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Giant resonant radiative heat transfer between nanoparticles

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We show that periodic multilayered structures allow to drastically enhance near-field radiative heat transfer between nanoparticles. In particular, when the two nanoparticles are placed on each side of the multilayered structure, at the same interparticle distance the resulting heat transfer is more than five orders of magnitude higher than that in the absence of the multilayered structure. This enhancement takes place in a broad range of distances and is due to the fact that the intermediate multilayered structure supports hyperbolic phonon polaritons with the key feature that the edge frequencies of the Type I and Type II Reststrahlen bands coincide with each other at a value extremely close to the particle resonance. This allow a very high-k evanescent modes resonating with the nanoparticles. Our predictions can be relevant for effective managing of energy at the nano-scale.
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Giant resonant radiative heat transfer between nanoparticles
Yong Zhang1,2, Hong-Liang Yi1,2,*, He-Ping Tan1,2, and Mauro Antezza3,4,
1School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, P. R. China
2Key Laboratory of Aerospace Thermophysics, Ministry of Industry and Information Technology, Harbin 150001, P.
R. China
3Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F- 34095 Montpellier, France
4Institut Universitaire de France, 1 rue Descartes, F-75231 Paris, France
We show that periodic multilayered structures allow to drastically enhance near-field
radiative heat transfer between nanoparticles. In particular, when the two nanoparticles are
placed on each side of the multilayered structure, at the same interparticle distance the resulting
heat transfer is more than five orders of magnitude higher than that in the absence of the
multilayered structure. This enhancement takes place in a broad range of distances and is due
to the fact that the intermediate multilayered structure supports hyperbolic phonon polaritons
with the key feature that the edge frequencies of the Type I and Type II Reststrahlen bands
coincide with each other at a value extremely close to the particle resonance. This allow a very
high-k evanescent modes resonating with the nanoparticles. Our predictions can be relevant for
effective managing of energy at the nano-scale.
Since the pioneering work of Polder and van Hove
[1] it is well known that when two objects are brought in
proximity to each other (i.e., in the near-field regime), the
radiative heat transfer (RHT) between them may be
significantly enhanced [2]. This is caused by the
tunneling effect of evanescent modes, as surface plasmon
polaritons (SPPs) or surface phonon polaritons (SPhPs)
[39]. This effect has several implications in various
technologies for near-field energy conversion [10,11] and
data storage [12] as well as active thermal management
[13] at nanoscale with, transistors [14,15], thermal
rectifiers [1619], memories [20,21].
For these reasons, huge efforts have been done to
find new configurations where this effect can be further
enhanced or modulated. Recently, it has been shown that,
due to the presence of many-body interactions, strong
exaltation effects of heat flux are possible [2226]. Focus
has been done on the RHT between two particles in the
presence of one or two plates [2731], where the two
particles lie on the same sides of the plate, and a
significant amplification of RHT is shown at a long-range
distance. Note that in the context of Forster/Resonant
energy transfer [32], the transmission configurations
where the two dipoles are placed on each side of a slab
with finite thickness have also been adopted theoretically
and experimentally in [33,34]. Among the possible
strategies, multilayers have been extensively studied
because mutual interactions of surface polaritons at
multiple interfaces inside multilayers [3538] provide
exotic features including tuning of near-field thermal
radiation.
FIG. 1 Scheme of the RHT between two nanoparticles in the
presence of a multilayered slab. The nanoparticles lie on the
opposite sides of the intermediate structure with a total
thickness of h. The top surface of the slab coincides with the
coordinate origin. The periodic multilayered structure consists
of multiple layers of alternating materials.
In this Communication, we study the RHT between
nanoparticles which are separated by a multilayered
structure, as shown in Fig. 1. Remarkably, we show that
the particle to particle heat transfer can be up to more than
five orders of magnitude higher than that in the absence
of the intermediate structure. We explain this effect in
terms of a new hyperbolic channel having peculiar
features.
We consider two particles which are isotropic
spherical with radius R = 5nm and modeled as simple
radiating dipoles, which is valid when the center-to-
center distance L > 3R [2730]. We choose the
nanoparticles made of silicon carbide (SiC), with its
dielectric function described by the Drude-Lorentz model
[39]:
 
2 2 2 2
() LT
ii
      
   
with
high-frequency dielectric constant
= 6.7, longitudinal
optical frequency ωL = 1.83 × 1014 rad/s, transverse
optical frequency ωT = 1.49 × 1014 rad/s, and damping
= 8.97 × 1011 rad/s. The periodic multilayered structure
consists of multiple layers of alternating materials,
namely film 1 and film 2. Each material has an individual
thickness hi (i = 1, 2) and if we have N layers the total
thickness of the complete multilayered structure is
 
12
( 1) ( 1) 2h N h N h  
. We choose film 1 made by
SiC and film 2 made by a polar dielectric NaBr, with its
dielectric function given as [40]:
 
2 2 2 2
2( ) +
L T T i
     
 
, where
= 2.6,
ωL = 0.39 × 1014 rad/s, ωT = 0.25 × 1014 rad/s and = 2.6
× 1011 rad/s. The coordinate system is defined such that
the xy-plane coincides with the surface of the topmost
layer and the z-direction is orthogonal to the surface. Both
the two nanoparticles are put in proximity to the surface
at a particle-surface distance of d. The vertical positions
of the two particles are thus given by z1 = d and z2 =hd,
respectively.
We assume that the whole system is initially
thermalized at a given temperature T, and we then give
the top particle a tiny temperature increment ΔT. The only
net heat exchanged by bottom particle is thus with the top
one. When ΔT0, we get the RHT conductance
between these two particles which is the quantity we are
going to discuss in this Communication. Note that the
heat exchange between the top article and the
intermediate structure will not affect the energy
transferred between the two particles.
According to the framework of fluctuational
electrodynamics (FE), the conductance
between two
identical nanoparticles at a temperature of T can be
conveniently expressed in terms of the Green’s function
(GF) describing the system as [28],
(1)
where k0 = ω/c and
denoting the free-space wave-
vector and the particle’s electric frequency-dependent
polarizability, respectively. In the limit R<< δ (with δ
being the skin depth of the given material),
is written
in the well-known Clausius-Mossoti form
 
3
( )=4 ( ) 1 ( ) 2R
     
 

. Notice that the
expression of
predicts a nanoparticle resonance
frequency ωnp corresponding asymptotically to the
condition ε(ω) + 2 = 0, which for SiC gives ωnp = 1.756
× 1014 rad/s.
 
,nT
denotes the derivative with
respect to T of the Bose-Einstein distribution
 
 
1
, exp 1
B
n T k T


. denotes the GF.
For the two isolates particles, the well-known free-space
GF reads,
0
023
0
00
= 0 0 ,
400
ik L a
eb
kL b





G
(2)
where
0
22a ik L
and
22
00
1b k L ik L  
. In the
presence of an intermediate multilayered slab, Eq. (2) is
replaced by the transmitted GF [33,41],
 
0 2 0 1
+
tr tr tr
0
()
==
4
d,
zz
sp
sp
i k z h k z
itt
e k k

 


G M M
(3)
where
k
and
22
00
=
z
k k k
are the lateral wave-
vector to the surface and the z component of the wave-
vector in vacuum, respectively. ts and tp are the Fresnel
transmission coefficients of the total structure associated
with the two polarizations. For the exact solution, they
take the form of
 
,1 1 ,1 1
2
01 12 10 12
= 1 ,
zz
ik h i k h
t t t e r r e
where we denote 0, 1, and 2 the vacuum region above the
film 1, the film 1 and the region below the film 1,
respectively. One can refer to Refs. [35,42] for the
detailed calculation method of t. The transmission
matrices
tr
s
M
and
tr
p
M
are defined as,
1
tr 2
1
1A 0 0
2
11
= 0 A 0
2
0 0 0
s
z
k









M
(4a)
21
tr 1
2
02
10
A 0 ( )
2
1
= 0 A 0
2
( ) 0 ( )
zx
pz
x x z
kik J k d
k
kik J k d J k d k k

 









M
(4b)
where Jn is the cylindrical Bessel function of order n.
1 0 2
A = ( ) ( )
xx
J k d J k d

,
2 0 2
A = ( ) ( )
xx
J k d J k d

and dx is the lateral center-to-center distance between the
two nanoparticles along the x-axis. Note that by setting
the transmission coefficients to one, viz., ts = tp = 1, the
transmitted GF in Eq. (3) degrades to the free-space GF,
viz. G0. When both the two particles locate in the upper
half-space, the total GF is the sum of the free space GF
and the reflected GF Gref, which has already been
discussed in Ref. [27,28].
Let us start the discussion of the results by
illustrating the main finding of our work. The two
particles are put in proximity to the intermediate
multilayered slab at a distance of d = 50nm, while the
total thickness of the multilayered varies from 100nm to
100μm. In Fig. 2(a), we plot the conductance at 300K
between the two SiC nanoparticles for a layer number of
N = 3, 11 and 21. The ratio R between and the
conductance in the absence of the structure 0 is shown
in Fig. 2(b). Meanwhile, for comparison we also present
the results for the single SiC slab and for the reflection
model from Ref. [28].
For a single SiC slab (N = 1), due to the coupling of
the evanescent waves on the up and down interfaces we
can also observe an enhancement of RHT. However, with
an increase in the thickness the coupling effect of surface
waves fade gradually, and eventually vanishes when h =
4.0μm. As we adopt the multilayered structures (N > 1),
a large enhancement RHT is achieved. More specifically,
FIG. 2 (a) RHT conductance Φ between the two particles as a
function of the interparticle distance L (2d + h). (b) The ratio R
between and the conductance in the absence of the structure
0. The temperature is T = 300 K. The particles are placed in
proximity to the surface at a distance of d = 50 nm. h varies
from 100nm to 100μm. Point A in panel (a) denotes the
conductance at 2 μm which equals to that at 0.2μm in the
absence of the intermediate structure. All the slabs are of equal
thickness.
as shown in Fig. 2(b) we observe maximum amplification
with values of 6000, 133000, 361000 at thicknesses of
0.95, 1.68 and 1.95μm for N = 3, 11 and 21, respectively.
That is to say, at the same interparticle distance the
resulting heat transfer in the presence of the multilayered
structure with N = 21 is more than five orders of
magnitude higher than that in the absence of the
multilayered structure. Furthermore, the amplification
persists to large distances of 6.8, 20.2 and 38.6μm,
indicating a great enhancement RHT from near-field to
far-field. In another view of point, in Fig. 2(a) we denote
point A the conductance at 2 μm which equals to that at
0.2μm in the absence of the intermediate structure, which
indicates that the same amount of heat flux can be
achieved at a long distance as in the near-field regime.
This means that the intermediate structure could act as a
passive amplifier for the heat signal detection.
Since for a large thickness evanescent waves
between the adjacent interfaces could not couple with
each other anymore, the RHT from the top particle is
101
105
109
1013
1017
N = 21
EMA
Ref. [28]
(10-32W K1)
Vacuum
N = 1
N = 3
N = 11
A
0.2 2 20110 100
10-12
10-9
10-6
10-3
100
4.0 38.6
6.8
(2.05, 3.60)
(1.78, 1.33)
Ratio, /0 ( 105 )
L (m)
(1.05, 0.06)
20.2
invisible to the other side of the slab, hence a suppression
of RHT as depicted in Fig. 2. When the two particles
locate above a SiC bulk, assisted by the propagation of
the surface waves an amplification of RHT is observed
[28]. Although both the direct and reflected RHT between
the two particles are included, the maximum
amplification is 400 at 20μm which is three orders of
magnitude less than those with our multilayered structure.
This confirms the superiority of our transmission
configuration with an intermediate multilayered structure.
Note that the above results are obtained exactly based on
the multilayered transmission coefficients. In the periodic
multilayered media as considered here, the periodicity is
far smaller as compared to the relevant thermal
wavelength at the chosen temperature. The effective
medium approach (EMA) can be used to describe it as a
homogenous but uniaxial material. The in-plane and our-
of-plane permittivities are thus approximately given by
12
= (1 )ff
 

and
 
12
=1 (1 )ff
 

,
where f is the filling ratio of the SiC defined as h1/(h1+h2).
The EMA results for f = 0.5 are presented in Fig. 2 for N
→ ∞, which agree well with those of N = 21 as L < 40μm,
and predict an enhancement of RHT even when L is up to
100μm.
The origin of this striking enhancement in RHT
conductance can be understood with an analysis of
optical property of the multilayered structure by EMA
(for more details, see Ref. [43]) and the frequency-wave-
vector dependence of the transmission coefficient tp. In
Fig. 3(a) we plot the dielectric function for f = 0.5 by
EMA. We observe two Reststrahlen bands, viz., Type I
band at [1.74241.826] 1014 rad/s, and Type II band at
[1.4951.742]1014 rad/s, indicating that the
multilayered structure supports the hyperbolic phonon
polaritons (hPhPs). Interestingly, the edge frequencies of
these two bands connect with each other at ωh =
1.7421014 rad/s. With h = 1μm and N = 101, Im(tp)
distributions by the exact model are plotted in Fig. 3(b).
Remarkably, we notice that both the slopes of the bright
bands in these two zones approach to zero around the
1.7421014 rad/s, indicating very-large-momentum
extraordinary rays, viz., high-k evanescent modes, at a
frequency extremely close to the SiC nanoparticle
resonance at ωnp. We stress that the frequency-matched
high-k evanescent modes are indeed responsible for the
giant enhancement of RHT between nanoparticles. To
identify the hPhPs more clearly, we plot the dispersion
lines obtained from
 
 
 
 
artan +artan 1k h n
  


 

[35] in which
||
=i
 
. Series of bright bands with
strong reflection appear in the two spectral ranges and are
in good agreement with the bright bands in Im(tp). This is
also the evidence of the validity of our exact approach.
FIG. 3 (a) Dielectric function of the multilayered structure by
EMA. Types I and II Reststrahlen bands are shaded in yellow
and blue, respectively. (b) Distributions of Im(tp) obtained by
the exact model for N = 21 and f = 0.5. The purple and blue
dotted lines mark the dispersion relations for Type I and Type
II hPhPs, respectively. (c) The absolute
x
E
-field amplitudes
as a function of ω and z-position for the exact model. h = 1m.
In addition to the transmission coefficient, we adopt
the generalized 4 × 4 matrix formalism [44,45] to
calculate the electric field distributions. SPhPs are
excited in the Otto geometry (for more details, see Ref.
[43]). The absolute
x
E
-field amplitudes are shown as a
function of frequency and z-position for the exact model
in Fig. 3(b). Remarkably, we see that this field
enhancement peaks at 1.7421014 rad/s penetrates
throughout the multilayered structure, indicating a fluent
transmission channel for the evanescent waves around
the particle resonance at ωnp.
FIG. 4 (a) Spectral conductance ω. (b) The trace of the GF
and the polarizability of the SiC particles. The absolute square
of the zz element |Gzz|2 of (c) the reflection GF for the two
positions separated at L = 2.05μm above the SiC bulk, and the
transmission GF for the two positions separate by a multilayer
structure with (d) N = 1, and (e) N =21. h is 1.95μm for the
transmission model and d = 50nm. The three vertical yellow
dotted lines in panels (a) and (b) mark the three resonances at
ωh, ωnp, and ωs, respectively. And also we denote them as
horizontal white, blue and pink dotted lines in panels (c)-(e).
To further interpret the underlying physics of the
above results and confirm that the frequency-matched
high-k modes are indeed responsible for the giant
enhancement of RHT, we show the spectral conductance
ω in Fig. 4(a). For the transmission configuration, the
total thickness h is chosen as 1.95μm. We note that the
SPhPs of the single SiC slab or the SiC bulk exhibit
resonance at ωs = 1.786×1014 rad/s corresponding
asymptotically to the condition ε(ω) + 1 = 0. Now we
have three characteristic frequencies in this system,
namely, ωh, ωnp, and ωs. Notice that the higher of the ω
at ωnp is, the larger total conductance is obtained as shown
in Fig. 2(a). While for the single slab or the bulk, a second
peak with small value emerges at ωs. However, the
contribution from this peak to the RHT between
nanoparticles is limited. Physically, one can expect that
the heat flux results from the interaction between the
waves emitted from the particles and those from the slab
or surface. Mathematically, the formulation for the
conductance expressed in Eq. (1) explicitly includes these
two waves’ characteristics in the polarizability
and
the GF, respectively. More specifically, we separate the
conductance into two quantities, viz.,
 
2
4
0Imk


and
*
Tr

as shown in Fig. 4(b). For the bulk or single
film, the trace of the GF only peaks at ωs. However the
value of the polarizability at ωs is far lower than that at
ωnp. Hence, as we multiply these two quantities, the
amplification contributed from the peak on the GF is
weakened. Meanwhile, due to the very small value of
*
Tr

at ωnp, the particle resonance’s contribution to
the RHT is also limited. As a result, no matter the
transmission or the reflection configuration is used, the
amplification for the single SiC is very limited. As the
multilayered structure with N = 3 is used, we see in Fig.
4(a) that an additional peak emerges at a frequency very
close to ωnp on the
*
Tr

, and the peak at ωs increases
to a bigger value than the case with the bulk or single film.
By further increasing N to 21, due to the increasingly
prominent effect of hPhPs the peak of
*
Tr

at ωs
almost disappears but the other peak is significantly
enhanced at ωh. Consequently, due to the highly matched
peak frequencies of the nanoparticle and multilayered
structure, we obtain a significant thermal conductance as
shown in Fig. 4(a) for N =21. We thus call this effect as
resonant RHT.
To interpret the behavior of
*
Tr

, we show the
absolute square of the zz element |Gzz|2 with respect to kx
and ω in Figs. 4(c)-4(e) for the cases with the bulk and
the multilayered structure with N =1, 21. It is shown that,
the largest value of kx for the |Gzz|2 contour is truncated.
This origins from the evanescent feature of the surface
wave. The exponential
 
0 2 0 1
()
zz
i k z h k z
e  
in the GF
expressed in Eq. (3) introduces a cutoff [33] at kρ ≈ 1/(2d).
We see that although the largest kx of N =21 is less than
that of the bulk, owing to the resonance at a frequency
closer to ωnp the conductance is much larger than that of
the bulk as shown in Fig. 4(a).
We note that f is assumed to be 0.5 in the above
10-5
10-1
103
107
1011 SiC bulk, Reflection
N = 1
N = 3
N = 21
Vacuum
(1040 W K1 rad-1 s)
h = 1.95 μm
d = 50 nm
h
np
s
1.6 1.7 1.8 1.9
100
103
106
109
TrGG* (10-8 )
(rad/s)
10-3
10-1
101
103
105
107
k4
0 [Im()]2 (1029)
discussion. Since f would change the optical properties of
the multilayered structure, one can expect that it would
make impact on the RHT. We find that the amplification
of RHT is most prominent for f = 0.5 where the epsilon-
near pole and epsilon-near zero frequencies coincide with
each other (for more details, see Ref. [43]). In addition,
we find that when dx > 0, viz., a non-zero lateral distance,
due to the interference between surface waves, the RHT
conductance reveals an interesting oscillating and non-
monotonic behavior (for more details, see Ref. [43]). This
oscillation RHT is an example of an electromagnetic
resonator in the context of heat radiation [29].
In conclusion, we have demonstrated that a giant
resonant radiative heat transfer between nanoparticles
could be achieved by inserting a multilayered structure.
The physical mechanisms have been identified and
elaborated to be the excitation of high-k evanescent
modes at the frequency close to the nanoparticle
resonance. Due to this quasi-monochromaticity and
resonant effect, our structure could be exploited to
considerably improve the efficiency of near-field energy
conversion devices.
Acknowledgements - This work was supported by the
National Natural Science Foundation of China (Grant No.
51706053, 51776054). M. A. acknowledges support from
the Institute Universitaire de France, Paris France (UE).
__________________________
*yihongliang@hit.edu.cn
mauro.antezza@umontpellier.fr
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Vanadium dioxide (VO2) exhibits a phase-change behavior from the insulating state to the metallic state around 340 K. By using this effect, we experimentally demonstrate a radiative thermal rectifier in the far-field regime with a thin film VO2 deposited on the silicon wafer. A rectification contrast ratio as large as two is accurately obtained by utilizing a one-dimensional steady-state heat flux measurement system. We develop a theoretical model of the thermal rectifier with optical responses of the materials retrieved from the measured mid-infrared reflection spectra, which is cross-checked with experimentally measured heat flux. Furthermore, we tune the operating temperatures by doping the VO2 film with tungsten (W). These results open up prospects in the fields of thermal management and thermal information processing.
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