Page 184 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Particle Statistics: Counting Particles
µ
defines the chemical potential as the change of lnZ
with respect to
N . We find that small changes dlnZ reveal the significant parameters of
the system
∂ lnZ ∂ lnZ µ
dln Z = -------------dN + -------------dβ = – ---------dN E– dβ (5.30)
∂ N ∂ β k T
B
5.2.3 Fermi-Dirac Statistics
The important difference for Fermi-Dirac statistics is that we constrain
each energy level to be occupied by only one particle. This changes the
sum in (5.25) to give

1
Q = ∏ ∑ exp ( – ( βE + α)n ) = ∏ ( 1 + e ( – α + βE i ) ) (5.31)
i
i
i n i = 0 i
We now take logarithms

lnQ = ∑ ln 1 +( e ( – α + βE i ) ) (5.32)
i
and thus obtain the average occupation number by inserting (5.32) into
(5.20)

∂
1 lnZ 1 βe ( – α + βE i ) 1
n = – ---------------- = --- -------------------------------– ( – α + βE i ) = ------------------------- (5.33)
i
β ∂
E
α +
i β 1 + e e βE i + 1
called the Fermi-Dirac distribution function after its inventors. The defi-
nition of the chemical potential follows the same procedure as in (5.29).
Pauli The reason why, in the Fermi-Dirac distribution, each energy level may
Principle be occupied by at maximum one particle, is given by the Pauli principle.
The Pauli principle does not allow two electrons to occupy the same
quantum state. They must differ at least in one quantum number. This
single fact gives rise to the orbital structure of atoms, and hence to the
large variety of molecules that can be formed.

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