Correlated 2D Electron Aspects of the Quantum Hall Effect
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1 Correlated 2D Electron Aspects of the Quantum Hall Effect
2 Magnetic field spectrum of the correlated 2D electron system: Electron interactions lead to a range of manifestations 10? = 4? = 2 Resistance (arb. units) 5 "stripes" "mixed?" composite fermions magnetic field (kg)
3 Lectures Outline: I. Introduction: materials, transport, Hall effects II. III. IV. Composite particles FQHE, statistical transformations Quasiparticle charge and statistics Higher Landau levels V. Other parts of spectrum: non-equilibrium effects, electron solid? VI. Multicomponent systems: Bilayers
4 Outline: I. Introduction: materials, transport, Hall effects A. General 2D physics B. Materials MBE C. Measurements quantum Hall effect D. Correlations E. Fractional quantum Hall effect II. III. IV. Composite particles composite fermions Quasiparticle charge and statistics Higher Landau levels V. Other parts of spectrum: non-equilibrium effects, electron solid? VI. Multicomponent systems: Bilayers
5 I. Introduction: materials, transport, Hall effects A. General 2D physics k y k x k F = (2?n) 1/2
6 I. Introduction: materials, transport, Hall effects A. General 2D physics
7 I. Introduction: materials, transport, Hall effects A. General 2D physics Higher Landau levels have more nodes Magnetic length l 0
8 I. Introduction: materials, transport, Hall effects A. General 2D physics 1
9 I. Introduction: materials, transport, Hall effects A. General 2D physics What are the sources of scattering in a real 2D electron system? How do you make a real 2D electron system? Molecular beam epitaxy
10 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy In ultra-high vacuum, sources provide material that is evaporated onto a heated substrate Heated substrate Ga Al Si As The material is deposited at ~ monolayers / second
11 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy Loren Pfeiffer Ken West 2D electron gas forms at interface of AlGaAs/GaAs in MBE grown crystal Si doping 2D electrons in high purity AlGaAs/GaAs interface AlGaAs GaAs
12 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy AlGaAs AlGaAs AlGaAs AlGaAs GaAs GaAs GaAs The material is deposited at ~ monolayers / second 15 monolayers Shuttering different sources layers the materials
13 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy a) layering: electrons from Si layer reside at AlGaAs/GaAs interface ionized dopants isolated from electron layer Doping modulation b) Energy level diagram: electron wavefunction traverses interface plane, has finite z-extent only lowest bound state used
14 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy (<1nm in normal metal)
15 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy Historical landmarks of 2DEG mobility in GaAs. Annotated with the specific MBE innovation that caused the mobility improvement. Electron Mobility (cm 2 /Vsec) UHV cryopump bake 1986 Al, Ga, As SourcePurity 1982 LN2 Shielded Sources Sample loadlock Single interface Undoped setback million cm 2 /Vsec Sample structure Si-modulation-doping Stormer-Gossard-Dingle Bulk Temperature (K)
16 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy Scattering mechanisms: interface roughness alloy scattering ionized impurities residual disorder systematic disorder
17 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy Numerous tricks used to provide clean layer interfaces to reduce the scattering probabilities
18 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy Multiple uses for MBE samples: Electronic and structural
19 I. Introduction: materials, transport, Hall effects C. Measurements Hall effect Cleave piece of wafer Diffuse contacts into wafer piece Cool down: He4 He3 Dilution process Apply B-field orthogonal to sample
20 I. Introduction: materials, transport, Hall effects B. Materials molecular beam epitaxy characterizing the electron system Voltage R Metal contacts to 2D electrons current Measure scattering length (mean-free-path) Piece of MBE wafer direct current from one contact to another with magnetic field applied perpendicular to layers Scattering length is curved path length 300?m mean-free-path in best samples: <1nm in normal metal
21 I. Introduction: materials, transport, Hall effects C. Measurements Hall effect R xy I xx B-field R xx R xy B-field
22 I. Introduction: materials, transport, Hall effects C. Measurements Hall effect R xy I xx B- field R xx R xy B-field
23 I. Introduction: materials, transport, Hall effects C. Measurements Hall effect With increasing B, degeneracy of LL increases and Fermi level is swept through spectrum (constant density n) Landau levels resolved when? c?>>1 B Shubnikov-deHaas oscillations
24 I. Introduction: materials, transport, Hall effects C. Measurements quantum Hall effect At extended states, the Hall voltage increases Localization of single electrons between extended states produces plateaus in Hall resistance
25 I. Introduction: materials, transport, Hall effects C. Measurements quantum Hall effect Minima at n = 2, 4, 6,.. show activated transport R = R 0 e (-? /kt)? = cyclotron gap R H = (1/?)(h/e 2 ) (Spin gaps at filling factors 1,3,5,..)
26 I. Introduction: materials, transport, Hall effects C. Measurements quantum Hall effect R H = (1/?)(h/e 2 ) Integral quantum Hall effect represents single particle localization process
27 I. Introduction: materials, transport, Hall effects C. Measurements quantum Hall effect Edge conduction No backscattering along same edge
28 I. Introduction: materials, transport, Hall effects C. Measurements quantum Hall effect - samples must have sufficiently low disorder that the Landau levels can be resolved (? c?>>1) - decreasing disorder further will unveil correlations R H = (1/?)(h/e 2 )
29 I. Introduction: materials, transport, Hall effects D. Correlations Correlations
30 I. Introduction: materials, transport, Hall effects D. Correlations Magnetic field quenches kinetic energy: If low intrinsic disorder, correlations manifest At high magnetic fields, electron orbits smaller than electron separation
31 I. Introduction: materials, transport, Hall effects D. Correlations Correlation effects may be seen IF: 1) Temperature is low enough 2) Disorder is low enough Potential Energy Electron gas (or liquid) x Fluctuations in potential
32 I. Introduction: materials, transport, Hall effects D. Correlations Correlation effects may be seen IF: 1) Temperature is low enough 2) Disorder is low enough Electron gas Potential Energy x Fluctuations in potential Lower density = Larger influence of disorder on electron gas
33 I. Introduction: materials, transport, Hall effects D. Correlations Correlation effects may be seen IF: 1) Temperature is low enough 2) Disorder is low enough Electron gas Potential Energy x Fluctuations in potential Disorder can destroy correlations
34 I. Introduction: materials, transport, Hall effects D. Correlations Correlation effects may be seen IF: 1) Temperature is low enough 2) Disorder is low enough Electron gas Potential Energy x Fluctuations in potential MINIMIZE DISORDER and USE HIGH ELECTRON DENSITY
35 I. Introduction: materials, transport, Hall effects D. Correlations Correlation effects may be seen IF: 1) Temperature is low enough 2) Disorder is low enough Electron gas Potential Energy x Fluctuations in potential MINIMIZE DISORDER and USE HIGH ELECTRON DENSITY
36 I. Introduction: materials, transport, Hall effects D. Correlations Back to transport measurements: Higher mobility (lower disorder) samples produced AlGaAs/GaAs heterostructures, modulation doped new quantum Hall state found at fractional filling factor 1/3 (note: high densities, high B fields) 1 ½ 1/3
37 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect Higher mobility (lower disorder) samples produced new quantum Hall state found at fractional filling factor: shouldn t be there Energy? = 1? = 1/3
38 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect Higher mobility samples show more FQHE states 5 4 1/3 1) a hierarchy of fractions observed: New quantum numbers at 1/3, 2/5, 3/7, 4/9,. R xx (arb. units) 3 2 3/7 2/5 Filling factor? = p / (2p+1) magnetic field (kg)
39 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect Higher mobility samples show more FQHE states 5 1) a hierarchy of fractions observed: 4 1/3 New quantum numbers at 1/3, 2/5, 3/7, 4/9,. Filling factor? = p / (2p+1) 2) The minima display activated transport: R = R 0 e (-?/kt)? here should reflect the Coulomb energy R xx (arb. units) /5 3/ magnetic field (kg) e 2 /?l 0
40 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect How can this all be explained?
41 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect The Laughlin wave function Describes an incompressible quantum liquid at?=1/3
42 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect - pictures Single electron in the lowest Landau level
43 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect - pictures Single electron in the lowest Landau level Filled lowest Landau level
44 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect - pictures Uncorrelated? = 1/3 state
45 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect - pictures Uncorrelated? = 1/3 state Correlated? = 1/3 state
46 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect The Laughlin wave function and its excitations Add one flux quantum
47 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect?= 1/3 quantum liquid uniform density The Laughlin liquid and its excitations??= 1/3 + B add one flux quantum??= 1/3 - B substract one flux quantum Non-uniform charge densities Added charges +, - 1/3e Energy required to change charge is??~ e 2 /?l 0 l 0 the magnetic length
48 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect Laughlin state describes: 1) New incompressible liquid state 2) Excitations of the liquid, charge 1/m 3) Consistency with experimental results at 1/3, 1/5 (found later) Fractionally charged excitations
49 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect 10 Higher mobilities result in more fractional quantum Hall states Resistance (arb. units) magnetic field (kg)? = 2? = 1
50 I. Introduction: materials, transport, Hall effects E. The Fractional quantum Hall effect Even higher mobilities result in even more fractional quantum Hall states Pan,et al. PRL 04
51 I. Introduction: materials, transport, Hall effects Summary: 2D electron samples show 2D physics in transport: ShubnikovdeHaas oscillations integer quantum Hall effect: resolved Landau levels with localization between centers of Landau levels low disorder 2D electron systems show fractional quantum Hall effect correlations of electrons as described by the Laughlin wave function what about many fractional quantum Hall states?
52 Outline: I. Introduction: materials, transport, Hall effects II. Composite particles composite fermions A. Experiments - hierarchy of fractions B. Composite fermions and their Landau levels C. Composite fermions and experiments D. Fermi surface picture - SAW E. Other Fermi surface experiments F. Composite fermion effective mass G. Other composite fermions III. IV. Quasiparticle charge and statistics Higher Landau levels V. Other parts of spectrum: non-equilibrium effects, electron solid? VI. Multicomponent systems: Bilayers
53 II. Composite particles composite fermions A. Experiments - hierarchy of fractions 5 4 1/3 With lower disorder samples, more fractional states observed R xx (arb. units) 3 2 2/5 3/ magnetic field (kg) Better samples, larger activation energies
54 II. Composite particles composite fermions A. Experiments hierarchy of fractions 5 FQHE states found at filling factors??= p/(2p+1), p=1, 2, /3 Or more generally, at filling factors??= p/(2np ± 1) and at?= 1 - p/(2np ± 1) This includes series of 2/3, 3/5, 4,7 and the series around filling factor 1/4 R xx (arb. units) /7 2/ magnetic field (kg) 1/3 filling factor understood from Laughlin state: what are the others?
55 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: magnetic field electrons From previous description of FQHE liquid we saw that a correlation hole is energetically favorable spot for electron to reside: this associates flux line(s) to the electrons - example 1/3
56 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: magnetic field electrons Any number of magnetic flux may be associated with charge to produce a quasiparticle of the flux/charge composite Filling factor 0 1 1/2 1/3
57 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: magnetic field electrons Given that flux may be associated with charge, now examine the statistics of these quasipartles: An electron wave function upon particle exchange gains phase change FERMIONS e -i?
58 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: magnetic field electrons Given that flux may be associated with charge, now examine the statistics of these quasipartles: An electron with one associated flux quantum obeys bosonic statistics FERMIONS BOSONS e -i? e -i2?
59 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: magnetic field electrons Given that flux may be associated with charge, now examine the statistics of these quasipartles: An electron with two associated flux quanta obeys fermionic statistics FERMIONS COMPOSITE FERMIONS e -i? e -i?
60 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: magnetic field electrons J. Jain examined the correlated 2DES using the quasiparticle, the composite fermion, with the rational that the FQHE states are due to Landau levels of this quasiparticle COMPOSITE FERMIONS e -i?
61 II. Composite particles composite fermions B. Composite fermions and their Landau levels Composite particles of charge and magnetic flux: 5 4 1/3 Composite fermion at its filling factor of 1 R xx (arb. units) magnetic field (kg) 3/7 2/5
62 II. Composite particles composite fermions B. Composite fermions and their Landau levels 5 4 1/3 R xx (arb. units) / magnetic field (kg)
63 II. Composite particles composite fermions B. Composite fermions and their Landau levels 5 +B(?=1/2) 4 1/3 R xx (arb. units) / magnetic field (kg)
64 II. Composite particles composite fermions B. Composite fermions and their Landau levels R xx 1 2/3 3/5 4/7 5/9 6/11 7/13 8/15 3/7 4/9 5/11 6/13 7/15 8/17 2/5 1/3 9/17 9/ M agnetic field in Tesla By enumeration, the hierarchy of observed fractions now become integer Landau levels for this quasiparticle true filling factor 1/3 1 2/5 2 3/7 3 4/9 4 composite fermion filling factor
65 II. Composite particles composite fermions B. Composite fermions and their Landau levels While this picture developed, new findings at an odd location Filling factor 1/2 : should be the center of the Landau level non-localized electrons following the classical Hall trace metallic? 1/2 3/2 5/2
66 II. Composite particles composite fermions C. Composite fermions and experiments Surface acoustic wave experiments: Measure conductivity over short distances Propagating sound wave applies electric field: electrons respond to this field ?m? sound
67 II. Composite particles composite fermions C. Composite fermions and experiments? SAW SAW device launches longitudinal wave with E-field in direction of propagation: conducting layer can short this piezoelectric field, changing the propagation properties SAW measures conductivity of the 2DES at the wavelength and frequency of the SAW? xx (q,? ) <=>? v/v SAW? down to 0.2? m
68 II. Composite particles composite fermions C. Composite fermions and experiments Results: At low frequencies, large wavelengths see all the features of a standard transport measurement SAW measures conductivity of the 2DES at the wavelength and frequency of the SAW? xx (q,? ) <=>? v/v
69 II. Composite particles composite fermions C. Composite fermions and experiments Results: At higher frequencies, smaller wavelengths (<3?m), see new features at ½ filling factor? = 1/2 Marks a new quantum number
70 II. Composite particles composite fermions C. Composite fermions and experiments? = 1/2 Results: The smaller the wavelength, the larger the feature at 1/2 Feature corresponds to enhanced conductivity
71 II. Composite particles composite fermions C. Composite fermions and experiments Results: At 3GHz, 1?m?, dominant feature is enhanced conductivity at ½? = 1/2
72 II. Composite particles composite fermions C. Composite fermions and experiments For higher frequencies, smaller wavelengths, the enhanced conductivity grows 1/2 What is causing this?
73 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 at? = 1/2 k y Halperin, Lee, and Read (1993): Quasiparticle composite fermions produce not only FQHE by filling Landau levels, they also form true filled Fermi sea at filling factor ½ k x k F = (2) 1/2 k F electrons Near ½, quasiparticles move in effective magnetic field B effective = B applied - B (1/2)
74 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 at? = 1/2 k y Halperin, Lee, and Read (1993): Quasiparticle composite fermions produce not only FQHE by filling Landau levels, they also form true filled Fermi sea at filling factor ½ k x k F = (2) 1/2 k F electrons Near ½, quasiparticles move in effective magnetic field B effective = B applied - B (1/2) Away from ½ the quasiparticles move in cyclotron orbits with radius R c = h k F / 2?e B effective
75 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 Halperin, Lee, and Read (1993): Quasiparticle composite fermions produce not only FQHE by filling Landau levels, they also form true filled Fermi sea at filling factor ½ Near ½, quasiparticles move in effective magnetic field B effective = B applied - B (1/2)
76 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 Wavevector dependence of conductivity derived in HLR For SAW wavelength less than the composite fermion mean-free-path, enhanced conductivity observed
77 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 Width of enhanced conductivity at ½ used to extract k F of composite fermions FWHM
78 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 H.L.R. : Commensurability of composite fermion cyclotron orbit and potential (SAW or lithographically defined) should be observed as has been done for electrons Need composite fermion meanfree-path > 2?R c
79 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 H.L.R. : Explicit predictions for SAW results if commensurability present Use large SAW q and large composite fermion m.f.p 1/2
80 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 For large SAW q, large composite particle m.f.p. must also consider denstiy inhomogeneities 1/2
81 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 Commensurability experimentally observed
82 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 10 GHz SAW secondary resonance primary resonance SAW??
83 II. Composite particles composite fermions D. Fermi surface picture: composite fermions form Fermi surface at filling factor 1/2 Experimental magnetic field positions of resonances for different SAW wavevectors can measure k F.? B ~ k F q SAW k F = (2) 1/2 k F electrons
84 II. Composite particles composite fermions E. More experiments on Fermi surfaces: focusing direct current from one contact to another with magnetic field applied perpendicular to layers Use composite fermion cyclotron radius to focus into contacts Smet: PRL 94
85 II. Composite particles composite fermions E. More experiments on Fermi surfaces: antidots Encircle holes made in 2D gas with cyclotron orbits and resistance increases as with electrons Kang, et al. PRL 93
86 II. Composite particles composite fermions E. More experiments on Fermi surfaces: ballistic shorting As nanostructure channel is defined, the B effective ballistic charge transport is enhanced for both electrons and composite fermions
87 II. Composite particles composite fermions F. Composite fermion effective mass H.L.R.: Cyclotron energy gaps of the composite fermions use the effective mass of the quasiparticle
88 II. Composite particles composite fermions F. Composite fermion effective mass in transport measurements the activation energies of the series of fractions (4/9, 3/7, 2/5, 1/3) corresponding to composite fermion Landau levels 4,3,2,1 indeed increase linearly with B effective. The effective mass derived from this is ~ 0.8m e, almost a factor of 10 larger than the free GaAs mass. A mass divergence toward ½ is expected Du, et al. PRL 93
89 II. Composite particles composite fermions F. Composite fermion effective mass Further transport measurements support this Du, et al. PRL 95
90 II. Composite particles composite fermions F. Composite fermion effective mass 10 GHz SAW This effective mass picture is compared with the results from SAW measurements: The quasiparticle cyclotron orbit frequency must be greater than the SAW frequency to observe resonances As effective mass increases, the cyclotron frequency will drop
91 II. Composite particles composite fermions F. Composite fermion effective mass The observation of SAW resonances is not consistent with a diverging composite fermion effective mass See theory of effective mass, Simon et al GHz SAW
92 II. Composite particles composite fermions G. Other composite fermions Composite fermions and their Fermi surfaces expected at other even denominator filling factors (1/4, 3/4, 3/2, 3/8, ) Features observed in transport Resonances in SAW observed, positions in B effective must be adjusted for the active composite particle number.
93 II. Composite particles composite fermions G. Other composite fermions 1/4 Composite fermions and their Fermi surfaces expected at other even denominator filling factors (1/4, 3/4, 3/2, 3/8, ) Presumably observed in transport Resonances in SAW observed, positions in B effective must be adjusted for the active composite particle number. 3/2 3/4
94 G. Statistical transformations and vortex picture: Composite bosons R xx 1 2/3 3/5 4/7 5/9 6/11 7/13 8/15 3/7 4/9 5/11 6/13 7/15 8/17 2/5 1/3 9/17 9/ M agnetic field in Tesla Composite fermions
95 Summary: 2D electron samples show 2D physics: shubnikovdehaas oscillations integer quantum Hall effect: resolved Landau levels with localization between centers of Landau levels low disorder 2D electron systems show fractional quantum Hall effect correlations of electrons as described by the Laughlin wave function composite fermions explain series of fractional quantum Hall states statistical transformations an important part of the magnetic field spectrum in 2D
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