Page 302 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities
and the thin upper layer of thickness as i =
wave equation holds d 2 . In each material the
2
1 ∂ u
2 yi
,
∇ u – ------------------ = 0 i = 12 (7.140)
yi
c βi ∂t 2
We assume here that the wave travels in the x-direction, that z points into
the depth of the half-space, and (through hindsight!) that we only have
transverse vibrations. Showing that this is so requires more space than
available. On the free surface of the layer we have zero traction, hence

∂u y2
---------- = 0 (7.141)
∂z
z = – d
Between the thin layer and the half-space we require displacement conti-
nuity, hence
u = u (7.142)
y2 y1
z = 0 z = 0
In addition, we also require that the stress field is continuous at this point,
so that

∂u ∂u
2 y2 2 y1
ρ c ---------- = ρ c ---------- (7.143)
1 β1
2 β2
∂z ∂z
z = 0 z = 0
The wave is assumed to have a shape fz() in the z-direction that travels
in the x-direction, so that we make the following ansatz for the displace-
ment
,
u = f z()exp ( ωt – k x) i = 12 (7.144)
yi i l
which we insert into (7.140) to obtain
2
∂ f z() 2  c l  2
1
------------------ – ( k β) f z() = 0 β = -------  – 1 (7.145a)
c 
l
1
2
∂z β2

Semiconductors for Micro and Nanosystem Technology 299