Page 132 - Science at the nanoscale

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Low-Dimensional Nanostructures
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6.2 ELECTRON TRANSPORT PROPERTIES IN LOW
For bulk 3D materials, the electric current in a material is pro-
portional to the voltage across it, and the material is said to be
“ohmic”, i.e. it obeys Ohm’s law (V = IR). A microscopic view
suggests that this proportionality (V ∝ I) comes from the fact
that an applied electric field superimposes a small drift velocity
on the free electrons in a metal. For ordinary currents, this drift
velocity is on the order of mm per second, which is much slower
than the speed of the electrons ∼ a million metres per second. The
electron speeds are themselves small compared to the speed of
transmission of an electrical signal down a wire, which is of the
order of the speed of light, 300 million metres per second. The
current density (electric current per unit area, J = I/A) can be
expressed in terms of the free electron density as:
J = nev
where n is the free electron density, e the electron charge, and v
the electron drift velocity. From Ohm’s law, and expressing resis-
tance in terms of conductivity σ or resistivity ρ (R = ρL/A),
V
V
(6.17)
=
J =
ρL
RA
ρ
A
which is Ohm’s law expressed in terms of current density J and
electric field E.
6.2.1 DIMENSIONAL SYSTEMS A d = E = σE (6.16) d ch06
2D Electron Transport
Electrons in a large block of material are free to travel in any
direction, forming a 3D “electron gas”. If we create a thin slab
of the material, the electrons can still travel freely in the plane of
the slab, but their motion in the third dimension is restricted. The
wave function of an electron in this dimension is represented by
a standing wave. The situation is analogous to the “particle-in-
the-box” concept introduced in Chapter 3, whereby a particle is
confined between two rigid walls of infinite potential energy from
which it cannot escape. The motion of the electron in the third
dimension is quantized and can be represented by a “ladder” of