2D electron gas conductivity
In degenerate n-type semiconductors at low temperature T <<
(εF -εo ) ,
electron states under Fermi energy εF
are filled and states with energy ε >
εF are empty (see Fig.1).
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
εk + (Px2 +
where εk 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.
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 semi-classically. These fluctuations
lead to fluctuations of εo(x,y).
Therefore the whole plane (x,y) is divided into filled by electrons
conducting regions with εF >
εo and isolated regions with
εF < εo . For small
εF 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
εF . Drag the line by mouse
to change εF.
Correlation radius Rc of the random potential will be
2D contour plot
Drag the black line on the right by mouse to change
εF. 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
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.
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.)
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 Quantum-Hall Transition"
Rudolf A. Romer, cond-mat/0106004
- "QHE Today"
V. J. Goldman, cond-mat/9907153
- "The Quantum Hall Effect: Novel Excitations and Broken Symmetries"
Steven M. Girvin, cond-mat/9907002
- "Scaling theory of the integer quantum Hall effect"
B. Huckestein, Rev. Mod. Phys. 67, 357-396 (1995)
- "Integral quantum Hall effect for nonspecialists"
D. R. Yennie, Rev. Mod. Phys. 59, 781-824 (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"
- "Composite Fermions in Fractional Quantum Hall Systems"
John J. Quinn, Arkadiusz Wojs, cond-mat/0001426
- "Fermi liquids and Luttinger liquids"
H.J. Schulz, G. Cuniberti, P. Pieri, cond-mat/9807366
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updated 9 March 2002