Page 292 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities
so that we can write that
2
2
∂ ψ ∂ φ
ε = – ------------ + -------- (7.116a)
11
∂x∂z 2
∂x
2
2
∂ ψ ∂ φ
ε = – --------- + ------------ (7.116b)
13 2
∂z ∂z∂x
2
2
2
2
∂ ψ ∂ φ ∂ ψ ∂ φ
ε = --------- + ------------ = – --------- + ------------ (7.116c)
31 2 2
∂x ∂x∂z ∂z ∂x∂z
2
2
∂ ψ ∂ φ
ε 33 = ------------ + -------- (7.116d)
∂z∂x ∂z 2
We can write the stress as σ = C:ε in tensor notation or use the more
compact Voight engineering notation. For silicon, which possesses cubic
crystal symmetry,
σ 11 C 11 C 12 C 12 ε 11
σ 22 C 12 C 11 C 12 0
σ C C C ε
33 12 12 11 33
= ⋅ (7.117a)
σ C 0
23 44
σ C 2ε
31 44 31
σ C 0
12 44
C ε + C ε
12 33
11 11
C ε + C ε
12 11
12 33
C ε + C ε
11 33
12 11
= (7.117b)
0
2C ε
44 31
0
Substituting (7.116a)-(7.116d) into (7.117a) gives

Semiconductors for Micro and Nanosystem Technology 289