8 Controlling Thermal Radiation with Surface Waves
309
We illustrate schematically in Fig. 8.15b the regions for which one can expect
frustrated modes. The discussion is performed using the real part of the index which
is a fair approximation when dealing with low lossy materials. From Fig. 8.15b it
is obvious that due to the frustrated internal reflection the number of contributing
modes for the heat flux increases, but is still limited to κ < n ≈ ω/c. This discussion
is valid for media with a real part of the refractive index and a small imaginary part
responsible for the emission and absorption. For polar materials such as SiC for
example, there is a frequency range in the IR where the dielectric permittivity can
have a negative real part between the longitudinal and the transverse frequencies ω L
and ω T . Note in Fig. 8.15c, that in the so-called reststrahlen region ω T < ω < ω L
no optical phonons can be excited. Within this frequency band the permittivity is
negative so that the material behaves effectively like a metal, i.e., the reflectivity is
close to one.
8.2.3.2 Surface Modes
Another kind of evanescent mode is responsible for the tremendous increase of the
heat flux at nanoscale, the so-called surface phonon polariton [61]. We now consider
two SiC interfaces across a gap with a gap distance d smaller than the attenuation
length of the surface mode i.e., 1/Im
ω 2 /c 2 − κ 2
SPhP
where κ SPhP is the surface
phonon polariton wavevector. In that case, the two surface modes are coupled. This
coupling removes the two-fold degeneracy and produces two branches. They can be
found by solving the equation [102]
−Im(r p )
2
+ Re(r p )
2
+ 2iIm(r p )Re(r p )
e
−2Im(k z0 d)
= 1.
(8.30)
It is of upmost importance in this context to realize that when this condition is
satisfied, the denominator of the transmission factor approaches zero and therefore,
the transmission factor for these modes is almost 1. In other words, the enhancement
due to the resonance compensates the exponential decay across the gap. Hence,
entails that these modes have a large contribution to the heat transfer. These modes
are schematically illustrated in Fig. 8.16a. Figure 8.16b shows the dispersion relation
of the surface phonon polariton. Here, the key feature is the fact that the dispersion
relation becomes almost flat for very large values of the wavevector. This entails that
the number of modes becomes very large at the asymptote frequency. It is important
to clarify a technical point here. When plotting the dispersion relation, different
choices can be made. A usual choice consists in taking κ complex and ω real. This
choice leads to a dispersion relation where the flat asymptote has been replaced by
a backbending. This suggests (incorrectly) that the density of states is no longer
large in the presence of losses. It has been shown in [2] that when dealing with the
density of states, the relevant choice is real κ and complex ω. This choice leads to
a flat asymptote of the dispersion relation. This asymptote yields the divergence of
309
We illustrate schematically in Fig. 8.15b the regions for which one can expect
frustrated modes. The discussion is performed using the real part of the index which
is a fair approximation when dealing with low lossy materials. From Fig. 8.15b it
is obvious that due to the frustrated internal reflection the number of contributing
modes for the heat flux increases, but is still limited to κ < n ≈ ω/c. This discussion
is valid for media with a real part of the refractive index and a small imaginary part
responsible for the emission and absorption. For polar materials such as SiC for
example, there is a frequency range in the IR where the dielectric permittivity can
have a negative real part between the longitudinal and the transverse frequencies ω L
and ω T . Note in Fig. 8.15c, that in the so-called reststrahlen region ω T < ω < ω L
no optical phonons can be excited. Within this frequency band the permittivity is
negative so that the material behaves effectively like a metal, i.e., the reflectivity is
close to one.
8.2.3.2 Surface Modes
Another kind of evanescent mode is responsible for the tremendous increase of the
heat flux at nanoscale, the so-called surface phonon polariton [61]. We now consider
two SiC interfaces across a gap with a gap distance d smaller than the attenuation
length of the surface mode i.e., 1/Im
ω 2 /c 2 − κ 2
SPhP
where κ SPhP is the surface
phonon polariton wavevector. In that case, the two surface modes are coupled. This
coupling removes the two-fold degeneracy and produces two branches. They can be
found by solving the equation [102]
−Im(r p )
2
+ Re(r p )
2
+ 2iIm(r p )Re(r p )
e
−2Im(k z0 d)
= 1.
(8.30)
It is of upmost importance in this context to realize that when this condition is
satisfied, the denominator of the transmission factor approaches zero and therefore,
the transmission factor for these modes is almost 1. In other words, the enhancement
due to the resonance compensates the exponential decay across the gap. Hence,
entails that these modes have a large contribution to the heat transfer. These modes
are schematically illustrated in Fig. 8.16a. Figure 8.16b shows the dispersion relation
of the surface phonon polariton. Here, the key feature is the fact that the dispersion
relation becomes almost flat for very large values of the wavevector. This entails that
the number of modes becomes very large at the asymptote frequency. It is important
to clarify a technical point here. When plotting the dispersion relation, different
choices can be made. A usual choice consists in taking κ complex and ω real. This
choice leads to a dispersion relation where the flat asymptote has been replaced by
a backbending. This suggests (incorrectly) that the density of states is no longer
large in the presence of losses. It has been shown in [2] that when dealing with the
density of states, the relevant choice is real κ and complex ω. This choice leads to
a flat asymptote of the dispersion relation. This asymptote yields the divergence of