Supporting Information for “Active Radiative Thermal Switching with Graphene
Plasmon Resonators”
Ognjen Ilic,
1
Nathan H. Thomas,
2
Thomas Christensen,
3
Michelle C. Sherrott,
1
Marin Soljaˇci ́c,
3
Austin J. Minnich,
2
Owen D. Miller,
4
and Harry A. Atwater
1
1
Department of Applied Physics and Materials Science,
California Institute of Technology, Pasadena, CA 91125, USA
2
Department of Engineering and Applied Science,
California Institute of Technology, Pasadena, CA 91125, USA
3
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
4
Department of Applied Physics and Energy Sciences Institute, Yale University, New Haven, CT 06511, USA
2
I. BEYOND THE LOCAL RESPONSE
APPROXIMATION (LRA) CONDUCTIVITY
The broadband spectrum of thermal radiation implies
that achieving an OFF state of the proposed thermal
switch necessitates strong spectral separation between
plasmonic resonances of the emitter and the absorber
side. Consequently, the two graphene structures must
have significantly different carrier concentrations. For ex-
ample, for low mobilities in the sheet-sheet case (Fig. 2),
the OFF states are realized for Fermi levels at the respec-
tive ends of the allowed range, namely
E
off
1(2)
=
E
min(max)
.
Because
E
max
E
min
, the dispersion of the resulting
mode will lie well inside the region of Landau damping
for the lower-doped graphene sheet. To fully assess the
contribution of these high-momentum plasmons (that is,
high-momentum relative to the Fermi momentum of the
lower-doped graphene), we adopt the full nonlocal con-
ductivity (i.e. incorporate momentum-dispersion in ad-
dition to frequency-dispersion); moreover, we include the
temperature-dependence of the nonlocal conductivity by
exploiting an integral identity due to Maldague [1, 2]. In
Fig. S1, we compare the dispersion relation and the spec-
tral dependence of Φ(
ω
), for LRA and RPA conductivi-
ties. We find negligible difference in the integrated heat
transfer (e.g. for mobility
μ
= 10
3
cm
2
/Vs we obtain
<
1% difference between the RPA and the LRA results),
for
d
= 100 nm separation. This observation is in accor-
dance with similar considerations of nonlocal effects in
the context of graphene RHT [3], and demonstrates the
validity of a local-response treatment.
II. EXPRESSIONS FOR THE RHT INVOLVING
DIPOLAR STRUCTURES
In the dipolar limit, the radiative heat transfer between
a resonator and its environment can be calculated using
the relevant Green’s functions [4, 5]:
H
=
∫
∞
0
d
ω
[Θ(
ω,T
1
)
−
Θ(
ω,T
2
)]
×
×
2
ω
2
μ
0
0
π
∑
i
=
x,y,z
Im[
α
i
(
ω
)]Im
[
←→
G
(
ω,
r
0
)
]
ii
(1)
where
α
is the polarizability of the resonator (given by
Eq. (3)of the main text), and
←→
G
is the Green tensor at
the resonator position
r
0
. For the case of two dipolar
resonators in the near-field, we have:
[
←→
G
(
ω
)
]
ii
=
1
16
π
2
k
2
R
6
α
i
−
α
i
6
R
2
i
R
2
+
9
R
2
i
R
4
∑
j
α
j
R
2
j
(2)
where
R
is the distance between the dipoles, and
k
=
ω/c
. For two parallel disks with polarizabilities
α
1
,
2
=
FIG. S1. Comparison of the LRA and the RPA graphene con-
ductivity models. (a) Effective dispersion for the OFF state
for the 2-sheet configuration of Fig. 2 for the case of mobility
μ
=10
3
cm
2
/Vs and
E
off
1(2)
=
E
min(max)
. (b) The corresponding
spectral transfer function Φ(
ω
).
α
(1
,
1
,
0)
T
, separated along the
z
axis a distance
d
apart,
we have
[
←→
G
(
ω
)
]
xx
=
1
16
π
2
k
2
d
6
α
(
ω
)
[
←→
G
(
ω
)
]
yy
=
1
16
π
2
k
2
d
6
α
(
ω
)
[
←→
G
(
ω
)
]
zz
= 0
(3)
Substituting Eq. (3) into Eq. (1), we recover the expres-
sion for the spectral heat transfer function, namely
Φ(
ω,d
) =
2
8
π
3
1
d
6
Im(
α
1
)Im(
α
2
)
(4)
In a similar manner, for the case when the second ob-
ject is an extended planar interface, the Green tensor is
given by:
←→
G
=
ik
8
π
∫
∞
0
s
s
z
r
s
−
s
2
z
r
p
0
0
0
r
s
−
s
2
z
r
p
0
0
0
2
s
2
r
p
e
2
iκd
d
s
(5)
where
s
≡
q/k
,
s
z
≡
κ/k
;
k
,
q
,
κ
are the total, parallel,
and the perpendicular wave-vectors, respectively;
r
p
and
r
s
are the Fresnel reflection coefficients for the planar
interface, for the
p
(TM) and
s
(TE) polarization, re-
3
FIG. S2. Schematic for the NF-RHT calculation between a
planar structure and a disk in a parallel (a) and a perpendic-
ular (b) orientation.
spectively; and
h
is the height above the planar interface
(see Fig. S2).
For a disk in a parallel configuration (Fig. S2a), we
write its polarizability as
α
=
α
(1
,
1
,
0)
T
, and the spec-
tral transfer function Φ becomes
Φ
‖
(
ω,h
) =
ω
3
4
π
2
c
3
2Im(
α
)Re
[
∫
∞
0
s
s
z
(
r
s
−
s
2
z
r
p
)
e
2
iκh
d
s
]
(6)
where the factor of 2 accounts for the two identical polar-
izations
x,y
. For a disk in a perpendicular configuration
(Fig. S2b), the polarizability is
α
=
α
(0
,
1
,
1)
T
. Simi-
larly, the transfer function simplifies to
Φ
⊥
(
ω,h
) =
ω
3
4
π
2
c
3
Im(
α
)
×
×
(
Re
[
∫
∞
0
s
s
z
(
r
s
−
s
2
z
r
p
)
e
2
iκh
d
s
]
+
+ Re
[
∫
∞
0
s
s
z
2
s
2
r
p
e
2
iκh
d
s
])
(7)
In the nonretarded limit (
q
k
), we note
κ
≈
iq
,
s
z
≈
iq
, which allows us to further approximate the integrals
above
Re
[
∫
∞
0
s
s
z
(
r
s
−
s
2
z
r
p
)
e
2
iκh
d
s
]
≈
∫
∞
0
(Im(
r
s
)+
+
q
2
k
2
Im(
r
p
))
e
−
2
qh
d
q
k
Re
[
∫
∞
0
s
s
z
2
s
2
r
p
e
2
iκh
d
s
]
≈
∫
∞
0
2
q
2
k
2
Im(
r
p
)
e
−
2
qh
d
q
k
(8)
As the optical response in the near field is dominated
by surface plasmons, we focus on
r
p
, the
p
-polarized re-
flection coefficient. In this limit, we can relate the parallel
(6) and the perpendicular (7) transfer function:
Φ
⊥
(
ω,h
)
≈
3
2
Φ
‖
(
ω,h
)
≈
3
4
π
2
Im(
α
)
∫
∞
0
q
2
Im(
r
p
)
e
−
2
qh
d
q
(9)
FIG. S3.
(a) Thermal switching between a disk and a
graphene stack, in a parallel configuration. RHT enhance-
ment
h
on
/h
bb
(left) and the switching ratio
η
=
h
on
/h
off
(right) for a disk-sheet (solid) and a disk-stack (dashed) con-
figuration. The disk is parallel to the sheet/stack. Here,
T
=300K,
R
=25nm,
d
=100nm,
δ
=10nm. Note that the
radiative thermal conductance
h
on(off)
is normalized to the
disk area (
A
=
πR
2
).
III. POLARIZABILITY MODEL FOR A
GRAPHENE RESONATOR; FIRST-ORDER
APPROXIMATION TO RTC
Resonator
ζ
1
∆
1
∆
2
1
/ζ
3
1
1.940
0.924
0.117
1.686
0.859
0.154
1.200
0.683
0.270
(x) 1.118
0.939
0.631
(y) 2.981
0.842
0.027
TABLE S1. Dimensionless eigenfrequencies (
ζ
1
) and normal-
ization factors (∆
1
) associated with Eq. (3) of the main text.
All resonators have the same area
A
, and only the lowest en-
ergy (and the strongest) mode is considered. Characteristic
length scale is defined as
L
≡
√
A
. For the ellipse, the aspect
ratio is 2:1.
For two resonators of the same shape, size, and Fermi
level (such as in an ON state), we can simplify the ex-
4
pression for the radiative thermal conductance:
h
(
T
) =
1
A
∫
∞
0
dω
∑
j
∈
ˆ
r
∂
Θ
∂T
(
ω
)
λ
j
8
π
3
d
6
[Im(
α
)]
2
(10)
where
∂
Θ
∂T
(
ω
) =
k
B
(
~
ω
k
B
T
)
2
e
~
ω/k
B
T
(
e
~
ω/k
B
T
−
1
)
2
(11)
Im(
α
) = 2
L
3
∑
ν
∆
ν
Im
(
1
ζ
ν
−
ζ
(
ω
)
)
(12)
where
ζ
(
ω
) = 2
i
0
ωL/σ
(
ω
). We can further simplify (12)
as:
Im(
α
) = 2
L
3
∑
ν
∆
ν
ζ
I
(
ζ
ν
−
ζ
R
)
2
+
ζ
2
I
(13)
where
ζ
≡
ζ
R
+
iζ
I
(with implicit dependence on fre-
quency). Substituting back into (10), we get
h
(
T
) =
(2
L
3
)
2
8
π
3
Ad
6
∑
j
∈
ˆ
r
λ
j
×
∑
ν
∫
∞
0
d
ω
∂
Θ
∂T
(∆
j
ν
)
2
ζ
2
I
[
(
ζ
ν
−
ζ
R
)
2
+
ζ
2
I
]
2
(14)
Assuming slowly variying
∂
Θ
∂T
relative to the squared
Lorentzian peaked at
ω
j
given by
ζ
j
ν
=
ζ
R
(
ω
j
ν
), we can
further approximate
h
(
T
) =
(2
L
3
)
2
8
π
3
Ad
6
∑
j
∈
ˆ
r
λ
j
∑
ν
(∆
j
ν
)
2
∂
Θ(
ω
j
ν
)
∂T
I
j
ν
(15)
where we define
I
j
ν
≡
∫
∞
0
d
ω
ζ
2
I
[
(
ζ
ν
−
ζ
R
)
2
+
ζ
2
I
]
2
(16)
To approximate (16), we assume the dominant con-
tribution to graphene conductivity for a resonantly-
coupled system comes from intraband transitions. Hence,
we express the optical conductivity as
σ
'
σ
intra
=
ie
2
E
F
/
(
π
~
2
(
ω
+
iγ
)). Defining
β
≡
2
π
0
~
2
e
2
L
E
F
, we can
further express
ζ
R
=
βω
2
,
ζ
I
=
βωγ
, and the resonant
frequencies at
ω
j
ν
'
√
ζ
j
ν
/β
. Correspondingly, in the in-
traband (Drude) approximation,
I
j
ν
is given by
I
j
ν
'
(
e
2
E
F
2
π
0
~
2
L
)
2
∫
∞
0
d
ω
(
ωγ
)
2
[
(
ω
j
2
ν
−
ω
2
)
2
+ (
ωγ
)
2
]
2
=
1
β
2
π
4
γ
1
(
ω
j
ν
)
2
=
π
4
γβζ
j
ν
(17)
Substituting back into (15), we have
h
(
T
) =
1
A
∑
j
∈
ˆ
r
∑
ν
λ
j
2
∂
Θ(
ω
j
ν
)
∂T
L
6
d
6
1
γ
(
∆
j
ν
ω
j
ν
2
πζ
j
ν
)
2
(18)
where we (re)emphasize:
ω
j
ν
'
√
ζ
j
ν
/β
β
≡
2
π
0
~
2
e
2
L
E
F
γ
=
ev
2
F
μE
F
To find the optimal Fermi level that maximizes heat
transfer for a given resonator size, temperature, and mo-
bility, we take the derivative of Eq. (18) with respect to
E
F
. Expressing
β
and
γ
in terms of
ω
j
ν
and substituting
back into (18), we observe the following dependence
h
(
T
)
∝
∑
j
∑
ν
∂
Θ(
ω
j
ν
)
∂T
(
ω
j
ν
)
4
(19)
Further assuming that the heat transfer is dominated
by a single mode (or a set of degenerate modes) at fre-
quency
ω
0
, the desired functional dependence is equiv-
alent (under substitution
x
=
~
ω/k
B
T
, and using (11))
to
f
(
x
) =
x
6
e
x
(
e
x
−
1)
2
(20)
which peaks at
x
0
that satisfies
x
0
coth
(
x
0
2
)
= 6
→
x
0
≈
5
.
9694
(21)
Specifically, this implies that assuming complete control
of the doping level, the optimal frequency of the peak
spectral heat transfer between identical graphene res-
onators will (in this approximation) be given by
~
ω
0
≈
6
k
B
T
.
Assuming the dominant response arises from
mode(s) with oscillator strength ∆
1
and frequency
ζ
1
(i.e.
ν
= 1), we can further approximate the ON state radia-
tive thermal conductance (and the corresponding Fermi
levels
E
1
,
2
=
E
ON
) as
h
ON
≈
116
.
23
(
0
k
5
B
T
4
2
π
~
2
e
3
v
2
F
)
1
A
λ
S
2
μ
L
7
d
6
∆
2
1
ζ
3
1
(22)
E
ON
≈
71
.
27
(
π
0
k
2
B
T
2
e
2
)
L
ζ
1
(23)
where
λ
S
≡
∑
j
∈
ˆ
r
λ
j
is the sum of all numerical prefac-
tors for the mode(s) associated with ∆
1
and
ζ
1
. Figure
S4 compares the approximated results from Eqs. (22)
and (23) to the full numerical calculations for graphene
disks. We observe that the approximated expressions be-
come more accurate for larger disk resonators where the
5
FIG. S4. (a) Optimal Fermi levels
E
on
1
=
E
on
2
, (b) size-
dependence, and (c) the corresponding maximum radiative
thermal conductance
h
ON
for graphene disks (
L
≡
√
πR
2
):
comparison between the actual (full numerical evaluation of
Eq. (10)) and the approximated result from Eqs. (22) & (23).
The shaded regions in (a) and (b) correspond to a range of
mobility values (i.e. the y-axis in (c)). Note that the
∝
R
5
de-
pendence in (b) applies to the
h
on
/h
bb
enhancement, whereas
h
on
∝
R
7
, in accordance with Eq. (4) of the main text.
optimal Fermi levels are higher (and the assumption of
intraband-dominated conductivity more valid).
Finally, these approximations allow us to directly com-
pare different resonator shapes. Using the values from
Table S1, we immediately see that two disks should have
the weakest ON state thermal conductance compared to
two squares, triangles, or (2:1) ellipses of the same area.
Full numerical simulations for these shapes indeed con-
firm this conclusion.
IV. RADIATIVE THERMAL SWITCHING:
SEPARATION DISTANCE, SUBSTRATE
FIG. S5. Switching ratio for the sheet-sheet configuration as
a function of mobility, for several separation distances
d
. We
observe similar trends for the sheet-stack configuration.
FIG. S6. Relative dielectric permittivity for SiC [6], SiN [7],
and SiO
2
[8] in the relevant frequency range.
6
[1] Maldague, P. F. Many-body Corrections to the Polariz-
ability of the Two-dimensional Electron Gas.
Surf. Sci.
1978
,
73
, 296–302.
[2] Duppen, B. V.; Tomadin, A.; Grigorenko, A. N.;
Polini, M. Current-induced Birefringent Absorption and
Non-reciprocal Plasmons in Graphene.
2D Mater.
2016
,
3
, 015011.
[3] Yu, R.; Manjavacas, A.; Garc ́ıa de Abajo, F. J. Ultrafast
Radiative Heat Transfer.
Nat. Commun.
2017
,
8
.
[4] Mulet, J.-P.; Joulain, K.; Carminati, R.; Greffet, J.-J.
Nanoscale Radiative Heat Transfer between a Small Parti-
cle and a Plane Surface.
Appl. Phys. Lett.
2001
,
78
, 2931–
2933.
[5] Novotny, L.; Hecht, B.
Principles of Nano-Optics
; Cam-
bridge University Press, 2006.
[6] Spitzer, W. G.; Kleinman, D.; Walsh, D. Infrared Proper-
ties of Hexagonal Silicon Carbide.
Phys. Rev.
1959
,
113
,
127–132.
[7] Cataldo, G.; Beall, J. A.; Cho, H.-M.; McAndrew, B.;
Niemack, M. D.; Wollack, E. J. Infrared Dielectric Prop-
erties of Low-stress Silicon Nitride.
Opt. Lett.
2012
,
37
,
4200.
[8] Kitamura, R.; Pilon, L.; Jonasz, M. Optical Constants of
Silica Glass from Extreme Ultraviolet to Far Infrared at
Near Room Temperature.
Appl. Opt.
2007
,
46
, 8118.