Elucidating the Hall Effect
By Philip F. Schewe, AIP Public Information .
The Hall effect is named after Edwin Hall, who in 1879 observed that electrons moving longitudinally along a metal strip (under the influence of an electric field) will, if also subject to a magnetic field perpendicular to the plane of the strip, be deflected toward the side of the strip. Because of this, an excess of charge will build up one side of the strip. This Hall voltage is proportional to the strength of the magnetic field. That is, a plot of Hall voltage (or equivalently the electrical resistance of the material to the sideways current flow) versus field strength would be linear. All of this can be explained in terms of classical physics.
Later, the Hall effect would be studied in a very different setting. This time the electrons are those moving in the two-dimensional world at the interface between two semiconductors. As with many other quantum phenomena the act of confinement (the two-dimensional electron gas, or 2DEG, stuck between the semiconductors) led to quantization. A plot of Hall resistance versus field strength was no longer linear: it had become a staircase. In other words, nature would not permit just any resistance, but only allowed certain resistances dictated by fundamental quantum principles. The specific choice of semiconductor did not play a part. Klaus von Klitzing discovered this "quantum Hall effect" in 1980 and won the physics Nobel Prize in 1985. So exacting is the quantization of resistance (better than a part in many millions) that von Klitzing's experiment has since been used to define the unit of resistance.
Stormer and Tsui carried this research further. At even colder temperatures and higher magnetic fields, they discovered steps within the steps. This "fractional quantum Hall effect" (FQH) was at first hard to explain. Robert Laughlin surmised that the electrons were combining with the flux quanta of the magnetic field. Electrons are fermions, spin-half particles, and normally do not like to condense into a shared quantum state, but in combination with the flux quanta they would become bosons, spin-zero or spin-one states, which are not averse to sharing a quantum state. This is analogous to what happens in low-temperature superconductors when, first, electrons pair up (into Cooper pairs, which are bosons) and then, second, condense into the shared superconducting quantum state in which all the electrons in the supercurrent act as an ensemble. One side effect of Laughlin's conjecture was that the FQH electron ensembles could have fractional charges. That is, the ensembles acted as if they were particles (quasiparticles) with an electrical charge which was a non-integral multiple of the basic electron charge. In 1997 this hypothesis was experimentally verified in Israel and France.
Goldman and Su, Science, 17 February 1995.