Environmental Engineering Reference
In-Depth Information
Here
m
denotes the index of vibrational modes,
f
=
1
/
[exp(
ω/
k
B
T
)
−
1]isthethermaldistributionofphononsin
thehot(cold)reservoir.
τ
m
(
k
)isthephonontransmissioncoe
cient
throughallthesystem,
ω
m
(
k
)and
v
m
(
k
)arethefrequencyandgroup
velocity of the mode
m
. Since the group velocity can be canceled by
v
m
=
∂ω
m
/∂
k
, Eq. 4.1 becomes
∞
ω
d
=
π
ω
(
f
hot
−
τ
m
(
ω
J
f
cold
)
).
(4.2)
2
0
m
Note that within this approach the electronic contribution to the
heat flux is calculated in a similar way as the phonon case with
the substitutions
ω
→
−
μ
and
f
hot(cold)
→
E
g
hot(cold)
,where
E
μ
and
are the electronic energy and the chemical potential,
g
hot(cold)
is the Fermi-Dirac distribution function, which is also temperature
dependent. The thermal conductance is given by
K
=
J
/δ
T
in the
limit
δ
T
=
T
hot
−
T
cold
→
0. Therefore, at
T
=
(
T
hot
+
T
cold
)
/
2we
have
∞
d
ω
ω
∂
f
(
ω
,
T
)
∂
T
1
2
π
K
=
τ
m
(
ω
)
ω
m
m
∞
2
π
k
B
T
2
m
2
2
e
ω/
k
B
T
(
e
ω/
k
B
T
ω
=
d
ω
ω
−
1)
2
τ
m
(
ω
),
(4.3)
ω
m
with
ω
m
denoting the cutoff frequency of the mode
m
. Introducing
the scaled frequency variable
x
=
ω/
k
B
T
gives the expression as
below
∞
−
1)
2
τ
m
k
B
T
x
.
h
m
k
B
T
x
2
e
x
(
e
x
K
=
dx
(4.4)
x
m
For the moment, let us assumed that the adiabatic contact
between the two thermal reservoirs and the quantum structure is
perfect, so that
τ
m
=
1. For this idealized case, Eq. 4.4 shows
that the quantity
k
B
T
/
h
plays the role of the quantum unit of the
thermal conductance, similar to the role of
e
2
/
h
as the quantum
electrical conductance for one-dimensional wires. In the limit
T
→
0, the contribution of the thermal conductance stems from several
acoustic modes with nonzero cutoff frequency, i.e.,
x
m
=
0.
After the integration of Eq. 4.4, we have
K
N
A
K
0
,where
N
A
denotes the number of acoustic modes with zero cutoff frequency.
=
