Environmental Engineering Reference
In-Depth Information
the defective graphene ribbons calculated by the non-equilibrium
Green's function [52]. This is because that the long wavelength
phonons with
ω
→
0 are hardly scattered by inhomogeneities in T-
shapedquantumstructure.Thiswillleadtoanimportantconclusion
that all the thermal conductance has the universal quantum values
in the limit
T
0, regardless of the geometric parameters of
inhomogeneities.Whenthefrequencyisincreased,thetransmission
coe
cient is reduced due to the interface scattering, and such the
reduction is more evident for larger
W
II
. From Fig. 4.2a, it is also
seen that for a given frequency, the phonon transmission exhibits
a periodic pattern as a function of the transverse width of T-shaped
structure. However, for larger
b
(Fig. 4.2b), when the frequency is
addeduptoacertaindegreewherethe1
th
modecanbejustexcited
in the T-shaped structure, the periodic pattern of the transmission
coe
cient is destroyed strongly with some irregular oscillations.
This is mainly due to the coupling effect between the higher modes
being excited in the scattering region and the incident zero mode.
In general, the more modes couple with the incident mode, the
more complex is the transmission coe
cient. As a result, the
transmissioncoe
cientbecomesmorecomplicatedwithincreasing
the frequency (not shown). An important feature in Fig. 4.2 is
that the transmission coe
cient presents many dips at certain
frequency, which are clearly distinguished from the dips originating
from the van singularity of the phonon branches. Especially, some
dips are of
τ
0
=
0 indicating that the resonant reflection behavior
occursinthiscase.Similarphenomenoncanalsobeobservedforthe
electron transport [86] or for higher order phonon modes (
m
→
1)
[57]. As discussed above, the resonant reflection can be understood
as a result of the resonant coupling between the incoming zero
mode and many higher order modes being excited in T-shaped
structure.
In terms of Eq. 4.3, one can study and discuss the thermal
conductance under the influence of external factors, i.e., temper-
ature and structural parameters, as also shown by Li
et al.
[58].
For the T-shaped junction example, although the transmission
coe
cient
τ
m
has many irregular oscillations as mentioned above,
these oscillations can be not observed in the thermal conductance,
andinsteadtheoverallthermalconductanceincreasemonotonously
≥
