5.3 Elasticity
119
(b)
(c)
(d)
(a)
Fig. 5.29 Deformation of a square (a). (b) Pure hydrostatic deformation ( xx = yy = 0.2, xy = 0), (c) pure shear
deformation ( xx = yy = 0, xy = 0.2), and (d) mixed deformation ( xx = yy = 0.1, xy = 0.1)
U =
1
2
∂u l
∂ x k
C klmn
∂u n
∂ x m
d
3 r ,
(5.50)
where C is the (macroscopic) tensor of the elastic coefficients. 21 components of this tensor can be
independent. For crystals with cubic symmetry the number of independent constants is reduced to 3.
An exchange k ↔ l and m ↔ n does not matter, only six indices have to be considered (x x, yy, zz,
yz, xz, and x y). The strain components i j are symmetrized according to
i j =
1
2
∂u j
∂ x i
+
∂u i
∂ x j
.
(5.51)
The strains xx are along the main axes of the crystal as visualized in Fig. 5.29.
The stresses
8
σ kl are then given by
σ kl = C klmn mn .
(5.52)
The inverse relation is mediated by the stiffness tensor S.
kl = S klmn σ mn .
(5.53)
Typically, the strain components e i j or e i are used with the convention x x → 1, yy → 2, zz → 3,
yz → 4, xz → 5, and x y → 6 (Voigt notation):
e i j = i j (2 − δ i j ) .
(5.54)
Then, σ m = C mn e n with the C i j being the elastic constants. The x, y, and z directions are the main
axes of the cubic solid, i.e. the 100 directions.
For zincblende material, the stress–strain relation reads
9
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
σ 1
σ 2
σ 3
σ 4
σ 5
σ 6
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
C 11 C 12 C 12 0
0
0
C 12 C 11 C 12 0
0
0
C 12 C 12 C 11 0
0
0
0
0
0 C 44 0
0
0
0
0
0 C 44 0
0
0
0
0
0 C 44
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
e 1
e 2
e 3
e 4
e 5
e 6
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(5.55)
Values of the compliances for several semiconductors are given in Table 5.2. The inverse relation is
given by the matrix
8 The stress is a force per unit area and has the dimensions of a pressure.
9 C 11 = C 1111 , C 12 = C 1122 and C 44 = C 1212 = C 1221 = C 2121 = C 2112 .