Page 135 - Science at the nanoscale

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6.2. Electron Transport Properties in Low Dimensional Systems
where e is the elementary charge and h is Planck’s constant. In the
integer quantum Hall effect, ν takes on integer values (ν = 1, 2, 3,
etc.). However, in the fractional quantum Hall effect, ν can occur
as a vulgar fraction (ν = 2/7, 1/3, 2/5, 3/5, 5/2 etc.).
The quantisation of the Hall conductance has the important
property of being extremely precise. Actual measurements of the
Hall conductance have been found to be integer or fractional mul-
2
tiples of e /h to nearly one part in a billion. This phenomenon,
referred to as “exact quantisation”, has been shown to be a sub-
tle manifestation of the principle of gauge invariance.
allowed for the definition of a new practical standard for elec-
trical resistance — the resistance unit h/e , or approximately
25812.8 ohms, is referred to as the von Klitzing constant R K , after
Klaus von Klitzing, the discoverer of exact quantisation. In 1980,
von Klitzing made the unexpected discovery that the Hall conduc-
2
tivity was exactly quantised, and for this finding, von Klitzing
was awarded the 1985 Nobel Prize in Physics.
The 2D electron gas in a GaAs-AlGaAs heterojunction has a
Fermi wavelength which is a hundred times larger than in a metal.
This makes it possible to study a constriction with an opening
comparable to the wavelength, and much smaller than the mean
free path for impurity scattering. Such a constriction is called a
quantum point contact. In 1988, the Delft-Philips and Cambridge
groups reported the discovery of a sequence of steps in the con-
ductance of a constriction in a 2D electron gas, as its width W
was varied by means of a voltage on the gate. The experimen-
2
tal step size is twice e /h because spin-up and spin-down modes
are degenerate.
6.2.2 1D Electron Transport 2 3 It has 125 ch06
In a 1D quantum wire, electrons are now quantum mechanically
confined in two dimensions, and can only travel freely in one
dimension. In 1957 Rolf Landauer showed that the electrical con-
ductance (G = 1/R) of a 1D quantum wire where electrons travel
2 K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
3 B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988); Phys. Rev. B 43, 12431 (1991).