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e dN
ρ(ε) = V dε
n(ε)
√
8πe
=
2m 3 ε n(ε),
(5.6)
h 3
where ρ(ε) dε is the charge per unit volume of conduction electrons
with total energy between ε and ε + dε. Assuming the potential
energy is zero everywhere, the total energy ε is given in terms of
the electron velocity v by
ε = 2
1 mv
2 .
(5.7)
301
5.2. The incident current density
by ε. In the following we deduce the one-dimensional properties
from the three-dimensional properties, making use of the planar
symmetry.
The density of energy states for a nearly free electron within the
solid is given in three dimensions (3.73) as
√
dN
8πV
=
2m 3 ε,
(5.4)
dε
h 3
where V is the volume, h is Planck’s constant, and m is the electron mass. This is the number of available states per unit energy
interval dε. Each electron energy level has two spin states with the
same total energy ε. To properly account for this, we have multiplied the right-hand side of (3.73) by two. The Pauli exclusion
principle permits, at most, one electron occupying a given state.
The expectation value of the occupation number of a state of energy ε is governed by Fermi-Dirac statistics, and is given by
−1
ε − ζ
n(ε) = exp
+ 1
,
(5.5)
kT
where k is Boltzmann’s constant, and T is the absolute temperature. The energy ζ is commonly referred to as the chemical potential per atom, and alternatively as the Fermi energy. It is easy
to verify that 0 ≤ n(ε) ≤ 1, consistent with the Pauli exclusion
principle. It follows that the average charge density within the
material is given as a function of total energy ε by