122
5 Mechanical Properties
C i j =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
C 11 C 12 C 13 0
0
0
C 12 C 11 C 13 0
0
0
C 13 C 13 C 33 0
0
0
0
0
0 C 44 0
0
0
0
0
0 C 44
0
0
0
0
0
0
1
2
(C 11 − C 12 )
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(5.64)
Experimental values for wurtzite materials are given in Table 5.3. The relation of the elastic tensor of
wurtzite and zincblende materials, in particular viewed along the 111-direction has been discussed
in [403, 404].
The bulk modulus of the wurtzite crystal is given by
B =
(C 11 + C 12 ) C 33 − 2 C
2
13
C 11 + C 12 + 2 C 33 − 4 C 13
.
(5.65)
5.3.3 Biaxial Stress
In heteroepitaxy (cf. Sect. 12.2.6), a biaxial stress situation arises, i.e. layered material is compressed
(or expanded in the case of tensile strain) in the interface plane and is expanded (compressed) in the
perpendicular direction.
12 Here, we assume that the substrate is infinitely thick, i.e. that the interface
remains planar. Substrate or wafer bending is discussed in Sect. 5.3.5.
The simplest case is epitaxy on the (001) surface, i.e. e 1 = e 2 = . The component e 3 is found
from the condition σ 3 = 0 (no forces in the z direction). All shear strains are zero. For zincblende
material it follows
100
⊥ = e 3 = −
C 12
C 11
(e 1 + e 2 ) = −
2 C 12
C 11
.
(5.66)
In Fig. 5.31 the ratio ⊥ // is depicted for GaAs and various crystal orientations; the formulas for other
orientations are more involved [409]:
110
⊥ = −
2C 12 − C 0 /2
C 11 + C 0 /2
(5.67)
111
⊥ = −
2C 12 − 2C 0 /3
C 11 + 2C 0 /3
.
(5.68)
Table 5.3 Elastic constants (in GPa) of some wurtzite semiconductors
Material
C 11
C 12
C 13
C 33
C 44
Refs.
GaN
391
143
108
399
103
[405]
AlN
410
149
99
389
125
[406]
ZnS
124
60.2
45.5
140
28.6
[407]
ZnO
206
118
118
211
44
[408]
12 In a large amount of literature and in previous editions of this book, this situation has been labelled ‘biaxial strain’.
However, there is in-plane and out-of-plane strain, thus the strain along all directions is non-zero. Since the out-of-plane
stress is zero (for zero ambient pressure), it is actually a ‘biaxial stress’ situation.
5 Mechanical Properties
C i j =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
C 11 C 12 C 13 0
0
0
C 12 C 11 C 13 0
0
0
C 13 C 13 C 33 0
0
0
0
0
0 C 44 0
0
0
0
0
0 C 44
0
0
0
0
0
0
1
2
(C 11 − C 12 )
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(5.64)
Experimental values for wurtzite materials are given in Table 5.3. The relation of the elastic tensor of
wurtzite and zincblende materials, in particular viewed along the 111-direction has been discussed
in [403, 404].
The bulk modulus of the wurtzite crystal is given by
B =
(C 11 + C 12 ) C 33 − 2 C
2
13
C 11 + C 12 + 2 C 33 − 4 C 13
.
(5.65)
5.3.3 Biaxial Stress
In heteroepitaxy (cf. Sect. 12.2.6), a biaxial stress situation arises, i.e. layered material is compressed
(or expanded in the case of tensile strain) in the interface plane and is expanded (compressed) in the
perpendicular direction.
12 Here, we assume that the substrate is infinitely thick, i.e. that the interface
remains planar. Substrate or wafer bending is discussed in Sect. 5.3.5.
The simplest case is epitaxy on the (001) surface, i.e. e 1 = e 2 = . The component e 3 is found
from the condition σ 3 = 0 (no forces in the z direction). All shear strains are zero. For zincblende
material it follows
100
⊥ = e 3 = −
C 12
C 11
(e 1 + e 2 ) = −
2 C 12
C 11
.
(5.66)
In Fig. 5.31 the ratio ⊥ // is depicted for GaAs and various crystal orientations; the formulas for other
orientations are more involved [409]:
110
⊥ = −
2C 12 − C 0 /2
C 11 + C 0 /2
(5.67)
111
⊥ = −
2C 12 − 2C 0 /3
C 11 + 2C 0 /3
.
(5.68)
Table 5.3 Elastic constants (in GPa) of some wurtzite semiconductors
Material
C 11
C 12
C 13
C 33
C 44
Refs.
GaN
391
143
108
399
103
[405]
AlN
410
149
99
389
125
[406]
ZnS
124
60.2
45.5
140
28.6
[407]
ZnO
206
118
118
211
44
[408]
12 In a large amount of literature and in previous editions of this book, this situation has been labelled ‘biaxial strain’.
However, there is in-plane and out-of-plane strain, thus the strain along all directions is non-zero. Since the out-of-plane
stress is zero (for zero ambient pressure), it is actually a ‘biaxial stress’ situation.