Environmental Engineering Reference
In-Depth Information
Figure 3.1
Schematic diagram of a phonon transport system. Phonons
(denoted by the circles) transport from the hot reservoir to the cold
reservoir as well as in the inverse direction. This results in a net thermal
current from the hot to the cold reservoir.
transportofquasi-particlesandcollectiveexcitations.Hereweshow
the Landauer formalism forphonon transport.
Figure 3.1 schematically shows a representative thermal-
transport system in which the heat carriers are phonons. The
dispersion relation
ω
(
k
) determines the intrinsic properties of
phonons. One phonon of frequency
ω
and wave vector
k
has energy
of
ω
and a group velocity of
∂ω/∂
k
This phonon will experience
scatterings during the transport process. Assuming that it has a
transmissionprobabilityof
(
ω
,
k
),thethermalcurrentcontributed
by this phonon mode
I
(
ω
,
k
)
is
I
(
ω
,
k
)
=
ω
∂ω
∂
k
ω
ω
(
,
k
)[
f
(
,
T
H
)
−
f
(
ω
,
T
C
)],
(3.5)
where
f
is the Bose-Einstein distribution function,
T
H
and
T
C
represent the temperature of the hot and cold thermal reservoirs,
respectively. The total thermal current
I
is the summation of the
contributionof all the phonon modes:
dk
2
π
d
ω
2
π
ω
(
ω
)[
f
(
ω
,
T
H
)
−
I
= ∫
I
(
ω
,
k
)
= ∫
f
(
ω
,
T
C
)].
(3.6)
Here the transmission function
(
ω
)isdefinedas
(
ω
)
=
k
,
∂ω/∂
k
>
0
(
ω
,
k
).
(3.7)
Notethatonlythephononmodesofpositivegroupvelocitiesare
counted in the definition of
(
ω
). The thermal conductance
σ
(
T
)is
given by the temperature derivative of
I
:
