112
5 Mechanical Properties
surroundings (see Table 3.8) with the four neighbors being
69 Ga or
71 Ga. The natural isotope mix
is an ‘alloy’
69 Ga x
71 Ga 1−x As with x = 0.605. The configurations with T d symmetry contribute
one peak each, the lowest (
71 Ga surrounding) and highest (
69 Ga surrounding) energy transitions.
The configurations with C 3v and C 2v symmetry contribute each with 2 and 3 nondegenerate modes,
respectively.
The vibrations of impurity complexes have been discussed in [387].
5.2.8 Phonons in Alloys
In an alloy of the type AB 1−x C x , the phonon frequencies will depend qualitatively and quantitatively on
the ternary composition [388]. For the binary end materials AB and AC clearly TO and LO frequencies
exist. The simplest behavior of the alloy is the one-mode behavior (Fig. 5.20d) where the mode frequencies vary continuously (and approximately linearly) with the composition. The oscillator strength
(LO–TO splitting, (9.86)) remains approximately constant. In many cases, the two-mode behavior is
observed where the LO–TO gap closes (accompanied by decreasing oscillator strength) and a localized
vibrational mode and a gap mode occur for the binary end materials (Fig. 5.20a). Also, a mixed-mode
behavior (Fig. 5.20b, c) can occur.
The masses of the three constituent atoms will be M A , M B , and M C . Without limiting the generality
of our treatment, we assume M B < M C . From the considerations in Sect. 5.2.7 on LVM and gap modes,
the condition
M B < M A , M C
(5.47)
for two-mode behavior can be deduced. This ensures a LVM of atom B in the compound AC and a gap
mode of atom C in the compound AB. However, it turns out that this condition is not sufficient, e.g.
Na 1−x K x Cl fulfills (5.47) but exhibits one-mode behavior. From a modified REI
4 model (for k ∼ 0
modes) it has been deduced that
(a)
(b)
400
450
600
550
500
GaP
0
20
40
60
80
100
27
Al Ga
28
Si Ga
11
B P
14
N P
10
B P
12 C Ga
-1
Fig. 5.18 a Numerical simulation of a linear chain model for GaP (M 1 = 31, M 2 = 70). Energy of local vibrational
modes (dashed (solid) line): substitution on P (Ga) site) in units of the optical phonon frequency at ( m = 45.4 meV,
cf. Fig. 5.10a). The grey areas indicate the acoustic and optical phonon bands. Solid squares are experimental data
(from [381]), scaled to the theoretical curve for the 27 Al Ga LVM frequency. b Differential transmission spectrum of GaP
structure (nitrogen-doped layer on zinc-doped compensated substrate) against pure crystal (T = 77 K). Data from [386]
4 Random element isodisplacement.
5 Mechanical Properties
surroundings (see Table 3.8) with the four neighbors being
69 Ga or
71 Ga. The natural isotope mix
is an ‘alloy’
69 Ga x
71 Ga 1−x As with x = 0.605. The configurations with T d symmetry contribute
one peak each, the lowest (
71 Ga surrounding) and highest (
69 Ga surrounding) energy transitions.
The configurations with C 3v and C 2v symmetry contribute each with 2 and 3 nondegenerate modes,
respectively.
The vibrations of impurity complexes have been discussed in [387].
5.2.8 Phonons in Alloys
In an alloy of the type AB 1−x C x , the phonon frequencies will depend qualitatively and quantitatively on
the ternary composition [388]. For the binary end materials AB and AC clearly TO and LO frequencies
exist. The simplest behavior of the alloy is the one-mode behavior (Fig. 5.20d) where the mode frequencies vary continuously (and approximately linearly) with the composition. The oscillator strength
(LO–TO splitting, (9.86)) remains approximately constant. In many cases, the two-mode behavior is
observed where the LO–TO gap closes (accompanied by decreasing oscillator strength) and a localized
vibrational mode and a gap mode occur for the binary end materials (Fig. 5.20a). Also, a mixed-mode
behavior (Fig. 5.20b, c) can occur.
The masses of the three constituent atoms will be M A , M B , and M C . Without limiting the generality
of our treatment, we assume M B < M C . From the considerations in Sect. 5.2.7 on LVM and gap modes,
the condition
M B < M A , M C
(5.47)
for two-mode behavior can be deduced. This ensures a LVM of atom B in the compound AC and a gap
mode of atom C in the compound AB. However, it turns out that this condition is not sufficient, e.g.
Na 1−x K x Cl fulfills (5.47) but exhibits one-mode behavior. From a modified REI
4 model (for k ∼ 0
modes) it has been deduced that
(a)
(b)
400
450
600
550
500
GaP
0
20
40
60
80
100
27
Al Ga
28
Si Ga
11
B P
14
N P
10
B P
12 C Ga
-1
Fig. 5.18 a Numerical simulation of a linear chain model for GaP (M 1 = 31, M 2 = 70). Energy of local vibrational
modes (dashed (solid) line): substitution on P (Ga) site) in units of the optical phonon frequency at ( m = 45.4 meV,
cf. Fig. 5.10a). The grey areas indicate the acoustic and optical phonon bands. Solid squares are experimental data
(from [381]), scaled to the theoretical curve for the 27 Al Ga LVM frequency. b Differential transmission spectrum of GaP
structure (nitrogen-doped layer on zinc-doped compensated substrate) against pure crystal (T = 77 K). Data from [386]
4 Random element isodisplacement.