Environmental Engineering Reference
In-Depth Information
Figure 4.1
Schematic diagram the T-shaped quantumstructure.
propagating along the
x
direction as follows:
N
ξ
[
A
n
e
ik
n
(
x
−
x
ξ
)
B
n
e
−
ik
n
(
x
−
x
ξ
)
]
ψ
ξ
(
x
,
y
)
η
n
(
y
),
=
+
(4.28)
n
=
0
ξ
where
x
ξ
is the reference coordinate in region
along the
x
axis,
η
n
(
y
)representstheorthonormaltransversewavefunctionofmode
n
inregion
,
A
n
,and
B
n
areconstantstobedeterminedbymatching
the boundary conditions. When the stress-free boundary condition
(
n
·∇
ψ
=
0) is employed at the edges with
n
being the vector
perpendicular to the edge, the transverse function
η
n
(
y
)canbe
expressed by
ξ
2
/
W
ξ
cos(
n
π
y
/
W
ξ
),
if
n
=
0
η
n
(
y
)
=
1
0
.
(4.29)
/
=
W
ξ
,
if
n
ψ
=
However, when the hard wall boundary condition (
0) is
employed at the edges, itshould beexpressed as
η
n
(
y
)
2
0, (4.30)
which is similar to the case of the electron propagation. It is
worthnotingthatthestress-freeboundaryconditionoftheacoustic
phonon allows the propagation of an mode with
=
/
W
ξ
sin(
n
π
y
/
W
ξ
), if
n
=
0 as one expect
for elastic waves, while the hard wall boundary condition does not.
Because of many similarities in the derived process for these two
ω
=