Continuum percolation

2D electron gas conductivity

2D gaz We consider two-dimensional (2D) electron gas, e.g. in MOSFET on silicon or GaAs/AlGaAs HEMT. Electron motion perpendicular to the layer (z axis) is quantized, therefore electron energy is
    e = ek + (Px2 + Py2)/2m*
where ek is the energy of the k-th quantum state, Px,y are components of momentum along the layer, m* is the effective mass of electron.
In degenerate n-type semiconductors at low temperature T << (eF -eo) , electron states under Fermi energy eF are filled and states with energy e > eF are empty (see Fig.1).
Potential V(x,y) fluctuates due to the presence of disorder. If De Broglie wave length of electron l is much less, then typical scale of fluctuations, then electron motion can be treated semi-classically. These fluctuations lead to fluctuations of eo(x,y). Therefore the whole plane (x,y) is divided into filled by electrons conducting regions with eF > eo and isolated regions with eF < eo . For small eF values only the most deep valleys are filled. They are isolated and current is absent. The structure is conducting only when there is infinite extending cluster. Percolation threshold is equal to <V> for infinite systems.

Percolation in random potential landscape
Drag mouse to rotate 3D image (with "shift" to zoom it). The white line (in the blue bar to the right) corresponds to the average <V> value. The yellow line corresponds to the Fermi energy eF . Drag the line by mouse to change eF. Correlation radius Rc of the random potential will be explain later.
2D contour plot
Drag the black line on the right by mouse to change eF. Percolation regions are marked by the white color (or by the black, if "burn" is turned on). Turn on "burn" to test, that percolation threshold is equal to <V>.

Percolation in random potential landscape

Let us take some V'. We "paint" regions with V(r) < V' in white and all the rest in black colors (unfortunately different colors are used in the upper applets :) Percolation threshold on white is V' = Vc value at which white infinite cluster appears. White space portion pc is similar to the percolation threshold in site percolation.
saddle In plane there is necessarily percolation in white space or in black one. (The only exeption is a saddle point shown in Fig.2, but a small devitation of V' leads to percolation in white or in black.) Therefore from symmetry with respect to replacement of white and black (random potential V(r) is symmetric with respect to <V> too) it follows, that Vc = <V> and pc = 1/2. (In 3D space percolation in white and black may take place simultaneously. In accordance with Monte-Carlo computations pc = 0.16.)
It is evident, that if correlation radius of random potential Rc = 0, then neighboring points are uncorrelated and we get site percolation problem (remember, that for square lattice pc = 0.5927). With increasing of Rc we shall pass by step from pc = 0.5927 to pc = 0.5.

Anderson localization

In quantum mechanics electron is reflected even if it moves over potential barrier. The quantum contribution reduces the whole current. This contribution increases with decreasing of temperature. In accordance with scaling theory of localization (Abrahams, Anderson, Licciardello, Ramakrishnan, 1979) this leads to Anderson localization of wave function of 2D electron as T -> 0. See also E.Abrahams, S.Kravchenko, M.Sarachik "Metallic behavior and related phenomena in two dimensions" Rev.Mod.Phys. 73,251,(2001)

Electron localization in high magnetic field. "Shoreline" percolation

The same "Random Hills" picture can illustrate one of ideas used in explanation of the Integer Quantum Hall effect (IQHE). You can read much more about electron localization, edge states (chiral Luttinger liquid) and composite fermions in the lectures referenced below.
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updated 9 March 2002