capri spring school current topics in quantum hall research allan macdonald university of texas at...
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Capri Spring School
Current Topics in
Quantum Hall Research
Allan MacDonaldUniversity of Texas
at Austin
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1.QHE – Incompressible States
2.QHE – Edge States & Line Junctions
3.QHE – Bilayer Spontaneous Coherence& Counterflow Superfluidity
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I
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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)
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Two-Dimensional Electron Gas
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Ga As
ultra highvacuum
heated cells
high qualityGaAs substrate
Al
Molecular Beam Epitaxy
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Integer Quantum Hall Effect
xy/(h/e2)
xx
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Cyclotron Orbits
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Landau Levels
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Lowest Landau Level
Orbit Center
Ladder Operator
Bottom of Ladder
Analytic Wavefunctions
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Incompressible States & Streda Formula
Compressibility
Edge Current
Conductance and
LL degeneracy
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Fractional Quantum Hall Effect
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Haldane Pseudopotentials
Center of Mass
& Relative
2-particle states
Haldane Pseudopotentials
Details Hardly Matter!
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Laughlin Wavefunction
FQHE Hamiltonian
LLL Wavefunctions
COM & Relative for each pair
Hard-core modelE=0 Eigenstates
Laughlin Wavefunction
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Fractionally Charged Quasiparticles
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Composite Fermions
Flux Attachment =1/3 = 1
= 2/5
Fractionally Charged
Quasiparticles
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Thermodynamic Stability?
Hard Core Model
Chemical Potentialvs. Density
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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
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II
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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)
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Quantum Hall Edge States
Skipping Orbits
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Edge States
X = k l2 kF1kF0
i = kF/2π
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Field Theory of QH Edge
Hamiltonian
More on V later
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Field Theory of QH Edge
Creation & Annihilation
Free Chiral Bosons
Filling Factor
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Field Theory of QH Edge
Conjugate Variable
Local Fermi Wavevector
Chiral DensityWave
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Edge Magnetoplasmons
Frequency Domain:Wassermeier et al. PRB (1990)
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Time Domain: Ashoori et al. PRB (1992)
ns
Magnetoplasmons in time Domain
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First Quantization Bosonization
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Bosonization by Example
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Luttinger Liquids
3D
E
k
1D
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Density of States Anomaly
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Spin-Charge Separation
Alexi Tsvelik
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Tunneling DOS Calculation
Fermi Golden Rule
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Tunneling DOS Calculation
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Tunneling into Edge
TunnelingGrayson, Chang et
al.PRL 1998,2001
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Noise: Glattli et al. PRL (1997); Heiblum et al. (1997)
Edge State Measurements
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But … what’s this??
voltmeter
0
Roddaro et al. (Pisa) PRL 2003, 2004
2DEG Hall Bar
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and … what’s this??
Roddaro et al. (Pisa) PRL 2003, 2004, 2005
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Quantum Hall Line Junction
Quantum Hall Condensate
Quantum Hall Condensate
X=0
X=L/4
X=L/2
X=3L/4
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Magnetoplasmons in Line Junction Systems
Safi Schulz PRB 1995,1999
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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
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Line Junction Systems – Split Gate
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Line Junction Systems – CEO
Kang et al. Nature 2002
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Corner Line Junctions
Grayson et al.2004, 2005
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Interaction Parameter Theory
Hartree-FockEnergy
Functional
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Interaction Parameter TheorySimple Chiral Edge
X = k l2
ε(k)
’
= δk/2π
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Interaction Parameter TheorySimple Chiral Edge
X = k l2
ε(k)
’
= δk/2π
Attraction to NeutralizingBackground
EMP Velocity
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Quantum Hall Domain Walls
Baking Bread
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Sine-Gordon ModelKang et al. Nature 2002
Sine-Gordon Model
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Fun with 2D Electrostatics
Co-Planar appox.
Conformal transformatio
n
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Smooth Edge Model
= 1 = 1=2/3 = 1 = 1 = 1/3
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III
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III – Bilayer Condensates Reference
J.P. Eisenstein and A.H. MacDonald
Nature 432, 7018 (2004).
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superfluid helium
superconductor
Bose-Einstein Condensates (BECs)
BEC of sodium atoms
Durfee & Ketterle, Optics Express 2, 299 (1998)
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References
Eisenstein and AHM - cond-mat/0404 Nature Dec (2004)
Abolfath, Radzihovsky & AHM – PRB (2004)
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History of Superconductivity
Kammerlingh Onnes 1911
Bardeen-Cooper-Schrieffer (BCS) 1957
Brian Josephson 1962
Bednorz and Mueller 1986
T
ρ
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Electrons polarize nearby ions creating surplus of positive charge
Attractive e-e Interactions
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Pairs of electrons behave like bosons coherent many-body wavefunction
Order Parameter is ClassicalEnergy Barriers are Large
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Electron-Electron Pairs
Cooper Pairs
Order Parameter
Superflow
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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
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AlGaAs AlGaAsGaAs
100 A
Quantum Well
e-
energ
y
Ga
Al
Si
BilayersAlGaAs
GaAs
100 A
AlGaAs
Double Quantum Well
-
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III
QHE for =1/2 + 1/2
Spontaneous Phase Coherence
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QH Bilayers
Easy-PlaneFerromagnetsExcitonic BECs
(Josephson Junctions)
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Excitons – Elementary Excitations of Intrinsic Semiconductors
e-
h
h
e
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… also Keldysh JETP 1968
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Electron-Hole Pairs(n’,n)=(,) =Ferromagnetism(n’,n)=(c,v) = Excitonic BECs
(n’,n)=(TopLayer,BottomLayer) Order Parameter
Counterflow Superflow
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Excitonic BEC and Superfluidity?
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3D
Ec + EV2D Bilayer
Ec + EV
2D Bilayer in Field
Exciton Condensation in Semiconductors
Keldysh 1964
Lezovik 1975
Kuramoto 1978
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BCS Nambu-Gorkov & PHT
Attractive Interactions Repulsive Interactions
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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
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• 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
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Electrons and Holes in the QH Regime
Add magnetic field
Particle-hole transformation
Assemble Bilayer
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How to detect an excitonic BEC
No Odd Channel Resistivity
1996
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e-
e-
e-e- e- e- e- e-e- e-
Independent
Contacts
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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
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I
Ivoltmeter
voltmeter0
0
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15
10
5
0
-5
-10
Hal
l Vol
tage
1050Magnetic Field
Electron-hole Pair Current
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Superflow in Electron-Electron Bilayers
Kellogg and Eisenstein cond-mat/0401521
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Superflow in Electron-Electron Bilayers
Kellogg and Eisenstein cond-mat/0401521
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Topological Charge =
Electric Charge
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Vortex-Flow Dissipation
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Collective Dynamics & Dissipation
Ferromagnets vs. Josephson Junctions vs. Bilayers
J.J. Dynamics
Thin Film Ferromagnet
Dynamics
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Joglekar TDHFA+SCBA
Joglekar PRL (2002)
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Collective Dynamics
Ferromagnets vs. Josephson Junctions vs. Bilayers
J.J. Dynamics
Thin Film Ferromagnet Dynamics
+Ist
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=1, QlB=0.838, V0/(e2/lB)=1.5 , N=36, Symmetric Disorder
localdensity
super-current:
pseudo-spin
d/lB=0.5
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Collective Spin Transport
I
Easy Plane Free Magnet
Perpendicular Easy-Axis Pinned Magnet
Konig AHM et al. PRB (2003); PRL (2001)
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Easy-Plane Current-Driven Dynamics
Easy-Plane Fero = Superconductor = Quantum Hall Bilayer
LL Dynamics
Uniaxial Anisotropy
Current Driven
MicromagneticExchange
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Spin Supercurrent
I
Konig AHM et al. PRB (2003); PRL (2001)
Super Spin Current
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Nunez+AHM, cond-mat/0403710
Spin-TransferTheory
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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
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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??
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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