5.3 Elasticity
121
Fig. 5.30 Elastic constants as a function of ionicity for various semiconductors with diamond or zincblende (circles) and
wurtzite (squares) structure. Constants are normalized by the modulus C 0 = e 2 /d 4 , d being the average nearest-neighbor
distance. a Bulk modulus, B ∗ = (C 11 + 2C 12 )/(3C 0 ), (b, c) shear moduli, b C ∗
S = (C 11 − C 12 )/C 0 , c C ∗
44 = C 44 /C 0 .
Solid lines are a simple model as discussed in [402]. Adapted from [403]
generally depends on the normal direction n of a strain. It is equivalent to 1/S 11 of (5.57a).
For isotropic material Y and the Poisson ratio ν are related to the elastic constants of cubic
material by
Y = C 11 −
2 C
2
12
C 11 + C 12
(5.61a)
ν =
C 12
C 11 + C 12
.
(5.61b)
For isotropic materials also Lamé’s constants λ and μ are used. They are given by
10 C 11 = λ + 2μ,
C 12 = λ and C 44 = μ (note that C 0 according to (5.58) is zero).
The bulk modulus B (inverse of the compressibility),
1
B
= −
1
V
∂ V
∂ p
,
(5.62)
for the zincblende crystal is given as
B =
C 11 + 2 C 12
3
.
(5.63)
We note that Y , ν and C i j of typical materials are both positive. Materials with negative Poisson ratio
are called auxetic [398–400]. Also materials with negative compressibility are possible [401].
Beyond the dependence of the elastic constants on the bond length (as materialized in the phonon
frequencies in Fig. 5.15), they depend on the ionicity. In Fig. 5.30, the elastic constants of various
zincblende semiconductors are shown as a function of the ionicity f i . The values for the elastic constants
are normalized by e
2
/a
4 , a being the average nearest-neighbor distance.
For wurtzite crystals, five elastic constant are necessary for the stress–strain relation that reads
11
10 For an isotropic material, C i jkl = λ δ i j δ kl + μ (δ ik δ jl + δ il δ jk ).
11 (C 11 − C 12 )/2 = C 1212 , C 44 = C 1313 = C 2323 .
121
Fig. 5.30 Elastic constants as a function of ionicity for various semiconductors with diamond or zincblende (circles) and
wurtzite (squares) structure. Constants are normalized by the modulus C 0 = e 2 /d 4 , d being the average nearest-neighbor
distance. a Bulk modulus, B ∗ = (C 11 + 2C 12 )/(3C 0 ), (b, c) shear moduli, b C ∗
S = (C 11 − C 12 )/C 0 , c C ∗
44 = C 44 /C 0 .
Solid lines are a simple model as discussed in [402]. Adapted from [403]
generally depends on the normal direction n of a strain. It is equivalent to 1/S 11 of (5.57a).
For isotropic material Y and the Poisson ratio ν are related to the elastic constants of cubic
material by
Y = C 11 −
2 C
2
12
C 11 + C 12
(5.61a)
ν =
C 12
C 11 + C 12
.
(5.61b)
For isotropic materials also Lamé’s constants λ and μ are used. They are given by
10 C 11 = λ + 2μ,
C 12 = λ and C 44 = μ (note that C 0 according to (5.58) is zero).
The bulk modulus B (inverse of the compressibility),
1
B
= −
1
V
∂ V
∂ p
,
(5.62)
for the zincblende crystal is given as
B =
C 11 + 2 C 12
3
.
(5.63)
We note that Y , ν and C i j of typical materials are both positive. Materials with negative Poisson ratio
are called auxetic [398–400]. Also materials with negative compressibility are possible [401].
Beyond the dependence of the elastic constants on the bond length (as materialized in the phonon
frequencies in Fig. 5.15), they depend on the ionicity. In Fig. 5.30, the elastic constants of various
zincblende semiconductors are shown as a function of the ionicity f i . The values for the elastic constants
are normalized by e
2
/a
4 , a being the average nearest-neighbor distance.
For wurtzite crystals, five elastic constant are necessary for the stress–strain relation that reads
11
10 For an isotropic material, C i jkl = λ δ i j δ kl + μ (δ ik δ jl + δ il δ jk ).
11 (C 11 − C 12 )/2 = C 1212 , C 44 = C 1313 = C 2323 .