Capri Spring School Current Topics in Quantum Hall Research Allan MacDonald University of Texas at...

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 – 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 – 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


Creation & Annihilation

Free Chiral Bosons

Filling Factor


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


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





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 – 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


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





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

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


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





Capri Spring School Current Topics in Quantum Hall Research Allan MacDonald University of Texas at...

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