mshop.physics.harvard.edu · thesis advisor author bertrand i. halperin yaroslav tserkovnyak spin...

Spin and Charge Transfer in SelectedNanostructures

A thesis presented

by

Yaroslav Tserkovnyak

to

The Department of Physics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Physics

Harvard University

Cambridge, Massachusetts

February 2003

Thesis advisor Author

Bertrand I. Halperin Yaroslav Tserkovnyak

Spin and Charge Transfer in Selected Nanostructures

Abstract

The general theme in this thesis is the interplay between electron spin and charge in

nanoscale transport phenomena. The main presentation is divided into three indepen-

dent chapters. In chapter 2, we propose a mechanism that explains the excess damping

and dynamic exchange interactions which are observed in ferromagnet/paramagnet

hybrids. A moving ferromagnetic magnetization emits spin current into adjacent con-

ductors, exerting a relaxation torque and transferring angular momentum out of the

ferromagnet. This spin angular momentum can scatter back, relax in a nonmagnetic

spacer, or be absorbed by a second ferromagnet. In the first case, the macroscopic

magnetization dynamics is not affected; in the second case, the magnetization motion

is nonlocally damped by spin-flip scattering processes in the spacer; and in the latter

case, the two ferromagnets become dynamically coupled by an exchange of itinerant

spins, resulting in collective excitation modes. This relaxation and coupling can be

large and, in some cases, dominant over other mechanisms in ultrathin films and

nanoparticles.

Chapter 3 is devoted to studying electronic transfer in tunnel-coupled quantum

wires of exceptional quality, fabricated at the cleaved edge of a GaAs/AlGaAs bilayer

heterostructure. Tunneling between such wires depends on the one-electron confine-

ment profiles along the wires as well as on electron-electron interactions in the system.

Abstract iv

Observed oscillations in the differential conductance, as a function of bias voltage and

applied magnetic field, provide direct information on the shape of the confining po-

tential; superimposed modulations indicate the existence of two distinct excitation

velocities, as expected from spin-charge separation. Another interesting interplay be-

tween the finite size and electron-electron interactions occurs at low energies (voltage

and temperature), when the measured tunneling exponent is determined not only

by the strength of the electron-electron interactions but also by the extent of the

tunneling region compared to the scale set by the applied voltage and temperature.

Finally, in chapter 4 we perform a Monte Carlo study of non-Abelian statistics of

quasiholes in the Moore-Read (MR) quantized Hall state. First, a general framework

for numerical adiabatic braiding of quasiholes in fractional quantum Hall systems

is developed employing Metropolis Monte Carlo method. We then investigate, in

some detail, the MR state which is believed to occur in nature as an incompressible

quantum Hall state at filling factors 5/2 and 7/2 (corresponding to the first excited

Landau level). The non-Abelian statistics of MR quasiholes is demonstrated explic-

itly, confirming the results predicted by conformal field theories.

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vCitations to Previously Published Work . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Magnetoelectronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Magnetoelectronic Dc Circuit Theory . . . . . . . . . . . . . . 81.2.2 Magnetization Dynamics: Spin Pumps and Spin Sinks . . . . 111.2.3 Dynamic Ferromagnetic Exchange . . . . . . . . . . . . . . . . 16

1.3 One-Dimensional Conductors . . . . . . . . . . . . . . . . . . . . . . 221.3.1 Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Elementary Excitations: Spinons and Holons . . . . . . . . . . 231.3.3 Spin-Charge Separation . . . . . . . . . . . . . . . . . . . . . 261.3.4 Luttinger Model . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.5 Luttinger-Liquid Conjecture . . . . . . . . . . . . . . . . . . . 291.3.6 Tunnel-Coupled Double Wires . . . . . . . . . . . . . . . . . . 31

1.4 Quasihole Statistics in Quantized Hall States . . . . . . . . . . . . . . 331.4.1 Laughlin and Standard Hierarchy States . . . . . . . . . . . . 331.4.2 Paired (Moore-Read) and Parafermion States . . . . . . . . . 361.4.3 Monte Carlo Approach . . . . . . . . . . . . . . . . . . . . . . 39

2 Dynamic Phenomena in Magnetic Multilayers 412.1 Enhanced Gilbert Damping . . . . . . . . . . . . . . . . . . . . . . . 42

2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1.2 Single Ferromagnetic Films . . . . . . . . . . . . . . . . . . . 452.1.3 Precession-Induced Spin Pumping . . . . . . . . . . . . . . . . 552.1.4 Spin Backflow in F/N and N/F/N Structures . . . . . . . . . 59

v

Contents vi

2.1.5 Damping in F/N1/N2 Trilayers . . . . . . . . . . . . . . . . . 672.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.2 FMR-Operated Spin Battery . . . . . . . . . . . . . . . . . . . . . . . 742.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.2.2 Functionality and Maximum Loads . . . . . . . . . . . . . . . 752.2.3 Nuclei Polarization and the Overhauser Field . . . . . . . . . 822.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.3 Dynamic Exchange Coupling . . . . . . . . . . . . . . . . . . . . . . . 842.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.3.2 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . 852.3.3 Experimental Procedure and Results . . . . . . . . . . . . . . 892.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.4 Precessional Stiffness of Spin Valves . . . . . . . . . . . . . . . . . . . 942.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.4.2 Dynamic Exchange . . . . . . . . . . . . . . . . . . . . . . . . 962.4.3 Angle-Dependent Stiffness . . . . . . . . . . . . . . . . . . . . 1002.4.4 Zero-Temperature Switching Current . . . . . . . . . . . . . . 1032.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3 Tunneling between Parallel Quantum Wires 1063.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.2 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.2.1 Fabrication of the Samples . . . . . . . . . . . . . . . . . . . . 1083.2.2 Measurement on an Isolated Tunnel Junction . . . . . . . . . 110

3.3 Description of the Experimental Results . . . . . . . . . . . . . . . . 1113.3.1 Dispersions of Elementary Excitations in the Wires . . . . . . 1113.3.2 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143.3.3 A Dip in the Tunneling Conductance . . . . . . . . . . . . . . 118

3.4 Theory and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.4.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . 1213.4.2 Interference Pattern . . . . . . . . . . . . . . . . . . . . . . . 1233.4.3 Asymmetry due to Soft Boundaries . . . . . . . . . . . . . . . 1243.4.4 Modulation due to Spin-Charge Separation . . . . . . . . . . . 1303.4.5 Upper Crossing Point . . . . . . . . . . . . . . . . . . . . . . . 1403.4.6 Dephasing of the Oscillations . . . . . . . . . . . . . . . . . . 1413.4.7 Zero-Bias Anomaly . . . . . . . . . . . . . . . . . . . . . . . . 1443.4.8 Crossing Points . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.4.9 Direct Tunneling from the 2DEG . . . . . . . . . . . . . . . . 148

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Contents vii

4 Non-Abelian Braiding of Moore-Read Quasiholes 1534.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.2 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.3 Results for the Pfaffian Wave Function . . . . . . . . . . . . . . . . . 1594.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A Adiabatic Spin Pumping (Appendix to Chapter 2) 166

B Appendices to Chapter 3 170B.1 Independent-Mode Approximation . . . . . . . . . . . . . . . . . . . . 170B.2 Direct Tunneling from the 2DEG . . . . . . . . . . . . . . . . . . . . 172

Bibliography 174

Citations to Previously Published Work

Chapter 2 is based on a sequence of papers:

• “Enhanced Gilbert Damping in Thin Ferromagnetic Films,” Y. Tserkovnyak,A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett., 88:117601 (2002)

• “Spin Pumping and Magnetization Dynamics in Metallic Multilayers,” Y. Tserkovnyak,A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 66:224403 (2002)

• “Spin Battery Operated by Ferromagnetic Resonance,” A. Brataas, Y. Tserkovnyak,G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B, 66:060404(R) (2002)

• “Dynamic Exchange Coupling in Magnetic Bilayers,” B. Heinrich, Y. Tserkovnyak,G. Woltersdorf, A. Brataas, R. Urban, and G. E.W. Bauer, submitted to Phys.Rev. Lett. (preprint cond-mat/0210588)

• “Dynamic Stiffness of Spin Valves,” Y. Tserkovnyak, A. Brataas, and G. E. W.Bauer, submitted to Phys. Rev. B (preprint cond-mat/0212130)

In addition, the content of

• “Dynamic Exchange Coupling and Gilbert Damping in Magnetic Multilayers,”Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J. Appl. Phys., in press (forproceedings of the MMM’02 conference)

helped me to compose parts of section 1.2 of the Introduction. All this work is closelyrelated to the so-called magnetoelectronic circuit theory [19, 20] which we extendedand generalized in

• “Shot Noise in Ferromagnet–Normal Metal Systems,” Y. Tserkovnyak and A.Brataas, Phys. Rev. B, 64:214402 (2001)

• “Current and Spin Torque in Double Tunnel Barrier Ferromagnet-Superconductor-Ferromagnet Systems,” Y. Tserkovnyak and A. Brataas, Phys. Rev. B, 65:094517(2002)

• “Universal Angular Magnetoresistance and Spin Torque in Ferromagnetic/NormalMetal Hybrids,” G. E. W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, andA. Brataas, Phys. Rev. B, in press

Chapter 3 is derived from two articles:

• “Finite-Size Effects in Tunneling between Parallel Quantum Wires,” Y. Tserkovnyak,B. I. Halperin, O. M. Auslaender, and A. Yacoby, Phys. Rev. Lett., 89:136805(2002)

• “Interference and Zero-Bias Anomaly in Tunneling between Luttinger-LiquidWires,” Y. Tserkovnyak, B. I. Halperin, O. M. Auslaender, and A. Yacoby,submitted to Phys. Rev. B (preprint cond-mat/0302274)

Finally, we used the following article in writing chapter 4:

• “Monte Carlo Evaluation of Non-Abelian Statistics,” Y. Tserkovnyak and S. H.Simon, Phys. Rev. Lett., 90:016802 (2003)

Acknowledgments

I was very fortunate to have met exceptional pedagogues at each important stage

of my educational career. Without any of them, my anticipated completion of Ph.D.

requirements in physics would have probably being very unlikely if not impossible. I

would like to list them in the chronological order.

My high-school physics teacher in Ukraine Lyudmyla Zasedka was the first to show

me ways of having simple fun doing physics. She enticed my mind with the ideals of

elegant and beautiful physical reality and helped me to start seeing and appreciating

it.

During my first year in Ukrainian Medical University, guidance of biophysics pro-

fessor Alexander Chalyi have led me back to the light of a scientific pursuit that

started fading away during sleepless nights of memorizing medical terminology. The

following year I joined Physics Department of the University of British Columbia to

meet yet another extraordinary person. Julyet Benbasat encouraged and assisted me

in any imaginable way boosting my self-confidence and aspirations that still continue

to lead me not only in scientific but in various other pursuits.

Arne Brataas have guided me through my first scientific projects and publications

at Harvard University. With his help I stepped onto a qualitatively new level of having

fun in physics. Finally, I am particularly grateful to Bert Halperin, who will always

remain for me the golden standard of intellectual excellence and scientific conduct.

Looking for an advisor I found a mentor who despite his weight and respect in the

community was (and still is) constantly helping me to look for my own voice in

physics.

Dedicated to my parents Igor and Nina

I glimpse at window–gorgeous sight!Last breath of summer, all is quiet...

I look again–and days of pastLook back at me, they shake off dust,And start to stare, to cry, to laugh,

They make me drunk, they are to last!..

Chapter 1

Introduction

The questions: Why and When and What?The answers: Since—Forever—All.Emotions, thinking, sadness, hypeWill overwhelm defrosted minds.

New sun will come, new sun will fall,New sun will come again... and fall...

Oh, God! Please give us more control.

Please let us play with sun like ball,And eat the moon like cheese with holes,

We’ll toss a star like shiny toy,We’ll let it come and let it go.

And our hearts will rise to glow,And our minds will cease to blow.

But while we wait for it and hope,Let those things enjoy their role,

Let’s dream and cry, let’s love and hate,The sun will rise, but stars won’t fade.

1

Chapter 1: Introduction 2

1.1 Preliminaries

Approximating electrons as spinless fermions can be a reasonable starting point

in understanding electronic transport phenomena in some systems. While it is justi-

fiable in many cases, a large number of interesting effects require properly taking into

account the electron’s spin degree of freedom.1 It is certainly the case, e.g., when one

deals with structures which are known to exhibit magnetism.

Chapter 2 contains a compilation of our work on spin-polarized electron transport

in magnetic multilayers in the dynamic regime, i.e., when one or more of the (mon-

odomain) ferromagnetic magnetizations are driven by, e.g., an applied microwave

field or a current bias. Spin polarization of the electric current in magnetic hybrids

arises due to exchange splitting of conduction bands corresponding to different spin

orientations. For example, it is easy to understand the so-called spin-valve effect in

ferromagnet/normal-metal/ferromagnet structures, where the relative orientation of

the two magnetic layers determines the total resistance of the trilayer. Such systems

found important technological applications as the realignment of the magnetizations

between the parallel and antiparallel configurations can be accompanied by a (giant)

magnetoresistance [6, 15, 97] of several hundred per cent when the current is passed

along the layered direction. A basic result that inspired most of the work in chapter 2

is that a moving ferromagnetic magnetization injects spins into adjacent conductors.

1A good example of this is Kondo’s explanation of increasing low-temperature conductivity ofsome bulk metals (e.g., gold) with decreasing temperature, according to which the conductivitybecomes dominated by spin-dependent electron scattering on paramagnetic impurities (e.g., ironatoms). Similar physics also leads to a zero-bias anomaly in transport through Coulomb-blockadedodd electron-number quantum dots: The spin-degenerate state of the unbalanced electron (playingthe role of a paramagnetic inclusion in the Kondo picture) opens a quantum conductance channelthrough the quantum dot at zero temperature.


This leads to the magnetization damping as the process results in energy transfer be-

tween the coherent magnetization motion and the spin-accumulation buildup which

can eventually relax to the lattice as heat, see section 2.1. This spin accumulation

can also be detected by another ferromagnet, resulting in a long-range dynamic ex-

change coupling which has already been observed experimentally, as we discuss in

section 2.3. (See also section 2.4.) In addition, the spin accumulation can be drawn

out of the system as a pure spin current. The latter possibility is used in section 2.2

to formulate a concept of the spin battery operated by the ferromagnetic resonance

(FMR).

Relevance of the electron spin is also apparent in spin-charge separation of one-

dimensional (1D) conductors: Unlike the higher-dimensional (i.e., 2D and 3D) elec-

tron gases which can usually be well described by Landau’s Fermi-liquid theory,2 1D

systems need a special treatment because of the phase-space constraints for electron

scattering. Low-energy properties of gapless 1D electron gases are in general de-

scribed by Luttinger-liquid theory [45, 138]. One of the fundamental results of the

latter is that the spectral function of electrons in 1D has two singular peaks: one

corresponding to the spin- and the other to the charge-density waves.3 In chapter 3

we investigate consequences of spin-charge separation in conductance measurements

on cleaved-edge semiconductor double quantum wires weakly coupled via a tunnel

barrier. Simultaneous and independent control of energy and momentum of tunneled

electrons allows to study dispersion relations of elementary excitations in the sys-

tem. We mostly focus on certain peculiarities of the observed electron wave-function

2At least in the case of repulsive electron interactions and at low magnetic fields.3This should be contrasted to Fermi liquids which have a single quasiparticle peak in the spectral

function, similarly to the free-electron gases.


diffraction patterns, decoding of which gives a wealth of information about electron

confinement along the wires and about electron-electron interactions. In particular,

we show a direct evidence of spin-charge separation observed via existence of two

distinct velocities in one-particle correlation functions of individual wires.

Chapter 4 deals with quantum Hall physics of 2D electron gases (such as formed

in a GaAs/AlGaAs inversion layer or a quantum well). A strong magnetic field

applied perpendicular to a 2D electron gas is assumed to split spin-up and spin-

down Landau levels by a gap corresponding to exchange-enhanced Zeeman energy.

Available conduction electrons are taken to occupy only the lowest of the two spin

bands and the excitations to the other band via spin flips are disregarded. The

main focus of the chapter is the non-Abelian transformation of a degenerate ground

state when a number of quasiholes are adiabatically braided. We develop a Monte

Carlo method to explicitly study quasihole statistics corresponding to this braiding

and apply it to the Moore-Read (MR) wave function [89] for the ν = 1/2 fractional

quantum Hall (spin-polarized) state. The MR state is thought to describe the 5/2 and

7/2 plateaus, which are the only even-denominator plateaus observed to date in single-

layer samples. This ν = 1/2 state is particularly intriguing as its generalizations result

in both even- and odd-denominator filling fractions, the latter of which are believed

to be competitive with the standard hierarchies [46, 47] in the first excited Landau

level.

In the remainder of the Introduction, we set the stage for the aforementioned

discussions. In Sec. 1.2 we briefly recall some basic results that stimulated the rapid

growth of the field of magnetoelectronics before discussing our work on dynamic phe-


nomena in magnetic hybrids in Secs. 1.2.1, 1.2.2, and 1.2.3. Chapter 2 contains a

thorough account of that work. Introductory material for chapter 3 is presented in

Sec. 1.3 which reviews transport properties of one-dimensional conductors within the

Hubbard model and the Luttinger-liquid picture. Sec. 1.4 contains some basic mate-

rial concerning statistics of charged excitations in incompressible fractional quantum

Hall liquids, including both the standard-hierarchy and paired states. The latter is

studied in detail in chapter 4.


1.2 Magnetoelectronics

Magnetoelectronic circuits have recently attracted considerable interest due to

their potential for nonvolatile magnetic random-access memories (MRAM) and sen-

sors as well as for fundamental studies of spin transport in magnetic and nonmagnetic

devices.

The spin transfer in ferromagnet/paramagnet hybrids causes a number of exciting

phenomena. Equilibrium spin currents across a thin normal-metal (N ) layer separat-

ing two ferromagnetic (F ) films explain the oscillating RKKY-type exchange coupling

[118] as a function of the spacer thickness. This nonlocal exchange interaction may

stabilize an antiparallel equilibrium configuration in magnetic multilayers [43] which

display a giant magnetoresistance (GMR) [6, 15, 97]. For thicker N spacers or tun-

neling barriers, this static ferromagnetic exchange vanishes but the F layers can still

be made to communicate by driving a dc current through the system. For exam-

ple, depending on the direction of the current flow perpendicular to a layered F/N/F

spin-valve structure, a (spin) torque can be exerted on the magnetizations. When one

magnetization is fixed, this torque favors either parallel or antiparallel configurations,

and may lead to a switching of the other magnetization [119, 120, 91, 65]. Transport

in the dynamic regime of moving magnetization directions has traditionally attracted

relatively little attention, however. We will discuss our contribution to this field in

chapter 2.

Recently [130] we launched the idea of spin pumping in excited/driven magnetic

nanostructures: A moving ferromagnetic magnetization emits spins into adjacent

conductors, exerting a relaxation torque and transferring an angular momentum out


of the ferromagnet. The spin-pumping concept has proven to be fruitful in under-

standing both the nonlocal damping mechanism [130, 131], see Sec. 2.1, and dynamic

exchange interaction [51, 129], see Secs. 2.3 and 2.4, in hybrid F/N systems. The

functionality of magnetic devices, such as MRAM, is strongly affected by relaxation

characteristics of the magnetic media. A thorough understanding of the underly-

ing fundamental processes is a key to establishing ways to control the device speed,

operation threshold bias, and power consumption. Since the nonlocal relaxation and

coupling can be large and, in some cases, dominant over other mechanisms in ultrathin

films, controlling them by choosing the right composition, geometry, and magnetic

configuration of hybrid F/N systems might help to engineer next-generation mag-

netic devices. In addition, by discussing a spin-battery concept [21] in Sec. 2.2, we

demonstrate that the spin-pumping mechanism can also be used to propose novel

devices.

We review the key results of our work in the following three sections as follows. In

Sec. 1.2.1 the magnetoelectronic circuit theory [19] is generalized to treat intermetallic

interfaces in diffuse systems [9]. The conductance matrix defined in Sec. 1.2.1 is a

central concept for the developments of the entire chapter 2. Sec. 1.2.2 previews our

theory on the spin pumping in magnetic hybrids, which will be elaborated in detail

in Sec. 2.1 and used for formulating the spin-battery concept in Sec. 2.2. Finally,

we discuss the origin of the long-ranged dynamic exchange in ferromagnetic bilayers

in Sec. 1.2.3. A detailed theory (as well as an experimental confirmation) of this

phenomenon is presented in Sec. 2.3 which, in turn, serves as a basis for the theory

of the dynamic stiffness of spin valves in Sec. 2.4.


1.2.1 Magnetoelectronic Dc Circuit Theory

Magnetoelectronics achieves new functionalities by incorporating ferromagnetic

materials into electronic circuits. The GMR, i.e., the dependence of the electrical

resistance on the relative orientation of the magnetizations of two ferromagnets in an

F/N/F spin valve, can serve a useful purpose in magnetic read heads of high infor-

mation density magnetic storage systems. Usually, such a device is viewed as a single

bit, the magnetization vectors being either parallel or antiparallel. Early seminal

contributions by Slonczewski [119] and Berger [11] revealed new physics and techno-

logical possibilities of noncollinearity, which triggered a large amount of experimental

and theoretical research. An important example is the nonequilibrium spin-current

induced torque (briefly, spin torque) which one ferromagnet can exert on the mag-

netization vector of a second magnet through a normal metal. This torque can be

large enough to dynamically turn magnetizations, which is potentially interesting as

a low-power switching mechanism for MRAM. The spin torque is also essential for

novel magnetic devices such as the spin-flip transistor [19, 146] and for the Gilbert

damping of the magnetization dynamics in thin magnetic films [130, 131].

Recently, two theoretical approaches have been developed which address charge

and spin transport in diffusive noncollinear magnetic hybrid structures. The “magne-

toelectronic circuit theory” [19, 20] is based on the division of the system into discrete

resistive elements over which the applied potential drops, and low-resistance nodes

at quasiequilibrium (as in Fig. 1.1a). The electrical properties are then governed by

generalized Kirchhoff’s rules and can be computed easily. Each resistor is thereby

characterized by four material parameters, the spin-up (σ =↑) and spin-down (σ =↓)


Figure 1.1: Different realizations of perpendicular spin valves: (a) Highly resistivejunctions like point contacts and tunneling barriers limit the conductance. (b) Spinvalve in a geometrical constriction amenable to the scattering theory of transport.(c) Magnetic multilayers with transparent interfaces. θ is the angle between magne-tization directions.

conductances

gσσ =∑mn

[δmn − |rσ

mn|2], (1.1)

as known from the scattering theory of transport [28], as well as the real and imaginary

part of the “mixing conductance”

g↑↓ =∑mn

[δmn − r↑mn(r↓mn)∗

], (1.2)

where rσmn is the reflection coefficient between mth and nth transverse modes of an

electron with spin σ in the normal metal at a contact to a ferromagnet.

An alternative approach was proposed by Waintal et al. [140] who studied the

random-matrix theory of transport in noncollinear magnetic systems as sketched in

Fig. 1.1b. Their formalism did not require the assumption of highly resistive elements,

but the algebra of the 4 × 4 scattering matrices in spin space appeared so complex

that analytical results were obtained in limiting cases only.

Both theories are not valid in the limit of transparent interfaces in a diffuse envi-

ronment (see Fig. 1.1c) like the perpendicular spin valves, studied thoroughly by the

Michigan State University collaboration [102, 7] and others [39]. These studies pro-


vided a large body of evidence for the two-channel (i.e., spin-up and spin-down) series

resistor model and a wealth of accurate transport parameters like the interface re-

sistances for various material combinations, in the regime of collinear magnetization

configurations. Theoretically, transport through transparent interfaces in a diffuse

environment has been studied for collinear magnetizations by Schep et al. [111]. Un-

der the condition of isotropy of scattering by disorder, it was found that the circuit

theory is recovered if we renormalize the interfacial conductance parameters (1.1) for

each spin σ as

1

gσσ=

1

gσσ− 1

2

(1

NN

+1

NFσ

), (1.3)

where NFσ and NN are the number of transverse modes of the bulk materials on

both sides of the F/N contact. Physically, in Eq. (1.3) half of the Sharvin contact

resistances (i.e., N−1) is subtracted from the Landauer-Buttiker result of scattering

theory. This correction is large for intermetallic interfaces and essential to obtain

agreement between experimental results and first-principles calculations [111, 125,

147]. In Ref. [9] we further showed that the entire magnetoelectronic circuit theory

for the noncollinear transport [19, 20] can be recovered if, in addition to Eq. (1.3),

we also correct the mixing conductance for the (half of the) normal-metal contact

resistance:

1

g↑↓=

1

g↑↓− 1

2NN

. (1.4)

The tildes in Eqs. (1.3) and (1.4) denote the renormalized conductances, which reduce

to the bare values, Eqs. (1.1) and (1.2), when the numbers of transverse channels in

the normal metal, NN , and the ferromagnet, NFσ, sufficiently exceed the contact con-

ductances gσσ′ . In the following, we will omit tildes on the conductance parameters,


for notational simplicity, while it is implied throughout the rest of the thesis that we

are using gσσ′ parameters defined in Eqs. (1.3) and (1.4).

1.2.2 Magnetization Dynamics: Spin Pumps and Spin Sinks

Consider an F/N bilayer as in Fig. 1.2. Without a voltage bias, no spin or charge

currents flow when the magnetization of the ferromagnet is constant in time. When

the magnetization direction starts precessing (as, e.g., under the influence of an ap-

plied magnetic field), a spin current Ipumps is pumped out of the ferromagnet into the

N layer [130, 131]. When the ferromagnetic film is thicker than its transverse spin-

coherence length λsc, d > λsc = π/|k↑F−k↓F |, k

↑(↓)F being the spin-dependent Fermi wave

vectors, this current depends on the interfacial mixing conductance g↑↓ = g↑↓r + ig↑↓i

by

Ipumps =

~4π

(g↑↓r m× dm

dt− g↑↓i

dm

dt

). (1.5)

Here the time-dependent order parameter of the ferromagnet is a unit vector m(t),

assuming a monodomain magnet with a spatially uniform magnetization at all times.

A detailed derivation of Eq. (1.5) based on the scattering-matrix theory is given in

Appendix. A. Alternatively, this result can be derived in the framework of magneto-

electronic circuit theory using only energy and angular-momentum conservation, as

explained in Sec. 2.1.3.

The total spin current, Is, across the F/N interface also has a backflow contri-

bution, Ibacks , in addition to the pumped current, Ipump

s , see Fig. 1.2. The total spin

transfer in the steady state,

Is = Ipumps − Iback

s , (1.6)


µµ

Figure 1.2: Schematic view of the F/N bilayer. Precession of the magnetizationdirection m(t) of the ferromagnet F pumps spins into the adjacent normal-metallayer N by inducing a spin current Ipump

s . This leads to a buildup of the normal-metal spin accumulation which either relaxes by spin-flip scattering or flows backinto the ferromagnet as Iback

s . The N layer here is a film of the same cross section asthe magnetic layer F ; the spin accumulation is position (x) dependent.

is determined self-consistently by the spin accumulation close to the interface. A finite

value of the spin current, Is, would indicate the presence of a spin-sink mechanism

in the normal metal by, e.g., spin-orbit coupling, when the spin angular momentum

is transferred from the electron system to the lattice. In the case of small spin

relaxation, the dynamically created spin accumulation in the normal metal may serve

as a spin-battery device, as discussed in Sec. 2.2. In the opposite spin-flip scattering

regime, the angular-momentum loss of the ferromagnet by Is results in a Gilbert-type

damping of the magnetic precession, see Sec. 2.1.

The spin current out of the ferromagnet carries angular momentum perpendicular

to the magnetization direction, corresponding to a torque τ = −Is on the ferro-

magnetic condensate [119]. Disregarding interfacial spin flips, this torque is entirely


transferred to the magnetization, which is described by a generalized Landau-Lifshitz-

Gilbert (LLG) equation [40, 119]

dm

dt= −γm×Heff + α0m× dm

dt+

γ

MsVIs , (1.7)

where γ is the absolute value of the gyromagnetic ratio, Heff is the effective magnetic

field (including applied, anisotropy, and demagnetization fields), α0 is the dimension-

less intrinsic Gilbert-damping constant, Ms is the saturation magnetization of the

ferromagnet, and V is its volume. Referring to Eq. (1.5) we see that the real part

of the mixing conductance contributes to the damping just like intrinsic bulk con-

stant α0 which is thus smaller than the total Gilbert damping α = α0 + α′, whereas

the imaginary part of the mixing conductance contributes like an effective field. The

additional damping α′ is observable in, for example, FMR spectra. Although not nec-

essarily so for ferromagnetic insulators [53], the mixing conductance for intermetallic

F/N interfaces is to a good approximation real [146] and therefore

α′ = κγ~g↑↓

4πMsV. (1.8)

κ = 1 corresponds to the perfect spin-sink model, when all pumped spins relax in the

N layer and the backflow Ibacks vanishes. κ < 1 corresponds to a finite backflow [131].

The spin-sink capacity of clean normal-metal layers in contact with a ferromag-

netic film, as in Fig. 1.2, is usually governed by spin-orbit scattering processes at

impurities or defects. As the spin-flip probability ε = τel/τso (defined in terms of the

elastic scattering time τel and spin-orbit relaxation time τso) rapidly increases with

the atomic number Z, ε ∝ Z4 [1, 80], we expect a larger spin-sink effect [and therefore

κ in Eq. (1.8)] for heavier metals (impurities as well as hosts). The extent of the hy-

bridization of the conduction bands with p or d orbitals also plays an important role.


µµ

Figure 1.3: Same as Fig. 1.2, but now the normal-metal system is composed of abilayer N1/N2. Ferromagnetic precession pumps spins into the first normal-metallayer N1. The spin buildup in N1 may flow back into the ferromagnet F as spincurrent Iback

s1 , relax in N1, or flow into the second normal-metal layer N2 as spincurrent Iback

s2 . The spin accumulation in N2 is disregarded since the layer is assumedto be a perfect spin sink.

In particular, clean noble metals, Cu, Ag, and Au, with predominantly s conduction

electrons are poor spin sinks with correspondingly small ε, but Pd and Pt, whose con-

duction electrons have significant d character, have a high spin-orbit scattering rates

and are efficient spin sinks. We note that heavy or magnetic impurities can turn

an otherwise poor spin sink into a good one. The hierarchy of the Gilbert-damping

enhancement has been measured for normal-metal buffers of Cu, Ta, Pd, and Pt (in

order of increasing damping) [83], in agreement with the aforementioned arguments

[131]. In particular, the effect of spin pumping on the magnetization dynamics was

shown to be negligible in the case of Cu, while Pt was a nearly perfect spin sink,

resulting in a large increase of the Gilbert constant; this was also experimentally

confirmed by Ingvarsson et al. [58].


To further investigate the spin-sink effect of the normal-metal buffer, we next

consider a more complex system, consisting of a double N layer attached to the

ferromagnet, see Fig. 1.3. An interesting situation arises when the layer N1 is a bad

spin sink, such as Cu, and N2 is a perfect spin sink, such as Pt [131]. Mizukami

et al. [85, 84] experimentally studied the FMR line width in permalloy (Py)/Cu/Pt

composites as a function of Cu (N1 ) width L. Next to L, there are three relevant

length scales in the problem: the Fermi wave length, λF , the elastic scattering mean

free path, λel, and the spin-diffusion length, λsd. If L = 0, the damping enhancement

(1.8) is governed by g↑↓F/N2. A quantum-well modulation of the mixing conductance

sets in when L ∼ λF , i.e., when a thin N1 layer is sandwiched between F and N2. If

L > λF , the spin transport across the N1 spacer may be described by the diffusion

equation [131], provided that either the spacer or the interface is disordered. In this

regime, the spin backflow can be partitioned between the ferromagnet and layer N2,

see Fig. 1.3. The relevant effective mixing conductance g↑↓eff then has to account for

scattering at both interfaces, F/N1 and N1/N2, as well as in the N1 spacer. One

finds that

1

g↑↓eff=

1

g↑↓F/N1

+RN1 +1

gN1/N2

, (1.9)

where RN1 is the resistance (per spin, in units of h/e2) of the N1 layer, gN1/N2 is the

one-spin conductance of the N1/N2 interface, and we assumed that L < λsd. If in

addition L < λel, the bulk scattering, RN1, can be disregarded and the total resistance

is given simply by the sum of the interfacial contributions. λF thus sets the length

scale for a sharp drop in the Gilbert-damping enhancement, followed by an algebraic

decay with the effective mixing conductance (1.9) for L > λF . Such a damping drop


was reported in Ref. [84] for Py/Cu(L)/Pt hybrids.

The regime L > λF was studied in Refs. [85, 84], where a smooth algebraic decay

of the damping enhancement was measured for L < λsd followed by an exponen-

tial suppression for thicker Cu spacers, in excellent agreement with our theory [131].

Mizukami et al. [84] also offered an explanation for their measurements, using a

phenomenological theory due to Silsbee et al. [115]. The theoretically calculated

damping profiles, α(L), as a function of L in Refs. [131] and [84] are barely distin-

guishable. In our opinion, the phenomenological approach employed in the latter

has severe limitations, however. In particular, the separation between localized and

conduction electron spins [115, 84] is not justified for itinerant ferromagnets like the

transition metals. Furthermore, this separation leads to wrong results for insulating

ferromagnets which, we believe, can generate a spin current into adjacent nonmag-

netic conductors in the same way as a conducting ferromagnet [131], in contrast to

the prediction of Ref. [115].

The generality of our approach is demonstrated in the following by showing its

validity for a qualitatively different system of two magnetic layers separated by a

normal-metal film with negligible spin flip. We find that the same mechanism (namely

the spin pumping) is responsible for the damping enhancement in coupled ferromag-

nets as well as the F/N structures discussed so far.

1.2.3 Dynamic Ferromagnetic Exchange

In F/N/F spin valves, the spin pumping causes qualitatively different effects in

addition to the ones just described. Consider a system shown in Fig. 1.4. In the


µµ

2

2

1

1

Figure 1.4: Two ferromagnetic films F1 and F2 connected by a normal-metal spacerN of width L. When the magnetization directions m1 and m2 precess, spin currentsIpumps1 and Ipump

s2 are pumped into the normal metal. The spin-dependent chemical-potential imbalance in N causes the backflow of spin currents Iback

s1 and Ibacks2 .

following we take L > λF , so that quantum coherence (in particular the static ex-

change coupling) can be disregarded, but L < λel, so that the spins of electrons are

transferred between the two ferromagnets ballistically.4

The total spin current pumped into the normal metal,

Ipumps =

~4π

(g↑↓1 m1 ×

dm1

dt+ g↑↓2 m2 ×

dm2

dt

), (1.10)

has contributions from both Fi/N interfaces, see Eq. (1.5). Here the small imaginary

part of the mixing conductance [146] is again disregarded. This pumping of spins

4The ferromagnetic exchange interaction discussed in this section is quite different from the staticRKKY coupling. The latter is oscillatory in thickness L of the paramagnetic spacer with period ofthe Fermi wave length and, for realistic disordered structures, rapidly decaying after only severalperiods of oscillation. This static coupling requires quantum coherence as well as the differencebetween spin-up and spin-down reflection coefficients across the spacer. The dynamic exchange, onthe other hand, needs only the asymmetry in spin-up and spin-down reflection, but does not requireformation of quantum-well states. In particular, we show the dynamic exchange to be long ranged:it decays as 1/L for λel < L < λsd and exponentially for L > λsd.


induces a normal-metal spin accumulation [19, 20]

µs =

∫dεTr[σf(ε)] , (1.11)

where σ is the Pauli matrix vector and f(ε) is the 2× 2 matrix distribution function

at a given energy ε of N layer. In a nearly collinear configuration, the spin accumu-

lation µs is (approximately) perpendicular to the magnetizations. This simplifies the

expression for the bias-driven spin current to [20, 131, 21]

Ibacks =

g↑↓1 + g↑↓24π

µs . (1.12)

A steady state is established when the two spin currents, Eqs. (1.10) and (1.12), cancel

each other: Ibacks = Ipump

s . The spin accumulation is then given by µs = 4π(g↑↓1 +

g↑↓2 )−1Ipumps . In general, Is2 = −Is1, by the conservation of angular momentum, but

the net spin current Isj = Ipumpsj − Iback

sj across a single interface does not vanish. For

the left (j = 1) interface we find

Is1 =~4π

g↑↓1 g↑↓2

g↑↓1 + g↑↓2

(m1 ×

dm1

dt−m2 ×

dm2

dt

). (1.13)

The normal-metal layer in our model scrambles the incoming spin current and divides

it back over both ferromagnets. [For very transparent interfaces the scrambling is

only partial, but the present treatment is still adequate if we properly renormalize

the conductance parameters as in Eqs. (1.3) and (1.4).] The second ferromagnet can

thus cause damping of the precession in the first magnetic film, and vice versa. A

ferromagnetic layer can therefore serve as an efficient spin sink, just like a normal

metal with strong spin-flip scattering.

We first discuss the implications of the spin transfer (1.13) when the second fer-

romagnet, F2, can be approximated to be stationary, m2 ≈ 0. In the FMR measure-


ments, this is the case when one ferromagnet is in resonance, whereas the resonance

frequency of the second ferromagnet is sufficiently different because of different mag-

netic anisotropies. The spin torque τ = −Is1 on the first ferromagnet (F1 ) then has

the form of the Gilbert damping when added as a source term to the LLG equation

(1.7). In this case, the dynamic coupling of the ferromagnetic layers simply leads to

an enhancement of the Gilbert-damping parameter with respect to its intrinsic value,

exactly like in the case of single F films. The damping enhancement, following from

Eq. (1.13), of the F1 layer therefore reads

α′ =γ~g↑↓1 g

↑↓2

4πMsV (g↑↓1 + g↑↓2 )(1.14)

(with γ, Ms, and V of F1 ). Eq. (1.14) satisfactorily explains the increased viscous

damping observed in Fe/Au/Fe spin valves [136]. The physical nature of the damping

in single F films, Eq. (1.8), versus spin valves, Eq. (1.14), is quite clear. The angular

momentum is first driven out of the ferromagnet via spin pumping. In the former case

(single ferromagnet), the subsequent spin-orbit processes in the normal metal relax

injected spins and, as a result, slow down the coherent motion of the ferromagnet.

In the latter case (spin valve), the spin angular momentum is transferred from ferro-

magnet F1 and into F2. This spin current is absorbed by asynchronously driving the

magnetization dynamics of F2. Therefore, both the normal-metal and ferromagnetic

spin sinks act as an external brake to slow down the precession of the resonantly

excited magnet. The spin-sink efficiency of the normal metal is characterized by the

spin-flip probability ε and its thickness L [131], which is lumped into the parameter κ

in Eq. (1.8). It follows from Eq. (1.14) that the efficiency of the adjacent ferromagnet

F2, on the other hand, only depends on the F2/N mixing conductance (as long as


the second ferromagnet is thicker than its transverse spin-coherence length λsc5).

In the following we consider dynamics of the coupled F1/N/F2 system when both

magnetizations are allowed to precess, i.e., when the ferromagnet resonance conditions

are close to each other. By augmenting the LLG equation (1.7) for m1 by the spin

current source term (1.13) and with small variables ui = mi−hi, where |ui| 1 and

ui ⊥ hj, we obtain the linearized expression (assuming a circular precession)

u1 = ω1h1 × u1 + α0h1 × u1 + α′1 (h1 × u1 − h2 × u2) , (1.15)

where ωi is the resonance frequency of ferromagnet Fi, hi is the unit vector in the

direction of its effective magnetic field, and α′i = γi~g↑↓1 g↑↓2 /[4πMsiVi(g

↑↓1 + g↑↓2 )], sub-

script i labeling corresponding quantities of Fi. The dynamics of the magnetization

direction m2 is obtained by exchanging subscripts 1 ↔ 2 in Eq. (1.15).

For a spin valve in the parallel configuration, hi = h, and identical resonance

frequencies, ωi = ω, the quantity u = u1/α′1 + u2/α

′2 (which, up to a scaling factor,

is the rf component of the total angular momentum) is affected only by the intrinsic

bulk damping, u = ωh × u + α0h × u, while the difference ∆u = u1 − u2 relaxes

according to ∆u = ωh × ∆u + αh × ∆u with the enhanced (Gilbert) damping

constant α = α0 + α′1 + α′2. The dynamic coupling in the antiparallel configuration

as well as in the parallel configuration when the resonance frequencies have a large

mismatch, ∆ω (α0+α′1+α

′2)ω, favors relaxation of each magnetization towards the

equilibrium configuration. In this case, the two modes corresponding to excitations of

5The transverse component of the spin angular momentum precesses inside the ferromagnet andeventually decoheres on the scale of λsc. If the ferromagnetic layer is thicker than λsc, the reflectionof the transverse spin component on the normal-metal side will thus not depend on the ferromagneticthickness. This is easily achievable for transition-metal ferromagnets which have λsc comparable tothe Fermi wave length.


either ferromagnet decouple, each having enhanced damping parameter αi = α0 +α′i.

This picture explains in detail the FMR profiles measured on the Fe/Au/Fe spin

valves, not only in both symmetric and very asymmetric limits discussed herein but

also in the intermediate regime of closely matched but different resonance frequencies

in the parallel alignment [51], as we explain in Sec. 2.3.


1.3 One-Dimensional Conductors

One-dimensional (1D) electronic systems are a very fertile ground for studying

physics of interacting many-body systems. In one dimension, the elementary ex-

citations are collective spin and charge modes, the spectrum of which is strongly

influenced by the Coulomb interaction. An electron entering such a system must,

therefore, decompose into the corresponding eigenmodes, resulting in a suppression

of the tunneling density of states. This suppression was detected in a variety of

experiments, such as tunneling from metal contacts into carbon nanotubes [17] and

resonant tunneling in one dimension [3]. A unique feature of interacting electrons

in one dimension, described by Luttinger-liquid (LL) theory [45], is the decoupling

of the spin and charge degrees of freedom, each of which propagates with a different

velocity determined by the Coulomb interaction. The most direct experimental verifi-

cation of this phenomenon to date was reported in Ref. [132]. Issues pertaining to the

decoherence and relaxation of the elementary excitations of the LL, however, remain

a challenge both theoretically and experimentally. In chapter 3 we will investigate

manifestations of the LL behavior in a pair of cleaved-edge semiconductor quantum

wires coupled by a tunnel barrier.

1.3.1 Hubbard Model

Much of the initial success in understanding properties of 1D systems was achieved

my means of the Hubbard model [74]. The Hubbard model naturally reduces to the

Luttinger-liquid picture at low energies, but it also applies at larger energies (i.e.,

long-time and long-distance asymptotics) where the curvature of electronic dispersions


becomes important. While LL theory thus does not entirely supersede the Hubbard

model, it is hard to use the latter for certain purposes (as, e.g., understanding one-

particle spectral properties necessary for our theoretical constructions in chapter 3)

since it is still unclear how the physical electron decomposes into its fundamental

elementary excitations, the so-called spinons and holons. Let us nevertheless briefly

recall key results of the Hubbard model as it gives a simple picture of spin-charge

separation, which is not immediately obvious within the LL description.

The 1D Hubbard model is described by the Hamiltonian

HHub = −t∑

<i,j>s

c†i,scj,s +∑i,s

U

2(ni,s − 1/2)(ni,−s − 1/2)− µni,s (1.16)

expressed in terms of the annihilation operators ci,s for electrons in Wannier orbitals at

site i with spin s; ni,s = c†i,sci,s is the corresponding particle number, t is the (positive,

real-valued) hopping amplitude, U is the repulsion of two electrons on the same site,

and µ is the chemical potential. < i, j > denotes summations over nearest neighbors

only (equally spaced with distance a between the neighboring sites). The filling factor

n = Nelectrons/Nsites is fixed and periodic boundary conditions are assumed for a finite

chain. We will restrict our discussion to the physical case of on-site repulsion, U > 0,

and filling of only the lowest Hubbard band, n ≤ 1.

1.3.2 Elementary Excitations: Spinons and Holons

The 1D Hubbard model (1.16) has been solved exactly by Bethe Ansatz [74]. In

this section we briefly recall the nature and spectrum of elementary excitations at

n < 1, when the system is metallic,6 after reviewing the Bethe Ansatz solution.

6At exactly the half filing, n = 1, a gap opens for putting an extra electron into the Hubbardchain and the system thus becomes a Mott insulator.


Each of the N ! permutations, Q, of N particles defines a quadrant in spatial

coordinates, such that 0 ≤ xQ1 ≤ xQ2 ≤ . . . xQN≤ L (L = Na is the total length

of the chain and N ≡ Nelectrons). The Bethe Ansatz postulates that in each of these

quadrants characterized by Q, the wave function is given by

ψ(x1, . . . , xM , xM+1, . . . , xN) =∑

P

A[Q,P ] exp

(i

N∑j=1

kPjxQj

), (1.17)

where M down spins occupy sites x1, . . . , xM and N−M up spins sites xM+1, . . . , xN .7

The summation above is performed over all permutations P of N particle indices.

The problem of solving for the eigenstates of Eq. (1.16) thus reduces to finding N !2

coefficients A[Q,P ], and the approach would therefore be impractical if one needed

to explicitly evaluate the corresponding wave functions. Lieb and Wu [74], however,

have demonstrated that the N numbers ki in Eq. (1.17) can be determined from the

coupled equations [u = U/(4t)]

2πIj = Lkj − 2M∑

β=1

arctan

(sin kj − Λβ

u

), (1.18)

2πJα = 2N∑

j=1

arctan

(Λα − sin kj

u

)− 2

M∑β=1

arctan

(Λα − Λβ

2u

), (1.19)

Ij =

integer

half − odd integer

if M =even

odd

, (1.20)

Jα =

integer

half − odd integer

if N −M =odd

even

, (1.21)

(1.22)

which, in turn, can be used to determine the excitation spectrum of the system as

7We can fix the total spin along z axis, Sz, since the total spin-operator, S, commutes with theHamiltonian (1.16).


the total energy and momentum are respectively given by

E = −2tN∑

i=1

cos(kia) and P =N∑

i=1

ki . (1.23)

Eqs. (1.17)-(1.23) give the exact energy, momentum, and the corresponding wave

function of the 1D Hubbard model. ki are the momenta characterized by the spatial

degrees of freedom of the electrons and Λα are the so-called “rapidities” that describe

the spin state of the particles. ki are not equally spaced as in the case of free particles

(i.e., U = 0) but the integers (or half-odd integers) Ii and Jα are. The ground state

is obtained by occupying levels with minimum |Ii| and |Jα| so that the distributions

of qi = 2πIi/L and pα = 2πJα/L are given by step functions Θ(kF↑ + kF↓ − qi) and

Θ(kF↓ − pα), respectively. In the ground state (at zero magnetic field), kF↑ = kF↓ =

kF , so that qi has a doubled Fermi wave vector.

Elementary excitations are obtained by making a hole either in the Ii or in the Jα

distributions. In the former case one obtains a charge-e spinless “holon” and in the

latter a spin-1/2 neutral “spinon.” Removing a physical electron would affect both

channels but the exact form of the corresponding representation of real holes in terms

of holons and spinons is not known to date.

In the weak-coupling limit (i.e., u 1), the dispersions of holons and spinons are

given by

ε(h)(q) = 4t cos(qa/2)− 2t cos(kFa) , (1.24)

ε(s)(p) = 2t [cos(pa)− 2t cos(kFa)] . (1.25)

The physical hole states can, in principle, be constructed as a continuum of holon-

spinon pairs.


1.3.3 Spin-Charge Separation

The Hubbard model provides a transparent interpretation of the spin-charge sep-

aration which is a common property of 1D liquids, as discussed below. Consider a

section of a 1D Hubbard chain with a nearly half-filled band and a strong on-site

repulsion. The ground state of this system is known to have a single-occupied chain

of Wannier states with the antiferromagnetic spin order:

· · · ↑↓↑↓↑↓↑↓↑↓↑↓ · · · (1.26)

Removing a physical electron introduces a hole:

· · · ↑↓↑↓↑ O ↑↓↑↓↑↓ · · · (1.27)

which after moving to the left will leave a locally-broken antiferromagnetic order:

· · · ↑↓ O ↑↓↑↑↓↑↓↑↓ · · · (1.28)

After a long time, the hole surrounded by two oppositely-oriented spins (the holon)

completely decouples from the spinon formed by two adjacent up spins:

· · · ↑↓ O ↑↓↑↓↑↓↑↑↓ · · · (1.29)

Spinons thus appear as Block walls in the underlying local antiferromagnetic order,

while holons are holes that do not disrupt the spin alternation along the lattice. The

same (qualitative) picture also holds away from the half filling at strong coupling.

This gives a physical interpretation of the two types of quantum numbers, Ii and Jα,

in the exact solution of Lieb and Wu, which can be associated with the dynamics of

the holons and spinons, respectively.


1.3.4 Luttinger Model

In the case of the linear dispersion of a 1D free-electron gas, the elementary

electron-hole excitations of the system can be described by bosonic operators. Re-

markably, the bosonic picture can also be generalized to treat low-energy properties

of interacting electrons in 1D. Let me first briefly review the key assumptions and

results of the Luttinger model (LM) and then we will recall how they generalize to

the case of a general 1D gapless liquid.

The linearized noninteracting Hamiltonian of a free-electron gas with Fermi ve-

locity vF is conveniently written in terms of the annihilation operators for the right

(left) movers a+ (a−):

H0 = vF

∑k

(k − kF )a†+,ka+,k − vF

∑k

(k − kF )a†−,ka−,k , (1.30)

where the summation is performed over the wave vector k running near the two Fermi

points, ±kF , within a window of width 2Λ, i.e., the first (second) sum goes over k in

the range [±kF − Λ,±kF + Λ]. (For simplicity, we have suppressed the spin degree

of freedom here.) This is a valid approximation when we are interested only in the

electronic states close to the Fermi points (as, e.g., when we consider low-temperature

thermodynamic properties). In the LM, we send the cutoff Λ to infinity, see Fig. 1.5.

The reason for adding additional occupied states in the LM is obtaining the com-

mutation relations

[ρα(−q), ρα′(q′)] = δαα′δqq′

αqL

2π(1.31)

(L being the length of the 1D particle box) for the density operators

ρ±(q) =∑

k

a†±,k+qa±,k . (1.32)


k

EF

εk

Figure 1.5: Single-electron spectrum in the LM. The grey area shows occupied statesand the black shows the states added to make the model solvable.

In addition, the kinetic energy H0 obeys a simple commutation relation with the

density operator:

[H0, ρα(q)] = αvF qρα(q) . (1.33)

Acting by ρ±(q) upon an eigenstate, therefore, increases (decreases) energy by ±vF q.

This is not surprising since ρ±(q) changes the total momentum by ±q and we assumed

the linear dispersion relation with the slope vF . Furthermore, it is possible to write

the Hamiltonian (at a fixed number of particles and up to an overall constant) in

terms of the density operators

H0 =πvF

L

∑q 6=0,α=±

ρα(q)ρα(−q) . (1.34)

Hamiltonian (1.34) is certainly consistent with above commutations. The equivalence


of Eqs. (1.30) and (1.34) is known as Kronig’s identity.

1.3.5 Luttinger-Liquid Conjecture

Consider now the interacting case with the total Hamiltonian given by

H = H0 +Hint (1.35)

in terms of the kinetic energy (as before, but now explicitly including the spin degree

of freedom and not fixing the particle number)

H0 =π

L

∑q 6=0,α=±,ν=ρ,σ

vFνα(q)να(−q) +π

2L

[vF (N+,ν +N−,ν)

2 + vF (N+,ν −N−,ν)2],

(1.36)

where

ρα(q) =1√2

[ρα,↑(q) + ρα,↓(q)] , Nα,ρ =1√2

[Nα,↑ +Nα,↓] ,

σα(q) =1√2

[ρα,↑(q)− ρα,↓(q)] , Nα,σ =1√2

[Nα,↑ −Nα,↓] , (1.37)

and an interaction term

Hint =1

L

∑q,ν

Vν [2ν+(q)ν−(−q) + ν+(q)ν+(−q) + ν−(q)ν−(−q)] . (1.38)

Hint is the “forward” scattering Hamiltonian. In a realistic system, other interaction

terms will of course be present, such as the “backward” and Umklapp scattering.

It can be shown, however, that in the (physical) case of a repulsive potential, the

latter two interactions are irrelevant in the renormalization-group sense (i.e., they

renormalize downward) and we therefore disregard them.8 Further assuming spin-

rotationally invariant interactions, we set Vσ = 0. Physically, Vρ =∫dxV (x), in

8A strong enough attractive backward scattering would open up a spin gap and the Umklappscattering can make the system insulating, invalidating the LL description.


terms of the two-body potential V (x); it can be thought of as the strength of the

short-range interaction V (x) = Vρδ(x). (Details of the interactions do not matter as

we focus on the long wave-length asymptotics only.)

Because the total Hamiltonian (1.35) is bilinear in ν’s, one can perform a diago-

nalizing canonical transformation to obtain

H =π

L

∑q 6=0,α,ν

vν να(q)να(−q) +π

2L

∑ν

[vν

Kν

(N+,ν +N−,ν)2 + vνKν(N+,ν −N−,ν)

2

],

(1.39)

where

Kν =1√

1 + 2Vν/(πvF )(1.40)

and

vν = vF/Kν . (1.41)

The two velocities, vρ > vσ = vF (for a repulsive potential), correspond to charge-

and spin-density waves, respectively, and determine the positions of singular peaks

in the spectral function. Kρ < 1 (Kσ = 1) governs the long-time behavior of the

one-particle Green’s functions and, in particular, the power-law suppression of the

tunneling conductance at low temperature and bias, as follows. The single-particle

density of states

N (ε) = |ε− εF |α (1.42)

has a power-law singularity at the Fermi energy εF with the exponent

α =1

4

(Kρ +K−1

ρ − 2)

(1.43)

which is finite for a nonvanishing interaction strength. The tunneling conductance

of an infinite 1D liquid has correspondingly a zero-bias power-law tunneling anomaly


with the same exponent. The non-Fermi properties of the Luttinger model are also

apparent from the power-law singularity of the momentum distribution function

nk =1

2− const · sign(k − kF )|k − kF |α (1.44)

near the Fermi wave vector kF [which is determined by electron density n: kF =

(π/2)n, as usual]. The Fermi liquid, in contrast, has a finite density of states N (ε)

at εF and a step-like discontinuity of nk at kF .

Kν and vν are the main parameters determining all (low-energy) thermodynamic

properties of the system. Haldane conjectured in early 1980’s that any gapless 1D

electron system will asymptotically exhibit the LL properties at low energies (i.e.,

close to the Fermi surface) which will be determined by two parameters, Kν and vν ,

for each degree of freedom. This conjecture is presently supported by a large number

of case studies, including the Hubbard model.

1.3.6 Tunnel-Coupled Double Wires

In chapter 3 we present a detailed experimental investigation and theoretical ex-

planation of a set of interference patterns in the nonlinear tunneling conductance

between two parallel wires that were reported in Ref. [132]. A sketch of the tunneling

geometry is shown in Fig. 1.6. The interference appears because the tunneling process

is coherent to a very high degree, and is due to the finite length of the tunnel junc-

tion. As explained in Sec. 3.4.2, a wealth of information can be extracted from the

interference: The pattern itself encodes microscopic details of the potentials in the

wires, while the structure of its envelope reflects the presence of two distinct excita-

tion velocities per electron mode in the data, as expected from spin-charge separation.


Figure 1.6: Schematics of the circuit. A wire of length L runs parallel to a semi-infinite wire. Boundaries of the wires are formed by two gates above. The energyand momentum of the tunneling electrons are governed by the applied voltage andmagnetic field, respectively.

The decay of interference may also yield information on decoherence processes of the

elementary excitations in 1D systems.

In addition, low-energy anomaly of the Luttinger liquids in the wires is studied by

temperature and voltage dependence of the tunneling conductance, which reveals an

intricate interplay between the finite length of the tunnel junction and the electron-

electron correlation in and between the wires. This is discussed in Sec. 3.4.7.


1.4 Quasihole Statistics in Quantized Hall States

Clean 2D electron gases (2DEG) in strong magnetic field yield a series of incom-

pressible states at certain values of the magnetic field strength, corresponding to

discrete fractional filling factors ν of the Landau level(s). Experimentally, the values

of ν are measured as plateaus in the transverse conductance, σxy = νe2/h, as a func-

tion of magnetic field. Let us for definiteness consider fillings of a single spin-polarized

Landau level, i.e., ν ≤ 1. Laughlin [72] has explicitly constructed a series of trial wave

functions at ν = 1/p, p odd, which were shown to overlap exceptionally well with the

exact ground states at respective fillings. In much of the later work, these Laughlin

wave functions were used as a starting point for generalizations [46, 47] aimed at ex-

plaining the growing number of observed fillings. Apart from ν = 1/p (e.g., 1/3, 1/5,

and 1/7), several other fractions (e.g., 4/5, 2/3, and 3/5), all odd-denominator, were

experimentally found in 1980’s, see, e.g., Refs. [135, 27]. Subsequently [144], ν = 1/2,

the only even-denominator plateau, was observed in the first excited Landau level

(i.e., ν = 2 + 1/2). Non-Abelian statistics of the elementary excitations of this state

are the main focus of our work presented in chapter 4.

1.4.1 Laughlin and Standard Hierarchy States

Since early days of the fractional quantum Hall (FQH) effect, understanding statis-

tics of elementary excitations in a given quantized state played a key role in deriving

its “daughter states,” i.e., constructing new incompressible states in the vicinity of a

known FQH plateau. This forms the basis of the standard hierarchy schemes [46, 47]

which generalized Laughlin’s 1/p to any odd-denominator rationals q/p via continued


fractions

ν =1

p+ α1

p1+α2

...pn−1+ αn

pn

, (1.45)

where p=1, 3, 5, . . ., αi = ±1, and pi = 2, 4, 6, . . .. Setting n = 0 recovers the Laughlin

fractions.9

The quantum statistics of a system of identical particles describe the effect of

adiabatic particle interchange on the many-body wave function. While all the funda-

mental particles are thought to be either bosonic or fermionic, collective excitations,

such as FQH quasiparticles (or quasiholes) can be anyonic (i.e., having fractional

statistics) as well as obeying more general statistics. Arovas et al. have explicitly

demonstrated in Ref. [2] using Berry’s adiabatic theorem that the Laughlin quasiholes

obey fractional statistics. In order to show this, it suffices to integrate the differential

equation for the Berry’s phase γ:

dγ(t)

dt= i 〈ψ(t) |dψ(t)/dt〉 , (1.46)

where ψ(t) is the (time-dependent, normalized) wave function of a (nondegenerate)

state with a number of quasiholes localized at given positions. Using the Laughlin

wave function10

ψL =∏i<j

(zi − zj)p exp

(−∑

k

|zk|2

4a20

)(1.47)

9An alternative approach was suggested by Jain [60], who showed that all odd-denominatorfractions can be understood as the usual integer-valued quantum Hall plateaus [139] of compositefermions. The latter consist of electrons with even number of flux quanta, Φ0 = hc/e, attached; theflux attachment “absorbs ” a certain amount of the total magnetic flux passing through the 2DEG,without affecting the (fermionic) statistics of composite particles.

10In the following, the positions of electrons and quasiholes are complex-valued variables, z, withthe real (imaginary) part given by the x (y) coordinate, i.e., z = x + iy. An important length scaleis set by the magnetic length a0 =

√Φ0/(2πH), H being the magnetic field.


with several quasiholes added at positions wj:

ψqh =∏ij

(zi − wj)ψL , (1.48)

where zi are electron coordinates, one can see that the phase accumulated by adiabatic

braiding of one quasihole around another equals 2π/p, in the case of a large separation

[on the scale of the magnetic length a0]. It directly follows from (assuming ψqh is

properly normalized)

δγj = −i∑

i

〈ψqh(wj)|(zi − wj)−1|ψqh(wj)〉δwj

= −i∫dzρ(z, wj)(z − wj)

−1δwj , (1.49)

after substituting Eq. (1.48) into Eq. (1.46). Neglecting dependence of the electron

density ρ on the position of the quasihole wj, one finally obtains

γj ≈ 2π〈Nj〉 . (1.50)

The total phase γj obtained by integrating Eq. (1.49) over a loop in the complex

plane of wj thus approximately equals to the expectation value of the total number

of electrons, Nj, inside the loop multiplied by 2π. (See Ref. [2] for a justification

of this approximation.) We thus arrive at a one-to-one correspondence between the

Landau filling factor and the quasihole statistics in the case of the Laughlin states,

since removing a quasihole from inside of the integration loop would lead to a particle

number increase (and correspondingly phase γj/2π increment) of ν = 1/p.11

11This, in turn, follows from the conductance quantization σxy = νe2/h and the fact that aquasihole at position wj is created by passing a localized flux quantum hc/e through this point.The former is accompanied by a total outward charge flow of νe, using Faraday’s law and the abovevalue of the transverse conductance σxy.


The hierarchy states [47] in general have (Abelian) anyonic statistics which can

be derived iteratively starting from the fermionic statistics of the first integer-filling

plateau (in particular reproducing the above result for ν = 1/p fillings at the first

step of the inductive construction).

1.4.2 Paired (Moore-Read) and Parafermion States

The only observed single-layer quantum Hall state which has an even-denominator

filling factor, ν = 1/2, is also believed to possess quasiparticles with non-Abelian

statistics. Physically, the ν = 1/2 state is commonly thought to be a p-wave paired

state of composite fermions with two flux quanta attached to bare electrons. This is

the so-called Pfaffian constructed by Moore and Read [89].

The Pfaffian is a ground state of a three-body Hamiltonian

H = U∑

i<j<k

Pijk(3Nφ/2− 3) , (1.51)

where U is a positive (real-valued) constant. For convenience we wrote the Hamilto-

nian on a sphere [46], with the total magnetic flux Nφ (in units of hc/e) created by

a magnetic monopole placed at the center. In the lowest Landau level, one-electron

states have orbital momenta of Nφ/2, and the Hamiltonian (1.51) is taken to be pro-

portional to the projection operator Pijk onto a (unique) multiplet of maximum total

angular momentum (i.e., 3Nφ/2−3) for each triplet of electrons. Since the closest ap-

proach of three particles on the sphere corresponds to the state of maximum possible

total angular momentum for the three, the Hamiltonian (1.51) thus makes it costly for

triplets to cluster together. Although this seems to only crudely model the physical

two-body Coulomb interaction, the densest zero-energy ground state of this Hamil-


tonian is known to give a good overlap with wave functions obtained numerically by

exact diagonalization of a finite system.12

The highest-density (zero-energy) Pfaffian at a fixed magnetic field corresponds

to the ν = 1/2 filling of the lowest (spin-polarized) Landau level. The quasiholes of

this state appear in pairs, each of the pairs created by adding a single flux quantum,

hc/e, each of the quasiholes thus corresponding to the superconducting flux quantum,

hc/2e. The quasiholes carry charge e/4 (which is 1/2 of the value νe corresponding

to the half filling; the extra factor of 1/2 being due to a paired nature of the state)

and can be thought of as vortices of the BCS superconductor of composite fermions.

Unlike their Laughlin counterparts, the Moore-Read quasiholes have a trivial relative

statistics, as we discuss in chapter 4, in the case of a single pair of quasiholes present.

If two or more, n, pairs of the quasiholes are formed (by slowly increasing the mag-

netic field), the braiding of the quasiholes can no longer be described by the anyonic

statistics. The reason for this is that the corresponding ground state acquires a non-

trivial degeneracy of 2n−1. Braiding of a pair of two of the 2n quasiholes will in general

result in a transformation of this ground state, which is noncommutative when two

different pairs are braided in series. The corresponding representation of the braid

group can be understood as the SO(2n)×U(1) spinor representation with generators

of the braid group mapping into representations of the π/2 rotations around the axes,

see chapter 4.

If the three-body Hamiltonian is added to the usual Haldane’s [46] two-body

12Using reasonable pseudopotential parameters for ν = 2 + 1/2 in the first excited Landau level.


interaction

HH =M∑

M ′=1

UM ′

∑i<j

Pij(Nφ −M ′) (1.52)

(where M and M ′ are positive odd integers) which penalizes the closest approach of a

pair of electrons (i.e., projects electron doublets onto a large total angular momentum

with UM ′ > 0), the densest zero-energy state has Landau-level filling factor

ν =1

M + 3, (1.53)

where unphysical value M = −1 reproduces the result for vanishing pair-interaction

potential.13

A straightforward generalization of the previous discussion is to consider a k + 1-

body interaction similar to Eq. (1.51), with k ≥ 2. The densest zero-energy states of

the k + 1-body interaction gives a series of fillings

ν =k

(M + 2)k + 2, (1.54)

where M is odd, corresponding to adding a two-body repulsion, as before. This result

was obtained in Ref. [107]. It was shown there that the wave functions at k > 2 can be

obtained as the correlation functions of conformal field theories of Zk parafermions.14

Counting of the quasihole degeneracy can correspondingly be performed using the

parafermion statistics of the quantum-mechanical particles. The problem becomes

progressively complex as k increases. In particular, it was shown that for k = 3, the

degeneracy of a state with 3n quasiholes is a Fibonacci number, F3n−2, where F1 = 1,

13The (densest zero-energy) wave function correspondingly acquires an additional Laughlin-Jastrow prefactor

∏i<j(zi − zj)M+1 to the usual ν = 1/2 Pfaffian.

14The Pfaffian at the k = 2 level, on the other hand, can be expressed in terms of the correlatorsof the Majorana fermions.


F2 = 2, F3 = 3, F4 = 5, and Fm = Fm−1 + Fm−2. The non-Abelian statistics of the

quasihole braiding is thus expected in this case as well as k = 2, for n > 1.

An interesting twist of this development is that the fractions (1.54) can coincide

with the values derived in the standard hierarchy schemes, see Eq. (1.45). For ex-

ample, setting k = 3 and M = 1 gives ν = 3/5 according to Eq. (1.54). The same

(odd-denominator) rational can be obviously obtained as a continued fraction (1.54).

In the lowest Landau level, the hierarchy states are most stable and the parafermion

states will not be competitive. It is nevertheless argued that the situation can be

different in the first excited Landau level: Exact diagonalization of a finite system

performed in Ref. [107] suggests that the parafermion 3/5 state gives a much better

overlap with the numerical ground state than the standard hierarchy state, using a

reasonable value for the pseudopotential in the first excited Landau level. This issue

as well as understanding braiding statistics of the parafermion states need further

investigation.

1.4.3 Monte Carlo Approach

Braiding statistics of the paired state with 2n quasiholes were studied in Ref. [93]

where it was shown that the braid group supports a continuous extension–the spinor

representation of SO(2n)×U(1)–which reduces to the original group by restricting

to π/2 rotations around the axes. The entire derivation of Ref. [93] was based on

the assumption that the Berry matrix of the conformal blocks vanishes,15 which still

lacks a proof. It was therefore desirable to perform a direct calculation of the statistics

15But not the explicit monodromy which thus becomes the sole contribution to the statistics.


using the known form of the Moore-Read wave function. We have managed to do

this, confirming results reported in Ref. [93], for the case of n = 1 and 2 [133], which

is discussed in chapter 4.16 The method used in chapter 4 is based on integrating

Berry’s equation (1.46) generalized to the degenerate case, where the wave-function

overlaps are evaluated using the Metropolis method.

16The case of n = 3 was also analyzed supporting the result derived within the conformal fieldtheory, but we are not going to discuss it in the thesis.

Chapter 2

Dynamic Phenomena in Magnetic

Multilayers

In this chapter we study dynamic properties of thin ferromagnetic films in contact

with normal metals and semiconductors. Moving magnetizations cause a flow of spins

into adjacent conductors, which relax by spin flip, scatter back into the ferromagnet,

or are absorbed by another ferromagnet. Relaxation of spins outside the moving mag-

netization enhances the overall damping of the magnetization dynamics in accordance

with the Gilbert phenomenology. Transfer of spins between different ferromagnets by

these nonequilibrium spin currents leads to a long-ranged dynamic exchange inter-

action and novel collective excitation modes. Our predictions agree well with recent

ferromagnetic-resonance experiments on ultrathin magnetic films and bilayers. In

addition, we propose a concept of spin battery operated by ferromagnetic resonance.

41

Chapter 2: Dynamic Phenomena in Magnetic Multilayers 42

2.1 Enhanced Gilbert Damping

A moving magnetization vector causes pumping of spins into adjacent nonmag-

netic layers. This spin transfer affects the magnetization dynamics similar to the

Landau-Lifshitz-Gilbert phenomenology. The additional Gilbert damping is signifi-

cant for small ferromagnets, when the nonmagnetic layers efficiently relax the injected

spins, but the effect is reduced when a spin accumulation build-up in the normal

metal opposes the spin pumping. The damping enhancement is governed by (and, in

turn, can be used to measure) the mixing conductance or spin-torque parameter of

the ferromagnet/normal-metal interface. Our theoretical findings are confirmed by

agreement with recent experiments in a variety of multilayer systems.

2.1.1 Background

Spin-polarized transport through magnetic multilayers is the physical origin of

many interesting phenomena such as giant magnetoresistance and spin–current-induced

magnetization reversal [39, 73, 91, 65, 142, 119, 120]. It has attracted attention in the

basic physics community and industry over the last decades, but there are still open

fundamental questions. So far, the main research activity has been focused on the dc

transport properties of these systems. Ac magnetotransport has drawn considerably

less attention than its dc counterpart. In a recent paper [130], we reported a novel

mechanism by which a precessing ferromagnet pumps a spin current into adjacent

nonmagnetic conductors proportional to the precession frequency, using a formalism

analogous to that for the adiabatic pumping of charges in mesoscopic systems [22].

We showed that spin pumping profoundly affects the dynamics of nanoscale ferromag-


nets and thin films, by rescaling fundamental parameters such as the gyromagnetic

ratio and Gilbert damping parameter, in agreement with experiments [82, 83].

The switching characteristics of a magnetic system depends in an essential way

on the Gilbert damping constant α. In magnetic field-induced switching processes,

for example, α governs the technologically important magnetization reversal time of

ferromagnetic particles. Its typical intrinsic value [14] α0 . 10−2 for transition-metal

ferromagnets is smaller than its optimal value of α & 10−1 for the fastest switching

rates [68, 99, 57]. The present mechanism allows engineering of the damping constant

by adding passive nonmagnetic components to the system and/or adjusting the geom-

etry to control spin flow and relaxation rates described below, thus helping to create

high-speed magnetoelectronic devices. Also, in spin–current-induced magnetization

reversal, the critical switching current is proportional to α [120].

For some time it has been understood that a ferromagnet/normal-metal (F/N )

interface leads to a dynamical coupling between the ferromagnetic magnetization and

the spins of the conduction-band electrons in the normal metal [115, 56, 119, 120, 11,

19, 20, 140]. More recently, several theoretical frameworks were put forward proposing

a mechanism for magnetization damping due to F/N interfacial processes [11, 141,

130]. This F/N coupling becomes important in the limit of ultrathin (.10 nm)

ferromagnetic films and can lead to a sizable enhancement of the damping constant.

Our theory is based on a new physical picture, according to which the ferromag-

netic damping can be understood as an adiabatic pumping of spins into the adjacent

normal metals [130]. This spin transfer is governed by the reflection and transmission

matrices of the system, analogous to the scattering theory of transport and interlayer


exchange coupling. The microscopic expression for the enhanced Gilbert damping and

the renormalized gyromagnetic ratio can be calculated by simple models or by first-

principles band-structure calculations without adjustable parameters. The present

theory therefore allows quantitative predictions for the magnetization damping in

hybrid systems that can be tested by experiments.

The Gilbert damping constant in thin ferromagnetic films has been experimentally

studied [52, 101, 82, 83, 85, 136] by measuring ferromagnetic-resonance (FMR) line

widths. In the regime of ultrathin ferromagnetic films, α was in some cases found to

be quite large in comparison with the bulk value α0, and sensitively depending on the

substrate and capping layer materials. For example, when a 20-A-thick permalloy

(Py) film was sandwiched between two Pt layers, its damping was found to be α ∼

10−1, but recovered its bulk value α ∼ 10−2 with a Cu buffer and cap [82, 83]. Heinrich

et al. [52] observed an enhanced damping of .20 A-thick Fe films when they were

grown on Ag bulk substrates but no significant change in the damping constant was

seen for films grown on GaAs even when the film thickness was reduced down to

several atomic monolayers [48]. We will demonstrate here that our theory explains

all these experimental findings well.

First we will study the situation when the normal-metal layers adjacent to the

ferromagnetic films are perfect spin sinks, so that the spin accumulation in the normal

metal vanishes [130]. Later this assumption will be relaxed and we will have to self-

consistently take into account the spin accumulation, which will enable us to explain

experimental findings for various F/N systems [52, 82, 83, 85] in a unified framework

based on the spin-pumping picture.


2.1.2 Single Ferromagnetic Films

The magnetization dynamics of a bulk ferromagnet is well described by the phe-

nomenological Landau-Lifshitz-Gilbert (LLG) equation [40, 71]

dm

dt= −γm×Heff + αm× dm

dt, (2.1)

where m is the magnetization direction, γ is the (minus) gyromagnetic ratio, and Heff

is the effective magnetic field including the external, demagnetization, and crystal

anisotropy fields. The second term on the right-hand side of Eq. (2.1) was first

introduced by Gilbert [40] and the dimensionless coefficient α is called the Gilbert

damping constant. For a constant Heff and α = 0, m precesses around the field vector

with frequency ω = γHeff . When damping is switched on, α > 0, the precession spirals

down to a time-independent magnetization along the field direction on a time scale

of 1/αω. Study of α in bulk metallic ferromagnets has drawn a significant interest

over several decades. Notwithstanding the large body of both experimental [14] and

theoretical [69, 75, 126, 110] work, the damping mechanism in bulk ferromagnets is

not yet fully understood.

The magnetization dynamics in thin magnetic films and microstructures is tech-

nologically relevant for, e.g., magnetic recording applications at high bit densities.

Recent interest of the basic physics community in this topic is motivated by the spin-

current induced magnetization switching in layered structures [119, 120, 91, 65]. The

Gilbert damping constant was found to be 0.04 < α < 0.22 for Cu/Co and Pt/Co

[5, 91], which is considerably larger than the bulk value α0 ≈ 0.005 in Co [113, 65].

Previous attempts to explain the additional damping in magnetic multilayer systems

involved an enhanced electron-magnon scattering near the interface [11] and other


mechanisms [141, 10], both in equilibrium and in the presence of a spin-polarized

current.

In this section we describe a novel mechanism for the Gilbert damping in normal-

metal/ferromagnet (N/F ) hybrids. According to Eq. (2.1), the precession of the

magnetization direction m is caused by the torque ∝ m × Heff . This is physically

equivalent to a volume injection of what we call a spin current. The damping occurs

when the spin current is allowed to leak into a normal metal in contact with the

ferromagnet. Our mechanism is thus the inverse of the spin-current–induced magne-

tization switching: A spin current can exert a finite torque on the ferromagnetic order

parameter, and, vice versa, a moving magnetization vector loses torque by emitting a

spin current. In other words, the magnetization precession acts as a spin pump which

transfers angular momentum from the ferromagnet into the normal metal. This ef-

fect can be mathematically formulated in terms of the dependence of the scattering

matrix of a ferromagnetic layer attached to normal metal leads on the precession of

m, analogous to the parametric charge pumping in nonmagnetic systems [22]. The

damping contribution is found to obey the LLG phenomenology. Enhancement of

the damping constant α′ = α−α0 can be expressed in terms of the scattering matrix

at the Fermi energy of a ferromagnetic film in contact with normal metal reservoirs,

which can be readily obtained by model or first-principles calculations. Our numer-

ical estimates of α′ compare well with recent experimental results [82, 83]. Earlier

experiments reported in Ref. [52] can also be understood by our model [131].

We consider a ferromagnetic film sandwiched between two paramagnetic layers

as shown in Fig. 2.1. Spin pumping is governed by the ferromagnetic film and the


µ µ µ µ

Figure 2.1: Ferromagnetic film (F ) adjacent to two normal metal layers (N ). Thelatter are viewed as reservoirs in common thermal equilibrium. The reflection andtransmission amplitudes r and t′ shown here govern the spin current pumped into theright lead.

vicinity of the N/F interfaces. The normal metal layers are, therefore, interpreted as

reservoirs attached to nonmagnetic leads. The quantity of interest is the 2×2 current

matrix in spin space

I =1

2Ic1−

e

~Is · σ (2.2)

for the charge (Ic) and spin flow (Is) from the magnetic film into adjacent normal

metal leads, where 1 is the unit matrix and σ the vector of Pauli spin matrices.

When no voltages are applied and the external field is constant, the charge current

vanishes. Two contributions to the spin current Is on either side of the ferromagnet

may be distinguished, viz. Ipumps and Iback

s . Ipumps is the spin current pumped into

the normal metal to be discussed below, whereas Ibacks is the current which flows

back into the ferromagnet. The latter is driven by the accumulated spins in the


normal metal and gives, e.g., rise to the spin-current–induced magnetization switching

[119, 120, 91, 65]. Here we model the normal metal as an ideal sink for the spin current,

such that a spin accumulation does not build up. This approximation is valid when

the spins injected by Ipumps decay and/or leave the interface sufficiently fast, e.g.,

when the dimensionless conductance of the N/F interface is smaller than h/(τsfδ)

[131]. Here, τsf is the spin-flip relaxation time and δ is the energy level spacing at the

Fermi surface of a normal metal film with a thickness which is the smaller one of the

geometrical film thickness and the spin-flip diffusion length.

The current I(t) pumped by the precession of the magnetization into the right and

left paramagnetic reservoirs, connected to the ferromagnet by normal metal leads (R)

and (L), may be calculated in an adiabatic approximation since the period of preces-

sion 2π/ω is typically much larger than the relaxation times of the electronic degrees

of freedom of the system. The adiabatic charge-current response in nonmagnetic sys-

tems by a scattering matrix which evolves under a time-dependent system parameter

X(t) has been derived in Refs. [25, 22]. Adopting Brouwer’s notation [22], the gen-

eralization to the 2× 2 matrix current (directed into the normal metal lead l = R or

L) reads

I(t)pump = e∂n(l)

∂X

dX(t)

dt, (2.3)

where the matrix emissivity into the lead l is

∂n(l)

∂X=

1

4πi

∑mnl′

∂smn,ll′

∂Xs†mn,ll′ + H.c. (2.4)

and s is the 2 × 2 scattering matrix of the ferromagnetic insertion. m and n label

the transverse modes at the Fermi energy in the normal metal leads and l′ = R,L.

Spin-flip scattering in the contact is disregarded. s depends on the magnetization


direction m of the ferromagnet through the projection matrices [19, 20]

u↑ =(1 + σ ·m

)/2 and u↓ =

(1− σ ·m

)/2 (2.5)

by

smn,ll′ = s↑mn,ll′u↑ + s↓mn,ll′u

↓ . (2.6)

The spin current pumped by the magnetization precession is obtained by identifying

X(t) = ϕ(t), where ϕ is the azimuthal angle of the magnetization direction in the

plane perpendicular to the precession axis. The resulting current is traceless, Ipump =

−(e/~)σ · Ipumps , i.e., charge current indeed vanishes, and

Ipumps =

~4π

(Arm× dm

dt− Ai

dm

dt

), (2.7)

where the interface parameters are

Ar =1

2

∑mn

[∣∣r↑mn − r↓mn

∣∣2 +∣∣t′↑mn − t′↓mn

∣∣2] , (2.8)

Ai = Im∑mn

[r↑mn(r↓mn)∗ + t′↑mn(t′↓mn)∗

]. (2.9)

Here, r↑mn [r↓mn] is the reflection coefficient for spin-up [spin-down] electrons in the

lth lead and t′↑mn [t′↓mn] is the transmission coefficient for spin-up [spin-down] electrons

into the lth lead. (See Fig. 2.1 for l = R.) Using unitarity of the scattering matrix

for each spin direction, we can summarize Eqs. (2.8) and (2.9) by

Ar + iAi = g↑↓ − t↑↓ , (2.10)

where gσσ′ is the (dc) conductance matrix and

t↑↓ =∑nm

t′↑mn(t′↓mn)∗ . (2.11)


The spin current (2.7) trivially vanishes for the steady state, i.e., when dm/dt = 0,

and for unpolarized contacts s↑mn,ll′ = s↓mn,ll′ .

Per revolution, the precession pumps an angular momentum into an adjacent

normal metal layer which is proportional to Ar, in the direction of the (averaged)

applied magnetic field, and decaying in time. At first sight, it is astonishing that a

pump can be operated by a single parameter varying in time, whereas the “peristaltic”

pumping of a charge current requires at least two periodic parameters [22]. However,

there are actually two periodic parameters (out of phase by π/2) hidden behind ϕ(t),

viz. the projections of the unit vector defined by ϕ in the plane perpendicular to the

axis of precession.

By conservation of angular momentum, the spin torque on the ferromagnet re-

sulting from the spin pumping into the nonmagnetic leads gives an additional term

to the LLG equation (2.1). After including this term, Eq. (2.1) remains valid, but

the gyromagnetic ratio and the damping constant are renormalized:

1

γ=

1

γ0

1 + [A(L)i + A

(R)i ]

γ0~MsV

, (2.12)

α =γ

γ0

α0 + [A(L)r + A(R)

r ]γ0~MsV

. (2.13)

Here, Ms is the magnetization and V is the volume of the ferromagnetic film; subscript

0 denotes the bulk values of γ and α; superscripts (L) and (R) denote parameters

evaluated on the left and right side of the F layer, respectively. Eqs. (2.12) and

(2.13) are the central result of this section. Ar and Ai affect, e.g., ferromagnetic

resonance experiments as a shift of the resonance magnetic field via A(L)i + A

(R)i ,

whereas A(L)r + A

(R)r increases the relative resonance line width.

From now on we focus on ferromagnetic films which are thicker than the spin-


coherence length λsc = π/|k↑F − k↓F |, where k

↑(↓)F are the spin-up and spin-down Fermi

wave vectors, i.e., thicker than a few monolayers in the case of transition metals. In

this regime, spin-up and spin-down electrons transmitted or scattered from one N/F

interface interfere incoherently at the other interface, t↑↓ vanishes and the mixing con-

ductance g↑↓ is governed by the reflection coefficients of the isolated N/F interfaces.

Ai = Img↑↓ vanishes for ballistic and diffusive contacts as well as nonmagnetic

tunnel barriers [19, 20]. First-principles calculations find very small Ai for Cu/Co

and Fe/Cr [146]. It is, therefore, likely that Ai may be disregarded in many systems.

If Ai does vanish on both sides of the ferromagnetic film, it follows from Eqs. (2.12)

and (2.13) that the resonance frequency is not modified γ = γ0 and the enhancement

of the Gilbert damping is given by α′ = [A(L)r + A

(R)r ]γ~/(4πMsV ).

The coefficient Ar can be estimated by simple model calculations [19, 20]. For

ballistic (point) contacts, ABr = (1+p)g with the polarization p = (g↑↑−g↓↓)/(g↑↑+g↓↓)

and the average conductance g = (g↑↑ + g↓↓)/2. For diffusive N/F hybrids, ADr = gN ,

the conductance of the normal metal part. A nonmagnetic tunneling barrier between

F and N suppresses the spin current exponentially. The magnetization precession of

a magnetic insulator can also emit a spin current into a normal metal, since g↑↓ does

not necessarily vanish because the phase shifts of reflected spin-up and spin-down

electrons at the interface may differ [146].

Let us now estimate the damping coefficient α′ for thin films of permalloy (Ni80Fe20,

Py), a magnetically very soft material of great technological importance. Mizukami et

al. [82, 83] measured the ferromagnetic-resonance line width of N /Py/N sandwiches

and discovered systematic trends in the damping parameter as a function of Py layer


thickness d for different normal metals. The spin polarization of electrons emitted by

Py has been measured to be p ≈ 0.4 in point contacts [123], the magnetization per

atom is f ≈ 1.2, and g factor gL ≈ 2.1 [82, 83]. The interface conductance of metal-

lic interfaces with Fe or Co is of the order of 1015 Ω−1m−2, with significant but not

drastic dependences on interface morphology or material combination [147]. This cor-

responds to roughly one conducting channel per interface atom. Assuming the Fermi

surface of the normal metal is isotropic, we arrive at the estimate α′ ≈ 1.1/d(A).

The factor 1/d does not reflect an intrinsic effect; a reduced total magnetization is

simply more sensitive to a given spin-current loss at the interface. Comparing with

the intrinsic α0 ≈ 0.006 of permalloy [98, 8, 82, 83] the spin-current induced damping

becomes important for ferromagnetic layers with thickness d < 100 A. We can refine

the estimate by including the significant film-thickness dependence of the magnetiza-

tion measured by the same group [82, 83]. We, therefore, improve our above estimate

as

α′(d) ≈ κ× 1.1

d(A)× f0

f(d), (2.14)

where f0 and f(d) are the atomic magnetization of the permalloy bulk and films. κ

is an adjustable parameter representing the number of scattering channels in units of

one channel per interface atom, which should be of the order of unity.

The experimental results for the damping factor α and the relative magnetization

f/f0 for N /Py/N sandwiches with N=Pt, Pd, Ta, and Cu are shown in the insets

of Fig. 2.2. Our estimate (2.14) appears to well explain the dependence of α on the

permalloy film thickness d (see Fig. 2.2) for reasonable values of κ. First-principles

calculations are called for to test these values.


10 100d

Py (Å)

0.0001

0.001

0.01

0.1

1

(α−α

0)f/f 0

20 40 60 80 1000

0.20.40.60.8

1

f/f0

PtPdTaCu

20 40 60 80 100

0

0.02

0.04

0.06

0.08

α

Figure 2.2: The lines show our theoretical result (9) with κ =1.0, 0.6, and 0.1; the datapoints are derived from the measurements [12] shown in two insets. Insets: MeasuredGilbert damping constant α (lower inset) and the relative atomic magnetization f/f0

(upper inset) in permalloy film of varied thickness dPy in a trilayer structure N /Py/N.

The lack of a significant thickness dependence of damping parameter of the Cu/Py

system requires additional attention. An opaque interface might be an explanation,

but it appears more likely that due to long spin-flip relaxation times in Cu, the

5 nm thick buffer layers in [82, 83] do not provide the ideal sink for the injected

spins as assumed above. This means that a nonequilibrium spin accumulation on Cu

opposes the pumped spin current and nullifies the additional damping when h/(τsfδ) is


comparable or smaller than the conductance g. For 5 nm Cu buffers, gδ/h ∼ 1013 s−1,

whereas 1/τsf ∼ 1012 s−1 [80]. It follows that Cu is indeed a poor sink for the injected

spins and the Gilbert damping constant is not enhanced. On the other hand, Pt, Ta,

and Pd are considerably heavier than Cu and, since 1/τsf scales as Z4 [1], where Z is

the atomic number, have much larger spin-relaxation rates and our arguments hold.

A physical picture of the effect of magnetization precession in layered systems has

been proposed earlier by Hurdequint et al. [86, 56, 55] in order to explain ferromag-

netic and conduction electron spin resonance experiments. These authors realized

that the precessing magnetization is a source of a nonequilibrium spin accumulation

which diffuses out of the N/F interfaces into the adjacent normal metal layers where

it can dissipate by spin-flip processes. Enhanced Gilbert damping in thin ferromag-

netic films in contact with normal metal has also been discussed by Berger [11] for a

ballistic N/F interface in a spin-valve configuration. His expression for the damping

coefficient (Eq. (20) in Ref. [11]) scales like ours as a function of layer thickness, but

differs as a function of material parameters. E.g., in contrast to our result, Berger’s

expression does not vanish with vanishing exchange splitting.

In conclusion of this section, we demonstrated that the Gilbert damping constant

is enhanced in thin magnetic films with normal metal buffer layers by a spin-pump

effect through the N/F contact. The damping is significant for transition metal films

thinner than about 10 nm. Recent experiments on permalloy films [82, 83] are well

explained.


2.1.3 Precession-Induced Spin Pumping

Consider an N/F/N junction as in Fig. 2.1 and Fig. 2.3 in the Appendix. Without

a voltage bias, no spin or charge currents flow when the magnetization of the ferro-

magnet is constant in time. When the magnetization direction starts precessing (as,

e.g., under the influence of an applied magnetic field), a spin current Ipumps is pumped

out of the ferromagnet [130]. This current into a given N layer depends on the

complex-valued parameter Ar + iAi (the “spin-pumping conductance”) by Eq. (2.7).

Note that the magnetization can take arbitrary directions; in particular, m(t) may

be far away from its equilibrium value. In such a case, the scattering matrix itself can

depend on the orientation of the magnetization, and one has to use A(m) in Eq. (2.7).

When the ferromagnetic film is thicker than its transverse spin-coherence length,

d > π/(k↑F − k↓F ), t↑↓ vanishes [125], the spin pumping through a given F/N inter-

face is governed entirely by the interfacial mixing conductance g↑↓ = g↑↓r + ig↑↓i , and

we can consider only one of the two interfaces. This is the regime we are focus-

ing on throughout this chapter. Note that the conductance matrix gσσ′ has to be

renormalized for highly transparent interfaces in columnar geometries (by properly

subtracting Sharvin resistance contributions from the inverse conductance parame-

ters), as discussed in Ref. [9].

As we showed above, the spin current (2.7) leads to a damping of the ferromagnetic

precession, resulting in a faster alignment of the magnetization with the (effective)

magnetic field Heff . In the derivation by the time-dependent scattering theory, the

pumped spins are entirely absorbed by the attached ideal reservoirs. In the following,

it is shown that Eq. (2.7) can be also derived for a finite system by observing that the


µ

δ

Figure 2.3: A ferromagnetic film F sandwiched between two nonmagnetic reservoirsN. For simplicity of the discussion in this section, we mainly focus on the dynamics inone (right) reservoir while suppressing the other (left), e.g., assuming it is insulating.The spin-pumping current Ipump

s and the spin accumulation µs in the right reservoircan be found by conservation of energy, angular momentum, and by applying circuittheory to the steady state Ipump

s = Ibacks .

enhanced rate of damping is accompanied by an energy flow out of the ferromagnet,

until a steady state is established in the combined F/N system. For simplicity, assume

a magnetization which at time t starts rotating around the vector of the magnetic

field, m(t) ⊥ Heff . In a short interval of time δt, it slowly (i.e., adiabatically) changes

to m(t+ δt) = m(t)+ δm. In the presence of a large but finite nonmagnetic reservoir

without any spin-flip scattering attached to one side of the ferromagnet, this process

can be expected to induce a (small) nonvanishing spin accumulation µs. For a slow

enough variation of m(t), this nonequilibrium spin imbalance must flow back into

the ferromagnet, canceling any spin current generated by the magnetization rotation,


since, due to the adiabatic assumption, the system is always in a steady state.

Let us assume for the moment that the spins are accumulated in the reservoir

along the magnetic field, µs ‖ Heff . Flow of Ns spins into the normal metal transfers

energy ∆EN = Nsµs/2 and angular-momentum ∆LN = Ns~/2 (directed along Heff).

By the conservation laws, ∆EF = −∆EN and ∆LF = −∆LN , for the corresponding

values in the ferromagnet. Using the magnetic energy ∆EF = γ∆LFHeff , where γ

is the absolute gyromagnetic ratio of the ferromagnet, we then find that Nsµs/2 =

γNs(~/2)Heff . It then follows that µs = ~γHeff = ~ω, where ω = γHeff is the Larmor

frequency of precession in the effective field: The spin-up and spin-down chemical

potentials in the normal metal are split by µs = ~ω, the energy corresponding to the

frequency of the perturbation. For a finite angle θ between µs and Heff , the same

reasoning would lead to µs = ~ω cos θ, which is smaller than the “energy boost” ~ω

of the time-dependent perturbation, thus justifying our initial guess.

We can employ magnetoelectronic circuit theory [19, 20] to derive an expression

for the backflow of spin current Ibacks which, as argued above, has to be equal to the

pumping current Ipumps = Iback

s :

Ibacks =

1

2π

(g↑↓r µs + g↑↓i m× µs

)=

~4π

(g↑↓r m× dm

dt− g↑↓i

dm

dt

). (2.15)

Here we used µs = ~ω and µs ⊥ m, since by the conservation of angular momentum,

the spin transfer is proportional to the change in the direction δm ⊥ m. We thus

recover Eq. (2.7) for the case of a single and finite reservoir. It is easy to repeat the

proof for an arbitrary initial alignment of m(t) with Heff . Furthermore, a straightfor-

ward generalization of this discussion to the case of the N/F/N sandwich structure

recovers our previous result [Eq. (2.7)].


The expressions for the adiabatic spin pumping are not the whole story, since

spin-flip scattering is an important fact of life in magnetoelectronics. In Ref. [130]

we only considered the extreme situation where the normal-metal layer is a perfect

spin sink, so that all spins injected by Ipumps relax by spin-flip processes or leave the

system; the total spin current through the contact was, therefore, approximated by

Is ≈ Ipumps and Iback

s ≈ 0. Here we generalize that treatment to self-consistently take

into account the spin build-up in the normal metal at dynamic equilibrium. We then

find the contribution to Is due to the spin–accumulation-driven current Ibacks back

into the ferromagnet, which vanishes in the absence of spin-flip scattering.

The spin current out of the ferromagnet carries angular momentum perpendicular

to the magnetization direction. By conservation of angular momentum, the spins

ejected by Is correspond to a torque τ = −Is on the ferromagnet. If possible inter-

facial spin-flip processes are disregarded, the torque τ is entirely transferred to the

coherent magnetization precession. The dynamics of the ferromagnet can then be

described by a generalized LLG equation [40, 71, 119]

dm


dt+

γ

MsVIs , (2.16)

where α0 is the dimensionless bulk Gilbert damping constant, Ms is the saturation

magnetization of the ferromagnet, and V is its volume. The intrinsic bulk constant

α0 is smaller than the total Gilbert damping α = α0 + α′. The additional damping

α′ caused by the spin pumping is observable in, for example, FMR spectra and is the

main object of interest here.


2.1.4 Spin Backflow in F/N and N/F/N Structures

The precession of the magnetization does not cause any charge current in the

system. The spin accumulation or nonequilibrium chemical potential imbalance µs(x)

[similar to Eq. (1.11), but spatially dependent now] in the normal metal is a vector,

which depends on the distance from the interface x, 0 < x < L, where L is the

thickness of the normal-metal film, see Fig. 1.2.

When the ferromagnetic magnetization steadily rotates around the z axis, m× m

and the normal-metal spin accumulation µs(x) are oriented along z, as depicted in

Fig. 1.2. There is no spin imbalance in the ferromagnet, because µs is perpendicular

to the magnetization direction m. As shown below, the time-dependent µs is also

perpendicular to m even in the case of a precessing ferromagnet with time-dependent

instantaneous rotation axis, as long as the precession frequency ω is smaller than the

spin-flip rate τ−1sf in the normal metal.

The spin accumulation diffuses into the normal metal as

iωµs = D∂2xµs − τ−1

sf µs , (2.17)

where D is the diffusion coefficient. The boundary conditions are determined by the

continuity of the spin current from the ferromagnet into the normal metal at x = 0

and the vanishing of the spin current at the outer boundary x = L:

x = 0 : ∂xµs = −2(~NSD)−1Is ,

x = L : ∂xµs = 0 , (2.18)

whereN is the (one-spin) density of states in the film and S is the area of the interface.


The solution to Eq. (2.17) with the boundary conditions (2.18) is

µs(x) =coshκ(x− L)

sinhκL

2Is

~NSDκ(2.19)

with the wave vector κ = λ−1sd

√1 + iωτsf , where λsd

def=√Dτsf is the spin-flip diffusion

length in the normal metal. In Sec. 2.2 we use arguments similar to those in the

present section to calculate the spin accumulation (2.19) generated by the precessing

magnetization. While the size of the effect and its relevance for spintronic applications

are detailed in Sec. 2.2, in this section we focus on the role of the spin accumulation

in the dynamics of the ferromagnetic magnetization.

We assume in the following that the precession frequency ω is smaller than the

spin-flip relaxation rate ω τ−1sf so that κ ≈ λ−1

sd . For a static applied field of 1 T,

typically ω ∼ 1011 s−1. The elastic scattering rate corresponding to a mean free path

of λel ∼ 10 nm is τ−1el ∼ 1014 s−1. Consequently, the derivation below is restricted to

metals with a ratio of spin-conserved to spin-flip scattering times εdef= τel/τsf & 10−3.

In practice [80], this condition is easily satisfied with higher impurity atomic numbers

Z (as ε scales as Z4 [1]). The high-frequency limit ω & τ−1sf , on the other hand,

is relevant for hybrids with little spin-flip scattering in the normal metal, and was

discussed in the context of the spin-battery concept [21]. Nevertheless, we will see

that a sizable Gilbert damping enhancement requires a large spin-flip probability

ε & 10−1 (thereby guaranteeing that ω τ−1sf ) unless the frequency is comparable

with the elastic scattering rate in the normal metal. The latter regime will not be

treated in this section.

Using relation D = v2F τel/3 between the diffusion coefficient D (in three dimen-

sions), the Fermi velocity vF , and the elastic scattering time τel, we find for the


spin-diffusion length

λsd = vF

√τelτsf/3 . (2.20)

An effective energy-level spacing of the states participating in the spin-flip scattering

events in a thick film can be defined by

δsddef= (NSλsd)

−1 . (2.21)

The spin–accumulation-driven spin current Ibacks through the interface reads [20]

Ibacks =

1

8π

[2g↑↓r µs(x = 0) + 2g↑↓i m× µs(x = 0)

+(g↑↑ + g↓↓ − 2g↑↓r

)(m · µs(x = 0))m

]. (2.22)

Substituting Eq. (2.19) into Eq. (2.22), we find for the total spin current [Eq. (1.6)]

Is = Ipumps − β

2

[2g↑↓r Is + 2g↑↓i m× Is +

(g↑↑ + g↓↓ − 2g↑↓r

)(m · Is)m

], (2.23)

where the spin current returning into the ferromagnet is governed by the “backflow”

factor β,

βdef=

τsfδsd/h

tanh(L/λsd). (2.24)

When the normal metal is shorter than the spin-diffusion length (L λsd), β →

τsfδ/h, where δ = (NSL)−1 is the energy-level splitting. In the opposite regime of

thick normal metals (L λsd), β → τsfδsd/h. Basically, β [Eq. (2.24)] is therefore

the ratio between the energy level spacing of the normal-metal film with a thickness

Lsf = min(L, λsd) and the spin-flip rate.

By inverting Eq. (2.23), we may express the total spin current Is in terms of the

pumped spin current Ipumps [Eq. (2.7)]

Is =

[1 + βg↑↓r +

(βg↑↓i )2

1 + βg↑↓r

]−1(1− βg↑↓i

1 + βg↑↓r

m×

)Ipumps . (2.25)


After substituting Eq. (2.7) into Eq. (2.25), we recover the form of Eq. (2.7) for the

total spin current Is, but with a redefined spin-pumping conductance Ar + iAi,

Is =~4π

(Arm× dm

dt− Ai

dm

dt

). (2.26)

A can be expressed in terms of the mixing conductance g↑↓ and the backflow factor

β by Ar

Ai

=

1 βg↑↓i (1 + βg↑↓r )−1

−βg↑↓i (1 + βg↑↓r )−1 1

×

[1 + βg↑↓r +

(βg↑↓i )2

1 + βg↑↓r

]−1 g↑↓r

g↑↓i

. (2.27)

It has been shown [146] that for realistic F/N interfaces g↑↓i g↑↓r , so that g↑↓ ≈ g↑↓r .

(The latter approximation will be implied for the rest of our discussion.) In this

important regime, Ai vanishes and the term proportional to Ar in Eq. (2.26) has the

same form as and therefore enhances the phenomenological Gilbert damping. This

can be easily seen after substituting Eq. (2.26) into Eq. (2.16): The last term on the

right-hand side of Eq. (2.16) can be combined with the second term by defining the

total Gilbert damping coefficient α = α0 + α′, where

α′ =

[1 + g↑↓

τsfδsd/h

tanh(L/λsd)

]−1γ~g↑↓

4πMsV(2.28)

is the additional damping constant due to the interfacial F/N coupling. Equa-

tion (2.28) is the main result of this section. When L → ∞, Eq. (2.28) reduces

to a simple result: α′ = gLg↑↓eff/(4πµ), where

1

g↑↓eff=

1

g↑↓+Rsd . (2.29)


Here, Rsd = τsfδsd/h is the resistance (per spin, in units of h/e2) of the normal-

metal layer of thickness λsd. [Which follows from the Einstein’s relation σ = e2DN

connecting conductivity σ with the diffusion coefficient D, and using Eq. (2.21).] It

follows that the effective spin pumping out of the ferromagnet is governed by g↑↓eff , i.e.,

the conductance of the F/N interface in series with diffusive normal-metal film with

thickness λsd [9].

The prefactor on the right-hand side of Eq. (2.28) suppresses the additional Gilbert

damping due to the spin angular momentum that diffuses back into the ferromagnet.

It was disregarded in Sec. 2.1.2 where the normal metal was viewed as a perfect spin

sink. Because spins accumulate in the normal metal perpendicular to the ferromag-

netic magnetization, the spin–accumulation-driven transport across the F/N contact,

as well as the spin pumping, is governed by a mixing conductance. This explains why

the other components of the conductance matrix do not enter Eq. (2.28).

We now estimate the numerical values of the parameters in Eq. (2.28) for transition

metal ferromagnets Fe, Co, and Ni, in contact with relatively clean simple normal

metals Al, Cr, Cu, Pd, Ag, Ta, Pt, and Au. For an isotropic electron gas, N =

k2F/(πhvF ). Using Eqs. (2.20) and (2.21), we find h/(δsdτsf) = 4

√ε/3Nch, where

Nch = Sk2F/(4π) is the number of transverse channels in the normal metal and ε is

the spin-flip probability at each scattering. In Ref. [146], g↑↓ was calculated for Co-Cu

and Fe-Cr interfaces by first-principles band-structure calculations. It was found that

irrespective of the interfacial disorder, g↑↓ ≈ Nch for these material combinations.

As shown in Ref. [9], g↑↓ has to be renormalized in such limit, making the effective


conductances about twice as large. We thus arrive at an estimate

α′∞α′

≈ 1 +[√ε tanh(L/λsd)

]−1, (2.30)

where α′∞ = γ~g↑↓/(4πMsV ) is the Gilbert damping enhancement assuming infinite

spin-flip rate in the normal metal τsf → 0, i.e., treating it as a perfect spin sink [130].

It follows that only for a high spin-flip probability ε & 10−2, the normal-metal film

can be a good spin sink so that α′ ∼ α′∞. This makes the lighter metals, such as Al,

Cr, and Cu, as well as heavier metals with only s electrons in the conduction band,

such as Ag, Au, and Ta less effective spin sinks since these metals have a relatively

small spin-orbit coupling, typically corresponding to ε . 10−2 [80, 13, 150]. Heavier

elements with Z & 50 and p or d electrons in the conduction band, such as Pd and

Pt, on the other hand, can be good or nearly perfect spin sinks as they have a much

larger ε & 10−1 [80]. This conclusion explains the hierarchy of the observed Gilbert

damping enhancement in Refs. [82, 83]: Pt has about 2 electrons per atom in the

conduction band, which are hybridized with d orbitals, and a large atomic number

Z = 78 and, consequently, leads to a large magnetization damping enhancement

in the N/F/N sandwich for thin ferromagnetic films. Pd which is above Pt in the

periodic table having similar atomic configuration but smaller atomic number Z = 46

leads to a sizable damping, but smaller than for Pt by a factor of 2. Ta is a heavy

element, Z = 73, but has only s electrons and the damping enhancement is an order

of magnitude smaller than in Pt. Finally, Cu is a relatively light element, Z = 29,

with s electrons only and does not cause an observable damping enhancement at all.

According to Eq. (2.30), a sufficiently thick active layer, L & λsd, is also required for

a sizable spin relaxation.


The limit of a large ratio of spin-flip to non-spin-flip scattering ε ∼ 1 deserves

special attention. In this regime, Eq. (2.30) does not hold, since by using the diffusion

equation (2.17) and boundary conditions (2.18) we implicitly assumed that ε 1. If

ε & 10−1, on the other hand, even interfacial scattering alone can efficiently relax the

spin imbalance, and such films, therefore, are good or nearly perfect spin sinks (so

that α′ ∼ α′∞), regardless of their thickness (in particular, they can be thinner than

the elastic mean free path).

Infinite vs vanishing spin-flip rates in the normal metal are two extreme regimes

for the magnetization dynamics in F/N bilayers. In the former case, the damping

constant α = α0 + α′∞ is significantly enhanced for thin ferromagnetic films, whereas

in the latter case, α = α0 is independent of the ferromagnetic film thickness. Experi-

mentally, the two regimes are accessible by using Pt as a perfect or Cu as a poor spin

sink in contact with a ferromagnetic thin film, as done in Refs. [82, 83] for N /Py/N

sandwiches. (Using the N/F/N trilayer simply increases α′ by the factor of 2, as

compared to the F/N bilayer, due to the spin pumping through the two interfaces.)

The measured damping parameter G = γMsα is shown in Fig. 2.4 by circles.

For the Cu/Py/Cu trilayer, our theory predictsG(d) = G0, while for the Pt/Py/Pt

sandwich

G(d) = G0 +(gLµB)2

2π~g↑↓S−1

d(2.31)

as a function of ferromagnetic film thickness d. The Py g factor is gL ≈ 2.1 [82, 83].

These expression agree with the experiments for G0 = 1.0 × 108 s−1 and g↑↓S−1 =

2.6×1015 cm−2 (see Fig. 2.4). Both numbers are very reasonable: G0 equals the bulk

value 0.7 − 1.0 × 108 s−1 for Py [98, 8], while g↑↓S−1 compares well with g↑↓S−1 ≈


20 40 60 80 100d [Å]

0

3

6

9

12

15

G [1

08 s−

1 ]

Pt−Py(d)−PtCu−Py(d)−Cu

Figure 2.4: Circles show measured (Refs. [82, 83]) Gilbert parameter G of a permalloyfilm with thickness d sandwiched between two normal-metal (Pt or Cu) layers. Solidlines are predictions of our theory with two fitting parameters, G0 and g↑↓–Py bulkdamping and Py/Pt mixing conductance, respectively, see Eq. (2.31).

1.6× 1015 cm−2 found in angular-magnetoresistance (aMR) measurements in Py/Cu

hybrids [9]. (We recall that here one has to use the renormalized mixing conductance

g↑↓, in the notation of Ref. [9].) In fact, since Pt has two conduction electrons per

atom, while Cu–only one, and they have similar crystal structures, we expect g↑↓ to be

larger in the case of the Py/Pt hybrid, justifying the value used to fit the experimental

data. We have thus demonstrated that the additional damping in ferromagnetic thin

films can be used to measure the mixing conductance of the F/N interface.


2.1.5 Damping in F/N1/N2 Trilayers

In this section we consider ferromagnetic spin pumping into a bilayer N1/N2

normal-metal system, see Fig. 1.3. It is assumed that the spins are driven into the

first normal-metal film (N1 ) of thickness L. While in N1, spins are allowed to diffuse

through the film, where they can relax, diffuse back into the ferromagnet, or reach the

second normal-metal layer (N2 ). N2 is taken to be a perfect spin sink: spins reaching

N2 either relax immediately by spin-flip processes or are carried away before diffusing

back into N1. We show that measuring the ferromagnetic magnetization damping as

a function of L in this configuration can be used to study the dc mixing conductance

of the two N1 film interfaces as well as the N1 spin-diffusion time.

The analysis in this section was inspired by experiments of Mizukami et al. [85],

who in a follow-up to their systematic study of Gilbert damping in N /Py/N sand-

wiches [82, 83], studied magnetization damping in Py/Cu and Py/Cu/Pt hybrids as

a function of Cu film thickness L. The measured damping parameter G is shown

by circles in Fig. 2.5. As shown in the preceding section, Cu is a poor sink for the

pumped spins, while Pt is nearly a perfect spin absorber, thus identifying the Cu film

with N1 and the Pt layer with N2.

We use the same notation as in the previous section to discuss the F/N1 spin

pumping with subsequent spin diffusion through N1. Similar to Eqs. (2.18), the

boundary conditions for the diffusion equation (2.17) in the normal metal N1 are


10 100 1000 10000L [Å]

0.6

0.8

1

1.2

1.4

1.6

1.8

G [1

0−8 s

−1]

Py−Cu(L)−PtPy−Cu(L)

Figure 2.5: Circles show the measurements by Mizukami et al. (Ref. [85]) of theGilbert damping in Py/Cu/Pt trilayer and Py/Cu bilayer as a function of the Cubuffer thickness L. Solid lines are our theoretical prediction according to Eqs. (2.35)and (2.36).

now:

x = 0 : ∂xµs = −2(~NSD)−1Is1 ,

x = L : ∂xµs = −2(~NSD)−1Is2 . (2.32)

Is1 and Is2 are the total spin currents through the left (x = 0) and right (x = L)

interfaces, respectively. Is1 (similarly to Is [Eq. (1.6)] in the previous section) includes

the pumped spin current (2.7) and the spin–accumulation-driven spin current (2.22)

contributions. Is2, on the other hand, is entirely governed by the N1→N2 spin–


accumulation-driven flow

Is2 =g

4πµs(x = L) , (2.33)

where g is the conductance per spin of the N1/N2 interface.

Solving the diffusion equation (2.17) with the boundary conditions (2.32), we

find the spin current Is1 as we did in the preceding section. The Gilbert damping

enhancement due to the spin relaxation in the composite normal-metal system is then

given by

α′ =

[1 + g↑↓

τsfδsdh

1 + tanh(L/λsd)gτsfδsd/h

tanh(L/λsd) + gτsfδsd/h

]−1γ~g↑↓

4πMsV. (2.34)

Setting g = 0 decouples the two normal-metal systems and reduces Eq. (2.34) to

Eq. (2.28) giving the damping coefficient of the F/N1 bilayer. From Eq. (2.34), we

get for the Py/Cu(L)/Pt trilayer

G(L) = G0 +

[1 + g↑↓

τsfδsdh

1 + tanh(L/λsd)gτsfδsd/h

tanh(L/λsd) + gτsfδsd/h

]−1(gLµB)2

2h

g↑↓S−1

d(2.35)

and for the Py/Cu(L) bilayer (putting g = 0)

G(L) = G0 +

[1 +

g↑↓τsfδsd/h

tanh(L/λsd)

]−1(gLµB)2

2h

g↑↓S−1

d. (2.36)

In the experiments, the permalloy thickness d = 30 A is fixed and the Cu film

thickness L is varied between 3 and 1500 nm as shown by the circles in Fig. 2.5.

Our theoretical results, Eqs. (2.35) and (2.36), are plotted in Fig. 2.5 by solid lines.

We use the following parameters: The bulk damping [98, 8] G0 = 0.7 × 108 s−1; the

spin-flip probability ε = 1/700 and the spin-diffusion length λsd = 250 nm for Cu

(which correspond to elastic mean free path λel =√

3ελsd = 16 nm), in agreement

with values reported in literature [80, 150, 61]; g↑↓S−1 = 1.6 × 1015 cm−2 from the

aMR measurements [9]; and gS−1 = 3.5 × 1015 cm−2 for the Cu/Pt contact, which


lies between values for the majority and minority carriers as measured and calculated

[147] for the Cu/Co interface. Figure 2.5 shows a remarkable agreement (within the

experimental error) between the measurements and our theory. It is important to

stress that while the profiles of the trends displayed in Fig. 2.5 reveal the diffusive

nature of spin transfer in the Cu spacer, they cannot be used to judge the validity of

a detailed mechanism for spin injection (relaxation) at the Py/Cu (Cu/Pt) interface.

The case of our spin pumping picture is strongly supported by the normalization

of the curves (in agreement with experiment), which are governed in our theory by

quantities known from other sources.

The trends in Fig. 2.5 can be understood as follows. Since Cu is a poor spin

sink, a Py/Cu contact with a single Cu film does not lead to a significant damping

enhancement. The small spin-flip ratio, ε 1, causes most of the spins transferred

into the normal-metal layer to be scattered back and relax in the ferromagnet be-

fore flipping their direction in the Cu buffer. This leads only to a small damping

enhancement, which saturates at L λsd and vanishes in the limit L λsd. The

situation changes after a Pt film, a very good spin sink, is connected to the bilayer: If

the normal-metal layer is smaller than the elastic mean free path, L λel, the spin

accumulation is uniform throughout the Cu buffer. The spin pumping will now be

partitioned. A fraction of the pumped spins reflects back into the ferromagnet, while

the rest get transmitted and subsequently relax in the Pt layer. The ratio between

these two fractions equals the ratio between the conductance of the Py/Cu contact

and the Co/Pt contact, g↑↓/g, and is of the order of unity. This results in a large

magnetization damping as a significant portion of the spin pumping relaxes by spin-


orbit scattering in Pt. When L is increased, less spins manage to diffuse through the

entire Cu buffer, and, in the limit L λsd, the majority of the spins scatter back

into the ferromagnet or relax in Cu not feeling the presence of the Pt layer at all. In

the intermediate regime, the spin pumping into the Pt layer has an algebraic fall-off

on the scale of the elastic mean free path and exponential one on the scale of the

spin-diffusion length.

It is important to emphasize that the strong dependence of damping on the Cu

layer thickness L in the Py/Cu/Pt configuration gives evidence of the spin accumula-

tion in the normal-metal system. This spin accumulation, in turn, indicates that an

excited ferromagnet (as in the FMR experiment discussed here) transfers spins into

adjacent nonmagnetic layers, confirming our claim [130]. Furthermore, this supports

our concept of the spin battery, see Sec. 2.2.

Before ending this section, it is illuminating to make a small digression and further

study Eq. (2.35) in the limit of vanishing spin-flip processes in the buffer layer N1.

Recalling our definitions for λsd and δsd [Eqs. (2.20) and (2.21)] and taking limit

τsf →∞, we find that Eq. (2.35) reduces to Eq. (2.31), only with g↑↓ replaced by g↑↓eff

defined in Eq. (1.9) [compare it with Eq. (2.29)]. The right-hand side of Eq. (1.9) is

simply the inverse mixing conductance of the N1 buffer in series with its two interfaces

(one with F and one with N2 ) [9]; in particular, when layer N1 is thick enough, the

total mixing conductance g↑↓eff is just the conductance of the diffusive normal-metal

spacer separating F and N2 [19, 20]. The spin pumping into layer N1 with the

subsequent diffusion and then spin absorption by the ideal spin sink N2 (as discussed

in this section) can thus be viewed as the spin pumping across an effective combined


scatterer separating the ferromagnet (F ) from the perfect spin sink (N2 ) [as done in

obtaining Eq. (2.31)]. This shows consistency of our approach.

2.1.6 Summary

Let us now summarize our results in Sec. 2.1, which serve as a basis for all the

following discussions in the present chapter. Ferromagnets emit a spin current into

adjacent normal metals when the magnetization direction changes with time. In

Secs. 2.1.2 and 2.1.3 we investigated a novel mechanism for this spin transfer based

on the picture of adiabatic spin pumping [130]. It was shown that our theory explains

the increased magnetization damping in ferromagnets in contact with normal metals

in measurements of the FMR line widths [82, 83, 85, 52, 136].

Whereas the spin pumping affects the magnetization dynamics, it also creates

a nonequilibrium magnetization in adjacent nonmagnetic films. We first calculated

this spin accumulation for F/N metallic multilayers in Sec. 2.1.4 and found that it

induces a spin backflow into the ferromagnetic layer that reduces the overall spin

pumping. This spin–accumulation-driven current is significant for light metals or

metals with only s electrons in the conduction band, which have a small spin-flip to

spin-conserving scattering ratio.

The picture of ferromagnetic spin pumping and subsequent spin diffusion in the

adjacent normal-metal layers was further applied to the F/N1/N2 configuration in

Sec. 2.1.5 in order to analyze recent experiments [85] on magnetization damping in

Py/Cu/Pt trilayers. We showed that our theory quantitatively explains the exper-

imental findings. Our analysis of the experiments by Mizukami et al. [82, 83, 85]


demonstrates that FMR of ultrathin ferromagnetic films in contact with single or

composite normal-metal buffers is a powerful tool to investigate interfacial transport

properties of magnetic multilayers as well as the spin relaxation parameters of the

normal-metal layers.


2.2 FMR-Operated Spin Battery

Precessing ferromagnets are predicted to inject a spin current into adjacent con-

ductors via Ohmic contacts, irrespective of a conductance mismatch with, for exam-

ple, doped semiconductors. This opens the way to create a pure spin source (spin

battery) by the ferromagnetic resonance. We estimate the spin current and spin bias

for different material combinations.

2.2.1 Background

The research field of magnetoelectronics strives to utilize the spin degree of free-

dom for electronic applications [145]. Devices made from metallic layered systems dis-

playing the giant [6] and tunnel magnetoresistance [81, 88] have been proven useful for

read-head sensors and magnetic random-access memories. Integration of such devices

with semiconductor electronics is desirable but difficult because a large resistivity mis-

match between magnetic and normal materials is detrimental to spin injection [112].

Spin injection into bulk semiconductors has been reported only in optical pump and

probe experiments [67], and with high-resistance ferromagnetic injectors [36, 94] or

Schottky/tunnel barriers [87, 151]. In these cases, the injected spin-polarized carriers

are hot and currents are small, however. Desirable are semiconductor devices with an

efficient all-electrical cold-electron spin injection and detection via Ohmic contacts at

the Fermi energy, just as has been realized by Jedema et al. for metallic devices [61].

We introduce a concept for dc spin-current injection into arbitrary conductors

through Ohmic contacts, which does not involve net charge currents. The spin source

is a ferromagnetic reservoir at resonance with an rf field. Pure spin-current injection


into low-density conductors should allow experimental studies of spintronic phenom-

ena in mesoscopic, ballistic, and nanoscale systems, which up to now has been largely

a playground of theoreticians like Datta and Das [29], whose spin transistor concept

has stimulated much of the present interest in spintronics.

The combination of a ferromagnet at the ferromagnetic resonance in Ohmic con-

tact with a conductor can be interpreted as a spin battery, with analogies and differ-

ences with charge batteries. For example, charge-current conservation dictates that a

charge battery has two poles, plus and minus. A spin battery requires only one pole,

since the spin current does not need to be conserved. Furthermore, the polarity is

not a binary, but a three-dimensional vector. The important parameters of a charge

battery are the maximum voltage in the absence of a load, as well as the maximum

charge current, which can be drawn from it. In the following we present estimates for

the analogous characteristics of the spin battery.

2.2.2 Functionality and Maximum Loads

Central to our concept is a precessing ferromagnet, which acts as a source of spin

angular momentum, when in contact with normal metals [130], see Fig. 2.6. This

spin injection can be formulated in analogy with the adiabatic pumping of charge

in mesoscopic systems [22]. When the ferromagnet is thicker than the ferromagnetic

coherence length (a few Angstrøms in transition metals such as Co, Ni or Fe), the

spin current emitted into the normal metal is determined by the mixing conductance

g↑↓, see Sec. 2.1.2. The mixing conductance governs the transport of spins that are

noncollinear to the magnetization direction in the ferromagnet [19, 20, 54] and is


! #"$&%('*)+"

,.-

/ 02143

5 687:9<;=57>? @A BC DFEHGJILKNM OQPSRUTV

Figure 2.6: Schematic view of the spin battery. Precession of the magnetization m(t)of the ferromagnet F emits a spin current Isource

s into the adjacent normal-metal layerN. The spin accumulation in the normal metal either relaxes by spin-flip scatteringor flows back into the ferromagnet, resulting in a net spin current Is = Isource

s − Ibacks .

also a material parameter proportional to the torque acting on the ferromagnet in

the presence of a noncollinear spin accumulation in the normal metal [130, 19, 20,

146, 125]. For most systems (with the exception of, e.g., ferromagnetic insulators

[53]) the imaginary part of the mixing conductance can be disregarded due to the

randomization of phases of spin-up and spin-down electrons in reciprocal space [146]

and this is assumed in the following. The spin current emitted into the normal metal

is then, simply [130]

Isources =

~4πg↑↓m× dm

dt. (2.37)

In our notation, the spin current is measured in units of mechanical torque. Eq. (2.37)

is a time-dependent correction to the Landauer-Buttiker formula for noncollinear

ferromagnetic/normal-metal hybrid systems [19, 20]. A simple physical picture can

be inferred from the following thought experiment [114]. Suppose we have a F/N in-

terface at equilibrium and switch the magnetization instantaneously. The mismatch

of the spin-up and spin-down chemical potentials leads to large nonequilibrium spin


currents on the length scale of the spin-diffusion length. A slower magnetization re-

versal naturally induces smaller spin currents. Eq. (2.37) represents the adiabatic

limit of the spin currents pumped by a slow magnetization dynamics. When the spin

current (2.37) is channeled off sufficiently rapidly, the corresponding loss of angular

momentum increases the (Gilbert) damping of the magnetization dynamics, as dis-

cussed in Sec. 2.1. Eq. (2.37) is the maximum spin current that can be drawn from

the spin battery.

Next, we need the maximum spin bias obtained when the load vanishes. When

the spin-flip relaxation rate is smaller than the spin-injection rate, a spin angular

momentum s (in units of ~) builds up in the normal metal. We can neglect spatial

dependence within the ferromagnet when the film is sufficiently thin. Under these

conditions, one finds that the component of the backflow spin current Ibacks , from

the normal metal to the ferromagnet, parallel to the instantaneous magnetization

direction m is canceled by an opposite flow from the ferromagnet. The component of

Ibacks perpendicular to m is [19, 20]

Ibacks =

g↑↓

2πN[s−m (m · s)] , (2.38)

where N is the one-spin density of states. We note that the mixing conductance in

Eqs. (2.37) and (2.38) ought to be renormalized in highly transparent junctions [9].

The relation between spin excess s and total spin current Is = Isources − Iback

s in a

normal diffusive metal is governed by the spin-diffusion equation [64]

∂s

∂t= D

∂2s

∂x2− s

τsf, (2.39)

where D is the diffusion coefficient in d dimensions, D = v2F τel/d. We solve the

diffusion equation with boundary conditions at x = 0, where (DS~)∂xs = −Is, and


at the end of the sample x = L, where the spin current vanishes, ∂xs = 0. S is the

cross section of the system.

The precession of the magnetization vector of a ferromagnet under a resonant rf

electromagnetic field applied perpendicularly to a dc magnetic field [117] can be used

to drive the spin battery. The magnitude of the spin current Isources and spin bias

∆µ = 2〈s〉t/N as a function of the applied field H0 follows from the LLG equation

(2.1), where magnetic anisotropies have been disregarded for simplicity. The spin bias

also has ac components. However, its frequency ω harmonics are strongly suppressed

when λsd/(ωτsf)1/2 < L < λsd, which can be easily realized when ωτsf > 1, e.g.,

τsf > ω−1 ∼ 10−11 s/H0 [T]. The dominant contribution to the spin bias is then

constant in time and directed along H0. The magnitude of the time-averaged spin

accumulation ∆µ = 2〈s(t)〉t/N in the normal metal close to the F/N interface then

reads

∆µ = ~ω0sin2 θ

sin2 θ + η, (2.40)

where the precession cone angle between H0 and m is θ, η = (τi/τsf) tanh(L/λsd)/(L/λsd)

is a reduction factor, and we have introduced the spin-injection rate τ−1i = g↑↓/(2π~NSL).

Large systems have a smaller injection rate since more states have to be filled.

The ratio of the injection and spin-flip relaxation times can be evaluated for a pla-

nar geometry. We consider a free-electron gas in contact with a metallic ferromagnet.

The mixing conductance is g↑↓ = κSk2F/(4π) (g↑↓ = κSkF/π) for spin injection into

three-(two-)dimensional systems. First-principles band-structure calculations show

that for combinations like Co/Cu or Fe/Cr κ remains close to unity [146]. The ratio

between the injection and spin-flip relaxation times in three (two) dimensions can be


calculated to be τi/τsf =√

8/3κ−1√ε(L/λsd) [τi/τsf = 2κ−1

√ε(L/λsd)]. ε, the ratio of

the elastic scattering rate and the spin-flip relaxation rate, is usually much smaller

than unity.

When the spin relaxation time is longer than the spin injection time and the

precession cone angle is sufficiently large, sin2 θ > η, the spin bias saturates at its

maximum value ∆µ0 = ~ω0. In this regime the spin accumulation does not depend

on the material parameters. It should be feasible to realize the full spin bias when

L λsd since η ≈√

8/3κ−1√ε(L/λsd), e.g., when L/λsd = 0.1,

√8/3κ−1

√ε = 0.1

the precession cone angle should be larger than 6 degrees. For small precession

cone angles θ ≈ H1/(αH0), so for, e.g., H0 = 1.0 T, α = 10−3 this requires a

H1 = 0.1 mT rf field with a resulting spin bias of ∆µ = 0.1 meV. For a smaller

precession angle, e.g., θ = 0.6 degrees the spin-bias is smaller, ∆µ = 1 µV, but still

clearly measurable. Epitaxially grown clean samples with even longer spin-diffusion

lengths and smaller spin-flip to non-spin flip relaxation ratios ε will function as spin-

batteries with smaller precession angles. The precession cone angle θ in FMR is

typically small, but θ > 15 degrees can be achieved for a sufficiently intense rf field

and a soft ferromagnet, e.g., permalloy [48]. The maximum dc spin current source is

|〈Isources 〉t| ≈ ~ω0κSk

2F sin2 θ/(4π), e.g. for a precession cone angle of θ = 6 degrees

the equivalent electrical spin current (e/~)|〈Isources 〉t| is 0.1 nA per conducting channel.

The total number of channels, Sk2F/(4π), is a large number since the cross sections

may be chosen very much larger than the Fermi wavelength thus ensuring that a large

spin current may be drawn from the battery.

Ferromagnetic resonance dissipates energy proportional to the damping param-


eter α of the magnetization dynamics. The power loss dE/dt = α~ω20Ns sin2 θ is

proportional to the volume of the ferromagnet through the number of spins in the

ferromagnet in units of ~, Ns. The power loss can be significant even for a thin film

ferromagnet, e.g., for a 10 monolayer thick Fe film with α ∼ 10−3, sin2 θ ∼ 10−2,

and ω0 ∼ 1011 s−1, the power loss per unit area is (1/S)dE/dt ∼ 0.1 W/cm2. The

temperature can be kept low by, e.g., immersing the sample in superfluid helium.

The heat transfer is then approximately 8 W/cm2K for small temperature gradients

and increases for larger temperature gradients [116], which appears sufficient for the

present purpose.

Schmidt et al. [112] realized that efficient spin injection into semiconductors by

Ohmic contacts is close to impossible with transition-metal ferromagnets since virtu-

ally all of the applied potential drops over the nonmagnetic part and is wasted for

spin injection. The present mechanism does not rely on an applied bias and does not

suffer from the conductance mismatch, because the smallness of the mixing conduc-

tance for a ferromagnet/semiconductor interface is compensated by the small spin

current that is necessary to saturate the spin accumulation.

Possible undesirable spin precession and energy generation in the normal-metal

parts of the system is of no concern for material combinations with different g factors,

as, e.g., Fe (g = 2.1) and GaAs (g = −0.4), or when the magnetic anisotropy modifies

the resonance frequency with respect to electrons in the normal metal. The optimal

material combinations for a battery depend on the planned usage. From Eq. (2.37) it

follows that the largest spin current can be achieved when the conductor is a normal

metal, whereas any material combination appears suitable when the load is small, as


long as the contact is Ohmic and the system is smaller than the spin-diffusion length.

Standard metals, like Al and Cu, are good candidate materials, since the spin-

diffusion length is very long, λsd ∼ 1 µm at low temperatures, and remains quite long

at room temperature [61, 62]. Indirect indications of spin accumulation in Cu can be

deduced from the absence of any enhancement of the Gilbert damping in FMR when

in contact with thin ferromagnetic films [82, 83, 130].

Semiconductors have the advantage of a larger ratio of spin bias to Fermi energy.

Let us first consider the case of GaAs. The spin-flip relaxation time in GaAs can be

very long, τsf = 10−7 s at n = 5× 1016 cm3 carrier density [66, 121]. These favorable

numbers are offset by the difficulty to form Ohmic contacts to GaAs, however. Large

Schottky barrier exponentially suppress the interfaces mixing conductance parameter

κ. InAs has the advantage of a natural accumulation layer at the surface that avoids

Schottky barriers when covered by high-density metals. However, the spin-orbit in-

teraction in a narrow gap semiconductor like InAs is substantial, which reduces τsf .

In asymmetric confinement structures, the spin-flip relaxation rate is governed by

the Rashba type spin-orbit interaction, which vanishes in symmetrical quantum wells

[63, 33]. The remaining D’yakonov-Perel scattering rate is reduced in narrow quasi-

one-dimensional channels of width d due to waveguide diffusion modes by a factor

of (Ls/d)2, where Ls = vF/|h(kF )| is the spin-precession length in terms of the spin-

orbit coupling Hso = h(k) · s [76], which makes InAs-based systems an interesting

material for a spin battery as well. In Si, the spin-flip relaxation time is long, since

spin-orbit interaction is weak. Furthermore, the possibility of heavy doping allows

control of Schottky barriers. So, Si appears to be a good candidate for spin injection


into semiconductors.

The spin bias can be detected noninvasively via tunnel junctions with an analyzing

ferromagnet having a switchable magnetization direction. A voltage difference of p∆µ

is detected for parallel and anti-parallel configurations of the analyzing magnetization

with respect to the spin accumulation in the normal metal, where p = (G↑−G↓)/(G↑+

G↓) is the relative polarization of the tunnel conductances of the contact. The test

magnetic layer need not be flipped. It is sufficient to reverse the direction of the dc

static magnetic field. The spin current, on the other hand, can be measured via the

drop of spin bias over a known resistive element.

2.2.3 Nuclei Polarization and the Overhauser Field

Spin-pumping into the normal metal can also have consequences for the nuclei

via the hyperfine interaction between electrons and nuclear spins [66]. An initially

unpolarized collection of nuclear spins can be oriented by a spin-polarized electron

current, which transfers angular momentum by spin-flop scattering. A ferromagneti-

cally ordered nuclear-spin system can lead to an Overhauser field [95] on the electron

spin. This effect does not affect the spin bias ∆µ, but induces an equilibrium spin

density in the normal metal s0 via the nuclear magnetic field, and can be exploited in

experiments where the the total spin-density s + s0 is an important parameter. The

electron-nuclear interaction can be included by adding [95, 117]

Inucs =

~sn

Tn

(2.41)

to the electron spin dynamics so that I → Isources − Iback

s + Inucs , where sn is the

nonequilibrium nuclear spin accumulation and Tn is the electron-nuclear relaxation


time. The nuclear spin dynamics is described by

dsn

dt= − sn

T ′n

+s

Te

, (2.42)

where T ′n ≤ Tn is the nuclear-spin relaxation time and Te is the nuclear-electron

relaxation. In steady state, sn = (T ′n/Tn)(Tn/Te)s. In the experimentally relevant

regime T−1e τ−1

i the electron-nuclear interaction (2.41) has a negligible effect on the

nonequilibrium spin accumulation s and thus Eq. (2.40) remains unchanged. Tn/Te =

8I(I + 1)εFnN/(9kBTne) for small polarizations, where εF is the Fermi energy of the

electron gas, kBT is the thermal energy, nN is the nuclear density and ne is the one-

spin electron density [95]. Using N = (3/2)ne/εF (N = ne/εF in two dimensions)

and Eq. (2.40) the relative enhancement of the dc nuclear spin polarization is

sn = nNT ′

n

Tn

2

3I(I + 1)

∆µ

kBT. (2.43)

for ∆µ kBT . The nuclear-spin polarization increases with the spin bias and by

lowering the temperature. The hyperpolarized nuclei, in turn, produce an effective

nuclear field that polarizes the equilibrium properties of the electron gas s0. In bulk

GaAs, the nuclear magnetic field is Bn = 5.3 T when the nuclei are fully spin-polarized

which should occur at sufficiently low temperatures [96].

2.2.4 Summary

In Sec. 2.2 we presented the new concept of a spin battery, which is a source of spin,

just as a conventional battery is a source of charge, and estimated its performance

for different material combinations.


2.3 Dynamic Exchange Coupling

A long-ranged dynamic interaction between ferromagnetic films separated by normal-

metal spacers is reported in this section, which is communicated by nonequilibrium

spin currents. It is measured by ferromagnetic resonance and explained by an adi-

abatic spin-pump theory. In FMR the spin-pump mechanism of spatially separated

magnetic moments leads to an appreciable increase in the FMR line width when the

resonance fields are well apart, and results in a dramatic line-width narrowing when

the FMR fields approach each other.

2.3.1 Background

Magnetic read heads and high-density nonvolatile magnetic random-access memo-

ries typically consist of F/N/F metal hybrid structures, i.e., magnetic bilayers which

are an essential building block of the so called spin valves. The static Ruderman-

Kittel-Kasuya-Yosida (RKKY) interlayer exchange between ferromagnets in magnetic

multilayers [43, 49, 23, 124, 118] is suppressed in these devices by a sufficiently thick

nonmagnetic spacer N or a tunnel barrier. In the following we study the largely un-

explored dynamics of magnetic bilayers in a regime when there is no discernible static

interaction between the magnetization vectors. Surprisingly, the magnetizations still

turn out to be coupled, which we explain by emission and absorption of nonequilib-

rium spin currents. Under special conditions the two magnetizations are resonantly

coupled by spin currents and carry out a synchronous motion, quite analogous to

two connected pendulums. This dynamic interaction is an entirely new concept and

physically very different from the static RKKY coupling. E.g., the former does not os-


cillate as a function of thickness and its range is exponentially limited by the spin-flip

relaxation length of spacer layers and algebraically by the elastic mean free path. This

coupling can have profound effects on magnetic relaxation and switching behavior in

hybrid structures and devices.

2.3.2 Theoretical Model

The unit vector m = M/Ms of the magnetization M(t) of a ferromagnet changes

its direction in the presence of a noncollinear external magnetic field. The motion of m

in a single domain is described by the LLG equation (2.1). The magnetization vector

can be forced into a resonant precession motion by microwave stimulation. This FMR

resonance is measured via the absorption of microwave power using a small rf field at

a frequency ω polarized perpendicular to the static magnetic moment as a function of

the applied dc magnetic field, see the right inset in Fig. 2.7. The absorption is given

by the imaginary part of the susceptibility χ‖ of the rf magnetization component

along the rf driving field. This FMR signal has a Lorentzian line shape with a

width ∆H = (2/√

3)αω/γ when defined by the inflection points (i.e., the extrema of

dχ‖/dH), see the left inset in Fig. 2.7.

When two or more ferromagnets are in electrical contact via nonmagnetic metal

layers, interesting new effects occur. Transport of spins accompanying an applied

electric current driven through a magnetic multilayer causes a torque on the magne-

tizations [119, 11], which at sufficiently high current densities leads to spontaneous

magnetization-precession and switching phenomena [91, 134, 42, 142]. Even in the

absence of an applied charge current, spins are injected into the normal metal by a


Figure 2.7: Dependence of the FMR resonance fields H1 (circles) and H2 (triangles)for the thin Fe film F1, and the thick Fe film F2, respectively, on the angle ϕ of theexternal dc magnetic field with respect to the Fe [100] crystallographic axis. Thesketch of the in-plane measurement in the left inset shows how the rf magnetic field(double-pointed arrow) drives the magnetization (on a scale grossly exaggerated foreasy viewing). In the right inset we plot the measured absorption peaks for layers F1and F2 at ϕ = 60 Deg.

ferromagnet with moving magnetization. This causes additional magnetic damping,

provided that the spin-flip relaxation rate of normal metal is high [130, 131]. The

present section focuses on the discovery of novel dynamic effects in F1/N/F2 struc-

tures in the limit when the spin-flip scattering in N is weak. Let us first sketch the

basic physics. A precessing magnetization mi “pumps” a spin current Ipumpsi ⊥ mi

into the normal metal [130, 131, 21]. We focus on weakly excited magnetic bilayers

close to the parallel alignment, so that Ipumpsi ⊥ mj for arbitrary i, j = 1, 2. The


spin momentum perpendicular to the magnetization direction cannot penetrate a fer-

romagnetic film beyond the (transverse) spin-coherence length λsc which is smaller

than a nanometer for 3d metals [125]. A transverse spin current ejected by one fer-

romagnet can therefore be absorbed at the interface to the neighboring ferromagnet,

thereby exerting a torque τ . Each magnet thus acts as a spin sink which can dissipate

the transverse spin current ejected by the other layer.

The theoretical basis of this picture is the adiabatic spin pumping mechanism

[130, 131, 21] and magnetoelectronic circuit theory [19, 20, 127, 128]. N is assumed

thick enough to suppress any RKKY [43], pin-hole [16], and magnetostatic (Neel-type)

[109] interactions. We consider ultrathin films with a constant magnetization vector

across the film thickness [50], which are nonetheless thicker than λsc and, therefore,

completely absorb transverse spin currents. In the experiments described below, N is

thinner than the electron mean free path, so that the electron motion inside the spacer

is ballistic. Precessing mi pumps spin angular momentum at the rate [130, 131, 21]

Ipumpsi =

~4πg↑↓mi ×

dmi

dt, (2.44)

where g↑↓ is the dimensionless “mixing” conductance [19, 20] of the F/N interfaces,

which can be obtained via ab initio calculations of the scattering matrix [146] or

measured via the angular magnetoresistance of spin valves [9, 38] as well as FMR

line widths of F/N and F/N/F magnetic structures [130, 131, 82, 83, 85, 136]. Note

that g↑↓ must be renormalized for the intermetallic interfaces considered here [9]. We

assume identical Fi/N interfaces with real-valued g↑↓, as suggested by calculations

for various F/N combinations [146].

Alloy disorder at the interfaces scrambles the distribution function [32]. Under


Figure 2.8: A cartoon of the dynamic coupling phenomenon. In the left drawing,layer F1 is at a resonance and its precessing magnetic moment pumps spin currentinto the spacer, while F2 is detuned from its FMR. In the right drawing, both filmsresonate at the same external field, inducing spin currents in opposite directions. Theshort arrows in N indicate the instantaneous direction of the spin angular momentum∝ mi × dmi/dt carried away by the spin currents. Darker areas in Fi around theinterfaces represent the narrow regions in which the transverse spin momentum isabsorbed.

the isotropy assumption and disregarding spin-flip scattering in the normal metal,

an incoming spin current on one side leaves the ballistic normal-metal node by equal

outgoing spin currents to the right and left [9]. On typical FMR time scales, this

process occurs practically instantaneously. The net spin torque at one interface is

therefore just the difference of the pumped spin currents divided by two:

τ 1 = (Ipumps2 − Ipump

s1 )/2 = −τ 2 . (2.45)

When one ferromagnet is stationary, see the left drawing in Fig. 2.8, the dynamics of

the other film, Fi, is governed by the LLG equation with a damping parameter αi =

α(0)i + α′i enhanced with respect to the intrinsic value α

(0)i by α′i = γ~g↑↓/(8πMsVi),

where Vi is the volume of Fi and both ferromagnets are assumed to have the same

magnetization Ms. Since g↑↓ scales linearly with the interface area, α′i is inversely


proportional to the film thickness.

When both magnetizations are allowed to precess, see the right drawing in Fig. 2.8,

the LLG equation expanded to include the spin torque reads

dmi

dt= −γmi ×Hi

eff + α(0)i mi ×

dmi

dt+ α′i

[mi ×

dmi

dt−mj ×

dmj

dt

],(2.46)

where j = 1(2) if i = 2(1). As a simple example, consider a system in the parallel

configuration, m(0)1 = m

(0)2 , with matched resonance conditions. In addition, assume

the resonance precession is circular. If we linearize Eq. (2.46) in terms of small

deviations ui(t) = mi(t)−m(0)i of the magnetization direction mi from its equilibrium

value m(0)i , we find that the average magnetization deviation u = (u1V1+u2V2)/(V1+

V2) is damped with the intrinsic Gilbert parameter α(0), whereas the difference ∆u =

u1 − u2 relaxes with enhanced damping constant α = α(0) + α′1 + α′2.

2.3.3 Experimental Procedure and Results

Measuring the spin torques requires independent control of the precessional motion

of the two F layers, with FMR absorption line widths of isolated films dominated by

the intrinsic Gilbert damping. Both conditions were met by high-quality crystalline

Fe(001) films grown on 4x6 reconstructed GaAs(001) substrates by Molecular Beam

Epitaxy [136, 32]. Fe(001) films were deposited at room temperature from a thermal

source at a base pressure of less then 2 × 10−10 Torr and the deposition rate was

∼ 1 ML/min. For the experiments discussed below, single Fe ultrathin films with

thicknesses dF = 11,16,21,31 ML were grown directly on GaAs(001) and covered by

a 20 ML protective Au(001) cap layer. The magnetic anisotropies as measured by

FMR are described by a constant bulk term and an interface contribution inversely


proportional to dF . The Fe ultrathin films grown on GaAs(001) and covered by

gold have magnetic properties nearly identical to those in bulk Fe, modified only by

sharply defined interface anisotropies. The in-plane uniaxial anisotropy arises from

electron hybridization between the As dangling bonds and the iron interface atoms.

These Fe films were then regrown as one element of a magnetic bilayer structure

and in the following referred to as F1 layers. They were separated from a thick

Fe layer, F2, of 40 ML thickness by a 40 ML Au spacer. The magnetic bilayers

were covered by 20 ML of protective Au(001). The complete structures are there-

fore GaAs/Fe(8,11,16,21,31)/40Au/40Fe/20Au(001), where the integers represent the

number of MLs. The electron mean free path in thick films of gold is 38 nm [32] and,

consequently, the spin transport even in the 40 ML (8 nm) Au spacer is purely ballis-

tic. The interface magnetic anisotropies allowed us to separate the FMR fields of the

two Fe layers with resonance-frequency differences that can exceed 5 times the FMR

line widths, see Fig. 2.7. Hence, the FMR measurements for F1 in double layers can

be carried out with a nearly static F2.

The FMR line width of F1 increases in the presence of F2. The difference ∆H ′

in the FMR line widths between the magnetic bilayer and single-layer structures is

nearly inversely proportional to the thin-film thickness dF [136], proving that ∆H ′

originates at the F1/N interface. Secondly, ∆H ′ is linearly dependent on microwave

frequency for both the in-plane (the saturation magnetization parallel to the film

surface) and perpendicular (the saturation magnetization perpendicular to the film

surface) configurations [136], strongly implying that the additional contribution to the

FMR line width can be described strictly as an interface Gilbert damping [50]. At the


FMR, the film precessions are driven by an applied rf field. When the resonance fields

are different, one layer (say F1 ) is at resonance with maximum precessional amplitude

while the other layer (F2 ) is off resonance with small precessional amplitude, see

Fig. 2.8. The spin-pump current for F1 reaches its maximum while F2 does not

emit a significant spin current at all. F2 acts as a spin sink causing the nonlocal

damping for F1. The N/F2 interface provides a “spin-momentum brake” for the F1

magnetization. The corresponding additional Gilbert parameter α′ for a 16 ML Fe

is significant, being similar in magnitude to the intrinsic Gilbert damping in isolated

Fe films, α(0) = 0.0044.

These assertions can be tested by employing the in-plane uniaxial anisotropy in

F1 to intentionally tune the resonance fields for F1 and F2 into a crossover which

is shown in the shaded area of Fig. 2.7. When the resonance fields are identical,

H1 = H2, the rf magnetization components of F1 and F2 are parallel to each other,

see the right drawing in Fig. 2.8. The total spin currents across the F1/N and

N/F2 interfaces therefore vanish resulting in zero excess damping for F1 and F2, see

Eq. (2.46), which is experimentally verified, as shown in Fig. 2.9. For a theoretical

analysis, we solved Eq. (2.46) and determined the total FMR signal as a function of

the difference between the resonance fields H2 −H1. The theoretical predictions are

compared with measurements in Fig. 2.9. The remarkable good agreement between

the experimental results and theoretical predictions provides strong evidence that the

dynamic exchange coupling not only contributes to the damping but leads to a new

collective behavior of magnetic hybrid structures.


Figure 2.9: Comparison of theory (solid lines) with measurements (symbols) close toand at the crossover of the FMR fields, marked by the shaded area in Fig. 2.7. Theleft and right frames show FMR signals for the field difference, H2 − H1, of -78 Oeand +161 Oe, respectively. The theoretical results are parameterized by the full setof magnetic parameters which were measured independently [136]. The magnitude ofthe spin-pump current was determined by the line width at large separation of theFMR peaks. The middle frame displays the effective FMR line width of magneticlayers for the signals fitted by two Lorentzians as a function of the external field.At H1 = H2, the FMR line widths reached their minimum values at the level ofintrinsic Gilbert damping of isolated films. The calculations in the middle frame didnot take small variations of the intrinsic damping with angle ϕ into account, whichresulted in deviations between theory and experiment for larger |H1−H2|. Note that∆H1 first increases before attaining its minimum, which is due to excitation of theantisymmetric collective mode.

2.3.4 Summary

In conclusion, we found decisive experimental and theoretical evidence for a new

type of exchange interaction between ferromagnetic films coupled via normal metals.

In contrast to the well-known oscillatory exchange interaction in the ground state,

this coupling is dynamic in nature and long ranged. Precessing magnetizations feel

each other through the spacer by exchanging nonequilibrium spin currents. When

the resonance frequencies of the ferromagnetic banks differ, their motion remains


asynchronous and net spin currents persist. However, when the ferromagnets have

identical resonance frequencies, the coupling quickly synchronizes their motion and

equalizes the spin currents. Since these currents flow in opposite directions, the net

flow across both F1/N and N/F2 interfaces vanishes in this case. The lifetime of the

arising collective motion is limited only by the intrinsic local damping. These effects

can be well demonstrated in FMR measurements.


2.4 Precessional Stiffness of Spin Valves

The dynamics of the magnetic order parameters of F/N/F spin valves and isolated

ferromagnets may be very different. Here we investigate the role of the nonequilibrium

spin-current exchange between the ferromagnets in the magnetization precession and

switching. We find a (low-temperature) critical current bias for a uniform current-

induced magnetization excitation in spin valves, which unifies and generalizes previous

ideas of Slonczewski and Berger. In the absence of an applied bias, the effect of the

spin transfer can be expressed as magnetic–configuration-dependent Gilbert damping.

2.4.1 Background

Modern magnetic storage media and prototype magnetic random-access memories

consist of F/N composites with information stored by switchable magnetic configu-

rations. Device performance is measured in terms of bit density as well as speed of

reading and writing information. Magnetization reversal is usually achieved by mag-

netic fields generated by electric currents. In small structures much energy is wasted

in the form of stray magnetic fields, which motivates consideration of other switching

mechanisms. An interesting effect inverse to the GMR is the spin torque exerted on

the magnetizations by an applied electric current which, at a critical current bias,

leads to magnetization switching [11, 119, 120], as has been experimentally confirmed

[91, 65, 142, 90].

Perpendicular spin valves, i.e., Fs/N/Fh trilayer pillar structures with layer thick-

nesses down to a few monolayers and lateral dimensions in the submicron region,

are ideal to study precession and switching phenomena in hybrid systems. When


reservoirs are attached on the outer sides, these spin valves can be biased by an

electric current perpendicular to the interface planes. Fs is a “soft” ferromagnetic

film with a magnetization that can be reversed easily, whereas Fh is a “hard” mag-

netic layer whose magnetization is pinned by an exchange bias, but also by surface

magnetic anisotropy or a resistance anisotropy [70]. For small enough systems, the

magnetic layers are monodomain ferromagnets characterized by two magnetization

vectors. The relevant variable is then the time-dependent magnetization of the soft

layer under applied magnetic fields and/or electric currents.

Slonczewski [119, 120] and Berger [11, 12] were the first to predict novel time-

dependent effects in spin valves. Both authors have realized that a current flowing

through a spin valve causes a spin transfer through the nonmagnetic spacer, inducing

spin torques on the ferromagnets. In addition, Berger predicted that the two ferro-

magnets interact via spin transfer even in the absence of an applied electric current,

resulting in a significant contribution to the Gilbert damping of the magnetization dy-

namics. He further demonstrated that a sufficiently large electric current can induce

coherent spin-wave emission in the ferromagnet, an idea which was later supported

experimentally [134]. The condition for spin-wave emission [11, 12] is similar to the

criterion for magnetization switching due to Slonczewski [119, 120], who treated the

Gilbert damping parameter as a phenomenological constant. In Refs. [11, 12], how-

ever, a nontrivial dependence of the Gilbert damping on the relative angle between

the magnetizations was predicted. Some of Berger’s and Slonczewski’s results as well

as the underlying theoretical models and methods were thus not consistent with each

other. In the following we offer an alternative theory, which both unifies and extends


the seminal work of these pioneers.

2.4.2 Dynamic Exchange

Based on the concept of adiabatic spin pumping, we demonstrated in Refs. [130,

131] that the magnetization motion of a ferromagnetic layer is damped by emitting

(pumping) spins into adjacent conductors. The presence of a second ferromagnet

can considerably affect the relaxation of the pumped spins, and therefore the mag-

netization dynamics, as discussed below. We combine adiabatic spin pumping with

magnetoelectronic circuit theory [19, 20, 9] to provide a self-contained framework for

spin transfer in spin valves. The main results presented here are the critical current

bias for a low-temperature magnetization instability and the configuration-dependent

Gilbert damping parameter. In terms of conductance parameters accessible to first-

principles calculations [147, 146] and combined with micromagnetic simulations, the

full range of the precession and switching dynamics can then be studied in principle.

We consider the system sketched in Fig. 2.10. The Fs/N/Fh trilayer is sandwiched

between two normal-metal contacts sustaining a current bias J . The soft layer Fs mag-

netization m will start moving from its equilibrium direction at a critical value Jc

(depending on the applied magnetic field). Thermal activation facilitates current-

induced magnetization switching [90], but we focus here on the low-temperature

regime. The generalized LLG equation for the magnetization direction m(t) of Fs

in the presence of a spin current Is flowing out of Fs reads

dm


dt+

γ

MsSdm× Is ×m , (2.47)

where we stripped off the right-hand side of the longitudinal component of Is [compare


m(t)

µ sN

I spump

I sbias

Ibacks1 Iback

s2

Fs Fh

M

θ

N

J

Figure 2.10: Schematics of a current-biased spin valve. The symbols are explained inthe text

it to Eq. (2.16), where it is implied that Is ⊥ m]. Fs is characterized by α0, its intrinsic

(dimensionless) Gilbert damping constant, Ms, its saturation magnetization, d, its

thickness, and S, its cross section. The spin current

Is = Ibiass + Iexch

s (2.48)

consists of Ibiass , driven by an applied current bias, and the dynamic-exchange cur-

rent Iexchs induced by the spin pumping. The latter has recently been shown to be

responsible for a dynamic coupling between the ferromagnets [51] and, according to

Refs. [11, 12], also determines the threshold for spin-wave emission. Alternatively,

one can interpret it as a “dynamic stiffness,” which stabilizes the relative magnetiza-

tion configuration of the spin valve against the torques exerted by Ibiass or an applied

magnetic field. In high-density metallic systems, the applied voltages and spin ac-

cumulations are safely smaller than the Fermi energies, which means that we are in

the linear-response regime and both spin currents may be calculated independently


of each other. Spin pumping in the outward direction, i.e., into the external connec-

tors, would simply increase the intrinsic damping coefficient α0 by a constant value

[130, 131], so we disregard it here for simplicity.

When the conductance parameters of the spin valve are symmetric, the bias-

induced spin transfer Ibiass is coplanar with the magnetization directions,

Ibiass = Ibias

s

m + M

2 cos θ/2, (2.49)

θ being the angle between m and M. This becomes clear by expanding the spin

current as Ibiass (m,M) = f11(cos θ)m + f22(cos θ)M + f12(cos θ)m × M and noting

that Ibiass (m,M) = Ibias

s (M,m), which implies f11 ≡ f22 and f12 ≡ 0. The electric

current corresponding to a given spin-current bias depends on θ and can be calculated

readily by circuit theory [19, 20, 9].

The spin current Ipumps pumped into the spacer by a time-dependent m(t) is present

even when J vanishes [130, 131]. When Fs is thicker than its transverse spin-coherence

length, Ipumps is given by Eq. (1.5). For simplicity, the conductance parameters of the

two F/N interfaces are taken to be identical in the following. The mixing conductance

is to a good approximation real-valued, i.e., g↑↓i g↑↓r , at least for transition-metal

ferromagnets [146]. Ipumps creates a spin accumulation µsN in N, which induces a

backflow spin current Ibacksi into both ferromagnets i = 1, 2. According to the cir-

cuit theory [19, 20, 9], making use of the zero-electric–current condition through the

interfaces,

Ibacksi =

1

4π

[2g↑↑g↓↓

g↑↑ + g↓↓mi (∆µsi ·mi) + g↑↓mi ×∆µsi ×mi

]. (2.50)

Here, gss is the (dimensionless) spin-s interface conductance, m1 = m, m2 = M,

and ∆µsi = µsN −µsF i is the spin-accumulation difference across the Fi/N interface.


Note that for intermetallic interfaces Sharvin contributions must be subtracted if

conductances are computed microscopically by scattering theory [9]. The time scale

of the magnetization dynamics, ∼ 10−11 s, is much larger than typical electron dwell

times in the metallic spacer. Assuming weak spin-flip scattering in N, the conservation

of spin then implies that

Ipumps = Iback

s1 + Ibacks2 . (2.51)

The dynamic stiffness is therefore given by the spin current flow into the hard layer:

Iexchs = Iback

s2 . (2.52)

The transverse component of the spin accumulation is absorbed in a ferromagnetic

layer on the scale of a Fermi wave length [19, 20, 125]; the longitudinal component, on

the other hand, can penetrate the ferromagnet on the scale of the spin-diffusion length

λsd. In order to find Ibacksi , we solve the diffusion equation for spin transport in the

ferromagnets, assuming that the spin current vanishes on the outer boundaries of Fs

and Fh. It is shown that the longitudinal spin-accumulation flow into a ferromagnetic

slab of thickness d is then governed by an effective conductance g∗ defined by

1

g∗=g↑↑ + g↓↓

2g↑↑g↓↓+

1

gsd tanh(d/λsd)(2.53)

in terms of

gsd =h

e2S

λsd

2σ↑σ↓

σ↑ + σ↓, (2.54)

where σs is the spin-s conductivity of the ferromagnetic bulk, so that the backflow

current, Eq. (2.50), can be written as

Ibacksi =

1

4π

[g∗mi (µsN ·mi) + g↑↓mi × µsN ×mi

]. (2.55)


g∗ → 0 when d λsd, i.e., when the spin-flip relaxation vanishes, or when the

ferromagnet is halfmetallic, so that it completely blocks the longitudinal spin flow

due to charge conservation. The parameter

ν =g↑↓ − g∗

g↑↓ + g∗(2.56)

characterizes the asymmetry of the absorption of transverse vs longitudinal spin cur-

rents. Let us estimate typical values of ν for sputtered Co/Cu and Py/Cu hybrids at

low temperatures, taking d = 5 nm. The principal difference between the two com-

binations is the spin-diffusion length in the ferromagnets: Co has a relatively long

λsd ≈ 60 nm, while λsd ≈ 5 nm is very short in Py [7, 35, 100]. Using known values

for spin-dependent conductivities [7, 35, 100], we thus find gsdS−1 ≈ 2.7 nm−2 for Co

and 16 nm−2 for Py. 2g↑↑g↓↓/(g↑↑+g↓↓)S−1 ≈ 20 nm−2 for Co/Cu interfaces [147, 146]

and we expect the value for Py/Cu to be similar. Finally, taking g↑↓S−1 ≈ 28 nm−2

for the Co/Cu interface [147, 146] and 15 nm−2 for Py/Cu [9], we find ν ≈ 0.98 for

Co/Cu and ν ≈ 0.33 for Py/Cu.

2.4.3 Angle-Dependent Stiffness

With ν the same for both layers, we solve for the spin accumulation in the normal

metal, µsN , in the absence of applied current, J = 0, by using spin conservation,

Eq. (2.51), and Eq. (2.55) for the backflow in terms of µsN . The dynamic stiffness is

then given by Eq. (2.52), and we arrive at

Iexchs =

1

2

[Ipumps − ν (Ipump

s ·M)M− νm cos θ

1− ν2 cos2 θ

]. (2.57)


−1 −0.5 0 0.5 1cosθ

0

0.2

0.4

0.6

0.8

1

α′(θ

)/α′

(0)

ν=0.99, s=99

ν=0.5, s=1

ν=0.9, s=9

Figure 2.11: Solid lines are our prediction for the precession-cone angle dependenceof the Gilbert damping parameter [Eq. (2.59)] and the dotted lines are Berger’s[Eq. (2.60)]. The lower (solid) line is representative for Co, while the upper forPy, assuming thickness of 5 nm. We expect Fe and Ni to be characterized by the twolower (solid) lines.

Semiclassically, this equation can be understood as a multiple scattering of spin cur-

rent between the interfaces at which the longitudinal part is reflected with probability

P ∝ 1 + ν and the transverse component with P ∝ 1− ν. We have taken the spacer

to be ballistic, so that µsN is uniform. Otherwise, the exchange current will be sup-

pressed by diffuse scattering in the interlayer N. It is straightforward to extend our

theory to take this into account by, e.g., solving the spin-diffusion equation in the

spacer and using the same boundary conditions, Eqs. (1.5) and (2.55), as above.

The magnetization dynamics (in the absence of an applied bias) is determined

by substituting Iexchs into the LLG equation, which thus has a damping term that

cannot be modeled by a constant effective Gilbert parameter. We now analyze the


configuration dependence of the damping in more detail, which is experimentally

accessible by the FMR line-width broadening at high rf intensities [48] (and therefore

finite “precession cones”). For m precessing around M,

m× Iexchs ×m =

g↑↓

8π

(1− ν

sin2 θ

1− ν2 cos2 θ

)m× dm

dt. (2.58)

The angular dependence of the additional Gilbert damping parameter due to the

exchange spin current then reads

α′(θ)

α′(0)= 1− ν

sin2 θ

1− ν2 cos2 θ, (2.59)

where α′(0) = γ~g↑↓r /(8πMsSd) is the damping enhancement in a collinear configu-

ration. Interestingly, this result bares similarity with Berger’s [12]

α′(θ)

α′(0)=

1

1 + s(1− cos θ), (2.60)

where s ∝ τsf , a characteristic spin-flip time, and his form of α′(0) is similar as well

[11]. Expressions (2.59) and (2.60) are compared in Fig. 2.11. The two functions are

barely distinguishable at small angles, but are qualitatively different in the antiparallel

alignment. We can rewrite Eq. (2.59) as

α′(θ)

α′(0)≈ 1

1 + [ν/(1− ν)] (1− cos θ), (2.61)

for small enough θ. The last equality reproduces Eq. (2.60) after identifying s =

ν/(1 − ν). As mentioned above, ν is close to 0.98 for cobalt and s should be of the

order of 100 (s = 333 is found in Ref. [12] for Co/Cu with Co 1.5 nm thick, which

remarkably would be quite similar to our estimate for this thickness), so that the

lower solid line in Fig. 2.11 represents the damping for Co.


The “precessional stiffness” is thus significantly reduced for angles which only

slightly deviate from the collinear configurations (we expect this conclusion to be also

true for Fe and Ni). Modeling of the magnetization dynamics with a constant damping

parameter is thus not allowed for sufficiently thin magnetic layers. For permalloy, on

the other hand, the precessional stiffness is expected to remain significant for all

angles, see the upper solid line in Fig. 2.11. This implies that the magnetization

reversal has higher energy-dissipation power, but can occur faster than in cobalt in

field-induced switching. If m moves away from M, i.e., only the relative angle θ

changes, then Iexchs ⊥ M and Eq. (2.57) reduces to Iexch

s = Ipumps /2. The “tilting

stiffness” has thus an angle-independent enhancement with respect to the intrinsic

Gilbert damping, which is given by the same expression as α′(0), i.e., the damping in

a collinear configuration.

2.4.4 Zero-Temperature Switching Current

Eqs. (2.47), (2.48), (2.49), and (2.57) completely determine the dynamics of m(t).

The exchange (2.57) induced by the spin pumping (1.5) causes relaxation towards an

equilibrium configuration, while the bias current (2.49) can either relax or excite a

perturbation from an equilibrium, depending on the sign of J . In the process of, e.g.,

switching, the trajectory of m(t) can become very complicated. While in this report

we outline the general formalism, a detailed numerical study of the magnetization

dynamics will be carried out elsewhere. In the remainder of the section we discuss the

critical current bias at which a collinear equilibrium configuration becomes unstable.

Near a collinear configuration, Eq. (2.57) simplifies to Iexchs = Ipump

s /2. Let m


precesses around M with the FMR frequency ω: m× dm/dt = ωm×M×m. The

total (projected) spin current in the Gilbert form then reads:

m× Is ×m =

[~g↑↓

8π+Ibiass

2ω

]m× dm

dt. (2.62)

An instability is reached when the effective Gilbert damping coefficient becomes neg-

ative. The critical bias is thus given by

Ibiass,c =

[g↑↓

4π+

2α0MsSd

~γ

]~ω . (2.63)

Neglecting the first term in the brackets of the above expression, we obtain result

analogous to Slonczewski’s [119, 120], while neglecting the second term, we get a

condition similar to Berger’s spin-wave emission criterion [11]. The spin-pumping

contribution (first term) is comparable with the intrinsic damping (second term)

for films with thickness d of several nanometers [130, 131, 136], with the former

dominating for very thin films.

2.4.5 Summary

We have developed a general theoretical framework for the low-temperature mag-

netization dynamics in small spin valves, unifying and extending pioneering work by

Slonczewski [119, 120] and Berger [11, 12]. The nonequilibrium spin torque induced

by the bias current and the enhanced Gilbert constant due to the spin pumping

must be treated on equal footing. When the memory magnetic element is sufficiently

thin (d < 10 nm), the nontrivial dependence of the damping on both the static

and dynamic configurations of the system can importantly modify the magnetiza-

tion dynamics. We derived the dependence of the Gilbert damping of Fs on the


precession-cone angle, which can also be measured by FMR [48]. Micromagnetic sim-

ulation codes should take these effects into account as the device and magnetic bit

dimensions decrease down to the nanometer scale.

Chapter 3

Tunneling between Parallel

Quantum Wires

We present theoretical calculations and experimental measurements which reveal

the Luttinger-liquid (LL) nature of elementary excitations in a system consisting of

two quantum wires connected by a tunnel junction at the edge of a GaAs/AlGaAs

bilayer heterostructure. The boundaries of the wires are important and lead to a

characteristic interference pattern in measurements on short junctions. We show that

the experimentally observed modulation of the conductance oscillation amplitude as

a function of the voltage bias can be accounted for by spin-charge separation of the

elementary excitations in the interacting wires. Furthermore, boundaries affect the LL

exponents of the voltage and temperature dependence of the tunneling conductance at

low energies. We show that the measured temperature dependence of the conductance

zero-bias dip as well as the voltage modulation of the conductance oscillation pattern

can be used to extract the electron interaction parameters in the wires.

106

Chapter 3: Tunneling between Parallel Quantum Wires 107

3.1 Background

Quasi-one-dimensional (1D) structures with gapless electronic excitations, such

as carbon nanotubes, quantum Hall edge states, and confined states at the edge of a

quantum well heterostructure (i.e., quantum wires), possess unique properties which

cannot be described by Landau’s Fermi-liquid theory. Even small electron-electron

interactions in a 1D confinement make inadequate the picture based on the existence

of long-lived fermionic quasiparticles which can be mapped onto single-particle states

in a free-electron gas. A powerful framework for understanding universal properties

of 1D electron systems was put forward by the formulation of Luttinger-liquid (LL)

theory [45]. (For a review see Ref. [138].) The spectral density, A(k, ω), of the

one-electron Green’s function in a Luttinger liquid is fundamentally different from

that of a Fermi liquid: While the latter has one quasiparticle peak, the former has

two singular peaks corresponding to the charge- and spin-density excitation modes

[79, 137].

Tunnel-coupled quantum wires of high quality created at a cleaved edge of GaAs/AlGaAs

double-quantum-well heterostructures appear to be an exceptional tool for probing

spectral characteristics of a 1D system [4, 26, 152]. It is achieved [4] by measuring

the differential conductance G(V,B) as a function of the voltage bias between the

wires, V , and magnetic field oriented perpendicular to the plane of the cleaved edge,

B, allowing for simultaneous control of the energy and momentum of the tunneling

electrons. In a recent article [132] we demonstrated that the picture of noninteract-

ing electrons can be used with great success to explain some of the most pronounced

features of the conductance interference pattern arising from the finite size of the


tunneling region. Taking electron-electron interactions into account was shown to

explain experimentally observed long-period oscillation modulations in the V direc-

tion, which can be understood as a moire pattern arising from spin-charge separation

of electronic excitations. In this chapter we use LL formalism to further investigate

an interplay between electron correlations and the finite length of the tunnel junc-

tion, which allows us to understand peculiarities of the oscillations and the zero-bias

anomaly in the measured tunneling conductance G(V,B).

3.2 Experimental Method

In this section we describe the means by which we measure the tunneling conduc-

tance through a single isolated junction between two parallel wires.

3.2.1 Fabrication of the Samples

The two parallel 1D wires are fabricated by cleaved edge overgrowth (CEO), see

Fig. 3.1 and Ref. [148]. Initially, a GaAs/AlGaAs heterostructure with two closely

situated parallel quantum wells is grown. The upper quantum well is 20 nm wide,

the lower one is 30 nm wide and they are separated by a 6 nm AlGaAs barrier about

300 meV high. We use a modulation doping sequence that renders only the upper

quantum well occupied by a two-dimensional electron gas (2DEG) with a density

n ≈ 2 · 1011 cm−2 and mobility µ ≈ 3 · 106 cm−2s−1. After cleaving the sample

in the molecular beam epitaxy growth chamber and growing a second modulation

doping sequence, two parallel quantum wires are formed in the quantum wells along

the whole edge of the sample. Both wires are tightly confined on three sides by


B

Figure 3.1: Illustration of the sample and the contacting scheme. The sample isfabricated using the CEO method. The parallel 1D wires (thick solid and dashedblack lines) span along the whole cleaved edge (right facet in the schematic). Theupper wire (UW) overlaps the 2DEG, while the lower wire (LW) is separated fromthem by a thin tunnel barrier. Contacts to the wires are made through the 2DEG.Several tungsten top gates can be biased to deplete the electrons under them: Weshow only G1, biased to deplete the 2DEG and both wires, and G2, biased to depleteonly the 2DEG and the upper wire. The magnetic field B is perpendicular to the planedefined by the wires. The depicted configuration allows the study of the conductanceof a tunnel junction between a section of length L of the upper wire and a semi-infinitelower wire (thick black lines).

atomically smooth planes and on the fourth side by the triangular potential formed

at the cleaved edge.

Spanning across the sample are several tungsten top gates of width 2 µm that lie

2 µm from each other (two of these are depicted in Fig. 3.1). The differential conduc-

tance G of the wires is measured through indium contacts to the 2DEG straddling

tungsten top gates. While monitoring G with standard lock-in techniques (we use an

excitation of 10 µV at 14 Hz) at T = 0.25 K, we decrease the density of the electrons

under the gate by decreasing the voltage on it (Vg). At Vg = V2D, the 2DEG depletes

and G drops sharply, because the electrons have to scatter into the wires in order to


pass under the gate. For V2D > Vg > VU the conductance drops stepwise each time a

mode in the upper wire is depleted [149]. In this voltage range, the contribution of

the lower wire to G is negligible because it is separated from the upper quantum well

by a wide tunnel barrier. When Vg = VU , the upper wire depletes and only the lower

wire can carry electrons under the gate. This last conduction channel finally depletes

at VL and G is suppressed to zero.

3.2.2 Measurement on an Isolated Tunnel Junction

The measurements are performed in the configuration depicted in Fig. 3.1. The

source is the 2DEG between two gates, G1 and G2 in Fig. 3.1, the voltages on which

are V1 < VL and VL < V2 < VU , respectively. The upper wire between these gates

is at electrochemical equilibrium with the source 2DEG. This side of the circuit is

separated by the tunnel junction we wish to study from the drain. The drain is the

2DEG to the right of G2 (the semi-infinite 2DEG in Fig. 3.1) and it is in equilibrium

with the right, semi-infinite, upper wire and with the whole semi-infinite lower wire in

Fig. 3.1. Thus, any voltage difference (V ) induced between the source and the drain

drops on the narrow tunnel junction between the gates. This configuration gives

us control over both the energy and the momentum of the tunneling electrons, as

explained below. An additional gate lying between G1 and G2 (not shown in Fig. 3.1)

allows us to deplete the 2DEG in the center of the source, thus reducing the screening

of the interactions in the wires by the 2DEG.

The energy of the electrons tunneling between the wires is given by eV , −e being

the electron charge. The tunneling process occurs along the whole length L of the


tunnel junction. Therefore, momentum is conserved to within an uncertainty of order

2π/L kF, where kF is a typical Fermi wave vector in the wires. We can shift the

momentum of the tunneling electrons with a magnetic field (B) perpendicular to the

plane defined by the wires. The value of the shift is ~qB = eBd, where d is the

center-to-center distance between the wires.

3.3 Description of the Experimental Results

In the experiment we measure the nonlinear differential tunneling conductance

G(V,B) through a junction between the parallel wires. The sample we report here

contains four top gates allowing us to vary the length of the junction L by choos-

ing different combinations of gates. We have studied in detail junctions with L =

2, 4, 6, 10 µm as well as symmetric junctions (L = ∞). The results presented here

are from junctions with L = 2, 6, 10 µm. Many of the effects that we measure rely

on the smallness of 1/L, while others (which we address here in detail) are present

only when L is finite.

3.3.1 Dispersions of Elementary Excitations in the Wires

By mapping out G(V,B) we determine the dispersion curves of the wires [4].

These are given by the curves that are traced by the main peaks as seen in Fig. 3.2.

We can understand their gross features employing a noninteracting electron picture

[4]: The peaks result from tunneling between a Fermi point in one wire and a mode

in the other wire. Since each occupied mode has two Fermi points, two copies of

the dispersion show up in the G(V,B) scan. All in all, for each pair of occupied


Figure 3.2: Plot of G(V,B) for a 10 µm junction.

modes in two wires we expect to observe four dispersions, because there are four

Fermi points involved: ±kiFu and ±kj

Fl. (Indices i and j label various modes in the

wires, u and l–the upper and the lower wires.) In reality, we observe only some of the

transitions, perhaps due to selection rules related to the shape of the wave functions in

the direction perpendicular to the cleaved edge. For example, by carefully studying

Fig. 3.2 one can distinguish dispersions of three modes in the upper wire and five

in the lower one, but only the following transitions seem to have a sizable signal:

|u1〉 ↔ |l1〉, |u3〉 ↔ |l2〉, and |u2〉 ↔ |l3,4,5〉, where the order in the list is of decreasing


Bi,j2 (defined below); |u3〉 is the 2DEG occupying the upper quantum well. (See

Ref. [4] for further explanation.)

The dispersions allow us to extract the densities of electrons in each mode, niu(l) =

(2/π)kiFu(l), as follows. Tunneling amongst each pair of occupied modes is enhanced

near V = 0 at two values of B > 0, where the two curves in G(V,B) cross. In the

first (in the following referred to as the “lower crossing point”), which occurs at Bi,j1 ,

the direction in which the electrons propagate is conserved in the tunneling process.

In the second (referred to as the “upper crossing point”), the Lorentz force exerted

by Bi,j2 exactly compensates for the momentum mismatch between oppositely moving

electrons and the direction of propagation of the tunneling electrons reverses. In wires

with vanishing cross section, these crossing points occur at

∣∣∣Bi,j1(2)

∣∣∣ =~ed

∣∣kiFu ∓ kj

Fl

∣∣ . (3.1)

In principle Eq. (3.1) can be used to extract the densities of the modes, regardless of

electron-electron interactions in the wires [26] or mesoscopic charging [18] that can

merely smear them at a finite voltage bias. In realistic wires that have a finite cross

section, finding the densities is hampered by the weak magnetic field dependence that

they acquire. This difficulty is overcome by a simple fitting procedure that we have

developed: We assume that all the modes in a wire have the same field dependence,

a reasonable assumption for our tight-confining potential in the growth direction of

the quantum wells. We then guess the B = 0 occupations of the modes in each wire,

niu(0) and nj

l (0), and calculate their field dependences. If the resulting dispersions do

not cross at Bi,j1(2), we adjust ni

u(0) and njl (0) and repeat the procedure. This is done

iteratively for all the crossing points that we see, because changing the occupation


of one mode affects the field dependence of all the other occupations in a wire. The

dispersion that we use is that of noninteracting electrons in a finite well, in the

presence of a magnetic field. Such a dispersion depends only on the width and depth

of the well and on the band mass of electrons in GaAs.

In every case we have studied, we see clear deviations of the measured dispersions

from the calculated noninteracting ones at a finite bias. In particular, we find that

the velocities of some excitations are enhanced relative to the Fermi velocities vFu(l).

The former are given by

vp =1

d

∂V

∂B

∣∣∣∣Bi,j

1(2)

(3.2)

(along the observed main peaks), while the latter can be obtained by the calculated

slope of the (noninteracting) dispersions at the Fermi points. This velocity enhance-

ment is thought to correspond to the charge-density modes and can be accounted for

by electron-electron interactions in the wires [26, 4].

The ability to determine the dispersion relations relies on the high quality of the

junctions to sustain momentum-conserving tunneling. Momentum relaxation ensues

as soon as invariance to translations is broken. The most obvious mechanism by

which this happens is the finiteness of L. We find that we indeed observe its effects.

The second mechanism is the disorder inherent to all semiconductor devices, some

effects of which seem to also be observed.

3.3.2 Oscillations

The most spectacular manifestation of the breaking of translational invariance is

the appearance of a regular pattern of oscillations away from the dispersion curves.


Figure 3.3: Nonlinear conductance oscillations at low field from a 2 µm junction. (b)shows the oscillations as a function of both B and V . (A smoothed background hasbeen subtracted to emphasize the oscillations.) The brightest (and darkest) lines, cor-responding to tunneling between the lowest modes, break the V -B plain into regionsI, II, and III. Additional bright lines in II arise from other 1D channels in the wiresand are disregarded in our theoretical analysis. Also present is a slow modulation ofthe strength of the oscillations along the abscissa. (a) Absolute value of the peak ofthe Fourier transform of S1−1/βG

(V, S1+1/β

)at a fixed V in region II as a function

of V . (See Sec. 3.4.3 for definition of S and other details.) Its slow modulation as afunction of V is easily discerned.

Figs. 3.3b and 3.4b are typical examples of the patterns that we measure at low

magnetic field. In this range of field, the lines that correspond to the dispersion

curves appear as the pronounced peaks that extend diagonally across the figures. In

addition to these we observe numerous secondary peaks running parallel to the main

dispersion curves. These side lobes always appear to the right of the wire dispersions,


Figure 3.4: Same as Fig. 3.3 but for a 6 µm junction. Note that the oscillations areapproximately three times faster than in Fig. 3.3, as expected from Eq. (3.3). For thisjunction, there are several additional side lobes present on the left of the principalpeaks, unlike in the case of the shorter junction in Fig. 3.3.

in the region that corresponds to momentum conserving tunneling for an upper wire

with a reduced density. As a result, we see a checkerboard pattern of oscillations

in region I, a hatched pattern in region II, and no regular pattern in region III (see

Figs. 3.3b and 3.4b for the definitions).

The interference pattern also appears near the upper crossing point at high mag-

netic field. A typical example is shown in Fig. 3.5.

The frequency of the oscillations depends on L. When L is increased from 2 µm,

Fig. 3.3, to 6 µm, Fig. 3.4, the frequency in bias (∆V ) and in field (∆B) increases

by about a factor of three. The period is approximately related to the length of the


Figure 3.5: G(V,B) near the upper crossing point for a 6 µm junction. In thismeasurement, a central 2 µm gate midway between G1 and G2 is biased to deplete allupper wire modes except the lowest one. One can see a pattern of oscillations aroundthe dispersion peaks.

junction through the formula

∆V L/vF = ∆BLd = φ0 , (3.3)

where φ0 = 2π~/e is the quantum of flux.

A close examination of the low-field oscillations reveals an interesting behavior of

their envelope. Notable is the suppression of G(V,B) near V = 0 which is indepen-

dent of field. Also visible are faint vertical gray stripes, where the amplitude of the

oscillations in the B direction is reduced. The modulation of the oscillation ampli-

tude, as a function of V , is shown at the top of figures 3.3 and 3.4 (in panels a). The


oscillatory part of G thus depends on V on two major scales: The faster scale (0.5 mV

for L = 2 µm) corresponds to the oscillations described by Eq. (3.3). The slower scale

(2 mV for L = 2 µm) governs the distance between the stripes of suppressed G(V,B)

parallel to the field axis, including the zero-bias suppression. Like the fast scale, the

slow scale is roughly inversely proportional to the lithographic length of the tunneling

region.

3.3.3 A Dip in the Tunneling Conductance

Prominent in all scans that have high enough resolution in V is a strong sup-

pression of the conductance near V = 0 at all magnetic fields. The width of this

conductance dip is of order of 0.1 mV, see Figs. 3.2 and 3.5. The size of the dip is

very sensitive to temperature, as depicted in Fig. 3.6, and it exists for T . 1.0 K.

3.4 Theory and Discussion

The 1D modes in the upper quantum well are coupled to the 2DEG via an elastic

1D-2D scattering which ensures a good electronic transfer between the extended and

confined states of the well [30]. In addition to tunneling between the confined states

in the wires, if the extended states have an appreciable weight at the edge, there

will be a direct transition from the 2DEG to the lower wire. With this in mind,

we separate the total current into two contributions, one due to tunneling between

1D bands and the other due to direct tunneling from the 2DEG. As explained in

Sec. 3.3.1, each of the wires carries several 1D modes. In our analysis and comparison

with the experiment, we will only consider the transition between the lowest 1D bands


Figure 3.6: Zero-voltage dip of the tunneling conductance G as a function of tem-perature on a log-log scale. The circles show measurements on a 6 µm junction atB = 2.5 T, the lines are a fit using G ∝ Tα for V = 0. The dashed line is theresult for α = αbulk(gl) while the solid line is the result for α = αend(gl), where thebest fit is obtained with gl = 0.59, see Sec. 3.4.9 for discussion. Insets: G(V ) forT = 0.24 K and T = 0.54 K (the temperature dependence was generated from V = 0point of such scans). The curves were calculated with Eq. (3.51) and using the abovevalue of gl extracted from the fit of the temperature dependence of the dip. [We ob-tained Fα(x) by convoluting the derivative of the Fermi distribution in the 2D leads,[1/(4kBT )]sech2[eV/(2kBT )], with the finite-temperature tunneling density of statesin the lower wire, see Eq. (5) in Ref. [17].] The dashed lines correspond to the αbulk

value of the exponent while the solid lines to αend.


-0.5 0 0.5x

Ul(x)

Uu(x)

ψ(x)

-eV

B

→

→

x0

Figure 3.7: Schematic picture of the theoretical model. The upper wire is formed bya potential well Uu(x) created by gates G1 and G2 (shown in Fig. 3.1) and the lowerwire is semi-infinite with the left boundary Ul(x) at gate G1. ψ(x) is an electron wavefunction in the upper wire. The energy and momentum of the tunneling electrons isgoverned by the voltage bias V and magnetic field B.

of the wires (i.e., the bands with the largest Fermi momentum), |u1〉 ↔ |l1〉, and the

direct tunneling from the 2DEG, |u3〉 ↔ |l2〉, which both have a strong signal, as

seen in Fig. 3.2. In each wire, the 1D modes interact with each other, but since the

bands have very different Fermi velocities, we treat them independently. This is a

reasonable approximation, as explained in Appendix B.1.

The geometry for our theoretical description is shown in Fig. 3.7. The poten-

tials Uu(x) and Ul(x) are felt by electrons in the upper and lower quantum wires,

respectively. The electrons in the upper wire are confined to a region of finite length

by potential gates at both its ends (see the source region in Fig. 3.1). One of these

gates (G1) causes the electrons in the lower wire to be reflected at one end, but the

other (G2) allows them to pass freely under it. The effective tunneling region is deter-


mined by the length of the upper wire, which is approximately the region |x| < L/2

in Fig. 3.7. The magnetic field, B, gives a momentum boost ~qB = eBd along the

x -axis for the electrons tunneling from the upper to the lower wire.

First, we develop a general formalism in Sec. 3.4.1. We then apply it to study the

conductance interference pattern in Sec. 3.4.2 and the zero-bias anomaly regime in

Sec. 3.4.7.

3.4.1 General Formalism

Let us first consider transport between two 1D bands in the wires. We use the

following model Hamiltonian to study the intermode tunneling in the system:

H =∑ν=u,l

Hν0 +

∑νν′=u,l

Hνν′

int +H1D−2D +Htun . (3.4)

Hν0 is the kinetic energy of the electrons [ν = u (l) labels the upper (lower) wire],

Hννint [Hul

int] describes spin-independent electron-electron interactions in (between) the

wires, H1D−2D is an effective Hamiltonian for the 1D-2D scattering of electrons in the

top quantum well, and Htun is the tunneling Hamiltonian:

Hν0 = vFν

∑s

∫dx[Ψ†

Rsν(−i∂x)ΨRsν −Ψ†Lsν(−i∂x)ΨLsν

], (3.5)

Hνν′

int =1

4π

∑ss′

∫dkVνν′(k) [2ρRsν(k)ρLs′ν′(−k)

+ ρRsν(k)ρRs′ν′(−k) + ρLsν(k)ρLs′ν′(−k)] , (3.6)

Htun = λ∑

s

∫dxΨ†

suΨsle−iqBx + H.c., (3.7)

where s and s′ are spin indices, Ψsν is the spin-s electron field operator, ΨRsν

and ΨLsν are the field operators for the right and left movers, respectively, Ψsν =


eikFνxΨRsν + e−ikFνxΨLsν , ρRsν(k) =∫dxeikxΨ†

RsνΨRsν is the density-fluctuation op-

erator for the spin-s right movers (and analogously for the left movers), and Vνν′(k) =∫dxeikxVνν′(x) is the Fourier transform of the two-particle interaction potential Vνν′(x).

WritingHint in terms of the interactions between electrons of fixed chirality in Eq. (3.6)

is possible after restricting electron correlations to small momentum transfer scatter-

ing, e.g., if V (k) ∝ exp(−rc|k|) with 1/rc kF. (By making this approximation

we disregard backward and Umklapp scattering processes, which are thought to be

unimportant in our cleaved-wire structure, see, e.g., Ref. [26].)

The 1D-2D scattering randomizes the direction of the 1D electrons in the top

quantum well with a mean free path l1D−2D ≈ 6 µm [30]. In infinite wires, this

weak scattering can be taken into account by rounding the 1D electron-gas spectral

function by a Lorentzian of half width Γ = 1/(2τ1D−2D), where τ1D−2D is the 1D-2D

scattering time.

If there were no interactions between the wires, i.e., Vul ≡ 0, low-energy spin

and charge excitations in each wire would propagate with velocities vsν = vFν and

vcν = vFν/gν , respectively. The parameters gν can be obtained by bosonization as

gν =

[1 +

2Vνν(0)

π~vFν

]−1/2

< 1 , (3.8)

in the case of repulsive interactions, Vνν(0) > 0. In the limit of a free-electron gas,

Vνν(0) = 0, gν = 1.

We treat tunneling between the wires to lowest order in perturbation theory.

Mesoscopic charging effects, such as discussed in, e.g., Ref. [18], are disregarded in


our analysis. The current (for electrons of each spin) is given by

I = e|λ|2∫ ∞

−∞dxdx′

∫ ∞

−∞dteiqB(x−x′)eieV t/~C(x, x′; t) , (3.9)

where C(x, x′; t) is a two-point Green’s function

C(x, x′; t) =⟨[

Ψ†l Ψu(x, t),Ψ

†uΨl(x

′, 0)]⟩

. (3.10)

In the limit of vanishing interactions between the wires, it reduces to

C(x, x′; t) = G>u (x, t;x′, 0)G<

l (x′, 0;x, t)−G<u (x, t;x′, 0)G>

l (x′, 0;x, t) (3.11)

expressed in terms of the one-particle correlation functions

G>ν (x, t;x′, t′) = −i

⟨Ψν(x, t)Ψ

†ν(x

′, t′)⟩, (3.12)

G<ν (x, t;x′, t′) = i

⟨Ψ†

ν(x′, t′)Ψν(x, t)

⟩. (3.13)

Note: Throughout this chapter, as in the above equations, the correlation functions

are defined for electrons with a fixed spin orientation and the spin index is therefore

omitted.

The results of this section also hold for direct 2DEG-1D tunneling, if we define

Ψsu(x, t) as the field operator for the 2DEG at the edge of the upper quantum well.

3.4.2 Interference Pattern

As discussed in Sec. 3.3.2, the breaking of translational invariance due to the

finite size of the tunneling junction can result in an oscillatory dependence of the

conductance G on voltage bias V and magnetic field B. In this section we discuss

in detail this behavior that arises due to interference of electrons tunneling through


a finite-sized window. We show that our theoretical framework can quantitatively

explain the conductance oscillations observed near the crossing points.

In the following, we mainly focus on the analysis of very distinct interference

patterns measured at low magnetic fields (as in Figs. 3.3 and 3.4). In Sec. 3.4.5 we

briefly comment on the conductance near the upper crossing point at high fields (as in

Fig. 3.5). It appears likely that while in the former regime the translational invariance

is broken due to the finiteness of the tunneling region only, in the latter case some

other mechanisms can also play a prominent role.

In the actual experiments, several 1D electron modes are occupied in the wires.

Here we consider only tunneling between modes which have the lowest energy of

transverse motion, and hence the largest Fermi momentum along the wire, namely

|u1〉 and |l1〉. These modes have densities that differ by only a few percent (see

Ref. [4]). We thus make a simplifying approximation vFu = vFl = vF, which is

justified by the measured dispersion slopes [4].

3.4.3 Asymmetry due to Soft Boundaries

In Ref. [132] we showed that the observed asymmetry of the secondary oscilla-

tion peaks on the two sides of the main dispersion curves (see Figs. 3.3, 3.4) can

be explained within a noninteracting electron picture and assuming a soft confining

potential Uu(x) for the upper wire. Here we will employ the model developed there

to quantitatively study the form of Uu(x).

Using the phenomenological tunneling Hamiltonian (3.7), we express the current


through the junction at zero temperature

I ∝ sgn(V )∑m

|M(n, qB, V )|2 (3.14)

in terms of the tunneling matrix element

M(n, qB, V ) =

∫dxψ∗n(x)e−iqBxϕkl

(x) (3.15)

between the upper-wire state ψn and the lower-wire state ϕkl, the energy of which is

lower by eV . The summation in Eq. (3.14) is over the integers [sgn(V )− 1]/2 < m <

e|V |L/(π~vF) denoting the offset of the ψn index n = nF + sgn(V )m with respect

to the state ψnFjust below the Fermi energy of the upper wire (and linearizing the

dispersions near the Fermi points, assuming e|V | is not too large). The current

(3.14) can be expressed by a single sum because the states of the confined upper wire

are discrete, while the states in the lower wire ϕkl(x) = e±iklx can be indexed by a

continuous wave vector kl. (As in Ref. [132], it is assumed that the left boundary

Ul(x) of the lower wire lies outside the tunneling region.) Since the Zeeman energy

in GaAs is small, we ignore the spin degrees of freedom.

We argued [132] that for practical purposes of understanding our measurements,

the sum in Eq. (3.14) can be replaced by an integral

I ∝∫ eV

0

dε|M(EFu + ε, qB, V )|2 (3.16)

labeling states in the upper wire by energy EFu + ε with respect to the Fermi energy

EFu. For the conductance obtained by differentiating the current, this approximation

will smear out the δ-functions appearing when the chemical potential of the upper wire

crosses each discrete energy level. Physically, such a smearing can be caused by 1D-

2D scattering and finite temperature. But even at low temperatures and vanishing


scattering, the result obtained by integration [Eq. (3.16)] will not be far off from

that found by summation [Eq. (3.14)] as the dominant contribution to the oscillation

pattern near the lower crossing point comes from differentiating the summand in

Eq. (3.14) [or correspondingly the integrand in Eq. (3.16)], as explained below.

We linearize the dispersions about the Fermi wave vectors kFν , so that kl is given

by (kl−kFl)vF = (ε−eV )/~. The wave vector inside the upper wire similarly depends

on energy: (ku− kFu)vF = ε/~. The matrix element squared |M(EFu + ε, qB, V )|2 can

then be written as a sum of contributions due to tunneling between right movers and

between left movers,

|M(EFu + ε, qB, V )|2 = |M(κ+)|2 + |M(κ−)|2 , (3.17)

where κ± = ku−kl± qB = ∆kF + eV/(~vF)± qB and ∆kF = kFu−kFl. The tunneling

matrix element

M(κ) =

∫dxeiκxψu(x)e

−ikFux (3.18)

is determined by the form of the bound-state wave function ψu(x) at the Fermi level

of the upper wire. We wrote the right-hand side of Eq. (3.17) as an incoherent sum

of the contributions of the two chiralities. This is an approximation we make by

disregarding additional interference arising due to the reflection of electrons in the

lower wire under gate G1 (i.e., by the potential Ul in Fig. 3.7). Taking the latter into

account does not considerably affect our results.

|M(κ±)|2 do not depend on energy ε, and the current (3.16) can, therefore, be

written as [132]

I ∝ V[|M(κ+)|2 + |M(κ−)|2

]. (3.19)


The differential conductance G = ∂I/∂V corresponding to the current I ∝ V |M(V )|2

becomes G ∝ |M(V )|2 + V ∂|M(V )|2/∂V . If, for example, the oscillatory component

of |M(V )|2 has the form sin(const · V ), the amplitude of the second term in the

conductance will be 2πN times larger than the amplitude of the first term after N

periods of oscillation. The dominant contribution to the oscillatory component of the

conductance near the lower crossing point is thus

G ∝ V∂

∂V

[|M(κ+)|2 + |M(κ−)|2

]. (3.20)

If the upper wire confining potential Uu is smooth enough so that the states at

the Fermi energy can be evaluated by the WKB approximation, the form of M(κ)

[Eq. (3.18)] can be studied both numerically and analytically [132]. In the region

between the classical turning points,

ψu(x) =1√ku(x)

eikFuxe−is(x) , (3.21)

where ku(x) = kFu[1−Uu(x)/EFu]1/2 and s(x) =

∫ x

0dx′[kFu−ku(x

′)]. In the stationary-

phase approximation (SPA), M(κ) is evaluated near positions x± (x+ > x−) where

ku(x±) = kFu− κ and the integrand in Eq. (3.18) has a stationary phase. In the case

of a symmetric potential, Uu(x) = Uu(−x), the SPA gives

M(κ) ∝ Θ(κ)√U ′

u(x+)

cos[κx+ − s(x+)− π/4

], (3.22)

where Θ(κ) is the Heaviside step function, the prime in U ′u denotes the derivative.

The SPA approximation (3.22) diverges for small values of κ and we have to resort

to a numerical calculation of the integral in Eq. (3.18) [132]. Fig. 3.8 shows the

calculated |M(κ)|2.


−5 0 5 10 15κL/2π

0

0.1

0.2

0.3

0.4

0.5

|M(κ

)|2 [arb

. uni

ts]

β=8 (numeric)β=8 (SPA)square well

Figure 3.8: |M(κ)|2 obtained using wave function ψu for the 100th WKB state inthe potential well [Eq. (3.23) with β = 8] of the upper wire. The solid line is thenumerical calculation, the dotted line is the SPA approximation [Eq. (3.22)] and thedashed line shows the result for the square-well confinement, for comparison.

We study the profile of confinement Uu(x) by measuring the period ∆κ of the

|M(κ)|2 oscillations as a function of κ. In a square well of length L, this period is given

by 2π/L. In a soft confinement, the interference stems from the oscillations of the

electron wave function near the classical turning points, so that ∆κ ≈ 2π/(x+− x−).

For a potential of the form

Uu(x) = EFu

∣∣∣∣2xL∣∣∣∣β (3.23)

(where β characterizes the ratio between the total length of the upper wire and the

extent of its boundaries),1 2x+/L ≈ (2κ/kF)1/β for κ > 0 assuming that ∆kF kF =

1If we assume β = L/l in Eq. (3.23), where l is a fixed length, then for large L, Uu(x) takes theform which is independent of L at the boundaries: Uu(x) ≈ EFu exp[−2(L/2− |x|)/l]. Note: α wasused in Ref. [132] instead of β in Eq. (3.23); in this chapter α denotes the LL exponent, see, e.g.,Eq. (3.48).


(kFu + kFl)/2 [132] (x− = −x+ for a symmetric potential), and the period is therefore

given by ∆κ ≈ (2π/L)(kF/2κ)1/β. Experimentally, we extracted ∆κ by measuring the

distance between oscillation zeros in region II of the interference shown in Fig. 3.3.

In order to reduce the statistical uncertainty, the conductance was averaged along

lines of constant κ+, separately for positive and negative bias. In terms of variable

S = ~κ+/ed (which reduces to magnetic field B at zero voltage and vanishing ∆kF),

∆S ≈ 2π~edL

(~kF

2edS

)1/β

. (3.24)

This ∆S is compared with the data in Fig. 3.9 for several values of β (we extract

kF ≈ 1.5 · 108 m−1 using measured electron densities [4]). At each β shown in the

figure, the distance L was found by the best (least-square) fit of the curve (3.24) to

the measurements. The lithographic length for the junction was Llith = 2 µm and

the width d = 31 nm. Such fitting allows us to extract two quantities, L/Llith =

1.45 ± 0.1 and β = 8 ± 2, characterizing the extent of the 1D confinement and the

sharpness of the potential-well boundaries, respectively. It appears that the effective

length of the upper wire [defined as the distance between the classical turning points,

see Eq. (3.23)] L is actually about a micron longer than the lithographic length.

This conclusion is relatively insensitive to the fitting procedure, as ∆S in Eq. (3.24)

approaches 2π~/(edL) for S & ~kF/(2ed) if the exponent 1/β is small. The difference

between L and Llith can be due to significant screening of the tungsten gates (which

are positioned 0.5 µm above the junction) by the 2DEG in the upper quantum well,

as viewed by the upper-wire electronic bands.

As a consistency check for the result of the fit in Fig. 3.9, we performed an analysis

of the conductance oscillations that takes into account the dependence on β. Accord-


ing to the S−1/β scaling of the oscillations’ period, see Eq. (3.24), and the S1/β−1

fall-off of the their amplitude [which follows from Eq. (3.22), also see Ref. [132]], the

function S1−1/βG(V, S1+1/β

)is periodic in S1+1/β. Fourier analyzing it at a fixed V ,

and setting β = 8, we obtained a main peak, the position of which depends very

weakly on V , which corresponds to a length of L = 2.81±0.02µm, in agreement with

the result of the fit. In Fig. 3.3a we plot the absolute value of the main peak, which

is seen to decay on a scale of a few mV. We discuss this decay in Sec. 3.4.6. For com-

parison, we also Fourier analyzed the data in Fig. 3.4b. For that we found that one

has to use a larger value, β = 21.5, in order to obtain a relatively voltage-independent

position of the peak. This value of β is reasonable because it gives approximately

the same boundary profile for a 6 µm (upper) wire as β = 8 gives for a 2 µm wire.

Again we obtained a reasonable length (L = 7.3 ± 0.3 µm) that varied only weakly

as a function of V . The height of the main peak in this case is shown in Fig. 3.4a

where it is seen to decay on a faster scale than for the shorter upper wire. The ratio

of the scales is approximately the ratio of the upper-wire lengths.

3.4.4 Modulation due to Spin-Charge Separation

In the following we describe how electron-electron interactions in the wires and

between them affect the oscillation pattern. We show our theoretical results for

G(V,B) near the lower crossing point of the |u1〉 ↔ |l1〉 transition and compare them

to measurements on 2 µm and 6 µm junctions, Figs. 3.3 and 3.4. In particular, we find

that the difference in the velocities of the charge- and spin-excitation modes in the

double-wire system can account for the observed G(V,B) suppression stripes running


0 0.2 0.4 0.6 0.8S [Tesla]

0.04

0.06

0.08

0.1

∆S [

Tes

la]

V<0V>0β=4, L/L

lith=1.70

β=8, L/Llith

=1.40β=12, L/L

lith=1.32

2πh/(edLlith

)

Figure 3.9: Period of (faster) oscillations in region II of Fig. 3.3 as a function ofS = ~κ+/(ed). Circles show measurements at positive and negative bias and thecurves are fits using Eq. (3.24) at several values of β. The best overall fit is reachedat β ≈ 7.67 and L/Llith ≈ 1.41, where Llith = 2 µm.

parallel to the B-axis.

As a starting point, let us consider the case when the interwire interactions are

vanishingly small Vul Vll and the interactions in the two wires are the same,

Vuu = Vll, so that gl = gu = g, as defined in Eq. (3.8). For positive voltages V > 0,

the current (3.9) is then given by

I = e|λ|2∫ ∞

−∞dxdx′

∫ ∞

−∞dteiqB(x−x′)eieV t/~G>

u (x, t;x′, 0)G<l (x′, 0;x, t) . (3.25)

At low magnetic field, the conductance has two main contributions, corresponding

to the two edge-state chiralities. The two contributions give bright conductance peaks

and side lobes with opposite slopes, as described in Sec. 3.4.2 and Ref. [132]. Let us

discuss tunneling between the right movers (current due to tunneling between the


left movers at field B equals tunneling between the right movers at field −B). We

assume that the electron density in each wire varies slowly on the length scale set

by the respective kF (except for unimportant regions very close to the boundaries).

The zero-temperature Green’s functions entering Eq. (3.25), in this regime, can be

written as [122, 79, 137]

Gu,l(x, t;x′, 0) = ± 1

2πΦu,l(x, x

′)1

(z − vFt± i0+)12

1

(z − vct± i0+)12

×[

r2c

z2 − (vct∓ irc)2

] 12γ

, (3.26)

where vc = vF/g, γ = (g+g−1−2)/4, z = x−x′, and rc is a short distance cutoff (i.e.,

1/rc is a momentum-transfer cutoff in the electron-electron interactions). Here, Gu is

the G> Green’s function (3.12) for the upper wire and Gl is the G< Green’s function

(3.13) for the lower wire. The function Φν is defined by Φν(x, x′) = ψν(x)ψ

∗ν(x

′),

in terms of the WKB wave functions ψν(x) for right-moving electrons at the Fermi

energy in wire ν in a confining potential Uν(x) which must be chosen self-consistently

to give the correct electron density. Here we assume that ψu(x) and Uu(x) are given

by Eqs. (3.21) and (3.23), while ψl(x) = eikFlx.

Several additional approximations are implied in using Eq. (3.26) to calculate the

tunneling current (3.25): (1) The weak 1D-2D scattering is neglected, (2) The voltage

is small enough so that one can linearize the noninteracting electron dispersions about

the Fermi-points and use LL theory (i.e., we disregard the curvature), (3) t vcL, so

that the discreteness of the energy levels of the upper wire due to electron confinement

within a well of length L and their reflection at the boundaries does not considerably

modify the LL Green’s function for an infinite wire [the confinement, however, is


manifested in the form of the wave function ψu(x); effects due to the discreteness

are discussed in Sec. 3.4.3 in the regime of noninteracting electrons, and they are

believed to be small]. The last approximation breaks down for very low voltages

(and, correspondingly, long times) in the regime of the zero-bias anomaly, which is

treated separately in Sec. 3.4.8.

Substituting Green’s functions (3.26) into integral (3.25), we obtain for the tun-

neling current

I ∝∫ ∞

−∞dxdx′ei(qB−kFl)(x−x′)ψu(x)ψ

∗u(x

′)h(x− x′) , (3.27)

using the definition

h(z) = −∫ ∞

−∞dt

eieV t/~

(z − vFt+ i0+)(z − vct+ i0+)

×(

rc

z − vct+ irc

)γ (rc

z + vct− irc

)γ

. (3.28)

The integrand in Eq. (3.28) has a simple analytic form: it has two first-order poles at

t = z/(vF + i0+) and t = z/(vc + i0+), and two branch cuts starting with singularities

at t = (±z + irc)/vc. The contour of integration can be deformed leaving two nonva-

nishing contributions: h(z) = h1(z) + h2(z). The first contribution, h1(z), is due to

integration around the poles:

h1(z) =2πieieV z/(~vc)

(vc − vF)(z + i0+)

(r2c

r2c + 2izrc

)γ

− 2πieieV z/(~vF)

(vc − vF)(z + i0+)

×(

r2c

r2c + z2[1− (vc/vF)2] + 2izrcvc/vF

)γ

, (3.29)

and the second contribution, h2(z), is due to integration around the branch cuts. For


z > 0, for example,

h2(z) = 2i sin(γπ)e−eV rc/(~vc)

∫ ∞

z/vc

−∫ −z/vc

−∞

dt

×(

rc

(vct)2 − z2

)γeieV t/~

(z − vct− irc)(z − vFt− ircvF/vc). (3.30)

In our system we expect that [4] g ≈ 0.7, so that γ ≈ 0.03 1. Therefore, since

rc ∼ 30 nm (the width of the wires) and z < L ≈ 2 − 6 µm, the terms of the form

(· · · )γ in Eq. (3.29) can be safely ignored (except for the regime of extremely low

voltages, which will be discussed in Sec. 3.4.8). Furthermore, h2(z) h1(z), so that

we arrive at an approximation

h(z) ≈ −2πieieV z/(~vF) − eieV z/(~vc)

(vc − vF)(z + i0+). (3.31)

Substituting this into Eq. (3.27), we can now evaluate the current. One notices that

after making the approximation (3.31), the current (3.27) becomes the same as if there

were no electron-electron interactions but different Fermi velocities in the two wires,

given by vF and vc. Using Eqs. (3.27) and (3.31), we, finally, get for the conductance

G = ∂I/∂V :

G(V,B) ∝ 1

vc − vF

[1

vF

|M(κF)|2 − 1

vc

|M(κc)|2], (3.32)

where κF,c = qB + ∆kF + eV/(~vF,c) and M(κ) is given by Eq. (3.18).

If the excitation velocities in the wires are nearly the same, vF ≈ vc = v, we can

approximate the conductance (3.32) by

G ∝ ∂

∂ηη |M(η, V )|2 = |M(κ)|2 + V

∂

∂V|M(κ)|2 , (3.33)

where M(η, V ) = M(qB + ∆kF + ηV ) and η = e/(~v). This reproduces Eq. (3.19).

One can refine the form of the second term on the right-hand side of Eq. (3.33) (which


can be much larger than the first term, see Sec. 3.4.3), using approximation (3.22),

when the difference between velocities vF and vc becomes appreciable (which is the

case for g ≈ 0.7):

G ∝ Θ(κ)

U ′u(x

+)· sin[eV x+(1/vF − 1/vc)/~]

1/vF − 1/vc

cos 2[κx+ − s(x+)

]. (3.34)

Here, κ and x+ are defined using velocity v = 2(1/vF + 1/vc)−1, G stands for the

second contribution to the conductance in Eq. (3.33). At low bias, G→ 0 linearly in

V and the term G ∝ |M(κ)|2 governs the conductance. This contribution is further

suppressed as V α (at zero temperature) in the zero-bias anomaly regime discussed in

Sec. 3.4.8.

We can generalize the preceding discussion of this section to include interactions

between the wires, i.e., Vul 6= 0. Since the quantum wires are closely spaced, the

interwire interactions can be sizable. Furthermore, because the Fermi velocities in

modes |u1〉 and |l1〉 are similar, the excitations in the coupled wires can propagate

with velocities quite different from those in the isolated wires. When we take Vul

into account, the dominant part of Green’s function (3.10) becomes (assuming weak

interactions, in the spirit of the preceding discussion) [152]

C(x, x′; t+ i0+) ∝ − ΦuΦ∗l (x, x

′)

(z − vFut)12 (z − vFlt)

12

× 1

(z − vc−t)12+θr(z − vc+t)

12−θr

, (3.35)

where

vc± ≈vcu + vcl

2± Vul(0)

π~√

1 + r2 . (3.36)

Here, r = π(vcu − vcl)/2Vul(0) and θr = 1/(2√

1 + r2) is finite for nonvanishing in-

teractions between the wires, vcn are the charge-excitation velocities in the isolated


wires. [Note that there appears to be a sign error in Ref. [152] in the expression for

the velocities vc± in the physical case of repulsive interactions Vul(0) < 0.]

For a symmetric double-wire system, vFu = vFl = vF, Vuu ≡ Vll, and vcu = vcl,

so that r = 0 and θr = 1/2. (In this case, vc+ and vc− become the velocities of the

symmetric and antisymmetric charge excitations, respectively.) Green’s function then

reduces to C ∝ −ΦuΦ∗l (z − vFt)

−1(z − vc−t)−1 and we reproduce our main result of

this section, Eq. (3.32), after replacing vc with the antisymmetric charge-excitation

velocity vc−. This is natural as tunneling in a symmetric biwire can only excite the

antisymmetric modes at low magnetic fields.

In addition to the structure studied in Sec. 3.4.3 for the system of noninteracting

electrons, we now show that the electron-electron interactions in the wires lead to a

modulation of the conductance oscillations along the voltage axis [132]. This modu-

lation suppresses the contribution G [Eq. (3.34)] to zero in stripes parallel to the field

axis. The distance between them is:

∆Vmod =π~vc−vF

ex+(vc− − vF). (3.37)

The ratio between ∆Vmod and the period

∆V =2π~vc−vF

ex+(vc− + vF)(3.38)

due to the wave-function oscillations near the turning points [compare to Eq. (3.3)]

∆Vmod

∆V=

1

2

vc− + vF

vc− − vF

=1

2

1 + g−1− g−

(3.39)

can be used as an independent measure of the interaction parameter g− = vF/vc−.

From Figs. 3.3b and 3.4b, we find that

g− = 0.67± 0.07 , (3.40)


hvFπeVL/(2 )

Bq

L/(

2π)

Figure 3.10: The differential conductance interference pattern near the lower crossingpoint calculated by Eq. (3.32) for tunneling between right movers (and similarlyfor left movers) using a smooth confining potential for the upper wire, Eq. (3.23).vc− = 1.4vF, ∆kF = 4π/L, β = 8. We used the numerically found |M(κ)|2, alsoshown in Fig. 3.8. The figure has to be compared to experimental Fig. 3.3.

similarly to the value for gl obtained by the zero-bias anomaly in Sec. 3.4.9. Also,

from Eq. (3.34) it follows that the oscillation pattern [of the principal term G(V,B)]

gains a π phase-shift across each suppression strip. Such phase shifts can also be seen

in experimental Figs. 3.3b and 3.4b.

Finally, we compare the interference pattern predicted by our theory, Eq. (3.32),

with the experiment, Figs. 3.3b, 3.4b. G(V,B) calculated using a smooth confining

potential [Eq. (3.23) with β = 8] for the upper wire is shown in Fig. 3.10. Many


hvFπeVL/(2 )

Bq

L/(

2π)

Figure 3.11: Same as Fig. 3.10 but with ∆kF = 10π/L and β = 22, describing a longerjunction with a similar boundary profile. |M(κ)|2 was correspondingly recomputed(now putting 300 electrons per spin in the upper wire). The figure has to be comparedto experimental Fig. 3.4.

pronounced features observed experimentally–the asymmetry of the side lobes, a slow

fall-off of the oscillation amplitude and period away from the principal peaks, an

interference modulation along the V -axis, π phase shifts at the oscillation suppression

stripes running parallel to the field axis–are reproduced by the theory.

In Fig. 3.11, we repeat the calculation using β = 22, which defines potential (3.23)

with a similar boundary profile near the turning points of a three-times longer wire.

(Here by length we mean the distance between the classical turning points, which,


as explained in Sec. 3.4.3, can be somewhat different from the lithographic length.)

Again an agreement between the predicted (Fig. 3.11) and measured (Fig. 3.4) oscil-

lation patterns is apparent. In Fig. 3.11 a few weak side lobes also appear to the left

of the main dispersion peaks, unlike in Fig. 3.10 where they appear strictly to the

right. In addition, the interference modulation in the voltage direction has sharper

features in Fig. 3.11. These trends are expected for longer junctions as the boundaries

become steeper on the scale set by the total length.

Tunneling between 1D channels with different Fermi velocities can also yield an in-

terference modulation similar to that described in this section even when the electron-

electron interactions are vanishingly small. It is thus important to emphasize that we

suggest the spin-charge separation picture to explain this modulation relying on the

experimental result (see Ref. [4]) that the densities of modes |u1〉, |l1〉 and, therefore,

the corresponding Fermi velocities are nearly identical.

Using Eq. (3.35) we also studied various possible scenarios when the interactions

in the two wires differ. For example, in a situation when the upper wire is perfectly

screened, so that Vuu, Vul ≡ 0, there are still two velocities present in the system, vF

and vcl, but the interference pattern is qualitatively very different from that shown

in Fig. 3.10 and observed experimentally [see Figs. 3.3 and 3.4]. Since a considerable

weight of the charge-excitation contribution to the tunneling strength is shifted to

velocity vF (which is now also the charge-excitation velocity in the upper wire), the

oscillation pattern does not exhibit the pronounced vertical suppression stripes, but

rather a much weaker modulation. The same conclusion also holds for intermediate

regimes of relative screening in the two wires, when the system is not symmetric and


the two charge-excitation velocities significantly differ. The pronounced suppression

stripes are, therefore, present only if most of the charge-excitation tunneling weight

is peaked at a single velocity vc− (which is guaranteed only when the system is nearly

symmetric).

Taking into account 1D-2D scattering in the upper quantum wire will smear out

the oscillation pattern by its convolution with a Lorentzian in the B-direction, simi-

larly to Eq. (3.52) below. The corresponding effect is, however, small because of the

high quality of our wires, which have a long scattering length [30] l1D−2D ≈ 6 µm.

3.4.5 Upper Crossing Point

In practice, since the fields necessary to reach the upper crossing point are quite

large (e.g., 7 T for the |u1〉 ↔ |l1〉 transition), even atomic-scale disorder in the

junction can lead to a significant variation δqB of the momentum transfer along the

tunneling region. In particular, δqB = eBδd can be comparable with 2π/L, the

reciprocal wave vector of the upper wire. This can significantly broaden the principal

dispersion peaks. Furthermore, Zeeman splitting becomes about a per cent of the

Fermi energy at these high fields and results in somewhat different dispersions for

different spin modes. Away from the main peaks, however, we still expect to see side

lobes due to stationary phases at the ends of the junction, similarly to the regime

of low magnetic fields discussed above (with possibly a faster decoherence in the

V direction than just due to the dispersion curvature studied in Sec. 3.4.6). Such

oscillations [with about the period (3.3)] are indeed observed experimentally, as can

be seen in Fig. 3.5. Because of the mentioned complications, we, nevertheless, do not


pursue a detailed analysis of the conductance near the upper crossing point in this

chapter.

3.4.6 Dephasing of the Oscillations

It is evident from Figs. 3.3b and 3.4b that the interference decays as |V | is in-

creased. A more quantitative analysis of this decay is shown in Figs. 3.3a and 3.4a,

where the amplitude of the oscillations is plotted as a function of voltage. It is clear

that the measured modulation has a fast-decaying envelope, which can not be ex-

plained by the analysis of section 3.4.4. (See, for example, Eq. (3.34) which predicts

that the modulation is roughly periodic.)

One scenario for the dephasing occurs even in the case of noninteracting elec-

trons considered in Sec. 3.4.3, when we take the finite curvature of the single-particle

dispersions into account. Let us return to the form of the current in Eq. (3.16):

I ∝∫ eV

0

dε[|M(κ+)|2 + |M(κ−)|2

]. (3.41)

Correcting our previous results to take into account nonlinear dispersion near the

Fermi points, we now write κ± = [k2Fu + 2mε/~2]1/2 − [k2

Fl + 2m(ε− eV )/~2]1/2 ± qB.

[Using Eq. (3.41) we still imply low enough bias V , so that the density of states in the

wires are relatively constant on the energy scale of e|V |.] Expanding this expression

to lowest order in curvature, we further obtain

κ± = ∆kF +eV

~vF

± qB +eV (eV − 2ε)

2~2v2FkF

. (3.42)

[Eq. (3.19) can be recovered by neglecting the last term above.] The current (3.41)


then becomes

I ∝∫ eV/2

−eV/2

dε

∣∣∣∣M (∆kF +

eV

~vF

+ qB −εeV

~2v2FkF

)∣∣∣∣2 + (qB → −qB) . (3.43)

It is easy to see now that the contribution to the conductance obtained by dif-

ferentiating the integrand in Eq. (3.43) will be suppressed when the argument κ of

the tunneling matrix amplitude M(κ) changes by the full period of oscillations ∆κ

upon energy ε variation between the integration limits ±eVsup/2. We thus arrive at

the condition for the suppression voltage Vsup:

∆κ =(eVsup)

2

~2v2FkF

. (3.44)

Approximating ∆κ ≈ 2π/L and translating it into the oscillation period in the bias

direction e∆V = ~vF∆κ, one finally obtains

Vsup

∆V=

√LkF

2π. (3.45)

Using density 100 µm−1 for the lowest bands in the wires [4], we find Vsup/∆V ≈ 7

(≈ 12) for the 2 µm (6 µm) junction. An implicit assumption in the derivation is that

we are still close enough to the Fermi level so that higher-order corrections should not

modify the result significantly [in particular, for the calculation of the matrix element

(3.18) it can still be reasonable to use the wave function ψu(x) at the Fermi energy].

The result of the numerical calculation using Eq. (3.43) and matrix element M(κ)

plotted in Fig. 3.8 (using parameters characteristic for the 2 µm sample) is shown

in Fig. 3.12. Notice that when the voltage exceeds Vsup ≈ 7∆V , so that the pattern

starts dephasing due to the finite curvature, a beating pattern appears. It defers from

the data in several important aspects: First of all, the lines of suppressed G(V,B)


hvFπeVL/(2 )

Bq

L/(

2π)

Figure 3.12: The differential conductance interference pattern near the lower crossingpoint calculated by Eq. (3.43), within the noninteracting electron picture, using thematrix element M(κ) shown in Fig. 3.8 (for β = 8). See text for further details.

are not equidistant. In addition, Vsup, which determines the distance between the

zero-bias suppression stripe and the next one on either positive- or negative-voltage

side, is about twice larger than the period we observe in Fig. 3.3a and four times

larger than that in Fig. 3.4a, which in both cases is given by about 3∆V . This hints

that the source of the beating in the experimental data is not the curvature of the

dispersions, but rather the spin-charge separation mechanism discussed in Sec. 3.4.4.

Another important difference between Eq. (3.41) and the experiment is that the

decay of the oscillations is much stronger in the latter. It might therefore be necessary

to consider both the curvature and electron interactions in order to understand the


fast decay of the conductance oscillation amplitude with increasing voltage. Taking

into account the curvature while bosonizing excitations of the interacting electrons

[45, 138] leads to higher-order terms in the Hamiltonian. Physically this corresponds

to interactions between bosonic excitations which therefore acquire a finite life time.

The singularities of the spectral densities will correspondingly be rounded, in turn

smearing the conductance interference pattern. Further complications may arise from

the electron backscattering which was entirely disregarded: While the low-energy

properties of the system are not affected by the backscattering (apart from rescal-

ing of certain parameters) since it renormalizes downward in the case of repulsive

interactions, the story at a finite energy could be different. The reason for this is a

slow (logarithmic) renormalization flow of the backscattering strength. If a signifi-

cant backscattering is present in the original Hamiltonian, it could therefore be still

considerable at a finite energy. A detailed study of these effects however lies beyond

this chapter’s scope.

3.4.7 Zero-Bias Anomaly

3.4.8 Crossing Points

It is enlightening to further study tunneling between 1D channels at low bias when

the magnetic field is tuned to match two Fermi-points of the wires (see Sec. 3.3.1).

The zero-bias properties are similar near the two crossing points and, for definiteness,

we choose to discuss the upper crossing, where the magnetic wave vector qB is close to

kFu +kFl and the field changes the chirality of the tunneling electrons: The tunneling

is amongst the left movers of the upper wire and the right movers of the lower wire.


For the |u1〉 ↔ |l1〉 transition, this point is located at B ≈ 7 T, see Fig. 3.2. The

results are straightforward to apply to the regime of the lower crossing point, as well.

For clarity, we start by making a series of simplifying assumptions which will be

dropped in subsequent generalizations: First, we set the upper-wire and interwire

interactions, Vuu and Vul, to zero. Physically, this corresponds to a regime where

the Coulomb interactions in the upper wire are perfectly screened by the 2DEG.

Secondly, we further simplify the model by assuming a square-well confinement for

the electronic states in the upper quantum wire and an infinitely-steep reflecting

left boundary for the electrons in the lower wire, i.e., Uu(x) [Ul(x)] is constant for

|x| < L/2 [x > −L/2] and infinite otherwise. As we showed in the previous sections,

both of the above assumptions are not very realistic for the purpose of studying the

interference pattern. In the zero-bias anomaly regime, however, they can be a good

starting point, at least, for pedagogical reasons.

Electron states participating in tunneling near the crossing points [Eq. (3.1)] lie

close to the Fermi levels in both wires. It is therefore possible to calculate the correla-

tion functions analytically using LL theory, after the dispersion relations in the wires

are linearized. At the upper crossing point, we only need to retain Green’s functions

of the left movers of the upper wire and the right movers of the lower wire. At zero

temperature these are given by

G>u (x, t+ i0+;x′, 0) = − 1

4L

e−ikFuze−Γ|z|/vF

sin π2L

(z + vFt)L→∞= − 1

2π

e−ikFuze−Γ|z|/vF

z + vFt(3.46)


for |x|, |x′| < L/2, and G>u vanishing otherwise, and

G<l (x′, 0;x, t+ i0+) = − 1

2π

e−ikFlz

(z − vFt)12

1

(z − vclt)12

×[

r2c

z2 − (vclt− irc)2

] gl+g−1l

−2

8

×[

z′2 − z2

z′2 − (vclt)2

] gl−g−1l

8

, (3.47)

for x, x′ > −L/2, and vanishing otherwise, where z = x−x′, z′ = x+x′+L, and rc is

a small distance cutoff. As specified above, Eq. (3.46) [Eq. (3.47)] contains only the

component for the left (right) movers in the upper (lower) wire; we have thus omitted

terms proportional to eikFz, eikFz′ , and e−ikFz′ which do not contribute constructively

to tunneling near the upper crossing point. The last factor in the expression for G<l

is due to the closed boundary at x = −L/2 [31, 77, 34].

For sufficiently large voltages, eV 2~vF/(glL), the tunneling electrons do not

feel the junction boundaries on the time scale set by the voltage. In particular,

the left boundary of the lower wire does not affect the dynamics and, effectively,

electrons directly tunnel into the bulk of the lower wire: The last term in Eq. (3.47)

is close to unity and can, therefore, be omitted. Terms of the form 1/(z±vt)ϑ entering

Eqs. (3.46) and (3.47) are dominated by the long-t behavior in the integral [Eq. (3.25),

the voltage is assumed to be positive] if eV ~ max(vq,Γ), where q = qB−(kFu+kFl).

The conductance is then suppressed as a power law

G(V ) ∝ V α (3.48)

with the exponent αbulk = (gl + g−1l − 2)/4. This result is easy to generalize for the


case of unscreened interactions in the upper wire:

αbulk =∑ν=u,l

gν + g−1ν − 2

4. (3.49)

If the interwire interactions Vul are also significant, the elementary excitation modes

in the wires become coupled and αbulk has a more complicated form than that in

Eq. (3.49) [26]. Interference oscillations discussed in Sec. 3.4.2 can modulate the

power-law current suppression (3.48), setting an upper voltage bound, eV < e∆V ≈

2π~vF/L, for the validity of Eq. (3.48). It would therefore be hard to observe the

exact power-law voltage dependence (3.48) with the exponent (3.49) in the regime

when eV 2~vF/(glL) (see, however, Sec. 3.4.9).

If eV 2~vF/(glL), electrons effectively tunnel into the end of the lower wire

and the current suppression is governed by processes in the lower wire outside the

tunneling region. In particular, details of the interactions in the finite upper wire do

not play a role. The last term in Eq. (3.47) now also contributes to the exponent of

the long-t asymptotic, and α in Eq. (3.48) is given by

αend =g−1

l − 1

2. (3.50)

The upper wire, in this case, can be viewed as a point contact and the tunneling expo-

nent is determined entirely by the properties of the lower wire outside the tunneling

region.

At a finite temperature, the time scale relevant for the discussion above is set by

max(eV, kBT ). The power law (3.48) should now be replaced with

G(V, T ) ∝ TαFα

(eV

kBT

), (3.51)


where Fα(x) is a known scaling function with properties Fα(0) = const and Fα(x) ∝

xα in the limit of x 1 [17]. At low temperatures the conductance yields a low bias

dip extending to voltages eV ∼ kBT with G(V = 0) ∝ Tα.

In Sec. 3.4.2 we showed that the conductance G(V,B) exhibits a characteristic

interference pattern due to wave-function oscillations near the gates confining the

tunneling region. We can easily read out the profile of this pattern for the current

(3.25) using the correlation functions (3.46), (3.47) in the low-energy regime consid-

ered in this section (namely t z):

G(B) ∝∫ ∞

−∞dk

Γ/vF

k2 + (Γ/vF)2|M(k − q)|2 , (3.52)

where M(κ) is the tunneling matrix element, Eq. (3.18).

The present discussion also holds for the lower crossing point, where the electrons

do not change their chirality upon tunneling. To directly apply the above results to

this regime (for definiteness, assuming we now consider the transition between the

right-moving electrons), we only need to redefine the distance from the crossing point

in the field direction: q = qB + kFu − kFl (and analogously for the transition between

the left movers).

3.4.9 Direct Tunneling from the 2DEG

It is straightforward to generalize the main results of the preceding section to the

regime of direct tunneling from the 2DEG. Eq. (3.25) stays valid in this case, but now

G>u is Green’s function for the 2DEG near the edge of the upper quantum well. We

calculate this correlation function and discuss its limiting behavior at low energies

in Appendix B.2. The 2DEG density of states is finite at the Fermi energy and,


therefore, the long-t behavior of the one-particle Green’s function is G>(t) ∝ 1/t. If

max(eV, kBT ) ~vFkF,2D, where ~kF,2D is the 2DEG Fermi momentum and vF is

the lower of the Fermi velocities of the 1D band and the 2DEG, the temperature and

voltage dependence of the differential conductance are governed by the exponents

(3.49), with gu = 0, or (3.50), depending on the relation between max(eV, kBT ) and

2~vF/(glL). Because in this regime we tunnel directly from the 2DEG, interactions

in the 1D modes of the upper quantum well do not play a role, and both αbulk and

αend are determined only by the interaction constant gl of the lower wire. While

the field dependence of the conductance for the direct 2DEG–lower wire tunneling

is different from Eq. (3.52) (in particular, the conductance does not exhibit a strong

oscillation pattern), the low-energy properties stay similar to the case of the 1D-1D

tunneling. In spite of a complicated dependence of G(V,B) on magnetic field, the

zero-bias anomaly is pronounced in the data for tunneling either between different

1D bands or between the 2DEG and the 1D bands.

As described in Sec. 3.3.3, we measured the zero-voltage conductance dip at tem-

peratures 0.2 < T < 2 K on a junction of length L = 6 µm at B = 2.5 T. It can

be seen in Fig. 3.2 that at this magnetic field, the conductance is dominated by di-

rect tunneling from the 2DEG, |u3〉 ↔ |l2〉. Since ~vFkF,2D/kB ∼ 100 K T , the

temperature dependence of the zero-bias dip can be used to extract the value of the

interaction constant gl for the band |l2〉. The data points and the (best) theoretical

fitting curves are shown in Fig. 3.6; we find

gl = 0.59± 0.03 . (3.53)

The transition point between the two lines in the plot is consistent with an estimate


2~vF/(glLkB) ≈ 0.5 K for the second 1D mode of the lower wire, |l2〉.

As a consistency check, we plot in the insets to Fig. 3.6 curves calculated using

Eq. (3.51) (taking both αend and αbulk for the exponent). gl and the overall pro-

portionality constants were independently obtained from the power-law temperature

dependence of the bottom of the dip, i.e., G(V = 0, T ), so that at this point we do not

have any remaining fitting parameters. The results show reasonable agreement with

the data: When max (eV, kBT ) > 2~vF/(glL) the data is consistent with α = αbulk

while when max (eV, kBT ) < 2~vF/(glL) it is more consistent with α = αend. Thus,

in particular, there is a crossover between αend and αbulk in the data for G(V ) at

T = 0.24 K. For voltages V ∼ 1 meV that are comparable to the Fermi energies of

the modes participating in tunneling, the power-law behavior (3.51) is replaced by

a more complex structure modulated by the dispersions in the wires and the upper

well, see Fig. 3.2.

3.5 Summary

We have presented a detailed experimental and theoretical investigation of tun-

neling between two interacting quantum wires of exceptional quality fabricated at

the cleaved edge of a GaAs/AlGaAs heterostructure. The study focused on revealing

electron-electron interaction effects on the conductance interference pattern arising

from the finite size of the tunneling region and the conductance suppression at low

bias.

In the analysis of the data the finiteness of the junction plays a central role.

Breaking translational invariance, the boundaries give rise to secondary dispersion


peaks in dependence of the conductance on voltage bias and magnetic field. Smooth

gate potentials result in a strongly asymmetric interference profile, while the Coulomb

repulsion in the wires leads to a spin-charge separation which, in turn, modulates the

conductance oscillation amplitude as a function of voltage bias.

An interplay between the electron correlations in the wires and the finiteness of

the junction length also results in different regimes of the zero-bias anomaly. At

the lowest voltages, the upper wire is effectively a point-contact source for injecting

electrons into the semi-infinite lower wire. On the other hand, at higher voltages,

electrons effectively tunnel between the bulks of the two wires along the length of the

junction.

Using the temperature dependence of the zero-bias dip, we found the value of

the interaction parameter gl = vFl/vcl for band |l2〉 in the lower wire to be 0.59 ±

0.03. From the ratio between the long (due to spin-charge separation) and slow (due

to upper-wire confinement) scales of the conductance oscillations, we also extracted

the interaction parameter g− = vF/vc− corresponding to the antisymmetric charge-

excitation mode in the lowest bands |u1〉 and |l1〉 of the biwire to be 0.67± 0.07.

While g− and gl have similar numerical values, these quantities should be con-

trasted: gl is the interaction parameter (3.8) of the channel |l2〉 in the lower wire,

which is screened by other 1D states in the wires as well as the 2DEG of the upper

quantum well. g−, on the other hand, is a parameter characterizing the (antisym-

metric) charge mode in the coupled |u1〉 and |l1〉 channels of the two wires, which is

relatively weakly screened by the 2DEG since the latter has a smaller Fermi velocity

(being, nevertheless, still larger than the Fermi velocity of |l2〉) [4]. This can explain


why g− and gl are comparable while |l2〉 has about half the Fermi velocity of |u1〉 and

|l1〉. [The interwire interaction would only enhance the mismatch as it reduces vc−,

see Eq. (3.36)].]

Similar values for the interaction parameter g, in the range between 0.66 and

0.82, were found in Ref. [3] for single cleaved-edge quantum wires by measuring the

temperature dependence of the line width of resonant tunneling through a localized

impurity state. Spectral properties of the same double-wire structure as reported here

were investigated in Ref. [4], also indicating comparable values of g, about 0.75, for

various intermode transitions. An interaction parameter g ≈ 0.4 was found for GaAs

quantum-wire stacks in resonant Raman scattering experiments [108]; the smaller

value of g there can be attributed to much lower electron densities and no screening

by the 2DEG as in our measurements.

Chapter 4

Non-Abelian Braiding of

Moore-Read Quasiholes

We develop a general framework to (numerically) study adiabatic braiding of

quasiholes in fractional quantum Hall systems. Specifically, we investigate the Moore-

Read (MR) state at ν = 1/2 filling factor, a known candidate for non-Abelian statis-

tics, which appears to actually occur in nature. The non-Abelian statistics of MR

quasiholes is demonstrated explicitly for the first time, confirming the results pre-

dicted by conformal field theories.

153

Chapter 4: Non-Abelian Braiding of Moore-Read Quasiholes 154

4.1 Background

The quantum statistics of a system of identical particles describe the effect of

adiabatic particle interchange on the many-body wave function. All fundamental

particles belong to one of two classes: those that have their wave function unaffected

by particle interchange (bosons) and those whose wave function gets a minus sign

under permutation (fermions). In two dimensions, it is known that a number of ex-

otic types of statistics can exist for particle-like collective excitations. For example,

elementary excitations of the Laughlin fractional quantum Hall (FQH) states exhibit

“fractional” statistics: The phase of the wave function is rotated by an odd fraction

of π when two Laughlin quasiparticles (or quasiholes) are interchanged [47, 2]. Even

more exotic statistics can exist when a system with several excitations fixed at given

positions is degenerate [89]. In such a case, adiabatic interchange (braiding) of ex-

citations can nontrivially rotate the wave function within the degenerate space. In

general, these braiding operations need not commute, hence the statistics are termed

“non-Abelian”. Remarkably, the Moore-Read (MR) state, a state which is commonly

believed [103] to describe observed FQH plateaus at ν = 5/2 and 7/2 (which corre-

spond respectively to half filling of electrons or holes in the first excited Landau level),

is thought to have such non-Abelian elementary excitations [89]. Other possible phys-

ical realizations of non-Abelian statistics have also been proposed [107, 92, 59]. States

of this type have been suggested to be attractive for quantum computation [37].

In Ref. [2], in order to establish the nature of the statistics of the Laughlin quasi-

holes, a Berry’s phase calculation was performed that explicitly kept track of the

wave-function phase as one quasihole was transported around the other. Although


approximations were involved in this calculation, it nonetheless established quite con-

vincingly the fractional nature of the statistics. Unfortunately, it has not been possible

to generalize this calculation to explicitly investigate statistics of the MR quasiholes

[89]. Although there has been much study of the statistics of the MR quasiholes in

the framework of conformal field theories (CFT), it would be desirable to perform

a direct calculation analogous to that of Ref. [2]. The purpose of this chapter is to

provide such a calculation, albeit numerically. Furthermore, the approach developed

here is readily applicable to other FQH systems which are not easily accessible to

analytic investigations.

4.2 Monte Carlo Method

The evolution operator of a many-body system described by a Hamiltonian H(λ)

is in principle determined by the Schrodinger equation. In general, H(λ) itself can

change in time through dependence on some varying parameter λ(t). In such a case,

let us define ϕi(t) at a given time t to be an orthonormal basis for a particular

degenerate subspace, requiring that this basis is locally smooth as a function of t. If

λ is varied adiabatically (and so long as the subspace does not cross any other states),

then the time-evolution operator maps an orthonormal basis of the subspace at one

t onto an orthonormal basis at another t. A solution of the Schrodinger equation,

ψi(t) = Uij(t)ϕj(t), is simply given by [143]

(U−1U)ij = 〈ϕi |ϕj 〉 ≡ Aij(t) . (4.1)


Since the matrix A is anti-Hermitian, U(t) is guaranteed to be unitary if its initial

value U(0) is unitary. Note that if we vary λ so that the Hamiltonian returns to its

initial value at time t, i.e., H(λ(t)) = H(λ(0)), the corresponding transformation of

the degenerate subspace can be nontrivial, i.e., ψi(t) 6= ψi(0) [143].

We explicitly demonstrate that this is the case for the MR state with at least four

quasiholes. The analysis is done in spherical geometry [46]: N electrons are positioned

on a sphere of unit radius, with their coordinates given by (u1, v1), . . . , (uN , vN), using

the spinor notation (i.e., u = eiφ/2 cos θ/2 and v = e−iφ/2 sin θ/2 in terms of the usual

spherical coordinates). A monopole of charge 2S = 2N + n − 3 in units of the flux

quanta Φ0 = hc/e is placed in the center of the sphere, giving rise to 2n quasiholes

which are put at (u1, v1), . . . , (u2n, v2n). Using the gauge ~A = (Φ0S/2π)φ cot θ, the

MR wave function [89] is then given by

ψPf = PfΛ(a,b,...)(α,β,...)ij

∏i<j

(uivj − viuj)2 , (4.2)

where PfΛ(a,b,...)(α,β,...)ij is the Pfaffian [89] of the N ×N antisymmetric matrix 1

Λ(a,b,...)(α,β,...)ij = (uivj − viuj)

−1 ×

[(uiva − viua)(uj vα − vjuα) ×

(uivb − viub)(uj vβ − vjuβ) × · · · + (i↔ j)] .

Pfaffian wave functions (4.2) were first constructed in Ref. [89] as CFT conformal

blocks. This MR state is the exact ground state for a special three-body Hamiltonian

[41] and is also thought to pertain for realistic two-body interactions in the first ex-

1Computationally, the evaluation of the Pfaffian is not very expensive since (PfΛ)2 = DetΛ, thematrix determinant. The sign of the square root can be obtained by enforcing appropriate linearrelations for the overcomplete basis of Pfaffian wave functions [93].


cited Landau level [103]. The presence of quasiholes in the ground state is dictated

by the incommensuration of the flux with the electron number. Physically, the MR

state can be thought of as p-wave BCS pairing of composite fermions (CF’s) at zero

net field with quasiholes being the vortex excitations2 [89, 104]. Each quasihole has

charge e/4 and corresponds to half a quantum of flux (because of the paired order

parameter [89]). Eq. (4.2) describes a state with quasiholes created in two equal-size

groups: (ua, va), (ub, vb), . . . and (uα, vα), (uβ, vβ), . . .. Different quasihole groupings

realize a space with degeneracy 2n−1 [93, 106]. (Even though there are 2n!/2(n!)2

ways to arrange 2n quasiholes into 2 groups of n, the resulting wave functions are not

all linearly independent.) In the presence of finite-range interactions, the exact degen-

eracy may be split by an amount exponentially small in the large vortex separation

[104]. In this case, infinitely slow braiding will not exhibit non-Abelian statistics, al-

though for a very wide range of intermediate time scales, such statistics should apply

[104]. The effects of disorder on the statistics are only partially understood [104].

Consider an orthonormal basis ϕi, with i = 1, . . . , 2n−1, for the subspace with 2n

quasiholes, which is locally smooth when parameterized by the quasihole coordinates.

In order to determine the braiding statistics, we find the transformation ϕi → Uijϕj

under the evolution operator after two of the quasiholes are interchanged while the

others are held fixed. The unitary matrix Uij is obtained by first solving Eq. (4.1) and

then projecting the final basis onto the initial one. (Since we require ϕi to be only

locally smooth, the basis itself can nontrivially rotate after the quasiholes return to

their original positions). Eq. (4.1) is integrated numerically: The differential equation

2See, however, A. Wojs, Physical Review B, 63, 125312 (2001), where it was suggested, based onnumerical diagonalization, that the MR state be understood as a Laughlin bosonic state of paired(bare) electrons.


is discretized and the wave-function overlaps (the right-hand side of the equation)

are evaluated using the Metropolis Monte Carlo method. The computational errors

are easily evaluated by varying the number of operations. We aim the calculation

at addressing the following questions: (1) What is the Berry’s phase accumulated

upon quasihole interchange due to the enclosed magnetic flux and due to the relative

statistics? (2) What is the transformation matrix for the ground-state subspace

corresponding to the braiding operations? In the following, we will first describe the

numerical method, then present the results, and compare them to CFT predictions

[89, 93].

In order to integrate Eq. (4.1) numerically, the quasihole interchange is performed

in a finite number of steps. If U (l) is the value of the transformation matrix at the

lth step, then at the next step

U (l+1) = U (l)[1 + A(l)/2][1− A(l)/2]−1 , (4.3)

where A(l)ij = 〈ϕ(l+1)

i +ϕ(l)i |ϕ

(l+1)j −ϕ(l)

j 〉/2. Our choice of the finite-element scheme (4.3)

will become clear later. In practice, in general we do not know an orthonormal basis

for the MR states (4.2) in an analytic form, but we can numerically orthonormalize a

set of 2n−1 linearly-independent Pfaffian wave functions ψPfi. Let B(l)ij = (ψ

(l)Pfi, ψ

(l)Pfj)

denote the normalized overlaps of different states. [It is implied here and throughout

the chapter that (ψ(k)Pfi, ψ

(l)Pfj) ≡ 〈ψ

(k)Pfi|ψ

(l)Pfj〉/‖ψ

(k)Pfi‖‖ψ

(l)Pfj‖ is evaluated numerically.] We

then easily show that

A(l) = (V (l))†W (l)V (l+1)/2− H.c. , (4.4)

where W(l)ij = (ψ

(l)Pfi, ψ

(l+1)Pfj ) and V (l) is defined by (V (l))†B(l)V (l) = 1, constructing

an orthonormal basis ϕ(l)i = V

(l)ji ψ

(l)Pfj. We require V (l) to be locally smooth as a


function of the quasihole coordinates: The basis can continuously transform while

the quasiholes are moved, but, e.g., sudden sign flips are not allowed.

According to Eq. (4.4), A(l) is anti-Hermitian, so that the transformation U (l+1)

is guaranteed to be unitary if U (l) is unitary. This explains our choice (4.3) for

discretizing Eq. (4.1). Another feature preserved by our numerical scheme is that

making a step forward, ψ(l)Pfi → ψ

(l+1)Pfi , followed by a step backward, ψ

(l+1)Pfi → ψ

(l)Pfi,

results in a trivial transformation. We start at U (0) = 1 and find U (ns) after performing

ns + 1 steps for braiding of two quasiholes (ns is increased to convergence). Because

ψ(ns)Pfi is some nontrivial linear combination of ψ

(0)Pfi, we, finally, have to project the

transformation onto the initial basis: U (ns) → U (ns)OT , where O = (V (0))†ΩV (ns)

and Ωij = (ψ(0)Pfi, ψ

(ns)Pfj ). The resulting unitary transformation matrix U then gives

a representation of the braid group for quasihole interchanges. In the following, we

describe our numerical experiments.

4.3 Results for the Pfaffian Wave Function

The space describing 2n = 2 MR quasiholes is nondegenerate, so non-Abelian

statistics cannot occur. There is, nevertheless, a Berry’s phase accumulated from

wrapping these quasiholes around each other. Our calculation of this phase for the

MR state is analogous to the one performed in Ref. [2] for the Laughlin state, except

that our calculation is numerical and therefore requires no mean-field approximation.

Let us first briefly recall results for the Laughlin wave function at filling factor ν = 1/p.

In the disk geometry, the Berry’s phase χ corresponding to taking a single quasihole

around a loop is given by 2π for each enclosed electron, i.e., χ = 2π〈N〉, where 〈N〉 is


0 0.2 0.4 0.6 0.8 1cosξ

−0.1

0

0.1χ/

π

N=22

N=23

N=24

N=25

N=26

o

o

ξ

Figure 4.1: Berry’s phase χ for looping one MR quasihole around the equator withanother quasihole fixed at a zenith angle ξ. N = 4, 8, 16, 32, 64 is the number ofelectrons. The dashed line, χ/π = −1/8, shows a naive prediction. For cos ξ ≈0, the two quasiholes approach each other very closely and we see strong finite-size oscillations in the Berry’s phase. For larger N and cos ξ (i.e., larger quasiholeseparation in units of the magnetic length), χ appears to be converging toward zero.χ(− cos ξ) = −χ(cos ξ).

the expectation number of enclosed electrons [2]. Therefore, when another quasihole is

moved inside the loop, the phase χ drops by 2π/p which implies fractional statistics of

the quasiholes. In spherical geometry [46], the same result holds unless the south and

north poles (which have singularities in our choice of gauge) are located on different

sides of the loop. In the latter case, the Berry’s phase is given by χ = π〈Nin −Nout〉,

where Nin(out) is the number of electron inside (outside) the loop. If a single Laughlin

quasihole is then looped around the equator, its Berry’s phase vanishes, but if another

quasihole is placed above or below, the phase becomes χ = ±π/p. We check our

Monte Carlo method by reproducing these results numerically. The charge of the MR

(ν = 1/2) quasihole is e/4, so that by analogy with the Laughlin state one might


naively expect that the Berry’s phase for looping one quasihole around the equator

with another fixed above or below it is given by χ = ±π/8 [41] (with an extra factor of

1/2 due to MR quasiholes corresponding to only half of the flux quantum). In Fig. 4.1

we show numerical calculation of χ for a MR system having 2 quasiholes, one looped

around the equator and the other held fixed. If the two quasiholes approach each other

too closely, we see strong finite-size oscillations in the Berry’s phase. However, for

larger separation, χ appears to be converging towards zero, which was first predicted

in Ref. [105] and can be well understood using the plasma analogy [44].

Even though the relative statistics of two MR quasiholes are trivial, they do pick

up a phase due to their wrapping around the electrons, analogous to what occurs

in the Laughlin case. Fig. 4.2 shows that as the size of the system increases, the

phase accumulated by interchanging two quasiholes (filled symbols) or braiding one

around the other (open symbols) can be well approximated by assuming the wave

function rotates by π for each enclosed electron (compare to 2π for the Laughlin

state), when the poles are not separated by the loop (and the effect of the pole

singularities is analogous to that in the Laughlin state). Even for systems consisting

of only 4 electrons, this approximation stays quite good if we correct the average

electron density for the charge pushed out by one localized quasihole (see dashed

lines in Fig. 4.2). This method of correcting the average density also works for the

Laughlin state on the sphere.

We now turn to 2n = 4 MR quasiholes, which is the simplest case when statistics

can be non-Abelian (the ground state has degeneracy 2). While the above results

for 2 quasiholes are anticipated by the plasma analogy [44], one may need deeper


0 0.2 0.4 0.6 0.8 1cosξ

−15

−10

−5

0

5

10

15

χ/π

oo

o

o

ξ

ξ

Figure 4.2: For χ > 0 (χ < 0) filled symbols show the phase accumulated by inter-changing two quasiholes around a circle with opening angle ξ centered on the equator(north pole), for various N as in Fig. 4.1. The straight dashed lines in the upperhalf are 0.5(N + 1/4)(1 − cos ξ), corresponding to the expectation of the number ofelectrons enclosed by the loop. The +1/4 accounts for the charge pushed out by oneof the quasiholes. For χ < 0, the dashed lines are −0.5(N + 1/4) cos ξ, i.e., one halfof the number of electrons inside minus one half the number outside the loop. Opensymbols, corresponding to a similar calculation with one quasihole moving and theother fixed at the center of the circle, almost overlay the filled symbols, confirmingthe trivial relative statistics.

CFT [89, 93] arguments in order to understand the following findings. In the cal-

culation, we first fix all quasiholes on the equator and then interchange an adjacent

pair of them around a circle with different opening angles ξ centered on the equator.


0 0.2 0.4 0.6 0.8 1cosξ

0

5

10

15

χ/π

N=22

N=23

N=24

N=25

Figure 4.3: Same as the upper half of Fig. 4.2, but now with four quasiholes present,two of which are fixed on the equator, at φ = ±3π/4, and two interchanged, withinitial and final positions at φ = ±ξ on the equator. The straight dashed linesare 0.5(N + 3/4) cos ξ − 1/4. Here, +3/4 accounts for the average electron-densitycorrection for the charge localized at 2n − 1 quasiholes. The additional phase offsetof −1/4 reflects the Abelian part of the braiding statistics, in agreement with thepredictions of Refs. [89, 93].

Parameterizing a unitary matrix U by

U = eiχ

eiη cos β/2 ie−iε/2 sin β/2

ieiε/2 sin β/2 e−iη cos β/2

, (4.5)

we plot in Figs. 4.3 and 4.4 the results (in a convenient basis) for the transformation

U1 corresponding to the braiding operation on one of the quasihole pair. Due to

the rotational symmetry around the vertical axis, knowing U1 we can deduce other

transformations U2, U3, and U4 (for interchanges of pairs ordered along the equator)

by rotating and projecting the initial basis and correspondingly transforming U1.

It is then easy to show that U1 = U3 and U2 = U4 due to the form (4.2) of the

wave function. Furthermore, we find numerically that U2 ≈ F †U1F , where F =

(σz −σx)/√

2, σ’s being the usual Pauli matrices. This approximation is good within


0 0.2 0.4 0.6 0.8 1cosξ

0

0.1

0.2

0.3

0.4

angl

e/π

η

β

Figure 4.4: Parameters η and β defining transformation matrix (4.5) for the sameoperations as χ shown in Fig. 4.3. The dashed line shows 1/4, an approximation usedfor η in the text. Similarly β can be approximated as zero [so that ε in Eq. (4.5) isnot defined]. These approximations become better with larger system size and forintermediate cos ξ when the quasiholes remain further apart. The symbol conventionis the same as in Fig. 4.3. Lines interpolate Monte Carlo results.

a few percent for smaller systems and is even better for larger ones.

According to Fig. 4.4, we see that apart from the Abelian phase χ, U1 can be

approximated by U1 ≈ diag(1+ i, 1− i)/√

2, with the disagreement becoming smaller

for larger systems. Using F , we can then construct all other matrices Ui. After

performing the above approximations, we find that the unitary transformations cor-

responding to the braid operators realize the right-handed spinor representation of

SO(2n)×U(1) (restricted to π/2 rotations around the axes) as predicted in Ref. [93]

using CFT. In addition to the usual relations required of a representation of the braid

group on the plane, on the sphere the generators must obey an additional relation.

For the case of 2n = 4, for example, we expect to have U1U2U3U3U2U1 = 1. One can

easily show that (for general n) the relevant representation of the braid group pre-

dicted in Ref. [93] satisfies this additional relationship up to an Abelian phase. (The


failure of the Abelian phase to satisfy this law is related to the gauge singularities,

and will be discussed elsewhere.)

4.4 Summary

We formulated a numerical method to study braiding statistics of FQH excitations

and applied it to perform the first direct calculation of the non-Abelian statistics in the

MR state. Our findings confirm results previously drawn within the CFT framework.

Appendix A

Adiabatic Spin Pumping

(Appendix to Chapter 2)

Here we present a detailed discussion of spin pumping into normal-metal layers

by a precessing magnetization direction m of an adjacent ferromagnet. A schematic

of the model is displayed in Fig. 2.3. The ferromagnetic layer F is a a spin-dependent

scatterer that governs electron transport between two [left (L) and right (R)] normal-

metal reservoirs.

The 2 × 2 operator Il for the charge and spin current in the lth lead (l = L,R)

can be expressed in terms of operators aαm,l(E) [bαm,l(E)] that annihilate a spin-α

electron with energy E leaving [entering] the lth lead through the mth channel:

Iαβl (t) =

e

h

∑m

∫dEdE ′ei(E−E′)t/~

×[a†βm,l(E)aαm,l(E

′)− b†βm,l(E)bαm,l(E′)]. (A.1)

When the scattering matrix sαβmn,ll′(t) of the ferromagnetic layer varies slowly on the

166

Appendix A: Adiabatic Spin Pumping (Appendix to Chapter 2) 167

time scales of electronic relaxation in the system, an adiabatic approximation may

be used. The annihilation operators for particles entering the reservoirs are then

related to the operators of the outgoing states by the instantaneous value of the scat-

tering matrix: bαm,l(E) = sαβmn,ll′(t)aβn,l′(E). In terms of aαm,l only, we can evaluate

the expectation value⟨Iαβl (t)

⟩of the current operator using 〈a†αm,l(E)aβn,l′(E

′)〉 =

fl(E)δαβδmnδll′δ(E−E ′), where fl(E) is the (isotropic) distribution function in the lth

reservoir. When the scattering matrix depends on a single time-dependent parameter

X(t), then the Fourier transform of the current expectation value Il(ω) =∫dteiωtIl(t)

can be written as

Il(ω) = gX,l(ω)X(ω) (A.2)

in terms of a frequency ω- and X-dependent parameter gX,l [25]:

gX,l(ω) = −eω4π

∑l′

∫dE

(−∂fl′(E)

∂E

)×

∑mn

(∂smn,ll′(E)

∂Xs†mn,ll′(E)− H.c.

). (A.3)

Equation (A.2) is the first-order (in frequency) correction to the dc Landauer-Buttiker

formula [24]. At equilibrium fR(E) = fL(E), Eq. (A.2) is the lowest-order nonvanish-

ing contribution to the current. Furthermore, at sufficiently low temperatures, we can

approximate −∂fl(E)/∂E by a δ-function centered at Fermi energy. The expectation

value of the 2 × 2 particle-number operator Ql(ω) [defined by Il(t) = dQl(t)/dt in

time or by Il(ω) = −iωQl(ω) in frequency domain] for the lth reservoir is then given

by

Ql(ω) =

(e

4πi

∑mnl′

∂smn,ll′

∂Xs†mn,ll′ + H.c.

)X(ω) , (A.4)

where the scattering matrices are evaluated at the Fermi energy. Because the prefactor


on the right-hand side of Eq. (A.4) does not depend on frequency ω, the equation is

also valid in time domain. The change in particle number δQl(t) is proportional to

the modulation δX(t) of parameter X and the 2×2 matrix current (directed into the

normal-metal leads) reads

Il(t) = e∂nl

∂X

dX(t)

dt, (A.5)

where the “matrix emissivity” into lead l is

∂nl

∂X=

1

4πi

∑mnl′

∂smn,ll′

∂Xs†mn,ll′ + H.c. . (A.6)

If the spin-flip scattering in the ferromagnetic layer is disregarded, the scattering

matrix s can be written in terms of the spin-up and spin-down scattering coefficients

s↑(↓) using the projection matrices u↑ =(1 + σ ·m

)/2 and u↓ =

(1− σ ·m

)/2

[19, 20]:

smn,ll′ = s↑mn,ll′u↑ + s↓mn,ll′u

↓ . (A.7)

The spin current pumped by the magnetization precession is obtained by identifying

X(t) = ϕ(t), where ϕ is the azimuthal angle of the magnetization direction in the

plane perpendicular to the precession axis. For simplicity, we assume that the mag-

netization rotates around the y axis: m = (sinϕ, 0, cosϕ). Using Eq. (A.7), it is then

easy to calculate the emissivity (A.6) for this process:

∂nl

∂ϕ= − 1

4π[Arσy + Ai(σx cosϕ− σz sinϕ)] , (A.8)

where Ar(Ai) = Re(Im)[g↑↓ − t↑↓], as explained in Sec. 2.1.2. Expanding the 2 × 2

current into isotropic and traceless components,

I =1

2Ic −

e

~σ · Is , (A.9)


we identify the charge current Ic and spin current Is. Comparing Eqs. (A.5), (A.8),

and (A.9), we find that the charge current vanishes, Ic = 0, and the spin current

Is = (Ai cosϕ,Ar,−Ai sinϕ)~4π

dϕ

dt(A.10)

can be rewritten as Eq. (2.7). Because the spin current transforms as a vector, it is

straightforward to show that Eq. (2.7) is also valid in the case of the general motion

of the magnetization direction.

Even though the mathematics of our scattering approach to adiabatic spin pump-

ing is entirely analogous to the charge-pumping theory developed in Ref. [22], there

are some striking differences in the physics. In the case of a spin-independent scat-

terer as in Ref. [22], the average charge-pumping current has the same direction in the

two leads, by charge conservation: the charge entering the scattering region through

either lead must leave it within a period of the external-gate variations. Whereas the

particle number of the two reservoirs must (on average) be conserved also here, the

total conduction-electron spin angular momentum is not conserved. In fact, as we

explained in Ref. [130] for a symmetric system shown in Fig. 2.1, a precessing ferro-

magnet loses angular momentum by polarizing adjacent nonmagnetic conductors. In

this respect, the phenomenon looks more similar to a spin “well” or “fountain”: An

excited ferromagnet ejects spins in all directions into adjacent conductors by losing

its own angular momentum, rather than transfers (“pumps”) spins from one lead to

the other. The angular momentum has to be provided, of course, by the applied

magnetic field.

Appendix B

Appendices to Chapter 3

B.1 Independent-Mode Approximation

In our analysis we treat different 1D bands in the wires as independent and disre-

gard interband interactions. While this is a convenient approximation for theoretical

investigations that has been commonly assumed in the context in previous works

[4, 132, 26, 152], it needs to be further justified. Tunneling into multimode 1D wires

was considered in Ref. [78]. It was shown that low-energy tunneling into the edge of a

semi-infinite wire with N bands is governed by exponents αi such that the differential

conductance is given by G ∝∑N

i=1 |ti|2V αi . In the independent-mode approximation

with interactions described by Hamiltonian (3.6) for each mode, these exponents are

given by Eq. (3.50) with the parameter g describing interactions in each mode. On

the other hand, in a more realistic picture one deals with an interaction Hamiltonian

Hint =V0

2

N∑i,j=1

∫ ∞

0

dxρi(x)ρj(x) (B.1)

170

Appendix B: Appendices to Chapter 3 171

which takes into account the interband coupling. Here, V0 is the zero-momentum

Fourier component of the interaction potential V (x) = V0δ(x) and ρi is the electron

density in the ith band. The exact form of the potential is not important as we are

only interested in the long–wave-length quantum fluctuations [78].

The exponents are given by [78] αi =∑N

l=1 γ2il(sl/vi − 1), where vi is the Fermi

velocity of the noninteracting 1D electron gas at the density of the ith mode, sl is

the velocity of the lth soundlike excitation in the presence of the potential V (x),

and γil characterizes coupling between the ith and lth noninteracting modes after the

interaction potential V (x) is switched on. In the case of a single transverse mode with

spin degeneracy, N = 2, γ2il = 1/2, and s1 = vF

√1 + 2V0/(π~vF), s2 = vF are the

charge- and spin-excitation velocities, respectively. For a general N , the velocities sl

are given by roots of the equation

N∑i=1

vi

s2l − v2

i

=π~V0

(B.2)

and the coefficients γil are given by

γ2il =

vi

(s2l − v2

i )2

[N∑

j=1

vj

(s2l − v2

j )2

]−1

. (B.3)

In our system [4], the Fermi velocities of the highest occupied bands are very different

(e.g., the highest transverse mode has twice the velocity of the next lower-lying mode).

Furthermore, since the interaction V0 . max(~vi) is not too large, the correction to

the exponents αi due to the interband coupling is expected to be relatively small.

One can accommodate for this correction by slightly renormalizing the interaction

constants g, viewing it as a mutual interband screening [78].

Also, it is safe to disregard intermode transitions as they are determined by the

Fourier components of the interaction with a large wave vector k ∼ kF, which are


small for a smooth long-range potential [78]. The weak backscattering within each

spin-degenerate mode can be further renormalized downward at low energies in the

physical case of repulsive interactions [122].

B.2 Direct Tunneling from the 2DEG

In order to describe the V and B dependence of the conductance for direct 2DEG–

lower wire tunneling, we approximate Green’s function of the top quantum well by

the edge Green’s function of a 2D electron gas occupying a half plane y > 0 with

x extended from −∞ to ∞. We assume the potential is V (x, y) = 0 for y > 0 and

V (x, y) = ∞ for y < 0. Therefore, we find

iG>(x, y, t;x′, y′, 0) =1

π2

∫ ∞

−∞dpeip(x−x′)

∫ ∞

0

dk

× sin(ky) sin(ky′)Θ(ε)e−iεt/~ , (B.4)

where ε = ~2(p2 + k2 − k2F)/(2m) is the energy and k2

F is the Fermi wave vector of

the 2DEG. Θ(ε) is the Heaviside step function. When we calculate the tunneling

current, y and y′ run from 0 to ξ, the width of the tunnel junction (i.e., the extent

of the 1D mode of the lower wire in the direction perpendicular to the cleaved edge).

We set (y, y′) → ξ/2 and approximate sin(kξ/2) ≈ kξ/2 assuming kF < 1/ξ. In the

frequency domain, Green’s function G>(z, ω) =∫∞−∞ dteiωtG>(z, t), with z = x − x′,

then becomes

iG>(z, ω) =ξ2

2π~

∫ ∞

−∞dpeipz

∫ ∞

0

dkk2δ(ε− ω)Θ(ω) . (B.5)


In the limit of small positive frequencies it reduces to

iG>(z, ω → 0+) = mξ2

2π~

∫ kF

−kF

dpeipz√k2

F − p2 = m(ξkF)2J1(kFz)

2~kFz, (B.6)

where J1 is the first-order Bessel function of the first kind. In particular, since J1(x) ∝

x when x → 0, the density of states is finite at the Fermi energy and G>(t) ∝ 1/t

as t → ∞. Furthermore, from the low-energy form of the 2DEG Green’s function

[Eq. (B.6)] it follows that the relevant range of z in integral (3.9) is 1/kF rather than

1/max(q,Γ/vF) as in the case of the 1D-1D tunneling.

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