Download development of high refractive index poly(thiophene)

DEVELOPMENT OF HIGH REFRACTIVE INDEX POLY(THIOPHENE) FOR THE
FABRICATION OF ALL ORGANIC 3-D PHOTONIC MATERIALS WITH
A COMPLETE PHOTONIC BAND GAP
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Matthew J. Graham
December 2006
DEVELOPMENT OF HIGH REFRACTIVE INDEX POLY(THIOPHENE) FOR THE
FABRICATION OF ALL ORGANIC 3-D PHOTONIC MATERIALS WITH
A COMPLETE PHOTONIC BAND GAP
Matthew J. Graham
Dissertation
Approved:
Accepted:
_________________________________
Advisor
Dr. Stephen Z. D. Cheng
______________________________
Department Chair
Dr. Mark D. Foster
_________________________________
Committee Member
Dr. Gustavo A. Carri
______________________________
Dean of the College
Dr. Frank N. Kelley
_________________________________
Committee Member
Dr. Thein Kyu
______________________________
Dean of the Graduate School
Dr. George R. Newkome
_________________________________
Committee Member
Dr. Darrell H. Reneker
______________________________
Date
_________________________________
Committee Member
Dr. Alexei P. Sokolov
ii
ABSTRACT
The field of photonics hopes to harness light to supercede in performance many of
the functions carried out by electronics. To accomplish this, the flow of light can be
controlled by means of a photonic band gap (PBG) the same way electronic band gaps
can control the flow of electrons. PBGs, through the coherent backscattering of radiation,
create frequency ranges in which light propagation is forbidden. A PBG is created when
a wave propagates through a periodic array of materials with sufficient refractive index
(n) contrast (n1/n2) where the dimensionality of the periodicity defines the dimensionality
of the PBG. The n contrast required to open a PBG increases as the dimensionality
increases. Currently, only inorganic materials have a sufficiently high n to open a
complete 3-D PBG.
The goal of this project is to fabricate a polymeric material with a complete 3-D
PBG, to bring the tailorable physical, electrical, and optical properties of polymeric
materials to 3-D PBG materials.
The first step was to develop a polymer with a
sufficiently high n. Because of its conjugated nature and the presence of a heavy sulfur
atom in its repeat unit, poly(thiophene) (PT) is predicted to have one of the highest
polymeric refractive indices with n = 3.9 at 700 nm1, but the reported n value for PT is
1.4 at 633 nm.2 This discrepancy is because the potential needed to electrosynthesize PT,
iii
the only method available to synthesize thick and high quality PT films, is higher than its
degradation potential. It was found that by polymerizing thiophene with an optimized
monomer concentration, proton trap concentration, and reaction temperature in a strong
aprotic Lewis acid solvent, the polymerization potential could be reduced below the
degradation potential of PT. The resultant PT film had a maximum n of 3.36, which is
sufficiently high to open a 3-D PBG.
Photonic templates were then constructed using a combination of Colvin’s
method3 with monodisperse spheres and mechanical annealing.
High n PT was used to
infiltrate the templates, and the templates were removed, leaving a polymeric inverse opal
with the possibility of a complete 3-D PBG.
iv
ACKNOWLEDGEMENTS
I would like to acknowledge the direction given by my adviser Dr. Stephen Z. D.
Cheng. Additionally, I would like to acknowledge Dr. Shi Jin for his help in teaching me
the fundamentals of electropolymerization, and Dr. Kwang-Un Jeong for instructing me
on a number of characterization techniques. Finally, I would like to thank Kurt Eyink of
the Airforce Materials and Manufacturing Directorate and Jason Ge for determination of
the poly(thiophene) optical constants.
v
TABLE OF CONTENTS
Page
LIST OF TABLES…………………………………………………………………….
xi
LIST OF FIGURES…………………………………………………………………… xii
CHAPTER
I. INTRODUCTION……………...………..………………………………………
1
II. BACKGROUND………………………………………………………………...
8
2.1 Introduction………………………..………………………………....……….
8
2.2 A Brief History Concerning The Understanding of Light…………….……..
8
2.3 Nature of Light…………….………………………………………………...
12
2.4 Monochromatic Light in a Vacuum…….…………………………………...
13
2.5 Monochromatic Light in a Dielectric………………….…………………….
19
2.6 Scattering of Monochromatic Light From Random Dielectric Scatterer...….. 28
2.7 Scattering of Monochromatic Light From Periodic Dielectric Scatterer...….. 36
2.8 Polychromatic Light………..………………………………………………... 40
2.9 Scattering of Polychromatic Light From Periodic Dielectric Scatterers……. 43
2.10 Development of the Photonic Band Gap……………………………………. 50
2.11 3-D Photonic Band Gap Structures………………………………………….. 55
2.12 Fabrication of Photonic Band Gap Materials…………….…………………. 56
vi
2.13 Material Refractive Indices …………………………………………………
58
2.14 Summary…………………………………………………………………….
59
III. EXPERIMENTAL………………………………………………………………
61
3.1 Poly(thiophene) Synthesis……...……..…………………………………….
61
3.1.1
Materials………………….……………………………………………... 61
3.1.2
Electrical Setup………….………………………………………………
62
3.1.2.1 Electrochemical Cell…...…………………………………………….. 62
3.1.2.2 Working Electrodes………...………………………………………… 63
3.1.2.3 Counter Electrode…………...………………………………………... 64
3.1.3
Temperature Bath……………………………………………………….
64
3.1.4
Electrochemical Synthesis Procedure………….………………………..
65
3.2 Colloidal Templates…….…………………………………………………...
66
3.2.1
Materials………….……………………………………………………... 66
3.2.2
Procedure…….………………………………………………………….
66
3.3 3-D Photonic Band Gap Material……….…………………………………... 67
3.3.1
Materials……….………………………………………………………... 68
3.3.2
Procedure….…………………………………………………………….. 68
3.4 Equipment and Characterization………….…………………………………
70
3.4.1
Atomic Force Microscopy………….…………………………………… 70
3.4.2
Density Measurements………………………………………………….
71
3.4.3 Four Point Conductivity Measurements……………….………………... 71
3.4.4
Fourier Transform Infrared Spectroscopy……….……………………… 72
3.4.5
Optical Diffraction…………….………………………………………… 73
vii
3.4.6 Optical Microscopy…………………………………………………….. 74
3.4.7
Potentiostat/Galvanostat………….……………………………………... 74
3.4.8
Profilametry……………………………………………………………..
75
3.4.9
Scanning Electron Microscopy……..……………………………...……
75
3.4.10 Solid State Carbon-13 Nuclear Magnetic Resonance…….……………..
76
3.4.11 Ultra-violet-Visible Spectroscopy…………………….………………...
76
3.4.12 Variable Angle Spectrographic Ellipsometry………….……………......
77
3.4.13 X-ray Diffraction………….………………………………………...…..
78
IV. DEVELOPING HIGH REFRACTIVE INDEX POLY(THIOPHENE)…….….. 79
4.1
Introduction………………………………………………………….………...
79
4.2
Molecular Design of High Refractive Index Dielectrics…………….………... 79
4.3
Candidates for High Refractive Index Dielectrics………………………….… 82
4.4
Synthetic Techniques for Poly(thiophene)………….………………………… 84
4.5
Optimization of the Oxidative Electrochemical Polymerization of
Poly(thiophene)…….…………………………………………………………. 87
4.5.1
Solvent Determination………………….………………………………....
4.5.2
Optimization of Monomer Concentration in BFEE………….………...…. 91
4.5.3
Optimization of Proton Trap Concentration……………….……………… 96
4.5.4 Polymerization of Poly(thiophene) at Optimized Monomer and Proton
Trap Concentrations……………………………………………….……....
89
98
4.5.5
Optimization of Poly(thiophene) Synthesis Reaction Temperature…....…. 104
4.5.6
Post-Synthesis Thermal Annealing……………….…………………...….. 111
4.6
Summary……………………….……………………………………………… 114
viii
V.
GROWING THE COLLOIDAL TEMPLATE…………...……………………. 115
5.1 Introduction………………….……………………………………………….. 115
5.2 Colloidal Self-Assembly………………….………………………………….. 116
5.3 Determining Colloidal Structure…………………….……………………….. 119
5.4 Determining Conditions for Optimal Thickness………….………………….. 133
5.5 Mechanical Annealing the Colloidal Structure……………….……………… 139
5.6 Optimizing the FCC Structure…………………………….………………….. 141
5.7 Summary………………………………………………………………….…... 146
VI. FABRICATING THE ORGANIC 3-D PHOTONIC MATERIAL…..………… 147
6.1
Introduction……………………………………….………………………….. 147
6.2
Optimizing Poly(thiophene) Synthesis at Reduced Charge Collection Rates.. 148
6.3 Fabrication of the Poly(thiophene) Inverse Opal………………….…………. 152
6.4
Summary……………………………….……………………………………... 170
VII. SUMMARY…………...………………………………………………………... 171
REFERENCES……………………………………………………………………….. 175
APPENDIX FRINGE CALCULATIONS…...……...……………………...……….. 182
A.1 Obtaining Refractive Index Values………………………………………….. 182
A.2 Obtaining Film Thicknesses…………………………………………………. 187
ix
LIST OF TABLES
Table
Page
2.1 Refractive Index Contrast Threshold for Selected 3-D Photonic Structures……
56
2.2 Refractive Indices of Selected Organic and Inorganic Materials……………….
58
4.1
Predicted Refractive Index Values for Conjugated Polymers…………………..
83
4.2
Actual Refractive Index Values for Conjugated Polymers……………………... 83
x
LIST OF FIGURES
Figure
1.1
Page
The cross section of a holey fiber optic cable…………………………………...
3
1.2 The simulation of a waveguide with a 90° angle………………………………..
4
1.3
Schematic of an LED……………………………………………………………
5
2.1
Physical origin of the umbra and penumbra based on the rectilinear
propagation of light……………………………………………………………... 10
2.2
The conceptual description of light as a function length scale…………………. 13
2.3
A schematic diagram showing key features of scattering from a group of
dense but optically distinct objects……………………………………………...
31
2.4
A schematic of light being emitted from two scatterers………………………...
36
2.5
A schematic of light being reflected from two scatterers………………………. 38
2.6 A schematic of elastic scattering in a periodic structure, subtracting the
incident vector from the resultant vector gives the reciprocal vector which is
dependent on the periodicity of the scatterers…………………………………..
39
2.7 A 2-D direct lattice and its corresponding first order reciprocal lattice………...
39
2.8
2.9
The summation of two waves of slightly different frequency results in a high
frequency wave modulated by a low frequency wave…………………………..
41
The Brillouin zone constructed from the reciprocal lattice of a square
2-D structure…………………………………………………………………….
43
2.10 The dependence of wave number on frequency at the edge of a Brillouin zone.. 49
xi
2.11 The difference in periodicity experienced by light traveling in different
directions in a FCC unit cell……………………………………………………. 53
2.12 The individual directional PBGs for X and Y form a complete PBG when
they overlap……………………………………………………………………..
54
4.1
Chemical formula for the Grignard coupling synthesis of PT………………….. 84
4.2
Chemical formula for the oxidative coupling synthesis of PT………………….
85
4.3
Chemical formula for the cathodic reduction synthesis of PT………………….
86
4.4 Chemical formula of oxidative electrochemical synthesis of PT………………. 86
4.5
A proposed mechanism for the oxidative electrochemical synthesis of PT…….
88
4.6 A graph indicating that the PT oxidative synthesis initiation voltage for
acetonitrile was 1.6V while for BFEE it was 1.3V……………………………... 90
4.7
The dependence of PT DP on thiophene concentration in BFEE……………….
92
4.8
The absolute conductivity of PT synthesized at room temperature in
BFEE as a function of thiophene concentration. ………………………………. 93
4.9
FTIR spectrum of a PT film polymerized in BFEE solvent with a
thiophene concentration of 50 mmole…………………………………………..
94
4.10 A structural depiction of a saturated unit in a PT trimer………………………..
94
4.11 The formula for water reacting with BFEE…………………………………….. 95
4.12 Structural depictions of (a) DTTP, (b) TTTP, and (c) TTTM…………………..
96
4.13 The relative conductivity of PT synthesized at room temperature in BFEE
with a thiophene concentration of 50 mmole as a function of proton trap……...
98
4.14 FTIR spectra of PT films synthesized at room temperature in BFEE and
50 mmole thiophene with and without a proton trap……………………………
99
4.15 WAXD of the PT film synthesized at room temperature in BFEE and
50 mmole thiophene with and without the proton trap…………………………. 100
4.16 The WMA of PT films synthesized at room temperature in BFEE and
50 mmole thiophene with and without the proton trap…………………………. 101
xii
4.17 Refractive index dispersion curve for the PT optimized at room temperature…. 102
4.18
13
C-NMR results for PT polymerized at conditions optimized for room
temperature……………………………………………………………………… 103
4.19 The optimization of monomer concentration and current density for
the synthesis of PT at -50°C with respect to the WMA………………………… 106
4.20 The WMA of PT films with and without a proton trap and with a proton
trap at -50°C…………………………………………………………………….. 108
4.21 The FWHH of PT films for a range of monomer concentration and charge
collection rates at -50°C………………………………………………………… 109
4.22 Refractive index dispersion curve for PT optimized at -50°C………………….. 110
4.23 The red-shift of the WMA during thermal annealing as a function of time……. 113
5.1
The ABC stacking of spheres for a FCC structure……………………………… 116
5.2
Calculated photonic band structure of the FCC structure………………………. 118
5.3
Transmission mode optical microscopy of a colloidal template of 290 nm
PS spheres………………………………………………………………………. 121
5.4
Transmission mode optical microscopy of a colloidal template fabricated
from 290 nm PS spheres………………………………………………………... 122
5.5
AFM of the colloidal crystal fabricated from 269 nm diameter PS spheres……. 124
5.6
Optical diffraction from a colloidal template fabricated from 269 nm diameter
PS spheres………………………………………………………………………. 126
5.7 Reflection UV-Vis spectroscopy of 310 nm diameter PS spheres……………... 128
5.8
AFM height image of the minority phase………………………………………. 129
5.9
The optical diffraction from the minority phase………………………………... 130
5.10 Image of fast to slow growth transition by optical microscopy………………… 134
5.11 The 2nd derivative of a fringe pattern from a colloidal template of 290nm
PS spheres………………………………………………………………………. 137
5.12 The linearized extrema fit to determine film thickness………………………… 138
xiii
5.13 AFM height images through time of a colloid template being mechanically
annealed………………………………………………………………………… 140
5.14 PBG width as a function of the ratio between the cylinder radius and the
lattice parameter………………………………………………………………… 142
5 15 AFM image of 269 nm diameter PS spheres after heating for 30 minutes
at 80°C………………………………………………………………………….. 143
5.16 Geometric description of the cylinder connecting fused spheres………………. 144
5.17 UV-Vis transmission spectrum of template aged at room temperature………… 145
6.1
The optimization of reaction temperature and monomer concentration for the
synthesis of PT at a charge collection rate of 0.05 mA/cm2 with respect to the
WMA…………………………………………………………………………… 150
6.2: SEM of a PT inverse opal………………………………………………………. 155
6.3
SEM of a PT inverse opal at higher magnification……………………………... 158
6.4
SEM image of a PT inverse opal between cracks………………………………. 159
6.5 Origin of triple spots in the FFT representation of figure 6.4a…………………. 161
6.6
SEM of the top of a PT inverse opal……………………………………………. 162
6.7
SEM of the side of a PT inverse opal…………………………………………... 164
6.8 Reflection mode optical microscopy of the top of a PT inverse opal…………... 165
6.9 Reflection mode optical microscopy of the bottom of a PT inverse opal………. 166
6.10 Wavelengths of reflection from the PT and from the inverse opal structure…… 168
6.11 The UV-Vis transmission spectrum of the [111] zone of an inverse opal……… 169
A.1 The fringe spectrum from a weakly absorbing film and the 2nd derivative
analysis to magnify the maxima and minima of the fringe pattern…………….. 186
A.2 The linearized extrema fit to determine film thickness…………………..……... 190
xiv
CHAPTER I
INTRODUCTION
The field of photonics is concerned with the interaction of radiation and matter.
Light is an oscillating electromagnetic field transmitted in packets called photons. The
interplay of light and matter can be controlled by photonic band gaps (PBGs) the same
way electronic band gaps are used to control the flow of electrons. PBGs, through the
coherent backscattering of radiation, create frequency ranges for which light propagation
is forbidden. A band gap forms when a wave propagates through a periodic array of
potentials. The symmetry and spatial relations of the array can be characterized by its
space group and corresponding Brillouin zone. The wavelength/frequency range whose
wave vectors connect a reciprocal lattice point to the edge of the Brillouin zone are
forced to be standing waves in order to maintain the translational symmetry of the lattice.
These waves are described as being in Bloch states. Frequencies between the boundaries
of the PBG are coherently backscattered causing a frequency notch in the intensity of the
transmitted radiation. As a result there is a dramatic modification of the density of states.
There are two major consequences of this. In the PBG, the density of states are vanishing
1
which results in the quantum mechanical suppression of optical behavior including
transmission. Consequently, the PBG has perfect lossless reflection of light. At the
edges of the PBG, the density of states is abnormally large enhancing the interaction
between light and matter thus promoting phenomena that rely on this interaction such as
emission. These two phenomena are at the core of why researchers are interested in PBG
materials and could profoundly impact several critical technological platforms including
signal transmission, lighting systems, and optical computing.
The use of fiber optic cables has dramatically increased the rate at which
information can be transferred.
The cost of sending a signal optically is largely
dependant on and inversely related to the distance the signal can be sent before it decays
and needs to be re-amplified by a repeater. Signal decay is caused by both reduced
intensity from material absorption and signal dispersion. Thus, the lower the absorption
of the material, the more cost efficient fiber optics become. Since PBGs can offer
lossless reflection and subsequent transmission in nearly lossless air, PBG holey fiber
optic cables have been developed.
These can make fiber optic cabling more cost
effective, thereby, making it financial feasible to extend networks to isolated or low
population density areas.
The focus on information transmission has not only been on being able to go
longer distances, but also to be able to transmit optical signals over shorter length scales
with more complex geometric constraints.
2
Figure 1.1: The cross section of a holey fiber optic cable. Reprinted with
permission from ref 4 Copyright 1998 American Association for the
Advancement of Science.4
To harden critical systems from electro-magnetic pulses, the Airforce is seeking
to replace many electrical systems with optical ones.
This is particularly true for
aircraft.5 The issue is the physical constraints of the aircraft, the optical fibers must be
bent at significant angles. Conventional fiber optic cables, which rely on total internal
reflection, cannot be significantly bent without having the light leakage. Intel determines
at what length scales it is more efficient to use optical signals versus electrical ones in
communicating between and across microchips. Recently, this assessment has revealed
that to send a signal from one end of a chip to the other it is more efficient to use optical
signals.5 To accomplish this, very high density waveguides that enable the light to be
directed around 90° angles will have to be constructed. Current optical transmission
systems cannot do this. Simulations and experimental evidence on radio frequency
3
radiation has indicated that a two dimensional PBG can steer light around a 90° corner
while maintaining >95% of the initial intensity.6 So, using PBG materials in signal
transmission could enable both of these technological advancements.
Figure 1.2: The simulation of a waveguide with a 90° angle. Reprinted with
permission from ref 6 Copyright 1996 American Physical Society. 6
Lighting systems account for 7% of all energy consumed in the U.S. and 20% of
worldwide electricity consumption.7 This translates to lighting systems based on the
conventional electrical grid being a market of $15 billion while the market for off grid
lighting, which accounts for >1.5 billion people, is $40 billion a year.8 Incandescent light
bulbs have efficiencies of about 5%, while most of the energy gets converted into heat.
Fluorescent lights can have efficiencies around ~20%.7 One possible technology that
could dramatically improve the efficency of lighting systems is light emitting diodes
(LEDs).
4
λ
-
-
+
+
-
+
+
-
+
Figure 1.3: Schematic of an LED.
Light emission from a material as a result of electrical current not heating was
first documented in 1907.7 The emission of light from charged species combining at an
interface can have internal quantum efficiencies of around 90%.7 This means that 90% of
electrons injected into a LED results in the formation of a photon. But external
efficiencies are only about 25%.7 This is because there is a relatively narrow cone where
light is emitted. The rest gets reabsorbed by the material before leaving the LED.
Suppression of this absorption will result in significantly increased external efficiencies.
This could be achieved using 2-D PBG materials. If the PBG is tuned to the wavelength
of emission, lateral emission is suppressed promoting emission perpendicular to the PBG.
It is calculated that properly coupling a PBG to the LED emission wavelength can result
in external efficiencies of ~90%.9
All optical switching offers the opportunity to dramatically increase information
processing speeds. To accomplish this, it is desired to be able to turn PBGs on and off. It
has been proposed that non-linear optics could be used to create all-optical switches.
5
Materials experience a change in refractive index as the percentage of electrons excited in
a material increases to a material specific level. So by varying the intensity of light
incident upon a material one can control its refractive index. Therefore, a material that
has a refractive index just sufficient to open a complete PBG can be used to create a
switch with a pump beam used to open and close the PBG. The fundamental physical
process limiting the speed of this switch is the rate at which the material can relax from
the excited state. It is well known that organics can relax from the excited state up to 100
times faster than inorganics.10 This is directly related to the dimensionality of the excited
state. Excited species in crystalline inorganics can randomly walk in 3-D, while
equivalent species in organics usually can only randomly walk in 1-D along the
conjugation direction. As a result, the rate at which the positively and negatively charged
species can find each other is dramatically higher in the organic systems. Hence why
organics are preferable for this type of application, but inorganics dominate the field of
higher dimensional PBG materials used for this and many other photonic applications.
This is because to open a complete 1-D PBG any periodic refractive index contrast will
work, to open a 2-D PBG one needs any 2-D periodicity and a sufficient refractive index
contrast, while for a 3-D PBG one needs a high symmetry 3-D periodicity and a high
refractive index contrast.11 Therefore, because of their high refractive index values and
the extensive knowledge base developed for creating precise structures on the necessary
lengths scales from the semiconductor industry, inorganics are predominately used in
high dimensional PBG materials, while organic materials, with significantly lower
refractive indices, are used more frequently in 1-D PBG materials. In fact, currently, no
6
organic material is reported in literature to have a sufficient refractive index to open a
complete 3-D PBG for commonly used structures.
The goal of this project is bring the range and tailorability of the electronic and
physical properties offered by organics to the field of high dimensional PBG materials.
The achievement of this objective is dependent on two things. First is the development of
organic materials with a sufficient refractive index to open a complete 3-D PBG. The
second is to develop the processing methods to precisely construct the structures needed
on the length-scale of light to open a complete 3-D PBG in the visible region. As such,
this dissertation is structured along those two threads. Chapter II presents the
foundational science upon which photonic band gap materials and this project are based.
Chapter III details the experimental techniques used in material synthesis and
characterization techniques used on the resultant materials and structures. Chapter IV
discusses the development a new synthetic approach to synthesize high refractive index
poly(thiophene) (PT). Chapter V deals with improvements to the technique of
constructing photonic templates on the length scale of light from monodisperse spheres.
Chapter VI then describes the process of combining the high refractive index PT with the
optimized FCC structure and the optical behavior of the resultant inverse opal material,
while Chapter VII summarizes these findings.
7
CHAPTER II
BACKGROUND
2.1 Introduction
This chapter lays the foundation of basic concepts for the rest of this dissertation.
This includes a brief history of the development of our understanding of light, a primer of
basic concepts of how light is described in vacuum, how light and materials interact, and
finally, the basic concepts behind photonic band gap (PBG) materials will be discussed.
2.2 A Brief History Concerning the Understanding of Light
Initial ideas about light focused on its relationship with sight. At first sight was
thought to be the product of probes or rays being sent out by the eyes to feel surfaces outof-reach. This was called tactile theory and had proponents including Ptolemy and
Euclid.13 Although this concept creates an intuitive sense of light by likening it to touch,
it failed to explain everyday phenomena such as the inability to see at night. Later
8
Aristotle among others, proposed that sight actually originated from rays leaving an
object (either through emission, reflection or refraction) and entering the eye. This was
called emission theory. This theory completely supplanted tactical theory when Abu Ali
al-Hasan Ibn al-Haitham, also known as Alhazen, published his work Kitab-at-Manazir
(Book of Optics). With the understanding that rays originate from the object being
perceived, the next major discussion was on the nature of the rays. This broke down into
two schools of thought, the corpuscular theory and the wave theory.13 The corpuscular
theory held that light traveled like particles with rectilinear trajectories through space.
This explained phenomena at large distances very well. For example, when light from a
relatively small source hits a large object, one can observe a sharp shadow. The sharp
shadow would be the product of rectilinear propagation. Additionally, when light from a
relatively large source hits an object, one observes two shadow rings. The outer lighter
ring, called the penumbra, is from light crossing from the opposite side of the source,
while the darker ring called the umbra is bound by the object edge blocking light from the
same source edge. Again, rectilinear propagation neatly accounts for this phenomena.
9
Figure 2.1: Physical origin of the umbra and penumbra based on the rectilinear
propagation of light.
Isaac Newton used this picture of the sharp umbra and penumbra shadows as
evidence that light travels in a rectilinear fashion which could be described by small
particles following Newtonian motion. Newton later went on to describe interference
patterns, called Newton rings, and Newton recognized that the repetition of the light and
dark regions suggested a periodic character to light. He posited that the particles
themselves might have some internal vibration, but held that rectilinear propagation could
only be properly described by particles. Even in Newton’s life time, Grimaldi’s
experimental results looking at a shadow’s edge at high magnification showed that the
shadow edge was not sharp but banded with light and dark regions.13 This suggested
that, in terms of light propagating in 3-D space, the corpuscular theory was incomplete.
The wave theory held that light consisted of waves traveling through space.
Bartholinus, through experiments with birefringent crystals, was able to determine that
10
the forward propagation of light was affected by its orientation to interfaces.13 This
suggested that light had a polarization perpendicular to its direction of propagation.
From this basis, Christiaan Huygens was first to put forth a mathematical description of
light as a transverse wave.13 Although this described how light propagated, including
interference phenomena, it did not illuminate the nature of light. In 1845, Faraday
showed that the polarization of light traveling through a material could be affected by
applying a magnetic field to the material.13 This suggested that light was somehow
related to magnetic fields, which had already been shown to be related to electric fields.
Faraday speculated that light had its origin in electromagnetic fields but could not derive
it. This set the stage for James Clark Maxwell. Starting with what were the fundamental
equations of electromagnetism (Coulomb’s law, Ampere’s law, and Farady’s law),
Maxwell showed that field equations allowed for high frequency electromagnetic waves
that traveled at a fixed velocity in vacuum. This offered a framework to understand the
nature of light, how it propagates in space, and how it behaves at interfaces.
Work continued on several interesting problems such as how atoms and
molecules interact and scatter light (Rayleigh), and how finite elements of varying shape
interacted with light (Mie). All of these considerations were worked on from the
perspective of light being perfectly described by Maxwell’s field equations. This
approach broke down when Rayleigh tried to explain black-body thermal radiation in
terms of classical wave equations. The results agreed well with long wavelength
emission but failed to capture short wavelength behavior. In fact, Rayleigh was able to
prove that is was impossible to truly describe black-body radiation with classical waves.13
In response to this issue, Planck developed a new way of understand the problem with
11
discrete energy levels instead of continuous energy levels as assumed by Maxwell’s field
equations. This approach was only meant to be used to describe the interaction between
light and matter, not to illuminated a new aspect to light. Einstein then used the concept
of discrete energy level particles in his theoretical development of the physical origin of
the photo-electric effect. Again, this application dealt with the interaction between light
and matter. The two apparently contradictory concepts of photons and electromagnetic
waves coexisted like the corpuscular theory and wave theory until Heisenberg introduced
the uncertainty principle.13 This allowed the two concepts to be reconciled as extreme
cases of well defined momentum for the electromagnetic description and well defined
position for the photon description.
2.3 Nature of Light
The historical development of the description of light has varied depending upon
the length-scale one is interested in. In general the description of light falls into three
length scales. When the path length (L) of light is significantly greater than the
wavelength (λ) of interest (L>> λ), we can use geometric optics to describe it. When the
path length is similar to the wavelength of interest (L~ λ) or when concepts of
polarization are needed, then the electromagnetic approach to light is appropriate. When
interested in atomic length scale phenomena, one needs to use the quantum description.
12
Quantum
Electromagnetic
Geometric
L>>λ
L~λ
L<¼λ
Figure 2.2: The conceptual description of light as a function length scale.
Since the quantum construction is the most general description so far, all
phenomena on longer length scales must be an out working of quantum mechanics. As
such, any phenomena occurring at longer length scales must have quantum level
consequences. Therefore, if one can force an unusual electromagnetic effect, one can
dramatically affect the quantum mechanical interaction of materials and light.
2.4 Monochromatic Light in a Vacuum
To get to the physical phenomena behind the PBG, the propagation of light needs
to be understood. The simplest case is light in a vacuum. The propagation of light in a
vacuum, because detailed information about absorption and emission processes is not
needed, can be is accurately described by Maxwell’s field equations. This development
13
closely follows the treatment in The Feynman Lectures.14 The four Maxwell equations
deal with all of electrodynamics in a self-consistent manner.
ρ
ε0
(2.1)
∇× E = −
∂B
(2.2)
∂t
∇⋅E =
∇⋅B = 0
c 2∇ × B =
j
ε0
(2.3)
+
∂E
∂t
(2.4)
In these equations, E is the electric field, ρ is the charge density, ε0 is the
dielectric permittivity in free space, B is the magnetic field, t is time, c is the speed of
light, and j is any currents in the material. To show that an electromagnetic wave can be a
solution to the field equations, the curl of equation 2.2 is taken.
∇ × (∇ × E ) = −
∂
∇× B
∂t
(2.5)
Then using the vector identity in equation 2.6 on the left side of equation 2.5 and
the equation 2.4 on the right side, equation 2.5 can be transformed into equation 2.7.
14
∇ × (∇ × E ) = ∇(∇ ⋅ E ) − ∇ 2 E
∇(∇ ⋅ E ) − ∇ 2 E = −
(2.6)
1 ∂2E
c 2 ∂t 2
(2.7)
Since in vacuum there is no charge, equation 2.1 can be rewritten as equation 2.8.
∇⋅E = 0
(2.8)
Then using equation 2.8 with equation 2.7 results in equation 2.9
∇2E =
1 ∂2E
c 2 ∂t 2
(2.9)
Equation 2.9 therefore describes the electric field dynamics in vacuum. To
determine if the wave description of light is valid, the solution of the wave equation must
also satisfy the field equations. Assuming light is a periodic transverse electric field, as
suggested by Newton’s rings, Bartholinus’s birfringent crystals, and Faraday’s
magnetically altering light polarization, it can be described as a forced harmonic
oscillator. It is known that light waves can be understood as being driven by an
oscillating electronic dipole. One might argue that in vacuum there are no dipoles, but
15
light traveling in a vacuum may have its origin outside of vacuum. This type of forced
oscillation can be described using Newtonian physics motion for a mass on a spring with
a harmonic driving force as seen in linear differential equation 2.10.
E
dx
d 2x
+ γE
+ Eω 02 x = E 0 cos(ωt − δ )
2
dt
dt
(2.10)
Here E is the electric field, γ is equivalent to any dampening in the system, t is
time, x is the displacement distance from equilibrium, and ω0 is the resonance frequency,
ω is the probe frequency, and δ is a phase shift of the wave. It is known that the steadystate solutions to this type of equations take the form of the forcing function.
E = E 0 cos(ω 0 t − δ )
(2.11)
Euler’s equation (equation 2.12) indicates that harmonic motion can be described
by a complex exponential.
eiθ = cos(θ ) + i sin (θ )
(2.12)
Using Euler’s equation, the electric field can be represented using complex exponentials
E = E 0 e i (ωt −δ ) (2.13)
16
So for the case of light traveling in the z direction polarized in the x direction the
steady-state solution will take the form of equation 2.14.
E x = E 0 e i (ωt − kz )
(2.14)
Here δ is replaced with k, the reciprocal of wavelength, and z, the distance
traveled in z direction. It is known that in wave functions that take the form of equation
2.15 that the ν represents the phase velocity of the wave.
E x = E 0 e i ( z −vt )
(2.15)
By a simple rearrangement the electric field equation one can see that the phase
velocity of light is a function of the ratio of the frequency and reciprocal wavelength of
the light.
Ex = E0e
⎛ ω ⎞
− ik ⎜ z − t ⎟
k ⎠
⎝
ν ph =
ω
(2.16)
(2.17)
k
To determine if this wave construction is a valid description of light, it needs to be
a solution of the wave equations derived above. For simplicities sake, we will continue
17
to work with light in a vacuum. In a vacuum, there is no charge, so ρ would be 0. So for
light traveling in the z direction polarized in the x direction, equation 2.9 can be written
as equation 2.18.
∂ 2 Ex
1 ∂ 2 Ex
=
∂z 2
c 2 ∂t 2
(2.18)
Plugging in equation 2.14 for the E-field in equation 2.18 results in equation 2.19.
− k 2 Ex = −
ω2
c2
Ex
(2.19)
Since Ex is on both sides of the equation, they can be canceled leaving equation 2.20.
k=
ω
c
=
1
λ
(2.20)
So in vacuum, the wave representation of the electromagnetic field is a valid
solution to the Maxwell field equations. This shows that the wave picture of light within
the context of electromagnetic fields is justified.
18
2.5 Monochromatic Light in a Dielectric
While traveling in vacuum, from the perspective of electrodynamics, the medium
does not affect the travel of light. This is not true while traveling through a medium
populated with electrons. The interaction of light and matter modifies how light travels.
To quantify this affect, one starts with a physical picture of how light and matter are
related. In electrodynamics, the rise and fall of electrons to various energy levels acts as
an oscillating dipole resulting in an oscillating electric field. This can be modeled as a
damped harmonic oscillator.
⎛ d 2x
⎞
dx
m⎜⎜ 2 + γ
+ ω 02 x ⎟⎟ = F = qe E
dt
⎝ dt
⎠
(2.21)
The damped harmonic oscillator has x standing for the displacement from
equilibrium, m is the mass of the object oscillating, γ is the dampening force, ω0 is the
natural frequency of the restoring force, qe as the charge of the electron, E is the electric
field, and F stands for force. The steady state solution to equation 2.21 is equation 2.22.
E = E 0 e i (ωt −δ ) (2.22)
Here E0 is the initial state of the electric field, ω is the frequency of the driving
force, t is time, and δ is a phase shift. It is assumed from here on out that the phase shift
19
is zero. Because of the equality, we know that the displacement on the left side of
equation 2.21 follows the same form.
x = x 0 e iωt
(2.23)
Here, x0 is the initial displacement. Using this information, we can plug equation
2.22 and 2.23 into equation 2.21.
d 2x
= −ω 2 x (2.24)
2
dt
dx
= iω x
dt
(2.25)
Equations 2.24 and 2.25 just show the 1st and 2nd derivatives of x. From there,
we can solve for x with respect to E.
x=
qe
E
m − ω + iγω + ω 02
(
)
2
(2.26)
From physics, we know that the strength of a dipole is a function of both the
charge and displacement.
20
p = xqe
(2.27)
By plugging equation 2.26 into equation 2.27 gives equation 2.28.
p=
q e2
E
m − ω 2 + iγω + ω 02
(
)
(2.28)
Macroscopically polarizibility is described by equation 2.29.
p = ε 0α (ω ) E
(2.29)
Here α is the material polarizability and ε0 is the dielectric permittivity of free
space. Comparing equations 2.28 and 2.29, it is evident that some detailed molecular
aspects can be added flush out the picture of polarizability.
α (ω ) =
q e2
mε 0 − ω 2 + iγω + ω 02
(
)
(2.30)
To more specifically characterize the complete polarizability of a molecule, the
polarizability contribution of each electron and electronic mode needs to be summed up.
α (ω ) =
q e2
fk
∑
2
ε 0 m h − ω + iγ k ω + ω 02k
21
(2.31)
Here h represents each electronic mode and f is correlated to the strength of each
mode. To recover the material polarizability from the molecular polarizability the
polarizability of each molecule needs to be multiplied by the concentration or density of
the electrons per unit area (N).
P = ε 0 Nα (ω )E
(2.32)
With this quantity now defined from a molecular perspective, we can now go
back and solve the Maxwell equations and determine the effect of the medium on light.
Specifically, the case of a dielectric material will be looked at. By that, it is meant that
all charges are bound. This case particularly affects how charges and currents are dealt
with. Despite the fact that the electrons are bound, polarization of dipoles allows for
volume charge. So for the case of a dielectric, we can use equation 2.33 to describe
material charge.
ρ pol = −∇ ⋅ P
(2.33)
Now, current is the amount of charge passing through an imaginary surface in the
material. So physically, this is the product of the charge of the electron (qe) times the
number of charged particles per unit volume (N), and their velocity (ν).
j = Nq eν
(2.39)
22
In the case of a dielectric, the organized fluctuation of dipole moments from
polarization constitute the current in the material. This is described by equation 2.40.
j pol =
dP
dt
(2.40)
We can plug in these descriptions for charge and current into the Maxwell
equations to give us the electrodynamics of a wave in a dielectric medium.
∇⋅E = −
∇⋅P
∇× E = −
∂B
∂t
∇⋅B = 0
c 2∇ × B =
(2.41)
ε0
(2.42)
(2.43)
⎞
∂⎛P
⎜⎜ + E ⎟⎟
∂t ⎝ ε 0
⎠
(2.44)
The new set of equations are approached the same as for the case of light in a
vacuum. First take the curl of equation 2.42.
∇ × (∇ × E ) = −
∂
∇× B
∂t
23
(2.45)
Then using the vector identity in equation 2.46 on the left side of equation 2.45
and the equation 2.44 on the right side, equation 2.45 can be transformed into equation
2.47.
∇ × (∇ × E ) = ∇(∇ ⋅ E ) − ∇ 2 E
∇(∇ ⋅ E ) − ∇ 2 E = −
(2.46)
1 ∂2P 1 ∂2E
−
ε 0 c 2 ∂t 2 c 2 ∂t 2
(2.47)
Then using equation 2.47 with equation 2.44 results in equation 2.48.
∇2E −
1 ∂2E
1
1 ∂2P
(
)
P
=
−
∇
∇
⋅
+
ε0
c 2 ∂t 2
ε 0 c 2 ∂t 2
(2.48)
The resultant equation indicates that E depends on P, but from equation 2.41, we
know that P is dependant on E. So for the case of light traveling in the z direction
polarized in the x direction the steady-state solution will take the form
E x = E 0 e i (ωt − kz )
(2.49)
It is known that in wave functions that take the form of equation 2.50 that the ν
represents the phase velocity of the wave.
24
E x = E 0 e i ( z −vt )
(2.50)
By a simple rearrangement the electric field equation one can see that the phase
velocity of light is a function of the ratio of the frequency and wave number of the light.
E x = E0e
⎛ ω ⎞
− ik ⎜ z − t ⎟
k ⎠
⎝
ν ph =
ω
(2.51)
(2.52)
k
Combining the conceptual definition of phase velocity with that of the
phenomenological definition, one obtains a phenomenologically based definition of the
refractive index
ν ph =
n=
c
n
(2.53)
kc
(2.54)
ω
The effect of the dielectric medium can be incorporated into the electric field
equation.
Ex = E0e
⎛ nz ⎞
iω ⎜ t − ⎟
c ⎠
⎝
25
(2.55)
Since the polarization of the light is in the x direction, P does not vary along x.
∇⋅P = 0
(2.56)
Because P is driven by E, the steady-state solution to P will have the same form as E.
PX ∝ e iωt
(2.57)
As a result, the 1st and 2nd derivatives of P are given by equations 2.58 and 2.59.
∂2P
= −ω 2 Px
∂t 2
(2.58)
∂ 2 Ex
= −k 2 E x
2
∂z
(2.59)
After simplifying equation 2.48, one gets equation 2.60.
− k 2 Ex +
ω2
c2
Ex = −
ω2
P
ε 0c 2 x
(2.60)
Now we can substitute our earlier definition of P based on E (2.32) into the righthand side of the equation. The E dependence then drops out and we can simplify the
26
result to equation 2.61. This equation satisfies the field equations, therefore, the
sinusoidally varying electric field is a valid description of the propagation of light in a
dielectric medium.
k2 =
k=
ω2
c2
ω
c
(1 + Nα (ω ))
(2.61)
1 + Nα (ω )
(2.62)
Rearranging equation 2.62 and substituting the equation for refractive index
posited above in equation 2.54 results in equation 2.63.
n 2 = 1 + Nα (ω )
(2.63)
n = 1 + Nα (ω )
(2.64)
This is a functional description of how a molecule can be designed to engineer the
refractive index. Earlier, the physical origin polarizability was described by equation
2.31.
q e2
fk
α (ω ) =
∑
2
2
2ε 0 m k ω k − ω + iγ k ω 2
(
27
)
(2.65)
From equation 2.31, it is easy to see that since qe, ε0, and m are universal
constants and that polarizability is related with the resonant oscillation frequencies of a
material and their relative strength. From equations 2.64 and 2.65, it is known that the
denser a material is the higher the refractive index and the closer to a resonance
frequency the higher the refractive index.
2.6 Scattering of Monochromatic Light From Random Dielectric Scatterers
Not only does the refractive index of a material affect the propagation of light, the
physical arrangement independent dielectric scatterers also modifies light propagation.
The monochromatic light waves from two separate scatters can be described as the
summation of two waves as seen in equation 2.66.14
(
)
E R e iφR e iωt = E1e i (ωt +φ1 ) + E 2 e i (ωt +φ2 ) = E1e iφ1 + E 2 e iφ2 e iωt
(2.66)
Here E is the electric field, ω is the frequency of light, t is time, and φ is the phase
term. Since the light is monochromatic, the frequency term can be factored out and
eliminated.
E R e iφR = E1eiφ1 + E2 eiφ2
28
(2.67)
To determine the absolute strength of the electric field, which is the intensity of
the light, the absolute value of the amplitude of the alternating field can be determined by
multiplying the imaginary alternating field equation by its complex conjugate.
(
)(
E R2 = E1e iφ1 + E 2 e iφ2 E1e − iφ1 + E 2 e − iφ2
)
(2.68)
Multiplying out the complex conjugate gives equation 2.69.
(
E R2 = E12 + E 22 + 2 E1 E 2 e i (φ1 −φ2 ) + e i (φ2 −φ1 )
)
(2.69)
It is known that the last term can be recast using the Euler formula.
e iθ + e − iθ = cos(θ ) + i sin (θ ) + cos(θ ) + i sin (θ )
(2.70)
Since we are only interested in the real component, we can ignore the imaginary
part resulting in equation 2.71.
e iθ + e −iθ = 2 cos(θ ) (2.71)
Plugging equation 2.71 into equation 2.69 gives equation 2.72.
E R2 = E12 + E 22 + 2 E1 E 2 cos(φ 2 − φ1 ) (2.72)
29
Equation 2.72 indicates that the absolute strength of the field varies as a function
of the phase difference between the two waves. This phase differential can be caused by
two phenomena. The first is a natural phase difference between the waves being emitted.
Second is the geometry of the scatterers. In the case of a forced harmonic oscillation, like
in reflection, the phase being emitted from each scatterer is in phase. This leaves only the
geometric arrangement to cause the phase difference. The arrangement of the scatters
can be one of two extremes. In one extreme, there is no correlation between the dielectric
scatters. Figure 2.3 shows a schematic of the scattering from randomly distributed, but
optically distinct dielectric scatterers.
30
kf
ki
kf
θ
rn
ki
θ
d2
r3
r4
r1
d1
r2
Figure 2.3: A schematic diagram showing key features of scattering from a group
of dense but optically distinct objects.
As indicated in figure 2.3, scattering from objects introduces a directional
component into the propagation of normally incident light. These changes in propagation
direction result in the phase differentials, so inducing constructive and destructive
interference. This influences the strength of the electric field at a given point. The phase
difference between waves scattered from two different scatterers is given by equation
2.73.
∆φ =
2π
λ
(d 2 − d1 )
(2.73)
Here λ is the wavelength of light, ∆ φ is the phase difference between the waves
while d1 and d2 are defined by equation 2.74 and equation 2.75.
31
d1 = − k i • (rn − r1 )
(2.74)
d 2 = k f • (rn − r1 )
(2.75)
The definitions of d1 and d2 can be plugged into equation 2.73 and terms can be
collected resulting in equation 2.76.
∆φ =
2π
λ
(k
f
+ k i ) • (rn − r1 ) (2.76)
In the case of diffuse backscattering, the dot product terms can be defined by
equations 2.77 and equation 2.78.
R = rn − r1
(2.77)
⎛θ ⎞
k f + k i = 2 sin ⎜ ⎟
⎝2⎠
(2.78)
In equation 2.71, θ is the angle of the reflected light with respect to the incident
light. Plugging these definitions into equation 2.69 and evaluating the dot product results
in equation 2.79.
∆φ =
2π
⎛θ ⎞
2 sin ⎜ ⎟ R cos(α )
λ
⎝2⎠
32
(2.79)
Here α is the angle between (kf+ki) and (rn-r1), and R is the distance between two
independent scatters. For a group of scatters, this can be taken in be the average ensemble
radius between scatters. This is directly related to the diffusion rate of the scatterers.
⎛ cl * ⎞
R 2 = 6 Dt = 6⎜
⎟t
⎝ 3 ⎠
(2.80)
Here D is the coefficient of diffusion, c is the speed of light, l * is the mean free
elastic scattering distance, and t is the random-walk time. Taking the root-mean-square
of this representation of R and plugging into equation 2.79 gives equation 2.81.
∆φ =
2π
2π
⎛θ ⎞
⎛ θ ⎞ ⎛ cl * ⎞
2 sin⎜ ⎟ 6⎜
2 sin⎜ ⎟ 2l * ct cos(α ) (2.81)
⎟t cos(α ) =
λ
λ
⎝2⎠
⎝2⎠ ⎝ 3 ⎠
Equation 2.81 can be plugged into equation 2.72 for a generalized expression of
the electric field scattered from random dielectric scatterers.
⎛ 2π
⎞
⎛θ ⎞
E R2 = E12 + E 22 + 2 E1 E 2 cos⎜⎜
2 sin ⎜ ⎟ 2l * ct cos(α )⎟⎟
⎝2⎠
⎝ λ
⎠
(2.82)
Even in the case of uncorrelated scatterers, there are conditions where the phase
difference is minimized resulting in enhanced backscattering. The phase difference is
going to be mininzed at small θ. Additionally, for dense arrangements of scatterers, (rn 33
r1) and (kf + ki) are nearly parallel to the surface so cos(α)~1. These simplifications result
in the phase difference being described by equation 2.83.
∆φ ≈
2π
λ
θR
(2.83)
In this construction, the criterion for the coherent backscattering is equation 2.84.
∆φ
<< 1
2π
(2.84)
Since λ, l * , c, t for a particular system is fixed, there must be a critical angle (θc )
less than which the phase difference becomes vanishing. The critical angle where
backscattering enhancement starts is given by equation 2.85.
θc ≈
λ
2l * ct
(2.85)
From this equation, it can be seen that the θc is maximized, resulting in the
greatest reflection, when the diffusion distance (ct) is minimzed. Here the minimum
diffusion distance bounded by the mean free elastic scattering distance.
ct = l *
(2.86)
34
Plugging equation 2.86 into equation 2.85, one sees that the maximum possible
angle for coherent backscattering is given by equation 2.87.
θ max =
λ
2l *
(2.87)
Plugging this angle back into 2.83 gives equation 2.88
∆φ =
2π
λ
θ max 2l* = 2π
(2.88)
The greatest field intensity can be seen by plugging 2.88 into 2.82.
E R2 = E12 + E 22 + 2 E1 E 2 cos(2π )
(2.89)
So to minimize the phase difference and thereby maximize the intensity reflected
from uncorrelated scatterers, the scatters should be densely packed with a mean free
elastic scattering distance i.e. the scattering cross section, similar to the random walklength of the scatterers.
35
2.7 Scattering of Monochromatic Light From Periodic Dielectric Scatterers
The extreme of randomly distributed scatters was described in section 2.6. The
other extreme of highly correlated scatterers will be described in this section. Starting
the same as for random scatterers, monochromatic light waves from two separate points
can be described as the summation of two waves as seen in equation 2.90.
E R2 = E12 + E 22 + 2 E1 E 2 cos(φ 2 − φ1 ) (2.90)
The phase difference is the function of two separate phenomena. The first is a
natural phase difference in the waves being emitted (α). The second is based on the
geometry of the scatterers.
d
θ
d sin(θ)
Figure 2.4: A schematic of light being emitted from two scatterers.
The geometric difference in phase is due to the distance differential between the
waves being emitted from two different points (equation 2.91).
36
φ 2 − φ1 =
2πd sin (θ )
λ
(2.91)
Combining these two phenomena, one obtains equation 2.93.
φ 2 − φ1 = α +
2πd sin (θ )
λ
= α + kd sin (θ )
(2.93)
In the case of a forced harmonic oscillation, the phase being emitted from each
scatterer will be the same, so α = 0. Here, the maxima in intensity being emitted occurs
when the waves emitted constructively interfere. This criteria is expressed in equation
2.94.
φ2 − φ1 = 2πm
(2.94)
Plugging in equation 2.94 in to equation 2.93 and setting α = 0, one can get an
expression for the intensity pattern from the light emitted from two scatterers.
mλ = d sin(θ )
(2.95)
This shows that given a limiting case of light from in-phase harmonic oscillators,
the intensity pattern of a scattered wave is determined solely by the geometric
37
arrangement of the scatterers. As such, one can probe a structure with a monochromatic
wave where the resultant intensity pattern is a map of the scattering structure.
k0
k
θ
d
d sin(θ)
d sin(θ)
Figure 2.5: A schematic of light being reflected from two scatterers.
Figure 2.5 presents a case where there is elastic scattering of a coherent wave
incident upon a set of scatterers. Since both scatterers are being driven by the same
incident wave, they both emit in-phase light, so α = 0. Again, the geometry of the
scatterers solely determines the intensity pattern of the emitted light. Since the intensity
will be maximized when the geometric difference in the distance traveled by the emitted
waves is a multiple of 2πλ, the expression for this pattern is given in equation 2.96.
mλ = 2d sin(θ )
(2.96)
The resulting expression is called Bragg’s law. Here m is the diffraction order, λ
is the wavelength of light used, d is the distance between scatterers, and θ is the angle at
which the light is incident to the surface. Using this approach one can determine the
periodicities in a structure, but to define the structure, one also needs the orientation of
the various periodicities.
38
Assuming elastic scattering, one can determine the orientation of a periodicity by
subtracting the incident wave vector (k0) from the reflected wave vector (k), as seen in
figure 2.6.15 The resulting wave vector is called a reciprocal vector.
k
∆k
k0
Figure 2.6: A schematic of elastic scattering in a periodic structure, subtracting the
incident vector from the resultant vector gives the reciprocal vector which is
dependent on the periodicity of the scatterers.
The reciprocal vector is directed perpendicular to the periodicity and its
magnitude is the reciprocal of the length of the periodicity. In this construction, the
orientation and the length of periodicities comprising a structure are captured.
(a)
(b)
Figure 2.7: (a) A 2-D direct lattice and (b) its corresponding first order reciprocal
lattice.
39
Figure 2.7 shows both a direct square lattice and it’s corresponding first order
reciprocal lattice. The points represent reciprocal vector end points, so the magnitude of
the vector connecting the origin to a reciprocal point is the reciprocal of the period length
and the vector orientation is perpendicular to the orientation of the periodicity. As such,
the reciprocal lattice completely represents the direct structure in frequency space.
2.8 Polychromatic Light
As previously explained, light can be thought of as a wave. Because of the
principle of superposition and the linear nature of the wave description, the intensity of
overlapping waves can simply be added together to get the resultant intensity. This is
true for light of a single wavelength/frequency and for light composed of a range of
wavelengths and frequencies, but certain complexities become apparent when adding
together waves of different frequencies.14 We have so far described the propagation of
light using equation 2.97.
E = E 0 e iωt
(2.97)
Here E is the electric field, E0 is the amplitude of the field, ω is the frequency of
the wave, and t is the time. Now in the case of a non-dispersive medium, we can add two
waves together using equation 2.98.
40
E1e iω1t + E 2 e iω2t = e
i (ω1 +ω 2 )t
2
i (ω1 −ω 2 )t
i ( − ω1 +ω 2 )t
⎛
⎜ E1e 2 + E 2 e 2
⎜
⎝
⎞
⎟
⎟
⎠
(2.98)
To clarify the affect of the two waves on the field strength, assuming wave
frequencies ω1 and ω2 are close, one can factor out the average of the two waves. The
result is that a wave of average frequency whose field strength is being modulated as a
function of the difference between the two constituent frequencies. This appears as a
high frequency wave being modulated by a lower frequency wave. Because the medium
is non-dispersive and the strength of the field only depends on the frequency difference,
the phase velocity υp of the modulated wave follows the standard description of phase
velocity found in equations 2.52 and 2.53.
νp =
ω
k
=
c
n
(2.99)
Figure 2.8: The summation of two waves of slightly different frequency results in
a high frequency wave modulated by a low frequency wave.
41
In dispersive media, the dependencies of the field strength become more complex.
This situation can be described by equation 2.100.
E1e
i (ω1t − k1x )
+ E2 e
i (ω2t −k2 x )
=e
i ((ω1 +ω2 )t −( k1 + k2 ) x )
2
⎛
⎜⎜ E1e
⎝
i ( ( ω1 −ω 2 ) t − ( k1 − k 2 ) x )
2
+ E2 e
i ( ( −ω1 +ω 2 )t − ( − k1 + k 2 ) x )
2
(2.100)
It can be seen that the field strength is a function of both frequency and the wave
number (k). So, the resultant modulated wave has a phase velocity that is a function of
both the difference in frequency and the difference in wave number as shown in equation
(2.101).
νM =
ω1 − ω 2
k1 − k 2
(2.101)
In this description, υM is the velocity of the wave modulation. In the limiting case
of small differences between the constituent frequencies and wave numbers, the phase
velocity of the modulated wave is described as the group velocity (υg) by equation 2.102.
νg =
dω
1
=
dk τ d
(2.102)
This result has several important consequences. First is that the sign of the group
velocity is determined by the slope of the dispersion of the material. If the refractive
42
⎞
⎟⎟
⎠
index of a material increases with wavelength, as in the abnormal dispersion, the group
velocity is negative. If the refractive index of a material decreases with wavelength, as
with normal dispersion, the group velocity is positive. If the refractive index varies very
rapidly with respect to wavelength, the group velocity can be superluminal. These cases
are important because the dwell time of the light, a measure of the extent to which light
interacts with a material, is inversely proportional the group velocity indicating that
through the group velocity one can tailor the interaction between the light and matter.16
2.9 Scattering of Polychromatic Light from Periodic Scatterers
The reciprocal lattice of the 2-D square lattice is shown in figure 2.9. It
completely captures the periodicity and orientation of the direct lattice in frequency
space. Each point corresponds to a reciprocal vector of the lattice. How the real lattice
interacts with polychromatic light can be understood using the concept of a Brillouin
zone.
k1
θ
∆k
Figure 2.9: The Brilliuon zone constructed from the reciprocal lattice of a square
2-D structure.
43
The Brillouin zone is constructed by drawing perpendicular bisectors of the
closest reciprocal points.15 The blue points of the reciprocal lattice are the closest points
and the lines are the perpendicular bisectors. The enclosed area is called the first
Brillouin zone. Perpendicular bisectors of the second closest reciprocal points, the black
ones, define the second Brillouin zone, and so on for each higher order of reciprocal
points. The physical significance of the Brillouin zone is more obvious when one
mathematically describes any vector (k1) going from the origin and ending on any point
on the Brillouin zone edge (1/2∆k). The magnitude of such a vector is the dot product of
the constituent vectors as shown in equation 2.103.15
(k )• ⎛⎜ 12 ∆k ⎞⎟ = k
v
1
v
⎝
⎠
1
1
∆k cos θ
2
(2.103)
This vector can also be geometrically described by equation 2.104.15
k1 cos θ =
1
∆k
2
(2.104)
Using the left side of equation 2.104 to simplify the right side of equation 2.103
results in equation 2.105 which describes all vectors connecting the origin to the Brillouin
zone edge.
(k )• ⎛⎜ 12 ∆k ⎞⎟ = 12 ∆k
v
1
v
⎝
⎠
44
2
(2.105)
From section 2.7, we know that a reciprocal vector can be described with equation 2.106.
v v
v
k − k 0 = ∆k
(2.106)
Equation 2.106 can be rearranged and both sides can be squared resulting in
equation 2.107.
(k ) = (k − ∆k )
v
2
v
v
2
(2.107)
0
Expanding the squared terms results in equation 2.108.
v
v
v v
v
k 02 = k 2 − 2 ∆k k + ∆k 2
( )( )
(2.108)
Because the scattering is elastic, the incident wave vector magnitude is equivalent
to the scattered wave vector magnitude.
k0 = k
(2.109)
This allows the k0 and k vectors to cancel out of equation 2.108 resulting in
equation 2.110.
( )( )
v
v v
∆k 2 = 2 ∆k k
45
(2.110)
Both sides of the equation can be divided by four giving equation. 2.111.15
2
1 v v
⎛ 1 v⎞
⎜ ∆k ⎟ = ∆k k
2
⎠
⎝2
( )( )
(2.111)
This is the exact same form as the vectors connecting the origin of the reciprocal
lattice to the Brillouin zone surface. This indicates that a Brillouin zone vector suffers
Bragg diffraction and gets reflected by the structure. Therefore, waves traveling in a
periodic medium are modified by the structure. Specifically, because of the translational
symmetry of the periodic structure imposes a translational symmetry on the wave. These
are called Bloch waves and a Bloch wave equation takes the form of equation 2.112.17
E (r ) = ∑
G
∑C
k −G
e i ( k − G )r
(2.112)
k
Here E is the electric field, r is the distance from the origin, C is the unperturbed
electric field strength , k is the wave vector, G is the magnitude of the reciprocal vector.
The dielectric constant (ε) of the periodic media varies spatially. This is described by
breaking the dielectric down into two components. The first is the volume fraction
weighted average of the dielectric constants of the constituent materials as seen in
equation 2.113.17 This is the average background dielectric.
ε 0 = φε 1 + (1 − φ )ε 2 (2.113)
46
The second component is given in equation 2.114 as the spatially varying part
captured by the Fourier coefficient (UG) of the reciprocal vector.17
ε (r ) = ε 0 + ∑ U G e iGr
(2.114)
G
The Fourier coefficient is given by equation 2.115.15
U G = Vc−1 ∫ dVε (r )e − iG⋅r
(2.115)
cell
Here, V is the volume of the unit cell. This can be recast as the Rayleigh-Gans
equation (2.116) which gives the Fourier coefficient for a reciprocal vector.17
UG =
3φ
(GR )3
(ε 1 − ε 2 )(sin (GR ) − GR cos(GR ))
(2.116)
Here φ is the filling fraction of the high dielectric component, G is the reciprocal
vector, R is half the thickness of the periodic high density domains. It can be seen that
the Fourier coefficient gets larger when there is a larger dielectric difference between the
structure and the background as well as for smaller unit cell volumes. Using this
definition of the Fourier coefficient and plugging equation 2.116 into equation 2.114
results in an infinite set of simultaneous equations to solve to determine the electric field.
47
To simplify the problem, one can limit the wave propagation in one dimension. This
reduces the propagation vectors to scalar gradients. Then by limiting the situation to 1-D
propagation along a high symmetry point of the lattice, the only equations one has to
solve are for the cases of G = 0 and G as the shortest reciprocal vector for the symmetry
point. These cases can be written as equations 2.117 and 2.118 respectively.17
⎛ 2
ω2 ⎞
ω2
⎜⎜ k − ε 0 2 ⎟⎟C k − 2 U G C k −G = 0
c ⎠
c
⎝
−
(2.117)
ω2
⎛
ω2 ⎞
2
⎟C
⎜
(
)
U
C
k
G
=0
+
−
−
ε
G k
0
2 ⎟ k −G
⎜
c2
c
⎠
⎝
(2.118)
For these two cases to be solved simultaneously, their determinants need to be
vanishing. As a result, the wave vectors for polychromatic light propagating through a
periodic structure are given by equation 2.119.17
k=
1
ω2
ω2
ω4
G2
+ ε 0 2 − G 2 ε 0 2 + U G2 4
G±
2
4
c
c
c
(2.119)
One can see in 2.119 that not all wave vectors are purely real. Non-real wave
vectors are suffering Bragg diffraction. This decreases the amplitude of the wave as it
propagates through the periodic structure. One can notice that a range of wavelengths get
reflected. The width of this wavelength range is directly dependent on the value of the
Fourier coefficient. As a side note, figure 2.10 shows that the dispersion around the
48
wavelengths reflected exhibits a wide range of group velocities. This includes a
discontinuous transition between a positive group velocity and a negative group velocity.
In this region, the wavelength of the electromagnetic envelope is divergent. As the group
velocity rapidly changes so does the dwell time of the light. This strongly modifies how
light and matter interact. This is an example of how using the electromagnetic wave
picture, situations can be created where the efficiency of quantum mechanical phenomena
Wave Number [ k ] (m-1)
like spontaneous emission can be strongly altered.
Frequency [ ω ] (s-1)
Figure 2.10: The dependence of wave number on frequency at the edge of a Brillouin
zone.
49
2.10 Development of the Photonic Band Gap
Physicists in the middle of the 20th century spent significant time and effort to
understand the metal-insulator transition in materials. Mott proposed that in crystalline
solids that columbic repulsion between strongly correlated electrons in single valent
systems can result in jammed electron transport.18 More specifically, repulsive
interactions between electrons dampen lattice site fluctuations, reducing electron
mobility, thereby, prohibiting electron transport across the sample. This picture
accurately explained the conductivity behavior of crystalline materials, but did not
address behaviors in conducting glasses such as vanishing conductivity as the
temperature approaches absolute zero.19 In 1958 P.W. Anderson published a paper
concerning the diffusion of electrons in a random lattice, equivalent to electronic
conduction in amorphous solids.19 He assumed that the conduction mechanism in these
solids was the jumping of electrons between sites and not by free carriers as in crystalline
metals. Anderson dealt with localized sites of varying energy (due to size confinement,
impurities, interactions with the outside environment, and imperfections in local atomic
structure) by randomly varying the energy levels of neighboring sites without reference to
any specific causes. The energy level at any particular site was given an energy
distribution band width. Electrons could couple and jump to neighboring sites through
columbic interactions. It was found that if the strength of the interaction between sites
fell off at a rate >1/r3 where r is the distance from a site and if the interaction strength is
less then the width of the energy gap between two given sites then there will be no
electron transport. Effectively the electron at the given energy is localized at that site. In
50
systems with columbic charge, all electronic states can localize. This led to the
development of the Ioffe-Regel criterion for the metal-insulator transition in amorphous
materials.21
l ≤
λ
2π
(2.120)
For electrons, λ is the Fermi wavelength of the electron and l is the elastic mean
free path of free carriers. The essence of this criterion is that as the distances that the
electron can probe for a new site is similar to the distance between sites, electron
transport comes to a halt. Pure examples of this transition in amorphous materials have
been difficult to observe experimentally due to electron-electron interactions and
electron-phonon coupling.22 In order to observe a pure Anderson transition without
particle interactions, the question started to be raised whether classical waves without
columbic charge could be localized using an Anderson-like mechanism.23-25 If so,
interaction free waves could be used to clearly demonstrate this mechanism. From this
premise, researchers moved from charged electrons to interactionless photons, but it has
been difficult to observe in this construction as well because photons do not interact with
single sites to promote localization.26, 27 Because light does not localize to a single site,
localization is evidenced by the coherent backscattering of the light. In terms of light, the
Ioffe-Regel criterion (equation 2.121) is redefined such that 3-D localization in a
disordered sample occurs when the mean free elastic scattering distance (l) is less then
the wavelength of light.28
51
l<
λ
2π
(2.121)
To achieve this criterion, the l needs to be significantly reduced. This may be
achieved through Mie scattering. As the diameter of a particle approaches the λ of the
light being scattered, the scattering cross section of the particle becomes significantly
larger than the geometric cross section of the particle. But to sufficiently reduce l, the
scatterers have to be densely packed. But to maintain the scattering strength of the Mie
resonance, the particles have to be greater than λ distance apart. As a result, to achieve
localization in a completely disordered sample the scatterers must be more than λ apart,
but have l < λ.28 This is practically impossible.
In the face of this, another approach was proposed. In periodic media, a range of
wavelengths can be forced to become a standing wave due to Bragg diffraction. Just
outside of this range, light can propagate through the material. These propagating waves
form a long wave modulated envelope (λ’) on the standing wave. At wavelengths very
close to the wavelengths suffering Bragg diffraction the envelope wavelength diverges.
This is when wave propagation effectively stops, and light is localized. As such a new
Ioffe-Regel criterion for light can be expressed as equation 2.123.28
l<
λ'
2π
(2.122)
52
This arrangement indicates that light can localize and open a photonic band gap
(PBG) when the group velocity deviates significantly from the phase velocity. This
picture explains the presence of a PBG in any one direction. This becomes more
complex when one is trying to control the propagation of a particular wavelength of light
in many dimensions at once. The mean free scattering length (l), i.e. the periodicity of
the structure, determines the center wavelength of light that gets reflected to form the
PBG, and the refractive index contrast (n1/n2), through the Fourier coefficient, determines
the range width of the wavelengths that get reflected.29 So to control the propagation of
light in more than one direction, scatterers need to have periodicities similar enough in
multiple dimensions that there is sufficient refractive index contrast to overcome any
mismatch in periodicities. An example of this is shown in figure 2.11.
X
Y
n1
n2
Figure 2.11: The difference in periodicity experienced by light traveling in
different directions in a FCC unit cell.29
53
In figure 2.11, the two vectors are propagating in two different directions through
a FCC unit cell. The length of periodicity in each direction is different. Therefore, the
center wavelength of light undergoing Bragg reflection will be different. Again, the
range of wavelengths reflected is determined by the Fourier coefficient for the structure
which is dependent on the refractive index contrast between the two materials in the
Wavelength
Wav
elength
structure.
Complete
Band Gap
Y Band Gap
X Band Gap
X Y
Periodicity
Figure 2.12: The individual directional PBGs for X and Y form a complete PBG
when they overlap.29
54
Figure 2.12 shows the length of periodicity versus the wavelength of light passing
through the structure. One can see at lengths corresponding to the periodicities
associated with the X and Y directions that directional PBGs exist. A complete PBG
between these two directions only exists for the narrow range of wavelength where the X
PBG and Y PBG overlap.29
This example indicates two important points. First is that the higher the level of
symmetry in the structure the better chance that a PBG exists. This is the reasoning
behind the rule of thumb that the more spherical the first Brillouin zone is the more likely
a PBG will be opened for a particular structure. Second is that since there is no 2-D nor
3-D structures with infinite symmetry, all 2-D and 3-D PBGs have a threshold refractive
index contrast that must be met before a complete PBG opens.
2.11 3-D Photonic Band Gap Structures
Unlike 2-D structures, not every 3-D structure can open a 3-D PBG at any
refractive index contrast. So, if one wants to control the propagation of light independent
of direction, one needs to use certain 3-D structures.30-44 Each structure has its own
threshold refractive index contrast (n1/n2). Table 2.1 contains a list of several of the most
common 3-D structures and their respective refractive index contrast thresholds.
55
Table 2.1: Refractive Index Contrast Threshold for Selected 3-D Photonic Structures
Structure
∆n
Inverse HCP
3.10
Inverse FCC
2.89
Ball/Stick BCC
2.80
Single Gyroid
2.28
Inverse Square Spiral
2.20
Diamond
1.87
The refractive index contrast thresholds are calculated, and because the
electromagnetic wave equations are scalable, these values are constant for all frequencies.
But, not all of these structures can be fabricated on the length-scale of visible light. This
is true of the diamond, inverse square spiral, single gyroid, and ball/stick BCC structures.
Therefore, despite their higher refractive index requirement, the most commonly used
structures for visible wavelength PBG materials are the inverse FCC and inverse HCP
structures.
2.12 Fabrication of PBG materials
Photonic band gaps (PBG) can be one, two, or three dimensional depending on
the number of directions that the propagation of light can be suppressed. The fabrication
of 1-D PBGs requires a periodicity in just one dimension and any refractive index
56
contrast is sufficient. This type of PBG is particularly amenable to fabrication techniques
used by organics.45-48
The fabrication of 2-D PBGs requires a structure with a 2-D periodicity and a
threshold refractive index contrast. With a sufficient refractive index contrast, any 2-D
structure will open a complete PBG. For this reason, 2-D PBGs have often been used for
proof of principal structures to explore phenomena of interest that occur at the PBG edge
like ultra-refraction, superprism, and lasing phenomena.49-57
Many of the more exotic applications of PBG materials will require 3-D PBG
materials. Due to limitations on the range of 3-D structures that can open a complete 3D PBG at any refractive index contrast, the development of 3-D PBG materials has been
slower. The construction of 3-D PBG materials started in the microwave regime by
drilling micron size holes in dielectric blocks.36 To open 3-D PBGs at shorter
wavelengths, semiconductor processing techniques were then used, particularly for
woodpile type structures.58,59 This approach is extremely costly and time consuming, so
only very thin structures of limit area could be fabricated. Consequentially, this approach
is rarely used. The two other more frequently used techniques to fabricate PBG materials
at visible light wavelengths are holographic lithography and colloidal crystallization.
Holographic lithography relies on the interference of coherent lasers in a
photopolymerizable media to record the resultant periodic intensity patterns.60-63 This
can rapidly create structures on the appropriate length-scales, but the 3-D structures are of
limited thickness and area. That is why the most common method of creating a visible
wavelength 3-D PBG material relies on the spontaneous self-assembly of monodisperse
spheres.64-68 Techniques like Colvin’s method create FCC structures (opals) where the
57
periodicity can be precisely varied by changing the diameter of the spheres assembled.3
Once one has a template, it can be infiltrated with a high refractive index material, and
then the spheres can be removed leaving behind an inverse FCC structure.69-71
2.13 Material Refractive Indices
The limited range of structures that can be built on the length scale of visible light
restricts the range of materials that can be used to fabricate PBG materials. A list of
common organic and inorganic materials is given in Table 2.2.
Table 2.2: Refractive Indices of Selected Organic and Inorganic Materials
Material
n
GaSb
Si
Sb2S3
GaAs
CdS
SiN
ITO
Poly(sulfone)
Poly(styrene)
Poly(atactic propylene)
Poly(isoprene)
Poly(methyl methacrylate)
Low Density Poly(ethylene)
5.26
3.88
3.74
3.66
2.50
2.02
1.80
1.63
1.59
1.51
1.51
1.49
1.49
Inverse FCC
Diamond
One can see that materials such as GaAs and Si have a sufficient refractive index
to open a complete 3-D PBG with an inverse opal structure. On the other hand, no
58
organic reported in the literature has met the refractive index requirement. Even for the
diamond structure which has the lowest refractive index contrast, conventional polymers
do not have sufficient refractive index to open a complete 3-D PBG. This is a significant
hurdle to the fabrication of completely organic 3-D PBG materials, and consequentially
reduces the range of physical and electronic properties available to this technology.
2.14 Summary:
It was shown that the wave picture of light satisfies the Maxwell electromagnetic
field equations in a vacuum, but in a dielectric, one needs to introduce a retardance into
the wave equation. This retardance is called the refractive index and causes the phase
velocity of light to slow down. Additionally, the refractive index can vary, particularly
strongly around resonance frequencies of electrons in the dielectric, as a function of
wavelength. The superposition of two electromagnetic waves traveling at different
speeds is a long wavelength envelope modulating a shorter wavelength wave. Bragg
reflection of light from a periodic structure can cause dramatic changes in this wave
envelope for a range of wavelengths, determined by the length-scale of periodicity and
the refractive index contrast of the structure, such that the resultant envelope wavelength
can become divergent. In this wavelength range, light is completely reflected from the
structure. This is a PBG. At the edge of the PBG, the group velocity thus the dwell time
of the light changes dramatically modifying the extent that light and matter interact. The
dimensionality of the PBG is dependent on the dimensionality of the periodicity. Since
no structure has infinite symmetry, there is a mismatch between the center wavelengths
59
being reflected depending on the direction light is traveling through the structure. In the
case of a set of high symmetry structures, this mismatch can be overcome by increasing
the refractive index contrast of the dielectric structure. So to open a complete 3-D PBG
for visible light, one needs to fabricate a 3-D photonically active structure from materials
of sufficient refractive index on the length-scale of visible light. Currently, no organic
has a sufficient refractive index to open a complete 3-D PBG for the structures that can
presently be constructed on the length scale of light.
60
CHAPTER III
EXPERIMENTAL
3.1 Poly(thiophene) Synthesis
The first step of this project was the synthesis of a high refractive index polymer.
The most promising candidate according to literature was poly(thiophene) (PT). The
polymerization of thick PT films to infiltrate photonic templates required synthesis
through electrochemical oxidative reduction. A detailed description of the materials and
procedure of this process is listed below.
3.1.1 Materials
Benzene free thiophene monomer with a 99.5% purity was purchased from Acros
Organics. The thiophene was redistilled to further purify it. The resultant thiophene was
colorless. Thiophene not immediately used was packed under an inert atmosphere and
stored away from light in a freezer. Redistilled boron trifluoride diethyl etherate (BFEE)
was purchased from Aldrich. It was stored at room temperature, packed in nitrogen, kept
61
from exposure to light and was used as received. The BFEE was colorless upon use
indicating the purity of the solvent. Proton traps (2,6-di-tert-butylpyridine 2,4,6-tri-tertbutylpyridine, 2,4,6-tri-tert-butylpyrimidine) where also purchased from Aldrich at purity
levels from 99% to 97%. To preserve efficacy, proton traps shipped in larger bottles
were reapportioned into smaller vials in a dry box then packed under nitrogen.
3.1.2 Electrical Setup
The electrical setup is critical to electrochemical polymerization. Electrode
materials and cell arrangement are known to affect the orientation and conjugation of the
resultant polymer. A description of the electrodes and cell setup are given below.
3.1.2.1 Electrochemical Cell
A 60 ml clear, wide-mouthed, bottle is used as the electrochemical cell. The
bottles were washed with soap and deionized water. After drying they are stored in a
120°C oven until use. A rubber #7 stopper was used to create an air proof seal for the
reaction vessel. A hole was drilled into the top of the stopper to allow electrodes and
argon purge gas into the cell. The electrical and gas system was passed through a glass
tube whose diameter was slightly larger than the diameter of the hole drilled into the
stopper to create an air tight seal around the glass tube. After the wires and gas system
were passed through the tube, both sides of the tube were then plugged with a fast setting
two part, marine epoxy. This setup allowed the electrical system into the cell without
62
contamination from the outside. The electrodes were then arranged parallel to each other
with a 5 mm space between the two. The electrodes were cut, so the areas would match.
3.1.2.2 Working Electrodes
Conducting silicon with both sides polished was obtained. Slides were cut to the
appropriate size using a diamond knife. Silicon slides were then ultrasonically cleaned in
soapy deionized water, deionized water, acetone, and isopropyl alcohol solutions
respectively for fifteen minutes each. They were then immersed in a 70% Sulfuric
acid/30% hydrogen peroxide solution heated to 75°C for 30 minutes. The slides were
then washed with copious amounts of deionized water. Slides were then stored in an
oven at 120°C. This type of electrode was used to make films for reflective variable
angle spectroscopic ellipsometry analysis to extract optical constants of the films. Since
both the front and back sides of the electrode are conductive, PT film would grow on
both sides. The ratio of front side thickness to backside thickness was 3:1. This made it
difficult to grow films of precise thickness.
Indium tin oxide (ITO) coated glass slides were cut to the appropriate size with a
diamond knife. The ITO slides were then ultrasonically cleaned in soapy deionized water,
deionized water, acetone, and isopropyl alcohol solutions respectively for fifteen minutes
each. The 70% Sulfuric acid/30% hydrogen peroxide solution was not used because it
etches the ITO. Since only the front side of the electrode is conductive, it limited film
growth to just the front surface. For this reason, films that need precise thicknesses were
63
made using ITO slides. All UV-Vis measurements and FTIR measurements were made
from films that were grown on ITO.
3.1.2.3 Counter Electrode
Nickel was used as the counter electrode due to its resistance to corrosion. Nickel
foil (0.125 mm thick 99.9+% purity) was purchased from Aldrich. The foil was cut to
match the area of the working electrode using side snips. The counter electrode was
polished to a mirror-like finish using 300 nm aluminum oxide grit. The nickel is then
washed with acetone to remove the grit while not leaving a fluid film. The counter
electrode was re-polished after every use.
3.1.3 Temperature Bath
The temperature of a electrosynthesis reaction can strongly affect the mechanical
and electrical properties of the resultant polymer. In cationic radical polymerizations,
lower temperatures are usually desired. These temperatures were achieved through the
use of temperature baths. For a 0°C reaction temperature, a water/ice bath was used to
fix the temperature. For temperatures below 0°C, a water/ethanol bath was utilized. The
appropriate water/ethanol ratio was used to tailor the freezing point of the solution to ±
2.5°C of the desired reaction temperature. The bath temperature was then reduced using
dry ice.
64
3.1.4 Electrochemical Synthesis Procedure
To reduce monomer concentration variation for each set of conditions, a master
batch of each reaction solution was made to dispense at each temperature. The master
batches were made within 48 hours of use and stored in a dry box under an argon blanket.
The proton trap was added to the master batches first to scavenge any acidic proton
impurities in the BFEE solvent. After addition to the solvent, the proton trap was
allowed to completely dissolve before addition of the monomer. The reaction solution
was then distributed into the reaction cells. The electrode system was then placed in the
reaction solution. The reaction mixture was kept under a blanket of argon to prevent
atmospheric humidity from coming in contact with the solvent. The electrochemical cell
was placed in the temperature bath and allowed to equilibrate to the desired temperature
for an hour. After setting the appropriate constants in the chronopotentiometry
polymerization software, the reaction was then started. After polymerization, no PT was
found floating in the solvent, and the solvent maintained it’s translucent yellow color.
This indicates that the short polymer chains did not migrate from the surface. The
polymer on the working electrode was immediately washed in acetone. The film was
then allowed to dry in the air. Conductivity measurements were made fifteen minutes
after washing. The films were then dedoped with ammonium hydroxide for at least 24
hours before any other characterization was carried out.
65
3.2 Colloidal Templates
High quality templates then had to be created to grow the PT film through. Below
is a description of the materials and procedures used to create photonically active
templates.
3.2.1 Materials
Colloidal crystals were used as photonic templates. The colloidal spheres were
purchased from Duke Scientific with sizes ranging from 269 nm to 320 nm. The standard
deviation from the reported diameters was less than 5%. The colloid was suspended in a
deionized water solution. The colloid was stored in a refrigerator between use. Before
use, the colloid was sonicated for 30 minutes to breakup any sphere aggregates that may
have formed.
3.2.2 Procedure
A measured volume of solution was then diluted down to the appropriate
concentration with ethanol in a graduated cylinder. Once calibrated, the desired template
thickness can be varied by changing the volume fraction of spheres. The spheres were
then mixed in the ethanol by sonication. The solution was then transferred to a clean
open 30 ml wide mouth bottles. The bottle was then placed on an isolation table hung
from the ceiling to dampen building vibrations. Cleaned ITO electrodes were then placed
66
in the bottle with the diluted colloidal suspension such that as the solvent evaporates, the
meniscus would sweep across the length of the electrode. The mouths of the bottles were
then covered with Kimwipes to prevent dust from getting into the colloidal solution
during evaporation. The ethanol was then allowed to completely evaporate. This
generally took about seven days. After which the electrode is removed and placed in a
vacuum oven at room temperature. The oven is then pumped down to pressures less than
0.1 inHg for 24 hours to finish the drying process. The crystal could then be exposed to
mechanical annealing to perfect the structure. This was accomplished by using a
potentiostat to pass a designed voltage wave profile through a piezoelectric element. The
amplitude/voltage of the wave determined the vertical displacement of the template,
while the frequency of the wave set the frequency of the vibration. Amplitudes varied
from 0.125 V to 2 V and frequencies varied from 10 Hz to 200 Hz.
3.3 3-D Photonic Band Gap Material
With the procedures to fabricate the high n PT and method to fabricate the
template set, the two now have to be combined to form the inverse opal structure. The
materials and procedures associated with this process are described below.
67
3.3.1. Materials
The materials used to synthesize the PT are the exact same as those used in
section 3.1.1. The photonic template was fabricated as described section 3.2.2.
3.3.2 Procedure
Samples were made to observe the morphology and optical properties of the
inverse opal. These templates were ~9 µm thick. This translated into ~20 unit cells
which is a compromise between having as many unit cells as possible to insure that there
would be 9 defect free ones together, and having as few unit cells as possible to reduce to
voltage required to enable monomer transport through the template to the electrode or
film surface. The charge collected was sufficient to fill the template, assuming the
template was constructed from hard spheres, plus 50% more for a polymerization with
efficiencies the same as without the template. UV-Vis spectra in transmission were taken
to determine the initial thickness of the film. The template then sat to enable the sphere
surfaces to relax into each other. The blue shift in film reflectivity, as detected by UVVis, resulting from the structure changes due to relaxation was monitored until the
desired interpenetration had been achieved.
Exposed portions of the working electrode were passivated with an ethyl acetate
film to ensure that the polymer grows through the template and not preferentially in areas
where the template had chipped off. Reaction solutions were prepared in a dry box under
an argon atmosphere at the optimized proton trap and monomer concentrations. The
68
proton trap was added first and allowed to completely dissolve before the thiophene was
added. The thiophene was redistilled before use and added to the BFEE solvent. The
reaction solution was then distributed into the reaction cell. The electrode system was
then placed in the reaction solution with the nickel counter electrode and the template
covered working electrode. The reaction mixture was kept under a blanket of argon to
prevent atmospheric humidity from coming in contact with the solvent. The template
was then allowed to soak in the reaction solution for 9 hours to allow the solution to
completely penetrate and fill the tortuous voids in the colloidal crystal template. A
water/ethanol temperature bath was prepared such that it would freeze at -50°C. Dry ice
was used to reduce the temperature. The electrochemical cell was placed in the
temperature bath and allowed to equilibrate at the desired temperature for an hour. After
setting the appropriate constants in the chronopotentiometry polymerization software, the
reaction was then started. For samples made to investigate the inverse opal morphology,
the charge collected was set to fill the template 90%. Because the efficiency of the
reaction in the presence of the template has not been studied, the actual filling fraction is
not known. For samples made for optical characterization, the film thickness was chosen
to overshoot the template by 50%. This was sufficient to completely fill the template and
form a continuous film of PT on top of the crystal. After polymerization, no PT was
found floating in the solvent, and the solvent maintained its translucent yellow color.
These indicate that short polymer chains did not migrate from the surface. The polymer
on the working electrode was immediately washed in acetone. The film was then
allowed to dry in the air. Dedoping results in a film volume reduction. This reduction is
usually less then 10%, but could significantly distort the inverse opal morphology if it
69
occurred quickly without the template to maintain the desired structure. Since dedoping
is a spontaneous process that will occur slowly without a dedoping agent like hydrazine
or ammonium hydroxide, the film was allow to sit in the air for a week. During this time,
a significant portion of dedoping occurs at a very slow rate allowing the chains in the film
to adjust and minimize film stresses while maintaining the macro-porous structure. After
the week, the film was submersed in ammonium hydroxide for three days to completely
finish the dedoping process. Afterward, the film was washed in water and then
submersed in tertahyrafuran (THF) for 24 hours to dissolve the polystyrene template.
The film was then place in a vacuum oven at ~250°C for five days.
3.4 Equipment and Characterization
Each step in the project required characterization from the PT film, to the
template, to the final inverse opal. The equipment used and how the samples were
characterized are listed below.
3.4.1 Atomic Force Microscopy
An atomic force microscope (AFM, Digital Instrument Nanoscope IIIa), in
tapping mode was employed to characterize the surface of colloidal crystals. This
enabled quantification of defect levels and variation of template height. The force of the
cantilever was small enough to limit its damage to the samples, while maintaining good
engagement with the surface. The scanning rate ranged between 1 to 3 Hz at a resolution
70
of 512 x 512 for areas ranging from 5x5 µm to 40x40 µm. The operation and resonance
frequency was ~290 kHz. The 100 µm scanner was calibrated with a standard grid for
both lateral size and height.
3.4.2 Density Measurements
A solution of water saturated with potassium iodine salt, purchased from Aldrich
and unmodified before use, was made. The material of unknown density is placed in a
vial partially filled with deionized water. It was checked to make sure no bubbles were
visible on the surface of the material. The potassium iodine salt solution was then added
to the vial until the material of unknown density floated in the middle of the mixed
solution. The density of the mixed solution was then measured using a second vial of
precisely known weight and volume. By subtracting the empty vial weight from the full
vial weight and dividing by the volume of the vial one can determine the density of the
unknown material.
3.4.3 Four Point Conductivity Measurements
A homemade inline four point probe was used to make relative conductivity
measurements. Gold wire with a 0.25 mm diameter and a purity of 99.9% was purchased
from Aldrich. Gold was used to minimize the working function between the PT film and
the electrode. Each of the probes was embedded in the dielectric holder 3 mm apart. A
potentiostat was used to create a 1mA current across the two outside probes, while a
71
multimeter was use to measure the voltage of the inside probes. Films were removed
from the conducting substrate and placed on insulating glass to remove the influence of
the conductive substrate. A minimal pressure was applied to the probe during
measurement to prevent the film from being punctured. Measurements were made 15
minutes after the synthesis of the film. This ensured that all the films had similar levels
of doping. The film thickness was then measured by profilametry. Using the current,
voltage, and thickness information one can calculate a relative conductivity. This is
relative because constants associated with the homemade probe were not calibrated.
Since conductivity measurements in this dissertation are only for comparison with each
other and not for PT measured with a different probe, the relative nature of the
measurement is acceptable.
3.4.3 Fourier Transform Infrared Spectroscopy
The Digilab Win-IR Pro FTS 3000 Fourier transform infrared spectrometer
(FTIR) was calibrated using a polystyrene standard sample. An aluminum masked was
used to ensure that the PT film completely covered the area being measured
An
equivalent number of background and sample spectra were collected. In most cases, the
number of spectra collected was 2048. A 4 cm-1 resolution was used. FTIR allowed for
quantitative analysis of the saturated thiophene units being incorporated into the film.
72
3.4.5 Optical Diffraction
Optical diffraction patterns were acquired using a 4mW 633 nm HeNe laser. A
screen was setup inline with the laser beam to collect the light reflected from the
template. A hole was punched into the screen large enough to allow the laser beam to
pass through without introducing an aperture diffraction pattern onto the optical
diffraction. The colloidal templates used were constructed from PMMA spheres with a
diameter of 300 nm. PMMA (n = 1.49) was used because glycerol (1.47) could then be
used as a refractive index matching agent. The refractive index matching agent
minimizes extraneous light being scattered from any surface roughness, and reduces decollimation of the laser beam due to structural defects. Because of the strong interaction
between the light and material at optical frequencies this can be a major concern, while it
is relatively unimportant at x-ray frequencies. The sample was placed in the center of a
large clean spherical flask and then immersed in glycerol. The beam size and diffraction
pattern was small compared to the curvature of the flask, so distortion was minimal. The
glass flask gave no diffraction itself. The resultant diffraction pattern from the template
was captured with a 7.1 megapixel digital camera.
For Kossel diffraction, a divergent laser beam is used. This was achieved by
passing the beam through a convex lens. Again, a screen was setup inline with the laser
beam to collect the light reflected from the template. A hole was punched into the screen
large enough to allow the laser beam to pass through without introducing an aperture
diffraction pattern onto the optical diffraction. Because the beam does not have to be
collimated, a refractive index matching fluid was not needed. The reflected image was
73
then captured using a 7.1 megapixel digital camera. Photoshop© was then used to invert
the intensity to make the pattern easier to see.
3.4.6 Optical Microscopy
Optical microscopy was used to characterize the wavelengths of light reflected by
the opal templates. An Olympus BH-2 Confocal Optical Microscope was used with an
attached camera. To measure the wavelengths reflected, an Ocean Optics USB 4000
spectrometer was attached to the microscope.
3.4.7 Potentiostat/Galvanostat
A model 273 E&G Princeton Applied Research was used to conduct the
electrosynthesis of PT. Specifically, the chronopotentiometry program was used. This
was chosen over chronocoulometry because it has been shown to produce denser, higher
quality films. The charge collection rate allowed control over how fast the films
developed, while the total charge collection time set the thickness of the film. The
working electrode was attached to the electrode upon which the film is deposited. The
counter electrode was attached to the nickel electrode. No reference electrode was used
for the synthesis of these materials. This is because the standard reference material
voltages have a temperature dependence and a solvent dependence. The combination of
both factors complicates the use of these standards. The lack of a standard will not affect
the quality of the resultant film using chronopotientiometry. The potentiostat was also
74
used to produce constant current for the four point probe conductivity measurements and
voltage waves of prescribed height and frequency for mechanical annealing.
3.4.8 Profilametry
A profilameter (Alpha-Step 500) was used to characterize the thickness of
synthesized poly(thiophene). It is located on an isolation table to limit the effects of
vibration in the measurement. The profilameter was calibrated using a VLSI Standards
Incorporated 880 QC lateral and height standard. The height measurement was found to
be ± 2 nm. Measurements were taken by making a small cut in the film to expose the
substrate where the thickness measurement was desired. The stylus was set to 2.5 mg
force. It was carefully brought into contact with the surface and then it was allowed,
following a straight line path, to map the height variation of the surface for a distance of 5
mm.
3.4.9 Scanning Electron Microscopy
Scanning Electron Microscopy (SEM, JEOL JSM-5310) was used to look at the
morphology of PT films and inverse opals. Films and substrates were attached to the
aluminum disk via conducting tape. A SPI Sputter was used to deposit a layer of
conductive silver onto the insulating film surface. The samples were then placed in the
lock-in chamber and pumped down. Electron beam was set to minimize any damage to
the sample while still allowing a clear image to be captured. In the case of inverse opal
75
morphology, the images were then analyzed using the 4/8/05 release of the Chirokov fast
Fourier transform pluggin for Adobe Photoshop©.
3.4.10 Solid State Carbon-13 Nuclear Magnetic Resonance
A solid-state carbon-13 nuclear magnetic resonance (13C-NMR, Chemagnetics
CMX 200) operating at 201.13 and 50.78 MHz for 1H and 13C nuclei was used to analyze
PT samples. The cylindrical sample rotor was made from single crystalline Al2O3
(sapphire) and has a diameter of 7.5 mm. The samples were spun in nitrogen gas at 4.5
kHz at the magic angle. The magic angle was optimized by the intensity calibration of
the aromatic carbon resonance of hexamethylbenzene.
13
C Cross Polarization/Magic
Spinning Angle (CP/MAS) NMR Spectra were collected to confirm that PT was formed.
The contact time was 2 ms, and each spectrum consisted of an accumulation of 5000
scans. The temperature of the solid state 13C-NMR experiment was controlled using a
REX-F900 VT unit covering a temperature range of 25°C to 200°C. The samples were
made into a powder with a total mass of ~100 mg.
3.4.11 Ultra-violet-Visible Spectroscopy
The Hewlett Packard 8453 UV-Vis spectrometer with deuterium and tungsten
lamps was turned on and allowed to stabilize for ten minutes. Three background spectra
were collected with the substrate and averaged together. The exposure time of the
background was the same as the exposure time of the sample. PT films characterized by
76
UV-Vis were approximately 150 nm in thickness. Thicker films absorbed too much
intensity to accurately determine wavelength of maximum absorption, while thinner films
(< 75 nm) showed a strong wavelength of maximum absorption dependence on
thickness.72-73 This is thought to be due to thiophene π-orbitals energetically overlaping
with d-orbitals in the working electrode causing greater chain alignment close to the
surface of the electrode. To determine the wavelength of maximum absorption of slightly
overlapping peaks, the trace was first smoothed using a boxcar function followed by a
second-derivative analysis. For curves that were significantly overlapped, the PeakFit
Version 4 software was used to deconvolute the peaks to determine the peak wavelength
and the full width at half height of the peak.
3.4.12 Variable Angle Spectrographic Ellipsometry
Optical constants were determined with a J. A. Woollam Variable Angle
Spectrographic Ellipsometer (VASE) and the WVASE software suite. The VASE was
calibrated with a single polished side silicon disk with a 10 nm native oxide layer. The
sample films, on conducting silicon substrates, ranged from 500nm to 3500 nm in
thickness. Measurements of ψ and ρ were taken from 500 to 800 nm in 5 nm steps on
both the front side and back side of the substrate. The thickness ratio of the front side
film and the back side film was ~3:1. After collection, the data was analyzed using the
WVASE software. The psemi model was used to describe the optical behavior of the
sample. A roughness layer was included to account for non-uniformities in the thickness
of the film. This is the most general model currently available. Each selected parameter
77
was optimized with respect to the mean square error (MSE) value. The iteration process
continued until the MSE was minimized. The optical constants were then taken from the
minimized model.
3.4.13 X-ray Diffraction
WAXD patterns of samples were obtained using an imaging system (Rigaku, RAXIS-IV) with an 18 kW rotating anode X-ray generator. Cu Kα radiation of 0.154 nm
was used as the x-ray source. The x-ray beam was line focused and monochromatized
using a graphite crystal. The beam size was controlled by a divergence slit of 0.5°. The
diffraction peak positions and widths were calibrated with silicon crystals of known
crystal size in the high 2θ-angle region (> 15°) and silver behenate in the low 2θ-angle
region. A 30 min exposure time was required for a high-quality pattern. In WAXD
experiments, background scattering was subtracted from the sample scans. To get
enough material for a good diffraction pattern, a single film was folded upon itself to
increase the amount of material in the beam line.
78
CHAPTER IV
DEVELOPING HIGH REFRACTIVE INDEX POLY(THIOPHENE)
4.1 Introduction
Currently, no organics reported in literature have a sufficient refractive index (n)
to open a complete 3-D photonic band gap (PBG) with an FCC inverse opal structure.
This chapter focuses on filling that hole by developing of an organic material with a
sufficient refractive index to open a complete 3-D PBG with an FCC inverse opal
structure. First, characteristics of high refractive index dielectrics will be discussed.
From these, poly(thiophene) (PT) was identified as a potentially high refractive index
candidate. The rest of the chapter focuses on the development of a synthetic approach to
synthesize high n PT.
4.2 Molecular Design of High Refractive Index Dielectrics
The fabrication of 3-D photonic band gap (PBG) materials from an FCC inverse
opal structure requires materials with refractive indices greater than n = 2.8.35 In the case
of the optimized FCC inverse opal structure, that threshold is reduced to n = 2.5. In
79
chapter II section 2.4, it was shown that for dielectrics, the refractive index is controlled
by two factors, the number of oscillators per unit volume which can be understood as the
electron density and the polarizability of the oscillators. These dependencies are captured
in equation 4.1.
n = 1 + Nα (ω )
(4.1)
Here N is the electron density, and α is the polarizability of the individual
oscillators. The polarizabilty of the individual molecules is given by equation 4.2.
α (ω ) =
q e2
ε 0 me
∑ −ω
k
fk
2
+ iγ k ω + ω 02k
(4.2)
Where qe is the charge of and electron, me is the mass of the electron, ε0 is the dielectric
permittivity of free space, f is the oscillator strength, ω is the frequency of the electric
field, γ characterizes oscillator dampening, and ω0 is the natural resonance frequency of
the oscillator. The natural resonance frequency can be understood from the Newtonian
description of mass on a spring. It is assumed in Newtonian motion that force (F) is
equal to mass (m) times acceleration (a), and for a spring, force is equal to a spring
constant (k) times the displacement distance (x) as shown in equation 4.3.
F = ma = m
d 2x
= −kx
dt 2
80
(4.3)
Dividing equation 4.3 by mass gives equation 4.4.
F d 2x
k
= 2 =− x
m dt
m
(4.4)
It is known that the solution for the harmonic motion of a mass on a spring can be
described with equation 4.6.
x = A cos(ω 0 t + δ )
(4.5)
The amplitude (A) of the wave and the phase ( δ ) of the wave are determine by
the initial conditions, but without knowing the initial conditions, one can express the
acceleration experienced by the mass with equation 4.6.
d 2x
= −ω 02 A cos(ω 0 t + δ ) = −ω 02 x
2
dt
(4.6)
The right hand side of equation 4.4 and the right hand side of equation 4.6 can be
set equal giving equation 4.7.
− ω 02 x = −
k
x
m
(4.7)
Solving equation 4.7 results in the resonance frequency being given by the spring
constant divided by the effective mass on the spring.
81
ω0 =
k
m
(4.8)
Plugging equation 4.8 back into the molecular polarizibility expression 4.2, one
gets equation 4.9.
q e2
α (ω ) =
∑
ε 0m k
fk
2
⎛ k ⎞
⎟
− ω 2 + iγ k ω + ⎜⎜
⎟
⎝ m ⎠ 0k
(4.9)
In this arrangement, it is obvious that the lower the effective spring constant for a
particular oscillation mode the higher the molecular polarizability. Physically, this means
that the electrons comprising a molecule should only be weakly correlated to any
particular nucleus. In summary, high refractive index dielectrics will have high electron
density with very weak correlation between the electrons and nuclei.
4.3 Candidates for High Refractive Index Dielectrics
Conjugated molecules have both characteristics desired for good high refractive
index dielectrics. The delocalized nature of the π-electrons allows them to have high
polarizablitiy, while rigid back bones allow them to pack tightly for high densities. As
such, the ultimate refractive index values for members of this class of materials have
82
been predicted from semi-empirical calculations. Table 1 is a list of refractive values
over a range of wavelengths for several conjugated polymers.1
Table 4.1: Predicted Refractive Index Values for Conjugated Polymers1
Polymer
n700nm
n1064nm
n2500nm
2.47
2.44
trans-Poly(acetylene)
Poly(para-phenylene vinylene)
2.28
2.04
1.95
Poly(para-phenylene)
2.13
1.97
1.89
Poly(thiophene)
3.90
3.04
2.77
Poly(thiophene vinylene)
3.56
2.66
2.42
Of these conjugated polymers, only PT and poly(thiophene vinylene) are
predicted to have a sufficient refractive index. Actual refractive index values for a subset
of the above listed polymers can be found in Table 2.
Table 4.2: Actual Refractive Index Values for Conjugated Polymers
Polymer
npred.
nexp.
trans-Poly(acetylene)
2.442.5 µm
2.33∞74
Poly(para-phenylene vinylene)
2.28700 nm
2.09633 nm75
Poly(thiophene)
3.9700 nm
1.4633 nm2
For trans-poly(acetylene) (PA) and poly(para-phenylene vinylene) (PPV), the
predicted refractive index values are relatively close to the actual measured refractive
index values. For the case of electrosynthesized PT, which is the only way to grow a
thick PT film through a photonic template, the predicted value is dramatically higher than
83
the actually achieved value. This discrepancy could be due to inaccuracies in the
formulation of the calculation, but the predicted and actual values for PA and PPV are
similar. Additionally, for six unit PT oligomers synthesized through a coupling reaction,
the measured refractive index was n = 2.15633 nm.76 These facts taken together point to an
issue involving the electrosynthesis of PT.
4.4 Synthetic Techniques for Poly(thiophene)
Several techniques have been developed to synthesize PT. One method was the
classic Grignard coupling route.77 Figure 4.1 shows the chemical formula for a Grignard
type of PT synthesis.
Br
S
Br
Mg/THF
Ni(bipy)Cl2
S
S
n
Figure 4.1: Chemical formula for the Grignard coupling synthesis of PT.
The issue with this approach is that it is a homogenous reaction. This means that
the polymer and the monomer need to be in the same phase. As the degree of
polymerization (DP) approaches six it is very difficult to keep the polymer in solution.
By the time a DP of ten is reached, the underivatized polymer will not stay in solution
independent of the solvent used. The result is high quality, but low molecular weight
powders that because they are infusible and insoluble, are not useful for the fabrication of
PBG materials.
84
Another technique is to use oxidative coupling.78 This approach uses a
homogenous solution of monomer and oxidizing agent. Figure 4.2 shows the chemical
formula for the oxidative coupling synthesis of PT.
FeCl3
S
S
CHCl3
S
x+
n
Figure 4.2: Chemical formula for the oxidative coupling synthesis of PT.
The DP of oxidative coupling is limited for the same reason as the Grignard
coupling approach. As the DP of PT approaches 10 the polymer fails out solution
effectively removing it from the chemical reaction and limiting the molecular weight.
The resultant polymer is in a powder form and is not useful for this application.
Still another approach is the cathodic electrochemical reduction route.79 This
approach has several benefits over the previous approaches. It is a heterogeneous
polymerization, such that even as the polymer falls out of solution, the polymerization
can continue. Figure 4.3 shows the chemical formula for the cathodic reduction synthesis
of PT. This route yields polymer chains with much higher DP then the coupling reactions.
The problem with the cathodic approach is that the resultant polymer is in the insulating
state.
85
Br
Br
S
+ 2ne
Ni (PPh3)Br2
S
S
n
Figure 4.3 Chemical formula for the cathodic reduction synthesis of PT.
In the insulating state, the charge can not diffuse through the film thus limiting the
thickness of the film. At a practical level, although the quality of the film is very high,
the film thickness is limited to about 10 nm. This is significantly thinner than what is
required to fabricate a PBG material.
The fourth approach is electrochemical oxidative polymerization.80 This route,
similar to the cathodic route, is heterogeneous. Figure 4.4 shows the chemical formula for
the oxidative electrochemical synthesis of PT. As a result, relatively high DP polymer
chains can be synthesized.
-
2n
S
(2+x)ne
S
S
n
x+
+ 2nH2
Figure 4.4: Chemical formula of oxidative electrochemical synthesis of PT.
Additionally, this route results in PT that is doped therefore conductive. This
allows charge to transport through the film. Consequentially, the films can grow much
thicker, on the order of tens of microns. This combination of being able to make high DP
chains in the form of thick films makes oxidative electrochemical polymerization the
synthetic route of choice for this application.
86
4.5 Optimization of the Oxidative Electrochemical Polymerization of Poly(thiophene)
The mechanism of the oxidative electrochemical polymerization is mostly settled
upon, but due to difficulty of isolating intermediate species, there is still some
disagreement. This mechanism is illustrated in Figure 4.5.80 The initiation step is the
removal of an electron resulting in a cationic free radical. Because the sulfur atom can
stabilize the radical, the free radical spends more time at the α-carbon than the β-carbon.
This leads to preferential α- α linkages which promote conjugation.81,82 The initiation of
the monomer requires 1.6 V. The activated species diffuse along the surface until
happening upon another species. If it is a non-activated species, the activated species
may transfer the reactive center to the other species. If it is another activated species,
they can couple through the ejection of two protons. This new species can then be reactivated to react again. Through this process the molecular weight of the polymer is
built-up.
The issue confronting the oxidative electrosynthesis of PT is that the degradation
potential of PT is 1.4 V while the monomer initiation potential is 1.6 V.83-85 Therefore, as
the monomer is being activated to continue polymerization, the polymer is being
degraded. As a result, the films have extremely poor physical and electrical properties.
This is referred to as the “Polythiophene Paradox.”
It has been found that the activation voltage of the higher degree of
polymerization (DP) species is less than lower molecular weight species. For example,
the monomer oxidation voltage is 1.6 V while the dimer oxidation voltage is 1.2V and the
trimer oxidation voltage is 1.1V as measured against Ag/AgCl in acetonitrile.80
87
Figure 4.5: A proposed mechanism for the oxidative electrochemical synthesis of PT.
It has been suggested that starting with dimers and trimers, which have an
activation voltage less then the degradation potential of the polymer chain, may be a
pathway to high quality, high molecular weight PT. This approach has one issue. The
presence of the sulfur atom stabilizes the free radical on the α carbon in the thiophene
monomer ring. Because of this stabilization the free radical has been found to spend 95%
of the time at this location and the remaining 5% at the β carbon.81 As a consequence, for
two initiated monomer units reacting there is a 94.75% chance of an α-α linkage, a 4.99%
chance of an α-β linkage, and a 0.26% chance of a β-β linkage. As the length of the
chain increases the energy difference between the electron being on the α or β becomes
88
smaller and eventually becomes equivalent around a DP of 6 at room temperature.86,87
As a consequence, an initiated monomer unit reacting with a thiophene hexamer there is a
90.48% chance of an α-α linkage, a 4.76% chance of an α-β linkage, and a 4.76% chance
of a β-β linkage. For two initiated thiophene hexamer units reacting there is a 33.33%
chance of an α-α linkage, a 33.33% chance of an α-β linkage, and a 33.33% chance of a
β-β linkage. Since it is desirable to increase the conjugation length through α-α linkages,
this analysis indicates that the highest level of conjugation can be achieved by initiated
monomer units reacting with growing chains and not by growing chains reacting with
each other.88 This is why films from thiophene monomer, despite requiring higher
initiation voltages, have better electrical properties than those from bithiophene and
terthiophene monomers.89
It is known that activation potential of the monomer can be reduced if the cationic
character of the free radical is stabilized. It is also known that a cationic charge can be
stabilized by an environment with an anionic character. A good candidate for this is a
strong aprotic Lewis acid. The aprotic nature is essential to preventing the hydrogen
from disassociating and reacting with and saturating the monomer. This would cut the
conjugation in the chain and reduce the electrical properties of the resultant polymer.
4.5.1 Solvent Determination
It was found that Boron Trifluoride Diethyl Etherate (BFEE) could be a good
contender. Figure 4.6 is experimental data from literature supporting this.83 The y-axis is
current and the x-axis is voltage. The three lines indicate three different solvent mixtures.
89
The first is the standard acetonitrile. The second is the standard acetonitrile with an
aluminum trichloride supporting salt. It is known that the supporting salts can reduce the
voltage required for polymerization. The third is the BFEE solvent. Polymerization is
indicated by the production of current at a particular voltage as the electrons are taken
from the monomer.
C
Ct = 0.1 M
A: CH3CN
B: 3M AlCl3/CH3CN
C: BF3•Et2O (BFEE)
B
A
Figure 4.6: A graph indicating that the PT oxidative synthesis initiation voltage
for acetonitrile was 1.6V while for BFEE it was 1.3V. Reprinted from ref 83
Copyright 1997 American Chemical Society. 83
It can be seen that in acetonitrile that the polymerization does not start until 1.6V
which is significantly higher than the 1.4V PT degradation potential.83-85 The presence of
a supporting salt appreciably reduced the voltage required to initiate polymerization, but
at 1.4V the current was very low. This indicates that there would only be a very narrow
range of reaction conditions, particularly the charge collection rate, that would result in
higher quality films. In the case of BFEE solvent, significant current could be generated
90
below 1.4V enabling the possibility of high quality PT synthesized using a range of
reaction conditions.
4.5.2 Optimization of Monomer Concentration in BFEE
Dr. Shi Jin optimized the monomer concentration in the BFEE with respect to the
degree of polymerization (DP) and conjugation length. The larger the DP, the longer the
possible conjugation length in the polymer chain and the higher the density due to fewer
chain ends. Both of these will improve the refractive index of PT. Films were
synthesized in BFEE at room temperature at a charge collection rate of 1 mA/cm2 with
monomer concentrations ranging from 25 mmole to 2000 mmole. Because the resultant
polymer chains are insoluble, standard gel permeation chromatography could not be used
to characterize the relative DP of the chains. Instead Fourier transform infrared (FTIR)
spectroscopy was used. One can compare the area under the absorption peak located at
790 cm-1 for hydrogen attached to the β position to the area under the peak at 700 cm-1
associated with the hydrogen attached to the α position. Only the two thiophene units
that are at the ends of the PT chains or that have been subject to an α-β linkage will have
a α-hydrogen, while each end unit and monomer unit in the chain will have two βhydrogens. Assuming that the number of α-β linkages is negligible, which will be
checked later, and assuming that the absorptivity of the α and β hydrogens are the same,
dividing the area of the β hydrogen peak by the area of the α hydrogen peak will give the
DP of the PT chains.
91
DP =
Aβ
(4.10)
Aα
Analysis of the PT films showed that a maximum DP of ~70 was achieved at a
monomer concentration of 50 mmole, as seen in figure 4.7. The higher degrees of
polymerization and reduced degradation allowed for π-π interaction between chains
which translated into dramatically improved physical properties with tensile strengths of
135 MPa.90,91 This is greater than aluminum. PT synthesized from conventional solvents
was very brittle as a consequence of the material consisting of relatively short chains (DP
~30) with poor interchain interaction due to degradation from polymerization.
Degree of Polymerization
80
70
60
50
40
30
20
10
0
0
500
1000
1500
2000
Monomer Concentration (mmole)
Figure 4.7: The dependence of PT DP on thiophene concentration in BFEE.92
Since the conductivity of a film is related to the conjugation length of the chains,
it can be used to make inferences about the polarizability of the PT chains. The higher
the conductivity, the longer the conjugation length actually achieved.
92
Absolute
conductivity measurements of the films were taken to investigate the electrical quality of
the films as a function of monomer concentration.
-1
Conductivity (Scm)
100
80
60
40
20
0
0.0
0.5
1.0
1.5
2.0
Concentration (M)
Figure 4.8: The absolute conductivity of PT synthesized at room temperature in
BFEE as a function of thiophene concentration.92
In figure 4.8, one can see that the maximum conductivity of the PT films, which
occurred at a monomer concentration of 50 mmole, was ~100 Scm-1. This was lower
than expected. Regioregular 3-alkylthiophene has been reported to achieve
conductivities up to ~2000 Scm-1.72 Reduced interchain contacts due to the interference
of the alkyl chains should lower the conductivity of derivatized PT with respect to the
unmodified polymer. This indicated that there was some complicating factor with the PT
synthesized from the BFEE solvent that was degrading the electrical quality of the films.
93
0.6
Saturated Units
Absorbance
0.4
3200
3000
Wavenumber (cm-1)
2800
0.2
0.0
3500
3000
2500
2000
1500
1000
500
-1
Wavenumber (cm )
Figure 4.9: FTIR spectrum of a PT film polymerized in BFEE solvent with a
thiophene concentration of 50 mmole.
To investigate the origin of the electrical deficiency, FTIR spectroscopy was used
to determine the chemical composition of the synthesized PT film. Figure 4.9 shows the
resultant spectra. The peak at 790 cm-1 corresponds to the β attached hydrogens. The
absence of peaks at 820 cm-1 and 730 cm-1 indicates that there are few α-β linkages. The
key region is in the inset between 3200 cm-1 and 2800 cm-1. There are three bands at
2956, 2925, and 2853 cm-1, respectively. These correspond to the modes from saturated
thiophene units being incorporated into the chain.
Figure 4.10: A structural depiction of a saturated unit in a PT trimer.
94
Saturated thiophene units hurt the refractive index of PT in two ways. First, the
saturated unit is unconjugated, thereby, cutting the conjugation length and polarizability
of the molecule (see figure 4.10). Secondly, the saturated unit is non-planar. It’s
incorporation into the chain introduces a kink in the chain which effects the packing
density.
It was speculated that these saturated units were the source of the reduced
conductivity and polarizability. The question became, what is the origin of the saturated
units. FTIR analysis of freshly distilled monomer indicated negligible concentrations of
saturated units, pointing to saturation occurring during the reaction. BFEE is an aprotic
acid, so it was ruled out as a source of acidic protons.
BF3 ·Et2O + H2O → HBF3(OH) + Et2O
Figure 4.11: The formula for water reacting with BFEE.
As part of the reaction mechanism, it is known that acidic protons are released.
Additionally, the solvent would promote the dissociation of an acidic species including
water. It is known that moisture from the atmosphere can react with the solvent to form
an acidic species as seen in figure 4.11. A proton from the resultant acid species can be
released into the reaction solution and saturate the monomer.
95
4.5.3 Optimization of Proton Trap Concentration
To sequester protons released during polymerization and from BFEE reacting
with atmospheric moisture, a proton trap can be added to reaction solution. Cationic
syntheses often use this type of additive, so several have been developed. Proton traps
generally consist of a pyridine or pyrimidine core with tert-butyl groups on either side of
the nitrogen atom as seen in figure 4.12. The pair of electrons on the nitrogen atom are
basic, so they can capture the acidic protons while the tert-butyl groups sterically prevent
the nitrogen from reacting with anything else.
N
(a)
(b)
(c)
Figure 4.12: Structural depictions of (a) DTTP, (b) TTTP, and (c) TTTM.
For the purposes of this research, three specific proton traps were evaluated 2,6
di-tert-buytlpyridine (DTTP), 2,4,6, tri-tert-butylpyridine (TTTP), and 2,4,6 tri-tertbutylpyrimidine (TTTM)).93 The optimal proton trap concentration was determined by
comparing the relative conductivities of PT synthesized using a range of proton trap
concentrations at room temperature, at a charge collection rate of 1 mA/cm2, and a
monomer concentration of 50 mmole. DTTP was found to work best at a 50 mmole
96
concentration, but was highly unstable and would react quickly with atmospheric
moisture.92 As a result, it lost its efficacy quickly. It was also a liquid at room
temperature which made exact weight measurements problematic. TTTP was found to
work best at a 10 mmole concentration. It was found to be more stable than DTTP, and
was a solid at room temperature. TTTM was found to work best at a 10 mmole
concentration. It also was more stable than DTTP, and a solid at room temperature. For
reasons of efficacy and availability, TTTP was chosen as the proton trap that will be used
for the rest of research presented here. The graph showing this optimization is shown in
figure 4.13.
97
)
Conductivity (Scm
-1
200
150
100
50
0
0
40
80
Concentration (mmol)
120
Figure 4.13: The relative conductivity of PT synthesized at room temperature in
BFEE with a thiophene concentration of 50 mmole as a function of proton trap.
4.5.4 Polymerization of Poly(thiophene) at Optimized Monomer and Proton Trap
Concentrations
PT films where then synthesized using the optimized monomer and proton trap
concentrations. The reaction was carried out at room temperature, with a charge
collection rate of 1 mA/cm2. FTIR spectroscopy conducted on the film is shown in figure
4.14. The intensity of the three bands at 2956, 2925, and 2853 cm-1 is significantly lower
in the films synthesized with the proton trap. This indicates that fewer saturated units are
being incorporated into the chain. This should translate into longer conjugation length
and better packing. Both of which should improve the refractive index of the PT film.
98
w/o Proton Trap
w Proton Trap
3200
3000
Wavenumber (cm-1)
2800
Figure 4.14: FTIR spectra of PT films synthesized at room temperature in BFEE
and 50 mmole thiophene with and without a proton trap.
Wide angle x-ray diffraction (WAXD) was conducted on the PT film. The PT
films synthesized without the proton trap exhibited two diffuse halos at 18° and 27°
degrees as seen in figure 4.16. These two halos correspond to the zipper packing and π-π
stacking of the chain respectively.94 With the proton trap, the halos sharpened indicating
significantly improved packing. This is attributed to the reduced number of kinks in the
chain caused by saturated units which translated into improved density. Without the
proton trap, the PT film density was measured to be ρ = 1.495 g/cm3. With the proton
trap, the PT film density was ρ = 1.512 g/cm3. Again, this increase in density is
attributed to the reduced number of kinked saturated units being incorporated into the PT
chain.
99
w/ Proton Trap
w/o Proton Trap
0.5 nm
0.35 nm
S
S
S
S
S
S
S
S
S
0.5 nm
S
S
S
0.35 nm
10
15
20 25
2 θ (°)
30
35
Figure 4.15: (a) WAXD of the PT film synthesized at room temperature in BFEE
and 50 mmole thiophene with and without the proton trap. (b) Characteristic
packings of PT chains in the film.
Taken together, the improved chain regularity and reduced cuts in conjugation
due to saturated units should translate into increased conjugation length. Conjugation
length can be directly probed using Ultra-violet-Visible (UV-Vis) transmission
spectroscopy. The major absorption peak of dedoped conjugated films is related to the
electronic band gap between the valence and conduction band. The average energy
associated with the electronic band gap is inversely proportional to the average
conjugation length of the chain.88,95,96 So, the longer the conjugation length, the longer
the wavelength of maximum absorption (WMA) for the electronic band gap transition.
The WMA for PT films synthesized in BFEE without the proton trap was 490 nm. When
the optimized concentration of proton trap is added to the reaction solution, the WMA for
PT was 501nm. This is a 11 nm red-shift, as seen in figure 4.16, which is indicative of
increased conjugation.
100
---490
---501
w/o Proton Trap
w Proton Trap
400 450 500 550 600
Wavelength (nm)
Figure 4.16: The WMA of PT films synthesized at room temperature in BFEE and
50 mmole thiophene with and without the proton trap.
So the combination of BFEE solvent with optimized monomer concentration and
proton trap concentration resulted in PT films with higher density and polarizability.
Consequently, the refractive index of these film should be substantially higher than that
previously reported for PT. Variable angle spectrographic ellipsometry in reflection
mode was used to characterize the optical behavior of PT synthesized on a conducting
silicon substrate. Knowing that the highest refractive index values would be from the
abnormal refractive index dispersion at wavelengths slightly higher than the WMA, the
measurements were concentrated between 500 nm and 700 nm. The resultant data was
then analyzed using the psemi model, which was the most general model available. The
electronic band gap was fit as a Gaussian curve.
101
n (Real Refractive Index)
3
n
k
2
2.5
1.5
2
1.5
1
1
0.5
0.5
0
500
550
600
Wavelength (nm)
650
k (Imaginary Refractive Index)
2.5
3.5
0
700
Figure 4.17: Refractive index dispersion curve for the PT optimized at room
temperature.
After analysis, the optical constants of the model describing the optical behavior
of the material were extracted. In figure 4.17, the real (n) and imaginary (k) refractive
index of the PT film are plotted against wavelength. For the film measured, the WMA
was at 500 nm as seen in the values of k. The absorption in the material has significantly
dropped off by 600 nm and was close to 0 at 700 nm. The n values followed the expected
abnormal dispersion. The n reached its maximum value of 3.15 at approximately 575 nm
and fell to approximately 2.25 by 700 nm. The maximum value of n = 3.15 is the highest
n value reported for an organic molecule and is high enough for an organic molecule to
open a complete 3-D PBG in an inverse FCC opal structure. In fact, there is a 45 nm
102
range between 530 nm and 596 nm which the n value is sufficient to open a complete 3-D
PBG.
To ensure that the refractive index was not influenced by the presence of any
other organic that may have been contaminated the reaction or formed during the
synthesis 13C solid state NMR was conducted on the high n PT film. The samples were
spun in nitrogen gas at 4.5 kHz at the magic angle. The contact time was 2 ms, and each
spectrum consisted of an accumulation of 5000 scans. The temperature of the solid state
13
C-NMR experiment was controlled to be 25°C using a REX-F900 VT unit. The films
were cut into a powder with a total mass of ~100 mg.
1000
900
800
Intensity (A.U.)
700
600
500
400
300
200
100
0
60
80
100
120
ppm
140
160
180
200
Figure 4.18: 13C-NMR results for PT polymerized at conditions optimized for
room temperature.
103
The resultant spectra, seen in figure 4.18, showed only two peaks, one at 125.5
ppm and the other at 136.4 ppm. These values are the same as those found in literature
corresponding to the β and α carbons, respectively.97,98 The presence of no other peaks
indicates that only PT is present in the high n samples in any appreciable quantities.
4.5.5 Optimization of Poly(thiophene) Synthesis Reaction Temperature
The next step was to try and improve both the highest n value for the PT film and
to increase the range of wavelengths in which a complete 3-D PBG can be opened in an
inverse FCC opal. It is known that cationic radical polymerizations result in more regular
chains at lower temperatures, so the synthesis of PT was reoptimized at for -50°C.80 This
temperature was chosen because the BFEE solvent freezes at -58°C. For this
optimization, the monomer concentrations ranged from 10 mmole to 250 mmole, while
the charge collection rate was varied from 0.05 mA/cm2 to 1 mA/cm2. The proton trap
concentration was held constant for two reasons. First, the reaction mechanism should
release the same number of protons independent of temperature and the protons produced
by the solvent reacting with ambient water should result in the same number of protons
independent of temperature. So, the number of acidic protons in the solution should be
constant independent of temperature. Second, the proton trap reaction is an acid-base
reaction with very low energetic barriers, so whose efficiency should not be strongly
affected by the reaction temperature. Taken together, the same number of protons and
same proton trap efficiency, the proton trap concentration trap should remain the same.
The resultant films were then characterized by their WMA. This was chosen at the
104
central optimization parameter instead of conductivity because of its stability as a
function of time. The dopant level strongly effects the conductivity of a conjugated
organic. Although all PT synthesized using electropolymerization start with similar
doping levels, the dopant concentration spontaneously decays exponentially as a function
of time. The constants associated with this exponential decay vary strongly as a function
of film structure. Therefore, the conductivity measurements are not solely a function of
conjugation length, but also sensitively on the time after synthesis at which the
measurement was taken and less so on the structure of the film. This complicates
optimization with respect to conductivity. The WMA is measured after dedoping,
thereby, eliminating the time variable from the measurement making it a more reliable
parameter to optimize to. Additionally, since conjugation length and structure are linked
by the same causality, incorporation of saturated units, this one measurement should give
a good indication of the optical quality of the film. The results of the optimization matrix
are presented in figure 4.19. For clarity only the 10, 50, and 150 mmole concentrations
have been presented. These examples capture the best and worst cases with a transition
concentration.
105
530
Temperature ~ -50°C
[Proton Trap] ~ 10 mmole
Wavelength (nm)
520
[Monomer]
10 mmole
510
50 mmole
500
150 mmole
490
480
470
460
0
0.2
0.4
0.6
0.8
2
1
Current Density (mA/cm )
Figure 4.19: The optimization of monomer concentration and current density for
the synthesis of PT at -50°C with respect to the WMA.
It is clear from figure 4.19 that at lower temperatures the conjugation length is
maximized at lower monomer concentrations. This is thought to be a function of the
reduced diffusion rate at the electrode surface at lower temperatures. After the electrode
activates the monomer by taking an electron, the monomer diffuses until it can react with
another activated species. It is preferable that the synthesis of PT follow a chain growth
type build up of molecular weight where an activated monomer reacts with an activated
higher molecular weight species. This promotes longer stretches of α-α linkages to the
exclusion of α-β and β-β linkages. At temperatures near the freezing point of a solvent,
the diffusion rate drops dramatically, particularly for high molecular weight species,
106
increasing the time needed for species to diffuse to the electrode for activation. As such,
monomer units are preferentially activated. With few activated higher molecular weight
species, monomers and low molecular weight species will preferentially react with each
other. This results in a step growth type of molecular weight build up which will result in
more α-β linkages. By decreasing the charge collection rate, one can allow more time for
the higher molecular weight species to be activated by the electrode. This promotes a
more appropriate ratio of high and low molecular weight species being activated. If the
charge collection rate is too slow, the voltage required to carry out the reaction at the
specified rate will drop, preferentially activating the higher molecular weight species
which will react with each other resulting in more α-β linkages. From figure 4.19, one
can see that the optimal balance of monomer concentration and charge collection rate is
10 mmole thiophene at 0.5 mA/cm2. The resultant WMA was at 521 nm. This is a total
31 nm red-shift as seen in figure 4.20. This indicates that lower temperatures have
resulted in polymers with significantly higher levels of conjugation than at room
temperature.
107
---490
---501
--521
-521
w/o Proton Trap
w Proton Trap
Low T w Proton Trap
400 450 500 550 600
Wavelength (nm)
Figure 4.20: The WMA of PT films with and without a proton trap and with a
proton trap at -50°C.
To determine if the lower temperatures resulted in a more regular electronic
environment, the full width at half height (FWHH) of the absorption curves was
determined using the Peakfit software. The thinner the FWHH of an absorption peak the
more regular the electronic structure. The absorption peaks were fit using Gaussian
curves. The results can be found in figure 4.21. Again for clarity, only the 10, 50, and
150 mmole concentrations have been presented.
108
Temperature ~ -50°C
[Proton Trap] ~ 10 mmole
[Monomer]
10 mmol
F W H H (n m )
260
240
50 mmol
220
150 mmol
200
180
160
0
0.2
0.4
0.6
0.8
1
2
Current Density (mA/cm )
Figure 4.21: The FWHH of PT films for a range of monomer concentration and
charge collection rates at -50°C.
It can be seen that the lower monomer concentrations had narrower FWHHs.
Additionally, the FWHH was found to be narrowest at 0.5 mA/cm2. These conditions
correspond to the conditions needed to maximize the WMA. The optimized FWHH was
164 nm. This is significantly narrower than the FWHH of 186 nm for the optimized
conditions at room temperature. In summary, the low temperature optimization of the
monomer concentration and charge collection rate has resulted in PT films that are
significantly more conjugated and regular than equivalently optimized PT films
synthesized at room temperature. The refractive index of the low temperature PT was
then measured.
109
0.50
n
k
3.00
0.40
2.50
0.30
2.00
0.20
1.50
1.00
500
600
700
800
Wavelength (nm)
900
k (Imaginary Refractive Index)
n (Real Refractive Index)
3.50
0.10
1000
Figure 4.22: Refractive index dispersion curve for PT optimized at -50°C.
Figure 4.22 shows the refractive index dispersion of the low temperature PT film
analyzed from 500 nm to 700 nm using an ellipsometer around the absorption band, and
by a fringe interference pattern technique in the transparent region between 700 nm and
1000 nm. The fringe method is described in detail in Appendix I. It can be seen that the
maximum refractive index was n = 3.36. This is considerably higher than the n = 3.15
previously achieved. Additionally, the wavelength range where a complete 3-D PBG
FCC inverse opal can be opened expanded to between 500 nm and 635 nm. This is a 135
nm range. It is also noted that in the case of the low temperature PT, the k values are
lower than those observed for the room temperature PT despite the higher refractive
index. This is because the off-resonance refractive index is significantly higher for the
low temperature PT (n = 2.6) than for the room temperature PT (n = 2.25), so the
110
oscillator strength does not have to be as strong to achieve higher maximum refractive
indices. The higher off-resonance refractive index is the product of increased
polarizability. The low temperature k value is at a maximum at 510 nm indicating that for
the film measured, the WMA is red-shifted 10 nm from the room temperature PT. It is
noted at this point that the off-resonance refractive value of n = 2.6 is sufficient to open a
complete 3-D PBG for an optimized FCC inverse opal structure to be discussed in
Chapter 5.
4.5.6 Post-Synthesis Thermal Annealing
The heterogenous nature of the electrosynthesis of PT means that, although we
can improve chain packing through reaction conditions, the chains are far from the ideal
conformation. Conventional methods of improving chain conformation include solvent
or melt processing. It is well known that PT is insoluble, so solution processing of
underivatized PT is impossible. Additionally, PT thermally degrades before softening
eliminating any possibility of melt processing. The question became is there any postsynthetic processing which could improve the orientation of the thiophene units within
the PT chains. Gas phase calculations of PT chains indicate that the individual thiophene
units in the chain can rock without much hinderance through an angle of ~60° with
respect to one another and have an equilibrium angle of ~30°.82 Now, any angle outside
of co-planarity of the thiophene units reduces conjugation by equation 4.11.
η actual = η potiential cos(θ )
111
(4.11)
Here η is the level of conjugation and θ is the angle between ideally co-planar
units. The energy required for rocking the thiophene units is appreciably less then the
energy needed for any translational motion, such that the thermal energy required to
activate the rocking motion to allow the packing energies associated with neighboring
chains to force the thiophene units into a more favorable co-planar orientation to could be
below the degradation temperature of PT (~400°C in nitrogen). To test this, a PT film
with few saturated units and α-β linkages, but a low WMA was identified. The low
WMA, absent saturated units and α-β linkages, was speculated to be caused by poor
chain conformation. After the initial UV-Vis measurement, the PT film was placed in an
evacuated vacuum oven at 235°C. The film was allowed to thermally anneal for 120
hours at that temperature. Periodically UV-Vis data was collected to track any change in
WMA as a function of time. The absorption peaks were then analyzed to get the WMA.
During analysis, it was determined that there was a fine structure peak on the shoulder of
the electronic band gap peak. The presence of this fine structure has been noted in
literature.72 The WMA of the fine structure peak was also determined as a function of
annealing time. The results of this experiment have been plotted in figure 4.23.
112
520
515
580
510
575
505
570
Peak 1
Peak 2
500
565
Peak Wavelength (nm)
Peak Wavelength (nm)
585
T ~ 235°C
495
0
20
40
60
80
100
120
560
140
Annealing Time (hr)
Figure 4.23: The red-shift of the WMA during thermal annealing as a function of time.
Initially, the WMAs for the two peaks started at 500 nm and 578 nm. Upon
annealing the WMAs both quickly red-shifted and seemed to hold steady at values of 512
nm and 586 nm ,respectively. The electronic bang gap peak red-shifted 12 nm while the
fine structure peak red-shifted 8 nm. It is not known why one peak shifted further than
the other, but the presence of the shift upon annealing indicates that there was some
rearrangement in chain conformation to increase the average conjugation length in the
chains. Similar temperature affects have been observed by others in XRD experiments.94
113
4.6 Summary
It was determined that a high refractive index dielectric would have highly
delocalized electrons and relatively heavy atoms. This pointed to PT as a good candidate
to have a sufficiently high refractive index to open a complete 3-D PBG in a FCC inverse
opal structure. Literature indicated that PT did not exhibit the refractive index values
predicted by semi-empirical calculations. This opened up the possibility of an improved
route to the synthesis of PT. To address the “PT Paradox”, BFEE solvent was used. The
resultant saturation of monomer units by acidic protons was dealt with by introducing a
proton trap. The monomer concentration, proton trap concentration, charge collection
rate, and reaction temperature were optimized with respect to conjugation length. The
subsequent PT films had dramatically improved conjugation length as well as refractive
index values unprecedented for organics and sufficient to open a complete 3-D PBG in a
FCC inverse opal structure.
114
CHAPTER V
GROWING THE COLLOIDAL TEMPLATE
5.1 Introduction
To fabricate a photonic band gap (PBG) material, one needs to create a structure
on the length-scale of light that you want to be subject to the PBG. The desired
dimensionality of the photonic band gap dictates the type and dimensionality of the
periodicity of the structure. The most commonly used method to create 3-D PBG
templates for visible light is the colloidal self-assembly of monodisperse spheres whose
diameters are on the length-scale of light.64-68 This approach lends itself to this project
well. As long as the template substrate is relatively flat, the exact substrate material is
not critical. This allows templates to be grown on the working electrode, and then high
refractive index poly(thiophene) can be grown through the template. The template can
then be removed leaving behind a negative copy of the template or an inverse opal. To
accomplish this, a particular self-assembly method was chosen. Next, the exact structure
of the template was determined. Then process control was established to achieve the
115
target template thickness. Finally, with the structure determined and thickness fixed,
post-processing techniques will be discussed to optimize the template structure.
5.2 Colloidal Self-Assembly
Monodisperse spheres are known to spontaneously self-assemble into ordered
aggregates above a threshold concentration. These aggregates form either ABCABC or
ABAB hexagonal close packed structures. Both have a packing fraction of 0.7405 and a
coordination number of 12. The ABCABC structure is face centered cubic (FCC) which
has a Fm3m space group, while the ABAB hexagonal close packed (HCP) structure has a
P63/mmc space group.100,101 The difference between these two structures is graphically
represented in figure 5.1.
Figure 5.1: (a) The ABC stacking of spheres for a FCC structure. (b) The ABA stacking
of sphere for a HCP structure.
For the purposes of fabricating a photonic crystal, these two structures are not
equivalent, as seen in the band calculations for the two structures in figure 5.2. The PBG
116
of the FCC structure occurs between the 8th and 9th bands.35 The PBG of the HCP
structure occurs between the 17th and 18th bands.35 Since the instability of a PBG with
respect to disorder increases as the band numbers bounding the PBG increase, the FCC
structure PBG is favorably more stable than the HCP structure. Additionally, the FCC
structure requires a refractive index contrast of ~2.8 and 9 unit cells to open up a
complete PBG.35 The HCP structure requires a refractive index contrast of 3.1 and 17
units to open a PBG.35 Despite the difference in structure between the FCC and HCP
structures, the differences in the bands where the PBGs occur allow the PBG wavelengths
to overlap. Of the two, the lower refractive index contrast threshold, fewer unit cells, and
stability with respect to disorder makes the FCC structure preferred over the HCP
structure. Fortunately, the structures are not energetically equivalent, and the FCC
structure has been found via single occupied cell calculations to be slightly more stable
(∆G = 0.005kBT) than the HCP structure, but because the energy difference is small,
thermodynamically one would expect an approximately 50/50 mixture of the two
structures.104 Consequently, to create large single crystals of the FCC structure, one
should use a kinetically driven method to construct the template where the FCC is
kinetically preferred.105-107
117
(a)
(b)
Figure 5.2 (a) Calculated photonic band structure of the FCC structure. (b)
Calculated photonic band structure of the HCP structure. Reprinted with
permission from ref 35. Copyright 1998 American Physical Society.35
118
The most common method is Colvin’s method.3 In this approach, the colloid is
suspended in a volatile carrier fluid. A substrate is placed vertically in the colloidal
solution. As the solvent evaporates, the meniscus is swept down the substrate. The
capillary forces in the meniscus pull the spheres together into a close packed
arrangement. The thickness of the colloidal film is determined by the volume fraction of
the spheres and the contact angle between the solvent and substrate. The greater the
volume fraction of spheres the thicker the resultant film was. Additionally, as the contact
angle increases from 0° to 90° the resultant template becomes thicker. The rate of
evaporation did not have an effect on the thickness of the film, but strongly influenced
the regularity of the film. Experimentally, it has been reported that this method results in
80% to 90% of the structure being FCC while 10% is HCP or random HCP. Simulations
of this process indicate that solvent flow because of evaporation is faster through pores
not-blocked by spheres directly behind them. This causes a greater pressure differential,
therefore promotes film growth, at pores that lead to FCC formation over HCP
formation.108 This kinetic factor leads to the favored formation of the FCC over the HCP
structure despite the very small energy difference.
5.3 Determining Colloidal Structure
We know using Colvin’s method, the monodisperse spheres can self-assemble
into a regular structure but it is not known which of the two structures, FCC or HCP, are
actually formed. Because these are periodic structures on the length scale of light, they
can cause Bragg diffraction. The surface of a FCC structured film exposes the (111) set
119
of planes. For the HCP structure, the surface of the film exposes the (101) set of planes.
The d-spacing of the FCC (111) and HCP (101) planes are different, so one piece of
evidence indicating which structure was constructed is the wavelength of light being
reflected from the surface. With this in mind, optical microscopy was used to image a
template of polystyrene (PS) spheres (n = 1.59) with diameter (d) = 290 nm, as shown in
figure 5.3.
120
(a)
(b)
Figure 5.3: (a) Transmission mode optical microscopy of a colloidal
template of 290 nm PS spheres. (b) Reflection mode optical microscopy of
a colloidal template of 290 nm PS spheres.
121
(a)
(b)
Figure 5.4: (a) Transmission mode optical microscopy of a colloidal
template fabricated from 290 nm PS spheres. (b) Reflection mode optical
microscopy of a colloidal template fabricated from 290 nm PS spheres.
122
Figure 5.3 is an image in of the template (a) in transmission mode and (b) in
reflection mode. In transmission, one sees only a very large cracked green area. In a
reflection image of the exact same area, the color is red. This indicates that most likely
only one structure is present in this set of images of a large area. Figure 5.4 shows two
images, one (a) in transmission mode, and the other (b) in refection mode, with a
different color. The two colors indicate that there are at least two structures present in the
film. The second structure only occupies a very small fraction of the area. It was
calculated that the [111] zone of a FCC structure would reflect λ = 691 nm while the
[101] zone of a HCP structure would reflect λ = 647 nm. So for this sphere size, both
phase structures in reflection would appear red, but different shades of red, and green in
transmission mode. This does not explain the orange color in transmission and green in
reflection. First, the majority phase (the green phase in transmission) will be determined.
Second, the structure of the minority phase (orange in transmission) will be determined.
AFM in height mode was used to characterize the sphere packing in the majority
phase. The resulting image can be seen in figure 5.5a. One finds the spheres in a very
regular close packed arrangement. A fast Fourier transform (FFT) of the surface, as seen
in figure 5.5b, confirms the obvious six-fold symmetry of the surface. This is just a 2-D
image, so there is no way to know from the AFM whether this is a FCC or HCP structure.
123
(a)
(b)
Figure 5.5: (a) AFM of the colloidal crystal fabricated from 269 nm diameter PS
spheres. (b) FFT of the AFM image.
124
To determine the 3-D structure, diffraction experiments are needed. There are
two distinct types of diffraction experiments. Standard optical diffraction utilizes a
collimated beam of light. Because of the strong interaction between light and material,
structural defects readily de-collimate laser beams. De-collimation of the beam smears
the spots and can ruin the interpretation of the intensity pattern in diffraction. This
phenomena is relatively unimportant at x-ray frequencies, except at very high defect
levels, due to the weak interaction with materials. A refractive index matching fluid was
used to minimize the light being scattered from any surface roughness or from defects.
The other type of diffraction experiment is Kossel line analysis.109 In this experiment, the
beam is divergent not collimated. At the angles that satisfy Bragg’s condition, light gets
reflected. Since the beam is divergent the result is reflection cones, not diffraction spots,
and often results in sharper features, since de-collimation from defects does not strongly
affect the interpretation. To determine the 3-D structure of the majority phase structure,
both standard optical diffraction and Kossel line analysis experiments were conducted
using a 633nm laser.
125
(a)
(b)
Figure 5.6: (a) Optical diffraction from a colloidal template fabricated from 269
nm diameter PS spheres. (b) Kossel line image from a colloidal template
fabricated from 269 nm diameter PS spheres.
126
Figure 5.6a is the optical diffraction pattern. One observes a six point star
corresponding to the six-fold symmetry of a close packed unit cell. The fact that there is
only one set of diffraction points indicates that the film is single crystalline in nature.
Scanning across the template, the orientation of the star points did not change. This
suggests that large areas of the template are single crystalline. Figure 5.6b is the Kossel
line image. The result was three arcs separated by 120° with a faint circle in the middle.
Each arc sweeps out approximately ~60°. This corresponds to what is observed
specifically for a FCC structure.110 It is interesting to note that only three and not six arcs
are observed. This indicates that the FCC structure is not twinned.
To determine if one or both structures are present, one can use an optical
microscope in reflection mode with a spectrometer attached. If there are two peaks then
both structures are present, and since the scattering power for both structures is similar,
the relative intensities of the two peaks will give insight into the prevalence of one
structure over the other. If one peak is present, the wavelength of maximum reflection
will indicate which structure it is. This experiment was carried out with a structure
constructed from d = 310 nm PS spheres.
127
Figure 5.7: Reflection UV-Vis spectroscopy of 310 nm diameter PS spheres.
As can be seen in figure 5.7, only one peak is present. This indicates that one
structure dominates the template. Using the sphere diameter (d = 310 nm) and the
refractive index of PS (n = 1.59) it was calculated that the [111] zone of a FCC structure
would reflect λ = 739 nm while the [101] zone of a HCP structure would reflect λ = 692
nm. From the spectra it can be seen that the peak reflected light occurs at a λ ~ 735 nm.
Considering that standard organic refractive indices are determined at λ = 589 nm and
that due to dispersion, the literature value of n = 5.9 is a little high, there is very good
agreement between the observed peak at 735 nm and the calculated peak at 739 nm. This
is a strong indication that the majority phase is a FCC structure. Taking the AFM, optical
diffraction, Kossel line analysis, and reflectometry together, one can assign the majority
phase has an FCC structure.
128
(a)
(b)
Figure 5.8: (a) AFM height image of the minority phase. (b) The FFT of the AFM
height image of the orange phase.
129
(a)
(b)
Figure 5.9: (a) The optical diffraction from the minority phase. (b) Reflection
spectrum for the minority phase.
130
The minority phase was then characterized using AFM. Figure 5.8a is the AFM
height image. Figure 5.8a shows that the spheres are not close packed and there does not
appear to be any large areas with orientational order. Even with no apparent lateral
correlation between the spheres, all the spheres are at basically the same height which
indicates that there is still layer-like stacking. Figure 5.8b is the FFT of the AFM image.
In the low angle region, one sees a halo. The halo, as opposed to spots, indicates that
despite some spatial regularity, there is no orientational correlation. At higher angles is a
second peak. This peak is not a halo indicating that at smaller length-scales there is some
positional and orientational order. It is thought that the meniscus flowing down the
substrate imparts some very local orientation. Although this is just a 2-D image it
suggests that there is no real crystalline order in the minority phase.
To get information on the 3-D structure, optical diffraction and reflectivity
measurements were made on the minority phase. First, templates with larger domains of
the minority phase were made by quickly evaporating the carrier solution. Figure 5.9a
shows the optical diffraction pattern from a 633nm laser incident normal to the minority
phase surface. The result is just a halo. There is no obvious orientation in the pattern. It
is known from a collection of random scatters that the angle where enhanced scattering
starts can be given by equation 5.1.
θ=
λ
2l *
131
(5.1)
It can be seen for a given wavelength (λ), that the scattering angle (θ) is a function
solely of the mean free scattering distance ( l * ) which in a condensed template is
associated with the sphere diameter. So it is thought that the halo is associated with the
size of the sphere diameter. These suggests that there is no regular structure throughout
the sample.
The reflectometry of the minority phase can be seen in figure 5.9b is from a
sample of 310 nm spheres, as opposed to the 290 nm spheres for the optical microscopy
or the 269nm spheres used for AFM. One sees enhanced reflection around 500 nm which
accounts for the green color, but it is diffuse not sharp indicating that it is not the result of
Bragg diffraction. This again suggests that there is no order in the sample. Despite the
apparent lack of order, one can see a fringe pattern. This is from the interference of light
reflecting from the top and the bottom of the template, but is not a function of order
within the film. The presence of the fringe pattern indicates that there is less then a 10%
variation in film thickness in the sampled area. More variation than this and the fringe
pattern disappears. The AFM, optical diffraction, and reflectometry of the minority
phase, all indicate that it is random non-close packed sphere phase.
It has been determined that Colvin’s method can be used to grow templates with
very large single crystalline FCC templates. A second phase occupies a small fraction of
the template. The secondary phase was found to be a random agglomeration of spheres.
So in conclusion, Colvin’s method is a good technique to making an FCC template.
132
5.4 Determining Conditions for Optimal Thickness:
Not only is the quality of the FCC structure important, but so is the thickness of
the template. PBG band gap calculations have shown that for an FCC structure to open a
complete PBG, one needs nine unit cells.35 As a practical point, one wants to have a
thick template to ensure that a block of nine perfect unit cells exists in the template, but
because the material needs to be polymerized through the template, one wants a relatively
thin template. As such, assembly conditions need to be determined to strike a
compromise between these two needs. The number of sphere layers deposited on the
substrate follows equation 5.2.3
k=
Lϕ
(5.2)
fd (1 − ϕ )
Here k is the number of layers, L is the meniscus height, ϕ is the volume fraction
of the spheres, d is the diameter of the spheres, and f is a constant particular to each
system. Given a sphere size, one can only control template thickness by varying the
volume fraction of spheres or the meniscus height. It is noted that the rate of solvent
evaporation does not affect the template thickness. To test this, a template was grown
from a solution of d = 300 nm PMMA spheres in an ethanol carrier and deposited onto a
substrate. The evaporation rate was controlled by temperature. The template was first
grown at 25°C, and half way down, the temperature was dropped to 0°C. After drying,
optical microscopy was used to image the morphological transition from fast to slow
growth. AFM was used to characterize sphere packing changes after the transition.
133
5 µm
(a)
(b)
Figure 5.10: (a) Image of fast to slow growth transition by optical microscopy.
(b) AFM image of the fast to slow growth transition.
134
Figure 5.10a shows a distinct morphological change with growth rate. The fast
growth rate region appears red in the image, with cracks running parallel to each other
and perpendicular to the growth front at a meniscus. This type of cracking is due to
stresses building up in the template during drying. The slow growth rate region appears
yellow, with cracks changing from parallel to a dried mud crackle pattern. Generally
speaking, the angle between intersecting cracks in the slow growth region was ~120°.
These observations point to a change in the packing of the spheres. AFM images were
made across the fast/slow growth transition. One of these images is seen in figure 5.10b.
A black line has been drawn where this transition took place. To maintain the same
image orientation, the images were made along the parallel cracks. It should be noticed
that the height of the template on either side of the line stayed constant. The quality of
sphere packing changed dramatically at the transition. In the fast growth area, the
spheres only have local order with no specific orientation. In the slow growth region, the
spheres pack much more regularly. The orientation in this region is uniform and
continues across the crack. These observations explain the crack pattern in the
morphology. Without large organized areas, there are not regular cleavage planes. This
localizes the stress propagating the crack in a line perpendicular to the receding meniscus.
In the slow growth region, the organized structure gives rise to weaker cleavage planes.
As a result, cracks preferentially propagate along these planes. In conclusion, the growth
rate strongly affects the packing quality thus the color and crack morphology of the
template, but not the thickness.
135
This leaves the volume fraction of spheres and the meniscus height to control
template thickness. Meniscus height is a function the contact angle between the carrier
solution and the substrate, so reproducibility of the contact angle is key to controlling
template thickness. ITO glass was chosen as the substrate because the conductivity was
limited to one side, thus film growth is limited to one side as well. The ITO slides were
ultrasonically cleaned in soapy deionized water, deionized water, acetone, and isopropyl
alcohol solutions respectively for fifteen minutes each, so that the surface contamination
would not change the contact angle from sample to sample. This left thickness control
solely to the volume fraction of spheres. Even though the meniscus height is fixed, it is
not known, so one cannot a priori know the volume fraction of spheres needed to hit a
specific thickness for a particular system. Since the thickness varies linearly with the
volume fraction of spheres, from two sphere concentrations and thicknesses
measurements of the resultant films, determining the proper concentration becomes
straight forward.
136
1.E-05
8.E-06
Absorbance (a.u.)
6.E-06
4.E-06
2.E-06
0.E+00
-2.E-06
-4.E-06
-6.E-06
-8.E-06
-1.E-05
800
850
900
Wavelength (nm)
950
1000
Figure 5.11: The 2nd derivative of a fringe pattern from a colloidal template.
A non-destructive Fabry-Perot optical fringe method was used to determine the
thickness of the resultant templates.3 This fringe method is discussed in detail in the
Appendix. Normally incident transmission UV-Vis was used create the fringe pattern.
Due to the thicknesses needed to achieve at least nine unit cells, the intensity associated
with the fringes was small, so to magnify the extrema, the 2nd derivative of the
transmission spectra was taken. The results of this can be seen in figure 5.11. The
maxima and minima wavelengths for the fringe pattern can be fit using a modified
Bragg’s law via a least squares regression where the slope of the line is the thickness of
the template in nanometers.
137
6
0.45 v/v% [Spheres]
Ethanol
Fringe Order
5
4
3
2
y = 9172.1x + 0.0392
2
R = 0.9998
1
0
0.E+00
Fringe Maxima or Minima
~13 FCC unit cells
1.E-04
2.E-04
3.E-04
4.E-04
2neff (λ p − λ0 )
× 103
λ p λ0
5.E-04
6.E-04
Figure 5.12: The linearized extrema fit to determine film thickness.
After several iterations it was determined that a 0.45% volume fraction of spheres
yields a ~ 9.2 µm thick template as seen in figure 5.12. This translates into ~20 FCC unit
cells. This thickness was a thick enough to accommodate a few defects while still being
thin enough to polymerize through. This concentration was chosen to make the photonic
templates.
138
5.5 Mechanical Annealing the Colloidal Structure
In section 5.3, it was shown that there are regions where the spheres are randomly
arranged. To improve the order in the template, a mechanical annealing method was
developed. In this method, the condensed template immediately after drying was
vibrated by a piezoelectric element with a voltage profile of selected frequency and
amplitude. Using the potentiostat, the amplitude was varied between 0.125V and 4V
while the frequency was varied from 20Hz to 200Hz. Since the close packed structure is
of lower energy, when the spheres are given sufficient dynamic energy, they will seek
this lower energy configuration.100,101,104,111 To determine if the annealing improves the
order in the template, a disordered template was made and mechanically annealed.
Periodically, AFM images were taken of the template. Sequential images of template are
shown in figure 5.13.
139
Figure 5.13: AFM height images through time of a colloid template being
mechanically annealed.
The images clearly show that as the annealing progresses the disordered spheres
start to form small regions of local order. These merge to form larger grains. The
orientation of the individual grains are uncorrelated, but because grain boundaries are
higher energy, further annealing causes the largest grains to grow by absorbing the
smaller less stable grains. After annealing for a long period of time, large single crystal
grains develop. This shows that mechanical annealing can be used to improve the
140
regularity of the colloidal template. Similar results have been found for agitated randomly
arranged macroscopic spheres.112
5.6 Optimizing the FCC Structure
Calculations have shown that the hard sphere FCC filling fraction of 0.7405 was
not optimal for opening a PBG. Therefore the refractive index threshold of n = 2.8 for
the FCC structure is higher than absolutely necessary. It was calculated that if the
spheres are sintered together such that the spheres overlap slightly one can reduce the
refractive index contrast threshold. In calculations, the overlap was modeled as a
cylinder were as the overlap got the larger the radius of the cylinder got bigger. Figure
5.14, shows how the PBG size changes as a function of cylinder radius.35
141
∆n = 2.5
∆n == 2.9
2.8
Figure 5.14: PBG width as a function of the ratio between the cylinder radius and
the lattice parameter. Reprinted with permission from ref 35. Copyright 1998
American Physical Society. 35
For hard unconnected spheres with just their surfaces touching, the PBG opens at
a refractive index contrast of ∆n = 2.8. When the cylinder radius is ~14% of the lattice
parameter, the PBG size increases substantially. Consequently the refractive index
contrast drops to a minimum of ∆n = 2.5. This threshold is surpassed by the offresonance refractive index values for the low temperature poly(thiophene). If the
cylinder size increases much more, the PBG size drops precipitously. The question
becomes, can we control the structure of the template to achieve this more favorable
configuration.
It is known that the surface relaxation temperature of a polymeric material is
different from the bulk Tg. The depth of this region tends to be on the order of a few
142
nanometers.113 This fact can be used to tailor the structure of the template. By letting the
surface of PS spheres to relax into each other, the spheres will fuse slightly creating the
favorable connection without losing the spherical shape. .
Figure 5.15: AFM image of 269 nm diameter PS spheres after heating for 30
minutes at 80°C.
Figure 5.15 shows randomly arranged PS spheres after heating them for 30
minutes at 80°C. This is ~20°C less then the Tg of bulk PS, but high enough that if
possible, the spheres will fuse in a relatively short time frame. Randomly arranged
spheres were used as opposed to a regular array to better expose the connection between
spheres. It can be seen that the spheres have sintered together. Where three spheres form
a triangle, the PS filled in the hole while the spheres kept their distinct shape. Where
143
there are only two spheres touching, one can observe a neck forming. This neck is
tubular in shape like the cylinder used to describe the connection between the spheres.
This shows that the PS spheres can be fused together at temperatures below PS’s
conventional Tg without stretching into an ellipsoidal shape.
The next question is how can one control/monitor the fusing of the spheres. The
geometry of the situation is shown in figure 5.16.
rcylinder
doverlap
rsphere
rsphere - doverlap
Figure 5.16: Geometric description of the cylinder connecting fused spheres.
If the radius of the sphere is known and the thickness that the spheres overlap, one
can calculate the radius of the cylinder. The radius of the spheres is known, but the
overlap distance needs to be experimentally determined. As the spheres fuse, the lattice
parameter reduces such that the peak reflection wavelength will shift to lower
wavelengths. This shift can be used to back calculate the new lattice parameter, thus the
thickness of interpenetration. To show that this method works, a colloidal template was
grown with 290 nm diameter PS spheres. It was calculated that the [111] zone of a FCC
144
structure would reflect λ = 691 nm while the [101] zone of a HCP structure would reflect
λ = 647 nm. It has been previously shown that newly grown templates reflect the FCC
wavelength, so the template was allowed to sit for several weeks at room temperature. A
UV-Vis transmission spectrum was then taken at normal incidence. As seen in figure
5.17, the reflection peak was shifted to lower wavelengths.
1
Transmission (a.u.)
0.9
0.8
0.7
0.6
0.5
658 nm
0.4
0.3
0.2
0.1
0
500
HCP
600
FCC
700
800
900
1000
Wavelength (nm)
Figure 5.17: UV-Vis transmission spectrum of template aged at room temperature.
From the wavelength of maximum reflection (λp = 658 nm), one can back
calculate that the lattice parameter to be a = 390 and the overlap to be d = 7 nm. Using
this value the cylinder radius was determined to be rcylinder = 44.5 nm. Dividing the
cylinder radius by the lattice parameter one gets a value of 0.114. From figure 5.14, it
can be determined that the PBG would be ~6.4%. Through interpolation it can be
determined that the resultant threshold refractive index would be ∆n = 2.52. This is low
145
enough to open a complete PBG with the off-resonance refractive index values of PT.
So, using the surface relaxation of polymer spheres, the standard hard sphere packing
arrangement can be optimized yielding a structure that has a threshold refractive index
contrast below the off-resonance refractive index values of low temperature optimized
PT.
5.7 Summary
The self-assembly of monodisperse spheres whose diameters are on the lengthscale of light can be used to create an opal template. The template could have a FCC
structure, a HCP structure, or be a random agglomeration. It was determined through
AFM, optical diffraction, Kossel line analysis, and reflectometry that the majority phase
was FCC. The minority phase was determined through AFM, optical diffraction, and
reflectometry to be a random agglomeration of spheres. It was found that mechanically
annealing the template caused the random sphere agglomeration to organize. By
carefully controlling the contact angle between the carrier solution and substrate and the
volume fraction of spheres, one can grow templates sufficiently thick to open a complete
PBG. It was also found that by allowing the PS spheres to fuse together, one can achieve
an optimized structure that reduces the refractive index contrast threshold to ∆n = 2.52.
Taken together, this means one can build a high quality template to open a complete 3-D
PBG with the low temperature optimized PT.
146
CHAPTER VI
FABRICATING THE ORGANIC 3-D PHOTONIC MATERIAL
6.1 Introduction
This chapter focuses on combining the high refractive index poly(thiophene) (PT)
developed in chapter IV with the optimized colloidal crystal templates created in chapter
V to fabricate an FCC inverse opal material with a complete 3-D PBG. The first major
complication is how to synthesize PT that is not only high refractive index, but also can
create a negative copy of all the facets of an opal. Fortunately, electrochemical
polymerization has been used several times in literature to synthesize nano-structured
films for a range of conjugated polymers, even for the PT. 114-118 Using charge collection
rate values that showed good template morphology, the monomer concentration and
reaction temperature of the PT synthesis was re-optimized with respect to the wavelength
of maximum absorption (WMA). Using the new reaction conditions, PT films were
grown through colloidal templates. After dedoping and removal of the template, the
resultant inverse opal morphology and optical behavior was characterized.
147
6.2 Optimizing Poly(thiophene) Synthesis at Reduced Charge Collection Rates
In previous chapters, it has been shown that each constituent part of an organic
photonic crystal, the high refractive index material and high quality structure, could be
made. Combining these parts together presents new challenges. From the synthetic
perspective, the monomer will have to transport by means of a tortuous pathway through
the template to the working electrode surface to initiate polymerization. The rate of
polymerization will then strongly influence the voltages necessary to drive this transport
thus the quality of the PT film. From literature other researchers have optimized film
growth rates with respect to the morphology of the film growing through the template.114118
It was determined that a charge collection rate of 0.05 mA/cm2 results in a film that
faithfully replicates a negative of the template. This is 10 times slower than the
0.5 mA/cm2 optimal charge collection rate at low temperatures. To account for this, the
monomer concentration and reaction temperature were re-optimized at the new charge
collection rate. This optimization was accomplished without the presence of a colloidal
crystal. This was done because we needed to determine if the best possible film at the
charge collect rate would be sufficient to open a complete 3-D PBG, and the optical
characterization would be made more difficult with the opal structure present. The
temperature was varied between 0°C and -50°C because the BFEE solvent freezes at
-58°C. To reduce variation in monomer concentration between each temperature, a
master batch of each reaction solution was made to dispense at each temperature. The
master batches were made within 48 hours of use and stored in a dry box under an argon
blanket. The proton trap was added to the master batches first to scavenge any acidic
148
proton impurities in the BFEE solvent. The proton trap concentration was maintained at
10 mmole for this set of polymerizations. After addition to the solvent, the proton trap
was allowed to completely dissolve before addition of the monomer. The thiophene
monomer concentration was varied from 2.5 mmole to 25 mmole based on experience
from previous optimizations. For the 0°C temperature, a water/ice bath was used to fix
the reaction temperature. For temperatures below 0°C, a water/ethanol bath was used.
The appropriate water/ethanol ratio was used to tailor the freezing point of the solution to
± 2.5°C of the desired reaction temperature. The bath temperature was then reduced
using dry ice. ITO was used as the working electrode and nickel was used as the counter
electrode. Both were prepared as described in chapter II section 3.1.2.1. The counter
electrode was re-polished after every use. No supporting salts were used. The film
thickness was targeted to be 150 nm. This thickness was chosen because the dependence
of conjugation length on film thickness is minimized while still being able get a good
WMA measurement. The ability of the material to form a complete uniform film varied
with reaction conditions. Generally speaking, the higher the concentration of monomer
and temperature, the worse the morphology of the film. For conditions with good film
formation, the films were slightly thicker on the edges of the electrode and thinner in the
center. The thickness variation was normally less than 10%. After polymerization, the
films were immediately washed in acetone and dedoped with ammonium hydroxide for
24 hours. The electrical quality of the films were quantified by the WMA. The UV-Vis
measurements were taken from the center of the films while still attached to the ITO.
149
530
520
WMA (nm)
510
500
490
[Monomer]
2.5 mmole
5 mmole
10 mmole
15 mmole
20 mmole
25 mmole
480
470
460
450
0.05 (mA/cm2)
440
0
-10
-20
-30
-40
-50
-60
Temperature (°C)
Figure 6.1 : The optimization of reaction temperature and monomer concentration
for the synthesis of PT at a charge collection rate of 0.05 mA/cm2 with respect to
the WMA.
The WMA was plotted against the reaction temperature for each of the monomer
concentrations used in figure 6.1. Lines drawn through the points for a particular
concentration were generated by a least squares fit of the data, but are only meant as
guides for the eye. At 0°C, all monomer concentrations produced films with WMAs
around 505 nm. With decreasing reaction temperatures, the effect of concentration
became more evident. Films resulting from the 25 mmole concentration were shown to
have shorter conjugation lengths at lower temperatures with the WMA dropping to ~450
nm at -50°C. In the 20 to 15 mmole concentration range, the films showed little
dependence of conjugation length on reaction temperature maintaining a WMA ranging
150
from ~495 nm to ~505 nm. The films resulting from monomer concentrations ranging
from 10 mmole to 2.5 mmole were found to increase in conjugation length with
decreasing reaction temperature. This behavior can be understood from mechanistic
factors affecting the conjugation length, i.e. the ratio of conjugation promoting α-α
linkages versus conjugation cutting α-β and β-β linkages.88 This is why high
concentrations of monomer degrade the film properties. After initial stages of
polymerization, it is preferable that the activated monomer diffuse from the electrode
surface to attach to the end of a long growing chain rather than to react with another
monomer near the electrode surface increasing the concentration of bithiophene or other
initial products that then have a higher probability of reacting with each other in a nonfavorable arrangement.86,87 Lower temperatures improve the polymerization by
maximizing the effect of the energy difference between the α and β carbon as the energy
difference decreases with chain length. Lower temperatures reduce the electronic
fluctuations in the ring that enable the electron to spend equal time at either carbon site,
thus making the electron spend more time at the energetically favorable α position. By
this mechanism, α-α linkages are promoted at lower temperatures. This explains the
relationships seen between temperature, monomer concentration and WMA. At high
temperatures, monomer concentration in this range is not so important because activated
species can diffuse quickly away from the electrode surface to find a chain end as
apposed to building up activated monomer concentration near the electrode that will form
unfavorable bithiophene or other low molecular weight species. As the temperature
decreases, the monomer mobility drops significantly with increasing viscosity of the
solvent. The electric field pulls the monomer to the electrode, but after initiation there is
151
no strong force driving diffusion away from the surface. This results in the monomer
spending more time near the electrode surface effectively increasing the concentration of
initiated monomer at a given charge collection rate. At high monomer concentration this
can significantly increase the probability that two monomers will react forming low
molecular weight species. This in turn increases the concentration of species with low
degrees of polymerization which, despite lower temperatures, have a greater probability
of reacting with α-β or β-β linkages thereby breaking conjugation.88 For the middle
monomer concentrations, there is less initiated monomer build up such that their affect on
conjugation length can be offset by the increased probability of α-α linkages. At low
monomer concentrations, the probability of an activated monomer species finding each
other as opposed to high molecular weight species is low enough that the negative effects
caused by reduced mobility are overcome by the benefit of increased numbers of α-α
linkages.
6.3 Fabrication of the Poly(thiophene) Inverse Opal
With optimized reaction conditions identified, attempts were made to fabricate the
inverse opal. Colloidal templates were assembled onto ITO glass substrates. The
substrates were cleaned as prescribed in chapter III section 3.1.2.1. The colloidal
templates were fabricated with 290 nm diameter polystyrene spheres in a 0.45 vol%
solution with ethanol. The colloidal crystal grew on a vibration isolation table. The
templates assembled over the course of seven days. The resultant templates were dried in
vacuum for 24 hours and mechanically annealed. UV-Vis spectra in transmission mode
152
were taken to determine the initial thickness of the template film. The template was then
allowed to relax into each other to achieve interpenetration of the spheres. The blue shift
in film reflectivity as detected by UV-Vis, from the structure changes due to surface
relaxation, was monitored until the desired interpenetration had been achieved. Exposed
portions of the working electrode were passivated with an ethyl acetate film to ensure that
the polymer grew through the template and not preferentially in areas where the template
had chipped off. Reaction solutions were prepared a dry box under in argon atmosphere
at the optimized proton trap and monomer concentrations. The proton trap was added
first and allowed to completely dissolve before the thiophene was added. The thiophene
was redistilled before use and added to the BFEE solvent. The reaction solution was then
distributed into the reaction cells. The electrode system was then placed in the reaction
solution with the nickel counter electrode and the template covered ITO working
electrode. The reaction mixture was kept under a blanket of argon to prevent
atmospheric humidity from coming in contact with the solvent. The template was then
allowed to soak in the reaction solution for 9 hours to allow the solution to completely
penetrate and fill the tortuous voids in the colloidal crystal template. A water/ethanol
temperature bath was prepared such that it would freeze at -50°C. Dry ice was used to
reduce the temperature. The electrochemical cell was placed in the temperature bath and
allowed to equilibrate at the desired temperature for an hour. After setting the
appropriate constants into the chronopotientiametry polymerization software, the reaction
was started. For samples made to investigate the inverse opal morphology, the charge
collected was set to fill the template 50%. Because the efficiency of the reaction in the
presence of the template has not been studied, the actual filling fraction is not known.
153
For samples made for optical characterization, the film thickness was chosen to overshoot
the template by 50%. This was sufficient to completely fill the template and form a
continuous film of PT on top of the crystal. After polymerization, no PT was found
floating in the solvent, and the solvent maintained its translucent yellow color. This
indicates that short polymer chains did not migrate from the surface. The polymer and
working electrode were immediately washed in acetone after polymerization. The film
was then allowed to dry in the air. Dedoping results in a reduced film volume. This
reduction is usually less then 10%, but could significantly distort the inverse opal
morphology if it occurred quickly without the template to maintain the desired structure.
Since dedoping is a spontaneous process that will occur slowly without a dedoping agent
like hydrazine or ammonium hydroxide, the film was allow to sit in the air for a week.
During this time, a significant portion of dedoping would occur at a very slow rate
minimizing film stresses while maintaining the macro-porous structure. After the week,
the film was submersed in ammonium hydroxide for three days to completely finish the
dedoping process. Afterward, the film was washed in water and then submersed in
tertahyrafuran (THF) for 24 hours to dissolve the polystyrene template. The film was
then placed in an evacuated vacuum oven at ~250°C for five days. Samples meant to be
characterized for morphology were sputter coated with a thin lay of platinum and carbon.
After sputter coating, the sample morphology was characterized using scanning electron
microscopy (SEM).
154
10 µm
(a)
(b)
Figure 6.2: (a) SEM of a PT inverse opal. (b) FFT of the SEM image.
155
Figure 6.2a is the film at a relatively low magnification. This enables one to
observe very large areas of the film all at once. One sees that the PT preferentially grew
through the cracks in the template. This is expected, because transport of the monomer
through the cracks to the surface in comparison to through the crystal would enable faster
growth. In between the cracks one can see the inverse opal structure. One can ask
whether this area of the inverse opal is single crystalline or whether the areas bounded by
the cracks each contain a separate crystal orientation. Taking a Fast Fourier transform
(FFT) of the surface, as seen in Figure 6.2b, allows one to better understand and
summarize the surface structure. The confidence of a statistical analysis is critically
dependent on the sample size used. Dividing the area imaged (~3,575 µm2) by the area
of exposed unit cells in the (111) plane (~0.042 µm2) indicates that the sample size is
~85,000 unit cells. This is of sufficient size to draw statistically significant conclusions if
the signal is not extremely noisy. In the high angle region, one can observe six spots.
These spots have are slightly diffuse and have no corresponding second order spots. In
the low angle region, there is a relatively diffuse circle, but there is an increase in
intensity on the right and left hand side of the circle. The low angle diffuse circle with
some orientation can be ascribed to the large cracks transversing the image preferentially
from top to bottom. Note that intersections of cracks generally occur at angular multiples
of 60° which is a hold over from the six-fold symmetry of the FCC crystal structure. The
six spots in the high angle region are assigned to the honeycomb structure of the inverse
opal. The observation of only six and not a ring of spots whose number is a multiple of
six indicates that there is only one crystal orientation over the entire area supporting the
idea that the template was single crystalline. The diffuse nature of the spots and a lack of
156
higher order spots may indicate that there may have been little long range order in the
template, but this idea does not agree with the atomic force microscopy results from the
templates. It is thought that shrinkage and film stresses from the dedoping process may
have caused the film to form large wavelength, low amplitude waves in the sample. To
test this hypothesis, a magnified image was taken. If disorder caused the loss of the
higher order spots, than the FFT of the surface should not show higher order spots. On
the other hand, if the high order spots are missing because of long range fluctuations and
the image covers a relatively smaller area than the fluctuations, then the FT of the surface
should recover the higher ordered spots.
Figure 6.3a is an image of the inverse opal at higher magnification. In this image
it is easier to see that the PS spheres have been removed leaving behind the template.
The images bridges a crack. Again, this allows one to see if the cracks correspond to
grain boundaries in the film. A smaller crack was chosen to minimizes its impact on the
FFT analysis. One side of the crack occupies approximately one third of the image area
while the other side occupies approximately two thirds of the area of the image. Because
of this difference, spots associated with one side of the crack can be differentiated from
the other sided due to intensity differences in the FFT. The FFT image in figure 6.3b
shows six first order spots and twelve corresponding second order spots. Both orders are
sharper than the FFT in figure 6.2.
157
(a)
(b)
Figure 6.3: (a) SEM of a PT inverse opal at higher magnification. (b) FFT of
the SEM image.
158
2 µm
(a)
(b)
Figure 6.4: (a) SEM image of a PT inverse opal between cracks. (b) FFT of the
SEM image.
159
The fact that there is only one set of first, second, and very faint third order spots
indicates that there is only one crystal orientation, again suggesting that the cracks are
due to stress building up during the colloidal crystal drying process and are not
boundaries of uncorrelated crystal grains. As further evidence of this, one can observe a
large line dislocation in the upper middle portion of the image in figure 6.3a. This
dislocation crosses the crack and continues on the other side. This indicates that the
crystallization process that formed the line dislocation occurred before the crack formed.
The increased sharpness of the spots despite the reduced number of unit cells to average
out any disorder indicates that a large portion of the disorder observed is not based on
local disorder, but on long wavelength, low amplitude fluctuations probably caused by
residual film stresses from dedoping.
The same region of this inverse opal structure was investigated at even higher
magnification. The image in figure 6.4a is of a region between cracks. One can see the
regular honeycomb structure of the template. There is an apparent variation in hole size.
As the film grows around a sphere, the hole is the largest and the walls the thinnest
halfway up the sphere. It can be seen in figure 6.4a that there are areas where the holes
are smaller, the corresponding walls of thicker. This is caused by certain local sections of
the film growing faster than other areas as a consequence of variation in local monomer
transport. At this magnification one can see that the PT film faithfully copies the
template, and this replica survives the dedoping shrinkage. A FFT of the image was
calculated. One can clearly see the first, second, and third order spots. The spots are
very sharp to the point that one can resolve three spots where previously only one could
be seen. The two side spots are equidistant from the center spot. This suggests that the
160
two spots are related. It is thought that these correspond to the variation in wall
thickness. This seems to indicate that within the image one can identify two distinct
growth patterns of the film. The far left side of the image generally has a distinctly
different wall size than the rest of the image. This can result in three spots because the
correlation length difference between the inside and outside edge of the wall and the
corresponding neighboring wall is small for the thin walls. This results in one bright
spot. For the thicker walls, the correlation length difference for the inside and outside of
the walls is great enough to cause two distinct spots that straddle the location of the thin
wall spot as seen in figure 6.5.
Real
Thin Walls
Real
Thick Walls
FFT Thin
FFT Thick
Figure 6.5: Origin of triple spots in the FFT representation of figure 6.4a
161
100 nm
1 µm
(a)
(b)
Figure 6.6: (a) SEM of the top of a PT inverse opal. (b) Holes surrounding a
sphere in a FCC structure.
162
The previous images established that the surface of the film has a regular single
crystalline honeycomb like structure, but it is not proof of a 3-D inverse opal structure.
The growing PT film may have pushed the template from the surface just leaving an
impression of the template on the surface. Figure 6.6a shows an image where the film
had not completely formed around the top layer of spheres enabling one to see the layer
directly below the top layer. One sees an array of posts in a hexagonal arrangement with
three posts surrounding the position of each sphere. Three posts develop first because
ever sphere is surrounded by six holes as seen in figure 6.6b. Three of the holes allow
direct access through the second layer to the film growing in the third layer. The faster
transport of the monomer in these areas promotes faster film growth. For holes centered
on second layer spheres, the film must grow to cover the second layer spheres before the
posts can grow from these holes. The fact that three holes can be observed beneath each
sphere indicates that a complete three dimensional porous structure is being constructed.
The holes are also larger than would be expected if template was constructed from hard
spheres, which indicates that the spheres did sinter into each other.
To determine if the structure developed completely through the film, a portion of
the film was torn and lifted from the ITO electrode as seen in figure 6.7. The fact that
this is possible attests to the unique physical properties available to organic of photonic
crystals. Inverse opals made from inorganics would have shattered upon this kind of
manipulation. This allowed the SEM to investigate the morphology through the film. It
was observed that the PS spheres throughout the film had dissolved in the THF leaving
only the PT. One can also observe the regular stacking of the sphere layers. It is also
interesting that when looking at the bottom of the film, certain areas appear to be inverse
163
opals while other areas appear as flat films. This variation is not observed in the top of
the film. This suggests that in regions where the template does not adhere to the
electrode well, the growing PT film can lift template from the surface slightly, but, as
evidenced by the uniformity of the inverse opal on the surface, the interdigitated spheres
hold the loose template in place eventually forcing the film to grow through the template.
SEM
1 µm
25,000x
Figure 6.7: SEM of the side of a PT inverse opal.
164
(a)
(b)
Figure 6.8: (a) Reflection mode optical microscopy of the top of a PT inverse
opal. (b) Transmission optical microscopy of the top of PT inverse opal.
165
Figure 6.9: Reflection mode optical microscopy of the bottom of a PT inverse opal.
Samples were also made to observe the optical properties of the inverse opal.
These templates were ~9 µm thick. This translated into ~20 unit cell which is a
compromise between having as many unit cells as possible to insure that there would be 9
defect free ones, and having as few unit cells as possible to reduce to voltage required to
enable monomer transport through the template to the electrode or film surface. The
charge collected was sufficient to fill the template, assuming the template was
constructed from hard spheres, plus 50% more for a polymerization with the same
efficiency as one without the template in the way. This resulted in the film growing past
the template and forming a PT film on top. Figure 6.8a shows the top of an inverse opal
166
in refection mode. The top of the inverse opal appears green. The image is of lower
quality because of the relatively rough surface morphology of the film after growing
through the template. To compensate for this, long exposure times were required to
capture the image. Also note the black lines in the film. These correspond to cracks in
the template which allow for fast film growth. The resultant rough morphology inhibits
specular reflection causing them to look black in reflection mode. The film was also
observed in transmission mode. In figure 6.8b, one sees black areas and red areas. The
red areas are caused by the PT reflecting away the green light leaving only the red to be
observed. The black areas arise for two reasons. The first is that the PT in the cracks is
distinctly thicker than that which grew through the template, so it absorbs all the light.
The second set of areas are in the template regions. These are caused by the PT reflecting
away the green light, the inverse opal structure reflecting away the red light, and simple
roughness scattering eliminating any blue intensity leaving no intensity left. The bottom
of the inverse opal was also observed with reflection mode microscopy. Figure 6.9 is a
characteristic image showing predominately green areas and yellow areas. The green
areas are where the PT has pushed the template off of the electrode and started to grow.
The yellow areas were originally unexpected. The PT is known to reflect green while the
inverse opal structure is known to reflect red (see figure 6.10). Consulting a color wheel
indicates that for light mixing, the combination of red and green yields yellow. This
indicates that both reflection from the material and the inverse opal structure are
occurring simultaneously.
167
Red
Intensity (a.u.)
PC Peak
PT Peak
Green
Green + Red = Yellow
300
400
500
600
700
800
Figure 6.10: Wavelengths of reflection from the PT and from the inverse opal
structure.
In the areas of mostly green, one can see a yellow hue. This is evidence that
behind the thin PT film the inverse opal exists and contributes to the image. One should
also notice that a rainbow band appears on the far left hand side of the image. It
correlates with the long yellow band where the inverse opal is exposed to the surface on
the left side of the image as evidenced by the fact that the bend in the yellow is followed
by the bend in the rainbow. This band does not originate from the interference pattern do
to a space between the film and the bottom substrate, but originates from the film itself.
This is unusual because the film has the shape of a flat slab not that of prism. Without
the magnifying effect of the angle in the prism, a material needs to have a large
dispersion to resolve the different colors. PBGs have been shown to have large
dispersions outside of the PBG. This is a function of the modification of the density of
168
states caused by the PBG. To determine if a PBG really does exist, one needs
spectrographic data. Selected area transmission UV-Vis experiments were conducted.
To do this, a mask with a 5 mm diameter hole was made. This enabled just the best
section of the film to be tested. The optical behavior of the film with light propagating
perpendicular to the [111] zone of the PT inverse opal was examined.
log(Intensity) (%)
1
Calc. [111] PBG ~ 700 nm
0.1
0.01
0.001
0.0001
400
500
600
700
800
Wavelength (nm)
900
1000
Figure 6.11: The UV-Vis transmission spectrum of the [111] zone of an inverse opal.
The inverse opal PT exhibited a decrease in intensity across the entire spectrum
tested as seen in figure 6.11. This is due significantly to the rough morphology at the top
of the inverse opal structure. A dramatic attenuation of intensity was observed centered
at 699 nm. The scalar wave approximation (SWA) was used to calculate the placement
of a PBG for a n = 2.6 inverse opal structure with optimized interpenetrating spheres.17
The SWA approximation should be valid because we are dealing polarized light
169
propagating in 1-D along a line of high symmetry where the scattering is mainly
contributed to by a single set of planes. The refractive index approximation is taken from
the refractive index dispersion curve for the low temperature PT under optimized
conditions in chapter III at 700 nm. The result was that the PBG would be centered at
700 nm. This is strong evidence supporting the idea that an all-organic complete 3-D
PBG has been formed.
6.4 Summary
. Using charge collection rate values that showed good film morphology in
literature, the monomer concentration, and reaction temperature in the synthesis of PT
was re-optimized. The PT films at the newly optimized reaction conditions showed
conjugation lengths similar to PT films synthesized at higher charge collection rates.
Using the new reaction conditions, PT films were grown through the colloidal templates.
After removal of the template, it was determined that the PT films could faithful create a
negative copy of the opal structure. Film contraction during dedoping did not distort
local morphology, but seemed to contribute to long length-scale fluctuations in the film.
Optical microscopy indicated that light was being reflected from both the material
constructing the inverse opal and the inverse opal itself. UV-Vis transmission spectra
indicated a significant attenuation in intensity where the inverse opal PBG was calculated
to be in the [111] zone. This is strong evidence of a complete PBG, and supports the
claim that PT can be used to create the first all-organic PBG material with a complete
PBG.
170
CHAPTER VII
SUMMARY
Photonic band gaps (PBGs), through the coherent backscattering of radiation,
create frequency ranges in which the propagation of light is forbidden. A PBG is created
when a wave propagates through a periodic array of materials with sufficient refractive
index (n) contrast (n1/n2) where the dimensionality of the periodicity defines the
dimensionality of the PBG. The n contrast required to open a PBG increases as the
dimensionality increases. Currently, only inorganic materials have a sufficiently high n
to generate the complete 3-D PBG needed for many applications. The goal of this project
was to fabricate a polymeric photonic crystal with a complete 3-D PBG, and in doing so
bring the tailorable physical, electrical, and optical properties of polymeric materials to
high dimensional PBG crystals.
The first step was to develop a polymer with a sufficiently high n. It was
determined that a high refractive index dielectric would have highly delocalized electrons
and relatively heavy atoms. Because of its conjugated nature and the presence of a heavy
sulfur atom in its repeat unit, poly(thiophene) (PT) is predicted to have one of the highest
171
polymeric refractive indices with n = 3.9 at a wavelength of 700 nm.1 This suggested
that PT was a good candidate that could have a sufficiently high refractive index to open
a complete 3-D PBG for a FCC inverse opal structure, the most common structure for
visible wavelength PBG materials. Literature indicated that PT exhibited refractive index
values significantly below those predicted by semi-empirical calculations.2 This opened
up the possibility of an improved route to the synthesis of PT. To address the “PT
Paradox”, it was found that by polymerizing thiophene with a strong aprotic Lewis acid
solvent, BFEE, the initiation potential could be reduced below the degradation potential
of PT. Despite improved physical properties, saturation of monomer units by acidic
protons degraded the electronic properties. This was dealt with by introducing a proton
trap. The monomer concentration, proton trap concentration, charge collection rate, and
reaction temperature were optimized with respect to conjugation length as measured by
the wavelength of maximum absorption. The subsequent PT films had dramatically
improved conjugation length as well as refractive index values unprecedented for an
organic (n = 3.36) that were sufficient to open a complete 3-D PBG in a FCC inverse opal
structure.
The next step was to create photonic templates that PT could be polymerized
through to fabricate polymeric 3-D PBG crystals. The most common way to construct
these templates is through the colloidal crystallization of monodisperse spheres, but this
process is not without its difficulties. Using a combination of Colvin’s method3 and
mechanical annealing, high quality photonic templates have been fabricated. It was
determined through AFM, optical diffraction, Kossel line analysis, and reflectometry that
the majority phase of the template was the preferable FCC structure. The minority phase
172
was determined through AFM, optical diffraction, and reflectometry to be a random
agglomeration of spheres. It was found that mechanically annealing the template caused
the random sphere agglomerations to organize in to a hexagonally close packed structure.
By carefully controlling the contact angle between the carrier solution and substrate as
well as the volume fraction of spheres, one can grow templates sufficiently thick to open
a complete PBG. It was also found that by allowing the PS spheres to fuse together, one
can achieve an optimized structure that reduces the refractive index contrast threshold to
∆n = 2.52. Taken together this means one can build a high quality template to open a
complete 3-D PBG.
Using charge collection rate values that showed good film morphology in
literature, the monomer concentration, and reaction temperature in the synthesis of PT
were re-optimized. The PT films at the newly optimized reaction conditions showed
conjugation lengths similar to PT films synthesized at higher charge collection rates.
Using the new reaction conditions, PT films were grown through the colloidal templates.
High n PT infiltrated the templates, and the templates were removed leaving a polymeric
inverse opal with the possibility of a complete 3-D PBG. After removal of the template, it
was determined that the PT films could faithfully create a negative copy of the opal
structure. Film contraction during dedoping did not distort local morphology, but seemed
to contribute to large length-scale fluctuations in the film. Optical microscopy indicated
that light was being reflected from both the material constructing the inverse opal and the
inverse opal itself. Rainbow bands caused by the film are evidence of the highly
dispersive nature of the structure. UV-Vis transmission spectra indicated a significant
attenuation in intensity where the inverse opal PBG was calculated to be for the [111]
173
zone. This is strong evidence of a complete PBG, and supports the claim that PT can be
used to create the first all-organic PBG material with a complete 3-D PBG.
174
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181
APPENDIX
FRINGE CALCULATIONS
A.1 Obtaining Refractive Index Values
The optical interference pattern from thin films is caused by the constructive and
destructive interference of light reflecting off of the front and back of the film.99 The
exact wavelength of the maximum intensity and the range of wavelengths between each
intensity maximum is dependent on the optical thickness of the film. The optical
thickness of a material is the product of the thickness (d) and the refractive index (n).
OT = n ∗ d
(A.1)
The interference pattern of light being reflected from a film is described by
Bragg’s law.
mλ0 = 2n0 d sin (θ )
182
(A.2)
At normal incidence Bragg’s law simplifies to equation A.3.
mλ0 = 2n0 d
(A.4)
So to determine the refractive index, one can rearrange equation A.4
n0 =
mλ0
2d
(A.5)
Here the refractive index is a function of the film thickness (d), wavelength (λ),
and diffraction order (m). Of these, the film thickness can be determined experimentally,
and the wavelength of maxima and minima can be easily determined through UV-Vis
spectroscopy. The diffraction order though is not known, so another expression is needed
to relate the diffraction order to the other variables. For two sequential intensity maxima
at wavelengths λ01 and λ02, the corresponding refractive indices are n01 and n02, and the
diffraction order parameter is given by equation A.6
m1 =
n01λ02
2(n02 λ01 − n01λ02 )
(A.6)
Now we have a second expression, but have introduced to a new unknown
variable, the refractive index at a second wavelength. If one assumes that the refractive
183
index dispersion is close to zero between the two extrema wavelengths, n01 ≈ n02, then
expression A.6 can be simplified to expression A.7.
M=
λ02
2(λ01 − λ02 )
(A.7)
Assuming the refractive index dispersion is Cauchy-like, one can determine that
the effect of dispersion on M will take the form of equation A.8
Mλ 0 =
D
λ0y
+C
(A.8)
Here C, D and y are all constants where m is approximated by equation A.9
m≈
C
(A.9)
λ0
This expression for m can then be plugged back into equation A.5 allowing one to
calculate n0.
n0 =
C
2d
(A.10)
To get the value of C, one treats M thus m as continuous numbers and not discrete
orders. Then one fits the wavelengths of experimental extrema using a least squares
184
nonlinear regression with equation A.8. The quality of the fit can be attested to by
comparing agreement between the calculated continuous m values and the discrete
values. To demonstrate the quality of the fit, an example is given in figure A.1.
185
0.9
Transmission (A.U.)
0.85
Transmission
2nd Der.
0.8
0.75
0.7
0.65
0.6
0.55
0.5
650
700
750
800
850
900
950
1000
1050
Wavelength (nm)
(a)
4.4
4.2
Refractive Index
4
3.8
3.6
3.4
mcalc
m
13.0
13.5
14.0
14.6
15.1
15.5
16.0
16.5
17.0
17.5
17.8
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
3.2
3
2.8
2.6
2.4
n
2nd Der.
2.2
2
600
700
800
900
1000
1100
Wavelength (nm)
(b)
Figure A.1: (a) The fringe spectrum from a weakly absorbing film and the 2nd
derivative analysis to magnify the maxima and minima of the fringe pattern.
(b) The refractive index dispersion from the fringe pattern above.
186
Figure A.1a shows the transmission spectrum from a poly(thiophene) film.
Obvious fringes outside the absorbing region can be seen from 700 nm to 1050 nm. To
magnify the extrema to make picking out peak wavelengths easier, the 2nd derivative of
the transmission spectrum was taken. A nonlinear regression was used to fit the data
using equation A.8. The resulting calculated continuous diffraction order values are
shown in figure A.1b compared to the discrete values. One can see only minimal
deviation between the two indicating a good fit. Then using the film thickness
determined by profilametry, the corresponding refractive indices were calculated and
plotted in figure A.1b. These values are smoothly and slowly decreasing as expected for
a Cauchy-like dispersion far from the absorbing region. So using this method, one can
obtain refractive index values from weakly absorbing film with normally incident UVVis spectroscopy.99
A.2 Obtaining Film Thicknesses
One can attack the problem of determining the thickness of a thin film in a similar
manner.3 The optical interference pattern in thin films is a caused by interference of light
reflecting off of the front and back of the film. The exact wavelength of the intensity
maxima and the range of wavelengths between each intensity maximum is dependent on
the optical thickness of the film. The optical thickness of a material is the product of the
thickness (d) and the refractive index (n).
OT = n ∗ d
187
(A.11)
The interference pattern of light being reflected from a film is given by Bragg’s
law.
mλ0 = 2n0 d sin (θ )
(A.12)
Here m is the diffraction order, λ is the wavelength of light, n is the film refractive
index, d is the thickness of the film and θ is the incident angle of the light. At normal
incidence Bragg’s law simplifies to equation (A.13)
mλ0 = 2n0 d
(A.13)
For materials comprising multiple phases with different refractive induces whose
phase size is on the order of the wavelength of light being reflected or smaller, the
effective material refractive index is the volume weighted average of the constituent
phases. For a two phase material, equation A.14 gives the effective refractive index.
neff = φn12 + (1 − φ )n22
(A.14)
For a colloid crystal, the interference pattern at normal incidence is given by
equation A.15.
m0 λ m0 = 2neff d
188
(A.15)
Assuming no appreciable refractive index dispersion, the next subsequent fringe
is given by equation A.16.
(m0 + m1 )λ m
1
= 2neff d
(A.16)
Combining equations A.15 and A.16 can give A.17.
⎞
⎛
λ p λ1
⎟
p = d⎜
⎜ 2n (λ − λ ) ⎟
p ⎠
⎝ eff 1
(A.17)
This defines the film thickness in terms of measurable quantities. The first
maximum is diffraction order zero. For each subsequent maximum, the diffraction order
increased by one. The first minima is diffraction order 0.5. For each subsequent minima,
the diffraction order increased by one. To determine the thickness, plot p versus the
bracket term on the right-hand side of the equation. The resultant slope is the film
thickness.
To demonstrate the viability of this approach, a PS colloid crystal UV-Vis
transmission spectrum was taken. PS has a refractive index of n = 1.59. The packing
fraction of a close pack sphere structure is 0.7405. To magnify the spectrum extrema to
make picking-out peak wavelengths easier, the 2nd derivative of the transmission
spectrum was taken. Clear fringes could be seen in the wavelength range 650 nm to 850
nm. The extrema wavelength values were used to calculate the value of the far righthand term. These two were than plotted against each other in figure A.2.
189
8
7
Fringe Order
6
5
4
3
2
y = 9475.2x + 0.1004
1
R = 0.9998
2
Fringe
Fr
inge Maxima or Minima
0
0.E+00
2.E-04
4.E-04
6.E-04
2neff (λ p − λ0 )
× 103
λ p λ0
8.E-04
Figure A.2: The linearized extrema fit to determine film thickness.
The result was then least square fit with a straight line. The equation of the line
and R2 value are shown in figure A.2. The R2 value is close to one indicating that the line
describes the data well. The slope indicates that the thickness of the film is 9.48 µm. The
thickness of the film as determined by profilametry was ~9.5 µm. This is very close to
the value determined from the fringe calculation. So using this method, one can obtain
accurate thickness values from normally incident UV-Vis spectroscopy.3
190