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The Hall Effect: Classical & Quantum

The Hall Effect: Classical & Quantum. Tim Morgan. Outline. Classical Hall Effect The Physical Phenomenon Electrical Characterization Geometry Quantum Hall Effect Two Dimensional Electron Systems Integer Quantum Hall Effect Fractional Quantum Hall Effect. The Physical Phenomenon.

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The Hall Effect: Classical & Quantum

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  1. The Hall Effect: Classical & Quantum Tim Morgan

  2. Outline • Classical Hall Effect • The Physical Phenomenon • Electrical Characterization • Geometry • Quantum Hall Effect • Two Dimensional Electron Systems • Integer Quantum Hall Effect • Fractional Quantum Hall Effect Tim Morgan | 6.7.05 | UA | Hall Effect

  3. The Physical Phenomenon • Current through conductor • Magnetic field splits holes and electrons • Potential established • Equilibrium of electromagnetic forces reached Tim Morgan | 6.7.05 | UA | Hall Effect

  4. Fundamental Electricity & Magnetism • Electromagnetic Forces • F = qE • FL=q[v x B] • Voltage • V = E * L • Current Density • J = I/A • J = σE • J = n*e*v Basic electricity and magnetism knowledge will lead to an understanding of the Classical Hall Effect. Tim Morgan | 6.7.05 | UA | Hall Effect

  5. Foundational Relationships Equilibrium of Forces qE = q[v x B] Carrier Density p = JxBz/eEy Drift Velocity vd = μE Hall Coefficient -Ey = RH JxBz Drift Mobility μ = RH σ RH = 1/ep RH = VHw/BzIx Tim Morgan | 6.7.05 | UA | Hall Effect

  6. The Importance of the Hall Voltage Complete Characterization Measure Hall Voltage Calculate Values RH σ ρ, μ, vd p, n VH The measurement of the Hall voltage leads to the complete electrical characterization of a conductor or semiconductor material. Tim Morgan | 6.7.05 | UA | Hall Effect

  7. Geometrical Considerations • Various structures designed for different purposes C S C/S • Rectangle: • Simple production • GCF Small S contacts • Greek Cross: • π/2 invariant rotation • Interchangeable C/S contacts • Clover-leaf: • GCF ~ 1 • Minimizes heating and noise • Precise voltage drop Tim Morgan | 6.7.05 | UA | Hall Effect

  8. Geometry Correction Factor • Hall voltage decreases when current not confined perfectly • G = VH/VH∞ • Based on: • Hall Angle • Dimensions 1 G General trend for a rectangular Hall plate. ΘH J Eext EH ET l/w 3 Tim Morgan | 6.7.05 | UA | Hall Effect

  9. 2-D Electron Systems z MOSFET & Heterojunctions Vg Semiconductor y Oxide Metal x Gate Voltage & Band Bending E E Conduction E EF Vg EF EF Valence Vg EF EF Depletion M O S Depletion z z z Inversion Tim Morgan | 6.7.05 | UA | Hall Effect

  10. Integer Quantum Hall Effect • Drude Theory predicts: ρxy = B/nee • Experimental Verification: • Wakabayashi and Kawji • Hall conductivity: σxy = nee/B • von Klitzing et al. conductivity σxx -σxy RH ρxy = h/e2k, k = 1,2,3… VG VG Tim Morgan | 6.7.05 | UA | Hall Effect

  11. IQHE Implications • Two Dimensional System • ρxy = RH • Only need current and voltage for accurate mesaurement • Diagonal ρxx and σxx vanish at plateau • Hall voltage probes placed anywhere on sides Tim Morgan | 6.7.05 | UA | Hall Effect

  12. Fractional Quantum Hall Effect • IQHE explored more with ‘cleaner’ samples • High mobility (GaAs) • Lower temperatures • Plateaus of Hall resistivity at 1/3 and 2/3 of e2/h • Electron interactions • Delocalized when Coulomb potential is larger than impurity potential Tim Morgan | 6.7.05 | UA | Hall Effect

  13. Summary • Classical Hall Effect • Occurs in a 3-D conducting or semi-conducting sample • Fundamental E&M allows for full explaination • Useful in electrical characterization • Quantum Hall Effect • Discovered how the 2DES is formed • Learned the concept of the IQHE and FQHE Tim Morgan | 6.7.05 | UA | Hall Effect

  14. References • P. Blood, J. W. Orton. The Electrical Characterization of Semiconductors: majority Carriers and Electron States. Philips Research Laboratories. London. 1992. • D. C. Look. Electrical Characterization of GaAs Materials and Devices. Wright State University. Chichester. 1989. • www.eeel.nist.gov/ 812/fig4.htm (Clover-leaf image on slide 7) • D. Yoshioka. The Quantum Hall Effect. Springer. Berlin. 1998. (Diagrams based on images in book.) Tim Morgan | 6.7.05 | UA | Hall Effect

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