devices and materials for thz spectrosopy: ghz cmos
TRANSCRIPT
DEVICES AND MATERIALS FOR THZ SPECTROSOPY: GHZ CMOS CIRCUITS,PERIODIC HOLE-ARRAYS AND HIGH-FREQUENCY DIELECTRIC MATERIALS
By
DANIEL J. ARENAS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2009
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c© 2009 Daniel J. Arenas
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To my mother who worked so hard to bring me to this wonderful country
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ACKNOWLEDGMENTS
First, I would like to thank my advisor, Professor David B. Tanner for giving me
great opportunities in my graduate career. Thanks to him, I have been fortunate to work
in many projects and learn a great deal. His advice, teaching and contagious enthusiasm
for physics have been greatly valued and appreciated. Also I would like to thank my
supervisory committee: Professor Arthur F. Hebard, Professor Juan Nino, Professor David
Reitze and Professor Peter Hirschfeld.
As a graduate student, I have worked in many different projects and have had the
privilege to meet, collaborate and learn from many wonderful people. I would like to
thank Sinan Selcuk and his advisor Professor Art Hebard for letting me be part of the
periodic hole arrays project. Thanks to Jinho Lee, Takahisa Tokumoto and Professor
Stephen McGill in NHMFL for teaching me and helping me a great deal in ultrafast
optics. Professor Kenneth O for letting me be part of the 410 GHz circuit project, and
his students and my good friends Eunyoung Seok and Dongha Shim. Thanks to Professor
Lev Gasparov in UNF for his help in the Raman measurement and introducing me to
the optics field and teaching me when I was an undergrad. Thanks to Professor Tom
Pekarek in UNF for his invaluable advice in the last seven years. Once again thanks to
Professor Juan Nino for his teaching and patience in the bismuth pyrochlores project; and
his student Wei Qiu. Also immense thanks to Professor David Silverman for letting me be
part of his Co crystals project and his student Balu Avaru. Thanks to Jay Horton, Marc
Link, Ed Storch, Raymond Frommeyer, Bill Malphurs, Larry Phelps and Rob Hamersma
for all the help in designing, repairing and building equipment. I learned a great deal from
them. Thanks to Dr. Robert DeSerio and Charles Parks for their tutelage when I was a
teaching assistant.
To my colleagues in the lab. Naveen Margankunte, my lab senior, whom I learned so
much in our lab and from our trips to NHMFL. Dimitrios Koukis, who has been the best
teammate anyone could wish to work with. Thanks to Nathan Heston, my good friend
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and labmate for the last five years. Thanks to my seniors Andrew Wint, Kwangje Woo,
Minghan Chen and Haidong Zhang. Jungseek Hwang from whom I learned so much as
his time as postdoc in our lab. Also thanks for the help of the talented junior students in
my lab, Zahra Nasrollahi, and Kevin Miller. Thanks to Xiaoxiang Xi, who we should all
expect great things from. Thanks to my good friends, Wan Wu, Wei Chen, Rajiv Misra,
Shawn Allgeier, Rod Delgadillo, Charles “Chuckles” Perry, Lexy Kemper, Richard “wtf
is my driver” Ottens, Dan Sindhikara, Jeffrey “El Jefe” Hoskins, Gregorious Boyd and El
Joey Nicely.
Finally the biggest thank you goes to my mother, Rosario Orozco, for working so hard
to raise me and my brother and bring us to U.S.A. Y a mis abuelos: Gustavo y Fanny
Orozco. Los quiero mucho.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 DETECTION OF RADIATION FROM A 410 GHz CIRCUIT . . . . . . . . . . 18
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Detection of the Radiation Using an Interferometer . . . . . . . . . 21
2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Demonstration of the Operating Frequency . . . . . . . . . . . . . . 222.3.2 Estimate of the Radiated Power . . . . . . . . . . . . . . . . . . . . 22
2.4 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 PERIODIC HOLE ARRAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Enhanced Transmission Effect . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Surface Plasmons Theory . . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Dynamical Diffraction Theory . . . . . . . . . . . . . . . . . . . . . 303.2.3 Trapped Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Reflection and Transmission Studies of Periodic Hole Arrays . . . . . . . . 323.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . 323.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Ultra-Fast Optics Study of Periodic Hole Arrays . . . . . . . . . . . . . . . 373.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . 383.4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Considerations for Nonlinear Applications . . . . . . . . . . . . . . . . . . 393.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 RAMAN STUDY OF THE PHONON MODES IN BISMUTH PYROCHLORES 52
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.1 Tentative Assignment of the “Ideal” Modes . . . . . . . . . . . . . . 544.3.2 Tentative Assignment of the “Disorder” Modes . . . . . . . . . . . . 56
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
APPENDIX
A EXTRACTING ELECTROMAGNETIC WAVES AND THE OPTICAL CONSTANTSFROM MAXWELL’S EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . 71
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.2 Electromagnetic Waves and the Optical Constants . . . . . . . . . . . . . . 72
B BOUNDARY CONDITIONS. TRANSMITTANCE AND REFLECTANCE . . . 78
C RESPONSE FUNCTIONS AND KRAMERS KRONIG ANALYSIS . . . . . . . 82
D MICROSCOPIC MODELS FOR THE OPTICAL CONSTANTS . . . . . . . . . 89
D.1 The Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89D.2 Lorentzian Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
E INTERFEROMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
E.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95E.2 Properties of the Interferogram . . . . . . . . . . . . . . . . . . . . . . . . 97E.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
F SURFACE PLASMONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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LIST OF TABLES
Table page
4-1 Raman modes and tentative assignment of BMN, BZN, BZT and BMT. . . . . . 62
4-2 Comparison between IR modes and Raman modes. . . . . . . . . . . . . . . . . 62
4-3 Comparison between the Raman modes of BMN and other pyrochlores . . . . . 63
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LIST OF FIGURES
Figure page
2-1 Circuit Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2-2 Emission spectrum of the 410 GHz circuit. . . . . . . . . . . . . . . . . . . . . . 26
2-3 Observed emission peak for the 410 GHz circuit. . . . . . . . . . . . . . . . . . . 27
2-4 Emission comparison between the circuit and blackbody sources. . . . . . . . . . 27
2-5 Source compartment diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2-6 Source compartment. Collecting mirror and aperture. . . . . . . . . . . . . . . . 28
3-1 A periodic hole array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3-2 Transmittance spectra of two periodic hole arrays. . . . . . . . . . . . . . . . . . 41
3-3 R + T spectra for ZnSe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3-4 Predicted R + T from trapped modes theory. . . . . . . . . . . . . . . . . . . . . 42
3-5 R and T spectra. Dg = 6 µm (ZnSe substrate). . . . . . . . . . . . . . . . . . . 43
3-6 R + T spectra. Dg = 6 µm (ZnSe substrate). . . . . . . . . . . . . . . . . . . . . 43
3-7 R and T spectra. Dg = 8 µm (ZnSe substrate). . . . . . . . . . . . . . . . . . . 44
3-8 R + T spectra. Dg = 8 µm (ZnSe substrate). . . . . . . . . . . . . . . . . . . . . 44
3-9 Reflectance and transmittance spectra for two periodic hole-arrays on a quartzsubstrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3-10 Sum of the reflectance and transmittance spectra for the two periodic hole-arrayson a quartz substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3-11 Caricature of a sharp pulse transmitted through an array. . . . . . . . . . . . . . 46
3-12 Caricature of a broad pulse transmitted through an array. . . . . . . . . . . . . 46
3-13 Caricature of a pulse with timewidth comparable to the modes lifetime . . . . . 47
3-14 NHMFL Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3-15 Autocorrelation data for the reflected pulse from the silver film. . . . . . . . . . 49
3-16 Autocorrelation data for the periodic hole arrays. . . . . . . . . . . . . . . . . . 50
3-17 Comparison of the autocorrelated data for the reflected pulse from the silverfilms and the various hole-arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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4-1 Pyrochlore structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4-2 BMN Raman spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4-3 BZN Raman spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4-4 BMT Raman spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4-5 BZT Raman spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4-6 Comparison of Raman spectra between the four samples. . . . . . . . . . . . . . 69
4-7 Normal modes of a linear O-A-O molecule. . . . . . . . . . . . . . . . . . . . . . 70
A-1 Boundary conditions at an interface. . . . . . . . . . . . . . . . . . . . . . . . . 77
C-1 Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
E-1 Diagram of a basic interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . 99
E-2 Interference in an interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . 99
E-3 Caricature of an interferogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
E-4 Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
F-1 Surface plasmons at the interface. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
F-2 TE and TM polarization for a surface wave . . . . . . . . . . . . . . . . . . . . 107
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Abstract of dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
DEVICES AND MATERIALS FOR THZ SPECTROSOPY: GHZ CMOS CIRCUITS,PERIODIC HOLE-ARRAYS AND HIGH-FREQUENCY DIELECTRIC MATERIALS
By
Daniel J. Arenas
August 2009
Chair: David B. TannerMajor: Physics
This dissertation is composed of three main projects, linked together by the THz
region of the electromagnetic spectrum. In the first project, we detected the radiation
from a silicon CMOS circuit, using a fourier transform interferometer. At the time
of measurement, this 410 GHz circuit had the highest operating frequency for silicon
integrated technology. The measured radiated power from the 410 GHz circuits was
in the order of 0.01 µW. This circuit had radiated intensities comparable to those of
commercially available black-body sources in the 400 GHz region. The high power and
high emission per source area suggested possible spectroscopy applications.
We also studied the optical properties of periodic hole-arrays with resonant
frequencies in the THz region. Although the transmittance spectra of these structures
have been extensively studied, here we present reflectance measurements that allow the
analysis of the extinction/absorption spectra. The results were compared to predictions
from the trapped-mode theory on the ohmic losses of these systems. Our results did not
support the prediction of a suppression of the R + T spectra at the resonant frequency.
Also, we studied the time-dependence of femtosecond pulses reflected from periodic hole
arrays with resonant frequencies in the NIR region. Our results show that if the trapped
modes theory is correct, then the lifetime of these modes are below 100 fs.
Finally, in the third project, we studied the Raman active modes of various bismuth
pyrochlores Bi3/2ZnNb3/2O7 (BZN), Bi3/2ZnTa3/2O7 (BZT), Bi3/2MgNb3/2O7 (BMN)
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and Bi3/2MgTa3/2O7 (BMT), which have earned recent attention for high-frequency
applications. The spectra of the four compositions are very similar, suggesting no major
structural differences among these materials. The spectra were compared to those of
other pyrochlores and specific discussions are offered for the assignment of each mode.
Although there are clear differences between the spectra of these samples compared to
other pyrochlores, these differences can be explained by the appearance of additional
modes due to the relaxation of the selection rules (caused by the displacive disorder in the
Bi pyrochlores). Some additional modes had frequencies close to modes in the IR data,
and others had frequencies close to optically inactive modes calculated by computational
work in the literature. The additional modes were tentatively assigned by this comparison.
Finally, the existence of additional modes in the Raman spectra of all four compounds
suggests no difference in the amount of disorder among these samples.
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CHAPTER 1INTRODUCTION
In this dissertation, I present three projects linked together by the THz and infrared
spectrum. For decades, infrared spectroscopy has been used for the characterization
and detection of compounds, and its uses are so widespread that no single review article
could do it full justice. To this date, infrared spectroscopy remains an important tool in
condensed matter physics for the study of many systems such as high-Tc superconductors
[1, 2], multiferroics [3], manganites [4], nanostructures [5], heavy fermions [6], and others.
The infrared spectrum is separated into three regions (with various boundaries
depending on the division scheme), the far-infrared (10 - 700 cm−1; 0.3 - 21 THz; 1000 - 15
µm), the mid-infrared (700 - 4000 cm−1; 15 - 2.5 µm), and the near-infrared (4000 - 14000
cm−1; 2.5 - 0.7 µm). The “THz” region of the spectrum is now used to specify the 0.3 - 3
THz range, although some authors may extend this definition to 30 THz [7]. The 0.3 - 3
THz region was once considered a poorly developed region of the spectrum due to the lack
of intense sources [8]. Blackbody (thermal) sources, the most common in spectroscopy,
have low intensity at these low frequencies. For a long time, the frequencies in the THz
region were considered too fast for solid-state circuits, and too slow for solid-state lasers
[9]. One of the first alternative ways to generate THz radiation began in the 60s with
the use of nonlinear crystals for difference frequency generation [10] and parametric
amplification [11–13]. However, judging from the literature, the explosion in THz research
and sources seemed to occur in the mid 80s with the use of femtosecond lasers to induce
THz radiation from various systems; such as photoconducting structures [14–20] and
electro-optic materials [21, 22]. The generation of THz radiation from quantum cascade
lasers was also an important field in the 90s and continues to be so [23, 24]. For the last
decade, the new exciting and powerful systems for THz radiation include free electron
lasers (FEL) [9], synchrotron sources [25, 26], and other systems that also use relativistic
electrons to generate THz radiation [27–29].
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The availability of new sources in the 0.3 - 3 THz region of the spectrum has
attracted great interest across many scientific fields. In bio-imaging, THz radiation is
desirable because it can penetrate organic materials without ionization (which is not the
case for the prevalent x-ray imaging). Also, THz radiation is absorbed by water, which
allows the distinction of differing water contents in cells [30], and hence the detection of
certain cancer cells [31, 32]. The high transmission of THz radiation through non-metallic
media such as shoes, clothes and cardboard, has also motivated the use of THz imaging for
detection of concealed weapons [33, 34]. Furthermore, there is continuous research in the
detection of dangerous materials, such as explosives [35, 36], chemical agents [37, 38], and
even illicit drugs [7, 39]. A review of these and other applications, as well as other THz
sources not mentioned here, is given by Siegel in reference [8].
Although these new THz sources have allowed new exciting research in this region,
their sizes are big and their costs are high. For widespread applications, it is desirable
to build more compact, and more importantly, cost-efficient sources and detectors. One
possibility is the use of mainstream silicon technology, such as CMOS (complimentary
metal-oxide semiconductors), to build fast circuits. The first project of my graduate
studies deals in this area and is presented in Chapter 2. This chapter shows the detection
of THz radiation from a 410 GHz CMOS circuit equipped with a patch antenna. The
circuit was designed by Dr. Kenneth O’s group in the Silicon Microwave Integrated
Circuits and Systems Research Group located at University of Florida Department of
Electrical and Computer Engineering. And, it was constructed at Texas Instruments.
When the circuit was constructed, there were no available high-frequency probes above
325 GHz [40] at the time. Our contribution to this project was to demonstrate the
operating-frequency of the circuit by measuring its electromagnetic radiation using a
Fourier Transform Interferometer. At the time of measurement, this circuit had the
highest operating frequency of any circuit fabricated with silicon integrated mainstream
technology; and, marks the first time for thirty years that a CMOS circuit is faster than
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the more expensive indium phosphide and gallium arsenide technologies. We also show
that at 410 GHz, the radiation from the 200 x 200 µm2 patch antenna of the circuit is
comparable to that of commercially available blackbody sources. The demonstration of
the operating frequency and the radiation power from the patch antenna suggests these
circuits could be used as detectors in the THz region. Furthermore, the high emission per
unit area of these circuits (about 1000 times more efficient than blackbody sources at 410
GHz) suggests that they could be used as cheap THz sources for spectroscopy applications.
Chapter 3 presents reflectance and transmittance data, as well as femtosecond
measurements, for periodic hole-arrays with resonant frequencies near the THz region.
The optical studies of periodic hole-arrays and other nanostructures are motivated by
applications involving the manipulation of light [41–46]. Periodic hole arrays exhibit an
effect where at resonant wavelengths, the transmittance through the metal films exceeds
the value predicted by diffraction and geometric optics [47]. The explanation of enhanced
transmission in periodic hole-arrays remains controversial and several additional theories
have been proposed. Among these theories is the Trapped Modes theory, which states
that at a resonant frequency near the diffraction threshold, the electromagnetic fields get
trapped in the sub-wavelengths structures for a characteristic lifetime [48, 49]. This theory
is fascinating because it suggests temporal manipulation of light and has proposed exciting
applications in nonlinear optics [50]. The nonlinear optics possibilities are important
because they could lead to optical signal processing, where the purpose is to control light
using light [44]. Many of the applications of these structures, including nonlinear optics, is
relevant to the infrared and THz region due to our ability to tune the resonant frequency
of these structures by changing the periodicity of the holes. In this work, we try to test
two predictions from the trapped modes theory. We measured the time characteristics of a
∼ 100 fs pulse reflected by periodic hole-arrays, to see if the lifetime of the trapped modes
was comparable to the pulse temporal width. The other prediction states that when EM
modes get trapped at the resonant frequency (redshifted from the diffraction threshold),
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there is an increase in ohmic losses due to the longer exposure time. Computations from
Selcuk et al. show this effect as a dip in the R + T spectra [49]. In this work we were able
to measure not only transmittance, but also reflectance, for periodic hole-arrays and tested
the prediction.
Chapter 4 is about the Raman study of phonon modes in bismuth pyrochlores.
Bismuth pyrochlores have earned recent attention for high-frequency applications thanks
to their low loss, high-permittivity and good temperature stability [51]. We present
the Raman spectra of four different bismuth pyrochlores: Bi3/2ZnNb3/2O7 (BZN),
Bi3/2ZnTa3/2O7 (BZT), Bi3/2MgNb3/2O7 (BMN) and Bi3/2MgTa3/2O7 (BMT). The
purpose of this work was to compare how the Raman active modes behaved across the
four samples and to compare them to other pyrochlore structures. Our results show that
spectra is overall similar for the four bismuth pyrochlores, but shows key differences to
other pyrochlore materials. The observation of more than the six Raman modes predicted
from the ideal pyrochlore structure confirmed the displacive disorder in the bismuth
pyrochlores. Comparison to the infrared data [52] and computational work on other
pyrochlores [53] allowed identification of the additional modes due to the relaxation of the
selection rules.
Besides the projects mentioned in this thesis, I’ve had the opportunity to study other
types of materials using IR spectroscopy [54, 55]. In our lab, it is common to collaborate
with groups interested in the optical properties of their samples. In this dissertation, the
appendices give a brief overviews of the optical constants. These appendices are meant
to provide a pedagogical introduction for the benefit of younger students who choose to
read this thesis. Appendix A first shows how the optical constants are obtained from
Maxwell’s equations and a propagating wave solution. It shows why they can be complex
functions, and how the optical constants are inter-related. Then, appendix B shows how
the boundary conditions at an interface between two media relate the optical constants
to reflectance and transmittance. Appendix C shows the derivation and the use of the
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Kramers Kronig relations. These relations are particularly useful when only single-bounce
reflectance measurements are available. After obtaining the optical constants of a system,
we have to compare our results to a model to gain further understanding and insight.
Appendix D shows the Drude and Lorentz models, the simple but broadly used models
for free electrons and bound charges. This section also shows how we would expect free
electrons and bound charges to behave at very small and very large frequencies based on
the models. These limiting behaviors are very useful when performing Kramers Kronig
Analysis. Appendix E shows a brief tutorial on interferometers and their properties.
Finally, Appendix F shows the derivation of surface plasmons as solutions of a propagating
wave localized in a surface.
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CHAPTER 2DETECTION OF RADIATION FROM A 410 GHZ CIRCUIT
2.1 Introduction
In the past few years, electronic circuits have appeared with operating frequencies in
the multi-GHz regime, and the trend to higher frequencies continues. The 30 - 300 GHz
spectrum has already been allocated to various mobile, satellite, and radio astronomy
applications, as well as wireless USBs, ethernets and HDTVs [40]. The 300 GHz - 3
THz spectrum (dubbed the THz region) has been extensively studied for use in radars,
remote sensing, advanced imaging and bio-agent and chemical detection [8, 56, 57].
To bring the price down of these applications, it is desirable to build the THZ circuits
using the main-stream silicon integrated technology CMOS (Complimentary metal-oxide
semiconductors) [40, 58]. A THz circuit could also be useful in spectroscopy applications,
since blackbody sources have low power in the THz region and their intensity dies off
at longer wavelengths. Furthermore, these circuits could later be designed with tunable
frequencies so that they can be used in spectroscopy without the use of interferometers.
In this work, we report THz radiation from a 410 GHz CMOS circuit. This circuit
has the highest operating frequency among those fabricated using cost-efficient silicon
integrated technology. It was designed by Dr. Kenneth O’s group in the Silicon Microwave
Integrated Circuits and Systems Research Group located at the UF Electrical and
Computer engineering department (UF-EEL). At the time of the circuit’s design and
construction (2007), high frequency probes could not be used to measure its output due
to the lack of probes available above 325 GHz [40]. Instead, the operating frequency of
the CMOS circuit was demonstrated by measuring its electromagnetic radiation from an
on-chip patch antenna using an interferometer. All optical measurements were performed
at the Tannerlab in UF-Physics. We estimated the 410 GHz radiation power of the circuit
at around 0.001 - 0.01 µW. At these frequency, the power from this circuit is comparable
to commercially available blackbody sources used in interferometers. And, the smaller
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area of the circuit’s patch antenna (∼ 4 x 10−8 m2), compared to the emission area from
the blackbody sources (∼ 8 x 10−5 m2), suggests that the emission per unit area of these
circuits can be 1000 times higher than blackbody sources.
2.2 Experimental Procedures
2.2.1 Circuit Design
As previously stated, all the credit in circuit design goes to our collaborators Dr.
Eunyoung Seok, Dongha Shim and P.I. Dr. Kenneth O from UF-EEL. The circuit was
built in NXP Semiconductors and Texas Instruments. Figure 2.4 shows a diagram of
their circuit. The push-push oscillator is equipped with a patch-antenna to extract the
signal. The resonant frequency of the circuit is determined by the capacitances of the
transistors M1 and M2 and the inductors L1 and L2. As all oscillators, the amplifier (i.e.
transistor) limits the maximum frequency of the fundamental. This frequency is referred
to as fmax and corresponds to the maximum frequency at which the transistor will give
an amplification higher than unity. For our circuit, this frequency is around 200 GHz. To
increase the operating frequency, the second harmonic of the fundamental was generated
by using a push-push oscillator architecture. In this design, the cross-coupled design of
the transistors in the oscillator core is such that their signals are 180 degrees out of phase.
Unlike the fundamental, the second harmonic signal generated due to nonlinearity of the
transistors is in phase:
JM1 = σ(1)Eeiωt + σ(2)(Eeiωt)2 + ... (2–1)
JM1 = σ(1)Eeiωt + σ(2)E2e2iωt + ... (2–2)
JM2 = σ(1)Eeiωteiπ + σ(2)(Eeiωteiπ)2 + ... (2–3)
JM2 = σ(1)Eeiωteiπ + σ(2)E2e2iωt(1) + ..., (2–4)
19
where σ(1) is the linear conductivity of the transistors (assumed the same for both), σ(2) is
the nonlinear term dependent on E2, and JM1 and JM2 denote the currents at transistors
1 and 2. At the output node, the fundamental signals are attenuated due to their phase
difference and the second harmonic is extracted through the patch antenna,
JM1 + JM2 = 2σ(2)E2ei2ωt. (2–5)
Two PMOS transistors (M3 and M4) are used to bias the core, and two quarter
wave transmission lines (T.L1 and T.L2) are used to isolate these components from the
fundamental and second harmonic. Details on the circuit’s and patch antenna design can
be found in references [40] and [59].
20
2.2.2 Detection of the Radiation Using an Interferometer
The operating-frequency of the circuit was demonstrated by measuring the radiation
from its 200 x 200 µm2 patch antenna. The 410 GHz radiation was measured using a
Bruker 113v fourier transform infrared spectroscopy (FTIR) system. As every other
FTIR, the Bruker 113v has a source, an interferometer and a detector (please refer to
Appendix E for a tutorial on interferometers). Widely used sources for the Bruker 113v
are globar (silicon carbide) resistive lamps and mercury arc lamps. For this experiment,
the circuit was placed in the source compartment, which later allowed us to compare the
radiation power from the 410 GHz circuit to the power of a mercury arc lamp. For the
interferometer, we used a 5 mil (127 µm thick) Mylar beamsplitter. The thickness of the
beamsplitter (and subsequent efficiency minima) was chosen to increase the sensitivity
around 410 GHz. The scanning speed of the interferometer was varied to distinguish 60 Hz
(and harmonics) noise. The resolution in our measurements was 0.1 cm−1, which is limited
by the maximum path length difference (10 cm) in the interferometer. For the detector, we
used a 4.2 K cooled, Si bolometer (HD-3, 1378) from Infrared Laboratories.
For absolute power measurements, the source and detector were removed from the
interferometer and the source was placed directly at the input of the bolometer. The S/N
was greatly improved by modulating the circuit at 100 Hz and using a lock-in amplifier. A
lock-in amplifier takes a signal modulated in frequency S(ω) and a reference sin(ωrt − φ)
and amplifies only the ωr component by performing the integral:
1
T
∫ T
0
S(ω) sin(ωrt− φ)dt, (2–6)
where T is an integration time larger than the period of the reference signal and
the period of other components in S(ω). Other frequencies are filtered due to the
orthogonality of sine and cosine functions:
∫ 2π
0
sin(ω1t− φ) sin(ω2t)dt = 0; if ω1 6= ω2 or φ =π
2. (2–7)
21
If we assume a white noise spectrum (this assumption and the length of T can
demand the use of low and high pass electronic filters), then the signal/noise ratio is
improved by attenuating the noise at frequencies other than ωr and keeping only the noise
in the bandwidth of the reference frequency. The absolute power radiated from the source
was estimated by using the readout from the lock-in setup and compared to the 1.10 x 104
V/W optical responsivity of the bolometer (as reported by Infrared Laboratories). The
power radiated from the antenna was also estimated by comparing the power spectrum of
the circuit to the spectrum from the mercury arc lamp.
2.3 Results and Discussion
2.3.1 Demonstration of the Operating Frequency
Figure 2-2 shows the results for the power spectrum of the circuit, and the background
spectrum (circuit off). The signal at 410 GHz confirms the operating frequency of
the circuit, although the fundamental of the circuit is also observed at 205 GHz. The
remaining baseline intensity above 10 cm−1 in the background seems like blackbody
radiation from the circuit being hotter than room temperature. Figure 2-3 shows the
spectrum of the 410 GHz circuit near the emission peak (the background has been
subtracted for this figure). The FWHM (Full Width Half Maximum) of the peak is around
0.1 cm−1 which corresponds to the resolution of our measurements. This suggests that we
cannot resolve the width of the emission peak with our interferometry setup.
2.3.2 Estimate of the Radiated Power
From the separate absolute power measurements using the lock-in setup, the radiation
power at 410 GHz was calculated to 0.01 µW. Figure 2-4 shows a comparison between
the power of the 410 GHz radiation (substracted from the background) to the power of
the mercury lamp at the same conditions. The spectra shows that the radiation power
of the circuit is comparable to that of a commercially available spectroscopy source for
the narrow 3 GHz band of emission. To estimate the power from a mercury lamp at 410
22
GHz at a frequency width of 3 GHz (our resolution), we use Rayleigh Jeans Formula for
spectral radiance at low frequencies:
B(T ) =2υ2kBT
c2, (2–8)
where B(T ) is the spectral radiance with SI units of W/m2Hz sr (sr refers to steradians,
the units of solid angle), kB is the Boltzmann constant equal to 1.38 x 10−23 J/K, and υ
is the frequency. Our mercury arc lamp has a temperature close to 5000 K, which yields a
spectral radiance of around 2.6 x 10−13 W/m2Hz sr for 410 GHz.
To estimate the radiation area from the blackbody source and the collected solid
angle, we refer to Fig. 2-5 and 2-6. The collecting mirror has a diameter of 70 mm and
it images the source into an aperture of varying diameter (Fig. 2-5). Since only the light
that passes this aperture goes into the interferometer, the effective radiation area from the
source is approximately the area of the aperture (which has a 10 mm maximum diameter
for the Bruker 113v):
Area = π(5 mm)2 = 8× 10−5 m2. (2–9)
To calculate the collected solid angle, we use the diameter of the collecting mirror and
its distance to the source (Fig. 2-6):
Ω = 2π(1− cos θ), (2–10)
where θ is the angle of the radiation cone and is equal to:
tan θ =35
240, (2–11)
which yields a solid angle Ω of 0.07 sr. Therefore the power radiated from the mercury
lamp in the frequency width (dυ) of 3 GHz is approximately:
23
P = B(υ) · dυ · Ω · A = 0.004 µW. (2–12)
The Bruker 113 v has a second source housing that allows the use of the same
collecting mirror and aperture by using a flipping mirror. For the second source, we
used a Globar (silicon carbide) lamp with a temperature around 1500 K. This yields an
estimated power of 0.001 µW for 410 GHz and our resolution. Comparing these powers
of the mercury and globar lamp emission and the 410 GHz circuit in Fig. 2-4 suggests
that the power of the circuit is higher than 0.001 µW, but lower than our first estimate
(0.01 µW) using the sensitivity of the detector reported by the company. The efficiency
in emission per unit area is very high for the 410 GHz source. The mercury arc lamp has
an area of approximately 8 x 10−5 m2 while the patch antenna (200 x 200 µm2) for the
410 Ghz circuit has an area in the order of 4 x 10−8 m2. This suggests that the 410 GHz
source is in the order of 1000 more efficient in emission per unit area. (The equivalent of
having a 106 K blackbody source).
2.4 Conclusions and Future Work
By measuring the circuit’s radiation using an interferometer, we have demonstrated
this circuit’s 410 GHz record-setting operating frequency. This is the highest operating
frequency for silicon integrated technology. The width of the emission peak could not
be resolved due to the resolution limit (0.1 cm−1; 3 GHz) of our interferometer. The
power radiated from the patch antenna is estimated to be between 0.001 and 0.01 µW
as measured by two techniques; direct power measurements using the bolometer, and
comparison to a blackbody source (mercury arc lamp) of known temperature. Also the
high emission per unit area of the source suggests that these circuits could have possible
spectroscopy applications (about 1000 times more radiation power per unit area than
blackbody sources).
24
Future work in this project should consist in designing and building circuits with
tunable frequencies. The experimental setup for interferometry described in this chapter
would be an excellent way to test and demonstrate the tunability of the circuit.
25
Figure 2-1. Circuit diagram for the push-push oscillator system with an on-chip patchantenna.
Figure 2-2. Emission spectrum of the 410 GHz circuit.
26
Figure 2-3. Observed emission peak for the 410 GHz circuit.
Figure 2-4. Emission comparison between the 410 GHz circuit and the mercury lamp andglobar lamp normally used in our Bruker 113v interferometer.
27
Figure 2-5. Diagram of the source compartment for the mercury lamp used in our Bruker113v interferometer. Not drawn to scale!
Figure 2-6. Dimensions for the collecting mirror and aperture used to calculate radiationarea from the source and the collected solid angle.
28
CHAPTER 3PERIODIC HOLE ARRAYS
3.1 Enhanced Transmission Effect
The transmission of a single subwavelength hole is expected to be very small based on
diffraction optics. Bethe et al. [60] showed that for wavelengths larger than the hole size a,
the transmittance falls off as:
limaλ¿1
T ∼(
a
λ
)4
. (3–1)
However, Ebbesen et al. [47] discovered that for a periodic array of such sub-wavelength
holes, the transmittance spectra can exhibit large transmittance peaks. This effect is even
more puzzling in that even in the geometric optics limit (λ ¿ a) the transmittance should
not exceed the open area fraction f of the holes. The open area fraction of square holes on
a square grid is given by:
f =
(a
Dg
)2
, (3–2)
where Dg is the hole-separation (See Fig. 3-1). Figure 3-2 shows two examples for
the enhanced transmittance effect for periodic hole arrays. The spectra shown are for
patterned silver films on a ZnSe substrate with two different periodicities and open area
fraction of 0.44 (Fig. 3-3 shows the transmittance and reflectance of the substrate.)
The spectra show that the location of the enhanced transmittance peak changes for
different periodicities of the sample. The enhanced transmission effect has received much
attention [47, 61–70] and has suggested many exciting applications in light manipulation
and nonlinear optics [44, 50, 71, 72]. However, the explanation of this effect remains
controversial. And, although the original [47] and still popular [62, 66, 73–78] explanation
by Ebbesen et al. attributed the effect to surface plasmons, other explanations have
been proposed. In the next sections, we will mention only two of the other theories:
dynamical diffraction and trapped modes theory (although the reader is suggested to see
29
references [79–83] for additional theories). Although I cannot offer insight into the validity
of these theoretical or computational arguments, the purpose of this work was to identify
predictions offered by these alternate theories and to then measure them.
3.2 Theories
3.2.1 Surface Plasmons Theory
The enhanced transmission effect was originally attributed to the interaction of
light with surface plasmons (SPs) in the perforated metal film. Surface plasmons are
electromagnetic waves confined to the interface between a positive dielectric and a
negative dielectric. The wave propagates along the surface and can only couple to light
when the surface has a periodic structure. (See Appendix F for a brief discussion on
surface plasmons). The mechanism proposed consists in excitation of a SP in the top
of the film, and a reemission of light by an SP at the bottom of the film. Ebbesen et
al. attributed the causal role of enhanced transmission to SPs due to two important
results: One, using angle dependent transmittance measurements, Ebbesen showed that
the frequency of enhanced transmission versus the wavevector k gives a dispersion curve
characteristic of surface plasmons; and, they showed that periodic hole arrays in Ge
(positive dielectric) did not show enhanced transmission.
3.2.2 Dynamical Diffraction Theory
M.M.J. Treacy and the supportes of dynamical diffraction theory [84] and [85] argue
that SPs do not play a causal role in enhanced transmission, and that the effect is linked
to diffraction. The inspiration for this theory is based on dynamical diffraction theory
for x-rays. Perfectly oriented crystals radiate x-ray light coherently and cause anomalous
effects at wavelengths close to the lattice constant of the crystal (λ ∼ d). Ewald [86]
created the coherent dynamical diffraction theory to explain these effects which were
unaccounted for by traditional kinematic diffraction theory. The theory consists in solving
Maxwell’s equations in a periodic media, where the periodicity of the dielectric function,
ε(~r), is written as:
30
ε(~r)−1 =∑
g
Fgei~g·~r (3–3)
where Fg are the amplitude of the Fourier components (~g) of the grating [84]. Treacy
et al. [85] argued that hole arrays are basically the same system except with negative
dielectric constants as opposed to the close to unity dielectric in x-ray diffraction. Their
theory and calculations also succesfully explain the enhanced transmittance effect. The
elegant part of this theory is that it argues that enhanced transmission, anomalous x-ray
diffraction in crystals, and even gaps in photonic crystals are all basically the same effect.
For discussions on photonic crystals, please reference [41, 87, 88].
3.2.3 Trapped Modes
The theory of trapped modes [48–50, 89–92] states that at a resonant frequency,
where the enhanced transmittance effect occur, the electromagnetic fields get trapped
in the vicinity of the holes and decay by emitting nearly monochromatic light. The
characteristic decay time is related to the inverse of the width of the transmittance
peak Γ. Similar to the dynamical diffraction theory, the trapped modes theory argues
that ET effect is purely geometric and not due to surface plasmons. Their calculations
consist on fully solving the time-dependent Maxwell’s equations for radiating boundary
conditions. Their results show a resonant frequency near the diffraction thresholds where
the electromagnetic modes get trapped and enhanced transmittance occurs.
There are several predictions offered by the trapped modes theory [49, 50]. We will state
here the ones relevant to this dissertation:
1. Ohmic losses and therefore absorption should increase at the resonant frequency dueto the “longer” exposure of the modes to dissipative processes. This ohmic losses (orabsorptive) peak should also depend on the open-area fraction of the array.
2. Trapping (or delay) of light inside periodic hole arrays occurs at the resonantfrequency.
3. The trapping of light in 2D structures can lead to useful applications in nonlinearoptics.
31
In the following subsections, we will explain each prediction in more detail and
compare them to our results.
3.3 Reflection and Transmission Studies of Periodic Hole Arrays
3.3.1 Motivation
As stated in the previous section, the trapped modes theory states that light
gets trapped in the vicinity of the holes at the resonant frequency, and then decays
by emission. They predict that at this frequency, electromagnetic fields are exposed
to energy-loss mechanisms for a longer time. Their computations [49] show a dip
in the spectra of the sum of the transmittance and reflectance near the resonant
wavelength (See Fig. 3-4). This dip is attributed to absorption due to ohmic losses,
and is therefore redshifted from the diffraction threshold (redshifted similarly to the
enhanced transmittance peak). As of 2006, when this project got started, there was not
much work done on the reflectance of periodic hole arrays, and therefore data on the
extinction or absorption of these samples was limited. The lack of reflectance data was
perhaps due to the higher difficulty of measuring reflectance instead of transmission. In
this work, we present reflectance and transmittance data for two sets of samples fabricated
by different methods and measured with different equipment at different spectra. For the
NIR-VIS region, we show data from silver films grown on quartz, and we also show data
for the FIR region for silver films grown on ZnSe.
3.3.2 Experimental Procedures
The hole arrays are fabricated on 100 nm thick silver films deposited on quartz
substrates for the NIR-VIS measurements and deposited on ZnSe substrate for the
Far-Infrared. The hole-arrays in the quartz samples are patterned by using e-beam
lithography and the sample dimensions were 300 X 300 µm2. Photolithography was used
to pattern the ZnSe samples which had dimensions of 2 x 2 cm. The films were fabricated
by collaborators in HebardLab, UF Department of Physics and an extensive account of the
fabrication process can be found on Ref. [93]. The reflectance and transmittance data of
32
the quartz samples were measured using a Zeiss microscope photometer for the NIR-VIS
region (0.6 - 2 µm; 5000 - 15000 cm−1). Both reflectance and transmittance measurements
were carried out with a normal incidence beam with an angular spread in the order of 10
degrees. For the ZnSe samples, we used a Bruker 113v Fourier spectrometer in the FIR
and MIR region (2 - 100 µm; 100 - 5000 cm−1). For the FIR region (50 - 500 cm−1), we
used a doped Si bolometer cooled at 4.2 K, a mercury source, and Mylar beamsplitters
of different thicknesses (3.5, 12, 23 µm). For the MIR region, we used a DTGS detector,
a Ge/KBr beamsplitter and a silicon carbide (globar) lamp. The overlap for the different
measurements was well within 1 %. For the reflectance measurements, the angle of
incidence is 8 degrees, and the angular spread for both reflectance and transmittance is
around 8 degrees. All measurements were made using non-polarized light.
3.3.3 Results and Discussion
FIR: Ag films on ZnSe.
Figure 3-3 shows the reflectance and transmittance of the ZnSe substrate and their
sum R + T spectra. As shown by this figure, the absorption in the ZnSe substrate is very
small for frequencies between 650 - 4000 cm−1 but becomes very strong for frequencies
below 650 cm−1. There is a strong phonon mode around 200 cm−1 and the thickness of the
material (∼ 1 cm) makes the transmittance very sensitive to small absorption coefficients.
The index of refraction of the ZnSe substrate ns was 2.4 and this value was confirmed by
both the reflectance and transmittance measurement. For normal incidence, the enhanced
transmittance peak occurs near the diffraction threshhold associated with the index of
refraction of the substrate:
λ = nsDg (3–4)
Then, we expect the first diffraction threshold for a Dg = 6 µm array and refractive index
of the substrate ns = 2.4, to be located around 14.4 µm (695 cm−1). The higher orders of
33
diffraction are given by:√
m2x + m2
yλ = nsDg, (3–5)
where m is an integer. This yields diffraction thresholds at 982 cm−1 for the degenerate
cases of mx=0, my=1 and mx=1, my=0 and 990 cm−1 for mx=my=1. The diffraction
thresholds for the air interface (n = 1) are located near 1666 cm−1 and 2357 cm−1. (For
reasons still unknown to this author, the enhanced transmission frequency seems to be
associated with the substrate-film interface and not the air-film interface.) Figure 3-5
show our results for the reflectance and transmittance of Dg = 6 µm hole arrays. The
diffraction thresholds and the strongest transmittance peak are observed near 700 cm−1,
and, the next threshold is located around 1000 cm−1. The reflectance data showed the dip
associated with the transmittance peak and the diffraction thresholds.
It has been predicted and observed that arrays with larger open area fractions f have
larger widths transmittance peaks [49, 94]. The open area fraction of these samples is
given by:
f =a2
D2g
. (3–6)
The larger widths in transmittance peaks are attributed to smaller lifetimes of the trapped
modes for the larger hole samples [49]. Our data supports previous results for the relation
between width and open area fraction. For the f = 0.25 sample (left of Fig. 3-5) the full
width half maximum (FWHM) is ∼ 40 cm−1 and 70 cm−1 for our f = 0.44 sample (right).
The main purpose of the reflectance measurements was to test the prediction of the
trapped modes theory that a dip occurs in the R + T spectra (corresponding to a peak in
the absorption) due to ohmic losses. The theory also states that smaller open area fraction
samples should have more pronounced dips since the modes are longer lived. Figure 3-6
shows our results for the R + T spectra of the Dg = 6 µm samples. For simplicity, we
plot R + T instead of absorption or extinction, because at wavelengths smaller than the
diffraction threshold nsDg, both absorption and diffraction losses are possible:
34
E = 1− (R + T ), (3–7)
where E is the extinction and is equal to the sum of absorption and diffraction losses.
Comparison of our data to Selcuk et al. [49] computations shows good agreement in the
overall shape of the R + T spectra. However, although our results show a dip in the
R + T spectra, this dip is located near the diffraction threshold and not redshifted at
the resonant frequency. Therefore, this dip in the R + T spectra could be attributed to
diffraction losses, and not to ohmic losses. Another set of samples with periodicity Dg
of 8 µm and open area fractions of 0.25 and 0.44 were also studied. The transmittance
peak of these samples are located near 540 cm−1 (Figure 3-7) where the absorption of the
ZnSe becomes noticeable. Therefore the analysis of these samples has to be more careful.
Figure 3-8 shows the R + T spectra for these two samples. The results for these arrays is
the same as for the previous samples. The overall shape of the R + T spectra agrees with
computation, but the dip is seen near the diffraction threshold and not redshifted near the
resonant frequency.
NIR: Ag films on quartz.
Figure 3-9 (right) shows the reflectance and transmittance spectra for a periodic hole
array on quartz with periodicity of 0.8 µm and open area fraction of 0.25. The index of
refraction of quartz is 1.4 and the absorption in this region is negligible. The diffraction
threshold for this periodicity is observed and expected to be around 9000 cm−1. The
reflectance data is also consistent with the transmittance data. The R + T spectra is
shown in Figure 3-10 (right). There is a small dip in R + T at the resonant frequency, but
similarly to the ZnSe samples, there is a much stronger dip near and blueshifted from the
diffraction threshold. Similar results were obtained for films with periodicity of 1 µm.
In conclusion, for various samples for two different fabrication and optical measurement,
the overall shape of the R + T spectra agrees with computations. However, we do not
have a direct observation of the trapped modes theory prediction due to the fact that
35
the R + T dip is redshifted. One possibility is that the broad angular width of the
incoming beams for reflectance and transmittance are masking the effects. Furthermore,
for the ZnSe samples in the FIR, the angle of incidence was 8 degrees for the reflectance
measurements. Although, the angle-dependent measurements for non-polarized light
showed no appreciable changes in the transmittance peak (not shown), the broad angular
spread of the beam may be broadening the transmittance peak and reflectance dip widths.
Further work should concentrate on decreasing the angular spread of both reflectance and
transmittance and trying both TE and TM polarizations for the reflectance measurements.
36
3.4 Ultra-Fast Optics Study of Periodic Hole Arrays
3.4.1 Introduction
The main prediction from the trapped modes theory is that light gets trapped inside
the periodic hole arrays and then is remitted with a characteristic decay time 2π/ Γ. To
test this prediction, we studied the time dependence of a transmitted or reflected pulse
from a periodic hole-array. Figure 3-11 shows a caricature of the reasoning. The pulse is
Gaussian in time, with a duration set by the properties of the laser source. If the lifetime
of the modes is larger than the pulse width, then the transmitted or reflected pulse will
be broadened with an exponentially decaying tail, characteristic of the mode lifetime.
However, if we use a pulse that is too long, then we expect the pulse to remain the same
(Fig. 3-12). If the lifetime of the modes is comparable to the duration of the pulse (Fig.
3-13), then the detection of this effect would rely on the faster decay of the Gaussian pulse
instead of an exponential decay process. To study pulses this short (100 fs), we had to use
autocorrelation because we cannot measure their temporal profile with equipment such
as streak cameras. In autocorrelation, a beamsplitter is used to split the beam, which are
then mixed in a nonlinear crystal to obtain the sum-frequency. This nonlinear process
makes signal to noise (S/N) a very important issue. Also, the autocorrelation of the pulse
must be symmetric and therefore, we lose information about the pulse (for example, we
would lose information on the sharp left side of the pulse in Fig. 3-11); however, the
autocorrelated pulse should still look broader.
The Dg = 6 µm samples (Fig. 3-5) on the ZnSe substrates had widths Γ ∼ 50 cm−1
(1.5 THz). The Q of these samples ∼ 10 and based on the width we expected the lifetimes
to be around 100 fs. However, femtosecond pulses in the FIR region are limited and are
only available for specialized free electron lasers in national labs such as Jefferson Lab
(Virginia, USA). In the NIR region, however, chirped pulse amplifiers (CPA) and optical
parametric amplifers (OPA) can achieve ∼ 100 fs pulses. A CPA-OPA setup is available
in the Ultrafast-Optics Cell in the National High Magnetic Field Lab (NHMFL). For the
37
NIR region, the samples could be constructed to have same quality factor Q ∼ 10, but the
resulting larger width (Fig. 3-9) results in lifetimes around 10 - 100 fs. We carried out the
ultra-fast optics measurements for four samples with resonant wavelengths near 1600 nm.
Our results show no broadening of the pulse and suggest that the lifetime of the trapped
modes in these samples is below 100 fs.
3.4.2 Experimental Procedures
The time dependence of the reflected pulse from four different hole arrays were
measured using an autocorrelation setup. The reflected pulse was measured instead of the
transmitted pulse to avoid possible pulse broadening from the quartz substrate. The pulse
reflected from the silver film was used as a reference. Figure 3-14 shows a diagram of the
setup. Femtosecond pulses with frequency 12500 cm−1 (375 THz; 800 nm) at a repetition
rate of 1 KHz were obtained from a Clark-MXR CPA-2001 chirped pulse amplifier. To
bring this frequency near the resonant frequency of our samples, we used a TOPAS 4/800
OPA and halved the frequency to 6250 cm−1 (188 THz; 1600 nm). The autocorrelation
of pulses from the OPA showed that the width of these pulses were around 140 fs. The
normal-incident reflected beam was measured by using a beamsplitter in the geometry
shown in Fig. 3-14. All four arrays measured were in the same silver film; and the film
was transversally displaced from the incident beam to change between different arrays
and to measure the silver film as reference. The autocorrelator setup used to measure
the time profile of the reflected pulse is shown in Fig. 3-14. A beamsplitter was used to
split the beam into two paths, and one path contained a movable retro-reflector used
to change the path length. Then, using a lens, the two beams were focused into a BBO
(βBaB2O4) nonlinear crystal. The sum-frequency (SF) generated beam was then focused
into a photodiode detector and this signal was used as the autocorrelated data. The
laser power incident into the sample was kept below 1 mW. The ∼ 1 mJ/cm2 high laser
fluence was necessary to obtain good S/N from the autocorrelation setup. After the laser
38
measurements, the samples were inspected through microscopy. The samples seemed intact
and still showed the enhanced transmission effect.
3.4.3 Results and Discussion
Figure 3-15 shows the autocorrelation data for the beam reflected from the silver
film. The FWHM of the autocorrelation pulse is around 200 fs, which gives a pulse
width of 140 fs. Figure 3-17 shows the results for the periodic hole arrays of varying
geometry compared to the silver pulse. The results show no broadening of the pulse or any
appreciable higher signal for the longer time delays. Therefore our results suggest that if
the EM modes do get trapped in our samples, they have a lifetime less than 100 fs.
3.5 Considerations for Nonlinear Applications
Recently, nano-structures have been proposed as exciting materials for nonlinear
optics applications [50]. In standard nonlinear materials, the interaction length of the
EM fields has to be many orders of magnitude higher than the wavelength, and therefore
the materials have to be thick. The trapped modes theory enthusiasts argue that for hole
arrays, there is no requirement for a long interaction length because the fields interact for
a long time before radiating. Their computations [50] report 105 enhancement of nonlinear
effects on periodic hole arrays filled with nonlinear materials, and the enhancement
is attributed to the longer interaction time of trapped modes [50]. Unfortunately, the
manuscript does not report the lifetime of the modes for their simulation.
Here, we present a simplistic (and perhaps naive) insight into how long the lifetime
of these modes should be for possible applications. A typical BBO crystal (like the one
used in this work to find the sum-frequency generation) is about 5 mm thick, and can
have efficiencies in the order of 10 % for second harmonic generation in the visible. This
interaction length of 5 mm is in the order of 1000 times the wavelength and translates to
an interaction time of around 16 ps. Based on this calculation, we would conclude that
the periodic hole-arrays we can build with resonant frequencies around the visible region
with Q factor of 10 and lifetimes less than a 100 fs would be terrible for nonlinear effects
39
even if the holes were to be filled with a good nonlinear material. For these materials to
be suitable for nonlinear applications, the Q factor would have to be increased from 10
to a 1000 or 10000. For possible MIR and FIR applications, the larger wavelengths would
require larger interaction lengths (or longer interaction time for the arrays). Therefore,
we would have to keep the requirements on the Q factor to increase by two orders of
magnitude.
Unfortunately, the two sets of samples that were fabricated by two different
techniques (e-beam and photolithography) were limited to a Q factor of 10. Perhaps
further advanced in sample preparation technology with time will improve these factors.
This chapter’s conclusion is that even if the trapped modes theory is correct, we currently
cannot make samples with lifetimes large enough to become competitive in nonlinear
applications.
3.6 Future Work
Future work in this project includes decreasing the angular spread for both reflectance
and transmittance measurements. Also, the dependence on TM or TE polarization of
the reflectance should be studied. For the femtosecond measurements, the transmitted
pulse should be studied as well. To increase the S/N significantly, a cross-correlation setup
should be used. In cross-correlation, the pulse reflected (or transmitted) from the array
would be mixed with a higher power reference pulse from the OPA. In autocorrelation,
the amount of power of both pulses is limited by the damage threshold of the samples;
however, in cross correlation we can increase the power of the reference pulse and increase
the S/N ratio.
40
Figure 3-1. Diagram of a patterned hole array. The white spaces denote empty holes. Thehole size is denoted as a and the hole-spacing as Dg.
Figure 3-2. Transmittance spectra for two silver films on a ZnSe substrate. The horizontalline represents 0.44, the open area fraction.
41
Figure 3-3. Reflectance and transmittance spectra for the ZnSe substrate (Left). The sum(R + T ) is shown in the right.
Figure 3-4. Sum of the reflectance and transmittance calculated by trapped modes theory.This figure is borrowed from reference [49].
42
Figure 3-5. Reflectance and transmittance spectra for two periodic hole-arrays withperiodicity Dg = 6 µm on a ZnSe substrate. a denotes hole size and Dg
periodicity.
Figure 3-6. Sum of the reflectance and transmittance spectra for the two periodichole-arrays on a ZnSe substrate with periodicity Dg = 6 µm.
43
Figure 3-7. Reflectance and transmittance spectra for two periodic hole-arrays withperiodicity Dg = 8 µm on a ZnSe substrate. a denotes hole size and Dg
periodicity.
Figure 3-8. Sum of the reflectance and transmittance spectra for the two periodichole-arrays on a ZnSe substrate with periodicity Dg = 8 µm.
44
Figure 3-9. Reflectance and transmittance spectra for two periodic hole-arrays on a quartzsubstrate.
Figure 3-10. Sum of the reflectance and transmittance spectra for the two periodichole-arrays on a quartz substrate.
45
Figure 3-11. Thought experiment for the time dependence of a pulse transmitted orreflected from a periodic hole-array. This is the case where the lifetime of themodes is larger than the pulse width.
Figure 3-12. Thought experiment for the time dependence of a pulse transmitted orreflected from a periodic hole-array. This is the case where the lifetime of themodes is much smaller than the pulse width.
46
Figure 3-13. Thought experiment for the time dependence of a pulse transmitted orreflected from a periodic hole-array. The lifetime of the modes is comparableto the pulse width.
47
Figure 3-14. Experimental setup at NHMFL for the autocorrelation of reflected pulsesfrom periodic hole-arrays. SF refers to the sum frequency pulse, and BBOrefers to the βBaB2O4 nonlinear crystal that generates the SF. BS refers tobeamsplitters.
48
Figure 3-15. Autocorrelation data for the reflected pulse from the silver film.
49
Figure 3-16. Autocorrelation data for the reflected pulse from four periodic hole-arrays.The hole size is denoted by a and the periodicity by Dg.
50
Figure 3-17. Comparison of the autocorrelated data for the reflected pulse from the silverfilms and the various hole-arrays.
51
CHAPTER 4RAMAN STUDY OF THE PHONON MODES IN BISMUTH PYROCHLORES
4.1 Introduction
Bismuth pyrochlores have been extensively studied for dielectric applications [95, 96],
but have earned recent attention for high-frequency filter applications thanks to their
low loss, high permittivity, and good temperature stability [51]. The pyrochlore structure
(Fig. 4-1) is described as consisting of interpenetrating networks of BO6 octahedra and
A2O′ chains [97] and it is assigned to the space group Fd3m. The nominal composition
can be written as A2B2O7 or as A2B2O6O′, with the latter formula differentiating the
oxygen in the A-O2′ chains. The pyrochlore family is fascinating because the A and B
sites can be occupied by a broad range of elements that can give rise to a great variety
of physical properties. In the bismuth pyrochlore, Bi1.5Zn0.92Nb1.5O6.92 (BZN), the A
site is mostly occupied by Bi and the B site by Nb; while Zn partially occupies both
sites. It is important to note that in the literature, cubic BZN is typically described
as having the expected nominal composition of Bi1.5Zn1.0Nb1.5O7. However, phase
refinement studies [98], have demonstrated partial substitution of Zn in the A2O′ network
(with a resulting oxygen deficiency as presented above) to satisfy the crystallochemical
balance between ionic bonding, lattice strain and charge balance. In addition, the BZN
structure has been shown to differ from an ideal pyrochlore structure through random
displacements of the A and O′ ions [98]. Many bismuth-based materials have been
investigated, but only the spectrum of BZN [99] was known until recently, when Chen
et al. investigated the infrared modes of BZN along with three additional systems:
Bi3/2ZnTa3/2O7 (BZT), Bi3/2MgNb3/2O7 (BMN) and Bi3/2MgTa3/2O7 (BMT) [52]. It is
of great interest to study the vibrational spectra of these materials, because they provide
unique material-dependent information about defects or impurities, the crystallographic
ordering, and the ordering and orientation of dipoles. However, infrared spectroscopy
can only detect those vibrational modes which have a net dipole moment change: in
52
contrast, Raman spectra show vibrational modes with net changes in polarizability. Thus,
infrared inactive modes can be Raman active and some Raman inactive modes can be
detectable in the infrared. Therefore, infrared and Raman spectra are both needed to give
a complete picture of the vibrational modes of a material. In this report, we present the
complementary Raman measurements and their mode assignments for the four ceramics
materials studied by Chen et al. [52, 100].
The main purpose of this work was to compare how the Raman active modes behaved
across four bismuth pyrochlores having different constituents, and to compare them with
other pyrochlore-structured materials. The results show that the Raman spectra are on
balance quite similar for the bismuth samples. Each sample shows more than the six
modes predicted for the ideal pyrochlore structure, confirming the displacive disorder in
the bismuth pyrochlores. The Raman modes we observed are assigned to specific normal
modes by reference to the literature [101–107]. Furthermore, the results were compared to
the work by Fischer et al. where the frequency of Raman, infrared, and optically inactive
modes of Cd2Nb2O7 were calculated by ab initio calculations [53]. This comparison offers
insight into the origin of the additional modes due to disorder. Our Raman spectra were
also compared to the infrared data by Chen et al. [52] and the comparison also suggests
that some of the extra modes are due to disorder.
4.2 Experimental Procedures
Disk-shaped samples (of 1 cm radius, 5 mm thickness) were prepared by conventional
solid-state powder processing techniques. The procedure outlined by Nino et al. [108, 109]
was used for sample processing. Room temperature Raman spectra were measured with
a T 64000 Jobin Yvon triple Raman spectrometer equipped with a liquid-nitrogen-cooled
back-illuminated CCD detector. We used the 488 nm and 501 nm lines of the Ar+ ion
laser to excite Raman scattering. The measurements were done with the laser power on
the sample not exceeding 6 kW/cm2 and with an accumulation time of 20 seconds. The
spectra were taken in the back scattering geometry; the scattered light was not polarized
53
while the incident light had vertical polarization. The utilization of the subtractive mode
allowed us to reach low frequencies (below 150 cm−1).
4.3 Results and Discussion
A factor group analysis of the ideal pyrochlore structure [110] yields six Raman active
modes (R) and seven infrared active (IR) modes:
Γ = A1g(R) + Eg(R) + 4F2g(R) + 7F1u(IR) + F1u + 4F2u + 2F1g + 3A2u + 3Eu, (4–1)
where the A and B cations are placed on an inversion center, and all of the six Raman
modes involve motion of oxygens atoms only. Figures (4-2, 4-3, 4-4, 4-5) show the
Raman spectra for each sample along with individual lorentzian oscillators used for
each fit. Figure (4-6) shows the four spectra with scaled and shifted intensities for ease
of comparison. The appearance of more than six Raman modes in all four samples
confirmed the additional disorder or ionic displacements from the ideal atomic positions
in the pyrochlore structure in the investigated bismuth based samples. However, it was
reasonable to expect that many of the modes from the ideal pyrochlore structure would
still be present, and the assignment of these “ideal” modes was done by referencing
previous literature on diverse pyrochlores [101–104, 106, 107, 111, 112].
Table I shows the frequencies of the various observed bands for the four samples along
with the assignment of modes. It is not trivial to assign each band to a specific stretching
or bending vibration, since both Vanderborre et al. [101] and Brown et al. [102] show
that there is mixing of different vibrations for a particular band. Their work estimates
the contribution of each vibrational mode to a observed mode by calculating the potential
energy distribution. Table I lists only the most significantly contributing vibrational mode.
4.3.1 Tentative Assignment of the “Ideal” Modes
The A1g, and one of the F2g modes of the ideal structure are assigned to observed
features at 530 and 420 cm−1 respectively. These modes are mostly B -O stretching and
O-B -O bending vibrations as presented by the normal coordinate analysis of Ref. [101].
54
For the A1g mode, which is mostly due to the completely symmetrical elongation of the
BO6 octahedron, the frequencies of the four compositions agreed well with each other
(standard deviation ∆ ∼ 2 %), and with other Bi samples: Bi2Hf2O7 (∼ 550 cm−1) [103]
and BiYTi2O7 (520 cm−1) [104]. The A1g mode for the Nb based pyrochlore Cd2Nb2O7
is at 509 cm−1 and predicted at 482 cm−1 [53]. This mode does not seem to vary greatly
between pyrochlores: 523 cm−1 (Y2Ti2O7), 489 cm−1 (Ti2Mn2O7), 511 cm−1 (In2Mn2O7),
512 cm−1 (Tb2Ti2O7), 489 cm−1 (Ti2Mn2O7), 498 cm−1 (La2Zr2O7), since the variance
in the A1g frequency for all the samples mentioned, including the bismuth pyrochlores, is
3 %.
There was, however, a systematic increase in frequency of 3 percent for the A1g mode
for BMT over BMN and BZT over BZN. In the A1g mode, the B atom does not move
and thus the frequency of the mode should only depend on the square root of the force
constant. The 3 percent increase in Ta samples suggests a force-constant ratio of 1.06 for
Ta over Nb. This result corroborates work by Wang et al. where it has been reported that
in the octahedron, oxygen binds tighter to Ta than to Nb as much as a 1.10 force-constant
ratio [113]. Furthermore, the widths of the modes were also lower for BZT and BMT (γ ∼70 cm−1) than for BZN and BMN (γ ∼ 100 cm−1). The results for A1g are corroborated
by the mode located around 428 cm−1 which had the same trend, with frequencies ∼ 3
percent higher for the Ta samples than the Nb samples, and widths smaller for BZT and
BMT (γ ∼ 45 cm−1) than for BZN and BMN (γ ∼ 88 cm−1). This mode was tentatively
assigned by comparing the F2g mode at Cd2Nb2O7 calculated at 441 cm−1 [53] and
observed at 422 cm−1, and other pyrochlores Gd2Ti2O7 (455 cm−1), Tb2Ti207 (452 cm−1),
In2Mn2O7 [102] (442 cm−1 )and the bismuth based YBiTi2O7 (451 cm−1).
The Eg mode had an opposite trend; frequencies were significantly higher (> 10
percent) for BZN and BMN (∼ 310 cm−1) than for BZT and BMT (∼ 345 cm−1). In the
literature, the band assigned to Eg can have a significantly varying frequency: 250 cm−1
(Cd2Re2O7) [106], 297 cm−1 (BiYTi2O7), 312 cm−1 (Y2Ti2O7), 327 cm−1 (Ti2Mn2O7),
55
330 cm−1 (Tb2Ti2O7), 346 cm−1 (In2Mn2O7), ∼ 360 cm−1 (Bi2Hf2O7) and 405 cm−1
(La2Zr2O7). This is an 11 % variance for the frequencies of the compounds mentioned.
Fischer’s work offers further insight into this band. For Cd2Nb2O7 they estimate 300
cm−1 for Eg, but they also predict an additional F2g mode at 332 cm−1 and an IR mode
F1u at 360 cm−1. For our Raman spectra, the large widths of the Eg mode (γ ∼ 130
cm−1) gives the possibility of seeing multiple modes in this band. This suggests that this
band could have both the Eg mode and the F2g mode predicted by Fischer. Also, the IR
spectra of these samples have a mode very near the Raman frequencies (see Table I and
II) with widths γ ∼ 80 cm−1, which makes it possible that this band contains a normally
Raman-inactive F1u mode appear in the spectrum due to disorder. The dependence on
the B mass of the Raman modes supports this suggestion (the possibility of observing
normally Raman inactive modes will be discussed further in the next section). As for
the 236 cm−1 band, the Nb samples also had higher frequencies by as much as 5 percent.
This band is predicted at 265 cm−1 and measured at 279 cm−1 in Cd2Nb2O7. For other
pyrochlores the F2g mode is reported at 211 cm−1(Gd2TI2O7), 220 cm−1 (Tb2Ti2O7), 240
cm−1 (Cd2Re2O7), 289 cm−1 (Tl2Mn2O7) and 292 cm−1 (In2Mn2O7).
4.3.2 Tentative Assignment of the “Disorder” Modes
For the low-frequency region, the 180 cm−1 bands in BMN, BZN and BZT are
interesting because they have frequencies lower than those found in some other pyrochlore-structure
materials. For other Bi based samples, Bi2Hf2O7 showed no modes below 250 cm−1, and
in BiYTiO7 no modes were reported below 300 cm−1. As for the other pyrochlores, for the
hafnates and zirconates, the observed lowest mode is at 298 cm−1 [101]; for the manganites
it is at 292 cm−1 [102]; and, the lowest reported modes are above 210 cm−1 for Y2Ti2O7,
BYTi2O7 [104] and Cd2Re2O7 [106]. An exception is Tb2Ti2O7, where there is a band at
173 cm−1 assigned to an F2g mode. In the BZN literature, this mode seems to be difficult
to assign: In Ref. [113] it is assigned as the same normal vibration F2g of 255 cm−1 (where
the 255 cm−1 band belongs to a Zn-O stretch, and the 180 cm−1 belongs to Bi-O stretch)
56
[113], while other work has assigned it to a F2g band separate than the 255 cm−1 F2g mode
[114].
We propose an alternative tentative assignment. We propose this band is a normally
Raman-inactive/IR active mode that appears in the Raman due to the displacive disorder
of the A site in the Bi pyrochlores. We can recall that based on symmetry, the selection
rules result in some vibrational modes being optically inactive. For an inversion-symmetric
center, the mutual exclusion rule states a mode can be IR active or Raman active, but
never both. From random displacement disorder, we can expect new previously inactive
modes to appear in both IR and Raman spectra, but also that normally IR-only modes
appear in the Raman spectra and vice versa. For our samples, the infrared data of these
samples [52] show bands with very close frequencies to those reported here for the Raman
(compare Table I and II). Furthermore, the width of the Raman bands in cm−1 are
(71 and 75) for BMN and BZN and 58 for BZT, and for the IR modes, the widths are
84, 84 for BMN and BZN and 68 for BZT. This suggestion is also corroborated by the
calculations on Cd2Nb2O7 by Fischer et al., which gives a F1u mode at 190 cm−1, and no
F2g modes below 265 cm−1.
Given the somewhat unusual assignment proposed here, an extended discussion is
presented. It is important to recall that since the A and B sites are placed at inversion
centers in the ideal pyrochlore structure, the mutual exclusion rule states the active
Raman modes are inactive in the IR and vice versa [115–117]. The mutual exclusion
rule is a general result from symmetry and group theory, but for descriptive purposes,
consider small vibrations in the linear O-A-O molecule shown in Figure (4-7): all modes
are either Raman active or IR active [118, 119]. The symmetric stretching mode is not
infrared active because the net change in dipole moment is zero. This mode however is
Raman active because the stretching of each bond yields a positive change in polarizability
(a positive change in polarizability results from an increase in bond length). A similar
analysis would show that Q2, the antisymmetric stretch, would be IR active but not
57
Raman active. The two bending modes of this molecule appear in the IR because a dipole
moment is created in the direction perpendicular to the bonds; but they do not appear
in the Raman because in this configuration, the change in polarizability is zero for small
changes perpendicular to the bond. Now consider the nonlinear O-A-O molecule: we
can see that Q1 and Q3 have a net dipole moment change in the z direction, and Q2 in
the y direction. As for the polarizability, Q1 and Q3 both have a net change in the bond
direction. While for Q2, the nonlinear molecule has a nonzero (δαyz)/(δQ2) component,
where Py = αyzEz (the dipole moment in the y direction generated by Ez). Therefore all
modes in the nonlinear molecule are both IR and Raman active.
We also observed modes at about 80 cm−1 and about 150 cm−1. These frequencies
are close to modes found in the infrared (see Tables I and II), which suggests that these
modes are also normally Raman inactive modes that appear due to the displacive disorder.
The results also agree with Fischer’s work, which predicts two normally optically inactive
modes (Eu) at 69 and 133 cm−1 for Cd2Nb2O7, as well as a 71 cm−1 IR active F1u mode.
The Raman bands above 600 cm−1 are perhaps the most difficult to assign. The
614 cm−1 band agrees well for our four samples (variance ∼ 2 %). In BZN this band has
been assigned as an F2g mode [113, 114]. For Tl2Mn2O7, In2Mn2O7 and La2Zr2O7 there
is an assigned F2g mode at 512, 548 and 590 cm−1 respectively [101, 102]. In Cd2Re2O7
there is no band in this region. In Y2Ti2O7 and Gd2Ti2O7 a band near 570 cm−1 has been
attributed to the F2g mode, but the band is not present for all preparation methods [104].
BiYTi2O7 shows two bands at 588 and 612 cm−1, where the first is assigned as F2g and
the second to leftover TiO2 rutile. For Tb2Ti2O7 there is a band near 582 cm−1, but it is
not known if this band or their 452 cm-1 band is the F2g mode. For Cd2Nb2O7, Ref [53]
calculates no F2g modes in this region, two optically inactive modes at 579 cm−1 (F2u)
and 617 cm−1 (F1g). For our samples, the suggestion that this band is not a normal F2g
mode is supported by the similar band found in the IR (see Table I and II). Our results,
58
compared to the ample literature above, can not distinguish whether this band is a F2g
mode, or a normally inactive mode, or both.
A strong peak near 760 cm−1 is found in all four samples. This mode also seems
challenging to assign. In BZN, it has been previously attributed to an overtone [114] and
also to a stretching mode of the Nb-O bond [113]. The overtone assignment is the most
common for other pyrochlores: La2Zr2O7 (743 cm−1) [101]; In2Mn2O7 (∼ 700 cm−1) and
Tl2Mn2O7 (∼ 750 cm−1); Cd2Re2O7 (∼ 700 cm−1). Based on lattice dynamic calculations,
Maczka et al. [107] suggest that no fundamental F2g stretching mode should exceed 600
cm−1 for T2Ti2O7 and therefore assign higher modes to overtones . However, Fischer’s
calculations predict a 883 cm−1 F2g mode for Cd2Nb2O7. In Bi2Hf2O7 and Bi2Ti2O7 the
authors associate modes in the 650 - 800 cm−1 region with octahedral B -O stretching
modes. And, in Ref [104] the modes around 700 cm−1 are left unassigned for Gd2Ti2O7,
Y2Ti2O7 and BiYTi2O7. Therefore, assigning this mode based on the literature is not
trivial. For BMN, BMT, BZN and BZT, the high intensity of this mode suggests it is
more likely a fundamental mode rather than a two-phonon scattering process (overtone).
It is also possible that this mode is a normally Raman silent mode as well. As for the
Nb-O stretch suggestion, the systematic increase in frequency for BMN over BMT and
similarly for BZN over BZT suggests that it would be an assymetric stretching mode,
where the B cation moves as well (based on the assumption from the A1g mode and Ref.
[[113]] that Ta bonds stronger than Nb). Then, the 3% increase from Nb samples over Ta
samples is expected from the reported force constant ratio (kTa/kNb = 1.10) for Ta over
Nb and the reduced mass ratio (µ) of the BO6 octahedron (1.15).
(ωTa)2
(ωNb)2=
kTa
kNb
· µNb
µTa
(4–2)
µTa =mTa · (6m0)
mTa + (6m0). (4–3)
59
However, we must be careful in this analysis, because it is simplistic in using a single-spring
model for interatomic forces and relying on modes arising from just one, very local,
vibration. This is indeed a good agreement, but perhaps fortunate for such a simplistic
model.
Lastly, modes around 800 cm−1 were observed for all samples. Also, a weak 860
cm−1 mode is observed in BZN and BMN but not in BMT or BZT. This mode is close to
the 880 cm-1 F2g mode predicted for Cd2Nb2O7 [53] and corroborates their calculation.
However, based on the low amplitude and high-frequency of the mode it could be argued
that the mode is an overtone as well. It is also interesting that the infrared spectra of
these samples also show an 850 cm−1 mode for Nb samples but not for Ta samples [52].
There is yet another interpretation for this mode then. Chen et al. argued that this mode
appears in the IR due to the vibration of the unequal bond length O-A-O bond (where the
unequality in bonds is due to the displaced A cation). They argue that the mode appears
in BZN and BMN and not in BZT and BMT, because the Ta samples have less displacive
disorder, due to decrease in the lone pair character of Bi3+ by lone-pair hybridization with
Ta. The argument is based in the larger electronegativity of Ta over Nb. However, the
Raman spectra of these four samples suggest there is no difference in disorder between the
Nb samples and the Ta samples. Both Nb and Ta samples had modes that appeared due
to the relaxation of the selection rules.
4.4 Conclusions
The Raman spectra of BZT, BMN, BZT and BMT are very similar, suggesting no
major structural differences among these materials. Comparison to the other pyrochlore
structured materials showed differences between BMN, BMT, BZN and BZT and other
non Bismuth pyrochlores, distinctly in the low and high frequency regions. We see
that the differences in the Raman spectra to other pyrochlores can be explained by the
relaxation of the selection rules due to the displacive disorder in the Bi pyrochlores. A
discussion was offered in the assignment of each mode by comparison to the literature, but
60
we pointed out that these assignments are not final for various authors may have different
views. For the additional modes, the comparison to the IR data and computational work
by Fischer et al. allowed the assignment of these normally Raman inactive modes, and
corroborated Ref. [53] calculations. Finally, the existence of additional modes in the
Raman spectra of all four compounds suggests no difference in the amount of disorder
between Nb and Ta samples.
4.5 Future Work
Future work on this project could consist in collaborations with computational
groups to calculate the vibrational modes of pyrochlores to aid in the assignment of the
vibrational modes. The assignment of modes based on references to the literature can be
challenging, as different authors can have different assignments. Furthermore, some papers
would claim ”perfect agreement” or ”great coincidence” with other pyrochlores but only
acknowledge papers with similar conclusions, and ignore papers with differing views. For
his first submission to a peer-review paper, the author of this dissertation was guilty in
not using enough references. Therefore, calculations specific to the BZN structure could
provide further insight into the assignment of these modes.
61
Table 4-1. The Raman modes of BMN, BZN, BMT and BZT are shown along with theirtentative assignment. The class assignments (i.e. F2g) are meant for comparisonto the ideal pyrochlore structure.
Raman frequencies (cm−1)BMN BMT BZN BZT Mode Assignment78 72 76 69 Eu and or F1u: O’-A-O’ Bend150 148 150 151 Eu: O-A-O Bend186 185 184 F1u: A-BO6 stretch236 223 255 243 F2g: A-O’ Stretch350 310 343 309 Eg and/or F2g: O-B -O Bend414 428 418 433 Eg: O-B -O Bend513 530 528 541 A1g: Symmetric BO6
elongation603 619 612 622 F2g and/or F1g: B -O Stretch781 759 762 742 B -O Stretch819 805 804 809 Overtone862 860 F2g or Overtone
Table 4-2. The infrared modes with close frequencies to those of the Raman are shown(Work from Chen et al.) ∗ Chen assigns this mode to O-A-O bending.
Comparison to IR frequencies (cm−1)BMN BMT BZN BZT Mode Assignment86 83 81 O’-A-O’ Bend
149 142 145 O-A-O Bend178* 178 178 192 A-BO6 stretch367 336 340 303 A - O stretch599 642 624 639 B - O stretch850 850 Short A - O bond stretch
62
Tab
le4-
3.C
ompar
ison
bet
wee
nth
eR
aman
modes
ofB
MN
and
other
pyro
chlo
res
Mod
esfr
eque
ncy
and
assi
gnm
ents
inlit
erat
ure
BM
NC
d 2N
b 2O
7
(cal
c.)
[53]
YB
iTi 2
O7
[104
]B
i 2T
i 2O
7
[103
]G
d 2T
i 2O
7
[104
]T
b 2T
i 2O
7
[107
]La 2
Zr 2
O7
[101
]In
2M
n 2O
7
[102
]T
l 2M
n 2O
7
[102
]C
d 2R
e 2O
7
[106
]
7869
(Eu)
and
71(F
1u)
76(L
ong
A-O
stre
tch)
n.o.
n.o.
n.o.
n.o.
n.o.
n.o.
100
150
133
(Eu)
n.o.
n.o.
n.o.
n.o.
n.o.
n.o
n.o.
150
186
190
(F1u)
n.o.
n.o.
n.o.
173
(F2g)
n.o.
n.o.
n.o.
180
236
265
(F2g)
220
(F2g)
∼23
021
1(F
2g)
210
(F2g)
238
(∗F
2g)
292
(F2g
289
(F2g)
240
(F2g
+E
g)
350
300
(Eg),
332
(F2g),
360
(F1u)
297
(F2g)
360
312
(F2g)
310
(F2g),
330
(Eg)
307
(F2g)
346
(Eg)
327
Eg
320
Eu
414
441
(F2g)
451
(F2g)
n.o.
451
(F2g)
452
(F2g?)
405
(Eg)
428
(F2g)
n.o.
460
(F2g)
513
482
(A1g)
523
(A1g)
550
519
(A1g)
512
(A1g)
490
(A1g)
510
(A1g)
510
(A1g)
510
(A1g)
603
617
(F1g)
588
(F2g),
600
(TiO
2
ruti
le)
∼60
0∼
560
(F2g)
582
(F2g?)
591
(F2g)
548
(F2g)
510
(F2g)
n.o.
781
n.o.
725
780
(B-O
stre
tch)
708
672
(ove
r-to
ne)
743
(ove
r-to
ne)
700
(ove
r-to
ne)
700
(ove
r-to
ne)
700
(ove
r-to
ne)
810,
850
883
(F2g)
n.o.
n.o.
n.o.
n.o.
n.o.
n.o.
n.o.
n.o.
63
Figure 4-1. Pyrochlore structure. Red atoms signify oxygen in the BO6 octahedra andblue atoms signify oxygen in the O’-A-O’ chain. The A and B cations aredepicted by yellow and green respectively, while the white atoms showvacancies.
64
Figure 4-2. Raman spectrum of BMN (black) with the fitting (red) to the data. Theindividual lorentzian oscillators (blue) used in the fit are also shown.
65
Figure 4-3. Raman spectrum of BZN (black) with the fitting (red) to the data. Theindividual lorentzian oscillators (blue) used in the fit are also shown.
66
Figure 4-4. Raman spectrum of BMT (black) with the fitting (red) to the data. Theindividual lorentzian oscillators (blue) used in the fit are also shown.
67
Figure 4-5. Raman spectrum of BZT (black) with the fitting (red) to the data. Theindividual lorentzian oscillators (blue) used in the fit are also shown.
68
Figure 4-6. Raman spectra of BMN, BZN, BMT and BZT. Their intensities have beenscaled (by 1, 1.14, 1.64, and 2.46, respectively) and shifted for ease ofcomparison
69
Figure 4-7. The normal modes of vibration of a linear O-A-O molecule are shown (top),along with the modes of a nonlinear molecule (bottom). The IR and Ramanmodes are labeled.
70
APPENDIX AEXTRACTING ELECTROMAGNETIC WAVES AND THE OPTICAL CONSTANTS
FROM MAXWELL’S EQUATIONS
Spectroscopy studies the interaction of light with materials. We can quantify the
optical properties by defining a set of optical constants, such as the refractive index N(ω),
dielectric function ε(ω) and the conductivity σ(ω). This appendix first shows how the
optical constants are obtained from Maxwell’s equations and a propagating wave solution.
It then shows why they can be complex functions, and how the optical constants are
inter-related.
A.1 Introduction
In this derivation, we will assume an isotropic medium. This is an approximation that
fails for many interesting systems, but it is useful for a derivation seeking insight into the
optical constants. We start with Maxwell’s equations in cgs units [120, 121]:
(CGS units)
∇× ~H − 1
c
∂ ~D
∂t=
4π
c~J (A–1)
∇× ~E +1
c
∂ ~B
∂t= 0 (A–2)
∇ · ~D = ρ (A–3)
∇ · ~B = 0 (A–4)
where ρ is the charge density and−→E and
−→B are the electric field and the magnetic
induction respectively. The electric displacement−→D , the magnetic field
−→H , and the current
density−→J are given by the (linear) material equations:
71
~D = ε ~E (A–5)
~B = µ ~H (A–6)
~J = σ ~E (A–7)
σ is the conductivity and ε and µ are the permittivity and permeability. For vacuum,
the conductivity σ is equal to zero, and the permittivity and permeability are equal to
one. These equations for the material parameters are a linear approximation that neglects
contributions to the current and polarization from higher powers of the electric field (i.e.
E2). For this chapter, we focus only on the linear regime.
A.2 Electromagnetic Waves and the Optical Constants
This section shows how electromagnetic waves are extracted from Maxwell’s
equations. The derivations shows the origin and meaning of the optical constants ε
and σ. These quantities govern the propagation of light in a material, as well as the
phenomena in the interface of two media. To obtain EM waves, we use the two Maxwell’s
equations (A–1 and A–2) that interrelate the electric and magnetic field. First we take the
curl of equation A–1:
~∇× ~∇× ~E =−1
c
∂(∇× ~B)
∂t, (A–8)
and insert equation A–2:
~∇(~∇ · ~E)− (∇)2 ~E =−µ
c
∂
∂t(1
c
∂ ~D
∂t+
4π
c~J). (A–9)
For neutrally charged materials, ρ is zero and equation A–9 simplifies to
72
∇2 ~E =µ
c2
∂
∂t(∂ ~D
∂t+
4π
c~J) (A–10)
∇2 ~E =µ ε
c2
∂2 ~E
∂t2+
4πµσ
c
∂ ~E
∂t. (A–11)
For vacuum, this equation becomes the simple and familiar wave equation
∇2 ~E − 1
c2
∂2 ~E
∂t2= 0, (A–12)
and its solution is an unattenuated propagating wave in the form
~E = Re[ ~E0ei(~k·~r−ωt)], (A–13)
where Re indicates taking the real part. For the rest of this work, we will remember that
we have to take the real part of the electric field at the end, and write
~E = ~E0ei(~k·~r−ωt), (A–14)
and, inserting this solution into the vacuum equation, A–12, we obtain the dispersion
relation in vacuum
k02 =
ω2
c2, (A–15)
where k0 is the propagation constant in vacuum and equals 2π/λ, and ω is the angular
frequency 2π/T (where T is the period). For a material with nonzero conductivity, the
third term in equation A–11 will attenuate the propagating wave. The solution will be a
complex propagation constant:
k2 =ω2
c2(εµ) + i
4πµσ
c2ω. (A–16)
From now on, we just define a complex dielectric function:
73
k2 =ω2
c2εµ, (A–17)
and now we don’t have to keep track of a real conductivity σ and a real permittivity ε. We
just define a complex dielectric function and a complex conductivity related by:
ε = 1 + i4πσ
ω, (A–18)
where their real and imaginary parts are referred to by subscripts 1 and 2:
σ = σ1 + iσ2 (A–19)
ε = ε1 + iε2 (A–20)
It is useful to define complex material parameters. Now we can use either the
dielectric function or the complex conductivity to describe a system, because one
property can be obtained from the other. More importantly, responses (i.e. currents
and polarization) are not always local in time, meaning they are dependent on the driving
field at previous times and can therefore have a different phase to the driving field. The
definition of complex conductivity sums up this statement. For a material influenced by an
electric field E0e−iωt:
Jeiωt = σ1E0eiωt + iσ2E0e
iωt = σ1E0eiωt + σ2E0e
iωt−π/2, (A–21)
the real part of the conductivity is associated with the current generated in phase with the
electric field, and the imaginary part describes a 90 degree-out-of-phase current.
The refractive index N is also a complex function, and it is defined using:
k =ω
cN(ω), (A–22)
and with this definition:
74
N(ω) =√
εµ = n(ω) + κ(ω). (A–23)
We can show that the real part of the refractive index describes refraction (the change
of wavelength and speed of light inside the material) while the imaginary part describes
the attenuation of the electric field, by inserting equation A–23 into the solution of the
attenuated propagating wave inside the material A–14:
~E = ~E0ei(N ~k0·~r−ωt) (A–24)
~E = ~E0e−κ(ω)( ~k0·~r)ei(n(ω) ~k0·~r−ωt), (A–25)
and compare to the vacuum solution:
~E = ~E0ei( ~k0·~r−ωt). (A–26)
The refractive index also gives a nice relation between the electric field and magnetic
field. First, we remember that:
~∇ · ~E0ei(~k·~r−ωt) = i~k · ~E = iN~k0 · ~E, (A–27)
which holds only for our our isotropic assumption of N. With this result, Maxwell’s
equations A–3 and A.1 tell us that for a homogeneous, neutrally charged medium, the
propagating wave solutions yields:
i~k · ~D = 0, (A–28)
and
i~k · ~B = 0. (A–29)
75
This means that the electric field and the magnetic field must be perpendicular to the
direction of propagation. To get the relation between E and H we use Maxwell’s equation
A–2 and obtain:
i~k × ~E =1
µc(iω) ~H = 0, (A–30)
and since ~k and ~E are perpendicular to each other, we see that the magnitudes of E and
H are related by
Nk0E0 =1
c(ω)H0, (A–31)
which by definition of the propagation constant gives
µN(ω)E0 = H0. (A–32)
For reference, in SI units this relation is given by
√µ√ε(ω)E0 = H0 (A–33)
Therefore, the intensity per unit area will be proportional to the square of the electric
field. For ~k ∼ ~z
I(z) = Re[EH∗] ∼ E02e−2κ(ω)ω
cz, (A–34)
the exponential decay of the intensity is characterized by the absorption cofficient:
α = 2κ(ω)ω
c. (A–35)
76
Figure A-1. Boundary conditions at an interface. Electric fields for the incident, reflectedand transmitted propagating wave solutions at the interface between twomedia.
77
APPENDIX BBOUNDARY CONDITIONS. TRANSMITTANCE AND REFLECTANCE
The reflectance and transmittance at an interface between two media are related
to the optical constants by the condition that the transverse electric field and magnetic
field must be continuous. In this work, we will focus only on normal incident light. The
boundary conditions are:
Ei + Er = Et, (B–1)
and
Hi + Hr = Ht, (B–2)
where the subscripts i, r and t denote incident, reflected and transmitted respectively.
Using equations A–30 and A–32, which relate the magnetic field to the electric field, and
equations B–1 and B–2, we obtain
N1µ1Ei −N1µ1Er = N2µ2Et. (B–3)
Please note that subscripts 1 and 2 refer to the first and second medium (as in Fig. A-1),
and that the real and imaginary parts of N are given by n and κ as in equation A–23.
The negative term in N1µ1Er comes from equation A–30 and B–2 and the fact that the
reflected wave has a negative k value. Now we can solve for the reflection amplitude
coefficient:
r12 =Er
Ei
=N1µ1 −N2µ2
N1µ1 + N2µ2
. (B–4)
This quantity is complex, and allows for the reflected electric field to have a phase
difference from the incident electric field. This statement can be written as:
78
r12 = |r12|eiφ, (B–5)
where ψ is the phase change upon reflection and |r| is the magnitude of the electric field.
Similarly, through the same interface, the transmitted amplitude is:
t12 =Et
Ei
=2N1µ1
N1µ1 + N2µ2
. (B–6)
Most materials we study are nonmagnetic and have µ=1. From now on we will drop
the µ, but for materials with nonzero µ we can come back to equation B–4.
Nonmagnetic materials:
µ = 1. (B–7)
To relate r12 and t12 to transmittance and reflectance, we have to use the Poynting vector.
Amusingly, finding the transmittance between absorbing media is not a trivial endeavor
(as one can find equations for R and T in the literature that do not conserve energy). The
Poynting vector ~Sis given by:
~S = ~E × ~H∗ (B–8)
~S = ~E0ei~k·~r−iωt × (−~k∗ × ~E)ei~k∗·~r−iωt. (B–9)
The vector magnitude S will go as:
S = E0N∗E0e
i~k·~r−iωte−i~k∗·~r+iωt, (B–10)
and at the interface we then get:
S = E0N∗E0. (B–11)
79
For reflectance, the answer is easier because both media have the same refractive
index and the reflectance goes as the ratio of the electric fields:
R = r12r∗12 = |r12|2 =
|N1 − N2|2
|N1 + N2|2(B–12)
R =(n1 − n2)
2 + (κ1 − κ2)2
(n1 + n2)2 + (κ1 + κ2)
2 . (B–13)
For the transmittance, this author humbly gives the right answer only for an interface
between one non-absorbing medium n1 and an absorbing medium characterized by n2 and
κ2. To get the time-averaged flux (power), we have to get the real part of the Poynting
vector. Therefore T is given by
T =~St + ~S∗t~Si + ~S∗i
=n2
n1
t12t12∗. (B–14)
Using equation B–6 for real N1
t12 =Et
Ei
=2n1
n1 + N2
(B–15)
T =4n1n2
(n1 + n2)2 + (κ2)2. (B–16)
For one interface, the useful case is the simple case of the interface between air and a
sample with refractive index N :
r =Er
Ei
=(1− n)− iκ
(1 + n) + iκ(B–17)
R = |r|2 =(1− n)2 + κ2
(1 + n)2 + κ2. (B–18)
The transmittance equation for one interface is not so useful because samples have a
finite thickness, (and we wouldn’t place a detector in a medium). For finite thickness
80
samples, the boundary conditions have to be written for all interfaces. This means that
the measured reflectance and transmittance would be affected by internal reflections inside
the sample. The derivation for this expression can be found on Ref. [122]
t = t12t23eiδ[1 + (r12r23e
2iδ) + (r12r23e2iδ)2 + ...]. (B–19)
This infinite series reduces to
t =t12t23e
iδ
1− r12r23e2iδ, (B–20)
where δ is
δ =ω
cNd = n
ω
cd + i
α
2d. (B–21)
When dealing with multiple layer films, the equations for r and t become increasingly
complex and they are not trivial to solve. The reader is suggested to use references
[122, 123] and [124] for the matrix method.
Notice that there are two unknowns in the determination of the refractive index,
the real and imaginary part. Therefore, two separate measurements would be required
(i.e. T and R) to obtain the optical constants. However, certain samples can be almost
completely absorbing or completely reflective (i.e. metallic) so that only the reflectance
from the air-sample interface can be measured (We call this single-bounce reflectance).
In this case, the equation for single bounce reflectance is valid because there there are
no internal reflections. But, it leaves us with two unknowns and only one measurable
quantity. One technique to overcome this problem is Kramers Kronig Analysis, which
consists in a wide frequency measurement of the reflectance to obtain the phase change in
reflectance from equation B–5. The following section explains KK analysis.
81
APPENDIX CRESPONSE FUNCTIONS AND KRAMERS KRONIG ANALYSIS
The Kramers-Kronig relations refer to the inter-dependence of the real and imaginary
part of a linear response function (such as the permittivity and the conductivity) due to
causality. To derive the relations, first we define what a linear response function is:
X(t) =
∫ ∞
−∞G(t− t′)f(t′) dt′, (C–1)
X(t) is a response of the system (i.e. current or polarization), f(t − t′) is the external
stimulus (i.e. electric field), and G(t − t′) is the response function. Examples of response
functions are the conductivity and the susceptibility
J(t) =
∫ ∞
−∞σ(t− t′)E(t′) dt′, (C–2)
P (t) =
∫ ∞
−∞χ(t− t′)E(t′) dt′. (C–3)
The integrals in equations C–2 and C–3 state that responses are generally nonlocal in
time. Responses depend on stimuli applied at previous times, the same way the velocity of
a particle depends on previous forces (ie. a falling object’s dependence on how long it has
been falling). More importantly, these equations show that the response of a system won’t
always be in phase with the stimulus. This out of phase possibility is another justification
for the use of complex response functions. Equation C–1 can be rewritten in a simpler
form by writing the fourier transforms of X, G and f :
X(ω) = G(ω)f(ω), (C–4)
where
82
G(ω) =
∫ ∞
−∞G(t− t′)eiω(t−t′) d(t− t′). (C–5)
This Fourier transform equation implies that
G∗(−ω) = G(ω), (C–6)
which is an useful relation for the oddness and evenness of the real and imaginary parts of
complex functions. We will use this relation later.
Causality puts an important restriction on response functions. It states that a
response G at t cannot depend on future times:
G = 0 ; for t− t′ < 0. (C–7)
We will quickly derive that this restriction causes the Kramers Kronig relations:
G1(ω) =1
π
∫ ∞
−∞
G2(ω′)
ω − ω′dω′ (C–8)
G2(ω) =−1
π
∫ ∞
−∞
G1(ω′)
ω − ω′dω′. (C–9)
To derive these relations between the real and imaginary parts of a response function,
we first map G on the complex plane by rewriting equation C–5 using both real and
imaginary frequencies:
ω = ω1 + iω2, (C–10)
G(ω) =
∫ ∞
−∞G(t− t′)eiω1(t−t′)e−ω2(t−t′) d(t− t′). (C–11)
83
The exponential term with ω2 in equation C–11 imposes that for positive ω2, the term
(t − t′) must be positive as well. Similarly, for negative ω2, (t − t′) must be negative. But,
causality, C–7, states that G = 0 for negative (t − t′); therefore a response function that
obeys causality and equation C–11 is restricted to the upper half of the complex plane (see
Fig. C-1). The purpose of this long rant is to map G and use Cauchy’s theorem to relate
the value of G at a certain frequency ω0 to the other frequencies around it. First we use
Cauchy’s theorem for an analytic function G(ω) [120]:
∮G(ω)
ω − ω0
dω = 0, (C–12)
and Fig. C-1 shows the closed loop we will use in the integral. We draw a semicircle
around the frequency of interest, ω0, with an infinitesimal radius ε, and an outer semicircle
with radius R. We can draw the boundary of the integral this way because we know that
G(ω) is zero for negative imaginary frequencies (good old causality).
We evaluate the whole integral, which should be zero according to Cauchy’s theorem:
∮ θ=π
θ=0
G(Reiθ)
ReiθReiθ dθ+
∮ ε−ω0
−∞
G(ω)
ω − ω0
dω+
∮ θ=π
θ=0
G(ω0 + εeiθ)
εeiθiεeiθ dθ+
∮ ∞
ε+ω0
G(ω)
ω − ω0
dω = 0.
(C–13)
The integration along the outer semicircle should be zero for large enough R, meaning
response functions should go to zero at high enough frequencies (a nature friendly
condition supported by the tendency of x-rays and gamma rays to easily penetrate
matter). If we take the limit of ε to zero, the equation yields:
G(ω0) = P−1
π
∫ ∞
−∞
G(ω)
ω − ω0
dω (C–14)
where P denotes principal value, meaning that the integral does not use the value at ω0
(So if a program would be written to evaluate this integral, then the value G(ω0)∆ω,
where ∆ω is the spacing between data, would not be added to the sum). Notice that this
84
result relates the value of the response function G at ω0 to the value of G(ω) at all other
frequencies. Also notice the convenience that we only need real frequencies ω. The i term
in front of the integral is responsible for relating the real part of the response function, G1,
to the imaginary value, G2, at all other frequencies (and viceversa). This yields the KK
equations C–8 and C–9. Notice that this result is due to causality; if the integral in the
lower half of the complex plane were not zero, then we would have no interesting result.
To eliminate the negative frequencies in the KK equations, we use the properties of
the complex conjugate of G in equation C–6 and show that
G1(−ω) = G1(ω) (C–15)
G2(−ω) = −G2(ω). (C–16)
Using the even property of G1 and the oddness of G2, we obtain the more useful KK
relations:
G1(ω) =2
π
∫ ∞
0
ω′G2(ω′)
ω2 − ω′2dω′ (C–17)
G2(ω) =−2ω
π
∫ ∞
0
G1(ω′)
ω2 − ω′2dω′. (C–18)
Therefore, for the conductivity response function we get:
σ1(ω) =2
π
∫ ∞
0
ω′σ2(ω′)
ω2 − ω′2dω′ (C–19)
σ2(ω) =−2ω
π
∫ ∞
0
σ1(ω′)
ω2 − ω′2dω′. (C–20)
As a word of caution, we do not assume that the KK relations work for any complex
quantity and assume that we can equate the real and the imaginary parts by equations
C–8 and C–9 for any arbitrary complex function. For example, to get the KK relations
85
for the dielectric function ε, we have to relate it to the response function χ by ε = 1 + χ.
Then we obtain that:
ε1(ω)− 1 =2
π
∫ ∞
0
ω′ε2(ω′)
ω2 − ω′2dω′ (C–21)
ε2(ω) =−2ω
π
∫ ∞
0
ε1(ω′)
ω2 − ω′2dω′. (C–22)
These words of caution are most relevant for the derivation of the KK relations between
Reflectance R and the phase change φ. The starting response function is the reflection
amplitude r, where the response is Er and the stimulus is Ei
Er(ω) = r(ω)Ei(ω), (C–23)
r is then rewritten as
r = |r|eiφ, (C–24)
where |r|2 is the measurable single-bounce reflectance. We take the logarithm of equation
C–24 to separate `n r as the real part and φ as the imaginary part. The derivation of this
Kramers Kronig relation for the logarithmic function is difficult because r goes to zero at
infinite frequencies and ln n r would then ’blow up’. The reader is referenced to Wooten’s
Optical Properties of Solids [124] for the derivation. The result is:
`n|r(ω)‖ =2
π
∫ ∞
0
ω′φ(ω′)− ωφ(ω)
ω′2 − ω2dω′ (C–25)
φ(ω) =−2ω
π
∫ ∞
0
ln‖r(ω′)‖ − ln‖r(ω)‖ω′2 − ω2
dω′, (C–26)
and these are the important relations we use the most in this work. The main point is
that by measuring one quantity R over a broad spectra, we can obtain a second variable
86
φ that allows us to obtain the real and imaginary values of the optical constants shown in
the equation:
r = |r|eiφ =(1− n)− iκ
(1 + n) + iκ. (C–27)
Notice that the integrals in C–25 and C–26 span all frequencies. Experimentally, this
is impossible and requires that we estimate(guess) the behavior at the lower or higher
ends of the spectrum. Fortunately, the denominator in the KK integral, C–8, suggests
that G1 at a frequency ω depends mostly on G2 at frequencies close to ω. Furthermore, we
estimate the lower and higher ends of the spectrum by using physical models that explain
the behavior of the optical constants. The estimation for free electrons and bound charges
from the Drude and Lorentz model will be discussed in the next section.
87
Figure C-1. Closed loop integration around a frequency point ω0 in the complex plane.Causality requires G to be zero in the bottom half.
88
APPENDIX DMICROSCOPIC MODELS FOR THE OPTICAL CONSTANTS
The previous sections dealt with the derivation of the optical constants and their
relation to the behavior of light inside a material. Appendix A-C showed how to relate
optical constants to the reflectance. We now need physical models that predict the
behavior and values of the optical constants. These models are necessary to understand
the physics of a system from its optical properties (or else the optical constants are just
numbers to keep tally of). This section deals with two of the most useful models: The
Drude model and the Lorentzian model. These models are also useful because they allow
us to estimate the behavior of electrons at low and high driving frequencies. For models
for other systems, such as superconductors, the reader is encouraged to read Wooten’s and
Dressel’s books [120, 124].
D.1 The Drude Model
The Drude model describes the optical response of a free electron. By free, we
mean that there is no acting force (or potential) on the electron beside that of the
electromagnetic driving field (light). In the Drude model, there is also a damping term
that includes possible energy-loss mechanisms. The above statements are best summarized
in the ~F = m~a differential equation:
− e ~Eoe−iωt − γm~v = m
d~v
dt, (D–1)
where the first term is the EM field, and the second term represents the damping term
with a scattering rate γ equal to the inverse of the relaxation time τ . In the Drude model,
the force arising from the magnetic field is ignored due to its smaller value compared to
the electric force. To incorporate “not so free” electrons, the drude model replaces the
mass m by an effective mass m∗, to estimate small electron-lattice and electron-electron
interactions. The solution to equation D–1 using m∗ is:
89
~v(ω) =−e ~Eo
m∗
−iω + γe−iωt. (D–2)
The velocity gives us the current:
~J(ω) = −Neev(ω), (D–3)
where e is the charge of an electron and Ne is the number of conduction electrons per unit
volume. Therefore our complex conductivity σ is then:
σ(ω) =Nee2τ
m∗1− iωτ
(D–4)
σ(ω) =σ0
1− iωτ, (D–5)
where σ0 is the DC conductivity [125, 126]:
σ0 =Nee
2τ
m∗ (D–6)
High frequency (Relaxation) regime.
The Drude model is also very useful for Kramers Kronig Analysis, because we can
guesstimate how free electrons behave at low and high frequencies. For frequencies much
higher than the scattering rate (remember we are in CGS units) (ωÀγ):
ε1 = 1− 4πσ0τ
1 + (ωτ)2≈ 1− 4πσ0
ω2τ, (D–7)
and ε2 falls off even faster as 1/ω3. Equation D–7 can be rewritten as:
ε1 ≈ 1− ωp2
ω2, (D–8)
by defining the plasma frequency ωp:
90
ωp2 ≡ 4π
Nee2
m∗ = 4πσ0
τ. (D–9)
Good metals have plasma frequencies in the VIS-UV. From equations D–7 and D–9, we
can see that good conductors (metals) have negative ε1 at frequencies much lower than the
plasma frequency. For the refractive index, we use equation D–8 and ignore the fast dying
ε2. We obtain:
n(ω) ≈ 1− ωp2
2ω2(D–10)
and a κ that falls even faster as 1/ω6. Therefore, we get that the reflectance due to
conduction electrons falls off as 1/ω4 for high enough frequencies:
limω→∞
RDrude(ω) ≈ ωp2
4ω4. (D–11)
Low frequency (Hagen-Rubens) Regime:.
For frequencies much lower than γ, σ1 approaches its DC value while the imaginary
part approaches zero for low frequencies.
σ1 =σ0
1 + (ωτ)2≈ σ0 (D–12)
σ2 = σ0(ωτ)
1 + (ωτ)2≈ σ0(ωτ). (D–13)
As for the dielectric function, the real part approaches a constant negative value while the
imaginary part diverges:
ε1 ≈ −4πσ0τ = −(ωpτ)2 (D–14)
ε2 ≈ 4πσ0
ω, (D–15)
91
and the real and imaginary part of the refractive index both approximate:
n(ω) ≈ κ(ω) ≈ [ωp
2τ
2ω]1/2
. (D–16)
We then obtain the useful result:
limω→0
RDrude(ω) ≈ 1− [2ω
πσ0
]1/2
. (D–17)
D.2 Lorentzian Oscillators
The Lorentz model is a simple approximation for modeling bound charges. To obtain
it, we add to the free electron equation D–1, a restoring force (k ·~r) that keeps the electron
bounded:
− e ~Eoe−iωt − γm
d~r
dt−mω0
2~r = md2~r
dt2, (D–18)
and the solution is:
~r(ω) =−e ~Eo
m∗
(ω02 − ω2)− iωγ
e−iωt. (D–19)
Introducing the polarization definition (for isotropic media), ~P = χ ~E = −eNe~r · ~E, we
obtain for the dielectric function and the conductivity:
ε(ω) = 1 +ωp
2
(ω02 − ω2)− iωγ
(D–20)
σ(ω) =ω
4πi
ωp2
(ω02 − ω2)− iωγ
, (D–21)
where ωp is referred to as the oscillator strength frequency and equal to Ne e2/m∗. Notice
that the conductivity equation reduces to the Drude equation for ω0 = 0. Since, there
may be different electrons bounded with different restoring forces, we will introduce the
subscript j to denote the electrons (Nj) bounded with a characteristic frequency ω0j.
92
It is useful to see how Nj electrons behave at low and high frequencies. For
frequencies much lower than ωpj and 1/τj, the real part of the conductivity σ1 and ε2
drop to zero. The real part of the dielectric function goes as:
ε1(ω) = 1 +ωpj
2(ω0j2 − ω2)
(ω0j2 − ω2)2 + ω2γ2
j
≈ 1 +ωpj
2
ω0j2
= 1 + ∆εj. (D–22)
This equation shows that at zero frequency, the set of Nj oscillators contribute to the
dielectric of the material by
∆εj =ωpj
2
ω0j2. (D–23)
This is a very useful result. In the infrared region, we approximate all the contributions
from processes in the visible and UV by introducing the high-frequency dielectric constant
ε∞. The Lorentz model is then rewritten as:
ε(ω) = ε∞ +∑
j
ωpj2
(ω0j2 − ω2)− iωγj
. (D–24)
For KK analysis purposes, we see that the low frequency limit is:
limω→0
εLorentz = ε∞ +∑
j
∆εj = Constant. (D–25)
or more importantly, that the dielectric function will be real and constant in this limit.
Therefore the reflectance due to bound charges should stay a constant towards lower
frequencies.
For high frequencies, we can see from the conductivity term:
σ(ω) =ω
4πi
ωpj2
(ω0j2 − ω2)− iωγj
, (D–26)
that at high enough frequencies (ω >> ω0j) the limit will be the same as that of the Drude
model. For high enough frequencies, bound charges will give the same response as free
electrons:
93
limω→∞
RLorentz ∼ 1
ω4. (D–27)
This limit works only for frequencies at least higher than UV-VIS, since there are
processes such as band transitions that give rise to the ε∞ dielectric response. In our
Kramers Kronig programs, we typically assume the 1/ω4 dependence of the reflectance
only for frequencies above 106 cm−1.
94
APPENDIX EINTERFEROMETERS
E.1 Derivation
In a Michelson interferometer, an electromagnetic wave is split into two different
paths by a beam-splitter, and then recombined and directed into a detector (See Fig.
E-1). The intensity at the detector is measured as a function of path length to obtain the
interferogram. The Fourier transform of the interferogram then gives you the spectrum
(intensity vs frequency). By referring to Figure E-2, we can see that if the electric field
intensity at the source is E0 with propagation vector k, then the electric field after
recombination at the beamsplitter is:
Esum = E0rbtbeψb+π(e2ikd1 + e2ikd2), (E–1)
where the reflectance and transmittance amplitudes of the beamsplitter are rb and
tb, and the path length to the fixed and movable mirror are d1 and d2 respectively.
In this equation, we have assumed a very thin beamsplitter with no absorption, and
perfect mirrors (that change the phase by a factor of π). At the detector, we measure the
intensity of the electric field:
I = EsumE∗sum = E0RbTb(e
2ikd1 + e2ikd2)(e−2ikd1 + e−2ikd2), (E–2)
I = 2I0RbTb(1 + cos kx), (E–3)
where x is the path difference 2(d2 − d1), and I0 is the original source intensity:
I0 = E0E∗0 . (E–4)
If we use cgs units, then we can replace k with 2πυ(where frequency has units of cm−1).
The total intensity at the detector is the sum of intensity at all frequencies:
95
Itotal =
∫ ∞
0
2I0(ω)Rb(ω)Tb(ω)(1 + cos ωx) dω, (E–5)
assuming that the detector has a frequency-independent responsivity (a good approximation
for bolometers in the FIR). Notice from equation E–5, that the coherence (interference)
term comes from the cos(ωx) term. And, for a perfect beamsplitter:
Rb = Tb =1
2(E–6)
the whole beam (intensity I0) would be measured at the detector at a zero path difference.
In the incoherent limit (x approaches infinity), the total intensity is just the sum of
all intensities:
limx→∞
Itotal =
∫ ∞
0
2I0(ω)Rb(ω)Tb(ω) dω. (E–7)
This term represents the baseline of the interferogram (see Figure E-3). In practice,
we measure the interferogram and substract the baseline intensity:
Is =
∫ ∞
0
2I0(ω)Rb(ω)Tb(ω) cos ωx dω, (E–8)
where Is denotes the scaled intensity. Our goal is now to find the spectrum I(ω) from the
interferogram. Take the fourier transform of both sides of the above equation using ω2 as a
dummy variable:
∫ x=∞
x=−∞Is(x)e−iω2x dx =
∫ x=∞
x=−∞e−iω2x
∫ ∞
0
2I0(ω)Rb(ω)Tb(ω) cos ωx dω dx, (E–9)
rearrange
∫ x=∞
x=−∞Is(x)e−iω2x dx =
∫ x=∞
x=−∞2I0(ω)Rb(ω)Tb(ω)
∫ ∞
0
e−iω2x cos ωx dω dx, (E–10)
96
use the relation (where δ(ω) denotes the Dirac delta function)
∫ x=∞
x=−∞eiωxe−iω2x dx = δ(ω2 − ω), (E–11)
∫ x=∞
x=−∞Is(x)e−iω2x dx =
∫ ∞
0
2I0(ω)Rb(ω)Tb(ω)(δ(ω2 − ω)
2+
δ(ω2 + ω)
2) dω, (E–12)
the first delta function drops the dummy ω2 and the second delta function yields zero (for
there are no negative frequencies on the integral). Our final result relates the intensity
spectra to the measured interferogram:
2I0(ω)Rb(ω)Tb(ω) =
∫ x=∞
x=−∞Is(x)e−iωx dx. (E–13)
E.2 Properties of the Interferogram
Symmetry.
If the interferogram Is(x) is perfectly symmetric, then equation E–13 becomes a cosine
transform and I(ω) is real. However, if the interferogram is not perfectly symmetric (like
the noisy hand drawing in Figure E-2), then the spectrum will look like:
2I0(ω)Rb(ω)Tb(ω) =
∫ x=∞
x=−∞Ieven(x) cos(ωx) dx− i
∫ x=∞
x=−∞Iodd(x) sin(ωx) dx, (E–14)
which gives the spectrum a phase φ
2|I0(ω)|eiφRb(ω)Tb(ω) =
∫ ∞
x=−∞Is(x)e−iωx dx. (E–15)
Sample and reference measurements.
The measured interferogram always has the efficiency of the beamsplitter included.
However, when we measure reflectance or transmittance, we always measure a reference
97
(air for transmittance, and a mirror for reflectance). The ratio of the sample to reference
measurement eliminates the efficiency of the beamsplitters.
Rsample(ω)
Rmirror(ω)=
2|Isample(ω)|eiφRb(ω)Tb(ω)
2|Imirror(ω)|eiφRb(ω)Tb(ω). (E–16)
E.3 Resolution
The resolution in frequency is limited by the maximum path difference of the
interferogram. To conceptualize this statement, notice the poorly drawn wave with
frequency ω1 in Figure E-4, and the waves ω2, ω3 and ω4 which are exact replicas of poorly
drawn wave 1, but shrunk. For the two waves with distinct frequencies (ω1 and ω2), the
signals could be resolved easily because their sum will display interference. However,
for the waves with close frequencies (ω3 and ω4), we would have to scan farther to see
interference. If we scan too shortly, then we just see a sum of the same wave and we can’t
resolve them. This is analogous to a beat wave. For a source with two frequencies ω1 and
ω2, the interferogram will give you
cos(ω1x) + cos(ω2x) (E–17)
The beat wave is:
cos((ω2 − ω1)
2x), (E–18)
to see a minimum, you have to scan till:
(ω2 − ω1)
2xmax =
π
2. (E–19)
Therefore, your resolution (the ability to distinguish ω2 from ω1) is:
R =1
xmax
. (E–20)
98
Figure E-1. Diagram of a basic interferometer.
Figure E-2. Addition of the electric field from the two different paths. The amplitude ofthe electric field of the top mirror is shown as it progresses through the path.
99
Figure E-3. Caricature of an interferogram (intensity vs. path difference). The limit as xgoes to infinity gives the baseline.
Figure E-4. Caricature of the resolution between two waves. A longer scan is needed toobserve destructive interference between two waves of closer frequency.
100
APPENDIX FSURFACE PLASMONS
Surface plasmons are the solution to Maxwell’s equations for a propagating wave
confined within an interface between two media. This wave is confined in the interface by
a e−α1z term in the top and a eα2z term at the bottom (see Fig. F-1):
Etop = (Etx, E
ty, E
tz)e
i(kxx−ωt)e−α1z, (F–1)
Ebot = (Ebx, E
by, E
bz)e
i(kxx−ωt)eα2z, (F–2)
and similarly for the magnetic field:
H top = (H tx, H
ty, H
tz)e
i(kxx−ωt)e−α1z, (F–3)
Hbot = (Hbx, H
by, H
bz)e
i(kxx−ωt)eα2z. (F–4)
These forms of the electric and magnetic field have the properties:
∂ ~E
∂t=
∂
∂x(Ex, Ey, Ez)e
(ikxx−ωt)e−α1z = −iω ~E (F–5)
∂ ~H
∂x=
∂
∂x(Hx, Hy, Hz)e
(ikxx−ωt)e−α1z = ikx~H (F–6)
∂ ~H
∂y= 0 (F–7)
∂ ~H top
∂z= −α1
~H top (F–8)
∂ ~Hbot
∂z= α2
~Hbot. (F–9)
101
The propagating surface wave can have two different polarizations. One is Transverse
Electric (TE), where the only component of the electric field is transverse to the direction
of propagation. And the second is Transverse Magnetic (TM), where instead, the electric
field is in the plane of incidence, meaning that it has a component along the direction of
propagation and a component normal to the surface (see Fig. F-2). We’d like to explore
how these surface waves behave for these two cases. For TM, where the electric field is in
the plane of incidence,
EtopTM = (Et
x, 0, Etz)e
i(kxx−ωt)e−α1z, (F–10)
and for TE, the electric field is perpendicular:
EtopTE = (0, Et
y, 0)ei(kxx−ωt)e−α1z, (F–11)
the sum of these two should give us the total electric field
~E = ~ETE + ~ETM . (F–12)
To derive the dispersion relation for these waves, we have to set the boundary
conditions across the interface. Let’s first look at TM polarization:
TM polarization.
The boundary conditions across an interface tells us that the electric field parallel to
the surface should be continuous. At z = 0
Etop = (Etx, 0, E
tz) H top = (0, H t
y, 0) (F–13)
Ebot = (Ebx, 0, E
bz) Hbot = (0, Hb
y, 0), (F–14)
102
Etop = (Ex, 0, Etz) H top = (0, Hy, 0) (F–15)
Ebot = (Ex, 0, Ebz) Hbot = (0, Hy, 0), (F–16)
since the boundary conditions state that:
Etx = Eb
x (F–17)
H ty = Hb
y. (F–18)
Now, lets relate the electric field to the magnetic field by using Maxwell’s A–1 with no
surface current:
∇× ~H =ε
c
∂ ~E
∂t. (F–19)
To solve this equation we can use:
∇× ~H =ε
c
∂ ~E
∂t, (F–20)
∇× ~H = ~x(∂Hz
∂y− ∂Hy
∂z) + ~y(
∂Hx
∂z− ∂Hz
∂x) + ~z(
∂Hy
∂x− ∂Hx
∂y), (F–21)
∇× ~HTM = −∂Hy
∂z~x +
∂Hy
∂x~z (F–22)
∇× ~ETE = −∂Ey
∂z~x +
∂Ey
∂x~z, (F–23)
∇× ~HTE = ~y(∂Hx
∂z− ∂Hz
∂x) (F–24)
∇× ~ETM = ~y(∂Ex
∂z− ∂Ez
∂x). (F–25)
103
Using the previous relations, we begin with top TM waves:
∇× ~H topTM = α1Hy~x + ikxHy~y (F–26)
ε1
c
∂ ~E
∂t= −iε1ωEx~x− iε1ωEtop
z ~z, (F–27)
then, bottom:
∇× ~HbotTM = −α2Hy~x + ikxHy~y (F–28)
ε2
c
∂ ~E
∂t= −iε2ωEx~x− iε2ωEbot
z ~z, (F–29)
and equate x components for both:
α1Hy = −iωε1Ex (F–30)
α2Hy = iωε2Ex (F–31)
7→ α2
α1
= −ε2
ε1
. (F–32)
Since α1 and α2 are positive, equation F–32 shows that the propagating wave for
the surface is only possible for the interface between a negative dielectric (metal) and a
positive dielectric (i.e. air or quartz):
ε2
ε1
= −1. (F–33)
As a side note, notice that we are assuming that the dielectric constant for the
metal is negative and REAL. This assumption is valid for frequencies below the plasma
frequency, but higher than the scattering rate.
TM Dispersion relation.
To obtain the dispersion relation, we use equation A–11 we obtained back in
Appendix A, and use the properties for the curl of ETM listed above:
∇2 ~E =ε
c2
∂2 ~E
∂t2, (F–34)
104
(−k2x + α2
1)~Etop =
−ε1ω2
c2
∂2 ~Etop
∂t2, (F–35)
(−k2x + α2
2)~Ebot =
−ε2ω2
c2
∂2 ~Ebot
∂t2, (F–36)
Substracting equation F–35 from F–36 and a lot of algebra gives:
α21 −
ε22
ε21
=ω2
c2(ε1 − ε2) (F–37)
α21 =
ω2
c2
−ε1
ε1 + ε2
, (F–38)
and the dispersion equation:
kx =ω
c
√ε1ε2
ε1 + ε2
. (F–39)
TE polarization.
With the above results, we can show that there are no surface plasmons for TE
polarization. We begin with Maxwell’s A–2:
∇× ~E =−1
c
∂ ~H
∂t(F–40)
Referencing equations F–23 through F–25, and remembering that Ey and Hx are
continuous:
∇× ~Etop = α1Ey~x + ikxEy~y (F–41)
∇× ~Ebot = −α2Ey~x + ikxEy~y (F–42)
1
c
∂ ~H top
∂t=
iω
cHx~x +
iω
cH top
z ~z (F–43)
105
1
c
∂ ~Hbot
∂t=
iω
cHx~x +
iω
cHbot
z ~z, (F–44)
we obtain:
Hx = iα2cEy
ω(F–45)
Hx = −iα1cEy
ω(F–46)
H topz =
ckxEy
ω(F–47)
Hbotz =
ckxEy
ω. (F–48)
Since α is positive, Hx only makes sense if Ey is zero. This combined with equations
F–46 through F–48 result in every component being zero:
~ETE = ~HTE = 0. (F–49)
Therefore, a propagating surface wave with TE polarization is not allowed.
106
Figure F-1. Electric fields for a confined surface wave propagating in the x direction. Topand bottom solutions.
Figure F-2. Diagrams for the electric and magnetic field components of TM and TEpolarizations of a surface wave.
107
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BIOGRAPHICAL SKETCH
Daniel Arenas was born in Colombia in 1983. He moved to the U.S.A after graduating
from high school. He attended Edison Community College in Naples, FL and then
transferred to the University of North Florida in Jacksonville. By chance, he took some
upper-level physics classes, loved them, and ended up switching his major to physics.
After getting his B.S., he was accepted into University of Florida in 2004 and joined the
TannerLab in his second year. In the last five years, he has met wonderful people and
made great friends.
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