Page 124 - Science at the nanoscale

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RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Low-Dimensional Nanostructures
114
Conduction
Conduction
Conduction
Band
Band
Band
E gap
Some electrons have
High
At absolute
Temperature
energy above the Fermi
zero, 0 K
level.
Fermi
Level
f(E)
f(E)
f(E)
Valence Band
Valence Band
Valence Band
1.0
1.0
1.0
No electrons can be above the valence band
At high temperatures, some
electrons can reach the conduction
at 0 K, since none have energy above the
Fermi level and there are no available energy
band and contribute to electric current.
states in the band gap.
Schematic diagrams of the Fermi-Dirac function applied to
Figure 6.3.
the band structure of a semiconductor at different temperatures.
Density of States
6.1.2
Our discussion so far assumes that there is a uniform availability
of states for electrons in either the valence or conduction band.
The situation is more complicated for real solids and we need to
define a density of states (DOS) function g(E) to describe the
availability of states for electrons to occupy at different energies.
The electron population depends upon the product of the Fermi-
Dirac function (probability that a given state will be occupied) and
the electron density of states. The number of electrons per unit ch06
volume with energy between E and E + ∆E is given by:
n(E)∆E = g(E) f (E)∆E (6.4)
To find out how many ways there are to obtain a particular
energy in an incremental energy range dE (the differential limit of
∆E), we use the approach of the quantum mechanical ‘particle in
a box’. The energy for an infinite walled 3D box (from Eq. 3.30) is:
2
2
2
(n + n + n )h 2
y
z
x
E = (6.5)
8mL 2