission coefficient becomes:

T = Tr(t†1K

†t†2t2Kt1

). (6.14)

Using Eqs. (6.10) and (6.11) we reduce it to:

T = cos4 ΘTr(K†K

). (6.15)

The trace of the square of the multiple reflection kernel contains the information of

the order parameter orientations and accounts for the dependence of the transmis-

sion on their relative orientations. Straightforward calculation leads to:

|Λ|2 Tr(K†K

)= (2 + 4(1 + n1 · n2)

cos(2ν − δL

)cos2 Φ sin2 Θ + 8 cos4 Φ sin4 Θ

), (6.16)

where we have used δL to denote the phase shift associated with the translation of

the antiferromagnetic layers and Λ is defined in Eq. (C.41). From Eq. (6.16) we

read off the dependence on the angle between the orientations of the order parameter

that enters via n1 · n2 ≡ cos θ.

We see how this simple argument leads us to a finite AGMR ratio. Its precise

value depends on the parameters Θ and Φ, and, when summing over momenta

perpendicular to the current direction, also on their momentum dependence. To

further illustrate magnetoresistive, and, in the next section, spin torque effects, we

consider a specific model of an antiferromagnet in the remainder of this section.

6.3.1 Elementary Local Spin Model

To understand the magneto-resistive effect in ferromagnets, a simple picture is given

as follows. The basic idea is to consider the spin up and spin down channels as classi-

121

Figure 6.1: (a) Effective resistance arrays that represents a parallel configuration in aconventional GMR device. (b) same for antiparallel.

Figure 6.2: (a) Effective resistance arrays that represents a parallel configuration in aAFM-GMR device. (b) same for antiparallel. No GMR effect can be observed from theclassical system.

cal parallel channels. In this picture the difference between parallel and anti-parallel

resistances emerges from the differences in the “effective” circuits that represent the

different situations.

For up channel the situation parallel situations implies that the electron must

go through two high resistances (2R), for channel down the same situation implies

two low resistances (2r), so the parallel addition of this two resistances implies:

1R↑↑

= 12R + 1

2r . The opposite configuration has one large and one small resistor in

each channel, leading to a net resistance: 1R↑↓

= 2R+r . The magneto-resistance ratio

is then:

MR =R↑↓ −R↑↑

R↑↓=

(r −R

r +R

)2

. (6.17)

When we try to use the same ideas to describe the antiferromagnetic situation

we face the following problem. The difference between the spins ups and spin down

channels vanishes as we increase the number of alternating layers. From the classical

point of view we get no magneto-resistance at all. However its clear that in a system

where the alternating layers have such small separation we need a quantum-transport

approach to describe the effects. pick a highly idealized model in order to include

122

Figure 6.3: (a) Scattering process for right-going incoming electrons. (b) same forleft-goers. Both processes are included in the S matrix.

quantum effects at the most elementary level. We consider a 1-channel system with

point spin-like scatterers located on a lattice:

V (x) = JN∑

i=1

Ωi · ~τ δ(x − xi) (6.18)

and we calculate the transmission coefficients for different configurations of the

Ωi

. To do that we first calculate the scattering matrix for a single scatterer

located at the origin and pointing along Ω.

It is an easy matter to prove that:

S =

Γ − 1 Γ

Γ Γ − 1

, (6.19)

where:

Γ =1

1 − λ21 +

λ

1 − λ2Ω · ~τ , (6.20)

with λ = −iJ/~v, where v = ~k/M is the velocity of the free electron1. If the

scatterer is at x0 we need to use the translated scattering matrix:

S =

e2ikx0(Γ − 1) Γ

Γ e−2ikx0(Γ − 1)

. (6.21)

Finally we can calculate the scattering matrix of an arbitrary array of scatterers

by composing the scattering matrices of the series. This is done using the series of

1It is an easy task to prove the identity 2Æà = à + Æ, from which unitarity of S follows.

123

reflections between two scatterers. For two scatterers with scattering matrices S1

and S2 we obtain a composite scattering matrix S12 given by:

S12 =

r1 + t′1r2 (1 − r′1r2)

−1 t1 t′1 (1− r2r′1)

−1 t′2

t2 (1 − r′1r2)−1 t1 r′2 + t2 (1 − r2r

′1)

−1 r′1t′2

(6.22)

We start by composing two consecutive layers of opposite spin orientation separated

by a distance x0 with e2ikx0 ≡ eiφ we can write the respective scattering matrices

as:

S1 =

e−iφ

(Γ(Ω) − 1

)Γ(Ω)

Γ(Ω) eiφ(Γ(Ω) − 1

)

(6.23)

S2 =

eiφ

(Γ(−Ω) − 1

)Γ(−Ω)

Γ(−Ω) e−iφ(Γ(−Ω) − 1

)

. (6.24)

The multiple reflection kernel then becomes:

(1− r′1r2

)−1=(1 − ei2φ (Γ(Ω) − 1) (Γ(−Ω) − 1)

)−1(6.25)

and after some elementary manipulations we obtain:

(1− r′1r2

)−1=

1 − λ2

1 − λ2(1 − ei2φ

)

1 (6.26)

and both transmission coefficients become:

t12 = t′12 =1

1 − λ2(1 − ei2φ

) , (6.27)

and are spin independent. All the spin dependence is canceled due to the alternating

structure of the spin lattice. This correspond to the basic naive picture of spin echo,

what one spin does to the electrons is ”un-done” by the subsequent spin. This effect

breaks down in the presence of boundaries, as is shown in the behavior of reflection

coefficients. The reflection coefficients do depend on the the spin orientation:

r12 =λe−iφ

1 − λ2ω(φ, λ)

(λ1 + Ω · ~τ

)(6.28)

r′12 =λe−iφ

1 − λ2ω(φ, λ)

(λ1− Ω · ~τ

)(6.29)

124

Figure 6.4: The model heterostructure for which we perform our calculations.

where,

ω(φ, λ) = 1 − eiφ

1 − λ2(1 − ei2φ

) . (6.30)

it is from this dependence on the direction of the reflection coefficients that all the

effects we are discussing emerge. Note that the symmetry requirements are satisfied

explicitly by this result.

6.4 Tight-Binding Non-equilibrium Calculation

We analyze a two-dimensional single-band lattice model intended to illustrate generic

qualitative features of spintronics in antiferromagnetic metal circuits. The model

has near-neighbor hopping, transverse translational invariance, and spin-dependent

on-site energies, as illustrated in Fig. 6.4:

Hk = −t∑

〈i,j〉,σ

c†k,i,σ ck,j,σ + h.c.

+∑

i,σ,σ′

[(ǫi + ǫk)δσ,σ′ − ∆iΩi · ~τσ,σ′

]c†k,i,σ ck,i,σ′ . (6.31)

Here, k denotes the transverse wave number, t the hopping amplitude and ǫk the

transverse kinetic energy. The second term in Eq. (6.31) describes the exchange

coupling ∆i of electrons to antiferromagnetically ordered local moments Ωi = (−)in

that alternate in orientation within each antiferromagnet. In the paramagnetic

regions of the model system ∆i = 0. The on-site energies ǫi are allowed to change

across a heterojunction to represent band-offset effects.

We use the non-equilibrium Greens function formalism to describe the trans-

port of quasiparticles across the magnetic heterostructure. The essential physical

125

0.88

0.9

0.92

0.94

0.96

0.98

1

0 0.5 1 1.5 2 2.5 3

T(θ

)/T

(0)

θ

N=15,M=16N=15,M=17N=25,M=16N=25,M=17

Figure 6.5: Landauer-Buttiker conductance as a function of the angle θ betweenthe magnetization orientations Ωi on opposite sides of the paramagnetic spacerlayer. There is a sizable giant magnetoresistance effect, with larger conductance atsmaller θ and weak dependence on layer thicknesses. These results were obtainedfor ∆/t = 1 and ǫi = 0.

properties of the system are encoded in the real time Greens function [75, 155],

defined by the ensemble average, G<σ,i;σ′,j(k; t, t

′) = i〈c†k,i,σ(t) ck,j,σ′(t′)〉, from which

the (spin) current and (spin) density are evaluated. To evaluate the strength of the

model’s AGMR, we calculate the transmission coefficient as a function of the angle

θ between orientations Ωi on opposite sides of the spacer. In Fig. 6.5 the trans-

mission coefficient is shown for specific values of the number of layers N and M , in

the first and second antiferromagnet. The fact that there must be a AGMR effect

can be seen by taking the limit of zero width for the paramagnetic region. In this

case the resistance is greater when θ is zero since this arrangement interrupts the

periodic pattern of exchange fields. The AGMR effect can generally be traced to the

interference between spin-current carrying electron spinors reflected by the facing

layers. (This is also the origin of the spin transfer effect to be discussed later.) At

the paramagnetic spacer layer thicknesses studied here, the model AGMR depends

on the orientation of the layers opposite the spacer in the usual way, i.e. the resis-

tance is highest for θ = π and lowest for θ = 0. Also, we find that the AGMR ratio,

defined as the absolute difference between the maximum and minimum value of the

126

transmission coefficient normalized to the minimum, saturates as a function of the

length of the antiferromagnetic elements.

The main point of these calculations is to demonstrate by explicit calculation

that AGMR in antiferromagnetic metal circuits can in principle have a magnitude

comparable to GMR in ferromagnetic metal circuits. It is instructive to compare

these numerical results with qualitative pictures of AGMR in an effort to judge their

robustness. The simplest picture of transport in a magnetic system is the bulk two-

channel transport Julliere picture [32]in which magnetoresistance arises ultimately

from the difference between the majority-spin and minority-spin resistivities of bulk

material. For bulk antiferromagnets the resistivity is spin independent, so this effect

cannot explain the AGMR that appears in our numerical calculations.

The difference between parallel and anti-parallel configurations amounts to

merely a shift by 1 period of the spin-density wave in the second anti-ferromagnet.

That such a shift can give rise to AGMR is seen explicitly in Eq. (14). The sign of the

AGMR for a given channel depends on the phase shift acquired in the paramagnetic

spacer region by the electron. One must integrate over all such channels in the

transport window, and the total AGMR is the sum over each channel’s value of

AGMR. Coherent interference effects are critical to seeing this effect, and we expect

the AGMR ratio to vanish as the spacer thickness becomes much larger than the

phase coherence length. As we explain in the discussion section, this will not be

a problem in practice. We also expect that the AGMR effect will be very weak

when the magnetization also varies in the plane parallel to the antiferromagnet-

paramagnet interface.

127

6.4.1 Transmission through oscillating 1D exchange fields

Next, in our attempt to shed some light into the problem of AFM/AFM transport,

we use the following model for the AFM:

H = − ~2

2m∇2 + J cos

(2π

λsdwx

)Ω · ~τ (6.32)

with atomic units, and the scaling of coordinates x = (λsdw/π)z we can write the

Schrodinger equation as:

d2Ψσ

dz2+ (a − 2qσ cos(2z)) Ψσ = 0, (6.33)

where a =(

λsdwπ

)22E and qσ =

(λsdw

π

)2Jσ. This equation correspond to two sepa-

rated Mathieu equations, for spin up and spin down, whose solutions are well known

as the Mathieu functions, mc and ms corresponding to cos and sin, respectively for

q = 0 . The general solution is:

Ψσ(z) = Aσmc(a, qσ , z) +Bσms(a, qσ , z). (6.34)

6.4.2 Spin Filter Effect suppression

The first step, just like in the FM case is to evaluate the spin filtering effects of a

single antiferromagnetic layer. For that purpose we consider a AFM slad in between

two PM metals. The potential then is:

V (x) =

0 x < 0

J cos(

2πλsdw

x)

Ω · ~τ 0 < x < L

0 x > L

(6.35)

The explicit solutions are then:

Ψσ(x) =

ασ exp (ikx) + βσ exp (−ikx) x < 0

Aσmc(a, qσ ,π

λsdwx) +Bσms(a, qσ ,

πλsdw

x) 0 < x < L

γσ exp (ikx) x > L

(6.36)

128

0 5 10 15 20

0

2

4

6

8

10

12

14

EF

Singlet Transmission

0 5 10 15 20

Triplet Transmission

0 0.5 1 1.5 2

10 Im k

0 1 2 3

Re k

Figure 6.6: The Transmission coefficient of an oscillating exchange field. from left to right,the first and second panel displays the absolute values of the singlet and triplet transmissionsas a function of the width of the antiferromagnetic slab (horizontal axis, in units of the λsdw)and the fermi energy (vertical axis). The color code is black for small and white for large.The third and fourth panels displays the imaginary and real parts, respectively, of the Blochmomentum at given energy (vertical axis)

.

By imposing standard conditions of continuity we can solve for the transmission

transmission matrix. Since we have a separate equation for spin up and down it

is clear that we can define to independent transmission amplitudes. We define the

singlet transmission as the average of this two variables and the triplet transmission

as half the difference. It is clear that this later quantity correspond to a measure of

the effective spin filter effect. The results of the magnitude of the singlet and triplet

transmissions are plot in the following figure (6.6).

The main feature to rescue from this figure is the cancelation of the triplet

transmission for slabs widths that corresponding to full periods of the spin density

wave. This is due to the spin-echo effect. This is in agreement with the delta-

function model described earlier, and the tight-binding numerical calculation.

6.5 Current-driven switching of an antiferromagnet

To address the possibility of current-induced switching of an antiferromagnet we

evaluate spin transfer torques in the second antiferromagnet. The spin transfer

129

torque originates from the contribution made by transport electrons to the exchange-

correlation effective magnetic field and is given[153] by Γ = ∆iΩi × 〈si〉/~, where

〈si〉 is the nonequilibrium expectation value of the quasiparticle spin. (In effect,

the presence of a bias voltages separates a transport window from the quasiparticle

system and redirects its spin-density contribution, creating two-subsystems that can

mutually precess. The torque acts on the spins of quasiparticles outside the trans-

port window while the orientations of transport spins are instantaneously fixed by

the transport bias voltages and the generally non-collinear exchange fields through

which they travel.) In our model system we distinguish the spin-torque component

in the plane spanned by n1 and n2 and the component out of this plane. In Fig. 6.7

we show the in-plane and out-of-plane transport-induced spin torques. As antici-

pated the in-plane spin transfer torque in this model is exactly staggered (for any

∆/t value) and is therefore extremely effective in driving order-parameter dynamics.

We have checked numerically that staggered in-plane spin-transfer torques that do

not decay also occur in continuum toy models of an antiferromagnet with piece-wise

constant and sinusoidal exchange fields. These persistent spin torques are a generic

property of antiferromagnetic circuits and related to the absence of spin-splitting

in the Bloch bands. The staggered in-plane spin-transfer is produced by an out-of-

plane spin density that is exactly constant in our lattice model antiferromagnet and

exactly periodic in a continuum model antiferromagnet.

The effect can be understood qualitatively as follows. Transport through an

antiferromagnet-paramagnet interface will tend to be dominated by the top layer

spin. When these spin-polarized electrons enter the second antiferromagnet the

exchange field in the top layer induces a precession to an orientation that has an

out-of-plane component. Exchange fields in subsequent layers produce a periodic

oscillation which leaves the out-of-plane spin density at a non-zero average value.

The out-of-plane spin-density in the paramagnetic and upstream ferromagnetic lay-

130

ers has to be understood in terms of reflection from the downstream material, just

as in the ferromagnetic case. While this simple explanation does not fully capture

the effect since it does not capture the difference between in-plane and out-of-plane

spin densities, we believe that it has some qualitative validity and can use it as a

guide in anticipating the influence of the elastic, inelastic, and spin-dependent scat-

tering that is not included in our model calculation. It is clear for example that as

in the ferromagnetic case, the antiferromagnetic spin-torque effect will occur only if

the width of the paramagnetic spacer layer is less than a spin-coherence length. On

the basis of the picture explained above, we expect that the torque will act over the

portion of the antiferromagnet that is within an inelastic scattering length of the

interface with the paramagnetic spacer, compared to the full volume effect in the

absence of scattering and the Fermi wavelength attenuation scale that applies for

ferromagnets. It is also reduced when the antiferromagnetic order parameter has

zero spatial average in planes perpendicular to the current direction, which is the

case when antiferromagnetic domains are present.

Since the exchange-interactions that stabilize the antiferromagnetic will nor-

mally be very strong compared to the transport-induce spin torques, the magne-

tization dynamics of each antiferromagnetic element will be coherent and respond

only to the staggered component of each spin-torque. In Fig. 6.8 we show the total

staggered torque acting on the downstream antiferromagnet, as a function of the

angle θ. Clearly, the out-of-plane component of the torque is small compared to

the in-plane component. Since the angular dependence of the spin transfer torque is

Γ ∼ g(θ) sin(θ), the value for g(π) can be extracted by evaluating ∂θΓ at θ = π. This

quantity is shown in Fig. 6.9, and we will see that the critical current for reversal is

inversely proportional to this quantity.

Having demonstrated the presence of spin transfer torques in a heterostruc-

ture containing two antiferromagnetic elements, we estimate the critical current for

131

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40 50

Γ out

i

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0 10 20 30 40 50

Γ in

Figure 6.7: Local spin-transfer torques in the down-stream antiferromagnet. Thein-plane spin transfer is staggered and therefore effective in driving coherent orderparameter dynamics. The out-of-plane spin-transfer component is ineffective be-cause it is not staggered. These results were obtained for ∆/t = 1, ǫi = 0, θ = π/2,N = 50, and M = 50.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

Γ in,

Γ out

θ

ΓinN=30,M=30

ΓoutN=30,M=30

ΓinN=31,M=31

ΓoutN=31,M=31

ΓinN=32,M=32

ΓoutN=32,M=32

Figure 6.8: Total spin transfer torque action on the downstream antiferromagnet,as a function of θ. We used the parameters ∆/t = 1 and ǫi = 0.

132

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30

M g

(θ=

π)

M

N=50N=51N=52N=53

Figure 6.9: Derivative of the total spin transfer torque per unit current, Mg(θ = π),acting on the downstream antiferromagnet with respect to the angle θ at θ = π asa function of M . We used the parameters ∆/t = 1 and ǫi = 0.

switching the second antiferromagnet assuming that the first is pinned. To illustrate

our ideas, we use the crystalline anisotropy energy density for Cr [150, 156], given

by

E(n) = K1(z · n)2 +K2(x · n)2(y · n)2 , (6.37)

where n is a unit vector in the direction of the staggered moment and Q is taken

to be in the z direction. The first term favors a staggered moment that is either

parallel or perpendicular to the ordering vector Q and changes sign at the spin flop

transition [150]. The term proportional to K2 captures cubic anisotropy in the plane

perpendicular to Q.

As we have seen, the spin transfer torques act cooperatively throughout the

entire antiferromagnet. We can focus our description on a single domain, character-

ized by the orientation of one ferromagnetic layer within the antiferromagnet since

all layers will have definite relative orientations when the order parameter dynamics

is spatially coherent. The order parameter equation of motion (for the downstream

ferromagnet for example) is therefore

dn2

dt= n2 ×

[− γ

Ms

∂E(n2)

∂n2

]+ g(θ)ωj n2 × (n1 × n2) − α n2 ×

dn2

dt. (6.38)

Here γ ≃ µB/~ denotes the gyromagnetic ratio, and Ms ≃ µB/a3 denotes the

133

saturated staggered moment density, where a ≃ 0.3 nm denotes the lattice constant

of Cr. The term containing ωj ≡ γ ~j /(2eaMs), with j the current density and

e the electron charge, describes the in-plane spin transfer torque. We neglect the

out-of-plane component because, as we have seen, it averages to a small value. The

last term in Eq. (6.38) describes the usual Gilbert damping, with a dimensionless

damping constant for which we take the typical value α = 0.1 [156]. The anisotropy

constants are given by K1 = 103 J m−3 and K2 = 10 J m−3 [156]. Since the

out-of-plane component of the spin torque competes with the anisotropy, whereas

the in-plane component competes with the damping term, it turns out that (even

in ferromagnets) the in-plane component of the spin torque is most important in

determining the critical current for current-driven switching, providing a second

justification for the neglect of this term. (Of course both terms can be calculated

using standard techniques for any specific atomic and magnetic arrangement.) A

linear stability analysis of Eq. (6.38) shows that for the optimal situation n1 = −x,

the fixed point n2 = x becomes unstable if j exceeds

jc =eαa

g(π)~(K1 + 8K2) ≃ 105A cm−2, (6.39)

where we have taken a value for for g(π) (g(π) ≃ 0.05) from our toy model nu-

merical calculations. In practice g(π) will depend on the specific materials com-

binations in the circuit. This critical current is smaller than the typical value for

current-switching of a ferromagnet primarily because the spin transfer torques act

cooperatively throughout the entire antiferromagnet and also because of the absence

of shape anisotropy in antiferromagnets. Using the model of Eq. (6.38) we also find

that, depending on the applied current, the staggered moment n2 can relax to stable

fixed points at n2 = ±y or completely reverse its direction.

134

6.6 Discussion and conclusions

The most obvious potential application of this effect is for purely antiferromagnetic

spin valve structures, like those illustrated schematically in Fig. 6.4. Say for example

that the circuit consists of perfectly epitaxial materials including commensurate an-

tiferromagnets with moment alteration in the current direction and facing moments

that are originally parallel. Because of the antiferromagnetic spin-torque effect, a

high current can make this arrangement unstable. If we assume that the one of

the two antiferromagnets is free and the other is pinned, a high current can cause

a transition to a configuration in which the facing moments are antiparallel. This

transition may be detected with the AGMR effect.

There are obvious challenges that make the scenario we have outlined less

easy to realize than in the ferromgnetic case, even taking away the body of knowledge

on ferromagnetic metal spintronics that has been built up over the past two decades.

One trivial difference is that shape anisotropy can no longer be used to pin one of

the ferromagnets. More challenging is the difficulty of realizing antiferromagnetic

material in which the magnetization orientation of the surface layer is fixed. This

aspect of antiferromagnet material physics figures prominently in efforts to increase

the strength of exchange bias effects in coupled ferromagnet/antiferromagnet sys-

tems and to achieve a quantitative understanding of the behavior of exchange bias.

Indeed, exchange bias might provide a useful tool for studying spin-torque effects in

antiferromagnets. At a ferromagnet-antiferromagnet interface the spin-orientation

of the layer of an antiferromagnet that is in contact with the ferromagnet is variable

because of surface roughness, domain structure in the antiferromagnet, and because

of the influence of the ferromagnet on the moment arrangement within the antifer-

romagnet [157]. As a corollary of our ideas, we expect that a strong current will also

alter the magnetic microstructure of the antiferromagnet in a hybrid heterostructure

containing one pinned ferromagnetic and one antiferromagnetic element. Using the

135

same methods as presented in the present paper, we have explicitly checked analyt-

ically and numerically that spin torques occur in such hybrid heterostructures, and

we will report in more detail about these findings in a future publication. Hence,

we expect that current-driven antiferromagnetic order parameter dynamics could

in this case be observed by comparing exchange bias properties before and after

application of a large current perpendicular to the interface.

The toy model calculations we have performed to date are all for disorder free

epitaxially matched antiferromagnetic and paramagnetic elements. We expect the

AGMR will be weakened by disorder, and in particular that the property that the

spin transfer torque acts throughout the volume of the antiferromagnet will apply

only in this idealized disorder free case. We do not however expect that the disorder

and electron-lattice scattering that is present at room and elevated temperatures

in real materials will completely destroy the effect, but instead limit spin-torques

to within one mean-free-path of the paramagnet-antiferromagnet interface. Using

the approximate[158] universal expression for the product of resistivity ρ and mean-

free-path ℓ

ℓρ ≈ 10−5µΩcm2 (6.40)

and taking ρ ≈ 10µΩcm for the resistivity of a typical antiferromagnetic metal

gives ℓ ≈ 10nm. Films with a thickness of 10nm will consist typically of 50 atomic

layers, close to the number chosen for our model calculations and comparable to the

film thicknesses used in ferromagnetic metal spintronics circuits. We do not expect

scattering to be a major obstacle to realizing this effect. Indeed, other phenomena

relying on phase-coherent interference such as oscillatory exchange coupling and

oscillatory GMR have been seen experimentally in ferromagnetic metallic multilayers

[159].

The materials combinations that will exhibit the effects we have in mind

most strongly depend on a large variety of considerations and could be identified by

136

a combination of experimental and theoretical work which follows in the footsteps

of the successful ferromagnetic metals materials research of recent years.

137

Chapter 7

Conclusions and Outlook

The End

And in the end

the love you take

is equal to the love you make

Lennon-McCartney

7.1 Conclusions

This thesis focuses on a class of effects in ferromagnetic metals that are interest-

ing from a basic science point of view and are likely to be exploited to qualita-

tively improve magnetic information storage technology based on magnetism. The

general topic is the ability of transport currents to influence magnetization statics

and dynamics in a ferromagnetic metal. These effects can sometimes be under-

stood in terms of action/reaction torques related to the approximate conservation

of total spin angular momentum, and for that reason are commonly referred to as

spin-transfer torques. The starting point was a thorough analysis of the effect of

currents on the spin wave spectrum of a ferromagnet (Chap. 3). We have worked

138

with a formalism that can handle spin momentum transfer effects in quite general

situations. We have shown that from a microscopic point of view spin transfer

effects are associated with the response of a magnetic hetero-structure to the non-

equilibrium exchange fields generated by spin-polarized current-carrying electrons

running across it. By performing a direct calculation of the non-equilibrium density

matrix those fields can be spatially resolved. This approach can be used to charac-

terize the spatial response of a variety of systems and geometries to a current. The

evaluation of the non-equilibrium density matrix is achieved, within this formalism

from the non-equilibrium real-time Green’s function formalism [153]. The method is

compatible with self-consistent and time-dependent mean-field descriptions of mag-

netically ordered states, and electron-electron interactions can be included at the

self-consistent level, using either Hartree or local density approximations. Based on

those ideas we study the effects that electronic current have on spin textures (spin

waves, domain walls, etc) with a characteristic length bigger than the electronic

length scales. The spatially extended spin transfer effect gives rise to an overall

drift of the structures in response to the current, the so-called spin wave Doppler

Effect.// As we have seen, spintronic effects in ferromagnetic metals can often be

qualitatively understood in terms of conservation of total spin. However, in an anti-

ferromagnetic metallic material, such as chromium, the magnetic order alternates by

definition on a microscopic scale such that there is no net magnetic moment. This

makes the application of spin conservation impossible. The fact that spintronics is

nevertheless possible with antiferromagnets is therefore somewhat unexpected. A

preliminary model study of spin dependent transport in antiferromagnets has nev-

ertheless revealed that i) a giant magnetoresistive effect is possible, and ii) it is

very likely that nanostructures containing antiferromagnetic elements will exhibit

current-induced magnetization dynamics (Chap. 6). In particular it turns out that,

contrary to the ferromagnetic case, the spin transfer torques act throughout the en-

139

tire free antiferromagnet to cooperatively switch it, a result of the special symmetries

of the antiferromagnetic state. This implies that the critical current for inducing

collective magnetization dynamics is likely to be lower in antiferromagnetic metal

nanostructures than in ferromagnetic spin valves. Hence, nano-circuits containing

antiferromagnetic elements are very attractive for applications to high-density in-

formation storage based on current-induced magnetization reversal. Another reason

for this is that antiferromagnets produce almost no magnetic fields, which makes

miniaturization possible without unwanted magnetic interactions between the an-

tiferromagnets. Moreover, the energy barriers separating the lowest energy states

of an antiferromagnet are typically lower than in ferromagnets, leading to further

reduction of the critical current. The predictions above are in agreement with recent

experiments in ferromagnetic/antiferromagnetic hybrids systems [24].

7.2 Outlook

In this work we have discussed several effects, all having in common the importance

of the electronic spin on them. Spin transfer effects have been discussed (Chap.

5)from a very general point of view that has already allowed for generalizations to

different areas like antiferromagnetic spintronics and quantum Hall bilayer physics.

There is already empirical evidence for the detection of spintronic effects in antifer-

romagnetic structures (Chap.6), along the lines predicted by work included in this

work [24]. The potential for those effects is great, as discussed in the main text. The

possibilities provided by domain wall motion effects (Chap. 4) in ferromagnetic sys-

tems, are under deep study at several laboratories around the world. These effects

have great potential in the implementation of magnetic random access memories.

The details of the advantages of such an implementation and of the actual architec-

tures that are being tried is beyond the scope of this work. This is an active field

of research, one still pregnant with possibilities. are many open issues, that might

140

turn out just to be loose ends that need to be tied down, or might unravel to reveal

entire new research fields. Just to name a few issues, that are in the short term “to

do” list related to the topics of this work:

• Analysis of the thermal effects on domain wall motion (stochastic dynamics of

the domain wall location). These effects are likely to be crucial at the critical

point where the domain wall is just un-pinned. Motion in this regime is likely

key to understanding the basic mechanism of the pinning.

• Spin transfer torques noise effects in spin valves. Quite related to the above.

These issues are of great relevance in technological applications.

• Detailed microscopic studies of the exchange bias effects in antiferromagnetic-

ferromagnetic hybrid systems and its dependence on currents.

• Effects of spin decoherence in antiferromagnetic dynamics. This is a key sub-

ject for the feasibility of implementing technologies based on the ideas de-

scribed in this work.

• etc.

141

Appendix A

Basic calculations

A.1 Introduction

During the present work we are going to use repeatedly some general simple manip-

ulations. In the present appendix we are going to show, and prove them, in order to

be able to use them in the text without breaking the continuity of the more relevant

and interesting arguments.

A.2 Pauli Spin Matrices

The Pauli spin matrices are:

σ1 = τx =

0 1

1 0

(A.1)

σ2 = τy =

0 −i

i 0

(A.2)

σ3 = τz =

1 0

0 −1

, (A.3)

142

together with the identity matrix they form a basis for the 2 by 2 matrices. Some

useful relations are:

ǫijk σiσj = iσk Lie Algebra (A.4)

σiσj + σjσi = 2 δij Anticonmutation (A.5)

from which the following expression can easily be derived:

tr(σi) = 0 (A.6)

tr(σiσj) = 2 δij (A.7)

tr(σiσjσk) = i ǫijk, (A.8)

σασβ = δαβ + iǫαβγσγ , (A.9)

σασβσγ = iǫαβγ + δαβσγ − δαγσβ + δβγσα, (A.10)

Another useful set of relations is:

σ · σ = 3, (A.11)

(σ · a) (σ · b) = a · b + ia × b · σ, (A.12)

(A.13)

We also can write, quite generally:

Γ(as + at · σ) = γs(as,at) + ~γt(as,at) · σ, (A.14)

where,

γs(as,at) =1

2(Γ(as + at) + Γ(as − at)) (A.15)

~γt(as,at) =at

2(Γ(as + at) − Γ(as − at)) , (A.16)

and in particular:

1

e− jΩ · ~τ=

1

2

1

e− j+

1

e+ j

+

1

2Ω · ~τ

1

e− j− 1

e+ j

(A.17)

143

A.3 Discrete Green’s functions

Some basic matrix manipulations

Inverse of a 2 × 2 Blocks Matrix The inverse of a 2 × 2 Block matrix can be

found simply by writing:

HLL HLR

HRL HRR

−1

=

GLL GLR

GRL GRR

(A.18)

which by direct multiplication reduces to:

HLLGLL +HLRGRL = 1

HRLGLL +HRRGRL = 0

HLLGLR +HLRGRR = 0

HRLGLR +HRRGRR = 1

The second and third equation in this list imply: GRL = −gRRHRLGLL and

GLR = −gLLHLRGRR, where we have defined gµν ≡ H−1µν . Back in the first

and fourth equations imply:

GLL = (HLL −HLR gRR HRL)−1

GRL = −gRRHRL (HLL −HLR gRR HRL)−1

GLR = −gLLHLR (HRR −HRL gLLHLR)−1

GRR = (HRR −HRL gLLHLR)−1 (A.19)

Inverse of a 3 × 3 Block Matrix In [75] we are faced with a system split in three.

144

Figure A.1: The tunnelling part of the Hamiltonian dresses the propagation on one side(blue) with events of tunnelling to the other side (red). That can be represented by a selfenergy that in this simple case equals the amplitude of two tunnelling events from one sideto the other and then back.

Two electrodes and a device connected to both.

HLL HLI 0

HIL HII HIR

0 HRI HRR

−1

=

GLL GLI GLR

GIL GLL GIR

GRL GRI GRR

(A.20)

again by direct multiplication we can calculate the values of the reduced in-

verses. We have, the first row multiplication:

HLLGLL +HLIGIL = 1

HLLGLI +HLIGII = 0

HLLGLR +HLIGIR = 0, (A.21)

the second row:

HILGLL +HIIGIL +HIRGRL = 0

HILGLI +HIIGII +HIRGRI = 1

HILGLR +HIIGIR +HIRGRR = 0 (A.22)

and the third row:

HRIGIL +HRRGRL = 0

145

HRIGII +HRRGRI = 0

HRIGIR +HRRGRR = 1. (A.23)

Now, from the second equations of the first and third rows we obtain the

relations: GLI = −gLLHLIGII and GRI = −gRRHRIGII . This two relations

can be used in the second equation of the second row to find GII :

GII = (HII −HILgLLHLI −HIRgRRHRI)−1 (A.24)

Similar manipulations lead us to:

GRL = gRRHRIgLLHLI

A.3.1 Recursive Green’s Function Algorithm

In most applications we are only interested in the diagonal elements or in a particular

row or column of the Green’s function. It then would be a waste of computational

resources to implement a full calculation of the Green’s function (e.g. by calling

a packed subroutine to evaluate the full inverse). For system sizes of relevance

this wasteful approach would not only cause a major delay in the development

of the calculation but it can quickly become a limitation1. The recursive Green’s

function algorithm [79] is just a clever way to calculate only a few elements of the

Green’s function by solving a system of Dyson’s equations. The basic idea is simple.

Consider the system depicted in fig.(A.2). The Green’s function of the system is

the matrix GL1i1σ1,L2i2σ2 , where L1,2 stands for the layer index (increasing along the

vertical in the figure), i1,2 indexes the horizontal coordinate on each layer, and σ1,2

is the band index. Now suppose we start building the system layer by layer. Once

we add the first layer its Green’s function can be written down (using the equation

(A.19)) as:

gL1i1σ1,1i2σ2

≡ gL1,1 = (E −H1 − t1,0ΣL t0,1)

−1 , (A.25)

1Simply to store the Green’s function of a discrete 2D system for a linear dimension of ∼ 1000sites, with two bands (eg. spin bands), would require ∼ 1000 Gb.

146

Figure A.2: This cartoon represents a generic system whose Green’s function is going to becalculated using the recursive Green’s function algorithm. Note that the system can haveany shape, with varying width and can even have holes.

if we continue adding layers we can define:

gLL,L =

(E −HL − tL,L−1g

LL−1,L−1 tL−1,L

)−1. (A.26)

Here, HL correspond to the block in the Hamiltonian involving only hopping within

the L layer. We continue iterating until we reach the other end of the system. In

the same manner, starting from the other end we can add layers going down.

gRN,N = (E −HN − tN,N+1ΣR tN+1,N )−1 , (A.27)

if we continue adding layers we can define:

gRL,L =

(E −HL − tL,L+1g

RL+1,L+1 tL+1,L

)−1. (A.28)

So far we haven’t calculated the real Green’s function of the system. However using

eq.(A.24) we see that with the set of gR’s and gL’s we can calculate the final Green’s

function:

GL,L =(E −HL − tL,L+1g

RL+1,L+1 tL+1,L − tL,L−1g

LL−1,L−1 tL−1,L

)−1(A.29)

147

Note the efficiency of the algorithm by just estimating the number of operations.

The naive approach of direct inversion would involve O(NbandsNlayersNcol)3 whereas

the recursive algorithm involves only O(3Nlayers(NbandsNcol)3). However, the di-

agonal elements just calculated are the only ones that enter in the description of

equilibrium systems (for example, the density of states is related to the trace of

the spectral function, the non-hermitian part of the retarded Green’s function. For

non-equilibrium problems we also need to evaluate some non-diagonal elements of

the green’s function. The transport properties are related to the columns that relate

the contacts with the rest of the system. We can build those elements by using the

non-diagonal expressions on eq. (A.19) to get:

GL,1 = gRL,L(−tL,L−1)GL−1,1 (A.30)

A.4 Manipulations in Keldysh Space

A.4.1 Keldysh Rotations

The non-equilibrium Green’s functions can be sorted in the form of a 2 × 2 matrix.

This 2-dimensional space is often referd to as Keldysh space. We have:

G =

Gt −G<

G> −Gt

(A.31)

We can benefit from the relations in Eq. (3.26) by performing the following manipulations[73]:

G → G ≡ τ3G (A.32)

G → G ≡ LGL†, (A.33)

where2,

L =1√2

(τ0 − iτ2

). (A.34)

2While the τ -matrices are numerically the Pauli matrices this rotations act only on Keldysh-space and leave the spin space unchanged

148

The new explicit form for the non-equilibrium Green’s functions is:

G =

GR GK

0 GA

(A.35)

Basically what we have achieved is simply to reduce the number of unknowns using

the linear relation between Green’s functions, eq.(3.25). With this it is easier to

solve the Dyson’s equation:

G = G0 + G0 Σ G, (A.36)

performing the transformation on each matrix, we get:

G = G0 + G0 Σ G . (A.37)

The new Green’s function are:

G =

GR GK

0 GA

(A.38)

G0 =

gR gK

0 gA

(A.39)

Σ =

ΣR ΣK

0 ΣA

(A.40)

where,

Σ = LΣτ3L†. (A.41)

A.4.2 Lehmann Spectral Representation

The expectation value of the equal-time commutation relation:

[ψ(x), ψ†(y)

]

±δ(x0 − y0) = δ(x− y), (A.42)

lead us in spectral representation to:

i

∫dω′

2π

(G>(ω,k) −G<(ω,k)

)= 1. (A.43)

149

As usual we can define the spectral function by:

A(1, 1′) = i(GR(1, 1′) −GA(1, 1′)) = i(G>(1, 1′) −G<(1, 1′)) (A.44)

In terms of the spectral density we obtain the sum rule:

∫dω′

2πA(ω,k) = 1 (A.45)

From the relation3:

Gr(1, 2) = Θ(1, 2)(G>(1, 2) −G<(1, 2)

)(A.46)

we can obtain:

Gr(ω,k) =

∫dω′

2π

A(ω′,k)

ω − ω′ + iε, (A.47)

analytical in the upper half-plane of ω. In the same way we have:

Ga(ω,k) =

∫dω′

2π

A(ω′,k)

ω − ω′ − iε, (A.48)

analytical in the lower half-plane of ω. In similar fashion we can derive the Lehmann

representation of the Feynmann causal operators:

GF (ω,k) = i

∫dω′

2π

G>(ω′,k)

ω − ω′ + iε− G<(ω′,k)

ω − ω′ − iε

(A.49)

GF (ω,k) = i

∫dω′

2π

G>(ω′,k)

ω − ω′ − iε− G<(ω′,k)

ω − ω′ + iε

(A.50)

The above relations can be summarized by defining the functions G1 and G2 in the

complex plane:

G1(z,k) = i

∫dω′

2π

G>(ω′,k)

z − ω′(A.51)

G2(z,k) = i

∫dω′

2π

G<(ω′,k)

z − ω′. (A.52)

3The Θ-function used here is defined, in terms of the usual θ-function, as Θ(1, 2) ≡ θ(t1 − t2).

150

The Lehmann representation becomes:

Gr(ω,k) = G1(ω + iε,k) − G2(ω + iε,k) (A.53)

Ga(ω,k) = G1(ω − iε,k) − G2(ω − iε,k) (A.54)

GF (ω,k) = G1(ω + iε,k) − G2(ω − iε,k) (A.55)

GF (ω,k) = G2(ω + iε,k) − G1(ω − iε,k) (A.56)

G>(ω,k) = G1(ω + iε,k) − G1(ω − iε,k) (A.57)

G<(ω,k) = G2(ω + iε,k) − G2(ω − iε,k) (A.58)

151

Appendix B

Spin Transfer torques in

piece-wise constant

ferromagnets

B.1 Introduction

In this appendix we present a brief calculation of the spin-torques exerted on a fer-

romagnet due to an incoming spin current. In section (B.2) we present an explicit

form for the spin conservation law, in a system with an exchange energy and arbi-

trary scalar potential. The precession of the spin density around the exchange field

is manifested in this law as a source that modifies the usual conservation law. In

section (B.3) we illustrate how a ferromagnetic slab with constant magnetization

acts as spin filter, polarizing a, spin unpolarized, incoming current in the direction

of the exchange field. Finally, section (B.4) is used to show the action of the slab

over an incoming current, originally spin polarized along a direction different from

the exchange field in the slab. It is shown that under that circumstances a net spin

torque is exerted over the slab. The direction of this spin torque is shown to be in

152

agreement with the expected behavior.

B.2 Spin current conservation

In this section we are going to derive a spin current conservation law, for a system

described by a Hamiltonian of the for:

H =p2

2m+

1

2~∆ · ~τ . (B.1)

The wave function Ψ solution of the equation:

i∂Ψ

∂t= HΨ, (B.2)

defines the local average values of the spin 〈~s(~r)〉 = Ψ†(~r) ~s Ψ(~r), with ~s = 12~τ . We

can evaluate the time derivative of the local average spin,

d〈~s(~r)〉dt

=∂Ψ†(~r)

dt~s Ψ(~r) + Ψ†(~r) ~s

∂Ψ(~r)

dt(B.3)

using the Hamiltonian we get the spin conservation equation,

d〈sj(~r)〉dt

= ∇iJij +

1

2

[~∆ × 〈~s(~r)〉

]j

(B.4)

where the spin current is defined by,

Jij =

1

2mIm(Ψ†(~r)τj∇iΨ(~r)

), (B.5)

This equation shows that the spin dynamics has mainly two effects, one is the natural

convective flow of the spin, represented by the conservative term, and the other is

the expected precession around the order parameter field ~∆. On the other hand,

if we look for the effects of this precession on the order parameter we see that the

reaction torque must be locally equal to Γ = −~∆× 〈~s(~r)〉 and the total torque over

the volume of the sample must be equal to Γtot = −~∆×〈~s〉, ~s being the total average

spin. Simple integration gives, in the stationary regime,

Γtotj =

∮dSiJ

ij , (B.6)

153

where the integration runs over the entire surface of the system. The torque is,

then, equal to the difference of the outgoing and incoming spin currents. This result

[8, 91] is valid whenever the system is described by a Hamiltonian with the features

of the one just described. The presence of a spin-orbit interaction term would spoil

the conservation equation in a way to be described later in these notes.

B.3 Spin filter effect

A basic element implicit in the discussion of the spin transfer effect is the fact

that a current passing through a single domain ferromagnet will evolve into a spin

current with polarization along the magnetization of the ferromagnet. The following

discussion is a short digression about that idea. The system under consideration is

made of a normal metal-ferromagnet-normal metal sandwich.

The normal metals are described simple as a free electron gas and the ferro-

magnet is treated in mean field theory. The only dimension of interest is, of course,

the width of the ferromagnetic layer L. The Hamiltonian of the normal metals is (i

refers to the different layers):

Hi =p2

2m⊗ 1 (B.7)

and the Hamiltonian for the ferromagnet:

HF =p2

2m⊗ 1 +

1

2~∆ · ~τ (B.8)

We solve Schrodinger equation for stationary states,

HΨ = EΨ, (B.9)

Choosing the axis of the magnetization ~∆ as the quantization axis for the spin

operators, we can write the eigenfunctions in each part of the system as:

Ψi = r+i

1

0

eikx+ r−i

0

1

eikx+ l+i

1

0

e−ikx+ l−i

0

1

e−ikx (B.10)

154

for the normal metals, and

ΨF = r+F

1

0

eik+x + r−F

0

1

eik−x + l+F

1

0

e−ik+x + l−F

0

1

e−ik−x

(B.11)

for the ferromagnet. The obvious notation is made that l states correspond to

left movers and respectively r states to right movers. The upper index ± refers

to the spin projection. In the above equations we write k =√

2mE, and k± =√2m(E ± |~∆|). The different wave vectors k± give different modulations for the

up/down wave functions accounting for the precession of the average spin. The

different amplitudes are related by the boundary conditions at the ends of the fer-

romagnet, demanding continuity of both Ψ and ∇Ψ, and the boundary conditions

at distances far away from it. The later ones are given by the following picture: a

given spin current is incoming from the left. This left the values of r+0 and r−0 to be

independent parameters and forces the relation l+1 = l−1 = 0. The remaining ampli-

tudes are determined by the boundary conditions. The condition Ψ(0+) = Ψ(0−)

implies: r+0 + l+0

r−0 + l−0

=

r+F + l+F

r−F + l−F

(B.12)

The condition ∇Ψ(0+) = ∇Ψ(0−) reduces to: kr+0 − kl+0

kr−0 − kl−0

=

k+r

+F − k+l

+F

k−r−F − k−l

−F

(B.13)

Now at the other end, Ψ(L−) = Ψ(L+) r+1 e

ikL

r−1 eikL

=

r+F e

ik+L + l+F e−ik+L

r−F eik−L + l−F e

−ik−L

, (B.14)

and finally, ∇Ψ(L+) = ∇Ψ(L−) kr+1 e

ikL

kr−1 eikL

=

k+r

+F e

ik+L − k+l+F e

−ik+L

k−r−F e

ik−L − k−l−F e

−ik−L

, (B.15)

155

The set of 8 equations above can be solved for the 8 unknowns: l±0 , l±F , r±F and r±1 in

terms of r±0 . The system can be written in terms of ~X = (l+0 , l−0 , r

+F , r

−F , l

+F , l

−F , r

+1 , r

−1 )

and ~S = (r+0 , r−0 , r

+0 , r

−0 , 0, 0, 0, 0) as:

Γ ~X = ~S −→ ~X = Γ−1~S (B.16)

where Γ is given by:

Γ =

−1 0 1 0 1 0 0 0

0 −1 0 1 0 1 0 0

1 0 q+ 0 −q+ 0 0 0

0 1 0 q− 0 −q− 0 0

0 0 Q1+ 0 Q2

+ 0 −1 0

0 0 0 Q1− 0 Q2

− 0 −1

0 0 q+Q1+ 0 −q+Q2

+ 0 −1 0

0 0 0 q−Q1− 0 −q−Q2

− 0 −1

, (B.17)

where we introduce the convenient notation q± = k±/k, Q1± = ei(k±−k)L, and

Q2± = e−i(k±+k)L.

Inverting the matrix Γ and solving the linear system we obtain:

l±0 =(−1 + q2±)

den(±)(Q1

± −Q2±)r±0 (B.18)

r±F = 2(1 + q±)

den(±)Q2

±r±0 (B.19)

l±F = 2(−1 + q±)

den(±)Q1

±r±0 (B.20)

r±1 = 4q±Q

1±Q

2±

den(±)r±0 (B.21)

where we have introduced the notation: den(±) = −(−1 + q±)2Q1± + (1 + q±)2Q2

±.

The equations above provide a complete solution for the problem of scattering of a

spin polarized current by a ferromagnetic obstacle. To understand the problem of

156

spin filter we need to focus on the spin current of the spin current at the right of

the magnet. The spin current along the x-axis is:

~J =k

2m

Re(r+1 (r−1 )†)

Im(r+1 (r−1 )†)

|r+1 |2 − |r−1 |2

(B.22)

Using the above expressions we can evaluate the spin current, first let’s cal-

culate r+1 (r−1 )†.

r+1 (r−1 )† = 16q+q−den(+)†den(−)

|den(+)|2|den(−)|2 (r+0 )†r−0 (B.23)

Now,

|den(±)|2 = (1 + q±)4 − 2(1 + q±)2(1 − q±)2 cos(2k±L) + (1 − q±)4, (B.24)

and defining K±,± = (1 ± q+)2(1 ± q−)2

den(+)†den(−) = ei(k+−k−)LK++−ei(k++k−)LK+−−e−i(k++k−)LK−++e−i(k+−k−)LK−−

(B.25)

the oscillating behavior of those quantities led us to the conclusion that the collective

effect of all the electrons participating in the spin transport, all with different ener-

gies in a window between EF and EF ±Vbias, will average out the components of the

spin current perpendicular to the collective magnetization. Along the magnetization

however the outgoing spin current is:

Jz =k

2m

(16q2+

|den(+)|2 |r+0 |2 −

16q2−|den(−)|2 |r

−0 |2)

(B.26)

a term that clearly survives the averaging process. We should note that the need to

average over a energy window is only a consequence of the oversimplification made

by considering a single channel problem. In a multichannel system the average is

performed automatically by the simultaneous superposition of the different channels’

contributions (as illustrated in figure B.1). Then a spin current polarized along any

157

Figure B.1: The top figures represent, by the use of real space “trajectories” of electrons,two different dynamical behaviors corresponding to different channels. The bottom figuresdisplay the spin dynamics associated with those different channels. This different “preces-sion rates” lead to a cancelation when summing over a great number of channels. This“averaging” processes takes place over a width proportional to λsc = π/(kF

↑ − kF

↓ ).

axes will end up polarized alon the axis of the collective magnetization. In the

incoming current is nor spin polarized (i.e. it is best described by a density matrix

proportional to 1 in spin space), it is easy to show that the density matrix of the

outgoing current represent a polarized one. So we have prove the spin filter effect in

the sense that a ferromagnet polarized an unpolarized current, and in the sense that

it reorient the polarization of a current to make it polarized along the magnetization

axis.

B.4 Spin transfer

To calculate the torque exerted by the electrons participating in the transport on the

collective magnetization we well may use the conservation law for spins. However

since our ultimate goal is to calculate the effect of the spin-orbit interaction on

158

the efficiency of the spin transfer, and in the case of spin orbit there is no such a

conservation law, we are going to calculate the torque directly from the expression

~Γtot = −~∆ × 〈~s〉 where the average means a spatial average over the ferromagnetic

system, again the average values of the are given by the solution of the transmission

problem. Inside the ferromagnet the wave function is:

Ψ(x) =

r+F e

ik+x + l+F e−ik+l

r−F eik−x + l−F e

−ik−l

=

Ψ1(x)

Ψ2(x)

(B.27)

and the local spin average is as usual 12Ψ†(x)~τΨ(x) and in terms of the components

we have:

〈~s(x)〉 =

Ψ†1Ψ2 + Ψ1Ψ

†2

i(Ψ†1Ψ2 − Ψ1Ψ

†2)

Ψ†1Ψ1 − Ψ2Ψ

†2

(B.28)

let focus on the term Ψ†1Ψ2 whose real and imaginary components give us the average

spin along the axes perpendicular to the magnetization.

Ψ†1Ψ2 = (r+F e

ik+x + l+F e−ik+x)† ∗ (r−F e

ik−x + l−F e−ik−x) (B.29)

= (r+F )†r−F e−i(k+−k−)x + (r+F )†l−F e

−i(k++k−)x + (l+F )†r−F ei(k++k−)x + (l+F )†l−F e

i(k+−k−)x(B.30)

Using the integral: ∫ L

0eiaxdx =

eiaL − 1

ia(B.31)

we can calculate the sum over space of Ψ†1Ψ2:

〈Ψ†1Ψ2〉 = (r+F )†r−F

e−iδL − 1

−iδ+(r+F )†l−F

e−i∆L − 1

−i∆+(l+F )†r−F

ei∆L − 1

i∆+(l+F )†l−F

eiδL − 1

iδ(B.32)

where we have introduced the symbols δ = k+ − k− and ∆ = k+ + k−. Now, we

have:

(r+F )†r−F = 4(1 + q+)(1 + q−)

|den(+)|2|den(−)|2 eiδL(e−iδLK++ − e−i∆LK+− − ei∆LK−+ + eiδLK−−

)(r+0 )†r−0

159

(r+F )†l−F = 4(1 + q+)(−1 + q−)

|den(+)|2|den(−)|2 ei∆L

(e−iδLK++ − e−i∆LK+− − ei∆LK−+ + eiδLK−−

)(r+0 )†r−0

(l+F )†r−F = 4(−1 + q+)(1 + q−)

|den(+)|2|den(−)|2 e−i∆L

(e−iδLK++ − e−i∆LK+− − ei∆LK−+ + eiδLK−−

)(r+0 )†r−0

(l+F )†l−F = 4(−1 + q+)(−1 + q−)

|den(+)|2|den(−)|2 e−iδL

(e−iδLK++ − e−i∆LK+− − ei∆LK−+ + eiδLK−−

)(r+0 )†r−0

Again we use the fact that the total effect correspond to a sum over a window of

energies and this will average out all the oscillating terms. The only survivors of

this average process are the terms without exponential terms:

〈Ψ†1Ψ2〉 = −4

(1 + q+)(1 + q−)

|den(+)|2|den(−)|2K++(r+0 )†r−0−iδ

+ 4(1 + q+)(−1 + q−)

|den(+)|2|den(−)|2K+−(r+0 )†r−0−i∆

+

+ 4(−1 + q+)(1 + q−)

|den(+)|2|den(−)|2K−+(r+0 )†r−0

i∆− 4

(−1 + q+)(−1 + q−)

|den(+)|2|den(−)|2 K−−(r+0 )†r−0

iδ(B.33)

The last equation can be simplified by introducing the symbol K±,± = (1±q+)3(1±q−)3 and so we get:

〈Ψ†1Ψ2〉 =

−4i

|den(+)|2|den(−)|2 (r+0 )†r−0

(K++ −K−−

δ+

K+− −K−+

∆

)(B.34)

= −2gi(r+0 )†r−0 (B.35)

For an incoming spin current polarized along the axis n = (θ, φ) the entering spinor

is: r+0

r−0

=

cos θ

2 eiφ

2

− sin θ2 e

−iφ2

(B.36)

then (r+0 )†r−0 = −12 sin θe−iφ, and in that way we get:

〈Ψ†1Ψ2〉 = gi sin θe−iφ (B.37)

Then the components of the average spin are:

〈sx〉 = g sin θ sinφ, 〈sy〉 = −g sin θ cosφ (B.38)

the above equation can be written in vectorial terms:

〈~s⊥〉 = g~∆ × n (B.39)

160

The last equation reproduces the expected Sloncewski term since ~Γtot = −g~∆ ×(~∆ × n).

We have proved then that in general the average spin will be given by an

form like:

〈~s〉 = g~∆ × n+ α~∆ (B.40)

We still need to find an expression for α, its contribution being zero for the present

case it could be of importance in the case with spin orbit interaction.

161

Appendix C

Some Scattering Matrix

Properties in magnetic systems

C.1 The general properties of the FM scattering matrix

It is clear that the scattering matrix for a ferromagnet must satisfy relations that are

similar to those of an antiferromagnet. We use the same notation that the used in

that case. We denote the asymptotic wave functions traveling to the right (x→ ∞)

and to the left (x→ −∞) by

Ψ−∞(x) = |−∞R〉 eikx + |−∞L〉 e−ikx ; (C.1)

Ψ∞(x) = |∞R〉 eikx + |∞L〉 e−ikx , (C.2)

where |∞R〉 , · · · and |∞L〉 , · · · are the spinor coefficients of the right and left goers,

respectively. The scattering matrix expresses the outgoing spinors in terms of the

incoming spinors:

|−∞L〉

|∞R〉

= S

|−∞R〉

|∞L〉

with S in turn expressed in terms

of 2×2 transmission and reflection matrices S =

r t′

t r′

. We choose the direction

162

of the Zeeman field in the antiferromagent, n, to be the spin quantization axis.

Invariance under simultaneous rotation of n and quasiparticle spins allows us to

write each transmission and reflection matrix in the scattering matrix as a sum of

a triplet and a singlet parts

S = Ss + Stn · τ . (C.3)

Because the system is invariant under the space inversion symmetry operation, the

components of this transformed scattering wave functions must be related by the

same scattering matrix. This condition imposes the following symmetry constraint

on S:

S =

0 1

1 0

S

0 1

1 0

. (C.4)

This relation forces the elements to be related by r′ = r and t′ = t. With those

symmetry constraints we now write the conditions for unitary scattering SS† = 1

which are eight equations:

|rs|2 + |rt|2 + |ts|2 + |tt|2 = 1, (C.5)

(rtrs + rsrt) + (ttts + tstt) = 0, (C.6)

(rsts + tsrs) + (rttt + ttrt) = 0, (C.7)

(rtts + tsrt) + (rstt + ttrs) = 0, (C.8)

In view of equation (C.5) we can invoke the following parametrization:

rs = sin Θ cos Φ exp (iνrs) (C.9)

rt = sin Θ sin Φ exp (iνrt) (C.10)

ts = cos Θ cos Φ exp (iνts) (C.11)

tt = cos Θ sin Φ exp (iνtt) (C.12)

Then, the rest of equations (C.6,C.7,C.8) become:

sin 2Φ cos2 Θ(tan2 Θ cos (νrs − νrt) + cos (νts − νtt)

)= 0, (C.13)

163

sin 2Θ cos2 Φ(tan2 Φ cos (νrt − νtt) + cos (νrs − νts)

)= 0, (C.14)

sin 2Φ sin 2Θ (cos (νrt − νts) + cos (νrs − νtt)) = 0, (C.15)

Those equations can be used to reduce the number of parameters in the scattering

matrix. Before that, we consider some limiting cases that arise from those equations:

• Pure singlet scattering Φ = 0, in this regime we have cos (νrs − νts) = 0,

• Pure triplet scattering Φ = π/2, in this regime we have cos (νrt − νtt) = 0,

• Pure transmission Θ = 0, in this regime we have cos (νts − νtt) = 0,

• Pure reflection Θ = π/2, in this regime we have cos (νrs − νrt) = 0,

from now on we assume that we are in a generic situation, away from those limiting

cases. From Eq. (C.15) we have:

νrt − νts = νrs − νtt + (2n+ 1)π (C.16)

Which back into Eq. (C.13) implies:

νts − νtt = (2m+ 1)π

2, (C.17)

If we write νts = ν + δ and νtt = ν − δ we have δ = (2m+ 1)π4 . On the other hand,

we have:

νrs − νrt = −(2m+ 1)π

2− (2n+ 1)π (C.18)

Then if we write νrs = η+ ǫ and νrt = η− ǫ we obtain ǫ = −(2m+ 1)π4 − (2n+ 1π

2 .

With all this we go back to Eq.C.14 and have:

η − ν = (2k + 1)π

2(C.19)

Collecting all these results we have a general parametrization for the scattering

phases in terms of a single phase and some integers:

exp (iνts) = exp (iΣ) (−1)k , (C.20)

164

exp (iνtt) = −i exp (iΣ) (−1)k+m , (C.21)

exp (iνrs) = −i exp (iΣ) (−1)m−n , (C.22)

exp (iνrt) = − exp (iΣ) (−1)n , (C.23)

C.2 Composed Transmission of an AFM and FM hybrid

We have the following composition problem. To the left there is a ferromagnet with

scattering matrix:

SFM =

eiδ (rs + rt n1 · τ) ts + tt n1 · τ

ts + tt n1 · τ e−iδ (rs + rt n1 · τ)

. (C.24)

while in the antiferromagnet,

SAFM =

e−iδ (rs + rt n2 · τ) t′s

ts eiδ (rs − rt n2 · τ)

. (C.25)

The transmission amplitude for this system is given by:

T = Tr(t†1K

†t†2t2Kt1

). (C.26)

If we use the parametrization we get:

t†2t2 = cos2 ΘAFM, (C.27)

t1t†1 = cos2 ΘFM, (C.28)

Then, the transmission becomes:

T = cos2 ΘAFM cos2 ΘFMTr(K†K

). (C.29)

As in the AFM/AFM case we can evaluate the trace of the reflection kernel

squared directly. Its dependence on the relative orientation of the order parameters

165

of the upstream ferromagnet and downstream antiferromagnet is clear. For a single

channel we obtain the following expression:

T = cos2 ΘAFM cos2 ΘFM .Λs + Λp n · m

Γs + Γp n · m + Γd ( n · m)2(C.30)

The dependence on the angular MR is given by the specific relation between the

coefficients of rational function of n · m and the scattering matrix parameters. For

completeness they are described next:

Λs =5

2− 1

2(cos 2Θ1 + cos 2Θ2 − cos 2Θ1 cos 2Θ2) ; (C.31)

Λp = −4 cos χ R T (C.32)

where we have introduce the phase χ = 2δ+νAFM+νFM, the joint reflection proba-

bility (in an inchoherent process) R = sin Θ1 sin Θ2 and the joint singlet and triplet

weights S = cos Φ1 cos Φ2.T = sin Φ1 sin Φ2. The denominator is more cumbersome;

Γs = 1 + R(4RS

2 + R3γ2 − 4S(1 + γR2)

)cosχ+ 2Rγ cos 2χ; (C.33)

Γp = 4 R T[2R S − cosχ

(1 − γR2

)]; (C.34)

Γd = 4 R2

T2 (C.35)

where γ = 1 + sin2 Φ1.

C.3 Outline of a proof of the periodicity of the trans-

verse spin density

In this Appendix we proof that the out-of-plane spin density is constant and equal

in the left lead, spacer, and right lead of a heterostucture containing two antifer-

romagnets separated by a paramagnetic spacer. The proof that the out-of-plane

spin density is periodic in the antiferromagnets proceeds along the same lines, but

is much more involved.

166

The general manipulations are cumbersome when the two antiferromagnetic

layers are misaligned. However the polar representation introduced in Sec. 6.2 re-

duces most manipulations to standard trigonometry. We are interested in the spin

densities in the regions at the left, the center (in between the scatterers), and the

right, for a wave incoming from the left. We use the notation∣∣±∞R/L

⟩for the states

at ±∞, moving to the left and right respectively and∣∣0R/L

⟩for the states at the

center of the system. We need to find the combined scattering matrix of an antiferro-

magnetic element, a paramagnetic element, and a second antiferromagnetic element

that has been translated with respect to the first and rotated in spin-orientation.

We first note the following behavior of scattering matrices under translation by x0:

T(x0)S =

e2ikx0r t′

t e−2ikx0r′

. (C.36)

The spin-dependent scattering matrix S12 for two scatterers described by S1 = r1 t′1

t1 r′1

and S2 =

r2 t′2

t2 r′2

is

S12 =

r1 + t′1r2Kt1 t′1Kt′2

t2Kt1 r′2 + t2Kr′1t′2

. (C.37)

where we have defined the multiple reflection kernel K = (1− r′1r2)−1. Using this

composition rule, along with the translation property and the results explained in

Sec. 6.2 for the scattering matrix of a single spatially coherent antiferromagnet

inEq. (6.10) with the constraints in Eq. (6.11)], allows us to infer general properties

of spin dependent transport through two antiferromagnets.

For the situation of an incoming beam from the left we write all amplitudes in terms

of |−∞R〉:

|0R〉 = Kt1 |−∞R〉 ;

|0L〉 = r2Kt1 |−∞R〉 ;

167

|−∞L〉 =(r1 + t′1r2Kt1

)|−∞R〉 ;

|∞R〉 = t2Kt1 |−∞R〉 , (C.38)

which solves the scattering problem at all the positions in the system.

With the explicit form of the wave functions we evaluate the densities (and spin

densities) at any position in the system.

Sα−∞(x) = 〈−∞R|Sα| −∞R〉 + 〈−∞L|Sα| −∞L〉 +〈−∞R|Sα| −∞L〉 e−2ikx + 〈−∞L|Sα| −∞R〉 e2ikx

,

Sα0 (x) = 〈0R|Sα|0R〉 + 〈0L|Sα|0L〉 +〈0R|Sα|0L〉 e−2ikx + 〈0L|Sα|0R〉 e2ikx

,

Sα∞(x) = 〈∞R|Sα|∞R〉 + 〈∞L|Sα|∞L〉 +〈∞R|Sα|∞L〉 e−2ikx + 〈∞L|Sα|∞R〉 e2ikx

.

We split our result in spatially dependent and independent parts. First we focus on

the spatially dependent spin density in the center of the system. It is of the form:

〈0R|Sα|0L〉 e−2ikx + h.c.

=

⟨−∞R|t†1K†Sαr2Kt1| −∞R

⟩e−2ikx + h.c..

The expectation value becomes a trace when summed over all incoming channels,

while the fact that the transmissions are spin independent allows us to factor them

out of the trace. We find:

〈0R|Sα|0L〉 e−2ikx

∼ |t1|2

Tr(K†Sαr2K

)e−2ikx

.

The trace itself can be simplified:

Tr(K†Sαr2K

)= rs2Tr

(K†SαK

)+ rt2n

β2Tr

(K†SαSβK

).

We calculate explicitly the traces with the aid of Eq. (6.11). We evaluate

them projecting the expression along the perpendicular axis using n⊥ = n1×n2 and

168

find that

|Λ|2 Tr(K† (n⊥ · S)K

)= 4RS sin (χ) sin2 θ, (C.39)

and

|Λ|2 Tr(K† (n⊥ · S) (n1,2 · S) K

)= −4iRS sin (χ) sin2 θ, (C.40)

where we have introduced the denominator:

|Λ(θ,Θ,Φ)|2 = 1 +R4 + 4R2S2 + 4R(1 +R2)S cosχ

+ 4RT (2RS + (1 +R2) cosχ) cos θ

+ 4R2T 2 cos2 θ , (C.41)

with R = sin2 Θ, T = sin2 Φ and S = cos2 Φ characterizing the joint reflection

amplitudes and the joint triplet and singlet relative weights of the reflection of the

antiferromagnets, and χ =(2ν − δL

)the phase shift associated with the reflections.

Their identity up to a factor −i compensates the identity of the rs,t up to a factor i,

and their net contribution cancels. So there is no spatially dependent part. Hence

the out-of-plane spin density is constant in the spacer. 1.

Now, we focus on the constant parts of each expression.

Sα−∞ = 〈−∞R|Sα| −∞R〉 + 〈−∞L|Sα| −∞L〉 ;

Sα0 = 〈0R|Sα|0R〉 + 〈0L|Sα|0L〉 ;

Sα∞ = 〈∞R|Sα|∞R〉 + 〈∞L|Sα|∞L〉 ,

These expressions can be reduced to expressions involving only |−∞R〉. We obtain:

Sα−∞ =

⟨Sα +

(r†1 + t

†1K

†r†2t

′†1

)Sα(r1 + t′1r2Kt1

)⟩;

Sα0 =

⟨t†1K

†(Sα + r

†2S

αr2

)Kt1

⟩;

Sα∞ =

⟨t†1K

†t†2S

αt2Kt1

⟩.

1This equality between this two forms, seems odd, since apparently involve the equality of alinear and a bilinear form of the pauli matrices. However since the reflection kernel also has spintriplet terms there is no contradiction.

169

where the expectation value 〈·〉 = 〈−∞R| · |−∞R〉. Summing over the incoming

unpolarized current those expectation values become a trace:

Sα−∞ = Tr

((r†1 + t

†1K

†r†2t

′†1

)Sα(r1 + t′1r2Kt1

));

Sα0 = Tr

(t†1K

†(Sα + r

†2S

αr2

)Kt1

);

Sα∞ = Tr

(t†1K

†t†2S

αt2Kt1

).

We take the difference:

Sα0 − Sα

∞ = Tr(t†1K

†(Sα + r

†2S

αr2

)Kt1

)− Tr

(t†1K

†t†2S

αt2Kt1

), (C.42)

which can be written as:

Sα0 − Sα

∞ = |t1|2Tr(K†(Sα + r

†2S

αr2 − t†2S

αt2

)K). (C.43)

This is easily proven to cancel when projected on the out-of-plane direction, by

making use of the relations in Eqs. (C.39) and (C.40).

170

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Vita

Alvaro S. Nunez was born in Santiago de Chile, Chile in May 11th 1976. He took

the Bs. Sc. Physics degree from the Facultad de Ciencias Fısicas y Matematicas

de la Universidad de Chile. He is married to Viviana Jeria, and has a 2 year old

daughter named Penelope Millaray Nunez-Jeria.

Permanent Address: UT at Austin, Physics Department,

1 University Station C1600,

Austin, TX 78712

This dissertation was typeset with LATEX2εby the author.

186