228
8 Transport
Fig. 8.2 Scheme of microscopic carrier scattering mechanisms
μ ion.imp. =
2
7/2
(4ππ 0 r )
2
π 3/2 Z 2 e 3
√
m ∗
(kT )
3/2
N ion
1
ln(1 + b) − 1/(1 + 1/b)
,
(8.14)
with b = 4(k/l D )
2
= 8 m
∗ E (l D /)
2 . In the Thomas-Fermi screening model
l
2
D = 4π
e
2
0 r
N (E F ) =
3
π
1/3 4 m
∗ e
2
0 r 2 n
1/3
.
(8.15)
The formula (8.14) is valid only for b 1, i.e. small carrier densities. A similar formula from [720] is
μ ion.imp. =
128
√
2π ( 0 r )
2
(kT )
3/2
m ∗1/2 Z 2 N ion e 3
ln
24 m
∗
0 r (kT )
2
n e 2 2
−1
.
(8.16)
For large ionized impurity (and carrier) density (b 1), the mobility is given by [555]
μ ion.imp. =
4 e
3 1/3 π 2/3 h
n
−2/3
,
(8.17)
the value of the pre-factor being about 3 × 10
14 (Vs)
−1 .
The scattering time depends like τ ∝ (E/kT )
s on the kinetic energy; for moderate or weak scattering
s = 3/2, for very strong scattering, s = −1/2 [714].
For typical substitutional impurities, the charge of the scattering center is |Z | = 1; in oxides, oxygen
vacancies may have Z = 2. At high impurity densities, impurity clusters may form with |Z | > 1; this
will have a strong influence on the scattering rate since it proportional to Z
2 . The decrease of mobility
for N D > 10
20 cm
−3 (Fig. 8.3a) is attributed to such effect which can be described with an effective
impurity clustering charge Z D (Fig. 8.3b) [722, 723].
8 Transport
Fig. 8.2 Scheme of microscopic carrier scattering mechanisms
μ ion.imp. =
2
7/2
(4ππ 0 r )
2
π 3/2 Z 2 e 3
√
m ∗
(kT )
3/2
N ion
1
ln(1 + b) − 1/(1 + 1/b)
,
(8.14)
with b = 4(k/l D )
2
= 8 m
∗ E (l D /)
2 . In the Thomas-Fermi screening model
l
2
D = 4π
e
2
0 r
N (E F ) =
3
π
1/3 4 m
∗ e
2
0 r 2 n
1/3
.
(8.15)
The formula (8.14) is valid only for b 1, i.e. small carrier densities. A similar formula from [720] is
μ ion.imp. =
128
√
2π ( 0 r )
2
(kT )
3/2
m ∗1/2 Z 2 N ion e 3
ln
24 m
∗
0 r (kT )
2
n e 2 2
−1
.
(8.16)
For large ionized impurity (and carrier) density (b 1), the mobility is given by [555]
μ ion.imp. =
4 e
3 1/3 π 2/3 h
n
−2/3
,
(8.17)
the value of the pre-factor being about 3 × 10
14 (Vs)
−1 .
The scattering time depends like τ ∝ (E/kT )
s on the kinetic energy; for moderate or weak scattering
s = 3/2, for very strong scattering, s = −1/2 [714].
For typical substitutional impurities, the charge of the scattering center is |Z | = 1; in oxides, oxygen
vacancies may have Z = 2. At high impurity densities, impurity clusters may form with |Z | > 1; this
will have a strong influence on the scattering rate since it proportional to Z
2 . The decrease of mobility
for N D > 10
20 cm
−3 (Fig. 8.3a) is attributed to such effect which can be described with an effective
impurity clustering charge Z D (Fig. 8.3b) [722, 723].