Environmental Engineering Reference
In-Depth Information
basic, elementary building blocks for handling phonons, namely the
conceptual realization and its possible operation of some function
thermal (phonon) devices mentioned above [64]. According to our
previous works, here we present a review of the low temperature
ballistic thermal transport contributed by phonons in quantum
structures, of particular focus on the effects of the boundary con-
ditions, dimensions, and inhomogeneities on the ballistic thermal
conductance. In Section 4.2, we introduce the Landauer formalism
for calculating the ballistic thermal conductance in nanoscale
systems and present the necessary details of the scattering-matrix
method within the framework of elastic continuum model. In
Section 4.3, we discuss numerically the ballistic thermal-transport
propertiesoftheelasticwavesinboth2Dand3Dmodels.InSection
4.4, weprovide a summary of thestudy.
4.2 Formalism
This section is divided into three parts. First we embark on the
Landauerformulaforthephononthermalconductanceinthelinear
response limit, analogous to the electrical conductance formula.
Then we present a continuum elastic model to describe the ballistic
phonon transport. Finally, we give the scattering-matrix method of
calculating the phonon transmission coe
cient for quite a general
geometric structure shown in Fig. 4.1.
4.2.1
Landauer Formula for the Thermal Conductance
We assume that a two-terminal system is connected to hot and
cold reservoirs with a small temperature difference
δ
T
.Here
δ
T
=
T
hot
−
T
cold
and
δ
T
T
hot
(
T
cold
), that
T
hot
and
T
cold
denote the
temperature of the hot and cold reservoirs, respectively. So we can
adoptthemeantemperature
T
=
(
T
hot
+
T
cold
)
/
2asthetemperature
of the hot and cold reservoirs. According to the Landauer theory
[20-22, 25], the heat flux
J
from the hot reservoir to the cold
reservoir can beexpressed by
∞
dk
2
π
ω
m
(
k
)
v
m
(
k
)(
f
hot
−
=
f
cold
)
τ
m
(
k
).
J
(4.1)
0
m
