sorin melinte- quantum hall effect skyrmions: nuclear magnetic resonance and heat capacity...

UNIVERSITE CATHOLIQUE DE LOUVAIN

Faculte des Sciences Appliquees

Unite de Physico-Chimie et de Physique des Materiaux

QUANTUM HALL EFFECT SKYRMIONS:

NUCLEAR MAGNETIC RESONANCE

AND

HEAT CAPACITY EXPERIMENTS

Dissertation presentee en vue de l’obtention du grade de

‘Docteur en Sciences Appliquees’par

Sorin MELINTE

Promoteur:

Prof. Vincent BAYOT

Septembre 2001

UNIVERSITE CATHOLIQUE DE LOUVAIN

Faculte des Sciences Appliquees

Unite de Physico-Chimie et de Physique des Materiaux

QUANTUM HALL EFFECT SKYRMIONS:

NUCLEAR MAGNETIC RESONANCE

AND

HEAT CAPACITY EXPERIMENTS

Dissertation presentee en vue de l’obtention du grade de

‘Docteur en Sciences Appliquees’par

Sorin Melinte

Membres du jury:

Jury: Prof. Vincent Bayot, promoteurDr. Mladen HorvaticProf. Jean-Paul IssiProf. Jean-Pierre MichenaudProf. Mansour ShayeganProf. Piotr Sobieski, president

Septembre 2001

Închinare

mamei mele, Ilinca

în memoria lui Nistor, tatal meu,surghiunit prin Siberia comunista

lui Carmen si Eusebiu

parintelui meu duhovnicesc, Nicodim

Oration

Il n‘y a en verite qu’une seule gehenne pour ceux qui ont malvecu: l’anonymat, l’obscurite et l’effacement definitif, qui duLethe, la riviere d’Oubli, les conduit au fleuve sans sourires, puisles emporte au large, a l’ocean sans limite et sans fond qui char-rie tout ce qui n’a servi a rien, tout ce qui n’a rien fait, tout cequi est reste sans gloire et inconnu.

Plutarque

V. Bayot

C. Berthier

J.-M. Beuken

J.-P. Colinge

S.M. Girvin

E. Grivei

M. Horvatic

J.-P. Issi

L.-P. Levy

J.-P. Michenaud

I. Pop

M. Shayegan

Abstract

Of all the new physics generated by the highly perfect two-dimensional (2D)electron systems, the quantum Hall effects (QHEs), integer and fractional,with their richness and complexity are perhaps the most active and excit-ing. The fractional QHE is the manifestation of a new state of electronmatter - a peculiar, uniform density incompressible liquid phase, observedat low temperature (T ) and in the presence of a strong magnetic field (B)perpendicular to the 2D electron layers. Part of the intellectual fascinationwith the QHE phases stems from the challenge of finding new concepts todescribe their properties.

The energy levels available to an electron confined to a 2D layer in aperpendicular magnetic field are known as Landau levels. Each Landaulevel can accommodate many electrons; dividing the number of electronsper unit sample area by the degeneracy of a Landau level defines the fillingfactor ν. At a precise value of the magnetic field (ν = 1), the 2D electronscondense into a ferromagnetic QHE ground state: the electronic system isan itinerant ferromagnet with a quantized Hall resistivity. The low-energyelectron-spin dynamics of this QHE ferromagnet is extremely unusual be-cause of the subtle interplay between Coulomb interaction among electronsand Zeeman coupling of electronic spins to the external B. The elemen-tary excitations are spin-textured objects, called Skyrmions, which displaycomplex equilibrium behavior reflecting liquid, crystalline and glassy phases.Qualitatively novel physics arises, moreover, because these topological exci-tations carry electrical charge and possess an effective spin larger than thatof a single electron.

Thermodynamic measurements on 2D electron systems in the fractionalQHE regime were long considered futile because the effects were thought tobe too small to observe. This thesis aims to revise this pessimistic outlookby reporting measurements of heat capacity and standard nuclear magneticresonance down to very low T . Nuclear magnetic resonance (NMR) andcalorimetric measurements in the mK temperature range are a tour de force

v

of experimental technique and confirm the existence of QHE Skyrmions. Inmultiple-quantum-well GaAs/AlxGa1−xAs heterostructures, we found a dra-matic enhancement of the nuclear spin-lattice relaxation rate around ν = 1,that cannot be explained at present without express consideration of Skyrmi-ons being accommodated in the ferromagnetic QHE ground state. The elec-tron spin polarization peak (detected in NMR Knight shift measurements)and the observation of nuclear heat capacity of GaAs quantum wells are con-sequences of the strong, Skyrmion-mediated hyperfine coupling between thenuclear spin system and 2D electrons in the vicinity of ν = 1. The B- andT -dependencies of the nuclear-spin lattice relaxation rate around ν = 1 seemto be influenced by the inhomogeneity of the 2D electron system. The quan-titative analysis of the nuclear spin-lattice relaxation rate measurements ismore difficult as the interplay between the electron-electron interaction anddisorder is not well understood.

Compelling evidence for QHE Skyrmions comes from tilted magneticfield studies. Introducing an in-plane magnetic field causes a spin phasetransition in the electronic system: Skyrmion-like excitations transform tosingle spin-flip excitations above a critical Zeeman energy. We report ontilted magnetic field heat capacity and nuclear magnetic resonance measure-ments that may elucidate the discrepancies in the literature concerning therange of Zeeman energies over which Skyrmions are the stable excitationsof the ν = 1 ground state. We found a critical Zeeman energy of ∼ 0.04 (inunits of Coulomb energy), consistent with Hartree-Fock calculations whichtake into account the finite thickness of the electron layers.

Remarkably, the heat capacity (measured as a function of temperaturenear ν = 1) displays a sharp peak at very low T , suggestive of a Skyrmionsolid-to-liquid phase transition. We performed quasi-adiabatic thermal ex-periments revealing that the mechanism responsible for the peak in heatcapacity vs temperature is a dramatic enhancement of nuclear spin diffusionacross the quantum well-barrier interface.

Finally, QHE excitation gap measurements allow microscopic propertiesof Skyrmions (such as the number of encompassed reversed spins) to beestablished and explored. Our results reveal that the spin of a thermallyactivated Skyrmion–anti-Skyrmion pair at ν = 1 is ∼ 9, which corroborateswith both Skyrmion spin measured in conventional single-layer 2D electronsystems and theoretical predictions.

vi

Preface

În sudoarea fetii tale îti vei manca painea ta, pana te vei întoarceîn pamantul din care esti luat: caci pamant esti si în pamant tevei întoarce.

Întaia carte a lui Moise

To promise at the beginning a nice story is most courtly and fashion-able. This thesis explores the physics of two-dimensional electron systemsexhibiting the quantum Hall effects (integer and fractional) - an extremelyrich set of phenomena with truly fundamental implications. The quantumHall effect (QHE) is a large subject with undefined frontiers. This bookintroduces the readers to basic experimental and theoretical aspects of thequantum Hall effect. It cannot and do not survey all important and excitingtopics in this field. This thesis is intended to cover the subject of QHESkyrmions. It does so by providing a close examination of two experimentsof vital importance as, I believe, they prevented Skyrmions to fade awayand furnished ample evidence for their intriguing behavior. It also attemptsto show that the growth of high-quality, multiple two-dimensional electronsystems is one of the main factors that have made possible the rise of novellow-temperature spectroscopy methods, which have tried, and still try, tooffer deep insights into the QHE physics. The data included in this thesishas been collected on two GaAs/AlxGa1−xAs multiple-quantum-well het-erostructures containing two-dimensional electron layers; the stages in theexperimental procedure and data analysis were largely the same for bothsamples studied.

Chapter 1 sets the scene. Here I present the samples and basic magneto-transport experiments. The treatment that I have given to the integralQHE is necessarily perfunctory; for neither the oscillating density of statesnor the presence of disorder can be detached from the explanation of the

vii

phenomenon. I hope that I have not entirely omitted anything that is es-sential to its comprehension. Traditionally, the discussion of the fractionalQHE follows after the discussion of the integer QHE. The perspectives onthe fractional QHE are various. Many of them were inspired by the art-ful Laughlin’s argument. An overview of the fractional QHE is given fromthis point of view. The book is cursory in the treatment of few key topics:(1) QHE excitation gaps, (2) finite thickness corrections and Landau levelmixing, (3) mixed-spin QHE ground states and spin-reversed quasiparticles,and (4) QHEs in tilted magnetic fields.

I’m asking in Chapter 2 what is meant by QHE ferromagnets and whatQHE Skyrmions are. Although much of the detail of the physics has beenstripped away to get to the essentials, I tried hard to make the answer tothese questions clear and the presentation responsible. This chapter repre-sents my attempt to understand the seminal ideas and first steps in buildingup the theory of QHE ferromagnets. It does not try to replace the existingbooks and review articles on this subject, filled with worthy cogitations andwisdom. It rather tries to present a thorough account of the theoreticalformalism necessary for understanding the following Experimental Work.

Chapter 3 focuses on heat capacity experiments. Heat capacity mea-surements of two-dimensional electron systems are among the most chal-lenging experiments because of the very small electron contribution. Themethod of ac calorimetry is commonly applied to systems with small heatcapacities which are difficult to isolate thermally and exhibit small signal-to-background variations. Thereby, it is perfectly suited to the particularcase of two-dimensional electron systems. This technique has been origi-nally developed by Dr. Joseph Kung Wang for heat capacity measurementsof two-dimensional electron systems in the integer QHE regime and it wasintroduced to me by Prof. Vincent Bayot and Dr. Eusebiu Grivei. The dis-covery of the ”giant” heat capacity of QHE ferromagnets sprang, ironically,from our experimental study of the heat capacity of multiple-quantum-wellGaAs/AlxGa1−xAs heterostructures in the integer QHE regime at dilutionrefrigerator temperatures. In general, heat capacity is a bulk property withcontribution from all components of a given thermodynamic system. Experi-mentally, this feature is considered a disadvantage in the study of a samplecontaining the lattice, the addenda as well as the two-dimensional electronlayers of interest. However, one may elicit useful knowledge about the elec-tronic system, even though the nuclear spin system gives the dominant con-tribution to the measured heat capacity. Prof. Vincent Bayot, Dr. EusebiuGrivei, and myself have adapted the relaxation-time heat capacity techniquefor specific application when the nuclear spin heat capacity of GaAs quan-

viii

tum wells is orders of magnitude larger than the heat capacity of the two-dimensional electron system. The observation of the nuclear heat capacity ofGaAs quantum wells around Landau level filling factor ν = 1 necessarily im-plies a strong coupling between the nuclear spins and the lattice; we presenthere compelling evidence that this coupling is provided by QHE Skyrmi-ons. Heat capacity experiments are accompanied by thermal measurementsperformed on the nuclear spin system of a GaAs/Al0.3Ga0.7As multiple-quantum-well heterostructure. To our knowledge, these experiments are theonly ones showing signs of a Skyrmion solid-to-liquid phase transition.

In Chapter 4, standard nuclear magnetic resonance (NMR) spectroscopypartake in the understanding of QHE Skyrmions. Convincing experimentalevidence for the existence of Skyrmions is locked into optically pumped NMRexperiments which probe the two-dimensional electron spin polarization anddynamics in the QHE regime. For the standard NMR broadened the horizonsopened by the optically pumped NMR technique, the results presented inthis thesis are weighted against the fundamental work of Prof. Sean Barrettand collaborators (summarized in Appendix B). The organization and styleI adopted for this chapter are those of a research article: the Introduction,Theory, Experimental methods, Results and discussion, and the Conclusionsections follow in logical order. I have tried to present enough material thatany reader will be able to achieve some degree of comprehension on the ap-plication of standard NMR to the study of two-dimensional electron systemsconfined to GaAs quantum wells. At the time of writing the Conclusion,it is clear that several aspects of NMR experiments described in this thesisneed to be understood better. Future work, as I remark in the Epilogue,is desirable. The NMR study presented here was performed at GrenobleHigh Magnetic Field Laboratory. I thank Prof. Vincent Bayot, Dr. ClaudeBerthier, Prof. Laurent-Patrick Levy, and Prof. Mansour Shayegan for initi-ating this project. Throughout the entire process of ferreting out the NMRfundamentals, I received helpful insights, invaluable friendship, and encour-agement from Dr. Mladen Horvatic.

Let me include a few words on the process of writing this book. I wouldnever have succeeded in attaining the modest level of scientific writing with-out the extensive and patient help received from Prof. Mansour Shayegan,Dr. Mladen Horvatic, and Prof. Vincent Bayot. The debts that I owe tomany scholars are enormous. In presenting the inception of QHE Skyr-mions, I have been largely dependent on the work of Prof. Steven Girvin.Chapter 3, on heat capacity, is an extensively re-written version of threePhysical Review Letters articles, all of them co-authored with Prof. VincentBayot, Dr. Eusebiu Grivei, and Prof. Mansour Shayegan. Parts of the ma-

ix

terial presented in the Chapter 4, on nuclear magnetic resonance, have beenpublished in a recent issue of Physical Review B. Writing this book was acollaborative effort. The friends who have given me helpful criticism andadvice are too numerous to be recorded by name. Perhaps the best advice,deserving all praise, was the following: The average Ph. D. thesis is nothingbut a transference of bones from one graveyard to another.

x

Contents

Închinare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiOration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

1 The Quantum Hall Effect 11.1 Idealized two-dimensional electron systems . . . . . . . . . . . 1

1.1.1 The free electron . . . . . . . . . . . . . . . . . . . . . 31.1.2 Idealized quantum wells . . . . . . . . . . . . . . . . . 31.1.3 Landau levels . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 The idealized two-dimensional density of states . . . . 5

1.2 Realistic two-dimensional electron systems . . . . . . . . . . . 71.2.1 Sample fabrication . . . . . . . . . . . . . . . . . . . . 71.2.2 Electrons in GaAs quantum wells . . . . . . . . . . . . 121.2.3 The Landau level diagram . . . . . . . . . . . . . . . . 15

1.3 Magnetotransport: Experimental details . . . . . . . . . . . . 161.4 Fundamental aspects of the QHE . . . . . . . . . . . . . . . . 20

1.4.1 Measuring the QHE . . . . . . . . . . . . . . . . . . . 201.4.2 Integral QHE . . . . . . . . . . . . . . . . . . . . . . . 251.4.3 Fractional QHE . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Interlude: Ground state theory at ν = 1 . . . . . . . . . . . . 331.6 Supplemental aspects of the QHE . . . . . . . . . . . . . . . . 35

1.6.1 Finite thickness corrections . . . . . . . . . . . . . . . 351.6.2 Landau level mixing . . . . . . . . . . . . . . . . . . . 361.6.3 Spin and the fractional QHE . . . . . . . . . . . . . . 371.6.4 QHEs in tilted magnetic fields . . . . . . . . . . . . . 39

xi

1.7 Magnetotransport: Results and discussion . . . . . . . . . . . 431.7.1 Measurements in tilted magnetic fields . . . . . . . . . 431.7.2 QHE excitation gap measurements . . . . . . . . . . . 43

1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2 Quantum Hall Effect Skyrmions 512.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.1 Symmetry and ferromagnetism . . . . . . . . . . . . . 522.1.2 Dimensionality and spin waves . . . . . . . . . . . . . 542.1.3 The abomination of topology . . . . . . . . . . . . . . 57

2.2 QHE ferromagnetism at ν = 1 . . . . . . . . . . . . . . . . . . 652.2.1 Spin-wave excitations . . . . . . . . . . . . . . . . . . 652.2.2 Topological excitations (QHE Skyrmions) . . . . . . . 682.2.3 Further aspects of QHE Skyrmions . . . . . . . . . . . 722.2.4 Electron spin polarization at ν = 1 . . . . . . . . . . . 76

2.3 QHE ferromagnetism near ν = 1 . . . . . . . . . . . . . . . . 792.3.1 Electron spin polarization near ν = 1 . . . . . . . . . . 792.3.2 Quantum treatment of a Skyrmion . . . . . . . . . . . 812.3.3 Skyrmion lattices at zero temperature . . . . . . . . . 832.3.4 Skyrmion lattices at finite temperatures . . . . . . . . 862.3.5 Skyrmions and the nuclear spin-lattice relaxation rate 86

2.4 Interpretation of the ν = 1 QHE excitation gap . . . . . . . . 892.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3 Heat Capacity Evidence for Skyrmions 933.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.1 Small-sample calorimetry . . . . . . . . . . . . . . . . 933.1.2 Thermal equilibrium in multiple-QW samples . . . . . 963.1.3 Samples for heat capacity experiments . . . . . . . . . 983.1.4 Heat capacity: Experimental details . . . . . . . . . . 101

3.2 Calorimetry in the integer QHE regime . . . . . . . . . . . . 1063.2.1 The steady-state, ac-temperature calorimetric method 1073.2.2 Results and discussion . . . . . . . . . . . . . . . . . . 109

3.3 The ”holy” nuclear heat capacity . . . . . . . . . . . . . . . . 1143.3.1 The time-constant calorimetric method . . . . . . . . 1143.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 1193.3.4 Disappearance of Skyrmions . . . . . . . . . . . . . . . 1223.3.5 Ramifications . . . . . . . . . . . . . . . . . . . . . . . 124

3.4 Skyrmion lattices . . . . . . . . . . . . . . . . . . . . . . . . . 129

xii

3.4.1 Heat capacity measurements at very low temperatures 1293.4.2 The variable nuclear thermal coupling model . . . . . 1323.4.3 Results of quasi-adiabatic thermal experiments . . . . 1343.4.4 The crystal of Skyrmions and the nuclear spin diffusion136

3.5 The nuclear spin-lattice relaxation rate . . . . . . . . . . . . . 1403.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4 Skyrmions Probed by NMR 1454.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.2 Principles of NMR . . . . . . . . . . . . . . . . . . . . . . . . 149

4.2.1 General aspects of NMR . . . . . . . . . . . . . . . . . 1494.2.2 Interactions of nuclei with electrons. NMR in metals. 1534.2.3 NMR in QHE systems . . . . . . . . . . . . . . . . . . 157

4.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . 1614.3.1 Experimental setup and the sample . . . . . . . . . . 1614.3.2 Experimental technique . . . . . . . . . . . . . . . . . 163

4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 1744.4.1 Untilted magnetic field data . . . . . . . . . . . . . . . 1744.4.2 Tilted-magnetic field electron spin polarization . . . . 185

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Epilogue 189

A [Appendix] The Nuclear Spin System in GaAs QWsand the Schottky Heat Capacity 193

B [Appendix] Optically Pumped NMR Observations 195

Bibliography 199

xiii

List of Symbols

Here are the main symbols and acronyms used in the present thesis. Consis-tency with traditional usage has been maintained as far as possible. Vectorsare printed in bold-faced fonts.

2DES two-dimensional electron systemA areaA, A magnetic potentialA, A magnetic vector potentialAeff effective hyperfine coupling constanta lattice constantac alternating currentB magnetic field (induction)B⊥ perpendicular magnetic field componentB magnetic flux densityC heat capacityDn nuclear spin diffusion constantD(EF ) density of states at the Fermi levelD 1D: one-, 2D: two-, 3D: three-dimensionalDOS density of statesE energyE0 ground state energyEQW

0 bottom energy of the ground subbandEQW

1 bottom energy of the first excited subbandE∗ cyclotron energyEC Coulomb energyEF Fermi energyExc exchange energyEZ Zeeman energye absolute value of electron’s chargee∗ quasiparticle charge

xv

ez unit vector for the z-directionf frequencyFWHM full width at the half maximumg∗ electron’s effective g-factor in bulk GaAsge electron’s g-factor in vacuumg ratio of the Zeeman energy to the Coulomb energyH Hamiltonian operatorh Planck constantI electric current (rms value of the alternating current i)I magnitude of the nuclear spinj electric current density vectorK thermal conductanceKS Knight shiftkB Boltzmann constantk wavevectorkF Fermi wavevectorlB magnetic lengthL LagrangianLL Landau levelm∗ electron’s effective mass in bulk GaAsme electron’s mass in vacuumM magnetization vectorm magnetic momentMBE molecular beam epitaxyn areal electron densityN number of particlesNMR nuclear magnetic resonanceNSS nuclear spin systemOPNMR optically pumped nuclear magnetic resonanceP electric powerP electron spin polarizationq wavevectorQHE quantum Hall effectQW quantum wellR electrical resistanceR0 zero magnetic field resistanceRH Hall resistanceRL longitudinal resistance

xvi

rs ratio of the Coulomb energy to the cyclotron energyr space vectorRj jth site on the latticeR radiusRF radio frequencyrms root-mean-squares spin quantum numberSdH Shubnikov-de HaasT (lattice) temperatureT−1

1 nuclear spin-lattice relaxation rateT−1

2 nuclear spin-spin relaxation rateTn nuclear spin temperaturet time variableV electric potential differenceVd strength of the disorder potentialVe−e potential energy for electron-electron interactionsV volumevs versusw quantum well widthw0 rms width of the subband wave functionzj complex coordinate of jth particleχ magnetic susceptibility∆ν QHE excitation gap at LL filling factor ν∆n nuclear-spin energy level spacingε static dielectric constant of bulk GaAsΓ0 disorder broadeningγ gyromagnetic ratioκ thermal conductivityµ0 zero magnetic field mobilityµB Bohr magneton∇ gradient operatorν Landau level filling factorω0 nuclear Larmor pulsationωC cyclotron angular frequencyΦ magnetic fluxΦ0 magnetic flux quantumφ (in-plane) orientation angleΨ[z] many-body total wave function

xvii

ψ0(z) ground subband wave functionρ(z) density profile along z-directionρ electrical resistivityρ0 zero magnetic field electrical resistivityρs electron spin stiffnessτ time constantτ0 transport scattering timeσ electrical conductivityσ standard deviationθ sample’s tilt angleε infinitesimal quantityϑ step function

xviii

List of Figures

1.1 QHE magneto-transport measurement geometry . . . . . . . 21.2 Single-particle DOS vs energy. Spinless 2D electrons . . . . . 61.3 Multiple-QW heterostructure growth sequence (sample M242) 81.4 Multiple-QW heterostructure growth sequence (sample M280) 101.5 Calculated self-consistent charge distribution within the QW

confining potential (sample M242) . . . . . . . . . . . . . . . 141.6 Energy spectrum for 2D electrons confined to GaAs QWs . . 161.7 Photographs of the investigated specimen in magneto-transport

studies (sample M280) and Bayotron mixing chamber . . . . 191.8 Magnetoresistance of sample M280 . . . . . . . . . . . . . . . 211.9 Magnetotransport survey (sample M280) . . . . . . . . . . . . 221.10 Magnetotransport survey (sample M242) . . . . . . . . . . . . 231.11 Low-magnetic field magnetotransport overview (sample M280) 241.12 Single-particle DOS vs energy. Real 2DES . . . . . . . . . . . 281.13 Construction of Laughlin quasiparticles . . . . . . . . . . . . 321.14 Numerical simulation for the Landau level crossing . . . . . . 401.15 Tilted magnetic field transport overview (sample M242) . . . 411.16 Tilted magnetic field transport overview (sample M280) . . . 421.17 QHE excitation gap analysis (sample M280) . . . . . . . . . . 451.18 QHE excitation gap analysis (sample M242) . . . . . . . . . . 461.19 Re-entrant behavior of the ν = 4/3 QHE state (sample M280) 48

2.1 Schematic representation of a spin wave . . . . . . . . . . . . 532.2 Schematic representation of a XY vortex . . . . . . . . . . . 582.3 Illustration of a Skyrmion spin texture . . . . . . . . . . . . . 632.4 Numerically calculated spin-wave dispersion of the QHE fer-

romagnet at Landau level filling factor ν = 1 . . . . . . . . . 672.5 Skyrmion’s magnetization profile . . . . . . . . . . . . . . . . 732.6 Hartree-Fock calculations for Skyrmions at Landau level fill-

ing factor ν = 1. Bare Coulomb interactions . . . . . . . . . . 74

xix

2.7 Hartree-Fock calculations for Skyrmions at Landau level fill-ing factor ν = 1. Realistic Coulomb interactions . . . . . . . . 75

2.8 Predicted temperature dependence of the 2D electron spinpolarization at Landau level filling factor ν = 1 . . . . . . . . 78

2.9 Skyrmions and the 2D electron spin polarization near ν = 1 . 802.10 Qualitative phase diagram for Skyrmion crystal states . . . . 852.11 Predictions on the nuclear spin-lattice relaxation rate around

Landau level filling factor ν = 1 . . . . . . . . . . . . . . . . . 882.12 Measured QHE excitation gap at ν = 1 (sample M280) . . . . 90

3.1 Thermal diagram for multiple-QW samples . . . . . . . . . . 973.2 Photographs of an investigated specimen in heat capacity

studies (sample M280) and tilting stage . . . . . . . . . . . . 1003.3 Carbon paint thermometers for heat capacity measurements . 1033.4 Thermal response of multiple-QW samples in ac heat capacity

experiments at low temperatures (samples M242 and M280) . 1083.5 The measured heat capacity in the integer QHE regime (sam-

ples M242 and M280) . . . . . . . . . . . . . . . . . . . . . . 1103.6 Line shape of the electronic heat capacity in the integer QHE

regime (sample M242) . . . . . . . . . . . . . . . . . . . . . . 1113.7 Time dependence of the lattice temperature in relaxation heat

capacity experiments (sample M242) . . . . . . . . . . . . . . 1153.8 The nuclear heat capacity around ν = 1 (samples M242 and

M280) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.9 Angular dependence of the nuclear heat capacity near ν = 1.

I (samples M242 and M280) . . . . . . . . . . . . . . . . . . . 1203.10 Angular dependence of the nuclear heat capacity near ν = 1.

II (sample M280) . . . . . . . . . . . . . . . . . . . . . . . . . 1223.11 Disappearance of the nuclear contribution to the measured

heat capacity at high tilt angles (sample M280) . . . . . . . . 1243.12 The measured heat capacity near ν = 1 as a function of the

Zeeman energy (sample M280) . . . . . . . . . . . . . . . . . 1253.13 Heat capacity measurements at intermediate tilt angles (sam-

ple M280) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.14 QHE excitation gap and heat capacity measurements at Lan-

dau level filling factor ν = 4/3 as a function of the Zeemanenergy (sample M280). . . . . . . . . . . . . . . . . . . . . . . 128

3.15 The temperature dependence of the heat capacity near ν = 1(sample M242). Untilted magnetic field data . . . . . . . . . 129

xx

3.16 The temperature dependence of the heat capacity near ν = 1(sample M242). Tilted magnetic field data . . . . . . . . . . . 130

3.17 The temperature dependence of the heat capacity near ν = 1(sample M280) . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.18 The numerically calculated Schottky effect in GaAs QWs(sample M242) . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.19 The measured temperature of the lattice during the quasi-adiabatic thermal experiments (sample M242) . . . . . . . . . 135

3.20 Temperature as a function of time during a heat capacityexperiment in the quasi-adiabatic regime (sample M242) . . . 138

3.21 The determined nuclear spin-lattice relaxation rate in calori-metric experiments (sample M242) . . . . . . . . . . . . . . . 142

4.1 Pictorial illustration of the NMR setup and the theoreticalNMR line shape . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2 Schematic diagram of the spin-echo sequence . . . . . . . . . 1644.3 Typical 71Ga NMR spectra at low temperatures . . . . . . . . 1654.4 Low-temperature 71Ga NMR spectra at ν = 1 . . . . . . . . . 1664.5 NMR spectra comparing the line shapes of 71Ga and 69Ga . . 1684.6 NMR determination of the low-temperature 71Ga nuclear spin-

lattice relaxation rate . . . . . . . . . . . . . . . . . . . . . . 1704.7 The measured low-temperature 71Ga nuclear spin-spin relax-

ation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.8 The measured 2D electron spin polarization in the extreme

quantum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.9 Temperature dependence of the electron spin polarization at

Landau level filling factor ν = 1, 1/2, and 1/3 . . . . . . . . . 1764.10 Comparison between NMR and OPNMR electron spin polar-

ization results near ν = 1. . . . . . . . . . . . . . . . . . . . . 1774.11 Temperature dependence of the electron spin polarization

peak at Landau level filling factor ν = 1 . . . . . . . . . . . . 1784.12 Magnetic field (Landau level filling factor) dependence of the

71Ga nuclear spin-lattice relaxation rate . . . . . . . . . . . . 1814.13 Tilt angle dependence of the electron spin polarization . . . . 186

xxi

List of Tables

1.1 Compilation of various structural and electronic properties ofsamples M242 and M280 . . . . . . . . . . . . . . . . . . . . . 11

1.2 Physical parameters of stoichiometric GaAs . . . . . . . . . . 121.3 Estimates of relevant energy scales for samples M242 and M280 341.4 Zeeman energy contribution to the QHE excitation gap due

to thermally-activated spin reversed quasiparticles . . . . . . 381.5 The measured QHE excitation gaps (sample M280) . . . . . . 47

2.1 Calculations of the Skyrmion excitation energy which takeinto account both the finite thickness of the electron layerand the Landau level mixing. . . . . . . . . . . . . . . . . . . 76

3.1 Landmark papers describing calorimetric techniques for small-sample heat capacity measurements . . . . . . . . . . . . . . 94

3.2 Calorimetric techniques for heat capacity measurements inthe QHE regime . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.3 The low-temperature heat capacity of multiple-QW samplesand details of addenda in heat capacity measurements . . . . 99

3.4 Various properties of materials used for the thermal anchoringof the sample in heat capacity experiments . . . . . . . . . . 105

3.5 Amplitude of the heat capacity magneto-oscillations in theinteger QHE regime (samples M242 and M280) . . . . . . . . 112

4.1 Details of NMR coils . . . . . . . . . . . . . . . . . . . . . . . 1624.2 The measured line width of NMR spectra . . . . . . . . . . . 171

A.1 Isotope table for GaAs QWs’ nuclei . . . . . . . . . . . . . . . 194

B.1 The measured 71Ga nuclear spin-lattice relaxation rate nearν = 1 (OPNMR) . . . . . . . . . . . . . . . . . . . . . . . . . 196

B.2 The measured QWs’ line width near ν = 1/3 (OPNMR) . . . 198

xxiii

Chapter 1

The Quantum Hall Effect

Si iarasi am vazut sub soare ca izbanda în alergare nu este acelor iuti si biruinta a celor viteji, si pîinea a celor întelepti,nici bogatia a celor priceputi, nici faima pentru cei învatati, cacitimpul si întamplarea întampina pe toti.

Ecclesiastul

1.1 Idealized two-dimensional electron systems

It is by experiment that quantum Hall effects (QHEs), integer and fractional,were discovered 1. Simple electrical measurements of real semiconductor de-vices containing two-dimensional electron systems (2DESs) have uncovereda fascinating low-temperature behavior of the Hall resistance when a strongmagnetic field is applied perpendicular to the 2DES. For a 2DES devicecut into a standard Hall bridge [Fig. (1.1)] and placed in a magnetic fieldperpendicular to the sample’s surface (i.e., lying along the z-axis), the Hallresistance (RH) is the ratio of the Hall voltage, measured across the 2DESdevice in the y-direction, to the current applied in the x-direction. Thehallmark experimental feature is that the Hall resistance has plateaus overa finite range of values of the magnetic field, quantized to RH = h/(υe2).The quantum number υ is either an integer or a simple rational fraction withodd denominator. The QHE is sufficiently reproducible that it provides an

1The integer QHE was discovered in 1980 by von Klitzing et al. [108]. The very firstobservation of the fractional QHE was reported two years later by Tsui et al. [102].

1

2 CHAPTER 1. THE QUANTUM HALL EFFECT

invariable reference standard of electrical resistance linked to fundamentalphysical constants: h/e2 = 25812.805 Ω. Advocacy of the quantization ofRH is based upon general grounds invoking Landau quantization of the elec-tron orbits, electron localization, and electron-electron interactions. Whythese still unfolding phenomena are so peculiar and amazing is partly dueto the intimate connection between theoretical interpretations of the ob-served quantization to various topological and field theories. We have tolook first at some preliminary aspects of the QHE before to come to thepart of interest here: the domain of spin-related phenomena in 2DESs.

Figure 1.1: Sample geometries for magneto-transport measurements. (Top)Crude Hall bridge defined by cleaving the sample to appropriate size. Thecurrent contacts are S and G, while the potential probes are A, B, C, and D,yielding RH = |VBD|/ISG ≡ VH/I and RL = |VCD|/ISG ≡ VL/I. (Bottom)van der Pauw, square-shaped sample. In a first experiment, A and B arethe current contacts and C and D are the potential probes yielding thelongitudinal resistance RL = |VCD|/IAB ≡ VL/I. In a second experiment,A and C are the current contacts and B and D are the potential probes.Consequently, the Hall resistance RH = |VBD|/IAC ≡ VH/I is obtained.

1.1. IDEALIZED TWO-DIMENSIONAL ELECTRON SYSTEMS 3

1.1.1 The free electron

Let no one think that properties of an electron are easy to understand.The properties of this elementary particle we are most familiar with arethe mass me and the charge −e. For small-scale phenomena the electronbehaves quantum mechanically and obeys the principles of relativity; thequantum mechanics of the free electron was captured by Dirac in his famousrelativistic wave equation. He predicted, for the electron, the existence ofnew internal degrees of freedom: the electron has a spin angular momentum~s, defined by the spin quantum number s = 1

2 . It turns out that spinis connected to statistics and electrons obey Fermi-Dirac statistics. Theelectron possess a spin magnetic moment

me = −geµBs = γe~s, (1.1)

where ge is the free electron Lande g-factor, µB = e~/(2me) is the Bohrmagneton, and γe is the electron gyromagnetic ratio. To the lowest order inthe fine structure constant α = e2/(~c), quantum electrodynamics predictsan ”anomaly” for the free electron magnetic moment: ge = 2[1 + α/(2π)].The g-factor for the free electron characterizes the (Zeeman) coupling ofthe electron’s spin to a magnetic field B; the spin Hamiltonian of the freeelectron is Hs = −me ·B.

1.1.2 Idealized quantum wells

We know how to solve few, simple quantum mechanics problems for such anelementary particle. In the following, we shall restrict our area of inquiryto the particular case when the electron motion has a two-dimensional (2D)character. In order to understand this point fully, let’s consider the motionof an electron confined to a one-dimensional (1D) infinite square quantumwell (QW). The potential energy is given by

Vb(z) =

∞ for |z| > w/2,

0 for |z| < w/2,

where w is the QW width. The eigenenergies of the 1D Schrodinger equation− ~2

2me

d2

dz2+ Vb(z)

ψ(z) = Eψ(z), (1.2)

describing the quantum mechanical bound motion along the z-axis, arequoted in all textbooks on quantum mechanics

ErQW =

~2π2(r + 1)2

2mew2, (1.3)


where r = 0, 1, . . . is a non-negative integer. The discreteness of the energyspectrum is the key reason for the 2D behavior of electrons confined to QWs.The normalized even and odd wave functions for |z| < w/2 are

ψr(z) =

(2w

)1/2 cos[(r + 1)πz

w

], r = 0, 2, 4 . . .(

2w

)1/2 sin[(r + 2)πz

w

], r = 1, 3, 5 . . .

It is worth noting that there is at least one bound state, irrespective ofthe height of the confining barrier, i.e., infinite or finite. The quantummechanical motion in the xy-plane is described by the Schrodinger equation

− ~2

2me

d2

dx2+

d2

dy2

ψ(x, y) = Eψ(x, y). (1.4)

The energy spectrum is continuous and the wave functions are the well-known plane waves. In polar coordinates, the complete energy levels are

Er,k‖ = ErQW + E

k‖QW =

~2π2

2mew2(r + 1)2 +

~2

2mek2‖, (1.5)

where Ek‖QW is the electron’s kinetic energy in the xy-plane and k‖ is the

magnitude of the in-plane wavevector. Obviously, one may associate anelectric subband, which represents the kinetic energy arising from the in-plane motion of the carrier, with each of the QW bound states. In short, anelectron in the ground electric subband (r = 0) of an infinitely deep QW hasan energy E0

QW = ~2π2/(2mew2) and it is described by the wave function

ψ0(z) = (2/w)1/2 cos (πz/w). What is meant here by 2D electrons is thatelectrons have quantized energy levels in one direction (z-direction), but arefree to move in two dimensions (xy-plane).

1.1.3 Landau levels

Let’s focus now on how the energy spectrum of an electron splits into Landaulevels (LLs) in the presence of a magnetic field. Consider the Hamiltonianfor the free electron in a magnetic field

Horbital =1

2me(p + eA)2, (1.6)

where p is the canonical momentum and A is the magnetic vector potential.The vector potential can be chosen to lie in the plane perpendicular to themagnetic field so that the contributions to the Hamiltonian from motion

1.1. IDEALIZED TWO-DIMENSIONAL ELECTRON SYSTEMS 5

along and perpendicular to the magnetic field separate. Since the motionalong the magnetic field direction is unchanged, only the 2D motion of theelectron, in the plane perpendicular to the magnetic field, remains of inter-est 2. In order to fix these preliminary ideas in a specific way, you haveto imagine that the magnetic field is applied along the z-axis (B = Bez)and the motion in the xy-plane is the 2D motion of the electron in theground electric subband due to the 1D confinement in the z-direction. Thexy-plane is taken to be the complex plane with z being a complex coordi-nate related to the position vector (x, y) via z = (x+ iy)/lB. The magneticlength lB = [~/(eB⊥)]1/2 is the natural length scale in the problem 3. Theeigenenergies of Horbital [Eq. (1.6)] are

EN = ~ωc(N + 1/2); ωc = eB⊥/me. (1.7)

Here ωc is the cyclotron frequency, N = 0, 1, ... is the LL index, and theenergy zero coincides with the bottom of the ground electric subband. Theenergy spectrum is discrete, the manifold of states with energy EN [Eq. (1.7)]constitutes the Nth LL, and E∗ = ~ωc is the cyclotron energy. The eigen-functions are of an especially simple form in the lowest (N = 0) LL

ψN=0m (z) =

zm

(2π2mm!)1/2exp

(− |z|2

4

), (1.8)

where m is the angular momentum index. The magnetic field couples tothe electron’s orbital motion and quantizes its kinetic energy into massivelydegenerate LLs. The energy eigenvalues [Eq. (1.7)] do not depend explicitlyon the angular momentum index. Including the spin degeneracy, one obtainsthe same degeneracy per unit area for all LLs: 2eB⊥/h.

1.1.4 The idealized two-dimensional density of states

The preceding discussion has been restricted to a single electron. We con-sider next the density of states (DOS) of a quantum collection of N inde-pendent 2D electrons of areal density n. First, it makes sense to examinethe DOS at zero magnetic field. Since we are ultimately interested in thedescription of 2DESs confined to QWs, we remark that the DOS could bedecomposed into contributions which arise from different electric subbands.

2Note that the Lagrangian of the problem writes as L2D = (me/2)vµvµ−evµAµ, wherev is the velocity of the particle and µ = x, y.

3The magnetic length is, in fact, the radius of the cyclotron orbit. We have explic-itly indicated that it is the perpendicular component of the magnetic which controls theelectron’s orbital motion.


Figure 1.2: The flat DOS (per unit area) at zero magnetic field (panel a)collapses into a series of massively degenerate Landau levels when a perpen-dicular magnetic field B is applied to the 2DES (panel b). A disorder-free,spin-degenerate situation is illustrated.

It is not difficult to obtain the DOS per unit area associated with one electricsubband [panel (a) to Fig. (1.2)]

D0 =me

π~2, (1.9)

where we have taken into account the spin degeneracy. Thus, each electricsubband contributes the same constant quantity to the total DOS

D02D(E) = D0

∑r

ϑ(E − ErQW ), (1.10)

where ϑ(x) is the step function. At zero magnetic field, DOS exhibits jumpsof finite amplitude D0 whenever the energy passes through the edge of anelectric subband [panel (a) to Fig. (1.2)]. In the ground state, i.e., at zero

1.2. REALISTIC TWO-DIMENSIONAL ELECTRON SYSTEMS 7

temperature (T = 0), the electrons occupy all states with energy E lessthen a limiting value EF (the Fermi energy). If n is known and all electronscan be accommodated into the ground electric subband, using the simpleformula EF = n/D0, the Fermi energy could be readily obtained.

Consider now the DOS of a 2DES confined to a QW when a magneticfield is applied perpendicular to the plane of the 2D electron layer. Assumingthat only the ground electric subband is populated, the DOS per unit area(including the spin degeneracy) is

D2D(E) =2eB⊥h

∑N

δ(E − EN ), (1.11)

whit δ(x) denoting the Dirac delta function. The fact that DOS have δ-likesingularities centered at the LLs energy eigenvalues [panel (b) to Fig. (1.2)]witnesses for the remarkable properties of 2DESs placed in a finite, homoge-nous perpendicular magnetic field.

1.2 Realistic two-dimensional electron systems

1.2.1 Sample fabrication

Modern QHE experiments are performed almost exclusively on 2DESs con-fined to GaAs/AlxGa1−xAs heterostructures. The samples investigated inthis thesis were constructed by means of molecular beam epitaxy (MBE).Initiated in the early 1970s, MBE is presently an extremely versatile tech-nology for the fabrication of multilayer structures based on lattice-matchedsemiconductors such as GaAs and AlxGa1−xAs. In this technique, the crys-talline structure is obtained via reactions between thermal molecular beamsof elemental species and a substrate surface maintained at a high tempera-ture in ultrahigh vacuum. Since it is inherently a slow growth process andelectrically active impurities are incorporated to the growing crystal withseparate beams, extreme dimensional control over both compositional vari-ations and doping profile can be achieved. It is known that MBE producesatomically smooth interfaces and the conduction and valence bands exhibitabrupt steps at a common interface between GaAs and AlxGa1−xAs. Theconduction band of GaAs sits lower in energy and the steps are ≈ 300 meVfor an Al mole fraction of x = 0.3. In the physical formation of 2DESsdescribed below only the conduction band profile will be involved.


Figure 1.3: Design for the multiple-QW sample M242 is shown schematically.


In the conventional (simplest) approach for the realization of a 2DES (see, forexample, Fig. (1.11) in Ref. [86]), the essential part of the MBE grown struc-ture consists of an undoped GaAs layer, followed by an undoped AlxGa1−xAsspacer layer and then a Si-doped AlxGa1−xAs region. To maintain a con-stant Fermi level throughout the structure, excess electrons, donated by theremote Si impurities, find their way to the conduction band of GaAs. Thischarge transfer creates strong internal electric fields that cause significantbending of the conduction band in the vicinity of the GaAs/AlxGa1−xAsinterface, which, in turn, acts as a confinement potential for electrons. A2DES results as the carrier motion on the direction perpendicular to theinterface is quantized. After donating their electrons, the impurity atomsare left positively charged, the net charge of the sample thus being zero 4.

The single-QW arrangement could be viewed from the substrate upwardas a combination of an inverted interface (GaAs on top of AlxGa1−xAs) and anormal interface (AlxGa1−xAs on top of GaAs). For modulation-doped QWheterostructures, dopants are placed on both sides of the QW [Fig. (1.3)].Single QW samples and conventional heterointerfaces provide a unique realmfor standard magneto-transport measurements. For thermodynamic studieson 2DESs 5, however, one has to seek for a method that will enhance themeasurable signal. One straightforward way to solve this problem is toconstruct multiple-QW heterostructures [90].

The two samples used in this thesis were grown by MBE on semi-insulating, (100)-oriented GaAs substrates at Princeton University. It canbe easily seen in Figs. (1.3) and (1.4), that samples M242 and M280 are sim-ilarly structured. They are composed of one hundred GaAs QWs boundedon each side by AlxGa1−xAs barriers which are δ-doped with Si near theircenters. A complicated sequence of layers was grown on the substrate preced-ing the multiple-QW system and following it. The rationale behind choosingsuch a ponderous design is to fashion 2DESs with very high mobilities [74].

For the purposes of this thesis, I mention here some important character-istics of the heterostructure design. From the top downward, the structure iscomposed of a GaAs cap layer followed by a AlxGa1−xAs layer. This regionconsists of two δ-doped Si layers: the majority of the impurities are placed

4Although the parent donors are spatially separated from the 2DES by the spacerlayer, they unavoidable contribute to the electron scattering. The remote ionized-impurityscattering is one of the mechanisms responsible for the finite low-T mobility of the 2DES.This (zero-magnetic field) low-T mobility is a major physical variable which assesses thequality of the sample.

5Determination of the DOS via specific heat measurements is an example.


Figure 1.4: Design for the multiple-QW sample M280 is shown schematically.


in a plane far from the 2DES and close to the surface. The δ-layer situatedcloser to the QW provides electrons for the 2DES, while the heavy-dopedδ-layer mainly passivates the surface states at the GaAs-air interface. TheQWs have been slightly asymmetrically δ-doped by placing the donors nearthe center of the barriers. Asymmetric doping has been used to compensatefor the migration of Si along the growth direction [65]. The amount of Sidopant was chosen (1) to be enough to pull the conduction band edge closeto the Fermi level at the location of the ionized impurities and (2) to attainthe same doping level for all QWs.

Table 1.1: Essential structural and electronic properties of samples M242and M280. Here, dB is the thickness of the barriers; other symbols aredefined in the text [see also Figs. (1.3), (1.4), and (1.5)]. The thicknesses (wand dB) and the composition (x) were determined from calibrated growthrates; w0 is the rms width of the self-consistently calculated subband wavefunction in each QW. The zero energy coincides with the bottom of theQW. The energy levels E0

QW and E1QW for sample M242 are results of self-

consistent calculations, whereas for sample M280 they are estimates basedon Eq. (1.3).

w w0 dB E0QW EF E1

QWSample

Ax

meVn [1011 cm−2]

M242 250 65 1850 0.3 10.5 15.5 31 1.40± 0.02

M280 300 71 2500 0.1 6 9 25 0.86± 0.02

While δ-doping technique was used to minimize the remote impurity scatter-ing, sample quality also depends on the unintentionally incorporated impuri-ties. Such impurities could be introduced by the substrate itself; subsequentmigration with the growth front results in contamination of the interfaceswere the 2DES is formed. To combat this an AlAs/GaAs ”gettering” super-lattice is grown first. In such a superlattice, migrating impurities tend tobe trapped at the AlAs/GaAs or GaAs/AlAs interfaces. Important struc-tural and electronic properties of M242 and M280 samples are summarizedin Table (1.1). Note that the measured density is essentially the same as thenominal Si-doping. In other words, in these samples practically all electronsfrom the donors are transferred to the QWs.


1.2.2 Electrons in GaAs quantum wells

The GaAs crystal has a zinc blende type of structure. Representative prop-erties of bulk GaAs at room temperature are given in Table (1.2) accordingto the work of Blakemore [12]. In GaAs, there are 8 valence electrons perunit cell which contribute to the chemical bonds. The other electrons of eachkind of atom are ”frozen” in closed shell configurations and they do not con-tribute to the electronic properties investigated in this thesis. The valenceelectrons hybridize to form tetrahedral bonds between one kind of atom (sayGa) and its four nearest neighbors (As). In other words, the orbitals of everyatom (s-like or p-like) hybridize with the orbital of the neighboring atom,thus producing two levels: one bonding and one antibonding. Along theGa-As bonding chain, the orbitals have most of the charge density shiftedtowards As ion. The bonding s-orbitals are always occupied by 2 valenceelectrons per unit cell. The remaining 6 valence electrons per unit cell oc-cupy the three p-bonding orbitals. Because there is a large number of unit

Table 1.2: Unit cube size, nearest-neighbor distance between similar atoms,atomic density, molecular weight, and crystal density of bulk GaAs [12].

Parameter Symbol Value

Length of side of unit cube Ac 5.653 A

Nearest-neighbor distance dc =√

2Ac/2 3.997 A

Atomic density 8/A3c 4.428× 1022 cm−3

Molecular weight Mc 145 amu

Crystal density ρc 5.317 g · cm−3

cells, bonding and antibonding levels broaden into bands. The top of thevalence band in GaAs occurs at the center of the Brillouin zone (Γ point). Inthe absence of the spin-orbit coupling (see below), the three valence bands(which originate from p-bonding orbitals) are degenerate at the Γ point.The bands originating from the antibonding orbitals are all empty, the low-est lying s-band forming the conduction band of the material. In GaAs thebottom of the conduction band occurs at the Γ point and the bandgap isabout 1.52 eV at low T . In short, for GaAs, at the Γ point of the con-duction band, the s-like contribution dominates the total electron chargedensity distribution, which has an antibonding character with most of the

1.2. ELECTRONIC PROPERTIES 13

charge localized on the As ion. In the following, I would like to remark onsome important properties of electron-doped GaAs QWs.

The effective mass and the effective Lande g-factor

The overall effects of the GaAs band structure, i.e., the fact that the electronexperiences the periodic potential of the lattice instead of moving in thevacuum, are embodied in the use of an effective mass instead of the freeelectron mass 6. The relevant effective mass of 2D electrons in our samplesis assumed to be the isotropic conduction electron mass in bulk GaAs, m∗ =0.067me.

One detail still needs to be clarified. What happens to the g-factor ofthe electron? Up to now we considered the electron spin completely in-ert dynamically, but in GaAs electrons move in an effective electric fieldthat results from the built-in potential of the non-symmetric zinc-blendestructure of the underlying crystal. As a result, an electron experiences anadditional potential proportional to the scalar product of its spin magneticmoment with the vector product of its velocity and the crystalline electricfield. This additional interaction is referred to as the spin-orbit couplingand it leads to a renormalization of the g-factor of the free electron. In bulkGaAs, conduction electrons have an isotropic effective g-factor: g∗ = −0.44.In GaAs QWs, the quantum confinement renders the g-factor of 2D elec-trons anisotropic and modifies the values of the components of the g-factoralong (g‖) and perpendicular (g⊥) to the growth axis of the heterostruc-ture. Experimental studies in GaAs/Al0.3Ga0.7As systems revealed that forw & 120 A there is no observable g-factor anisotropy [72]. It has beenalso verified that for w & 200 A the g-factor is close to g∗ = −0.44 [72].Therefore, in our samples, the effective g-factor of 2D electrons could beconsidered isotropic and equal to g∗ = −0.44 in all experimental situations.

The electronic charge distribution

The problem of charge density distribution in GaAs QWs demands eluci-dation. If one considers a 2DES in the presence of an external potential,the spatial electronic distribution is described in general by the Schrodingerequation for the wave function (containing the confinement potential) and

6The effective mass will replace the free electron mass in Eqs. (1.2), (1.3), (1.4), (1.5),(1.6), (1.7), and (1.9). The constant DOS for 2DESs in GaAs QWs is m∗/(π~2) = 2.8×1010 cm−2 ·meV−1.


Figure 1.5: The numerically calculated charge distribution ρe(z) (solidcurve) within the QW confining potential (dotted curve) for sample M242.Open circles describe a Gaussian fit to ρe(z), defining the rms width w0 ofthe 2DES. [Courtesy of S. Shukla]

the Poisson equation for the potential (containing the wave function throughthe charge density). These have to be solved together self-consistently. Therealistic confinement potential energy is represented by the conduction bandedge of the host material. The quantum mechanical description of sampleM242 is summarized in Fig. (1.5), which shows the part of the band diagramcontaining the resulting conduction band profile after the charge transfer hastaken place 7. Table (1.1) includes the eigenenergy solutions. The secondsubband is unoccupied and energetically distant from the ground subband,

7The modelling assumed a perfect symmetry with respect to the center of the QW andemployed the measured electron density.

1.2. ELECTRONIC PROPERTIES 15

and may be safely ignored. The ground subband wave function (ψ0) hasa finite spatial extent in the z-direction. The finite thickness of the 2DEScould be simply estimated by representing the exact wave function in termsof a Gaussian distribution of rms width w0 [Table (1.1)]. However, in orderto simplify the calculations, it is usually assumed that the 2DES is ideal,i.e., the 2DES is placed in a sheet of zero thickness, surrounded by anhomogeneous medium with a dielectric constant ε = 13. This value is closeto the dielectric constant of bulk GaAs.

1.2.3 The Landau level diagram

The effective Lande g-factor and the effective mass dictate the energy spec-trum of 2DESs in the presence of an external magnetic field. If we properlyinclude m∗ and g∗ into the description of 2DESs in GaAs QWs, the single-particle Hamiltonian writes as

H1−e =1

2m∗ (p + eA)2 + |g∗|µBs ·B. (1.12)

Its energy eigenvalues are given by

EN,σ = ~ωc(N + 1/2) + σ|g∗|µBB; ωc = eB⊥/m∗. (1.13)

In the presence of a perpendicular magnetic field, the electron energies arelabelled by two quantum numbers [the electric subband index (r) and the LLindex (N)] plus the spin variable σ = ±1

2 . Each electric subband developsinto a series of spin-split LLs counted from the subband edge. Assumingthat only the ground electric subband is occupied, the DOS per unit area is

D(E) = nφ

∑N,σ

δ(E − EN,σ), (1.14)

where nφ = eB⊥/h is the degeneracy of one spin-split LL. The nominalnumber of filled LLs at any magnetic field is called LL filling factor ν, andis given by ν = n/nφ. We shall not extend the discussion of LLs other thanto remark that an accurate description of 2DESs in GaAs QWs at B 6= 0should also take into account the correlation-exchange effects between elec-trons. Exchange effects (which keep electrons with antiparallel spin apart)were included into the diagram depicted in Fig. (1.6), which shows only theLLs relevant for the experiments presented in this thesis. In the presence ofa perpendicular magnetic field, one may talk about electric subbands sepa-rated in energy by both exchange and Zeeman effects and then, further splitinto LLs.


This point of view of two spin bands is particularly useful for understandingseveral experimental results, and we shall adopt it here. In Fig. (1.6) theenergy gap between (N = 0, ↑) LL and (N = 0, ↓) LL is exchange enhancedand it equals the sum of EZ and Exc, whereas the energy gap between(N = 0, ↑) LL and (N = 1, ↑) LL is E∗.

Figure 1.6: The energy spectrum of 2D electrons confined to GaAs QWs inthe presence of a quantizing perpendicular magnetic field is schematicallyshown. Orbital (E∗), exchange (Exc), and Zeeman (EZ) effects are included.

1.3 Magnetotransport: Experimental details

All electrical resistance data discussed in this thesis were collected in anOxford Instruments Kelvinox 300 Dilution Refrigerator (Bayotron) with ameasured base temperature of 8 mK. The design of the base of the mixingchamber, made in Stycast epoxy, is illustrated in Fig. (1.7). Samples aremounted in vacuum by means of a (dual in-line multiple pin) DIP-socketwhich is fixed onto the mixing chamber very near to the copper cold fin-ger. At zero magnetic field, while monitoring the RuO2 mixing chamberthermometer, typical temperatures that can be maintained continuously inthis apparatus are below 20 mK. The Bayotron is presently used with asuperconducting solenoid designed to provide a maximum central magneticfield of 15 T at 4.2 K and 17 T at 2.2 K. The spatial homogeneity of themagnetic field (over a 10 mm diameter spherical volume) is 1 part in 103.The dilution refrigerator is equipped with a tilting stage (see for details theexperimental section in Chapter 3) which can be rotated in situ, so the sam-

1.3. MAGNETOTRANSPORT: EXPERIMENTAL DETAILS 17

ples can be rotated from the horizontal to vertical (perpendicular to parallelto the applied magnetic field). The tilt angle θ is defined as being the an-gle between the direction of the applied magnetic field and the normal tothe sample plane. Unless otherwise stated, the magnetic field was appliedperpendicular to the plane of the 2D electron layers.

The geometry of our measured specimens differs from the traditionalgeometry employed for Hall resistivity measurements. The so-called vander Pauw geometry, as illustrated in Fig. (1.1), has been used throughoutthe present electrical transport experiments [106]. Each sample was cleaved(along the natural cleavage directions 〈110〉 and 〈110〉) into a small square(approximately 2 to 3 mm on a side) from the MBE grown wafer. In thestandard procedure of fabricating ohmic contacts to the 2DES, the samplewas first cleaned in trichlorethylene, acetone, methanol, and blown dry withN2 gas. Then it was held face up in a vacuum chunk and metal contactswere deposited by hand with a soldering iron, under a microscope. Metalcontacts consisted of eutectic mixture of In : Sn (≈ 50% Sn) and wereapposed on the corners and at the centers of the edges of the sample 8.The sample and alloy contacts were then annealed in an oxide-reducinghydrogen atmosphere N2 : H2 (≈ 5% H2) at typically 440 − 450C for 15-25 minutes to form ohmic contacts to the buried electron layers. With careunder a microscope, the sample was then held in the vacuum chunk and25 µm-diameter gold wires, which serve as electrical leads, were solderedto the ohmic contacts. The next step consisted of mounting the sampleon a DIP-header, by soldering the electrical leads to the pins with indium.The DIP-header supporting the sample fits either into the standard DIP-socket mounted on the tilting platform or into the DIP-socket fixed onto themixing chamber [Fig. (1.7)]. The sample was placed in close proximity to thecalibrated RuO2 chip resistor (with known magnetoresistance corrections)employed as a thermometer. The thermometer was attached to the samplethrough a copper strip, which was glued to the DIP-header with GE7031varnish. An example of the final result of this procedure is shown in Fig. (1.7)for a sample cleaved from the M280 wafer.

In the context of magneto-transport measurements on multiple-QW sam-ples we make two crucial remarks. First, the carrier concentration was notchanged through a persistent photoconductive effect in any of experimentspresented in this thesis 9. Second, in order to ensure a good well-to-well ho-

8It is very important that the metal for the contacts touches the edge of the sample toprevent Corbino-geometry contacts.

9Modulation-doped conventional heterointerfaces are usually sensitive to the exposureof light. A red light emitting diode is commonly used to vary the electron density through


mogeneity, samples were cooled slowly (over several hours) from room tem-perature to the liquid helium temperature. In such state-of-the-art multiple-QW heterostructures we expect that the average electronic density varies byonly a few percents from layer to layer. In a typical specimen with maxi-mum size 3 × 3 mm, the density fluctuations across the sample are below2%. Despite the fragility of the sought-after behavior, experimental reasonsdictated the use in magneto-transport experiments of samples cleaved nearthe edge of MBE grown wafers. The quality of specimens coming from themarginal parts of the wafer is slightly worse than those cleaved from thecentral part. Furthermore, the density across the 2-inch GaAs wafer typ-ically varies by ±15 % from the nominal values calibrated near the wafercenter, resulting in a slightly larger density of samples cut from the edge ofthe wafer than those taken from the central part.

For magneto-transport measurements a standard ac lock-in techniquehas been used. By means of the ohmic contacts a constant low-amplitude,low-frequency ac current is driven through the sample. The voltage in phaseis recorded, with a lock-in amplifier across contacts at different positions.Two measurements are made: the voltage drop in the direction parallelto the current flow (VL) and the voltage difference across the current flow(VH). Voltages VL and VH were recorded and divided by the applied currentI, to obtain the longitudinal resistance (RL = VL/I) and the Hall resistance(RH = VH/I). The longitudinal resistance could be optionally scaled toobtain the longitudinal resistivity, while the Hall resistivity is exactly equalto the measured Hall resistance 10. In a first series of experiments, thetransport coefficients RL and RH are measured as a function of B, at fixed T .As our samples were not immersed in the 3He/4He mixture, care was takento avoid heating of the 2DES above the bath temperature by the excitationcurrent. We used excitation currents of 1-100 nA rms at frequencies <100 Hz, and the sweeping magnetic field rate was kept below 0.1 T/min toensure the thermal equilibrium condition. The second series of experimentsperformed at fixed B, in which RL is measured as a function of T , will bedescribed later on.

a persistent photoconductive effect.10In two dimensions, the Hall resistivity ρH equals RH , independent of the sample

geometry. The longitudinal resistivity is ρL = (π/ ln 2)RL for van der Pauw square-shapedsamples. We assume that an adequate annealing of ohmic contacts to all 100 buried 2Delectron layers was obtained for our measured specimens. Though the details of the currentpath may be intricate, we consider that each multiple-QW structure behaves as a stackof one hundred well-separated 2DESs (resistors) connected in parallel. Accordingly, thelongitudinal resistance is either given for all electrically contacted 2D electron layers (asmeasured) or quoted per single layer. The Hall resistance is always given per one layer.

1.3. MAGNETOTRANSPORT: EXPERIMENTAL DETAILS 19

Figure 1.7: (Top) View of the Bayotron mixing chamber. (Bottom) Exper-imental configuration for magneto-transport measurements (sample M280).


1.4 Fundamental aspects of the QHE

1.4.1 Measuring the QHE

To go any further toward understanding the QHE it is useful to briefly looknow at the ordinary Hall effect, discovered in 1879. In the Drude theoryof the electrical resistivity of a simple metal, an electron is accelerated bythe applied electric field for an average time τ0, the mean free time (aliasthe transport scattering time), before being scattered to a state which hasaverage velocity zero. In the free electron model, the electrical resistivityat zero magnetic field is ρ0 = m∗/(ne2τ0). The carrier mobility at zeromagnetic field is traditionally defined as µ0 = (neρ0)−1. In the presence ofthe magnetic field the electron’s path is curved, due to the Lorentz force,and the resistivity tensor writes as(

ρL −B/(ne)B/(ne) ρL

)(1.15)

The longitudinal resistivity ρL has the same significance as the conventionalnotion of the electrical resistivity of an elemental material. The Hall resis-tivity is zero in the absence of a magnetic field and increases linearly withmagnetic field according to ρH = B/(ne). These expressions have beenextensively compared to the experiment and they are often good approx-imations. These simple results have been obtained within a semiclassicaltheory with quantum mechanics entering very indirectly through the valuesof τ0 and m∗. Only few decades ago they were expected to remain valid atlow-T for 2DESs in the presence of very high magnetic fields. The pioneer-ing paper of Fowler et al. [37] was the first to establish experimentally theexistence of quantum corrections to the longitudinal resistivity of a 2DES ina perpendicular magnetic field. These corrections produce the well knownShubnikov-de Haas (SdH) oscillations.

Let’s turn now our attention to low-T magneto-transport experimentson samples M280 and M242. Figure (1.8) displays a detailed synopsis ofthe evolution of magnetoresistance with decreasing T in sample M280. Fig-ures (1.9) and (1.10) show low-T traces of the transport coefficients RL andRH , in magnetic fields up to 15 T, for samples M280 and M242, respec-tively. The detailed structure observed for transport coefficients in M280sample, at low magnetic fields (B ≤ 1 T), is presented in Fig. (1.11). Con-centrating on the low-B regime (B . 0.2 T) [Fig. (1.11)], we first note thatthe Hall resistance is proportional to the magnetic field, same as in theclassical Hall effect. The longitudinal resistance presents SdH oscillations,

1.4. FUNDAMENTAL ASPECTS OF THE QHE 21

with the characteristic periodicity in 1/B⊥, as expected. The SdH struc-ture becomes stronger with decreasing T . We have observed only one setof oscillations, indicative of one subband of 2D electrons. As B is rampedup, under the condition of Landau quantization, EF moves through succes-sively LLs. The period in 1/B⊥ of magnetoresistance oscillations is givenby e~/(m∗EF ) = (2/n)(e/h). The electron density determined from theposition of RL minima (h/e = 4.137× 10−11T · cm2) agrees with the densitydeduced from the slope of RH . By assigning the filling factor ν according toν = 2πl2Bn, one can explicitly label the minima displayed by the longitudinalresistance. At the lowest investigated temperatures, the LL spin-splitting isreadily visible at ν = 15 in sample M280. In the present experiments, thezero magnetic field electrical resistivity is T -independent below 2 K. Theestimated low-T zero magnetic field electron mobility (µ0) in our samples isgiven in Table (1.3).

Figure 1.8: Overview of the B dependence of the longitudinal resistance RL

at θ = 0 and indicated temperatures (sample M280). Note that the widthsand depths of the RL minima dramatically increase for T → 0.


Figure 1.9: Longitudinal and Hall resistance [RL (left axis) and RH (rightaxis)] vs magnetic field at T = 50 mK and θ = 0 (sample M280). Thevertical lines indicate some prominent integral and fractional filling factors.


Figure 1.10: Longitudinal and Hall resistance [RL (right axis) and RH (leftaxis)] vs magnetic field at θ = 0 and indicated temperatures (sample M242).The vertical lines indicate some prominent integral and fractional fillingfactors where the QHE is observed.


Figure 1.11: Longitudinal and Hall resistance [RL (left axis) and RH (rightaxis)] vs magnetic field at T = 50 mK and θ = 0 revealing the Shubnikov-deHaas [SdH] regime (B . 0.3 T) and the quantum Hall effect [QHE] regime(B & 0.3 T), and the spin-splitting collapse at ν = 15 (sample M280). Thedashed line indicates the expected classical behavior for the Hall resistivity.


Let’s move now to the high magnetic field region. Figures (1.9) and (1.10)display two astonishing aspects which form the hallmark of the QHEs. First,the longitudinal resistance fall essentially to zero over wide ranges of B. Forexample, measurements of the T -dependence of RL at ν = 1/3 (B ∼ 11 T)on M280 sample clearly show the development of a zero-resistance state[Figs. (1.8) and (1.10)]. The second aspect, even more astonishing than thefirst, is that concurrent with the vanishing RL, there are plateaus in theHall resistance. Note that with decreasing T , the B-ranges over which thelongitudinal resistance is essentially zero become larger and larger, whereasthe width of the plateaus tends to its maximal value, which correspondsto perfectly abrupt steps in RH . Close examination of the values of RH

at these plateaus reveals that all can be described by a universal formulaRH = h/(υe2), with υ either an integer or a simple rational fraction withodd denominator. The richness of the phenomenon is evidenced by thelarge number of observed fractions 11. Another important feature of theexperimental data is that the higher the µ0, the more prominent is thefractional QHE. This point is very clear from the data shown in Fig. (1.9)(high-mobility sample) and Fig. (1.10) (low-mobility sample).

1.4.2 Integral QHE

Based on phenomenological grounds, it seems clear that what is required forthe observation of QHEs is an energy gap ∆, separating the ground stateand its current carrying excitations. In this context, one important aspect tobe established is the criterion for the delineation of the quantum-transportregime, separating it from the classical regime. In general, two conditionsshould be fulfilled simultaneously for the macroscopic observation of quan-tum effects in 2DESs: Γ . ∆ and kBT . ∆. Here, Γ = ~/τs (in units ofenergy) is associated to the single-particle scattering time 12. Since mea-surements are typically performed at dilution refrigerator temperatures, the

11Note the weak resolution of the RL minima at fractions in sample M242. This isattributed to the poorer quality of the sample used in the present experiments comparedto the previously investigated sample [8] (cleaved near the center of the MBE grown wafer).

12In the transport theory of 2DESs one must deal with two different characteristic times- a single-particle scattering time τs and a transport scattering time τ0. The single-particlescattering time is a measure of the time for which an electronic momentum eigenstate canbe defined in the presence of scattering. The transport scattering time is given by µ0 andcould be related to a disorder broadening parameter Γ0 which is an informative measureof the degree of disorder in real samples. For the 2DESs studied here, estimates of Γ0,based on the expression Γ0 = ~/τ0, are of the order of 1 K. We note that τ0 could be twoorders of magnitude larger than τs [25].


condition kBT . ∆ is trivially satisfied. As the presence of a quantizingmagnetic field is a necessary premise for the observation of QHEs, we ex-pect that ∆ is in some way related to B. In fact, we have already met ∆ inthe single-particle energy level calculations, deguised either in the cyclotronsplitting or in the Zeeman splitting. One of the most transparent illustra-tions of this general criterion for macroscopic quantum behavior in 2DESscan be found in the occurrence of SdH oscillations; magneto-quantum oscil-lations in RL will appear when the magnetic field is large enough such asΓ . E∗. For even larger magnetic fields, the Zeeman splitting will becomeresolved when Γ . EZ .

In combination with the single-particle quantization effects, disorder isthe essential ingredient for observing the integer QHE. In the most simpletheory of the integer QHE, one follows the one-electron analysis and looks atthe effects of disorder on the energy spectrum of 2D electrons in a quantizingmagnetic field 13. In a real 2DES, LLs are broadened into a set of energylevels due to the finite amount of the disorder present in the sample; theDOS consists of a series of ”non-ideal δ-functions” centered at the discreteLL energies [Fig. (1.12)]. For mathematical convenience, in the present work,the shape of each LL is considered to be a Gaussian [114]. The rms width ofeach LL is denoted by ΓG. (The full width at the half maximum is 2.36ΓG

and may be compared to Γ0 = ~/τ0.) If only the ground electric subband isoccupied, the Gaussian DOS is

DG = nφ

∑N,σ

(2π)−1/2Γ−1G exp −(E − EN,σ)2/(2Γ2

G). (1.16)

More generally, the DOS could be obtained from a self-consistent theorywhere the impurity-induced level broadening and screening determine eachother [68]. Specifically, when EF lies at the middle (edge) of a Landau levelscreening is strong (weak) and the screened potential is short ranged (longranged). It should be noted that self-consistent DOS calculations becomeintractably complex when many LLs are occupied.

13There is a glaring discrepancy between von Klitzing’s discovery and the localizationtheory [E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, ScalingTheory of Localization: Absence of Quantum Diffusion in Two Dimensions, Phys. Rev.Lett. 42, 673 (1979).] Localisation theory predicts that weak disorder is sufficient tolocalize non-interacting electrons in 2D at B = 0 and zero temperature. The occurrenceof QHEs, which necessarily implies the existence of delocalized states, gave new impetusto the study of the 2D metal-to-insulator transition at B = 0.


The crucial point is that some states within the broadened LLs will be lo-calized and the others will be extended. It turns out that there is exactlyone extended state per LL, so that the filling factor ν equals the number ofoccupied extended states. These extended states are located at EN,σ. Thebasic picture of the integral QHE is as follows. The Hall resistance is quan-tized whenever EF lies in a mobility gap, i.e., a region of localized states 14.When EF = EN,σ, the Hall resistance jumps from one quantized plateau tothe next 15. As long as EF moves through a region of localized states, thelongitudinal resistance keeps its value, which is essentially zero. The peaksobserved in the RL curves correspond to maximal electron scattering, whichoccurs when EF coincides to EN,σ.

One may succinctly summarize the integral QHE in the followingway [27]. As EF passes through the ”critical” energy EN,σ there is a”insulator-metal-insulator” transition (at T = 0), with the 2DES metal-lic precisely at EF = EN,σ. The ”insulator-metal-insulator” transition canbe understood as a percolation transition. As the percolation level is ap-proached, the edge states on the two sides of the sample will begin takingdetours deeper and deeper into the bulk and begin to communicate witheach other. When the percolation transition occurs edge states become partof the bulk and the sample is a normal metal.

Finally, we call attention to the fact that in transport effects the local-ized and the extended states are both of crucial meaning. For the equi-librium properties, such as specific heat, the difference between localisedand extended states is of no importance, i.e., equilibrium properties dependprimarily on the total DOS [26, 70].

14A short note on the edge states is in order here. To understand edges it is necessaryto realize that they are normal metals with dissipation. In a real sample, the mobilitygap collapses in a complicated way near the edge of the sample, so as to make the edge anormal metal with a well defined chemical potential. The detailed nature of the edge isunimportant, as long as the net result of a QHE gedanken experiment, in which the 2DESis bent into a loop, is the transfer of electrons from the local Fermi level at one edge tothe local Fermi level at the other, without dissipating energy. Laughlin’s gauge argumentfor the integral QHE [66] relates the quantized Hall resistance to the charge of electronstransferred in a gedanken experiment.

15The dependence of the plateau width upon the electronic mobility suggests that QHEplateaus originate from a mobility gap.


Figure 1.12: Single-particle DOS vs energy of a 2DES in a quantizing mag-netic field. (Top) Disorder free 2DES. (Bottom) Realistic 2DES. The shadedregions consist of localized states. Gaussian broadening of LLs was assumed.

1.4.3 Fractional QHE

What is common to the integer and the fractional QHE is the formation ofplateaus in the Hall resistance concomitantly with a vanishing longitudinalresistance as T → 0. The fact that RL has an activated behavior with T inthe plateau region indicates the existence of a finite excitation gap betweenthe QHE ground state and its excited states. The explanation of the frac-tional QHE can not be sustained without basing it upon the existence ofa many-particle excitation gap, whose physical origin differs fundamentallyfrom that of the integer QHE excitation gap. Previously, in explaining theinteger QHE, we started with a single-particle Hamiltonian. Building upon


the non-interacting electron picture, we considered the effect of the disorderpotential on the energy spectrum. With the formation of disorder broadened(spin-split) LLs we arrested the necessary (single-particle) excitation gap.Then, we argued that whenever the chemical potential lies within a mobilitygap which occurs at integer filling factor ν = i, the Hall resistance dependsonly on i through h/(ie2). At low filling factors the 2DES enters a regimewhere, in the absence of disorder, its ground state is determined entirelyby electron-electron interactions 16. For ν < 1 the meaning of this is clearenough: the kinetic energy is an irrelevant constant and the Hamiltonianhas only one energy scale set by the Coulomb interaction 17. Therefore,for understanding the fractional QHE one should first consider the effect ofinteractions between electrons. The many-particle Hamiltonian relevant forthe fractional QHE was given by Laughlin in his celebrated paper on theAnomalous Quantum Hall Effect: An Incompressible Liquid with Fraction-ally Charged Excitations [67]. With a priori unjustifiable omission of theZeeman term, it writes as

HFQHE =1

2m∗

N∑i=1

[pi + eAi(ri)

]2 +12

N∑i6=j=1

Ve−e(ri − rj), (1.17)

where N is the number of electrons, the potential is the Coulomb repulsionenergy Ve−e(r) = e2/(ε|r|) (in CGS), and the disorder, which would enterthrough an additional potential energy term, is ignored. Laughlin proposeda many-electron (spin-polarized) variational wave function of the form

ΨLm(z1, z2, . . .) = ΨL

m[z] = Pm[z]N∏

k=1

e−14|zk|2

=N∏i<j

(zi − zj)mN∏

k=1

e−14|zk|2 .

(1.18)

The Laughlin wave function and plasma analogy formalism describe stablestates of the 2DES, for which m is related to the filling factor throughν = 1/m. The value ofm in Eq. (1.18) can be deduced by noticing that Pm[z]must change sign when any pair of particles are interchanged. The Pauli

16The strength of the disorder potential in our samples is much smaller than EC . Onlysuch low-disorder (clean) samples are promising hunting grounds for the fractional QHE.

17Coulomb repulsion between electrons causes them to condense into highly correlatedmany-body ground states. The sequence of fractional QHE ground states terminates in ainsulating state (presumably a Wigner crystal).


exclusion principle requires that m is a positive odd integer (m = 1, 3, 5, . . .),and since ν = 1/m, the fractional QHE may occur at odd-denominatorfilling factors (ν = 1/3, 1/5, . . .). Thus, the most striking feature of thetheory is the implication that, for an ideal, spinless 2DES with no impurityscattering, there exists at T = 0 a sequence of fractional QHE ground statesat ν = 1/m, with m an odd integer. Experimentally, the fractional QHEoccurs at a large number of other filling factors. Theoretical studies haveshown that the ν = 2/3 fractional QHE state is an electronic conjugate ofthe ν = 1/3 fractional QHE state. Invoking the particle-hole symmetry inthe lowest LL, the state at ν = 1 − 1/m is composed of holes behavingas electrons at ν = 1/m. Furthermore, with the inclusion of spin, theparticle-hole symmetry dictates that fractional QHE states at ν and 2 − νare identical. Thus, the fractional QHE state at ν = 4/3 is expected tohave the same phenomenology as the ν = 2/3 fractional QHE state. Notethat all prominent fractional quantum numbers (less than unity) observedin our samples consists of two hierarchies of continued-fraction filling factorsderived from 1/3 and 2/3

i

2i+ 1=

13,25, . . .

i

2i− 1=

23,35, . . . ,

where i is a positive integer.By the internal logic of his argument, the logic of the trial wave function,

Laughlin was led to the conclusion that any elementary charged excitation inthe fractional QHE has to carry a fractional charge and lies at a finite energyabove the many-body ground state 18. The quasiparticle bands, separated inthe disorder-free case by the (many-particle) excitation gap ∆ν , are broad-ened in the presence of disorder into a continuum consisting of two bandsof extended states separated by a band of localized ones. For the fractionalQHE state at ν = 1/m, the Hall resistivity is related by gauge invariance tothe charge of the quasiparticles e∗ = e/m through RH = h/(ee∗). Wheneverthe chemical potential lies in the localized state quasiparticle band, RH isquantized to RH = h/[(1/m)e2]. This result remains valid for arbitrary frac-tional quantum numbers, i.e., when 1/m→ p/q. The theoretical excitation

18What is necessary to explain the fractional QHE is a downward cusp in the total energyof the system at particular fractional filling factors. The existence of a cusp in the totalenergy implies a discontinuity in the chemical potential, which implies, in turn, that thereis a gap in the spectrum of charge carrying excitations. At densities where the chemicalpotential has a discontinuity the system is said to be incompressible.


gap for a pair of one free quasielectron and one free quasihole at ν = p/q is

∆tν∼= Cq

e2

εlB. (1.19)

This should be compared with the QHE excitation gap determined in trans-port experiments 19. The prefactor Cq depends on the denominator q and ismodel dependent. For ν = 1/3, 2/3, 4/3, . . . theories have been convergingtoward a value of C3 ≈ 0.05 − 0.1, whereas for ν = 2/5 or 3/5 theoreticalestimates give C5 ≈ 0.015 − 0.030. Estimates for the QHE excitation gaps(∆t

ν) at prominent fractions in sample M280 are given in Table (1.5).The construction of Laughlin quasiparticles is very easy to visualize and

predicts a remarkably successful phenomenology. Imagine piercing the in-compressible 2DES at filling factor ν = 1/m with a infinitely thin solenoidlocated at a point we take to be the origin, as illustrated in Fig. (1.13).Then, the flux through the solenoid is adiabatically changed from 0 to Φ0.Once an entire flux quantum has been added, the Hamiltonian has, up toa gauge transformation, evolved back to its value at zero flux. Thus, thesolenoid threaded by Φ0 may be gauged away; the state generated by thisprocess (if the ground state has no degeneracy) is an exact excited state ofthe original Hamiltonian. Furthermore, one can show that this state hascharge e/m added to an area surrounding the origin. When magnetic flux isadded through a disk of radius R of the system, an azimuthal electric fieldEφ(t) = − 1

2πRdΦdt is generated by virtue of Faraday’s law. This electric field

drives a radial current jr(t) = σHEΦ(t) = − 1m

e2

hdΦ

(2πR)dt , where σH = 1m

e2

his the quantized Hall conductance. As the time derivative of the chargeaccumulated inside the disk is −

∮jr(t)dr, the total charge added for one

flux quantum increase is e/m. Since electric neutrality has to be preserveda charge −e/m also appears outside the disk. To be precise, threading oneflux quantum generates one quasiparticle plus one quasihole. The charge ofthe quasihole is e/m. The change in total energy when one flux quantum

19Thermodynamic (∆µν ) and transport (∆ν) excitation gaps are related by ∆µ

ν = m∆ν

at filling fraction ν = 1/m. This follows from the fractionally charged nature of the quasi-particle/quasihole pairs which control the current response of the 2DES at fractional fillingfactors. Since the addition of a real electron to the ν = 1/m state generates m quasiparti-cles, thermodynamic gaps are m times larger than transport gaps. Thermodynamic gapsare usually inferred from the measured magnetization jumps [112]. As soon as the 2DEShas two compressible regions (alias the edges), separated by a incompressible region per-colating through the sample, a deep minimum is observed in the longitudinal resistance.The change in the potential across the incompressible region is ∆µ

ν , which is related tothe magnetization jump by the relation ∆Mν = νA∆µ

ν /Φ0. For samples investigated inthis thesis, we expect that 70% of the sample area is incompressible at ν = 1/3.


is added to the system is 1/m times smaller than the energy necessary toremove an electron.

Figure 1.13: Illustration of the Laughlin’s gedanken experiment leading tofractional charge quasiparticle states at fractional LL filling factor ν = 1/3.

We are now in a position to consider (at a qualitative level) the nature of col-lective modes in the QHE. Collective excitations of the QHE states relatedto the charge degree of freedom are represented by density excitations. Inthese excitations a particle (or quasiparticle) is promoted to an empty stateleaving behind an oppositely charged hole (or quasihole). For a wavevectorq there is a displacement between the centers of the two cyclotron orbitsthat represent the single-particle states, given by lq = ql2B. Collective modes(labelled by q) are constructed with such a pair as a basis. The collec-tive excitations can be regarded thus, as magnetic excitons with an averagedipole length lq. The inter-Landau-level excitations involve the transfer ofelectrons from one LL to next higher LL, whereas in intra-Landau-level ex-citations, electrons do not change Landau level quantum number. Theseintra-Landau-level excitations reside lower in energy and are of principalinterest in the fractional QHE.

1.5. INTERLUDE: GROUND STATE THEORY AT ν = 1 33

1.5 Interlude: Ground state theory at ν = 1

If we digress now, it is in order not to lose information. We list the relevantenergy scales for the QHE in Table (1.3). In vacuum, neglecting the quantumelectrodynamics corrections, EZ = E∗, but this is no longer valid for 2Delectrons in GaAs QWs. The cyclotron splitting, due to the small effectivemass of electrons in GaAs, is increased by a factor of me/m

∗ ≈ 14, whereasthe applied magnetic field scarcely couples to the spins due to the strongspin-orbit interaction which reduces the electron g-factor by ge/|g∗| ≈ 5times. The resulting Zeeman splitting, is thus some 70 times smaller thanthe cyclotron splitting. There is a net decoupling of the scales of orbitaland spin energies. At low temperatures, it is possible for the 2DES to bein a regime where the orbital motion is fully quantized (kBT ~ωc) butthe low-energy spin fluctuations are not completely frozen out (EZ ≈ kBT ).The low energy spin dynamics of the 2DES is particularly interesting atν = 1. Since the Coulomb energy scale dominates at ν = 1, the systemspontaneously polarizes at T = 0 to minimize its energy. The 2D electronspin polarization is complete as the kinetic energy has been quenched by themagnetic field 20.

The 2DES at ν = 1 is our incompressible quantum fluid prototype.Clearly, itinerant ferromagnetism turns out to be a fundamental issue atν = 1 and we shall have occasion later on to study it more carefully 21.Here we intend to complete the specification of the Laughlin state at ν = 1by inclusion of the spin degree of freedom. While the Coulomb force isspin independent, exchange effects lead to additional ”complications” inthe QHE. Pauli exclusion principle requires that the total many-body wavefunction Ψ[z], which is the product between the orbital ΨL

1 [z] and the spinpart |Ξs〉, changes sign under the simultaneous interchange of both space

20The reader should be persuaded that there is no kinetic energy cost associated withthis electron spin polarization. The situation is different for a conventional ferromagnetor Fermi gas of electrons. Please think at the criterion for a fully polarized ground statein a 3D homogeneous electron gas at zero magnetic field. The ground state energy forferromagnetic alignment of electron spins is

N»3

522/3EF −

3

4π21/3(e2kF )

–.

The first term is the kinetic energy and the second term represents the Coulomb exchangeenergy. Only once the exchange term dominates the kinetic term the ferromagnetic groundstate is self-consistent.

21The electron spin stiffness, which reflects the ordering of the electron spins, is anenergy scale closely related to EC (see Chapter 2).


and spin coordinates. At ν = 1 the symmetric spin part is

|Ξs〉 = | ↑1↑2 . . . ↑N 〉, (1.20)

and the antisymmetric orbital part writes as

ΨL1 [z] =

N∏i<j

(zi − zj)N∏

k=1

e−14|zk|2 . (1.21)

Table 1.3: Estimates of relevant energy scales for samples M242 and M280are given. They are based on values of the perpendicular magnetic fieldcomponent at Landau level filling factor ν = 1 [5.9 T for M242 sample and3.6 T for M280 sample] and electron mobility µ0 values at zero magneticfield [3 × 105 cm2 · V−1 · s−1 for M242 sample and 7 × 105 cm2 · V−1 · s−1

for M280 sample]. Once the different energy scales have been identified, itis important to convert them to the same units in order to compare themeasily. Here, all energies are given in K, |g∗| = 0.44 is the effective g-factor ofelectrons in bulk GaAs, m∗ is the effective mass of electrons in bulk GaAs,µB is the Bohr magneton, ε is the dielectric constant in bulk GaAs, B⊥ is theperpendicular component of the applied magnetic field B, lB = [~/(eB⊥)]1/2

is the magnetic length, and ωc = eB⊥/m∗. Rough estimates of important

dimensionless parameters are also given.

Energy Scale Expression M242 M280

Zeeman energy a EZ = |g∗|µBB 1.8 1.1

Coulomb energy b EC = e2/(εlB) 121 95

Electron spin stiffness ρs = e2/(16√

2πεlB) 3 2

Cyclotron energy c E∗ = ~ωc 118 72

Disorder broadening Γ0 = ~/τ0 0.7 0.3

rs EC/E∗ 1 1

g EZ/EC 0.01 0.01

a Expressed in K, EZ ≈ 0.3×B [T].b Expressed in K, EC ≈ 50× (B⊥ [T])1/2.c Expressed in K, E∗ ≈ 20×B⊥ [T].

1.6. SUPPLEMENTAL ASPECTS OF THE QHE 35

Remarkably, at ν = 1, the Coulomb exchange energy per particle can becalculated exactly. It equals −(π/8)1/2EC . Consequently, a quasiparticle-quasihole Laughlin pair at ν = 1 causes a loss of exchange energy equalto Exc = (π/2)1/2EC . However, as spin and charge degrees of freedom areconnected at ν = 1, we need a more sophisticated mathematical languageto describe the lowest energy charged excitations in the system 22.

1.6 Supplemental aspects of the QHE

1.6.1 Finite thickness corrections

In our discussion thus far, we have ignored the finite spread of the electronwave function perpendicular to the 2DES plane. One of the most commonapproximate forms assumed for the calculated charge profile in a QW isthe Gaussian distribution |ψ0(z)|2 = (2π)−1/2w−1

0 exp [−z2/(2w20)], where

w0 is the rms width. In our samples, w0 is about a factor of 2 smallerthan lB at ν = 1. When w0 is comparable to the magnetic length, the finitethickness corrections are expected to have a substantial effect on the physicalquantities of interest, such as the QHE excitation gap or the ground-stateenergy. The finite thickness of the 2DES softens the short-range divergenceof the bare Coulomb interaction and hence, reduces both the cohesive energyof the ground state and the QHE excitation gap. In order to ascertain thedetrimental influence of the finite spread of the electron wave function on theQHE excitation gap and ground-state energy, the bare Coulomb interactionis replaced by a more realistic force law of the form [23] where the effect ofthe density profile in the z-direction is integrated out

Vw(r2D) =∫ ∫ ∞

−∞

e−(z21+z2

2)/2w20

2πw20

1√r22D + (z1 − z2)2

dz1 dz2. (1.22)

Here, r2D is the 2D vector separating the two electrons, bare Coulomb in-teractions correspond to w0 = 0, and the energy is expressed in units of EC .While the reduction of the ground-state energy still leaves the incompress-ible quantum-fluid state energetically favorable over the crystalline state,the reduction of the QHE excitation gap is important for comparison withthe experiment [17]. Theoretical results for the QHE excitation gap at ν = 1

22Note that, when only the charge degree of freedom is taken into account, the dispersionrelation for the collective neutral mode at ν = 1 for large q is E(q) = E∗ +Exc− e2/(ql2B).At small q, E(q) = E∗ + e2q/2.


in sample M280, which take into account the finite thickness of the 2DES,will be presented in the next chapter.

1.6.2 Landau level mixing

The fractional QHE is observed at relatively large B and believed to occur inthe limit B →∞. In this limit, since the Coulomb energy (∼ B1/2) is smallcompared to the cyclotron energy (∼ B), the LL mixing could be safelyneglected. The degree of LL mixing can be expressed by the dimensionlessparameter rs, which is the ratio of the Coulomb energy to the cyclotronenergy. As rs ∝ B−1/2, the degree of the LL mixing raises by decreasingthe magnetic field. In order to make the discussion more quantitative andfor further reference, we note that in our samples rs at ν = 1 is of theorder of unity (rs & 1). Clearly, rs can not act as a small parameter for aperturbation expansion. The LL mixing is critical for real 2DESs and mustbe incorporated into realistic theories.

In the presence of LL mixing several aspects of the fractional QHE be-come questionable. Among the most worrisome are the existence of thefractional charge, the validity of the particle-hole symmetry, and the mag-nitude of the QHE excitation gap. Laughlin offered a lucid explanation forthe exactness of the fractional quantization in the presence of extreme LLmixing. The failure of the particle-hole symmetry between the fractionalQHE states at ν and 1−ν due to LL mixing, was first investigated by Yosh-ioka [18]. He computed the QHE excitation gaps at ν = 1/3 and 2/3 andfound a weak effect (≈ 10%), which is rather inconsistent with the largeasymmetry (∆1/3/∆2/3 ≈ 3) usually observed in experiments.

A real breakthrough in the subject came with Monte Carlo studies [76],which calculate the QHE excitation gap at various filling factors by takinginto account both the LL mixing and the finite thickness of the 2DES. (Weshall tackle the case of filling factor unity in Chapter 2). Results obtained atν = 1/3 by Melik-Alaverdian et al. [76] show that the LL mixing has a weakereffect on the spin-reversed quasiparticle excitation gap but has a strong effecton the spin-polarized quasiparticle excitation gap. Interestingly, when onefirst includes the finite thickness of the 2DES, the additional correction dueto the LL mixing is essentially the same for both spin-polarized and spin-reversed quasiparticle excitation gaps.


1.6.3 Spin and the fractional QHE

Now how did the electron spin and the fractional QHE come to be con-nected? In the limit of large Zeeman energy, all electrons have their spinsaligned with the magnetic field and the spin degree of freedom is irrelevantfor the fractional QHE. Halperin was the first to point out, in a prescientpaper [48], that for 2DESs in GaAs QWs, in most of the experimental cases,EZ EC , E∗ and hence, it makes sense to re-examine the assumption ofcomplete electron spin polarization in the lowest LL. He went on to pro-pose a family of generalized Laughlin wave functions that could incorporatereversed spins. The spin-unpolarized QHE state at ν = 2/(m+ p) writes as

Ψm,m′,p[z] =∏i<j

(zi − zj)m∏α<β

(zα − zβ)m′∏j

∏α

(zj − zα)p

∏j

e−14|zj |2

∏α

e−14|zα|2 ,

(1.23)

where Latin and Greek indices correspond to electrons with different spinstates. This wave function is characterized by the exponents m, m′ (positiveintegers) and p (non-negative integer) describing the relative angular mo-mentum between species of the same spin and different spin, respectively.In particular, a completely unpolarized wave function was constructed atν = 2/5 (m = 3, m′ = 3, p = 2). Considering the spin-up electronsand spin-down electrons as two different species of particles and using atwo-component classical plasma approach, Chakraborthy and Zhang [16]calculated the ground state energy for the ”two-spin” fractional QHE stateat ν = 2/5. With this result a very intriguing possibility to observe a spin-reversed fractional QHE ground state was established. It indicated that, al-though in the limit of large Zeeman energy the fully polarized ground stateis favored energetically, there always exists a possibility for a transition toa not fully polarized state at lower Zeeman energy.

In addition to the ground-state spin configuration, spin should play arole in determining the spectrum of excited states in QHE systems. Forinstance, it was predicted that, while the ν = 1/3 QHE ground state is spinpolarized even for vanishing Zeeman energy, its lowest-energy elementaryexcitations are spin-reversed quasiparticles at low B [17]. Theorists alsoinquired how the QHE excitation gap will vary with the total magneticfield. For example, at low B, the lowest-energy excitations at ν = 1/3were identified to be pairs of spin-reversed quasielectrons and spin-polarizedquasiholes. Accordingly, the QHE excitation gap rises linearly because of thedominant Zeeman energy contribution. AsB is increased further, a crossover


point is reached beyond which a fully polarized quasihole – quasielectron pairis energetically favored, and a B1/2 dependence is obtained.

We hasten to point out that the standard approach used in magneto-transport experiments to gain access to the spin degree of freedom is thetilted magnetic field technique, introduced by Fang and Stiles in their studyof the g-factor in Si inversion layers [35] and first applied to the QHE byHaug et al. [49]. In essence, the QHE excitation gap ∆ν is measured at fixedν for various tilt angles. A straightforward interpretation of the results isoffered if we write

∆ν = ∆0ν − |g∗|µBB∆S, (1.24)

where ∆S (in units of ~) is the difference in spin of the quantum fluid statebefore and after the excitation of the quasielectron-quasihole pair. It isassumed that the total magnetic field couples to the spin degrees of freedom,while the perpendicular magnetic field component B⊥ controls the orbitaldynamics. The term ∆0

ν is the contribution to the gap from all non-Zeemansources and, in this model, depends only upon B⊥. Thus, the quantity ∆0

ν

which is in general unknown, is an irrelevant constant in QHE excitationgap studies at fixed ν as a function of the tilt angle.

Table 1.4: Zeeman energy contribution to the excitation gap of a spin-polarized QHE liquid | ↑↑ . . . ↑〉 for various combinations of one quasielectronand one quasihole with different spin-1

2 polarizations.

Quasihole Quasielectron ∆S

(1) (↑) spin polarized (↑) spin polarized 0

(2) (↑) spin polarized (↓) spin reversed -1

(3) (↓) spin reversed (↑) spin polarized -1

(4) (↓) spin reversed (↓) spin reversed -2

Let’s examine the tilted magnetic field dependence of the QHE excitationgap in some detail, confining ourselves to the case when the quasiparticlesare fundamentally spin-1

2 objects [Table (1.4)]. One can distinguish fourcases: (1) both the quasielectron and quasihole spins are polarized just likethe parent fluid, (2) the quasielectron spin is reversed but the quasiholeremains polarized, (3) reverse of case 2, and (4) both quasiparticle spins arereversed. For a spin-polarized QHE liquid, ∆S = 0 for case 1, ∆S = −1


for both cases 2 and 3, and ∆S = −2 for case 4, implying that ∆ν remainsthe same or it may increase when the tilt angle is increased. On the otherhand, if the QHE liquid is partially spin-polarized it is possible for the netspin to increases upon quasielectron – quasihole pair excitation (∆S > 0).In this case ∆ν will decrease when the tilt angle is increased, and eventuallyvanishes. The above discussion is well-adapted to the purposes of this thesis;it will be of great service for the case at hand of tilted magnetic field QHEexcitation gap measurements.

1.6.4 QHEs in tilted magnetic fields

The assumption that the parallel component of the magnetic field enters theproblem only through the Zeeman energy term is valid only if the 2DES isinfinitely thin. Surely, in real experimental situations, the 2DES has a finitewidth. The introduction of a magnetic field component parallel to the 2DESwill influence the ground subband wave function ψ0(z) of the confined 2DES.Tilting the sample in a magnetic field, squeezes the wave function in thez-direction, thereby making the 2DES more two-dimensional. Variationalcalculations suggest that, for samples studied here, ψ0(z) will almost halveits z-extension in a parallel field component of 20 T [49].

The situation is, in fact, more complex then it appears at first sight.For the particular situation of a magnetic field perpendicular to the 2DES,the quantum mechanical problem for an electron was separated into out-of-plane and in-plane motions and simple expressions for the eigenenergieswere obtained. For tilt angles θ 6= 0, the motions in the confinement planeand perpendicular to it are coupled. Additionally, assuming an arbitrary 1Dconfinement potential, the model is not analytically soluble. The resultingeigenenergies are labelled by one quantum number plus the spin variable (jand σ), instead of r, N , and σ. Theoretically, the single-electron problemin tilted magnetic fields was investigated by several authors [18]. Halonenet al. [47] considered the influence of subband-LL coupling on the fractionalQHE excitation gaps.

In the following, we briefly remark on how the LL separation dependswith the tilt angle in samples M242 and M280. Elaborate energy levelnumerical calculations were performed by Jungwirth and MacDonald. Theseauthors went beyond the Hartree approximation by including the exchangeand correlation effects within the local density approximation scheme andtake into account the possibility of an in-plane magnetic field.


Figure 1.14: Numerical calculation for the Landau level (LL) crossing be-tween the lowest spin-down LL and the first spin-up LL in tilted magneticfields [sample M242 (dashed curve) and sample M280 (dash-dotted curve)].[Courtesy of T. Jungwirth]

Referring to Fig. (1.6), we are interested in the crossing between the ”secondLL for majority spins” (j = 1, σ = 1) and the ”first LL for minority spins”(j = 0, σ = −1). The energy separation between these two levels is ∆01

and its evolution with the tilt angle is shown in Fig. (1.14) for ν = 1. Thetheory predicts that the tilt angle where this LL crossing occurs is slightlyhigher than θ = 70 for both M242 and M280 samples. This value is weaklydependent on the form of the exchange correlation potential and is ratherinsensitive for 0.8 . ν . 1. Due to this possible LL crossing, great care hasto be exercised when analyzing tilted-magnetic field experiments at high θ.


Figure 1.15: Longitudinal resistance RL vs filling factor (ν ≤ 2) at T =50 mK and indicated tilt angles (sample M242). Curves are offset for clarity.


Figure 1.16: Longitudinal resistance RL vs filling factor (ν ≤ 2) at T =50 mK and indicated tilt angles (sample M280). Curves are offset for clarity.

1.7. MAGNETOTRANSPORT: RESULTS AND DISCUSSION 43

1.7 Magnetotransport: Results and discussion

1.7.1 Measurements in tilted magnetic fields

The following magneto-transport measurements describe the evolution of theQHE with the tilt angle for ν . 2 in our samples. Figures (1.15) and (1.16)show magnetoresistance traces taken at T = 50 mK and various tilt anglesin samples M242 and M280, respectively. A rather complex structure ofthe fractional QHE for ν . 2 has been observed in previous tilted magneticfield studies on single heterojunctions [21, 31, 32]. In the present data,several fractional QHE states are readily detectable: ν = 2/3, 4/3, and5/3 in sample M280 [Fig. (1.16)], and ν = 2/3 and 4/3 in sample M242[Fig. (1.15)]. Only very small changes occurred by tilting the samples aroundν = 5/3. The RL minima at ν = 2/3 and ν = 1 became deeper in the caseof tilted samples. With increasing the tilt angle we observe the developmentof a peak in the longitudinal resistance at ν . 0.8 instead of the weakminimum displayed at θ = 0. We believe that the weak structure in RL for2/3 . ν . 1 is simply ”consumed” by the widening of the adjacent QHEstates. This behavior is consistent with the standard model of disorder-driven QHE transitions [58]. On the other hand, for unpolarized QHE states(such as ν = 4/3), the increased total magnetic field is the main factor, ofmore significance than the effect of B‖ on the z-extent of the electron wavefunction. This is clearly reflected in our measurements which reveal that theν = 4/3 QHE state weakens with increasing the tilt angle and it is absentfor θ & 65. In sample M280 we found, in fact, a re-entrant behavior forthe ν = 4/3 QHE state, as it re-emerges for θ & 72 [21]. The systematicdifference observed in the behavior of ν = 2/3 and 4/3 QHE states, suggeststhe destruction of the electron-hole symmetry in the fractional QHE in tiltedmagnetic fields. The disparity between ν = 4/3 and ν = 5/3 states, believedto be identical if they occur at the same perpendicular magnetic field in the(fully polarized) hierarchical model for the fractional QHE, is also evident.

1.7.2 QHE excitation gap measurements

We address here another important aspect of the QHE, namely, how themeasured transport QHE excitation gaps compare quantitatively with thetheoretical predictions. The investigation of integer an fractional QHE exci-tation gaps has been one of the major objectives in the early studies of theQHE. The angular dependence of the excitation gaps has been intensivelyused to unravel the spin configurations of various fractional QHE statesas a function of the Zeeman energy. As the Zeeman energy is increased,


spin-unpolarized fractional QHE states undergo transitions ↑↓ → l → ↑↑,where ↑↓, l, and ↑↑ signify zero, partial, and full polarization. The ν = 4/3fractional QHE state provided the first indications for a spin-unpolarizedQHE ground state. Compelling evidence came from experiments of Clark etal. [21], which mapped out the destruction of the ν = 4/3 fractional QHE,followed by its re-emergence upon increasing the tilt angle.

The values of the QHE excitation gaps are usually determined from theT -dependence of the longitudinal resistance at QHE minima 23. The frame-work which is useful in analyzing and interpreting the activation energy datahas been excellently summarized by Boebinger et al. [13]. To extract theQHE excitation gap ∆ν (expressed in K) at filling factor ν we use the ex-pression RL(T ) ∝ exp (−∆ν/2T ) to produce Arrhenius plots [lnRL vs 1/T ].Then, we perform linear fits to the midsection of these plots [Figs. (1.17)and (1.18)]. Physically, this procedure is equivalent to assume that the con-duction mechanism is by thermal activation of charged quasiparticles acrossthe QHE excitation gap 24. Arrhenius plots of data always develop a down-ward curvature at high T , as a consequence of a high density of excitations.Additionally, the data can show an upward curvature at low T , so that thefull plot is S shaped 25. The upward curvature is attributed to hoppingconduction via localized states near the Fermi level. Meaningful fits usingthe sum of two conduction mechanisms are not possible due to the reduceddynamic range of the data. Figure (1.18) shows results from an experimenton sample M242 in which the T -dependence of the longitudinal resistancewas determined at ν = 4/3. This was accomplished in two equivalent ways- either by systematic isothermal magnetic field sweeps to pick off the RL

minimum, or by fixing the magnetic field at the minimum, and changing thetemperature. Our raw data was mainly collected by sweeping the mixingchamber temperature. In the present experiments, the RL(T ) dependenciesare measured over a range of temperature within the practical limit of ourdilution refrigerator: 40 mK . T . 2 K. We note that deviations in RL

from a simple activated behavior mainly originate from the following facts:(1) the weakly developed minimum resides on top of a weakly T -dependentbackground, (2) the fixed B differs from the exact value corresponding to

23We assume that the nuclear magnetization of GaAs QWs does not affect the QHEexcitation gaps through the hyperfine interaction (see Chapter 4).

24The longitudinal conductivity σL [and, consequently ρL; due to ρL = σL/(σ2L +

σ2H) ≈ σL/σ2

H ∝ σL] is expected to be proportional to the number of thermally activated,electrically charged quasiparticles.

25At low T , for samples mounted in vacuo and in good thermal contact to the 3He/4Hemixture this upward curvature is conspicuously lacking.

1.7. THE EXCITATION GAP 45

Figure 1.17: Arrhenius plots of the longitudinal resistance RL for QHEstates at (a) ν = 1 and (b) ν = 1/3 (squares) and ν = 2/3 (solid curve).Dashed lines represent fits to the activated regions of the data, from whichthe excitation gaps ∆ν were determined.


Figure 1.18: (a) RL vs B at various fixed temperatures. (b) Arrhenius plotsof RL at ν = 4/3 [discrete T (circles) and continuous T (solid curve)].


Table 1.5: The measured transport gaps ∆ν for prominent QHE states atvarious tilt angles are given in K (sample M280). Theoretical estimates ∆t

ν

[Eq. (1.19) with C3 = 0.1 and C5 = 0.03] are also included.

Landau level filling factor ν

1/3 2/5 2/3 1 4/3

∆ν(θ = 0) 4.8 1.2 1.1 20 0.4

∆ν(θ = 30) 4.6 - 1.6 22 0.4

∆ν(θ = 45) - - 2.2 26 0.3

∆ν(θ = 60) - - 2.9 30 -

∆tν 16 4.8 12 124 8

the longitudinal resistivity minima 26, (3) errors in the temperature readinginduced either by too fast warming up of the mixing chamber either bytoo strong level of sample’s excitation. The uncertainty in the measuredtransport gaps is ±10% at low θ but increases to ±30% at high θ. InTable (1.5) we have listed the measured excitation gap of several QHE statesat various tilt angles for sample M280. The most consequential amongthe investigated QHE excitation gaps is that at ν = 1. During the late1980s, QHE excitation gap studies at ν = 1 clearly established that theCoulomb energy plays a dominant role, leading to a substantially largerQHE excitation gap than the single-particle Zeeman splitting [83, 104]. Thisaspect, of relevance to the nature of the QHE excitation gap at ν = 1, willbe addressed separately, in the next chapter. The first topic to be addressedhere is the evolution of the QHE excitation gaps at the thirds with θ. Datacollected at ν = 2/3 shows that the QHE excitation gap rises with increasingZeeman energy, whereas at ν = 1/3 the QHE excitation gap stays constantupon increasing the Zeeman energy. This different angular dependence forν = 1/3 and 2/3 is explained by the theory of subband-LL coupling for thefractional QHE in tilted magnetic fields [47]. A dissimilar evolution with thetilt angle is equally observed for ν = 4/3 and 5/3 fractions. We tentativelyconclude that ν = 1/3, 2/3, and 5/3 states preserve the same configuration↑↑ for all tilt angles, whereas the configuration for ν = 4/3 at low θ is ↑↓.

26Hysteresis effects in the superconducting magnet cause ±0.02 T unrepeatability in themagnetic field values.


Figure 1.19: (a) RL vs filling factor (1 ≤ ν ≤ 2) at T = 50 mK and selectedtilt angles (sample M280). Panel (b) shows the B-dependence of ∆4/3. Thegrey region indicates the B-range where the ν = 4/3 fractional QHE isabsent. The corresponding tilt angles are given on the top axis.


The reentrant behavior of the ν = 4/3 state is displayed in Fig. (1.19). De-struction of the 4/3 structure for θ & 65 is consistent with ↑↓ as increasingthe Zeeman energy forces the system into a different spin state (presumably,l). The 4/3 state collapses from ∆4/3 = 440 mK to zero over the magneticfield range 2.8 − 6.5 T, and it is absent for 6.5 . B . 8.8 T. Our observa-tions are too scant to speculate on the detailed nature of the reentrant 4/3state 27.

We now contrast the measured QHE excitation gaps at the thirds withthe theoretical predictions. The observed 3∆ is smaller by a factor of 4than the unbound quasiparticle-quasihole pair-creation energy gap 3∆t. Itis generally believed that this discrepancy stems from the combined impactof three important factors: the finite thickness of the 2DES, the LL mixing,and the ubiquitous presence of disorder. Since no calculation currently treatsthese three corrections simultaneously, we must consider them in cumula-tive succession. Existing calculations suggest that upon adequate inclusionof the finite thickness and LL mixing corrections, the QHE excitations gapswill be reduced by 50%. These corrected theoretical gaps overestimate ourexperimental gaps by a factor of 2. This large offset between theoretical andexperimental values is rather disturbing and incorporation of disorder intotheory remains a challenging task. Under the empirical assumption that dis-order leads to a magnetic field-independent gap reduction, all theoreticallyvalues need to be shifted down rigidly. Even though a rigid displacement∆Γ of about 3 K is able to simultaneously describe the measured QHE exci-tations gaps at 1/3, 2/3, and 4/3, it should be noted that this simple modelis questionable when ∆Γ ≈ ∆ν , for which thermal excitations do not yielda simple exponential behavior. Finally, we remark on the measured QHEexcitation gaps at ν = 2/5 and 3/5. Experimentally, we find at θ = 0 that∆2/5/∆1/3 ≈ ∆3/5/∆2/3 ≈ 0.25, compared with an expected ratio 5∆/3∆of 0.3 from simple theoretical considerations. In spite of the quantitativeinconsistency with the theoretical predictions, it is a pivotal aspect thatthe measured QHE excitation gaps compare well with those observed inconventional heterointerfaces. The realization of high-quality multiple-QWheterostructures is a magnificent development which made possible the birthof quantum Hall effect Skyrmions.

27There are indications that the scenario proposed by Clark et al. [21] directly appliesto our measurements. To complete the QHE picture for 1 < ν < 2, one may assign thedaughter states 8/5 and 7/5. The 7/5 state with e∗ = e/5 is absent on increasing Buntil the parent 4/3 transition in which e∗ changes to e/5. The proposed l 4/3 state is(14/15 ↑, 2/5 ↓). At high B, only the parent 5/3 and the reemergent 4/3 state remain.


1.8 Summary

In this chapter we have introduced the fabrication and physics of two-dimensional electron systems, along with some properties of electron-dopedGaAs/AlxGa1−xAs multiple-quantum-well heterostructures, interesting intheir own right.

We have detailed several basic magneto-transport experiments revealingthe quantization of the Hall effect. At certain integer (ν = i; i = 1, 2, . . .)and fractional (ν = p/q; p = 1, 2, . . . ; q = 3, 5, . . .) Landau level fillingfactors plateaus develop in the Hall resistance together with minima in thelongitudinal resistance. The value of the Hall resistance at the plateaush/(νe2) depends only on fundamental constants of the nature. I gave asuccinct overview of the rich phenomenology of the quantum Hall effects(QHEs) and briefly traced the history of the development of our theoreticalunderstanding of these remarkable low-temperature phenomena.

Many properties of the QHE states have been probed by magneto-transport measurements in recent years, and collectively, they make a cogentevidence for the electron-electron interaction playing an important role atodd Landau level filling factors (e.g., ν = 1). The results obtained in tiltedmagnetic fields in the fractional QHE regime add a new twist and are con-sistent with the observation of unpolarized (or partially polarized) fractionalQHE states. It is evident from our measurements that the spin polarizationof various fractional QHE states (e.g., ν = 4/3) varies with the Zeemanenergy, which favors fully polarized ground states.

The discussion initiated here will be continued in the next three chapters,which will address more particularly the physics of two-dimensional electronsystems around Landau level filling factor one.

Chapter 2

Quantum Hall EffectSkyrmions

Piara ziua în care m-am nascut si noaptea care a zis: un pruncde parte barbateasca s-a zamislit! Ziua aceea sa se faca întunericsi Domnului din cer sa nu-i pese de ea si lumina sa n-o mai lu-mineze. . . Sa se întunece stelele revarsatului zorilor ei; sa asteptelumina si nimic sa nu vina si sa nu mai vada genele aurorei. . .

Cartea lui Iov

2.1 General formulation

At Landau level filling factor ν = 1, the two-dimensional electron system(2DES) condenses into a fully ferromagnetic aligned quantum Hall effect(QHE) ground state: all electron magnetic moments line up to point in themagnetic field direction. Since spins are bound to mobile charges, the 2DEScould be viewed as an itinerant ferromagnet with a quantized Hall resistiv-ity: a QHE ferromagnet. What distinguishes the QHE ferromagnet fromother conventional ferromagnets is the physical nature of its low-lying ex-cited states. If the QHE ferromagnet is excited energetically, disturbing theelectron spins away from their mutually-parallel arrangement, a Skyrmion(charged spin-texture) results.

51

52 CHAPTER 2. QUANTUM HALL EFFECT SKYRMIONS

Skyrmions are not the only type of elementary excitations of the ν = 1 QHEferromagnet. We have seen in the precedent chapter that various elementaryexcitations related to charge degrees of freedom exist at ν = 1. This chapterwill be concerned with the nature of elementary excitations associated tospin degrees of freedom: What is the character of spin excitations (collective,topological,. . . )? Are they neutral or charged? Are they separated from theground state by an energy gap or not? The answer to these questions isbewildering as it matters if the system is precisely at ν = 1 or near ν = 1.

A brief survey of some central ideas of the theory of ferromagnetismis a necessary prerequisite for inquiring about spin excitations in the QHEregime and, in particular, QHE Skyrmions. The reason is that the theoryof ferromagnetism embraces exotic structures spawned by the alliance ofsymmetry and dimensionality. Additionally, these magnetic structures canbe strongly affected by topological constraints that arise from the globalsymmetry of magnetic moments. If we want to understand QHE Skyrmions,we must now scrutinize the part played in the theory of ferromagnetism byideas such as Symmetry, Dimensionality, Topology, . . .

2.1.1 Symmetry and ferromagnetism

Of all distinct forms of symmetry, the broken symmetry has, perhaps, themost profound implications in QHE physics. Here is the preface that PhilipW. Anderson, Nobel laureat who fabricated the concept of broken symmetry,placed before the description of the ”discovery”: Our interest is often focusedon the set of low-energy excited states of a system as the physically mostfundamental property of it. Relationships among symmetry and elementaryexcitation spectra, which can be lumped together under the name of ”theoryof broken symmetry”, play a vital role in the theory of almost all forms ofquantum condensation 1.

The relationship of broken symmetry to elementary excitation spectrumis the essence of Goldstone’s theorem 2: Any system with a broken continuoussymmetry has Goldstone modes. These gapless collective modes are space-time dependent oscillations in the order parameter field whose non-vanishingground state expectation value defines the broken symmetry. This theorem isvery general and its proof is rigorous. For a useful illustration of Goldstone’stheorem, let us take a matter about which none of us feel the slightest doubt:the isotropic ferromagnet. At zero magnetic field, isotropic ferromagnets are

1P.W. Anderson, Concepts in Solids, (World Scientific, Singapore, 1997), p. 175-182.2J. Goldstone, Field Theories with ”Superconductor” Solutions, Nuovo Cimento 19,

154 (1961).

2.1. GENERAL FORMULATION 53

described by the rotationally invariant Hamiltonian

Hisotropic = −J2

∑i6=j=1

si · sj , (2.1)

where J > 0 is the ferromagnetic exchange integral, si denotes the spinoperator of ith particle, the interaction between every pair of particles iscounted once, which accounts for the factor 1

2 , and the double sum runsover the indices i and j separately, excluding the value i = j. Above itsordering temperature T , the system has SO(3) symmetry and no permanentmagnetic moment. But when cooled below its ordering temperature, itbecomes spontaneously magnetized in a particular direction, determined byrandom spin fluctuations [Fig. (2.1)(a)]. The fully ferromagnetic groundstate has all spins aligned along the same particular direction and is clearlynot rotationally invariant: instead, it possesses SO(2) symmetry. In otherwords, the ground state ”spontaneously breaks” the continuous rotationalsymmetry. Indeed, there are infinitely many alternative ground states, inwhich all the spins are lined up together in different directions [Fig. (2.1)(b)].

Figure 2.1: (a) Ferromagnetic ground state; below the ordering temperatureall spins are aligned. (b) An alternative possible ground state. (c) Longwavelength excitation, involving rotation of neighboring spins.


This situation is accompanied by the presence of gapless collective modesby virtue of Goldstone’s theorem. As the total spin operator has a non-vanishing expectation value in the ground state, we can easily identify theorder parameter as being the local magnetization (alias the local spin den-sity). Thus, Goldstone modes correspond to space-time dependent oscil-lations in the local magnetization field. More precisely, Goldstone modesare spin-wave excitations for which the energy approaches zero in the longwavelength limit (λ→∞). Figure (2.1)(c) shows a spin wave of long wave-length and gives us a further bit of useful physical insight. Over regions ofsize a λ one has approximately a ground state with magnetization in aparticular direction. But corresponding to the existence of the spin waveexcitation, the spins will rotate from region to region, with a characteristicspatial period of order λ. Now it requires very little energy to bring aboutthis excitation provided the spin-spin forces are of finite range. In this caseone can always find a large enough λ such that the energy required to excitethe mode is essentially zero. The Goldstone’s theorem, so transparent inthis case, has profound consequences for QHE systems. In short, a brokencontinuous symmetry (rotational, translational,. . . ) signals the presence ofgapless elementary excitations.

2.1.2 Dimensionality and spin waves

Ferromagnetism in electron systems occurs not because of direct dipolarinteraction between electron magnetic moments, but rather derives from acombination of electrostatic forces, electron’s kinetic energy, spin, and statis-tics. It was Heisenberg who first realized that electron ferromagnetism isintrinsically a quantum many-body effect, and proposed the scenario thatspin-independent Coulomb interaction and Pauli exclusion principle gener-ate direct exchange interaction between electronic spins. Soon after, thephenomenon of exchange coupling among itinerant electrons was suggestedas having importance for magnetism by Bloch. The collective electron treat-ment of ferromagnetism enables, for example, an immediate interpretationfor the non-integral values of the atomic moments of ferromagnetic elemen-tal metals such as iron. In ferromagnetic elemental metals where a narrowd band is overlapping with a much larger s band, the ferromagnetism maybe attributed to the electrons in the partially filled band corresponding tothe d electron states. Nevertheless, the derivation of ferromagnetism from areasonable model of itinerant electrons is a challenging problem in theoreti-cal physics. A unique possibility is offered by the QHE ferromagnet at ν = 1to understand electron itinerant ferromagnetism without the complications


imposed by the band theory. The simple reason is that the lowest Landaulevel behaves like a band of zero width [53].

The fact that the lowest energy band in the single-electron spectrum isdispersionless, is not the only difference between the ferromagnetism of the2DES at ν = 1 and, let’s say, the ferromagnetism of a piece of bulk iron. Itis well known that the low-lying excited states of conventional ferromagnetshave spin-wave character and determine their thermodynamic properties atlow T . For example, the spontaneous magnetization at zero magnetic fielddepends on the form of the band, the magnitude of the exchange inter-action, and must be calculated on the basis of Fermi-Dirac statistics. Inthe spin-wave approximation, the magnetization approaches the saturationas T 3/2 (the so-called T 3/2-Bloch law) 3. For spin-wave phenomena, bothreduced dimensionality and finite magnetic fields become paramount. Forinstance, Mermin and Wagner rigorously demonstrated that there can beno spontaneous magnetization at any finite T for isotropic two-dimensional(2D) short-range spin models 4. With respect to spin-wave phenomena, fer-romagnetism in 2DESs at ν = 1 deviates radically from the ferromagnetismof bulk iron: in the former case, electrons experience a finite magnetic fieldB and their dynamics is two-dimensional.

In order to make the concept of spin waves in QHE ferromagnets moretangible, it is useful to elaborate first on ”standard” 2D spin-wave excita-tions. If JH > 0 is the ferromagnetic exchange integral, si denotes the spinoperator at the site Ri on the 2D lattice, and the Zeeman interaction is takeninto account, then the 2D insulating isotropic (Heisenberg) ferromagnet isdescribed by the Hamiltonian

HH = −JH

2

N∑i6=j=1

si · sj − EZ

N∑i=1

szi . (2.2)

Here we consider that the magnitude of spin is s, B lies along z-axis, EZ isthe Zeeman splitting, and the total number of spins is N . The ground state

3The ferromagnetic ideal survived the discovery of amorphous metals in 1960s, becauseferromagnetic exchange coupling does not depend in any fundamental way on the existenceof a crystalline lattice. Details may change in the amorphous state: the T -dependence ofthe spontaneous magnetization is modified owing to a distribution of exchange interac-tions; yet the critical behavior is not fundamentally altered. We also note that disorderin QHE systems plays a role similar to the band dispersion in the band theory of ferro-magnetism.

4N.D. Mermin and H. Wagner, Absence of Ferromagnetism and Antiferromagnetismin One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17, 1133(1966).


is the one with all spins aligned along z-axis and parallel to each other

HH| ↑1↑2 . . . ↑N 〉 = E0H| ↑1↑2 . . . ↑N 〉, (2.3)

and its energy E0H is

E0H = −JH

2

N∑i6=j=1

s2 − EZ

N∑i=1

s. (2.4)

To construct the low-lying excited states one starts by examining a state|Rj〉 differing from the ground state only in that the spin at site Rj has itscomponent along the B-axis reduced from s to s−1. This state is generatedby the so-called local spin-lowering operator s−(Rj)

|Rj〉 =1√2ss−(Rj)| ↑1↑2 . . . ↑j . . . ↑N 〉. (2.5)

By making a superposition of these states, we obtain a spin-wave state ofwavevector q (lying inside the Brillouin zone)

|q〉 ≡ S−q | ↑1↑2 . . . ↑N 〉 =1√N

N∑j=1

e−iq·Rj |Rj〉

=1√N

1√2s

N∑j=1

e−iq·Rjs−(Rj)| ↑1↑2 . . . ↑N 〉.

(2.6)

The excitation energy of the state |q〉, generated from the ground state byS−q [the Fourier transform of the local spin-lowering operator s−(Rj)] is

EH(q) = sJH

∑p

(1− e−iq·Rp

)+ EZ . (2.7)

Here∑

p represents the sum over the nearest-neighbor lattice vectors. Fora square 2D lattice of constant a, at small magnitude q of the wavevector,the dispersion of the collective mode is quadratic

EH(q) ≈ sJHq2a2 + EZ . (2.8)

If we wish to pursue the analysis further, and use Eq. (2.8) to calculatethe T -dependence of the magnetization at B = 0, we shall arrive at theremarkable result that at any finite T so many spin waves are excited thatthe magnetization is completely eliminated. The density of states of a branch


of excitations with energy EH(q) scales as q · dq and the occupancy of thelow-lying modes diverges logarithmically at q → 0. Any B 6= 0 can makethe frequency of the q = 0 spin-wave mode finite. Consequently, the T -dependence of the magnetization is essentially T ln (kBT/EZ), where EZ

acts as a cutoff energy. In the above example, the probability of lowered spinbeing found at a particular site Rj in the state |q〉 is 1/N ; i.e., the loweredspin is distributed with equal probability among all the spins. We shallsee soon that, although the concept of a spin wave in QHE ferromagnets isvery different from that of a single reversed spin distributed coherently overa large number of otherwise aligned spins, the dispersion of the spin-wavemode at small q is still quadratic.

2.1.3 The abomination of topology

The Kosterlitz-Thouless phase transition

According to the Mermin-Wagner theorem the conventional long range or-der (positional spin alignment) at finite T for rotationally invariant 2D spinmodels is prevented by thermal fluctuations. However, the Mermin-Wagnertheorem does not rule out the possibility of some other type of ordering orfinite-T phase transition in 2D systems. During the early 1970s, Kosterlitzand Thouless showed that a phase transition occurs at a finite tempera-ture (TKT) in 2D XY -ferromagnets with short-range interactions 5. Theseauthors considered the Hamiltonian for a system of planar spins with unitmodulus on a square 2D lattice

HKT = −JKT

∑〈ij〉

si · sj = −JKT

∑〈ij〉

cos (φi − φj), (2.9)

where JKT > 0 is the ferromagnetic exchange integral,∑

〈ij〉 means thatthe sum is taken over all pairs of nearest-neighbor sites, and sj is the spinoperator at the site j, represented by sj = cosφj ·ex +sinφj ·ey, with φj be-ing the in-plane orientation angle. The interesting physics is brought aboutby the vortices of XY spins. An isolated vortex in the XY -model, shownin Fig. (2.2), has several remarkable properties: (1) around the center ofthe vortex the spin scan rotate along closed contours with an angle 2mπ,

5J.M. Kosterlitz and D.J. Thouless, Ordering, Metastability, and Phase Transitions inTwo-Dimensional Systems, J. Phys. C: Solid State Phys. 6, 1181 (1973). At any finite Tthe total magnetization is zero. Yet, a clear qualitative change happens within the systemat T = TKT, signaled by different behaviors of the spin-spin correlation function belowand above TKT.


with m an integer number (the so-called vorticity or strength of the vor-tex), (2) the total magnetic moment of the system containing such a vortexstructure is zero, (3) the spin ordering clearly differs from random order-ing, and (4) moving along a vortex contour the angle deviations betweennearest-neighbor spins is small. The energy associated with the variation ofthe orientation angle in such a vortex structure can be written as

Evortex =JKT

2

∑〈ij〉

(φi − φj)2. (2.10)

For an isolated vortex of radius R, it turns out that

Evortex ≈ πJKT ln (R/a), (2.11)

Figure 2.2: Schematic illustration of an isolated vortex in the XY model.We can take the center of the vortex to be located on a dual lattice whosesites lie at the centers of the squares of the original lattice. The minimalcontour around the center of the vortex passes through only four sites. Thestrength of the vortex is m = 1.


where a is the vortex core radius (alias the lattice constant). At low T , thefree energy variation per single vortex is

δFvortex ≈ (πJKT − 2kBT ) ln (R/a). (2.12)

This quantity depends logarithmically on the size of the system, so that theconcentration of free vortices at low T is zero in the thermodynamic limit.The critical temperature associated with the creation of a single vortex isTKT ≈ [π/(2kB)]JKT, as δFvortex < 0 for T > TKT. The immediate questionthat comes in mind is what kind of phase exists below TKT? A very cleverway to see this is to draw an analogy between the system of vortices and a2D system of electrical charges, and notice that a pair of opposite charges -a dipole - has a finite total energy, because at large distances the electricalfields from each charge almost cancel. Thus, it is presumed that in the low-Tphase, vortices can only appear in bound pairs, with vorticities of differentsign (vortex – anti-vortex pairs) 6. As the density of vortex – anti-vortexpairs raises with the temperature, the pairs begin to ”screen” each other andeventually unbind. In the high-T phase, the system behaves like a plasmacomposed of an equal number of free, screened charge of each sign. Thisis the essence of the Kosterlitz-Thouless phase transition 7. This particularphysics - pertaining to phase transitions in 2D systems with (magnetic)short-range interactions, will appear a little while further, in our discussionof QHE ferromagnetism near ν = 1.

The nature of this type of phase transition is topological because themost prominent feature of the low-T phase is the presence of non-trivialtopological order, rather than the existence of a conventional order param-eter. As pointed out by Kosterlitz and Thouless, the idea of topologicalordering appeared for the first time in the dislocation theory of lattice melt-ing. In this theory, it is assumed that a liquid close to the freezing pointhas a local structure similar to that of a solid. In the liquid phase, there isa concentration of free dislocations, which can move under the influence ofan arbitrary small shear stress. Below the melting point, there are no freedislocations; the dislocations are coupled in pairs, so that the dislocations inthe same pair are described by equal and opposite Burgers vectors. There-fore, the solid-to-liquid phase transition is, in fact, a Kosterlitz-Thoulessphase transition between a state with dislocation pairs and a state with freetopological defects 8.

6As a consequence, below TKT the total magnetization of the system is indeed zero.7An important remark here is that TKT decreases when the number of particles per

unit area decreases.8The theory of lattice melting is much easier to apply in two dimensions than in tree


Classical treatment of a Skyrmion

Before sketching the topological nature of the Skyrmion, it is the breadthgenerality of the classical theories for the ferromagnetic state that I wishto stress. Let me say immediately that one of the main reasons for thecontinued interest in ferromagnetism is a rare opportunity of simultane-ously describing complex phenomena occurring in the same system both inclassical and quantum-mechanical terms. The classical approach takes thequantum mechanical concept of a spin and treats it as if it were a classicalvector. While the quantum theory proceeds from the formulation of thespin Hamiltonian, the principal assumption of the classical theory is thatthe ferromagnetic state is unambiguously describable 9 by the magnetiza-tion vector M. In general, the magnetization as a function of space andtime M(r, t) is a solution of the phenomenological equation

dMdt

= γM×Beff − ηM× dMdt

. (2.13)

The precession and the motional damping of the vector M are described bythe gyromagnetic factor γ and the relaxation constant η, respectively [59].The effective magnetic field in Eq. (2.13) is the variational derivative of theenergy functional with respect to the vector M

Beff =δE[M]δM

. (2.14)

This equation has the integral of the motion M2 = constant, consistent withthe assumption that the length of the vector M in a ferromagnet representsits equilibrium characteristic. In the ground state the value of M is thespontaneous total magnetization M0. The focus here is on the dynamicsof the magnetization when the time scales involved are such that M main-tains its ground state magnitude during the motion. Equation (2.13) mayhave special solutions which are generated by its mathematical structureand related to the global symmetry of M(r). Such solutions are usuallytopologically specific ones and they are called topological solitons [59]. In

since a dislocation is associated with a point rather than a line.9We shall assume that the total magnetization of the thermodynamic system M(r; B)

is single valued for any r and B. Systems for which M(r; B) is not single valued aresaid to exhibit hysteresis; most ferromagnetic systems have this property. Hysteresis isassociated with a magnetic heterogeneity of the sample, the separate regions being knownas domains. The analysis we shall develop is generally applicable within a single domain.This assumption is valid for the ν = 1 QHE ground state as long as it behaves like ahomogeneous and isotropic quantum liquid.


order to understand the situation with topological solitons, a unit vectorm(r) = M(r)/M0 is introduced which maps the ferromagnet space onto thesurface of a unit sphere.

Let’s specialize now to the case of isotropic 2D ferromagnets. In termsof the unit vector field m(r), the energy functional is given by

E[m(r)] =ρs

2

∫∂am

µ(r)∂amµ(r) d2r, (2.15)

where ρs is the ferromagnetic spin stiffness which has units of energy, Greekindices refer to the directions in the spin space (µ = x, y, z), and Latinindices refer to the directions in the coordinate space. The energy functionalE[m(r)] (the so-called minimal gradient energy) express the cost due tothe loss of Coulomb exchange when the spin orientation varies with theposition and it is the first term in the Taylor expansion of dipolar magneticinteraction which preserves the rotational symmetry. In close analogy toSkyrme’s theory 10, the dynamics of the isotropic ferromagnet could bestudied within the Lagrangian formalism, using the scalar fields mµ(r) withthe constraint m(r) · m(r) = 1. The static configurations of an infinitecontinuum system can be obtained by solving Euler-Lagrange equations thatfollow from the Lagrangian

L = −ρs

2

∫∂am

µ(r)∂amµ(r) d2r + λL

∫[mµ(r)mµ(r)− 1)] d2r, (2.16)

where λL is a Lagrange multiplier that enforces the length constraint. Theunit magnetization vector field obeys a non-linear equation 11

∇2m(r)−[m(r) · ∇2m(r)

]m(r) = 0. (2.17)

Notice that the constraint introduced a completely new twist to the problem.Without the constraint, Eq. (2.17) reduces to the much simpler Laplace

10In particle physics, the goal of the field theory is to explain the rich diversity ofexperimental phenomena with a very small number of fundamental fields that propagateunder the influence of a highly symmetric Lagrangian. The Skyrmion was born with afamous conjecture by Skyrme - that a particular class of non-linear field theories is relevantto the description of elementary particles such as mesons. The mesonic field obeys a non-linear differential equation which has solutions that approximate to travelling waves. Inaddition, the equation has a static solution which could be interpreted as a ”particle”(Skyrmion). [T.H.R. Skyrme, A Non-Linear Field Theory, Proc. Roy. Soc. (London) Ser.A 260, 127 (1961).]

11This equation is, of course, identical to the one deduced from Eqs. (2.13 - 2.15) withinthe assumption that η = 0 and dM

dt= 0.


equation. The ”standard” Skyrmion solution of Eq. (2.17) is

mx(r) =2λx

λ2 + r2=

2λr cos (θ − φ)λ2 + r2

my(r) =2λy

λ2 + r2=

2λr sin (θ − φ)λ2 + r2

mz(r) =r2 − λ2

λ2 + r2, (2.18)

where (r, θ) are the polar coordinates in the coordinate plane, φ is a con-stant that controls the in-plane spin orientation, and λ is the size of theSkyrmion [95]. This is illustrated in Fig. (2.3), in which spins have beenjoined to sites of a square lattice. The magnetization varies smoothly fromdown (at the center) to up (at infinity) as function of radius and angle withm(r) lying in the xy plane at a distance λ from the center of the Skyr-mion. The energy corresponding to this solution is E1

Sk = 4πρs and doesnot depend on parameters λ and φ. In few words, borrowed from the math-ematical language, the standard Skyrmion is a field configuration m(r) thatwraps around the O(3) order parameter sphere exactly once and thus, it isdescribed by a spatially extended topological charge that integrates to unity.This can be more ample enunciated, using concepts of ordinary topology, asfollows. A quantity called topological charge density is defined as

ρtop(r) =18πεuvm(r) ·

[∂um(r)× ∂vm(r)

]. (2.19)

Since the Jacobian converting area in the plane into the solid angle on thesphere is 4π times the topological charge density, ρtop(r) plays a key rolefor the wrapping of the plane onto a sphere. Associated to the topologicalcharge density is the topological charge given by

Qtop =18π

∫εuvm(r) ·

[∂um(r)× ∂vm(r)

]d2r. (2.20)

The topological charge must be always an integer since it counts the numberof times the compactified plane is wrapped around the O(3) order parametersphere by the mapping m(r). The Skyrmion configuration of Eq. (2.18)wraps the plane around the O(3) order parameter sphere exactly once andhenceforth, has Qtop = 1. The importance of the topological charge for theQHE will become clear shortly, but one aspect is already worth noting at thisstage of discussion 12. The topological charge is a topological invariant, i.e.,

12The corresponding anti-Skyrmion has Qtop = −1. It is of the same form but withmy(r) 7→ −my(r) and ρtop(r) 7→ −ρtop(r). Skyrmions with higher topological charge may

exist. Their energy is EQtop

Sk = 4πρs|Qtop| [95].


Figure 2.3: Vector representation of a Skyrmion spin texture on a squarelattice. Unlike a XY vortex there is no singularity at the origin because thespins are able to rotate out of the plane. The spin is down at the center andgradually turns up at infinite radius. [Courtesy of N.R. Cooper]


it remains stable against smooth continuous distortions of the field m(r).For example, spin waves could travel through the Skyrmion and Qtop doesnot change, i.e., the energy of the Skyrmion remains the same.

The topological phase

Before we go on, it is necessary for me to be relieved of a certain anxiety.This difficulty is brought about by the quantum treatment of spin dynamicsand can be elucidated by the following argument [39, 97]. Let’s take theLagrangian formalism as a convenient formulation of the laws of dynamicsand look at a spin s = ~sm precessing under the influence of a magnetic fieldB. Here s is the spin length and the unit vector m denotes the direction ofspin. The aforementioned difficulty arises from the fact that the descriptionof the system will be given in terms of the time-dependent history of the unitvector m.

The Lagrangian that reproduces the precession of the quantum spinshould contain (1) the Zeeman interaction term γ~smµBµ and (2) a termthat enforces the length constraint λL(mµmµ − 1). Here γ is the gyro-magnetic factor, µ refers to the directions in the spin space, and λL is theLagrange multiplier. When the orientation of a spin is moved adiabaticallyaround a closed loop, the system picks up a Berry’s phase 13. Henceforth, anew term should be added to the Lagragian to account for this quantum ef-fect. Defining in the spin space a velocity vector ~sm and a vector potentialA which satisfies m = ∇m ×A, the Berry’s phase term in the Lagrangiancan be expressed as 14

−~s mµAµ. (2.21)

To sum up, the Lagrangian of the precessing spin turns out to be

Lprecession = −~smµAµ + γ~smµBµ + λL(mµmµ − 1). (2.22)13The adiabatic principle of Ehrenfest stipulates that any system adiabatically ”trans-

ported” around a closed circuit C in the parameter space will return in its original state(of energy E). Its internal clocks will register the passage of time t; this is the origin ofthe dynamical phase factor: exp (−iEt/~). Additionally, the system records its historyin a geometrical way, expressed by a geometrical phase factor: exp [iΓ(C)]. [M.V. Berry,Quantal Phase Factors Accompanying Adiabatic Changes, Proc. Roy. Soc. (London) Ser.A 392, 45 (1984).] The Berry’s phase writes as Γ(C) = −2π

RC ρtop(r) d2r.

14There is a close analogy between the Berry’s term in the quantum Lagrangian of aprecessing spin and the Lagrangian of a charged particle moving under the influence of areal magnetic vector potential A. If the particle is massless, its velocity is v, and we setthe electrical charge to −1, then the Lagrangian reduces to the first order time derivativeterm −v ·A. Thus, the Lorentz force is disguised in the Magnus force, proportional tothe vector product of the spin velocity and the local spin magnetization [39, 97].

2.2. QHE FERROMAGNETISM AT ν = 1 65

This Lagrangian correctly reproduces both the Berry’s phase and the equa-tion of motion for the quantum spin whose Hamiltonian is given by

Hprecession = −γ~smµBµ. (2.23)

We shall see soon how this weird physics fits into the theory of QHE ferro-magnetism.

2.2 QHE ferromagnetism at ν = 1

2.2.1 Spin-wave excitations

The spin-wave mode of the QHE ferromagnet at ν = 1 could be derivedwithout great mathematical efforts if one uses the convenient formulationof quantum mechanics within the subspace of the lowest Landau level 15.In this formalism, the Hamiltonian for N interacting 2D electrons can beexpressed solely in the terms of the projected density operator ρk as

H =12

∑k

[V (k)ρkρ−k − ne−

12|k|2l2B

], (2.24)

where V (k) is the Fourier transform of the electron-electron interaction po-tential V (k) =

∫Ve−e(|r − r′|)e−ik·|r−r′| d2r. An explicit microscopic wave

function at ν = 1, which factorizes into the Laughlin spatial wave functionfor the filled spin-polarized Landau level ΨL

1 [z] and the fully symmetric spin

15For a 2DES with area A and N particles, the Fourier transforms of the charge andtotal spin density operator, projected to the lowest Landau level, have the form

ρq =1√A

NXj=1

e−iq·rj =e−

14 |q|

2l2B√

A

NXj=1

τq(j),

Sµq =

1√A

NXj=1

e−iq·rj sµ(rj) =e−

14 |q|

2l2B√

A

NXj=1

τq(j)sµ(rj).

Here τq(j) = e−iqlB

∂∂zj

− i2 q∗lBzj

is the so-called magnetic translation operator, whichtranslates the jth particle a distance l2B(ez × q). The dimensionless complex coordinatefor the jth particle is zj , lB =

p~/(eB) is the magnetic length, and µ refers to directions

in the spin space. Note that, because of the itinerant character of the electrons, themagnitude of the wavevector q could be arbitrary big. The projected spin and chargedensity operators do not commute. This implies that within the lowest Landau level, thedynamics of spin and charge are entangled [39].


wave function |Ξs〉, was given in the previous chapter

Ψ[z] =N∏i<j

(zi − zj)N∏

k=1

e−14|zk|2 | ↑1↑2 . . . ↑N 〉. (2.25)

In the standard procedure [see Eq. (2.6)], low-lying excited states, labeledby the wavevector q, are obtained from the many-body ground state Ψ[z]as

Ψq[z] ≡ S−q Ψ[z] =N∑

j=1

e−iq·rjs−(rj)Ψ[z], (2.26)

where S−q is the Fourier transform of the local spin-lowering operator s−(rj)and overline indicates projection onto the lowest Landau level. The energyof the spin-wave mode is given by

Esw(q) =1

(2π)2

∫e−

12|k|2l2BV (k)

(1− cos [ez × q · k]l2B

)d2k. (2.27)

For pure Coulomb interactions [Ve−e(|r − r′|) = e2/(ε|r − r′|)] and in thepresence of Zeeman splitting (EZ = |g∗|µBB), the dispersion of the spin-wave mode at ν = 1 has the following compact analytical form

Esw(q) = |g∗|µBB +e2

εlB

(π

2

)1/2[1− exp (−q2l2B/4)I0(−q2l2B/4)

], (2.28)

where I0 is the modified Bessel function of the first kind [95]. This is illus-trated in Fig. (2.4) using the estimates for sample M242 given in Table (1.3).The asymptotic limits of Eq. (2.28) are

Esw(q)q→0= |g∗|µBB +

e2lB4ε

(π

2

)1/2

q2, (2.29)

Esw(q)q→∞= |g∗|µBB +

e2

εlB

(π

2

)1/2

= EZ + Exc. (2.30)

For q → 0 the spin-wave mode energy has a quadratic q-dependence (similarto 2D isotropic ferromagnets on a lattice [Eq. (2.8)]). Since Exc EZ ,Esw(q → ∞) is a constant of the order of the Coulomb energy [EC =e2/(εlB)]. The physical picture of a spin flip process is uncanny. At small q,simultaneously with a flipped spin, a bound quasiparticle–quasihole pair iscreated. At large q, when a spin is flipped, the quasiparticle is concomitantly


Figure 2.4: Dispersion relation of the spin-wave mode at ν = 1 [Eq. (2.28)].Parameters EZ = 1.8 K and Exc = (π/2)1/2EC = 153 K correspond tosample M242 [Table (1.3)]. Surprisingly, in the light of the strong magneticfields, the physics associated with the spontaneous magnetization at ν = 1is experimentally accessible because EZ is weak compared to EC , and caneven be tuned to zero [75, 92].

moved far away from the quasihole. Interestingly, the Lagrangian formalismcould be used to obtain the dispersion relation of the spin-wave mode atν = 1. This highlights the field-theoretical approach as truly valuable forthe discussion of ν = 1 QHE ground state. We have previously seen thatwhen the spin adiabatically precesses under the influence of a magnetic field,its Lagrangian contains a Berry’s phase term. For a 2DES with an electrondensity n the Berry’s phase term is

LBerry = −~sn∫mµ(r) · Aµ[m(r)] d2r, (2.31)


where µ describes the directions in the spin space, s = 12 is the spin length,

and m(r) is the local magnetization unit vector field. The Coulomb andZeeman terms are

LCoulomb = −ρs

2

∫∂am

µ(r)∂amµ(r) d2r, (2.32)

LZeeman = ~sn|g∗|µB

∫mµ(r)Bµ d2r. (2.33)

In the long wavelength limit, provided that mz(r) ≈ 1 and m(r) ·m(r) = 1,the spin-wave Lagrangian Lsw = LBerry + LCoulomb + LZeeman leads to thedispersion relation

Esw(q) = |g∗|µBB +ρs

nsq2. (2.34)

By comparing the expression for the spin-wave mode [Eq. (2.29)] to thedispersion relation from the above equation, we can identify the 2DES spinstiffness at ν = 1 as

ρs =1

16√

2πe2

εlB. (2.35)

Estimates of the spin stiffness at ν = 1 for our measured samples are givenin Table (1.3). Skyrmions may exist in planar conventional ferromagnets.Ordinary ferromagnets, however, have a large spin stiffness; consequently,Skyrmions are irrelevant for their observable properties.

2.2.2 Topological excitations (QHE Skyrmions)

The way lies now open to develop a Lagrangian formalism for topologicalexcitations of the ν = 1 QHE ground state. To find what a QHE Skyrmionis, we must include the manifest connection between the spin and chargeat ν = 1. In a self-consistent description, each electron at ν = 1 experi-ences a strong local exchange field produced by the rest of electrons. As itpropagates under the exchange field, its spin adiabatically follows the localorientation of the (slowly spatially varying) spin-textured background. Thisadiabatic spin dynamics gives rise to a geometrical phase for closed paths.The non-trivial spin background in which the electron moves is characterizedby a unit topological charge and produces a Berry’s phase which is equalto the change in the Aharonov-Bohm phase produced by the insertion ofone quantum of flux Φ0. For incompressible QHE ground states at fillingfactor ν, Laughlin argued that additional flux Φ into the system creates anextra charge Q = −eνΦ/Φ0 [67]. Consequently, the real charge of a spintexture with topological charge Qtop = 1 at ν = 1 is sharply quantized to the


elementary electron charge. Spatial spin fluctuations may produce a modu-lation of the topological charge density. Because of the equivalence betweenthe topological and real charge, we should not lose sight of the effect ofphysical charge density fluctuations which cost Hartree energy. The Hartreeenergy is the next leading term in the spin-texture energy functional givenby Eq. (2.15). Armed with this new knowledge, we can write the effectiveLagrangian that describes the long-wavelength and low-energy physics ofthe QHE ferromagnet at ν = 1 as

Leff = LBerry + LZeeman + LCoulomb + LHartree. (2.36)

If δρ(r) represents the deviation of the charge density from the uniformbackground density, the Hartree term in the effective Lagrangian is given by

LHartree = −12

∫d2r

∫δρ(r)

1ε|r− r′|

δρ(r′) d2r′. (2.37)

Skyrmions are static solutions of Leff and can be obtained by energy min-imization of the dominant (Coulomb) term as described previously. Theminimal gradient energy of the single Skyrmion is E1

Sk = 4πρs = Exc/4.Therefore, forgetting for an instant the Zeeman energy, the QHE excitationgap at ν = 1 is Exc/2. It is two times smaller than the quasiparticle-quasihole Laughlin pair creation energy [see Sec. (1.5)] and it is finite aslong as the spin stiffness is finite.

Quantum Hall effect Skyrmions are quasiparticles characterized by afinite size, charge, spin, and statistics. Let’s peer first into the size and spinof Skyrmions. The Zeeman and Hartree terms in the effective Lagrangiancontrol the stability of individual Skyrmions towards shrinking or growingin size. The Hartree energy prefers to expand the Skyrmion size, as for largeSkyrmions δρ(r) is small. The Zeeman energy tends to shrink Skyrmionssince it promotes polarization along the magnetic field direction and triesto stop the in-plane polarization. The scale invariance of E1

Sk is broken bythe presence of Zeeman and Hartree terms, whose competition yields anoptimal size for Skyrmions. The ratio g between the Zeeman splitting andthe Coulomb energy is an important, experimentally accessible parameter,which controls the size of the Skyrmion.

These qualitative considerations are followed by some relevant quantita-tive results. First, for a vanishing small g, Sondhi et al. [95] proposed the


following formulae for the size λ and energy EλSk of the Skyrmion

(λ/lB

)3 ∝ Cl1

lB(g| ln g|)−1,

EλSk − E1

Sk ∝ e2

εlB

(Cl

2

lB

)1/3

(g| ln g|)1/3,

(2.38)

where Cl1 and Cl

2 are positive constants with units of length. One can eas-ily see that the Skyrmion size diverges in the asymptotic limit of g → 0.Henceforth, the number of flipped spins within a Skyrmion diverges in thelimit of vanishing g. At finite Zeeman coupling, the optimal size λ (λ lB)of a Skyrmion could be obtained by considering the finite-size corrections toE1

Sk in a system with linear dimension R

EλSk − E1

Sk = CE1

(λ

R

)2

+ CE2

(lBλ

), (2.39)

where CE1 and CE

2 are positive constants with units of energy [80]. MinimizingEλ

Sk given by Eq. (2.39) with respect to λ gives λ ∝ R2/3l1/3B . Accordingly,

the number of reversed spins scales as λ2 ln (R/λ). Finally, of great im-portance with regard to the experiments, theory also predicts that above acritical g value (gc) Skyrmions evolve into single spin-flips. In this limit, thesize of Skyrmions approaches lB. For an ideally thin, disorder-free 2DES,and with no Landau level mixing, calculations [95] give a lower bound ofgc = 0.054.

One may ask how relevant are Skyrmions for real, disordered 2DESs inGaAs-based heterostructures. Because Skyrmions carry physical electricalcharge, unlike the case of a regular ferromagnet, a scalar potential mayinduce the formation of Skyrmions in the ground state even at T = 0. Theenergy of a Skyrmion in a random disorder potential Vd(r) is

EλSk(r0) =

∫ρ(r− r0, λ)Vd(r) d2r, (2.40)

where r0 is the position of the Skyrmion, ρ(r, λ) = −eνρtop(r, λ), andρtop(r, λ) = [λ/(r2 + λ2)]2/π is the topological charge density for the stan-dard Skyrmion of size λ. The optimal size of the Skyrmion could be obtainedby adding this (no less interesting) term to Eλ

Sk [Eq. (2.39)] and subsequentminimization with respect to λ. Few other comments are in order here. Inthe previous chapter we have seen that it is essential for the observation ofthe QHE that charged quasiparticles are localized by disorder. In the T → 0


limit, it is believed that Skyrmions are pinned by the disorder and do notcontribute to the charge transport in the ν = 1 QHE plateau 16. It is alsoexpected that Skyrmion – anti-Skyrmion pairs will be thermally activated 17

and hence exponentially rare at low T .We take up now the subject of Skyrmion statistics. The effective La-

grangian [Eq. (2.36)] may also contain the Hopf term [87], reflecting thatthe ground state does not have time-reversal symmetry and parity invari-ance. The Hopf term is given by

LHopf = −NHopf

32π

∫εijkBiFjk d

2r, (2.41)

where εijk is the completely antisymmetric tensor, the prefactor NHopf is atopological invariant, Bi is an auxiliary gauge field, and Fij = ∂iBj−∂jBi =m(r) · [∂im(r)× ∂jm(r)]. The total spin of the Skyrmion has contributionsfrom both the Berry term and the Hopf term in the effective Lagrangian. Inthe long wavelength limit the contribution due to the former dominates. Thestatistics of the Skyrmion, on the other hand, is completely determined bythe prefactor NHopf . This is found to be equal to ν and the statistical phaseof the Skyrmion is νπ. Consequently, Skyrmions at ν = 1 obey Fermi-Diracstatistics 18.

2.2.3 Further aspects of QHE Skyrmions

While the field theoretic approach [95] is very successful in qualitatively ex-plaining the long-wavelength and low-energy physics of QHE ferromagnets,it fails when it comes to quantitative comparison with experiments. The

16Even given the fact that the Skyrmions are pinned by the disorder, the question ariseswhether the existence of gapless spin waves leads to dissipation and hence destroys theQHE in the limit of g → 0. It is likely that dissipationless flow survives the presence ofspin waves, since they are neutral and hence, irrelevant to charge transport.

17At finite temperatures, the T -dependence of Skyrmion density may exhibits a non-Arrhenius behavior due to the presence of disorder.

18A priori, similar charged spin-texture excitations may occur around other fractionalfilling factors with incompressible ferromagnetic many-body ground states. For instance,at ν = 1/3, Skyrmions are anyons and obey fractional statistics. However, calculationsof Skyrmion energies near ν = 1/3 show that Skyrmions are present only for very smallZeeman couplings, approximately 10 times smaller compared to those at ν = 1. [R.K.Kamilla, X.G. Wu, and J.K. Jain, Skyrmions in the Fractional Quantum Hall Effect, SolidState Commun. 99, 289 (1996).] I should mention that Skyrmions are also expected atodd integer filling factors such as ν = 3. Sadly, calculations near ν = 3 also reveal thatSkyrmions are relevant only for very small g, approximately 10 times smaller comparedto g at ν = 1.


failure of the classical field theory to predict the correct Skyrmion physicsat short length scales originates in the truncation of the gradient expansionof the energy functional E[m(r)]. On the other hand, Hartree-Fock calcula-tions [23, 36] are in essence self-consistent mean-field calculations with theorder of the gradient expansion taken to infinity and hence more successfulat predicting the small-size Skyrmion physics.

In a nutshell, the Hartree-Fock formalism used in Skyrmion studies isas follows. The form of the wave functions is essentially dictated by thesymmetry of the classical Skyrmion solutions which are invariant under theaction of Lz±Sz for the Skyrmion (anti-Skyrmion). Here Lz and Sz denotethe z-components of the total orbital and spin angular momentum, respec-tively. This property, combined with the lowest Landau level occupancyrequirement, uniquely picks the form of the Hartree-Fock wave function

|ΨHF〉 =∞∏

m=0

(uma

+m + vmb

+m+1

)|0〉, (2.42)

where |0〉 is the vacuum state, a+m creates a spin-down electron and b+m+1

creates a spin-up electron in themth angular momentum state. The Hartree-Fock procedure involves the minimization of the average energy with respectto the variational parameters um, vm, subject to the constraint that thewave function is normalized. The expectation value of the total spin operatorin the state described by Eq. (2.42) describes a spin-texture with Qtop = 1provided that um varies slowly with m from um=0 = 1 to um→∞ = 0. Thetotal number of reversed spins 19 in this wave function is K =

∑∞m=0 |um|2.

The K = 0 Hartree-Fock quasihole is obtained by choosing um = 0 for all m.The spin of the K = 0 Hartree-Fock quasihole is one half, consistent witha single spin flip. It represents the change in the total spin of the systemwhen it is introduced to the fully spin-polarized Landau band. Similarly,Skyrmions (or anti-Skyrmions) involving K > 0 reversed spins in the elec-tronic ground state have a spin K+1/2. Under the particle-hole symmetry,the spin of the quasiparticle-quasihole Hartree-Fock pair at ν = 1 is 2K+1,where the positive integer K gives the number of reversed spins within theSkyrmion (or anti-Skyrmion) and K = 0 corresponds to the spin-1

2 quasi-particle (or quasihole).

I next give two representative examples of Hartree-Fock calculations.Firstly, microscopic Hartree-Fock calculations revealed that the absolute

19Please note that the quantization of the total number of reversed spins (i.e., the factthat the Skyrmion quantum number K is a positive integer) is not captured nor by thisvariational wave function neither by the field theoretic approach.


value of the z-component of the spin is less than unity at the center of theSkyrmion. The solid curve in Fig. (2.5) shows the Hartree-Fock magneti-zation profile of the K = 3 Skyrmion 20. This should be contrasted withthe result of the classical field theory in which the spin is constrained to bereversed at the center of the Skyrmion [dash-dotted curve in Fig. (2.5)].

Figure 2.5: The radial distribution of mz(r), from Hartree-Fock calculationsfor a K = 3 Skyrmion, is shown by the solid curve. [Courtesy of M. Abol-fath]. The dash-dotted curve is the result of the classical field theory for thead hoc choice of λ/lB such as mz(r) = 0 at the same distance r, measuredfrom the center of the Skyrmion, as in the Hartree-Fock theory.

20The number of reversed spins K is given by K = 14πl2

B

R[1−mz(r)] d2r − 1

2.


Secondly, the Hartree-Fock theory describes well the evolution of both Skyr-mion spin and excitation energy with g. Using the Hartree-Fock approxima-tion and taking into account the z-extent of the 2DES, Cooper [23] studiedSkyrmions at ν = 1 in the presence of a finite Zeeman coupling. Defininga critical Zeeman energy by the value g at which the transition betweenthe charged spin-texture and the spin-1

2 quasiparticle occurs, Cooper foundgc = 0.054 for bare Coulomb interactions [w0 = 0, Fig. (2.6)].

Figure 2.6: Energy gap ∆t1 [in units of e2/(εlB)] for the creation of a Skyr-

mion – anti-Skyrmion pair (solid curve) and total spin-1 quasiparticle –quasihole pair (dashed line) as a function of g = EZ/[e2/(εlB)] from Hartree-Fock calculations at ν = 1 and w0 = 0 (bare Coulomb interactions). Thevertical arrow indicates gc = 0.054 above which a transition from Skyrmi-ons to single spin-flip excitations is predicted. The spin of a single chargeexcitation is shown in the inset. [Courtesy of N.R. Cooper]


When the finite thickness of the confined state of electrons is introduced,the critical Zeeman energy is reduced to gc ≈ 0.046 (for sample M242,w0/lB = 0.6) and gc ≈ 0.047 (for sample M280, w0/lB = 0.5). Only theresult for M280 sample is shown here [Fig. (2.7)]. Notice that at each finiteg, when the finite thickness of the 2DES is included, the energy and the spinof the Skyrmion are closer to those of the spin-1

2 quasiparticle.

Figure 2.7: The solid curve is the energy gap ∆t1 to create a Skyrmion –

anti-Skyrmion pair [in units of EC = e2/(εlB)] as a function of g = EZ/EC ,from Hartree-Fock calculations with w0/lB = 0.5 (sample M280) at ν = 1.The dashed line is the Hartree-Fock result for the total spin-1 quasiparticle– quasihole pair. The vertical arrow indicates gc = 0.047 above which Skyr-mions transform to single spin-flip excitations. The spin of a single chargeexcitation is shown in the inset. [Courtesy of N.R. Cooper]

I should also mention here calculations of the Skyrmion excitation energywhich take into account both the finite thickness of the 2DES and the Lan-


Table 2.1: The predicted gc below which the single spin flip excitation isunstable to Skyrmion formation and the percentage decrease of the QHEexcitation gap at ν = 1 [∆(∆t

1)], relative to the value calculated at w0 = 0and rs = 0 [60, 76]. We recall that rs is the ratio of the Coulomb energy tothe cyclotron energy and w0 parametrizes the z-extent of the 2DES.

gc ∆(∆t1) gc ∆(∆t

1)

w0 = 0 w0 = lB

rs = 0 0.061 0% 0.039 & 40%

rs = 1 0.052 & 30% 0.035 & 55%

dau level (LL) mixing. The LL mixing has the same qualitative effect asthe finite thickness of the 2DES on the stability of Skyrmions: it lowersthe Skyrmion excitation energy 21 and reduces gc. Interestingly, these cor-rections do not work ”coherently” [60, 76], i.e., the order in which thesecorrections are included is important [Table (2.1)]. For example, when the2DES is assumed infinitely thin (w0 = 0), the QHE excitation gap at ν = 1is & 30% smaller for rs ≈ 1 compared to its rs = 0 value. A further & 25%reduction of the QHE excitation gap occurs, due to the inclusion of a finitewidth of one magnetic length. On the other hand, considering that rs = 0,with the introduction of a finite width of one magnetic length the energygap drops & 40%. Then, the additional reduction of the energy gap due toLL mixing is only & 15% for rs ≈ 1.

2.2.4 Electron spin polarization at ν = 1

Various thoughtful approaches have been used to study the T -dependenceof the electron spin polarization P at ν = 1 [19, 43, 50, 53, 87]. Our start-ing point is P(T ) calculated for a non-interacting 2DES with the chemicalpotential in the middle of the Zeeman gap

P(T ) = tanh[EZ/(4kBT )

]. (2.43)

At high temperatures, according to Eq. (2.43), the 2DES behaves like aCurie paramagnet, i.e., P(T ) ∝ 1/T . A well-justified microscopic theory

21In the absence of LL mixing, the particle-hole symmetry around ν = 1 ensures thatthe energy required to create a Skyrmion is equal to that for creating an anti-Skyrmion.When the LL mixing is included into calculations this equality can no longer be assumed.


is the Hartree-Fock approximation, which represents the analog for QHEsystems of the band theory in itinerant electron magnetism, and yields

P(T ) = tanh[E↑

HF/(2kBT )]. (2.44)

Here E↑HF is the Hartree-Fock energy of spin-up electrons, measured from

the chemical potential (E↑HF = −E↓

HF = Exc/2). Because of the exchange-enhanced spin splitting, P is much larger at finite T than in the non-interacting case. Hartree-Fock approximation grossly overestimates the elec-tron spin polarization: for particular parameters of Fig. (2.8), the theorypredicts that the 2DES is essentially 100% polarized below 10 K [53]. Thesimplest many-body theory beyond the Hartree-Fock approximation whichreflects the presence of the spin waves excitations in itinerant ferromagnetsis one which includes a self-energy insertion consisting of a ladder sum ofrepeated interactions between electrons of one spin and holes of oppositespin [53]. The spin-wave theory predicts a strong suppression of the elec-tron spin polarization at much lower T because it includes the effects ofthermally excited spin waves. Calculations by Kasner and MacDonald [53],for a static screening wavevector 0.01l−1

B , are shown by the dash-dotted curvein Fig. (2.8). The detailed T -dependence of P is as follows. At low T , thereduction of P is dominated by the long-wavelength spin-wave contributionwhich gives [1+ C(T )T ln 1− e−EZ/(kBT )], where C(T ) depends weakly ontemperature. At higher temperatures, this suppression crosses over from anan activated T -dependence [1−C(T )T exp −EZ/(kBT )] for kBT < EZ , toan approximately linear temperature dependence [1 − C(T )T ln (kBT/EZ)]for kBT > EZ . This linear T -dependence was interpreted as the analog ofthe T 3/2-Bloch law in 3D ferromagnets for QHE systems. Although the re-sults shown in Fig. (2.8) are extremely insensitive to the choice of the screen-ing wavevector, progress in the diagrammatic technique approach includingtemperature and frequency-dependent screening has also been reported [50].Kasner et al. [54] substantiated the spin wave theory, by appreciating therole of the finite thickness of the 2DES [54]. The electron spin polarizationdecreases with T more quickly when the finite width of the 2DES is takeninto account. A complementary method to compute P(T ) was suggested byRead and Sachdev [87], in the context of a continuum quantum field theoryfor the QHE ferromagnet at ν = 1. Their approach belongs to the class of theso-called large-N methods 22. By considering ρs as a parameter, they used

22In their work, the method under consideration found double application as it permitsalso to calculate the nuclear spin-lattice relaxation rate [see Ref. [99] and Sec. (2.3.5)below].


Figure 2.8: The T -dependence of P at ν = 1 and g = EZ/EC =2.2 K/136 K = 0.016. The dashed curve illustrates P(T ) for non-interactingelectrons [Eq. (2.43)]. The dotted curve represents the spin-1

2 Brillouin func-tion. The dash-dotted line shows the result of the spin-wave theory [53].

SU(N) and O(N) symmetries in the N = ∞ limit and exploit dimensionalanalysis to compute P(T ) at ν = 1. The ρs = 0 limit of the SU(N = ∞)model yield the s = 1

2 Brillouin function, which is displayed in Fig. (2.8) bythe dotted curve. The ρs = 0 limit of the O(N = ∞) model yield the s = 1Brillouin function (not shown here). Working along the lines suggested byRead and Sachdev [87], Green [43] obtained results for P(T ) at ν = 1 in aweakly disordered QHE ferromagnet and showed that the effect of disorderis to reduce ρs, i.e., P decays more quickly with increasing T . Finally, wewish to stress out a feature shared by all calculations: the expected P(T )at ν = 1 for a disorder-free 2DES, lies always above the lower bound set bythe Eq. (2.43) [dashed curve in Fig. (2.8)]. That is, the predicted P(T ) isessentially saturated at very low-T (P ≈ 1 at T = 0.1 K).

2.3. QHE FERROMAGNETISM NEAR ν = 1 79

2.3 QHE ferromagnetism near ν = 1

Knowledge of the ground state precisely at ν = 1 and its low-energy exci-tations is essentially simpler than the knowledge of the ground state nearν = 1. The ground state near ν = 1 can be only approximately determinedbecause the QHE ferromagnet is non-colinear 23. Colinear order means thatthe electronic spins are all aligned, parallel or antiparallel, to the same di-rection everywhere in space; non-colinear order means that the directionof magnetization varies from point to point in space. Magnetic order nearν = 1 is no longer colinear forasmuch the ferromagnetic ground state wasdiluted with magnetic impurities (Skyrmions) that altered the balance of ex-change interactions. The electron spin stiffness is no longer the only relevantenergy scale near ν = 1.

2.3.1 Electron spin polarization near ν = 1

Without referring to any model, we can immediately settle a fundamentalprediction about the electronic ground state at ν 6= 1. This prediction,which concerns the electron spin polarization, naturally emerges from theproperties of QHE Skyrmions we already presented. For filling factors ν & 1(or ν . 1) there will be a finite density of Skyrmions (or anti-Skyrmions) inthe electronic ground state, all with the same charge −e (or e) and carryingthe same number of reversed spins K. We assume here that there is electron-hole symmetry around ν = 1. One can easily prove that

P(ν) =Sz(ν)N/2

=

1− 2K(1− ν)/ν, for ν . 1,

1− 2(K + 1)(ν − 1)/ν, for ν & 1.

In this picture, the lowest spin-up Landau level is completely filled at ν = 1[N = Nφ = Anφ]. The 2DES is fully polarized at zero temperature P = 1[Sz(ν = 1) = N/2]. Reduction of the magnetic field so that N = Nφ + 1,creates one Skyrmion, Sz(ν & 1) = Nφ/2− (K+1/2), and the electron spinpolarization is reduced. Alternatively, increasing the magnetic field so thatN = Nφ−1, creates one anti-Skyrmion, Sz(ν . 1) = Nφ/2−(K+1/2), whichalso reduces the electron spin polarization. The above equation suggeststhat a simple way to determine K would be to measure the electron spin

23Non-colinearity can generally be traced to competing interactions. For example, imag-ine a triangular lattice with classical spins and with two types of exchange interactions.Several structures (ferromagnetic, antiferromagnetic, and spiral) may occur, depending onthe sign and magnitude of the exchange coupling constants.


Figure 2.9: The ν-dependence of the theoretical spin polarization for non-interacting electrons is shown by the dashed curve. The dotted curve repre-sents the ν-dependence of P for |ν − 1| ≤ 0.1, when non-interacting K = 3Skyrmions are accommodated in the electronic ground state [36]. The solidcurve displays P(ν) for a square Skyrmion lattice at g = 0.015 [14].

polarization as a function of ν in the vicinity of ν = 1. The predictedreduction of P for K = 3 Skyrmions at ν 6= 1 is shown by the dotted line inFig. (2.9). The rapid decrease in the electron spin polarization is expected topersist only in the QHE plateau |ν−1| ≤ 0.1. Brey et al. [14] obtained resultsfor P(ν) using a Hartree-Fock approximation [solid curve in Fig. (2.9)].These authors noted that the slope of the spin polarization curve decreasesin magnitude away from ν = 1, indicating that Skyrmions/anti-Skyrmionsshrink into single reversed-spin quasiparticles as they become more dense.Recently, Nederveen and Nazarov [82] pointed out that the presence of afinite density of Skyrmions – anti-Skyrmions in the QHE plateau at ν = 1would manifest as a rounding of the spin polarization peak. We also note


that P(ν) obtained for K = 0 corresponds to the independent (spin-flip)electron model

P(ν) =

1, for ν ≤ 1,2ν − 1, for 1 ≤ ν ≤ 2.

When ν = 2, there is an equal number of spin-up and spin-down electronsand P = 0. The free electron picture is displayed by the dashed curve inFig. (2.9). If the non-interacting electron picture is a good first approxima-tion for the 2DES in the quantum limit, it should describe well the measuredP(ν) in the limit of T → 0 and very large g. These considerations will serveas a main frame for the discussion of our NMR data.

How the presence of Skyrmions near ν = 1 is reflected in heat capacityand NMR experiments, is the theme of the following two chapters. The cen-tral aspect is that for filling factors away from ν = 1 some of the electron spinfluctuations are orders of magnitude lower in frequency than the electronicZeeman splitting. These nearly gapless modes allow a strong coupling to theQWs nuclei, observed as a dramatic enhancement of the nuclear spin-latticerelaxation rate near ν = 1 in both heat capacity and NMR experiments. Ina disorder-free 2DES, to understand this behavior, we must conceive howSkyrmion dynamics involves low frequency spin fluctuations.

To understand the properties of spin excitations of the QHE ferromagnetnear ν = 1 is merely the most arduous task. Our additional labours could bedistilled into the following questions: What are the symmetries of the groundstate containing Skyrmions? What is the nature of the multi-Skyrmionsystem at T = 0? It is a solid or liquid? What is the character of phasetransitions, if any? These questions have an answer, quite complicated itis true, and in part still very hypothetical, but yet deserving respect as itprovides an explanation of important experimental facts.

2.3.2 Quantum treatment of a Skyrmion

Returning to the analysis of one Skyrmion [parametrized in Eq. (2.18)],we recall that there are two degeneracies in the classical field theory. Theenergy E1

Sk does not depend on (1) position of the Skyrmion and (2) in-plane orientation of components of the spin. These degeneracies are sourcesof nearly gapless excitations which license the nuclear spin-lattice coupling.The way to see this is to treat the Skyrmion quantum mechanically [39, 97].

First, as the Skyrmion has a distinguishable location, a translationaldegree of freedom [r0(t)] is added to the system. The spin field of a movingSkyrmion is defined as m(r, t) = m[r− r0(t)] and the rate of change of the


local spin orientation is

mλ(r, t) = −rµ0

∂

∂rµmλ[r− r0(t)].

In the first order approximation, the effective Lagrangian for the transla-tional degree of freedom reduces to

Lm = −~s∫

mλ(r, t)Aλ[m(r, t)] n(r) d2r, (2.45)

where s = 12 and we have taken into account our new-found knowledge that

the charge density is non-uniform. As discussed in Sec. (2.1.3), this La-grangian is equivalent to that of a massless charge −e moving in a uniformmagnetic field B. The addition to the Lagrangian of an inertial term pro-portional to |m|2 would induce a mass for the Skyrmion, and permit it tomake cyclotron orbits. Therefore, like the massive degeneracy associated tothe 2D dynamics of a charge in the presence of a uniform magnetic field, theeigenstates of the Hamiltonian describing a moving Skyrmion are degeneratein energy and therefore capable of relaxing the nuclei.

Secondly, since the Skyrmion has a distinguishable orientation, the spinsymmetry is broken with respect to rotations of the non-colinear spin con-figuration around the magnetic-field direction and a rotational U(1) degreeof freedom [φ] is introduced in the system 24. The effective Lagrangian forthe rotational degree of freedom is

Lφ = ~Kφ+~2

2Uφ2, (2.46)

where K and U are phenomenological parameters [39]. The correspondingHamiltonian is

Hφ =U

2(−i ∂

∂φ−K)2. (2.47)

Its eigenfunctions are ψSkm (φ) = 1√

2πeimφ and its eigenvalues are εSk

m =U2 (m − K)2. The expectation value of the canonical angular momentumoperator conjugate to φ, −i ∂

∂φ , is K. Thus, for each Skyrmion, K and φ arecanonically conjugate. Skyrmions carry an integer-valued quantum numberK, which physical meaning is the number of overturned spins. The actualorientation angle is completely uncertain because |ψSk

m (φ)|2 = 1. This im-plies that the U(1) rotational symmetry, broken in the classical solution, is

24The global U(1) symmetry of the colinear ferromagnet means that the ground state

is invariant (up to a phase) under the transformation defined by R(φ) = e−i~ φSz .


restored in the quantum solution because of quantum fluctuations in thecoordinate φ. However, in the thermodynamic limit of an infinite number ofSkyrmions coupled together, it is possible for the U(1) rotational broken sym-metry to survive quantum fluctuations. If this happens, then the excitationspectrum is gapless and the nuclear spin-lattice relaxation is allowed.

2.3.3 Skyrmion lattices at zero temperature

Moving away from ν = 1, more and more Skyrmions are present in theelectronic ground state and their interaction becomes important. Theoreti-cal calculations suggest that the ground state of the 2DES near ν = 1 is aSkyrmion crystal [14, 24]. The qualitative understanding of various typesof Skyrmion lattices and multi-Skyrmion ground states has drifted into theview that there are two competing interactions in the system: (1) a repul-sive, long-range Coulomb part (because Skyrmions are charged) and (2) ashort-range contribution related to the U(1) degrees of freedom. The latterterm, the so-called magnetic interaction, favors antiferromagnetic arrange-ment of U(1) ”spins”. Since the magnetic interaction is of short range, theCoulomb interaction gives the dominant energy scale for small Skyrmiondensities and a triangular lattice is preferred at very small values of |ν − 1|.The orientational degree of freedom of the Skyrmion is then frustrated 25.In the triangular phase the orientation angle differs by 120 between neigh-boring Skyrmions. At higher Skyrmion densities, the energy gained fromthe XY antiferromagnetic ordering outweighs the Coulomb energy cost anda square lattice structure is rather stabilized. The orientation angle is thenshifted by 180 between neighboring Skyrmions. Further lattice types mayalso be possible at intermediate Skyrmion densities 26. The main theoreticalprediction is that the non-colinear ground state breaks both translationaland U(1) rotational symmetry [24]. Associated to these broken symmetries(by virtue of Goldstone’s theorem) there are two Goldstone modes, namely,lattice vibrations and XY spin waves. The Goldstone mode associated withthe broken translational symmetry is just the ordinary magneto-phononmode of the crystal and has a power-law dispersion. More precisely, thedispersion relation is E(q) =

(2πµlBe2/ε

)1/2(qlB)3/2, where µ is the shear

modulus and q is the magnitude of the wavevector q. At long wavelengths,

25The key distinction between ferromagnetic and antiferromagnetic interactions is thatthe latter are subject to topological frustration (one can devise networks where all theexchange bonds can not be satisfied simultaneously), whereas the former are not.

26These considerations pertain to a (mean-field) classical model. At the highest Skyr-mion densities, quantum fluctuations are expected to give rise to a quantum-melted state.


the magneto-phonon mode is essentially decoupled from the XY spin-wavemode. Thus, ignoring the positional degrees of freedom, the ground statetheory of the QHE ferromagnet near ν = 1 reduces to the subtle physics ofU(1) degrees of freedom, on which we now focus. A phenomenological XYlattice antiferromagnetic model was introduced in Ref. [24] for Skyrmioncrystals

Hmulti−Skyrmion =U

2

∑j

(−i ∂∂φj

−K)2 − J∑〈ij〉

cos (φi − φj), (2.48)

where single indices label sites on the lattice and∑

〈ij〉 means that the sumis taken over all pairs of nearest neighbors 〈ij〉. Here U accounts for theenergy cost when the −i ∂

∂φj’s eigenvalue deviates from the optimal value

K. The stiffness to relative rotations of neighboring Skyrmions is J < 0,consistent with the antiferromagnetic XY order. This model is equivalent tothe model of bosons hopping on a lattice 27, in which the boson number onthe ith site is mapped to the number of flipped spins in the ith Skyrmion. Inthe boson language, the orientation-dependent interaction term correspondsto the boson hopping and favors the long range phase coherence of bosons,and U is the strength of the on-site interaction between bosons. For U J ,the system is an insulator and there is an excitation gap of the order of U .For U J , the system is a ”Bose superfluid”, the U(1) rotational symmetryis not restored by quantum fluctuations, and the corresponding Goldstonemode has a linear dispersion at small wavevectors

E(q) = (U J)1/2qa, (2.49)

where a is the Skyrmion lattice constant. If this situation occurs, then anexcitation gap is not produced and the nuclear spin-lattice relaxation rate isdramatically enhanced 28. The parameters J and U can be fixed by fittingthe microscopic Hartree-Fock calculations. Using the boson language, theboson interaction parameter U is given by the inverse boson compressibility

U = −12

(∂〈K(g, ν)〉

∂g

)−1

. (2.50)

The boson hopping parameter can be extracted then from the collectivemode spectrum. These considerations allow an estimate of the finite range

27M.P.A. Fisher, P.B. Weichman, G. Grinstein, and D.S. Fisher, Boson Localization andthe Superfluid-Insulator Transition, Phys. Rev. B 40, 546 (1989).

28In the ”Bose superfluid” phase, the number of overturned spin is uncertain and so isthe Zeeman energy cost to flip an electron spin.


in the g-ν phase diagram where it is possible for the 2DES to support gap-less elementary excitations [Fig. (2.10)]. It is worth noting here that theslight distortion in static positions of Skyrmions, in response to a disorderpotential, will give a small random contribution to the stiffness J . Thisrandomness is expected to produce a region of Bose-glass phase.

Figure 2.10: Qualitative, zero-temperature g-ν phase diagram for Skyrmioncrystal states [24]. The filled circles indicate the values ofK along the squareSkyrmion lattice region boundary. Non-colinear magnetic order survivesquantum fluctuations in the region approximately indicated by the shading.

2.3.4 Skyrmion lattices at finite temperatures

It is generally accepted that the qualitative distinction between differentmulti-Skyrmion magnetic phases at T = 0 will survive at finite T . If Skyr-mion positions were fixed to the ideal lattice at all temperatures, the above


discussed XY lattice antiferromagnetic model would capture the long wave-length physics. In the Skyrmion lattice, however, the positions are not fixedand the lattice itself can melt. In the melted phase, long length-scale corre-lations vanish, but we expect that short length-scale correlations to containcrystal-like features. Therefore, even at temperatures over the melting tem-perature, the multi-Skyrmion system could be analyzed in quite the samespirit as at T = 0. For example, above the lattice melting temperature,Goldstone modes of the 2DES will be overdamped modes derived from zero-temperature coherent modes.

At high T , the multi-Skyrmion system loses its quasi-long range orderin position and orientation. From a phase transition viewpoint, one expectstwo Kosterlitz-Thouless phase transitions as T is lowered. A rough upperbound estimate for the critical temperature of the Kosterlitz-Thouless phasetransition associated with the loss of magnetic quasi-long range order is [24]

TmagneticKT ≈ π

2kBρs ≈ 1 K, (2.51)

where U = 0 and J ≈ ρs. The theoretical estimate for the critical tempera-ture of the solid-to-liquid Kosterlitz-Thouless phase transition is [24]

TmeltingKT =

a2µ

4π≈ 0.1 K, (2.52)

Thus, the magnetic disordering transition is expected to occur at a higher Tthan the lattice melting transition 29. However, despite the fact the collec-tive modes are largely decoupled, the topological excitations (vortices anddislocations) may be coupled, leading to an interplay between magnetic andmelting transitions. Dislocation-driven simultaneous transitions and vortex-driven simultaneous transitions are two physically possible scenarios [100].Therefore we can surmise that near ν = 1, at least one Kosterlitz-Thoulessphase transition is expected as T is lowered.

2.3.5 Skyrmions and the nuclear spin-lattice relaxation rate

If one is to pick out a single experimental feature that epitomizes Skyrmionvita this would be the nuclear spin-lattice relaxation rate T−1

1 . Sharp struc-tures in T−1

1 as a function of T would be strong indications for phase transi-tions in the multi-Skyrmion system 30. I discuss here two significant theoreti-

29The Skyrmion lattice spacing is about 103 larger than the GaAs lattice spacing.30The nuclear spin-lattice relaxation in GaAs QWs and the general modelling of T−1

1

in QHE systems will be reviewed in Chapter 4.


cal achievements concerning T−11 (T, ν). I shall start with the bosons hopping

model [24] and then proceed to a quantum non-linear field model [44, 87].Calculations by Cote et al. [24] in the ordered phase of the multi-

Skyrmion system predict a remarkable evolution of T−11 with T and ν. Ac-

cording to this theory, when the Skyrmion lattice is pinned by the disorder,the magneto-phonon spectrum is gapped and does not contribute to T−1

1 .But the XY spin-wave spectrum do contribute to the nuclear spin-latticerelaxation rate, and in the limit of zero temperature, T−1

1 has a linear rela-tionship with both |ν− 1| and T . It is to be noted that, taking into accountthe combined effect of disorder and interactions, the electron spin dynamicswas theoretically studied in the past (well before ideas on QHE Skyrmi-ons were put forward) and quantitative predictions for T−1

1 were obtainedaround ν = 1 [5]. These calculations emphasized the importance of disorderand electron-electron interactions to the nuclear spin-lattice relaxation rate.Of paramount importance, they do not predict a slowing down of the nuclearspin-lattice relaxation near ν = 1 [dotted and solid curves in Fig. (2.11)].This is in sharp contrast with the bosons hopping model prediction thatT−1

1 → 0 when |ν − 1| → 0 [dashed lines in Fig. (2.11)]. We should alsoremark that recent theoretical studies questioned how good the predictedlinear T -dependence actually is. We now turn to this aspect.

A weighty attack on the problem was opened by Read and Sachdev [87];considering the scaling behavior of 2D Heisenberg models, these authorssucceeded in determining the T -dependence of the T−1

1 at ν = 1. Pursuingthe argument further, Green [44] pointed out the quantum critical nature offluctuations of the Skyrmion lattice and arrived at similar results near ν = 1.The different scaling regimes of 2D Heisenberg models could be understoodas follows. Recall that, according to the Mermin-Wagner theorem, when aspin-wave expansion is attempted for a ferromagnet at or below its criti-cal dimension, the occupation of low-frequency modes is found to diverge.However, the constraint, fixing the magnitude of the local spin, restricts thisdivergence. The interplay between the divergence and constraint give riseto a finite-T correlation length. The dynamics of the ferromagnet is verydifferent on the length scales greater or less than the correlation length. Onlength scales less than the correlation length the ground state is orderedand fluctuations are purely classical, i.e., the characteristic frequency ofspin fluctuations is much smaller than T . Accordingly, this regime is calledrenormalized classical and occurs at T ρs. Here ”renormalized” meansthat the T = 0 parameters such as the spin stiffness are renormalized com-pared to their mean-field values as a result of quantum fluctuations at shortwavelengths. At length scales greater than the correlation length the system


is disordered and fluctuations are overdamped. This is the quantum criticalregime and it is the high-T scaling regime (T ρs) in the phase diagram.The crossover between renormalized classical and quantum critical regimesoccurs at T ∼ ρs.

Figure 2.11: The dashed line pictorially describes the ν-dependence of T−11

in the bosons hopping model [24]. Calculations by Antoniou and Mac-Donald [5] are illustrated for two different values of the ratio between theexchange-enhanced Zeeman splitting (EZ +Exc) and the Landau level widthΓ. The ratio (EZ +Exc)/Γ is either 0.85 (solid curve) or 0.15 (dotted curve).

In short, we expect the following T -dependence of T−11 in the QHE plateau

at ν = 1. Starting from the high temperatures, T−11 is constant with T in

the quantum critical region. If the temperature is decreased, the systemcrosses to the renormalized classical regime. In this regime T−1

1 increasesexponentially fast with decreasing T . At even lower temperatures, preciselyat ν = 1, T−1

1 is expected to display an activated behavior. This should becontrasted with the T−1

1 ∝ T law expected near ν = 1 at T → 0. Finally, at

2.4. INTERPRETATION OF THE ν = 1 QHE EXCITATION GAP 89

T = 0 the nuclear spin-lattice relaxation rate is zero.

2.4 Interpretation of the ν = 1 QHE excitation gap

Looking back at this edifice, we may briefly examine its ground-plan. The2DES at ν = 1 has no genuine charge excitations. The number of reversedspins within a charge excitation at ν = 1 varies with g. Provided that gis small, the low-lying charged elementary excitations are Skyrmions, en-compassing many spin reversals. At large g, they are single spin flips. Ac-cordingly, the meaning of the QHE excitation gap at ν = 1, probed by themagnetotransport, varies with g. At small g, it represents the energy costfor the creation of a Skyrmion – anti-Skyrmion pair, whereas at large g itrepresents the energy cost for the creation of a widely separated quasiparticle– quasihole pair.

Investigations of the Skyrmion physics by magnetotransport were firstreported by Schmeller et al. [89]. These authors used tilted magnetic fieldsto tune g and measured the QHE excitation gap at ν = 1 (∆1) on single-QW samples. Noteworthy, in the tilted-magnetic field technique g increasesas the tilt angle θ increases and its minimal value is recorded at θ = 0.Henceforth, the tilted-magnetic field technique is particularly suitable forSkyrmion studies at large g. A series of measurements at ν = 1 concentratedon the range of small g. To achieve the g → 0 regime several ingeniousexperimental schemes have been proposed [75, 92]. The principal task ofmagneto-transport experiments [75, 89, 92] is the direct measurement of thenumber of reversed spins within a Skyrmion. The total spin of the thermallyactivated Skyrmion – anti-Skyrmion pair at ν = 1 can be extracted from thechange in ∆1 produced by tilting the total magnetic field B. It is assumedthat an in-plane magnetic field component couples only to the spin degreesof freedom, while the perpendicular magnetic field component B⊥ controlsthe orbital dynamics. This implies that the Zeeman contribution to theenergy gap ∆1 for creating a pair of elementary excitations with total spin2K + 1 (in units of ~) from the fully polarized ground state is 31

(2K + 1)|g∗|µBB = ∆1 −∆01. (2.53)

The term ∆01 is the contribution to the gap from all non-Zeeman sources and,

in this model, depends only upon B⊥. It follows that ∂∆1/∂B (evaluated

31The traditional (i.e., pre-Skyrmion) view of the ν = 1 QHE excitation gap haveassumed K = 0.


at ν = 1, where B⊥ is fixed) is just (2K + 1)|g∗|µB. Since the g∗ and µB

are known, measuring ∂∆1/∂B determines the quantum number K.

Figure 2.12: Measured QHE excitation gap at ν = 1 (∆1) vs Zeeman energy,both in units of e2/(εlB) (sample M280). The corresponding tilt angles areindicated on top axis. The dotted and dashed lines correspond to 2K+1 = 9and 2K + 1 = 1, respectively. The uncertainty in the measured ∆1 is ±6%.

I present here QHE excitation gap measurements at ν = 1 in tilted mag-netic fields, carried out on M280 sample. To our knowledge, this studywas performed up to the largest yet reported g ≈ 0.05. The value of theQHE excitation gap at ν = 1 was determined from the T -dependence ofthe longitudinal resistance in the thermally activated regime, as describedin the precedent chapter. The evolution of ∆1 with tilt angle is presented

2.4. THE ν = 1 QHE EXCITATION GAP 91

in Fig. (2.12) by plotting ∆1 vs Zeeman energy, both expressed in units ofe2/(εlB). Figure (2.7) shows the calculated Hartree-Fock energy gap for thecreation of a widely separated Skyrmion – anti-Skyrmion pair as a functionof g for sample M280. This calculation take into account the finite thicknessof the electron layers. The relevant parameter is w0/lB = 0.5, where w0

represents the rms width of the self-consistently calculated subband wavefunction in each QW. Similar to results for single QWs [89], there is anoverall qualitative agreement between the measured and calculated gaps inFig. (2.7). At θ = 0, ∆1 = 20 K, comparable to the measured ∆1 in high-quality conventional single-layer 2DESs. Note that ∆1(θ = 0) is only 16%of the theoretical value for an ideally thin, disorder-free 2DES, and with noLandau level mixing. The discrepancy originates from the inadequacies ofthis simplistic model for the 2DES. It is obvious from Fig. (2.7) that thefinite z-extent of 2D electron layers causes a dramatic reduction of the cal-culated ∆1, reducing the discrepancy between theory and experiment to afactor of 4. Even with the inclusion of both disorder and LL mixing, thecalculated excitation gap remains 2 times larger than the experimental one.More important, assuming that the slope 2K + 1 = ∂∆1/∂(|g∗|µBB) givesthe spin of a pair of charged excitations at ν = 1, both experiment andtheory give K ∼= 4 for g ≈ 0.012 and K ∼= 0 in the limit of large g. The con-tent of Figs. (2.7) and (2.12) demonstrates very satisfactorily a convergencebetween the theoretical description of Skyrmions and their experimentallydetermined properties. However, given the experimental uncertainty in themeasured gaps, the absence of quantitative agreement with theoretical val-ues and the fact that ∆1 is expected to slowly approach the single spin-flipdependence (2K + 1 = 1), prohibit an accurate determination of gc basedon transport measurements.

One of the objectives of heat capacity and NMR experiments to be de-scribed in the following two chapters is to clarify the problem of the rangeof g over which Skyrmions are the relevant excitations of the ν = 1 QHEground state, by exploring the large-g limit. In brief, tilted-magnetic fieldheat capacity measurements on M280 sample show evidence for a transitionfrom Skyrmions to single spin-flip excitations at gc ≈ 0.04. Our NMR datain tilted-magnetic fields, collected on M242 sample, reveal the presence ofSkyrmions in the electronic ground state for g . 0.022 and their absence forg & 0.037. These experimental values are in good agreement with Hartree-Fock predictions which take into account the finite thickness of the 2DES:gc ≈ 0.046 (for M242 sample) and gc ≈ 0.047 (for M280 sample).


2.5 Summary

In this chapter we have studied the QHE ferromagnet at ν = 1 and intro-duced the concept of QHE Skyrmions [95]. The Skyrmion is a spin texturewith spins down at the origin but up at infinity; at intermediate distancesthe spins lie in the XY plane and have a vortex like configuration. Theimportance of Skyrmions for QHE systems arises from the discovery thatthey describe the lowest energy charged-excitations of the QHE ferromag-nets. Therefore, Skyrmions dominate many of the low-T properties of thesesystems, and we have paid a particular attention to the T -dependence of theelectron spin polarization at ν = 1.

Properties of this excitation, such as its energy and spatial extent, aredetermined by the ratio g of the single-particle Zeeman energy, which limitsthe number of spin flips within an excitation, to the Coulomb energy whichfavors local ferromagnetic ordering. The spin of the thermally activatedSkyrmion – anti-Skyrmion pairs was deduced from the g dependence of theQHE excitation gap at ν = 1. Our results reveal that the spin of a pair ofcharged excitations at ν = 1 is ∼ 9 (g = 0.012 in sample M280). Althoughconsistent with the measured Skyrmion size, Hartree-Fock calculations [23],even after Landau level mixing and finite-thickness corrections, overestimatethe QHE excitation gap at ν = 1 over the entire range of our data.

We also addressed the properties of QHE ferromagnets near ν = 1,including the ν-dependence of the low-T electron spin polarization. Theground state of the 2DES at filling factors slightly away from ν = 1 is acrystal of Skyrmions. This means that the system has non-colinear order.Associated with this is a new U(1) degree of freedom in which the spinsrotate about the magnetic field direction. Because Skyrmions contain spinslying in the XY plane, such rotations do represent new states. Since therotation is about the magnetic field direction, this collective mode does nothave a Zeeman gap, but rather is a gapless Goldstone mode. Therefore,the 2DES strongly couples to the nuclear spin system and the nuclear spin-lattice relaxation rate at ν . 1 and ν & 1 increases by a factor of 103 overthe zero magnetic field value. We shall use the formalism developed in thischapter to analyze the following heat capacity and NMR experiments.

Chapter 3

Heat Capacity Evidence forSkyrmions

Pentru ca Însusi Domnul, întru porunca, la glasul Arhangheluluisi întru trambita lui Dumnezeu, Se va pogorî din cer, si cei mortiîntru Hristos vor învia întai. Dupa aceea, noi cei vii, care vom firamas, vom fi rapiti, împreuna cu ei, în nori, ca sa întampinampe Domnul în vazduh, si asa pururea vom fi cu Domnul.

Sf. Apostol Pavel

3.1 Introduction

3.1.1 Small-sample calorimetry

Hardly one can find elsewhere in solid-state physics the need for high-resolution small-sample calorimetry more imperatively expressed than inthe area of two-dimensional electron systems (2DESs). I have in mind notonly the low strength of the measured signal, resulting from the tiny mass ofthe specimen and the small number of electrons, but also the heinous regimeof low temperatures (T ∼ 100 mK) and high magnetic fields (B ∼ 10 T), fre-quently of principal interest. The low-temperature small-sample calorime-try in high magnetic fields has seen a number of breakthroughs both indesign concept and instrumentation over the years before the discovery ofthe quantum Hall effect (QHE). In the experiments which form the object

93

94 CHAPTER 3. HEAT CAPACITY EVIDENCE FOR SKYRMIONS

Table 3.1: Landmark papers describing calorimetric techniques for small-sample heat capacity measurements. Definitions are given in the text.

Year Definition Method

1963 C = P∆t/∆T adiabatic heat-pulsea

1968 C = P/(√

2ωH∆Tac) steady-state, ac-temperatureb

1972 C = Pτ/∆T weak-link time-constantc

1977 C = (Pτ/∆T )[1− exp (−∆t/τ)] weak-link heat-pulsed

a Morin and Maita, Ref. [81]. b Sullivan and Seidel, Ref. [98].c Bachmann et al., Ref. [6]. d Fagaly and Bohn, Ref. [34].

of this chapter we have employed three calorimetric techniques. Influentialpapers describing these methods are listed in Table (3.1). The ”adiabaticheat-pulse” technique make use of the classical definition of the heat capac-ity: C = P∆t/∆T . The temperature of the sample rises by an amount ∆Twhen the sample (thermally isolated from its surroundings) is subjected toa small heat pulse of power P and duration ∆t. In the ”steady-state ac-temperature” calorimetry, an ac heating signal [P cos (ωHt)] is applied at thefrequency fH = ωH/(2π) to the sample, which is thermally connected to itssurroundings. In the first-order approximation, the rms amplitude of the re-sulting temperature oscillations ∆Tac is simply related to the heat capacity:C = P/(

√2ωH∆Tac). Another appropriate technique for high-resolution

small-sample heat capacity measurements, referred to as ”weak-link time-constant” method, is also contingent upon the thermal link connecting thesample to the heat sink. A constant power P is applied to the sample so thatthe steady-state temperature of the sample rises by an amount ∆T . Then,the power is turned off and the temperature of the sample decays back tothe sink temperature with a characteristic time constant τ . The total heatcapacity 1 is given by C = Pτ/∆T .

While these preliminary remarks may suffice to determine the principlesaffixed to calorimetric methods used in this study, they do not tell the wholestory. They do not reflect the fact that, frequently, in real calorimetricsystems two time constants characterize the thermal dynamics of the sample:τext describes the thermal coupling between the sample and its surroundings,

1If the pulse duration is short, a suitable formula is C = (Pτ/∆T )[1− exp (−∆t/τ)].

3.1. INTRODUCTION 95

whereas τint describes the thermal relaxation within the sample. Simplereflection upon the most common experimental situations is sufficient toclarify the physical meaning of τext and τint. For a first orientation as wellas for the later considerations, let’s consider relaxation-time heat capacitymeasurements. In this case the sample is connected to a heat sink througha heat leak. We assume that the heat leak has a finite thermal conductanceK and zero heat capacity, the heat capacity of the sample is C, and the heatsink has a heat capacity Cs (C Cs). To start an experiment, the sampleis heated by a constant power P so that the temperature of the samplerises by an amount ∆T . Then, after the steady-state has been attained,the power is turned off. If the sample is a perfect thermal conductor, thetemperature of the sample will decay back to the sink temperature with asingle characteristic time constant τ (alias τext). One can easily see that Cis obtained from two independent measurements: (1) τ and (2) K = P/∆T .That is, the external time constant is determined by C and K: τ = C/K.

But when the sample is, for example, a poor thermal conductor, onehas to carefully consider the heat flow through it, and to take into ac-count its internal relaxation time τint. The internal thermal relaxation mightlead to a markedly non-exponential temperature decay and the conventionalrelaxation-time method (described above) need to be corrected 2. Let’s bemore specific and suppose that, in the example under consideration, the heatis carried by the phonons. Then, τint can be estimated from the thermaldiffusivity data: τint ≈ L2

θ/Dθ. Here Dθ = 13vθΛθ is the thermal diffusivity,

vθ is the speed of sound, Λθ is the phonon mean free path, and Lθ is thelargest dimension of the sample [115]. While it is fairly evident that forsuccessful heat capacity experiments τint should be smaller than τext, it isless obvious that the experimental values of τext and τint will force upon usthe adoption of one calorimetric technique or another [Table (3.2)].

The above example does not encode all forms of τint. Low temperaturescombined with high magnetic fields challenge us with a truly dangeroustrap: the nuclei might couple to the degrees of freedom whose heat capacitywe would like to measure (phonons or/and electrons). Traditionally, thiscoupling is described by the nuclear spin-lattice relaxation time T1. Whenheat is put into the sample, T1 plays the role of an internal time constant,i.e., it describes the establishment of the thermal equilibrium between thenuclear spin system (NSS) and the rest of the sample (the ”lattice”) 3. Since

2A notorious experimental signature of the τint-effect is an overshoot in the sample’stemperature when heat is turned on suddenly.

3In the nuclear magnetic resonance terminology (e.g., nuclear spin-lattice relaxationtime), the ”lattice” denotes all degrees of freedom of the sample excluding the nuclei.


Table 3.2: Calorimetric methods for heat capacity measurements in the QHEregime. In principle, provided that more restrictive experimental conditionsare fulfilled [40], any of these methods can be used for accurate (≈ 1%) heatcapacity measurements. In our experiments the uncertainty is ±10%.

Method τext Condition

adiabatic heat-pulse 104 − 105 s τext > ∆t > τint

weak-link time-constant 10− 103 s τext > τint

steady-state, ac-temperature 0.01− 1 s τext > 1/ωH > τint

at low temperatures, T1 could be very long, τint-effects might appear inrelaxation-time heat capacity measurements. A main theme of this chapteris the behavior of the heat capacity of multiple-quantum well (QW) sampleswhen the nuclei couple to the ”lattice” in the experimental time-scale. Themodel for the calorimetric system, the equations governing the heat transfer,and the expressions for C and T1 will be discussed in detail.

3.1.2 Thermal equilibrium in multiple-QW samples

Multiple-QW heterostructures are unique examples of multiple-subsystemssamples, as they are composed of the lattice (phonons) 4, the two-dimensional (2D) electron layers, and the NSS of QWs and barriers[Fig. (3.1)]. Since we are able to perceive now the dangers that beset low-Theat capacity experiments on multiple-subsystems samples, before gettingdown to the details, it is instructive to further remark on the process ofthermal equilibrium in our measured specimens [69].

Surprising thermal effects may happen in multiple-QW samples becausenuclei can be divided in two groups. The first group is represented by thenuclei located in the QWs. The second group is formed by the nuclei locatedin the barriers (and the GaAs substrate). In most of the cases, the couplingbetween the NSS and the lattice is so weak that, a fast refrigeration will leavethe nuclei at the starting T . A weird situation occurs when the couplingbetween the QWs’ nuclei and the lattice is strong enough, but the couplingbetween the barriers’ nuclei and the lattice is weak. In this case, a fast

4The crystal lattice is the regular array of sites in the three-dimensional space wherethe individual atoms are supposed to lie when the whole specimen is in its ground state.


refrigeration will pull the QWs’ nuclei to low T while leaving the barriers’nuclei at the starting T . Finally, it is possible to refrigerate all nuclei inthermal equilibrium with the lattice down to very low temperatures 5.

Figure 3.1: Thermal diagram of a multiple-QW sample connected to a heatsink. Subsystems are linked to each other by means of thermal resistances.

In general, heat capacity is a bulk property with contribution from allcomponents of a given thermodynamic system. Henceforth, depending on thestrength of the nuclear spin-lattice coupling at low T , one might measure:(1) the lattice (phonons) and the electronic heat capacity, (2) the lattice(phonons), the electronic, and the nuclear spin heat capacity of GaAs QWs,(3) the lattice (phonons), the electronic, and the nuclear spin heat capacityof GaAs QWs and barriers. An estimate of various heat capacities (to be

5At low-T , it is advantageous in thermal studies to attribute a separate nuclear spintemperature and heat capacity to each nuclear spin subsystem [Fig. (3.1)].


made shortly) will reveal that the heat capacity of a multiple-QW samplemight vary five orders of magnitude! The magnitude of the heat capacity,in turn, will critically constraint the choice of the experimental method[Table (3.2)].

3.1.3 Samples for heat capacity experiments

For the purpose of heat capacity measurements, samples were prepared byetching and polishing the material. The thinning of the excess bulk GaAssubstrate was necessary to remove the indium present on the back of thesamples from their molecular-beam-epitaxy block mount, and to improvethe 2DES signal-to-background ratio 6. For heat capacity experiments onM242 heterostructure we used a 7× 7 mm2 piece of the wafer thinned downto ≈ 65 µm. Heat capacity experiments on M280 heterostructure werecarried on a 7 × 10 mm2 piece of the wafer which was thinned down to ≈160 µm. Unfortunately, the expedient of using large area samples makes thedensity not entirely homogenous within a 2D electron layer. Typical densityinhomogeneities across our measured specimens are ±3%. The transversevariation of density (from layer to layer) is ±2%.

As stated before, in a multiple-QW heterostructure, various types ofheat capacities are encountered, each having its own T -dependence. Weestimate in the following the lattice (phonons) Cl, the electronic Ce, andthe nuclear Cn heat capacities of our measured specimens. According tothe Debye model [115], Cl shows a T 3-dependence at low temperatures.The phonon contribution to the specific heat of bulk GaAs amounts to cl ≈(4π4/5)3kB(8/A3

c)(1/ρc)(T/θD)3 = 26.9 (T/θD)3 [J/(gK)] for T < θD/20, inview of the crystal (ρc) and atomic density (8/A3

c) listed in Table (1.2). HereθD ≈ 345 K is the zero-T ”elastic” Debye temperature of GaAs [12]. As-suming that the Al0.1Ga0.9As and Al0.3Ga0.7As layers have the same phononspecific heat as the bulk GaAs, the estimated low-temperature Cl of oursamples (including the GaAs substrate) is

Cl =

1.1× 10−8T 3 (J/K) for M242

3.9× 10−8T 3 (J/K) for M280,

where T is expressed in K. For non-interacting 2D electrons, in the limitof low T such that kBT EF , Ce has a linear T -dependence and it isdetermined by the electronic density of states (DOS) at the Fermi levelD(EF ): Ce = (π2/3)k2

BTD(EF ) (J/K) per electron. Given the fact that6The thinning process of multiple-QW heterostructures is described in Ref. [110].


m∗/(π~2) = 0.24 × 1014 [1/(m2K)] (including the spin degeneracy), theestimated Ce of our samples at zero magnetic field is

Ce =

6× 10−12T (J/K) for M242

8× 10−12T (J/K) for M280,

where T is expressed in K. When the sample is placed in a magnetic field,besides lattice and electronic heat capacities, there might be an excess heatcapacity due to nuclear spins. In the high-T limit, the Schottky nuclear heatcapacity of Ga and As atoms in the QWs (see Appendix A) is estimated at

CQWn =

1.9× 10−11B2 T−2 (J/K) for M242

3.3× 10−11B2 T−2 (J/K) for M280,

whereas for the Schottky nuclear heat capacity of Ga and As atoms in thebarriers we obtain

CBn =

14.1× 10−11B2 T−2 (J/K) for M242

27.5× 10−11B2 T−2 (J/K) for M280.

Here T is expressed in K and B is given in T. The Schottky nuclear heatcapacity of Ga and As atoms in the barriers is counted by making use of theratio between the width of the barriers and the QWs’ width. Estimates ofCl, Ce, C

QWn , and CB

n (evaluated at T = 100 mK) are listed in Table (3.3).At a glance, one can see that C might vary by up to five orders of magnitude.

Table 3.3: Estimates of lattice (Cl), electronic (Ce), and nuclear (CBn , CQW

n )heat capacities (in pJ/K) for samples M242 and M280 at T = 0.1 K. HereCl is the phonon contribution to the heat capacity of the entire sample(including the GaAs substrate). The nuclear heat capacity of Ga and Asatoms is evaluated at B = 10 T. The thermal conductance of the heat leakKext (in pW/K) is estimated at T = 0.1 K (B = 0).

Sample Cl Ce CQWn CB

n Kext

M242a 10 0.6 2× 105 1× 106 60

M280b 40 0.8 3× 105 3× 106 60

a The heat leak assumed here is a 60 µm-diameter, 10 mm-long NbTi wire.b The heat leak considered here consists of forty 10 µm-diameter GY-70graphite fibers of length 10 mm.


Figure 3.2: (Top) Photograph of the Bayotron rotating platform. (Bottom)Experimental configuration for heat capacity measurements (sample M280).


3.1.4 Heat capacity: Experimental details

Over a four-year period from 1992 to 1996, Prof. Vincent Bayot and Dr.Eusebiu Grivei were heavily involved in bringing to fruition an apparatusfor heat capacity measurements at dilution refrigerator temperatures. Atthe very beginning of my thesis they offered me a wonderful opportunityto learn about low-T specific heat experiments. As the calorimetric systemsometimes requires manual adjustments of data collecting parameters (T ,B, time delays, ...), it stands to reason that a connection deepened betweenmyself and the low-T calorimetry.

Figure (3.2)(top) shows our rotating stage which permits in situ tiltingof the sample. The rotating platform is made from Stycast epoxy and itaccommodates a standard dual in-line multiple pin (DIP)-socket. If neces-sary, the tilting stage is attached to the mixing chamber and it is thermallyanchored to the cold finger of the Bayotron. A carefully done wiring ofthe tilting stage is an absolute must to ensure a well cooling of the sam-ple and to minimize thermal drifts. This was accomplished with insulated80 µm-diameter copper wires which were thermally anchored by wrappingthem around the pins of the DIP-socket. The wires were further cementedwith GE 7031. The sample holder with an actual sample mounted for heatcapacity measurements is shown in Fig. (3.2)(bottom). The DIP-headersupporting the sample fits either into the DIP-socket fixed onto the mixingchamber or on the standard DIP-socket mounted on the tilting platform.The sample could be positioned accurately and reproducibly in situ over arange 0− 90 within ±0.1, which represents the mechanical stability of therotating platform. The tilt angle is precisely determined by measuring theHall resistance on a 2DES device fixed on the rotating stage.

For heat capacity measurements, the M280 sample was tilted in situ sothat an angle 0 . θ . 77 formed between B and the normal to the sampleplane. For M242 sample, the experimental arrangement did not include therotating platform: C was measured while B was applied either perpendicularto the 2DES plane (θ = 0) or at a fixed angle of θ = 30± 2.

Thermometry

For the measurement of heat capacity, a heater and a thermometer shouldbe placed in good thermal contact with the sample. In our experiments,the thermometer and the heater are carbon paint resistors deposited on thesubstrate side of the sample. The carbon resistors were calibrated between≈ 15 mK and ≈ 1 K against a RuO2 resistance thermometer at B = 0.


It is important to note that carbon resistors were very stable, as they didnot change their resistance R over time or thermal cycling between roomtemperature and our lowest accessible temperatures. We found that theyagreed with the original calibration within 1 mK. We also checked that theircalibration was negligibly affected by the magnetic field 7. Figure (3.3)(a)shows R vs T for one of the thermometers used in our heat capacity studies.(The shape is characteristic for low-T resistance thermometers). The datawere fitted by a polynomial [open circles in Fig. (3.3)(b)] of the form

lnT =∑

i

Ci(lnR)i, (3.1)

where Ci are constants. (The method of least squares was utilized and thesame weight was given to each calibration point.) A plot of temperaturedeviations ∆Tth = Tth − T vs T is shown in the inset to Fig. (3.3)(b). HereT is the measured temperature and Tth is the temperature calculated fromthe observed resistance R and a 7th degree polynomial of the form given byEq. (3.1). The fractional deviation of the temperatures ∆Tth/T was below±2% for 20 mK . T . 1 K. In the event of unsatisfactory performance bya carbon paint resistor, the carbon paint could be easily removed from thesample upon immersion in its base solvent (isopropanol).

We now attempt to quantify the heat capacity of carbon paint resis-tors. Following Ref. [110], the paint is dabbed using a thin copper wire,in two spots of total mass 0.50 ± 0.25 mg. Measurements of the specificheat of graphite at low temperatures provide an experimental value about6× 10−7 J/(gK) at T = 400 mK, and show that well-ordered graphite doesnot have a Schottky anomaly 8. Our results seem to indicate that, in the in-vestigated T -range, the heat capacity of the carbon paint resistors employedin the present experiments is dominated by the electronic contribution. Theestimated heat capacity of the carbon paint is consequently expected torange between 30 and 110 pJ/K at T ≈ 100 mK, consistent with the totalheat capacity signal. Based on estimations shown in Table (3.3), we con-clude that in our experiments the heat capacity of carbon paint resistors isnegligibly small when the measured C & CQW

n , but it dominates C whenthe sample’s heat capacity reduces to ≈ (Cl + Ce).

7While keeping the mixing chamber temperature constant, the temperature reading ofthe carbon resistors showed < 3% monotonous deviation in the investigated B-range.

8M.S. Dresselhaus, G. Dresselhaus, K. Sugihara, I.L. Spain, and H.A. Goldberg,Graphite Fibers and Filaments, (Springer, Berlin, 1988), p. 106.


Figure 3.3: Calibration data for carbon paint thermometers (see text).


Heat leak

For sample M242, the thermometer was electrically connected with four8 µm-diameter NbTi wires. The electrical leads of the heater consisted ofone 60 µm-diameter NbTi wire and one 8 µm-diameter NbTi wire. TheseNbTi wires (45% Nb by weight) also served as thermal link to the heat sinkand mechanical support; they remained superconducting in the investigatedB-range. Further, the sample M242 has been mechanically supported by aperfluoroalkoxy-insulated 250 µm-diameter copper wire. For sample M280,the thermometer and heater were electrically connected with graphite fiberbundles which also served as thermal link to the heat sink and mechanicalsupport. We have used ex-polyacrylonitrile (GY-70) graphite fibers; theyhave high Young’s modulus and their low-T thermal conductivity is small,comparable with that of NbTi filaments. Further, the sample M280 hasbeen mechanically supported by a small rubber O-ring shaped in the formof a crown and a fine nylon tread [Fig. (3.2)(bottom)]. Various propertiesof materials used for sample’s thermal anchoring are listed in Table (3.4).

The external thermal conductance could be easily adjusted by varyingthe number of NbTi filaments (or graphite fibers) in order to obtain a con-venient time constant for the estimated C in the T and B-range understudy. The thermal conductivity κ of graphite fibers is obtained 9 from thesaturated low-T electrical resistivity ρ by virtue of Wiedemann-Franz law:κρ = L0T , where L0 = π2k2

B/(3e2) = 2.45 × 10−8 (WΩ/K2). The thermal

conductivity of NbTi wires was measured down to very low T and the fol-lowing empirical law 10 was derived: κ ≈ 2×10−2 T 2 (W ·K−1 ·m−1). Lowerbound estimates of the thermal conductance Kext of the actual heat leaksin M280 and M242 sample calorimeters are given in Table (3.3).

In the discussion of the heat capacity of the thermal link, I want to putaside as really relevant the fact that both NbTi wires and GY-70 graphitefibers display Schottky heat capacity anomalies. The carbon fibers displaya Schottky heat capacity anomaly at low magnetic fields (B ∼ 0.1 T), prob-ably due to the presence of magnetic impurities such as Fe. At the presentthere is no explanation in terms of a microscopic model for the origin ofthe hump in the heat capacity of graphite fibers at low-B. The nuclear-spincontribution of Nb atoms dominates the heat capacity of NbTi wires at high

9V. Bayot, Conduction et Desordre dans le Graphite, Ph. D. Thesis, UniversiteCatholique de Louvain, 1991, p.96.

10J.R. Olson, Thermal Conductivity of Some Common Cryostat Materials between 0.05and 2 K, Cryogenics 33, 729 (1993).


magnetic fields (B ∼ 10 T). The estimated Schottky nuclear heat capacityof Nb atoms at B = 10 T and T = 0.1 K in our M242 sample calorimeter is1.3× 105 pJ/K, which is comparable to CQW

n [Table (3.3)].

Table 3.4: Various properties of materials used for the thermal anchoring ofthe sample in heat capacity experiments: mass density (kg ·m−3), thermalconductivity (W ·m−1 ·K−1) at T = 0.1 K, specific heat (J · kg−1 ·K−1) atT = 4.2 K, and electrical resistivity (Ω ·m) at T = 4.2 K.

Material Thermalconductivity

Mass density Specificheat

Resistivity

NbTi 0.2× 10−3 6.2× 103 0.87 → 0

GY-70 0.2× 10−3 1.9× 103 0.25 9× 10−6

The simplified model of the calorimeter system

To keep the model for the calorimetric system at a manageable level of com-plexity, it is necessary to make few simplifying assumptions: (1) the heatleak has finite thermal conductance and zero heat capacity 11, (2) the ther-mometer and the heater are gifted with ethereal qualities, i.e., they are inperfect thermal contact with the sample and have zero heat capacities 12, (3)τint is negligible when the sample’s heat capacity is dominated by the latticecontribution (alternatively, τint is very long when sample’s heat capacity isdominated by the nuclear-spin contribution), (4) there is no heat transfer be-tween the sample and its surroundings by radiation, (5) C(T +∆T ) = C(T )and κ(T + ∆T ) = κ(T ) for small ∆T 13, and (6) at any T and B, electronsare in thermal equilibrium with the lattice, i.e., the electron spin-latticerelaxation time is the shortest time scale of the system.

11The approximation of zero heat capacity for the heat leak is discussed in Sec. (3.1.4).12The heat capacity of carbon paint resistors is discussed in Sec. (3.1.4).13In all our calorimetric experiments, the temperature differences are below 10% of the

mean temperature. At a given T and B, various finite heat capacities and thermal con-ductances are considered as being constant within the periods of heating and T -response.


3.2 Calorimetry in the integer QHE regime

Before exploring the domain of filling factors close to unity, we stop alongthe integer QHE to survey heat capacity measurements performed in thisregime. We have seen in Chapter 1 that D(EF ) of a 2DES in a perpen-dicular, quantizing magnetic field exhibits 1/B-periodic oscillations due tothe formation of disorder-broadened Landau levels (LLs); D(EF ) is max-imum at half-integral ν and it is minimum at integral ν. The oscillatingD(EF ) induces oscillations in many physical properties of the 2DES suchas electrical resistivity, magnetization, thermal conductivity, and specificheat. Thermodynamic quantities (such as heat capacity) probe the totalDOS, as for thermal equilibrium properties the difference between the lo-calized and extended electronic states is of no importance. Quantitativeanalysis of various measurements [30, 112] and, in particular, of heat capac-ity experiments [42, 110], paint an extremely complex picture for the DOS.Specifically, the B-dependence of the LL width, the functional form of theLLs, and the number of localised states between two adjacent LLs remainlargely unsettled.

We successfully carried out heat-capacity measurements in the integerQHE regime at dilution refrigerator temperatures using the ac calorimet-ric method [9]. Our results are qualitatively consistent with heat capac-ity data reported early on [42], suggesting that the total DOS consists ofGaussian peaks at the unperturbed LL positions superposed on a constantbackground. It should be noted that information about the DOS in sam-ples M242 and M280 could be also extracted from existing magnetizationand thermal conductivity data [8]. Since it is of notoriety that differenttechniques yield different results for DOS, one expects that the comparisonof results of various techniques on the same samples will provide a consis-tency test for quantitative determinations of the DOS. Why this thesis willskip over such a quantitative data analysis? Firstly, uttermost care must betaken in comparing the observed magneto-oscillations, as different aspectsof the DOS dominate the line shape of different thermodynamic quantities.For example, the maxima of oscillations appear in magnetization (or latticethermal conductivity) when EF is between the LLs, whereas they appearin the heat capacity when EF is at the center of the LLs. The LLs widthsestimated from the magneto-oscillations of heat capacity and magnetiza-tion/thermal conductivity are therefore not directly comparable. Secondly,as a complete synopsis of the T -dependence of the electrical resistivity inthe integer QHE plateaus is lacking at the time of the writing, it was notpossible to learn much from magnetotransport about the DOS between LLs.

3.2. CALORIMETRY IN THE INTEGER QHE REGIME 107

3.2.1 The steady-state, ac-temperature calorimetric method

In the ac calorimetry [98] the heater is driven by an ac voltage vH =VH

√2 cos (ωHt/2), where VH stands for the rms amplitude of the cos-

inusoidal voltage. The corresponding heat generated in the sample isP (t) = P + P cos (ωHt), where P = V 2

H/RH is the peak power and RH

is the heater’s resistance. This heating signal raises the temperature of thesample above Ts, where it oscillates about a steady-state offset temperature.The temperature difference between the sample and the heat sink could bewritten as ∆T (t) = ∆Tdc + ∆Tac

√2 cos (ωHt− φ), where ∆Tdc is the con-

stant offset in the sample’s temperature, ∆Tac is the rms amplitude of theresulting temperature oscillations, and φ is the phase shift with respect toheating oscillations. Assuming that both heat and temperature variationsare small and linearizable 14, the thermal equation for the system is

C∂[∆T (t)

]∂t

= P (t)−Kext∆T (t). (3.2)

In the steady state, the dc component of the heating is dissipated through thethermal link (∆Tdc = P/Kext) and the ac component of the heating probesthe heat capacity. In the first-order approximation, the heat capacity of thesample is given by

C = P/(√

2ωH∆Tac). (3.3)

Diagnostic procedure

The approximation that ∆Tdc is constant throughout the sample requiresthat τext > 1/ωH > τint. In physical terms this condition means that theheat flux is cycled into the sample quickly enough that the thermal responseis measured before the heat dissipates to the exterior, but slowly enoughthat the temperature response is uniform within the sample 15. One alwaysmeasures the frequency profile of the calorimeter to determine an operatingfrequency within a ”plateau” region where neither τext nor τint dominates thethermal response. When the ac calorimeter has a plateau over at least onedecade in frequency together with sharp low-frequency and high-frequencyrolloffs, a good choice for the measuring angular frequency is (τintτext)−

12 .

14In practice, the constant rise in temperature corresponds to a drop in the ther-mometer’s resistance. The corresponding drop in the thermometer’s resistance at ωH

was checked to be proportional to V 2H at ωH/2.

15The phase shift φ is close to π/2 for this intermediate frequency range, it is essentiallyzero for small ωH , and it approaches π/4 for large ωH .


But when the plateau is only a ”broadly rounded hump” and the time con-stants of the calorimeter cannot be easily identified, the measuring frequencyis guessed according to few practical rules. It should be as high as possibleto avoid 1/f noise in the electronics and low-frequency limitations of thelock-in detector. The lower limit of the measuring frequency is indicated bythe onset of deviations from the linear behavior describing the calorimeterresponse at low frequency [110]. The upper limit on frequency is dictatedby the need for thermal equilibrium in the sample. The thermal relaxationlength λθ = (2Dθ/ωH)1/2 should be greater than the largest dimension ofthe sample.

Figure 3.4: Calorimeter response fH∆Tac (or 1/C) vs fH = ωH/(2π) forsamples M280 (a) and M242 (b). The arrows indicate the measuring fH andthe dotted lines show the linear thermal response at low frequencies [110].


In practice, for any set of external parameters (T and B), we optimize thefrequency for heat capacity measurements by scanning fH∆Tac or, equiv-alently, 1/C as a function of fH = ωH/(2π). In Fig. (3.4), the productbetween fH and ∆Tac [panel (a), sample M280] and 1/C [panel (b), sampleM242] are plotted vs fH to determine the bounded range in the frequencydomain where C is given by Eq. (3.3). The similar thermal response observedat low frequency is not unexpected as the estimated Kext and C are nearlyidentical for samples M242 and M280. In our experiments, plateau signa-tures are observed around fH ≈ 20 Hz, which was chosen as the measuringfrequency 16. Fortunately, in the integer QHE regime, over the investigatedT - and B-range, the changes in heat capacity are small and a strong signalcould be maintained with only a single operating frequency. In the B-rangenear ν = 1, however, C increases dramatically, and the plateau is expectedat much lower frequencies. As the lower limit of the frequency range of thelock-in amplifiers (PAR 124A) used in this study is 2 Hz, no plateau signa-tures were found in a scan of fH∆Tac vs fH near ν = 1 and a heat-pulsetechnique (see below) was used for measuring heat capacity.

3.2.2 Results and discussion

The heat capacity data presented in Fig. (3.5) demonstrate the oscillatorybehavior of Ce for a 2DES in the integer QHE regime. We note that heatcapacity oscillates with the correct phase; i.e., it exhibits minima at integerν. The average density of the 2DESs, inferred from the position of theminima, is consistent with the magneto-transport data. The uncertaintyin the background subtraction usually precludes the in-depth analysis ofthe 1/B oscillations; only the amplitudes of magneto-oscillations are welldetermined and can be used to gain insight into the DOS. Even thoughC is dominated by the lattice and addenda contributions, up to ≈ 10%oscillations - coming from the 2DES, are clearly observed. It should benoted that the linear T -dependence observed at half integers in Fig. (3.5),namely, the scaling of the measured C with the temperature, correspondsto the usual Fermi-liquid-like behavior arising from intra-Landau-level con-

16As discussed above, τint can be estimated from the thermal diffusivity data: τint ≈L2

θ/Dθ. The phonon mean free path Λθ of bulk GaAs depends on the doping. For n-type GaAs, Λθ is ≈ 300 A at room temperature and it rises when T is lowered. For thehighest doping it saturates at low-T about 0.5 × 10−3 m, whereas for the lowest dopingit saturates at low-T about 10−3 m. Henceforth, taking vθ ≈ 3000 m/s, for sample M280(Λθ ≈ 10−2 m) the estimated τint ranges between 10−3 s and 2 × 10−2 s. The lattervalue is very close to τext, determined from the empirical analysis of the thermal signal,explaining why in our experiments the theoretical plateau is conspicuously lacking.


Figure 3.5: [(a) and (b)] Heat capacity C vs B in the integer QHE regime.


tribution to Ce when EF lies within a broadened Landau level. Because theC line shapes are similar for materials M242 and M280, we mainly discussbelow the results for sample M242 [Fig. (3.5)(a)] and, in particular, thoseobtained at T = 92 mK [Fig. (3.6)].

Figure 3.6: Line shape of Ce in the integer QHE regime after the subtractionof the background as a Schottky term (sample M242 at T = 92 mK).

In order to further shine light on the experimental findings, we shall use aGaussian model of the DOS [Eq. (1.16)], in which, neglecting the overlappingbetween LLs, the DOS peaks are given by [eB⊥/(hΓG)][1/(

√2π)]. Thus, we

obtainΓG[eV] = 1.5× 10−4 n2D

∆Ce[pJ/K]A[cm−2] B[T] T [K], (3.4)

where A is the area of the sample and n2D = 100 is the number of 2Delectron layers. The results for ΓG are given in Table (3.5). For sampleM242 at T = 92 mK and ν = 5/2 the amplitude of heat capacity oscillationsis ∆Ce = 3.9 pJ/K, implying that the LL rms width is ΓG(ν = 5/2) =0.4 meV. Similarly, at T = 92 mK and ν = 7/2 (ν = 9/2), where ∆Ce =2.6 pJ/K (∆Ce = 0.8 pJ/K), we infer ΓG(ν = 7/2) = 0.45 meV [ΓG(ν =9/2) = 1.1 meV]. Strictly speaking, an amplitude of oscillation measures


the difference between Ce with EF at the peak and with EF between peaks.If the DOS of the 2DESs, and therefore Ce, is not negligible between LLs,so that peak amplitudes are offset, then ΓG obtained here is only an upperbound 17.

Table 3.5: The measured ∆Ce (pJ/K) at half-integer filling factors are givenat various temperatures for samples M242 and M280. The uncertainty in∆Ce is less than 0.2 pJ/K. For sample M280 note that, mainly due to thestrong B-dependence of the background [Fig. (3.5)(b)], there is no percep-tible difference between ∆Ce measured at 68 and 50 mK. Estimates of ΓG

(meV) [Eq. (3.4)] are also included.

ν = 5/2 ν = 7/2 ν = 9/2

∆Ce ΓG ∆Ce ΓG ∆Ce ΓG

M242 (92 mK) 3.9 0.40 2.6 0.45 0.8 1.1

M242 (62 mK) 2.9 0.35 2.2 0.35 - -

M280 (68 mK) 5.2 0.19 2.8 0.25 - -

M280 (50 mK) 5.2 0.14 2.8 0.18 - -

Comparison with previous heat capacity data [110] gives a LL width atν = 5/2 (and 7/2) about five times smaller for our sample. This is consistentwith the lower disorder in our sample as evidenced by the presence of minimain Ce at odd ν down to ν = 5. Furthermore, the lower values of ΓG in sampleM280 than those determined in sample M242 [Table (3.5)], correspond tonarrower, better-defined Landau levels, indicating an improved quality ofthe 2DES in sample M280. It is worth noting that the disorder broadeningΓ0 computed from the electrical resistivity data at B = 0 is in sharp conflictwith the inferred Gaussian LL width at ν = 5/2 and 7/2. (Γ0 is one orderof the magnitude smaller than ΓG [26].)

17We recall that the short-range scattering theory [3], predicts that the LL width followsthe law ΓSR(B) = ~[2ωc/(πτ0)]

1/2, i.e., at any B the LL width is uniquely described byτ0. For sample M242, ΓSR(B) [meV] = 0.25×B1/2 (with B in T). The calculated ΓSR(B)at ν = 5/2, 7/2, and 9/2 are smaller but comparable to the Gaussian LL widths obtainedfrom the amplitude of heat capacity oscillations [Table (3.5)].


Qualitative analysis of the measured Ce line shape also yields roughestimates of the exchange-enhanced g factor [g(B)] at odd filling factors.In Fig. (3.6), we see that Ce has a weak minimum near ν = 5. The LLwidth near ν = 5 must be comparable to the spin-splitting between LLsEZ = g(B)µBB, as the LLs are spin-resolved. Our upper bound of ΓG(ν =9/2) = 1.1 meV then implies g(B) . 15.

In agreement with previous heat capacity experiments [109, 110, 111],two important conclusions can be drawn from our data: (1) the zero-magnetic field mobility does not seem to parametrize the LL broadening veryeffectively and (2) there is a g factor enhancement in the LL spin-splittingof a 2DES. As in Refs. [42, 110], self-consistent numerical calculations andmultiple-parameter fits are necessary to determine Ce, and thereby D(EF ),for arbitrary B. Our preliminary analysis (not shown here) suggests thatCe(B) could be reasonable well fitted in the Gaussian model with ΓG(B)varying periodically in 1/B. In optimal fits to our data the peak heights inthe B-dependence of the DOS and Ce are proportional to the inverse of theLandau level width.

Finally, we wish to remark on the sudden and dramatic decrease ofthe amplitude of Ce oscillations in the N = 2 Landau level (4 ≤ ν ≤ 6)[Fig. (3.6)]. Since Shubnikov-de Haas oscillations of the longitudinal resis-tance persist to much larger filling factors in the present samples, the in-triguing behavior of Ce suggests that some previously unanticipated physicsis at work. Recently, Koulakov, Fogler, and Shklovskii 18 have proposedthat in a clean 2DES in the N = 2 and higher LLs the uniform electronliquid may be unstable against the formation of charge density waves. Theyfurther suggest that near half filling of the LL the charge density wave is aunidirectional ”strip phase” having a wavelength of order of the cyclotronradius. In this strip phase the electron density in the uppermost LL alter-nates between zero and full filling. At ν = 5/2 this implies that there arestripes of the incompressible states at ν = 4 and ν = 5. It is plausible toconsider that, if the stripes are coherent over the macroscopic size of oursamples, they provide a intrinsic source of inhomogeneity which ”smoothsaway” heat capacity oscillations in the higher LLs (N ≥ 2).

18A.A. Koulakov, M.M. Fogler, and B.I. Shklovskii, Charge Density Wave in Two-Dimensional Electron Liquid in Weak Magnetic Field, Phys. Rev. Lett. 76, 499 (1996).


3.3 The ”holy” nuclear heat capacity of GaAsQWs

The interest in heat capacity magneto-oscillations has been superseded bythe heat capacity data around ν = 1. At low temperatures (T . 300 mK)and in the B-range near ν = 1, C unexpectedly increased by orders ofmagnitude compared to the low-B data. The observation of a ”giant” low-Theat capacity shifts our attention to the situation when the NSS contributesto the measured heat capacity. We shall see that, indeed, our heat capacityobservations in the vicinity of ν = 1 are quantitatively described by a simplemodel which takes into account the coupling between the NSS and the lattice.Further consequences of the nuclear spin-lattice coupling near ν = 1 willreward us with fundamental knowledge on the low-lying excitations of the2DES.

3.3.1 The time-constant calorimetric method

To facilitate these developments, it is useful now to establish the method-ology for the analysis of the heat capacity experiments in the practicallyimportant case when the QWs’ nuclei contribute to the measured C. Theessential key to our time-constant calorimetric technique is that the thermalconductance associated to the nuclear spin-lattice relaxation process is givenby KQW

n = CQWn /T1. Here CQW

n is the Schottky heat capacity of the QWs’nuclei that couple to the lattice and T1 is the nuclear spin-lattice relaxationtime which governs the heat transfer between the NSS of GaAs QWs andthe lattice.

In our thermal relaxation experiments, a constant power P is first in-jected in the lattice. After a period of time (t) much longer than τext, thetemperature of the nuclei that couple to the lattice (TQW

n ) is equal to lat-tice’s temperature T [t < 0 in Fig. (3.7)]. Consequently, we determine theexternal heat leak as Kext = P/∆T1, where ∆T1 is the difference betweenthe steady-state temperature of the lattice (T at t < 0) and the temperatureof the heat sink (Ts). At t = 0 the power is turned off and the thermal relax-ation of the nuclei then occurs through 1/K = 1/Kext + 1/KQW

n accordingto

CQWn

∂[∆TQW

n (t)]

∂t= K∆TQW

n (t), (3.5)

3.3. THE ”HOLY” NUCLEAR HEAT CAPACITY 115

Figure 3.7: The lattice temperature T vs time t in a relaxation heat capacityexperiment (sample M242). The two-parameter exponential fit to the Tdecay at t > 0 (full line) gives τ and ∆T2. The inset shows the thermal circuitof the sample with thermal conductances Kext and KQW

n . The constantpower injected in the lattice at t < 0, together with ∆T1, determines Kext.


where ∆TQWn (t) = TQW

n (t) − Ts. Introducing the time constant τ =CQW

n /K, the formula

TQWn (t)− Ts = ∆T1 exp (−t/τ), (3.6)

describes the evolution of TQWn with t. Since in our case CQW

n is much largerthan any other contribution to the measured C, at t = 0 the temperaturedrops from Ts + ∆T1 to Ts + ∆T2, and for t > 0 the same amount of heatflows through KQW

n and Kext

K[TQW

n (t)− Ts

]= KQW

n

[TQW

n (t)− T (t)]

= Kext

[T (t)− Ts

]. (3.7)

Equations (3.6) and (3.7) give

T (t)− Ts =K

Kext∆T1 exp (−t/τ), (3.8)

which can be used to analyze experimental data as illustrated in Fig. (3.7).Finally, we obtain the heat capacity C ≈ CQW

n as

C = τP

∆T1

(∆T2

∆T1

), (3.9)

and the nuclear spin-lattice relaxation time as

T1 = τ

(1− ∆T2

∆T1

). (3.10)

Equation (3.9) shows that a correction factor must be introduced to thenaively expected C = Pτ/∆T1 result as soon as a sharp temperature dropis observed at the beginning of the thermal relaxation curve, which couldbe still modeled by a single-exponential T -decay for t > 0. Equation (3.10)tells us that T1 is also propitiously measured in the above-described thermalrelaxation method. This model, usually described as the ”lumped” τint-effect, has been discussed in the literature [6, 91] in the context of a poorthermal contact between the sample and the substrate which has the heaterand the thermometer. To our knowledge, this is for the first time when T1 issimply determined from the analysis of cooling curves exhibiting the lumpedτint-effect.


Limitations of the time-constant calorimetric method

The modified time-constant calorimetric method suffers from the generalproblems of thermal relaxation calorimetry 19, and it is, in fact, applicableonly in very restrictive conditions 20. First, attention has to be paid tothe asymptotic limit ∆T2/∆T1 → 0. In order to get a decent measure ofτ [6], one needs 0.1 < K/Kext, which corresponds to 0.1 < ∆T2/∆T1. Thesecond limiting case, when ∆T2/∆T1 → 1 (K/Kext → 1), reduces to theconventional case introduced in Sec. (3.1.1), in which, for a single dominantrelaxation time, one can retrieve only one parameter: the total heat capacityC. Therefore, in order to reduce the inaccuracy in the determination of thetwo right terms of Eqs. (3.9) and (3.10), we extract C and T1 only in therange 0.1 < ∆T2/∆T1 . 0.8. In practice, since KQW

n (and hence K) isusually strongly T - and B-dependent, for a given Kext, thermal relaxationcurves of the form shown in Fig. (3.7) are observed only in a limited T -and B-range. Clearly, one major difficulty in doing this type of calorimetricexperiments is the suitable choice of the heat conductance of the thermallink. We have checked that changing Kext by up to a factor of 5 has noeffect on the measured values of C and T1.

It may happen that, even though the nuclear-spin lattice coupling doesnot provoke a lumped τint-effect, it leads to a pronounced non-exponentialtemperature decay for t ≥ 0 (the so-called ”distributed” τint-effect). Thisproblem has been solved by a number of authors [4, 6]; the relaxation curveis usually approximated as a sum of two exponentials. Consequently, theanalysis is more involved. Andraka et al. [4] discussed in detail the nuclear-spin lattice coupling in relaxation calorimetry from the point of view of thedistributed τint-effect. In order to obtain the quantities of interest (C, CQW

n ,and T1), they propose a general method to analyze thermal relaxations withtwo dominant time-constants 21. Occasionally, we have observed coolingcurves displaying an abnormally high initial slope compared to the rest ofthe T -decay, indicative of the distributed τint-effect. In this case, we simplyobtain an estimate of C from the single (largest) time constant as C =Pτ/∆T1.

19There is one basic limitation of the time-constant calorimetric method. Near phasetransitions, the simple exponential T -decay becomes complicated. In this case, not onlyT (t) must be measured but also the time derivative T (t) must be known to obtain C.

20We note that in our experiments, accurate determinations of time constants below10 s are unfeasible.

21These authors scrupulously take into account, in addition to the nuclear-spin heatcapacity, the finite heat capacity of the lattice Cl in their model for the heat flow. ForCl = 0, rearrangement of Eqs. 5 and 6 in Ref. [4] leads to Eq. (3.7).


Figure 3.8: C vs B at θ = 0 for T = 60 mK (•) and T = 100 mK (),showing orders of magnitude enhancement near ν = 1 over the low-B data.


3.3.2 Results

As a function of increasing B, in addition to oscillations associated withthe 2DESs’ oscillating DOS at the Fermi level [Fig. (3.5)], a ”giant” low-Theat capacity is observed around ν = 1 [Fig. (3.8)]. Relative to its low-Bmagnitude (∼ 10 pJ/K), C exhibits up to ∼ 105-fold enhancement nearν = 1. As shown in Fig. (3.8), the θ = 0 data are qualitatively similarfor both investigated samples. A key point in the structure of Fig. (3.8)is that C exhibits broad maxima at ν ≈ 0.8 and ν ≈ 1.2, and decreasesrapidly for ν & 4/3 and ν . 2/3. The fact that C maxima appear nearly atthe same ν in both samples is a prima facie evidence of two-dimensionalityplaying an important role for the anomalous C behavior. Figure (3.8) alsocaptures the evolution of C vs B with temperature in samples M242 andM280. In contrast to the low-B data where C decreases with decreasing T ,in the vicinity of ν = 1, C increases with decreasing T . For this particularrange of temperatures (T & 60 mK), a noteworthy feature of heat capacitydata is that C(T ) ∝ T−2 at maxima [see Sec. (3.4)].

The results presented so far suggest that the presence of the ”giant”heat capacity may well be a symptom of anomalous behavior of the 2DESsnear ν = 1. To verify this conjecture we repeated the experiments in tiltedmagnetic fields. Figure (3.9) displays the evolution of C vs B⊥ with the tiltangle in samples M242 and M280. For each sample, heat capacity maximaare seen at the same B⊥, making the 2DESs a necessity in explaining theanomalous C behavior.

3.3.3 Discussion

Both the very large magnitude of C and the T−2 dependence point to thenuclear Schottky effect which results from the entropy reduction of the NSSwith decreasing T when the thermal energy kBT is much larger than thenuclear-spin energy level spacing ∆n (see Appendix A). The observation ofthe Schottky effect requires good coupling of the NSS to the lattice in orderto reach thermal equilibrium in the time scale of the experiment. Ordinarily,this coupling is provided by electron spin-flip excitations and further relax-ation to the lattice. Consequently, while the effect is commonly observed inmetals, it usually remains undetected in high purity materials with low free-carrier density [22]. At first sight, because of their high purity and the lowfree-carrier density in the GaAs QWs, our multiple-QW heterostructuresshould not be good candidates for the observation of a nuclear Schottkyeffect. While this is supported by the low-B data [Fig. (3.5)] where only the


Figure 3.9: C vs B⊥ at indicated tilt angles. (Top) Sample M242 andT = 100 mK. (Bottom) Sample M280 and T = 60 mK. The dashed curvesrepresent the calculated nuclear heat capacity of GaAs QWs (see text) forθ = 30 (M242) and θ = 46 (M280).


lattice and 2D electron layers contribute to the measured C, surprisingly aSchottky behavior is observed near ν = 1. Why? This thesis proposes thefollowing answer: the contribution of GaAs QWs’ nuclei to the measuredC near ν = 1 is attributed to a Skyrmion-induced, strong coupling of theNSS to the lattice. First and foremost, the key role of Skyrmions for theobservation of the Schottky behavior is supported by the absence of thenuclear spin contribution to our measured C for ν & 1.3 and ν . 0.7. Atthese filling factors Skyrmions are no longer present in the ground state ofthe 2DES. Secondly, the angular dependence of heat capacity clearly showsthat the heat capacity anomaly relates to ν (rather than total B) and thus,reflects intrinsic properties of the 2DES.

We now turn to a more quantitative interpretation of the data. Thedashed curves in Fig. (3.9) clearly indicate that the calculated CQW

n (seeAppendix A) is semiquantitatively consistent with both the size and theoverall ∼ B2 dependence of the experimental data. A useful, dimensionlessparameter in describing heat capacity data is the ratio ξ of the measured Cto CQW

n . This ratio provides a measure of the QWs’ nuclei which stronglycouple to the lattice, and hence signals the presence of low-energy spinexcitations in the 2DES associated to Skyrmions. This manoeuvre allowsus to jettison C vs B graphs [Fig. (3.9)(bottom)], and to replace them by ξvs ν plots [Fig. (3.10)]. Presented this way, the θ = 0 heat capacity dataare highly symmetric around ν = 1. We see in Fig. (3.10) that ξ showsmaxima of the order of unity at ν ≈ 0.8 and ν ≈ 1.2. While the decrease inξ very near ν = 1 is attributed to the decreasing density of Skyrmions [14],its decrease very far from ν = 1, i.e., ν & 1.3 and ν . 0.7, is related tothe 2DES approaching fillings where the Skyrmions are no longer relevant.The non-vanishing ξ at ν = 1 could arise from density inhomogeneitiesacross the measured specimens, resulting in a finite density of Skyrmions atnominal filling factor ν = 1. More precisely, we have found that up to 70%of the QWs’ nuclei may contribute to the measured C at ν = 1 in the timescale of the experiment. It should be mentioned that for sample M280, ξ atmaxima reaches values above one [Fig. (3.10)], implying that the measuredheat capacity exceeds that of the QWs’ nuclei. Besides the experimentalaccuracy (±10%) and uncertainty in well-width (±10%), the barriers’ nucleimay enhance the measured C because of the penetration of the electron wavefunction into the Al0.1Ga0.9As barriers and because of nuclear spin diffusionacross the QW-barrier interface in the time scale of the experiment.


Figure 3.10: ξ = C/CQWn vs ν at T = 60 mK and indicated tilt angles (sam-

ple M280). The ξ-envelope for θ = 0 (dotted line), is shown for comparison.

3.3.4 Disappearance of Skyrmions

We now address the important theoretical prediction that a transition fromSkyrmions to single spin-flip excitations is expected above a critical gc [95],and present heat capacity measurements performed as g was tuned by tiltingthe sample (M280) in the magnetic field [77]. These measurements revealthe absence of the nuclear spin contribution of GaAs QWs to the measuredheat capacity as g exceeds a critical value gc ≈ 0.04. This absence suggeststhe suppression of the Skyrmion-mediated coupling between the lattice andthe nuclear spins as the spin excitations of the 2DES make a transition fromSkyrmions to single spin-flips above gc. Figures (3.10) and (3.11) capturethe evolution of ξ vs ν with tilt angle at T = 60 mK. As seen in Fig. (3.10),


ξ vs ν at θ = 46, is nearly identical to the θ = 0 data 22. On the otherhand, at θ = 71, the data show a significant asymmetry with respect tothe ν = 1 position and a broadening of the ν > 1 peak. For ν < 1, ξ atν ≈ 0.9 is reduced by a factor of 2 when compared to the θ = 0 value andit vanishes for ν . 0.8. Most remarkable, however, is that the magnitudeof ξ at the ν > 1 peak is comparable to the θ = 0 data, implying a stillstrong coupling of the nuclei to the lattice. This is a particularly noteworthyobservation as it highlights that the heat capacity is a very sensitive probeof the low-energy spin excitations, and therefore Skyrmions. We recall thatthis is a regime where both calculations and transport data reveal a smallSkyrmion size (K . 3) and a very weak dependence of ∆1 on g [Fig. (2.11)].

When θ is further increased above 71 only by few degrees [Fig. (3.11)],the nuclear heat capacity decreases dramatically for all investigated ν. Forθ & 74, the nuclear heat capacity is no longer detectable up to the higheststudied tilt angle (77). To bring into focus the evolution of the couplingbetween the NSS of GaAs QWs and the lattice with θ (and g), we plot ξ atν > 1 and ν < 1 maxima vs g [Fig. (3.12)]. The coupling due to low-energyelectron spin excitations is progressively suppressed for g & 0.035 and it van-ishes in the range 0.037 . g . 0.043. We believe that this behavior providesevidence for the transition from Skyrmions to single spin-flip excitations atgc ≈ 0.04 in our sample. This transition is quite abrupt: the disappearanceof the nuclear spin contribution of GaAs QWs to the measured heat capacityoccurs in a narrow g-range. The measured gc is smaller than the theoreticalgc = 0.054 calculated for the Skyrmion to single spin-flip transition for anideal 2DES [23]. We recount that several factors, however, are expected toreduce gc for a real 2DES, including the finite thickness of the electron layer,LL mixing, and disorder. Indeed, calculations by Cooper [see Secs. (1.6.1)and (2.2.3)] reveal that taking into account the finite z-extent of the 2DESalone leads to gc = 0.047, closer to our experimental value. The inclusion ofthe LL mixing will further push down gc to ≈ 0.042 [76]. After consideringthe role of disorder, we may conclude that the agreement between theoryand experiment is remarkable. It is worth emphasizing that calculationsare usually performed at ν = 1 while the heat capacity data of Figs. (3.10)and (3.11) provide values for gc in the full ν-range (0.8 . ν . 1.2) whereSkyrmions – anti-Skyrmions are expected in the ground state of the 2DES.In particular, we observe that gc depends on ν; it increases from gc ≈ 0.037at ν = 1.2 to gc ≈ 0.043 at ν = 0.9. The different gc observed at ν < 1 and

22We note that in sample M242, ξ vs ν at θ = 53 (the highest investigated tilt angle)is nearly identical to the θ = 0 data.


Figure 3.11: ξ = C/CQWn vs ν at T = 60 mK for 71 . θ . 74 (sample

M280). The dashed lines are guides to the eye.

ν > 1 could be brought into connection with the absence of the particle-holesymmetry around ν = 1, favored by the tilted magnetic fields. Finally, wealso recall the possibility of a LL crossing at high tilt angles in our sam-ple. This LL crossing might be responsible for the asymmetry of the heatcapacity data with respect to the ν = 1 position at θ ≈ 72.

3.3.5 Ramifications

We report here an anomalous and unexpected behavior for the measuredheat capacity at intermediate tilt angles (50 . θ . 66) which shows ev-idence for the complex dependence of Skyrmions on θ and g. As depictedin Fig. (3.13), we observe a reduction of ξ in a narrow ν-range comparedto the θ = 0 data. The vertical arrows in Fig. (3.13) point to the total


magnetic field at which ξ is most significantly reduced. The anomaly movesto higher ν with small increases in θ (from ν = 0.81 at θ = 50 to ν ≈ 1 atθ = 66), and shows an increasing intensity when compared to the θ = 0

data-envelope, e.g., at θ = 50, ξ at ν = 0.81 is reduced by a factor of ≈ 3,while at θ = 66, ξ at ν ≈ 1 is essentially zero. The observed reductionof ξ shows that low-energy electron spin excitations are strongly affectedor even suppressed, which would signal the disappearance of Skyrmions forlimited ν- or B-ranges dependent on tilt angle. It is worth noting that theheat capacity anomaly occurs for |ν − 1| . 0.2 and 0.02 . g . 0.03. This isprecisely the g-ν domain where, according to theory [Fig. (2.9)], the 2DESattains different subtle phases near ν = 1, some of which do not supportSkyrmions. Even though the exact behavior of the anomaly might be spe-cific to our heterostructure, it reveals the subtle influence of θ on the spinexcitations of the 2DES near ν = 1 whose description will require furthertheoretical and experimental work.

Figure 3.12: ξ = C/CQWn at ν > 1 (•) and ν < 1 () maxima is plotted vs

g = |g∗|µBB/[e2/(εlB)] at θ = 0, θ = 46, and θ & 71 (sample M280).


Figure 3.13: ξ = C/CQWn vs ν at T = 60 mK and indicated tilt angles (sam-

ple M280). The ξ-envelope for θ = 0 (dotted line), is shown for comparison.


We now complete the discussion of the heat capacity results obtained inthe presence of an in-plane magnetic field component by comparing themwith tilted magnetic field magneto-transport data. Recently, Kronmulleret al. [61] discovered that magnetotransport near ν = 2/3 is influenced bythe nuclear polarization of nuclei inside the GaAs QWs through the Fermicontact hyperfine interaction [1]. The question arises if the nuclear spins arealso involved in the physics of the ν = 4/3 fractional QHE state, which isthe electronic conjugate of the ν = 2/3 fractional QHE state. Examiningthe heat capacity data of Figs. (3.10), (3.11), and (3.13) we observe abroadening of the ν > 1 peak, starting with θ ≈ 61. Figure (3.11) showsthe reduction of the ν > 1 peak for θ & 72. Therefore, these data seemto support the idea that a nuclear heat capacity is observed near ν = 4/3for a finite range of tilt angles. Why? A possible escape from this enigmais provided by the tilted magnetic field magneto-transport data, exhibitinga re-entrant behavior of the ν = 4/3 fractional QHE state. The evolutionof ∆4/3 with B is summarized in Fig. (3.14)(a), whereas the evolution ofξ with B near ν = 4/3 is presented in Fig. (3.14)(b). Our tilted-magneticfield heat capacity data near ν = 4/3 uncover a rapid nuclear spin-latticerelaxation in the B-range where the ν = 4/3 fractional QHE state is absent.This behavior signals the presence of nearly gapless spin fluctuations in the2DES when the ν = 4/3 fractional QHE state undergoes a ↑↓→l spin phasetransition (i.e., from a spin-unpolarized ground state to a partially polarizedground state). These nearly gapless spin fluctuations permit then the strongcoupling of the 2DES to the GaAs QWs’ nuclei.

Under the given context of tilted magnetic field magneto-transport andheat capacity experiments near ν = 4/3, we take the opportunity to notethat Kronmuller et al. [61] observed that sweeping B upwards and down-wards produces a rapidly observable hysteresis in the longitudinal resistancenear ν = 2/3. While it is now widely accepted that the hysteresis in themagnetoresistance near fractional QHE states is provoked by transitions be-tween different spin configurations of the 2DES, several concerns (such asthe role of the nuclear spins) are not clearly established [20, 33]. For ex-ample, the nuclear spins may be responsible for the slow relaxation of themagnetoresistance in the vicinity of various fractional filling factors [33].Further research is necessary before we can decide that the onset of a van-ishing excitation gap at ν = 4/3 in our samples suffices for the observation ofboth nuclear heat capacity and hysteretic behavior. Searching for T - and θ-dependent hysteretic behavior near ν = 4/3 in samples M242 and M280 mayshed light on the participation of the nuclear spins in spin phase transitionsof unpolarized fractional QHE states.


Figure 3.14: (a) ∆4/3 vs B (sample M280). Panel (b) displays the B-dependence of ξ = C/CQW

n near ν = 4/3 at T = 60 mK. The grey regionsindicate the B-range where the ν = 4/3 fractional QHE is absent. Thecorresponding tilt angles are given on the top axis.

3.4. SKYRMION LATTICES 129

3.4 Skyrmion lattices

3.4.1 Heat capacity measurements at very low temperatures

At very low temperatures (T . 60 mK) and in a narrow ν-range (ν ≈ 0.8and ν ≈ 1.2), C is very large and the observed external time constant τext

reaches extremely large values which exceed 104 s. Therefore, in the timescale of our experiments, the sample is in the quasi-adiabatic regime 23.

Figure 3.15: C vs T at B = 7 T (ν = 0.81) is shown in the main figure ina log-log plot (sample M242). The dashed line shows the T−2 dependenceexpected for the Schottky model, and Tc is marked by the vertical arrows(see text). The inset shows a linear plot of C vs T at B = 6.7 T (ν = 0.85).

23All data discussed in this section were taken on sample M242.


Figure (3.15) shows the T -dependence of C at θ = 0 for ν = 0.81 andν = 0.85. We observe that the C ∝ T−2 behavior is followed only downto ≈ 60 mK. For T . 60 mK, C increases faster with decreasing T , andexhibits a remarkably sharp peak at Tc before decreasing at very low T .Figure (3.16) displays the T -dependence of C at θ = 30 for two fillingfactors in the neighborhood of ν = 0.8. Our tilted magnetic field dataindicate that the peak in C vs T could be associated to the 2DES, as it isobserved in the same range of filling factors (i.e., around ν = 0.8 and 1.2).

Figure 3.16: C vs T at θ = 30 and indicated values of ν (sample M242).The T−2 dependence expected for the Schottky effect is shown as a dashedline, and Tc is marked by the vertical arrows. The ν-dependence of Tc atθ = 0 () and 30 (•) is shown in the inset. The lines are guides to the eye.


A prominent feature of data taken at θ = 30 is that Tc sensitively dependson ν. The inset to Fig. (3.16) shows that the peak temperature Tc is max-imum at ν ≈ 0.8 and ν ≈ 1.2 and decreases as ν → 1. The sharp peak inC vs T is not observed in sample M280, which might signal that it occursat lower temperatures in this lower density heterostructure; the SchottkyT -dependence [C(T ) ∝ T−2] is followed down to T ≈ 30 mK. We note thatfor sample M280 we found a marked non-Schottky behavior of the heat ca-pacity at very low temperature at θ = 71 and ν = 1.22 [Fig. (3.17)], whichis reminiscent of the sharp peak in C vs T observed in sample M242.

Figure 3.17: C vs T at θ = 0 (ν = 0.88, filled circles) and θ = 71 (ν = 1.22,open circles) is shown in a log-log plot (sample M280). The dashed line showsthe T−2 dependence expected for the Schottky model.


The deviation of C from the T−2 dependence at T ∆n/kB and the shapeof the observed peak are clearly not consistent with the Schottky modelwhich predicts a smooth maximum in C at T ∼ ∆n/(2kB) [∼ 2 mK in ourcase, solid curve in Fig. (3.18)]. Instead, the shape and sharpness of thispeak are suggestive of a phase transition in the 2DES. It is tempting to as-sociate the sharp peak in C vs T observed at low T with the crystallizationof Skyrmions. The fact that our observed Tc decreases as ν → 1 is consis-tent with the decreasing Skyrmion density which should reduce the meltingtemperature of the Skyrmion solid. However, the details of the Skyrmionliquid-to-solid transition are largely unknown. While it is expected that aSkyrmion liquid-to-solid transition would affect ∆n only very weakly, howit would affect the nuclear spin dynamics is unclear.

The question arises what is the physical mechanism that affects the NSSand gives rise to the anomalous C-peak. The substantial enhancement of Cat low T could be interpreted as an indication that either (1) more nucleicouple to the lattice, or (2) the entropy of the coupled NSS decreases fasterwith decreasing T than what is expected from the Schottky model. Picture(1) relies on a modified coupling of the NSS to the lattice manifesting as anenhanced nuclear spin diffusion so that a larger number of nuclei contributeto the heat capacity near Tc. Picture (2), on the other hand, relies ona Skyrmion-induced nuclear spin polarization which reduces the entropy ofthe NSS. This is reminiscent of the dynamic nuclear polarization of the NSS,for example when nuclear spins interact with spin-polarized paramagneticimpurities [1]. While in the Skyrmion-liquid state there is no preferentialorientation of the electron spins, the transition to a pinned Skyrmion solidcould possibly induce a local preferential orientation of the electron spinswhich, in turn, would polarize the NSS and thereby reduce the entropy.

We have performed quasi-adiabatic thermal experiments [10] revealingthat the mechanism responsible for the peak in C vs T is a dramatic en-hancement of nuclear spin diffusion across the QW-barrier interface. Whileonly the nuclear heat capacity of the QWs’ atoms is observed away from Tc,an increasing number of nuclei in the barriers contribute to the measuredheat capacity when T → Tc.

3.4.2 The variable nuclear thermal coupling model

While at high temperatures (T & 60 mK) the measured value and T -dependence of C are consistent with the calculated Schottky nuclear heatcapacity of Ga and As atoms in the QWs [solid curve in Fig. (3.18)], at lowerT , C exceeds the calculated value by a factor of up to ∼ 10 at Tc. The peak


value of C appears consistent with the Schottky nuclear heat capacity of theheterostructure if the nuclei of the barriers are also included [dotted curvein Fig. (3.18)]. This observation suggests that the peak in C vs T might

Figure 3.18: Measured C vs T at B = 8.5 T and θ = 30 (sample M242).The curves represent the calculated Schottky nuclear heat capacity of theone hundred GaAs QWs (full curve) and the 100-period GaAs/Al0.3Ga0.7Asheterostructure (dotted curve).

come from the contribution of barriers’ nuclear spins to the measured C.Since the nuclear spin-lattice relaxation rate is extremely weak in the bar-riers at such low T , such an interpretation necessarily implies a variablethermal coupling between the QWs’ nuclei and the barriers’ nuclei. Theabove observations can be cast into a more concrete model as schematicallyshown in the inset to Fig. (3.19)(a). Here KQW

n and KBn are variable thermal


conductances which depend on both ν and T . They are normally negligiblysmall so that one measures only the heat capacity of the lattice and the2DESs 24. Near ν = 1, however, KQW

n becomes large and the QWs’ nucleidominate the measured C, while near ν = 1 and T = Tc both KQW

n and KBn

become large so that C of the ”entire” heterostructure is measured.

3.4.3 Results of quasi-adiabatic thermal experiments

We have critically tested the above conjectured model by performing quasi-adiabatic thermal experiments which utilize another noteworthy feature ofthe heat capacity data; as seen in Fig. (3.16) inset, Tc sensitively dependson ν. In the framework of our model, this strong dependence allow us totune KB

n by varying B in order to cross the region near Tc where KBn is

maximum. If the barrier nuclei are not in thermal equilibrium with therest of the sample before the crossing, then the lattice T should be greatlyaffected as Tc is approached and the barrier nuclei thermally couple to theQWs’ nuclei and the lattice. This is the key to understanding our results.

In these experiments, we first fix B at 8.5 T (ν = 0.77) where the peak inC occurs at Tc = 42 mK. Starting from T ∼ 60 mK, we cool down the coldfinger to base temperature (∼ 10 mK) and the lattice slowly cools downto T ∼ 32 mK by waiting for about 50 hours. At this temperature, themeasured heat capacity is much smaller than C observed at Tc, implyingthat the coupling to the barriers (KB

n ) is quite small. Then, in order tofurther improve the adiabatic conditions, the weak heat flow that resultsfrom the T -difference between the lattice and the cold finger is reduced byincreasing the cold finger temperature to ≈ 32 mK. Under these conditions,we observe that the measured T is stable within . ±0.2 mK over periodslonger than τext, meaning that the sample is in the quasi-adiabatic regime.

In a first experiment, whose results are shown in Fig. (3.19)(a) by opencircles, we started from such a quasi-adiabatic condition and swept B from8.5 to 7.7 T, at a rate of 0.01 T/min. We observe that T rises from 31.7 mKto 36.2 mK. B was then swept back to 8.5 T at the same rate; this ledto a rise of T to 39.5 mK. While the increase in T with increasing B maybe explained by the adiabatic magnetization of the nuclei coupled to thelattice, according to

Tn ∝ B, (3.11)

the T rise observed when B is lowered from 8.5 to 7.7 T is puzzling.

24The sharpness of the peak in C vs T implies that KBn → 0 fast as T deviates from Tc.


Figure 3.19: (a) T vs B during the first () and the second (•) quasi-adiabatic experiments. The full and the dotted lines are guides to the eye;the arrows show the direction of evolution. The dashed curve shows theB-dependence of Tc. (Inset) Schematic description of the coupling betweendifferent parts of the sample. (b) T and B vs time in the second experiment.B-sweeps (gray areas) separate hold times at the end of which T is recorded(1 to 7) and summarized in (a) together with the final T at B = 8.5 T (8).


In order to better understand the results of the above experiment, we per-formed a similar experiment, starting from T = 32.6 mK, but here we sweptB in steps of 0.2 T from 8.5 to 7.3 T, with a hold time of 3 hours betweensteps. The evolution of T during this second experiment is presented inFig. (3.19)(b) and summarized in Fig. (3.19)(a) by close circles. We notethat in each step, except for steps 2 → 3 and 3 → 4, the lattice T risesduring the B-sweeps and then decays to approximately the same value asthe one at the end of the previous step. These T rises are due to eddyingcurrents heating of the 2DES and the lattice during the B-sweeps. The heatis evidently slowly absorbed by the nuclei during and following the sweep viainternal relaxation. Because of the very large heat capacity of the nuclearspins, this heating (due to eddy currents) does not lead to an appreciableincrease in the lattice T at the end of the step. The key feature of thisexperiment, however, is that during the 2 → 3 and 3 → 4 steps, the latticeT increases significantly while the sample is in quasi-adiabatic conditions.Note in particular that, during the 2 → 3 step, the lattice T rises even afterthe B-sweep is completed. Indeed, it appears that near Tc the lattice isheated internally.

3.4.4 The crystal of Skyrmions and the nuclear spin diffusion

We now detail the predictions of the model for these particular experiments.We expect that during the initial cool-down at B = 8.5 T, when T decreasesbelow Tc = 42 mK, Tn in the barriers (TB

n ) remains close to 42 mK and Tn

of the QWs (TQWn ) decreases down to ≈ 32 mK due to strong coupling

to the lattice. Next, decreasing B below 8.5 T (steps 1 → 7) in adiabaticconditions has two distinct consequences: (1) TB

n and TQWn are reduced

due to adiabatic demagnetization of the nuclei [Eq. (3.11)], and (2) thecoupling between the QWs and the barriers will increase dramatically whenthe dashed (Tc vs B) curve in Fig. (3.19)(a) is crossed. While the former isessentially monotonic inB, the latter is not and implies that, near the dashedcurve in Fig. (3.19)(a), Tn should equalize over the entire GaAs/Al0.3Ga0.7Asheterostructure. Since the barriers’ nuclei were initially warmer than theQWs’ nuclei, we expect a rise in lattice T near this crossing, as observedexperimentally. Finally, we attribute the T rise during the final (7 → 8)step to the adiabatic magnetization of the nuclei [Eq. (3.11)] coupled to thelattice.

Beyond the good qualitative description of the experiments in Fig. (3.19),the above model appears to provide a reasonable quantitative account of thedata also. Nuclear demagnetization [Eq. (3.11)] reduces TB

n from ≈ 42 mK


to ≈ 36 mK when B is lowered from 8.5 to 7.3 T. Since the barriers areabout 10 times thicker than the QWs, the rise in TQW

n from steps 1 → 7 inFig. (3.19) reduces TB

n only weakly (≈ 0.5 mK). Therefore, TBn and TQW

n

should merge towards a final value close to ≈ 36 mK. This prediction is ingood agreement with experiments that give T ≈ 37 mK at B = 7.3 T 25. Thefinal increase of T from ≈ 37 mK to ≈ 43 mK when B increases from 7.3 to8.5 T is accounted for by the adiabatic magnetization of the heterostructure’snuclei [Eq. (3.11)]. Lastly, the fact that C(Tc) reaches the value expectedfor the entire heterostructure is consistent with our model.

Our data also shed light on the thermal conductances KQWn and KB

n asthey are related to the internal time constant (τint) observed in our experi-ments. Figure (3.20) presents a typical heat pulse/relaxation trace obtainedduring heat capacity experiments near Tc. In the T -range (T . 1.2 Tc) whereC exhibits a maximum, we measured C using the quasi-adiabatic technique,ensuring that the measured C is not affected by possible changes in theinternal time constant of the system. We recall that C = ∆Q/∆T where∆T is the temperature increase resulting from the applied heat ∆Q = P∆t,after internal relaxation is completed. The relaxation follows an exponentialdecay characterized by τint. The inset to Fig. (3.20) shows that τint exhibitstwo different, but remarkably constant, values above and below 30 mK:∼ 1600 s and ∼ 900 s, respectively. We note that this crossover temperaturecorresponds to the T below which C is reduced back to a value close to thenuclear heat capacity of the 100-QWs [full curve in Fig. (3.18)].

In order to explain these observations, we consider three possible mech-anisms responsible for the observed τint. Heat diffusion in the nuclear spinsystem of the barriers is governed by the one-dimensional diffusion equation

∂TBn

∂t= Dn

∂2TBn

∂z2 , (3.12)

where Dn ∼ 10−17m2/s is the nuclear-spin diffusion constant in GaAs 26

and the z-axis is along the growth direction. Therefore, it gives an internaltime constant

τd ∼r2

Dn, (3.13)

where r is the distance over which diffusion takes place 27. Since each barrier25Complementary experiments allow us to estimate the T increase expected from the

Eddy-current heating during the B-sweeps. Our estimate is ≈ 1 mK for the total T riseduring steps 1→7 or 7→8.

26In our estimate here, we use Dn for GaAs since Dn for Al0.3Ga0.7As is not known.27Note that the actual boundary conditions used to solve Eq. (3.12) may add a prefactor


is surrounded by two QWs, we take r = 925A (half the barrier thickness)and Eq. (3.13) gives τd ∼ 800 s. The two other mechanisms involved in theheat transfer from the lattice to the barrier nuclei, i.e., nuclear spin-latticerelaxation in the QWs and diffusion across the QW-barrier interfaces, resultin two additional time constants: T1 and τi, respectively. When C ≈ CQW

n

(T . 30 mK), the diffusion across the QW-barrier interface is negligible andτint should be determined by T1 only. On the other hand, when C exceedsCQW

n (30 . T . 50 mK) at least a fraction of the barriers’ nuclei couple tothe QWs’ nuclei. This implies that τd and τi should increase τint.

Figure 3.20: T vs time during a heat capacity experiment in the quasi-adiabatic regime at B = 8.5 T and θ = 30 (sample M242). The heat pulseis followed by a relaxation characterized by an exponential decay (full curve)giving τint ≈ 1500 s. The inset shows τint vs T at the same B and θ.

This is consistent with the rise in τint observed above 30 mK [Fig. (3.20)

to the right term of Eq. (3.13) which gives only a rough estimation of τd.


inset]. Moreover, we note that the increase of τint above 30 mK (∼ 700 s)is comparable to the estimated τd ∼ 800 s, implying that τi is rather smallonce the barrier nuclei do couple to those in the QWs 28.

We now remark on the physical origin of the peak in C vs T . Opticallypumped nuclear magnetic resonance (OPNMR) experiments [7, 103] showthat nuclear spin diffusion from the QWs into the barriers is extremely weak,except when optical pumping broadens the NMR (Knight shift) peak of theQWs which then overlaps with the NMR peak of the barriers. The spectraloverlap allows spin diffusion which is driven by nuclear magnetic dipole-dipole coupling. Therefore, the enhancement of the nuclear spin diffusionacross the QW-barrier interface near Tc could originate from a broadeningof the QWs’ NMR line. In the Skyrmion-liquid phase, ”motional averaging”of the local spin polarization of the 2DES produces a single Knight shiftpeak (see for details Chapter 4 and Appendix B). On the other hand, theabsence of motional averaging in the Skyrmion-solid phase should induceboth positive and negative Knight shifts depending on the local spin polar-ization of the 2DES. Since above and below the peak in C vs T the diffusionacross the QW-barrier interface is very weak, this implies that in both theliquid and the solid Skyrmion phases, the Knight shift peak(s) do not havesignificant overlap with the NMR peak of the barriers.

One possible explanation for the C vs T peak is that the critical slowingdown of the spin fluctuations in the 2DES, associated with the Skyrmionliquid-to-solid phase transition, could induce a broadening of the QWs’ NMRpeak 29. This broadening would induce spectral overlap of the QWs’ andbarriers’ NMR peaks and hence allow spin-diffusion across the QW-barrierinterface only near Tc.

The fact that τint is constant in the T -range where C > CQWn near Tc

implies that the strength of the coupling to the barriers remains constantwhile C varies strongly. Therefore, we deduce that the smaller value of Con the sides of the peak in C vs T comes from a reduced fraction of thesample in which the barriers’ nuclei couple to the QWs’ nuclei. The bestexplanation for this is that in a magnetically ordered Skyrmion state, thereare regions around Skyrmions where the electronic magnetization is perpen-dicular to the external magnetic field and the Knight shift vanishes. Thenuclear spins outside the QWs can come into equilibrium with the nuclei in

28It is worth remarking that the parts of the sample for which τd τext do not con-tribute to the measured C in the T and B range we are discussing. Since τd & 105 s forr > 10−6 m [Eq. (3.13)], the GaAs substrate contributes negligibly to C.

29The critical slowing down causes the electron motion time scale to pass through theNMR time scale.


these regions. The size of these regions is expected to peak at the Skyrmionsolid-to-liquid phase transition due to the critical slowing down of spin fluc-tuations. Hence, specific heat measurements suddenly see the nuclei outsidethe QWs when a Skyrmion lattice forms. The complexity of spin textures inthe 2DES, together with the fact that the phase transition is not observeddirectly but through its effect on the nuclear spin-lattice relaxation, does notallow a simple analysis of the data in the framework of the existing theoriesthat describe the melting and magnetic disordering of the Skyrmion lattice.It is worth emphasizing that values of the Skyrmions’ moment of inertiaand stiffness to relative rotations to neighboring Skyrmions for the multi-Skyrmion system extracted from calculations [24] suggest that the system isclose to criticality. The constant thermal relaxation rate for T . 50 mK isin accord with the predicted T dependence of T−1

1 in the quantum criticalregime [44]. To sum up, our results provide a semiquantitative phenomeno-logical description of the peak in C vs T that may originate from a Skyrmionsolid-to-liquid phase transition.

3.5 The nuclear spin-lattice relaxation rate

We now briefly discuss the T−11 results of our calorimetric experiments 30.

Figure (3.21)(a) summarizes the T - and ν-dependence of T−11 in sample

M242, measured at θ = 30. Interestingly, T−11 decreases when T is lowered

and, at a given T , a common T−11 is observed for 0.7 . ν . 1.3. That is, the

nuclear-spin lattice relaxation is as rapid at ν = 1 as on the either side ofν = 1 down to very low temperatures, consistent with the observation of alarge nuclear spin heat capacity at ν = 1 (0.3 . ξ . 0.7). In Fig. (3.21)(b) weshow Arrhenius plots of T−1

1 for various filling factors, revealing an activatedT -dependence.

The situation is rather subtle and a concise presentation of OPNMRT−1

1 results is appropriate at this juncture (see for details Chapter 4 andAppendix B). OPNMR experiments at ν . 1 and ν & 1 show no T -dependence for T−1

1 in the temperature range 1.5 K . T . 4.2 K. Inthis T -range, the nuclear-spin lattice relaxation is rapid on either side ofν = 1 (T−1

1 ≈ 0.05 s−1). On the other hand, at ν = 1, OPNMR measure-ments found a slow nuclear-spin lattice relaxation (T−1

1 < 0.01 s−1), whichstrongly depends on the temperature [Table (B.1)]. This peculiar behaviorof T−1

1 (T, ν) has been attributed to Skyrmion induced low-lying spin-flip

30An in-depth analysis of T−11 calorimetric data awaits completion at the time of the

writing.

3.5. THE NUCLEAR SPIN-LATTICE RELAXATION RATE 141

excitations in the 2DES [7, 103]. Clearly, this assignment is fully consistentwith existing calculations [24].

Our low-T T−11 calorimetric data open new vistas on the many-Skyrmion

ground state at T → 0. One of the central queries here is to understand thestriking ν-dependence of T−1

1 near ν = 1, i.e., the conspicuous absence ofthe slowing down of the nuclear-spin lattice relaxation at ν = 1. We believethat, due to a finite density of Skyrmions at nominal filling factor ν = 1,T−1

1 describes the average response of different spin-textured domains (withthe shortest time scales for nuclear spin-lattice relaxation) in the experi-mental time window. It is simple to imagine that, under the exceptionalcircumstance of Skyrmions present in the electronic ground state at ν = 1and of an inhomogeneous 31 2DES, T−1

1 at nominal filling factor ν = 1 couldbe constructed from averaging over the T−1

1 contributions at ν = 1 ± ε.Recounting that T−1

1 varies linearly with |1 − ν| [24], it is easy to see thatthe ν-averaged T−1

1 will have essentially no dependence on the filling factoraround ν = 1 and a fast nuclear spin-lattice relaxation is expected even forT → 0.

We now attempt to interpret the T -dependence of T−11 [Fig. (3.21)] 32.

At low-T (T . 60 mK), the experimentally determined T−11 never becomes

smaller than ≈ 1 × 10−3 s−1 for all investigated filling factors. This sat-uration value is close to the T−1

1 observed near the integer filling factors,in electron spin resonance experiments [28]. At higher temperatures, forfilling factors where Skyrmions are the expected ground state of the 2DES,T−1

1 shows an activated T -dependence with a very small activation energy≈ 0.3 K, consistent with a phonon-assisted mechanism through very lowenergy spin-flip excitations in the 2DES [57, 105]. Our speculative inter-pretation for this behavior is the following. The fact that Skyrmions mightbe in a magnetically ordered liquid state at low temperatures [24], togetherwith disorder pinning, introduces a gap (∆RR) in the energy spectrum ofthe 2DES. The essential physics will still be that the spin fluctuations havestrong spectral density at frequencies far below the Zeeman gap [39]. Thepresence of ∆RR, in turn, induces an activated T -dependence for T−1

1 whenT . ∆RR [28, 105]. At high-T (T & 0.3 K), the experimentally determinedT−1

1 saturates at ≈ 0.1 s−1. We shall see in the next chapter that T−11 ’s

determined near ν = 1 in calorimetric experiments for T & 0.3 K are

31The inhomogeneity could simply be due to a non-uniform Skyrmion density.32We neglect here the step in the thermal relaxation rate at T = 30 mK [inset to

Fig. (3.20)].


Figure 3.21: (a) T−11 vs B (bottom axis) and ν (top axis) at θ = 30

and indicated temperatures (sample M242). (b) Arrhenius plots of T−11 at

ν = 1.1 (•) and ν = 0.77 (), revealing an activation energy ∆RR = 0.32 K.

3.6. CONCLUSION 143

approximately equal to T−11 ’s observed in our standard NMR studies at

T = 1.5 K.

3.6 Conclusion

In this chapter we described heat capacity measurements down to very lowtemperature (T & 20 mK) and Landau level filling factor (ν & 0.5). As afunction of increasing magnetic field, B, in addition to oscillations associatedwith the 2DESs’ oscillating density of states at the Fermi level, we observea dramatic increase of the low-T heat capacity in the range 0.5 . ν . 1.5.For ν ≈ 0.8 and ν ≈ 1.2, C exhibits a striking T -dependence, including aremarkably sharp peak at very low temperature (Tc) suggestive of a phasetransition in the 2DES. We interpret these unexpected observations in termsof the Schottky model for the nuclear-spin heat capacity of Ga and Asatoms which couple to the lattice via the 2DESs’ low-energy spin-textureexcitations (Skyrmions).

Heat capacity measurements on sample M280 uncover the disappearanceof the nuclear spin contribution to the heat capacity as the ratio g betweenthe Zeeman and Coulomb energies exceeds a critical value gc ≈ 0.04. Thisdisappearance suggests the vanishing of the Skyrmion-mediated coupling be-tween the lattice and the nuclear spins as the spin excitations of the 2DESmake a transition from Skyrmions to single spin-flips above gc. This gc issomewhat smaller than gc = 0.054 expected from theoretical calculations foran ideal 2DES. The discrepancy likely comes from corrections to gc becauseof finite layer-thickness, Landau level mixing, and disorder in a real sam-ple. Furthermore, our tilted magnetic field heat capacity experiments revealthe subtle and critical influence of the in-plane magnetic field componenton the ground and excited states of 2DESs near ν = 1 in GaAs/AlGaAsheterostructures.

We discussed the physical origin of the sharp peak in C vs T in termsof the possible Skyrmion solid-to-liquid phase transition. The sharp peak inC vs T near ν = 1 can not possibly be due to the tiny amount of entropychange in the Skyrmion lattice itself. Rather it is due to the nuclei in thethick barriers between the GaAs quantum wells. Quasi-adiabatic thermalmeasurements on sample M242 show that the state of the confined 2D elec-trons profoundly affects the nuclear-spin diffusion near Landau level fillingfactor ν = 1. Our experiments provide quantitative evidence that a dra-matic enhancement of nuclear-spin diffusion from the GaAs quantum wellsinto the AlGaAs barriers is indeed responsible for the sharp peak in the tem-


perature dependence of heat capacity near ν = 1. While only the nuclearheat capacity of the QWs’ atoms is observed away from Tc, an increasingnumber of nuclei in the barriers contribute to the measured heat capacitywhen T → Tc.

Chapter 4

Skyrmions Probed by NMR

Fiti treji, privegheati. Potrivnicul vostru, diavolul, umbla,racnind ca un leu, cautand pe cine sa înghita, caruia statiîmpotriva, tari în credinta, stiind ca aceleasi suferinte îndurasi fratii vostri în lume.

Sf. Apostol Petru

4.1 Overview

In recent years, the development of novel low-temperature spectroscopicmethods has made available a new set of tools for the study of spin-relatedphenomena in the quantum Hall effect (QHE) regime. A principal task ofthese spectroscopic experiments [2, 7, 62, 78] is the direct measurement of thespin polarization P of two-dimensional (2D) electrons. In nuclear magneticresonance (NMR), P is determined from the (Knight) shift of NMR lineswhich is a direct measure of the additional effective magnetic field created bythe electronic spins at the position of the nucleus. Unfortunately, the NMRsignal from a conventional single quantum well (QW), containing a two-dimensional electron system (2DES), is typically too weak to be observed.

145

146 CHAPTER 4. SKYRMIONS PROBED BY NMR

The reason for this is that the NMR signal is proportional to the product ofthe number of nuclear spins and their average polarization; the number ofnuclei located in a single QW, which interact with 2D electrons, is small andtheir nuclear polarization is tiny even at very low temperatures. To solvethis problem, the first NMR measurements of P have been performed usingan optically pumped NMR (OPNMR) technique to increase the nuclear spinpolarization prior to the NMR detection [7, 103]. We found that for multiple-QW samples standard NMR provides enough sensitivity for detection ofNMR signals at low temperatures T . 10 K and in magnetic fields B &4 T. This chapter describes standard gallium NMR measurements on sampleM242 in the QHE regime 1 and focuses on findings around Landau levelfilling factor ν = 1.

As described in the preceding chapters, the key to understanding most ofthe phenomenology of the 2DES near ν = 1 is tied to viewing its propertiesdictated by the presence of Skyrmions - charged objects with an underly-ing spin texture encompassing many reversed spins. What are the issuespertaining to Skyrmions that NMR measurements endeavor to elucidate?

The most fervently pursued goal is the 2D electron spin polarization peakat ν = 1, in P(T, ν) experiments [7]. It is generally accepted that the 2Delectron spin polarization peak at ν = 1 unequivocally indicates the presenceof Skyrmions in the electronic ground state. In principle, measurements ofP(T, ν) could easily determine the dependence of the size of Skyrmions onZeeman energy [Sec. (2.3.1)]. According to theory [95], the number of re-versed spins within a Skyrmion varies with the ratio g = EZ/EC betweenthe Zeeman (EZ = |g∗|µBB) and the Coulomb energy [EC = e2/(εlB)].Above a critical g value (gc) it is expected that Skyrmion-like excitationsevolve into single spin-flips [95]. Here |g∗| = 0.44 is the g-factor of elec-trons in bulk GaAs, µB is the Bohr magneton, ε ≈ 13ε0, lB =

√~/(eB⊥)

is the magnetic length, and B⊥ is the perpendicular component of the ap-plied magnetic field. A very interesting aspect of heat capacity experimentsdescribed in the precedent chapter is the disappearance of the nuclear spincontribution to the measured heat capacity of sample M280 at gc ≈ 0.04,which is interpreted as evidence for a transition from Skyrmions to singlespin-flip excitations. At the risk of repetition, we recall that this gc is some-what smaller than gc = 0.054 expected from theoretical calculations foran ideally thin, disorder-free 2DES with no Landau level mixing [23, 95].Quite puzzling, optical experiments involving nuclear spins [63] revealed asignificantly smaller value for gc, close to ≈ 0.01.

1We observed a similar behavior in sample M280. These data are not included here.

4.1. OVERVIEW 147

One of the objectives of NMR experiments described below is to clarify theproblem of the range of g over which Skyrmions are the stable excitationsof the ν = 1 QHE ground state. To this end, we studied the evolution of Pnear ν = 1 in tilted magnetic fields up to 17 T, corresponding to an increaseof EZ and g by a factor of 3. The signature of Skyrmions, i.e., the 2Delectron spin polarization peak at ν = 1, is observed for g . 0.022, whereasfor g & 0.037 this feature disappears.

Well-established NMR probes of the electronic properties of bulk ma-terials, such as the nuclear spin-lattice relaxation rate T−1

1 , may also pro-vide crucial information about Skyrmions 2. For example, a rapid nuclearspin-lattice relaxation near ν = 1 would reflect the presence of low-energyelectronic spin excitations associated to Skyrmion dynamics. We have usedthe standard NMR technique to measure the low-T 71Ga T−1

1 around ν = 1.In doing so, besides assessing the fast nuclear-spin lattice relaxation nearν = 1, new phenomena in the 2DES were exposed, as described below.

The present NMR experiments reveal important differences around ν = 1as compared to OPNMR results [7, 103]. The latter technique observesa well developed (approaching full polarization P = 1) 2D electron spinpolarization peak at ν = 1 together with a strong suppression of T−1

1 due tothe single-particle spin (Zeeman) gap. In our NMR data, the 2D electronspin polarization peak is significantly smaller and no variation of T−1

1 aroundν = 1 is detected. The latter observation confirms the presence of rapidlyrelaxing nuclei even at ν ≈ 1 detected in heat capacity experiments onsamples from the same wafer [9]. We discuss to what extent these resultscan be explained in terms of electron density inhomogeneities across thesample.

The organization of the present chapter is as follows. Section (4.2) isdevoted to essentials of NMR. First, some general elements of NMR arepresented. Then, we delineate the basic theoretical formalism useful in un-derstanding the present experiments. Section (4.3) is devoted to the exper-imental method. First, a brief account of the experimental setup is given[Sec. (4.3.1)], followed by a discussion of how data were collected and ana-lyzed [Sec. (4.3.2)]. Section (4.4) contains our results and interpretation ofthe data. In Sec. (4.4.1) we specialize on our untilted magnetic field data.We discuss and compare these data to OPNMR results. In Sec. (4.4.2) weconsider electron spin polarization results in tilted magnetic fields approach-ing the large Zeeman energy limit. Conclusions are given in Sec. (4.5).

2Commonly, the geometrical arrangement of atoms in a crystal is termed the ”lattice”.In NMR, ”lattice” refers to all degrees of freedom of the sample, except the nuclear ones.


Figure 4.1: (Top) (a) The NMR probe tuning/matching circuit. (b) Illus-tration of the sample and the RF coil. (Bottom) Comparison of Gaussian[Eq. (4.6)] and Lorentzian [Eq. (4.5)] line shapes for ∆ω = 2.354 and ω0 = 0.

4.2. PRINCIPLES OF NMR 149

4.2 Principles of NMR

4.2.1 General aspects of NMR

NMR refers to the case of a sample containing Nn nuclei of spin I and mag-netic moment ~γnI (where γn is the nuclear gyromagnetic ratio). Usually, itis assumed that the nuclear spin system is immersed in an applied static mag-netic field (which direction defines the z-direction in the laboratory frame,B = Bez) and its energy level populations are described by the Boltzmanndistribution. The interaction energy between the applied static magneticfield and the nuclear magnetic moment is En = −~γnmIB = −~ω0mI ,where mI = I, I − 1, . . . ,−I and ω0 = γnB is the nuclear Larmor pul-sation. The thermal equilibrium condition for the nuclear magnetizationM ≡ (Mx,My,Mz) is Mz = M0 and Mx = My = 0, where

M0 =Nnγ

2n~2I(I + 1)3kBT

B. (4.1)

NMR measurements are basically the manipulation of the nuclear spinsin the externally imposed static magnetic field B, by a time-varying radio-frequency (RF) magnetic field BRF. The heart of the NMR probe is a tunedLC resonant circuit with the sample contained in the RF coil (a conventionalsolenoid) [Fig. (4.1)]. The NMR coil produces the driving BRF by means ofwhich the energy separation between the nuclear energy levels is measured.If BRF’s frequency is close to ω0, transitions between nuclear energy levelswill occur and the nuclear spin system will absorb electromagnetic energy.

A simple, classical way of looking at the NMR is the following. Thenuclear magnetization vector precesses around the applied static magneticfield according to the well-known formula

dMdt

= γnM×B. (4.2)

The angle between M and B is constant in time (t); its variation takes placeonly with emission or absorption of energy. A superposed BRF, applied at a90-degree angle to M, causes a forced precession of the nuclear magnetizationwith decreasing latitude if its frequency is close to ω0. In a nuclear inductionexperiment, BRF is turned off once the nuclear magnetization is pointing inthe xy-plane. Then, the free precession of the in-plane nuclear magnetizationinduces a measurable voltage in the NMR coil.


The phenomenological equations of Bloch

An ensemble of free nuclear spins is just an idealized test ground for NMR.Next, we need to address the question of NMR detection in realistic, exper-imental situations. Several physical quantities are important in describingthe phenomenon of magnetic resonance in solids, but two of them are inti-mately related to the microscopic details of the nuclear spin system.

First, in the presence of an applied static magnetic field, the estab-lishment of thermal equilibrium between the nuclear spin system and thesubstance containing the nuclei will require an exchange of energy betweennuclei and their surroundings (the so-called ”lattice”). That is, the nuclearspin system does not instantaneously reach thermal equilibrium with the lat-tice when the external static magnetic field is switched on. The rate of theprogress of the longitudinal nuclear magnetization Mz towards its thermalequilibrium value is T−1

1 , the nuclear spin-lattice relaxation rate.

Secondly, if there were no interactions among the nuclear spins, therewould be no decay of the in-plane nuclear magnetization in a nuclear in-duction experiment. However, the spin precession is a function of the localmagnetic field. For each single nucleus, because of nuclear dipole-dipoleinteractions, the local magnetic field is composed of the external static mag-netic field plus contributions from magnetic fields induced by the rest ofnuclear magnetic moments. The randomness from the dipole-dipole interac-tions is the main source of the decay of the in-plane nuclear magnetization.The nuclear spin-spin relaxation rate T−1

2 represents the rate of the decayof transverse components of the nuclear magnetization (Mx and My).

In the phenomenological theory of Bloch [1], the components of the nu-clear magnetization relax proportionally with their deviation from the ther-mal equilibrium values

dMz

dt=M0 −Mz(t)

T1,

dMx

dt= −Mx

T2,

dMy

dt= −My

T2.

(4.3)

Combination of Eqs. (4.2) and (4.3) leads to Bloch equations in the labora-


tory system

dMz

dt= γn(M×B)z +

M0 −Mz(t)T1

,

dMx

dt= γn(M×B)x −

Mx

T2,

dMy

dt= γn(M×B)y −

My

T2.

(4.4)

The macroscopic approach to NMR consists of solving the Bloch equa-tions in the presence of a perturbing, linearly polarized RF field BRF =2B1 cosωt, applied along the x-axis. Let’s consider that we are looking atan absorption-type NMR experiment and we are interested in the powerabsorbed in the RF coil, which is given by Pabs = Mx (dBx/dt). If theRF excitation is a small perturbation, the response of the system may beassumed proportional to it and can be written as Mx = 2B1[χ′(ω) cosωt +χ′′(ω) sinωt], where χ′(ω) and χ′′(ω) are the real and imaginary parts of theRF susceptibility χ(ω) = χ′(ω)− iχ′′(ω). One can easily find that the powerabsorbed by the nuclear spin system is proportional to the imaginary partof the complex susceptibility: Pabs = 2B2

1χ′′(ω)ω. From the Bloch equations

we obtain χ′′(ω) = (π/2)χ0ω0I(ω), where χ0 = M0/B is the static nuclearsusceptibility and I(ω) is the shape function (normalized to the unity) ofthe NMR spectrum

I(ω) =T−1

2

π

1(ω0 − ω)2 + T−2

2

. (4.5)

The NMR line shape and T−12

In general, the function I(ω) is a bell-shaped curve with a maximum at ω0.As we can see in the calculations of the NMR spectrum using Bloch equa-tions, T−1

2 is responsible for the broadening of the signal which would be aδ-function if there were no nuclear spin-spin interactions. A noteworthy fea-ture is that the exponential decay of the transverse magnetization results ina Lorentzian shape of the NMR spectrum. While Lorentzian line shapes arecommonly found in experiments performed at high T , Gaussian line shapesare characteristic for low temperatures. If T−1

2 = σ, the NMR spectrum ofGaussian shape is described by

I(ω) =1

σ√

2πexp

[− (ω − ω0)2

2σ2

]. (4.6)


Accordingly, the decay of the transverse magnetization M+ = Mx + iMy is

M+(t) = M+(0) exp (−σ2t2/2). (4.7)

It is most convenient to characterize NMR line widths by specifying theirrelation with T−1

2 . Our definition for the full width at the half maxi-mum (FWHM) of the NMR signal (in the ω-domain) is ∆ω = 2T−1

2 fora Lorentzian line shape and ∆ω = 2.36T−1

2 for a Gaussian line shape. Atthe resonant frequency ω0, the quality factor Q is the ratio between ω0 and∆ω. The Lorentzian and Gaussian line shapes are shown for comparison inFig. (4.1)(bottom) for ∆ω = 2.354 and ω0 = 0. In order to characterize aline shape to the full, one can study its variances (moments) defined as

(∆ω)n =∫

(ω − ω0)nI(ω) dω. (4.8)

For example, the second moment of a Gaussian curve is (∆ω)2 = σ2.

Nuclear dipole-dipole couplings

A variety of seemingly effects, including the nuclear spin diffusion and thenuclear spin-spin relaxation, are results of the same basic interaction, namelythe dipole-dipole couplings of nuclear spins. We wish to elaborate now morefully on these concepts as they are frequently used in this thesis.

A typical value of the nuclear spin-spin relaxation time in solids is≈ 100 µs, whereas the nuclear spin-lattice relaxation time can be equalto seconds, minutes, hours or more. Consequently, the establishment ofthermal equilibrium between the nuclear spin system and the lattice can bethought as proceeding in two steps 3. First, the nuclear spin system reachesan internal equilibrium at a temperature TS with a characteristic rate T−1

2 .Secondly, TS tends towards the lattice temperature T with a characteristicrate T−1

1 .We take up now the subject of the nuclear spin diffusion. In the presence

of nuclear dipole-dipole interactions it is possible to transport the nuclearmagnetization by flip-flop transitions 4. If we confine ourselves to an ac-count of flip-flop transitions of nearest neighbors, we obtain a nuclear spin

3In the preceding chapter we have assigned a temperature TS to the nuclear spin systemwithout questioning if it is conceptually correct.

4The dipolar Hamiltonian for Nn interacting, like nuclei is

Hd =~2γ2

n

2

NnXi6=j=1

»Ii · Ij

r3ij

− 3(Ii · rij)(Ij · rij)

r5ij

–, (4.9)


diffusion constant Dn ≈Wa2, where W is the probability (per unit time) ofa flip-flop transition of a pair of nearest identical nuclei and a is the distancebetween them. A rough estimate of W for a cubic crystal is W ≈ T−1

2 /30.The time required to transport the nuclear magnetization over a distancer is of the order of r2/Dn. Over distances long compared to the few lat-tice spacings, the nuclear magnetization evolves according to a macroscopicdiffusion equation 5.

4.2.2 Interactions of nuclei with electrons. NMR in metals.

Since straight resonance experiments in conventional 2DESs are hardly prac-ticable one may well ask what is the incentive for studying QHE systems byNMR. The reason is that electron-nucleus interactions result in remarkable,measurable effects such as the shift of the magnetic resonance, which, inturn, yield valuable insights into the properties of 2D electrons. In the fol-lowing we shall discuss the conceptual framework for NMR studies of QHEsystems, i.e., the interactions of nuclei with electrons.

The Knight shift (KS)

The general expression of the Hamiltonian for a nucleus at the position Rj

which interacts with its surroundings is [1]

Hj(t) = −γn~I ·Bj(t), (4.10)

where Bj(t) = 〈Bj〉 + δBj(t) is the time-varying, total magnetic field atthe position of the nucleus, 〈Bj〉 represents its average value, and δBj(t)stands for its fluctuation. The first term in the expression of Bj(t) is re-sponsible for the nuclear level splitting, whereas the second one induces atime dependence in the expectation values of various nuclear properties.In NMR measurements, transitions are induced at the resonance frequency

where rij is the distance between nuclei i and j. To count the interaction between everypair of nuclei only once there is a factor 1

2and the double sum runs over the indices i and

j separately, excluding the value i = j. The homonuclear dipolar Hamiltonian containsterms proportional to I+

i I−j + I−i I+j , which connect (mI , mI) to states (mI − 1, mI + 1)

or (mI + 1, mI − 1) by two simultaneous spin flips. These ”flip-flop” transitions conservethe total Zeeman energy of the nuclear spin system and act as a means of transportingthe nuclear magnetization through the lattice.

5Such a diffusion equation was used in conjunction with magneto-thermal experi-ments on sample M242 involving the ”cold” nuclei of GaAs QWs and the ”hot” nuclei ofAl0.3Ga0.7As barriers. There, we have tacitly supposed that the static applied field in thez-direction is homogeneous and T−1

1 is essentially zero.


ωn = γn〈Bj〉. Henceforth, as γn is known, the measured resonance frequencygives 〈Bj〉. The small difference between 〈Bj〉 and B, due to the electronicenvironment, defines the average value of the magnetic hyperfine field 〈Bhf

j 〉and the magnetic hyperfine shift tensor K as

〈Bj〉 = B + 〈Bhfj 〉 = B[1 + K]. (4.11)

The magnetic hyperfine shift has contributions from both the motion ofelectrical charges and from the magnetic moments associated to the electronspin. For the sake of simplicity, here we shall limit the discussion to theKnight shift, which originates from the isotropic Fermi contact interaction,i.e., the interaction between the nuclei and the conduction electrons insidethe nuclear volume. The Fermi contact interaction contribution to 〈Bhf

j 〉 is

〈BFj 〉

CGS= −8π3γe~〈Se(Rj)〉, (4.12)

where Se(Rj) is the total electron spin operator at the position Rj of thenucleus. In the presence of an applied static magnetic field along the z-axis,the Knight shift KS is obtained as

〈Bj〉z = B + 〈BFj 〉z = B[1 +KS ]. (4.13)

In a metal, the components of Se(Rj) could be expressed in terms of one-electron wave functions φk(r) = Uk(r)e−ik·r, where Uk(r) = (V/Ω)−1/2uk(r)is a function with the periodicity of the lattice normalized to the sample’svolume, uk(r) is the periodic part of the Bloch state of wavevector k normal-ized to the atomic volume, V is the volume of the sample, Ω is the atomicvolume, and r is the electron coordinate. Only the z-component of Se(Rj)has a non-zero average value, which is given by

〈Se(Rj)〉z =12

∑k

|Uk(Rj)|2[n(Ek,↑)− n(Ek,↓)], (4.14)

where n(Ek,↑) represents the probability of finding a spin-up electron in astate of energy Ek,↑ and it is given by the Fermi-Dirac distribution. As theaverage value of the Fermi contact interaction is taken over all electronicvariables, the electrons in complete shells contribute nothing to it. Assum-ing that the expectation values of the hyperfine couplings are the same forall the unfilled orbitals near the Fermi surface and χe is the electron spinsusceptibility, the (fractional) Knight shift writes as

KS ≡∆BB

=8π3χe|UkF

(Rj)|2. (4.15)


The Knight shift is given as a relative value, namely the NMR line shift di-vided by the reference line position (in % or ppm) [101]. When quoting thevalue of the Knight shift, the reference with respect to whichKS is computedshould be specified. The fractional Knight shift is nearly T -independentin common metals and, provided that the hyperfine field coupling con-stant Ahf [usually defined by Ahf = (8π/3)|ukF

(Rj)|2γe~] is known, itis ordinarily used to measure the electron spin susceptibility per site χe asKS = Ahf χ

e/(γe~).

The nuclear spin-lattice relaxation rate (T−11 )

In metals, the spin-flip scattering of conduction electrons by the nucleusconstitutes an important channel for nuclear-spin lattice relaxation. In thisprocess both spins of the conduction electron and the nucleus are flipped.The Fermi contact interaction Hamiltonian HF

j (t) = −γn~I ·BFj (t) contains

terms of the form I∓Se±(Rj , t) which produce simultaneous electronic- and

nuclear-spin flips. Here Se± = Se

x ± iSey are the (time-dependent) transverse

components of the total electron spin operator at the nuclear position Rj .A simple calculation, assuming non-interacting electrons, shows that T−1

1 ina common 3D metal is

T−11

CGS=64π3

9γ2

nγ2e~3|UkF

(Rj)|4[D(EF )]2kBT, (4.16)

where D(EF ) is the DOS at the Fermi level for one direction of the electronspin only. Korringa’s law tells us that T−1

1 is directly proportional to thetemperature, independent of the applied static magnetic field, and simplyrelated to the fractional Knight shift

TK2S =

~4πkB

(γe

γn

)2

T−11 . (4.17)

Korringa’s law is usually taken as a representative of metallic behavior. It isexpected to be valid for metals whose dominant contribution to the magnetichyperfine shift and T−1

1 comes from s-state electron-nucleus interactions. Itshould be noted that electron-electron interactions affect χe and D(EF ).Hence, the Korringa relation is modified to [113]

TK2S =

~4πkB

(γe

γn

)2[D0(EF )D(EF )

χe

χe0

]2

T−11 , (4.18)

where D0(EF ) and D(EF ) are the DOS at the Fermi surface for an idealFermi gas and for the real metal, respectively, and χe

0 and χe are the corre-sponding electron susceptibilities.


A convenient expression for T−11 caused by the Fermi contact interaction,

which takes into account the interactions between electrons, follows fromthe microscopic description of the nuclear spin-lattice relaxation processes.Nuclear transitions are induced by the fluctuations of the local hyperfinefield, which directly reflect electronic spin fluctuations [113]

T−11 =

γ2n

2

∫ ∞

−∞[cos (ωnt)]〈δBF

+(Rj , t)δBF−(Rj , 0)〉 dt, (4.19)

where 〈〉 means average over the states of the electronic system, repre-sents the symmetric product, and δBF

±(Rj , t) ∝ δSe±(Rj , t). The fluctuation-

dissipation theorem relates the Fourier transform of the local hyperfine fieldcorrelation function to the dissipative part of a generalized, frequency de-pendent magnetic susceptibility χ+− = χ′+− + iχ′′+− as [113]∫ ∞

−∞[cos (ωnt)]〈δBF

+(Rj , t)δBF−(Rj , 0)〉 dt ∝ −

χ′′+−(Rj ,Rj , ωn)1− e~ωn/(kBT )

. (4.20)

Without sacrificing any of the informational content of Eq. (4.19), onecan take the limit ~ωn kBT and use the spatial Fourier transform ofχ′′+−(Rj ,Rj , ωn) to obtain

T−11 ∝ kBT

~ωn

[1

(2π)2

∫ ∞

−∞χ′′+−(q, ωn) dq2

]. (4.21)

Thus, the problem of deducing T−11 is reduced to that of calculating

χ′′+−(q, ωn). Note that in the above formula we have explicitly considered a2D Fourier transform to point out that the theory also applies to 2DESs [5].

The electric field gradient (EFG)

No mention has been made so far of another important property of GaAsQWs’ nuclei, namely the electrical quadrupole moment Q, which gives ameasure of the lack of sphericity in the distribution of the electric chargeinside the nucleus. The Coulomb interaction of the electrical quadrupolemoment with the surrounding electron charge distribution is described bythe Hamiltonian

HQ =eQ

6I(2I − 1)

∑α,β

Vαβ

[32(IαIβ + IβIα)− δαβI

2

], (4.22)

where α and β denote the directions in an arbitrary frame of reference(X,Y, Z), Iα and I2 are operators associated to the angular momentum of


the nucleus, Vαβ = ∂2V (Rj)∂α∂β are the components of the local electric field

gradient (EFG) tensor, and V (Rj) is the electrostatic potential created bythe electrons at the position of the nucleus Rj . The EFG tensor is symmetricand by a change of the frame of reference can be diagonalized. In terms ofthe principal axes (x, y, z), we have

HQ =eQ

6I(2I − 1)

∑α

Vαα(3I2α − I2). (4.23)

Furthermore, as∑

α Vαα = 0, the EFG tensor is completely described bytwo parameters. (As a consequence, for lattice sites of cubic symmetry, i.e.,spherically symmetric charge distributions with Vxx = Vyy = Vzz, the EFGvanishes and there are no quadrupole interactions.) Generally, one choosesto describe the EFG tensor by the largest principal component Vzz and theasymmetry parameter η = (Vxx − Vyy)/Vzz. Within this convention, thequadrupole Hamiltonian writes as

HQ =hνQ

2

[I2z −

I2

3+η

3(I2

x − I2y )], (4.24)

where νQ is the quadrupole frequency

νQ =3eQVzz

2I(2I − 1)h. (4.25)

In the presence of an external static magnetic field, which direction makes anangle ϑ with the principal axis of the EFG tensor, the first order quadrupolecoupling leads to the following nuclear energy levels

En,Q = −~γnmIB +eQVzz

4I(2I − 1)

(3 cos2 ϑ− 1

2

)[3m2

I − I(I + 1)] (4.26)

where we have assumed that η = 0. The quadrupole coupling shifts thebare nuclear Zeeman energies by an amount which is related to both νQ

and ϑ. For I = 3/2 and ϑ = 0 the NMR spectrum is composed of acentral line at the frequency ω0/(2π) and two satellites, equally shifted tohigher [ω0/(2π) + νQ/2] and lower [ω0/(2π)− νQ/2] frequency. The centralresonance is the m = −1/2 ↔ 1/2 transition, whereas the satellites are them = ±1/2 ↔ ±3/2 transitions.

4.2.3 NMR in QHE systems

KS in GaAs quantum wells

We have seen in the preceding section the basic theory behind NMR experi-ments and how NMR can be used to probe the electron spin polarization of


a metallic system. In this section we point out some usage and definitionsspecific to NMR studies in QHE systems. We reiterate that NMR uses thenuclear magnetic moment ~γnI to measure the additional local magneticfield Bhf (Rj , t) produced by the electrons at the position Rj of the nucleus.The hyperfine coupling Hamiltonian [1] is

Hhfj (t) = −~γnI ·Bhf (Rj , t). (4.27)

The average value of the hyperfine field 〈Bhfj 〉 is measured as the shift of the

NMR line [compared to the reference line position in the absence of Bhfj ]

KS(Rj) [s−1] = −γn

2π〈Bhf

j 〉z, (4.28)

where it is assumed that the applied static magnetic field lies along the z-direction. We remark here that the shift definition given in Eqs. (4.28) and(4.31) differs from the usual one [ Eqs. (4.11) and (4.15)]. That is, for 2DESand QHE, the NMR shift is a measure of the local electron spin polariza-tion and not of the electron spin susceptibility per site; the absolute valueof the shift [given by Eq. (4.28) in frequency units] measures directly thetime-averaged value of Bhf (Rj , t). The hyperfine field includes contribu-tions from several sources, the strengths of which can be quite different. ForGaAs, calculations show that at the Γ point of the first conduction band, thes-like contribution dominates the total electron charge-density distribution,which is localized on the nuclei [88]. Thus, the isotropic Fermi contact inter-action gives the dominant contribution to the hyperfine field. The hyperfinecoupling Hamiltonian [Eq. (4.27)] reduces to

HFj (t) ∝ I · Se(Rj , t)ρe(Rj), (4.29)

where ρe(Rj) is the local value of the electron density and Se(Rj , t) is theelectron spin density at the nuclear site. The corresponding NMR line shift,called the Knight shift as in simple metals, directly measures the local spinpolarization P(Rj)

KS(Rj) ∝ P(Rj)ρe(Rj). (4.30)

For multiple-QW samples, the nuclei in the barriers, where the electrondensity is vanishingly small, will provide the reference line position. Strictlyspeaking, some non-zero electron density regions also exist in barriers closeto QWs and Si-doping layers. However, these regions cover only small per-centage of the volume of the barriers, so they contribute to the negligiblepart of the NMR signal from barriers and are not expected to modify its


peak position. This is indeed confirmed experimentally, as the barriers’ peakposition depends neither on the temperature nor on the recovery time tR[see Sec. (4.3.2) below], and it is found at the same position as the signalfrom the substrate.

To infer the global polarization P(T, ν) from the measured Knight shift,we shall assume a spatially homogeneous electron spin polarization. Thevalidity of this assumption resides in the fast dynamics of electrons alongthe QWs over the NMR observation time [93]. Thus, under the simplestassumption of uniform electron density (ρe = n/w), the nuclei in GaAsQWs provide a single NMR line whose shift measures directly the electronpolarization

KS(T, ν) = Acn

wP(T, ν). (4.31)

The above relation defines the effective hyperfine coupling constant Ac. Infact, as the electron density is not constant across the QW, its z-dependencesomewhat affects the NMR line shape 6. However, as much as the ρe(z) isa rigid function, essentially independent of T and ν, the line shape will alsobe rigid and we can use the shift of the peak position in Eq. (4.31) to deduceP(T, ν) [93]. The difference between the actual ρe(z) and the average valuen/w, will be taken into account by Ac.

The value of Ac defined in Eq. (4.31) is experimentally determined fromthe NMR shift obtained at low T , high B, and at filling factors (such asν = 1/3) where the 2DES is fully polarized (P = 1). Using the valuefor 71Ga determined in this way in OPNMR experiments [55, 64], Ac

∼=4.5 × 10−13 cm3/s, we calculate from Eq. (4.31) a maximum shift in oursample of KP=1

S = 24 kHz and deduce the spin polarization of 2D electronsas P(T, ν) = KS(T, ν)/KP=1

S . We have an uncertainty of ±10% in oursamples’ w, resulting in a similar uncertainty in our calculated KP=1

S . Theexperimental uncertainty of the measured KS(T, ν) is about ±10− 20%.

T−11 in GaAs quantum wells

The Fermi contact interaction is responsible for the nuclear spin-lattice re-laxation in common metals, such as copper [1, 94]. This relaxation mecha-

6In previous OPNMR electron spin polarization studies [55, 64] ρe(z) was approximatedby a sinusoidal shape and an intrinsic hyperfine shift for the nuclei in the center of the QWwas deduced. Corrections to P [Eq. (4.31)] arising from realistic approximations to ρe(z)are negligible [93]. We also note that our definition of P [Eq. (4.31)] makes no distinctionbetween the nuclei close to the GaAs/Al0.3Ga0.7As interface and those located near thecenters of the QWs. Such a distinction may be important when NMR spectra are sensitiveto dynamical processes.


nism involves a simultaneous spin flip of both spins of the conduction elec-tron and the nucleus, and energy conservation requires that there must beclosely spaced spin ↑ and ↓ electron levels with energy comparable to thenuclear Zeeman gap. For a 2DES, however, in the presence of a quantiz-ing magnetic field and in the absence of disorder, the single-particle energyspectrum is discrete. The flip-flop process under discussion is forbidden andno Korringa relaxation is expected. Adding to the puzzle is the fact thatthe nuclear spin-lattice relaxation in pure insulating crystals is negligiblesmall through direct interaction between the nuclear spins and the latticedegrees of freedom. The nuclear spin-lattice relaxation actually observed inhigh-purity bulk GaAs has been interpreted as ”magnetic relaxation”, owingto nuclear spin diffusion to paramagnetic centers, which serve as intermedi-ates between nuclear spins and lattice 7. The nuclear spin diffusion has noB-dependence and may add a constant background to T−1

1 (T,B), dictatedby the relaxing impurities. In the presence of high magnetic fields, how-ever, paramagnetic impurities, if there are any, should be frozen into theirlowest spin state; thus, the nuclear spin-lattice relaxation by paramagneticimpurities is expected to be suppressed.

Central to any discussion of the nuclear spin-lattice relaxation inelectron-doped GaAs QWs under the QHE regime is the existence of nearlygapless electron spin-flip excitations 8. Our understanding of a finiteT−1

1 is in terms of additional degrees of freedom present in real 2DESs.These degrees of freedom could be generally ascribed to impurities (disor-der) [5, 52, 105] and for the particular case of ν . 1 or ν & 1, also to thepresence of Skyrmions in the electronic ground state [24, 44]. Calculationsof T−1

1 in the QHE systems, which properly take into account the effects ofdisorder and electron-electron interactions, appeared well before the begin-ning of the Skyrmion era [5, 105]. By invoking the presence of Skyrmions,recent theories [24, 44, 87] explain the observed one thousand fold enhance-ment of T−1

1 at |ν − 1| ∼ 0.1, compared to the zero magnetic field value.The theoretical predictions for T−1

1 (T ) near ν = 1 have been summarized inChapter 2.

7An interesting nuclear spin-lattice relaxation process occurs in conventional ferromag-nets such as cobalt. Here, the nuclear spin-lattice relaxation is by nuclear spin diffusionaway from the domain walls and ultimately to the lattice by coupling with the conductionelectrons. Consequently, the T -dependence of the observed T−1

1 does not follow Korringa’slaw.

8We note that at B = 0 our 2DES is metallic-like. A simple calculation of T−11 , based

on the analogy with the 3D free electron gas and on the assumption that the Fermi contactinteraction is the only relaxation mechanism yields T−1

1 = 9π3~A2c(n/w)2(kBT/E2

F ). Forsample M242 at T = 1 K, we obtain T−1

1 ≈ 4× 10−4 s−1.

4.3. EXPERIMENTAL METHODS 161

We wish now to complete the discussion of T−11 by examining the inte-

ger QHE regime, where the discreteness of the 2D single-electron spectrummanifests in an interesting way in the B- and T -dependence of the nuclearspin-lattice relaxation rate [105]. Because the energy conservation in the flip-flop process can be fulfilled at any finite T by the participation of a phonon,an activated T -dependence of T−1

1 is expected at low temperatures. How-ever, at high T , because the number of thermally excited phonons increaseslinearly as the temperature is increased, the phonon-assisted T−1

1 should fol-low the usual Korringa’s law. On the other hand, since the disorder leads toa broadening of Landau levels (with DOS displaying maxima at half-integerfilling factors), one expects that T−1

1 undergoes magneto-quantum oscilla-tions, similar to that of the longitudinal magnetoresistance. This oscillatorybehavior of T−1

1 has been observed via electron spin resonance (ESR) of 2Delectrons 9 by Berg et al. [11]. The experimentally determined T−1

1 neverbecome smaller than T−1

1 = 1× 10−3 s−1, whereas the value of T−11 at max-

ima (half-integer filling factors) is about 4 × 10−3 s−1. This value is oneorder of magnitude larger than that expected at zero magnetic field.

4.3 Experimental methods

4.3.1 Experimental setup and the sample

In order to increase the NMR signal, two pieces of the multiple-QW sample,each ≈ 25 mm2 in area, were placed together (tightly wound with Teflontape) into the RF coil [Fig. (4.1)(top)]. Thus, the total number of QWs’nuclei (Ga or As) contained in our measured specimen was ≈ 3×1018, whichwas sufficient for the observation of 69,71Ga NMR signals at temperaturesbelow 10 K and magnetic fields above 4 T with our NMR apparatus [79].The experimental setup consists of a single-coil duplexing scheme, i.e., thesame coil is used for transmitting the RF pulses and receiving the nuclearinduction signals. The RF coils were conventional solenoids, constructedby hand from copper wire wound around a metallic slab whose cross-section

9Coupling as they do the 2DES and the nuclear spin system of GaAs QWs, the nuclear-electron interactions can manifest themselves in the study of either system. The electronicspin splitting is not solely due to the Zeeman effect, but also influenced by the nuclear-spinpolarization, which acts via hyperfine interaction on the electronic spins. In thermody-namic equilibrium, the nuclear polarization is negligible even at very low T . However, thenuclear spins may be dynamically polarized. In the presence of a large nuclear polariza-tion, the ESR line will be shifted. The T−1

1 of nuclei could be observed via ESR [28] of2D electrons because a non-equilibrium nuclear polarization results in a non-equilibriumESR line position, which relaxes towards the equilibrium with a characteristic rate T−1

1 .


was slightly larger than the sample’s. By potting the coils in epoxy resin, thesample could easily be inserted and replaced. This configuration also min-imizes the magneto-acoustic ringing [38]. We list in Table (4.1) importantparameters of RF coils used in the present study.

Table 4.1: Parameters of two coils used in NMR experiments are given. Allsample coils have an internal cross-section of 1 × 5 mm2. Typically, RFcoils have a quality factor of about . 100 at room temperature, measurednear the bottom end of the resonance frequency [f0 = ω0/(2π)] range. Thecorresponding B-range is given for 71Ga. Note that with the use of both Gaisotopes we were able to cover an extended range of B with the same coil.

Coil f0-range (MHz) B-range (T) Number of turns Length (mm)

# 1 49 - 101 3.8 - 7.8 24 7.5

# 2 108 - 228 8.3 - 17.5 10 6.5

The NMR spectra were recorded on a custom built pulsed NMR spectrom-eter. The temperature control was provided by a standard variable temper-ature insert (VTI) down to 1.5 K (in pumped liquid 4He or in a gas flowabove 4.2 K), and by a 3He/4He dilution refrigerator down to 0.05 K. TheNMR system is equipped with a superconducting magnet with maximumfield strength of 17 T (in pumped liquid 4He). The magnetic field spatialhomogeneity over the dimensions of the sample is better than 10 ppm andits temporal stability is few ppm per hour. An important advantage of ourstandard VTI setup was the possibility of varying in situ the tilt angle θbetween B and the normal to the plane of the 2D electron layers, togetherwith the low-T tuning and matching of the RF coil (the so-called “bottomtuning”) 10. This enabled us to vary at will three parameters: T , θ, andB (above 1.5 K and up to 17 T). For very low-T measurements, the RFcoil with fixed tuning/matching was mounted into the mixing chamber ofthe dilution refrigerator. This ensured good thermal contact to the sampleand kept optimal NMR sensitivity, but restricted each low-T experiment tosingle T -dependence at fixed θ and B.

10The resonant frequency of the circuit containing the coil is tuned by adding in seriesa variable capacitor Ctune. The impedance can be adjusted to 50 Ω (real) by adding inparallel a capacitor Cmatch [Fig. (4.1)(top)] or inductor [38].


4.3.2 Experimental technique

Essential to pulsed NMR experiments, is the detection of the free inductiondecay (FID). By applying a strong RF pulse, an alternating RF magneticfield is produced along the axis of the RF coil. This RF magnetic field, whosefrequency satisfies the condition for nuclear resonance, turns the nuclearmagnetization out of the z-direction. When the nuclear magnetization hasturned by 90-degrees, one speaks of a π/2-pulse. The RF magnetic field isswitched off at the end of the time interval corresponding to the π/2-pulseand the NMR coil detects the transient voltage due to the precessing nuclearmagnetization around z-axis. That is, the FID experiment probes the freeprecession of the transverse nuclear magnetization following a π/2-pulse.It is however clear that the transverse nuclear magnetization will decaywith a rate T−1

2 due to variations in the local magnetic field experiencedby the nuclei. The origin of these variations may be inhomogeneities of theapplied static magnetic field or internal inhomogeneities, such as the dipolarmagnetic fields of neighboring nuclei. If T−1

2 is very large, it is difficult tosee the NMR signal because the FID will disappear before the instrumentaldead time (& 2 µs) is over.

Spin echoes are commonly used to reform the NMR signal some timeafter the π/2-pulse in order to avoid the instrumental dead times whichalways corrupt the start of the FID. The spin echo sequence (π/2−τ−π−τ−echo) [15, 46] is pictorially illustrated in Fig. (4.2). The result of a π/2-pulseis a non-equilibrium configuration of nuclear magnetic moments. After theπ/2-pulse, the nuclear spins precess freely with different frequencies imposedupon them by the existing inhomogeneities. As they dephase with time, themacroscopic magnetization of the nuclear spin system is lost. However,application of a π-pulse after a time τ causes all nuclear spins to come backtogether at the same rate that they precessed prior to the π-pulse. After atime 2τ all nuclear spins are again in phase and the echo is produced.

NMR measurements

The first obvious problem with the NMR in multiple-QW samples is to sep-arate the signal of QWs from the barriers’ signal, which is about 7 timeslarger, and from the even stronger signal of the substrate 11. In our experi-ments, since the nuclear polarization builds up because of nuclear spin-lattice

11Note that in OPNMR this is done automatically, as one ”pumps” only the QWs’nuclei [1, 7]. We also stress out that standard NMR experiments differ from OPNMRmeasurements in that they do not require the realization of a highly polarized nuclearspin system through nuclear dynamic polarization.


Figure 4.2: (Top) Pulse sequence used to detect the NMR spectra. (Bottom) Schematic diagram of the spin-echosequence [π/2)x′ − τ − π)y′ − τ − echo] in the rotating frame. The spin echo is essentially two FID’s back-to-back.


relaxation processes, the unwanted signals can be suppressed by exploitingthe difference between the T−1

1 of QWs’ nuclei and T−11 of nuclei located in

the GaAs substrate and barriers. At low temperature, T−11 of the substrate

is extremely small due to the absence of mobile electrons and dopants.

Figure 4.3: 71Ga NMR spectra taken at (a) f0 = 73.915 MHz with tR =256 s (top) and 2 s (bottom), and at (b) f0 = 192.052 MHz with tR = 128s (top) and 32 s (bottom). For short recovery times (lower spectra) thecontribution is essentially from nuclei in QWs, while for longer times (upperspectra) barriers’ signal becomes stronger than the one from QWs.

For barriers’ nuclei, T−11 is increased because of the δ-doping; the largest

T−11 is normally expected for QWs’ nuclei due to 2D electrons (except for

the gapped phases of the 2DES at very low T ). Thus, at the beginning ofour pulse sequence, the nuclear magnetization of the Ga isotope of interest isset to zero by a comb of π/2-pulses. After the magnetization has recoveredduring time tR, we perform standard spin-echo or FID read out and obtain


spectra by Fourier transforming [Figs. (4.3), (4.4), and (4.5)]. At short tR(1 − 10 s), the signal contains mainly the QWs’ contribution; for longerrecovery times (& 10 s), it is dominated by the barriers.

Figure 4.4: 71Ga NMR spectra measured at ν = 1, θ = 0, and indicatedtemperatures. For short recovery times (lower spectra) the contribution ismainly from nuclei in QWs, while for longer times (upper spectra) barriers’signal becomes dominant. The NMR frequency shift is given relative to theposition of the barriers’ line.


The quantity called ”KS of Ga nuclei in QWs” is here defined as the differ-ence between the peak positions of QWs and barriers. As the same NMRshift data were obtained both by FID and by the spin-echo sequence, onlythe latter technique is used in the present work.

The complete pulse sequence used for the present measurements is de-picted in Fig. (4.2)(top). The π/2-pulse lasts for times of the order of 1 µswhich provide uniform irradiation over a wide spectral range. Typical sep-aration between the π/2- and π-pulses was τ ∼= 100 µs. The pulse spacingwithin the comb was set to 3 ms. The signal-to-noise is improved by aver-aging over several NMR scans and a ”phase cycling” is executed during theaccumulation of spectra to eliminate spurious effects.

A brief comment on the possibility of heating the sample is in orderhere. First, the 2DES is extremely susceptible to heating by eddy currentsaround ν = 1 12. Secondly, a direct coupling between the RF pulses andthe 2DESs might be possible [64]. In our experiments, any variation of thetemperature of the 2DES during the measurement is easily detected throughthe T -dependence of the measured NMR shift. The amount of RF heating isdetermined by the RF-pulse power and the repetition rate of the spin-echosequence. Values of these parameters are chosen to avoid any possible RFheating. That is, our standard test to definitely exclude the problems ofheating consists of checking that results do not change when the RF-pulsepower is reduced by an order of magnitude.

The quadrupole splitting of the resonance

Our measurements were carried out using both Ga isotopes. As expectedfrom Eq. (4.28), we indeed find that 71KS/

69KS = 71γn/69γn, and for sim-

plicity all the results are normalized to 71Ga isotope using the 71γn/69γn =

1.27 ratio.

12We note that if the electrons are heated by the Joule effect, no spin depolarizationis expected at ν = 1 as the electron heating is minimal (the longitudinal resistance isessentially zero). The ohmic heating, however, effectively depolarizes the spins of theelectronic systems for small variations of the filling factor away from ν = 1 by increasingthe 2DES temperature over the lattice temperature. This phenomenon, which is known inmagneto-transport experiments as the breakdown of the QHE, has a single-particle natureand it is independent of the existence of Skyrmions [62].


Figure 4.5: Set of NMR spectra comparing the line shapes of 71Ga and 69Ga,measured at T = 1.5 K, ν = 3/4, and θ = 0. The lower (upper) spectraare taken with short (long) recovery time tR, respectively. The quadrupolesplitting is better observed at short tR, as shoulders to the central QW line.The NMR frequency shift is given relative to the barriers’ line position.

Typical 69Ga and 71Ga NMR spectra are illustrated in Fig. (4.5). Forthe QW line we can clearly observe the quadrupole splitting [Figs. (4.3)(a)and (4.5) (bottom traces)]. On both sides of the main line appear satelliteswhose splitting for two isotopes scales with known electrical quadrupole mo-ments, 69Q/71Q = 1.6 [Table (A.1)]. Such quadrupole effects can occur onlyif the local symmetry is not cubic. One may suspect that breaking of thecubic symmetry is induced by the slight mismatch of the lattice constantsof GaAs and Al0.3Ga0.7As, where the barriers impose their lattice constantto the in-plane lattice constant of QWs. The quadrupole coupling is clearlyresolved at high temperatures [78], indicating that the effect is homogeneousover all QWs in the sample. The strain-splitting ratio in GaAs QWs wasmeasured to be (4.7 ± 0.4) × 10−6 kHz−1 for 75As [45]. From our exper-


imental conditions [Fig. (4.3)(a), bottom trace] we deduce an estimate of9× 10−5 for the actual strain in our sample, close to residual value 6× 10−5

found in Ref. [45]. At low T , the satellite splittings are obscured by the line-broadening, but they do contribute to the total width of the QWs’ resonanceline. Note that no quadrupole splitting appears in the barriers’ line.

71Ga T−11 measurements

The same RF pulse sequence described above is used for T−11 measurements

and is called the saturation recovery method, i.e., we measure the NMRsignal intensity as a function of tR. A typical result, presented in Fig. (4.6),singles out the difficulty to define the saturation value of the NMR signal atlong tR. The point is that at very long recovery times the barriers’ signalprevails in the NMR spectrum and thus “pollutes” QWs’ signal. In order tominimize this effect we analyzed only the (integral over the) left half of theQWs’ resonance line. Our analysis of the nuclear spin-lattice relaxation rateis based on Eq. (4.3) for all investigated T - and ν-ranges. Fits to the Blochrelaxation formula [Eq. (4.3)] reproduce well the measured recovery of thenuclear magnetization, suggesting a single dominant source of relaxation.That is, within our experimental accuracy determined by the signal to noiseratio and the possibility to separate the signal of QWs from the partiallyoverlapping barriers, we did not find evidence for a “multi-exponential” re-laxation. We also mention that for tR & 10 s, the nuclear spin diffusion couldbe considered as an efficient process for the thermal relaxation, ensuring ahomogeneous spin temperature across the QWs. Hence, a single T−1

1 whichdoes not vary along z-axis is a satisfactory approximation in describing thegrowth of QWs’ nuclear magnetization in our NMR experiments.

Reliable T−11 measurements become impossible at higher temperatures;

the NMR shift becomes small and the overlap with the barriers’ line is toostrong, leading to efficient nuclear spin diffusion between QWs and barrierswhich modifies the NMR line shape 13. Finally, we wish to point out thatfor the gapped phases of the 2DES at very low T , T−1

1 can be extremelysmall and the time for nuclear magnetization to reach its equilibrium valuebecomes prohibitively long for T−1

1 measurements.

13We observed that the barriers’ signal grows proportional to T−11 in the QWs. The

question should be raised then as how the contribution of the barriers’ nuclei to themeasured heat capacity near ν = 1 is usually not observed. We believe that the barriers’nuclear contribution remains hidden in heat capacity experiments because the thermalrelaxation is essentially completed when the nuclear spin diffusion into the barriers setsin.


Figure 4.6: Recovery time tR dependence of the NMR signal measured atT = 1.5 K, ν = 1 and θ = 0. The dashed curve is an one-parameterexponential fit to the data [Eq. (4.3)] used to determine T−1

1 .

The NMR line shape

At this point of the discussion we would like to briefly remark on the NMRline shape. The QWs’ line shape is observed at short recovery times; it israther symmetric and, neglecting the weak contribution from the barriers,can be approximated to a Gaussian. At intermediate times, the NMR lineshape has significant contributions from both QWs and barriers. Conse-quently, the separation of the overlapping QWs’ and barriers’ peaks is moreinvolved and less reliable. The barriers’ signal dominates the NMR spec-tra in the limit of very long recovery times and has the expected Gaussianshape. The FWHM values of the QWs and barriers’ Gaussian resonancesare summarized in Table (4.2). The FWHM value of the NMR signal ofbarriers that emerges from our analysis is ∆fB = 3.75 ± 0.28 kHz, in goodagreement with 3.5 kHz obtained in OPNMR experiments. The broaden-


ing of the barriers’ signal is attributed to the dipolar coupling between thenuclear spins. We determine for the NMR signal of QWs a FWHM value∆fQW ≈ 15 kHz, four times bigger than ∆fB. This value is also three timeslarger than ∆fQW observed in OPNMR experiments. The observed broad-ening is likely to originate from the quadrupole interaction of the nuclei withthe EFG as ∆fQW is approximately equal to our experimental quadrupolesplitting.

Table 4.2: The measured FWHM (in kHz) of the NMR signal of barriers(∆fB) and QWs (∆fQW ) at T = 1.5 K and θ = 0 is given for selectedB-values (in units of T).

B 4.7 5.7 6.7 7.6 9 11 13 15

∆fB 3.8 3.5 3.9 3.8 3.5 3.4 3.9 4.4

∆fQW 12.9 14.9 15.3 15.4 14.3 15.5 15.3 16.7

As discussed in Appendix B, one can obtain useful information about the 2Delectron spin dynamics from the measurements of the QWs’ line width as afunction of T [55, 64]. In the present study, our attention was not directedto the changes in the QWs’ line width with the temperature. An obviousdifficulty we have confronted is that the analysis of the QWs’ line shape inour sample can only be made if the quadrupole effects are properly takeninto account, which is laborious or rather impossible.

Finally, we note that insight into the dependence of NMR line shape onthe nuclear polarization profile along the z-axis [ρn(z)], ρe(z), and electronspin polarization and dynamics around ν = 1 has been attained recently bySinova et al. [93]. These authors have found a tremendous sensitivity of thespectral line shape to ρn(z) which explains the inhomogeneous broadening(asymmetry) of the QWs’ line shape observed in OPNMR experiments. Ingeneral, ρn(z) not only depends on the experimental technique but alsoon microscopic mechanisms (such as the nuclear spin diffusion and nuclearspin-spin relaxation). The question should be raised then as to how thescenario described in Ref. [93] accounts for the nearly symmetrical QWs’line shape observed in our experiments [bottom traces in Fig. (4.4)]. Itcould simply mean that the standard NMR favors a flatter ρn(z) comparedto the OPNMR method.


71Ga T−12 measurements

Using a conventional (π2 − π) spin-echo method, T−1

2 could be easily deter-mined: the echo intensity is monitored in a sequence of (π

2 − π) spin-echomeasurements, each (π

2 − π) spin-echo measurement taken with a differentvalue of the total time delay (t), separating the π/2 pulse from the echo.Our definition of T−1

2 is based on the assumption of a Gaussian decay of theecho intensity

Iecho(t) ∝ exp[− 1

2

(t

T2

)2]. (4.32)

Figure (4.7)(a) displays a typical time dependence of the echo intensitywhich directly yields T−1

2 . Under our experimental conditions, 71T−12 does

not seem to depend neither on B nor on T (including dilution refrigeratortemperatures). We determined 71T−1

2 = 1.38 ± 0.15 ms−1 [Fig. (4.7)(b)].It is noticeable that 71T−1

2 was previously estimated from the FWHM ofthe barriers’ signal in OPNMR measurements: a Gaussian FWHM of 3.5kHz gives 71T−1

2 = 9.32 ms−1. This calculated 71T−12 is usually expected to

be larger than the one obtained from spin-echo experiments. Interestinglyenough, spin-echo standard NMR measurements reveal a 71T−1

2 about 7times smaller than the value estimated from the FWHM of the barriers’signal in OPNMR measurements.

The present 71T−12 value allows an estimate of the nuclear spin diffusion

coefficient for 71Ga nuclei of 71Dn ≈ 0.74×10−13 cm2/s. Given the fact that71Dn/

75Dn = 0.7, we infer for 75As nuclei 71Dn ≈ 1.05×10−13 cm2/s, in ex-cellent agreement with previously reported 75Dn ≈ 1×10−13 cm2/s [84]. Wecould confidently assert that in our sample it takes about 10 s to transportthe nuclear magnetization over a distance r = 100 A. Finally, we mentionthat such T−1

2 measurements may prove fruitful in studying the motion ofSkyrmions 14.

14The key feature of T−12 experiments is that each (π

2−π) spin-echo measurement probes

the irreversible dephasing of the transverse nuclear magnetization during the time intervalt. A good example of physical situation where one can take advantage of this feature ofthe experimental procedure is the QHE ferromagnet near ν = 1, where Skyrmions areaccommodated in the electronic ground state. Consider first the situation where Skyrmion-size fluctuations are ”frozen” over the (varying) time window t/2, separating the π/2pulse from the π pulse. In this case, a static contribution (due to the Skyrmions) to thelocal magnetic field is experienced by the nuclei. This time-independent inhomogeneityis eliminated in a (π

2− π) spin-echo measurement, i.e., the π-pulse refocuses all nuclei

which dephased during the time interval t/2. On the contrary, if the fluctuations in thesize of Skyrmions are fast over the time interval t/2, the result is a time-dependent localmagnetic field inhomogeneity. In this case, only a fraction of the nuclei which dephased


Figure 4.7: (a) Typical time dependence of the echo intensity in a 71T−12

experiment. Referring to Eq. (4.32), we use a semilogarithmic plot of Iecho

vs t2 to extract 71T−12 (inset). (b) 71T−1

2 vs B at T = 1.5 K and θ = 0.


4.4 Results and discussion

4.4.1 Untilted magnetic field data

The electron spin polarization

I shall describe our results in the light of previous OPNMR experiments [7,55, 103] and start with the electron spin polarization data collected at θ = 0

as a function of ν and T . I must stress that our method has the greatadvantage that P can be measured at fixed θ and T over a broad ν-range.Therefore, Fig. (4.8) is the first fairly complete picture of the electron spinpolarization in the fractional QHE regime at θ = 0 and T = 1.5 K, fromNMR measurements. Also displayed in Fig. (4.8) are selected low-T datapoints at ν = 1, 1/2, 1/3, where full polarization is expected for T → 0.We observe a weak maximum at ν = 1 followed by a monotonous increaseof P as ν decreases, without any pronounced structure for ν . 1/3. Theincrease of P is roughly linear in B, however, the behavior changes at verylow temperatures, as illustrated by the T = 0.1 K data points. The T -dependence of P taken at these filling factors [Fig. (4.9)] confirms that forboth ν = 1/2 and 1/3 the full polarization is reached for T → 0.

On the other hand, our measurements revealed an unexpected T -dependence of P at ν = 1 [Figs. (4.9)(b) and (4.10)(a)]. The electron spinpolarization is only slightly increased as we cool down the sample from 1.5 Kto 0.1 K [P(ν = 1) . 0.4], meaning that the corresponding electron spin po-larization peak remains very weak. Figure (4.11) shows the significant effectof the temperature on the electron spin polarization peak 15. At T = 4.2 Kthe electron spin polarization peak is completely lost 16. In OPNMR exper-

during the time interval t/2 will be refocused by the application of the π pulse and theecho intensity would be drastically reduced. By measuring T−1

2 , let’s say, as a functionof T , it is therefore possible to address the microscopic motion of Skyrmions. It is worthpointing out that ”frozen” Skyrmions might be too strongly perturbed by π/2 pulses.To observe the dynamics of Skyrmions then one should devise modified pulse sequencesand/or use small ”tip-angle” RF pulses.

15For T & 1.5 K, note that the amplitude of the electron spin polarization peak is smalland comparable to the measurement inaccuracy. In principle, the electron-hole symmetryin the lowest Landau level is reflected in a symmetric shape of the electron spin polarizationpeak [7, 103]. Given the experimental uncertainty, this issue cannot be addressed by thepresent experiments.

16The very pronounced T -dependence of the electron spin polarization peak is similar tothe behavior observed in magneto-absorption experiments on highest-quality single-QWheterojunctions [see Ref. [73] and Fig. 3 (sample B) in M.J. Manfra, B.B. Goldberg, L.N.Pfeiffer, and K.W. West, Optical Determination of the Spin Polarization of a QuantumHall Ferromagnet, Physica E 1, 28 (1997)].

4.4. RESULTS AND DISCUSSION 175

iments on comparable samples (in our case g = 0.014 at ν = 1 and θ = 0),the electron spin polarization peak is clearly visible already at T ≈ 4 K [7, 56]and at T = 1.5 K it reaches nearly full polarization with its flat ±4% ν-widetop [Fig. (4.10)(b)]. We note that at ν ≈ 1.2 and T = 1.5 K both NMR andOPNMR techniques find P ≈ 0.20− 0.25 [Fig. (4.10)(b)], i.e., considerablyless than unity. Finally, an interesting feature of the P(T, ν) data shownin Fig. (4.11) is its strong T -dependence observed at ν = 3/4 compared tovery small variations measured at ν = 0.9.

Figure 4.8: 71Ga KS (left axis) and P (right axis) vs B at θ = 0. Thecorresponding ν is given on the top axis. Knight shift data were collectedat T = 1.5 K on 71Ga () and on 69Ga (•), which is renormalized to 71GaKS by the ratio 71γ/69γ = 1.27. Data points at T = 0.1 K are denoted byfilled triangles (N). The QHE state at ν = 1 corresponds to B = 5.7 T.


Figure 4.9: P (right axis) and 71Ga KS (left axis) vs T . (a) Open diamondsdenote P(T ) at B ≈ 14.8 T and θ = 40 (ν = 1/2). The solid curve is asingle-parameter fit to the data (see text). (b) Open circles denote P(T ) atB ≈ 17 T and θ = 0 (ν = 1/3). The dashed curve is a two-parameter fitto the data. Open squares denote P(T ) at B ≈ 5.7 T and θ = 0 (ν = 1).


Figure 4.10: (a) KS vs T at ν = 1 in NMR () and OPNMR () experi-ments [7]. OPNMR data is taken at θ = 28.5 and B = 7.05 T. (b) NMR(N) vs OPNMR () KS(ν) results at T = 1.5 K. OPNMR experiments ob-tain KS(ν) by tilting the sample in a fixed magnetic field B = 7.05 T. AllNMR data is taken at θ = 0. Curves are calculations explained in the text.


Figure 4.11: 71Ga KS (left axis) and P (right axis) vs B at θ = 0 andT = 1.5 K (M), 2.2 K (), and 4.2 K (O). The corresponding ν is given onthe top axis. A closed triangle (N) is used for the 0.1 K data point at ν = 1.

P(T ) at ν = 1/2 and ν = 1/3

We pause here to briefly discuss our measured P(T ) at ν = 1/2 andν = 1/3 [Fig. (4.9)]. As a first remark we mention that, due to the ex-perimental uncertainty 17, in the vicinity of ν = 1/3, the low-T limit of P

17Note that the data points at low T [Fig. (4.9)(b)] have a larger and asymmetric errorbar, showing possible greater consistency with KP=1

S = 24 kHz. In our experiments, thereare two factors which introduce errors near ν = 1/3 and underestimate KS at low T . Whilethe broadening of the QWs’ resonance line is responsible for the larger uncertainty, theasymmetry in the error bar is due to the excessively long tR necessary for the observation


seems to be . 10% lower than the saturation value observed at ν = 1/2[Fig. (4.9)]. The T -dependence of P at ν = 1/3 is shown by open circlesin Fig. (4.9)(b). As expected, we observe that P(T → 0; ν = 1/3) → 1.Although there is a significant scatter at low-T [78], P(T ; ν = 1/3) agreeswell with OPNMR results [55, 64, 78]. The raw Knight shift data is fitted toKsat

S (ν = 1/3) tanh(∆1/3/4kBT ), yielding KsatS (ν = 1/3) = 20± 2 kHz and

∆1/3 = 1.6EZ , in agreement with the OPNMR result ∆1/3 = 1.82EZ [55, 64]and the theoretical estimate ∆1/3 ≈ 2EZ [71]. Note that, according to the-ory, the contribution of single-particle excitations to P dominates at low-Tnear ν = 1/3 [71], in contrast to ν = 1 where the collective (spin-wave)modes are expected to control P at finite T [54].

In Fig. (4.9)(a) we present the measured T -dependence of the electronspin polarization at ν = 1/2 in our sample. We note that P reaches the fullpolarization value as T → 0, implying that the ground state of the 2DES isfully spin polarized at filling factor one half. Near half-integer filling factors(ν = 1/2, 1/4, . . . ), a large body of experimental and theoretical results canbe cast in a surprisingly simple picture of non-interacting composite fermions(CFs) [29, 51] with the same charge and spin as electrons 18. An importantissue in the physics of CFs is the spin polarization of the 2DES at half-integer filling factors. In an effort to understand the spin polarization of the2DES close to ν = 1/2, Park and Jain [85] introduced a phenomenologicalparameter, the CF polarization mass m∗

p, which is proportional to the ratioof the cyclotron and Coulomb energies. These authors obtained an estimatefor m∗

p at ν = 1/2

m∗p/me

∼= 0.60√B⊥ [T], (4.33)

where me is the electron’s mass in vacuum. The parameter m∗p, combined

with a parabolic dispersion law for CFs at ν = 1/2, uniquely determines theelectron spin polarization at any given T and B. Quite remarkably, our T -dependence of P at ν = 1/2 supports the non-interacting CF picture. Let’sassume that CFs have a g-factor roughly the same as electrons and considerparabolic bands occupied by n CFs with mass m∗

p. Hence, the density ofstates D∗

±(E) (for spin-up and spin-down CFs) is D∗±(E) = D∗ϑ(E±EZ/2),

where D∗ = m∗p/(2π~2), and ϑ is the step function. Making use of Fermi-

of the saturated peak position of the barriers.18In the CF model, the Coulomb interaction of one electron with all others is replaced

with a Chern-Simons gauge field, equivalent to attaching an even number (2m) of fluxquanta to each electron. In the mean field approximation the gauge field equals theexternal magnetic field at ν = 1/(2m) and the system of interacting electrons is replacedby one of non-interacting CFs in zero average magnetic field.


Dirac distribution we find

P(T,B) =2D∗kBT

n

EZ

2kBT− tanh−1

(1 +

exp( nD∗kBT )

sinh2( EZ2kBT )

)− 12

. (4.34)

Depending on the strength of the magnetic field, this model predicts eitherpartially or completely polarized Fermi sea of CFs in the T → 0 limit,i.e., P(T = 0) = minD∗EZ/n, 1. Taking m∗

p as a fitting parameter, inFig. (4.9)(a) we show the best fit of Eq. (4.34) to the data by the solidcurve. This curve indeed provide a reasonable description of the data atlow temperatures (T . 2 K), which are of experimental relevance for thephysics of CFs. At higher T , the measured P falls below the calculatedP(T,B). The deduced m∗

p = 1.7me value is found to be in good agreementwith the polarization mass predicted by Eq. (4.33): m∗

p/me = 2.0 at B =B⊥ = 11.4 T.

The nuclear spin-lattice relaxation rate

Figure (4.12)(a) presents the ν-dependence of T−11 measured at T = 1.5 K.

At this temperature essentially no variation is found over the investigatedν-range, with the exception of ν = 1/2 and ν = 1/3 points where T−1

1 issignificantly smaller. In particular, the value at ν = 1 is the same as valuesnearby, i.e., we observe no decrease of the nuclear relaxation rate. This isin contrast with OPNMR results [Fig. (4.12)(b) and Appendix B] reportingstrong reduction of T−1

1 (for 1.5 K . T . 4.2 K) in the vicinity of ν = 1,which is indeed expected as a signature of a gapped QHE state at ν = 1.

On the other hand, the NMR value of T−11 at ν = 1 is in good agreement

with our heat capacity experiments performed on a sample from the samewafer [9]. These measurements revealed that up to 70% of QWs nuclei havelarge T−1

1 as they contribute to the measured heat capacity within the ex-perimental time scale. Furthermore, the T−1

1 (ν) found in these calorimetricmeasurements [Fig. (3.21)] corroborates with our NMR data. This behaviorcould be accounted for by sample inhomogeneities and disorder which in-troduce localized states in the gap and ”compressible islands” in the 2DES,leading to additional relaxation channels for the QWs nuclei 19. The areaoccupied by the compressible regions can be estimated by comparing the

19Since T−11 is expected to be very different between compressible and incompressible

parts of the sample, standard NMR observes first the nuclei with large T−11 , located near

the compressible regions.


Figure 4.12: (a) 71Ga T−11 vs B at T = 1.5 K and θ = 0. The corresponding

ν is given on the top axis. The uncertainty in our measured T−11 is±20%. (b)

NMR (•) vs OPNMR () T−11 (ν) results near ν = 1. OPNMR experiments

obtained T−11 (ν) by tilting the sample in a fixed magnetic field B = 7.05 T.


jump in the electron orbital magnetization in the integer and the fractionalQHE regime [112]. This yields an upper bound of only 10% of the totalarea of the sample, however, by the mechanism of nuclear spin-diffusion,the percentage of nuclei affected by rapid relaxation may be much larger.In any case, we have searched on purpose at long recovery times tR for theNMR signal arising from slow-relaxing nuclei in incompressible regions nearthe predicted P = 1 position. Nothing was detected; if some slow relaxingNMR signal existed, it must involve a very small amount of the observednuclei. In our mind, this apparent discrepancy could indicate the presenceof a finite, ν-dependent density of magnetic states in the gap which provideefficient relaxation channels for the QWs nuclei. It is at present impossibleto know the origin and the nature of these states: they could be extrinsicas magnetic impurities, or intrinsic as formation of local moments or moreexotic states in the disorder potential.

Regarding the comparison of OPNMR [103] and NMR T−11 data, we

also remark that both techniques are in reasonable agreement away fromν = 1 [Fig. (4.12)(b)]. Recall that within the disorder-free description ofthe 2DES, the increase of T−1

1 away from the ν = 1 is attributed to theSkyrmion-mediated nuclear-spin lattice relaxation. For example, at T =1.5 K and ν ≈ 0.85 one has T−1

1 (OPNMR) ≈ 0.05 ± 0.01 s−1 [103] andT−1

1 (NMR) = 0.076±0.015 s−1. Moreover, at ν ≈ 0.9 both techniques revealan approximately T -independent T−1

1 between T = 1.5 K and T = 4.2 K(not shown here).

Discussion

We begin the examination of the θ = 0 data by recalling that, accordingto previous theoretical and experimental work [7, 14], the electron spin po-larization peak at ν = 1 [Fig. (4.10)(b)] signals the presence of Skyrmionsin the QHE ground state. Theoretical modelling of the electron spin polar-ization around ν = 1 [solid curve in Fig. (4.10)(b)], by Brey et al. [14], hasfound convincing agreement with OPNMR results. Our NMR data agreewith this numerical simulation for fillings away from ν = 1 (i.e., ν & 1.2and ν . 0.8), but disagree at ν ≈ 1. The etiology of this discrepancy isdiscussed in detail below.

Going a step further, one of the crux findings of OPNMR experimentsis the T -dependence of P at ν = 1 [Fig. (4.10)(a)], which received outstand-ing theoretical justification. Recent work by Song et al. [96], performed bya similar OPNMR technique, has uncovered a similar T -dependence of Pat ν = 1. As small variations in ν have a dramatic effect on P, the most


prosaic reason for small discrepancies between theory and (OPNMR) exper-imental data is the difficulty of precisely locating ν = 1 in the investigatedsamples [53]. Even admitting this exceptional circumstance, none of the ex-isting theoretical results can be trivially reconciled with our measured P(T )at ν = 1 in the T → 0 limit [open squares in Figs. (4.9)(b) and 4.10(a)].In other words, while full polarization of the 2DES is always predicted atlow-T , our measured P(T ) at ν = 1 falls well below the limit imposed byP(T ) for localized, non-interacting electrons [dashed line in Fig. (4.10)(a)].Clearly, a subtle physical mechanism is at work here. It is a significantpoint that, at high-T (T & 6 K), our P(T ) data corroborates with previousOPNMR P(T ) measurements (and theory).

Let’s turn now our attention to T−11 measurements. Our data [closed

circles in Fig. (4.12)] manifestly disagree with the OPNMR results aroundν = 1 [Fig. (4.12)(b)]. Moreover, since T−1

1 → 0 as T → 0 at ν = 1/3but does not at ν = 1, our nuclear spin-lattice relaxation rate data seemto contradict magneto-transport observations, which find a large QHE ex-citation gap at ν = 1 compared to that at ν = 1/3. Nevertheless, at leastat a qualitative level, one might understand this apparent inconsistency. Itis quite simple to imagine how sample’s inhomogeneities tend to smear theminimum in T−1

1 (ν) at ν = 1. Physically, it is plausible to consider that theT−1

1 at nominal filling factor ν = 1 in an inhomogeneous 2DES could be con-structed from averaging over the T−1

1 contributions at ν = 1±ε. Recountingthat T−1

1 increases linearly with |1− ν| due to the presence of Skyrmions, itis easy to see that the ν-averaged T−1

1 will have essentially no dependenceon the filling factor around ν = 1 and a fast nuclear spin-lattice relaxationis observed down to very low T . At ν = 1/3 the variations in the electrondensity will leave T−1

1 essentially unchanged because the elementary exci-tations are not Skyrmion-like and there is no reason for an enhancement ofT−1

1 near ν = 1/3. Consequently, a slowing down of the nuclear spin-latticerelaxation is observed as T → 0 at nominally filling factor ν = 1/3.

Let’s come back at P(T, ν) measurements. It may appear at the firstsight that sample’s inhomogeneities are a good candidate for explaining ourpuzzling observations at ν = 1. It has been suggested recently [82] that thepresence of disorder modifies the shape of the polarization peak at ν = 1.These results [82] have been obtained in the particular case of weak, micro-scopic disorder but naturally lead to a more general idea of inhomogeneities,short-range and/or long-range, governing both the magnitude and shape ofthe spin polarization peak at ν = 1. Motivated by the NMR and the heatcapacity T−1

1 results, we already mentioned the possibility of long-rangeelectron density inhomogeneities. Indeed, to minimize the contamination


during the MBE growth the wafer was not rotated, and our measured speci-mens present some electron density gradients. A conservative estimate givesthe maximum density variation of ±5% over the entire sample surface. Werecall here that the peak in P(ν) observed in OPNMR is a quite robustcharacteristic, i.e., it is rather wide around ν = 1, covering more than 5% at1.5 K. Thus, even if we admit that our NMR sample contains a span of 10%in the electron density, significant percentage of QW nuclei should observehigh electron spin polarization and should have a small T−1

1 . We shouldadd that the same conclusion is reached for any kind of static microscopicelectron density inhomogeneity of the same order of magnitude 20.

One might object that nuclei in the high-P domains are particularlydifficult to observe by NMR using saturation-recovery method, because oftheir very slow relaxation rate. This is true only at low-T ; at T ≈ 4.2 K,both in high-P and in low-P domains, nuclei are relaxing fast enough tobe easily observable. Therefore, at high-T , both NMR and OPNMR areexpected to give similar results on similar samples. However, in contrast tothe OPNMR results, no electron spin polarization peak is observed by NMRat T = 4.2 K [Fig. (4.11)], and T−1

1 (NMR) is one order of magnitude largerthan T−1

1 (OPNMR) [Fig. (4.10)(b)].A possible mechanism capable of diminishing the P(ν = 1) polarization

peak and hiding slow relaxation by some sort of average over ν might be adynamical (and necessarily short-range) inhomogeneity. That is, the systemis composed of spin-textured domains, whose ν-distribution moves rapidlyon the time scale of NMR data acquisition (i.e., faster than 10 µs). Whyand how should such hypothetical situation be realized in our NMR caseand be absent in OPNMR experiments is not clear to us.

The discrepancy between the OPNMR and NMR results at ν = 1 mayalso be induced by the differences between the two experimental techniques.One cannot rule out the possibility that the electron spin polarization peakobserved in OPNMR experiments might be affected by the optical pump-ing process. Hyperfine interaction with oriented nuclei changes the effectiveelectron Zeeman splitting; this Overhauser shift can significantly alter the

20The presence of significant static microscopic disorder can be readily seen in magneto-transport measurements: for example, the QHE excitations gaps would be markedly re-duced. The measured QHE excitation gaps on samples from the same wafer are compara-ble to those measured in high-quality conventional single-layer 2DESs, clearly discountingthe presence of strong, random disorder. One fundamental difference between NMR andmagneto-transport measurements should be mentioned here. Transport QHE excitationgaps are known to be sensitive to the percolation structure of the incompressible regionthrough the sample. While in transport experiments the compressible part of the sampleis ”quiet”, in NMR (which is a thermodynamic measurement) it does matter.


magnitude of the effective electron Zeeman energy (see Fig. 6 in Ref. [63])and it decays with a very long time constant after the optical pumping pro-cess is stopped [107]. Therefore, in OPNMR measurements, 2D electronsmight experience an ”artificial” g, leading to a larger electron spin polariza-tion. Finally, we note that a study of the P(ν = 1) peak as a function ofthe g factor has been performed using polarized magneto-reflectivity 21. Inthese experiments, for QWs where |g| < 0.4, P(ν = 1) saturates at low Tat values which are well below 100%, suggesting a strong spin mixing of thetwo spin sublevels of the zeroth LL at small values of EZ .

4.4.2 Tilted-magnetic field electron spin polarization

As aforementioned, the determination of the critical value gc of the Zeemanto Coulomb energy ratio g = EZ/EC above which Skyrmions are convertedto single spin flips is an important test for the Skyrmion picture near ν = 1.We have used the tilted-magnetic field technique to tune g and examine theeffect of an increasing Zeeman energy on the ν = 1 electron spin polarizationpeak. Figure (4.13)(a) includes results for P at three different total magneticfield values, measured by varying θ. In this experiment, for each B, the firstdata point is measured at θ = 0 while the largest investigated tilt anglecorresponds to ν = 2. To emphasize the behavior of P around ν = 1, thedata are displayed in Fig. (4.13)(b) as a function of ν. In this figure we alsoincluded the data taken at θ = 0 by varying B [from Fig. 4.8)], as wellas the numerically calculated P(ν) curve for the non-interacting electronmodel [96] [dashed curve in Fig. (4.13)(b)]. The main sources of uncertaintyin ν are geometrical errors which arise from the angle measurement ±1.

We first focus on the two nearly identical P(ν) curves measured at B =17 T and 14.8 T. The electron spin polarization drops to zero when ν → 2,as expected in the independent electron model, and closely mimics the non-interacting electron picture for 1.2 . ν . 2 [dashed line in Fig. (4.13)(b)].The key feature of these two curves is the absence of P(ν ≈ 1) peak whichis replaced by a monotonic ν-dependence. In contrast to these data, thepolarization peak at ν = 1 is clearly seen in the B = 9 T data (whereg = 0.022), as well as in already presented θ = 0 data (where g = 0.014).

Regarding the low ν-range of our T = 1.5 K data, we mention that

21V. Zhitomirsky, R. Chughtai, R.J. Nicholas, and M. Henini, Spin Polarization of 2DElectrons in the Quantum Hall Ferromagnet: Evidence for a Partially Polarized Statearound Filling Factor One, in Workbook of the 14th International Conference on theElectronic Properties of Two-Dimensional Systems, (Prague, July 30 - August 3, 2001),p. 17.


Figure 4.13: (a) Tilt angle dependence of 71Ga KS (left axis) and P (rightaxis) measured at T = 1.5 K and B = 9 T (3), 14.8 T (), and 17 T ().(b) Data points shown in panel (a) are displayed here as a function of ν.Open and filled triangles denote θ = 0 data points [Fig. (4.8)]. The dashedcurve is P(ν) at T = 0 for the independent-electron model (see text).


because of different T -dependencies at various filling factors 1/3 . ν . 2/3,P(ν) curves exhibit an apparent minimum between ν = 2/3 and ν = 1/3.In fact, from our studies [78] we know that the 2DES at ν = 1/2 is fullypolarized in the low-T limit [see in Fig. (4.13)(b) the T = 0.1 K data pointmeasured at θ = 0].

Before discussing the tilted-magnetic field results, we make two impor-tant remarks about the data presented in Fig. (4.13). First, as in the presentstudy tilt angles up to 80 were used, we warn the reader that strong tiltof the sample with respect to the magnetic field could induce significantmodifications of the electronic wave function, which in turn may influencethe lowest-energy excitations of the 2DES [see for details Sec. (1.6.4)]. Thiseffect is not included in our analysis. Secondly, a crossing between the firstspin-down Landau level and the second spin-up Landau level is predicted 22

to occur in our sample for θ & 74 [see for details Sec. (1.6.4)]. For B = 17 T,the highest investigated B, θ & 74 corresponds to ν & 1.2. Thus, this as-pect does not concern the discussion of spin properties of the ν = 1 QHEstate. Furthermore, in our experimental P(ν) data, up to B = 17 T, we didnot find signatures of a Landau level crossing. It is the overall agreementbetween the experimental data measured at B = 17 T and the theoreticallypredicted P(ν) for a non-interacting 2DES, which first deserves attention.The 2D electron spin polarization at ν = 2 is essentially zero at all B-valuesup to the highest investigated tilt angle, and at B = 17 T P(ν) is approach-ing this point along the prediction for the free electrons. At ν = 1 andT = 1.5 K, in the presence of total magnetic field B = 17 T, we actu-ally find the 2DES polarized only to ≈ 70%. However, it is clear that highmagnetic fields and low temperatures “push” the observed P(ν) dependencetowards the independent electron picture in the full ν-range [Fig. 4.13(b)].In particular, this is corroborated by T = 0.1 K data points at ν = 1/3 and1/2, given by filled symbols.

We next discuss the effect of large Zeeman energy on the 2D electron spinpolarization around ν = 1. For g ≈ 0.022 [open diamonds in Fig. 4.13(b)],P(ν) clearly displays a maximum at ν = 1, which we associated to the pres-ence of Skyrmions in the electronic ground state. This is also suggested byour very low-T tilted-field heat capacity studies on a sample from the samewafer. In these experiments, the nuclear contribution of Ga and As atomsto the measured heat capacity near ν = 1, up to the highest investigated tiltangle θ ≈ 53 (g ≈ 0.023), indicates that the nuclear spin-lattice relaxationoccurs via Skyrmions. Disappearance of the P(ν ≈ 1) peak in NMR data

22T. Jungwirth and A.H. MacDonald (private communication).


taken in magnetic fields of 14.8 T or higher is an experimental evidencefor a Skyrmion to single spin-flip transition near gc ≈ 0.037. The data aresomewhat affected by the finite temperature at which the experiment wasperformed and the range of investigated, fixed B is not fine enough to conveya precise value for gc. Nevertheless, given the fact that, in our experiments,P near ν = 1 slowly approaches the low-T saturation value when T is low-ered below ≈ 1.5 K, we believe that gc ≈ 0.037 is a reasonable upper boundestimate for the existence of Skyrmions near ν . 1. Note that numericalHartree-Fock calculations for the present sample, taking into account onlythe finite-thickness of the 2D electron layers, predict gc = 0.046 in reasonableagreement with the experiment [23]. Based on calculations which incorpo-rate both finite-width and Landau-level mixing corrections [60], we expect afurther reduction of gc, improving the agreement between the experimentaldata and theory.

4.5 Conclusion

A primary objective of this work was to obtain information about theν = 1 QHE state from a standard NMR measurement in a multiple-QW,GaAs/Al0.3Ga0.7As sample. Present 2D electron spin polarization data sup-port the Skyrmion picture around ν = 1 at low magnetic field, and providean upper bound estimate of the critical Zeeman energy (0.037 in units ofCoulomb energy) for the existence of Skyrmions. This value is in goodagreement with both theory and previous heat capacity measurements. Inthe high magnetic field and low temperature limit, the observed P(ν) ap-proaches the predicted P(ν) for non-interacting 2D electrons over the fullrange of Landau level filling factors 1/3 . ν . 2. Our data at θ = 0 alsoreveal that at ν = 1/2 and ν = 1/3 the 2DES is fully polarized as T → 0.

We also discuss the observed difference between the optically pumpedNMR data [7] and our NMR measurements. At ν = 1, we observe bystandard NMR a smearing out of the Skyrmion polarization peak, whichis complete at T = 4.2 K, and no reduction of the nuclear spin-lattice re-laxation rate at low temperatures. The possibility that this is due to theinhomogeneity of the electron density is analyzed, and we conclude that ob-served features could be accounted for only if the electron density patternis not static. Even if this hypothesis were true, it remains unclear why itis absent from low-T OPNMR data. The discrepancy between our resultsand those obtained by optically pumped NMR might also be due to theexperimental technique.

Epilogue

Şi am plans mult, fiindca nimeni n-a fost gasit vrednic sa de-schida cartea, nici sa se uite în ea.

Apocalipsa Sf. Ioan Teologul

It is a truism that principal results of modern physics reach far beyondthe domain in which they are won. The theoretical idea of Skyrmions wasfirst advanced in particle physics. The new views with regard to quantumHall effect (QHE) ferromagnets and pseudomagnets 23, which have risenas a result of vigorous theoretical and experimental research on Skyrmions,revolutioned our deeply held ideas about ferromagnetism – an old condensedmatter physics paradigm. Yet nobody dares, perhaps, to prophesy that thereis no conceivable experience which can refute the reality of QHE Skyrmions.

Praising the fruits of this thesis is not a cheerful task. For me, the tri-umphs of this work, if any, were the triumphs of the faith. Gallant effortssometimes have only a puny outcome; my joy was ruined by countless neg-ative results of experimental enterprise. In the preceding chapters, in themain, I have confined my account to our positive results of research, andhave not dealt in detail with problems which are not yet completely clearedup. The impression might thus be given that the measurements presentedin this thesis left a definitive mark on the physics of QHE Skyrmions. Thatis by no means the case.

23The pseudospin degree of freedom is represented by discrete quantum numbers suchas the electronic spin or the Landau level index.

189

190 EPILOGUE

While several issues concerning QHE Skyrmions are now settled, therestill remain difficult problems, in particular, those related to the effect ofdisorder. The basic theoretical understanding of the role of disorder for theinteger QHE is rather well-established. Nevertheless, describing the complexbehavior in disordered two-dimensional (2D) systems of strongly correlatedelectrons has been a beckoning, yet elusive goal for the 2D physics com-munity. Both microscopic and macroscopic inhomogeneities, always presentin macroscale (centimeter) samples, are potentially serious sources of error.Inescapable, high ideals such as Skyrmions are besmirched by the existenceof inhomogeneities or swamped by the bulk effects. One might seek toresolve this in measurements of thermodynamic properties on conventional(single quantum well) samples. I believe that clever experiments on multiple-quantum-well (QW) samples, however, will continue to guide our ceaselessunderstanding of the QHE.

Additional research is needed to refine many of the topics addressed inthis thesis. In the following, I tentatively formulate a prospectus of futureexperimental work. I shall start with ac heat capacity experiments. In atleast one important aspect, the present ac calorimetric study remains in-complete. The usefulness of our method is revealed by the fact that theheat capacity of two-dimensional electron systems (2DESs) could be mea-sured over a large range of magnetic fields and tilt angles. Therefore, I claimthat tilted-magnetic field ac heat capacity measurements, through a detailedstudy of the shape of heat capacity oscillations in the integer QHE regime,would provide relevant information on the functional form of both Landaulevel broadening and spin-splitting. Our heat capacity experiments in theinteger QHE regime indicated a finite density of states between Landau lev-els. This aspect, inherent to the presence of disorder, should be exploredexperimentally more thoroughly.

Of all difficult moments of the Skyrmion crusading era, quasi-adiabaticheat capacity measurements were the most impressive. These measurementshad cost thousands liters of liquid helium, they had tried the patience ofseveral people, and they had taught me that one could shine only scrimplight on Skyrmion crystals. I hope that future heat capacity experimentswill elucidate the enigma of the Skyrmion liquid-to-solid phase transition.Based on our present knowledge, high-density multiple-QW heterostructuresare believed the most promising candidates for the observation of the heatcapacity peak at low temperatures. It should be noted that several im-pediments have left nuclear spin-lattice relaxation rate (T−1

1 ) calorimetricmeasurements with limited capacity to contribute new insights into the Skyr-mion physics. A consistent framework for this type of experiments currently

EPILOGUE 191

exists, and future developments in this direction are desirable.Another possible continuation of the present heat capacity work is the

study of two spatially separated 2DESs. The separation between layers canbe made small enough so that the inter-layer and intra-layer Coulomb in-teractions are comparable in strength, i.e., strong correlations between thelayers are bring about. Bilayer 2DESs display complex physical phenomena,which thus far have been revealed and studied only by means of standardmagneto-transport methods. It seems to be no reason to doubt that heatcapacity measurements on bilayer 2DESs will be among the most excitingfuture research perspectives. On the experimental side, effort in this direc-tion was already directed, by the growth of multiple-double-QW samples.On such a sample, we have discovered evidence that Skyrmions are presentin the electronic ground state at total Landau level filling factor νt = 2(ν = 1 in each layer), consistent with a fully polarized QHE state in whichthe spins in each layer are aligned parallel to the magnetic field. An in-teresting question is what happens around νt = 2 as the layer separationis reduced or the sample is tilted in the magnetic field. On the theoreticalside, it was shown that at νt = 2 a canted antiferromagnetic phase couldbe observed under realistically obtainable experimental conditions 24. Ad-ditionally, in strong magnetic fields, double-layer 2DESs can form at νt = 1an unusual broken-symmetry state with spontaneous interlayer phase co-herence. Whenever the system has spontaneous interlayer phase coherenceand is incompressible, its low-lying excitations are expected to be interestingobjects 25, called Merons. The Meron is essentially one-half of a Skyrmion.

The present standard nuclear magnetic resonance (NMR) experimentsfurnish a new test of the 2D electron spin polarization in the extreme quan-tum limit. We have focused our efforts on comparing the spin polarizationdata with existing theoretical models and with a series of experiments. Itis necessary to insist here that NMR results at ν = 1/2 support the non-interacting composite fermion picture. The most intriguing results of ourobservations remain the spin polarization and T−1

1 at ν = 1. The subtle in-terplay between electron-electron interactions and disorder in the fractionalQHE regime prompted us to consider in detail the implications of sample’sinhomogeneities on thermodynamic properties of the QHE ferromagnet at

24S. Das Sarma, S. Sachdev, and L. Zheng, Canted Antiferromagnetic and Spin-SingletQuantum Hall States in Double-Layer Systems, Phys. Rev. B 58, 4672 (1998).

25Moon K., H. Mori, K. Yang, S.M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka,and S.-C. Zhang, Spontaneous Interlayer Coherence in Double-Layer Quantum Hall EffectSystems: Charged Vortices and Kosterlitz-Thouless Phase Transitions, Phys. Rev. B 51,5138 (1995).

192 EPILOGUE

ν = 1, and ask which of various sources of disorder is responsible for theobserved behavior. We argued that our NMR data cannot be reconciledwith theory (and partially with optically pumped NMR and heat capacitymeasurements) without express consideration of short-range, dynamical in-homogeneities across the sample. Experimental work is now underway at theGrenoble High Magnetic Field Laboratory; NMR measurements in the frac-tional QHE were extended towards very low-T and new, interesting aspectsoccurred 26. Since I have said not much about NMR in the integer QHEregime, I should mention that measurements on sample M280 suggest theexistence of interesting spin phenomena in the high Landau levels (ν = 2 andν = 4). We probably need to fortify our understanding of the role of nuclearspins in the integer QHE regime with direct, NMR T−1

1 measurements.In Chapter 1, we have followed a common trend in the presentation

of transport coefficients, and displayed the longitudinal resistance resultson an arbitrary scale. Seldom is the definition of the actual resistivityused because this entails determining a geometric value for scaling the re-sistance measurement. Such geometric factors are particularly difficult toascertain for our investigated multiple-QW specimens. The subtle and crit-ical nature of the magnetotransport in 2DESs is beautifully reflected inhysteretic and anisotropic behaviors, spikes and nonlinearities in the longi-tudinal resistance, and metastable fractional QHE states. These effects areusually observed in measurements where the current flow is precisely con-trolled through lithographic definition of the sample geometry. It should benoted that some of these anomalies are expected to be present in our sam-ples. Systematic magneto-transport studies on lithographically patternedHall bridges will certainly help to brighten the existing tableau of spin orpseudospin spatial ordering at various filling factors such as ν = 4/3 orν = 2.

This thesis outlines a strategy to improve the links between differentlevels of experimental description. Our heat capacity, NMR, and magneto-transport experiments uncovered only a very small part of the world ofQHEs. These phenomena are rich enough in changing hues and patterns toallure us to explore them in all directions.

26N. Freytag, Y. Tokunaga, M. Horvatic, C. Berthier, M. Shayegan, and L.-P. Levy,Observation of a New Phase Transition between Fully and Partially Polarized QuantumHall States with Charge and Spin gaps at ν = 2/3, preprint, cond-mat/0105590.

Appendix A

The NSS in GaAs QWs andthe Schottky Heat Capacity

Consider the case of a sample containing nuclei of spin I and nuclear gy-romagnetic ratio γn. Let’s suppose that the sample is placed in a staticmagnetic field B. Then, the nuclear energy levels are equidistantly spacedby ∆n = ~γnB. Here γn/(2π) is given in units of MHz/T and B inunits of T. Alternatively, we can express ∆n in units of K as ∆n =α [K]γn/(2π)[T−1]B [T], where α = (h/kB)× 1 MHz = 4.8× 10−2 mK.

The structure of nuclear levels is reflected in the heat capacity in avery interesting way [41]. At ∆n kBT all levels will be well populated,whereas at ∆n kBT the upper levels will be scarcely occupied. For ∆n ≈kBT transitions between levels take place in appreciable amounts. Thisrapid change in the internal energy corresponds to large specific heat whichbecomes zero at both high and low temperatures. There is an intriguingpossibility of a hump (Schottky effect) in the measured C when ∆n ≈ kBT ,which in general will be superimposed on lattice and other contributions.

Let’s focus now on the nuclear spin system (NSS) of GaAs quantumwells (QWs) and quantitatively explore the Schottky effect. Electromagneticproperties of GaAs QWs’ nuclei are listed in Table (A.1). It is convenientto invoke the canonical partition function for a I = 3

2 single nucleus

Z =m=3/2∑

m=−3/2

exp(m∆n

kBT

), (A.1)

where m = 3/2, 1/2, −1/2, −3/2. Provided that m∆n/(kBT ) is a smallnumber, it is permissible to make a linear expansion of the Boltzmann ex-

193

194 Appendix A: Schottky Heat Capacity

Table A.1: Published nuclear data for 71Ga, 69Ga, and 75As: spin I,gyromagnetic factor γn/(2π) (in units of MHz/T), αγn/(2π) (in units of10−4 K/T), natural abundance (in %), and electric quadrupole momentQ (in units of 10−24 cm2).

Isotope I γn/2π αγn/2π Natural abundance Q

69Ga 3/2 10.218 4.90 60.4 0.17871Ga 3/2 12.982 6.23 39.6 0.11275As 3/2 7.2919 3.50 100 0.3

ponential. In this limit, the average energy of a single nucleus writes as

〈E〉 ≈ 14

m=3/2∑m=−3/2

m∆n

(1− m∆n

kBT

), (A.2)

and the Schottky nuclear specific heat per nucleus is given by

C0 =54γ2

n~2

kB

(B

T

)2

. (A.3)

For a single 69Ga atom we obtain 69C0 = 0.3963 ×10−29(B[T])2(T [K])−2 J/K. Averaging over the two Ga isotopes givesGaC0 = 1.25 × [69C0]. The specific heat of a single As atom can beexpressed as AsC0 = 0.51× [69C0]. Consequently, provided that ∆n kBT ,the Schottky nuclear heat capacity of Ga and As atoms in the QWs (CQW

n )is estimated at ([GaC0]+ [AsC0]) times the total number of GaAs molecules.

Appendix B

OPNMR Observations

Sensitivity poses a persistent challenge to nuclear magnetic resonance ex-periments in the quantum Hall effect (QHE) regime, so spectroscopists haveresorted to schemes such as optical pumping in order to enhance the measur-able signals. We summarize here some salient aspects of optically pumpednuclear magnetic resonance (OPNMR) measurements in the QHE regime,pioneered by Barrett and collaborators at Bell Laboratories and Yale Univer-sity [7, 55, 56, 64, 103]. We start with a cursory outline of the experimentaltechnique.

The samples used in OPNMR experiments are electron-doped multiple-quantum well (QW) GaAs/Al0.1Ga0.9As heterostructures. The NMR signalis increased by optical pumping as follows. Polarized electrons are excitedin the conduction band by illuminating the sample with polarized light andthe strong hyperfine coupling ensures the transfer of this polarization tothe nuclei. Since the incident photons have an energy tuned to the GaAsbandgap only the nuclei located in the GaAs QWs will be polarized by thisprocess 1. The bandgap of the barriers is larger than photons’ energy andhence, there are no photo-excited electrons in Al0.1Ga0.9As regions. This op-tical pumping strongly enhances the nuclear spin polarization (by as muchas a factor of 100), allowing NMR signals of the GaAs QWs to be inves-tigated by direct, radio-frequency detection. Unfortunately, together withthis advantage one has the disadvantage that OPNMR signals exhibit quitecomplicate dependences on the pumping wavelength, light polarization, andpower. Even though the initial excitation process is well understood, the

1Note that, due to the local nature of the electron-nucleus interaction, the QWs’ nuclearpolarization profile in OPNMR measurements has a similar profile to the electron chargedensity.

195

196 Appendix B: OPNMR Observations

recombination process, i.e., the relaxation of the electronic spins, is muchmore complicated and may have an effect on the nuclear spin system of GaAsQWs. Note also that in OPNMR experiments the two-dimensional electronsystem (2DES) is observed while nuclei are strongly polarized, well beyondtheir small equilibrium value. OPNMR measurements were carried out withthe timing sequence SAT-τL-τD-DET, where SAT represents a train of π/2-pulses which sets to zero the nuclear magnetization of the nuclear isotope ofinterest, τL is a period during which the sample is optically excited, τD is aperiod of no optical excitation, and DET represents the direct detection ofthe free induction decay.

Table B.1: 71Ga nuclear spin-lattice relaxation rate T−11 as a function of

temperature T and Landau level filling factor ν, as seen by the OPNMR(reproduced with permission after Ref. [7]). OPNMR measurements foundminima in T−1

1 at ν ≈ 1 and ν ≈ 2/3 indicating energy gaps for electronicexcitations in both integer and fractional QHE states. The T -independentT−1

1 at intermediate ν-values (2/3 . ν . 1) suggest a manifold of low-lyingelectronic states with mixed spin polarization.

ν = 1.01 ν = 0.88 ν = 0.66

T−11 (T = 4.2 K) [s−1] 0.008 0.04 0.023

T−11 (T = 2.1 K) [s−1] 0.0008 0.05 0.015

In Chapter 4, we guide the discussion of our NMR data on the basis ofthe following four OPNMR observations. First, OPNMR experiments haveprovided perspicuous evidence for the existence of finite-size Skyrmions nearLandau level filling factor ν = 1. More precisely, it was found in theseexperiments that the low-temperature (T ) electron spin polarization P(ν)drops precipitously on either side of ν = 1, which is evidence that the chargedexcitations of the ν = 1 QHE ground state invoke many spin reversals.OPNMR data revealed that the low-T electron spin polarization approachesthe full polarization (P = 1) at ν = 1, thereby providing additional credenceto the Skyrmion picture at ν = 1 [Fig. (4.12), panels (a) and (b)]. Of furtherinterest to us is the fact that the electron spin polarization peak at ν = 1 isreadily observed already at T = 4.2 K.

Secondly, the nuclear spin-lattice relaxation rate T−11 (T, ν) of the 71Ga

nuclei located in the GaAs QWs has been measured for 1.5 K . T . 4.2 K

Appendix B: OPNMR Observations 197

in the vicinity of ν = 1 [Fig. (4.12)(c)]. These OPNMR measurements[Table (B.1)] strongly support the Skyrmion-mediated nuclear spin-latticerelaxation as the underlying mechanism responsible for ν-dependence of T−1

1

near ν = 1 in disorder-free 2DESs.

The third observation merely records the fact that OPNMR experimentsstudied the electron spin polarization of the 2DES around ν = 1/3. For ex-ample, OPNMR measurements revealed that the T -saturated P(ν) drops oneither side of ν = 1/3. The observed depolarization is quite small, consis-tent with an average of 0.1 spin flips per quasihole (or quasiparticle). Moreimportant, the low-T (P = 1) limit of OPNMR Knight shift (KS) data atν = 1/3, measured for several samples, was successfully used to determinethe effective hyperfine coupling constant Ac = (4.5 ± 0.2) × 10−13 cm3/s,given by the relationship KS = AcPn/w. Here w is the QW width and n isthe electron density.

Finally, OPNMR experiments also addressed the dynamics of the 2DESby observing cross overs from the ”motionally narrowed” regime to the”frozen” regime 2. This evolution is reminiscent of the behavior seen inNMR studies where the motion of the nuclei freezes out [1]. In particular,OPNMR spectra provided evidence that spin-reversed charged excitations ofthe ν = 1/3 QHE ground state are localized at low temperatures (T . 0.5 K)over the NMR time scale of about 40 µs. Information about the electron dy-namics has been drawn from the T -dependence of the NMR line width. Thisfourth observation is summarized, for convenience, in Table (B.2). When the2DES crosses from the motionally-narrowed to the frozen regime, the widthof the QWs’ resonance increases dramatically. Qualitatively similar trendswere recently uncovered in low-T OPNMR measurements near ν = 1 [56].

2What are the ”motionally narrowed” and ”frozen” regimes? The NMR spectrum canhave different line shapes, depending on the dynamics of the 2DES. An inhomogeneouselectron spin polarization will tend to broaden the NMR spectrum. Let’s suppose that thedensity of the electronic system is uniform, but the sample is divided into two regions, withdifferent electron spin polarizations. If the dynamic time of these regions is much slowerthan the experimental NMR time scale, then we shall obtain a broadened NMR spectrum(eventually composed of two peaks). However, if the dynamic time of these regions is muchfaster than the experimental time scale, then the nuclei will feel the average polarizationof the two regions, and the NMR spectrum will tend to show a single (narrow) peak. Theformer is the so-called frozen regime and the latter is the motionally-narrowed regime.

198 Appendix B: OPNMR Observations

Table B.2: The percentage increase of the QWs’ line width near ν = 1/3observed in OPNMR experiments, relative to the value of 5.2 kHz measuredat T = 1.5 K and ν = 1/3 (reproduced with permission after Ref. [55]).

ν T [K]

1.5 0.9 0.7 0.5 0.3

0.33 0% 4% 4% 3% 5%

0.29 2% 12% 20% 36% 32%

0.27 12% 21% 45% 69% 53%

It is probably worth adding here that Sinova et al. [93] addressed thequestion of how the average P is related to the NMR line shape. In themotionally-narrowed regime, where P could be considered homogeneous, theposition of QWs’ line in the NMR spectrum (what is usually understoodby the Knight shift) could be considered as a direct measurement of P.However, in the frozen regime, the QWs’ peak in the NMR spectrum is nolonger a good measure of P. Instead, in order to avoid dynamic effects, oneshould measure the first momentum of the QWs’ line shape which will giveP up to a proportionality constant.

Bibliography

[1] Abragam A., Principles of Nuclear Magnetism, (Oxford UniversityPress, New York, 1961).

[2] Aifer E.H., B.B. Goldberg, and D.A. Broido, Evidence of SkyrmionExcitations about ν = 1 in n-Modulation-Doped Single Quantum Wellsby Interband Optical Transmission, Phys. Rev. Lett. 76, 680 (1996).

[3] Ando T. and Y. Uemura, Theory of Oscillatory g Factor in a MOSInversion Layer under Strong Magnetic Fields, J. Phys. Soc. Japan 37,1044 (1974).

[4] Andraka B. and Y. Takano, Simultaneous Measurements of Heat Ca-pacity and Spin-Lattice Relaxation Time in High Magnetic Field at LowTemperature, Rev. Sci. Instrum. 67, 4256 (1996).

[5] Antoniou D. and A.H. MacDonald, Nuclear-Spin Lattice Relaxation andSpin-Wave Collective Modes in a Disordered Two-Dimensional ElectronGas, Phys. Rev. B 43, 11686 (1991).

[6] Bachmann R., F.J. DiSalvo Jr., T.H. Geballe, R.L. Greene, R.E.Howard, C.N. King, H.C. Kirsch, K.N. Lee, R.E. Schwall, H.-U.Thomas, and R.B. Zubeck, Heat Capacity Measurements on Small Sam-ples at Low Temperatures, Rev. Sci. Instrum. 43, 205 (1972).

[7] Barrett S.E., G. Dabbagh, L.N. Pfeiffer, K.W. West, and R. Tycko,Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAsQuantum Wells near Landau Level Filling ν = 1, Phys. Rev. Lett. 74,5112 (1995).

[8] Bayot V., M.B. Santos, and M. Shayegan, Oscillations in the Ther-mal Conductivity of a GaAs/AlGaAs Heterostructure in the FractionalQuantum Hall Regime, Phys. Rev. B 46, 7240 (1992).

199

200 BIBLIOGRAPHY

[9] Bayot V., E. Grivei, S. Melinte, M.B. Santos, and M. Shayegan, GiantLow Temperature Heat Capacity of GaAs Quantum Wells near LandauLevel Filling ν = 1, Phys. Rev. Lett. 76, 4584 (1996).

[10] Bayot V., E. Grivei, J.-M. Beuken, S. Melinte, and M. Shayegan, Crit-ical Behavior of Nuclear-Spin Diffusion in GaAs/AlGaAs Heterostruc-tures near Landau Level Filling ν = 1, Phys. Rev. Lett. 79, 1718 (1997).

[11] Berg A., M. Dobers, R.R. Gerhardts, and K. v. Klitzing, Magneto-Quantum Oscillations of the Nuclear Spin-Lattice Relaxation near aTwo-Dimensional Electron Gas, Phys. Rev. Lett. 64, 2563 (1990).

[12] Blakemore J.S., Semiconducting and Other Major Properties of GaAs,J. Appl. Phys. 53, R123 (1982).

[13] Boebinger G.S., H.L. Stormer, D.C. Tsui, A.M. Chang, J.C.M. Hwang,A.Y. Cho, C.W. Tu, and G. Weimann, Activation Energies and Local-ization in the Fractional QHE, Phys. Rev. B 36, 7919 (1987).

[14] Brey L., H.A. Fertig, R. Cote, and A.H. MacDonald, Skyrme Crystalin a Two-Dimensional Electron Gas, Phys. Rev. Lett. 75, 2562 (1995).

[15] Carr H.Y. and E.M. Purcell, Effects of Diffusion on Free Precession inNMR Experiments, Phys. Rev. 94, 630 (1954).

[16] Chakraborty T. and F.C. Zhang, Role of the Reversed Spins in theCorrelated Ground State for the FQHE, Phys. Rev. B 29, 7032 (1984).

[17] Chakraborty T., P. Pietilainen, and F.C. Zhang, Elementary Exci-tations in the Fractional QHE and the Spin-Reversed Quasiparticles,Phys. Rev. Lett. 57, 130 (1986).

[18] Chakraborty T. and P. Pietilainen, The Quantum Hall Effects: Integraland Fractional, (Springer-Verlag, 1995), 2nd ed.

[19] Chakraborty T., P. Pietilainen, and R. Shankar, Spin Polarizationsat and about the Lowest Filled Landau Level, Europhys. Lett. 38, 141(1997).

[20] Cho H., J.B. Young, W. Kang, K.L. Campman, A.C. Gossard, M. Bich-ler, and W. Wegscheider, Hysteresis and Spin Transitions in the Frac-tional QHE, Phys. Rev. Lett. 81, 2522 (1998).

BIBLIOGRAPHY 201

[21] Clark R.G., S.R. Haynes, A.M. Suckling, J.R. Mallett, P.A. Wright, J.J.Harris, and C.T. Foxon, Spin Configurations and Quasiparticle Frac-tional Charge of Fractional QHE Ground States in the N = 0 LandauLevel, Phys. Rev. Lett. 62, 1536 (1989).

[22] Collan H.K., M. Krusius, and G.R. Pickett, Specific Heat of Antimonyand Bismuth between 0.03 and 0.8 K, Phys. Rev. B 1, 2888 (1970).

[23] Cooper N.R., Skyrmions in Quantum Hall Systems with Realistic ForceLaws, Phys. Rev. B 55, R1934 (1997).

[24] Cote R., A.H. MacDonald, L. Brey, H.A. Fertig, S.M. Girvin, andH.T.C. Stoof, Collective Excitations, NMR, and Phase Transitions inSkyrme Crystals, Phys. Rev. Lett. 78, 4825 (1997).

[25] Das Sarma S. and F. Stern, Single-Particle Relaxation Time versusScattering Time in an Impure Electron Gas, Phys. Rev. B 32, 8442(1985).

[26] Das Sarma S. and X.C. Xie, Strong-Magnetic Field DOS in WeaklyDisordered 2DESs, Phys. Rev. Lett. 61, 738 (1988).

[27] Das Sarma S. and A. Pinczuk [editors], Perspectives in Quantum HallEffects, (Wileys, New York, 1997).

[28] Dobers M., K. v. Klitzing, J. Schneider, G. Weimann, and K. Ploog,Electrical Detection of NMR in GaAs/AlGaAs Heterostructures, Phys.Rev. Lett. 61, 1650 (1988).

[29] Du R.R., A.S. Yeh, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, and K.W.West, Fractional QHE around ν = 3/2: Composite Fermions with aSpin, Phys. Rev. Lett. 75, 3926 (1995).

[30] Eisenstein J.P., H.L. Stormer, V. Narayanamurti, A.Y. Cho, A.C. Gos-sard, and C.W. Tu, Density of States and de Haas-van Alphen Effectin 2DESs, Phys. Rev. Lett. 55, 875 (1985).

[31] Eisenstein J.P., H.L. Stormer, L.N. Pfeiffer, and K.W. West, Evidencefor a Spin Transition in the ν = 2/3 Fractional QHE, Phys. Rev. B 41,7910 (1990).

[32] Engel L., S.W. Hwang, T. Sajoto, D.C. Tsui, and M. Shayegan, Frac-tional QHE at ν = 2/3 and ν = 3/5 in Tilted Magnetic Fields, Phys.Rev. B 45, 3418 (1992).

202 BIBLIOGRAPHY

[33] Eom J., H. Cho, W. Kang, K.L. Campman, A.C. Gossard, M. Bich-ler, and W. Wegscheider, Quantum Hall Ferromagnetism in a 2DES,Science 289, 2320 (2000).

[34] Fagaly R.L. and R.G. Bohn, A Modified Heat Pulse Method for Deter-mining Heat Capacities at Low Temperatures, Rev. Sci. Instrum. 48,1502 (1977).

[35] Fang F.F. and P.J. Stiles, Effects of a Tilted Magnetic Field on a 2DES,Phys. Rev. 174, 823 (1968).

[36] Fertig H.A., L. Brey, R. Cote, A.H. MacDonald, A. Karlhede, and S.L.Sondhi, Hartree-Fock Theory of Skyrmions in Quantum Hall Ferromag-nets, Phys. Rev. B 55, 10671 (1997).

[37] Fowler A.B., F.F. Fang, W.E. Howard, and P.J. Stiles, Magneto-oscillatory Conductance in Silicon Surfaces, Phys. Rev. Lett. 16, 901(1966).

[38] Fukushima S.M. and S.B.W. Roeder, Experimental Pulse NMR: A Nutsand Bolts Approach, (Addison-Wesley, Massachusetts, 1981), p. 380.

[39] Girvin S.M., in Topological Aspects of Low Dimensional Systems, Pro-ceedings of the Les Houches Summer School of Theoretical Physics,Session LXIX, 1998, edited by A. Comtet, T. Jolicoeur, S. Ouvry, andF. David, (Springer-Verlag and EDP Sciences, 1999), p. 53.

[40] Gmelin, E., Modern Low-Temperature Calorimetry, ThermochimicaActa 29, 1 (1979).

[41] Gopal E.S.R., Specific Heats at Low Temperatures, (Plenum Press, NewYork, 1966).

[42] Gornik E., R. Lassnig, G. Strasser, H.L. Stormer, A.C. Gossard,and W. Wiegmann, Specific Heat of Two-Dimensional Electrons inGaAs/AlGaAs Multilayers, Phys. Rev. Lett. 54, 1820 (1985).

[43] Green A.G., Disorder and the Quantum Hall Ferromagnet, Phys. Rev.B. 57, R9373 (1998).

[44] Green A.G., Quantum-Critical Dynamics of the Skyrmion Lattice,Phys. Rev. B 61, R16299 (2000).

BIBLIOGRAPHY 203

[45] Guerrier D.J. and R.T. Harley, Calibration of Strain vs NuclearQuadrupole Splitting in III-IV Quantum Wells, Appl. Phys. Lett. 70,1739 (1997).

[46] Hahn E.L., Spin Echoes, Phys. Rev. 80, 580 (1950).

[47] Halonen V., P. Pietilainen, and T. Chakraborty, Subband-Landau-LevelCoupling in the Fractional QHE in Tilted Magnetic Fields, Phys. Rev.B 41, 10202 (1990).

[48] Halperin B.I., Theory of the Quantum Hall Conductance, Helv. Phys.Acta 56, 75 (1983).

[49] Haug, R.J., K. von Klitzing, R.J. Nicholas, J.C. Maan, and G.Weimann, Fractional QHE in Tilted Magnetic Fields, Phys. Rev. B36, 4528 (1987).

[50] Haussmann R., Magnetization of Quantum Hall Systems, Phys. Rev. B56, 9684 (1997).

[51] Heinonen O. [editor], Composite Fermions, (World Scientific, Singa-pore, 1998).

[52] Iordanskii S.V., S.V. Meshkov, and I.D. Vagner, Nuclear-Spin Relax-ation and Spin Excitons in a Two-Dimensional Electron Gas, Phys.Rev. B 44, 6554 (1991).

[53] Kasner M. and A.H. MacDonald, Thermodynamics of Quantum HallFerromagnets, Phys. Rev. Lett. 76, 3206 (1996).

[54] Kasner M., J.J. Palacios, and A.H. MacDonald, Quasiparticle Proper-ties of Quantum Hall Ferromagnets, Phys. Rev. B 62, 2640 (2000).

[55] Khandelwal P., N.N. Kuzma, S.E. Barrett, L.N. Pfeiffer, and K.W.West, Optically Pumped NMR Measurements of the Electron Spin Po-larization in GaAs Quantum Wells near Landau Level Filling Factorν = 1/3, Phys. Rev. Lett. 81, 673 (1998).

[56] Khandelwal P., A.E. Dementyev, N.N. Kuzma, S.E. Barrett, L.N. Pfeif-fer, and K.W. West, Spectroscopic Evidence for the Localization of Skyr-mions near ν = 1 as T → 0, Phys. Rev. Lett. 86, 5353 (2001).

[57] Kim J.H., I.D. Vagner, and L. Xing, Phonon-Assisted Mechanism forQuantum Nuclear-Spin Relaxation, Phys. Rev. B 49, 16777 (1994).

204 BIBLIOGRAPHY

[58] Kivelson S., D.H. Lee, and S.C. Zhang, Global Phase Diagram in theQuantum Hall Effect, Phys. Rev. B 46, 2223 (1992).

[59] Kosevich A.M., B.A. Ivanov, and A.S. Kovalev, Magnetic Solitons,Phys. Rep. 194, 117 (1990).

[60] Kralik B., A.M. Rappe, and S.G. Louie, Variational Monte Carlo Cal-culation of the Spin Gap in the ν = 1 Quantum Hall Liquid, Phys. Rev.B 56, 4760 (1997).

[61] Kronmuller S., W. Dietsche, K. v. Klitzing, G. Denninger, W. Wegschei-der, and M. Bichler, New Type of Electron Nuclear-Spin Interactionfrom Resistively Detected NMR in the Fractional QHE Regime, Phys.Rev. Lett. 82, 4070 (1999).

[62] Kukushkin I.V., K. v. Klitzing, and K. Eberl, Spin Polarization of Two-Dimensional Electrons in Different Fractional States and around FillingFactor ν = 1, Phys. Rev. B 55, 10607 (1997).

[63] Kukushkin I.V., K. v. Klitzing, and K. Eberl, Enhancement of theSkyrmionic Excitations due to the Suppression of Zeeman Energy byOptical Orientation of Nuclear Spins, Phys. Rev. B 60, 2554 (1999).

[64] Kuzma N.N., P. Khandelwal, S.E. Barrett, L.N. Pfeiffer, and K.W.West, Ultraslow Electron Spin Dynamics in GaAs Quantum WellsProbed by Optically Pumped NMR, Science 281, 686 (1998).

[65] Lanzillotto A.-M., M.B. Santos, and M. Shayegan, Silicon Migrationduring the MBE of δ-Doped GaAs and Al0.25Ga0.75As, J. Vac. Sci.Technol. A 8, 2009 (1990).

[66] Laughlin R.B., Quantized Hall Conductivity in Two Dimensions, Phys.Rev. B 23, 5632 (1981).

[67] Laughlin R.B., Anomalous Quantum Hall Effect: An IncompressibleQuantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett.50, 1395 (1983).

[68] Li Q., X.C. Xie, and S. Das Sarma, Calculated Heat Capacity and Mag-netization of 2DESs, Phys. Rev. B 40, 1381 (1989).

[69] Lounasmaa O.V., Experimental Principles and Methods below 1 K,(Academic Press, London, 1974), p. 277.

BIBLIOGRAPHY 205

[70] MacDonald A.H., H.C.A. Oji, and K.L. Liu, Thermodynamic Propertiesof an Interacting Two-Dimensional Electron Gas in a Strong MagneticField, Phys. Rev. B 34, 2681 (1986).

[71] MacDonald A.H. and J.J. Palacios, Magnons and Skyrmions in Frac-tional Hall Ferromagnets, Phys. Rev. B 58, R10171 (1998).

[72] Malinowski A. and R.T. Harley, Anisotropy of the Electron g Factor inLattice-Matched and Strained-Layer III-IV Quantum Wells, Phys. Rev.B 62, 2051 (2000).

[73] Manfra M.J., E.H. Aifer, B.B. Goldberg, D.A. Broido, L.N. Pfeiffer,and K.W. West, Temperature Dependence of the Spin Polarization of aQuantum Hall Ferromagnet, Phys. Rev. B 54, R17327 (1996).

[74] Manoharan H.C., New Particles and Phases in Reduced-DimensionalSystems, Ph.D. Thesis, Princeton University (1998).

[75] Maude D.K., M. Potemski, J.C. Portal, H. Henini, L. Eaves, G. Hill,and M.A. Pate, Spin Excitations of a Two-Dimensional Electron Gasin the Limit of Vanishing Lande g Factor, Phys. Rev. Lett. 77, 4604(1996).

[76] Melik-Alaverdian V., N.E. Bonesteel, and G. Ortiz, Skyrmions Physicsbeyond the Lowest Landau-Level Approximation, Phys. Rev. B 60,R8501 (1999).

[77] Melinte S., E. Grivei, V. Bayot, and M. Shayegan, Heat Capacity Evi-dence for the Suppression of Skyrmions at Large Zeeman Energy, Phys.Rev. Lett. 82, 2764 (1999).

[78] Melinte S., N. Freytag, M. Horvatic, C. Berthier, L.-P. Levy, V. Bayot,and M. Shayegan, NMR Determination of Two-Dimensional ElectronSpin Polarization at ν = 1/2, Phys. Rev. Lett. 84, 354 (2000).

[79] Melinte S., N. Freytag, M. Horvatic, C. Berthier, L.-P. Levy, V. Bayot,and M. Shayegan, Spin Polarization of Two-Dimensional Electrons inGaAs Quantum Wells around Landau Level Filling ν = 1 from NMRMeasurements of Gallium Nuclei, Phys. Rev. B 64, 085327 (2001).

[80] Moon K., H. Mori, K. Yang, S.M. Girvin, A. H. MacDonald, L. Zheng,D. Yoshioka, and S.-C. Zhang, Spontaneous Interlayer Coherence inDouble-Layer Quantum Hall Effect Systems: Charged Vortices andKosterlitz-Thouless Phase Transitions, Phys. Rev. B 51, 5138 (1995).

206 BIBLIOGRAPHY

[81] Morin F.J. and J.P. Maita, Specific Heats of Transition Metal Super-conductors, Phys. Rev. 129, 1115 (1963).

[82] Nederveen A.J. and Yuli V. Nazarov, Skyrmions in Disordered Het-erostructures, Phys. Rev. Lett. 82, 406 (1999).

[83] Nicholas R.J., R.J. Haug, K. von Klitzing, and G. Weimann, ExchangeEnhancement of the Spin Splitting in a GaAs/AlGaAs Heterojunction,Phys. Rev. B 37, 1294 (1988).

[84] Paget D., Optical Detection in High-Purity GaAs: Direct Study of theRelaxation of Nuclei close to Shallow Donors, Phys. Rev. B 25, 4444(1982).

[85] Park K. and J.K. Jain, Phase Diagram of the Spin Polarization ofComposite Fermions and a New Effective Mass, Phys. Rev. Lett. 80,4237 (1998).

[86] Prange R.E. and S.M. Girvin [editors], The Quantum Hall Effect,(Springer-Verlag, New York, 1987).

[87] Read N. and S. Sachdev, Continuum Quantum Ferromagnets at FiniteTemperature and the QHE, Phys. Rev. Lett. 75, 3509 (1995).

[88] Richardson S.L., M.L. Cohen, S.G. Louie, and J.R. Chelikowsky, Elec-tron Charge Densities at Conduction-Band Edges of Semiconductors,Phys. Rev. B 33, 1177 (1986).

[89] Schmeller A., J.P. Eisenstein, L.N. Pfeiffer, and K.W. West, Evidencefor Skyrmions and Single Spin Flips in the Integer QHE, Phys. Rev.Lett. 75, 4290 (1995).

[90] Shayegan M., J.K. Wang, M.B. Santos, T. Sajoto, and B.B. Goldberg,Fractional QHE in a High-Mobility GaAs/AlGaAs Multiple-Quantum-Well Heterostructure, Appl. Phys. Lett. 54, 27 (1989).

[91] Shepherd J.P., Analysis of the Lumped τ2 Effect in RelaxationCalorimetry, Rev. Sci. Instrum. 56, 273 (1985).

[92] Shukla S.P., M. Shayegan, S.R. Parihar, S.A. Lyon, N.R. Cooper, andA.A. Kiselev, Large Skyrmions in an Al0.13Ga0.87As Quantum Well,Phys. Rev. B 61, 4469 (2000).

BIBLIOGRAPHY 207

[93] Sinova J., S.M. Girvin, T. Jungwirth, and K. Moon, Skyrmion Dynam-ics and NMR Line Shapes in Quantum Hall Ferromagnets, Phys. Rev.B 61, 2749 (2000).

[94] Slichter C.P., Principles of Magnetic Resonance, (Springer-Verlag,Berlin, 1991).

[95] Sondhi S.L., A. Karlhede, S.A. Kivelson, and E.H. Rezayi, Skyrmionsand the Crossover from the Integer to Fractional QHE at Small ZeemanEnergies, Phys. Rev. B 47, 16419 (1993).

[96] Song Yi-Qiao, B.M. Goodson, K. Maranowski, and A.C. Gossard, Re-duction of Spin Polarization near Landau Level Filling Factor ν = 3 inGaAs/AlGaAs Quantum Wells, Phys. Rev. Lett. 82, 2768 (1999).

[97] Stone M., Magnus Force on Skyrmions in Ferromagnets and QuantumHall Effect Systems, Phys. Rev. B 53, 16573 (1996).

[98] Sullivan P. and G. Seidel, Steady-State, ac-Temperature Calorimetry,Phys. Rev. 173, 679 (1968).

[99] Timm C., S.M. Girvin, P. Henelius, and A. W. Sandvik, 1/N Expansionfor Two-Dimensional Quantum Ferromagnets, Phys. Rev. B 58, 1464(1998).

[100] Timm C., S.M. Girvin, and H.A. Fertig, Skyrmion Lattice Melting inthe Quantum Hall System, Phys. Rev. B 58, 10634 (1998).

[101] Townes C.H., C. Herring, and W.D. Knight, The Effect of ElectronicParamagnetism on NMR Frequencies in Metals, Phys. Rev. 77, 852(L)(1950).

[102] Tsui D.C., H.L. Stormer, and A.C. Gossard, Two-Dimensional Mag-netotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48, 1559(1982).

[103] Tycko R., S.E. Barrett, G. Dabbagh, L.N. Pfeiffer, and K.W. West,Electronic States in GaAs Quantum Wells Probed by Optically PumpedNMR, Science 268, 1460 (1995).

[104] Usher A., R.J. Nicholas, J.J. Harris, and C.T. Foxon, Observation ofMagnetic Excitons and Spin Waves in Activation Studies of a Two-Dimensional Electron Gas, Phys. Rev. B 41, 1129 (1990).

208 BIBLIOGRAPHY

[105] Vagner I.D. and T. Maniv, Nucler Spin-Lattice Relaxation: A Micro-scopic Local Probe for Systems Exhibiting the QHE, Phys. Rev. Lett.61, 1400 (1988).

[106] van der Pauw, L.J., A Method of Measuring Specific Resistivity andHall Effect of Discs of Arbitrary Shape, Philips Research Reports, 13,1 (1958).

[107] Vitkalov S.A., C.R. Bowers, J.A. Simmons, and J.L. Reno, ESR Detec-tion of Optical Dynamic Nuclear Polarization in GaAs/AlGaAs Quan-tum Wells at Unity Filling Factor in the QHE, Phys. Rev. B 61, 5447(2000).

[108] von Klitzing K., G. Dorda, and M. Pepper, New Method for High Accu-racy Determination of the Fine-Structure Constant based on QuantizedHall Resistance, Phys. Rev. Lett. 45, 494 (1980).

[109] Wang J.K., J.H. Campbell, D.C. Tsui, and A.Y. Cho, Heat Capac-ity of the Two-Dimensional Electron Gas in GaAs/AlGaAs Multiple-Quantum-Well Heterostructures, Phys. Rev. B 38, 6174 (1988).

[110] Wang J.K., ac Calorimetric Study of the Heat Capacity of Two-Dimensional Electrons in Magnetic Field, Ph.D. Thesis, Princeton Uni-versity (1990).

[111] Wang J.K., D.C. Tsui, M.B. Santos, and M. Shayegan, Heat Capac-ity Study of Two-Dimensional Electrons in GaAs/AlGaAs Multiple-Quantum-Well Heterostructures in High Magnetic Fields: Spin-SplitLandau Levels, Phys. Rev. B 45, 4384 (1992).

[112] Wiegers S.A.J, M. Specht, L.-P. Levy, M.Y. Symmons, D.A. Ritchie,A. Cavanna, B. Etienne, G. Martinez, and P. Wyder, Magnetizationand Energy Gaps of a High-Mobility 2D Electron Gas in the QuantumLimit, Phys. Rev. Lett. 79, 3238 (1997).

[113] Winter J., Magnetic Resonance in Metals, (Clarendon Press, Oxford,1971).

[114] Zawadski W. and R. Lassnig, Specific Heat and Magneto-Thermal Os-cillations of 2DESs in a Magnetic Field, Solid State Commun. 50, 537(1984).

[115] Ziman J.M., Principles of the Theory of Solids, (Cambridge UniversityPress, Cambridge, 1972), p. 353.

sorin melinte- quantum hall effect skyrmions: nuclear magnetic resonance and heat capacity...

Documents

d electron systems

d electron layers

thenuclear spin system

single electron

physicochimie et

existence of qhe skyrmions

spintextured objects

d electrons