Page 293 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
2
2
2 σ 11 σ 22 σ 33 σ 23 σ 31 σ 12 2 T = 2 2
⁄
C ( – ∂ ψ∂x∂z + ∂ φ∂x ) + C ( ∂ ψ∂z∂x + ∂ φ∂z )
⁄
⁄
⁄
12
11
2
2
2
2
2
2
⁄
C ( – ∂ ψ∂x∂z + ∂ φ∂x ) + C ( ∂ ψ∂z∂x + ∂ φ∂z )
⁄
⁄
⁄
12
12
(7.118)
2
2
2
2
2
2
⁄
C ( – ∂ ψ∂x∂z + ∂ φ∂x ) + C ( ∂ ψ∂z∂x + ∂ φ∂z )
⁄
⁄
⁄
12 11
0
2
2
2
⁄
2C ( – ∂ ψ∂z + ∂ φ∂x∂z)
⁄
44
0
Elasticity for We know that Love waves (through hindsight!) are pure shear waves in
Love Waves the plane of the surface of the solid, so that we choose a displacement
according to
(
,
,
,
u = ( 0 u xz) 0) (7.119)
2
In this case we do not need a scalar and vector potential, since the strain
is now
T
T ∂u ∂u
2
ε = ε 11 ε 22 ε 33 2ε 23 2ε 31 2ε 12 = 000 -------- 0 -------- 2 (7.120)
∂z ∂x
which clearly is a pure shear deformation. The stress is
0 0
σ C C C
11 11 12 12 0 0
σ C C C
22 12 11 12 0 0
σ C C C
33 12 12 11 ⋅ ∂u 2 ∂u 2
= -------- = C -------- (7.121)
σ 23 C 44 ∂z 44 ∂z
σ 31 C 44 0 0
∂u
σ 12 C 44 ∂u 2 C -------- 2
--------
∂x 44 ∂x
We see that for Raleigh and Lamb waves, two of six stress components in
silicon are zero if the displacement occurs in a plane, and in the Love
290 Semiconductors for Micro and Nanosystem Technology