I would like to thanks Seagate Technology Inc. for .... Motion of a Rigid Domain Wall Driven by an External Mag- ..... applications, the first one led the revolution in hard-disk technologies of the ... capacity was 10 EB (1 EB 1018B ) worth of data. ...... spin current is equal to the total current and we recover the result of BJZ [101].
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 d´ıa con arcilla, con trabajo, con fuego, y aprisionada tienes la agilidad del aire, si no fuera porque eres una semana de a´mbar, si no fuera por que eres el momento amarillo en que el oto˜ no sube por las enredaderas y eres a´ un el pan que la luna fragante elabora paseando su harina por el cielo oh, bien amada, yo no te amar´ıa! En tu abrazo yo abrazo lo que existe, la arena, el tiempo, el a´rbol 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 peov
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 paciencia y todas las alegrias que me han dado, a mi peque˜ na familia: Viviana y Penelope, Fogata de amor y gu´ıa, Raz´ on 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 heterostructures, 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 modifications 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 antiferromagnets 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 nanostructures 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
27
Basic Considerations in Non-Equilibrium Field Theory. . . . ix
3.4
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
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
Current induced Domain wall dynamics . . . . . . . . . . . . . . . .
71
4.3.1
Introduction
71
4.3.2
Numerical Solution of the Landau-Lifshitz equation in the
4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . .
presence of a current
. . . . . . . . . . . . . . . . . . . . . .
4.3.3
Hamiltonian form of Landau-Lifshitz equation
4.3.4
Bloch Domain Wall
4.3.5
74
. . . . . . . .
77
. . . . . . . . . . . . . . . . . . . . . . .
83
Motion of a Rigid Domain Wall Driven by an External Magnetic 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
5.4
Microscopic Theory of Spin Transfer . . . . . . . . . . . . . . . . . . 5.3.1
Quasiparticle Spin Dynamics . . . . . . . . . . . . . . . . . . 102
5.3.2
Collective Magnetization Dynamics: . . . . . . . . . . . . . . 102
5.3.3
Spin-Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Toy-Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4.1
5.5
99
Effect of spin-orbit interaction
. . . . . . . . . . . . . . . . . 106
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
6.4
Elementary Local Spin Model . . . . . . . . . . . . . . . . . . 121
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 ferromagnets
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.
2.1
Schematic band diagrams the spin transport in a parallel (a) and antiparallel
6
(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.
2.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
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. . . . . . . .
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. .
4.1
36
44
Cartoon of the torques driving the magnetization dynamics, (a) the ~ eff × usual ferromagnetic precession is driven by a torque of the form H
~ and (b) a dissipation torque driving the magnetization toward its M, 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 dotted line illustrates momentum transfer: the electron scatters off the domain wall and gives the domain wall a momentum kick. 4.4
. . . . .
72
Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 director vector. Here Q is infinity (no in-plane anisotropy) and the domain wall responds as a straight line with velocity X˙ =
J . 1+α2
As the domain moves the components in the hard plane precess. . .
xiv
78
4.5
Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 ). The third dimension 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 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.
4.6
. . . . . .
79
Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 ). The third dimension is the dimensionless z-component of the magnetization vector. (rest of panels) A 2D representation of the different coordinates of the director vector. Q is even smaller approaching the critical situation and the wiggles become stronger. . . . . . . . . . . . . . . . . . . . . . .
xv
80
4.7
Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 ). The third dimension is the dimensionless z-component of the magnetization vector. (rest of panels) A 2D representation of the different coordinates of the director vector. Q is small enough as to stop the motion of the domain wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8
The definition of the polar angles used as independent fields in the theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
81
82
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
˙ for the domain wall as a function of Q and 4.10 Average velocity hXi the driving field h. The color code represents the relative value of ˙ hXi/(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 ˙ as function of the anisotropy param4.11 Left panel: average velocity hXi 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: hXi/J cr (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 p 2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˙ = J 2 − Jcr hXi xvi
89
˙ as function of the anisotropy param4.12 Left panel: average velocity hXi 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: hXi/J cr (Q) as a function of J/Jcr (Q) for several values of Q. Above the criti-
5.1 5.2
cal current all the curves collapse into the dashed line described by p 2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˙ = J 2 − Jcr hXi Spin transfer mechanism . . . . . . . . . . . . . . . . . . . . . . . . . .
91 97
(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.
5.7
Transport spin density per unit current in the case without spin-orbit inter-
. . . . . . . . . . . . . 108
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.
6.3
. . . . . . . . . . . . . . . . . . . . 122
(a) Scattering process for right-going incoming electrons. (b) same for leftgoers. 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 ˆ i on opposite sides of the paramagthe magnetization orientations Ω 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, M g(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 represented 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 macroscopic number of constituents. The physical behavior of a system with many constituents 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 reality 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 microscopic 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 descriptions 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 behavior 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 preferred “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, superconductors, etc. The whole system of many particles can be accurately described by an account of the order parameter and quasi-particle excitations around the Fermisurface. 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 defined only in the immediate vicinity of the fermi-surface. Though they look like electronic excitations, they are, actually, also collective excitations involving the excitation 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 thousands 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 manifestations are apparent to every schoolchild yet its origins are rooted deep in quantum mechanics and relativity; its applications 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 interaction” [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 interaction 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 interactions 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 interaction 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 resistance that can be attained by manipulating the relative orientation of magnets in heterostructures of nanoscale dimension [7]. Spin Transfer Torques (ST) are associated 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 interplay 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 general 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 inter1
This 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 shipped capacity was 10 EB (1 EB 1018 B ) 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 manipulated 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 introduction 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
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present some formal results concerning the formalisms appropriate to describe theoretically 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 relation, (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 antiferromagnetic elements. The analysis is carried on by means of generic symmetry arguments and also by direct calculations using the formalism described in Chap. ( 3).
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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 example) resulting from the sensitivity of magnetization orientation to external fields, combined with the strong magnetization-orientation dependence of the spin potentials 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 fundamental effects in semiconductor systems stands as a major success of late 20th century
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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 antiferromagnets, 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 setting [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, 1
See http://www.almaden.ibm.com/st/magnetism/ms/
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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 represent tunneling 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≡
N↑ (EF ) − N↓ (EF ) J↑ (EF ) − J↓ (EF ) = . J↑ (EF ) + J↓ (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/Al2 O3 /F e), 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 Julli`ere[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 = 4πe2 NL NR |T|2 .
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(2.2)
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 Julli`ere’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↑↓ 2P1 P2 = . R↑↑ 1 − P1 P2
(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 interdiffusion 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 2 It must be emphasized that the argument leading to this figures is incomplete. The TMR dependence on the polarizations of the ferromagnets is complicated by several effects, most deviations come from the spin dependence of the tunneling rates, which can be expected since in the vicinity of the tunnel barrier the states in the metals are modified. This modification in the states also induced a non-trivial change in the polarization right at the interface with the subsequent change in the transport properties. Similar arguments indicate that the TMR must depend on the tunnel barrier width. Nevertheless experiments with F e/Al2 O3 /F e tunneling junctions [34] have reported TMR ranging from 30% (at 4.2 K) to 18% (at room temperature). TMR’s as high as 50% at room temperature have been found by several groups.
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Figure 2.2: Types of magnetoresistance. (a) AMR results from bulk spin-polarized scattering within a ferromagnetic metal. (b) Colossal magnetoresistance (CMR) results from interactions predominantly between adjacent atoms in certain crystalline perovskites. (c) GMR results from interfacial spin-polarized scattering between ferromagnets separated by conducting spacers in a heterogeneous magnetic material. (d) Tunneling magnetoresistance (TMR) in magnetic tunnel junctions results from spin filtering as spin-polarized electrons tunnel across an insulating barrier from one ferromagnet to another. (e) Anomalous MR from domain wall effects has been observed in single-crystal ferromagnetic Fe whiskers and patterned magnetic wires. (f) Ballistic MR (BMR) is another type of domain wall effect in the limit of very narrow constrictions where the conductance may be quantized. Figure taken 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 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. 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 semiconductors. 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-
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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 spintronics 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, 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).
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 injection 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 remarkable 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 sensibility 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.
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Figure 2.5: Some remarkable spintronics effects that have been found in DMS. (a) Shows how magnetism can be turned on or off [46], in a field effect device by just adjusting the gate voltage. Left Panel: When holes are depleted from the (Ga:Mn)As layer it becomes paramagnetic. Right Panel: When the gate is adjusted to increment the hole population the system becomes ferromagnetic. (b) The use of a DMS in a LED lead the the generation of polarized light [47].
Diluted Magnetic Semiconductors Diluted Magnetic Semiconductors are ternary alloys created by doping suitable nonmagnetic 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 paramagnetic 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 metallic 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-
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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 ”normal” 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 contribute 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 extrinsic 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 currents propagating along the transverse directions. A remarkable feature of this phe-
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nomenon is that such a transverse response does not require breaking time-reversal symmetry by means of a magnetic field or ferromagnetism. These predictions motivated 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 impact 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 concerning 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.
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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 canonical 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 quantum 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 mesoscopic 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 independent 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 to independent reservoirs, with different chemical potentials, µi . Each reservoir attempts to restore the system to equilibrium by interchanging electrons. A current flows through the system 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. ~ j , where This means that the current density at that element is given by ~ji = σij E 22
~ is the net electric field at the volume element. To calculate the net current flow E 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 LandauerButikker 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 1 As the system size increases the solution of the quantum mechanical problem becomes more and more 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 of the “corrections” to the semiclassical picture becoming small. The difference is qualitative rather than quantitative: the very concept of conductivity is ill defined for small system. This qualitative difference is the main content conveyed by the word “emergent”.
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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=
X
eT(Ek )vk ,
(3.1)
k∈T
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): 2e I= ~
Z
µL
dET(E).
(3.2)
µR
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 opposes 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 quantized 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 associated 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 processes? 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. Assuming 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 2 P Ii and Vi , respectively, the relation Ii = eh 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 properties 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 formalism. This formalism was developed during the 60’s by several people, among them Schwinger[60], Keldysh[61], and Kadanoff and Baym [62]. Those approaches differed in their mathematical methods but they convey, essentially, the same physical content. The Kadanoff-Baym formalism established differential equations for various 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 semiclassical 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 2
This is called the Wigner representation. In the density-matrix (or Green’s function) we can always perform the change of variables: ˆ (x1 , x2 ) ≡ ρ ˆ (X, x), ρ
(3.4)
ˆ (X, k) ∼ X = (x + x )/2 and x = x − x . For homogenous systems it is clear that ρ Rwhere ˆ (X, x) and we can build the semiclassical approximation regarding the system as a set of dx e ρ 1
2
1
2
ikx
wave-packets with position X and momentum k.
26
relevant variables to be describable by wave-packets, it is particularly useful in handling 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 parameter, 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 Hamiltonian. 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 =
1 β.
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: hOi = Tr {ρ(H)O} .
(3.7)
On the other hand if a non-equilibrium situation is induced by applying a timedependent potential: H(t) = H + δH(t),
(3.8)
we need to explicitly calculate the time evolution of the operators: hO(t)i = Tr ρ(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 ). In the expression above U corresponds to the time evolution operator. Z t H(t) . U(t, t0 ) = T exp −i
(3.10)
(3.11)
t0
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 noninteracting 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 = −∞)i = S(−∞, ∞) |Ψ(t = ∞)i .
(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 i ≡ |Ψ(t = −∞)i, are
simplified into purely time ordered structures as follows3 : hΨ(t = −∞)|T O|Ψ(t = −∞)i ≡
hΨ(t = −∞)|T O|Ψ(t = ∞)i . hΨ0 |S(−∞, ∞)|Ψ0 i
(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 timeordered 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 = hT c(1)c† (2)i. 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. 3 The denominator of the following expression, known as the vacuum polarization bubble, plays a very important role in the formalism, it can be shown to be responsible for the cancelation of the unconnected diagrams.
29
Figure 3.2: Contour Ct is a closed-time contour [60]. The contour has two branches, one going from the past to the future and another in the opposite direction. The contour ordering operator reorders its arguments in such a way that operators acting at times previous in the 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 ). With this a simpler expression for O(t) is given by: Z O(t) = TCt exp −i dτ δH(τ ) O0 (t) ,
(3.14)
(3.15)
Ct
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: ¯ t)] = Z[J(x, t), J(x, where, SK
Z
¯ ¯ DΨ(x, t)D Ψ(x, t) exp iSK Ψ(x, t), Ψ(x, t) ,
(3.16)
∂ ∇2 ¯ dt dxΨ(x, t) i + − V(x) Ψ(x, t) ∂t 2m ZC∞ Z ¯ t) + Ψ(x, ¯ + dt dx Ψ(x, t)J(x, t)J(x, t) . (3.17)
¯ Ψ(x, t), Ψ(x, t) =
Z
Z
C∞
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 plane
that configures the equilibrium situation at t0 , another going from the past to the future and another in the opposite direction. The contour ordering operator reorder its arguments in 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: ¯ ′ , t′ )) G(1, 1′ ) = Tr ρ(t0 )TC∞ (Ψ(x, t)Ψ(x 1 δ2 = Z J, J¯ . i2 δJ¯(x, t)δJ(x′ , t′ ) ¯ 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 δ2n ¯ Z J, J (3.19) i2n δJ¯(1) · · · δJ¯(n)δJ(1′ ) · · · δJ(n′ ) ¯ J,J=0 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−− = hT Ψ1 Ψ†2 i 12 hΨ1 Ψ† i t1 > t2 2 = † ∓hΨ Ψ i t < t , 1 2 2 1
(3.20)
iG++ = hTe Ψ1 Ψ†2 i 12 ∓hΨ† Ψ1 i t1 > t2 2 = hΨ Ψ† i t < t , 1 2 1 2
(3.21)
= hΨ1 Ψ†2 i, iG−+ 12
(3.22)
iG+− = ∓hΨ†2 Ψ1 i. 12
(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+− GF G+ GT G< ≡ ≡ G −+ −− > G G G− GF¯ G GT¯ 32
(3.24)
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 functions 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: 1 −1 1 1 1 1 1 + 1 GA + 1 GK , G = GR 2 2 2 1 −1 −1 −1 1 1
or using the spinors in Keldysh space ξ =
1 1
and η =
1 −1
1 1 1 Gµν = GR ξµ η ν + GA η µ ξ ν + GK ξ µ ξ ν , 2 2 2 33
(3.27)
we can write: (3.28)
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: Z
C∞
†
dxdtJ(x, t)Ψ (x, t) ≡
Z
∞ −∞
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 different directions on integrations along the path. This separation allow us to write symbolically: δ2 Z Gµν = − ¯ δJµ δJν
(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) = η µ Jµ , 1 1 Jc (t) ≡ (J+ (t) + J− (t)) = ξ µ Jµ , 2 2
(3.31) (3.32)
and the source term becomes: Z
J∆ Ψ†c + Jc Ψ†∆ ,
(3.33)
with the corresponding fields: Ψ†∆ (t) ≡ Ψ†+ (t) − Ψ†− (t) = η µ Ψ†µ , 1 1 † Ψ†c (t) ≡ Ψ+ (t) + Ψ†− (t) = ξ µ Ψ†µ . 2 2 34
(3.34) (3.35)
Again, we can invert these relations, using: 1 Jµ = Ψ†c ξ µ + J∆ η µ , 2 1 Ψ†µ = Ψ†c ξ µ + Ψ†∆ η µ , 2
(3.36) (3.37)
with the corresponding changes in the derivatives: δ δJµ δ δJc δ δJ∆
1 δ δ ξµ + ηµ , 2 δJc δJ∆ δ = ξµ , δJµ 1 δ = . η 2 µ δJµ =
(3.38) (3.39) (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: δ2 Z GA = − ¯ , δJ∆ δJc δ2 Z , GR = − ¯ δJc δJ∆ δ2 Z GK = − ¯ , δJc δJc
3.3.3
(3.41) (3.42) (3.43)
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 expresses 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 series 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 double line without indexes will denote the matrix in Keldysh space. To first order we obtain: (1)
G
′
(1, 1 ) =
By doing the separation:
R
Z
dx2
CK dτ →
Z
(0)
CK
(0)
dτ2 GCK (1, 2)V (2)GCK (2, 1′ ).
R∞
−∞ dt −
R∞
−∞ dt
(3.44)
we can write the term in terms
of standard functions. The different combinations of positions of 1 and 1′ give rise to four corrections: Z (1) (1) (0) (0) G++ (1, 1′ ) G+− (1, 1′ ) G++ (1, 2) G+− (1, 2) = × dt2 (3.45) (1) (1) (0) (0) G−+ (1, 1′ ) G−− (1, 1′ ) G−+ (1, 2) G−− (1, 2) (0) (0) V (2) 0 G++ (2, 1′ ) G+− (2, 1′ ) × (0) (0) ′ ′ G−+ (2, 1 ) G−− (2, 1 ) 0 −V (2) or, in the more compact notation in Keldysh space:
ˆ (1) (1, 1′ ) = G ˆ (0) (1, 2)Vˆ (2)G ˆ (0) (2, 1′ ). G
(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: ¯ = −i log Z[J, ¯ W[J, J] J],
36
(3.47)
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) δW δJ¯(1)
¯ c (1), ≡ Ψ
(3.48)
≡ Ψc (1),
¯ or vice-versa, they can of course be regarded as functions of the sources J and J, the sources be regarded as functions of the fields. We perform the usual Legendre transformation into the vertex generating functional: Z ¯ ¯ ¯ c (1)J(1) , Γ[Ψc , Ψc ] = W[J, J] − d1 Ψc (1)J¯(1) + Ψ
(3.49)
with the consequence:
δΓ ¯ δΨ(1) δΓ δΨ(1)
= −J(1)
(3.50)
¯ = −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: Z δ2 W δ2 Γ δJ(1) ¯ c (3)δΨc (1) d3 = − δJ(2) . δJ¯(2)δJ(3) δΨ
(3.52)
Finally, identifying the first element inside the integral as the connected 1-body Green’s function, we conclude: Z G(1, 2)Γ(2, 3)d2 = δ(1 − 3), ¯ we obtain: and similarly (by taking the derivative with respect to J) Z Γ(1, 2)G(2, 3)d2 = δ(1 − 3), 37
(3.53)
(3.54)
both expressions are known as Dyson Field Equations. Γ is known as the 1-particle irreducible (1PI) vertex function, Γ(1, 2) ≡
δ2 Γ ¯ c (1)δΨc (2) . δΨ
(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 : Z
d3 Γ(1, 3)σ 3 G(3, 2) = σ 3 δ(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: Z
˜ 2) = σ1 δ(1 − 2). ˜ 3)σ 1 G(3, d3 Γ(1,
(3.57)
This is the same equation that could have been reached from performing a Keldyshrotation on both the Green’s function and the vertex. Note that the usual Keldyshspace representation of the generating functional also holds: 0 ΓA ˜ = . Γ ΓR ΓK 4
(3.58)
here the bold-face in Γ is a reference to its matrix character, and it not the vertex generating functional
38
with the inverse relation: Γµν =
1 R 1 1 Γ ξ µ η ν + ΓA η µ ξ ν + ΓK ξ µ ξ ν , 2 2 2
(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)
g¯t(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: < gt g (3.64) G0 = > g g¯t Assuming that Γ(0) satisfies the Dyson equation with G0 we obtain: < γ γ t . Γ(0) = γ > γt¯
(3.65)
It becomes simple to treat this equation in the “physical time” representation: 0 gA ˜0 = ; G (3.66) gR gK A 0 γ ˜ (0) = , Γ (3.67) R K γ γ 39
from which, using eq.(3.57), we obtain5 γA = γR =
gA gR
−1
−1
,
(3.68)
,
(3.69)
γ K = −γ R gK γ A ≡ 0, directly. Going back to the single time representation: 1 1 R + γA R − γA < γ − γ γ γ t 2 = 2 Γ(0) = 1 1 > R A R + γA γ γt¯ γ − γ − γ 2 2 0 (ω − H0 ) = 0 − (ω − H0 )
(3.70)
(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 Keldyshspace 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: ΣT Σ< , Σ= > Σ ΣT¯
(3.73)
The product γ R gK γ A can be related to the difference between the inverse of the advanced and retarded greens functions, i.e. ∝ the infinitesimal η. 5
40
whose elements satisfy the usual constraint on Keldysh-space: ΣT + ΣT¯ = Σ< + Σ> , leading to the same physical representation: 0 ΣA ˜ = , Σ ΣR ΣK
Equation (3.57) reads: −1 0 gA − ΣA 0 GA −1 σ1 = σ1, R K R K R G G −Σ −Σ g
(3.74)
(3.75)
(3.76)
whose components imply that:
GA = GR =
−1 −1 A A g −Σ , −1 −1 R R g −Σ ,
GK = GR ΣK GA .
The, by now usual, rotation in Keldysh space (eq. (3.28)) leads us to: 1 R G< = G − GA − GR ΣK GA . 2
(3.77) (3.78) (3.79)
(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 + ΣR GR , GA = gA 1 + ΣA GA , G≶ = 1 + GR ΣR g≶ 1 + GA ΣA + GR Σ≶GA , Gt = 1 + GR ΣR gt 1 + GA ΣA + GR Σ¯t GA , G¯t = 1 + GR ΣR g¯t 1 + GA ΣA + GR Σt GA , 41
(3.81) (3.82) (3.83) (3.84) (3.85)
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 physical 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 transmission probability and the current bias characteristic using the ideas described in the section above. Some physical considerations are in order to help us better understand 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 decoupled 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 current, 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 distribution 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 potential difference 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: X X X H = tri rj r†ri ,σ rrj ,σ + tli lj l†li ,σ llj ,σ + tij d†i,σ dj,σ hri ,rj i,σ
+
X
ri ∈∂R
¯ j∈∂ R,σ
e tri j r†ri ,σ dj,σ +
hli ,lj i,σ
d†j,σ rrj ,σ
+
X
li ∈∂L
¯ j∈∂ L,σ
hi,ji,σ
e tli j l†li ,σ dj,σ + d†j,σ llj ,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. tri rj , tli lj and tij are the hopping parameters in the right lead, left lead, and system. h·i corresponds in this equation to a sum over nearest neighbors, while ∂L is the region ¯ (and similarly for R) in the of the left lead in contact with the system the region ∂ L 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 ¯ ¯li (t), d ¯i (t)]} , Z[Ji (t), Ji (t)] = D 2 lli D 2 rri D 2 di exp {iSK [rri (t), lli (t), di (t), r ¯ri (t), l
(3.88)
where the action can be separated in several pieces:
SLEADS =
Z
C∞
SK = SLEADS + SSYSTEM + SCOUPLING + SSOURCES (3.89) X X ¯li (t) i∂t δli lj + tli lj llj (t) dt r ¯ri (t) i∂t δri rj + tri rj rrj (t) + l ri
li
Z
dt
SCOUPLING =
Z
dt
+
Z
X
dt
SSYSTEM =
C∞
C∞
X
¯ j∈∂ R,σ
X
li ∈∂L
¯ j∈∂ L,σ
SSOURCES =
Z
¯i (t) (i∂t δij + tij ) dj (t) d
(3.91)
i
ri ∈∂R
C∞
(3.90)
¯li ,σ (t)dj,σ (t) + d ¯j,σ (t)llj ,σ (t) e tl i j l
dt C∞
¯j,σ (t)rrj ,σ (t) e tr i j r ¯ri ,σ (t)dj,σ (t) + d
X
¯i (t)di (t) ¯i (t)Ji (t) + J d
(3.92)
(3.93)
i
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. Z Z ¯i (t) (i∂t δij + tij ) δ(t − t′ ) − Σij (t, t′ ) dj (t′ ) dt dt′ d Seff = C∞ ZC∞ X ¯i (t)di (t). ¯i (t)Ji (t) + J d + dt (3.94) C∞
i
Finally, we can integrate the fermions definitively and so we are left with a path integral involving the currents only: Z Z Seff = dt C∞
dt′ ¯ Ji (t)Gij (t, t′ )Jj (t′ ) C∞
45
(3.95)
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< = GR Σ< 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: R/A
ΣR/A = ΣR
R/A
+ ΣL
(3.97)
separated in contributions from each lead. For each lead we have the self energy contribution: R/A
ΣR,L,L′ =
X
R/A
tL,L1 gR,L
L1 ,L2 ≤0 R/A
ΣL,L,L′ =
X
1 ,L2
tL2 ,L′
R/A
tL,L1 gL,L
1 ,L2
tL2 ,L′
(3.98) (3.99)
L1 ,L2 ≥N
The imaginary part of the self energy is: A ΓR/L = i ΣR − Σ R/L R/L
(3.100)
It follows that,
Σ< = i (nL ΓL + nR ΓR ) .
(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, D E † ′ ′ G< (k; t, t ) = i Ψ (t ) Ψ (t) ′ ′ λL,λ L λ′ kL′ λkL D E † ′ ′ G> (k; t, t ) = −i Ψ (t) Ψ (t ) ′ ′ λL,λ L λ′ kL′ λkL and so on. The observables of interest are, the electron density: Z 2i X dE X < NL = − GλL,λL (k; E) Aδ 2π λ k
(3.102) (3.103)
(3.104)
and the current density: Z dE X 2e < tλL1 ;λ′ L2 G< (3.105) JL = i λ′ L2 ,λL1 (k; E) − tλL2 ;λ′ L1 Gλ′ L1 ,λL2 (k; E) A 2π k λλ′ L1 ,L2
Since for this system all the derivations apply, we can also find the following relation: Z 2e X dE T r ΓR GR ΓL GA (nR − nL ) , J= (3.106) A 2π k 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. Fern´ andez-Rossier, M. Braun, A. S. N´ un ˜ez, 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 currentcarrying quasiparticles. 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 . 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 manipulate collective properties rather than vice-versa. Of particular importance is the theoretical 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 magnetization 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 equation
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 1
More precisely, J~ ≡
e ~N
P
~ k
∂ǫ(~ k) ∂~ k
h
i
n~↑k − n~↓k , 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 lowenergy 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=−
X
r,r′ ,σ
X † c†rσ tr,r′ + µ cr′ σ + U cr↑ cr′ ↑ c†r↓ cr′ ↓
(4.2)
r
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 =
Z
β
dτ
0
+ U
Z
X rr′
β
dτ 0
¯ r ∂τ − µ − tr,r′ Ψr′ Ψ
X
ψ¯r↑ ψr↑ ψ¯r↓ ψr↓
(4.3)
r
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]: 2 1 1 ¯ ¯ r σ · Ω r Ψr 2 , Ψr Ψr − Ψ ψ¯r↑ ψr↑ ψ¯r↓ ψr↓ = 4 4 52
(4.5)
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 = −
Z
Z
D 2 Ψr D [∆c , ∆s , Ω] exp β 0
Z
β
dτ 0
X rr′
¯ r ∂τ − µ − tr,r′ Ψr′ Ψ
X1 ¯ r (i∆cr + ∆sr σ · Ωr ) Ψr dτ ∆2cr + ∆2sr − Ψ U r
!
(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 zˆ-axis.3 For the other fields we obtain: U ¯ −i∆c = − hΨ r Ψr i 2 U ¯ ∆s = hΨr σ3 Ψr i. 2
(4.7) (4.8)
Given these solutions, the effective action for the electrons is a non-interacting one: Seff =
Z
β
dτ 0
X rr′
¯ r (∂τ + i∆c + ∆s σ3 − µ) δr,r′ − tr,r′ Ψr′ . Ψ
(4.9)
From this expression, the right hand side of equations (4.7) and (4.8) can be evaluated, given values for ∆c and ∆s . The resulting equations can be solved selfconsistently to obtain unique values for 2
This method has the advantage of keeping the symmetry unbroken in the Lagrangian. The symmetry is only broken when the saddle point equations are solved. Different methods common in the literature break the symmetry already at the Lagrangian level[107]. The above complicates the expressions for the low-energy effective Lagrangian[105]. 3 This is the crucial step in describing the broken symmetry state. Detailed analysis of this issue is 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
q
1 − L2r + Lr
(4.10)
If we fix the magnitude of the exchange fields we get the following expression for the free energy: Z = +
Z
D 2 Ψr D [∆c , ∆s , Ω] exp
∆s
Z
0
β
dτ
X r
Z
β
dτ 0
¯ r (σ · Ωr (τ ) ) Ψr Ψ
!
X rr′
¯ r ∂τ − µ + i∆cr − tr,r′ Ψr′ Ψ (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: Ψ r = Rr Φ r
(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 zˆ-axis.) σ · mr = Rr σ3 R†r 4
(4.13)
Here, the SU(2) stands for spin rotations and the U(1) correspond to consider rotations around the axis of the local magnetization as the identity. The special unitary group, denoted SU(N), is the group of unitary matrices of range NxN with unit determinant. SU(N) is a subgroup of the unitary group U(N), including all NxN unitary matrices. The notation A/B stands for the quotient group between the groups A and B.
54
Again, following the notation of [105], we use the identity: µ ¯ r R†r R ¯ Φ r+eµ Φr = Φr exp (∂µ − iAr ) Φr .
(4.14)
With this transformation the action becomes Srotated = −
Z
β
dτ
0
X
¯r Φ
∂τ − A0r − (i∆c + µ) − 2t
q
!
r
∆ s σ3
1 − l2r + lr · σ
X i
cos −i∂i − Air
Φr
(4.15)
A0r = −R†r ∂τ Rr
(4.16)
Air = iR†r ∂i Rr ,
(4.17)
where we have defined:
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 = − SExchange =
t 2
Z
Z
β
dτ 0
Sl2 =
jνµr Aνµr
(4.18)
¯ r cos (−i∂µ ) Φr + c.c. Aνµr 2 Φ
(4.19)
µνr
β
dτ
0
Sl = −∆s ∆s 2
X
Z
X µνr
β
dτ
0
Z
0
X
lνr jν0r
(4.20)
νr
β
dτ
X
l2r j30r
(4.21)
r
The spin-density and spin-current are defined as: ¯ r σ ν Φr , jν0r = Φ
(4.22)
¯ r sin (−i∂µ ) σ ν Φr + c.c., jνµr = tΦ
(4.23)
55
By integrating out the electronic degrees of freedom we obtain an effective action: Seff [mr , Lr ] = hSBerryPhase i + hSExchange i 1 2 1 + hSl i + hSl2 i − hSBerryPhase i − hSl2 i − hSBerryPhase Sl i(4.24) 2 2 Let’s neglect the fluctuations around the long-wave-length excitations, that is to say, let’s make lr = 0. We have: 1 2 Seff [mr ] = hSBerryPhase i + hSExchange i − hSBerryPhase i 2
(4.25)
where we have explicitly: hSBerryPhase i = − hSExchange i = 2 i hSBerryPhase
=
Z
Z
β
dτ
0
β
dτ
0
Z
0
X r
X1 µνr
β
dτ
Z
β
0
4
hjνµr iAνµr
(4.26)
hKi Aνµr 2
(4.27)
dτ ′
X ′ ′ hjνµr jνµ′ r′ iAνµr Aνµ′ r′
(4.28)
rr′
To evaluate the single particle averages is straightforward. In equilibrium, the electronic 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 zˆ-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: hjνµr i = δ3,ν δ0,µ M
(4.29)
The kinetic energy expectation value hKi 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 magnetic 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 phenomenological Landau-Lifshitz equations. The square of the non-abelian gauge field Aνµr 2 is proportional to the square of the gradient: X
Aνµr 2 = (∂µ Ω)2
(4.30)
ν
which together with the usual expressions for the dynamical equations of a spin system lead us to: ∂Ω ∂Ω = ρs Ω × ∇2 Ω + αΩ × ∂t ∂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 renormalization 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 = ρs k2 that follows from the equation for the magnetization orientation dynamics.. The inclusion of external fields and magnetic anisotropy can be accomplished by adding precession terms to this equation. ∂Ω ∂Ω = ρs Ω × ∇2 Ω + Ω × H eff + αΩ × ∂t ∂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 perhaps simpler and more direct. One advantage of the derivation presented above is that we can immediately understand the influence of a transport current on longwavelength magnetization dynamics. When a current is driven across a ferromagnet, 57
Figure 4.1: Cartoon of the torques driving the magnetization dynamics, (a) the ~ eff × M, ~ and (b) usual ferromagnetic precession is driven by a torque of the form H 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 polarization.The expectation value for the current gets modified to account for the spin current hjνµr i = δ3,ν δ0,µ M + δ3,ν Jµs
(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 ∂Ω = J si Ω × (Ω × ∇i Ω) + ρs Ω × ∇2 Ω + Ω × H eff + αΩ × ∂t ∂t
(4.34)
The relation Ω × (Ω × ∇i Ω) ∼ ∇i Ω, 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 ~ 1 enters takes place when a spin current coming from a magnet spin polarized along M ~ 2 . In this circumstance there is an imbalance in a magnet spin polarized along M ~ 2) between the incoming and the outgoing transverse component (with respect to M 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 magnetization. We consider again a magnet with charge current ~j, and spin current
59
J~ . We assume that current flows in the x ˆ direction and, importantly, that the spin ˆ current is locally parallel to the magnetization orientation J~ (x) = js Ω(x). It can be shown that this is the case in a wide range of situations. ˆ ~ where S0 is the average spin polarizaThe spin density reads S(x) = S0 Ω(x) tion. We focus on the slab centered at x and bounded by x − dx and x + dx. Spins
ˆ − dx) and leave at the rate js Ω(x ˆ + dx). are injected into the slab at the rate js Ω(x
ˆ Therefore, there must be a spin The resulting spin current imbalance is 2dxjs ∂x Ω. transfer to the local magnetization: ~ dS(x) dt
ˆ = js ∂x Ω
(4.35)
ST
ˆ 2 = 1 at every point of the space we obtain: Now, using |Ω| ~ dS(x) ˆ ˆ ˆ × (∂x Ω(x) × Ω(x)) = js Ω(x) dt
(4.36)
ST
which is exactly the same result obtained in [101]. Including this term in the Landau Lifshitz equation and solving for small perturbations around the homogeneous ground state (spin waves) results into the spin wave Doppler shift discussed in previous sections. In conclusion, this argument demonstrates that the spin-wave Doppler shift and spin transfer torques are different limits of the same physical phenomena, 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 h i3 ~ i (τ ) and higher. The resulting actionScl (∆ ~ cl )+SSW , where the first term order δ∆ is the classical approximation to the effective action and the fluctuation correction is: SSW =
1 X δ∆a (Q)Kab (Q)δ∆b (−Q) 2βN
(4.38)
Q,a,b
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 ret (~ propagator: Dab 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 spindiagonal 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 ˆ = n, we can write
K||
Kab (Q) = 0 0
0
0
Kyy Kyz 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 yˆ, zˆ to +ˆ z ± iˆ y chiral representations. The diagonal elements are then 4U 1 2 3 1 + 3 U Γ(±~ q , ±ω)
ret (~ q , ω) = D±
(4.40)
where Γ(~ q , ω) is the Lindhard function evaluated with the spin-split mean-field bands:
n~↑ − n~↓ 1 X k k+~ q Γ(~ q , ω) = ↑ ↓ N + ω + i0+ ~ ǫ~ − ǫ~ k
k
(4.41)
k+~ q
h i where n~σ is shorthand for the Fermi-Dirac occupation function nF ǫ~σ for the quasik
k
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: ret D± (~ q , ω) =
4U ∆ 1 3 ω ± ρq 2
(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 = ±ρq 2 . 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 ferromagnet 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 leading order in ω to show that yˆ and zˆ 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 quasiparticle 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 function Γ:
i ∂Γ 1 X ∂ǫ(~k) h ↑ ↓ = n − n ~k ~k ∂qi q=ω=0 N ∆2 ∂ki ~k
63
(4.43)
Since this expression uses the easy direction x ˆ as the spin-quantization axis, the x (spin) component of the spin current is: i e X ∂ǫ(~k) h ↑ J~ ≡ n~ − n~↓ k k ~N ∂~k ~
(4.44)
k
so that
~ ∂Γ = Ji ∂qi q=ω=0 e∆2
(4.45)
The quantity Ji , the component of the spin current polarized along the magnetization direction n = x ˆ and flowing along the i axis, is the difference between the current carried by majority and minority carriers. In equilibrium there is no current 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 majority 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 = ρq 2 −
2U ~ q~ · J~ 3∆ e
(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 ∆ =
2U 3 (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 ∆ =
2U 3 n
and ρ ≃ ω=
~2 2m ,
leading to
~2 2 ~ ~ ~2 2 q − q~ · j = q − ~~ q · ~vD 2m en 2m
(4.47)
where we have expressed the current as ~j = en~vD with ~vD the drift velocity, generalizing the half-metallic simple Doppler shift result to non-parabolic bands. 64
4 j=0 8 −2 j=5 10 Acm 9 −2 j=1.1 10 A cm
ω(q) (µeV)
3 2 1 0
0
2
4 6 −1 q (µm )
8
10
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 + ρq 2 −
2U ~ q~ · J~ 3∆ e
(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 spinwave 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
pq~ , xq~
1 2M~q
i ω2
−i ω2 K~q 2
pq~ xq~
(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
0 −i
+ 2U ~ J~ · q~ 3∆ e ρ~ q · q~ 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 fluctuation 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 electronic 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. According 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) =
i h i 2π X h ↑ n~ − n~↓ δ ǫ~↑ − ǫ~↓ + ω k k+~ q k k+~ q N
(4.51)
~k
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) ∝
X
~k,~k ′ ,ν,ν ′
h i ′ ′ Sν,ν ′ (~k, ~k′ ) n~νk − n~νk′ δ ǫ~νk − ǫ~νk′ + ω
(4.52)
where Sν,ν ′ (~k, ~k′ ) ≡ |h~k, ν|S (−) |k~′ , ν ′ i|2 is a matrix element between disorder broadened 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
X
Sν,ν ′
ν,ν ′
×
Z
dǫ
Z
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: X γ(ω) ≃ ω n2 Sν,ν ′ Nν (ǫF )Nν ′ (ǫF ) (4.54) ν,ν ′
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 damping 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 excitations and borrow results from master equation results for oscillators weakly coupled to a bath. For magnetization in the ‘↑’ direction, magnon creation is accompanied 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: # " ∂n ~ · ~vν (~k)τν (ǫν ) − g~kν = n~νk − eE ~k ∂ǫ ǫ=ǫν
(4.55)
~ k
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 spinσ . We reach this conclusion even though the phase space dependent drift velocities vD
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 10β A cm−2 is vd =
j en
≃ 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 = ω0 b†0 b0 +
X
ω(~ q )bq†~ bq~ + b†0
q~6=0
X gq~ √ bq~ + h.c. V q~6=0
(4.56)
where bq~ is the annihilation operator for the spin wave with momentum q~ and gq~ 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: 2π γ(J~ ) = ~
Z
d~ q |gq~ |2 δ(ω0 − ωq~ ) (2π)3
(4.57)
Now we use ω0 − ωq~ = ρq 2 − a~ q · J~ . After a straightforward calculation we obtain: γ(J~ ) =
g2 a|J~ | 4π ρ2
70
(4.58)
where we have approximated gq~ ≃ 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 4.3.1
Current induced Domain wall dynamics 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. The dashed-dotted line illustrates the electron transferring its spin angular momentum to the domain wall, leading to motion. The dotted line illustrates momentum transfer: the electron scatters off the domain wall and gives the domain wall a momentum kick.
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 equations 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 contributions, 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 ferromagnet. 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 electrons 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 addition 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 interaction 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 magnetic 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 δE ∂M = −γM × + αM × ∂t δM ∂t
(4.59)
E = Edemag + Euniaxial + Eexchange
(4.60)
Where E is given by:
dx 2πMx2 , Z K = − dx 2 Mz2 , M0 Z A = dx 2 |∇M |2 . M0
Edemag = Euniaxial Eexchange
Z
(4.61) (4.62) (4.63)
The competition between the uniaxial anisotropy and the exchange contributions q to the energy set the domain wall width λ = K A . Also the uniaxial energy sets a scale for simple precession around the easy axis. The frequency associated with this
75
scale is
K M0 .
These scales can be used to define units of time and length and the
equations of motion become: ∂Ω δE˜ ∂Ω = −Ω × + αΩ × , ∂t δΩ ∂t where E˜ =
R
(4.64)
ǫdx: ǫ = |∇Ω|2 − Ω2z +
1 2 Ω Q x
(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
= J Ω × (Ω × ∇x Ω) .
(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 evolution of the magnetization field in a ferromagnet: ~ ~ ∂M γ ~ δE α ~ ∂M + =− M × M× ~ ∂t M M ∂t δM
(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 = E˙ total =
Z
R
~ δE ∂ M αM dV · =− ~ ∂t γ δM
dV E, its rate of change is: Z
~ ∂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 ~ we can always describe the fields with polar describing conserve the norm of M, ~ = (cos φ sin θ, sin φ sin θ, cos θ). We can then write the LL equations angles. Let M as: ∂θ ∂t ∂φ sin θ ∂t sin θ
γ δE ∂φ − α sin2 θ M δφ ∂t γ δE ∂θ +α M δθ ∂t
= −
(4.69)
=
(4.70)
5 At this point it seems adequate to mention the existence of a packed set of numerical routines OOMMF [131] that handle the difficulties of magnetostatic effects in some restricted geometries.
77
Ωz Ωz
Ωy
Ωx
J = 0.4
Q=
α = 0.1
Figure 4.4: Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 director vector. Here Q is infinity (no in-plane anisotropy) and the domain wall responds as J a straight line with velocity X˙ = 1+α 2 . As the domain moves the components in the hard plane precess.
78
Ωz Ωz
Ωy
Ωx
J = 0.4
Q = 1/ 0.3
α = 0.1
Figure 4.5: Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 ). The third dimension 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 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
Ωz Ωz
Ωy
Ωx
J = 0.4
Q = 1/ 0.5
α = 0.1
Figure 4.6: Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 ). The third dimension is the dimensionless z-component of the magnetization vector. (rest of panels) A 2D representation of the different coordinates of the director vector. Q is even smaller approaching the critical situation and the wiggles become stronger.
80
Ωz Ωz
Ωy
Ωx
J = 0.4
Q = 1.0
α = 0.1
Figure 4.7: Exact solution of the Landau-Lifshitz equations for the parameters indicated. 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 ). The third dimension is the dimensionless z-component of the magnetization vector. (rest of panels) A 2D representation of the different coordinates of the director vector. Q is 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 the theory. By changing variables to Φ = cos θ (the projection of the magnetization along the easy axis) we obtain, using sin θ θ˙ = −∂t Φ, and (δθ E)/ sin θ = −δΦ E: p γ δE + α 1 − Φ2 φ˙ M δφ γ δE α ˙ φ˙ = − −√ Φ M δΦ 1 − Φ2
Φ˙ =
(4.71) (4.72)
where the Hamiltonian structure is evident when the damping terms are neglected. . The action is: S=
Z
γ dtd3 x φΦ˙ − E(Φ, φ) M
(4.73)
and the dissipative function[132] is: 1 R=− 2
Z
α dtd3 x E˙ = 2
Z
dtd3 x
˙2 Φ + (1 − Φ2 )φ˙ 2 1 − Φ2
!
(4.74)
The equations of motion can be written as: δS δR = , δΦ δΦ˙
and
82
δS δR = δφ δφ˙
(4.75)
4.3.4
Bloch Domain Wall
We now focus on discussion of a Bloch domain wall. The exchange energy is: Eexchange =
A (∇Mx )2 + (∇My )2 + (∇Mz )2 , 2 M
or in terms of the canonical coordinates, (∇Φ)2 2 2 Eexchange = A + (1 − Φ )(∇φ) . 1 − Φ2
(4.76)
(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 = 2πM 2 sin2 θ cos2 φ = 2πM 2 (1 − Φ2 ) cos2 φ using λ =
p
(4.79)
A/Ku as the unit of length we obtain for the total energy per unit
transverse area; Etotal =
p
AKu
Z
dx
1 (∇Φ)2 2 2 2 2 2 + (1 − Φ )(∇φ) + (1 − Φ ) + ) cos φ (1 − Φ 1 − Φ2 Q (4.80)
A static field is a stationary point of such a functional. The minimization condition on the variable φ is; ∇2 φ −
1 sin(2φ) = −∇ log(1 − Φ2 ) 2Q
(4.81)
and for the variable Φ ∇2 Φ = −Φ
(∇Φ)2 1 − Φ(1 − Φ2 )(1 + (∇φ)2 + cos2 φ) 2 1−Φ Q
(4.82)
A Bloch domain wall consists of a soliton solution of those equations of the form: π φ = ± , 2 Φ = ± tanh(x) 83
(4.83) (4.84)
1
2
My 0
1
E Mz
-1 -5 -4 -3 -2 -1 0
x/ λ
1
2
3
4
0
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
x/ λ
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 the energy 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 describe 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 Magnetic Field
The first simple study we can make of domain wall motion is a rigid one driven by and external magnetic field along the −ˆ z direction. The energy functional becomes: (∇Φ)2 1 2 2 2 2 2 E(Φ, φ) = + (1 − Φ )(∇φ) + (1 − Φ ) + (1 − Φ ) cos φ + hΦ (4.85) 1 − Φ2 Q and the action is; S=
Z
˙ − E(Φ, φ) , dtdx Φφ 84
(4.86)
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) π φ(x, t) = + p(t) 2
(4.87) (4.88)
where Φ0 (x) = tanh(x). The action can be calculated now in terms of the new ”‘collective”’ coordinates X and p. ˙ x Φ0 (x − X) Φ˙ = −X∂
(4.89)
φ˙ = p˙
(4.90)
We obtain: Seff =
Z
˙ + 2 cos2 (p) + h dt −2Xp Q
The dissipation function is:
Z
dx Φ0 (x − X)
R = α X˙ 2 + p˙2
(4.91)
(4.92)
Then the equations of motion are: δS δR = δX δX˙ δS δR = δp δp˙
−→ p˙ + h = αX˙
(4.93)
1 −→ −X˙ + sin(2p) = αp˙ Q
(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: Z h α 1 dp p˙ = + sin(2p) −→ =t 1 + α2 1 + α2 Q Λ + Γ sin(2p) 85
(4.95)
0.015
1.0
1
0.01
0.8
0.6
Q
0.5 0.005
0.4
0
0.2
0
0.2
0.4
0.6
0.8
1
0.0
h ˙ for the domain wall as a function of Q and the Figure 4.10: Average velocity hXi ˙ driving field h. The color code represents the relative value of hXi/(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
Below the WL the integral is: 1 t= √ log 2 2 Γ − Λ2
! √ Λ tan p + Γ − Γ2 − Λ2 √ Λ tan p + Γ − Γ2 − Λ2
(4.96)
which implies:
Γ tan p + Λ
p tanh( Γ2 − Λ2 t) +
r
Γ2 −1=0 Λ2
(4.97)
which after the transient (as t −→ ∞) implies again, X˙ = h/α. Above the WL we have: Γ tan p = − + Λ
r
1−
p Γ2 tan( Λ2 − Γ2 t) Λ2
(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 currentcarrying quasiparticles modifies the LL equation in the following way: ~ ~ ∂M γ ~ δE α ~ ∂M ~ × (M ~ × ∂k M). ~ =− M × + M× + Jk M ~ ∂t M M ∂t δM
(4.99)
~ is preserved we can write the above as: Using the fact that the norm of M ~ ~ ∂M γ ~ δE α ~ ∂M ~ + =− M × M× + J~ · ∇ M ~ ∂t M M ∂t δM
(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: Z ˙ + J~ · φ∇Φ − γ E(Φ, φ) S = dtd3 x φΦ M 87
(4.101)
In the case of a Bloch domain wall the action is: Z ˙ + J φ∂x Φ S = dtdx Φφ 1 (∇Φ)2 2 2 2 2 2 + (1 − Φ )(∇φ) + (1 − Φ ) + (1 − Φ ) cos φ , (4.102) − 1 − Φ2 Q and within the rigid approximation becomes (notice that the current is coupled to p whereas the field was coupled to X.): Z 1 2 Seff = 2 dt pX˙ + J p − sin p Q
(4.103)
Then the equations of motion are: δS δR = δX δX˙ δS δR = δp δp˙
−→ −p˙ = αX˙
(4.104)
1 −→ X˙ + J − sin(2p) = αp˙ Q
(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
0.015
1.0
5
5
4 0.01
4
0.5
3
2 0.005
2
1
1
0
0.0
0
Q
. Jcr
3
0
1
2
3
4
5
0
J
1
2
3
4
J/J cr
˙ as function of the anisotropy parameter Figure 4.11: Left panel: average velocity hXi 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: hXi/J cr (Q) as a function of J/Jcr (Q) for several values of Q. Above p the critical current all the curves collapse into the dashed line described by 2. ˙ = J 2 − Jcr hXi
89
5
S=
Z
dtdx −(pX˙ − J p)∂x Φ −
(∇Φ)2 1 + (1 − Φ2 ) + (1 − Φ2 ) sin2 p 2 1−Φ Q
,
(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 1 ˆ Φ(x) = Φ(x/Σ) where Σ2 = (1 + Q sin2 p) > 1 we get: Z (∇Φ)2 ′ 2 ˙ S = dtdx −(pX − J p)∂x Φ − Σ + (1 − Φ ) , 1 − Φ2
(4.107)
The solution is then a scaled domain wall Φ(x) = tanh(Σx), moving with collectives coordinates minimizing the action: Z Seff = dt −pX˙ − J p − 2Σ(p) and the dissipation function becomes: Z p˙2 R = α dt ΣX˙ 2 + Σ
(4.108)
(4.109)
The equations of motion are then, δS δR = δX δX˙ δS δR = δp δp˙
−→ −p˙ = αΣ X˙
(4.110)
dΣ p˙ −→ X˙ + J − 2 =α dp Σ
(4.111)
The static point condition X˙ = p˙ = 0 implies, again, a relation between p and J given by: J =
1 dΣ2 Σ dp
and therefore we have that the critical current is given by: 1 dΣ2 Jcr = Max Σ dp
(4.112)
(4.113)
with the asymptotic behavior √ 2 Q as Q → 0 Jcr −→ 1 as Q → ∞ Q 90
(4.114)
0.015
5
1.0
1
5
4 4 0.01
0.75 3
3
Q
0.5
0.5 0.005
2
0.25 1
p 4
p 2
1
0.0 0
0
2
1
2
J
3
4
5 0.0
0 0
1
2
˙ as function of the anisotropy parameter Figure 4.12: Left panel: average velocity hXi 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: hXi/J cr (Q) as a function of J/Jcr (Q) for several values of Q. Above p the critical current all the curves collapse into the dashed line described by 2. ˙ = J 2 − Jcr hXi
91
3
4
5
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. N´ un ˜ez and Allan H. MacDonald, Spin Transfer Without Spin Conservation, “The Proceedings 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 materials have been cleanly identified and systematically studied in magnetic nanostructures. 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 quasiparticles 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 introduce 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 considerations, 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 specific 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 spinorbit interactions are present. Although we can employ any time-dependent mean field theory of a metallic ferromagnet we expect that for magnetic metals most applications 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 magnetization. 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 semiconductors. 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 δF2 δF2 = −Ω2 × + αΩ2 × Ω2 × + Γ2ST . dt δΩ2 δΩ2
(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 = γout Ω1 · 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 magnetization. The basic questions are then, “Is there an in plane torque?” and “How 1
This is in spite the fact that we can also write a similar expression for the torque in this magnet:
e
e
Γ1ST = γout Ω2 × Ω1 + γin Ω1 × (Ω2 × Ω1 ) ,
(5.2)
This pinning can be implemented in several ways. In some works the pinned ferromagnet is just make large enough. Another strategy uses a more complex hetero-structure including an antiferromagnetic 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 pinned and free ferromagnetic layer. Because of conservation of total spin angular momentum, 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.
big is it?”. In other words what is the physics describing γin . To understand the current 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 direction. 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 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. 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 collective 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 second 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 memory 2 ). 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 characterized 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 equilibrium by an exchange field that is approximately proportional to the magnetization magnitude and points in the majority-spin direction. The spin-orientation of the singly occupied majority-spin orbitals is the collective-coordinate, the magne2
For 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 spin transfer torque. The red line is described by the magnetization vector as it flips from the north to the south pole in response to the current. The change in precession sense at the equator correspond to the change in the effective exchange field. This field is proportional to 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 collective degree of freedom is the spin-orientation of singly occupied orbitals. Right panel: Quasi~ that brings majority and minority spins into particles experience a strong exchange field ∆ equilibrium. 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 window of width eV at the Fermi energy can differ from the magnetization orientation. The spin-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 electrons in a narrow transport window with width eV centered on the Fermi energy, and can be evaluated by solving the quasiparticle Schroedinger equation for electrons 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 1~ + V (~r) − ∆(~ r ) · ~τ , 2m 2
(5.4)
~ r ) are arbitrary scalar and exchange potentials and ~τ is the Pauli where V (~r) and ∆(~ spin-matrix vector. In the local-spin-density approximation[138] (LSDA) of SDFT, ~ r ) = ∆0 (n(~r), m(~r))m(~ ∆(~ ˆ r ) where m ˆ is a unit vector, m ~ = mm ˆ is the total spindensity 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 i dsα,j (~r) 1 h~ i = ∇i Jα,j (~r) + ∆ × ~sα (~r) dt ~ j
(5.5)
where the spin current tensor for orbital α is defined by, i Jα,j (~r) =
1 Im Ψ†α (~r)τj ∇i Ψα (~r) . 2m
(5.6)
This equation exhibits the separate contributions to individual quasiparticle spin dynamics from convective spin flow, the source of the conservative term, and precession ~ Both sides of Eq.(5.5) vanish when the quasiparticle around the exchange field ∆. 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. i dmj (~r) X 1 h~ i = ∇i Jα,j (~r) + ∆ × m(~ ~ r) dt ~ j α 102
(5.7)
i where Jα,j is the contribution to the spin-current from orbital α. The main point
~ is proportional to m we wish to make here is that (in the LSDA) ∆ ~ 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 ˆ ˆ ∂m ∂m ~ + αm ˆ ×H ˆ× =m , eff ∂t ∂t
(5.8)
where α is the damping parameter, ˆ ~ (~r) ≡ δEMM [m] H eff ˆ r) δm(~
(5.9)
is the effective field that drives the long-wavelength collective dynamics of an elecˆ is the micromagnetic energy functional. trically isolated sample, and EMM [m]
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~ as we now explain. The slow dynamics of the tribution to the exchange field ∆ 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 ~ Transport electrons will in general make a contribution to instantaneous value of ∆. the spin-density that is small but perpendicular to the magnetization 3 . We define this transport contribution to the spin-density as m ~ tr . Because it is perpendicular to the magnetization, its contribution to the exchange-field experienced by all quasiparticles ~ δH ST = ∆0 (n, m)
m ~ tr m
(5.10)
~ . It follows produces a spin-torque that can be comparable to that produced by H eff that the influence of a transport current on magnetization dynamics is captured by ~ ~ ~ replacing H eff in Eq.(5.8) by Heff + δHST . 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 ~ ˆ The net correction δH just a correction in EMM [m]. ST to the effective dynamics under non-equilibrium configurations can be separated into two contributions. One ˆ “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 3
The transport orbitals will also, in general, contribute to the total spin-density component in the direction of the magnetization. This effect alters the exchange field along the magnetization direction and does not produce a spin torque. 4 Note that, this “effective damping”, can compensate the intrinsic damping[8, 9], signaling the instability 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-magneticsemiconcutor 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 noncollinear 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: h
i ~ r) × m ∆(~ ~ tr (~r) = −~∇i Jjtr,i (~r) j
(5.11)
where Jjtr,i 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 ~ (~r) = δH ST
ˆ ∇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 spintransfer 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 m ~ tr (~r) for a toy model containing a ferromagnetic 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 our calculations we apply periodic boundary conditions in the transverse yˆ direction. A spintransfer torque is present when the two magnetization directions are not aligned. The inset shows the Fermi surfaces of the two ferromagnets in which are identical in the absence of spin-orbit coupling and and indicates the transverse channel ky range over which one of the two Schroedinger equation solutions is an evanescent spinor. The Schroedinger equation solutions for electrons incident from x → −∞ can be solved by elementary but tedious calculations in which the spinors and their derivatives are required satisfy appropriate continuity conditions at the interfaces.
107
Figure 5.6: Right movers Fermi Surface in a Rashba System. The dashed lines are describing the behavior of the imaginary part of the wave-vector of the state with the Fermi energy, of course, for the usual states depicted in the inset of Fig.(5.5) the imaginary part is zero, 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 course does not affect their “Right mover” status, since in the presence of spin-orbit interaction the 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 magnitude. 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 ~ − ∆(x) · ~s + {λ(x), p~ × zˆ} · ~s, 2m
~ correspond to the local exchange vector, and λ is non-zero only on the where ∆ region with spin-orbit coupling. For this model we evaluated m ~ tr (~r) in a currentcarrying system using the Landauer-B¨ uttiker 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 spintransfer. 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 5
To ensure Hermiticity we write HSO = ({λ, ~ p} × zˆ) · ~s, where the symbol {·, ·} denotes the operator anticonmutator .
109
transport electrons should be approximately perpendicular to the transfer plane. It can be verified that this is indeed true by directly evaluating m ~ tr (~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 interference between incident and reflected quasiparticle waves that complicates the spin-transfer torques and also because it fails to account for electrons that are described 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 m ~ tr (~r) that is in the transfer plane. In Fig.(5.7(a)) we plot values of m ~ tr (~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 zˆ direction and the pinned magnet to be in the zˆ − x ˆ 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 argument, 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 approximates 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 transfer 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
(a)
(b) Sy
Sy Sx
θ
θ
(c)
Figure 5.7: (a) Transport spin density per unit current in the case without spin-orbit interaction. mtr y is the component perpendicular to the transfer plane (“non-conservative”)while mtr x is the smaller component in the transfer plane (“conservative”) that is contributed by evanescent spinors. Both components are normalized to the maximum mtr y which occurs for θ = π/2. (b)Non-equilibrium spin density per unit current perpendicular to the transfer plane for different spin-orbit interaction strengths. It follows that from these results that the spin-transfer torque is reduced in efficiently and altered in angle dependence by spin-orbit interactions. (c)Spin transfer efficiency, g, normalized by the ST efficiency in the absence of spin-orbit coupling, as a function of the spin-orbit strength, for several widths of the free magnet. The spin-transfer effect becomes weak when the spin-orbit splitting is comparable with 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 possible importance of incoherent nanomagnet magnetization dynamics in metal spintransfer 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 bilayers and to circuits that contain antiferromagnetic metals. In the case of quantum Hall bilayers, the collective coordinates of interest are the interlayer phase and population 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 ferromagnetism) and pseudospin transfer torques have been invoked to explain the sudden drop in interlayer conductance with bias voltage seen in experiment[149]. We anticipate 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 scattering 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. N´ un ˜ez, Rembert Duine, and A.H. MacDonald, Antiferromagnetic Metals Spintronics, 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 antiferromagnetic 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 antiferromagnet 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 conservation 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 picture 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 orientations 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 element 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 = rs 1 + rt n · ~τ , where n is the order parameter orientation 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 favorable 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 system 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 ferromagnets. 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 antiferromagnet the quasiparticles satisfy a Schroedinger equation with an exchange Zeeman field with oscillatory spatial dependence in the direction of the order param116
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 symmetry 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 i eikx + |−∞L i e−ikx ; Ψ∞ (x) = |∞R i eikx + |∞L i e−ikx ,
(6.1) (6.2)
where |∞R i , · · · and |∞L i , · · · are the spinor coefficients of the right and left goers, respectively. The scattering matrix expresses the outgoing spinors in terms of the |−∞L i |−∞R i =S with S in turn expressed in terms incoming spinors: |∞R i |∞L i r t′ . We choose the direction of 2×2 transmission and reflection matrices S = t r′ 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 + St n · τ .
(6.3)
Now, the operation space inversion-time reversal symmetry transform the
117
wave functions into: ˜ −∞ (x) = iσy |−∞∗L i eikx + iσy |−∞∗R i e−ikx ; Ψ ˜ ∞ (x) = iσy |∞∗L i eikx + iσy |∞∗R i e−ikx , Ψ
(6.4) (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 related 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′t τz = rs + rt τz ;
(6.7)
ts − tt τ z = ts + tt τ z ;
(6.8)
t′s − t′t τz = t′s + t′t τz .
(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 rs + rt n · τ t′s . S= (6.10) ts rs − rt n · τ
However the parameter space is further constrained by unitarity. This allows us to write rs = ieiν sin Θ cos Φ ; rt = eiν sin Θ sin Φ ; t′s = ei(ν−ξ) cos Θ ; ts = ei(ν+ξ) cos Θ , 118
(6.11)
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: −iξ 1 sin Θ Q cos Θ e Φ . S = eiν (6.12) iξ cos Θ e 1 sin Θ Q−Φ
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 ferromagnetic 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, indeed 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 noncollinear antiferromagnetic elements. In an array for multiple non-collinear antiferromagnets, each one will fail to induce spin polarization, however the multiple reflection process at each interface will lead to a non-trivial spin-current configuration. Most importantly, for two antiferromagnets with respective staggered moment 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 outof-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 antiferromagnets 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 paper 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 i = t2 Kt1 |−∞R i ,
(6.13)
where the multiple reflection kernel is defined by K = (1 − r′1 r2 )−1 . The net transmission coefficient becomes: T = Tr t†1 K† t†2 t2 Kt1 .
Using Eqs. (6.10) and (6.11) we reduce it to: T = cos4 ΘTr K† K .
(6.14)
(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 transmission on their relative orientations. Straightforward calculation leads to: |Λ|2 Tr K† K = (2 + 4(1 + n1 · n2 ) 2 cos 2ν − δL cos 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 classi121
Figure 6.1: (a) Effective resistance arrays that represents a parallel configuration in a conventional GMR device. (b) same for antiparallel.
Figure 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. 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: 1 R↑↑
=
1 2R
+
1 2r .
The opposite configuration has one large and one small resistor in
each channel, leading to a net resistance: R1↑↓ = is then: R↑↓ − R↑↑ = MR = R↑↓
2 R+r .
The magneto-resistance ratio
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 for
left-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) = J
N X i=1
ˆ i · ~τ δ(x − xi ) Ω
(6.18)
and we calculate the transmission coefficients for different configurations of the n o ˆ i . To do that we first calculate the scattering matrix for a single scatterer Ω
ˆ located at the origin and pointing along Ω.
where:
It is an easy matter to prove that: Γ−1 Γ , S= Γ Γ−1 Γ=
1 λ ˆ 1+ Ω · ~τ , 2 1−λ 1 − λ2
(6.19)
(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: e2ikx0 (Γ − 1) Γ . S = 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 1
It is an easy task to prove the identity 2Γ† Γ = Γ + Γ† , 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: −1 −1 ′ ′ ′ ′ ′ r1 + t1 r2 (1 − r1 r2 ) t1 t1 (1 − r2 r1 ) t2 S12 = −1 −1 ′ ′ ′ ′ ′ t2 (1 − r1 r2 ) t1 r2 + t2 (1 − r2 r1 ) r1 t2
(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:
ˆ −1 e−iφ Γ(Ω)
ˆ −1 eiφ Γ(−Ω)
S1 = S2 =
ˆ Γ(Ω)
ˆ Γ(−Ω)
ˆ Γ(Ω)
ˆ −1 eiφ Γ(Ω)
ˆ Γ(−Ω) . − i φ ˆ e Γ(−Ω) − 1
The multiple reflection kernel then becomes: −1 −1 = 1 − ei2φ (Γ(Ω) − 1) (Γ(−Ω) − 1) 1 − r1′ r2 and after some elementary manipulations we obtain: 2 1−λ −1 1 1 − r1′ r2 = 1 − λ2 1 − ei2φ
(6.23)
(6.24)
(6.25)
(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: λe−iφ ˆ · ~τ ω(φ, λ) λ1 + Ω (6.28) r12 = 1 − λ2 λe−iφ ′ ˆ · ~τ r12 = ω(φ, λ) λ1 − Ω (6.29) 1 − λ2 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 +
X
c†k,i,σ ck,j,σ + h.c.
hi,ji,σ
Xh
i,σ,σ′
i ˆ i · ~τσ,σ′ c† (ǫi + ǫk )δσ,σ′ − ∆i Ω 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 ˆ i = (−)i n coupling ∆i of electrons to antiferromagnetically ordered local moments Ω 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 transport of quasiparticles across the magnetic heterostructure. The essential physical 125
1
N=15,M=16 N=15,M=17 N=25,M=16 N=25,M=17
0.98
T(θ)/T(0)
0.96 0.94 0.92 0.9 0.88 0
0.5
1
1.5
θ
2
2.5
3
Figure 6.5: Landauer-Buttiker conductance as a function of the angle θ between ˆ i on opposite sides of the paramagnetic spacer the magnetization orientations Ω 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. 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 ) = ihck,i,σ (t) ck,j,σ′ (t )i, 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 ˆ i on opposite sides of the spacer. In Fig. 6.5 the transθ between orientations Ω 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 resistance 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 twochannel 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 antiferromagnetparamagnet 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: ~2 2 ∇ + J cos H=− 2m
2π ˆ · ~τ x Ω λsdw
(6.32)
with atomic units, and the scaling of coordinates x = (λsdw /π)z we can write the Schrodinger equation as: d 2 Ψσ + (a − 2qσ cos(2z)) Ψσ = 0, (6.33) dz 2 2 2 λsdw 2E and q = Jσ. This equation correspond to two sepawhere a = λsdw σ π π
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) = Aσ mc(a, qσ , z) + Bσ ms(a, qσ , z).
6.4.2
(6.34)
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: 0 ˆ · ~τ V (x) = J cos λ2π x Ω sdw 0
xL
The explicit solutions are then: ασ exp (ikx) + βσ exp (−ikx) π π Ψσ (x) = Aσ mc(a, qσ , λsdw x) + Bσ ms(a, qσ , λsdw x) γ exp (ikx) σ
128
(6.35)
0 −G¯t
(A.31)
We can benefit from the relations in Eq. (3.26) by performing the following manipulations[73]: ˇ ≡ τ 3G G → G
(A.32)
ˇ → G ≡ LGL ˇ †, G
(A.33)
where2 ,
2
1 L = √ τ 0 − iτ 2 . 2
(A.34)
While the τ -matrices are numerically the Pauli matrices this rotations act only on Keldyshspace and leave the spin space unchanged
148
The new explicit form for the non-equilibrium Green’s functions is: GR GK G= 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 . The new Green’s function are:
G
0
G
gR
gK
0
gA
ΣR ΣK
G = G0 =
where,
Σ =
R
0
K
G
A
ΣA
Σ = LΣτ 3 L† .
A.4.2
(A.37)
(A.38)
(A.39)
(A.40)
(A.41)
Lehmann Spectral Representation
The expectation value of the equal-time commutation relation: h i † ψ(x), ψ (y) δ(x0 − y0 ) = δ(x − y), ±
lead us in spectral representation to: Z dω ′ i G> (ω, k) − G< (ω, k) = 1. 2π 149
(A.42)
(A.43)
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: Z
dω ′ A(ω, k) = 1 2π
(A.45)
From the relation3 :
we can obtain:
Gr (1, 2) = Θ(1, 2) G> (1, 2) − G< (1, 2) r
G (ω, k) =
Z
dω ′ A(ω ′ , k) , 2π ω − ω ′ + iε
(A.46)
(A.47)
analytical in the upper half-plane of ω. In the same way we have: Ga (ω, k) =
Z
dω ′ A(ω ′ , k) , 2π 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: > ′ Z dω ′ G (ω , k) G< (ω ′ , k) GF (ω, k) = i − 2π ω − ω ′ + iε ω − ω ′ − iε > ′ Z G (ω , k) G< (ω ′ , k) dω ′ − GF¯ (ω, k) = i 2π ω − ω ′ − iε ω − ω ′ + iε
(A.49) (A.50)
The above relations can be summarized by defining the functions G1 and G2 in the complex plane: Z dω ′ G> (ω ′ , k) G1 (z, k) = i 2π z − ω ′ Z dω ′ G< (ω ′ , k) G2 (z, k) = i . 2π z − ω ′ 3
(A.51) (A.52)
The 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 ferromagnet 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 arbitrary 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 1~ + ∆ · ~τ . 2m 2
(B.1)
The wave function Ψ solution of the equation: i
∂Ψ = HΨ, ∂t
(B.2)
defines the local average values of the spin h~s(~r)i = Ψ† (~r) ~s Ψ(~r), with ~s = 12 ~τ . We can evaluate the time derivative of the local average spin, dh~s(~r)i ∂Ψ† (~r) ∂Ψ(~r) = ~s Ψ(~r) + Ψ† (~r) ~s dt dt dt using the Hamiltonian we get the spin conservation equation, i dhsj (~r)i 1 h~ = ∇i Jij + ∆ × h~s(~r)i dt 2 j
(B.3)
(B.4)
where the spin current is defined by, Jij =
1 Im Ψ† (~r)τj ∇i Ψ(~r) , 2m
(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 ~ On the other hand, the expected precession around the order parameter field ∆. if we look for the effects of this precession on the order parameter we see that the ~ × h~s(~r)i and the total torque over reaction torque must be locally equal to Γ = −∆ ~ si, ~s being the total average the volume of the sample must be equal to Γtot = −∆×h~
spin. Simple integration gives, in the stationary regime, I tot Γj = dSi Jij , 153
(B.6)
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 ferromagnet 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 ⊗1 2m
(B.7)
and the Hamiltonian for the ferromagnet: HF =
1~ p2 ⊗1+ ∆ · ~τ 2m 2
(B.8)
We solve Schrodinger equation for stationary states, HΨ = EΨ,
(B.9)
~ as the quantization axis for the spin Choosing the axis of the magnetization ∆ operators, we can write the eigenfunctions in each part of the system as: 1 0 1 0 Ψi = ri+ eikx + ri− eikx + li+ e−ikx + li− e−ikx (B.10) 0 1 0 1 154
for the normal metals, and 1 0 1 0 ΨF = rF+ eik+ x + rF− eik− x + lF+ e−ik+ x + lF− e−ik− x 0 1 0 1 (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± = q ~ 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 ferromagnet, 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 r0+ and r0− to be independent parameters and forces the relation l1+ = l1− = 0. The remaining amplitudes are determined by the boundary conditions. The condition Ψ(0+ ) = Ψ(0− ) implies:
r0+ + l0+ r0− + l0−
=
rF+ + lF+ rF− + lF−
The condition ∇Ψ(0+ ) = ∇Ψ(0− ) reduces to: kr0+ − kl0+ k+ rF+ − k+ lF+ = kr0− − kl0− k− rF− − k− lF−
Now at the other end, Ψ(L− ) = Ψ(L+ ) r1+ eikL rF+ eik+ L + lF+ e−ik+ L = , r − eikL r − eik− L + l− e−ik− L 1
F
(B.13)
(B.14)
F
and finally, ∇Ψ(L+ ) = ∇Ψ(L− ) kr1+ eikL k+ rF+ eik+ L − k+ lF+ e−ik+ L = , − ikL − ik − L − −i k − L kr1 e k− rF e − k− lF e 155
(B.12)
(B.15)
The set of 8 equations above can be solved for the 8 unknowns: l0± , lF± , rF± and r1± in ~ = (l+ , l− , r + , r − , l+ , l− , r + , r − ) terms of r0± . The system can be written in terms of X 0 0 1 F F F F 1 ~ = (r + , r − , r + , r − , 0, 0, 0, 0) as: and S 0 0 0 0 ~ =S ~ −→ X ~ = Γ−1 S ~ ΓX where Γ is given by: 1 0 1 0 0 0 −1 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 Γ= , 0 1 2 0 Q 0 Q 0 −1 0 + + 0 1 2 0 0 Q 0 Q 0 −1 − − 1 2 0 0 q Q 0 −q Q 0 −1 0 + + + + 1 2 0 0 0 q − Q− 0 −q− Q− 0 −1
(B.16)
(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: 2) (−1 + q± (Q1± − Q2± )r0± den(±) (1 + q± ) 2 ± Q r = 2 den(±) ± 0 (−1 + q± ) 1 ± = 2 Q± r0 den(±) q± Q1± Q2± ± = 4 r den(±) 0
l0± =
(B.18)
rF±
(B.19)
lF± r1±
(B.20) (B.21)
where we have introduced the notation: den(±) = −(−1 + q± )2 Q1± + (1 + q± )2 Q2± . 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: Re(r1+ (r1− )† ) k J~ = Im(r1+ (r1− )† ) 2m |r1+ |2 − |r1− |2
(B.22)
Using the above expressions we can evaluate the spin current, first let’s cal-
culate r1+ (r1− )† . r1+ (r1− )† = 16q+ q−
den(+)† den(−) (r + )† r0− |den(+)|2 |den(−)|2 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− )L K++ −ei(k+ +k− )L K+− −e−i(k+ +k− )L K−+ +e−i(k+ −k− )L K−− (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 energies 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: 2 2 16q+ 16q− k + 2 − 2 Jz = |r | − |r | 2m |den(+)|2 0 |den(−)|2 0
(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 figures display the spin dynamics associated with those different channels. This different “precession 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/(k↑F − k↓F ). 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 = −∆ ~ × h~si 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: Ψ1 (x) rF+ eik+ x + lF+ e−ik+ l = Ψ(x) = Ψ2 (x) rF− eik− x + lF− e−ik− l
(B.27)
and the local spin average is as usual 12 Ψ† (x)~τ Ψ(x) and in terms of the components we have:
Ψ†1 Ψ2 + Ψ1 Ψ†2
h~s(x)i = i(Ψ†1 Ψ2 − Ψ1 Ψ†2 ) Ψ†1 Ψ1 − Ψ2 Ψ†2
(B.28)
let focus on the term Ψ†1 Ψ2 whose real and imaginary components give us the average spin along the axes perpendicular to the magnetization. Ψ†1 Ψ2 = (rF+ eik+ x + lF+ e−ik+ x )† ∗ (rF− eik− x + lF− e−ik− x )
(B.29)
−k− )x = (rF+ )† rF− e−i(k+ −k− )x + (rF+ )† lF− e−i(k+ +k− )x + (lF+ )† rF− ei(k+ +k− )x + (lF+ )† lF− ei(k+(B.30)
Using the integral:
Z
0
L
eiaL − 1 eiax dx = ia
(B.31)
we can calculate the sum over space of Ψ†1 Ψ2 : hΨ†1 Ψ2 i = (rF+ )† rF−
e−iδL − 1 e−i∆L − 1 ei∆L − 1 eiδL − 1 +(rF+ )† lF− +(lF+ )† rF− +(lF+ )† lF− −iδ −i∆ i∆ iδ (B.32)
where we have introduced the symbols δ = k+ − k− and ∆ = k+ + k− . Now, we have: (rF+ )† rF− = 4
(1 + q+ )(1 + q− ) iδL −iδL + † − i∆L K + eiδL K −i∆L e e K − e K − e ++ +− −+ −− (r0 ) r0 |den(+)|2 |den(−)|2 159
(1 + q+ )(−1 + q− ) i∆L −iδL −i∆L i∆L K + eiδL K e e K − e K − e (r0+ )† r0− ++ +− −+ −− |den(+)|2 |den(−)|2 (−1 + q+ )(1 + q− ) −i∆L −iδL + † − −i∆L i∆L K + eiδL K (lF+ )† rF− = 4 e e K − e K − e ++ +− −+ −− (r0 ) r0 |den(+)|2 |den(−)|2 (−1 + q+ )(−1 + q− ) −iδL −iδL i∆L K + eiδL K −i∆L (lF+ )† lF− = 4 e K − e K − e e (r0+ )† r0− ++ +− −+ −− |den(+)|2 |den(−)|2 (rF+ )† lF− = 4
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: (r0+ )† r0− (r0+ )† r0− (1 + q+ )(1 + q− ) (1 + q+ )(−1 + q− ) K + 4 K + ++ +− |den(+)|2 |den(−)|2 −iδ |den(+)|2 |den(−)|2 −i∆ (r0+ )† r0− (−1 + q+ )(−1 + q− ) (r0+ )† r0− (−1 + q+ )(1 + q− ) K K + 4 − 4 (B.33) −+ −− |den(+)|2 |den(−)|2 i∆ |den(+)|2 |den(−)|2 iδ
hΨ†1 Ψ2 i = −4
The last equation can be simplified by introducing the symbol K±,± = (1 ± q+ )3 (1 ± q− )3 and so we get: hΨ†1 Ψ2 i
=
−4i (r + )† r0− |den(+)|2 |den(−)|2 0
= −2gi(r0+ )† r0−
K++ − K−− K+− − K−+ + (B.34) δ ∆ (B.35)
For an incoming spin current polarized along the axis n ˆ = (θ, φ) the entering spinor is:
r0+ r0−
φ i 2 cos e = θ −i φ − sin 2 e 2 θ 2
(B.36)
then (r0+ )† r0− = − 12 sin θe−iφ , and in that way we get: hΨ†1 Ψ2 i = gi sin θe−iφ
(B.37)
Then the components of the average spin are: hsx i = g sin θ sin φ,
hsy i = −g sin θ cos φ
(B.38)
the above equation can be written in vectorial terms: ~ ×n h~s⊥ i = g∆ ˆ 160
(B.39)
~ × The last equation reproduces the expected Sloncewski term since ~Γtot = −g∆
~ ×n (∆ ˆ ).
We have proved then that in general the average spin will be given by an form like: ~ ×n ~ h~si = g∆ ˆ + 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 i eikx + |−∞L i e−ikx ; Ψ∞ (x) = |∞R i eikx + |∞L i e−ikx ,
(C.1) (C.2)
where |∞R i , · · · and |∞L i , · · · are the spinor coefficients of the right and left goers, respectively. The scattering matrix expresses the outgoing spinors in terms of the |−∞L i |−∞R i =S with S in turn expressed in terms incoming spinors: |∞R i |∞L i r t′ . We choose the direction of 2×2 transmission and reflection matrices S = t r′ 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 + St n · 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)
(rt¯rs + rs¯rt ) + (tt¯ts + ts¯tt ) = 0,
(C.6)
(rs¯ts + ts¯rs ) + (rt¯tt + tt¯rt ) = 0,
(C.7)
(rt¯ts + ts¯rt ) + (rs¯tt + tt¯rs ) = 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, 163
(C.13)
sin 2Θ cos2 Φ tan2 Φ cos (νrt − νtt ) + cos (νrs − νts ) = 0,
sin 2Φ sin 2Θ (cos (νrt − νts ) + cos (νrs − νtt )) = 0,
(C.14) (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: π νts − νtt = (2m + 1) , 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 − (2n + 1)π 2
(C.18)
Then if we write νrs = η + ǫ and νrt = η − ǫ we obtain ǫ = −(2m + 1) π4 − (2n + 1 π2 . 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 , 164
(C.20)
C.2
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)
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 ) t s + t t n1 · τ
ts + tt n1 · τ . e−iδ (rs + rt n1 · τ )
while in the antiferromagnet, e−iδ (rs + rt n2 · τ ) t′s . SAFM = ts eiδ (rs − rt n2 · τ )
(C.24)
(C.25)
The transmission amplitude for this system is given by: T = Tr t†1 K† t†2 t2 Kt1 .
(C.26)
If we use the parametrization we get: t†2 t2 = cos2 ΘAFM ,
(C.27)
t1 t†1 = cos2 ΘFM ,
(C.28)
Then, the transmission becomes: T = cos2 ΘAFM cos2 ΘFM Tr 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: 5 1 − (cos 2Θ1 + cos 2Θ2 − cos 2Θ1 cos 2Θ2 ) ; 2 2 = −4 cos χ R T
Λs =
(C.31)
Λp
(C.32)
where we have introduce the phase χ = 2δ +νAFM +νFM , the joint reflection probability (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 4RS2 + R3 γ 2 − 4S(1 + γR2 ) cos χ + 2Rγ cos 2χ; Γp = 4 R T 2R S − cos χ 1 − γR2 ; Γd = 4 R2 T2
(C.33) (C.34) (C.35)
where γ = 1 + sin2 Φ1 .
C.3
Outline of a proof of the periodicity of the transverse 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 antiferromagnets 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 reduces 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 antiferromagnetic 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′ e2ikx0 r . T(x0 )S = (C.36) −2ikx ′ 0 t e r
The scattering matrix S12 for two scatterers described by S1 = spin-dependent ′ ′ r t r t 1 1 and S2 = 2 2 is t1 r′1 t2 r′2
S12 =
r1 + t′1 r2 Kt1
t′1 Kt′2
t2 Kt1
r′2 + t2 Kr′1 t′2
.
(C.37)
where we have defined the multiple reflection kernel K = (1 − r′1 r2 )−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 i: |0R i = Kt1 |−∞R i ; |0L i = r2 Kt1 |−∞R i ; 167
|−∞L i =
r1 + t′1 r2 Kt1 |−∞R i ;
|∞R i = t2 Kt1 |−∞R i ,
(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) = h−∞R |S α | − ∞R i + h−∞L |S α | − ∞L i + n o h−∞R |S α | − ∞L i e−2ikx + h−∞L |S α | − ∞R i e2ikx , Sα0 (x) = h0R |S α |0R i + h0L |S α |0L i + n o h0R |S α |0L i e−2ikx + h0L |S α |0R i e2ikx ,
Sα∞ (x) = h∞R |S α |∞R i + h∞L |S α |∞L i + n o h∞R |S α |∞L i e−2ikx + h∞L |S α |∞R i 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: n
o h0R |S α |0L i e−2ikx + h.c. = E D −∞R |t†1 K† S α r2 Kt1 | − ∞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: n
o n o h0R |S α |0L i e−2ikx ∼ |t1 |2 Tr K† S α r2 K e−2ikx .
The trace itself can be simplified: Tr K† S α r2 K = rs2 Tr K† S α K + rt2 nβ2 Tr K† S α S β K . 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
and
|Λ|2 Tr K† (n⊥ · S) K = 4RS sin (χ) sin2 θ,
(C.39)
|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 + 4R2 S 2 + 4R(1 + R2 )S cos χ + 4RT (2RS + (1 + R2 ) cos χ) cos θ + 4R2 T 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−∞ = h−∞R |S α | − ∞R i + h−∞L |S α | − ∞L i ; Sα0 = h0R |S α |0R i + h0L |S α |0L i ; Sα∞ = h∞R |S α |∞R i + h∞L |S α |∞L i , These expressions can be reduced to expressions involving only |−∞R i. We obtain: E α ′ S α + r†1 + t†1 K† r†2 t′† S r + t r Kt ; 1 1 2 1 1 D E = t†1 K† S α + r†2 S α r2 Kt1 ; D E = t†1 K† t†2 S α t2 Kt1 .
Sα−∞ = Sα0 Sα∞ 1
D
This equality between this two forms, seems odd, since apparently involve the equality of a linear and a bilinear form of the pauli matrices. However since the reflection kernel also has spin triplet terms there is no contradiction.
169
where the expectation value h·i = h−∞R | · |−∞R i. Summing over the incoming unpolarized current those expectation values become a trace: α ′ r†1 + t†1 K† r†2 t′† S r + t r Kt ; 2 1 1 1 1 = Tr t†1 K† S α + r†2 S α r2 Kt1 ; = Tr t†1 K† t†2 S α t2 Kt1 .
Sα−∞ = Tr Sα0 Sα∞
We take the difference: Sα0 − Sα∞ = Tr t†1 K† S α + r†2 S α r2 Kt1 − Tr t†1 K† t†2 S α t2 Kt1 ,
(C.42)
which can be written as: Sα0 − Sα∞ = |t1 |2 Tr K† S α + r†2 S α r2 − t†2 S α t2 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 ´ ´n ˜ez was born in Santiago de Chile, Chile in May 11th 1976. He took Alvaro S. Nu the Bs. Sc. Physics degree from the Facultad de Ciencias F´ısicas y Matem´ aticas de la Universidad de Chile. He is married to Viviana Jeria, and has a 2 year old ´n ˜ez-Jeria. daughter named Penelope Millaray Nu
Permanent Address: UT at Austin, Physics Department, 1 University Station C1600, Austin, TX 78712
This dissertation was typeset with LATEX 2ε by the author.
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