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) Orbit Radius: r = [(mv)/(|q|B)] = [p/(q|B|]) Orbit Frequency: ω = 2πf = (|q|B)/m Orbit Energy: K = (½)mv 2 = (q 2 B 2 R 2 )/2m A Momentum (p) Filter!! A Mass Measurement Method!

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 back/front surfaces  Hall Voltage sign is reversed for holes

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

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

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

Finally... B-field I V H

The Classical Hall effect The Lorentz force deflects j x however, an E-field is set up which balances this Lorentz force. Steady state occurs when E y = v x B z = V y /l y But j x = nev x  R xy = V y / i x = R H B z × (l y /A x ) where R H = 1/ne = Hall Coefficient l y is the transverse width of the sample & A x is the transverse cross sectional area of the sample. That is R H depends the on sample shape Slope is related to R H & sample dimensions Magnetic field (tesla) AxAx lyly l Hall Resistance R xy

Page 10 Phys 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

The surface current density s x = v x σ q, (σ = surface charge density) Again, R H = 1/σ e. But, now: R xy = V y / i x = R H B z since s x = i x /l y. & E y = V y /l y. That is, R xy does NOT depend on the sample shape of the sample. This is a very important aspect of the Quantum Hall Effect (QHE) The 2Dimensional Hall effect

The Integer Quantum Hall Effect The Hall Conductance is quantized in units of e 2 /h, or The Hall Resistance R xy = h/(ie 2 ) where i is an integer. The quantum of conductance h/e 2 is now known as the “Klitzing” !! 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 the 1985 Nobel Prize.

The Royal Swedish Academy of Sciences awarded The 1998 Nobel Prize in Physics jointly to Robert B. Laughlin (Stanford), Horst L. Störmer (Columbia & Bell Labs) & Daniel C. Tsui, (Princeton) The Fractional Quantum Hall effect 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.