Page 122 - Science at the nanoscale

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Low-Dimensional Nanostructures
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defined by:
k
where k is the Boltzmann constant. The Fermi level plays an
important role in the band theory of solids. In p-type and n-type
doped semiconductors, the Fermi level is shifted by the dopant
impurities. The Fermi level is referred to as the electron chemical
potential in the chemistry context.
6.1.1
Energy Distribution Functions
We next introduce the important statistical mechanics concept of
energy distribution functions. The distribution function f (E) is
the probability that a particle is in energy state E. f (E) is a gen-
eralization of the ideas of discrete probability to the case where
energy E can be treated as a continuous variable. Three dis-
tinctly different distribution functions are found in nature — the
Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distribution
functions; they are summarised in Fig. 6.2.
The Maxwell-Boltzmann distribution function is a classical
function used to describe a system of identical but distinguish-
able particles, e.g. ideal gas molecules, giving the well-known
Maxwell distribution of molecular speeds.
Maxwell-
1
Boltzmann
particles, e.g. Molecular speed
E/kT
Ae
distribution
(classical) f(E) = T F = E F Identical but distinguishable (6.2) ch06
1 Identical indistinguishable particles
Bose-Einstein f(E) = E/kT with integer spin (bosons), e.g.
(quantum) Ae −1
Thermal radiation, specific heat
1 Identical indistinguishable particles
Fermi-Dirac f(E) = E/kT with half-integer spin (fermions),
(quantum) Ae +1 e.g. Electrons in a metal,
conduction in semiconductor
Figure 6.2. The three distinctly different energy distribution functions
found in nature — the Maxwell-Boltzmann, Bose-Einstein, and Fermi-
Dirac distribution functions. The term A in the denominator of each dis-
tribution is a normalization term which may change with temperature.