Page 289 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
persion surface. Thus this rather simple idealization of the crystal surface
qualitatively reproduces the features of surface phonons, namely
• Exponential amplitude decay into the depth of the crystal, and
• A range of allowed phonon frequencies terminated by the optical and
acoustical branches of the dispersion diagram.

Surface We now consider what happens when an acoustic wave in a solid travels
Acoustic close to the surface in the long wavelength limit. Recall that the wave
Waves (SAW)
equation for a 3D solid is

2
∂ u
ρ-------- = ∇• σ (7.110)
2
∂t
which, together with the elastic constitutive equation for a crystal with
cubic symmetry and the strain-displacement relation

σ = C:ε (7.111a)
1 T
ε = --- ∇u +( ( ∇u) ) (7.111b)
2
results in the wave equations for the displacement field

2
∂ u 1
ρ---------- =
2
∂t
(7.112a)
2 2 2 2 2
∂ u 1  ∂ u 2 ∂ u   ∂ u 1 ∂ u 
3
1
44 
C ---------- + ( C 12 + C ) ----------------- + ----------------- + C  ---------- + ----------
44
11
2
2
2
∂x  ∂x ∂x 2 ∂x ∂x 3  ∂x ∂x 
1
1
1 2 3
2
∂ u 2
ρ---------- =
2
∂t
(7.112b)
2 2 2 2 2
∂ u 2  ∂ u ∂ u   ∂ u ∂ u 
3
2
2
1
C ---------- + ( C + C ) ----------------- + ----------------- + C  ---------- + ----------
44 
11 2 12 44 2 2
∂x 2  ∂x ∂x 1 ∂x ∂x 3  ∂x 1 ∂x 
2
2
3
286 Semiconductors for Micro and Nanosystem Technology