Page 126 - Science at the nanoscale

8:11
RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Low-Dimensional Nanostructures
116
The number of states per unit volume n is:
3/2

N

8π
(2mE)
(6.9)
=
n =
3
3
h
L
3
The final DOS as a function of energy g(E) is the derivative of
this population n with respect to energy:
3/2
dn
√
4π (2m)
(6.10)
E
=
g(E) =
3
dE
h
This 3D DOS function g(E) represents the number of electron
states per unit volume per unit energy at energy E. This expres-
sion can be applied to bulk 3D materials, and is independent of
the dimension L.
From Eq. (6.4), the carrier density n in a 3D bulk semiconductor
can be obtained by integrating the product of the 3D DOS function
g(E) and the probability density function f (E) over all possible
states, from the bottom of the conduction band E c , to the top of
the conduction band:
∞
∞
Z
Z
(6.11)
g(E) f (E)dE
n =
n(E)dE =
Ec
Ec
The integral in Eq. (6.11) is illustrated by the shaded area in
Fig. 6.5. f (E) is a step function with a gentle tail for T > 0. g(E) is
√
E function. Hence
the 3D DOS function which has the form of a
although the DOS increases with energy (i.e. there are more avail-
able states at higher energy), the probability of occupation drops
sharply so only the bottom of the conduction band is occupied by ch06
electrons. The actual location of the top of the conduction band
does not need to be known as the Fermi function goes to zero at
higher energies. This is why the upper limit of the integrals in
Eq. (6.11) is replaced by infinity.
Substituting the expressions for g(E) and f (E) in Eq. (6.11),
we obtain the expression for career density n 0 (where subscript
0 indicates that the system is at thermal equilibrium):
∞
Z
4π(2m ) 1
e
∗ 3/2 p
n 0 = 3 E − E c E−E dE (6.12)
h F
Ec 1 + e kT