Environmental Engineering Reference
In-Depth Information
denotes the phonon wavevector) by
dk
y
2
π
ω
m
(
k
)
v
xm
(
k
)
∂
f
(
ω
m
(
k
),
T
)
dk
x
2
π
σ
ballistic
(
T
)
/
w
=
,
∂
T
m
v
x
>
0
(3.69)
where
v
xm
(
k
)
=
∂ω
m
(
k
)
/∂
k
x
is the phonon group velocity along the
direction (i.e., the transport direction) and
f
is the Bose-Einstein
distribution function [42].
Similarly,ballisticthermalconductanceperwidthcontributedby
electrons as a function of temperature (
σ
ballistic
(
T
)
/
w
) is related to
the energy dispersion of electrons (
ε
m
(
k
), in which
m
is the index of
electronic bands and
k
denotes the electron wave vector) by
dk
y
2
π
ballisti
c
(
T
)
/
w
=
2
m
dk
x
2
π
ε
m
(
k
)
−
μ
]
v
xm
(
k
)
∂
f
FD
(
ε
m
(
k
),
T
)
∂
T
e
σ
[
,
v
x
>
0
(3.70)
where the extra factor of 2 accounts for the spin degeneracy of
each band,
μ
k
x
is the
electron group velocity along the
x
direction (i.e., the transport
direction) and
f
FD
is the Fermi-Dirac distribution function [42].
The ballistic thermal conductance contributed by electrons and
the ballistic electronic conductance satisfy the Wiedemann-Frantz
law. Both of them will change if the Fermi-level of grapheme is
shifted.Theelectronicthermalconductanceisimportantonlyatlow
temperatures and discussed in detail in Ref. [42]. Here we focus on
the thermal conductance contributed by phonons.
Oncewegetthephonondispersionofgraphene,ballisticthermal
conductance can be easily calculated by the Landauer formula. The
phonon dispersion can be calculated using the empirical potential
as in Ref. [42] or using the more accurate first-principles methods.
Fig. 3.10 presents the phonon dispersion of graphene obtained by
DFT calculations. Each unit cell of graphene contains two carbon
atoms, thus the phonon dispersion of graphene includes six phonon
branches: (i) out-of-plane acoustic (ZA) and out-of-plane optical
(ZO) phonons, which correspond to vibrations along the
z
direction
(i.e., perpendicular to the graphene plane); (ii) transverse acoustic
(TA) and transverse optical (TO) phonons, which correspond to
in-plane transverse vibrations; (iii) longitudinal acoustic (LA) and
is the Fermi energy,
v
xm
(
k
)
=
∂ε
m
(
k
)
/∂
