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The Classical Hall effect

The Classical Hall effect. Reminder: The Lorentz Force. F = q [ E + ( v  B )]. Lorentz Force: Review. Velocity Filter : Undeflected trajectories in crossed E & B fields: v = (E/B). Cyclotron motion : F B = ma r  qvB = (mv 2 /r).

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The Classical Hall effect

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  1. The Classical Hall effect

  2. Reminder:The Lorentz Force F = q[E + (v B)]

  3. Lorentz Force: Review Velocity Filter:Undeflected trajectories in crossed E&Bfields: v = (E/B) Cyclotron motion: FB = mar  qvB = (mv2/r) Orbit Radius: r = [(mv)/(|q|B)] = [p/(q|B|]) A Momentum (p) Filter!! Orbit Frequency: ω = 2πf = (|q|B)/m A Mass Measurement Method! Orbit Energy: K = (½)mv2 = (q2B2R2)/2m

  4. Standard Hall Effect Experiment  Current from the applied E-field e+ v e- v Lorentz force from the magnetic field on a moving electron or hole E field Top view—electronsdrift from back to front e- leaves +/– charge on back/front surfaces Hall Voltage sign is reversed for holes

  5. t d I I Electrons flowing without a magnetic field semiconductor slice _ +

  6. I B-field When the magnetic field is turned on .. qBv

  7. high potential I low potential As time goes by... qE qBv = qE

  8. V H I B-field Finally...

  9. The Classical Hall effect 1400 Slope is related to RH & sample dimensions 1200 1000 800 600 400 Hall Resistance Rxy ly l 200 0 Ax 0 2 4 6 8 10 Magnetic field (tesla) The Lorentz force deflects jxhowever, an E-field is set up which balances this Lorentz force. Steady state occurs when Ey = vxBz = Vy/ly But jx = nevx  Rxy = Vy / ix = RH Bz × (ly /Ax) where RH = 1/ne = Hall Coefficient lyis the transverse width of the sample & Ax is the transverse cross sectional area of the sample. That is RH depends the on sample shape.

  10. Semiconductors:Charge Carrier Density via Hall Effect • Why is the Hall Effect useful? It can determine the carrier type (electron vs. hole) & the carrier density nfor a semiconductor. • How? Place the semiconductor into external B field, push current along one axis, & measure the induced Hall voltageVH along the perpendicular axis. The following can be derived: • Derived from the Lorentz force FE = qE = FB = (qvB). n = [(IB)/(qwVH)] Hole Electron + charge – charge Solid-State Physics

  11. The 2Dimensional Hall effect The surface current density sx = vxσ q,(σ = surface charge density) Again, RH = 1/σ e. But, now: Rxy = Vy / ix = RH Bz since sx = ix /ly . & Ey = Vy /ly. That is, Rxydoes NOT depend on the sample shape of the sample. This is a very important aspect of the Quantum Hall Effect (QHE)

  12. The Integer Quantum Hall Effect Very important: For a 2D electron system only First observed in 1980 by Klaus von Klitzing Awarded the 1985 Nobel Prize. The Hall Conductance is quantizedin units ofe2/h,or The Hall ResistanceRxy = h/(ie2)where iis an integer. The quantum of conductanceh/e2 is now known as the “Klitzing” !! Has been measured to 1 part in 108

  13. The Fractional Quantum Hall effect The Royal Swedish Academy of Sciences awardedThe 1998 Nobel Prize in Physics jointly to Robert B. Laughlin (Stanford), Horst L. Störmer (Columbia & Bell Labs) & Daniel C. Tsui, (Princeton) The 3 researchers were awarded the Nobel Prize for discovering that electrons acting together in strong magnetic fields can form new types of "particles", with charges that are fractions of an electron charge. Citation: “For their discovery of a new form of quantum fluid with fractionally charged excitations.” Störmer & Tsui made the discovery in 1982 in an experiment using extremely high magnetic fields very low temperatures. Within a year Laughlin had succeeded in explaining their result. His theory showed that electrons in high magnetic fields & low temperatures can condense to form a quantum fluid similar to the quantum fluids that occur in superconductivity & liquid helium. Such fluids are important because events in a drop of quantum fluid can give deep insight into the inner structure & dynamics of matter. Their contributions were another breakthrough in the understanding of quantum physics & to development of new theoretical concepts of significance in many branches of modern physics.

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