Continuum percolation
2D electron gas conductivity

We consider twodimensional (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
ε =
ε_{k} + (P_{x}^{2} +
P_{y}^{2})/2m_{*}
where ε_{k} is the energy of the
kth quantum state, P_{x,y} are components of momentum along
the layer, m_{*} is the effective mass of electron.

In degenerate ntype semiconductors at low temperature T <<
(ε_{F} ε_{o }) ,
electron states under Fermi energy ε_{F}
are filled and states with energy ε >
ε_{F} are empty (see Fig.1).
Potential V(x,y) fluctuates due to the presence of disorder.
If De Broglie wave length of electron λ
is much less, then typical scale of fluctuations, then electron
motion can be treated semiclassically. These fluctuations
lead to fluctuations of ε_{o}(x,y).
Therefore the whole plane (x,y) is divided into filled by electrons
conducting regions with ε_{F} >
ε_{o} and isolated regions with
ε_{F} < ε_{o} . For small
ε_{F} 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
ε_{F} . Drag the line by mouse
to change ε_{F}.
Correlation radius R_{c} of the random potential will be
explain later.

2D contour plot
Drag the black line on the right by mouse to change
ε_{F}. 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' =
V_{c} value at which white infinite cluster appears. White space
portion p_{c} is similar to the percolation threshold in site
percolation.

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 V_{c} = <V> and p_{c} = 1/2.
(In 3D space percolation in white and black may take place simultaneously.
In accordance with MonteCarlo computations p_{c} = 0.16.)

It is evident, that if correlation radius of random potential
R_{c} = 0, then neighboring points are uncorrelated and we
get site percolation problem (remember, that for square lattice
p_{c} = 0.5927). With increasing of R_{c} we
shall pass by step from p_{c} = 0.5927 to
p_{c} = 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.
 "Percolation, Renormalization and the QuantumHall Transition"
Rudolf A. Romer, condmat/0106004
 "QHE Today"
V. J. Goldman, condmat/9907153
 "The Quantum Hall Effect: Novel Excitations and Broken Symmetries"
Steven M. Girvin, condmat/9907002
 "Scaling theory of the integer quantum Hall effect"
B. Huckestein, Rev. Mod. Phys. 67, 357396 (1995)
 "Integral quantum Hall effect for nonspecialists"
D. R. Yennie, Rev. Mod. Phys. 59, 781824 (1987)
 "The Composite Fermion: A Quantum Particle and Its Quantum Fluids"
Jainendra K. Jain, Physics Today, April 2000
 "Theories of the Fractional Quantum Hall Effect"
R.Shankar, condmat/0108271
 "Composite Fermions in Fractional Quantum Hall Systems"
John J. Quinn, Arkadiusz Wojs, condmat/0001426
 "Fermi liquids and Luttinger liquids"
H.J. Schulz, G. Cuniberti, P. Pieri, condmat/9807366
Contents
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updated 9 March 2002