2. The Classical Hall effect

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

4. Electrons flowing without a magnetic field t d semiconductor slice + _ II

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

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

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

8. Page 7 Phys 320 - Baski Solid-State Physics Why is the Hall Effect useful? It can determine the carrier type (electron vs. hole) & the carrier density n for a semiconductor. How? Place the semiconductor into external B field, push current along one axis, & measure the induced Hall voltage V H along the perpendicular axis. The following can be derived: Derived from the Lorentz force F E = qE = F B = (qvB). n = [(IB)/(qwV H )] Semiconductors: Charge Carrier Density via Hall Effect HoleElectron + charge – charge

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

10. r  K  2 mv  q B R 2 Lorentz Force: Review EBEB The Velocity Filter: Undeflected trajectories in crossed E & B fields: v = Cyclotron motion: F B = ma r  qvB = (mv 2 /r) mv p qB m  2  f  1 2 2 2m Orbit radius: Orbit frequency: Orbit energy: momentum (p) filter mass detection

11. Hallresistance-R xy (ohms) The classical Hall effect Lorentz force likes to deflect j x However, E-field is set up which balances Lorentz force Balance occurs when E y = v x B z = V y /l y But j x = nev x (or i x = nev x A x )  R xy = V y / i x = R H B z × (l y /A x ), where R H = 1/ne Where l y is transverse width of sample and A x is the transverse cross sectional area of the sample, i.e. depends on shape of sample 0246810 200 0 1200 1000 800 600 400 1400 Slope related to R H and sample dimensions Magnetic field (tesla) AxAx lyly

12. Surface current density s x = v x  q, where  is surface charge density Again, R H = 1/  e However, now: R xy = V y / i x = R H B z since s x = i x /l y and E y = V y /l y i.e. R xy does NOT depend on the shape of the sample. This is a v. important aspect of the QHE The 2D Hall effect

13. The integer quantum Hall effect Hall conductance quantized in units of e 2 /h, or Hall resistance R xy = h/ie 2, where i is an integer. The quantity h/e 2 is now known as the "Klitzing" 1 Klitzing  25,813  Has been measured to 1 part in 10 8 Very important: For a 2D electron system only First observed in 1980 by Klaus von Klitzing Awarded Nobel prize in 1985

14. The Royal Swedish Academy of Sciences has awarded The 1998 Nobel Prize in Physics jointly to Professor Robert B. Laughlin, Stanford University, California, USA, Professor Horst L. Störmer, Columbia University, New York and Lucent Technologies' Bell Labs, New Jersey, USA, and Professor Daniel C. Tsui, Princeton University, Princeton, New Jersey, USA. The fractional quantum Hall effect The three researchers are being 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 electron charges. Citation: "for their discovery of a new form of quantum fluid with fractionally charged excitations." Electrons in New Guises Horst L. Störmer and Daniel C. Tsui made the discovery in 1982 in an experiment using extremely powerful magnetic fields and low temperatures. Within a year of the discovery Robert B. Laughlin had succeeded in explaining their result. Through theoretical analysis he showed that the electrons in a powerful magnetic field can condense to form a kind of quantum fluid related to the quantum fluids that occur in superconductivity and in liquid helium. What makes these fluids particularly important for researchers is that events in a drop of quantum fluid can afford more profound insights into the general inner structure and dynamics of matter. The contributions of the three laureates have thus led to yet another breakthrough in our understanding of quantum physics and to the development of new theoretical concepts of significance in many branches of modern physics.