HAL Id: hal-03191119
https://hal.archives-ouvertes.fr/hal-03191119
Submitted on 25 May 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Polariton topological transition eects on radiative heat
transfer
Cheng-Long Zhou, Xiao-Hu Wu, Yong Zhang, Hing-Liang Yi, Mauro Antezza
To cite this version:
Cheng-Long Zhou, Xiao-Hu Wu, Yong Zhang, Hing-Liang Yi, Mauro Antezza. Polariton topological
transition eects on radiative heat transfer. Physical Review B: Condensed Matter (1978-1997),
American Physical Society, 2021, 103, pp.155404. �10.1103/PhysRevB.103.155404�. �hal-03191119�
PHYSICAL REVIEW B 103, 155404 (2021)
Polariton topological transition effects on radiative heat transfer
Cheng-Long Zhou,
1,2
Xiao-Hu Wu ,
3
Yong Zhang,
1,2
Hong-Liang Yi ,
1,2,*
and Mauro Antezza
4,5,†
1
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
2
Key Laboratory of Aerospace Thermophysics, Ministry of Industry and Information Technology, Harbin 150001, People’s Republic of China
3
Shandong Institute of Advanced Technology, Jinan 250100, Shandong, China
4
Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-34095 Montpellier, France
5
Institut Universitaire de France, 1 rue Descartes, F-75231 Paris, France
(Received 26 October 2020; revised 25 February 2021; accepted 19 March 2021; published 6 April 2021)
Twisted two-dimensional bilayer anisotropy materials exhibit many exotic physical phenomena. Manipulating
the “twist angle” between the two layers enables the hybridization phenomenon of polaritons, resulting in fine
control of the dispersion engineering of the polaritons in these structures. Here, combined with the hybridization
phenomenon of anisotropy polaritons, we study theoretically the near-field radiative heat transfer (NFRHT)
between two twisted hyperbolic systems. These two twisted hyperbolic systems are mirror images of each other.
Each twisted hyperbolic system is composed of two graphene gratings, where there is an angle ϕ between these
two graphene gratings. By analyzing the photonic transmission coefficient as well as the plasmon dispersion
relation of the twisted hyperbolic system, we prove the enhancement effect of the topological transitions of the
surface state at a special angle [from open (hyperbolic) to closed (elliptical) contours] on radiative heat transfer.
Meanwhile the role of the thickness of dielectric spacer and vacuum gap on the manipulating the topological
transitions of the surface state and the NFRHT are also discussed. We predict the hysteresis effect of topological
transitions at a larger vacuum gap, and demonstrate that as the thickness of the dielectric spacer increases, the
transition from the enhancement effect of heat transfer caused by the twisted hyperbolic system to a suppression.
DOI: 10.1103/PhysRevB.103.155404
I. INTRODUCTION
Since the pioneering work of Polder and Hove [1], when
two objects are placed at a distance comparable to or shorter
than the characterized thermal radiation wavelength, i.e., in
the near-field regime, the forward and backward evanescent
waves can couple with each other and open paths for photons
to tunnel through [2–4]. This phenomenon is called photon
tunneling and there are more tunneling photons than propa-
gating photons, resulting in a radiative heat transfer that can
be orders of magnitude higher than the blackbody limit by
Planck’s law. The huge radiative heat flux in the near field
opens the door to various applications like thermophotovoltaic
devices [5–8], thermal rectification [9,10], information pro-
cessing [11], and noncontact refrigeration [12]. Among these,
the coupling of surface polaritons plays a decisive role in
huge heat transfer enhancement [4]. In particular, the near-
field radiative heat transfer (NFRHT) can be far ahead of the
blackbody limit, either theoretically or experimentally, via the
resonant coupling of surface phonon polaritons (SPhPs) [13]
or surface plasmon polaritons (SPPs) [14]. Moreover, the de-
velopment in fabrication of metamaterials results in extensive
studies of the coupling of surface polaritons for NFRHT be-
tween metamaterials in theory, such as hyperbolic polaritons
[15], magnetoplasmon polaritons [16], ellipse polaritons [17],
nonreciprocal polaritons [18], and nonreciprocal hyperbolic
*
Corresponding author: yihongliang@hit.edu.cn
†
Corresponding author: mauro.antezza@umontpellier.fr
polaritons [19]. Various types of surface polaritons have been
extensively studied for their ability to dominate the photon
tunneling and greatly modulate the near-field heat transfer.
Moreover, since the hybridization effect of polaritons can
greatly change the optical performance of the system, contin-
uous efforts have been devoted to exploring the influence of
hybridization of polaritons on heat transfer. For example, sur-
face plasmons in graphene can couple with hyperbolic phonon
polaritons in an hBN film to form hybrid polaritons that can
assist photon tunneling [20]. The hybridization effect between
surface plasmon polaritons of graphene and semiconductor
can well alleviate the suppression effect of the decoupling
characteristics of the asymmetry system on the near-field
radiation heat transfer [21]. At a strong magnetic field, the
coupling of magnetoplasmon polaritons from graphene to sur-
face phonon polaritons from SiO
2
can reach a high thermal
magnetoresistance to impede near-field radiative heat transfer
[22].
These strategies have largely explored the influence of
the hybridization effect of polaritons on near-field thermal
radiation [20–27]. Nonetheless, up to now, the influence of
the hybridization of anisotropic polaritons on the near-field
radiation remains elusive. Recently, since the pioneering work
of Hu et al., twisted bilayer two-dimensional (2D) materials,
in which one layer is rotated with respect to the other, enable
the hybridization phenomenon of polaritons, resulting in fine
control of the dispersion engineering of the polaritons in these
structures [28,29]. Inspired by this concept, we explore heat
transfer based on the twisted hyperbolic system, in which
an interesting mechanism of hybridization of anisotropic
2469-9950/2021/103(15)/155404(13) 155404-1 ©2021 American Physical Society
ZHOU, WU, ZHANG, YI, AND ANTEZZA PHYSICAL REVIEW B 103, 155404 (2021)
polaritons may be introduced for NFRHT. The twisted hy-
perbolic system is composed of stacked uniaxial supporting
hyperbolic polaritons. The uniaxial hyperbolic metasurfaces
can be demonstrated in several material platforms, from Waals
materials [30] to black phosphorus [31,32] and other 2D ma-
terials. Moreover, the surface characteristics of objects can
be anisotropic when they are patterned into metasurfaces or
gratings. Especially, by patterning a single layer of graphene
sheet into ribbons, the closed circular dispersion of graphene
plasmons is opened to become hyperbolic, leading to a strong
NFRHT [33]. Since the strong NFRHT of graphene nanorib-
bons, continuous efforts have been devoted to exploit the heat
transfer phenomenon of graphene nanoribbons. For example,
the hyperbolic characteristics of a monolayer graphene grating
can be adjusted by magnetic field [34] and current [19]to
modulate its near-field radiation heat transfer. Since graphene
nanoribbons that can provide robust hyperbolic polaritons in a
board frequency band, we consider densely packed graphene
nanoribbons as the platform of choice.
In this work, we theoretically investigate the influence of
hybridization of anisotropic hyperbolic polaritons on NFRHT
between two bilayer graphene nanoribbons. Through tailored
interlayer coupling in twisted bilayer, the hyperbolic polari-
tons undergoes topological transitions at transitional angles,
that is, the polaritons change markedly in nature from hyper-
bolic (open) to elliptical (closed). We investigate the effects
of the topological transitions of hyperbolic polaritons on the
radiative heat flux and, in particular, what happens to the
hyperbolic modes of the twisted structure. In addition, the
effects of the thickness of dielectric spacer and vacuum gap
on the NFRHT are discussed at the end.
II. THEORETICAL ASPECTS
Let us consider a system composed of two twisted hyper-
bolic systems brought into close proximity with a vacuum
gap size of d
0
as sketched in Fig. 1(a). We can form a
twisted hyperbolic system by coupling two identical graphene
nanoribbons, as sketched in Fig. 1(a), separated by thickness
d
s
of the dielectric spacer. Each graphene strip has width
W and an air gap G separates neighboring strips. We define
the graphene strips of the twisted hyperbolic system above
the vacuum gap as G
1
and G
2
from top to bottom, and the
graphene strips of the twisted hyperbolic system below the
vacuum gap as G
3
and G
4
from top to bottom. The upper
structure is the emitter (G
1
and G
2
) with a higher tempera-
ture T
1
and the bottom structure (G
3
and G
4
) is the receiver
with a lower temperature T
2
. The temperatures are set to be
T
1
= 310 K and T
2
= 300 K, respectively. Here, α and β are
defined perpendicular to periodical directions of the first layer
and the second layer in the one bilayer, respectively. Here, the
first layer represents the graphene strips close to the vacuum
gap, and the second layer represents the strips away from the
vacuum gap. When the second layer in one bilayer system
can be rotated with respect to the y axis by an angle of ϕ
from 0° to 90°, as shown in Figs. 1(b) and 1(c), an interesting
effects of twisted periodicities is achieved. For simplicity, we
assume that the emitter and receiver are mirror images of each
other and the background materials to be vacuum. On the
other word, we rotate simultaneously an angle of ϕ between
FIG. 1. (a) Schematic of near-field heat transfer between two
twisted hyperbolic systems. The twisted hyperbolic system is com-
posed of two identical graphene nanoribbons, whose width and air
gap are W and G, respectively. The structure above the vacuum gap
(G
1
and G
2
) is the emitter with a higher temperature T
1
(310 K)
and the structure below the vacuum gap (G
3
and G
4
) is the receiver
with a lower temperature T
2
(300 K). (b) Top view of the crystalline
structure without rotation. (c) Top view of the twisted hyperbolic
system at a rotation angle ϕ of with respect to y axis.
G
1
and G
2
and an angle of ϕ between G
3
and G
4
. Here, we
strictly compliance the emitter and receiver are mirror images
of each other, that is, keeping G
2
parallel to G
3
and G
1
parallel
to G
4
. The nonmirrored system and dielectric backgrounds
can be easily captured by generalizing the formulation in this
work. In practical applications, we note that the twisted bi-
layer graphene grating can be implemented experimentally in
the following procedure: the bottom graphene grating can be
fabricated by electron beam lithography or chemical etching
on the basement [35]. Then, the dielectric spacer between
two graphene gratings can be prepared through chemical va-
por deposition [29]. Through mechanical exfoliation, the top
graphene sheet is transferred onto the dielectric spacer [36].
Last, the top graphene grating also can be fabricated by elec-
tron beam lithography or chemical etching on the dielectric
spacer. By separately controlling the direction of the electron
beam or the direction of chemical etching during the manufac-
turing process of the top and bottom gratings, the twist angle
in the bilayer twisted grating system can be realized.
The optical conductivity of the graphene nanoribbon array
of the present work is derived using a well-known effective
medium theory (EMT) [37–39] that holds when the unit-
cell period P = W + G is much smaller than the operating
wavelength, i.e., P λ
0
. Meanwhile, in Ref. [33], EMT is
valid only when the vacuum gap is larger than the period of
the ribbons. When the gap distance is several times greater
than the period, EMT can predict the real heat flux well, as
quantitatively shown in Ref. [33]. According to Ref. [33],
when the vacuum gap d
0
is 50 nm and the period P is 10 nm,
the error by EMT is less than 6.69% compared with rig-
orous coupled-wave analysis (RCWA). As the gap distance
increases, the accuracy of EMT can be further improved [33].
However, for the unit-cell period of 10 nm, EMT may fail to
155404-2
POLARITON TOPOLOGICAL TRANSITION EFFECTS ON … PHYSICAL REVIEW B 103, 155404 (2021)
predict the NFRHT when the gap distance is less than 50 nm,
and the results predicted by EMT will far exceed those calcu-
lated by RCWA [33]. Therefore, for the unit-cell period of
10 nm, the vacuum gap greater than 50 nm is the limit of
validity for this approximation [33]. In this paper we strictly
ensure that the vacuum gap is greater than 50 nm. The EMT is
based on modeling the strip near-field coupling as an effective
capacitance [33],
σ
eff
=
σ
xx
0
0 σ
yy
=
σ
xx
=
Pσ
C
σ
G
(W σ
C
+Gσ
G
)
0
0
W
P
σ
G
, (1)
where σ
C
=−iωε
0
P/(πln{csc[0.5π (P−W )/P]}) is the effec-
tive strip conductivity taking into account near-field coupling
and nonlocality. In the hyperbolic regime, the metasurface acts
as a metal for one transverse field polarization (Im[σ
yy
] > 0)
and as a dielectric for the other one (Im[σ
xx
] < 0). ω is the
frequency, and ε
0
is the vacuum permittivity. σ
G
is the optical
conductivity of the graphene, and it is given by the well-
known random phase approximation. Following Ref. [33], the
conductivity can be written as a sum of intraband (Drude) and
interband contributions, i.e., σ
G
= σ
D
+ σ
I
, respectively,
σ
D
=
i
ω + i/τ
2e
2
k
B
T
π ¯h
2
ln
2 cosh
μ
2k
B
T
, (2a)
σ
I
=
e
2
4¯h
G
¯hω
2
+ i
4¯hω
π
∞
0
G (ξ ) − G(¯hω/2)
(¯hω)
2
− 4ξ
2
dξ
,
(2b)
where G(ξ ) = sinh(ξ/k
B
T )/[cosh(E
f
/k
B
T ) + cosh(ξ/k
B
T )].
The conductivity depends explicitly on the temperature T and
the chemical potential μ. The relaxation time τ is fixed at
10
–13
s[33].
The radiative heat flux between two twisted hyperbolic
systems can be calculated based on the fluctuation-dissipation
theory [19],
q =
∞
0
q
ω
(
ω
)
dω =
∞
0
[
(
ω, T
1
)
−
(
ω, T
2
)
]
dω
×
∞
−∞
∞
−∞
ξ
(
ω, k
x
, k
y
)
8π
3
dk
x
dk
y
, (3)
where (ω, T ) = ¯hω/[exp(¯hω/k
B
T ) − 1] is the mean energy
of a Planck oscillator at angular frequency ω. ξ (ω, k
x
, k
y
)is
the photonic transmission coefficient (PTC) that describes the
probability of photons excited by thermal energy, which can
be written as
ξ (ω, k
x
, k
y
)
=
Tr [(I − R
∗
2
R
2
− T
∗
2
T
2
) D (I − R
1
R
∗
1
− T
∗
1
T
1
)D
∗
], k < k
0
Tr [(R
∗
2
− R
2
) D (R
1
− R
∗
1
)D
∗
]e
−2|k
z
|d
, k > k
0
(4)
for propagating (k < k
0
) and evanescent (k > k
0
)waves
where k =
k
2
x
+ k
2
y
is the surface parallel wave vector and
k
0
= ω/c is the wave vector in vacuum. k
z
=
k
2
0
− k
2
is
the tangential wave vector along the z direction in vacuum
and
∗
signifies the complex conjugate. The 2 × 2matrix
D is defined as D = (I − R
1
R
2
e
2ik
z
d
) which describes the
FIG. 2. (a) Radiative heat flux q as a function of rotation angle
ϕ at different dielectric permittivity of spacer. (b) Heat transfer co-
efficient ratio η between hyperbolic system and nonrotating system
as a function of rotation angle ϕ at different dielectric permittivity
of spacer. The parameters are d
0
= 50 nm, d
s
= 1nm, T
1
= 310 K,
T
2
= 300 K, μ = 0.4eV,W = 5.9nm,andP = 9.84 nm.
usual Fabry-Perot-like denominator resulting from the mul-
tiple scattering between the two interfaces of receiver and
emitter. The reflection matrix R is a 2 × 2matrixinthe
polarization representation; the calculation process of R can
be found in Appendix A.
III. HYPERBOLIC HYBRIDIZATION GUIDED HEAT
TRANSFER BETWEEN TWO BILAYER GRAPHENE
GRATING SYSTEMS
We first discuss the radiative heat flux (RHF) between the
two bilayer graphene grating systems as a function of the
rotation angle ϕ by fixing the vacuum gap d
0
to the value of
50 nm as shown in Fig 2. We fix the thickness of dielectric
spacer to d
s
= 1 nm. Here, it is worth noting that the thickness
of the dielectric gasket cannot be reduced indefinitely. If the
spacing between two graphene layers is within an atomic lat-
tice (0.35 nm), quantum coupling will occur between the two
graphene layers [40]. Their conductivity description based on
the random phase approximation no longer is applicable and
should be modified accordingly to accommodate nonlocal and
other tunneling effects [40]. However, for hypothetical twisted
atomic bilayers, if their interlayer distance is relatively large,
namely, thickness of the dielectric spacer is larger than 0.35
nm, the interlayer quantum coupling would disappear [40].
Thus, in the following sections, we always keep the thickness
of the dielectric spacer greater than 0.35 nm. The chemical
potential μ is fixed at 0.4 eV. Here, the chemical potential
can be set by nitric acid chemical doping [41] or an external
bias voltage [42]. In the nature, the width of graphene stripe
should be a multiple of the lattice constant a of graphene
(a = 0.246 nm). We consider that the width W is 24a (5.9 nm)
and the unit-cell period P is 40a (9.84 nm), which can truly
describe graphene grating. Here, besides, we also illustrate
the impact of the dielectric permittivity ε
s
of the spacer with
respect to the rotation angle ϕ on the results.
We can see in the Fig. 2(a) that the presence of the rotation
angle significantly modifies the RHF between the two bilayer
graphene grating systems. In most of the cases, the RHF is
higher than the case without rotation angle, which means that
this torsion can enhances effectively the heat transfer. It can
be seen more intuitively through the heat transfer coefficient
155404-3
ZHOU, WU, ZHANG, YI, AND ANTEZZA PHYSICAL REVIEW B 103, 155404 (2021)
FIG. 3. (a) Spectral RHF as a function of the frequency for a
vacuum gap d
0
= 50 nm and a chemical potential μ = 0.4eV. The
different lines correspond to different rotation angles. (b) Contour
plots of energy transfer function (ω), as a function of frequency
with different rotation angles.
ratio in Fig. 2(b) that the twisted system can enhance the
radiative heat transfer of the system, and the enhancement
would increase as the dielectric permittivity of spacer in-
creases. However, as shown in Fig. 2(a), we find that the RHF
exhibits a nonmonotonic dependency versus the rotation an-
gle, especially for a lower dielectric permittivity of spacer. For
the dielectric permittivity of a spacer of 1, when the rotation
angle reaches 40°, the maximum of the RHF increases from
28.3kWm
–2
for φ = 0
◦
to around 33.5kWm
–2
for φ = 40
◦
(η = 1.18), and decreases rapidly as the rotation angle further
increases. In addition, it can be seen that although the dielec-
tric permittivity of the spacer between two graphene gratings
could tune the RHF of a metastructure, the dependency of
RHF versus the rotation angle under different dielectric per-
mittivity has been constant. In order to conveniently reveal
the physical mechanism of a twisted hyperbolic system in
NFRHT, we fixed the dielectric permittivity of the spacer to
1 in the subsequent analysis.
To get further insight into the role of the twisted hyperbolic
system, we show in Fig. 3(a) the spectral RHF for a gap
d
0
= 50 nm, the dielectric permittivity of the spacer is 1, and
different rotation angles. This spectral RHF is defined as the
RHF per unit of frequency. It can be seen that a new peak
appears as the twisted hyperbolic system emerges (ϕ>0°).
In addition, when the rotation angle increases to 20°, the new
peak begin to dominate the spectral RHF, that is, the new peak
becomes the maximum of the spectral RHF. The new peak of
the spectral RHF is blueshifted upon increasing the rotation
angle from 0.018 eV/ ¯h for ϕ = 10
◦
up to around 0.145 eV/ ¯h
for ϕ = 90
◦
. Notice also that the maximum of the spectral
RHF also increases drastically with the rotation angle, reach-
ing a maximum of 0.31 nW m
–2
rad
–1
satϕ = 40
◦
.Asthe
rotation angle further increases, the maximum of the spectral
RHF gradually decreases to the value of 0.21 nW m
–2
rad
–1
s
at ϕ = 90
◦
.
In order to visualize the NFRHT of the rotation angle,
in Fig. 3(b), we calculate the energy transfer function (ω),
given by (ω) = q
ω
/[(ω, T
1
) − (ω, T
2
)]. As the rotation
angle increases, the peak of energy transfer function gener-
ated by the twisted hyperbolic system increases drastically,
reaching a maximum of 1.5 × 10
13
. This also explains that
the spectral RHF in Fig. 3(b) increases as the rotation angle
increases at the lower rotation angle range. However, the peak
of energy transfer function generated by higher rotation angle
is excited only by higher photonic energy, which corresponds
to the blueshifted peak of spectral RHF shown in Fig. 3(a).
Unfortunately, the contribution of spectral RHF with higher
frequency (higher photonic energy) to RHF is negligible, due
to the exponentially decaying mean energy of a Planck os-
cillator (ω, T) at room temperature. Therefore, in Fig. 3(a),
when the rotation angle is 90°, it is difficult to observe the en-
hanced spectral RHF excited by the ultrahigh energy transfer
coefficient, which is in agreement with the declining trend of
the RHF at the larger rotation angle range in Fig. 2.
IV. UNDERLYING PHYSICS OF THE HYPERBOLIC
HYBRIDIZATION
It is well known that the photonic transmission coefficient
(PTC) and dispersion relation can visually reveal the physical
mechanism of the effect of surface state on NFRHT. We thus
employed these two physical quantities to explore the under-
lying physical mechanisms for this twisted hyperbolic system
to the NFRHT. For a frequency of 0.15 eV/¯h, the lower panels
of Figs. 4(b)–4(f) show the distributions of the photonic trans-
mission coefficient and the dispersion relation with rotation
angle ϕ = 0
◦
, 30°, 60°, 80°, and 90°, respectively. As we all
know, different from conventional circles, the surface state
of anisotropic SPPs supported by graphite gratings graphs
as a hyperbola, where the dominant contribution to the op-
tical response of graphene comes from the scattering of free
electrons. As shown in Fig. 4(b), the regime featured with
PTC for the bilayer graphene grating system is axisymmetric
hyperbolic lines. The dispersion relation of effective graphene
gratings can well reflect this phenomenon, and it is given
by Eq. (A11). Moreover, as the two bodies are brought into
proximity, the evanescent field of HSPPs associated with two
graphene gratings can interact with each other, leading to the
two dispersion curves at the higher wave-vector region and
the smaller wave-vector region. However, due to the existence
of the attenuation length [δ
z
= 1/Im(k
z
)] [18], the surface
state dominated by dispersion at the higher wave-vector re-
gion is easily filtered by the vacuum gap, and cannot make a
corresponding contribution to the NFRHT. Therefore, in the
subsequent analysis, we only show the dispersion relationship
at the smaller wave-vector region, unless otherwise stated.
As shown in Figs. 4(b)–4(f), it can be clearly seen that
when the second individual graphene grating of a bilayer
system emerges with mechanical rotation, the bright branch
of the polaritons changes markedly—from hyperbolic (open)
to elliptical (closed). This photonic topological transition is
governed by the coupling of in-plane hyperbolic SPPs that
are individually supported by each layer of the bilayer sys-
tem [28]. Before the rotation angle at which the topological
transition arises, as the rotation angle increases, gradually the
hyperbolic bright branch would be flattened. After a topolog-
ical transition occurs on the surface state (i.e., the bright band
closes), the surface state gradually change from quasielliptical
plasmon to quasi-isotropic plasmon with the rotation angle
increase. Moreover, as its surface state changes from open to
closed, the topological transition yields diffraction less, and
low-loss directional SPPs become canalized, which in turn
produces a more intense surface state, i.e., a brighter and
155404-4
