5.2 Lattice Vibrations
115
Fig. 5.22 LO (solid lines)
and TO (dashed lines)
phonon energies of
Mg x Zn 1−x O in the
wurtzite structure (A 1
symmetry: blue lines, E 1
symmetry: red lines) and in
the rocksalt phase (F 1u
symmetry: black lines).
Experimental data are
shown as symbols. Adapted
from data of [391]
A 1
E 1
F 1u
Mg Zn O
x
1-x
700
600
500
400
Phonon energy (cm )
-1
Mg-concentration x
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 5.23 Oscillator
strength f (cf. (9.86)) of
alloy phonons in (Al,Ga)N
thin films on c-Al 2 O 3 .
Based on data from [392]
statistical distribution of mean ¯
M = 1 and a standard deviation
6
σ M . The density of states is displayed
for σ M / ¯
M = 0.1–0.5. The effects as shown in Fig. 5.24 are broadening of the peaks in the DOS,
broadening of the band edges, the development of band tails into the gap and eventually a closing of
the gap. This is a typical behavior that also exists for electronic states (cmp. Fig. 6.53).
5.2.10 Topological Edge States of the Diatomic Linear Chain
Here we treat a finite diatomic linear chain with N unit cells, i.e. A–B dimers. In case of periodic
boundary conditions, the rightmost B-atom is ‘connected’ to the leftmost A-atom, but now we chose
N to be finite.
The system of 2N equations of the type ω
2 u n = (∂U/∂u n )/M n (like (5.13a) or (5.13b)) is written
in matrix form and the eigenproblem M − ω
2 1 = 0 is solved numerically [393]. Also, we fix both
ends, i.e. the displacements u
A
0 and u
B
N are forced to be zero (Fig. 5.25). We set M 1 = M 2 and find two
quite different situations for the cases (i) C 1 > C 2 and (ii) C 1 < C 2 . Case (i) is similar to the bulk case
6 In order to avoid negative values for the masses, not a normal or Gaussian distribution is used but the Gamma distribution
with the parameters G(1/σ 2 , 1/σ 2 ).
115
Fig. 5.22 LO (solid lines)
and TO (dashed lines)
phonon energies of
Mg x Zn 1−x O in the
wurtzite structure (A 1
symmetry: blue lines, E 1
symmetry: red lines) and in
the rocksalt phase (F 1u
symmetry: black lines).
Experimental data are
shown as symbols. Adapted
from data of [391]
A 1
E 1
F 1u
Mg Zn O
x
1-x
700
600
500
400
Phonon energy (cm )
-1
Mg-concentration x
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 5.23 Oscillator
strength f (cf. (9.86)) of
alloy phonons in (Al,Ga)N
thin films on c-Al 2 O 3 .
Based on data from [392]
statistical distribution of mean ¯
M = 1 and a standard deviation
6
σ M . The density of states is displayed
for σ M / ¯
M = 0.1–0.5. The effects as shown in Fig. 5.24 are broadening of the peaks in the DOS,
broadening of the band edges, the development of band tails into the gap and eventually a closing of
the gap. This is a typical behavior that also exists for electronic states (cmp. Fig. 6.53).
5.2.10 Topological Edge States of the Diatomic Linear Chain
Here we treat a finite diatomic linear chain with N unit cells, i.e. A–B dimers. In case of periodic
boundary conditions, the rightmost B-atom is ‘connected’ to the leftmost A-atom, but now we chose
N to be finite.
The system of 2N equations of the type ω
2 u n = (∂U/∂u n )/M n (like (5.13a) or (5.13b)) is written
in matrix form and the eigenproblem M − ω
2 1 = 0 is solved numerically [393]. Also, we fix both
ends, i.e. the displacements u
A
0 and u
B
N are forced to be zero (Fig. 5.25). We set M 1 = M 2 and find two
quite different situations for the cases (i) C 1 > C 2 and (ii) C 1 < C 2 . Case (i) is similar to the bulk case
6 In order to avoid negative values for the masses, not a normal or Gaussian distribution is used but the Gamma distribution
with the parameters G(1/σ 2 , 1/σ 2 ).