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Supplementary Information for
Angle-selective perfect absorption with two-dimensional
materials
Linxiao Zhu1, Fengyuan Liu2, Hongtao Lin3, Juejun Hu3, Zongfu Yu4*, Xinran
Wang2*, Shanhui Fan5*
1Department
2National
of Applied Physics, Stanford University, Stanford, California 94305, USA
Laboratory of Solid State Microstructures, School of Electronic Science and
Engineering and Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China
3Department
of Materials Science and Engineering, Massachusetts Institute of
Technology, Massachusetts 02139, USA
4Department
of Electrical and Computer Engineering, University of Wisconsin-Madison,
Madison, Wisconsin 53706, USA
5Department
of Electrical Engineering, Ginzton Laboratory, Stanford University,
Stanford, California 94305, USA
*To whom correspondence should be addressed:
ZF Yu, Email: [email protected]; XR Wang, Email: [email protected]; SH Fan, Email:
[email protected]
Fermi energy determination from Raman spectra
We perform Raman measurement on the structures with graphene layer treated with
different chemical doping durations. We measure Raman spectra at five randomly
selected locations on each sample. Fig. S1 shows the extracted G and 2 D peak positions
of graphene Raman spectra. We determine the Fermi energy E F of graphene, by
comparing the G and 2 D peak positions we measured, with the characterized relation
between Fermi energy and G and 2 D peak positions from Ref. 1 (replotted in Fig. S1,
black dots). We estimate the Fermi energy as EF = 500 meV for the 5 -min doping
duration case, and EF = 300 meV for the 40 -s doping duration case.
Figure S1: Fermi energy determination from Raman spectra. (a) G peak positions of
graphene Raman spectra. The red and blue bands denote the positions of the G peak on
the 5 -min chemically doped graphene and 40 -s chemically doped graphene,
respectively. The black dots show the calibrated relation between G peak position and
Fermi level, replotted from Fig. 3b of Ref. 1. (b) 2 D peak positions of graphene Raman
spectra. The red and blue bands denote the positions of the 2 D peaks on the 5 -min
chemically doped graphene and 40 -s chemically doped graphene, respectively. The
black dots show the calibrated relation between 2 D peak position and Fermi level,
replotted from Fig. 3c of Ref. 1. In both a and b, the Raman spectra have been taken at
five randomly selected locations, as denoted by the bands.
Angle-selective perfect absorption in 2D material for s-polarization
In this section, we provide additional theoretical derivations to show that angle-selective
perfect absorption can be achieved in 2D material, for s-polarization.
We first derive the external decay rate of a bare structure, consisting of a lossless
spacer layer with thickness d and a perfect electric conductor (PEC) back reflector, as
shown in Fig. S2. The spacer material has a relative permittivity  , and relative
permeability  . Placing a sheet of 2D material atop the structure minimally changes the
field pattern inside the structure, and generally maintains the external decay rate. Thus, it
suffices to calculate the external decay rate of the bare structure without the 2D material.
Figure S2: Geometry for a bare structure with light incident in s-polarization. The
structure consists of a lossless spacer on top of a perfect electric conductor (PEC)
reflector. The spacer layer has thickness d .
In vacuum region, the electric field can be written as:
E y ,V = E0 [e
ikz ,V ( z  d )
 re
ikz ,V ( z d )
] e
ikx x it
,
(SI 1)
where r is the reflection coefficient, and k x and k z are wavevector components in x and
z directions, respectively. Here the subscript V denotes vacuum region. From Equation
(SI 1), the transverse component of magnetic field in vacuum region is:
H x,V =
k z ,V
0
E0 [e
ik z ,V ( z  d )
 re
ik z ,V ( z  d )
] e
ik x x it
.
In spacer region, the electric field can be described as:
ikz , S z
E y ,S = E0 [Ce
 Ce
ikz , S z
] e
ikx x it
,
where we have used the boundary condition at the spacer-PEC interface. Here C is a
coefficient, and the subscript S denotes spacer region. The transverse component of
magnetic field in the spacer region is:
H x,S =
k z ,S
0
E0 [Ce
ik z , S z
 Ce
ik z , S z
] e
ik x x it
.
By enforcing the continuity of transverse components of the electromagnetic
fields at the vacuum-spacer interface, the reflection coefficient can be solved as:
r=
where  s =
1  i s
,
1  i s
(SI 2)
  sin 2

cot ( d   sin 2 ) . At resonance frequency 0 ,  s = 0 .
  cos
c
Near the resonance frequency,  s can be linearized:
 s ( ) 
d s
d   sin 2
| = (  0 ) = (0   ) 
,
0
d
c   cos
(SI 3)
where c is the velocity of light, and the subscript s denotes s-polarization. Combining
Equations (SI 2) and (SI 3), we have
d   sin 2
1  i (  0 ) 
c   cos
r=
.
d   sin 2
1  i (  0 ) 
c   cos
(SI 4)
We now use coupled mode theory to re-express the reflection coefficient. We
consider a resonance of the bare structure, with normalized amplitude a . The resonance
has a resonance frequency 0 , and an external decay rate  e by external radiation
through the top surface. The coupled mode theory equations2, 3 are:
d
a(t ) = (i0   e )a(t )  2 e s (t ),
dt
s (t ) = s (t )  2 e a(t ),
where s and s represent the amplitude of incoming and outgoing waves. The reflection
coefficient is:
s ( ) 1  i (  0 )/ e
=
.
s ( ) 1  i (  0 )/ e
(SI 5)
By comparing Equations (SI 4) and (SI 5), we have the external decay rate:
 e,s =
c   cos

.
d   sin 2
(SI 6)
We now calculate the internal decay rate of a 2D material atop the structure. The
internal decay rate due to the absorption in the 2D material can be calculated as:
i =
Re ( )
| E ( ) z = d |2
2
1
 ,
1
dz [0 | E ( ) |2  0 | H ( ) |2 ] 2

0
4
d
(SI 7)
where  is the 2D conductivity of the 2D material. The total energy per unit area of the
resonance is:
1 d
dz [0 | E ( ) |2  0 | H ( ) |2 ] =| CE0 |2 0 d ,
4 0
at resonance
0
c
d   sin 2 =

2
 (2m  1) , where m is an integer. The energy
dissipation inside the 2D material at resonance is:
Re ( )
1
| E ( ) z = d |2 = Re ( )  4 | CE0 |2 .
2
2
Here, as the 2D material is atomically thick, it generally maintains the field pattern inside
the structure.
Using Equation (SI 7), the internal decay rate due to absorption in a 2D material
separated from a PEC mirror by a lossless spacer in s-polarization is:
 i,s =
For a small conductivity Re ( ) 
c 1
  Re ( ) Z 0 .
d 

1
 , the angle of incidence for
2   1 Z 0
critical coupling in s-polarization can be well described as:
c,s 

  1
Re ( ) Z 0 ,
2


where Z 0 = 0 / 0 is the vacuum impedance. In the main text, we consider the scenario
where the spacer is nonmagnetic, i.e.  = 1 , as it is closely related to our experimental
demonstration.
Angle-selective perfect absorption in 2D material for p-polarization
(SI 8)
Angle-selective perfect absorption for p-polarized light in 2D material can be achieved,
by placing the 2D material atop a structure, consisting of a lossless spacer layer and a
perfect magnetic conductor (PMC) back reflector. A schematic of the bare structure is
shown in Fig. S3. The spacer material has a relative permittivity  , and relative
permeability  .
Figure S3: Geometry for a bare structure with light incident in p-polarization. The
structure consists of a lossless spacer on top of a perfect magnetic conductor (PMC)
reflector. The spacer layer has thickness d .
In the vacuum region, the magnetic field can be described as:
H y ,V = H 0 [e
ikz ,V ( z d )
 re
ikz ,V ( z d )
] e
ikx x it
,
where r is the reflection coefficient. From Equation (SI 9), the transverse component of
electric field in vacuum region is:
Ex,V = 
k z ,V
 0
H 0 [e
ik z ,V ( z  d )
 re
ik z ,V ( z  d )
] e
ik x x it
.
In the spacer region, as the transverse component of magnetic field adjacent to the PMC
layer must be zero, we have:
ikz , S z
H y ,S = H 0 [Ce
 Ce
ik z , S z
] e
ikx x it
,
where C is a coefficient. Accordingly, the electric field in the spacer region is:
(SI 9)
Ex , S = 
Ez ,S = 
k z ,S
0
kx
0
H 0 [Ce
ik z , S z
] e
ik z , S z
] e
ik z , S z
 Ce
ik z , S z
 Ce
H 0 [Ce
ik x x it
ik x x  it
,
,
where the expression of E z , S will be used for evaluating internal decay rate.
By enforcing boundary conditions, the reflection coefficient can be solved as:
r=
where  p =
1  i p
1  i p
,
(SI 10)
  sin 2

cot ( d   sin 2 ) . At resonance frequency 0 ,  p = 0 .
  cos
c
Near resonance frequency, it suffices to linearize the expression of  p :
 p ( ) 
d p
d   sin 2
| = (  0 ) = (0   ) 
.
0
d
c   cos
(SI 11)
Combining Equations (SI 10) and (SI 11), we have
d   sin 2
1  i(  0 ) 
c   cos
r=
d   sin 2
1  i(  0 ) 
c   cos
(SI 12)
By comparing Equations (SI 12) and (SI 5), we have the external decay rate in ppolarization as:
 e, p =
c   cos

.
d   sin 2
We now calculate the internal decay rate of the 2D material. The total energy per
unit area of the resonance is:
1 d
dz[0 | E ( ) |2  0 | H ( ) |2 ] =| CH 0 |2 0 d ,

0
4
(SI 13)
at resonance

c
d   sin 2 =

2
 (2m  1) , where m is an integer. The energy
dissipation inside 2D material at resonance is:
2
 k 
Re ( )
| E ( ) z = d |2 = Re (  )  2 | CH 0 |2  x  ,
2
 0 
where   is the 2D conductivity of the 2D material in vertical direction.
Using Equation (SI 7), the internal decay rate due to absorption in the 2D material
can be calculated as:
c sin 2

 Re (  ) Z 0 .
d  2
 i, p =
The external decay rate as described in Equation (SI 13) monotonously decreases
as angle of incidence increases, and reaches zero at  = 90  . In contrast, the internal
decay rate shown in Equation (SI 14) increases as angle of incidence increases, and has a
finite value at  = 90  . Notice that at normal incidence  i , p = 0 . Thus, in this geometry,
there is always an incidence angle where the p-polarized light can be completely
absorbed in the 2D material. For a small 2D conductivity, the critical angle in ppolarization then can be well described as:
p 

  1
Re (  ) Z 0 .
2  3

Parasitic loss and angular spread
We calculate the absorptivity of the structures, using experimentally measured complex
refractive index of Ge23Sb7 S70 and tabulated complex refractive index of Au 4. Taking
into account these parasitic losses, the numerical calculation shows that the structure
(SI 14)
achieves perfect absorption, among which 66.8 % of incident s-polarized light is
absorbed inside graphene, as shown in Fig. S4a. Here, the structure consists of a
chemically doped graphene layer with EF = 500 meV, a 1.9 -µm-thick Ge23Sb7 S70 glass
layer and Au reflector. The assumed mobility for graphene of 750 cm2 V-1 s-1 is
consistent with the property of typical CVD-grown graphene5, 6. The parasitic loss in the
reflector may be eliminated by using a dielectric Distributed Bragg Reflector.
Figure S4: Influence of parasitic loss and angular spread. (a) Calculated absorptivity
for a structure without angular spread. The pink line shows the absorptivity for the whole
structure, and the cyan, grey and yellow lines show the absorptions inside graphene, the
metal layer and the chalcogenide glass (ChG), respectively. (b) Calculated absorptivities
in the presence of angular spread. Taking into account of a Gaussian distribution of
angular spread with standard deviation of 7.5 , the pink and cyan lines denote the
calculated absorptivities for the whole structure, and the part absorbed by graphene. The
green circles denote the measured absorptivity for the structure with graphene at
EF = 500 meV. In the calculations of a and b, the structure consists of a doped
graphene layer with EF = 500 meV and 750 cm2 V-1 s-1 mobility, on top of a 1.9 µmthick Ge23Sb7 S70 layer and Au reflector. The light is incident at 88 , in s-polarization.
The numerical calculation in Fig. S4a shows higher absorptivity in either the
whole structure or graphene, as compared with the measurement results (Figure 2 in main
text). The difference between the two can be accounted for by taking into account the
angular spread of the light incident on the structures in the measurement system. The
Fourier-transform infrared (FTIR) spectrometer combined with variable angle reflection
accessory has a relatively large angular spread7. Accordingly, measured absorptivity is an
average of the absorptivities weighed over a range of angles of incidence centered around
a nominal angle of incidence, which limits the highest absorptivity that can be measured.
We assume a Gaussian distribution for the angular spread, obeying a Gaussian weighing
function8:
f ( , N ,   ) =
1
2  
e
1    N
 
2  




2
,
where  N is the nominal angle of incidence as measured, and   is the angular spread of
the probing beam. Thus, the absorptivity AN for a Gaussian beam is:
where A( ) is the absorptivity at a single angle of incidence  .
Taking into account an angular spread with a   of 7.5 , Fig. S4b shows the
numerically calculated absorptivity at 88 angle of incidence for the structure with doped
graphene at EF = 500 meV (pink line), which agrees well with the measured
absorptivity (green circles). With the same angular spread, the peak absorptivity in
graphene is calculated to be 51.0 % , as shown in Fig. S4b cyan line, which closely
matches the graphene absorptivity extracted from measurement as 45.8 % at the same
angle of incidence. The angular spread here, which is obtained from a best fit to
simulations, is consistent with the angular spread of a similar measurement system in
Ref. 7. Therefore, we expect that our fabricated device may already achieve critical
coupling, where the majority of absorption is in graphene.
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