10. Appendix

The Discovery and Very Basics of the Quantum Hall Effect 577
Different approaches can be used to deduce a quantized value for the Hall
resistance. The calculation shown in Fig. 9.39, which led to the discovery of the
QHE, is simply based on the classical expression for the Hall effect. A quantized
Hall resistance h/e 2 is obtained for a carrier density corresponding to the
filling factor one. It is surprising that this simple calculation leads to the correct
result. Laughlin was the first to try to deduce the result of the QHE in
a more general way from gauge invariance principles [2]. However, his device
geometry is rather removed from the real Hall effect devices with metallic
contacts for the injection of the current and for the measurement of the electrochemical
potential.
The Landauer–Büttiker formalism, which discusses the resistance on the
basis of transmission and reflection coefficients, is much more suitable for analyzing
the quantum Hall effect [3]. This formalism was very successful in explaining
the quantized resistance of ballistic point contacts [4] and, in a similar
way, the quantized Hall resistance is the result of an ideal one-dimensional
electronic transport. In a classical picture this corresponds to jumping orbits of
electrons at the boundary of the device. In the future, the textbook explanation
of the QHE will probably be based on this one-dimensional edge channel
transport (see Fig. 9.40).
References
1 K. v. Klitzing, G. Dorda, M. Pepper: Phys. Rev. Lett. 45, 494 (1980)
2 R.B. Laughlin: Phys. Rev. B 23, 5632 (1981)
3 M. Büttiker: Phys. Rev. Lett. 57, 1761 (1986)
4 B.J. von Wees, H. van Houten, S.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven,
D. van der Marel, C.T. Foxon: Phys. Rev. Lett. 60, 848 (1988);
D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G.
Hasko, D.C. Peacock, D.A. Ritchie, G.A.C. Jones: J. Phys. C 21, L 209 (1988)