Capri Spring School Current Topics in Quantum Hall
- Slides: 94
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. QHE – Bilayer Spontaneous Coherence & Counterflow Superfluidity
I
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
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)
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 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 E=0 Eigenstates Laughlin Wavefunction
Fractionally Charged Quasiparticles
Composite Fermions Flux Attachment =1/3 = 1 Fractionally Charged Quasiparticles = 2/5
Thermodynamic Stability? Hard Core Model Chemical Potential vs. Density
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 – 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 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 Filling Factor Creation & Annihilation Free Chiral Bosons
Field Theory of QH Edge Local Fermi Wavevector Conjugate Variable Chiral Density Wave
Edge Magnetoplasmons Frequency Domain: Wassermeier et al. PRB (1990)
Magnetoplasmons in time Domain ns Time Domain: Ashoori et al. PRB (1992)
First Quantization Bosonization
Bosonization by Example
Luttinger Liquids 3 D 1 D E k
Density of States Anomaly
Spin-Charge Separation Alexi Tsvelik
Tunneling DOS Calculation Fermi Golden Rule
Tunneling DOS Calculation
Tunneling into Edge Tunneling Grayson, Chang et al. PRL 1998, 2001
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. (Pisa) PRL 2003, 2004
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 X=L/4
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 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 – CEO Kang et al. Nature 2002
Corner Line Junctions Grayson et al. 2004, 2005
Interaction Parameter Theory Hartree-Fock Energy Functional
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 ’ X = k l 2 = δk/2π
Quantum Hall Domain Walls Baking Bread
Sine-Gordon Model Kang et al. Nature 2002 Sine-Gordon Model
Fun with 2 D Electrostatics Co-Planar appox. Conformal transformation
Smooth Edge Model
III
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 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 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
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
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 A Quantum Well Double Quantum Well
III QHE for =1/2 + 1/2 Spontaneous Phase Coherence
QH Bilayers Excitonic BECs (Josephson Junctions) Easy-Plane Ferromagnets
Excitons – Elementary Excitations of Intrinsic Semiconductors e- e h h
… also Keldysh JETP 1968
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?
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
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 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
How to detect an excitonic BEC 1996 No Odd Channel Resistivity
-- 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
0 I I voltmeter 0 voltmeter
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 +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 , NF=36, Symmetric Disorder d/l. B=0. 5
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 Exchange LL Dynamics Current Driven
Spin Supercurrent Konig AHM et al. PRB (2003); PRL (2001) Super Spin Current I
Spin-Transfer Theory Nunez+AHM, cond-mat/0403710
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 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 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
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