The Hall Effect

Outline

• Discovery and History of Hall Effect– Edwin Hall, 1879– Anomalous Hall Effect in 1950’s– Quantum Hall Effect 1980

• Overview/Review of Basic Hall Effect– Carrier types– Open vs. closed orbits

• Quantum Hall Effect Overview• Current Applications of Hall Effect in Research

Discovery: Edwin Hall

• Discovered in 1879• Observed small voltage perpendicular to

current in a material within a magnetic field.– The Lorentz force presses its charge

carriers against one side of the conductor creating a voltage

– The ratio of the voltage created to the amount of induced current is known as the Hall resistance

• Allows one to find the sign of the charge carriers

The Hall Voltage: This can be obtained from the derivations to in the next slides

Basic Overview/Review

Magnetoresistance

Hall coefficient relation

Force on each electron

From Ashcroft + Mermin eq 1.12

Basic Overview/Review

Cyclotron Frequency of Electron

With this relation we get two coupled equations.

That simplify to this

Hall Field Ey has no transverse current so jy=0

Hall Coefficient is simplified to this

Anomalous Hall Effect

Anomalous Hall Effect’s 1st Explanation

Anomalous Hall Effect Controversy

• Klaus von Klitzing made the unexpected discovery in 1980 that the Hall conductivity was exactly quantized

Quantum Hall Effect: History

• Originally predicted by Ando, Matsumoto, and Uemura in 1975

• Resistivity/Conductivity in Hall devices, under low temperatures and high magnetic fields, becomes quantized

Quantum Hall Effect: Conditions

• Need electrons confined in plane but free to move within it– Originally observed in heterojunction– Observed in FET– Recently observed in graphene

• Temperature of a few Kelvin– Graphene at room temperature

• Magnetic field on order of TeslaGraph from Novoselov et al. showing QHE in graphene at room temperature. Blue is ρxx. Red is σxy.

Quantum Hall Effect: Landau Levels

• Quantized energy levels of electron orbits in a material with a magnetic field

• Solving Hamiltonian with magnetic field in Landau gauge yields:

• Expression for energy state is quantized, directly proportional to B• Gives rise to De Haas-van Alphen and Shubnikov-de Haas effects –

peaks at distinct • In material with impurities, small differences in energy between states

of same Landau level

Quantum Hall Effect: Qualitative Explanation

• Effect seen by changing magnetic field and holding VH or J constant

• As B increases, increased degeneracy of states

• Two kinds of Landau levels– Localized: do not carry current– Extended: carry current

• If Fermi level is in localized Landau energy state, constant J, constant σxy

• If Fermi level is in extended Landau state, decrease in J, increase in σxy

Quantum Hall Effect: Streda Derivation

Lenz’s Law

Number of electron states in Landau level

Fractional Quantum Hall Effect

• Discovered 1982• Peaks seen at fractional values with odd

denominators• Due to electron electron interactions and impurities

in system

Hall Resistance vs. Magnetic field graph for heterojunction displaying fractional QHE. From Eisenstein and Stormer.

Applications

• Metrology– Obtaining an exact quantum of resistance

• Sensors– Detecting field changes– Effective use in:

• Cars• Keyboards• Calculating Flux Density

Metrology

• Obtaining a quantum of resistance has allowed for a definition of a new standard of electrical resistance.

• This quantum is known as the von Klitzing constant.• Can effectively calibrate resistance measurements.• Can find a precise determination of the fine

structure constant used in QED.

von Klitzing constant

Sensors

• Simplest form: analog transducer– Returns a voltage

• Can be used as a timing device for motors and discrete position state detectors.

A Hall sensor can be used in timing operations for tachometers and similar applications.

It can also be used to detect the position of a magnetic object in a system.