of 6
Probing the Band Structure of Topological Silicon Photonic Lattices
in the Visible Spectrum
Siying Peng
(
)
,
1
Nick J. Schilder,
2
Xiang Ni,
3,6
Jorik van de Groep,
4
Mark L. Brongersma,
4
Andrea Alù,
3,5,6
Alexander B. Khanikaev,
3,6
Harry A. Atwater,
1
and Albert Polman
2
,*
1
Applied Physics, California Institute of Technology Pasadena, California 91125, USA
2
Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG, Amsterdam, Netherlands
3
Department of Electrical Engineering, City College of City University of New York, New York 10031, USA
4
Geballe Laboratory for Advanced Materials, Stanford University, 476 Lomita Mall, Stanford, California 94305, USA
5
Photonics Initiative, Advanced Science Research Center, City University of New York,
85 St. Nicholas Terrace, New York, New York 10031, USA
6
Physics Program, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, USA
(Received 16 July 2018; published 21 March 2019)
We study two-dimensional hexagonal photonic lattices of silicon Mie resonators with a topological
optical band structure in the visible spectral range. We use 30 keV electrons focused to nanoscale spots
to map the local optical density of states in topological photonic lattices with deeply subwavelength
resolution. By slightly shrinking or expanding the unit cell, we form hexagonal superstructures and observe
the opening of a band gap and a splitting of the double-degenerate Dirac cones, which correspond
to topologically trivial and nontrivial phases. Optical transmission spectroscopy shows evidence of
topological edge states at the domain walls between topological and trivial lattices.
DOI:
10.1103/PhysRevLett.122.117401
Topological photonic materials guide light in unconven-
tional ways, enabling entirely new ways to route information
for communication and computing purposes
[1
3]
.
Topological phases of light supported by well-designed
nanostructures have the unique property that backscattering
induced by imperfections is topologically forbidden,
allowing lossless photon transport in large-scale optical
integrated circuits. Experimentally, topological properties of
light have been studied in various systems, such as one-
dimensional gratings
[4]
, two-dimensional lattices
[5
8]
,
and three-dimensional photonic crystals
[9]
. Using spectro-
scopic techniques at microwave frequencies, photonic band
structures and edge modes were mapped out and proven to
be topological
[6,9]
. In the infrared spectral range, coupled
ring resonators with artificially induced magnetic fields
show topological protection of edge states from defects
[8]
and spontaneous emission from chiral quantum dot states
was shown to be guided unidirectionally in a topological
silicon photonic crystal structure
[10]
. In the visible spectral
range, three-dimensional Floquet topological insulators
were demonstrated using evanescently coupled chiral wave-
guide elements arranged in a two-dimensional geometry
[7]
.
For this purpose, relatively large optical elements (
>
4
μ
m)
were used, spaced at a distance of
15
μ
m.
However, to date there has been no experimental
demonstration of topological photonic crystals that operate
in the visible spectral range, and that employ nanoscale
architectures compatible with planar silicon device inte-
gration. Recent research has focused on tailoring lateral
lattice symmetry to create topological protection, creating
a way to drastically shrink the volume necessary for the
optical elements. For example, topological protection was
theoretically shown by breaking the inversion symmetry of
a hexagonal lattice, by splitting the degeneracy of valley
degrees of freedom
[11]
.
Another elegant and practical geometry, recently
proposed theoretically, creates topological protection via
pseudo-time-reversal symmetry by rearranging the ele-
ments of a two-dimensional hexagonal lattice while pre-
serving its
C
6
symmetry
[12,13]
. By expanding or
shrinking the unit cells to create a hexagonal superstructure,
a band gap opens in the Dirac-type band structure for TM
waves, with harmonic photonic modes representing elec-
tronic orbital-like
p
- and
d
-type waves. Here, we exper-
imentally realize these lattices composed of nanoscale
silicon Mie resonators. By tailoring their geometry at the
nanoscale, locally shrinking or expanding the unit cell, we
create geometries in which a band gap opens at the Dirac
point, with controlled topological properties. The two-
dimensional hexagonal photonic lattice forms a unique
platform to exploit the properties of topological states of
light, as it enables direct measurements of the optical
density of states and photonic band structure by accessing
the third dimension
[14]
. Here, we use cathodolumines-
cence spectroscopy using a tightly focused electron beam
as a probe for topological lattices.
Fabrication of two-dimensional hexagonal lattices oper-
ating in the visible spectral range is challenging because of
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117401 (2019)
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=
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=
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=
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© 2019 American Physical Society
the high sensitivity of the optical band structure to nano-
scale lattice imperfections. Our photonic lattice geo-
metries are made on an ultrathin Si
3
N
4
membrane, to
create a nearly symmetric optical environment to study the
topological properties. To realize this, we first lifted off a
200-nm-thick Si film by chemical etching from a silicon-
on-insulator wafer, and then transferred it onto a 10-nm
thick Si
3
N
4
membrane that was spanned inside a Si hold
wafer. Electron beam lithography and reactive ion etching
was then used to fabricate hexagonal lattices of hexago-
nally shaped Si cylinders (incircle diameter 250 nm, height
130 nm) with a lattice constant of 450 nm. Full details of
the fabrication process are described in the Supplemental
Material
[15]
, which includes Refs.
[16
18]
. Figures
1(a)
,
1(c)
, and
1(d)
show a schematic of the fabricated geometry
and scanning electron microscopy (SEM) images of the
fabricated structures.
To probe the photonic lattice modes at deeply subwave-
length spatial resolution, we use angle-resolved cathodolu-
minescence imaging spectroscopy (ARCIS) using 30 keV
electrons as an excitation source
[19]
. The time-varying
evanescent field carried by the swift (
v
0
.
3
c
) electron as
it traverse the sample contains a broad spectral range from
the UV to the near-infrared that couples coherently to the
photonic lattice modes
[20]
. The electron beam is raster-
scanned over the lattice and the emitted light (cathodolumi-
nescence, CL) is collected at every pixel. In contrast to the
case ofincoherent CL offluorescent samples, in this coherent
excitation mode there is a well-defined phase relation
between the generated modal field and the electric field
carried by the incident electron. The strongly localized
character of the electron-induced polarization density allows
probing the
z
component of the local density of optical states
(LDOS) inside the photonic lattices at a spatial resolution of
10nmat 30keV
[21,22]
. We alsomeasurethe CL emissionas
a function of azimuth and zenith angles in well-defined
spectral bands, probing the in-planewavevectors and thereby
thephotonic band structure
[2]
. Hyperspectral angle-resolved
CL, combining spectral measurements and angle-resolved
measurements along the zenithal angular range was also
performed
[23]
. A schematic ofthe CL excitation geometry is
shown in Fig.
1(b)
. In some angle-resolved measurements the
sample was tilted by 7.5° to enable collection of radiation
normal to the sample (
k
¼
0
), which is otherwise not possible
because of the hole in the parabolic mirror.
Figure
1(e)
shows a CL spectrum taken by averaging over a
single Si cylinder (not arranged in a photonic lattice). A large
number of Mie resonances is observed, consisting of in-plane
and out-of-plane electric and magnetic dipolar and quad-
rupolar modes of the single Si cylinder, whose coherent
superposition is collected in the far field
[24
26]
.The
measured field intensities for the modes corresponding to
the two most intense resonant peaks, shown as insets, reveal
two characteristically different modal profiles. By tuning the
coupling between neighboring scatterers, photonic lattice
mode degeneracies can be controlled, leading to photonic
band inversion and topologically nontrivial dispersion. As a
first signature of this interaction, Fig.
1(f)
shows the CL
spectrum for the hexagonal lattice, with most of the resonant
modes split up, characteristic for coupled resonators
[27]
.In
order to probe the radiative local density of states inside the
photonic lattice, spatial CL maps were retrieved at 426, 664,
and 750 nm [Fig.
2(a)
]. Comparing the maps at 664 and
750 nm, a distinct difference in the modal profiles can be
observed: at 664 nm a field minimum is observed between
the facing hexagon edges, while a maximum is observed at
750 nm. These features represent bonding and antibonding-
type modes within the twocoupled Mie resonances
[23]
.Such
coupling is not clearly observed for the 426 nm resonance,
for which the fields are more localized inside the particle.
Angle-resolved CL spectroscopy was used to characterize
the dispersion of the collective photonic modes in momentum
FIG. 1. Hexagonal photonic lattice and CL measurements.
(a) Schematic of unit cell of expanded and shrunken lattices
composed of Si hexagons composed of Si pillars on a 10-nm-thick
Si
3
N
4
membrane, where
a
0
is the lattice constant,
d
is the diameter
of the Si pillars,
h
is the pillar height, and
r
is the distance of the Si
pillars from the center of the unit cell. (b) Excitation geometries:
the 30 keV electron beam is incident through the hole in the
parabolic mirror and light emitted by the excited lattice is collected
by a parabolic mirror; (top) spectral detection using a spectrom-
eter, (bottom) angle-resolved detection using a CCD imaging
camera. (c),(d) SEM images of the fabricated lattice. (e), (f)
Cathodoluminescence spectra averaged over a single Mie scatterer
(left, with measured mode profiles at the two most intense peaks
shown as insets) and averaged over a single Mie scatterer in a
hexagonal lattice (right, with inset of geometry).
PHYSICAL REVIEW LETTERS
122,
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space [Fig.
2(b)
], where the localized nature of our source
allows selective addressing of different sublattices. Threefold
symmetric bands are observed, rotated 60° between the two
nonequivalent lattice sites in the unit cell, as expected. In-
planewavevectors
k
wg
wereretrievedfromsphericalcontours
fitted to the angular profiles, assuming that the angular
emission is the result of diffractive outcoupling of the in-
plane guided modes by the periodically arranged resonant
Mie scatterers
[28]
. Indeed, the excitation and outcoupling of
guidedmodesmakeangle-resolvedCL ideallysuitedtoprobe
thebandstructureofanextendedstructurelikethetopological
photonic crystals under consideration, despite the use of a
localized source. The angular distribution of the scattered
light is given by a coherent sum of all scattering fields at a
distance
R
:
E
ð
ˆ
k
Þ¼
e
ikR
R
S
ð
ˆ
k
Þ
X
N
n
¼
1
A
n
e
ikr
n
;
ð
1
Þ
with
S
ð
ˆ
k
Þ
the form factor for an individual Mie resonator,
r
n
is the position of individual Mie resonator, and
A
n
the
structure factor. The angular scattering distributions become
narrower for increasing wavelength due to the increased
propagation distance in the photonic lattice which we ascribe
to the fact that silicon is less absorbing and scattering is
less pronounced at larger wavelengths, which effectively
increases the number of scatterers contributing to the lattice
sum. The band structure derived from these data is shown in
Fig. S2
[15]
, and shows good agreement with full-wave
simulations showing near-linear dispersive bands with a
Dirac point at 450 nm. See Supplemental Material
[15]
for a
brief description of how numerical simulations were per-
formed, which includes Ref.
[29]
.
Next, we induce topological band splitting by shrinking
(
a
0
=r
¼
3
.
3
) and expanding (
a
0
=r
¼
2
.
7
) the hexamer unit
cell, while preserving
C
6
symmetry, using Si nanopillars
with
d
¼
88
(lowest-order Mie resonance
λ
470
nm),
h
¼
200
, and
a
0
¼
455
nm
[12]
. The shrunk and expanded
lattices form a hexagonal superlattice with each individual
lattice site (supercell) composed of a hexamer consisting
of six Si cylinders. The corresponding SEM images are
shown in Fig.
3(a)
, together with the Brillouin zone of the
basic lattice symmetry (dashed line) together with the
folded Brillouin zone of the supercell lattice (solid line).
Hyperspectral angle-resolved CL measurements were per-
formed along the
Γ
-
K
and
Γ
-
M
directions to record angular
profiles with high spectral resolution (0.87 nm) for the
shrunk and expanded lattices [Figs.
3(b)
,
3(c)
]. The sample
was mounted on a 7.5° tilted sample holder, so that
radiation emitted along the
Γ
point could be collected
(see Supplemental Material
[15]
). The corresponding
numerical simulations of the band structures are also shown
in Figs.
3(b)
,
3(c)
.
Many interesting features can be observed in Fig.
3
.
Theory shows that, for the shrunk lattice, inter-lattice-site
coupling is enhanced, causing the Dirac cones to separate
and leading to a band gap. Such band separation is clearly
observed in Fig.
3(b)
. For the expanded lattice, intra-lattice-
site coupling is enhanced, introducing band hybridization
to open the band gap. The band gap for the shrunk lattice,
derived from a parabolic fit near the
Γ
point is 9 nm
(30 meV) and for the expanded lattice is 6 nm (20 meV)
[12]
. For the expanded lattice, theory shows band hybridi-
zation that introduces winding of the band (pseudospin
Chern number
¼
1
), and therefore the band gap of the
expanded lattice is topologically nontrivial
[12]
. The two
originally doubly degenerate Dirac cones each split in two
pairs of bands, as also clearly observed in the CL data. For
the
Γ
-
K
direction the bands are closely spaced for large
wave vectors, as seen in both simulations and experiment.
For the
Γ
-
M
direction a much larger separation between the
FIG. 2. Measured density of states and dispersion of hexagonal
photonic lattice. (a) SEM image and CL maps at 426, 664, and
750 nm (bandwidth 10 nm). (b) Angular CL emission distributions
ð
k
=k
0
Þ
for positions A in the SEM image at 650, 600, 550, 500,
and 450 nm, and for position B at 600 nm, where
k
is the in-plane
wave vector and
k
0
¼
ω
=c
(
k
¼
0
directed towards mirror hole).
The spacing between the overlaid concentric contours represents
the measurement bandwidth of 40 nm. The black spots in the center
correspond tothe
600
μ
m-diameterholeintheparabolicmirrorthat
spans 6.9° and where no light is collected.
PHYSICAL REVIEW LETTERS
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separated bands is observed both experimentally and in
simulations. Deviations of the parabolic band curvatures
between theory and experiment are attributed to slight
differences in the fabricated and modelled geometry.
As the topological modes at
k
¼
0
exhibit flat dispersion,
corresponding to zero group velocity, they are best
described by their quality factor
Q
. We find the radiative
Q
of the modes varies from
100
(for bright dipolar modes
at oblique incidence) to very large numbers (for dark
quadrupolar modes at normal incidence), since radiation
loss is absent in this latter case
[14]
. From the observed
broadening of the lines, we estimate that absorption and
scattering in the samples leads to a reduction of the quality
factor to
Q
total
65
, implying a lifetime reduction of 35%
for bright modes. Further modeling will reveal if absorption
or scattering is the dominant loss process.
By creating an interface between the expanded lattice
(Chern number
¼
1
) and the shrunk lattice (Chern
number
¼
0
), a pair of interface states appears inside the
band gap located at the domain interface
[12]
. The interface
modes are protected by pseudo-time-reversal symmetry,
and they propagate in opposite directions along the inter-
face, where the chirality of the Poynting vector is linked
with the momentum along the edge. We made adjacently
placed shrunk and expanded hexagonal photonic lattices as
schematically shown in Fig.
4(a)
; the corresponding SEM
image is shown in Fig.
4(b)
.
We performed angle-resolved CL with the electron beam
positioned at the edge. We found results quite similar to
those for bulk lattice states probed one or more lattice units
away from the edge. We attribute this to the fact that CL
probes extended lattice states, as shown also by the band
structure measurements in Fig.
3
, and therefore CL cannot
sensitively probe just the edge states alone. Confocal
optical transmission spectroscopy was then performed
using an unpolarized light beam from a halogen lamp that
was weakly focused onto the sample. The transmitted light
either in the bulk shrunk or expanded regions or centered at
the interface [Fig.
4(d)
] was collected with
600
nm spatial
resolution using a confocal microscope with a 50x objec-
tive. For the bulk expanded and shrunk regions a clear
band gap around 640 nm is observed in the transmission
spectrum, with sharp dips corresponding to the band edges.
For the edge region, a single deep minimum is observed,
slightly shifted from the higher band edge, consistent with
the existence of an interface state. For the shrunk lattice, the
strongest coupling (largest transmission dip) is observed for
the low-energy band, consistent with the (bright) dipolar
nature of this band
[12]
. The smaller coupling to the (dark)
FIG. 3. Measured and simulated photonic band structure of trivial and topological lattices. (Top) shrunk lattice; bottom: expanded
lattice. SEM images (a) and calculated and measured dispersion of the triangular lattice with six silicon pillars for the unit cell, along
Γ
-
K
(b) and
Γ
-
M
(c) directions (calculated dispersion overlaid over the measurements as dashed curves). The blue cones on either side of the
calculated band structures correspond to modes outside the light cone. Data are taken with the tilted sample holder. The black vertical
band in the CL data corresponds to the angular range of the hole in the parabolic mirror, where no data are collected. For the band
structures, quality factor was encoded into the color of the bands.
FIG. 4. Edge states between shrunk and expanded hexagonal
photonic lattices. (a) Edge geometry. (b) SEM image of edge
region. (c) Numerically calculated band structure, red lines
indicate edge states. (d) Optical transmission spectra taken in
bulk and edge regions, black dashed lines indicate the band gap.
PHYSICAL REVIEW LETTERS
122,
117401 (2019)
117401-4
quadrupolar band is also observed and is attributed to the
finite angular range (numerical aperture NA
¼
0
.
2
) used in
the measurements. In contrast, for the expanded lattice the
strongest coupling is observed for the high-energy band,
directly reflecting the band inversion in this geometry.
In summary, we have designed and fabricated two-
dimensional hexagonal photonic lattices of silicon Mie
resonators that possess topological band structure in the
visible spectral range. Conventional hexagonal lattices show
a double-conical Dirac band structure with the Dirac point
at 450
650 nm, depending on the geometry. By slightly
shrinking or expanding the hexagonal unit cell, we form
hexagonal superstructures for which a band gap opens and
the double-degenerate Dirac bands split. Good agreement is
observed between measurements and numerical simula-
tions. Coherent 30 keV electron beam excitation proves a
powerful tool to probe the radiative local density of optical
states and the dispersion of topological lattices. Using
optical scattering, we confirm the existence of interface
states between shrunk and topological lattices which,
according to theory, are topologically protected against
coupling between counterpropagating modes. This work
demonstrates the feasibility to realize practical silicon-based
topological geometries that are commensurate with planar
Si-based integrated optics technology, and to excite them
locally with electron beams.
This work was supported by U.S. Department of Energy
(DOE) Office of Science Grant No. DE-FG02-07ER46405
(fabrication), the Air Force Office of Scientific Research
under Grant No. FA9550-16-1-0019 (characterization) and
MURI Grant No. FA9550-17-1-0002, the National Science
Foundation, the research program of the Netherlands
Organization for Scientific Research (NWO), and the
European Research Counsel (Grant No. SCEON
695343). The authors thank Seyedeh Mahsa Kamali for
useful discussions on reactive ion etching, Sophie Meuret
and Toon Coenen for assistance with CL spectroscopy, and
Femius Koenderink for useful discussions. We thank the
Kavli Nanoscience Institute at Caltech for cleanroom
facilities and the AMOLF NanoLab Amsterdam for cath-
odoluminescence spectroscopy facilities. X. N. and A. K.
acknowledge that numerical calculations were supported
by the National Science Foundation (DMR-1809915).
A. K. and A. A. acknowledge support by the National
Science Foundation (EFRI-1641069). Competing financial
interest: A. P. is co-founder and co-owner of Delmic BV, a
company that produces commercial cathodoluminescence
systems like the one that was used in this work.
*
Corresponding author.
A.Polman@amolf.nl
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