Capri Spring School

Current Topics in

Quantum Hall Research

Allan MacDonaldUniversity of Texas

at Austin

1.QHE – Incompressible States

2.QHE – Edge States & Line Junctions

3.QHE – Bilayer Spontaneous Coherence& Counterflow Superfluidity

I

I – References on QHE

cond-mat/9410047Introduction to the Physics of the Quantum Hall Regime(figures available by e-mail request

The Quantum Hall Effect

(Richard Prange and Steven Girvin)

Two-Dimensional Electron Gas

Ga As

ultra highvacuum

heated cells

high qualityGaAs substrate

Al

Molecular Beam Epitaxy

Integer Quantum Hall Effect

xy/(h/e2)

xx

Cyclotron Orbits

Landau Levels

Lowest Landau Level

Orbit Center

Ladder Operator

Bottom of Ladder

Analytic Wavefunctions

Incompressible States & Streda Formula

Compressibility

Edge Current

Conductance and

LL degeneracy

Fractional Quantum Hall Effect

Haldane Pseudopotentials

Center of Mass

& Relative

2-particle states

Haldane Pseudopotentials

Details Hardly Matter!

Laughlin Wavefunction

FQHE Hamiltonian

LLL Wavefunctions

COM & Relative for each pair

Hard-core modelE=0 Eigenstates

Laughlin Wavefunction

Fractionally Charged Quasiparticles

Composite Fermions

Flux Attachment =1/3 = 1

= 2/5

Fractionally Charged

Quasiparticles

Thermodynamic Stability?

Hard Core Model

Chemical Potentialvs. Density

Outline

I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument,

Haldane Pseudopotentials, Laughlin State, Fractionally Charge Quasiparticles,Composite Fermions, Thermodynamic Instability

IIEdge States,Chiral Luttinger Liquids, Line Junctions, Experiment

Canonical Line Junction Models, Sine-Gordon Models

III Experiment, Bilayer Mean Field Theory,

Easy-Plane Ferromagnet Analogy, Josephson-Junction AnalogyCounterflow Superfluidity, Interlayer Tunneling

II

II – Quantum Hall Edge State References

Review:

A.M. Chang, Rev. Mod. Phys. 74, 1449 (2003)

Original Chiral Luttinger Liquid Paper:

X.G. Wen, Phys. Rev. B 41, 12838 (1990)

Quantum Hall Edge States

Skipping Orbits

Edge States

X = k l2 kF1kF0

i = kF/2π

Field Theory of QH Edge

Hamiltonian

More on V later

Field Theory of QH Edge

Creation & Annihilation

Free Chiral Bosons

Filling Factor

Field Theory of QH Edge

Conjugate Variable

Local Fermi Wavevector

Chiral DensityWave

Edge Magnetoplasmons

Frequency Domain:Wassermeier et al. PRB (1990)

Time Domain: Ashoori et al. PRB (1992)

ns

Magnetoplasmons in time Domain

First Quantization Bosonization

Bosonization by Example

Luttinger Liquids

3D

E

k

1D

Density of States Anomaly

Spin-Charge Separation

Alexi Tsvelik

Tunneling DOS Calculation

Fermi Golden Rule

Tunneling DOS Calculation

Tunneling into Edge

TunnelingGrayson, Chang et

al.PRL 1998,2001

Noise: Glattli et al. PRL (1997); Heiblum et al. (1997)

Edge State Measurements

But … what’s this??

voltmeter

0

Roddaro et al. (Pisa) PRL 2003, 2004

2DEG Hall Bar

and … what’s this??

Roddaro et al. (Pisa) PRL 2003, 2004, 2005

Quantum Hall Line Junction

Quantum Hall Condensate

Quantum Hall Condensate

X=0

X=L/4

X=L/2

X=3L/4

Magnetoplasmons in Line Junction Systems

Safi Schulz PRB 1995,1999

Outline

I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument,

Haldane Pseudopotentials, Laughlin State, Fractionally Charged Quasiparticles,Composite Fermions, Thermodynamic Instability

IIEdge States,Chiral Luttinger Liquids, Line Junctions, Experiment

Canonical Line Junction Models, Sine-Gordon Models

III Experiment, Bilayer Mean Field Theory,

Easy-Plane Ferromagnet Analogy, Josephson-Junction AnalogyCounterflow Superfluidity, Interlayer Tunneling

Line Junction Systems – Split Gate

Line Junction Systems – CEO

Kang et al. Nature 2002

Corner Line Junctions

Grayson et al.2004, 2005

Interaction Parameter Theory

Hartree-FockEnergy

Functional

Interaction Parameter TheorySimple Chiral Edge

X = k l2

ε(k)

’

= δk/2π

Interaction Parameter TheorySimple Chiral Edge

X = k l2

ε(k)

’

= δk/2π

Attraction to NeutralizingBackground

EMP Velocity

Quantum Hall Domain Walls

Baking Bread

Sine-Gordon ModelKang et al. Nature 2002

Sine-Gordon Model

Fun with 2D Electrostatics

Co-Planar appox.

Conformal transformatio

n

Smooth Edge Model

= 1 = 1=2/3 = 1 = 1 = 1/3

III

III – Bilayer Condensates Reference

J.P. Eisenstein and A.H. MacDonald

Nature 432, 7018 (2004).

superfluid helium

superconductor

Bose-Einstein Condensates (BECs)

BEC of sodium atoms

Durfee & Ketterle, Optics Express 2, 299 (1998)

References

Eisenstein and AHM - cond-mat/0404 Nature Dec (2004)

Abolfath, Radzihovsky & AHM – PRB (2004)

History of Superconductivity

Kammerlingh Onnes 1911

Bardeen-Cooper-Schrieffer (BCS) 1957

Brian Josephson 1962

Bednorz and Mueller 1986

T

ρ

Electrons polarize nearby ions creating surplus of positive charge

Attractive e-e Interactions

Pairs of electrons behave like bosons coherent many-body wavefunction

Order Parameter is ClassicalEnergy Barriers are Large

Electron-Electron Pairs

Cooper Pairs

Order Parameter

Superflow

Outline

I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument,

Haldane Pseudopotentials, Laughlin State, Fractionally Charged Quasiparticles,Composite Fermions, Thermodynamic Instability

IIEdge States,Chiral Luttinger Liquids, Line Junctions, Experiment

Canonical Line Junction Models, Sine-Gordon Models

III Experiment, Bilayer Mean Field Theory,

Easy-Plane Ferromagnet Analogy, Josephson-Junction AnalogyCounterflow Superfluidity, Interlayer Tunneling

AlGaAs AlGaAsGaAs

100 A

Quantum Well

e-

energ

y

Ga

Al

Si

BilayersAlGaAs

GaAs

100 A

AlGaAs

Double Quantum Well

-

III

QHE for =1/2 + 1/2

Spontaneous Phase Coherence

QH Bilayers

Easy-PlaneFerromagnetsExcitonic BECs

(Josephson Junctions)

Excitons – Elementary Excitations of Intrinsic Semiconductors

e-

h

h

e

… also Keldysh JETP 1968

Electron-Hole Pairs(n’,n)=(,) =Ferromagnetism(n’,n)=(c,v) = Excitonic BECs

(n’,n)=(TopLayer,BottomLayer) Order Parameter

Counterflow Superflow

Excitonic BEC and Superfluidity?

3D

Ec + EV2D Bilayer

Ec + EV

2D Bilayer in Field

Exciton Condensation in Semiconductors

Keldysh 1964

Lezovik 1975

Kuramoto 1978

BCS Nambu-Gorkov & PHT

Attractive Interactions Repulsive Interactions

Ec + EV2D eh Bilayer

Ec + EV

2D eh Bilayer in Field

Exciton Condensation in Bilayers

Lezovik 1975

Kuramoto 1978

Bilayer QH 1991 Ec + EV

2D ee Bilayer in Field

• WHAT? Spontaneous Interlayer Coherence

• WHY?Gain in Interlayer Correlation Energy exceeds loss in Intralayer Correlation Energy

T T

Disordered Ordered

B BTop LayerElectron

Bottom LayerElectron Cloud

0vuˆ XBXXTXX

X ccm

Mean-Field Theory Description

Electrons and Holes in the QH Regime

Add magnetic field

Particle-hole transformation

Assemble Bilayer

How to detect an excitonic BEC

No Odd Channel Resistivity

1996

e-e- e-

e- e- e- e-

e- e-

e-

e-

e-

e-e- e-

e- e- e- e-

e- e-

e-

e-

e-e- e-

e-

e-e- e-

e-

e-

e-

e-e-

e-e-

e-e- e-

e- e- e- e-

e- e-

e-

e-

e-e- e-

e- e-

e-

e-

e-

e-

e-e-

e-

e-e-

e-

e-

e-

e-e-

e-e-

e-

e-

e-

e-

e- e- e- e-e- e-

e- e-

e-e-

e-

e-e-

e-

e-

e-

e-e-

e-e-

e-

e-

e-

e-

e-e- e- e- e- e-e- e-

Independent

Contacts

300

200

100

0Tu

nn

elin

g C

on

du

cta

nce

(

10-9

-1

)

-5 0 5Interlayer Voltage (mV)Interlayer Voltage

0

Tun

nelin

g ra

te

0

Weak to Strong Coupling Transition

I

Ivoltmeter

voltmeter0

0

15

10

5

0

-5

-10

Hal

l Vol

tage

1050Magnetic Field

Electron-hole Pair Current

Superflow in Electron-Electron Bilayers

Kellogg and Eisenstein cond-mat/0401521

Superflow in Electron-Electron Bilayers

Kellogg and Eisenstein cond-mat/0401521

Topological Charge =

Electric Charge

Vortex-Flow Dissipation

Collective Dynamics & Dissipation

Ferromagnets vs. Josephson Junctions vs. Bilayers

J.J. Dynamics

Thin Film Ferromagnet

Dynamics

Joglekar TDHFA+SCBA

Joglekar PRL (2002)

Collective Dynamics

Ferromagnets vs. Josephson Junctions vs. Bilayers

J.J. Dynamics

Thin Film Ferromagnet Dynamics

+Ist

=1, QlB=0.838, V0/(e2/lB)=1.5 , N=36, Symmetric Disorder

localdensity

super-current:

pseudo-spin

d/lB=0.5

Collective Spin Transport

I

Easy Plane Free Magnet

Perpendicular Easy-Axis Pinned Magnet

Konig AHM et al. PRB (2003); PRL (2001)

Easy-Plane Current-Driven Dynamics

Easy-Plane Fero = Superconductor = Quantum Hall Bilayer

LL Dynamics

Uniaxial Anisotropy

Current Driven

MicromagneticExchange

Spin Supercurrent

I

Konig AHM et al. PRB (2003); PRL (2001)

Super Spin Current

Nunez+AHM, cond-mat/0403710

Spin-TransferTheory

Transport Orbitals

eV

Condensate Orbitals

K = X l2

Coherent Edge Transport

Empl 10-6 eVΔQP 10-3 eV

G e2/h

Δt 10-9 eV

V* 10-6 Volts

Excitonic BEC does occurs in Bilayer QH Systems Excitonic BEC does lead to dramatic collective transport

Challenges for Theory

Height, width and field-dependence of zero-bias tunneling peak??

Hall and Longitudinal Resistivity at Finite T??

Outline

I 2DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument,

Haldane Pseudopotentials, Laughlin State, Fractional Charge Quasiparticles,Composite Fermions, Thermodynamic Instability

IIEdge States,Chiral Luttinger Liquids, Line Junctions, Experiment

Canonical Line Junction Models, Sine-Gordon Models

III Experiment, Bilayer Mean Field Theory,

Easy-Plane Ferromagnet Analogy, Josephson-Junction AnalogyCounterflow Superfluidity, Interlayer Tunneling