5.3 Elasticity
123
(a)
(b)
Fig. 5.31 Ratio − ⊥ // for GaAs under symmetric biaxial stress. The angle θ denotes the surface normal in the 110
azimuth (θ = 0: [001], θ = 90 ◦ : [110], the maximum of ⊥ // is for [111]). (b) is a three-dimensional visualization
For wurtzite crystals and pseudomorphic growth along [00.1] the strain along the epitaxial direction
(c-axis) is given by
⊥ = −
C 13
C 33
(e 1 + e 2 ) = −
2C 13
C 33
a ,
(5.69)
where ⊥ = c = (c − c 0 )/c 0 and a = (a − a 0 )/a 0 . For symmetric biaxial in-plane stress, the ratio
⊥ // is shown in Fig. 5.32 for GaN and varying angle θ of the c-axis against the epitaxy direction. For
the growth of wurtzite on wurtzite for θ = 0, the epitaxial strain is actually asymmetric in the interface
plane. For θ = 90
◦ , e.g. the epitaxy on m-plane substrate (cmp. Fig. 3.37) (c-axis lies in-plane), the
in-plane strains are e 1 = a and e 2 = c . For θ = 90
◦ , we find
⊥ = −
C 12 a + C 13 c
C 11
.
(5.70)
The situation for pseudomorphic growth in the (Al,Ga,In)N system has been discussed for various
interface orientations in [410] (cmp. also Fig. 16.14). The strains ⊥ along the epitaxy direction and
c along the c-direction are depicted for Al 0.17 Ga 0.83 N/GaN and Mg 0.3 Ga 0.7 O/ZnO in Fig. 5.33. The
different behavior of the nitride and the oxide system, e.g. regarding the sign change of c , is due to the
fact that a is negative (positive) for Al x Ga 1−x N/GaN (Mg x Ga 1−x O/ZnO) ( c < 0 for both cases) [411].
We note that pseudomorphic growth and biaxial stress of rhombohedral/trigonal (e.g. Al 2 O 3 ) and
monoclinic (e.g. β-Ga 2 O 3 ) thin films has been discussed as well [412–414]. A general treatment for
all crystal symmetries and orientations has been provided in [415].
5.3.4 Three-Dimensional Stress
The strain distribution in two-dimensional or three-dimensional objects such as quantum wires and
dots (see also Sect. 14) is more complicated.
A simple analytical solution for the problem of a strained inclusion is only possible for isotropic
material parameters [416].
The solution for a sphere can be extended to yield the strain distribution of an inclusion of arbitrary
shape. This scheme applies only for isotropic materials and identical elastic properties of the inclusion
123
(a)
(b)
Fig. 5.31 Ratio − ⊥ // for GaAs under symmetric biaxial stress. The angle θ denotes the surface normal in the 110
azimuth (θ = 0: [001], θ = 90 ◦ : [110], the maximum of ⊥ // is for [111]). (b) is a three-dimensional visualization
For wurtzite crystals and pseudomorphic growth along [00.1] the strain along the epitaxial direction
(c-axis) is given by
⊥ = −
C 13
C 33
(e 1 + e 2 ) = −
2C 13
C 33
a ,
(5.69)
where ⊥ = c = (c − c 0 )/c 0 and a = (a − a 0 )/a 0 . For symmetric biaxial in-plane stress, the ratio
⊥ // is shown in Fig. 5.32 for GaN and varying angle θ of the c-axis against the epitaxy direction. For
the growth of wurtzite on wurtzite for θ = 0, the epitaxial strain is actually asymmetric in the interface
plane. For θ = 90
◦ , e.g. the epitaxy on m-plane substrate (cmp. Fig. 3.37) (c-axis lies in-plane), the
in-plane strains are e 1 = a and e 2 = c . For θ = 90
◦ , we find
⊥ = −
C 12 a + C 13 c
C 11
.
(5.70)
The situation for pseudomorphic growth in the (Al,Ga,In)N system has been discussed for various
interface orientations in [410] (cmp. also Fig. 16.14). The strains ⊥ along the epitaxy direction and
c along the c-direction are depicted for Al 0.17 Ga 0.83 N/GaN and Mg 0.3 Ga 0.7 O/ZnO in Fig. 5.33. The
different behavior of the nitride and the oxide system, e.g. regarding the sign change of c , is due to the
fact that a is negative (positive) for Al x Ga 1−x N/GaN (Mg x Ga 1−x O/ZnO) ( c < 0 for both cases) [411].
We note that pseudomorphic growth and biaxial stress of rhombohedral/trigonal (e.g. Al 2 O 3 ) and
monoclinic (e.g. β-Ga 2 O 3 ) thin films has been discussed as well [412–414]. A general treatment for
all crystal symmetries and orientations has been provided in [415].
5.3.4 Three-Dimensional Stress
The strain distribution in two-dimensional or three-dimensional objects such as quantum wires and
dots (see also Sect. 14) is more complicated.
A simple analytical solution for the problem of a strained inclusion is only possible for isotropic
material parameters [416].
The solution for a sphere can be extended to yield the strain distribution of an inclusion of arbitrary
shape. This scheme applies only for isotropic materials and identical elastic properties of the inclusion