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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
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