2528 Chem.Soc.Rev.,2013, 42, 2528--2554 This journal is
c
The Royal Society of Chemistry 2013
Cite this: Chem. Soc. Rev., 2013,
42, 2528
Bottom-up assembly of photonic crystals†
Georg von Freymann,z
a
Vladimir Kitaev,z
b
Bettina V. Lotschz
c
and
Geoffrey A. Ozin*
d
In this tutorial review we highlight fundamental aspects of the physics underpinning the science of
photonic crystals, provide insight into building-block assembly routes to the fabrication of different
photonic crystal structures and compositions, discuss their properties and describe how these relate to
function, and finally take a glimpse into future applications.
1. Introduction
There are many examples of colour that are not chemical in
origin but rather physical in nature. By chemical we mean
colour that originates from the absorption of light by a chro-
mophore in molecular and nanoscale systems or solid-state
materials. Here the colour originates from excitation of an
electron between the ground and excited state of the chromo-
phore by the incident light. In contrast, colour that is physical
in nature stems from the way light is scattered and diffracted by
random or periodic structures exemplified by the red and blue
hues of the sky and the iridescent colours of opals, mother-of-
pearl, sea-mouse whiskers, beetle scales, iridophoric squid,
butterfly wings, feathers of peacocks and other birds (Fig. 1).
1
In this tutorial review, the focus of the presentation will be
mainly on the way light interacts with structures that are
periodic at the scale of the wavelength of light, called photonic
crystals. Historically, it was as long ago as 1887 that a one-
dimensional (1D) photonic crystal was studied by Lord Rayleigh
who showed the existence of high reflectivity of light over a
well-defined wavelength range known as the stop-band. Known
today as Bragg mirrors these 1D structures are used in optical
physics for enhancing the efficiency of solar cells, improving
the light extraction and colour purity of light emitting diodes,
and optimizing the performance of lasers.
Recently, photonic crystals have moved beyond 1D into the
realm of more complex 2D and 3D light-scale photonic lattices
and their practice, often stimulated by theory, has transitioned
rapidly from mainstream science to advanced technology.
There is ample evidence from published research of the last
decade that the ability of photonic crystals to control photons
can rival that of electronic crystals to manage electrons. This has
proven to be especially true for high refractive index contrast 3D
photonic crystals made of silicon that were envisioned theoretically
in 1987 by Yablonovitch
2
and John
3
to develop an omni-directional
photonic band gap offering complete control over the flow of light
in all three spatial dimensions.
Today it has been said that ‘‘photonic crystals have been the
classic underachievers: full of promise, sound in theory but
poor on implementation’’. This criticism has mainly come from
the optical physics community because of the difficulties
experienced of adapting top-down semiconductor engineering
Fig. 1 Photograph of Anna’s hummingbird (Calypte anna) showing highly
iridescent gorget and crown feathers (photo by Camden Hackworth). Adapted
from ref. 1 with permission from Royal Society Publishing.
a
Department of Physics and Research Center OPTIMAS, University of
Kaiserslautern, Erwin-Schro
¨
dinger-Straße 56, 67663 Kaiserslautern, Germany.
E-mail: georg.freymann@physik.uni-kl.de; Fax: +49 631 205 5226;
Tel: +49 631 205 5225
b
Department of Chemistry, Wilfrid Laurier University, 75 University Avenue West,
Waterloo, Ontario, N2L 3C5, Canada. E-mail: vkitaev@wlu.ca;
Fax: +1 519 746 0677; Tel: +1 519 884 0710 3643
c
Max Planck Institute for Solid State Research, Stuttgart, and Department of
Chemistry, University of Munich (LMU), Munich, Germany.
E-mail: b.lotsch@fkf.mpg.de; Fax: +49 711 689 1612; Tel: +49 711 689 1610
d
Department of Chemistry, University of Toronto, Toronto, Canada.
E-mail: gozin@chem.utoronto.ca; Fax: +1-416-971-201; Tel: +1-416-978-2082
† Part of the chemistry of functional nanomaterials themed issue.
‡ These authors contributed equally to this work.
Received 1st August 2012
DOI: 10.1039/c2cs35309a
www.rsc.org/csr
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techniques to fabricate photonic lattices with high enough struc-
tural and optical quality to enable the scale-up, manufacture,
economics and delivery of photonic crystal components and mod-
ules into industry that can profit from the monetizing potential
that photonic crystals foretell in markets that include solar cells,
light emitting diodes, lasers, displays, sensors and optical fibres.
Observing that top-down approaches to the fabrication
of photonic crystals often require complex and expensive litho-
graphic instrumentation, deposition equipment and clean-room
facilities, chemists realized there could be an easier way to
make photonic crystals using bottom-up building-block assembly
methods. The most popular chemical method today employs the
crystallization of silica or polystyrene microspheres in the form of
colloidal crystals, known as opals, and to use them as templates to
replicate their structure in the form of inverse opals (see Sections 3
and 4). This procedure was first used to make silica and titania
inverse opals that could display well defined stop bands.
This is more-or-less the story of how bottom-up assembly of
photonic crystals began and in the tutorial presentation that
follows we will describe the basic theoretical underpinnings and
experimental accomplishments of the field from its roots to its
branches. We will illustrate with some case histories how this
knowledge has blossomed forth into one of the most scientifically
vibrant areas of research with a multidisciplinary footprint that is
spawning a number of spin-off companies, promising to manu-
facture and deliver bottom-up photonic crystal products to
diverse markets from colour sensors for food and water quality
control to colour-coded anti-counterfeiting security features on
banknotes to full colour displays in the near future.
1.1. Colloidal photonic crystal milestones
If photonic crystals have been the classic underachiever, col-
loidal photonic crystals at least helped to demonstrate many of
the countless promises made from theory – some fundamental
ones even as ‘‘firsts’’. Prominently displaying colour from
structure, the gemstone opal is well known even outside the
scientific community. Looking for promising crystal structures
displaying complete photonic band gaps, theorists pretty early
realized that the diamond lattice is close to optimal. The
underlying face-centred-cubic lattice is the thermodynamically
preferred lattice of self-assembling colloidal photonic crystals.
The resulting optical properties have been studied in detail
early on. Unfortunately, this structure does not display any
form of complete photonic band gap. This could have been the
sudden death of its career as a photonic crystal material if not
for the seminal paper by K. Busch and S. John
37
that demonstrated
Georg von Freymann, Vladimir Kitaev, Bettina V. Lotsch and
Geoffrey A. Ozin
Georg von Freymann received his PhD degree from the Physics
Department, Universita
¨
t Karlsruhe (TH), Karlsruhe, Germany, in
2001. He was a Postdoctoral Researcher at Institute of
Nanotechnology, Forschungszentrum Karlsruhe, Karlsruhe,
Germany in 2002 and at the University of Toronto, Toronto,
Canada, till 2004. From 2005 to 2010 he headed a Junior
Research Group in the Emmy Noether-programme of the Deutsche
Forschungsgemeinschaft (DFG) at the Institute of Nanotechnology,
Karlsruhe Institute of Technology, Germany. Since 2010 he is full
professor for experimental/technical physics at University of
Kaiserslautern, Germany. His research interests include nano-
fabrication technologies, photonic crystals, photonic quasicrystals
and three-dimensional photonic metamaterials.
Vladimir Kitaev earned his MSc in Chemistry from Moscow State
University studying cholesteric polymer liquid crystals and PhD in
Chemistry from University of Toronto working with self-assembly of
polymers and surfactants and rheological and tribological properties of nanoscale films. Subsequently he joined Prof. G. A. Ozin for
research on monodisperse colloids, colloidal self-assembly, and photonic crystals. At present, he is an Associate Professor of Chemistry at
Wilfrid Laurier University. His current research interests include ligand-protected metal clusters, size and shape control of nanoparticles,
their self-assembly and optical properties, and chirality on nanoscale.
Bettina V. Lotsch studied Chemistry at the University of Munich (LMU) and the University of Oxford, and received her PhD from LMU
Munich in 2006. In 2007 she joined the group of Prof. Geoffrey Ozin at the University of Toronto as a Feodor-Lynen postdoctoral fellow
supported by the Alexander von Humboldt foundation. In 2009, BL was appointed associate professor at LMU Munich, and since 2011 she
additionally holds a group leader position at the Max Planck Institute for Solid State Research in Stuttgart. Her research interests include
‘‘smart’’ photonic crystals, functional and hierarchically porous framework materials, as well as two-dimensional materials for energy
conversion and storage.
Geoffrey Ozin studied at King’s College London and Oriel College Oxford University, before completing an ICI Postdoctoral Fellowship at
Southampton University. Currently he is the Tier 1 Canada Research Chair in Materials Chemistry and Nanochemistry, Distinguished
University Professor at the University of Toronto, and a Founding Fellow of the Nanoscience Team at the Canadian Institute for Advanced
Research. Internationally currently he is Distinguished Professor at Karlsruhe Institute of Technology (KIT). Over a four decade career he
has made innovative and transformative fundamental scientific and technological advances in the field of nanochemistry, a chemistry
approach to nanocrystals and nanowires, nanoporous materials, nanophotonic crystals and nanomotors that led to nanoscale constructs
capable of controlling electrons, photons, phonons and motion in unprecedented ways.
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the inverse structure actually possesses a complete photonic band
gap, if realized in a high-index-of-refraction material like silicon.
Colloidal self-assembly re-enters the stage as the most prominent
templating approach for photonic crystals with a complete photonic
band gap. Several firsts have been demonstrated in this system: The
first complete photonic band gap at the optical telecommunication
wavelength of 1.5 mm,
52
the first demonstration of enhanced and
suppressed spontaneous emission,
4
the first non-linear optical
experiments like all optical switching in transmittance for opals
5
and inverse opals
6
and the enhancement of third harmonic
7
and
second harmonic generation.
8
Without the need for a complete gap, several other ideas
have been first implemented with colloidal photonic crystals:
Graded structures for enhanced coupling efficiency,
9
binary
colloidal crystal architectures,
10
porous Bragg mirrors for sen-
sing applications,
67,69
slow photon enhanced photochemistry
and photocatalysis,
11
dynamic full colour displays, enhanced
light harvesting for solar-cells,
74
solid-state dye and polymer
lasers,
75
increased efficiency for lithium ion battery applica-
tions
12,13
as well as applications as a dual stationary phase and
detector in high-pressure liquid chromatography.
14
In this tutorial review we will start with the underlying
physical concepts behind the success of the bottom-up
assembled 3D colloidal photonic crystal, how the knowhow
was adapted to 2D photonic crystal architectures, expanded and
enriched to include 1D photonic crystal lattices, which will be
followed by a discussion of the materials and fabrication issues
of these different dimensionality structures and concluded by
an outlook to current and future applications.
Colloidal photonic crystals made by chemical methods may
be far from perfect photonic crystals, but for a myriad of
perceived and real applications even with their imperfections
they are still the gold standard.
2. Basic concepts
The interaction of light with materials as well as the propagation of
light within, are described by the macroscopic Maxwell’s equations
eD = r
eB =0
e E ¼
@
@t
B
e H ¼
@
@t
D þj
together with the constituent materials equations
D = e
0
E + P
B = m
0
(H + M)
Here, we use the following naming convention: D is the electric
displacement, B the magnetic induction, E the electric field, H
the magnetic field, r is the free charge density and j the free
current density, P the polarization and M the magnetization.
For simplification we will assume only linear optics and
isotropic materials in this tutorial so that the latter two
equations reduce to
D = e
0
eE
B = m
0
mH
From this set of equations we derive the master equation
describing the propagation of electric and magnetic fields. To
further simplify the discussion, we will assume that there are
no free charges and no free currents present in the materials
under consideration, that is r = 0 and j = 0. This is valid for all
dielectric materials mentioned throughout this tutorial review
and in the above described case of linear-optics.
2.1. Master equation and periodic structures
Combining the above equations provides us with the master
equation for the propagation of the magnetic field (the equation
for the electric field can be derived in an analogous fashion):
1
m
e
1
e
e H
¼
o
2
c
2
H
Here, we assumed that the magnetic field can be written in
the following form:
H = H(r,t)=H(r)e
iot
For all known natural materials the magnetic permeability m
is unity at optical frequencies, reducing this equation to the
following form of an eigenvalue equation:
e
1
e
e H
¼
o
2
c
2
H
The optical properties of the photonic crystal and its dispersion
relation are given by the eigenvalues on the right hand side. For
homogeneous and isotropic materials (e.g.,asolidblockofglass)
the permittivity e does not depend on the spatial coordinates and,
hence, the dispersion is solely described by the materials properties
alone (in this case we can move e to the right hand side of the
equation, giving us the well-known wave equation).
If we now introduce through periodic ordering a spatial
dependence of the materials properties (see Fig. 2), we can
control the eigenvalues and hence the optical properties of the
Fig. 2 Schematic representation of 1D, 2D and 3D photonic crystals. The electric
permittivity varies along one, two or three dimensions respectively. Adapted from
ref. 15.
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material through the distribution of the material: We control
the operator acting on the magnetic field on the left-hand side,
giving us the desired eigenvalues on the right-hand side.
Considering periodically arranged materials, i.e., imposing
translational symmetry into the structure,
e(r)=e(r + R)
with R being a lattice-translation vector, solutions of the master
equation will have the following form, well known from solid-
state physics:
H(r)=h
k
(r)exp(ikr)
with
h
k
(r)=h
k
(r + R).
This solution is known as the Bloch function. This ansatz
solves the master equation. As there are no analytically closed
solutions, this task is usually accomplished via numerical
methods, e.g., with the plane-wave expansion method, imple-
mented in the freely available MIT photonics band package
16
To discuss the essential properties of periodically structured
materials, we start with an intuitive approach to 1D systems,
which we gradually expand to higher dimensions whenever
necessary. We will end with the discussion of a full numerical
calculation of the bandstructure of an inverse opal.
2.2. Photonic bandstructure, band gaps and group velocity
For the beginning we assume light propagating in a homo-
geneous, isotropic material with an index of refraction of n. The
index of refraction and the permittivity are related via n ¼
ffiffi
e
p
.
The optical properties of this material are described by the
dispersion relation
o ¼
c
0
n
jkj
with the vacuum speed of light c
0
,wavevector|k|=k =2p/l,and
l being the vacuum wavelength of light. The energy of the
photons is given by E = ho. Plotting the dispersion relation for
forward propagation (positive k-values), results in a straight line.
For backward propagation (negative k-values) we get the same
dispersion properties (blue lines starting at the origin in Fig. 3).
The physics of our material will not change if we compose it
in a gedanken experiment from slabs with certain thickness a,
glued seamlessly together. This introduces a periodic structure
with a lattice constant a. As each unit cell has absolutely the
same properties as every other unit cell of our material, we can
limit our discussion to just one single unit cell, the primitive cell
or Wigner–Seitz cell, sufficient to construct the whole structure.
The lattice vectors R in real space, in which we construct our
sample and the ones of the reciprocal space (G), in which we
draw our dispersion relation, are closely related:
RG = m2p
m being a positive integer.
Hence, real-space periodicity results in periodicity in our
dispersion relation: This leads to several branches of the
original dispersion relation running through our primitive unit
cell in reciprocal space, called the 1st Brillouin zone, extending
from k = p/a to k =+p/a. We also can construct the exact same
picture, if we back-fold the dispersion bands starting at the
origin every time they hit the boundary of the Brillouin-zone.
This leads to the picture shown in Fig. 3 with the solid blue
lines. Interestingly, the crossing points of the dispersion lines
of the forward and backward propagating waves are located at
the frequency:
o ¼
c
0
p
ffiffi
e
p
a
:
This is nothing else but the Bragg-condition known from
X-ray diffraction, but at optical wavelengths. While the concept
of band structures is a very powerful one to describe materials
properties, not every reader might be familiar with it. A more
detailed introduction can be found in text books, e.g. in ref. 17.
So far, we have gained nothing but a more complicated
looking description of exactly the same physics as before.
Things become interesting, if we introduce different materials.
Let us first assume that each slab is now divided into
two materials with different permittivity, keeping the overall
periodicity exactly the same.
A forward propagating wave is now partially reflected at
every interface between the two materials, leading to forward
and backward travelling waves at the same time. The inter-
ference of forward and backward propagating waves leads to
the formation of a standing wave for frequencies fulfilling the
Fig. 3 Light lines and the formation of a band gap. For the calculation material
slabs of thickness 200 nm and refractive indices of 1.0 and 3.5 have been
assumed. The blue lines show the dispersion relation in an isotropic medium
with artificial periodicity.
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Bragg-condition, i.e., right at the edge of the Brillouin-zone.
Where will the nodes and antinodes of these standing waves be
located? Due to symmetry considerations, there are just two
possible solutions: The nodes and antinodes are either situated
in the low-refractive index and in the high-refractive index
material or the other way round. In-between-positions are
forbidden, as the physics has to stay the same under sample-
rotations in the laboratory. These two standing waves experi-
ence different effective refractive indices: The electric field for
the standing wave with the nodes in the low-index material
experiences mainly the high-index material, resulting in lower
photon energy according to the above equation. The other
standing wave experiences a reduced effective index and will
thus be found at higher photon energies. This leads to the
opening of a gap in the dispersion relation, the photonic band
gap. The energetic lower band is called the ‘‘dielectric band’’, as
its corresponding electric field is mainly concentrated in the
high-index material, e.g., silicon. The higher energetic band is,
hence, called the ‘‘air band’’, as its electric field is mainly
concentrated in the low index material, most of the time air.
The resulting dispersion relation or band structure is depicted
by the red curves in Fig. 3. For photons with energies inside the
band gap, no propagation inside the photonic crystal is
allowed. As can be seen directly from Fig. 3, we do not only
get the fundamental band gap, but also higher order gaps, at
least for one-dimensional systems. The centre position of the
gap is still perfectly described by the Bragg-condition. From the
above equation we directly conclude that we can shift the band
gap position by either changing the lattice constant or the
materials properties or both. Changing the high dielectric con-
stant material mainly influences the dielectric band while chan-
ging the low index material mainly shifts the air band.
Increasing the contrast between the two materials’ dielectric
functions widens the width of the band gap. In a 1D world, even
the slightest index contrast leads to a full photonic band gap –
this dramatically changes going to higher dimensions. Regard-
ing the proper terminology there is some confusion in the
community: We will use the term ‘‘band gap’’ to mark complete,
i.e., omnidirectional band gaps. In 1D systems every gap in the
bandstructure is automatically a complete band gap. For higher
dimensions, a gap might just occur for one particular direction
(for opals this is e.g. the [111] direction) but not for others. In this
case, the gap is called stop band or stop gap or even pseudo gap
just to distinguish it from complete band gaps. Unfortunately, in
the literature the term band gap is not clearly used and one has
to check, if really a complete gap is meant.
The phase fronts of plane waves propagating through a
photonic crystal experience different effective refractive indices
and, hence, propagate with different phase velocities, defined as:
n
phase
= o/k
The phase fronts in the dielectric band propagate generally
at lower velocities than the ones in the air band. Much more
important from an application’s point of view is the velocity
with which energy is transported. For absorption-free dielectric
materials in allowed bands, the energy transport happens with
the same velocity as the group velocity defined by the equation:
n
group
= qo/qk
The group velocity is directly proportional to the slope of the
bands. Its most remarkable property is that it vanishes at the
band edges. This is perfectly consistent with the formation of
standing waves, which do not transport any energy either. A
direct consequence of the vanishing group velocity is an
increased photon density and intensity, as well as an increased
interaction time between light and matter. This property is
highly interesting for amplifying non-linear effects, increasing
the efficiency of photocatalysis, and enhancing light-harvesting
in solar cells to name just a few applications.
If we extend our considerations to two dimensions, we will
find that slow light not only can be found at band edges, but
can also be found throughout the higher bands above the
fundamental gap (the gap at lowest frequencies).
Let us assume for the following discussion that the two materials
have again the same index of refraction, but are periodically
arranged along two dimensions in a simple square-lattice. The
dispersion relation is given by straight lines, which fold back at
the edges of the 1st Brillo uin-zone. For two dimensions this directly
leads to bands with low group velocity as depicted in Fig. 4. Here, k-
values along two different paths along the x-direction in two
different Brillouin-zones are plotted. For the green path inside the
1st Brillouin-zone, we get the already well known dispersion relation:
o
D
¼
c
ffiffi
e
p
jkj¼
c
ffiffi
e
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k
x
2
þ k
y
2
q
¼
c
ffiffi
e
p
k
x
which is just the straight line as expected. For the blue line in the
2nd Brillouin-zone, we get:
o
D
0
¼
c
ffiffi
e
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k
x
2
2p
a
2
s
Calculating the group velocities for both bands directly
shows that group velocities for the path following the blue line
Fig. 4 The reciprocal lattice of a 2D square-lattice. A path in the 1st (green) and
the 2nd Brillouin-zone (blue) are marked.
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