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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)

[3–9]. 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 [35–38] 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 x–y-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 ΔT→0, 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],

2

4*

0

0

4 , Im Tr ,

2

dn T k

(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 [1.7424–1.826] 1014 rad/s, and Type II band at

[1.495–1.742]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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