8 Controlling Thermal Radiation with Surface Waves
305
associated to each mode. The nanowire conductance is proportional to the number
of modes propagating through the nanowire weighted by a transmission factor.
We are now in a position to establish a link between the two systems as discussed
in Ref. [11]. The temperature plays the role of the voltage, the energy ω plays the
role of the quantized charge e, the thermal flux Φ plays the role of the current density.
The electromagnetic modes propagating between the two half spaces play the role of
the modes of the nanowire. Each electromagnetic mode is characterized by (ω, κ).
In order to analyse Eq. (8.19) as a sum over modes, we start by writing the energy
exchanged during a time interval Δt through an area L 2 . It is given by
ΦΔt L
2
=
∞
0
dω
(2π/Δt)
[Θ(ω, T 1 ) − Θ(ω, T 2 )]
j={s,p}
d 2 κ
(2π/L) 2 T j (ω, κ; d).
(8.20)
Now, we recognize the spatial mode spacing 2π/L in κ-space. Indeed, applying
periodic boundary conditions (also called Born von Karman conditions) to a system
with size L, we get exp(ik x L) = 1 so that k x = p2π/L where p is an integer.
Hence Ldk x /2π is the number of modes contained in dk x in an interval L. Similarly,
Δtdω/2π is the number of modes in dω in an interval Δt. Finally, it turns out that
dω
2π
d 2 κ
(2π) 2 is the density of modes per unit area and per unit time. We are now in a
position of interpreting the Eq. (8.19) as a sum over modes. It has been shown in
Ref. [96] that for a gap separating two identical interfaces, the factor T j (ω, κ) can
be interpreted as a transmission factor and is always smaller than 1. This has been
generalized to the case of a gap separating two arbitrary interfaces in Ref. [11]. We
can thus obtain an estimate of the total number of channels per unit surface which
effectively participate to the transfer of energy at a given frequency by summing over
all channels with a weighting factor given by the transmission factor:
N (ω, d) =
j={s,p}
d 2 κ
(2π) 2 T j (ω, κ, d).
(8.21)
In order to identify a linear conductance for each channel ( j, κ), it is necessary
to linearize, assuming a weak temperature difference, Θ(ω, T 1 ) − Θ(ω, T 2 ) =
∂Θ
∂ T
(T 1 − T 2 ). Assuming that a channel has a transmission factor independent of the
frequency, it is possible to find the thermal conductance g 0 associated to this channel [95, 103]:
∞
0
dω
2π
[Θ(ω, T 1 ) − Θ(ω, T 2 )] ∗ g 0 [T 1 − T 2 ],
(8.22)
where
g 0 =
∞
0
dω
2π
∂Θ
∂ T
=
π 2 k 2
B T
3h
,
(8.23)
is the quantum of thermal conductance [83]. We now analyse the linearized form of
the flux:
305
associated to each mode. The nanowire conductance is proportional to the number
of modes propagating through the nanowire weighted by a transmission factor.
We are now in a position to establish a link between the two systems as discussed
in Ref. [11]. The temperature plays the role of the voltage, the energy ω plays the
role of the quantized charge e, the thermal flux Φ plays the role of the current density.
The electromagnetic modes propagating between the two half spaces play the role of
the modes of the nanowire. Each electromagnetic mode is characterized by (ω, κ).
In order to analyse Eq. (8.19) as a sum over modes, we start by writing the energy
exchanged during a time interval Δt through an area L 2 . It is given by
ΦΔt L
2
=
∞
0
dω
(2π/Δt)
[Θ(ω, T 1 ) − Θ(ω, T 2 )]
j={s,p}
d 2 κ
(2π/L) 2 T j (ω, κ; d).
(8.20)
Now, we recognize the spatial mode spacing 2π/L in κ-space. Indeed, applying
periodic boundary conditions (also called Born von Karman conditions) to a system
with size L, we get exp(ik x L) = 1 so that k x = p2π/L where p is an integer.
Hence Ldk x /2π is the number of modes contained in dk x in an interval L. Similarly,
Δtdω/2π is the number of modes in dω in an interval Δt. Finally, it turns out that
dω
2π
d 2 κ
(2π) 2 is the density of modes per unit area and per unit time. We are now in a
position of interpreting the Eq. (8.19) as a sum over modes. It has been shown in
Ref. [96] that for a gap separating two identical interfaces, the factor T j (ω, κ) can
be interpreted as a transmission factor and is always smaller than 1. This has been
generalized to the case of a gap separating two arbitrary interfaces in Ref. [11]. We
can thus obtain an estimate of the total number of channels per unit surface which
effectively participate to the transfer of energy at a given frequency by summing over
all channels with a weighting factor given by the transmission factor:
N (ω, d) =
j={s,p}
d 2 κ
(2π) 2 T j (ω, κ, d).
(8.21)
In order to identify a linear conductance for each channel ( j, κ), it is necessary
to linearize, assuming a weak temperature difference, Θ(ω, T 1 ) − Θ(ω, T 2 ) =
∂Θ
∂ T
(T 1 − T 2 ). Assuming that a channel has a transmission factor independent of the
frequency, it is possible to find the thermal conductance g 0 associated to this channel [95, 103]:
∞
0
dω
2π
[Θ(ω, T 1 ) − Θ(ω, T 2 )] ∗ g 0 [T 1 − T 2 ],
(8.22)
where
g 0 =
∞
0
dω
2π
∂Θ
∂ T
=
π 2 k 2
B T
3h
,
(8.23)
is the quantum of thermal conductance [83]. We now analyse the linearized form of
the flux: