110
5 Mechanical Properties
u
2
0 =
n +
1
2
4
ρ V ω
.
(5.40)
The number of phonons with which a vibrational mode is populated is thus directly related to the
classical amplitude square.
Phonons act with a momentum k, the so-called crystal momentum. When phonons are created,
destroyed or scattered the crystal momentum is conserved, except for an arbitrary reciprocal-space
vector G. Scattering with G = 0 is called a normal process, otherwise (for G = 0) it is called an
umklapp process.
5.2.7 Localized Vibrational Modes
A defect in the crystal can induce localized vibrational modes (LVM). The defect can be a mass defect,
i.e. one of the masses M is replaced by M d , or the force constants in the neighborhood are modified to
C d . A detailed treatment can be found in [381]. LVM are discussed, e.g., in [382–384].
First we consider the LVM for the one-dimensional, monoatomic chain. If the mass at lattice point
i = 0 is replaced by M d = M + M ( M = M/M), the displacements are given by u i = AK
|i| , A
being an amplitude, with
K = −
1 + M
1 − M
,
(5.41)
and the defect phonon frequency ω d is
ω d = ω m
1
1 −
2
M
.
(5.42)
A real frequency is obtained for | M | < 1. ω d is then higher than the highest frequency of the bulk
modes ω m =
√
4C/M (5.11). For M < 0, i.e. the mass of the defect is smaller than the mass of the
host atoms, K is negative and |K | < 1. Thus, the displacement can be written as
u i ∝ (−|K |)
|i|
= (−1)
|i| exp (+ |i| log |K |) .
(5.43)
The numerical value of the exponent is negative, thus the amplitude decreases exponentially from the
defect and indeed makes a localized vibrational mode. For small mass M d M (5.42) yields approximately ω d =
√
2C/M d . This approximation is the so-called one-oscillator model. Since typically the
extension of the localized mode is only a few lattice constants, the picture of LVM remains correct for
impurity concentrations up to ∼10
18 –10
20 cm
−3 . For higher concentrations the concept of alloy modes
has to be invoked (cf. Sect. 5.2.8).
For the case of group-III or -V substitutional impurities in group-IV semiconductors the change in
force constants (treated below) can be neglected to some extent. For silicon (M = 28) and germanium
(M = 73) the effect of various substitutions is shown in Fig. 5.17.
Now, additionally the force constants left and right of the defect are replaced by C d = C + C
( C = C/C). The displacements are still given by u i = AK
|i| , now with
K = −
(1 + M ) (1 + C )
1 − M − 2 C
.
(5.44)
5 Mechanical Properties
u
2
0 =
n +
1
2
4
ρ V ω
.
(5.40)
The number of phonons with which a vibrational mode is populated is thus directly related to the
classical amplitude square.
Phonons act with a momentum k, the so-called crystal momentum. When phonons are created,
destroyed or scattered the crystal momentum is conserved, except for an arbitrary reciprocal-space
vector G. Scattering with G = 0 is called a normal process, otherwise (for G = 0) it is called an
umklapp process.
5.2.7 Localized Vibrational Modes
A defect in the crystal can induce localized vibrational modes (LVM). The defect can be a mass defect,
i.e. one of the masses M is replaced by M d , or the force constants in the neighborhood are modified to
C d . A detailed treatment can be found in [381]. LVM are discussed, e.g., in [382–384].
First we consider the LVM for the one-dimensional, monoatomic chain. If the mass at lattice point
i = 0 is replaced by M d = M + M ( M = M/M), the displacements are given by u i = AK
|i| , A
being an amplitude, with
K = −
1 + M
1 − M
,
(5.41)
and the defect phonon frequency ω d is
ω d = ω m
1
1 −
2
M
.
(5.42)
A real frequency is obtained for | M | < 1. ω d is then higher than the highest frequency of the bulk
modes ω m =
√
4C/M (5.11). For M < 0, i.e. the mass of the defect is smaller than the mass of the
host atoms, K is negative and |K | < 1. Thus, the displacement can be written as
u i ∝ (−|K |)
|i|
= (−1)
|i| exp (+ |i| log |K |) .
(5.43)
The numerical value of the exponent is negative, thus the amplitude decreases exponentially from the
defect and indeed makes a localized vibrational mode. For small mass M d M (5.42) yields approximately ω d =
√
2C/M d . This approximation is the so-called one-oscillator model. Since typically the
extension of the localized mode is only a few lattice constants, the picture of LVM remains correct for
impurity concentrations up to ∼10
18 –10
20 cm
−3 . For higher concentrations the concept of alloy modes
has to be invoked (cf. Sect. 5.2.8).
For the case of group-III or -V substitutional impurities in group-IV semiconductors the change in
force constants (treated below) can be neglected to some extent. For silicon (M = 28) and germanium
(M = 73) the effect of various substitutions is shown in Fig. 5.17.
Now, additionally the force constants left and right of the defect are replaced by C d = C + C
( C = C/C). The displacements are still given by u i = AK
|i| , now with
K = −
(1 + M ) (1 + C )
1 − M − 2 C
.
(5.44)