180
7 Electronic Defect States
and accordingly the density of holes in the valence band is
p =
E V
−∞
D h (E) f h (E) dE .
(7.2)
The energy of the top of the valence band is denoted by E V , the bottom of the conduction band as
E C . We assume here parabolic band edges, i.e. effective masses m h and m e for holes and electrons,
respectively. The density of states (per volume) in the conduction band D e and valence bands D h is
given by (6.74) and (6.75).
The statistical distribution function for electrons is denoted by f e is given in thermodynamical
equilibrium by the Fermi-Dirac distribution, (E.22),
f e (E) =
1
exp
E−E F
kT
+ 1
.
(7.3)
The distribution function for holes is f h = 1 − f e ,
f h (E) = 1 −
1
exp
E−E F
kT
+ 1
=
1
exp
−
E−E F
kT
+ 1
.
(7.4)
If several hole bands (hh, lh, so) are considered, the same distribution is valid for all hole bands in
thermal equilibrium.
If the Boltzmann distribution (E.23) is a good approximation, the carrier distribution is called
nondegenerate. If the Fermi-Dirac distribution needs to be invoked, the carrier ensemble is called
degenerate. If the Fermi level is within the band, the ensemble is highly degenerate.
If the Boltzmann approximation (E.23) cannot be applied, i.e. at high temperatures or for very small
band gaps, the integral over D f cannot be analytically evaluated. In this case the Fermi integral is
needed that is defined
1 as
F n (x) =
2
√ π
∞
0
y
n
1 + exp(y − x)
dy .
(7.5)
In the present case of bulk materials n = 1/2. For large negative argument, i.e. x < 0 and |x| | 1,
F 1/2 (x) ≈ exp(x), which is the Boltzmann approximation. F 1/2 (0) = 0.765 . . . ≈ 3/4. For large
argument, i.e. x 1, F 1/2 (x) ≈ (2/
√ π)(2/3)x
3/2 . Such fairly simple approximations are plotted in
Fig. 7.1 in comparison with the Fermi integral. For computations, analytical [556–559] or numerical
approximations [560, 561] are used.
The derivative of the Fermi integral is given by F
n (x) = n F n−1 (x), n > 0. For n = 0, i.e. a
two-dimensional system, the integral can be executed explicitly, F 0 (x) = (2/
√ π) ln[1 + exp(x)].
With the Fermi integral F 1/2 (7.10) and (7.11) the free-carrier densities can be written as
n = N C F 1/2
E F − E C
kT
(7.6)
p = N V F 1/2
−
E F − E V
kT
,
(7.7)
1 Equation (7.5) is restricted to n > −1. A form without restriction is F n (x) =
1
(n+1)
∞
0
y n
1+exp(y−x) dy. The factor
2/
√ π is often omitted but must be then added explicitly in, e.g., (7.6).
7 Electronic Defect States
and accordingly the density of holes in the valence band is
p =
E V
−∞
D h (E) f h (E) dE .
(7.2)
The energy of the top of the valence band is denoted by E V , the bottom of the conduction band as
E C . We assume here parabolic band edges, i.e. effective masses m h and m e for holes and electrons,
respectively. The density of states (per volume) in the conduction band D e and valence bands D h is
given by (6.74) and (6.75).
The statistical distribution function for electrons is denoted by f e is given in thermodynamical
equilibrium by the Fermi-Dirac distribution, (E.22),
f e (E) =
1
exp
E−E F
kT
+ 1
.
(7.3)
The distribution function for holes is f h = 1 − f e ,
f h (E) = 1 −
1
exp
E−E F
kT
+ 1
=
1
exp
−
E−E F
kT
+ 1
.
(7.4)
If several hole bands (hh, lh, so) are considered, the same distribution is valid for all hole bands in
thermal equilibrium.
If the Boltzmann distribution (E.23) is a good approximation, the carrier distribution is called
nondegenerate. If the Fermi-Dirac distribution needs to be invoked, the carrier ensemble is called
degenerate. If the Fermi level is within the band, the ensemble is highly degenerate.
If the Boltzmann approximation (E.23) cannot be applied, i.e. at high temperatures or for very small
band gaps, the integral over D f cannot be analytically evaluated. In this case the Fermi integral is
needed that is defined
1 as
F n (x) =
2
√ π
∞
0
y
n
1 + exp(y − x)
dy .
(7.5)
In the present case of bulk materials n = 1/2. For large negative argument, i.e. x < 0 and |x| | 1,
F 1/2 (x) ≈ exp(x), which is the Boltzmann approximation. F 1/2 (0) = 0.765 . . . ≈ 3/4. For large
argument, i.e. x 1, F 1/2 (x) ≈ (2/
√ π)(2/3)x
3/2 . Such fairly simple approximations are plotted in
Fig. 7.1 in comparison with the Fermi integral. For computations, analytical [556–559] or numerical
approximations [560, 561] are used.
The derivative of the Fermi integral is given by F
n (x) = n F n−1 (x), n > 0. For n = 0, i.e. a
two-dimensional system, the integral can be executed explicitly, F 0 (x) = (2/
√ π) ln[1 + exp(x)].
With the Fermi integral F 1/2 (7.10) and (7.11) the free-carrier densities can be written as
n = N C F 1/2
E F − E C
kT
(7.6)
p = N V F 1/2
−
E F − E V
kT
,
(7.7)
1 Equation (7.5) is restricted to n > −1. A form without restriction is F n (x) =
1
(n+1)
∞
0
y n
1+exp(y−x) dy. The factor
2/
√ π is often omitted but must be then added explicitly in, e.g., (7.6).