Capri Spring School Current Topics in Quantum Hall

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Capri Spring School Current Topics in Quantum Hall Research Allan Mac. Donald University of

Capri Spring School Current Topics in Quantum Hall Research Allan Mac. Donald University of Texas at Austin

1. QHE – Incompressible States 2. QHE – Edge States & Line Junctions 3.

1. QHE – Incompressible States 2. QHE – Edge States & Line Junctions 3. QHE – Bilayer Spontaneous Coherence & Counterflow Superfluidity

I

I

I – References on QHE cond-mat/9410047 Introduction to the Physics of the Quantum Hall

I – References on QHE cond-mat/9410047 Introduction 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

Two-Dimensional Electron Gas

Molecular Beam Epitaxy heated cells Ga Al As ultra high vacuum high quality Ga.

Molecular Beam Epitaxy heated cells Ga Al As ultra high vacuum high quality Ga. As substrate

Integer Quantum Hall Effect xx xy/(h/e 2)

Integer Quantum Hall Effect xx xy/(h/e 2)

Cyclotron Orbits

Cyclotron Orbits

Landau Levels

Landau Levels

Lowest Landau Level Orbit Center Ladder Operator Bottom of Ladder Analytic Wavefunctions

Lowest Landau Level Orbit Center Ladder Operator Bottom of Ladder Analytic Wavefunctions

Incompressible States & Streda Formula Compressibility Edge Current Conductance and LL degeneracy

Incompressible States & Streda Formula Compressibility Edge Current Conductance and LL degeneracy

Fractional Quantum Hall Effect

Fractional Quantum Hall Effect

Haldane Pseudopotentials 2 -particle states Center of Mass & Relative Details Hardly Matter! Haldane

Haldane Pseudopotentials 2 -particle states Center of Mass & Relative Details Hardly Matter! Haldane Pseudopotentials

Laughlin Wavefunction FQHE Hamiltonian LLL Wavefunctions COM & Relative for each pair Hard-core model

Laughlin Wavefunction FQHE Hamiltonian LLL Wavefunctions COM & Relative for each pair Hard-core model E=0 Eigenstates Laughlin Wavefunction

Fractionally Charged Quasiparticles

Fractionally Charged Quasiparticles

Composite Fermions Flux Attachment =1/3 = 1 Fractionally Charged Quasiparticles = 2/5

Composite Fermions Flux Attachment =1/3 = 1 Fractionally Charged Quasiparticles = 2/5

Thermodynamic Stability? Hard Core Model Chemical Potential vs. Density

Thermodynamic Stability? Hard Core Model Chemical Potential vs. Density

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally Charge Quasiparticles, Composite Fermions, Thermodynamic Instability II Edge 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 Analogy Counterflow Superfluidity, Interlayer Tunneling

II

II

II – Quantum Hall Edge State References Review: A. M. Chang, Rev. Mod. Phys.

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

Quantum Hall Edge States Skipping Orbits

Edge States i = k. F/2π X = k l 2 k. F 1

Edge States i = k. F/2π X = k l 2 k. F 1 k. F 0

Field Theory of QH Edge Hamiltonian More on V later

Field Theory of QH Edge Hamiltonian More on V later

Field Theory of QH Edge Filling Factor Creation & Annihilation Free Chiral Bosons

Field Theory of QH Edge Filling Factor Creation & Annihilation Free Chiral Bosons

Field Theory of QH Edge Local Fermi Wavevector Conjugate Variable Chiral Density Wave

Field Theory of QH Edge Local Fermi Wavevector Conjugate Variable Chiral Density Wave

Edge Magnetoplasmons Frequency Domain: Wassermeier et al. PRB (1990)

Edge Magnetoplasmons Frequency Domain: Wassermeier et al. PRB (1990)

Magnetoplasmons in time Domain ns Time Domain: Ashoori et al. PRB (1992)

Magnetoplasmons in time Domain ns Time Domain: Ashoori et al. PRB (1992)

First Quantization Bosonization

First Quantization Bosonization

Bosonization by Example

Bosonization by Example

Luttinger Liquids 3 D 1 D E k

Luttinger Liquids 3 D 1 D E k

Density of States Anomaly

Density of States Anomaly

Spin-Charge Separation Alexi Tsvelik

Spin-Charge Separation Alexi Tsvelik

Tunneling DOS Calculation Fermi Golden Rule

Tunneling DOS Calculation Fermi Golden Rule

Tunneling DOS Calculation

Tunneling DOS Calculation

Tunneling into Edge Tunneling Grayson, Chang et al. PRL 1998, 2001

Tunneling into Edge Tunneling Grayson, Chang et al. PRL 1998, 2001

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

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

But … what’s this? ? 0 voltmeter 2 DEG Hall Bar Roddaro et al.

But … what’s this? ? 0 voltmeter 2 DEG Hall Bar Roddaro et al. (Pisa) PRL 2003, 2004

and … what’s this? ? Roddaro et al. (Pisa) PRL 2003, 2004, 2005

and … what’s this? ? Roddaro et al. (Pisa) PRL 2003, 2004, 2005

Quantum Hall Line Junction X=3 L/4 Quantum Hall Condensate X=L/2 X=0 Quantum Hall Condensate

Quantum Hall Line Junction X=3 L/4 Quantum Hall Condensate X=L/2 X=0 Quantum Hall Condensate X=L/4

Magnetoplasmons in Line Junction Systems Safi Schulz PRB 1995, 1999

Magnetoplasmons in Line Junction Systems Safi Schulz PRB 1995, 1999

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally Charged Quasiparticles, Composite Fermions, Thermodynamic Instability II Edge 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 Analogy Counterflow Superfluidity, Interlayer Tunneling

Line Junction Systems – Split Gate

Line Junction Systems – Split Gate

Line Junction Systems – CEO Kang et al. Nature 2002

Line Junction Systems – CEO Kang et al. Nature 2002

Corner Line Junctions Grayson et al. 2004, 2005

Corner Line Junctions Grayson et al. 2004, 2005

Interaction Parameter Theory Hartree-Fock Energy Functional

Interaction Parameter Theory Hartree-Fock Energy Functional

Interaction Parameter Theory Simple Chiral Edge ε(k) ’ X = k l 2 =

Interaction Parameter Theory Simple Chiral Edge ε(k) ’ X = k l 2 = δk/2π

Interaction Parameter Theory Simple Chiral Edge Attraction to Neutralizing. Background ε(k) EMP Velocity ’

Interaction Parameter Theory Simple Chiral Edge Attraction to Neutralizing. Background ε(k) EMP Velocity ’ X = k l 2 = δk/2π

Quantum Hall Domain Walls Baking Bread

Quantum Hall Domain Walls Baking Bread

Sine-Gordon Model Kang et al. Nature 2002 Sine-Gordon Model

Sine-Gordon Model Kang et al. Nature 2002 Sine-Gordon Model

Fun with 2 D Electrostatics Co-Planar appox. Conformal transformation

Fun with 2 D Electrostatics Co-Planar appox. Conformal transformation

Smooth Edge Model

Smooth Edge Model

III

III

III – Bilayer Condensates Reference J. P. Eisenstein and A. H. Mac. Donald Nature

III – Bilayer Condensates Reference J. P. Eisenstein and A. H. Mac. Donald Nature 432, 7018 (2004).

Bose-Einstein Condensates (BECs) superconductor superfluid helium BEC of sodium atoms Durfee & Ketterle, Optics

Bose-Einstein Condensates (BECs) superconductor superfluid helium BEC of sodium atoms Durfee & Ketterle, Optics Express 2, 299 (1998)

References Eisenstein and AHM - cond-mat/0404 Nature Dec (2004) Abolfath, Radzihovsky & AHM –

References Eisenstein and AHM - cond-mat/0404 Nature Dec (2004) Abolfath, Radzihovsky & AHM – PRB (2004)

History of Superconductivity Kammerlingh Onnes 1911 ρ T Bardeen-Cooper-Schrieffer (BCS) 1957 Brian Josephson 1962

History of Superconductivity Kammerlingh Onnes 1911 ρ T Bardeen-Cooper-Schrieffer (BCS) 1957 Brian Josephson 1962 Bednorz and Mueller 1986

Attractive e-e Interactions Electrons polarize nearby ions creating surplus of positive charge

Attractive e-e Interactions Electrons polarize nearby ions creating surplus of positive charge

Order Parameter is Classical Energy Barriers are Large Pairs of electrons behave like bosons

Order Parameter is Classical Energy Barriers are Large Pairs of electrons behave like bosons coherent manybody wavefunction

Electron-Electron Pairs Cooper Pairs Order Parameter Superflow

Electron-Electron Pairs Cooper Pairs Order Parameter Superflow

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractionally Charged Quasiparticles, Composite Fermions, Thermodynamic Instability II Edge 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 Analogy Counterflow Superfluidity, Interlayer Tunneling

Bilayers Al. Ga. As Al. Ga. As - e- Ga energy Al Si 100

Bilayers Al. Ga. As Al. Ga. As - e- Ga energy Al Si 100 A Quantum Well Double Quantum Well

III QHE for =1/2 + 1/2 Spontaneous Phase Coherence

III QHE for =1/2 + 1/2 Spontaneous Phase Coherence

QH Bilayers Excitonic BECs (Josephson Junctions) Easy-Plane Ferromagnets

QH Bilayers Excitonic BECs (Josephson Junctions) Easy-Plane Ferromagnets

Excitons – Elementary Excitations of Intrinsic Semiconductors e- e h h

Excitons – Elementary Excitations of Intrinsic Semiconductors e- e h h

… also Keldysh JETP 1968

… also Keldysh JETP 1968

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

Electron-Hole Pairs (n’, n)=( , ) =Ferromagnetism (n’, n)=(c, v) = Excitonic BECs (n’, n)=(Top. Layer, Bottom. Layer) Order Parameter Counterflow Superflow

Excitonic BEC and Superfluidity?

Excitonic BEC and Superfluidity?

Exciton Condensation in Semiconductors Keldysh 1964 Lezovik 1975 Kuramoto 1978 3 D Ec +

Exciton Condensation in Semiconductors Keldysh 1964 Lezovik 1975 Kuramoto 1978 3 D Ec + E V 2 D Bilayer in Field

BCS Nambu-Gorkov & PHT Attractive Interactions Repulsive Interactions

BCS Nambu-Gorkov & PHT Attractive Interactions Repulsive Interactions

Exciton Condensation in Bilayers Bilayer QH 1991 Lezovik 1975 Kuramoto 1978 Ec + E

Exciton Condensation in Bilayers Bilayer QH 1991 Lezovik 1975 Kuramoto 1978 Ec + E V 2 D ee Bilayer in Field 2 D eh Bilayer in Field

Mean-Field Theory Description • WHAT? • WHY? T B Spontaneous Interlayer Coherence Gain in

Mean-Field Theory Description • WHAT? • WHY? T B Spontaneous Interlayer Coherence Gain in Interlayer Correlation Energy exceeds loss in Intralayer Correlation Energy T B Top Layer Electron Bottom Layer Electron Cloud Disordered Ordered

Electrons and Holes in the QH Regime Assemble Bilayer Add magnetic field Particle-hole transformation

Electrons and Holes in the QH Regime Assemble Bilayer Add magnetic field Particle-hole transformation

How to detect an excitonic BEC 1996 No Odd Channel Resistivity

How to detect an excitonic BEC 1996 No Odd Channel Resistivity

-- Independent e- e- Contacts eee- e- e - ee- e- - e e-

-- Independent e- e- Contacts eee- e- e - ee- e- - e e- e- e- ee- e- e- e- e- e- e - ee- e- e- ee-

Tunneling rate Weak to Strong Coupling Transition 0 0 Interlayer Voltage

Tunneling rate Weak to Strong Coupling Transition 0 0 Interlayer Voltage

0 I I voltmeter 0 voltmeter

0 I I voltmeter 0 voltmeter

Electron-hole Pair Current

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

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

Topological Charge = Electric Charge

Vortex-Flow Dissipation

Vortex-Flow Dissipation

Collective Dynamics & Dissipation Ferromagnets vs. Josephson Junctions vs. Bilayers J. J. Dynamics Thin

Collective Dynamics & Dissipation Ferromagnets vs. Josephson Junctions vs. Bilayers J. J. Dynamics Thin Film Ferromagnet Dynamics

Joglekar TDHFA+SCBA Joglekar PRL (2002)

Joglekar TDHFA+SCBA Joglekar PRL (2002)

Collective Dynamics Ferromagnets vs. Josephson Junctions vs. Bilayers J. J. Dynamics +Ist Thin Film

Collective Dynamics Ferromagnets vs. Josephson Junctions vs. Bilayers J. J. Dynamics +Ist Thin Film Ferromagnet Dynamics

super-current: f local density pseudo-spin =1, Ql. B=0. 838, V 0/(e 2/el. B)=1. 5

super-current: f local density pseudo-spin =1, Ql. B=0. 838, V 0/(e 2/el. B)=1. 5 , NF=36, Symmetric Disorder d/l. B=0. 5

Collective Spin Transport Konig AHM et al. PRB (2003); PRL (2001) Perpendicular Easy. Axis

Collective Spin Transport Konig AHM et al. PRB (2003); PRL (2001) Perpendicular Easy. Axis Pinned Magnet Easy Plane Free Magnet I

Easy-Plane Current-Driven Dynamics Easy-Plane Fero = Superconductor = Quantum Hall Bilayer Uniaxial Anisotropy Micromagnetic

Easy-Plane Current-Driven Dynamics Easy-Plane Fero = Superconductor = Quantum Hall Bilayer Uniaxial Anisotropy Micromagnetic Exchange LL Dynamics Current Driven

Spin Supercurrent Konig AHM et al. PRB (2003); PRL (2001) Super Spin Current I

Spin Supercurrent Konig AHM et al. PRB (2003); PRL (2001) Super Spin Current I

Spin-Transfer Theory Nunez+AHM, cond-mat/0403710

Spin-Transfer Theory Nunez+AHM, cond-mat/0403710

Coherent Edge Transport Empl 10 -6 e. V ΔQP 10 -3 e. V G

Coherent Edge Transport Empl 10 -6 e. V ΔQP 10 -3 e. V G e 2/h e. V Condensate Orbitals Transport Orbitals Δt 10 -9 e. V V* 10 -6 Volts K = X l 2

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

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 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractional

Outline I 2 DEG, QHE/FQHE, Landau Levels, Thermodynamic Argument, Haldane Pseudopotentials, Laughlin State, Fractional Charge Quasiparticles, Composite Fermions, Thermodynamic Instability II Edge 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 Analogy Counterflow Superfluidity, Interlayer Tunneling