1
Supporting Information
Dispersion Mapping in 3
-
Dimensional Core
-
Shell
Photonic Crystal Lattices Capable of Negative
Refraction in the Mid
-
Infrared
Victoria F. Chernow,
†
,
⊥
Ryan C. Ng,
*
,
‡
,
§
,
⊥
Siying Peng,
†
,
||
Harry A. Atwater,
†
Julia R. Greer
†
†
Division of Engineering and Applied Sciences, California Institute of Technology,
Pasadena,
CA 91125, United States
‡
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena,
CA 91125, United States
2
Exploring Varied Simulation Parameters
Here, we investigate the effect of the variation of various parameters such as effective index of
refraction, beam ellipticity, and shell
-
to
-
core position offset on AANR. We previously explored
the effects of the fill fraction (i.e. volume fraction/beam di
ameter) and the effective index of
refraction for polymer
-
Ge core
-
shell and carbon
-
Ge core
-
shell structures on AANR in PWE
simulations
, and now expand that analysis
.
1
Effective Index of Refraction
-
Polymer Core, Si Shell PhCs
:
Another relatively high index
dielectric material commonly utilized in PhC fabrication is silicon (n = 3.4189 at λ = 8.0 μm).
Silicon deposition methods are more ubiquitous and better optimized compared to Ge deposition,
so we also study the effect on aver
age AANR frequency and frequency range when core
-
shell
lattices are composed of polymer cores and Si shells.
A summary of
band structure and EFC
simulations
for such structures are summarized in Table S1.
Similar to
the case for polymer
-
Ge
and carbon
-
Ge co
re
-
shell structures, we observe
a monotonic increase in the average frequency
for AANR and a monotonic decrease in AANR frequency range with decreasing compound
refractive i
ndex of lattice beams (Figure S1
(a
-
b)). As the diameter size of the polymer core
in
creases and Si shell thickness decreases, the combined index of the whole PhC lowers and
reaches a threshold index below which AANR is no longer possible. Specifically, by simulating
EFCs, we again observe that below an index of n ~ 3.1, approximately equa
l to a polymer core of
b
core
= 400 nm and germanium shell of
t
shell
= 300 nm in lattices of volume fraction
f
= 0.23, an
AANR region does not exist. As was the case with polymer
-
Ge core
-
shell lattices, below an index
of 3.1, the periodic modulation effect
is too weak, and EFCs change shape from circular to square
-
like. Figure 31(c) shows the band 3 EFCs for a lattice with
b
core
= 500 nm and
t
shell
= 250 nm,
where n
beam
= 2.94, showing that in the frequency region below the light line (which falls at ω =
3
0.5
20(2π
c
/
a
) for this lattice) the contours have a square
-
like shape. Square
-
like EFCs imply that
these lattices are capable of self
-
collimation but not do not display AANR, when the compound
index of the PhC beams are below 3.1, irrespective of the materials
comprising the structure (Si
versus Ge).
Figure
S
1
.
Average AANR Frequency, AANR Frequency range, and equi
-
frequency contours for
polymer
-
silicon core
-
shell PhCs. (a) Average
AANR frequency
in the PhC when the compound
refractive index of the lattice is varied (polymer core to Si shell ratio changes). (b) AANR
frequency range, represented as a percentage difference, for PhCs with varying compound
refractive index (polymer core to Si shell ra
tio changes). (c) Slice of the equi
-
frequency surfaces
on the
k
y
= 0 plane for a PhC lattice with
b
core
= 500 nm and
t
shell
= 250 nm (n
beam
= 2.94).
Table
S
1
. Figures of merit derived from band structure and EFC calculations on core
-
shell
PhC lattices of v
aried polymer core diameter and Si shell thickness.
Core
beam
diameter,
b
core
(μm)
Shell
thickness
,
t
shell
(μm)
Refractive
Index,
n
beam
Absolute
Frequency
Range, Δω
(2πc/a)
Average
AANR
Frequency
(2πc/a)
AANR
Frequency
Range (%)
Average
AANR
Wavelength
(μm)
0.1
0.45
3.3996
0.0494
0.4526
10.9036
8.8376
4
0.2
0.40
3.3417
0.0491
0.4584
10.7066
8.7258
0.3
0.35
3.2453
0.0485
0.4678
10.3692
8.5498
0.4
0.30
3.1103
0.0474
0.4809
9.8474
8.3182
Because AANR with polymer
-
Si core
-
shell
structures necessitates a circular core diameter on
the order of 400nm, which is too small to achieve reliably with TPL DLW, we instead decided to
pursu
e polymer
-
Ge core
-
shell designs, though pyrolysis could enable dimensions in which such a
structure that
exhibits AANR with a Si shell would be possible.
The Effect of Beam Ellipticity:
From previous parameter sweeps
, we identified that a volume
fraction of
f
= 0.23, which translates to a circular beam diameter of 1.0 μm for a lattice with 4.0
μm periodicity, is appropriate for the fabrication of low index core, high index shell lattice
structures with AANR in the mid
-
infrared (7.7
-
8.62 μm range). Specifically
, we can aim to
fabricate lattices with a 500 nm diameter polymer core, and 250nm germanium shell, which should
display AANR between 8.4 μm
-
9.41 μm. While these parameters provide useful guidelines for
fabricating the desired core
-
shell nanolattice PhCs,
it is not realistically possible to fabricate beams
with perfectly circular cross
-
sections, as the volumetric pixel, or voxel, that is the building block
of the TPL DLW process is inherently elliptically shaped. While specialized writing schemes can
be em
ployed to generate beams that are largely cir
cular in cross
-
section (see reference 2
for more
details), fabricated beams will possess some degree of elliptical shape. To quantify the effect on
average AANR frequency and frequency range when the cross
-
secti
on of core
-
shell beams
becomes increasingly elliptical, we calc
ulated band structure and EFCs
varying the parameter of
ellipticity, ξ, while keeping cross
-
sectional area of beams constant (both full beam cross
-
sectional
area and respective core and shell c
ross
-
sectional areas are kept constant) for a lattice with a
polymer core of
b
core
= 500 nm and germanium shell of
t
shell
= 250 nm.
5
Ellipticity
is a measure of the compression of a circle along a diameter to form an ellipse, and
can be defined as
ξ
=
푎
−
푏
푎
, where
a
is the dimension of the semi
-
major axis of the total beam, and
b
is the dimension of the semi
-
minor axis of the beam. We varied the
semi
-
major axis from
a
=
0.5 μm to
a
=
0.83
3
̅
μm, and the semi
-
minor axis, which was treated as a function of
a
,
varied
from
b
=
0.5
2
/
a
= 0.5 μm to 0.3 μm. In this manner, beam cross
-
sectional area was kept constant
at 0.785 μm
2
, while ellipticity varied fr
om ξ = 0, for a circular cross
-
section, t
o a maximum of ξ =
0.64 (Table S2
).
Table
S
2
. Figures of merit derived from band structure and EFC calculations on core
-
shell
PhC lattices with beams of varying ellipticity and constant cross
-
sectional area.
Semi
-
major
beam axis,
a (μm)
Semi
-
minor
beam axis,
b (μm)
Ellipticity,
ξ
(
푎
−
푏
푎
)
Absolute
Fr
equency
Range, Δω
(2πc/a)
Average
AANR
Frequency
(2πc/a)
AANR
Frequency
Range (%)
Average
AANR
Wavelengt
h (μm)
0.5
0.5
0
0.0509
0.4507
11.3048
8.8753
0.5556
0.45
0.19
0.0494
0.4531
10.9054
8.8286
0.625
0.4
0.36
0.0479
0.4567
10.4933
8.7586
0.7143
0.35
0.51
0.0461
0.4629
9.9654
8.6415
0.8333
0.3
0.64
0.0446
0.4712
9.4696
8.4893
As the beam ellipticity
increases, we observe a monotonic increase in the average frequency
for AANR, and a monotonic decrease in
AANR frequency range (Figure S2
(a
-
b)). This is the result
of bands compressing, and band 3 specifically shifting to higher frequencies as beams becom
e
mor
e elliptical in shape. Figure S2
(c
-
d) shows the actual band structure in the Γ
-
N direction for
varying ξ. While the relative shape or concavity of band 3 does not change with ξ, we witness a
decrease in band separation between bands 3 and 4, as well a
s a migration of band 3 to higher
frequencies with increasing ellipticity.
6
Figure
S
2
.
Average AANR Frequency, AANR Frequency range, and evolution of band structure
for polymer
-
germanium core
-
shell PhCs with beams of varying ellipticity. (a) Average
AANR
frequency
in the PhC when beam ellipticity is varied (ratio of lattice beam semi
-
major and semi
-
minor axes changes). (b) AANR frequency
range, represented as a percentage difference, for PhCs
with varying beam ellipticity. (c) Band structure along the Γ
N
direction for the 3D core
-
shell PhC
with varying beam ellipticity. The grey diagonal line is the light line (90° relative to the surface
normal). (d) An enlarged subset of the band structure presented in
(c)
. The grey diagonal lines
denote the angular span
between 22.5° and 45°.
The Effect of Shell Offset Relative to Core Position:
The process of creating core
-
shell lattice
structures will involve deposition of high
-
index germanium onto low index polymer scaffolds. If
Ge deposition is conformal but anisot
ropic, we can approximate beam cross
-
sections as “egg
-
yolks,” where the Ge shell position is offset from the center position of the polyme
r core, as
depicted in Figure S3
(a). We study the effect of Ge shell offset on average AANR frequency and
frequency ra
nge by simulating 500nm polymer core, 250nm Ge shell lattices and displacing the
center position of the Ge shell relative to the polymer core center from 0 to 250nm in one
7
dimension, the X
-
direction, two dimensions, the X and Z
-
directions (Ge shell is offs
et from the
polymer core center by 0 to 250nm in both the X and Z directions), and in all three dimensions,
the X
-
Y
-
Z
-
directions (Fig
ure S3
(a)). We perform band structure and EFC simulations for all three
offset varieties, and report
the figures of merit i
n Table S3
.
Table
S
3
. Figures of merit derived from band structure and EFC calculations on polymer
-
Ge core
-
shell PhC lattices with Ge shell offset from the center core position in the X
-
direction, XZ
-
directions, and XYZ
-
directions.
Offset
Type
Offset
Value
(μm)
Absolute
Frequency
Range, Δω
(2πc/a)
Average
AANR
Frequency
(2πc/a)
AANR
Frequency
Range (%)
Average
AANR
Wavelengt
h (μm)
X
-
offset
0
0.0509
0.4507
11.3045
8.8752
0.05
0.0515
0.4493
11.4605
8.9037
0.1
0.0521
0.4471
11.6452
8.9471
0.15
0.0526
0.4445
11.8365
8.9995
0.2
0.0531
0.4417
12.0104
9.0556
0.25
0.0533
0.4392
12.1378
9.1075
XZ
-
offset
0
0.0509
0.4507
11.3045
8.8752
0.05
0.0510
0.4495
11.3466
8.8990
0.1
0.0507
0.4475
11.3212
8.9388
0.15
0.0500
0.4451
11.2231
8.9865
0.2
0.0490
0.4425
11.0718
9.0392
0.25
0.0479
0.4394
10.8989
9.1030
XYZ
-
offset
0
0.0509
0.4507
11.3045
8.8752
0.05
0.0503
0.4492
11.2058
8.9042
0.1
0.0486
0.4473
10.8718
8.9433
0.15
0.0461
0.4451
10.3610
8.9864
0.2
0.0435
0.4422
9.8432
9.0454
0.25
0.0425
0.4359
9.7517
9.1769
We observe that, as the degree of offset increases in all three tests, average AANR frequency
monotonically decreases, and the value of average AANR frequency does not vary significantly
8
between samples off
set in X, XZ, and XYZ (Figure S3
(b)). This is beca
use in general, offsetting
the Ge shell from the center of the polymer core shifts the position of AANR relevant band 3 to
smaller frequencies. In comparison, the degree of offset has a more complex effect on
AANR
frequency range (Figure S3
(c)). Increasing
the degree of X
-
offset appears to increase AANR
frequency range, while increasing XZ and XYZ
-
offset lowers the AANR frequency range. These
trends are the result of the offset parameter subtly changing the shape or curvature of band 3
depending on the type
of offset. While an offset of the Ge shell in X will maximally change the
average AANR wavelength by 2.6%, and an offset in XYZ will maximally alter average AANR
wavelength by 3.4%, these percentages translate to a change in AANR position of hundreds of
n
anometers, which can be significant when dealing with a
narrow characterization region, though
the AANR region still exists.
Figure
S
3
.
Examples of Ge
-
shell offset from the polymer
-
core center position in the X, XZ, and
XYZ dimensions, and the
effects of shell offset on average AANR Frequency and AANR
Frequency range. (a) Schematics depicting the cross
-
section of a core
-
shell beam when the Ge
-
shell of offset from the center position of the polymer in the X
-
direction, XZ
-
directions, and XYZ
-
9
direc
tions. Note that
the blue cross represents the new center position of the full beam following
shell offset. (b) Average frequency at which AANR is observed in the PhC when shell
-
offset
position is varied in 1, 2, and 3 dimensions. (c) AANR frequency range,
represented as a
percentage difference, for PhCs with varying shell
-
offset in X, XZ, and XYZ.
Methods
Sample Fabrication
:
Nanolattices were first fabricated out of the acrylate
-
based “IP
-
Dip”
photosensitive monomer, using a direct laser writing two
-
photon lithography (DLW TPL) system
developed by Nanoscribe GmbH. For this DLW TPL process, the 3D periodic cube
-
like bcc
archite
cture was created using a computer aided design (CAD) program which allowed for the
generation of beams possessing nearly circular cross
-
sections. The design was then imported as a
set of points describing the full 3D architecture to NanoWrite, proprietary
software which
interfaces with the Nanoscribe TPL DLW instrument to write the structure. To allow for
subsequent optical characterization with an infrared laser of ~75
-
100μm spot size, lattices were
designed to be 33 x 23 x 18 unit cells in x
-
y
-
z extent,
resulting in structures with dimensions of
approximately 130μm x 130μm x 100μm.
Nanolattice PhC samples were prepared by drop
-
casting the negative
-
tone photoresist “IP
-
Dip” (Nanoscribe GmbH) on a 500μm thick double sided polished silicon chip, to allow for
transmission measurements through the sample and substrate. After exposing the photoresist and
generating the 3D polymer scaffold, structures were developed for 30 minutes in propylene glycol
mono
-
methyl ether acetate (PGMEA) followed by a 5
-
minute rinse
in isopropyl alcohol. To
prevent lattice collapse or excessive shrinkage due to capillary forces during the drying stage,
lattices were dried via a critical point drying method using a Tousimis Autosamdri
-
815B, Series
B critical point dryer.
10
Figure
S
4
.
Scanning
electron microscopy image of the
(10
1
̅
)
lattice face of a bcc PhC lattice
after 45 min of oxygen
-
plasma etching
. The electron microscope image is taken at a 45
o
tilt with
approximate beam dimensions labeled in yellow
.
Following fabrication with
TPL DLW, lattices possess circular beams with dimensions on
the order of ~850
-
900 nm. Using simulation results as a guide, we know it is necessary to reduce
the polymer beam diameter to <600 nm. Lattices were dry etched with O
2
plasma using a Pie
Scientifi
c Tergeo Plus Plasma Cleaner in remote plasma cleaning mode. Typical etching
conditions included an oxygen flow rate of 15 sccm and 50 W power. In downstream or remote
plasma cleaning mode, oxygen plasma is generated outside sample chamber, which limits sa
mple
immersion in energetic plasma and reduces the extent to which lattices are sputtered by energetic
ions. Through this process, only gentle, isotropic chemical etching should take place on the lattice
beam surface by neutral
radicals
.
Using these plasma
etching parameters, we observed an average
11
beam etch rate of 10 nm/min
. However, at high pressures, even these mild etch conditions caused
anisotropic etching, resulting in the formation of elliptical beams as well as regions of over
-
etching
which could b
e approximated as tapering
(Figure S1)
.
The next step in the fabrication process for low
-
index core, high
-
index shell bcc PhC
lattices is the deposition of germanium onto the etched polymer scaffold. We deposit germanium
onto etched polymer lattices by spu
ttering with a germanium target (Kurt Lesker, Inc.) using a RF
power supply at 100 W, under 5 mTorr and 20 sccm argon in a magnetron sputter deposition
system (ATC Orion sputtering system, AJA International, Inc.). Base pressure is set to <4×10
-
6
Torr, and
target ramp up and ramp down times are precisely controlled at 10 W/min to prevent the
target
from
cracking. Prior to deposition on the sample, the target undergoes a 2
-
minute burn
-
in
process to remove contaminates and surface oxide. To improve the confor
mality of Ge deposition
and ensure material penetration through the lattice, sputtering was performed on samples mounted
at 90° on an SEM stub on the rotating chuck. With this configuration, the side faces of the lattice,
namely the (010) and (10
1
̅
)
surfaces, were directed towards the target, rather than the lattice top.
With the above mentioned sputtering parameters, we observed a deposition rate of 0
.375 nm/min
on the 3D lattice structure.
Each of the four sample side faces were directed towards the
target and
sputtered in increments of 2hrs and 46min, resulting in the deposition of approximately 250 nm
thick Ge shells on the polymer lattice beams.
Many polymers begin to degrade around 300
o
C as shown by thermogravimetric analysis
(TGA), which may l
ead to concern of degradation of the polymer.
Specifically,
for IP
-
Dip,
thermogravimetric analysis (see Figure
7b
in Supporting Information of reference
3
) shows
negligible loss of polymer in inert environment until above 400
o
C.
3,4
At the high powers used
in
two
-
photon lithography to crosslink IP
-
Dip, the polymer is sufficiently highly cross
-
linked such
12
that the deformation and degradation temperatures should occur at roughly the same temperature.
As a result, we do not expect deformation or degradation of
the polymer due to this sputtering
process.
In this work, sputter coating was chosen due to limitations in available deposition
techniques for Ge. Sputtering is a physical deposition process, which is difficult to optimize for a
3D architecture and may le
ad to non
-
conformalities or non
-
uniformities. It is worth noting that
with sputter coating, as the total beam thickness becomes larger, the structure is less open, meaning
that deposition deep into the lattice becomes more and more difficult. Should other
chemical
deposition techniques be available (e.g. atomic layer deposition or chemical vapor deposition) for
high index materials, this could greatly improve the deposition around the beams and reduce
experimental fabrication imperfections. While the primar
y concern is the effective index between
the polymer core and the high
-
index Ge shell, fabrication of these structures can be greatly
facilitated with a more consistent chemical deposition technique.
Focused ion beam (FIB) milling was used to assess the c
onformality and thickness of Ge
deposited onto polymer lattices. To be as minimally destructive towards the sample, only the edges
of the core
-
shell PhC were FIB milled, revealing beam cross
-
sections for imaging and
measurement with scanning electron micro
scopy
(SEM)
. Such cuts are approximately 10 μm wide
and span the height of the lattice, so that deposition can be assessed through all z
-
layers. We
examine Ge deposition and beam dimensions as a function of depth by iteratively FIB milling
through the latt
ice. We measure an average lattice period of 3.8 μm, likely from structure shrinkage
during TPL DLW. Subsequent cuts into the lattice reveal that offset in Ge shell/polymer core
position persists through the beam length, and dimensions of the full beam and
polymer core do
not vary significantly, remaining elliptical with similar average dimensions as described. FIB
13
cross
-
sectioning reveals that the Ge shell is conformal but non
-
uniform. An average of the polymer
core dimensions compared to the full core
-
she
ll beam dimensions provides determination of an
average Ge shell thickness of 255 nm, sufficient for observation of NR.
Figure
S
5
.
Schematic of
a side view of the experimental measurement set
-
up showing
the
orientation of incident QCL laser light relative to the PhC sample orientation.
It should be noted
that incident light is polarized along the [10
1
̅
] direction.
Measurement
:
Angle resolved spectr
oscopic measurements to selectively excite states at precise
momentum and wavelength is done in a setup consisting of a quantum cascade laser (QCL) with
an operating mode between 7.7
-
8.62 μm as the source, a series of ZnSe lenses to focus the laser
beam do
wn to spot size approximately 100
μm in diameter, and a germanium beam splitter to
direct half of the laser beam to a reference detector, and half to the sample
(Figure S2)
. Pyroelectric
sample and reference detectors are mounted on concentric rotation sta
ges allowing for the collect
14
of transmission and reflection spectra at distinct angles of incidence. Alignment of the laser beam
and the PhC sample is accomplished using a visible CCD camera and alignment markers. More
precise micron scale alignment is per
formed by transmitting QCL light through the sample, and
projecting the magnified image onto a mid
-
IR camera with MCT detector to determine sample
edges versus the sample center.
Measurements are taken between
θ = 22.5° to 45°
. Below 22.5
o
,
limitations in the set
-
up
(i.e. detector size) prevent measurements at smaller angles. In theory, measurements can be taken
at angles greater than 45
o
, though the limited lateral area of the as
-
fabricated samples limits
measurements at greater incident ang
les. Fabrication of larger lateral area lattices with similar
feature sizes may require stitching in two
-
photon lithography.
Simulation Parameters for Comparison to Experimental Band Structure
Figure 3 compares the experimentally measured band structure
to results generated in
PWE.
The simulated PhC
in Figure 3b accounted for
the structural features and lattice dimensions
determined through FIB and SEM measurements. The unit cell size was set to 3.8μm, and beam
ellipticity, taper, and Ge shell offset were
all
accounted for
as follows.
Beam ellipticity was
accounted by approximating the
polymer core as an elliptical beam with long axis and short axis
dimensions of 650nm and 450nm
,
respectively. Taper was introduced to polymer beams pointing
in the [010] dir
ection by changing the long axis dimension of their starting vertex to 250nm and
while keeping it at 650nm at the ending vertex, creating a core resembling an elliptical conical
frustrum. The germanium shell was described by embedding the polymer core in a
n elliptical beam
of Ge with long axis and short axis dimensions of 1.1
μm and 860
nm respectively. The Ge shell
beams were also offset from the polymer core beams by 200
nm in x, y, and z, yielding a
representative core
-
shell PhC for simulation. We
then
c
alculate the band structure for this PhC
15
approximation in the frequency range between ω = 0.441(2π
c
/
a
) and ω = 0.494(2π
c
/
a
) and angular
span between 22.5° to 45° along the ΓN direction
.
REFERENCES
(1)
Chernow, V. F.; Ng, R. C.; Greer, J. R. Designing Core
-
She
ll 3D Photonic Crystal Lattices
for Negative Refraction.
Proc. SPIE
2017
, 10112.
(2)
Meza, L. R. Design, Fabrication, and Mechanical Property Analysis of 3D Nanoarchitected
Materials. California Institute of Technology, 2010.
(3)
Liu, Y.; Wang, H.; Ho, J., Ng, R. C.; Ng, R. J. H.; Hall
-
Chen, V. H.; Koay, E. H. H.; Dong,
Z.; Liu, H.; Qiu, C
-
W.; Greer, J. R.; Yang, J. K. W. Structural Color Three
-
Dimensional
Printing by Shrinking
Photonic Crystals. Nature Comm. 2019, 10, 4340.
(4)
Sharipova, M. I.; Baluyan, T. G.; Abrashitova, K. A.; Kulagin, G. E.; Petrov, A. K.,
Chizhov, A. S.; Shatalova, T. B.; Chubich, D.; Kolymagin, D. A.; Vitukhnovsky, A. G.;
Bessonov, V. O.; Fedyanin, A. A. Effe
ct of Pyrolysis on Microstructures Made of Various
Photoresists by Two
-
Photon Polymerization: Comparative Study. Opt. Mater. Express
2021, 11, 371
-
384