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Outline: Introduction: materials, transport, Hall effects

Outline: Introduction: materials, transport, Hall effects Composite particles – FQHE, statistical transformations Quasiparticle charge and statistics Higher Landau levels Other parts of spectrum: non-equilibrium effects, electron solid? Multicomponent systems: Bilayers.

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Outline: Introduction: materials, transport, Hall effects

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  1. Outline: • Introduction: materials, transport, Hall effects • Composite particles – FQHE, statistical transformations • Quasiparticle charge and statistics • Higher Landau levels • Other parts of spectrum: non-equilibrium effects, electron solid? • Multicomponent systems: Bilayers • Vortex picture and statistical transformations • Early measurements of fractional charge • Noise measurements and fractional charge • Potential Statistical tests

  2. III. Quasiparticle charge and statistics A. Vortex picture and statistical transformations Composite bosons 1 2/3 3/5 2/5 1/3 3/7 4/7 4/9 5/9 5/11 6/11 6/13 7/15 7/13 8/17 8/15 9/17 9/19 Composite fermions

  3. III. Quasiparticle charge and statistics A. Vortex picture 2D electron system + B-field Induces vortex in 2DES vortex

  4. III. Quasiparticle charge and statistics A. Vortex picture 2D electron system + B-field Induces vortex in 2DES Near filling factor 1/3, the vortex charge is (1/3)e

  5. III. Quasiparticle charge and statistics electron A. Vortex picture + B-field vortex Now consider an electron Lowers energy to superpose vortex on electron

  6. IiI. Quasiparticle charge and statistics A. Vortex picture + B-field electron Vortex charge 1/3 Now consider an electron Lowers energy even more to superpose two vortices on electron

  7. III. Quasiparticle charge and statistics electron A. Vortex picture + B-field vortex Electron in triple vortex - 1/3 FQHE ground state Superpose three vortices to further lower energy Electron in triple vortex - 1/3 FQHE ground state Bosonic ground state

  8. III. Quasiparticle charge and statistics electron A. Vortex picture + B-field vortex Apply more B-field: get another vortex of +1/3 charge

  9. III. Quasiparticle charge and statistics electron A. Vortex picture + B-field vortex Decrease B-field: form a vortex/electron quasiparticle of -1/3 charge

  10. III. Quasiparticle charge and statistics electron A. Vortex picture + B-field vortex Add an electron: get three quasiparticles of -1/3 charge Vortex picture

  11. III. Quasiparticle charge and statistics A. Vortex picture and statistical transformations Composite bosons 1 2/3 3/5 2/5 1/3 3/7 4/7 4/9 5/9 5/11 6/11 6/13 7/15 7/13 8/17 8/15 9/17 9/19 Composite fermions Transformations occur as B-field is swept – rearrangement of ground states

  12. III. Quasiparticle charge and statistics B. Fractional charge measurement • Fractional charge and the fractional quantum Hall states: • fractional quantum Hall states are incompressible quantum liquids • The ground states at odd-denominator filling factors are bose condensates of bosonic quasiparticles • the charge carrying excitations are other quasiparticles • there is a finite energy required to produce these charge carrying excitations • this gap energy must be determined by the interaction or Coulomb energy • the nature of the excitations, or quasiparticles, implies a distinct similarity between charge and magnetic field • Can this fractional charge be measured?

  13. III. Quasiparticle charge and statistics B. Fractional charge measurement • Three sets of measurements – All point out the difficulties of examining these condensed states and their excitations • Narrow channel fluctuations • Charging an anitdot • Shot noise from a fractional quantum Hall state

  14. III. Quasiparticle charge and statistics B. Fractional charge measurement a) Narrow channel fluctuations FQHE in a narrow (2mm wide) channel: (Experimentally difficult) PRL ‘89

  15. III. Quasiparticle charge and statistics B. Fractional charge measurement a) Narrow channel fluctuations n = 2 Oscillations are observed in longitudinal resistivity near the minima of filling factors n = 2, 1, and 1/3 n = 1 n = 1/3

  16. III. Quasiparticle charge and statistics B. Fractional charge measurement a) Narrow channel fluctuations 1/3 1 2 3 The oscillation periods are different: n = 4,3,2, and 1 are one third that of the oscillation period at n = 1/3

  17. III. Quasiparticle charge and statistics B. Fractional charge measurement a) Narrow channel fluctuations • Accidental occurance of scattering site in channel • Site can support a magnetically bound state • Quasiparticles can tunnel from one edge, traverse the bound state site, and tunnel to other edge • Bound-state Bohr-Sommerfeld quantization condition: N flux quanta (h/e) enclosed • Transport through the bound state is resonant, with resistance period given by Mechanism: Resistance peaks therefore dependent upon charge A B If area of bound site is a, flux quantum is f, then oscillation period occurs for (DB/f)a=1 period: f=h/e*; DBe*a/h=1 period for e*=e, DB1, and for e*=e/3, DB2, Then DB2=3DB1 Resistance Rxx = [R/(1-R)]h/e2 With R the probability of scattering from one edge to the other

  18. III. Quasiparticle charge and statistics B. Fractional charge measurement a) Narrow channel fluctuations • Accidental occurance of scattering site in channel • Site can support a magnetically bound state • Quasiparticles can tunnel from one edge, traverse the bound state site, and tunnel to other edge • Bound-state Bohr-Sommerfeld quantization condition: Nh/e flux quanta enclosed • Transport through the bound state is resonant, with resistance period given by Mechanism: ALSO, if A B Resistance Rxx = [R/(1-R)]h/e2 With R the probability of scattering from one edge to the other Just right for the channel dimensions

  19. III. Quasiparticle charge and statistics B. Fractional charge measurement Bound states and Fermi level a) Narrow channel fluctuations Magnetically bound-state Simmons PRB ‘91

  20. III. Quasiparticle charge and statistics B. Fractional charge measurement a) Narrow channel fluctuations • Is this: • Tunneling through bound states in channel, which gives charge of quasiparticles, or • Coulomb blockade, which does not give charge of quasiparticles? Temperature dependence in experiments not consistent with this.

  21. III. Quasiparticle charge and statistics B. Fractional charge measurement b) Antidot “interferometer” Given these findings using an “accidental” bound state in the narrow channel, an artificial bound state or antidot was produced. Goldman Science ‘95

  22. III. Quasiparticle charge and statistics B. Fractional charge measurement A or B b) Antidot “interferometer” • Different mechanism proposed with the antidot • planned occurance of scattering site in channel • Quasiparticles can tunnel from one edge, traverse the bound state site, and tunnel to other edge • Either path A or B can be traversed: interference leads to periodic oscillations Mechanism: If area of bound site is a, flux quantum is f, then oscillation period occurs for (DB/f)a=1 period: f=h/e*; DBe*a/h=1 period for e*=e, DB1 for e*=e/3, DB2, Then DB2=3DB1 Is a about right for the bound state size? DB1 = 0.05T, f = 4x10-3 T-mm2, then a ~ 0.3 mm x 0.3 mm Aharanov-Bohm oscillations produce channel fluctuations

  23. III. Quasiparticle charge and statistics B. Fractional charge measurement b) Charging an antidot Oscillations observed and related to quasiparticle interference If area of bound site is a, flux quantum is f, then oscillation period occurs for (DB/f)a=1 period: f=h/e*; DBe*a/h=1 period for e*=e, DB1 for e*=e/3, DB2, Then DB2=3DB1

  24. III. Quasiparticle charge and statistics B. Fractional charge measurement b) “antidot charge interferometer” Further refinements of this device have occurred Goldman, Cond-mat/0502406

  25. III. Quasiparticle charge and statistics B. Fractional charge measurement b) “antidot charge interferometer” Cond-mat/0502406

  26. III. Quasiparticle charge and statistics B. Fractional charge measurement b) “antidot charge interferometer” Different periods observed for FQHE states and IQHE states

  27. III. Quasiparticle charge and statistics • B) Fractional charge measurement so far: • edge state tunneling to a central “defect”, natural or artificial • tunneling to and from magnetically bound state exposes charge of quasiparticles • Aharanov Bohm oscillations with change in B-field, and therefore number of flux quanta through the antidot • oscillation period ~ charge: period reflects the fractional charge • problem with both techniques: could have charging of the island, which gives non-specific tunneling conductance oscillation period due to larger Hall voltage in 1/3 versus filling factor 1 case (i.e. does not imply fractional charge)

  28. III. Quasiparticle charge and statistics C. Noise measurements and fractional charge Different type of quasiparticle charge measurement

  29. III. Quasiparticle charge and statistics C. Noise measurements Quasiparticles of fractional charge “pop” through quantum point contact – noise from this is measured Fractional quantum Hall liquids

  30. III. Quasiparticle charge and statistics C. Noise measurements Quantum shot noise as expected for integer filling, 1/3 FQHE state

  31. III. Quasiparticle charge and statistics C. Noise measurements • noise power shows appropriate charge for integer filling factors • noise power appears to support fractional charge at 1/3 state How can one test the statistics of a system???

  32. III. Quasiparticle charge and statistics D. Statistical tests Electron phase change p Quasiparticle phase change ??? Is it possible to experimentally test the statistics of a quasiparticle system? Presently under consideration are two avenues • Mach-Zehnder interference • Hanbury-Brown Twiss

  33. III. Quasiparticle charge and statistics D. Statistical tests

  34. III. Quasiparticle charge and statistics D. Statistical tests Examining interference for electrons in the QHE regime

  35. III. Quasiparticle charge and statistics D. Statistical tests Promising possibilities

  36. III. Quasiparticle charge and statistics D. Statistical tests 2) Hanbury Brown and Twiss

  37. III. Quasiparticle charge and statistics D. Statistical tests 2) Hanbury Brown and Twiss

  38. III. Quasiparticle charge and statistics: A. magnetic field spectrum shows interleaved bosonic and fermionic states B. vortex picture – magnetic field and charge contributions to quasiparticles C. fractional charge measurements – indirect measures of charge with open questions D. next step = statistical tests with quasiparticles: difficult to apply single particle (electron) methods to quasiparticles We will see a particularly interesting statistical state in the higher Landau levels

  39. Outline: • Introduction: materials, transport, Hall effects • Composite particles – FQHE, statistical transformations • Quasiparticle charge and statistics • Higher Landau levels • Other parts of spectrum: non-equilibrium effects, electron solid? • Multicomponent systems: Bilayers • Overview • 5/2 FQHE – the fraction that shouldn’t be there • 9/2 stripes and other things • Higher Landau level experimental issues

  40. IV. Higher Landau Levels A. Overview: Wavefunctions different in higher Landau levels Lowest: N=0 Different interactions energies and novel ground states: Exchange plays an important role Second: N=1 Third: N=2

  41. IV. Higher Landau Levels A. Overview: Energy and length scales (density 1x1011cm-2) Compare LLL to HLL n=7/2 and n=1/2 Coulomb 55 K 144 K energy Spin gap .35 K 2.4 K Magnetic length 230 Å 90 Å Effective Fermi 41 mm-1 110 mm-1 Wavevector Effective interaction energy scale much lower at 7/2

  42. IV. Higher Landau Levels A. Overview: Wavefunctions different in higher Landau levels. Also, filled inert lower Landau level leaves fewer electrons in the higher LL for screening Disorder has more severe consequence on higher Landau level physics Large disorder diminution of gaps in lowest Landau level: G ~ 2 K Similar absolute gap reduction may apply in HLL for intrinsically smaller gaps

  43. 1/7 1/5 1/3 ne IV. Higher Landau Levels A. Overview: Wavefunctions different in higher Landau levels & Lower effective density & Persistent disorder Smaller energy scales, more difficult to examine correlation effects Need lower temperatures and higher mobilities

  44. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: the fraction that shouldn’t be there According to composite fermion theory it is expected that at filling factors 1/2, 3/2, 5/2, etc. we should see Fermi surfaces forming This is true at 1/2 and 3/2, but at 5/2 it was found that at low temperatures a quantum Hall state exists

  45. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: the fraction that shouldn’t be there This is true at 1/2 and 3/2, but at 5/2 it was found that at low temperatures a quantum Hall state exists

  46. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: Upon tilting the sample in the B-field, the new state disappears Spin gap ~ total B-field Orbital gaps ~ orthogonal B-field

  47. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: • Two theoretical possibilities proposed: • Haldane-Rezayi: non-spin polarized state = d-wave pairing of composite fermions • Moore-Read: spin polarized state = p-wave pairing of composite fermions Tilted field results suggest that Haldane-Rezayi state is the likely candidate

  48. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: Much later, numerical studies by R. Morf indicated that the p-wave state (spin polarized) is energetically favorable Composite fermion theory developed to suggest that the system at high temperatures is a filled Fermi sea that condenses at low temps to the 5/2 FQHE Experimentally samples improved significantly

  49. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: Examine this transition from fermionic to bosonic system: can the Fermi surface at 5/2 be observed Sign of fermi surface formation is enhanced conductivity at even denominator filling factors observed using SAW

  50. IV. Higher Landau Levels B. 5/2 fractional quantum Hall effect: SAW results for “low” mobility system - small effect at 3/2 - no effect at 5/2 T=300mK

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