Continuum percolation. 2D electron gas conductivity

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
ε =
ε_{k} + (P_{x}^{2} +
P_{y}^{2})/2m_{*} where ε_{k} is the energy of the
k-th quantum state, P_{x,y} are components of momentum along
the layer, m_{*} is the effective mass of electron.
In degenerate n-type 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 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 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.
From symmetry it follows, that percolation threshold is equal
to <V> for infinite systems.
Percolation in random potential landscape

ε_{F} Use your fingers or mouse to control the model
(hold shift key or use mouse wheel to zoom it).
Canvas is matched to your browser window.
The black strip in the ε_{F} bar to the right
corresponds to the average <V> = 0 value.
Drag the bar to change ε_{F}.
Contents updated 21 Nov 2015