Supplementary Information
Vapor-deposited amorphous metamaterials as visible near-perfect
absorbers with random non-prefabricated metal nanoparticles
Yun Zhang1, Tiaoxing Wei1, Wenjing Dong1, Kenan Zhang1, Yan Sun1, Xin
Chen1&Ning Dai1
1
National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics,
Chinese Academy of Sciences, Shanghai 200083, China
Measurements: Scanning electron microscope images were obtained on FEI
Sirion200 to detect the surface coverage of the Au-NPs and their size distribution. To
characterize the absorption properties, Lambda 950 equipped with URA was used to
measure the reflectance with 2nm step and 5.0 slit varying the incident angle from 8
degree to 65 degree. The incident light is depolarized, so these peaks are the
maximum values in the absorptance spectra.
Numerical
simulation:
Numerical
simulations
were
performed
by
FiniteDifferenceTimeDomain (FDTD) with periodic boundary conditions along the X
and Y direction while perfect matched layers along the propagation of the incident
waves (Z direction). A normally incident plan waves was applied on the structure with
the polarization along with X direction. The permittivity of the Au nano-semispheres
was obtained from P. B. Johnson and R. W. Christye’s work in 1972 years1 while
ZnO is set to 4. To model Ag in the simulations, we use both the permittivity
extracted from our experiments and elsewhere1.
Theoreticalanalysis:
We can derive the effective permittivity, zeff, of the PMA by the transmission line
modeling2:
zeff  i  z
eff
II
zd  tan(kd  d )  zIIeff  tan(kII  heff )
zIIeff  zd  tan(kd  d )  tan(k II  heff )
,
(S1)
where i  1 . (zd, d, kd) are the (impedance, thickness, wave vector) of the dielectric
spacer layer, while ( z IIeff , heff,kII) are the (effective impedance, effective thickness,
wave vector) of the random layer of Au-NPs. We note that the bottom layer Ag film is
assumed to be a perfect conductor metal for convenient, that is the impedance of it is
set as zero. Besides, all materials are assumed to be nonmagnetic (𝜇=𝜇0)2.
For zeff, obtaining the properties of the random layer of Au-NPs, ( z IIeff , heff), are the
essential issue while it’s easier to get (zd, d). Here we give a simple estimation about
these factors which restrict z IIeff . As mentioned above, the optical response of these
Au-NPs can be described as dipoles to absorb and secondary scatter light when
having dimensions much smaller than the excited waves. Considering one Au-NP
with oblate spheroid shape (radius r and height h), although a slightly compress
deformation happened in real system, the dipole can be described by a static
polarizability as3:
4
3
 static    r 2  h 
 II   d
,
 II  L  ( II   d )
(S2)
whereεII and εd are the relative permittivities of the Au-NP and surrounding media,
respectively, and L is the shape factor. When the Au-NPs have dimensions about 50
nm or more, the static polarizability, αstatic, should be modified by long wave
approximation 
MLWA

static
k2
2
3
static
(1   i  k  
   static ) , k=nk0 (n is the
3
r
refractive index of the surrounding media, k is the wave vector)3,4. Importantly, due to
an effective image dipole induced when put the Ag film tens nanometers (the
thickness of dielectric spacer layer d) away from the Au-NP5, the polarizability, αMLWA,
should be further modified as an effective polarizability6, αm,
m 
 MLWA
1  Er   MLWA p
,(S3)
wherep is the dipole moment and Er is the reflected field at dipole location. Although,
in the real system, the dipole will be affected by other dipoles around and can be
modified by introducing a interaction factor4,7, the interactions among Au-NPs,
nevertheless, can be neglected due to the low surface coverage (22% here) and the
random distribution8.
Considering the real case, random distributed Au-NPs with a broader size range lay
on a flat surface, the absorption of these Au-NPs can be described as the a effective
thin film or meta-film with an effective thickness, heff, and impedance, z IIeff 3. We note
that for the PMAs, the absorption for a given size is proportional to the low surface
coverage of nanoparticles9, in our case we assume the absorption of the bare Au-NPs
layer is also proportional to the surface coverage. Since the total absorption cross
section, σabs,is linear to the imaginary part of the total effective polarizability3,  IIeff ,
we can derived the imaginary part of the  IIeff as an average effect
 abs
k
 Im( IIeff )  c   Im( m )  f (r ) ,
(S4)
r
wheref(r) is the area ratio versus the radius r roughly estimated from the SEM image
as shown in Fig. S3c (histogram).c is the surface coverage concentration.However,
according to the famous Kramers-Kroning relations, we can easy get the real part of
the effective permittivity of the arrays ( Re( IIeff ) ). As soon as the  IIeff was obtained,
the reflection coefficient (r) and transmission coefficient (t) under normal incidence
could be achieved7,10.We can derived the effective impedance of the Au-NPs through r
and t as11
zIeIff  
(1  r )2  t 2
,
(1  r )2  t 2
(S5)
with the effective thickness can be written as the average height of the Au-NPs,
heff=<h>. Thus, the effective impedance of the Au-NPs layer can be described as
zIIeff   IIeff  IIeff  H (c   Im( m )  f (r )) it’s worth to mention that αm is also related
N
to the permittivity and thickness of spacer layer, (zd, d), and the permittivity of the Ag
film. And according to Eq. S1, the effective impedance of the PMA can be described
as zeff  G( zI , zd , zIeffI , heff , d ) .
Fig. S1. a, TEM view of the Au-NPs transfer from the PMAs to copper grid. b, Top
view (60 degree off the surface normal of the top view) of the PMA surface. c,
Contacting-area fraction distribution of the Au-NPs sizes for a sample of 65 Au-NPs.
Fig. S2.a, Measured absorptance spectra for 8 degree incidence with (red solid line
with two maximum values denoted as peak b and c, respectively) and without (black
solid line denoted as peak a) random Au-NPs layer (i.e. from 3 nm Au thin film)
coated on ZnO/Ag structure for the random Au-NPs layer with 22% surface coverage
and size distribution shown in Fig. S1c, where the ZnO spacer layer is 30 nm in
thickness.b, c, Peak position and peak intensity of the peak a, b, c as a function of the
incident angle, respectively.d, The absorptance spectra for the 50nm ZnO spacer layer
with varying the incident angle from 8 degree to 65 degree. e, The absorptance spectra
for the 50nm Al2O3 spacer layer with varying the incident angle from 8 degree to 65
degree.
Fig. S3. a, AFM view of the PMA surface with a average roughness <10nm. b, The
reflectance spectra of the PMA measured by a Universal reflection accessory(URA,
red solid line, 8 degree incident) and the Integrate sphere accessory (ISA, black solid
line, normal incident) with the difference <1.5% in the valley at 596nm.
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