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Solid state midterm report Quantum Hall effect g Chienchung Chen

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OUTLINE Characteristic length Ballistic conductor Landauer formula Quantum Hall effect A straightforward idea Landau quantization Density of states SdH oscillations Origin of “zero” resistance Not really “zero” resistance What is the current? High magnetic field

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Characteristic length L : Conductor length L Ø : phase relaxation length L m : momentum relaxation length (mean free path) W : Conductor width λ : de Broglie wave length

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Characteristic length L >> L m >> L Ø → Classical conductor L << L m < L Ø → Ballistic conductor L m << L Ø << L → Localization nλ/2=W →cutoff frequency (sub bands) (transverse modes) Ballistic conductor All carriers with Fermi velocity, group velocity Contact resistance due to transverse modes Landauer formula Metallic sample M≈10 6

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Quantum Hall effect The discovery of the quantised Hall effect in 1980 won von Klitzing the 1985 Nobel prize. 2-dimension electron gas

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A straightforward idea

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Landau quantization : cycrotron frequency

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Density of states

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SdH oscillations Shubnikov-de Hass Oscillations

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Origin of “zero” resistance perturbation

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Edge state in equilibrium with Origin of “zero” resistance

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If the electrochemical potentials lies on bulk Landau levels then there is a continuous distribution of allowed states from one edge to the other. That is backscattering. This backscattering gives rise to a maximun in the longitudinal resistance.

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Not really “zero” resistance No Ohmic dissipation

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What is the current? The number of edge states (equal to the number of filled Landau levels in the bulk) plays the role played by the number of modes in a ballistic conductor.

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High magnetic field High magnetic field. How high? 1.An electron should be able to complete a few orbits before losing its momentum due to scattering. 2.The peaks in the DOS will be evident. That is, their energy spacing is much greater than the broadening caused by scattering. T=1.2K B=15T I=1μA L=400μm μ=100 m 2 /Vs >>B=100G

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