1
Three-dimensional single gyroid photonic crystals
with a mid-infrared bandgap
1
Siying Peng,
2
Runyu Zhang,
1
Valerian H. Chen,
1
Emil T. Khabiboulline,
2
Paul Braun,
1
Harry A.
Atwater
1. Applied Physics, Califor
nia Institute of Technology
2. Department of Materials Sci
ence and Engineering, University
of Illinois at Urbana-
Champaign
ABSTRACT:
A gyroid structure is a distinct morphology that is triply per
iodic and consists of
minimal isosurfaces containing no straight lines. We have desig
ned and synthesized amorphous
silicon (a-Si) mid-infrared gyroi
d photonic crystals that exhib
it a complete bandgap in infrared
spectroscopy measurements. Phot
onic crystals were synthesized b
y deposition of a-Si/Al
2
O
3
coatings onto a sacrificial polymer scaffold defined by two-pho
ton lithography. We observed a
100% reflectance at 7.5 μm for single gyroids with a unit cell
size of 4.5 μm, in agreement with
the photonic bandgap position predicted from full-wave electrom
agnetic simulations, whereas the
observed reflection peak shifted to 8 μm for a 5.5 μm unit cell
size. This approach represents a
simulation-fabrication-characterization platform to realize thr
ee-dimensional gyroid photonic
crystals with well-defined dimens
ions in real space and tailore
d properties in momentum space.
2
KEYWORDS:
photonic bandgap, three dimensional photonic crystals, mid-in
frared, gyroids,
Weyl points
TABLE OF CONTENTS GRAPHIC
Three-dimensional photonic crys
tals offer opportunities to prob
e interesting photonic states such
as bandgaps
1-8
, Weyl points
9, 10
, well-controlled dislocations and defects
11-13
. Combinations of
morphologies and dielectric consta
nts of materia
ls can be used
to achieve desired photonic states.
Gyroid crystals have interesti
ng three-dimensional morphologies
defined as triply periodic body
centered cubic crystals with minimal surfaces containing no str
aight lines
9, 10, 14-19
. A single gyroid
structure, such as the one s
hown in Fig. 1a, consists of isosur
faces described by
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ݔ
ሻ
ݏܥ
ሺ
ݕ
ሻ
݊݅ܵ
ሺ
ݕ
ሻ
ݏܥ
ሺ
ݖ
ሻ
݊݅ܵ
ሺ
ݖ
ሻ
ݏܥ
ሺ
ݔ
ሻ
ሻݖ,ݕ,ݔሺݑ
,
where the surface is constrained by
ሻݖ,ݕ,ݔሺݑ
. Gyroid structures exist in biological systems in
nature. For example, self-organizing process of biological memb
ranes forms gyroid photonic
crystals that exhibit the iridescent colors of butterfly’s wing
s
20
. Optical properties of gyroids could
vary with tuning of u(x,y,z)
21
, unit cell size, spatial symmetry
9
as well as refractiv
e index contrast.
Single gyroid photonic crystals, when designed with high refrac
tive index and fill fraction, are
predicted to possess among the wi
dest complete three-dimensiona
l bandgaps
9, 22
, making them
interesting for potential device
applications such as broadband
filters and optical cavities. In this
3
work, we demonstrate a synthesis approach for forming mid-infra
red three-dimensional gyroid
photonic crystals, and report expe
rimental measurements of the
bandgap for a single gyroid
structure at mid-infrared wavelengths.
Figure 1 Single gyroid structure and photonic bandstructure sim
ulation.
(a)Stacks of unit
cells of single gyroid structure
(b) bcc Brillouin zone. (c) ph
otonic band structure of a-Si single
4
gyroid from full wave simulations
, with unit cell size of 5 μm
in x, y, z directions. The color map
is proportional to logarithm
of the density of states.
RESULTS
To realize gyroid photonic crystals at mid-infrared wavelengths
, we utilize full-wave finite-
difference time-domain (FDTD) simulations to determine the dime
nsions and materials required
for crystal design (see Methods). The simulation shown in Fig.
1c reveals that a-Si single gyroid
crystals with a unit cell size
of 5μm and u(x,y,z)=1.1 (see Tab
le S1 for fill fraction) has a complete
bandgap (indicated by the dashed box) from 8μm to 10μm in all s
ymmetry directions of the bcc
Brillouin zone in Fig. 1b. In units of normalized frequency cal
culated by dividing unit cell size (a)
by wavelength (λ), the complete bandgap is between 0.5 and 0.6.
For constant refractive index at
mid-infrared wavelength, we can use the value of normalized fre
quency to deduce bandgap
position for crystals with different unit cell size. For exampl
e, for a=5.5μm single gyroid crystal
investigated in Fig.4b, the center of the bandgap can be inferr
ed by a normalized frequency,
resulting in a shift of the bandgap center by 0.6 μm. Therefore
, we identified a-Si as a suitable
material with its high refractive index and low loss at mid-inf
rared wavelength. Another suitable
candidate material is germanium, which has even higher refracti
ve index (n>4) and sufficiently
low loss in this wavelength regime.
To fabricate a-Si single gyroid s
tructures, we de
veloped a prot
ocol which incorporates multiple
steps
2, 12, 13, 23, 24
(see Methods for detailed description), as illustrated in Fig.
2. Two-photon
lithography was utilized to directly write a sacrificial polyme
r scaffold of gyroid photonic crystals
with unit cell sizes of 4.5, 5.1 and 5.5 μm on mid-infrared tra
nsparent silicon s
ubstrates. Each
sample is composed of 20x20x10 unit cells. We conformally depos
ited 40 nm thick aluminum
5
oxide coatings on the
polymer gyroids via a
tomic layer depositi
on (ALD) at 150
0
C. We then used
focused ion beam (FIB) milling to remove the crystal sides to f
acilitate polymer removal, yielding
a hollow inorganic aluminum oxide crystal after oxygen plasma c
leaning. Subsequently, the
structure was conformally coated and in-filled with a 100nm a-S
i layer at 350
0
C using chemical
vapor deposition (CVD). The polymer structure is not structural
ly stable at temperatures above
250
0
C, which are typically necessary for conformal deposition of hi
gh refractive index materials
such as a-Si. Therefore the hollow aluminum oxide crystal is a
critical intermediate structure to
provide a scaffold that can withstand high temperature. The fin
al structure consists of a 40nm
middle layer of aluminum oxide
and two 100nm/150nm a-Si layers
on both the inside and outside
of the aluminum oxide scaffold, corresponding to u(x,y,z) value
s of 1.1/1.05, 1.2, 1.25 and
1.35/1.37 for coated a-Si, Al
2
O
3
, in-filled a-Si and inner hollow
part respectively (see Table
S1 for
fill fraction values).
6
Figure 2 Fabrication procedures of gyroid photonic crystals.
(a) two photon lithography to
define a sacrificial polymer scaffold of the gyroid structure.
(b) atomic layer deposition of Al
2
O
3
to coat the polymer structure. (c) focused ion beam milling to
remove the sides of the structure,
followed by oxygen plasma to remove the polymer, leaving a holl
ow alumina structure. (d)
chemical vapor deposition of a-Si to coat and in-fill the hollo
w alumina structure.
We characterized the resulting a-Si single gyroid photonic crys
tals by Fourier transform infrared
spectroscopy (FTIR), shown in Fi
g. 3a (see Methods for a detail
ed description of the
measurements). SEM images of the characterized sample are shown
in Fig. 3b. The reflectance
spectrum of the a=4.5μm sample r
eveals a peak of 98% at 7.0 μm
as shown in Fig. 3c (red dashed
line), in agreement with the predicted reflectance peak center
of the 4.5μm trapezoidal structure in
7
the figure above (black dashed lin
e) (see the section of “Full
wave simulation on deformed
crystals” in Methods), as well as the band gap center for a cub
ical single gyroid
structure (gray
dashed line). The reflectance of the sample is normalized to th
e reflectance of an atomically smooth
gold mirror of 97% reflectance.
Additional 50nm coating and in-
filling of a-Si on the structure red
shifted the reflectance peak to 7.5 μm, giving rise to the 100%
reflectance peak shown in Fig. 3c
(red solid line). The reflectan
ce peak at 7.5μm is a direct man
ifestation of a photonic bandgap. The
transmittance spectrum shown in Fig. 4a has a wide 0% transmitt
ance band centered at 7.5 μm,
confirming the bandgap. The extin
ction (scattering+absorption)
is obtained by subtracting the
transmittance and reflectance percentages from 100%. Since the
FTIR collection angle is limited
to 16
0
-34
0
, that part of the reflected light that lies outside of this an
gular range is considered as
scattering here. These results reveal the photonic property of
the single gyroid structure, namely
the optical bandgap in the mid-infrared regime.
We also characterized the reflectance spectrum of the a=5.1 μm
period and a=5.5 μm period
samples and compared them with that of the a=4.5μm sample, as s
hown in Fig. 4b. We observed
a red shift of the ref
lection peaks by 0.6μm in wavelength for
the 5.5μm period sample, and a shift
of 0.4 μm in the 5.1μm sample relative to the 4.5μm sample, whi
ch is in agreement with the
increase of the bandgap wavelength predicted (see Fig. 1c and F
ig. 5). This observation can be
intuitively understood by considering the resonant coupling bet
ween the incoming beam and the
cavity of a periodic unit cell. The agreement between our exper
imental results and simulations for
three structures with different unit cell sizes confirms the mi
d-infrared bandgap feature of the
single gyroid photonic crystals.
8
Figure 3 FTIR characterization of bandgaps.
(a) Fourier transform infrared spectroscopy
experimental configuration. (b) SEM images of a hollow gyroid a
t (001) crystal orientation with
a-Si (150nm) /Al
2
O
3
(40nm) /a-Si (150nm) layers (c) reflectance spectrum from full
wave
simulations of a hollow gyroid at
(001) crystal o
rientation wit
h a-Si (100nm) /Al
2
O
3
(40nm) /a-Si
9
(100nm) layers (gray dashed line),
reflectance spectrum from fu
ll wave simulations of a
trapezoidal hollow gyroid at (001
) crystal orientation with a-S
i (100nm) /Al
2
O
3
(40nm) /a-Si
(100nm) layers (black dashed line), FTIR measurement of a hollo
w gyroid with a-Si (100nm)
/Al
2
O
3
(40nm) /a-Si (100nm) layers (red dashed line) and FTIR measurem
ent of a hollow gyroid
with a-Si (150nm) /Al
2
O
3
(40nm) /a-Si (150nm) layers (red dashed line).
10
Figure 4 FTIR characterization of single gyroids.
(a) reflectance, transmittance and
scattering+absorbance from a single gyroid structure with unit
cell size of 4.5
m and a total of
20x20x10 cells. (b) comparison of reflection spectra of samples
with unit cell sizes of 4.5μm,
5.1μm and 5.5μm (20x20x10 unit cells).
11
DISCUSSION
Several specific features arise in the reflectance spectra of t
hese single gyroid samples due
practical aspects of the chosen m
aterial compositions and defec
ts present in the crystal structures.
The bandgap center of the 4.5 μm period structure is shifted fr
om the predicted 8
m wavelength
inferred from Fig. 1c and Fig. 3c (black dashed line) to the ex
perimentally measured 7.0
m
wavelength in Fig. 3c (red dashed line). This is due to lower e
ffective refractive index of the a-
Si/Al
2
O
3
/a-Si heterostructure present in fabricated samples as compared
with an assumed
homogeneous dense solid a-Si cross-section for the photonic cry
stal elements in the simulations
given in Fig. 1c. The center of the bandgap calculated from ful
l wave simulation for the actual a-
Si (100nm) /Al
2
O
3
(40nm)
/a-Si (100nm) composition is at 7 μm, shown in Fig. 5. Another
minor
contribution comes from the lower
refractive index in the exper
imental a-Si as compared to those
of Palik
25
used in full wave simulations, shown in Fig. S1a, likely due t
o differences in the
deposition conditions.
The main crystal defect is a overall shape distortion of the la
ttice which is att
ributed to polymer
shrinkage, as shown in SEM images in Fig. 2 and Fig. 4, instead
of the cubic shape expected in an
ideal lattice. A trapezoidally shaped crystal is formed as the
result of the polymer scaffold
shrinkage. After direct laser wri
ting, the written IP-Dip photo
resist cross-links to form polymer
scaffolds. The unwritten photor
esist is then rem
oved in the pro
cess of developmen
t, during which
the top portion of the polymer s
caffold shrinks. The adhesion f
orce between the scaffold and the
rigid Si substrate prevents of t
he bottom lattice from shrinkin
g. This distortion results in a
decreasing unit cell size from the bottom to the top of the cry
stal, resulting in an overall trapezoidal
shape for the crystal. In an ideal cubic crystal, we expect a 1
00% reflectance peak with bandwidth
matching that of the predicted
bandgap, shown in Fig. 3c (black
dashed line). A narrowing of the
12
measured reflectance peak in Fig. 3c (red dashed line) is expec
ted when the trapezoidal shape is
taken into account in the deformed crystal simulation (see Meth
ods “Full wave simulation on
deformed crystals”), together with the a-Si/Al
2
O
3
/a-Si material composition. Intuitively, the
measured reflectance spectrum is a superposition of the scatter
ing from each crystal layer over the
range of momenta accessible by the 16
0
to 34
0
range of incident angles
of the FTIR configuration
shown in Fig. 3a and Fig. 5 (ora
nge lines). As the unit cell di
mension increases from top to bottom
of the trapezoidal shaped crystal, the reflectance peak center
of each layer is increasingly red
shifted. The width of the measured reflectance narrows as a res
ult of the effective inhomogeneity
and the lowered effective refractive index. The second reflecta
nce peak observed in Fig. 4b, at 6.1
μm for the a=4.5 μm sample, 6.5 μm for the a=5.1 μm sample and
6.8 μm for the a=5.5 μm sample,
respectively, can also be explai
ned by this effective inhomogei
neity . From reflectance simulations
of an ideal a-Si gyroid crystal in Fig. 3c (gray dashed line),
distinct photonic bands exist above the
band gap, manifested as sharp dips in the reflectance spectrum.
The states observed in the
reflectance spectra is equivalent of projected band structure o
n the (001) plane, spanned by Γ-H
and Γ-H’ symmetry directions. Ba
nd structure simulations in Fig
. 1c and Fig. 5 indicate photonic
bands above the band gap. A trapezo
idal shaped crystal gives ri
se to spectral inhomogeneity, and
consequently the photonic band f
eatures broadens
into one refle
ctance peak. Despite realistic
material compositions and sample defects, the essential physics
interpreted from these reflectance
spectra is not affected by the
above-mentioned nonidealities.
13
Figure 5 Photonic band structure simulation.
Photonic band structure of a single gyroid consists
of a-Si (100nm) /Al
2
O
3
(40nm) /a-Si (100nm) layers from full wave simulations, with un
it cell size
of 4.5μm in x, y, z directions. Bands in between the orange lin
es are accessible through FTIR
characterization shown in Fig. 3a
CONCLUSION
In conclusion, we experimentally observed a 100% reflective ban
dgap at mid-infrared
wavelengths in single gyroid photonic crystals with high refrac
tive index materials, fabricated
using two-photon lithography and
conformal layer deposition, co
nfirming photonic bandgap
predictions obtained from simul
ations. This mid-infrared bandga
p is also predictably tunable by
changing the unit cell size in the simulation design and fabric
ation. The synthesis/characterization
approach described here opens the door to design of more comple
x mid-infrared photonic crystals
with topological states, such as
Weyl points in double gyroid p
hotonic crystal with parity-breaking
symmetry, for which synthesis of
single gyroid photonic crystal
s establishes feasibility. Further
designs may also yield gyroid photonic structures whose surface
s exhibit topologically protected
states, suggesting the possibili
ty to synthesize these intrigui
ng structures to cre
ate unusual states
and phases of light.
14
METHODS
Full wave band structure simulation
Simulations were performed using Lumerical FDTD Solutions v8.15
.716. In simulations, bands
are excited by randomly placing dipoles inside the simulation r
egion that is defined by Bloch
boundary conditions in x, y and z directions. Randomly placed f
ield monitors record electric and
magnetic fields over time. Fourier transformation of the overal
l electric field versus time reveal
spectrum of the bands. By tuning phase of the Bloch boundary co
nditions, we were able to
calculate bands at wave vectors al
ong all high symmetry directi
ons in the Brillouin zone of a bcc
lattice. Palik
25
n, k data for a-Si are used in t
he simulation (shown in supple
mentary Fig. 1S).
Sample fabrication
Polymer gyroid structures were written in negative photoresist
IP-Dip using the Photonic
Professional GT system (Nanoscribe GmbH). 40 nm thick aluminum
oxide coatings on the
polymer gyroids were conformally deposited via atomic layer dep
osition at 150
0
C in a Cambridge
Nanotech S200 ALD System with H
2
O and trimethylaluminum (TMA) precursors. We used
focused ion beam milling with the FEI Nova 200 Nanolab at 30 kV
and 30 nA Ga beam condition
to remove the crystal sides to facilitate polymer removal. We e
tched out the polymers with oxygen
plasma using the March PX-500 plasma etcher, yielding a hollow
inorganic aluminum oxide
crystal. Then the structure is conformally coated and in-filled
with 100nm/150nm of a-Si at 350
0
C
using static chemical vapor deposition, with refilled silane as
the precursor at an average deposition
speed of 10nm/hour. Refractive index of both deposited a-Si and
Al
2
O
3
are measured and shown
in supporting information Fig. 1S.
15
FTIR characterization
The mid-infrared light is incident on the sample at incidence a
ngles from 16
0
to 34
0
after being
focused with a Cassegrain objective. The sample sits on an intr
insic double side-polished silicon
substrate with the (001) crystal
surface of the gyroid structur
e in parallel with the substrate surface.
Reflection and transmission spec
tra are collected from the same
range of angles with two identical
Cassegrain objectives on each side of the sample respectively.
Each incidence angle could excite
a corresponding wave vector in a specific symmetry direction in
the band structure. For three-
dimensional crystal structures,
the orientation of the crystal
could determine projection of the band
structure onto a specific symmetry plane.
Full wave simulation on deformed crystals
Simulations were performed using Lumerical FDTD Solutions v8.15
.716. To take into account
effects of polymer shrinkage dur
ing the structure developing pr
ocess, we performed full wave
simulation approximating the tra
pezoidal morphology shown in SE
M images in Fig. 2 and Fig. 4.
The simulated structure is infinitely periodic in the x and y d
irections and has a finite length of
three unit cells in the z direction, with the a-Si/Al
2
O
3
/a-Si material composition. The simulation
region has periodic boundary condition in the x and y direction
s and perfectly matched layer
(PML) absorbing boundar
y condition in the z
direction. Plane wa
ve source incidents on the
structure in the (001) direction as indicated in Fig. 3a. Frequ
ency domain field monitors are placed
above and below the structure to collect reflection and transmi
ssion spectra. We repeated the
simulation for a series of unit cell sizes ranging from 4.2 μm
to 4.7 μm with an increment of 0.1
16
μm. The arithmetic average of th
ese six reflectance spectra, sh
own as the black dashed line in Fig.
3c, is used to approximate the tra
pezoidal effect on the reflec
tance spectrum.
ASSOCIATED CONTENT
Supporting Information Available:
Optical constants of deposite
d materials (a-Si and Al
2
O
3
) and u(x,y,z) versus fill fraction.
ACKNOWLEDGMENTS
This work is part of the Light Material Interactions in Energy
Conversion Energy Frontier
Research Center funded by the U.S. Department of Energy, Office
of Science, Office of Basic
Energy Sciences under Award Number DE-SC0001293.The authors tha
nk George Rossman for
FT-IR assistance, the Kavli Nanoscience Institute at Caltech fo
r cleanroom facilities, the Lewis
Group ALD facility at Caltech, V.
W. Brar and F. Liu for insigh
tful discussions.
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25. Piller, H. Silicon (Amorphous) (a-Si). In
Handbook of Optical Constants of Solids
, Palik,
E. D., Ed. Academic Press: Burlington, 1997; pp 571-586.
19
Supporting Information
Optical constants of deposited materials
Figure S1 Optical constants at Mid-infrared wavelength.
(a) measured real part of refractive
index of a-Si (black solid line) and Palik’s real part of refra
ctive index of a-Si (blue dotted line)
(b) measured imaginary part of r
efractive index of a-Si (red so
lid line) Palik’s imaginary part of
refractive index of a-Si (blue dotted line) (c) measured real p
art of refractive index of Al
2
O
3
(black
solid line) and Palik’s real par
t of refractive index of Al
2
O
3
(blue dotted line) (d) measured
imaginary part of refractive index of Al
2
O
3
and Palik’s real part of refr
active index of (blue dotted
line)Al
2
O
3
Psi (Ψ) and delta (Δ) data were measured using IR-VASE Mark II
infrared variable angle
spectroscopic ellipsometer, from a-Si/Al
2
O
3
films deposited on an intrinsic silicon substrate.
Optical constants n and k were th
en obtained from psi (Ψ) and d
elta (Δ) using a three layer fitting
model.