244
8 Transport
Table 8.5 Fröhlich coupling constant α for various semiconductors. Data from [165]
GaSb
GaAs
GaP
GaN
InSb
InAs
InP
InN
0.025
0.068
0.201
0.48
0.022
0.045
0.15
0.24
3C-SiC
ZnO
ZnS
ZnSe
ZnTe
CdS
CdSe
CdTe
0.26
1.19
0.63
0.43
0.33
0.51
0.46
0.35
m p = m
∗
1 +
α
6
+ 0.025 α
2
+ · · ·
,
(8.39)
for α ≤ 1, with m
∗ being the band mass as defined in Sect. 6.9.2 and α the Fröhlich coupling constant
5
α =
1
2
e
2
2 m ∗
ω LO
1
∞
−
1
0
.
(8.40)
This process it called the polaronic effect and requires additional energy [786, 789]. Often, the polaron
mass is given as m p = m
∗
/(1 − α/6) which is the result of perturbation theory [789] and an approximation to (8.39) for small α.
For large coupling parameter, α 1, the polaron mass is given by [787]
m p = m
∗ 16
81 π 4 α
4
.
(8.41)
The energy of the electron is lowered due to the interaction with the lattice. The energy E 0 for k = 0
is given, relative to the uncoupled case, by
E 0 = −
α + 0.0098 α
2
+ · · ·
ω 0 , α ≤ 1
(8.42a)
E 0 = −
2.83 + 0.106 α
2
ω 0 , α 1
(8.42b)
Numerical results are reported in [790].
Polarons in semiconductors are typically ‘large’ or Fröhlich-type polarons, i.e. the coupling constant
is small (Table 8.5). The dressing with phonons (as the ion displacement is called in a quantummechanical picture) is then only a perturbative effect and the number of phonons per electron (≈α/2)
is small. If α becomes large (α > 1, α ∼ 6), as is the case for strongly ionic crystals such as
alkali halides, the polaron becomes localized by the electron–phonon interaction
6 and hopping occurs
infrequently from site to site.
8.7.2 Small Polarons
In a polaron, the charge carrier (electron or hole) sits in a potential well resulting from the ionic
displacements it created. In some materials, the shape and strength of this potential well is such that
the charge is confined to a volume of approximately one unit cell or less. In this case, the polaron is
5 This constant is part of the matrix element in the Hamiltonian of the electron–phonon interaction and is related to the
electric field created by LO phonons, as given in (9.29).
6 One can think about it in the way that the electron strongly polarizes the lattice and digs itself a potential hole out of
which it can no longer move.
8 Transport
Table 8.5 Fröhlich coupling constant α for various semiconductors. Data from [165]
GaSb
GaAs
GaP
GaN
InSb
InAs
InP
InN
0.025
0.068
0.201
0.48
0.022
0.045
0.15
0.24
3C-SiC
ZnO
ZnS
ZnSe
ZnTe
CdS
CdSe
CdTe
0.26
1.19
0.63
0.43
0.33
0.51
0.46
0.35
m p = m
∗
1 +
α
6
+ 0.025 α
2
+ · · ·
,
(8.39)
for α ≤ 1, with m
∗ being the band mass as defined in Sect. 6.9.2 and α the Fröhlich coupling constant
5
α =
1
2
e
2
2 m ∗
ω LO
1
∞
−
1
0
.
(8.40)
This process it called the polaronic effect and requires additional energy [786, 789]. Often, the polaron
mass is given as m p = m
∗
/(1 − α/6) which is the result of perturbation theory [789] and an approximation to (8.39) for small α.
For large coupling parameter, α 1, the polaron mass is given by [787]
m p = m
∗ 16
81 π 4 α
4
.
(8.41)
The energy of the electron is lowered due to the interaction with the lattice. The energy E 0 for k = 0
is given, relative to the uncoupled case, by
E 0 = −
α + 0.0098 α
2
+ · · ·
ω 0 , α ≤ 1
(8.42a)
E 0 = −
2.83 + 0.106 α
2
ω 0 , α 1
(8.42b)
Numerical results are reported in [790].
Polarons in semiconductors are typically ‘large’ or Fröhlich-type polarons, i.e. the coupling constant
is small (Table 8.5). The dressing with phonons (as the ion displacement is called in a quantummechanical picture) is then only a perturbative effect and the number of phonons per electron (≈α/2)
is small. If α becomes large (α > 1, α ∼ 6), as is the case for strongly ionic crystals such as
alkali halides, the polaron becomes localized by the electron–phonon interaction
6 and hopping occurs
infrequently from site to site.
8.7.2 Small Polarons
In a polaron, the charge carrier (electron or hole) sits in a potential well resulting from the ionic
displacements it created. In some materials, the shape and strength of this potential well is such that
the charge is confined to a volume of approximately one unit cell or less. In this case, the polaron is
5 This constant is part of the matrix element in the Hamiltonian of the electron–phonon interaction and is related to the
electric field created by LO phonons, as given in (9.29).
6 One can think about it in the way that the electron strongly polarizes the lattice and digs itself a potential hole out of
which it can no longer move.