Adiabatic Transport in the Fractional Quantum Hall Effect Regime

Abstract

The quantum Hall effect (QHE) is the phenomenon that the Hall conductance Gh is quantized in units of e²/h, as expressed by the formula

$$ {G_H} = \frac{p}{q}\frac{{{e^2}}}{h} $$

((1))

(p and q being mutually prime integers). The integer QHE (q = 1) was discovered 10 years ago by von Klitzing, Dorda, and Pepper [1] in the two-dimensional electron gas (2 DEG) confined to a Si inversion layer. The fractional QHE (q > 1 and odd) was first observed by Tsui, Störmer, and Gossard [2] in the 2 DEG at the interface of a A_xpGa_1-xAs/GaAs heterostructure. Microscopically the two effects are entirely different. The integer QHE, on the one hand, can be explained satisfactorily in terms of the states of non-interacting electrons in a magnetic field (the Landau levels). The fractional QHE, on the other hand, exists only because of electron-electron interactions [3]. Phenomenologically, however, the integer and fractional QHE are quite similar. In an unbounded 2 DEG this similarity is understood from Laughlin’s general argument [4] that: (1) The Hall conductance shows a plateau as a function of magnetic field (or Fermi energy) whenever the quasi-particle excitations in the bulk of the 2 DEG are localized by disorder; and that: (2) The value of G _H on the plateau is precisely an integer multiple p of ee*/h, where e* = e/q is the quasi-particle charge. (The product ee* appears because one e is needed to change the unit of conductance from Amperes per electron Volts to Amperes per Volts). Theory and experiment on the QHE in an unbounded 2 DEG have been reviewed in the books by Prange and Girvin [5] and by Chakraborty and Pietiläinen [6] (see also the article by MacDonald in the present volume).