Environmental Engineering Reference
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for the SV wave. In these expressions,
ω
Pn
=
n
π
v
L
/
W
III
and
ω
SVn
=
n
π
v
T
/
W
III
denote the cutoff frequency of the P and SV wave in the
outgoing lead, respectively. Thus, the total transmission coe
cient
can beexpressed as
τ
=
τ
Pm
+
τ
SVm
(4.88)
m
,
ω
Pm
<ω
m
,
ω
SVm
<ω
with
ω
Pm
=
m
π
v
L
/
W
III
and
ω
SVm
=
m
π
v
T
/
W
III
.
We stress that in this formalism, once we know how to handle
the simple T-shaped structure, the complicated problem can be also
done similarly. For instance, for the complicated quantum systems
such as the rough surface, catenoidal contacts, and so on, we firstly
subdivide all the system into very small segments so that each
segment can be regarded to have a uniform cross area. Then one
tries to calculate the scattering matrices of arbitrary two successive
segments by using the above procedures, so as to obtain the total
scattering matrix connecting the incoming lead to the outgoing
lead. As long as the length of each segment is small enough, such
the scheme is very consistent with analytic calculations, which is
also confirmed by Tanaka
et al.
[65]. In the following section,
we will numerically describe the properties of low temperature
ballisticthermaltransportbyphononsinlow-dimensionalquantum
structures.
4.3 Properties of Low-Temperature Ballistic Thermal
Transport by Phonons in Low-Dimensional Quantum
Structures
4.3.1
Properties of Ballistic Thermal Transport in 2D
Quantum Structures
Here, we consider a 2D quantum system in the
x
-
y
plane, in which
the inhomogeneities are embedded between two perfect semi-
infinite 2D quantum wires. In such a structure, we usually divide
the entire structure into three regions: the central scattering region
and the two lead regions, which are connected separately to the hot
and cold reservoirs. It is assumed that the contacts between the
two leads and reservoirs are perfect and the elastic scattering for
