Environmental Engineering Reference
In-Depth Information
related to thermal conductance (
σ
)by
κ
=
σ
/
L
S
,
(3.67)
where
L
and
S
are the length and the cross sectional area of the
sample, respectively. Once
L
becomes long enough, thermal trans-
port enters into the diffusive region and
σ
∝
1
/
L
Approximately,
σ
L
≈
σ
b
l
,where
σ
b
is the ballistic thermal conductance and
l
is the
effectivephonon mean free path. Then wehave
κ
≈
σ
b
l
/
S
.
(3.68)
The extremely high thermal conductivity may come from the
extraordinary high ballistic thermal conductance. Or it may be
caused by the superiorly long phonon mean free path. What is
the case in reality? It is important to answer this question, because
thesetwopossibilitiespointtworadicallydifferentdirectionsforthe
applications of graphene.
3.4.2.1 Ballistic thermal conductance of graphene
In the following, we will study the ballistic thermal conductance
of graphene, which is actually the upper limit to the thermal
conductance. As far as we know, two approaches are available
to calculate ballistic thermal conductance of 2D periodic systems
(graphene here). The first method uses the Landauer formula that
employs ballistic phonon transmission function calculated directly
fromphonondispersionbycountingtransportchannels.Thesecond
approach does not directly deal with the quasi-2D system. Instead,
thermaltransportinquasi-1Dsystemsofdifferentwidthsisstudied.
Afterknowingthesize-dependence,thermal-transportpropertiesof
the quasi-2D system are obtained by extrapolating results of the
finite-width systems to the infinite-width limit. Both approaches
are equivalent to each other and predict same results, as we will
demonstrate below.
The Landauer formula is applicable not only for 1D systems as
we have showed, but also for 2D and 3D systems. For a 2D system,
ballistic thermal conductance per width contributed by phonons
as a function of temperature (
σ
ballistic
(
T
)
/
w
) is related to phonon
ω
m
(
k
)inwhich
m
is the index of phonon bands and
k
dispersion (
