Optical Response in Planar Heterostructu

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Optical Response in Planar Heterostructures: From Artificial Magnetism to Angstrom-Scale Metamaterials
Citation
Papadakis, Georgia Theano
(2018)
Optical Response in Planar Heterostructures: From Artificial Magnetism to Angstrom-Scale Metamaterials.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z97H1GS1.
Abstract
The idea of expanding the range of properties of natural substances with artificial matter was introduced by V. G. Veselago in 1967. Since then, the field of metamaterials has dramatically advanced. Man-made structures can now exhibit a plethora of extraordinary electromagnetic properties, such as negative refraction, optical magnetism, and super-resolution imaging. Typical metamaterial motifs include split ring resonators, dielectric and plasmonic particles, fishnet and wire arrays. The principle of operation of these elements is now well-understood, and they are being exploited for practical applications on a global scale, ranging from telecommunications to sensing and biomedicine, in the radio frequency and terahertz domains. Accessing and controlling optical and near-infrared phenomena requires scaling down the dimensions of meta- materials to the nanometer regime, pushing the limits of state-of-the-art nano- lithography and requiring structurally less complex geometries. Hence, within the last decade, research in metamaterials has revisited a simpler, lithography- free structure, particularly planar arrangements of alternating metal and dielectric layers, termed hyperbolic metamaterials. Such media are readily realizable with well-established thin-film deposition techniques. They support a rich canvas of properties ranging from surface plasmonic propagation to negative refraction, and they can enhance the photoluminescence properties of quantum emitters at any frequency range.
Here, we introduce a computational approach that allows tailoring the dielectric and magnetic effective properties of planar metamaterials. Previously, planar hyperbolic metamaterials have been considered non-magnetic. In contrast, we show theoretically and experimentally that planar arrangements com- posed of non-magnetic constituents can be engineered to exhibit a non-trivial magnetic response. This realization simplifies the structural requirements for tailoring optical magnetism up to very high frequencies. It also provides access to previously unexplored phenomena, for example artificially magnetic plasmons, for which we perform an analysis on the basis of available materials for achieving polarization-insensitive surface wave propagation. By combining the concept of metamaterials’ homogenization with previous transfer matrix approaches, we develop a general computational method for surface waves calculations that is free of previous assumptions, for example infinite or purely periodic media. Furthermore, we theoretically demonstrate that hyperbolic metamaterials can be dynamically tunable via carrier injection through external bias, using transparent conductive oxides and graphene, at visible and infrared frequencies, respectively. Lastly, we demonstrate that planar graphene-based van der Waals heterostructures behave effectively as supermetals, exhibiting reflective properties that surpass the reflectivity of gold and silver that are currently considered the state-of-the-art materials for mirroring applications in space applications. The (meta)materials we introduce exhibit an order-of-magnitude lower mass density, making them suitable candidates for future light-sail technologies intended for space exploration.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
metamaterials; photonics; plasmonics; hyperbolic metamaterial; surface plasmon polariton; surface phonon polariton; transparent conductive oxide; graphene; van der Waals; spectroscopic ellipsometry; artificial optical magnetism
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Awards:
1. TomKat postdoctoral fellowship in Sustainability, Stanford University, CA, 2018
2. Marie Curie Individual postdoctoral fellowship, 2018
3. Best Student Paper, Metamaterials 2016 10th International Congress Metamaterials’ 2016, Greece (http://congress2016.metamorphose-vi.org/index.php/student-paper-competition)
4. Outstanding Poster Award, Metamaterials Science & Technology Workshop, Center for Metamaterials & Integrated Plasmonics, University of California San Diego, CA, 2015
5. Best Poster Award, Spring Material Research Society (MRS) Meeting (http://www.prolibraries.com/mrs/?select=session&sessionID=4504), Boston, MA, 2014
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Thesis Committee:
Vahala, Kerry J. (chair)
Faraon, Andrei
Daraio, Chiara
Schwab, Keith C.
Atwater, Harry Albert
Defense Date:
23 February 2018
Non-Caltech Author Email:
georgia.papadakis88 (AT) gmail.com
Funders:
Funding Agency
Grant Number
NSF Graduate Research Fellowship
UNSPECIFIED
American Association for University Women Dissertation Fellowship
UNSPECIFIED
Northrop Grumman Corporation
UNSPECIFIED
Record Number:
CaltechTHESIS:03082018-161639906
Persistent URL:
DOI:
10.7907/Z97H1GS1
Related URLs:
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URL Type
Description
DOI
Article adapted for Ch. 5
DOI
Article adapted for Ch. 5
DOI
Article adapted for Ch. 2
DOI
Article adapted for Ch. 6
DOI
Article adapted for Ch. 3
ORCID:
Author
ORCID
Papadakis, Georgia Theano
0000-0001-8107-9221
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
10764
Collection:
CaltechTHESIS
Deposited By:
Georgia Theano Papadakis
Deposited On:
09 Mar 2018 18:30
Last Modified:
08 Nov 2023 00:12
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Optical response in planar heterostructures:
From artificial magnetism to Angstrom-scale
metamaterials

Thesis by

Georgia Theano Papadakis

In Partial Fulfillment of the Requirements for the
degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2018
Defended February 23, 2018

Georgia Theano Papadakis
ORCID: 0000-0001- 8107-9221

iii
For Souzanna Papadopoulou &
for John and Katerina Papadakis,
with gratitude to Eleftherios Economou and to Pochi Yeh.

“Εµείς δεν τραγουδάµε για να ξεχωρίσουµε αδερφέ µου απ’τον κόσµο,
εµείς τραγουδάµε για να σµίξουµε το κόσµο.”
Και να αδερφέ µου, Γιάννης Ρίτσος
“We do not sing to distinguish ourselves from the world,
my brother, we sing to connect the world.”
The Fourth Dimension, Yannis Ritsos

ACKNOWLEDGEMENTS
Being admitted to Caltech in the graduate program of Applied Physics has
transformed my life path and my perspective. The completion of my PhD is
the result of the support that I received from numerous people to whom I will
always be grateful.
I thank Prof. Kiriaki Kiriakie, at the National Technical University of Athens
(NTUA) in Greece, for motivating me to pursue a PhD and for believing in
me. I also thank Prof. Ioannis Roumeliotis at NTUA for teaching me most of
what I know about electromagnetism, and for his support through these years.
By pursuing science, I have had the privilege to travel much and meet extraordinary people around the world. I have been lucky to conduct research in
Greece (NTUA), in Switzerland (CERN), in the Netherlands (AMOLF), and
at Caltech. The support I received from young, exceptionally bright friends
and colleagues has been instrumental. I thank Dr. Dimitris Calavrouziotis,
Dr. Loukas Gouskos, Maria Anastasiou, Prof. Kosmas L. Tsakmakidis, Dr.
Ruben Maas, Dr. Pankaj K. Jha, and Dr. Giulia Tagliabue for what they
taught me and for their support and friendship.
Large portions of the work I conducted at Caltech was supported by Northrop
Grumman Corporation, and particularly NG NEXT. I thank Dr. Luke Sweatlock, Dr. John Spargo, Dr. Philip Hon, and the staff at NG NEXT for supporting my research and investing in me. I also thank Prof. Prihena Narang
at Harvard for her support and for always believing in my research.
When I arrived at Caltech I knew very little about physics. I will always
remember the first years at Caltech as extremely eye-opening and creative
ones. I thank the Applied Physics Class of 2012 for many hours of studying
together, and particularly Magnus Haw, Teddy Albertson, Yu-Hung Lai, and
Matthew Fishman. I also thank Prof. Sandra Troian, the option representative
for the Applied Physics Class of 2012, for her advice and encouragement, and
for enjoyable conversations and exchange of ideas.
At Caltech I have made great friends. I thank Magnus Haw, Dr. Costas Sideris,
and Dr. Evan Miyazono for their compassion and friendship. I thank Dr. Gina
Panopoulou, Jane Herriman, Dr. Victoria Chernow, Dr. Samatha Johnson,
and Prof. Rebecca Saive for being exceptional friends and also role models

for women in tech, science, and engineering. I also thank my good friend Dr.
Christos Santis for his support and guidance. I am grateful to Dr. Panagiotis
Vergados and Dr. Christos Thrampoulidis for their unwavering support, and
for many empowering conversations and fun times. I have received invaluable
advice from Dr. Vasilios Christopoulos and I have enjoyed many fun times
with Marilena Dimotsantou, to whom I wish the best.
My officemates have made my time at work particularly enjoyful. I thank
Dr. Ragip Pala for his unwavering support, and I wish him the best. I thank
Dr. Ognjen Ilic for many long conversations and exchange of ideas and for
his advice and encouragement. I thank Prof. Howard Lee, who inspired me
with his determination and hard work as an exceptional experimentalist, and
I am grateful for what he taught me. I thank my friend and collaborator
Jeremy Brouillet for his hard work in our joint projects and for his support
and friendship. I also thank Ghazaleh Kafaie for great times in the office and
for her candid friendship.
My most enjoyable work at Caltech was conducted together with Dr. Artur
Davoyan and Dagny Fleischman. I thank Artur for his guidance, for teaching
me new things, and for being a good friend. His clarity of mind is limitless
and I envy his drive and talent in physics. I thank Dagny, who is an impressive material scientist, for being exceptionally professional while also being a
supportive friend and a wonderful source of optimism, and for believing in my
research.
As a graduate student I was very fortunate to collaborate with leaders in the
field of photonics. I thank Prof. Albert Polman at AMOLF for hosting me
in his group in 2014. I also thank Prof. Shankar Sundararaman at RPI and
Prof. Marin Soljacic at MIT for their support and collaboration through NG
NEXT in the years 2016-2017. I thank Prof. Nader Engheta at UPENN for
our collaboration, for his trust and support, and for being an academic role
model. I also thank Prof. Costas Soukoulis at Iowa State University and IESL
FORTH (Greece) for his support through the years.
I greatly appreciate the insightful feedback, honesty, and kindness of Prof.
Keith Schwab. I thank Prof. Kerry Vahala, Prof. Hyuck Choo, Prof. Andrei
Faraon, and Prof. Chiara Daraio, who took the time to be in my candidacy
and thesis committees. I also thank Prof. Amnon Yariv for our enjoyable
conversations and, most importantly, for introducing me to his former student,

vi
Prof. Pochi Yeh.
I reached out to Prof. Pochi Yeh during my second year at Caltech. He
invited me to his house to discuss my research, without knowing who I was
or what I was looking for. I am still in awe for his humbleness, I admire his
pure academic curiosity and his generosity in providing all he knows. Prof.
Yeh has been instrumental in my academic path; he has been my teacher and
supporter, and I will never be able to thank him enough.
I have been fortunate to be given a family of great kindness and generosity.
None of what I have done so far would have been possible without guidance
from my father, John. He has been my mental anchor and my source of
courage and perseverance. I thank my mother for providing me a role model
for women in science and for being my source of optimism. I thank my godmather Souzanna Papadopoulou for teaching me mathematics and motivating
my curiosity. But most importantly, I thank her for her generosity, for her
unwavering trust in me, and for being courageous in difficult times. I am sincerely indebted to Prof. Eleftherios Economou for noticing potential in me
and encouraging me to never stop learning. I also thank my siblings Paul,
Denise, Alison, and Sterg for their support and love.
I am grateful for the generous financial support for the work in this thesis,
mainly provided by the National Science Foundation, the American Association for University Women and NG NEXT at Northrop Grumman Corporation. I thank Christy Jenstad, Tiffany Kimoto, Jennifer Blankeship, Jonathan
Gross, and Lyann Lau for their support, advice, and enthusiasm through these
years.
Finally, with much gratitude, I thank my advisor, Prof. Harry A. Atwater,
for investing in me and giving me the opportunity to come to Caltech. I am
honored for the freedom he has given me and I appreciate his trust in me, his
support, and his encouragement. I am in debt to him for setting the bar high,
and I admire his ethics and his tireless enthusiasm for science.
Georgia Theano Papadakis
February 2018
Pasadena, CA

vii

ABSTRACT
The idea of expanding the range of properties of natural substances with artificial matter was introduced by V. G. Veselago in 1967. Since then, the field of
metamaterials has dramatically advanced. Man-made structures can now exhibit a plethora of extraordinary electromagnetic properties, such as negative
refraction, optical magnetism, and super-resolution imaging. Typical metamaterial motifs include split ring resonators, dielectric and plasmonic particles,
fishnet and wire arrays. The principle of operation of these elements is now
well-understood, and they are being exploited for practical applications on a
global scale, ranging from telecommunications to sensing and biomedicine, in
the radio frequency and terahertz domains. Accessing and controlling optical
and near-infrared phenomena requires scaling down the dimensions of metamaterials to the nanometer regime, pushing the limits of state-of-the-art nanolithography and requiring structurally less complex geometries. Hence, within
the last decade, research in metamaterials has revisited a simpler, lithographyfree structure, particularly planar arrangements of alternating metal and dielectric layers, termed hyperbolic metamaterials. Such media are readily realizable with well-established thin-film deposition techniques. They support a
rich canvas of properties ranging from surface plasmonic propagation to negative refraction, and they can enhance the photoluminescence properties of
quantum emitters at any frequency range.
Here, we introduce a computational approach that allows tailoring the dielectric and magnetic effective properties of planar metamaterials. Previously,
planar hyperbolic metamaterials have been considered non-magnetic. In contrast, we show theoretically and experimentally that planar arrangements composed of non-magnetic constituents can be engineered to exhibit a non-trivial
magnetic response. This realization simplifies the structural requirements for
tailoring optical magnetism up to very high frequencies. It also provides access to previously unexplored phenomena, for example artificially magnetic
plasmons, for which we perform an analysis on the basis of available materials for achieving polarization-insensitive surface wave propagation. By combining the concept of metamaterials’ homogenization with previous transfer
matrix approaches, we develop a general computational method for surface
waves calculations that is free of previous assumptions, for example infinite or
purely periodic media. Furthermore, we theoretically demonstrate that hyper-

viii
bolic metamaterials can be dynamically tunable via carrier injection through
external bias, using transparent conductive oxides and graphene, at visible
and infrared frequencies, respectively. Lastly, we demonstrate that planar
graphene-based van der Waals heterostructures behave effectively as supermetals, exhibiting reflective properties that surpass the reflectivity of gold and
silver that are currently considered the state-of-the-art materials for mirroring
applications in space applications. The (meta)materials we introduce exhibit
an order-of-magnitude lower mass density, making them suitable candidates
for future light-sail technologies intended for space exploration.

LIST OF PUBLICATIONS
G. T. Papadakis, A. Davoyan, P. Yeh, H. A. Atwater , “Phonons and excitons
for omnipolarization surface waves”, (in preparation) (2018)
G.T.P. developed the theoretical model together with A.D., carried out the numerical simulations and calculations, and lead the writing of the manuscript.
G. T. Papadakis, D. Fleischman, A. Davoyan, P. Yeh, H. A. Atwater , “Optical
Magnetism in Planar Metamaterial Heterostructures”, Nature Communications 9
(2018) , p. 296. DOI: 10.1038/s41467-017-02589-8.
G.T.P. developed the theoretical model together with A.D., carried out the numerical simulations and calculations, the experimental measurements and ellipsometric
fittings, and lead the writing of the manuscript.
G. T. Papadakis, P. Narang, R. Sundararaman, N. Rivera, H. Buljan, N. Engheta,
M. Soljacic, “Ultra-light Å-scale Optimal Optical Reflectors”, ACS Photonics 5
(2018), p. 384. DOI: 10.1021/acsphotonics.7b00609.
G.T.P. participated in developing the theoretical model, carried out the numerical
calculations of the photonic properties discussed in the manuscript, and participated
in the writing of the manuscript.
G. T. Papadakis H. A. Atwater, “Field effect-induced tunability in hyperbolic metamaterials”, Phys. Rev. B 92 (2015), p. 184101. DOI: 10.1103/PhysRevB.92.184101.
G.T.P. developed the theoretical model, carried out the numerical calculations and
the writing of the manuscript.
G. T. Papadakis, P. Yeh H. A. Atwater, “Retrieval of material parameters for
uniaxial metamaterials”, Phys. Rev. B 91 (2015), p. 155406. DOI: 10.1103/PhysRevB.91.155406.
G.T.P. developed the theoretical model, carried out the numerical calculations and
the writing of the manuscript.
H. W. Lee, G. T. Papadakis, S. P. Burgos, K. Chander, A. Kriesch, R. Pala, U.
Peschel, H. A. Atwater, “Nanoscale Conducting Oxide PlasMOStor”, Nano Lett. 14
(2014), p. 6463-6468. DOI: 10.1021/nl502998z.
G.T.P. participated in carrying out the numerical calculations, performed experimental characterization of conducting oxides, and paricipated in the writing of the
manuscript.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter I: Introduction: Metamaterials today . . . . . . . . . . . . . . . . . .
1.1 From atoms to materials, from resonators to metamaterials . . . . .
1.2 Negative refraction: The wire array and the split ring resonator . . .
1.3 Planar heterostructures: from photonic crystals to metamaterials . .
1.4 Hyperbolic metamaterials . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Scope and structure of the thesis . . . . . . . . . . . . . . . . . . . . 13
Chapter II: Metamaterials’ homogenization . . . . . . . . . . . . . . . . . . . 15
2.1 The concept of homogenization and S-parameter retrieval . . . . . . 15
2.2 Effective medium approximations and the Maxwell Garnett result for
1D metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 The Bloch wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 S-parameter retrieval for 1D metamaterials . . . . . . . . . . . . . . 23
2.5 A sanity check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Comparison between S-parameter retrieval and EMAs: µ , 1! . . . . 29
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Chapter III: Artificial magnetism in planar metamaterials . . . . . . . . . . . 33
3.1 Previous approaches: Artificial magnetism in 3D and 2D . . . . . . . 33
3.2 Parameter space for µ and . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Artificial magnetism in 1D . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Another sanity check: Impedance matching . . . . . . . . . . . . . . 39
3.5 Experimental method: Spectroscopic ellipsometry for metamaterials
41
3.6 Experimental verification of artificial magnetism in 1D . . . . . . . . 42
3.7 Implications of µ , 1 for bulk propagating modes . . . . . . . . . . . 45
3.8 Implications of µ , 1 for surface waves . . . . . . . . . . . . . . . . . 48
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter IV: Beyond plasmons: Omnipolarization surface waves . . . . . . . . 51
4.1 “Large permittivity begets high-frequency magnetism” . . . . . . . . . 51
4.2 Material requirements for surface-confined propagation beyond plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Generalized retrieval approach for surface wave computations . . . . 58
4.4 Figures of merit for surface-confined propagation . . . . . . . . . . . 64
4.5 Theory of plasmons and results for silver . . . . . . . . . . . . . . . . 64
4.6 Phonons and excitons . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Surface-confined waves on SiC on Si . . . . . . . . . . . . . . . . . . 72
4.8 Surface-confined waves on WS2 on Si . . . . . . . . . . . . . . . . . . 74
4.9 The case of a plasmon and a phonon . . . . . . . . . . . . . . . . . . 75

xi
4.10 The case of a plasmon and an exciton . . . . . . . . . . . . . . . . . 79
4.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter V: Active tunability in planar metamaterials . . . . . . . . . . . . . 83
5.1 Tuning the optical response . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Transparent conductive oxides as active components . . . . . . . . . 84
5.3 Tunable hyperbolic response at optical frequencies . . . . . . . . . . 87
5.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.5 Graphene as an active component . . . . . . . . . . . . . . . . . . . . 96
5.6 Tunable hyperbolic response at infrared frequencies . . . . . . . . . . 98
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter VI: Ultra-light van der Waals heterostructures as supermetals . . . . 104
6.1 What makes a perfect reflector? Beyond noble metals . . . . . . . . 104
6.2 Graphene-based van der Waals heterostructures . . . . . . . . . . . . 105
6.3 Computational approach and effective mass . . . . . . . . . . . . . . 106
6.4 Shinier than gold and silver! . . . . . . . . . . . . . . . . . . . . . . . 110
6.5 Loss tangent and quality factors . . . . . . . . . . . . . . . . . . . . . 111
6.6 Plasmonic propagation in vdW heterostructures . . . . . . . . . . . . 112
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Chapter VII: Summary & Outlook . . . . . . . . . . . . . . . . . . . . . . . . 115
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Appendix A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.1 Transfer Matrix Equations for an infinite bilayer heterostructure . . 138
A.2 Counting optical states with the reflection pole method . . . . . . . 139
A.3 Details on the electronic structure calculations in vdW heterostructures140

xii

LIST OF ILLUSTRATIONS

Number

Page

1.1

From atoms to materials, from resonators to metamaterials . . . . .

1.2

The next photonic revolution by N. Zheludev . . . . . . . . . . . . .

1.3

First demonstration of negative refraction . . . . . . . . . . . . . . .

1.4

1D photonic crystals, 1977 by Pochi A. Yeh et al. . . . . . . . . . . .

1.5

The photonic crystal and metamaterial regimes . . . . . . . . . . . .

1.6

Dispersion diagrams and equifrequency contours . . . . . . . . . . . .

1.7

Hyperbolic metamaterials . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Τhe concept of homogenization . . . . . . . . . . . . . . . . . . . . .

2.2

Maxwell Garnett and Bloch approaches . . . . . . . . . . . . . . . . .

2.3

Parameter retrieval for uniaxial structures . . . . . . . . . . . . . . .

2.4

Parameter retrieval flow chart . . . . . . . . . . . . . . . . . . . . . .

2.5

Wave parameters for a known system . . . . . . . . . . . . . . . . . .

2.6

Restored material parameters for a known system . . . . . . . . . . .

2.7

Effective dielectric permittivity in metallodielectric HMMs . . . . . .

2.8

Effective magnetic permeability in metallodielectric HMMs . . . . . .

3.1

SRR, fishnet and wire-based 3D and 2D magnetic metamaterials . .

3.2

Main motifs for 3D and 2D magnetic metamaterials . . . . . . . . . .

3.3

Parameter space for µ and . . . . . . . . . . . . . . . . . . . . . . .

3.4

Artificial magnetism in 1D: the concept . . . . . . . . . . . . . . . . .

3.5

Multilayer configurations for artificial magnetism . . . . . . . . . . .

3.6

Impedance matching sanity checks . . . . . . . . . . . . . . . . . . .

3.7

TEM images of fabricated multilayer magnetic metamaterials . . . .

3.8

Experimental approach and flow chart . . . . . . . . . . . . . . . . .

3.9

Experimentally measured eff and µeff . . . . . . . . . . . . . . . . . .

3.10

Comparison between model and raw experimental data . . . . . . . .

3.11

Bulk propagating modes in planar magnetic HMMs . . . . . . . . . .

3.12

Surface waves in planar magnetic HMMs . . . . . . . . . . . . . . . .

4.1

Permittivity resonances in polar dielectrics and excitonic materials .

4.2

Naturally occurring excitonic and polar dielectric materials

. . . . .

4.3

Guided modes in < 0 and > 0 material systems . . . . . . . . . .

4.4

SPP dispersion and high- materials dispersion . . . . . . . . . . . .

4.5

Definition of a photonic and a surface-confined mode . . . . . . . . .

4.6

Generalized retrieval approach for surface wave computations . . . .

xiii
4.7

Surface plasmon polaritons in Drude metals . . . . . . . . . . . . . .

4.8

Surface plasmon polaritons in silver . . . . . . . . . . . . . . . . . . .

4.9

Surface phonon polaritons and TE counterparts in polar dielectrics .

4.10

Surface-confined waves in excitonic materials . . . . . . . . . . . . . .

4.11

Omnipolarization surface-confined waves on SiC on Si . . . . . . . .

4.12

Omnipolarization surface-confined waves on WS2 on Si . . . . . . . .

4.13

Omnipolarization surface waves from plasmons and phonons . . . . .

4.14

Omnipolarization surface waves from plasmons and excitons . . . . .

5.1

Schematic of TCO-based tunable HMM . . . . . . . . . . . . . . . .

5.2

Complex dielectric permittivity of indium tin oxide . . . . . . . . . .

5.3

Extraordinary permittivity of a TCO-based tunable HMM . . . . . .

5.4

Ordinary permittivity of a TCO-based tunable HMM . . . . . . . . .

5.5

Tunable birefringence and dichroism . . . . . . . . . . . . . . . . . .

5.6

Tunable figure of merit of a TCO-based tunable HMM . . . . . . . .

5.7

Complex isofrequency contours at different gate biases . . . . . . . .

5.8

Three-dimensional tunable dispersion surface . . . . . . . . . . . . .

5.9

Sensitivity analysis of ENP condition . . . . . . . . . . . . . . . . . .

5.10

Schematic of graphene-based tunable HMM . . . . . . . . . . . . . .

5.11

Complex effective dielectric permittivity of graphene/SiO2 HMM . .

5.12

Hyperbolic and elliptical regimes of graphene/SiO2 HMM . . . . . . 100

5.13

Topological transitions at IR frequencies . . . . . . . . . . . . . . . . 101

6.1

Schematic of vdW heterostructures vs noble metals, meff and . . . 107

6.2

Reflectance: vdW heterostructures vs noble metals . . . . . . . . . . 111

6.3

Quality factors and loss tangent . . . . . . . . . . . . . . . . . . . . . 112

6.4

Plasmonic dispersion in vdW heterostructures . . . . . . . . . . . . . 113

7.1

Outlook: concepts and applications . . . . . . . . . . . . . . . . . . . 118

A.1

Low density of optical states in magnetic HMMs . . . . . . . . . . . 142

A.2

High density of optical states in magnetic HMMs . . . . . . . . . . . 143

xiv

LIST OF TABLES
Number
4.1

Page
Comparison of propagation characteristics between surface plasmon
polaritons in Ag and omnipolarization surface-confined waves in SiC
and in WS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

Breakdown voltage and carrier density in transparent conductive oxide gating schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Birefringent dielectric media . . . . . . . . . . . . . . . . . . . . . . .

6.1

List of the twenty most conductive elemental metals . . . . . . . . . 105

GLOSSARY OF ACRONYMS
IR Infrared
UV Ultraviolet
SRR Split ring resonator
HMM Hyperbolic metamaterial
EMA Effective medium approximation
TE Transverse electric polarization
TM Transverse magnetic polarization
ENZ Epsilon-near-zero
EMNZ Epsilon-and-mu-near-zero
ENP Epsilon-near-pole
RF Radio frequencies
SPP Surface plasmon polariton
RPM Reflection pole method
DOS Density of optical states
TMDC Transition-metal dichalgogenides
SPhP Surface phonon polariton
HWHM Half-width-half-maximum
TCO Transparent conductive oxide
ITO Indium tin oxide
MOS Metal/oxide/semiconductor
FOM Figure of merit
CNP Charge neutrality point
vdW van der Waals
PEC Perfect electrical conductor
DFT Density functional theory

Chapter 1

INTRODUCTION: METAMATERIALS TODAY
“All things are made of atoms; little particles that move around in perpetual motion,
attracting each other when they are a little distance apart, but repelling upon being
squeezed into one another. In that one sentece [...] there is an enormous amount of
information about the world.”
Richard P. Feynman, Six Easy Pieces, 1995
“I can’t see exactly what would happen, but I can hardly doubt that when we have
some control of the arrangement of things on a small scale we will get an enormously
greater range of possible properties that substances can have, and of different things
that we can do.”
Richard P. Feynman, Plenty of Room at the Bottom, 1959
1.1

From atoms to materials, from resonators to metamaterials

The electromagnetic response and optical properties of natural materials arise from
their electronic structure: the way that electrons interact with each other and with
the nuclei (Fig. 1.1a). For example, a material with nearly free electrons behaves like
a metal by screening any incident electric field, whereas when electrons are bound
to the nuclei materials act as transparent media (dielectrics) and fields are allowed
to propagate within them. The electromagnetic response of materials is typically
described through their dielectric and magnetic properties, namely the dielectric permittivity and magnetic permeability µ that enter Maxwell’s macroscopic equations
through the constitutive or material relations [1, 2]. Both and µ, together with
more complex electromagnetic parameters, for example chirality, bianisotropy, and
nonlinearities of materials, are all derived from their crystallographic arrangement
and electronic band structure.
Despite a plethora of naturally available materials, their range of electromagnetic
properties is limited. Listing the most desirable material responses that are unfound
in nature is a matter of personal preference and application of interest. For example,
near-unity material absorption at visible and near-infrared (IR) frequencies is desirable for solar harvesting, photovoltaic applications, sensing, and detection. Hence
this functionality is highly pursued with artificial matter [3–5]. With respect to

imaging technology, the diffraction limit restricts the resolution of an optical system
to λ/n, where λ is the free space wavelength and n is the index of refraction of
the transparent medium. Hence, an infinitely large index of refraction could allow
for perfect imaging resolution, however, the refractive indices of natural materials
range approximately from 0 to 4. Alternatively, the diffraction limit restrictions
can be alleviated with a negative index of refraction, also not observable in nature.
Specifically, a negative index allows for super-resolution imaging and has been one of
the most targeted functionalities in metamaterials research [6–10]. Another natural
limitation pertains to an asymmetry in the dielectric and magnetic response of natural materials; although their dielectric permittivity obtains positive, negative, and
near-zero values for frequencies up to the ultraviolet (UV) regime, the magnetic permeability µ of natural materials is unity for frequencies beyond the terahertz (THz)
range [2, 11, 12]. Hence natural materials do not interact strongly with magnetic
fields at high frequencies, limiting the degree of control of light-matter interactions
in the visible and IR regime.
The idea of expanding the properties of natural substances was introduced by V. G.
Veselago in 1967 [6]. In his original work, he discussed the implications of a hypothetical medium with simultaneously negative and µ. Veselago made speculations
about conducting ferromagnets with < 0 and µ < 0, however such properties have
not been observed in any naturally occuring material system, to date. Thirty-three
years later, a simultaneously negative and µ system composed of a periodic array of conducting split ring resonators and wires was experimentally realized by D.
R. Smith et al. [13]. At that time, the name “metamaterial” was established as
the general term for artificially constructed matter, made of periodically positioned
resonant subwavelength elements, often referred to as “meta-atoms”.
Rather than acquiring their macroscopic electromagnetic response from their chemical composition and electronic structure, as in natural materials (Fig. 1.1a), metamaterials obtain their properties from the collective response of the meta-atoms,
which are typically resonators that act as mesoscopic artificial electric and magnetic
dipoles. For incident wavelengths much larger than the metamaterial periodicity, the
metamaterial microstructure is not resolved by the incident electromagnetic fields.
Instead, electromagnetic radiation senses the net response of the composite (Fig.
1.1b). Hence the notion of an “effective medium” is typically introduced to describe
the response of such a system. As discussed in the pioneering work by J. B. Pendry
in et al. in [14], “Long wavelength radiation is too myopic to detect internal structure and, in this limit, an effective permittivity and permeability is a valid concept”.
By employing different meta-atoms, a plethora of new material functionalities have
been demonstrated within the last twenty years. Positive, zero, and negative values

(a) Natural materials
Diamond
lattice

(b) Artificial matter

(i) Van der Waals
heterostructures

electron
Spring-like
force
Nucleus
2 nm
2 nm

(iv) neff<0
(ii) Multilayer (iii) Chiral
metamaterial metamaterial metamaterial
(RF)
(visible)
(IR)

100 nm

λEM
Figure 1.1: (a) SuperSTEM image of a facetted nano-void in diamond, from the
SuperSTEM facility at the STFC Daresbury laboratory (UK). (b) (i) STEM
image of a stack of graphene and hBN bilayers with the layer sequence schematically shown to the left, from [15], (ii) TEM image of a Ge/Ag multilayer
metamaterial operating at optical frequencies [12] (iii) SEM image of helical
metamaterial operating at infrared frequencies, adapted from [16] (iv) picture
of a negative index metamaterial with unit cell size in the mm-scale, operating
at microwave frequencies, adapted from [17].

of effective permittivity and permeability at any frequency range are now a reality,
using accurate metamaterial design.
Initial experimental results in metamaterial research in the 2000’s pertained to the
microwave frequency range, with metamaterial elements in the millimeter (mm)scale; the photograph in Fig. 1.1b (iv) demonstrates a negative refractive index
composite system, adapted from from [13], which we will discuss in more detail in
what follows. Drastic technological progress in nanofabrication has allowed scaling the dimensions of metamaterial elements down to the micrometer (µm) and
nanometer (nm)-regime. In Fig. 1.1b (iii) we show a scanning electron microscopy
(SEM) image of a triple-helix metamaterial operating as a circular polarizer at IR
frequencies, realized using three-dimensional laser lithography [16]. Fig. 1.1b (ii)
shows a multilayer germanium/silver metamaterial operating at visible wavelengths,
adapted from [12], which will be discussed in detail in Chapter 3, realized with
electron-beam thin-film deposition. Impressively, as of 2012 and despite the dozens
of steps involved, it is now possible to stack two-dimensional materials with interlayer separation in the Å-regime. Fig. 1.1b (i) (right) shows a cross-section scanning
transmission electron microscopy (STEM) image of a van der Waals heterostructure
composed of six alternating bilayers of graphene and hexagonal boron nitride, as
illustrated on the left, adapted from [15]. The van der Waals interlayer separation

Figure 1.2: “The next photonic revolution: Metamaterials”, adapted from N.
Zheludev, Plenary talk, CIMTEC 2014, Optical functionalities on demand:
from metamaterials to metadevices
regime coincides with the interatomic separation distance in natural materials, as
shown in Fig. 1.1a. Hence, today we are able to access and control the arrangement
of material systems in the sub-nm regime. In this regime, electronic and electromagnetic degrees of freedom compete, paving the way for new classes of artificial
materials composed of layered combinations of two-dimensional (2D) materials in
van der Waals heterostructures. Namely, by controlling simultaneously the geometrical structure, for example by selecting appropriate layer sequencing, together with
controlling the electronic interactions between adjacent van der Waals layers, by
selecting appropriate 2D materials, we are in the position to design new Å-scale
(meta)materials with a bottom-up approach. We discuss this regime in Chapter 6
[18].
Since the 2000’s the field of metamaterials has dramatically advanced and transitioned from the mature research area of electrical engineering at microwave frequencies to the emerging field of nanophotonics, plasmonics, and the physics of light
coupled to nanostructures. Fig 1.2 is a histogram of the number of scientific pub-

lications in the field of lasers in the 1960’s and 1970’s in comparison to the field of
metamaterials up to 2012. The similar trends in the publication rates, together with
the dramatic technological impact of the lasers in the past, present, and future, show
the potential of metamaterials as an emerging field of both fundamental scientific
merit and technological promise.
1.2

Negative refraction: The wire array and the split ring resonator

It took thirty-three years for Veselago’s original proposal of a medium with negative refractive index to be experimentally demonstrated by D. R. Smith et al. [6,
13]. This delay is justified as building intuition about the metamaterial structural
requirements for yielding on demand values of and µ was, and often remains, a
challenging task. Successful demonstration of negative refraction was achieved with
the structure depicted in Fig. 1.3c, consisting of a periodic array of interspaced conducting nonmagnetic split ring resonators and continuous wires. The wire medium
and the split ring resonator are now widespread in metamaterial design and their
principle of operation deserves special attention.
The wire array was introduced by J. B. Pendy et al. in [19] in 1996, as a structure
that behaves effectively like a metal with plasma frequency controllable through the
metallic filling fraction of the wires. Hence the plasmonic response of this composite system could be tuned at will, leading to surface plasmonic propagation at
frequencies much lower than the plasma resonance of most noble metals that are
physically bound to the ultraviolet (UV) and visible regime. The principle of operation of the wire array is based on the fact that conducting channels composed of
wires yield a metallic response with eff < 0, and by placing metallic wires along all
three coordinate directions (Fig.1.3a) the response becomes isotropic.
The split ring resonator (SRR) was originally discussed by J. B. Pendry et al. in
[14] in 1999, as a way to achieve isotropic negative µeff . The search for negative
permeability is motivated by the lack of natural magnetism at high frequencies. As
stated by Pendry et al. in [14], “atoms and molecules prove to be a rather restrictive
set of elements from which to build a magnetic material. This is particularly true
at frequencies in the gigahertz range where the magnetic response of most materials
is beginning to tail off.”. The SRR, shown on the left of 1.3b, is composed of nonmagnetic metal and operates similar to a magnetic dipole; for an incident magnetic
field parallel to the axis of the rings, induced current can produce a magnetic field
that may either oppose (µeff < 1) or enhance (µeff > 1) the incident field. The gaps
between the rings in Fig.1.3b serve as control knobs for tuning the magnetic resonance at wavelengths much larger than the diameter of the rings. The smaller ring
has a gap oriented in the opposite side to the large ring in order to generate a large

(c)

(a)

(b)

Figure 1.3: (a) Schematic of an isotropic eff < 0 medium composed of a threedimensional periodic lattice of ultra-thin metallic wires, adapted from [19]. (b)
Left-planar view of a SRR, adapted from [13], right-schematic of an isotropic
µeff < 0 medium composed of SRRs, adapted from [14]. (c) Top-dispersion
curve for a periodic array of SRRs for magnetic field polarized along the SRR
axis. Bottom-dispersion curve for a directional negative refractive index neff
metamaterial composed of wires and SRRs and wires, adapted from [13].

capacitance in the small gap region and further lower the magnetic resonance. By
constructing a lattice of closely packed SRRs the axes of which are oriented along
all three coordinate directions, as shown on the right of 1.3b, one can achieve an
isotropic µeff < 0.
By combining a two-dimensional array of perioridically placed conducting wires
with layers of square arrays of SRR (as shown in Fig. 1.1b (iv)), D. R. Smith et al.
realized the first negative index metamaterial in 2000 [13]. A 2D wire array can be
modeled with an effective dielectric permittivity given by
eff = 1 −

ωp2
ω2

(1.1)

2 L(r) ) − 1/2,
where ω is the frequency of incident radiation, and with ωp = (αW

where αW is the wire spacing (Fig.1.3a), L(r) is the self-inductance of a single
wire and o is the dielectric permittivity of free space. Similarly, the SRR periodic
arrangement is modeled via
µeff = 1 −

π(r/αSRR ) 2
1 − 3l/µo Cr3 (πω) 2 + i(2lρ/ωrµo )

(1.2)

where r is smaller ring’s radius (Fig.1.3b (left)), αSRR is the SRR lattice parameter,
l is the distance between SRR layers, ρ is the resistance per unit length of the rings
measured around the circumference, C is the capacitance associated with the gaps
between the rings, and µo is the magnetic permeability of free space. By appropriately positioning the wire array and SRR lattice so that the resonant frequencies of
the two align, a negative index regime occurs (neff = eff µeff ).
For the SRR lattice alone, the frequency dispersion of the supported modes is shown
in Fig.1.3c (top). The band gap corresponds to the frequency regime for which
µeff < 0. By appropriately intersecting the wire array (inset in Fig.1.3c (bottom)),
the band gap is eliminated and the emerging passband, shown with the dashed line,
exhibits negative group velocity due to a simultaneously negative eff , which arises
from the plasmonic response of the wires. We note that the negative refraction of
the system introduced by Smith et al. is limited to the polarization depicted at
the inset of Fig.1.3c (bottom); for magnetic fields not alined with the SRR axes, or
for electric fields not aligned with the wires axes, the passband exhibits a positive
slope, and hence the group velocity is positive. The seminal works by D. R. Smith,
J. B. Pendry and co-authors shaped the roadmap for research in artificial matter
and established fundamental metamaterial elements that are now widespread in
metamaterial design.
1.3

Planar heterostructures: from photonic crystals to metamaterials

Following the realization of SRR metamaterials and wire arrays, a plethora of subwavelength structures have been reported to exhibit interesting and unprecedented
electromagnetic properties, c.f. Fig. 1.1b. Dielectric nanoparticles [20, 21], fishnet
structures [22–24], helical metamaterials [16], and nanowires [25] have been developed, operating at IR and even near-IR and visible frequencies. The experimental
realization of such three-dimensional (3D) and two-dimensional (2D) motifs calls
for high resolution nanofabrication methods including electron beam lithography,
nanoimprint, focused ion beam milling, and three-photon lithography. Extending
the properties of such 3D and 2D metamaterials to the visible and UV regime requires reducing their dimensions so that the unit cell does not exceed a threshold of
roughly tens to few hundreds of nanometers.
By contrast to the fabrication requirements of 3D and 2D unit cells of metamaterials,
the experimental realization of unpatterned, one-dimensional (1D) planar layered
media has been well established since the 1980s with thin-film deposition techniques.
Today, thin-film deposition systems include thermal and electron beam evaporation,
sputtering, molecular beam epitaxy (MBE) and atomic layer deposition (ALD), with
accuracy in layer thickness that can reach sub-nm scales in state-of-the-art systems.

(a)

(b)

1μm

Figure 1.4: (a) Band structure of a 1D photonic crystal: dark zones represent
the allowed bands. (b) SEM image of a 1D photonic crystal composed of
alternating layers of GaAs and Al0.20 Ga0.80 As, adapted from the seminal work
by P. A Yeh et al. in 1977 [1, 26].

The precursor of 1D planar multilayer metamaterials is the field of all-dielectric 1D
photonic crystals; structures that flourished within the 1970’s and 1980’s due to
their property to selectively reflect and transmit electromagnetic radiation [1, 26–
29].
Similarly to Bragg diffraction in crystals, where X rays are diffracted by the periodic energy potential of crystalline solids, 1D photonic crystals operate based on
diffraction of light. They are usually composed of a periodic alteration of two dielectric materials, typically exhibiting a low- and a high-refractive index. One of
the pioneering contributions to the field of 1D photonic crystals was by P. A. Yeh,
during his graduate school years at Caltech, when he developed a transfer matrix
formulation for layered media [1, 26], heavily used in this thesis and in the literature. The mathematical formulation of electromagnetic fields propagating in the
form of plane waves inside 1D photonic crystals is identical to the problem of an
electron propagating in a periodic potential, which is treated with Bloch’s theorem
[30]. Therefore, similar to the electronic bands and bandgaps in semiconductors, periodic layered media exhibit photonic bands and bandgaps. Figure 1.4 (a), adapted
from Yeh’s original thesis authored in 1978, shows the typical 1D photonic crystal structure (inset) and its characteristic band structure, where the dark zones
represent the allowed bands, ω is the frequency of the incident light, and β is the
in-plane propagation constant. In Fig. 1.4 (b) we show an SEM image of a Bragg
reflector composed of alternating layers of GaAs and Al0.20 Ga0.80 As, discussed in
Yeh’s thesis. Realizing that a simple periodic dielectric layered medium can support omnidirectional reflection [29], where Maxwell’s equations have no propagating

Photonic crystal
regime

Metamaterial
regime

100

λ=104Λ

λ=106Λ

λ=20Λ

λ=Λ

R, T

80
60
40
20

-10

-8

-6

-4

log{ω (in c/Λ)}

-2

Figure 1.5: Reflectance and transmittance spectra of an all-dielectric 1D layered medium composed of eleven alternating layers of refractive index n1 = 1.4,
n2 = 3.5.
solutions, led to a broad range of applications and photonic devices, with the most
well-known being the anti-reflection coating.
Contrary to the metamaterial limit, where the wavelength of light oughts to be much
larger than the unit cell periodicity (λ
Λ), photonic crystals operate at wavelengths comparable to the periodicity. This comparison in shown in the schematics
of Fig. 1.5. In this figure, regions of unity transmittance and reflectance correspond
to photonic bands and band gaps, respectively, for a 1D periodic layered medium
(frequency scale to be compared to Fig. 1.4 (a)). The band gaps disapear in the
metamaterial subwavelength regime, λ
Λ. In this regime, the field is too myopic to the internal structure of the composite and, instead, experiences an average,
homogeneous dielectric environment. In the particular example of Fig. 1.5, the
selected materials are purely dielectric (n1 = 1.4 and n2 = 3.5), and hence the dielectric environment seen by large-wavelength electromagnetic fields is transparent.
This explains the nearly perfect transmittance in the metamaterial regime depicted
in Fig. 1.5.

10
1.4

Hyperbolic metamaterials

As briefly mentioned above, all-dielectric multilayer metamaterials exhibit a purely
dielectric response in the deep subwavelength limit, where they are transmissive
(Fig. 1.5). By striking contrast, new and exciting effects arise from stacking metallic and dielectric layers to compose planar metallodielectric metamaterials, termed
hyperbolic metamaterials (HMMs). The field of HMMs has flourished within the
last six years [31–33], however the concept behind their fundamental properties was
discussed as early as in 2001 by I. V. Lindell et al. in [34], and in 2003 by D. R. Smith
and D. Schurig in [35]. A wide range of their exciting properties and functionalities
arise from their unnaturally anisotropic response. To our knowledge, HMMs are
the most anisotropic composite materials reported. In what follows, we introduce
HMMs and compare them to natural substances.
The frequency dispersion of electromagnetic waves propagating in a bulk isotropic
material is expressed through the equation
|~k| = nω/c

(1.3)

where ~k is the wavevector of the wave, n is the refractive index of the material
and c is the speed of light in vacuum. By contrast, many dielectric materials, for
example sapphire, hexagonal boron nitride (hBN), silicon carbide (SiC), and others,
are uniaxially anisotropic. In the absence of any magnetic properties, the dielectric
response of uniaxially anisotropic materials is described by the dielectric permittivity
tensor ~eff = diag{o , o , e }, where o and e are the ordinary and extraordinary

components of the tensor. The ordinary directions (x and y axes) refer to the
ones normal to the optical axis of the material, whereas the extraordinary direction
coincides with the optical axis (z axis). Ignoring the y-direction, the dispersion
equation of uniaxially anisotropic crystals is
kx2 kz2 ω 2
= 2
e
o

(1.4)

By fixing the frequency ω in the dispersion equations (1.3) and (1.4), we obtain
the angular dependence of the wavevector, or k-space of the material, which is also
termed equifrequency contour (EFC). The EFC shape determines fundamental electromagnetic properties of material systems and is useful in describing light-matter
interactions. As shown in Fig. 1.6a, the EFC of an isotropic material is a perfect
circle with radius nω/c. By contrast, uniaxially anisotropic dielectrics have elliptical
EFCs as depicted in Fig. 1.6b. Typically, dielectric materials occurring naturally
do not exhibit extreme anisotropy; the most uniaxually anisotropic dielectric sub√
stances are nematic liquid crystals with birefringence values ∆n = e − o ∼ 0.4.

11
(a) Isotropic materials

(b) Anisotropic dielectrics

kz(e)

kx(o)

Figure 1.6: Equifrequency contours for (a) an isotropic dielectric medium (Eq.
(1.3)), (b) an anisotropic uniaxial dielectric material (Eq. (1.4) with o e > 0)
and (c) for a hyperbolic medium (Eq. (1.4) with o e < 0).

By contrast, a hyperbolic medium is one for which the dielectric (or magnetic)
properties along different coordinate directions are opposite in sign, i.e., o e < 0
(µo µe < 0). A naturally occurring material that exhibits this response is hBN [36],
however, for hBN this response is physically bound to the IR regime and to a rather
narrowband frequency regime. In the case of a hyperbolic medium, the EFC is no
longer a closed surface as in Fig. 1.6b. By contrast, from Eq. (1.4), the EFC opens
up into a hyperbola, as shown in Fig. 1.6c.
The EFC diagram is useful, among others, in computing the group velocity and
density of optical states for bulk propagating modes in material systems. Namely,
given the dispersion equation ω(~k), the group velocity is [2]
v~g =

∂ω
∂~k

(1.5)

Hence, the group velocity vector is normal to the EFC and points in the direction
of its displacement as a function of increasing frequency [2, 37]. From Fig. 1.6a and
Eq.(1.3), we see that, for light propagation in isotropic materials, the direction of
the group velocity v~g coincides with the direction of propagation, i.e., the direction
of ~k. By contrast, due to the elliptical EFC shape of anisotropic dielectric media
(Fig. 1.6b), the propagation direction may differ from the group velocity, however
the two typically define an acute angle due to small birefringence values in naturally
occuring materials. The case of HMMs is of special interest as, in this case, v~g and ~k
can form an obtuse angle, a consequence of o e < 0, allowing for all-angle negative
refraction. Negative refraction and its implications have been widely reported within
the last decade using HMMs [36–40].
HMMs are also often utilized as metamaterial platforms for enhancing the luminescence properties of quantum emitters [41, 42]. The fundamental parameter re-

12
(a) Type I HMMs

(b) Type II HMMs

Figure 1.7: The physical implementation and corresponding equifrequency
contours for (a) type I HMMs, with metallic wires in a dielectric host, and for
(b) type II HMMs, with metallodielectric multilayers. Figure adapted from
[33].

sponsible for this functionality is the density of optical states (DOS), which is often
enhanced using HMMs. Particularly, the DOS is given by [30, 41, 43]
D(ω) =

L 2
2π

d3~k

(1.6)

Shell

where L3 is the the physical volume of the system and the quantity

3~
Shell d k ex-

presses the volume enclosed by two EFCs corresponding to frequencies ω and ω +dω,
respectively. Therefore, the open-shaped EFCs that HMMs exhibit (Fig. 1.6c) result in larger DOS compared to conventional isotropic or anisotropic dielectric media
(Fig. 1.6a and b, respectively). Based on the large DOS of HMMs, many interesting phenomena have been recently discussed in the literature, including topological
transitions in metamaterials [42], Purcell factor enhancement [41, 44] and directional
light emission [45].
The physical realization of HMMs requires composite structures with drastically
different dielectric properties along different coordinate directions, in order to induce
the o e < 0 response that we discussed above. To date, there are two well known
ways to realize HMMs; the first one is with metallic wires in a dielectric host, or,
vice versa, with dielectric wires in a metallic host. In this case, depicted in Fig.
1.7a, the out-of-plane (z) or extraordinary permittivity is negative, e < 0, while in
the ordinary directions (x and y), the response is dielectric with o > 0. This type
of hyperbolic response is typically referred to as type I and its three-dimensional
EFC is shown in Fig. 1.7a.
Type II HMMs are ones that support o < 0 while e > 0. This type of response is
easily achievable with metallodielectric multilayer metamaterials (Fig. 1.7b). The
hyperbolic response of metallodielectric multilayers can be interpreted as follows:
along the ordinary directions (x and y in Fig. 1.7b), the metallic layers dominate

13
as charge carriers are free to transport without encountering dielectric barriers,
and hence the system behaves as an effective metal with negative permittivity. By
contrast, in the extraordinary (z) direction, the dielectric layers act as barriers of
conduction and the system responds as an effective dielectric with positive permittivity. We note that there are certain frequency regimes at which type I hyperbolic
response can be observed with metallodielectric layered metamaterials [32, 33].
1.5

Scope and structure of the thesis

The scope of this thesis is to provide theoretical and computational tools to model
the response of planar heterostructures, mainly in the subwavelength limit and focusing on planar HMMs, together with providing approaches for characterizing them
experimentally. Until now, the response of planar HMMs has been mainly modeled
with effective medium approximations that date back to 1985 [46]. A large portion
of the thesis is focused in addressing certain properties of planar heterostructures
that previous EMAs or other approaches, discussed in detail in Chapter 2, tend
to omit [47]. Hence, we unveil and address in detail in Chapter 3 the magnetic
response of planar heterostructures, typically expressed through the effective mag~ eff . Until now, the consensus in metamaterials research was that
netic permeability µ

~ eff = I. By contrast,
planar metamaterials do not possess magnetic properties and µ
in Chapter 3 we propose a concept for tailoring the effective magnetic response
within planar, unpatterned, 1D multilayer structures [12]. We show theoretically
and experimentally that the magnitude and sign of the permeability tensor may be
engineered at will and we discuss its implications. In Chapter 4, inspired by our
results in Chapter 3, we focus on the implications of artificial magnetism on the surface waves that propagate on planar heterostructures. Previously, HMMs has been
considered to mainly support transverse-magnetic (TM) polarized surface plasmon
polaritons due to the metallic layers that support plasmons below their plasma
frequency. By inducing artificial magnetism in HMMs, we show that it is possible to guide transverse-electric (TE) polarized surface waves simultaneously with
their TM counterpart. We particularly focus on transition-metal dichalcogenides
and polar dielectric media and discuss regimes of simultaneously TM and TE, i.e.
unpolarized, surface wave propagation with the purpose of generalizing previously
polarization-dependent plasmonic properties of metals.
In Chapter 5, we switch gears and discuss possible means of inducing active tunability in planar arrangements, with the scope of actively controlling the dispersion
characteristics of metamaterials. Practical implications of these results pertain to
optical components, for example in optical switches, tunable polarizers, active displays and holography. We discuss the visible frequency range where transparent

14
conductive oxides (TCOs) can act as tunable materials for inducing active tunability in TCO-based planar HMMs [48]. At IR frequencies, we discuss graphene as the
tunable component, that, through carrier injection, can also tune the response of
graphene-based HMMs.
Finally, in Chapter 6 we transition to heterostructures composed purely of twodimensional materials, and demonstrate that doped graphite, composed of doped
graphene mono-layers, or graphene/hBN heterostructures can behave as “supermetals”, surpassing the perfect-electric-conductor response of noble metals at IR
frequencies [18].
We hope that the results of this thesis demonstrate the potential of artificial matter
as a field of study for designing novel optical materials.

15
Chapter 2

METAMATERIALS’ HOMOGENIZATION
“Essentially, all models are wrong, but some are useful.”
George Box, Science and Statistics, 1976
2.1

The concept of homogenization and S-parameter retrieval

The objective in metamaterials’ research is to design composite media for expanding
the properties of naturally available materials. Hence, it is often necessary to design
meta-atoms with complex geometrical arrangements that are drastically different
from the periodic arrangement of atoms in naturally occurring crystalline materials
(Fig. 1.1a). As can be inferred from the complex metamaterial shapes shown in
Fig.1.1b and in Fig. 2.1a, where we show typical SRRs, wire arrays, multilayers and
plasmonic nanoparticles, it is often computationally inefficient, if not impossible,
to attempt solving Maxwell’s boundary condition at each boundary of the problem.
An alternative approach is necessary for modeling complex structures and predicting
macroscopic phenomena. As mentioned in Chapter 1, provided that the wavelength
of the incident electromagnetic field is much larger than the unit of the metamaterial,
it is possible to assign effective parameters that describe the collective response of
a system [14].
This assignment of effective parameters, namely effective dielectric permittivity eff
and magnetic permeability µeff , is termed homogenization. There exists a variety of methods dedicated to metamaterials’ homogenization, ranging from effective
medium approximations (EMAs) initially developed in the early 1980’s [46, 49–52],
field-averaging approaches [53] and integral method [54–56], and S-parameter retrievals. Here, we will focus on S-parameter approaches, initially formulated in the
2000’s by Smith et al. in [57, 58] and by Chen et al. [59].
Conceptually, it is convenient to replace an inhomogeneous composite medium,
which typically consists of a collection of scattering objects, by a homogeneous
medium that exhibits identical scattering response to electromagnetic fields, namely,
the same complex transmission and reflection coefficients, t and r. This correspondence is demonstrated in Fig. 2.1a and b. The composite medium in Fig. 2.1a could
be, for example, an array of SRRs, or plasmonic nanoparticles, nanowire arrays or a
multilayer metamaterial, the meta-atoms of which could be arranged periodically or

16
(a)

(b)

medium

kz1

d Arbitrary composite

(c)

Homogenization

Equivalent effective
medium
(Zeff, keff)
(εeff, μeff)

(ε1, μ1)
ΗΤΕ
ΕΤΜ
kx

θin

keff

kz2 θ

kx
out

(ε2, μ2)

Figure 2.1: (a) An arbitrary composite medium could consist of any combination of meta-atoms in any arrangement, for example split ring resonators,
plasmonic nanoparticles, nanowires or multilayers. (b) An equivalent homogeneous medium that exhibits the same scattering properties as (a) and is
described with an effective permittivity eff and an effective permeability µeff .
(c) Schematic and definition of parameters with respect to Eqs. (2.3), (2.4).

aperiodically. This is the fundamental concept of the S-parameter retrieval approach
[57–60], which we explain in detail in what follows.
Let us start by examining the case of a finite slab of thickness d of an isotropic
and homogeneous material with dielectric permittivity and magnetic permeability
µ. Two equivalent parameters that can be introduced to describe the response of
µ and Z =
µ/,
the slab are the refractive index and the impedance, n =
respectively. By convention, we define the two linear polarizations as shown in Fig.
2.1c: transverse electric (TE) polarized waves, alternatively known as s-waves, have
an electric field vector transverse to the plane of incidence (Ey , 0). Similarly,
transverse magnetic (TM) polarized waves, or p-waves, have a magnetic field vector
transverse to the plane of incidence (Hy , 0).
The basis of the S-parameter retrieval lies on the fact that the complex transmission
and reflection coefficients of an incident electromagnetic field, namely t and r, can
be expressed analytically as functions of n and Z. In order to do this, we introduce
keff ≡ kz = n(ω/c)cosθ and ξ = Z p ω/c, where p = −1 for TE polarization while
p = 1 for TM polarization, and θ is the refraction angle inside the slab, as shown in

17
Fig. 2.1c, which can be calculated through Snell’s law in the case of homogeneous
substances. In terms of these quantities, the complex transmission and reflection
coefficients, as expressed by Menzel et al. in [60], are:

t(k, ξ) =

2k1 ξA
ξ(k1 + k2 )cos(kd) − i(ξ 2 + k1 k2 )sin(kd)

(2.1)

r(k, ξ) =

ξ(k1 − k2 )cos(kd) + i(ξ 2 − k1 k2 )sin(kd)
ξ(k1 + k2 )cos(kd) − i(ξ 2 + k1 k2 )sin(kd)

(2.2)

where k1,2 = α1,2 kz1,2 . We have kz1,2 = n1,2 (ω/c)cosθin,out where n1,2 =

1,2 µ1,2 ,

the refractive indices of the surrounding media, and 1 , µ1 , 2 , µ2 are the dielectric
and magnetic properties of the surrounding media, as shown in Fig. 2.1c. We define
αi = 1/µi for TE polarization and αi = 1/i for TM polarization. In Eq. (2.1),
the factor A amounts to A = 1 for TE polarization, whereas A =

1 µ2 /2 µ1 . By

inverting Eqs. (2.1, 2.2), we can obtain the expressions for k and ξ:

kd = ±arccos

k1 (1 − r2 ) + k2 (t/A) 2
± 2mπ
(t/A)[k1 (1 − r) + k2 (1 + r)]

ξ=±

k12 (r − 1) 2 − k2(t/A) 2
(r + 1) 2 − (t/A) 2

(2.3)

(2.4)

Eqs. (2.3), (2.4) hold for any angle of incidence, embedded in k1,2 . Additional
constraints must be imposed in order to select the correct sign and branch in Eq.
(2.3) and the correct sign in Eq. (2.4). For this, we use the physical requirements
for continuity and passivity. Namely, by starting the retrieval calculations from the
quasistatic limit ω → ∞, the branch m = 0 is selected. Subsequently decreasing
the frequency requires adjusting the branch number in order to ensure continuity
of the wavenumber k. Furthermore, we must ensure exponential decay for waves
propagating in the positive z direction. Assuming time-dependence of the form
~ ~r ) = E(~
~ r )ei(~k~r−ωt) , the imaginary part of k must be selected to be positive,
E(t,
which fixes the sign in Eq. (2.3). Finally a passive medium requires Re(Z) > 0,
fixing the sign in Eq. (2.4).
For complex geometries like the ones schematically presented in Fig. 2.1a, the
scattering properties t, r (also termed S-parameters, hence the name S-parameter
retrieval) are usually calculated with numerical methods, for example High Frequency Structure Simulator (HFSS) finite elements approaches, Lumerical Finite
Difference Time Domain (FDTD) simulators, Comsol Multiphysics or the Fourier
Modal method [60, 61]. In what follows, we will refer to the calculation of t and r

18
as the forward problem, in contrast to the inverse problem, being the calculation of
k and ξ through Eqs. (2.3), (2.4), given t, r (see Fig. 2.4).
In the case of inhomogeneous substances, the parameter k = keff is the effective
wavenumber of the metamaterial, the determination of which is the scope of homogenization. keff together with ξ allow decoupling of the effective dielectric permittivity eff from the magnetic permeability µeff . For the two linear polarizations
we have:
µ(kx , ω) =

(TE)
ξ(kx , ω)

(kx , ω) =

(TM)
ξ(kx , ω)

(2.5)

For TE (TM) polarization, the dielectric permittivity eff (magnetic permeability
µeff ) is obtained through the dispersion equation for isotropic materials:
kx2 + k 2 ω 2
= 2
eff µeff

(2.6)

where kx = n1,2 (ω/c)sinθ1,2 = k// is a conserved quantity that defines the angle
of incidence and frequency. For isotropic metamaterials, an effective index may
introduced, defined as:
neff = ±

kx2 + k(kx , ω) 2
ω/c

(2.7)

Eqs. (2.3), (2.4) are fundamental in understanding the S-parameter retrieval approach, and have been very widely used in order to obtain and report extraordinary
electromagnetic properties in 2D and 3D metamaterials. Examples include metallic
strip metamaterials [62], SRRs [13, 17, 57–59], fishnet structures [60] and others [37].
The method describe here is general and can be applied to obtain the effective parameters (eff and µeff ) along different coordinate directions for any inhomogeneous
slab, in the quasistatic limit. Despite its generality, the S-parameter approach must
be used with caution and the derived parameters eff and µeff have to be assessed
critically.
We briefly highlight that for the retrieval to be valid, in other words for the dielectric and magnetic effective parameters to be a useful macroscopic description
of the metamaterial, the system under consideration oughts to be in the subwavelength limit. Issues of non-locality and spatial dispersion, i.e. the dependence of
effective parameters on the wavenumber, arise when the metamaterial unit cell is
not deeply subwavelength [63]. In this regime, unphysical properties may occur, for
example negative Im(eff , µeff ). The conditions under which it is appropriate and
useful to introduce effective parameters for describing the macroscopic response of
metamaterials remains an area of active debate and research, see, for example [54,
63–66].

19
We stress that the parameters introduced here (eff , µeff ) are, in general, angle of
incidence dependent. By contrast, the material parameters of naturally occurring
material systems are, in general, angle independent, a property referred to as locality.
In order to obtain local material parameters from eff , µeff , further corrections are
necessary, depending on the symmetry of the metamaterial arrangement. We discuss
this in more detail in Section 2.4.
Additionally, to accurately employ the S-parameter retrieval in subwavelength composite systems, they ought to be centrosymmetric with respect to their center in z.
This is a requirement in order to ensure that the reflection, coming from incidence
from either side (1 or 2 in Fig. 2.1c) is the same, so that the effective parameters do
not depend on the incidence side, and avoid further non-locality complications (for
a detailed discussion see [60, 65]). Furthermore, the applicability of the retrieval is
limited to systems that maintain the polarization of incident light upon reflection
and transmission (i.e. in the absence of depolarization effects).
2.2

Effective medium approximations and the Maxwell Garnett result
for 1D metamaterials

Despite the extremely wide use and applicability of S-parameter retrieval approaches
for 3D and 2D metamaterials [13, 17, 37, 57–60, 62], planar 1D multilayer metamaterials are typically treated with effective medium approximations (EMAs). Particularly, planar HMMs, introduced in Section 1.4, constructed of subwavelength
metallic and dielectric layers are almost exclusively treated with EMAs [10, 31–33,
36, 38, 39, 41, 67–69]. The most commonly used approximation for modeling the
dielectric permittivity of a planar 1D multilayer is the Maxwell Garnett approximation, developed in 1985. We present it below [46].
First we note that a planar heterostructure exhibits uniaxial anisotropy, hence the dielectric permittivity is a diagonal tensor, ~eff = diag{x = o , y = o , z = e }, where

o and e are the ordinary and extraordinary components of the tensor, corresponding to the directions in-plane and along the optical axis (z direction), respectively.
Here we have assumed that the constituent materials of the planar heterostructure
are isotropic and non-magnetic, for simplicity.
Now let us constrain ourselves to a purely periodic, infinite heterostructure composed
of two materials with dielectric permittivities 1,2 and thicknesses l1,2 , as shown
schematically in Fig. 2.2. In order to compute o , we must consider electric fields
tangential to the heterostructure interfaces (x or y). The electric displacement field
at each layer of the heterostructure is given by Dx1,2 = 1,2 Ex1,2 , and hence the
average electric displacement over the heterostructure’s period Λ = l1 + l2 is

ΕΤΜ
ΗΤΕ

θin

kz inc

KB
an-1 cn-1 an cn
bn-1 dn-1 bn dn
l1 l2 l1 l2 l1 l2 l1 l2
ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2

(ε1, μ1)

z=(n-2)Λ z=(n-1)Λ z=nΛ z=(n+1)Λ

θout

kz out

(ε2, μ2)

Figure 2.2: Schematic of an infinite periodic bilayer medium with parameters
(1,2 , l1,2 ). KB is the Bloch wavenumber discussed in Section 2.3. an , bn ,
cn , dn are the forward and backward field amplitudes computed through the
transfer matrix for layered media [1]. kx is the in-plane wavenumber, which
remains conserved, and kz,inc/out are the incident and outgoing out-of-plane
wavenumbers.

hDx i =

1
l1 1 Ex1 + l2 2 Ex2
l1 + l2

(2.8)

The basic assumption behind the Maxwell Garnett result lies in the following: by
assuming l1,2
λ, where λ is the free space wavelength of the incident field, we can
make the approximation that the variation of the electric field Ex inside the heterostructure is extremely slow and can be neglected. By further employing Maxwell’s
boundary condition for the continuity of the tangential electric field across a boundary, we arrive at Ex1 = Ex2 = hEx i. Thus, from Eq. (2.8) and Maxwell’s constitutive
equation D = E, we obtain
o =

1
l1 1 + l2 2
l1 + l2

(2.9)

Analogously, the average value of Ez is
hEz i =

1 Dz1
Dz2
l1
+ l2
l1 + l2
1
2

(2.10)

By employing Maxwell’s boundary condition for the continuity of the normal component of the electric displacement and, again, assuming no field variation within
a period Λ of the heterostructure, we have Dz1 = Dz2 = hDz i. Using Maxwell’s
constitutive equation D = E, we obtain

21
1 l1 l2
e l1 + l2 1 2

(2.11)

Eqs. (2.9), (2.11) summarize the Maxwell Garnett result [46]. It is important to
emphasize that the validity of the Maxwell Garnett EMA is dependent upon the
accuracy of the assumption that the fields exhibit negligible variation within the
metamaterial unit cell. For this to be true, the index contrast between the two
materials that compose the heterostructure ( 1 − 2 ) ought to be small. Furthermore, EMA approaches, including the Maxwell Garnett result, typically pertain
to a bilayer arrangement that is infinite and purely periodic [46, 49–52]. Finally,
as seen from the analysis above, there has been no reference to magnetic effective
properties. Traditional EMAs for planar heterostructures typically assume µeff = 1
along all coordinate directions. This is justified with the argument that the absence
of any circular inclusions in a 1D arrangement of layers precludes the induction of
currents distributions that could resemble a magnetic dipole response. For more
details on the accuracy of this assumption, see Chapter 3.
In contrast to EMAs, the applicability of S-parameter approaches is not limited in
terms of the number of unit cells and constituent materials; the interior of the slab
of thickness d (see Fig. 2.1a) can be composed of any arrangement of meta-atoms,
as long as they are subwavelength and centrosymmetric, as mentioned in Section
2.1. Furthermore, by using an S-parameter approach that accounts for the finite
thickness of the slab, displayed with d in Fig. 2.1a, one can decouple the dielectric
response from the magnetic one, through Eqs. (2.5) and (2.6).For more details see
Chapter 3.
Despite the constraints under which the Maxwell Garnett EMA is valid, its simplicity
(see Eqs. (2.9), (2.11)) makes it the most widely used method for approximating
the response of a HMM [10, 31–33, 36, 38, 39, 41, 67–69].
2.3

The Bloch wavenumber

Another homogenization approach applicable to planar 1D heterostructures is the
Bloch method [69–71], formulated, among others, in [1]. In its most widely used
form discussed here, it is also constrained to purely periodic bilayer arrangements
that are unbound, similar to EMAs. In contrast to EMAs, the Bloch result holds
at any frequency range and is not limited to the quasistatic metamaterial limit. For
example, the Bloch approach is also widely used for determining the band structure
of 1D photonic crystals.
We start with a bilayer 1D arrangement that is periodic in the z-direction (see Fig.
2.2), with period Λ. This permittivity of this composite can be expressed through

22
(z + Λ) = (z). This dielectric environment is seen by a propagating electric field
equivalently to the way in which an electron wavefunction experiences a periodic
potential in crystalline solids. Hence the formulation of this problem is identical to
the Kronig-Penney model for the band theory of solids [1, 30]. Hence, solutions to
the wave equation are given by

EK (z, x) = EK (z)ei(kx x−ωt) eiKB z

(2.12)

where kx is the in-plane wavenumber. KB is the Bloch wavenumber that describes
wave propagation in the z direction, together with EKB (z). Using the Floquet
theorem, we have for EK (z)
EK (z + Λ) = EK (z)

(2.13)

Another formulation of wave propagation in the bilayer periodic layered medium
depicted in Fig. 2.2 is with the traditional transfer matrix approach [1]. Based on
it, the electric field is E(z, x, t) = E(z)ei(kx x−ωt) , where E(z) is expressed as a sum
of forward and backward propagating plane waves:

E(z) =

an eikz,1 (x−nΛ) + bn e−ikz,1 (x−nΛ) ,

nΛ − l1 < z < nΛ

cn eikz,1 (x−nΛ) + dn e−ikz,2 (x−nΛ) ,

(n − 1)Λ < z < nΛ − l1

(2.14)

The subscript n pertains to the layer order, as shown in Fig. 2.2. kz,1 and kz,2
are the out-of-plane wavenumbers at layers 1 and 2, respectively, given by kz,1,2 =

1,2 ωc2 − kx2 . The relative amplitudes an , bn , cn , dn are determined through Maxwell’s

boundary conditions in a matrix form. The details of the matrix formalism can be
found in [1]. Here, we briefly mention that wave propagation in consecutive unit
cells is expressed through

an−1

bn−1

A B

C D

an
bn

(2.15)

where A, B, C, D are the transfer matrix elements for the unit cell, presented in
Appendix A.1 for both TE and TM polarizations. In terms of this representation
and from Eq. (2.13) together with S, we obtain

an
bn

= eiKB Λ

an−1
bn−1

(2.16)

23
Using Eq. (2.15), we obtain

A B

C D

an
bn

−iKB Λ

an
bn

(2.17)

By solving Eq. (2.17) for the Bloch wavenumber, we get
cos(KB Λ) =

A+D

(2.18)

Eq. (2.18) expresses a homogenization. Similar to the effective wavenumber keff
that we introduced in Section 2.1 (Eq. (2.3)), the Bloch wavenumber KB is an
effective parameter introduced to simplify wave propagation inside the multilayer
heterostructure. The left hand of Eq. (2.18) refers to the homogenized system,
whereas the parameters A and D on right hand side pertain to the layered medium
and are functions of 1,2 and l1,2 .
Typically, in the case of HMMs, the Bloch wavenumber KB is translated into an
effective dielectric permittivity through the dispersion equation for uniaxial media
(see Eq. 1.4 in Chapter 1), by being treated as the out-of-plane (kz ) wavenumber
[69, 71]. The wavenumbers presented in Eqs. (2.3) and (2.17), pertaining to the Sparameter retrieval and to the Bloch approach, respectively, are useful for numerous
reasons apart from serving the purpose of homogenization, as we will discuss in more
detail in Chapter 4. In the limit of a purely periodic infinite medium, Eq. (2.18) is
identical to Eq. (2.3). For details see Chapter 4.
2.4

S-parameter retrieval for 1D metamaterials

The procedure presented in Section 2.1 describes a homogenization for obtaining
effective parameters eff and µeff for any composite system, at any angle of incidence
for either of the two linear polarizations. These results, formulated initially in
[60] are the generalization of previous approaches [57–59], which provided effective
parameters for a particular set of incidence direction and field polarization, i.e. for
~ H
~ ). Despite the generality of the S-parameter retrieval for
a particular set of (~k, E,
oblique incidence, the derived effective parameters do not hold the same meaning as
the dielectric permittivity and magnetic permeability of naturally existing materials.
Particularly, the angle of incidence and polarization dependence of the effective parameters hints that they do not provide direct information about the medium itself.
We will refer to the effective parameters derived from the S-parameter retrieval of
Section 2.1 [60] as wave parameters, emphasizing on the fact that they depend on the
wave polarization and angle of incidence, and we will use the symbols TE/TM (θin ),

24
µTE/TM (θin ) to refer to them. Their angle of incidence dependence typically arises
from intrinsic anisotropy that is not explicitly accounted for by the mere application
of the retrieval discussed in Section 2.1. In contrast to wave parameters, which are
scalars, we introduce the notion of material parameters, which are a global description of a system and are, thus, polarization independent and also independent of
the angle of incidence in the absence of spatial dispersion [72]. Hence, in their most
general form, material parameters are tensors with tensor elements that are independent of polarization, angle of incidence and total thickness of the medium [2].
This is, for example, the case for naturally occurring materials, the optical properties of which are global. By contrast, wave parameters of metamaterials do not
necessarily satisfy these condition [57, 65, 73]. We shall refer to material parameters
~ = diag{µxx , µyy , µzz }.
with ~ = diag{xx , yy , zz } and µ

The representation of the intrinsic electromagnetic properties of any system in terms
of tensorial dielectric permittivity and magnetic permeability, i.e. material parameters, is useful and general. From a theoretical point of view, it provides a concise
description of a system that is independent of the excitation conditions (angle of incidence, polarization, etc). From an experimental point of view, material parameters
are easily captured with conventional methods like spectroscopic ellipsometry; by
rotating a sample to probe different crystal orientations, all the tensor elements may
be obtained. By contrast, it is not straightforward nor meaningful to experimentally probe wave parameters, TE/TM (θin ), µTE/TM (θin ) with conventional material
characterization schemes.
The optimal way to obtain material parameters for metamaterials is to perform
ab initio electronic structure calculations of the meta-atoms arrangements, similar
to natural materials. However this requires very large scale computations and is
currently computationally intractable in terms of CPU performance and memory.
An alternative approach is to take into account the symmetry group of the metamaterial arrangement and to utilize the dispersion equations for its particular class
of anisotropy; isotropic, uniaxial, or biaxial (in the absence of magnetoelectric coupling). The dispersion equation, for a general bianisotropic medium with tensorial
~ is obtained through
dielectric permittivity ~ and tensorial magnetic permeability µ

Helmholtz’s equation

~k × µ
~ + ω 2~E
~ =0
~ −1 (~k × E)

(2.19)

In this chapter we present a method developed for determination of material parameters for metamaterials with uniaxial anisotropy [47]. We shall refer to this method
as material parameter retrieval, as opposed to the S-parameter retrieval described

25
(a)

(d)

(b)

optical axis

uniaxial slab

(c)

eff
out

Figure 2.3: (a) Three-dimensional representation of the composite uniaxial
slab that can be, among others, (b) a nanowire array or (c) a multilayer metamaterial. (d) Projection of (a) onto the xz plane and our convention for the
angle of incidence.

in Section 2.1 [60], which we will refer to as wave parameter retrieval. We are motivated to investigate uniaxial metamaterials due to their extraordinary properties in
the limit of parameters of opposite sign along different coordinate directions, which
we briefly discussed for the case of HMMs in Section 1.4. The two particular configurations that are of special interest are nanowire arrays (Fig. 2.3b) and multilayers
(Fig. 2.3c).
For a uniaxial medium, the permittivity and permeability tensors are of the form
~ = diag{µxx = µo , µyy = µo , µzz = µe },
~ = diag{xx = o , yy = o , zz = e }, and µ

respectively. The subscript -o pertains to the directions normal to the optical axis
(x and y here, see Fig. 2.3(a)), or ordinary direction, whereas -e pertains to the
optical axis, or extraordinary direction. Carrying out the algebra of Eq. (2.19) for
uniaxial permittivity and permeability tensors, for TE polarization (defined in Fig.
2.2) we obtain
kx2 + ky2
kz2
= ko2
o (ω, k)µe (ω, k) o (ω, k)µo (ω, k)

(2.20)

For TM polarization, the dispersion is:
kx2 + ky2
kz2
= ko2
e (ω, k)µo (ω, k) o (ω, k)µo (ω, k)

(2.21)

The dependence of the material parameters o , o , o , o on the wavenumber ~k
is referred to as spatial-dispersion or non-locality. Spatial dispersion may play
some role in systems that are not in the extreme deep subwavelength limit [72,
74].

Without loss of generality, we consider propagation in the xz plane (i.e.

26
Inverse problem

Forward problem
(ε1 , d1)
(ε2 , d2)
(ε3 , d3)
(ε4 , d4)

...

(ε5 , d5)

parameters of constituent
materials (εi, di, i=1,2,...)

Transfer matrix for
layered media [1]

tTE/TM(θin), rTE/TM(θin)

wave parameter retrieval
Eqs. (2.3)-(2.4) [60]

material parameter
retrieval
Eqs. (2.20)-(2.21)

wave parameters
εTE/TM(θin)
μTE/TM(θin)
keff, TE/TM(θin)

material parameters
εo, εe
μo, μe

repeat for different angles of
incidence θin- consistency check

Figure 2.4: Flow chart of material parameter retrieval for uniaxial metamaterials.
ky = 0). As shown in Fig. 2.2, kx is the tangential wavenumber, which is conserved and equal to kx = n1 ko sin(θin ) = n2 ko sin(θout ), where ko = ω/c. Considering the effective medium with effective parameters TE/TM (θin ), µTE/TM (θin ), we
have nTE/TM (θin ) = TE/TM µTE/TM , hence kx = nTE/TM ko sin(θeffTE/TM ), where
θeff is the effective refracted angle in the effective medium, as shown in Figs. 2.1,
2.3. With knowledge of nTE/TM through the material parameter retrieval, we easily obtain θeffTE/TM . Similarly to kx , the normal component of the wavevector is
kz = nTE/TM ko cos(θeff ). However kz is also the effective wavenumber, obtained
through Eq. (2.3). From the above, Eqs. (2.20), (2.21) yield
sin2 (θeffTE ) cos2 (θeffTE )
o µe
o µo
TE (θin )µTE (θin )

(2.22)

sin2 (θeffTM ) cos2 (θeffTM )
e µo
o µo
TM (θin )µTM (θin )

(2.23)

and

for TE and TM polarization, respectively.
Eqs. (2.22), (2.23) connect the wave parameters TE/TM , µTE/TM with the material
parameters o/e , µo/e . At normal incidence θin = 0 and kx = 0, hence sin(θeff ) = 0 and
o = effTE (θin = 0) = effTM (θin = 0) and µo = µeffTE (θin = 0) = µeffTM (θin = 0). This
is expected since at normal incidence the two linear polarizations are degenerate.
By having already retrieved the ordinary parameters o , µo , the wave parameter
retrieval may be applied at oblique incidence for TE (TM) polarization, for obtaining µe (e ), through Eqs. (2.22), (2.23) for non-zero cos(θeffTE/TM ). Repeating this

27
process for different angles of incidence allows us to determine the degree of spatial dispersion of the system under consideration. This procedure is schematically
outlined in Fig. 2.4.
2.5

A sanity check

To illustrate the validity of the described retrieval approach, let us first apply it
to an example case. We select a 50 nm slab of silver (Ag), for which we have a
priori knowledge of its parameters: o = e = Ag and µo = µe = 1. We model Ag
with a five-poles Drude-Lorentz [75] and calculate its transmission and reflection
coefficients for TE and TM polarization, using the transfer matrix approach [1]. By
applying the wave parameter retrieval (see terminology in Fig. 2.4), we obtain the
parameters TE/TM (k// ) and µTE/TM (k// ), presented in Fig. 2.5 as a function of the
incident angle since k// = ko sinθin , where ko = 2π
λ .
For TE polarization and for all incident angles, the effective parameters TE and
µTE are nearly angle independent; see Figs. 2.5a, b, e, f. Furthermore, we have
TE = Ag and µTE = 1 + i0. Although this is the expected result for both linear
polarizations, TM polarization yields a different response. As can be seen from Figs.
2.5c, d, g, h, TM is strongly dependent on the angle of incidence, which demonstrates
that it is not an intrinsic material parameter, as we know a priori that silver is
isotropic, and hence it has an angle independent dielectric response. Furthermore,
µTM drastically deviates from unity and is also strongly dependent on the angle of
incidence, leading to the same conclusion. Therefore, it is obvious that the wave
parameter retrieval does not yield the properties of silver.By contrast the parameters
TE/TM and µTE/TM depend on the illumination conditions (e.g., polarization, angle
of incidence), hence they do not provide direct information regarding the intrinsic
properties of the system under consideration.
This issue was addressed in our discussion in Section 2.4. By utilizing Eqs. (2.20),
(2.21), we obtain the material parameters for the example system, namely o/e and
µo/e , shown in Fig. 2.6. As expected, and contrary to the wave parameters, o/e
and µo/e are angle independent and exactly reproduce the optical response of silver:
o = e = Ag and µo = µe = 1 + i0.
This example serves as a sanity check for validating the accuracy of the material
parameter retrieval for planar systems, described in Section 2.4. A similar sanity
check can be applied to any homogeneous material slab with known parameters, as
long as some loss is involved in the system in terms if a non-zero imaginary part in
at least one of o/e , µo/e . This is necessary in order for the sign ambiguity in Eq.
(2.3) to be resolved (see discussion in Section 2.1).

28
(b) 3

-5

2.5

-15

Im(εTE)

Re(εTE)

(a) 5

-25

300

500

700

wavelength (nm)

Im(εTM)

Re(εTM)

-35

900

300

500

700

900

300

500

700

900

300

500

700

900

1.5
0.5

-45
300
1.2

500

700

wavelength (nm)

900

wavelength (nm)

(f) 8e-3

1.1

4e-3

Im(μTE)

Re(μTE)

700

wavelength (nm)

2.5

-25

-4e-3

0.9

-8e-3

300

(g) 50

500

700

wavelength (nm)

900

(h) 30

wavelength (nm)

Re(μTM)

Im(μTM)

500

(d) 3

-15

0.8

0.5
300

900

-5

(e)

1.5

-35
-45

k//
ko

20
10

20
15
10

300

500

700

wavelength (nm)

900

wavelength (nm)

Figure 2.5: Effective wave parameters for a slab of silver of thickness 50 nm.
(a) Re(TE (k// )), (b) Im(TE (k// )), (c) Re(TM (k// )), (d) Im(TM (k// )), (e)
Re(µTE (k// )), (f) Im(µTE (k// )), (g) Re(µTM (k// )), (h) Im(µTM (k// )), where
k// = ko sinθin . Black dotted lines in (a)-(d) correspond to the dielectric permittivity of silver Ag .

29
(a) 5

εo
εe
εAg

-15

2.5

Im(ε)

Re(ε)

-5

(b) 3

-25
-35
-45

1.5

300

500

700

wavelength (nm)

(d)

500

700

900

500

700

900

wavelength (nm)

8e-3
4e-3

Im(μ)

-4e-3

0.9
0.8
300

0.5
300

900

μo
μe

1.1

Re(μ)

k//
ko

-8e-3
500

700

wavelength (nm)

900

300

wavelength (nm)

Figure 2.6: Restored material parameters for a slab of silver of thickness 50
nm. (a) Re(o/e (k// )), (b) Im(o/e (k// )), (c) Re(µo/e (k// )), (d) Im(µo/e (k// )).

2.6

Comparison between S-parameter retrieval and EMAs: µ , 1!

By establishing (a) the difference between the nature of wave and material parameters in Section 2.4 and (b) the validity of the material parameter retrieval in Section
2.5, we proceed to examine multilayer configurations, which is the main topic of interest in the thesis. We study a layered system composed of Ag/SiO2 subwavelength
layers of thickness 20 nm each, terminated with SiO2 . As discussed in Section 1.4,
metallodielectric multilayer metamaterials exhibit a hyperbolic dispersion, arising
from their unusual property o e < 0. Here, we illustrate this and compare our
results to traditional EMAs.
As illustrated in Fig. 2.4, we start by using the transfer matrix approach for computing the transmission and reflection coefficients of our multilayer system of varying
number of layers for TE and TM polarization and angles of incidence from 0 to 90
degrees, hence k// ∈ [0, ko ], as in the previous example. By first obtaining the wave
parameters TE/TM , µTE/TM , which we omit presenting as they do not provide any
physical insight (see discussion in previous section), we obtain the effective tensorial

(e)

Maxwell-Garnett
Bloch EMA
-1
-2
-3
-4
-5 -1.8
-6
-7 -1.95
-8
-2.1
-9 12.96 12.99 13.03

Im(εo)

(a) 1

Re(εo)

SiO2 Ag

(o)

9 10 11 12 13 14 15 16 17 18 19 20

Re(εe)

12
10
-2
-4
-6

9.3

9.6

9.9

9 10 11 12 13 14 15 16 17 18 19 20

wavelength (in Λ)

18
16
14
12
10

# layers
51

wavelength (in Λ)

(d) 20

Im(εe)

wavelength (in Λ)

(b) 14

0.8
0.7 0.238
0.6
0.222
0.5
12.96 12.99 13.03
0.4
0.3
0.2
0.1
9 10 11 12 13 14 15 16 17 18 19 20

9.05

9.3

9.55

9 10 11 12 13 14 15 16 17 18 19 20

wavelength (in Λ)

Figure 2.7: Effective dielectric permittivity for a multilayer HMM composed
of Ag/SiO2 layers [47]. (a) Re(o ), (b) Im(o ), (c) Re(e ), (d) Im(e ), and
comparison with the Maxwell Garnett result [46] and a generalized non-local
Bloch approach [69].

dielectric permittivity of the metamaterial, ~eff = diag{o , o , e }, shown in Fig. 2.7.
As expected, the in-plane dielectric response is metallic, o < 0, due to the presence
of Ag, which is a plasmonic metal with Ag <0. By contrast, however, in the out-ofplane direction we have e > 0. This is justified as the dielectric (SiO2 ) layers act
as barriers of conduction, hence inducing a dielectric response [31, 47]. Therefore,
in the long wavelength limit we indeed obtain o e < 0 and this configuration is
hyperbolic. It is important to highlight the very weak dependence of the parameters
o , e on the number of layers. This property is fundamental for material parameters
to be valid, along with angle of incidence independence, which was justified in the
previous section.
In Fig. 2.7 we also compare our results with the Maxwell Garnett EMA, which is
the most widely used approach for approximating the response of HMMs, and with

31
2.2

Re(μo)

1.8

2.1

1.95
1.8
9.7

1.6

0.8
10.1

10.5

1.4

Im(μo)

(a)

1.2

(b) 2.2

Re(μe)

1.8
1.6

9 10 11 12 13 14 15 16 17 18 19 20

wavelength (in Λ)

(d) 1.2

2.1

1.95
1.8

9.7

0.8
10.1

10.5

1.4

0.1
9.7

10.1

10.5

0.6

9 10 11 12 13 14 15 16 17 18 19 20

wavelength (in Λ)
0.8
0.4
0.1
9.7

10.1

10.5

0.4
0.2

1.2

0.4

0.6

0.2

Im(μe)

0.6

# layers
51

1.1

9 10 11 12 13 14 15 16 17 18 19 20

wavelength (in Λ)

9 10 11 12 13 14 15 16 17 18 19 20

wavelength (in Λ)

Figure 2.8: Effective magnetic permeability for a multilayer HMM composed
of Ag/SiO2 layers [47]. (a) Re(o ), (b) Im(o ), (c) Re(e ), (d) Im(e ).

a generalized non-local EMA based on the Bloch approach (see Section 2.3) [69].
The two EMAs slightly differ from each other as the Bloch-based EMA accounts
for non-local effects, whereas the Maxwell Garnett result does not. Furthermore,
the material parameter retrieval results differ from both EMAs. This is justified
and expected because, as discussed in Sections 2.1, 2.2, 2.3, these EMAs pertain to
an infinite periodic system, hence the finite thickness of the stack in not taken into
account [1, 46, 47, 69]. By contrast, the basis of the S-parameter retrieval and its
generalization to obtain material parameters lies on accounting for finite thickness
effects. In fact, it is this additional information that allows decoupling the dielectric
permittivity from the magnetic permeability, which we present in Fig. 2.8 [12].
We note that, as the number of layers in increased (see colorbar in Fig. 2.7), our
results asymptotically approach the EMA results, along both the ordinary (in-plane)
and extraordinay (out-of-plane) directions. This is consistent with the fact that the
EMA results pertains to the infinite number of layers limit. We highlight, however,
that even for fifty-one layers (twenty-five unit cells), the dielectric permittivity of a
finite stack does not coincide with EMA results. Equivalent findings were reported
in [47] for Ag/SiO2 multilayers terminated by Ag, yielding the same conclusion: the

32
exact result produced by the material parameter retrieval discussed in this chapter
deviates from traditionally used EMAs for at least up to thirty layers. We refer to our
method as “exact” because it accounts for the finite thickness of the metamaterials
~ eff = I.
and does not a priori assume that µ

~ eff is concerned, we present in Fig. 2.8 the
As far as the magnetic permeability µ
~ eff = I, imposed in previous EMA
parameters µo , µe . By raising the constraint of µ

approaches that are widely used for approximating the response of HMMs [31, 32, 42,
76], we reveal that metallodielectric multilayer metamaterials exhibit a non-trivial
magnetic response! Both µo and µe drastically deviate from unity, as they exhibit
a resonance approximately at λ = 10Λ. To further emphasize on the physicality of
these two parameters, we point out that, similar to o , e , µo , and µe also exhibit
weak dependence on the number of layers.
2.7

Conclusion

In this chapter we presented a general method for retrieving the effective material
parameters for metamaterials with uniaxial anisotropy (Fig. 2.3). The effective
parameters we retrieve pertain to the diagonal elements of the permittivity and
~ eff = diag{µo , µo , µe }, respectively.
permeability tensors, ~eff = diag{o , o , e } and µ

The retrieved parameters are angle of incidence independent, hence they are local
material parameters. We apply our approach to planar metallodielectric metamaterials and arrive to the conclusion that their magnetic permeability deviates from
unity. This is in contrast to the previous consensus [31, 32, 42, 76]. The details of
our method can be supplementary found in [47].
The physical origin of the non-unity magnetic permeability shown in Fig. 2.8, along
with its implications deserve special attention. Hence, we devote Chapter 3 to
optical artificial magnetism with planar metamaterials and we present an analytical
model for understanding this rather surprising result, together with experimental
measurements on fabricated multilayer metamaterials. Additionally, we perform
impedance matching sanity checks that confirm the validity of our results (please
see Fig. 3.6 in Chapter 3).

33
Chapter 3

ARTIFICIAL MAGNETISM IN PLANAR METAMATERIALS
“Science is made up of mistakes, but they are mistakes which it is useful to make,
because they lead little by little to the truth.”
Jules Verne, A Journey to the Center of the Earth, 1864
3.1

Previous approaches: Artificial magnetism in 3D and 2D

In the optical spectral range, the magnetic response of most materials, given by the
magnetic permeability µ, is generally weak. This is famously expressed by Landau
and Lifshitz [2]: “there is no meaning in using the magnetic susceptibility from the
optical frequencies onward, and in discussing such phenomena, we must put µ = 1”.
By contrast, in the optical regime, materials possess a diverse range of dielectric
properties, expressed through the dielectric permittivity , which can be positive,
negative or zero.

(a)

(b)

(c)

(d)

Figure 3.1: Previous results on artificial magnetism with 3D and 2D metamaterials. (a) dielectric rings and nanoparticles, adapted from [21], (b) fishnet
metamaterials, adapted from [22], (c) SRRs, adapted from [11] , (d) high-
wire medium, adapted from [25].

(a)

Meff

(b)

Meff

Figure 3.2: Main design motifs for 3D and 2D magnetic metamaterials. Induced magnetization in (a) dielectric nanoparticles (3D metamaterials) (b) in
dielectric nanorods (2D metamaterials).

The weak magnetic response in natural materials has motivated a search for structures and systems that may exhibit magnetic properties arising from metamaterial
design. Specifically, engineered displacement currents and conduction currents can
act as sources of artificial magnetism when metamaterials are illuminated with electromagnetic fields [11]. Nonetheless, until now, the realization of such magnetic
metamaterials has required rather complex resonant geometries [11, 77, 78], including arrays of paired thin metallic strips [62, 79], split ring resonators [80–82], and
fishnet metamaterials [22], shown in Fig. 3.1 - structures that require sophisticated
fabrication techniques at optical frequencies. The principle of operation of such
magnetic metamaterials is outlined below.
A circulating electric current can create a magnetic dipole and is the key to inducing
magnetism in magnet-free systems. Based on this principle, induction coils generate
and induce magnetic flux, allowing to manipulate magnetic fields at radio frequencies
(RF). The same concept is widespread in metamaterials design [83, 84]; similarly to
the RF regime, by properly shaping metamaterial elements to produce a circulating
current flow, magnetic dipoles are induced. Dielectric nanoparticles [20, 21, 85–88]
and nanorods [25, 89] have been the building blocks for 3D and two 2D magnetic
metamaterial structures, respectively (Fig. 3.2).
3.2

Parameter space for µ and

Despite astonishing previous reports on magnetic metamaterials within the last
twenty years, it is east to see that harnessing artificial optical magnetism, namely at
frequencies above the terahertz range, remains a challenging task. The experimental
realization of 3D and 2D motifs requires multiple and complicated nanolithography
steps, with precision in the nm-scale. By contrast, planar heterostructures and in

35
particular metallodielectric multilayer HMMs, discussed in the previous chapter, are
straightforward to realize experimentally with well-established thin-film deposition
techniques (see discussion in Section 1.3). However, such heterostructures, along
with their all-dielectric counterparts, are generally considered non-magnetic [31, 32,
42, 46, 69, 76].
The rather simple geometrical structure of HMMs could drastically simplify the
means by which we harness artificial magnetism. Furthermore, their hyperbolic
dispersion constitutes them attractive candidates as a platform for investigating
magnetic effects. Particularly, in contrast to their assumed-to-be-trivial magnetic
response, their dielectric properties may be engineered at will along all coordinate
directions, yielding interesting hyperbolic and plasmonic effects. Exciting phenomena attributed to dielectric hyperbolic response, o e < 0, include but are not limited
to negative refraction [31, 36, 38–40, 90] without the need of a negative refractive
index, hyper-lensing [9], extreme enhancement in the density of optical states [42]
and interface-bound plasmonic modes [8, 71, 91–94]. Nevertheless, all these intriguiging physical effects and related applications are limited to TM polarization, while
phenomena related to TE polarized waves have remained largely unexplored. Utilizing the effective magnetic response (i.e. µeff = diag{µo , µo , µe } , I) is necessary
to harness and control arbitrary light polarization (TE as well as TM).
Namely, a uniaxial material with o < 0 while e > 0 supports negative refraction of
phase for TM polarization. On the other hand, for o > 0 while e < 0, the material
supports negative refraction of phase for the same polarization [2]. Furthermore,
a plasmonic metal or a metamaterial with negative permittivity, < 0, supports
surface plasmonic waves. These properties are shown in the first quadrant of Fig.
3.3 and have been demonstrated with HMMs [31, 36, 38–40, 71, 90, 91].
By duality, a uniaxial material system with a hyperbolic magnetic response of the
form µo < 0 while µe > 0 can allow for negative refraction of phase for TE polarization, demonstrated in the second quadrant of Fig. 3.3. Equivalently, if µo > 0
while µe < 0, negative refraction of energy would be supported for TE polarization
(fourth quadrant in Fig. 3.3). Similarly, a negative µ will yield magnetic plasmons for TE polarization, the magnetic counterpart of surface plasmon polaritons
(SPP), as demonstrated in the third quadrant of Fig. 3.3. Finally, gaining control over the magnetic permeability in planar systems can yield impedance-matched
epsilon-and-mu-near-zero (EMNZ) optical responses (Fig. 3.3-middle) [96]. While
it is straightforward to tailor the permittivity to cross zero in planar metamaterials
[97], a simultaneously EMNZ metamaterial at optical frequencies has not yet been
demonstrated. To date, none of the aforementioned exciting physical effects have

μe
Plasmonics
ε<0

TE Hyperbolic II
μo<0
kz

TM Hyperbolic II TM Hyperbolic I
εo<0
εe<0

ΕΜΝΖ
εoμo~0

Magnetic plasmon
μ<0
kz

TE Hyperbolic I
μe<0

μo

Figure 3.3: Parameter space for µ and in uniaxial structures. First quadrant: engineered dielectric response yields TM-polarization plasmonic surface
waves ( < 0) and hyperbolic metamaterials of type I (e < 0) or II (o < 0),
typically employed with metallic nanowires and metallodielectric multilayers,
respectively (HMM schematics adapted from [95]). Second and fourth quadrants: Engineerring µeff can yield TE-polarization hyperbolic response of type
II (µo < 0) or I (µe < 0), supporting negative refraction of phase and energy, respectively. Middle: and µ near zero (EMNZ) regime: phase diagram
demonstrating vanishing phase of incident light.

been demonstrated at high frequencies for TE polarization.
3.3

Artificial magnetism in 1D

Motivated to search for ways to induce artificial magnetism in planar systems, we
start by considering a single subwavelength dielectric slab of refractive index ndiel
and thickness d. When illuminated at normal incidence (z direction in Fig. 3.4a),
~ induces a macroscopic effective magthe displacement current J~d = iωo (n2 − 1) E
diel

~ eff = 1/2µo (~r × J~d )dS [2, 25, 53]. By averaging the magnetic field,
netization M

hHi =

R d/2

−d/2 H (z)dz, we use µeff ≃ 1 + Meff /(µo hHi) to obtain an empirical closed-

form expression for the magnetic permeability:

µeff ≃ 1 −

n2diel − 1
2n2diel

−1 +

ndiel πd/λ
tan(ndiel πd/λ)

(3.1)

37
By setting ndiel = 1, we recover the unity magnetic permeability of free space. From
Eq. (3.1), we see that µeff diverges when tan(ndiel πd/λ) = 0. This yields a magnetic
resonant behavior at free space wavelengths λ = ndiel d/ρ, with ρ = 1, 2, ... At these
wavelengths, the displacement current distribution is anti-symmetric, as shown in
Fig. 3.4b for ρ = 1, 2. This anti-symmetric current flow closes a loop in y = ±∞ and
~ eff that is opposite to the magnetic field of the incident
induces a magnetization M
wave (Fig. 3.4a), leading to a magnetic resonance. Eq. (3.1) enables estimating the
design parameters for enhanced magnetic response; in the long-wavelength limit,
only the fundamental and second order resonances, λ = ndiel d, ndiel d/2, play significant roles. In the visible and near IR, with layer thicknesses on the order of 10 − 100
nm, dielectric indices higher than ndiel ∼ 2 are required for strong magnetic effects
[98]. The same principle applies for grazing incidence, with the displacement current
inducing a magnetic response in the extraordinary or, out-of-plane (z) direction. So
far, we have shown that the circular shape designed to support a closed current loop
is not a requirement for magnetic metamaterials. A planar structure suffices, for
which the current loop closes in ± infinity.
In order to make this magnetic response significant, we extend this principle to
multilayer configurations. We first examine the case of two infinite parallel wires in
air, carrying opposite currents (Fig. 3.5a). Their net current distribution induces
~ eff ∝ ~r × J.
a magnetic moment which scales with their distance, as dictated by M
This is directly equivalent to a layered configuration composed of two high-index
dielectrics separated by air. Their displacement current distribution can be antisymmetric on resonance, as shown in Fig. 3.5b. By calculating their magnetic

Havg

(b)

Meff

Re(Jd) a.u.

(a)

ρ=1
ρ=2

-1
-2
-1/2

z (in d)

1/2

Figure 3.4: A 1D dielectric slab suffices for inducing artificial magnetism. (a)
~ avg is the average magnetic field which faces in the direction opposite to M
~ eff .
(b) Displacement current distribution at resonance for ρ = 1, ρ = 2, for a 90
nm slab of refractive index ndiel = 4.5.

38
permeability µeff [47], we confirm the magnetic character of this arrangement. As
shown with the black curve in Fig.3.5c, µeff strongly deviates from unity.
The planar geometry does not require that the two high-index layers be separated by
air; any sequence of alternating high-low-high refractive index materials will induce
the same effect. For example, replacing the air region with a layer of metal, with
nmetal < 1 at visible wavelengths, does not drastically change the magnetic response.
This is shown in Fig. 3.5c with the red curve for a separation layer of silver. Therefore, at optical frequencies, metals do not contribute significantly to the magnetic
response in this planar configuration. This is in contrast to the gigahertz regime,
where the conduction current in the metallic components of resonant structures has
been necessary for strong magnetic effects [79–82]. From the magnetic field distri-

(b)

Meff

Re(Jd) a.u.

(a)

-1

-2
-1/2

Re(μeff)

Re(Hx) a.u.

-1/2

Havg

Meff

-5

1/2

z (in d)
with air
with Ag

diel.
Ag
diel.

(c)

z (in d)

wavelength (in d)

Figure 3.5: Enhancing 1D artificial magnetism with multilayer configurations.
(a) Two infinite wires carrying opposite currents are equivalent to (b) two
dielectric layers (blue shaded regions) separated by air (pink shaded region)
in terms of their current distribution. Arrows in (b) indicate the direction of
J~d , which is anti-symmetric at resonance. (c) Effective permeability for two
dielectric layers separated by air and silver. Inset: tangential magnetic field
~ eff .
distribution at resonance: average magnetic field is opposite to M

39
bution shown in the inset of Fig. 3.5c, one can see that the average magnetic field
faces in the direction opposite to the magnetization, expressing a negative magnetic
response for the dielectric/silver unit cell.
3.4

Another sanity check: Impedance matching

Here we showed that planar configurations, consisting of either metallodielectric
multilayer HMMs or alternating high- and low-refractive index layers, possess magnetic properties. We are able to compute their magnetic permeability by utilizing
an S-parameter retrieval-based approach (discussed in Chapter 2 and in [47]). Contrary to previously used EMA-based approaches that assume an infinite periodic
arrangement [31, 32, 42, 46, 69, 76], our approach accounts for the finite thickness
of the metamaterial. By lifting the assumption of an infinite medium, both transmission T and reflection R coefficients can be computed and utilized. This allows
obtaining an effective wavenumber keff together with an effective impedance Zeff [58,
99]. These parameters are then used to decouple the effective permittivity from the
µeff
permeability through keff = eff µeff ωc and Zeff =
eff . By contrast, Bloch-based
approaches [1, 69] only consider a Bloch wavenumber KBloch , with no other information available for allowing decoupling µeff from eff (for details see Section 2.3).
Both the Maxwell Garnett EMA [46] and its Bloch-based generalizations (for example [69]) are based on the assumption that µeff = 1. For a more detailed discussion
see Chapter 2.
By letting the magnetic permeability µeff be a free parameter, instead of a priori
setting µeff = 1, we obtain magnetic resonances at wavelengths where magnetic
dipole moments occur, as demonstrated in Figs. 3.4b, Fig. 3.5 b, c. This confirms
the physicality of the non-unity µ; magnetic resonances arise at wavelengths where
the system supports loop-like current distributions.
Another way to establish the validity of the effective parameters is to perform an
impedance-matching sanity check in the low loss limit. Based on electromagnetic
theory, the impedance of a structure at normal incidence, Zeff =

µo
o , must be unity

at transmittance |T |2 maxima. As seen in Figs. 3.6a and b, the retrieved parameters o and µo accurately describe the scattering properties of both metallodielectric
and all-dielectric metamaterials of finite thickness, respectively. By contrast, not
accounting for a magnetic permeability leads to inaccurate prediction of transmittance maxima. This is seen both by our S-retrieval-based approach while setting a
priori the magnetic permeability to unity (Zµ=1 ), and with the traditional EMA;
both approaches fail to predict the resonances.
In what follows, we focus on metallodielectric multilayer metamaterials due to their

|T|2 ndiel=4 Λ
Ag
Zeff
Zμ=1
ZEMA

|T|2 , Zeff

(a)

0.5

25 layers
n =4

(b) 3
|T|2 , Zeff

2.5

diel

wavelength (in Λ)

|T|
Zeff
Zμ=1

n =4.5
diel

n =4.5

1.5

diel

17 layers
n =4.5

0.5

diel

wavelength (in Λ)

Figure 3.6: Impedance matching sanity check at normal incidence for (a) a
twenty-five layers dielectric/Ag metamaterial, for ndiel = 4 and for (b) a seventeen layers all-dielectric ndiel = 4.5/air metamaterial. The transmittance |T |2
calculation (blue solid line) was performed with the transfer-matrix formalism for the physical multilayer system in the lossless limit. The dielectric and
magnetic effective model (Zeff = µo o ) (red solid line) accurately captures
the resonances unlike the non-magnetic approach (Zµeff =1 ) (red dashed line)
and the Maxwell Garnett EMA (black solid line).

41
interesting hyperbolic response. We show experimentally that it is possible to induce
a significant additional magnetic response if the dielectric layers are composed of
high-index materials that are capable of supporting strong displacement currents at
optical frequencies.
3.5

Experimental method: Spectroscopic ellipsometry for metamaterials

In order to experimentally validate the theoretical findings discussed in the previous
sections, we fabricated metamaterials composed of SiO2 /Ag, TiO2 /Ag and Ge/Ag
alternating layers. These dielectrics, with refractive indices at optical frequencies
ranging from nSiO2 ≃ 1.5 to nTiO2 ≃ 2 and nGe ≃ 4 − 4.5, were selected in order to
demonstrate experimentally that as the refractive index is increased, the magnetic
effects are enhanced, as the empirical formula of Eq. (3.1) suggests. We prepared
the layered SiO2 /Ag, TiO2 /Ag metamaterials by electron-beam evaporation onto Ge
substrates. The Ge/Ag sample was deposited on a Si substrate to avoid interface
effects with the first Ge layer. All samples discussed in this work contain layers of 30
nm of Ag. Each Ag layer was deposited by first seeding a 2 nm AgO layer that was
reduced to Ag under vacuum, a process followed for obtaining smoother interfaces
[100]. Atomic force microscopy measurements indicated Ag roughness of 2.13 nm.
The thickness of the Ge, TiO2 , and SiO2 was aimed to be 40 nm. Deviations from
the aimed thicknesses were measured with TEM, varying ±20 nm. TEM images of
the fabricated samples are displayed in Fig. 3.7.
First we measured the optical constants of the individual constituent layers in a variable angle spectroscopic ellipsometer (VASE) system from J. A. Woollam. We also
determined their thicknesses with TEM imaging. We refer to this step as for forward
problem, as indicated in the schematic of Fig. 3.8. Specifically, it pertains to fitting

(a)

50 nm

SiO2/Ag

(b)

50 nm

TiO2/Ag

(c)

Ge/Ag

100 nm

Figure 3.7: Transmission electron microscopy (TEM) images of fabricated
samples. (a) SiO2 /Ag metamaterial, (b) TiO2 /Ag metamaterial, (c) Ge/Ag
metamaterial.

p plane p plane

θin
s plane
s plane

(n, k) of constituent
materials
with ellipsometry
on thin films

EMA fitting

εoMG, εeMG

Homogenization
ε and μ

metamaterial

substrate

Inverse problem

Forward problem

Effective slab
fitting

Thicknesses of
constituent layers
from TEM imaging

εo, εe
μo, μe

Comparison of relative error between
forward, EMA and effective slab fittings

Figure 3.8: Experimental approach - Spectroscopic ellipsometry for uniaxial
metamaterials. The forward problem involves fitting the experimental data
using the physical multilayer geometry whereas the inverse problem pertains to
the fitting of the effective parameters to the raw experimental data in a uniaxial
and Kramers-Kronig consistent model (Meta6 from WVASE (Woollam Co.)).

the experimental data using the physical multilayer structure. By experimentally
determining the thickness and optical properties of the metamaterials constituent
layers, we were able to homogenize them by assigning them effective parameters
(o , e , µo , µe ) [47], while taking into account fabrication imperfections. We then
performed ellipsometric measurements of the full metamaterial stacks and fitted the
experimental data with the effective parameters we computed, in a uniaxial and
Kramers-Kronig consistent model, while the total metamaterial thickness was held
to the value determined through TEM. The fittings were over-determined as the
number of incident angles exceeded the total number of fitted parameters. Additionally, we performed fittings using the EMA model for comparison. The VASE
system utilizes a Levenberg-Marquardt regression algorithm for the fittings. We use
the META6 model of WVASE to incorporate the uniaxial anisotropy.
3.6

Experimental verification of artificial magnetism in 1D

Here we display the fitted effective parameters o , e , µo , µe obtained through spectroscopic ellipsometry for the three samples discussed in Fig. 3.7. As expected
from the previous sections, these metamaterials are hyperbolic in terms of their
permittivity (in other words they are hyperbolic for TM polarization), i.e. o e < 0.
Importantly, we confirm their magnetic character as their magnetic permeability
deviates from unity along all coordinate directions.

(b)

800

900

wavelength (nm)

‫ﳌ‬

Re(εo)

-5

-10

400

600

800

wavelength(nm)

(d)

10
1000

Im(μe)

0.5

wavelength (nm)

-10

Im(εe)

-2

‫ﳌ‬

Re(μe)

100
50
-50
700

1.5

Re(εe)

Im(μo)

(c)

Im(εo)

SiO2/Ag
TiO2/Ag
Ge/Ag

Arg(TTE) (o)

Re(μo)

(a) 4

400

600

800

wavelength(nm)

1000

Figure 3.9: Experimentally determined (a) µo , (b) o , (c) µe , (d) e for
SiO2 /Ag-green, TiO2 /Ag-blue, Ge/Ag-red metamaterials. Solid lines-real
parts, dashed lines-imaginary parts. Shaded regions in (a) indicate the regime
of magnetic resonances in µo for the studied metamaterials. Asterisks in (a)
and (b) indicate the EMNZ wavelength for the Ge/Ag metamaterial. The
EMNZ condition is confirmed by a vanishing phase of the transmission coefficient at the EMNZ wavelength, shown in the inset of (a).

Namely, Fig. 3.9a shows that increasing the dielectric index redshifts the magnetic
resonance in the ordinary direction µo ; the SiO2 /Ag metamaterial supports a magnetic resonance in the long-wavelength ultraviolet (UV) regime (∼300 nm), whereas
the TiO2 /Ag and Ge/Ag metamaterials exhibit resonances in the blue (450 nm) and
red (800 nm) part of the spectrum, respectively. The enhanced absorption in Ge
at optical frequencies leads to considerable broadening of the Ge/Ag metamaterial
magnetic resonance, yielding a broadband negative magnetic permeability for wavelengths above 800 nm. We note that previously reported metallodielectric HMMs
have primarily featured dielectric layers with lower-refractive indices, such as LiF
[101], Al2 O3 [76, 102, 103] and TiO2 [42]. Fig. 3.9 shows that, for layer thicknesses below ∼50 nm, these lower-index dielectric/metal systems exhibit magnetic
resonances in the UV-short wavelength visible regime.
Furthermore, the presence of Ag induces a negative ordinary permittivity o (Fig.
3.9b), which, for the Ge/Ag metamaterial, becomes positive above 800 nm due to the
high-index of Ge. Notably, o crosses zero at 800 nm, similar to µo , as emphasized

50 nm SiO2
30 nm Ag
50 nm SiO2

30 nm Ag
50 nm SiO2

Ψ (o)

50 nm TiO2
30 nm Ag
50 nm TiO2
30 nm Ag
50 nm TiO2

60
40
36

(d)

100
-50
250

(f)

-50

225
175

50 nm Ge
30nm Ag
50 nm Ge

125

400

600

800

wavelength(nm)

1000

EMA

250

100

Δ (o)

Ψ (o )

(e)

(b)

Δ (o)

(c)

60º exp. * 60º fit 65º exp. * 65º fit 70º exp. * 70º fit

Ψ ( o)

(a)

55º exp. * 55º fit

Δ (o)

50º exp. *50º fit

400

600

800

30 nm Ag
50 nm Ge

wavelength(nm)

1000

Figure 3.10: Panel pairs ((a), (b)), ((c), (d)) and ((e), (f)) show the agreement
between raw experimental data, Ψ and ∆, and the ellipsometric fitting, for the
SiO2 /Ag, TiO2 /Ag and Ge/Ag metamaterial, respectively. Shaded regions
in (a), (b) emphasize the disagreement between experimental data and the
effective medium approximation (EMA).

with the asterisks in Figs. 3.9a, b. Thus, the Ge/Ag metamaterial exhibits an EMNZ
response at optical frequencies. The EMNZ condition is confirmed by transfermatrix analytical calculations of the physical multilayer structure. As shown in
the inset of Fig. 3.9a, the phase of the transmission coefficient vanishes at the
EMNZ wavelength, demonstrating that electromagnetic fields propagate inside the
metamaterial without phase advance [96].
By comparing µo and µe in Figs. 3.9a and c, respectively, one infers that increasing the dielectric index leads to enhanced magnetic anisotropy. The parameter µe
only slightly deviates from µo for the SiO2 /Ag metamaterial, while the deviation
is larger for the TiO2 /Ag one. For the Ge/Ag metamaterial, µe remains positive
beyond 800 nm, while µo < 0, indicating magnetic hyperbolic response for TE polarization. Furthermore, all three heterostructures exhibit hyperbolic response for
TM polarization, with o < 0 and e > 0 (Figs. 3.9b, d). Consequently, the Ge/Ag
metamaterial possesses double hyperbolic dispersion.
The ellipsometrically measured parameters Ψ and ∆ correspond to the relative

45
change of polarization state in amplitude and phase, respectively, of a reflected
beam off a sample. With respect to the complex reflection coefficients for TM and
TE polarization, rTM and rTE , Ψ and ∆ are defined as rTM /rTE = tan(Ψ)ei∆ .
Figs. 3.10a-f demonstrate excellent agreement between fitting and raw experimental
data. In Figs. 3.10a, b, we also provide a Maxwell Garnett EMA-based fit for the
SiO2 /Ag metamaterial. The EMA fails to reproduce the experimentally measured
features, in both Ψ and ∆ (grey-shaded regions in Figs. 3.10a, b), which correspond
to magnetic permeability resonances, as can be seen in Fig. 3.9a. These findings
are consistent with the results discussed in Fig. 3.6. Similar EMA-based fits for
the TiO2 /Ag and Ge/Ag metamaterials lead to large disagreement with the experimental data across the whole visible-near IR spectrum and are, thus, omitted. This
disagreement is expected, as the EMA approach is based on the assumption that
the electric field exhibits negligible or no variation within the lattice period [46],
which does not apply to high-index dielectric layers (see Section 2.2 for details).
It should be noted that the dielectric hyperbolic response o e < 0 is broadband in
planar systems, as seen in Figs. 3.9b, d. In contrast, the magnetic permeabilities
deviate from unity in a resonant manner along both coordinate directions µo and
µe , thereby making TE polarization-based phenomena more narrow-band in nature.
Here we established, theoretically and experimentally, that metallodielectric layered
systems may be described with an effective magnetic permeability that deviates from
unity across all coordinate directions. The purpose of introducing this parameter
is to build a simple and intuitive description for understanding and predicting new
phenomena, such as TE polarization response in planar systems. In what follows we
discuss how the non-unity and, in particular, the negative and anisotropic magnetic
response that we demonstrated (Fig. 3.9) manifests itself in the characteristics of
TE polarized propagating modes (Fig. 3.11) and surface waves (Fig. 3.12).
3.7

Implications of µ , 1 for bulk propagating modes

We utilize an example system of dielectric/silver alternating layers, similar to the one
we investigate experimentally. To emphasize that enhanced magnetic response at
optical frequencies requires high-index dielectrics, we let the refractive index of the
dielectric material ndiel vary. The calculations and full-wave simulations presented
here are performed in the actual, physical, multilayer geometry (Figs. 3.11a, d, e
and Fig. 3.12) and compared with the homogeneous effective slab picture (eff , µeff
- Figs. 3.11b, c). This helps assess the validity of our model and emphasize the
physicality of the magnetic resonances.
First, we perform transfer-matrix calculations for the example multilayer metama-

(a) 1

ndiel=1.5, λ=485 nm
ndiel=3.5, λ=655 nm

0.8

ndiel=2.5, λ=500 nm
ndiel=4.5, λ=825 nm

0.6
0.4
0.2

0.5

(b) 4

-2 -1 0 1

ETM

ε ε <0
-2 o e
-4

x(μm)

-2 -1 0 1

k// / ko

|E| (e)
a.u.

z(μm)

(d)
(c)

-5

-4

k// / ko

-2

-2
-4

k// / ko

kz / ko

k// / ko

Re(Ey)
a.u.

ETE

μ μ <0
-2 o e
-4
-5

x(μm)

Figure 3.11: Analytical calculations of (a) reflectance and (b), (c) isofrequency
diagrams for a metamaterial consisting of five alternating layers of dielectric
ndiel : 55 nm/Ag: 25 nm. Solid lines in (a) correspond to TE polarization
whereas dashed lines correspond to TM polarization. Solid lines in (b), (c) correspond to real parts whereas dashed lines correspond to imaginary parts. Vertical black lines in (b), (c) indicate the maximum free space in-plane wavenumber k// = ko . Color code is the same for panels (a)-(c). (d), (e): Numerical
simulation of a fifty-five layers dielectric (ndiel = 4)/Ag multilayer metamaterial. The surrounding medium has index nsur = 1.55, allowing coupling of
high-k modes. We increased the number of layers for clear visibility of field
localization inside the structure. Strong field localization is the consequence
of (d) dielectric hyperbolic dispersion for TM polarization (o e < 0) and (e)
magnetic hyperbolic dispersion for TE polarization (µo µe < 0).

47
terial and we show in Fig. 3.11a the angle dependence for TE and TM reflectance.
The strong angle dependence for TM polarization is well understood in the context
of an equivalent homogeneous material with anisotropic effective dielectric response
o e < 0. Bulk TM modes experience dispersion
kx2 + ky2
kz2
= ko2
e (ω, k)µo (ω, k) o (ω, k)µo (ω, k)

(3.2)

where ko = ω/c. This dispersion is hyperbolic, as shown with isofrequency diagrams
in Fig. 3.11b. Losses and spatial dispersion perturb the perfect hyperbolic shape
[32]. In contrast to the TM modes, TE bulk modes interact with the magnetic
anisotropy through the dispersion equation
kx2 + ky2
kz2
= ko2
o (ω, ~k)µe (ω, ~k) o (ω, ~k)µo (ω, ~k)

(3.3)

which is plotted in Fig. 3.11c. For small wavenumbers (k// /ko < 1) and small dielectric indices ndiel , the isofrequency diagrams are circular, in other words, isotropic.
This agrees well with our experimental results; as shown in Figs. 3.9a, c, for the
SiO2 /Ag metamaterial, ordinary and extraordinary permeabilities do not drastically deviate from each other. Increasing the dielectric index opens the isofrequency
contours, due to enhanced magnetic response in the ordinary direction (µo ), which
leads to magnetic anisotropy. We note that the displayed wavelengths are selected
at resonances of µo . Open TE polarization isofrequency contours for ndiel ≥ 2 are
also consistent with experimental results; as shown in Fig. 3.9 for TiO2 and Gebased metamaterials, increasing ndiel enhances the anisotropy. This also agrees well
with the picture of the physical multilayer structure, as shown in Fig. 3.11a; the
TE reflectance indeed exhibits extreme angle dependence for increasing dielectric
index. Strikingly, we observe a Brewster angle effect for TE polarization, which
is unattainable in natural materials due to unity magnetic permeability at optical
frequencies [104].
An open isofrequency surface can yield an enhancement in the density of optical
states relative to free space. Physically, this may lead to strong interaction between
incident light and a hyperbolic structure, and enhanced absorption when it is possible to couple to large wavenumbers from the surrounding medium [105, 106]. So
far, only TM polarization has been considered to experience this exotic hyperbolic
response in planar metallodielectric metamaterials, due to o e < 0 [32, 42, 76].
Based on the open isofrequency surfaces for both TE and TM polarizations in Figs.
3.11b, c, a high-index dielectric/metal multilayer metamaterial may exhibit distinct frequency regimes of double, that is, simultaneously TE and TM polarization,

48
hyperbolic-like response. To confirm this, we perform finite element simulations of
a (ndiel = 4)/silver multilayer metamaterial for both linear polarizations and set
the index of the surrounding medium to nsur = 1.55 to allow coupling to larger
wavenumbers. To facilitate visualizing the interaction between the fields and the
metamaterial, we consider a thick structure consisting of fifty-five layers. Without
loss of generality, we carry out the simulation in the low loss limit to unveil the
physics while avoiding side effects due to losses. Fig. 3.11d demonstrates the wellknown TM hyperbolic response since the electric field is strongly localized within
the multilayer. Switching the polarization to TE (Fig. 3.11e), we observe similar
hyperbolic behavior, which, however, cannot be attributed to dielectric anisotropic
response as the electric field only experiences the in-plane dielectric permittivity o
(Eq.(3.3)). The TE enhanced absorption is associated with the µo µe < 0 condition
[107]; the number of TE modes supported by this metamaterial in this frequency
regime is drastically increased. For a more technical discussion regarding the counting of optical states, see Appendix A.2.
3.8

Implications of µ , 1 for surface waves

Here we investigate surface wave propagation in our example system of a layered dielectric (ndiel )/silver metamaterial. In order to identify the structure’s eigenmodes,
we utilize the mode condition A(ω, k// ) = 0, where A is the 1-1 element of the transfer matrix for the layered system (see Eqs. (2.15)-(2.17) in the previous chapter) [1].
We implement this condition numerically using the reflection pole method [108]. In
order to ensure surface-confined propagation with fields decaying in air and in the
metamaterial, we impose an additional constraint for the waves to be located in the
optical band gaps of both bounding media. We discuss our methodology in detail
in Section 4.3.
Fig. 3.12a displays the dispersion for TM polarization. The identified surface waves
bear similarity to typical SPPs on metallic interfaces [92, 109] and to plasmonic
waves in metallodielectric waveguides and systems [93]. Their plasmonic nature is
evident as their dispersion asymptotically approaches the surface plasma frequency,
similar to SPPs. We show in Fig. 3.12c their field distribution (dashed lines), and
compare to SPPs on an equivalent silver slab (black dotted lines). Such TM surface
waves on metamaterial interfaces are often associated with an effective negative
dielectric response [71, 91, 94]. This is consistent with our effective dielectric and
magnetic model; as we showed experimentally in Fig. 3.9b, the ordinary permittivity
is negative o < 0.
Performing the same analysis for TE polarized waves, we find that TE surfacebound modes also exist (Fig. 3.12b). Their dispersion is parabolic, resembling that

(b)
405

ω/ωp

0.8
0.6

Ge Ag Ge

540

810

0.4

Re(k//)/ko

Re(BTE), Re(ETM), a. u.

0.4
-0.3
-1
-1

ETM

325

0.8

405

0.6

540

0.4

810

n diel=1.5

(c)

n diel=2.5

n diel=3.5

Re(k//)/ko

wavelength (nm)

325

wavelength (nm)

ω/ωp

(a)

n diel=4.5

BTE

SPP
ETM

BTE

SPP

ETM

BTE

SPP

z (nm)

Figure 3.12: (a) TM and (b) TE surface wave dispersion for a metamaterial
consisting of five alternating layers of dielectric ndiel : 55 nm/Ag: 25 nm. (c)
Field profiles (incidence from the left) and comparison to a surface plasmon
polariton (SPP) on an equivalent Ag slab (black dotted line). Calculations in
(c) correspond to a wavelength of 620 nm for ndiel = 2.5, 880 nm for ndiel = 3.5
and 1100 nm for ndiel = 4.5. Blue shaded regions in (c) indicate dielectric
layers whereas pink shaded regions indicate Ag.

of Tamm states in photonic crystals [94, 110]. However, here, we show that they
also exist in the subwavelength metamaterial limit and can coexist with typical TM
plasmonic surface waves. TE polarized Tamm states have been previously associated
only qualitatively with some arbitrary negative net magnetic response [94]. Here,
we confirm this hypothesis and explicitly connect the dispersion of Tamm states in
planar metamaterials to values of magnetic permeabilities that were experimentally
measured (Fig. 3.9). We further identify their physical origin, which is the strong
displacement current supported in high-index dielectric layers with a loop-like distribution on resonance. These TE surface waves emerge in the visible regime for
dielectric layers with refractive index ndiel ≥ 2 (Fig. 3.12b), at frequencies where

50
the metamaterial exhibits a negative effective magnetic response. For this reason,
these states may be seen as magnetic plasmons.
The frequency regimes in which double surface waves are supported demonstrate
the possibility of exciting TM polarized plasmonic modes simultaneously with their
TE counterparts in dielectric/metal pattern-free multilayers.
3.9

Conclusion

Here we have shown that non-unity effective magnetic permeability at optical frequencies can be obtained in 1D multilayer systems, arising from displacement currents in dielectric layers [12]. This makes it possible to tailor the magnetic response
of planar hyperbolic metamaterials, which have been previously explored only for
their dielectric permittivity features.
We experimentally demonstrated negative in-plane magnetic permeability in planar structures, which can lead to double hyperbolic metamaterials. This magnetic
character was verified by performing simple impedance matching sanity checks that
demonstrated the inadequacy of previously used EMA-based approaches for describing planar metamaterials of finite thickness, and the necessity to account for
the magnetic permeability parameter.
By studying bulk and surface wave propagation, we have identified frequency regimes
of a rather polarization-insensitive response. We reported the existence of TE polarized magnetic surface plasmons, attributed to negative magnetic permeability,
which are complementary to typical TM polarized surface plasmonic modes in materials with negative dielectric permittivity. The results reported here can open new
directions for tailoring wave propagation in artificial magnetic media in significantly
simplified layered systems. The reported findings may enable the generalization of
the unique properties of plasmonics and hyperbolic metamaterials, previously explored for TM polarized waves and negative permittivity media, for unpolarized
light at optical frequencies.
The topic of unpolarized surface waves is of special interest; at optical frequencies, only the dielectric permittivity obtains negative values in natural material
systems. Hence, typical SPPs are polarization dependent. Here we demonstrated
that an artificial µeff < 0 may be obtained with simple multilayer configurations,
utilizing materials with relatively large refractive indices, for example Ge at optical
frequencies. This realization motivates us to seek, in Chapter 4, for simplified material systems that can support omnipolarization surface-confined waves that can
compete with SPPs in terms of propagation distance and confinement.

51
Chapter 4

BEYOND PLASMONS: OMNIPOLARIZATION SURFACE
WAVES
“God made the bulk; the surface was invented by the devil.”
Wolfgang Pauli
4.1

“Large permittivity begets high-frequency magnetism”

Electromagnetic surface waves and their interaction with matter provide a path
for tailoring near-field optical phenomena. The rise of plasmonics holds promise
for advancing a broad range of applications in medical technology [111], chemistry
[112], lasers [113–115] and luminescence [116, 117], among others. Plasmons are
coherent electron oscillations that propagate on a metallic surface [109]. Their
intriguing properties originate from the special dispersion characteristics of Drude
metals, and their uniqueness lies in their large mode confinement. Particularly, the
frequency dispersion of a surface plasmon polariton (SPP) exhibits a characteristic
asymptotically increasing in-plane wavenumber (Fig. 4.4), which is unbound in the
lossless limit. Hence the modal wavelength of a surface plasmonic mode on a noble
metal interface can be up to ten times reduced compared to free-space wavelengths
[118]. For systems involving graphene, this confinement factor can even reach values
of hundreds [119–121]. Despite their extraordinary properties, surface plasmonic
waves are limited by polarization; they require an out-of plane electric field for their
excitation, which makes them relevant only for TM polarized fields.
Most generally, the excitation of a TM polarized surface wave requires a material with negative dielectric permittivity . Apart from plasmonic metals with
a broadband < 0 below their plasma frequency, this response is also found at
the Reststrahlen band of polar dielectric materials (red highlighted region in Fig.
4.1). Particularly, the permittivity of polar dielectric materials exhibits Lorentzianshaped resonances, typically at mid-far IR frequencies, as a consequence of lattice
vibrations, c.f. phonons, in their crystal structure. The Reststrahlen band’s < 0
allows for excitation of surface phonon polaritons (SPhP), which are similar to SPPs
but originate from bound charge oscillations in dielectrics, contrary to free charge
carriers in metals [122–124].
A TE-equivalent to the SPP or SPhP requires a material with µ < 0, however natural

52
magnetism typically vanishes at IR and visible frequencies [2, 11, 98] and magnetic
plasmons do not occur naturally [12]. Motivated by this natural asymmetry, in
Chapter 3 we demonstrated that layered 1D metamaterials can exhibit artificial
magnetic properties. As a consequence, we showed that, as an alternative to a
homogeneous material with µ < 0, one can engineer layered systems with effective
magnetic properties to support TE polarized surface waves simultaneously with
their TM (plasmonic) counterparts [12]. From previous surface waves reports in
layered media [94, 110, 125–128], it is unclear what material properties are required
for achieving simultaneously TE and TM surface wave excitation. By contrast, in
Chapter 3, we identified the origin of the artificial magnetic properties we report; it is
the strong displacement current loops in dielectric layers that induces non-vanishing
magnetic dipole moments.
~ therefore sysThe displacement current in a dielectric material is J~d = iωo ( − 1) E,
tems that contain high-permittivity materials will support a stronger displacement
current. As natural magnetism vanishes at high frequencies, high-permittivity materials provide an alternative to inducing artificial magnetism and its implications
[11, 12, 98], one of which is the appearance of TE surface waves. This realization
is nicely captured in the title of the seminal work by R. Merlin in [98]; “Large
permittivity begets high-frequency magnetism”.
In this chapter, we aim to alleviate the polarization dependence of SPPs and SPhPs,
by searching for high-permittivity material systems that can accommodate TM polarized surface waves simultaneously with their TE counterparts, with degree of
confinement similar to plasmons. In contrast to Chapter 3, where we focused on
the geometrical structure of multilayer metamaterials, here we focus on simple configurations of single layers and, therefore, we search the material requirements for
omnipolarization surface-confined wave propagation. In search for high-permittivity
materials, we resort to dielectric and semiconducting systems that exhibit resonances
at which the real part of their dielectric permittivity can reach very large values, on
the order of hundreds (Fig. 4.1).
Such resonances can be found at IR frequencies near the phonon energies of polar
dielectric materials, at the red side of their Reststrahlen band (see red curve in the
green highlighted region in Fig. 4.1). Materials with strong phonon resonances are,
among others, boron nitride (BN) [123, 130], silicon carbide (SiC) [124, 131, 132],
silicon dioxide (SiO2 ) [132], aluminum dioxide (Al2 O3 ) and titanium dioxide (TiO2 )
(see Fig. 4.2b). We emphasize that, in contrast to previous reports [123, 124, 131,
132] that focus on the Reststrahlen band and therefore on TM polarized SPhPs,
here, we focus on the large positive dielectric permittivity regime of polar dielectrics

53
Re(εphon.)
Im(εphon.)

ε>0

Re(εexc.)
Im(εexc.)

ε<0

γd

high-ε band

Reststrahlen
band (εphon.<0)

ωTO

ωLO

Figure 4.1: High permittivity resonances in polar dielectrics and excitonic materials. Red-Dielectric permittivity of a polar dielectric: ωTO and ωLO correspond to the transverse and longitudinal optical phonon energies, respectively
[122]. Blue-Dielectric permittivity of an excitonic resonance, for example in
TMDCs [129].

near their phonon energies. We demonstrate that, under certain conditions, polar
dielectrics can support highly-confined surface waves for both linear polarizations.
Apart from phonons, other quasiparticles that support permittivity resonances at
visible frequencies are excitons [134] in semiconducting materials. At their transition
energies, they exhibit considerably large values of dielectric permittivity, without the
negative permittivity regime of polar dielectrics, as shown with the blue curve in Fig.
4.1. Strong excitonic resonances can be found in typical semiconductors, for example
Si, Ge, GaAs, or InP, as well as in a new emerging classes of two-dimensional materials, namely transition-metal dichalcogenides (TMDCs), for example molybdenum
diselenide (MoS2 ) and tungsten disulfide (WS2 ) [128, 129] (see Fig. 4.2a).
The chapter is organized as follows. In Section 4.2, we outline a concept for surpassing the polarization dependence of plasmonics, and discuss material requirements
that can yield omnipolarization surface waves with confinement factors that can
compete with SPPs. In Section 4.3 we introduce a computational approach for surface wave calculations, inspired by the parameter retrieval that we introduced in
Chapter 2. Although here we focus on single-layer configurations, our method is
general and applies to any layered medium at any scale (c.f. metamaterial subwave-

(a)

WS2

visible: excitons

35
30

25
20
15
10

SiO2

500

600

700

Al2O3

TiO2

SiC

wavelength (nm)

150

1000

100

-500
12

800

500

12.5

IR: phonons

(b)

400

50
-50
10

wavelength (μm)

Figure 4.2: (a) Dielectric permittivity of Si, Ge [133], and WS2 at visible frequencies. The permittivity of WS2 was obtained by numerical fitting of the
experimental data presented in [129] (see Section 4.8). (b) Dielectric permittivity of SiO2 , Al2 O3 , TiO2 and SiC [124] at IR frequencies.

(a)

(b)

ε1>0

ε(ω)<0

d ε(ω), high>0
ε2>0

Figure 4.3: (a) A guided mode exists for TM polarization at the interface
between media with opposite dielectric permittivities. This mode can be either
a SPP or a SPhP on a plasmonic metal or a polar dielectric, respectively. (b)
A mode can be guided inside a slab of a dielectric material with (ω) > 0, if
it is bounded by media with lower dielectric permittivities, 1,2 < (ω). The
mode can be either TM or TE polarized.

length regime or thicker layers). It allows simultaneous determination of the band
structure and all the eigenmodes supported in arbitrary layered systems of finite
thickness. In Section 4.4 we introduce figures of merit for assessing the characteristics of surface waves. In Sections 4.5 and 4.6 we discuss the surface wave dispersion
characteristics in plasmonic metals, polar dielectrics, and excitonic materials. These
quasiparticles, namely plasmons, phonons, and excitons serve as a canvas of excitations for investigating omnipolarization surface wave propagation. In Sections 4.5,
4.7, 4.8 we consider realistic scenarios and show that certain materials, for example
SiC and WS2 at IR and visible frequencies, respectively, are promising candidates
for supporting omnipolarization, phase-matched surface wave propagation that resembles the SPP dispersion. We compare our results with typical SPPs on Ag.
Finally, in Sections 4.9, 4.10 we theoretically investigate the parameter space in
which combinations of plasmons with phonons and excitons, respectively, can yield
omnipolarization surface waves.
4.2

Material requirements for surface-confined propagation beyond plasmonics

We start our investigation by revisiting the boundary condition problem depicted in
Fig. 4.3a, namely surface wave propagation at the interface between two unbound
media with dielectric permittivities 1 > 0 and (ω). As is well known (see [109]), this
problem has surface propagating solutions only for (ω) < 0 and for TM polarization.
By considering a Drude metal, we model the dielectric permittivity (ω) with

56
m (ω) = ∞,m (1 −

ωp2
ω 2 + iωγm

(4.1)

where ωp is the plasma frequency, γm the free carrier collision frequency, and ∞,m
is the high-frequency limit of the permittivity. Solution to the boundary condition
problem of Fig. 4.3a yields the plasmonic dispersion

k//SPP (ω) =

(ω)1
ko
(ω) + 1

(4.2)

where k//SPP is the in-plane wavenumber of the SPP surface wave, while ko = ω/c
is the free-space wavenumber. By considering the permittivity of Eq. (4.1) in the
lossless limit, we display in Fig. 4.4 (black curve) the dispersion of a SPP mode,
which asymptotically approaches to the value of ∼ ωp / 2, as expected for 1 = 1
[109]. We note that the x-axis (k// ) in Fig.4.4 is normalized to ko in order to provide
a direct estimate of the confinement. Namely, the effective wavelength of a surface
wave is given by λeff = 2π/Re(k// ), hence the parameter Re(k// )/ko = λo /λeff
represents the number of mode wavelengths that fit into the free-space wavelength
of excitation, λo . Solutions for TE polarization do not exist for non-magnetic media;
a TE polarized magnetic surface plasmon would require µ(ω) < 0, however at high
frequencies, µ = 1 in natural materials (see Chapter 3, [11, 12, 98]).
Now let us consider another waveguiding problem, that depicted in Fig. 4.3b, assuming positive values of dielectric permittivity, 1,2 , (ω) > 0. In the absence of
frequency dispersion for the intermediate layer , f (ω), this problem is the 1D
version of a famous problem in silicon photonics; that of waveguiding optical power
at a Si waveguide on an SiO2 substrate. At the telecommunication frequency band
near λo ∼ 1.5 µm, we have nSi = 3.4 and nSiO2 = 1.55, and the dispersion of the
waveguide is bounded between nSiO2 ≤ k// (ω)/ko ≤ nSi . This regime of in-plane
wavenumbers is highlighted in Fig. 4.4 with blue color. The guided modes can be
either TE or TM polarized, or both, depending on the thickness of the intermediate
layer, d.
Similar conclusions can be made for any high- material slab between regions of lower
1,2 in Fig. 4.3b; assuming that the top region is air (1 = 1), the frequency dispersion
of k// (ω) will be bounded by the largest value of . Wave propagation inside the
2 +k 2 = (ω)k 2 , where
intermediate layer of thickness d in Fig. 4.3b is described via k//

kz is the out-of-plane wavenumber. As explained above, the quantity k// defines the
degree of confinement of an interface-localized wave. Hence, its maximum possible
value

nSiO2 nSi
0.7
0.71

ω/ωp

0.6
0.5
0.4

high-εI(ω)
high-εII(ω)
SPP

0.3
0.2

0.69

QΙ=10QΙΙ

0.67
0.65

Im(k//) (in ko)

Re(k//) (in ko)

(ω)1
Figure 4.4: Black line corresponds to the SPP dispersion k//SPP = (ω)+
ko ,
for a Drude material with (ω) < 0 (Eq. (4.1)),
√ while 1 = 1 (see Fig. 4.3a).
Red and cyan curves correspond to k//max = (ω)ko for the boundary condition problem in Fig. 4.3b, where (ω) is the high- permittivity of an excitonic
material or a polar dielectric, at visible and IR frequencies, respectively, with
γd,I = 0.1γd,II (Eq. (4.4)), or QI = 10QII . Inset shows the imaginary part of
k// , which corresponds to the losses along the propagation direction. We note
that here we considered ωp ∼ ωTO , with respect to Eqs. (4.1), (4.4).

k//max =

(ω)ko

(4.3)

highlights that the requirement for highly-confined mode is the large value of (ω).
As mentioned in Section 4.1, high-permittivity resonances are found in polar dielectric and excitonic materials, at IR and visible frequencies, respectively. The
permittivity dispersion of a polar dielectric material can be modeled with [124]

phon (ω) = ∞,d (1 +

2 − ω2
ωLO
TO
2 − ω 2 − iωγ )
ωTO

(4.4)

In Eq. (4.4), ωLO and ωTO correspond to the longitudinal and transverse phonon
energies, respectively, and γd is the inverse phonon lifetime (Fig. 4.1). We note that

58
solution to the boundary condition problem depicted in Fig. 4.3a for frequencies
ω at the Reststrahlen band of a polar dielectric material (phon < 0) will yield the
SPhP dispersion, similar to SPPs, as we discuss in what follows (see Section 4.6).
The permittivity of an excitonic material can also be modeled with Eq. (4.4), using
a very large offset ∞,exc to exclude the < 0 frequency regime (Reststrahlen band
in Fig. 4.1). For example, with Eq. (4.4), typical TMDCs [129] and semiconducting
systems are modeled well at visible frequencies.
By considering a Lorentzian-shaped permittivity resonance of the form in Eq. (4.4),
we select ωTO (Eq. (4.4)) close to ωp (Eq. (4.1)), and we display in Fig. 4.4 the
parameter k//max (Eq. (4.3)). As can be seen, due to the high- regime, k// can
compete with k//SPP , and the modes due to high- can, in principle, be as confined
as typical SPPs on Drude metals. However, as can be seen from Fig. 4.1, at
the high- regime of polar dielectrics or excitonic materials, losses (Im()) are also
increased, therefore dramatically decreasing the propagation distance of supported
modes. The losses in polar dielectric and excitonic materials are defined by the
inverse of the phonon and exciton lifetime, respectively, shown in Fig. 4.1 with the
generic parameter γd , which defines the width of the permittivity resonance. In
Fig. 4.4 we consider two different values of γd,I,II , with γd,I = 0.1γd,II . As can be
from the parameter Im(k//max ) displayed in the inset of Fig. 4.4, small values of
γd yield longer-distance mode propagation. It is useful to introduce the material
quality factor, defined as

ωTO
γd

(4.5)

From the two different values of γd,I,II in Fig. 4.4, we have QI = 10QII . It is clear
that the parameter Q ought to be large for large propagation distances and highconfinement factors. Hence, here we focus on high-Q polar dielectric and excitonic
materials at IR and visible frequencies, respectively, and seek for omnipolarization
surface-localized wave propagation.
Prior to delving into the details of designing material systems that support highlyconfined surface-localized wave propagation for both linear polarizations, we outline
in the following section our computational approach.
4.3

Generalized retrieval approach for surface wave computations

The problem of a wave propagating at a single interface between two media may
be analytically treated [109], however larger systems require semi-analytical or numerical approaches [93, 109, 135]. Although in the following sections we focus on
either single interfaces or single slabs, we note that interesting physical effects also

59
arise in multilayer heterostructures [12, 94, 110, 125–127]. For example, typical
photonic, leaky modes appear at the pass-bands of all-dielectric, infinite periodic
photonic crystals [1], while hybrid plasmonic modes arise in metallodielectric multilayer waveguides [93] and subwavelength hyperbolic metamaterials [12, 32, 71]. For
multilayer heterostructures that include more than two interfaces, numerical approaches are typically employed for computing the propagation constants of surface
waves [108, 136–144]. The majority of previous approaches is limited in applicability
in that they are either only able to handle a small number of interfaces [140] or an
infinite arrangement [1], or in that they are not able to distinguish between photonic
and surface-confined modes. By contrast, the method presented here applies to any
layered configuration and allows distinction between photonic and surface-confined
eigenmodes.
In Fig. 4.5 we distinguish between photonic modes and surface-confined ones. A
photonic mode is defined as one that exhibits sinusoidal propagation in the direction normal to the interface, inside the medium of interest (which can be either a
homogeneous material or a composite medium), as shown in Fig. 4.5a. Hence, a
photonic or leaky mode is not strongly confined to the interface between two media. Mathematically, assuming propagation of the form ∼ ei(k// x+keff z−ωt) , where
keff = kz , this translates to an out-of-plane propagation constant keff that is purely
real in all-dielectric media, or has Re(keff )
Im(keff ) in lossy media. By contrast,
a surface-confined mode is one that exponentially decays away from the waveguiding interface, as shown in Fig. 4.5b. In mathematical terms, this corresponds to
a purely imaginary keff in all-dielectric media, or to Im(keff )
Re(keff ) in lossy
media.
A schematic of the structures we investigate is displayed in Fig. 4.6a. We consider
a layered arrangement consisting of an arbitrary number of layers of non-magnetic
materials in an arbitrary sequence, with layer thicknesses that may be in the metamaterial subwavelength limit (di << λo , where λo is the free-space wavelength) or
thicker. Hence, we emphasize that our approach is generic and not limited to the
metamaterial regime. In order to compute the fields, we use the transfer matrix
formalism for layered media of finite thickness [1]. Similar to the transfer matrix for
an infinite and purely periodic medium, discussed in Section 2.3 and in Appendix
A.1, for a layered arrangement of finite number of layers, N, we have the transfer
~ connecting the incident and transmitted and reflected waves through
matrix M

Ao
Bo

m11 m12
m21 m22

AN
BN

(4.6)

(a) Photonic mode

(b) Surface-confined mode

|cos(keff d)|<1

|cos(keff d)|>1

Figure 4.5: Definition of (a) a photonic and (b) a surface-confined mode.

where Ao and Bo are the amplitudes of the incident and reflected wave, respectively,
as defined in Fig. 2.2. AN is the transmitted amplitude, while BN = 0 if waves are
launched only from the left side, with respect to Fig. 2.2. For details, see [1].
By letting the frequency ω and the in-plane wavenumber k// vary, we obtain the
elements m11 (k// , ω) and m21 (k// , ω), based on which the transmission and reflection
coefficients are defined as t = 1/m11 and r = m21 /m11 , respectively.
We seek for surface waves by first determining the full set of eigenmodes of the
structure, for which t → ∞, r → ∞. Alternatively, the eigenmodes are zeros of the
matrix element m11 (k// , ω). We employ the reflection pole method (RPM) [108],
which is based on the residue theorem of complex analysis, for detecting those zeros
in the complex plane. Zeros of the complex function m11 yield phase shifts of π,
which are detected by seeking for peaks of its derivative Arg(m11 ) per frequency
and wavenumber (Fig. 4.6b). As a result, we obtain pairs of (ω, k// ), that excite
eigenmodes. For more details regarding the RPM, see [108]. We note that that
this approach can be generalized to account for anisotropic materials by replacing
the traditional 2 × 2 transfer matrix [1] with a 4 × 4 formalism. See, for example
[145–147].
However, not all eigenmodes of a heterostructure constitute surface waves. For a
wave to be surface-localized, it is required to be located inside the optical band gap of
both bounding media [148], which, in the case studied here, are air and an arbitrary
layered heterostructure (Fig. 4.6a). A plane wave is located inside the band gap of
free-space when simply k// > ko . As discussed above (see Fig. 4.5), for the wave
to be inside the band gap of the arbitrary layered heterostructure, its out-of-plane
wavenumber keff oughts to have a non-zero imaginary part, assuring decay away
from the interface z = 0. Nevertheless, keff is not a always a well-defined quantity.
For an A-B-A-.. purely periodic infinite photonic crystal, wave propagation in the

61
z-direction is accurately expressed in terms of the Bloch wavenumber KB , discussed
in Section 2.3 and given by Eq. (2.18). However, the Bloch wavenumber KB is
inappropriate for any other system, for example a random aperiodic arrangement,
chirped or non-centrosymmetric multilayer.
In order to circumvent this limitation, we propose an alternative description of
the composite medium, for describing a complete surface wave detection scheme
that applies to any layered arrangement. Our approach originates from, but is
not limited to, metamaterials’ homogenization (see Chapter 2). Particularly, it
is based on the S-parameter retrieval, discussed in Section 2.1. As explained in
Section 2.1, an arbitrary composite system of finite thickness d and known scattering
properties, namely complex transmission and reflection coefficients, t and r (Fig.
4.6a), can be represented by an effective impedance and an effective out-of-plane
wavenumber, Zeff and keff , respectively, given by Eqs. (2.3) and (2.4). Typically,
based on the subwavelength size of the meta-atoms (thickness of layers in the 1D
case examined here), these effective parameters are translated to effective dielectric
µeff
permittivity and magnetic permeability through keff = eff µeff ωc and Zeff =
eff .
The conditions under which the assignment of effective parameters eff and µeff is
valid are complex [54, 74] and remain an area of literature debate [57, 63, 98].
We stress, however, that in contrast to the validity of eff and µeff , it is always valid
to describe any system in terms of the effective wavenumber keff and the effective
impedance Zeff (Eqs. (2.3) and (2.4), respectively). For the purpose of surface waves
computations, and for the distinction between photonic and surface-confined modes,
only the effective wavenumber keff is required. Particularly, keff is directly associated
with the scattering coefficients t and r (Eq. 2.3) [58, 60, 99], and hence it can be
used for describing an arbitrary heterostructure at any scale, not necessarily in the
metamaterial subwavelength limit, as long as t and r may be computed. As a sanity
check, we note that keff in Eq.(2.3) is identical to the Bloch wavenumber KB in Eq.
(2.18) in the special case of an unbound and purely periodic medium composed of
two alternating materials, when the cladding and substrate have the same optical
constants (1 = 2 , see Figs. 2.1, 2.2). In this case, Eq. (2.3) reduces to Eq.(2.18).
This is easily seen by expressing t and r in Eq. (2.3) in terms of transfer matrix
~ ) = 1, and the fact that m21 = m∗ , for purely
elements, using the identity det(M

dielectric materials (where m21 and m12 are purely imaginary), and m21 = −m12 ,
for dispersive materials [1].
To recap, for an excitation to be considered as a surface wave, three conditions must
be satisfied; first, it has to be an eigenmode of the structure, which we evaluate with
the RPM (m11 = 0); second, it has to be in the band gap of the surrounding medium

kx=k// z=0
(ε1 , d1)

keff=kz

(ε2 , d2)
(ε3 , d3)

d Determination of all Homogenization [47,60]
eigenmodes
Generalized band
RPM [108]
structure
keff = f(t,r)
m11(ωeig, k//eig)=0
|cos(keff d)|=1

(ε4 , d4)

...

(ε5 , d5)

(b)

Transfer matrix for layered media [1]
m11(ω, k//)
t(ω, k//), r(ω, k//)

m11 (a.u)

Arg(m11) (in π)

|d(Arg(m11))/dk//|(a.u)

ω3
ω2

ω1

1.5

Im(k//eig)

|cos(keff d)|

(a)

0.5

Re(k//eig)

-2
1.6

1.8

2.0

2.2

2.4

0.0
2.6

2.8

3.0

k// / ko
Figure 4.6: (a) Left-schematic of an arbitrary layered system, rightcomputational approach flow chart. (b) Combination of RPM [108] with our
generalized band structure calculation (inspired by [60]) for the distinction between different types of eigenmodes. At the eigenmode in-plane wavenumber
k//eig , m11 vanishes (black solid curve), Arg(m11 ) jumps by π (black dashed
11 )
curve), and its derivative dArg(m
resonates (green curve). The highlighted
dk//
region corresponds to a band gap, for which |cos(keff d)| > 1. For frequency ω1 ,
the mode is photonic, as it resides inside the band, whereas for ω2 the mode is
located at the band edge. For ω3 , the mode is a surface-confined, propagating
at the air/heterostructure interface.

63
(k// > ko ); and third, it has to be in the band gap of the planar heterostructure.
This third condition is implemented by introducing the notion of a generalized band
structure, applicable to any planar configuration. A band is a set of (ω, k// ) for
which

|cos(keff (ω, k// )d)| ≤ 1

(4.7)

In the case of lossless media, this condition yields a purely real keff , whereas for
systems with loss we have Re(keff )
Im(keff ). Surface waves exist in the exterior
of a band or at its edge, for which |cos(keff d)| ≥ 1. To demonstrate this, we present
in Fig. 4.6b an eigenmode of a planar structure. At the eigenmode’s in-plane
wavenumner k//eig , the matrix element m11 vanishes, therefore its phase Arg(m11 )
jumps by π. Taking the derivative of m11 with respect to k// , we obtain a peak, as
shown with the green curve. Its half-width-half-maximum (HWHM) corresponds to
the in-plane decay length through L = 1/2Im(k//eig ) [108]. In order to determine the
nature of the mode (c.f. photonic or surface-confined), we employ our generalized
band edge condition, |cos(keff d)| = 1 . The quantity |cos(keff d)| is shown for three
different frequencies ω1 , ω2 and ω3 . For ω1 , |cos(keff d)| < 1 at k//eig , and this mode
belongs to band, resulting in wave propagation inside the heterostructure, in other
words it is a photonic mode. For ω2 , the parameter |cos(keff d)| crosses unity at
k//eig and this mode is located exactly at the band edge. Finally, for frequency
ω3 , the mode is inside the band gap, highlighted in Fig. 4.6b with the top orange
shaded area, and the mode is forbidden from propagating inside the structure, hence
it as a surface wave. By retrieving keff , we are also able to estimate the degree of
confinement through the penetration depth t = 1/2Im(keff ).
The discussion in this section applies to any layered configuration. However, we note
that when the multilayer heterostructure is reduced to a single isotropic and homogeneous material slab with permittivity , the band condition (Eq. (4.7)) reduces
to k// ≤ k//max , where k//max is defined in Eq. (4.3). In what follows (Sections
4.6, 4.7, 4.8, 4.9, 4.10) we focus on modes located at the band edge of isotropic,
positive media, where Eqs. (4.3) and (4.7) reduce to k// = k//max = (ω)ko and
|cos(keff (ω, k// )d)| = 1, respectively. Despite the fact that such band edge modes
cannot classified as strictly surface-confined, there exist classes of materials for which
propagation is extremely confined, hence we refer to such modes as surface-confined
as well. In the following section we introduce an additional metric of confinement
that is independent of Eqs. (4.3) and (4.7).

64
4.4

Figures of merit for surface-confined propagation

In the following sections we evaluate realistic material systems and configurations
for excitation and detection of omnipolarization surface waves. In order to assess
the characteristics of these waves, we introduced figures of merit pertaining to their
propagation distance and confinement.
In absolute terms, the propagation distance of a surface wave is given by L =
1/2Im(k// ). In order to obtain a normalized quantity that can be used as a figure
of merit for comparison between different materials and different wavelength ranges
(c.f. visible and IR regimes), we divide the propagation distance by the effective
wavelength of the mode, λeff = 2π/Re(k// ). Hence, the effective propagation length
Leff is

Leff ≡

Re(k// )
= 4π
Im(k// )
λeff

(4.8)

and expresses the number of mode wavelengths that a wave propagates before it
decays.
Another quantity that deserves attention is the degree of confinement of a mode. In
absolute terms, this can be expressed in terms of the cross-sectional area occupied
by

Idz]2

a mode. In the one-dimensional case studied here, this area is given by A1D = R I 2 dz ,
where I is the intensity profile of the mode (E 2 (z) or B 2 (z)). By normalizing
this quantity to the diffraction limited spot, which is Ao,1D = λo /2, we obtain the
normalized cross-sectional area [149]
A1D
[ Idz]2 / I 2 dz
Aeff ≡
Ao,1D
λo /2

(4.9)

where the limits of integration are taken to be on the order of tens to hundreds of
free-space wavelengths away from region where the wave is localized.
Finally, another metric of confinement is the number effective wavelengths of the
mode that fit in the free-space wavelength of excitation, λo . This quantity is simply
given by λo /λeff = Re(k// )/ko .
4.5

Theory of plasmons and results for silver

We start by investigating SPPs in a Drude metal, which serves as a sanity check for
confirming the validity of the method we introduced in Section 4.3. We examine the
case of a semi-infinite slab of metal and a thin slab in air, for which we expect the
existence of one and two SPP modes, respectively, as a consequence of the number
of interfaces in each case.

(a)

0.8
0.7

SPP

0.5

Re(Et(z)) a.u.

ω/ωp

0.6

0.4
0.3

light line
TM

0.2
0.1

0.2

0.3

0.4

0.5

0.6

k// (in Λchar/2π)

(b)

0.8

anti-symmetric

SPP

0.7

symmetric

0.5
0.4

Re(Et(z)) a.u.

ω/ωp

0.6

-5
0 d

0 d

0.3

light line
TM

0.2
0.1

0.2

0.3

0.4

k// (in Λchar/2π)

0.5

0.6

Figure 4.7: Dispersion of a TM polarized surface plasmon polariton on a Drude
metal. (a) Semi-infinite case and (b) finite slab with thickness dm = λp /30. For
both (a) and (b) Λchar = λp /30. The free-space light line is shown with black.
Insets show field profiles. For case (b) the two modes are the anti-symmetric
(left) and symmetric (right) plasmonic modes.

66
We model the metallic Drude permittivity with Eq. (4.1). By performing the
transfer matrix, retrieval and RPM approaches described in Section 4.3, we show in
Fig. 4.7 the dispersion of a TM polarized SPP on (a) a semi-infinite slab of a Drude
metal and (b) on a finite slab with thickness dm = λp /30, where λp = 2πc/ωp . The
SPP dispersion asymptotically approaches the value of ∼ ωp / 2, as expected [92,
109, 118]. In the case of a finite and thin slab, the dispersion is composed of two
branches as expected, since there exist two interfaces to support SPP propagation.
The two branches correspond to the symmetric and anti-symmetric modes. The field
profiles in both semi-infinite and finite thickness cases demonstrate the good surfaceconfinement characteristics of SPPs. The symmetric mode is even as a function of
z, whereas the anti-symmetric mode is odd in z. Across the whole frequency range
displayed in Figs. 4.7 (a), (b), the permittivity of the metal is negative, m < 0.
2 = k 2 − k 2 < 0) and,
Hence, the out-of-plane wavenumber is imaginary (kz2 = keff
m o
//

based on our definition of a photonic band (Eq. 4.7), the SPP belongs in a band
gap region. Therefore, it is a surface-confined mode, which confirms the validity of
our photonic band criterion.
We proceed by studying SPP propagation on a semi-infinite slab of silver at UVvisible frequencies, near the plasma frequency of Ag, which is taken here to be
ωp,Ag = 9.6 eV [150]. The results of SPPs on silver serve as a reference point in
terms of confinement and propagation length, for the realistic cases we examine in
the following sections.
A schematic of the geometry we study is shown in Fig. 4.8a, where we include a
grating that allows excitation of SPPs, with k// > ko , from free space (k// ≤ ko ).
Namely, the grating’s period γ determines the in-plane wavenumber via

k// = 2π/γ

(4.10)

In order to directly compare the degree of confinement, in terms of number of modal
wavelengths λeff that fit in the free-space wavelength λo , between different cases,
we normalize the x-axis in the dispersion curve of Fig. 4.8b to ko . Hence, since
λo /λeff ≡ Re(k// )/ko , we see that the modal wavelength of SPPs on Ag can be up
to six times reduced compared to the free-space wavelength in this frequency range.
Furthermore, as expected, the dispersion of TM polarized SPP modes on Ag asymp√
totically approaches to the value of ∼ ωp,Ag / 2 (Fig. 4.8b). The field profile of the
SPP mode is shown in the inset of Fig. 4.8b and is highly localized at the air/silver
interface. As the frequency increases, the in-plane wavenumber k// increases, hence
SPPs have access to higher confinement. We calculate the normalized cross-sectional

(a)

k//

ETM

0.5

Aeff=0.6

0.4

=0.8

eff

0.7
0.6
0.5
0.4
0.3
0.2

0.3
0.2

eff

=1.22

200

400 600
γ (in nm)

800 1000

ω/ωp

Aeff=0.52

(c)

0.7
0.6
0.5
0.4

visible

ω/ωp

0.6

Re(Et(z)) a.u.

light line

0.7

ω/ωp

(b)

Silver

0.3
0.2

Re(k//) (in ko)

Leff (in λeff)

Figure 4.8: (a) Schematic of a semi-infinite slab of Ag and a grating with
period γ for SPP excitation. (b) Dispersion of surface plasmon polaritons
on a semi-infinite slab of Ag. The parameters for Ag were taken from [150]:
ωp = 9.6 eV and γm = 22.8 meV/cm in Eq. (4.7). The free-space light line is
the vertical line at k// /ko = 1. Top inset shows the grating period γ required
for excitation of SPPs. Bottom inset corresponds to the SPP field profile. Aeff
is the effective mode volume, as defined in Eq. (4.9). (c) Effective propagation
length of SPPs on Ag, as defined in Eq. (4.8).

area of the SPP modes by first calculating the fields at different frequencies, and the
results range from Aeff ∼ 1.2 to Aeff ∼ 0.5 for frequencies in the far UV and visible
regimes, respectively (see black points in Fig. 4.8b).
Plasmonic modes in silver can propagate for up to ∼ 50 times their effective wavelength λeff , as shown in Fig. 4.8c. The slight oscillations in the Leff (ω) curve
originate from our computational approach; as discussed in Section 4.3, the parameter Im(k// ) (which is inversely proportional to Leff , see Eq. (4.8)) is defined as the
HWHM of peaks in the derivative of m11 with respect to k// (see green curve in
Fig. 4.6) and the accuracy of the peaks detection scheme depends on computational
power.
Finally, we calculate the parameter γ, corresponding to the grating period required

68
for excitation of SPPs. This parameter is shown in the top inset of Fig. 4.8b and
ranges from 300 nm to 900 nm for excitation with visible light [8, 135, 151].
4.6

Phonons and excitons

So far we have established the validity of the scheme introduced in Section 4.3
for detecting photonic and surface-confined modes in layered systems, by accuratly
obtaining the SPP dispersion in plasmonic metals (Section 4.5). Here, we return to
the search for surface-confined waves that resemble the SPP dispersion but, contrary
to SPPs, are simultaneously TE and TM polarized. For this we investigate polar
dielectrics and excitonic materials, based on the concept discussed in Section 4.2.
We revisit the problem of a slab of thickness d of a polar dielectric or excitonic
material (Fig. 4.3b). Initially, we perform calculations in two extreme cases (a) an
thick slab of polar dielectric and excitonic material and (b) a thin slab, compared to
the free-space wavelength λo . In case (a), Fabry-Pérot resonances exist leading to
multiple photonic bands for both linear polarizations. By contrast, in case (b), due
to the small thickness of the slab, we obtain a single-mode for a single-polarization
per frequency.
We model the permittivity of both the polar dielectric and the excitonic material with Eq. (4.4). We select a characteristic length defined as Λchar,phon/exc =
πc/ωTO

phon/exc (ωTO ). For the thick slab, or case (a), we select dphon/exc =

10Λchar,phon/exc . By contrast, for the thin slab, or case (b), we set dphon/exc =
Λchar,phon/exc , which is small compared to the free-space wavelength, as phon/exc (ωTO )
is taken to be rather large (Fig. 4.1).
In Fig. 4.9 we present the results of our calculations for an arbitrary polar dielectric
material. For frequencies between ωTO and ωLO , i.e. in the Reststrahlen band
highlighted with red, we have phon < 0, hence an SPhP mode occurs for TM
polarization. The SPhP has two branches that correspond to the symmetric and
anti-symmetric modes. In the inset of Figs. 4.9a and b we display their field profiles.
These modes exhibit field profiles very similar to the SPP ones (Fig. 4.7b). Due
to the negative sign of phon in the Reststrahlen band, based on Eq. (4.3) and Eq.
(4.7), the SPhPs belong to a band gap region and are surface-confined, in direct
analogy to SPPs [122].
At frequencies slightly lower than ωTO , i.e. in the regime of high-phon , we obtain
a number of modes, as can be seen in the grey highlighted region of Fig. 4.9a.
Importantly, we obtain a mode that occurs for both linear polarizations right at the
band edge of the material, for which k// = k//max = phon ko . For these high-k//
modes, we present in the inset of Fig. 4.9a field profiles for both linear polariza-

(a) 1.5

1.3

1.2

Re(E

ω/ωTO

SPhP

1.4

1.1

k//max(ω)

light line

Re(Bt(z)) a.u.

0.9
0.8

(b) 1.5

-6
0 d

0.1

0.2

0.3

k// (in Λchar/2π)
k//max(ω)

light line

SPhP

1.4

Re(Et(z)) a.u.

1.3

ω/ωTO

-6

1.2

0.5

TE
anti-symmetric

-5

1.1

0.4

symmetric
Re(Et(z)) a.u.

0.7

Re(Et(z)) a.u.

Re(Bt(z)) a.u.

0.9
0.8
0.7

-5

0.1

0.2

0.3

k// (in Λchar/2π)

0.4

0.5

Figure 4.9: Dispersion of surface-confined and photonic modes in a polar dielectric
material. The characteristic length is defined as Λchar =
πc/ωTO phon (ωTO ). Mode dispersions for (a) a thick slab with thickness dph = 10Λchar and (b) for a thin slab with thickness dph = Λchar .
The free-space
light line is shown with black. Dashed line corresponds to
k//max = phon (ω)ko . Insets show field profiles for (a) an omnipolarization
mode located at the band edge (bottom) and a SPhP mode (top), and for
(b) an anti-symmetric (top left) and a symmetric (top right) SPhP mode, and
a highly-confined TE mode (bottom). The high- regime is highlighted with
(a) grey and (b) green, indicating omnipolarization and TE polarized wave
propagation, respectively. The Reststrahlen band is highlighted with red.

70
tions at a single frequency ω and wavenumber k// . As these modes coexist in ω
and k// for TE and TM polarizations, this allows single-frequency excitation and
phase-matched propagation of omnipolarization surface waves. By decreasing the
thickness of the slab, in Fig. 4.9b, we decouple to two polarizations and we only
obtain a TE polarized mode near the band edge. Based on Eqs. (4.3), (4.7), the TE
mode displayed in Fig.4.9b is classified as a photonic one, since k// < exc (ω)ko .
However, it continues to exhibit good confinement characteristics, which is evident
by the large values that k// obtains near the mode’s resonance, at ωTO . The confinement of these modes strongly depends on the quality factor of the polar dielectric, as
we show in the following sections. This TE polarized mode disappears for ω > ωTO
because phon < 0 in this frequency regime. Hence, the material becomes opaque for
TE polarization and no modes are supported.
In Fig. 4.10 we perform analogous calculations for an arbitrary excitonic material.
For a thick slab (Fig. 4.10a), we obtain a large number of photonic modes that, in
contrast to the polar dielectric material, continue to exist at frequencies above ωTO .
This happens because an excitonic resonance does not involve a region of exc < 0
in contrast to the Reststrahlen band of polar dielectrics (see Fig. 4.1). Importantly,
near ωTO , we obtain modes at the band edge, with k// = k//max = exc ko , for
both TE and TM polarizations, the field profiles of which are shown in the inset of
Fig. 4.10a at a single frequency and wavenumber. Their degeneracy allows singlefrequency excitation and phase-matched propagation of omnipolarization surface
waves. The degree of surface-confinement of these band edge modes depends on
the quality factor, Q, of the excitonic material, as we discuss in what follows. By
decreasing the thickness of the excitonic material slab, we decouple the two polarizations, the modes of which occur at different frequencies, as shown in Fig. 4.10b.
Namely, at frequencies below ωTO , the high-exc regime continues to yield a high-k//
TE polarized mode. In contrast to the polar dielectric material, for which the TE
polarized mode disappears for ω > ωTO , due to phon < 0 at the Reststrahlen band,
for the excitonic material we have exc > 0 even above ωTO (Fig. 4.1). Hence, the
TE mode continues to exist in the finite slab case (Fig. 4.10b), however it is no
longer degenerate with the TM polarized one.
To conclude this section, we investigated two extreme cases of slabs of excitonic and
polar dielectric materials; (a) a thick slab for which multiple photonic bands are
supported and (b) a thin slab compared to the free-space wavelength. In case (a),
for both polar dielectrics and excitonic materials, we obtain a mode for each linear
polarization that occurs at the band edge, for which k// = k//max = phon/exc ko
(Figs. 4.9a, 4.10a). The magnitude of k// for this mode depends on the maximum
value of the dielectric permittivity of the material, (ωTO ). Large values of near

(a) 1.6

light line

k//max(ω)

1.5
1.4

ω/ωTO

1.3
1.2
1.1
0.8
0.7

(b) 1.5

0.1

light line

0.2

0.3

-2

k// (in Λchar/2π)
k//max(ω)

0.4

Re(Et(z))
(z))a.u.
a.u.
Re(E

Re(Bt(z))

0.9

-2

0.5

1.4

ω/ωTO

1.3
1.2
1.1
Re(Bt(z)) a.u.

0.9
0.8
0.7

-5

0.1

0.2

0.3

k// (in Λchar/2π)

0.4

0 d

0.5

Figure 4.10: Dispersion of surface-confined and photonic modes in√a excitonic
material. The characteristic length is defined as Λchar = πc/ωTO exc (ωTO ).
Mode dispersions for (a) a thick slab with thickness dexc = 10Λchar and (b)
for a thin slab with thickness dexc = Λchar . The free-space
light line is shown
with black. Dashed line corresponds to k//max =
exc (ω)ko . Insets show
field profiles for (a) an omnipolarization mode located at the band edge, and
for (b) a highly-confined TE mode. The high- regime is highlighted with
(a) grey and (b) green, indicating omnipolarization and TE polarized wave
propagation, respectively.

72
ωTO can yield extremely confined surface modes with characteristics that can compete, or even surpass, the confinement factors of SPPs in noble metals. For thin
slabs, in case (b), the two modes no longer overlap in frequency and wavenumber
(Figs. 4.9b, 4.10b). The cutoff thickness at which the TE polarized and TM polarized band edge modes separate from each other depends on material properties
(ωTO , Q, etc). In Sections 4.7 and 4.8 we investigate high-Q polar dielectrics and
excitonic materials, respectively, and seek for this cutoff slab thickness that will
allow omnipolarization surface-localized wave propagation over large distances with
large degree of confinement in thin slabs.
4.7

Surface-confined waves on SiC on Si

We proceed by investigating surface-confined modes in polar dielectrics. In selecting
a material for our investigation, we seek record-high values of phon near the phonon
energy (see Fig. 4.1), simultaneously with low loss, to ensure high-Q (Eq. 4.5).
One of the polar dielectric materials with record-high quality factors is SiC, with
ωTO,SiC = 797 cm−1 and γd,SiC = 2.2474 × 1011 s−1 , hence QSiC = 668 [122]. Its
dielectric permittivity is presented in Fig. 4.2b.
In Section 4.6 we showed that the TE and TM polarized band edge modes of a thick
slab of polar dielectric material overlap in frequency and wavenumber (Fig. 4.9a).
By confining the two modes in an ultra-thin slab, the two polarizations no longer
exhibit overlapping dispersions curves (Fig. 4.9b). Here, we seek the cutoff thickness
for a slab of SiC, that ensures omnipolarization surface-confined wave propagation
near the band edge (k//max,SiC = SiC ko ). In order to investigate an experimentally
realistic scenario, we consider that the SiC slab lies on top of an intrinsic Si substrate,
with index nSi = 3.4 [152] (see schematic in Fig. 4.11a). We find that for SiC on Si,
this cutoff thickness is dSiC = 8πc/(ωTO,SiC

SiC (ωTO,SiC )) = 3.22 µm.

As can be seen from Fig. 4.11b, at this thickness of SiC, higher order Fabry-Pérottype propagation bands arise. These bands can be seen in Fig. 4.11b with the
grey shaded regimes for TE polarization and they were obtained with the method
discussed in Section 4.3. The dispersion curves that lie inside the propagation bands
correspond to photonic modes that exhibit weak confinement characteristics (See
Fig. 4.5a). In contrast, dispersion curves that lie in the exterior of the bands
constitute surface-confined waves (Fig. 4.5b).
Here we focus on the high-k// modes that are in close proximity to the band edge
of SiC, k//max,SiC (black dashed line in Fig. 4.11b). These modes lie in-between
propagation bands, hence they are surface-confined. From the field profiles in the
lower inset of Fig. 4.11b, we see that these modes are strongly localized inside the

k//

ETM BTE

(a)

SiC

Aeff=0.54e-1
Aeff=0.60e-1
Aeff=0.66e-1

air

-8

Re(k//) (in ko)

1.5

2.5

3 3.5

SiC

γ (in μm)

0 d

-8

ω/ωTO

Aeff=0.45e-1

Re(Et(z)) a.u.

0.99
0.98
0.97
0.96

Re(B (z)) a.u.

ω/ωTO

TE band TM

k//max, SiC(ω)

ω/ωTO

light line

ω/ωTO

(b)

0.98

0.975
0.97

0.965
0.96
0.955

10 20 30 40 50 60 70

Leff (in λeff)

Figure 4.11: (a) Schematic of a slab of SiC on Si, and a grating with period γ
for surface-confined waves excitation. (b) Dispersion
qof surface-confined waves
on a slab of SiC with thickness dSiC = 8πc/(ωTO,SiC SiC (ωTO,SiC )) = 3.22 µm
on Si. The parameters for SiC were taken from [122]: ωTO,SiC = 797 cm−1 ,
ωLO,SiC = 973 cm−1 and γd,SiC = 2.2474 × 1011 s−1 in Eq. (4.4). The refractive
index of Si is taken nSi = 3.4 The free-space light line is located at k// /ko = 1
(off-axis). The modes located at k// /ko = 3.4 are the band edge modes in
bulk Si. The dashed line corresponds to k//max,SiC = SiC (ω), the band edge
of SiC. Top inset shows the grating period γ required for excitation of the
surface-confined waves. Bottom inset shows the TE and TM field profiles at
a single-frequency ω and wavenumber k// . Aeff is the effective mode volume,
as defined in Eq. (4.9). (c) Effective propagation length of surface-confined
waves on SiC on Si, as defined in Eq. (4.8).

74
SiC slab, and exhibit negligible leakage into the Si substrate. Their confinement
factors are calculated as explained in Section 4.4. We obtain normalized mode
cross-sectional areas that range from Aeff ∼ 0.04 to Aeff ∼ 0.06 for both TE and
TM polarizations, as shown with the values displayed in Fig. 4.11b at different
frequencies. Importantly, this confinement is larger than in the SPP modes in Ag
(Aeff ∼ [0.5 − 1.2] in Fig. 4.8b). We note that the modes located at k// = 3.4ko
(y-axis in Fig. 4.11b) correspond to the photonic band edge of Si (k//max,Si =
Si (ω)ko ).
In an experiment, the excitation of these strongly surface-confined omnipolarization
modes requires grating periods in the range of γ ∼ 1 µm to γ ∼ 2.5 µm for both
TE and TM polarizations, as shown in the top inset in Fig. 4.11b. Importantly, as
a consequence of the high-Q of SiC, these modes have large propagation distances.
As shown in Fig. 4.11b, they propagate approximately 10 − 70 effective mode
wavelengths before decaying, similar to SPPs on Ag (Fig. 4.8c). This constitutes
them detectable in surface waves experiments, where detection is achieved in the
far-field through input and output gratings, see, for example, [151].
Previous reports have considered high-Q polar dielectric materials for SPhP propagation at the Reststrahlen band [122].

In contrast, here we showed that the

frequency regime where the permittivity of polar dielectrics obtains positive and
large values (Fig. 4.1) can yield omnipolarization surface-confined propagation with
characteristics similar to SPhPs, however accommodating both linear polarizations,
therefore alleviating the TM polarization limitation of SPhPs. We emphasize that
the requirement of high-Q is crucial for ensuring both a high degree of confinement
and a long propagation distance.
4.8

Surface-confined waves on WS2 on Si

Here, we perform an analysis similar to the one in the previous section, but for
a high-Q excitonic material at visible frequencies, instead of a polar dielectric at
IR frequencies. One class of materials that one could consider investigating is that
of traditional semiconductors, for example Ge or Si. However, the quality factors
of traditional semiconducting materials are rather low, which can be seen in Fig.
4.2a. In contrast, two-dimensional TMDCs have been recently investigated for their
optical properties [129], and they show considerably higher Q’s, compared to bulk
semiconductors. We select to investigate WS2 , the permittivity of which is plotted
in Fig. 4.2a. By performing a numerical fitting (with Eq. (4.4)) to the experimental
data reported by Li et al. in [129], we infer that QWS2 ∼ 113.2. Based on our fitting,
with respect to Eq. (4.4), we find that ωTO,WS2 = 1.996 eV, ωLO,WS2 = 2.009 eV,
γd,WS2 = 37.363 fs and ∞,WS2 = 17.

75
In order to simulate an experimentally realistic scenario, we assume that the WS2
is on top of a Si substrate, for which we set nSi = 3.4 [152] (see schematic in Fig.
4.12a). As discussed in Section 4.6, we seek for the minimum (cutoff) thickness
of a WS2 slab, for which the band edge modes for the two linear polarizations
continue to overlap. For the material parameters of WS2 , this thickness is dWS2 =
6πc/(ωTO,WS2

WS2 (ωTO,WS2 )) = 336 nm.

For this thickness, we obtain the desirable high-k// modes near the band edge of
WS2 (k//max,WS2 = WS2 ko ), as shown in Fig. 4.12b. Additionally, higher order
Fabry-Pérot-type modes with smaller k// also arise. However these higher order
modes are not particularly surface-confined and we do not analyze them further.
We focus on the large-k// modes near the band edge of WS2 (dashed line in Fig.
4.12b), the field profiles of which are shown in the inset of Fig. 4.12b. The fields
are strongly-localized in the WS2 slab and exhibit negligible leakage into the Si
substrate. We compute their normalized cross-sectional area that ranges from Aeff ∼
0.068 to Aeff ∼ 0.081 for both polarizations, surpassing the mode confinement of
SPPs on silver at visible frequencies by an order of magnitude (Fig. 4.8b). Their
propagation distance Leff is very large and reaches values as high as L = 800λeff
in the lower-frequency regime near ∼ 0.75ωTO,WS2 (Fig. 4.12c). Finally, we show
in the left inset of Fig. 4.12b the grating period, γ, required for single-frequency
simultaneous excitation of TE and TM surface-confined waves on WS2 on Si.
To conclude, in the present section and in the previous one, we discussed materials
with permittivity resonances and high-Q, as platforms for omnipolarization, phasematched surface-confined wave propagation. Our approach aimed to identify the
minimum thickness of a slab of a positive-permittivity material, for which there
exist modes near the band edge k//max = ko ). Utilizing high- materials yields
large degree of confinement, and focusing on high-Q material resonances can lead
to large propagation distances. Based on these requirements, we focused on slabs of
SiC and WS2 , as representatives of high-Q polar dielectrics and excitonic materials
at IR and visible frequencies, respectively.
4.9

The case of a plasmon and a phonon

In this section and in the following one, we investigate theoretically the scenarios of
combining a plasmon in a Drude metal (Fig. 4.7) with a phonon in a polar dielectric
(Fig. 4.9) and with an exciton in an excitonic material (Fig. 4.10), respectively.
Similar to the previous sections, we aim to identify regimes of omnipolarization
surface wave propagation. However, in contrast to the discussion in Section 4.2,
here, the origin of the overlap in the dispersions for TE and the TM modes is not

k//

ETM BTE

(a)

WS2

Si
k//max, WS2(ω)

light line

(b)

(c)

0.85

WS2

-3

-1
-3

Aeff=0.80e-1

0.8

Aeff=0.81e-1
3.5

4.5

0.9

0.9
0.85
0.8

0.8

Re(k//) (in ko)

0.95

ω/ωTO

Aeff=0.78e-1

Re(B (z)) a.u.

Aeff=0.76e-1

0.9

air

ω/ωTO

Re(Et(z)) a.u.

Aeff=0.68e-1

0.95

120

160

200

γ (in nm)

5.5

240

200

400

600

Leff (in λeff)

800

Figure 4.12: (a) Schematic of a slab of WS2 on Si, and a grating with period γ
for surface-confined waves excitation. (b) Dispersion of surface-confined
waves
on a slab of WS2 with thickness dWS2 = 6πc/(ωTO,WS2 WS2 (ωTO,WS2 )) =
336 nm on Si. The parameters for WS2 were fitted from data reported in
[129]: ωTO,WS2 = 1.996 eV, ωLO,WS2 = 2.009 eV, and γd,WS2 = 37.363 fs,
in Eq. (4.4). The refractive index of Si is taken nSi = 3.4. The free-space
light line is located at k// /ko = 1 (off-axis). The dashed line corresponds to
k//max,WS2 = WS2 (ω), the band edge of WS2 . Left inset shows the grating
period γ required for excitation of the surface-confined waves. Right inset
shows the TE and TM field profiles at a single-frequency ω and wavenumber
k// . Aeff is the effective mode volume, as defined in Eq. (4.9). (c) Effective
propagation length of surface-confined waves on WS2 on Si, as defined in Eq.
(4.8).

77
the band edge condition. In contrast, by assuming that the plasma frequency of
the Drude metal, which serves here as a substrate, is considerably larger than ωTO ,
the transverse optical phonon/exciton energy, the TM polarized SPP modes of the
Drude metal are hybridized and obtain a frequency dispersion similar to the TE
polarized band edge modes of the positive permittivity medium (polar dielectric or
excitonic material).
We start by considering a system composed of a thin slab of polar dielectric material, with thickness dphon = πc/ωTO

phon (ωTO ) on a Drude metal substrate. We

select the plasma frequency of the metal to be ωp = αωTO

1 + phon (ωTO ), such

that the surface plasma frequency of the plasmon is ωsp ≃ αωTO , and we set α = 2.5.
This selection of ωsp locates the SPP on the linear, lower side of the dispersion (Fig.
4.7a). The relevant dispersion curves that we are combining in this configuration
are: (i) the unbound SPP mode, shown in Fig. 4.7a, with (ii) the highly-confined
TE polarized mode due to the phonon resonance, shown in Fig. 4.9b. The frequency regime in which this scenario becomes relevant is the IR range at which
polar dielectric materials exhibit permittivity resonances (Fig. 4.2b).
A schematic of the geometry we study is shown in Fig. 4.13a, where we also include a grating that allows excitation of surface waves. As shown in Fig. 4.13b,
by interfacing the TM polarized SPP mode of the Drude metal with the highlyconfined band edge TE polarized mode of the polar dielectric, we position them in
the same frequency range, and obtain almost overlapping dispersions for TE and
TM polarization. However, here, the origin of the TM polarized mode is the plasmonic substrate, contrary to the previous sections. For completeness, we continue
to include in Fig. 4.13b the curve corresponding to the band edge of the polar
dielectric material, given by the equation k//max,phon =

phon (ω)ko ; as expected,

the omnipolarization surface waves occur in the high-phon frequency regime.
In the inset of Fig. 4.13b, we show the field profiles of the TE and the TM modes
for the same frequency and wavenumber. Although the TE and TM dispersions do
not exactly overlap, by selecting ωTE = ωTM and k//TE = k//TM we excite modes
for both polarizations, with field profiles that show similar degree of confinement.
The simultaneous excitation of modes for both polarizations is a consequence of
the losses introduced in the system. Namely, the high-phon regime is accompanied
by large values of Im(phon ) (see Fig. 4.1). Hence, the dispersions displayed in
Fig. 4.13b extend to a small range of frequencies, ±∆ω, and wavenumbers, ±∆k// ,
around their central curves, which allows the TE and TM dispersions to overlap, as
we emphasize with the purple highlighted regime.
In order to assess the feasibility of simultaneously excitating TE and TM polarized

(a)

k//

ETM BTE

phonon

Λchar

plasmon

light line

k//max, phon(ω)
ω/ωTO

(b) 1.06
1.04

1.05
0.95

ω/ωTO

1.02

0.2 0.4 0.6 0.8
γ (in λTO)

0.98

0.94
0.92

Re(Et(z)) a.u.

Re(Bt(z)) a.u.

0.96

-4

-4
0 d

0.1

0.2

0.3

k// (in Λchar/2π)

0.4

0.5

Figure 4.13: Omnipolarization surface waves from plasmons and phonons.
(a) Schematic ofq a thin slab of polar dielectric material with thickness
dph = πc/ωTO phon (ωTO ) on a Drude metal substrate, with ωp =

αωTO 1 + phon (ωTO ) (α = 2.5). (b) Dispersion of surface-confined modes for
TE and TM polarization, Λchar = dph . The free-space
light line is shown with
black. Dashed line corresponds to k//max,phon = phon (ω)ko , the band edge of
the polar dielectric. Lower inset shows field profiles for TE and TM polarization at the same frequency and wavenumber, ωTE = ωTM and k//TE = k//TM .
Top inset shows the grating period γ required for excitation of the modes.

79
surface-confined modes at a single-frequency and with a single grating, we calculate
in the inset of Fig. 4.13b, the grating period γ required for TE and TM mode excitation. As can be seen, the grating periods for the two polarizations are almost identical, and their differences lie within the limits of fabrication imperfections (±0.1λTO ).
The small differences in γTE and γTM , together with the aforementioned discussion
regarding the losses in the system at frequencies near the high-phon regime, ensure
that omnipolarization surface waves can be excited in the configuration of Fig. 4.13a
at a single-frequency. By selecting appropriately a high-Q polar dielectric material
and a Drude metal, the plasma frequency of which can be tuned to approximately
ωp = 2.5ωTO

1 + phon (ωTO ), the surface waves shown in Fig. 4.13 may arise at IR

frequencies, where the phonon energies of polar dielectric materials lie.
Here we combined a polar dielectric with positive permittivity phon together with
metal with negative permittivity m , to achieve omnipolarization surface wave propagation. In principle, another possibility is to combine two different polar dielectric
materials with slightly different frequencies of transverse and longitudinal phonons,
ωTO,1 , ωTO,2 and ωLO,1 , ωLO,2 , such as the Reststrahlen band of the first
(phon,1 < 0) overlaps with the high- regime of the second (phon,2
1). There is
an abundance of phonon resonances in different materials in the IR spectral range,
hence it is straightforward to find such a combination of positive and negative permittivities. However, the frequency regimes near ωTO,1 , ωTO,2 entail high loss (see
Fig. 4.1), hence combining two polar dielectric materials near resonance will lead
to very large damping constants of surface confined modes. Therefore, we do not
investigate further the case of two phonons.
4.10

The case of a plasmon and an exciton

In analogy the the case examined above, here we investigate a thin slab of an excitonic material on a Drude metal substrate, in order to combine the highly-confined
TE band edge mode due to the high-exc regime (Fig. 4.10b) with the TM polarized SPP due to the plasmon of the metallic substrate (Fig. 4.7a). The frequency
regime in which this scenario becomes relevant is the visible range at which excitonic materials exhibit permittivity resonances (Fig. 4.2a). We select a thickness
of dexc = πc/ωTO exc (ωTO ) for the excitonic material and a plasma frequency
of ωp = αωTO 1 + exc (ωTO ) for the metal. By setting α = 2.5, we ensure that
ωsp ≃ αωTO is well-above ωTO , hence, the plasmonic mode lies in the linear-regime,
on the lower-frequency side of the SPP dispersion (see Fig. 4.7a). A schematic of
the geometry we study is shown in Fig. 4.14a. We include a schematic of an input
grating that allows excitation of surface waves from free space, via k// = 2π/γ.
As can be seen in Fig. 4.14b, we obtain a TE mode due to the high-exc regime of

(a)

k//

ETM BTE

exciton

Λchar

plasmon

k//max, exc(ω)

light line

ω/ωTO

(b) 1.06
1.04

0.1 0.2 0.3 0.4 0.5 0.6
γ (in λTO)

0.98

Re(Bt(z)) a.u.

0.96

Re(Et(z)) a.u.

ω/ωTO

1.05
0.95

1.02

0.94

0.92

-4

-4
0 d

0.1

0.2

0.3

0.4

k// (in Λchar/2π)

0.5

0.6

Figure 4.14: Omnipolarization surface waves from plasmons and excitons. (a) Schematic√of a thin slab of an excitonic material with thickness d√
exc = πc/ωTO exc (ωTO ) on a Drude metal substrate, with ωp =
αωTO 1 + exc (ωTO ) (α = 2.5). (b) Dispersion of surface-confined modes
for TE and TM polarization, Λchar = dexc . The free-space
√ light line is shown
with black. Dashed line corresponds to k//max,exc =
exc (ω)ko , the band
edge of the excitonic material. Lower inset shows field profiles for TE and
TM polarization at the same frequency and wavenumber, ωTE = ωTM and
k//TE = k//TM . Top inset shows the grating period γ required for excitation
of the modes.

81
the excitonic material, and a TM polarized hybrid-SPP mode due to the m < 0
of the metal. The typical SPP dispersion that is linear in the low-ω regime (Fig.
4.7a) is hybridized here due to the strong frequency dispersion of exc (ω) near ωTO ,
and almost overlaps with the TE mode. For completeness, we continue to include
the curve corresponding to the band edge of the excitonic material, given by the
equation k//max,exc = exc (ω)ko ; as expected, the omnipolarization surface waves
arise in the high-exc frequency regime.
Although the TE and TM dispersions do not exactly overlap, by selecting ωTE = ωTM
and k//TE = k//TM we excite modes for both polarizations, the field profiles of
which are displayed in the lower inset of Fig. 4.14b and exhibit similar degrees
of surface-confinement. Similar to the comments made in the previous section, this
simultaneous excitation occurs due to the losses introduced in the systems we study.
Namely, near ωTO , the excitonic material exhibits high damping (large Im(exc )).
Hence, the dispersions displayed in Fig. 4.14b extend to a small range of frequencies,
±∆ω, and wavenumbers, ±∆k// , around their central curves, which results in the
overlap of the TE and TM modes dispersion curves, as emphasized with the purple
highlighted regime.
In the top inset of Fig. 4.14b we calculate the grating period γ required for TE
and TM mode excitation. The grating periods for the two polarizations are similar,
and their differences lie within the limits of fabrication imperfections (±0.1λTO ,
where λTO lies in the visible spectral range), ensuring simultaneously TE and TM
polarized mode excitation with a single grating.
4.11

Conclusion

In this chapter we studied the high-permittivity frequency regime of polar dielectric
materials and excitonic materials, at IR and visible frequencies, respectively. We
showed that use of high-Q materials can lead to surface-confined waves that occur
for both linear polarizations simultaneously, i.e. they can be excited at a singlefrequency (ωTE = ωTM ) and require a single grating for their excitation (k//TE =
k//TM ), hence they are also phase-matched.
We followed two different approaches; (i) the first one pertained to utilizing the
polarization degeneracy of the band edge modes of a bulk polar dielectric or excitonic
material. (ii) The latter approach entailed the combination of plasmonic metals with
polar dielectrics and excitonic materials, at IR and visible frequencies, respectively,
for omnipolarization surface waves. In this case, the TE and the TM modes have
different origins; namely, the TM mode is a hybrid-SPP mode due to the plasmonic
metal, whereas the TE mode originates from the dielectric’s band edge.

82
With regards to (i), we demonstrated theoretically that use of SiC at IR frequencies
and WS2 at visible frequencies can yield omnipolarization surface-confined waves
with higher degree of confinement and longer propagation distances, in comparison
to SPPs in Ag. In Table 4.1, we summarize these results and provide a comparison
between the cases we examined. We emphasize that SiC and WS2 were selected due
to their high-Qs (Eq. (4.5)), which is a crucial parameter for obtaining surface waves
with good confinement characteristics and large propagation distances. We note,
however, that, in contrast to the broadband nature of SPPs, the omnipolarization
surface waves discussed here are narrowband in nature, and physically bound to the
frequency range where phonons and excitons occur in naturally available materials.

Material
Semi-infinite Ag
(visible)
SiC on Si
(IR)
WS2 on Si
(visible)

TE polarization
TM polarization
Aeff N.A.
Aeff ∼ [0.5 − 1.2]
Leff N.A.
Leff ∼ [5 − 55]
Aeff ∼ [0.045 − 0.066] Aeff ∼ [0.045 − 0.066]
Leff ∼ [10 − 70]
Leff ∼ [10 − 70]
Aeff ∼ [0.068 − 0.081] Aeff ∼ [0.068 − 0.081]
Leff ∼ [10 − 800]
Leff ∼ [10 − 800]

Table 4.1: Comparison of figures of merit between SPPs in Ag (see Section
4.5) and omnipolarization surface-confined waves in SiC (see Section 4.7) and
in WS2 (see Section 4.8). Aeff is the cross-sectional area of the mode normalized to a diffraction limited spot Ao,1D = λo /2 (see (Eq. 4.9)). Leff is the
propagation length of the mode normalized to the effective wavelength of the
mode (see Eq. (4.8)).
Regarding the latter approach (ii), we showed that combination of plasmons and
excitons or plasmons and phonons, at visible and IR frequencies, respectively, can
yield omnipolarization surface-confined propagation. In search for realistic configurations that may support such surface waves, it is highly desirable to be able to
actively tune the plasma frequency of the plasmonic component to frequencies that
match phonon energies (IR) or exciton resonances (visible). In the following chapter, we perform detailed investigations for inducing such active tunability in planar
heterostructures in both visible and IR frequency regimes.

83
Chapter 5

ACTIVE TUNABILITY IN PLANAR METAMATERIALS
“This field is not quite the same as other fields in physics in that it will not tell us
more of fundamental physics but it is more like solid-state physics in the sense that
it might tell us much of great interest about the strange phenomena that occur in
complex situations.”
Richard P. Feynman, Plenty of Room at the Bottom, 1959
5.1

Tuning the optical response

The range of fundamental phenomena and applications achievable by metamaterials
can be significantly expanded by actively tuning their effective electromagnetic parameters to enable dynamic control over their optical response. Such active control
over metamaterials’ response can pave the way towards novel active optical components like holographic displays, improvement of liquid crystal display technology
with solid-state materials, tunable polarizers, sensors and switches, slow-light media,
and optical memories. Particularly, simultaneous control of phase and amplitude
of fields scattered or transmitted through metamaterial-based optical components
is challenging to achieve but highly desirable for holography and lidar technology
[153, 154].
In this chapter, we investigate means for tuning the response of planar metamaterials along all coordinate directions, by focusing on their fundamental material
parameters. By starting from intrinsic metamaterial properties (eff and µeff ), we
can directly predict and control their dispersion characteristics. Control over the
dispersion diagram or the EFC, introduced in Chapter 1 (see Section 1.4) is of interest from a more fundamental physics point of view as well (apart from tuning the
scattering properties of metamaterials).
Namely, the past decade has revealed a plethora of new phenomena in electronic
materials in condensed matter physics, and also in their photonic counterparts, arising from engineering their electronic band structure and photonic dispersion surface,
respectively. Classical phenomena such as Lifshitz transitions [155] and Van Hove
singularities that lead to extreme values of magnetoresistance arise from inducing
transitions in the Fermi surface of electronic systems. Quantum mechanical effects
such as those arising from Dirac-like dispersion surfaces, leading to topological Dirac

84
phases, are the result of band structure engineering, leading to three-dimensional
semimetals [156] and topological insulators [157]. Photonic analogues of topological
insulators, in other words photonic materials that support non-trivial topologically
protected states against back-scattering, have been realized with helical waveguides
in a honeycomb lattice [158], by engineering the dispersion characteristics of photonic crystals. The dispersion surface or EFC is also critical for engineering emission
with hyperbolic media, as investigated and demonstrated experimentally in [42] (see
discussion in Section 1.4). Particularly, in analogy to the Lifshitz transition in electronic systems, a transition from a closed EFC to an open, hyperbolic one, comes
with a singularity in the density of optical states. By controlling the dielectric properties of HMMs along different coordinate directions, namely the near-zero bands of
o,eff and e,eff , the frequency regime of topological transitions becomes tunable [48].
This chapter is structured as follows: in Sections 5.2, 5.3, 5.4 we discuss means
for inducing actively tunable response in planar metamaterials using transparent
conductive oxides (TCOs). In Sections 5.5, 5.6 we investigate the use of graphene as
an active medium. We show that field-effect-based gating, i.e. carrier accumulation
and depletion, using TCOs, yields a largely tunable response along the optical axis
of planar heterostructures, for frequencies below the UV range [48], while capacitorbased gating with graphene mono-layers leads to significant tuning along the in-plane
direction.
We note that alternative tuning mechanisms include modifying the complex dielectric function of component materials via phase transitions [159] and electromechanical deformations [160]. In contrast to these, use of the field-effect and electronic
gating via carrier injection are particularly attractive due to their robustness and
very low power dissipation in steady state, having potential to yield power-efficient
tunable metamaterials with ultrafast (fs) modulation speed. The field-effect has
been recently investigated for optical modulators [153, 161–164] by using the spectral tunability of the electronic properties in TCOs or transition-metal nitrides [165]
for modulating the modal effective index in waveguide configurations.
5.2

Transparent conductive oxides as active components

We start with the toy model shown in Fig. 5.1, demonstrating a metamaterial unit
cell composed of two mirror-symmetric metal/oxide/semiconductor (MOS) sequence
of layers. Instead of a traditional semiconductor, we investigate here TCO active
layers. The charge carrier density of TCOs can be controlled by the number of
oxygen vacancies when deposited, for example via RF sputtering deposition. More
importantly, however, their charge carrier density can be tuned actively as well [153,
154, 159], when incorporated in MOS devices and gating between the metal and the

85
TCO is applied, injecting carriers in TCO accumulation and depletion regimes, as
shown in Fig. 5.1 with light blue colors.

V+

metal
oxide
e region
TCO activ
ground
TCO back
e region
TCO activ

oxide
metal

V+
Figure 5.1: Schematic of TCO-based field-effect tunable hyperbolic metamaterial unit cell
Prior to delving into the details on the optical properties of the motif shown in
Fig. 5.1, we discuss the field-effect mechanism using TCOs. We focus on indium
tin oxide (ITO) as an example TCO, because it has been studied extensively in the
literature [153, 161, 166–169] and its electronic properties are fairly well characterized experimentally [153, 161, 167, 170]. We emphasize, however, that the results
discussed here can be extended to other TCOs, for example aluminum doped zinc
oxide [166] and gallium doped zinc oxide [164, 166], or transition-metal nitrides and
degenerately doped semiconductors [163].
The intrinsic parameter modulated via the field-effect, that is in turn used to induce
optical tuning, is the charge carrier density, Nacc , in the active layers of accumulation and depletion (light blue colors in Fig. 5.1), relative to the background charge
carrier density Nb . Ignoring detailed band bending effects, to first order, the charge
carrier accumulation can be schematically modeled via a uniform layer with increased carrier concentration relative to the background TCO, with thickness given
by the Debye length d. A simple electrostatic calculation on a metal/dielectric/TCO
interface, as depicted in Fig. 5.1, dictates that:

86
eNacc d
≤ Ebr
kdiel o

(5.1)

where V is the applied bias between the metal and the TCO, t is the thickness
of the dielectric and kdiel is its DC dielectric constant, Ebr is the corresponding
breakdown field and o is the dielectric permittivity of free space. To achieve tunability within the visible regime, high-carrier density is required, on the order of
1019 − 1021 /cm3 . Additionally, dielectric materials with very high breakdown fields
are necessary. Previous reports have shown that ITO can be heavily doped using RF
sputtering to yield a background carrier density, Nb , in the range 1019 − 1021 /cm3 .
Following previous experimental results [153, 161, 162, 167–170], we consider here a
background carrier density of Nb = 5 × 1020 /cm3 and active tunability of the carrier
density in the accumulation regimes of up to two orders of magnitude, assuming
high-strength dielectric materials for the oxide layer (see Fig. 5.1). We note that
here we only consider accumulation of carriers, however depletion can be achieved
by simply inversing the bias polarization. We compute, in Table 5.1, using Eq.
(5.1), the maximum achievable carrier concentration in the accumulation layer of
ITO, for various high-strength dielectric materials whose breakdown fields have been
reported. Different Debye lengths d for the ITO are also considered, consistent with
the range of parameters observed in previous experimental results [153, 161, 162,
168].
High-k dielectric quality is important for sustaining high carrier concentrations in
the accumulation layer. HfO2 is particularly attractive due to its simultaneously
large values of DC dielectric constant [171, 174, 176, 177] and its high breakdown
field. We note that, apart from Sire et al. in [171] and Yota et al. in [174], Kim
et al. in [178] have obtained breakdown fields as high as 36 MV/cm for large-area
(up to 100µm ×100µm) planar field-effect electrode geometries, while values in the
range 5 − 10 MV/cm have been broadly reported in the literature [179–181].
As far as the optical properties of TCOs at visible and IR frequencies are concerned, and particularly for the material of interest here, which is ITO, its dielectric
permittivity is described with the Drude model [153, 161, 167, 170, 182].

ITO = ∞ −

ωp2
ω 2 + iωγ

(5.2)

where ωp2 = Ni e2 /o meff , ∞ is the high-frequency dielectric permittivity and meff
is the effective mass. Ni is the carrier density, for i = b, acc for the background
and accumulation regions, respectively. Following previous experimental work [153,
161, 182], we take ∞ = 3.4, γ = 1.8 × 1021 /sec and meff = 0.35me , where me

87
Dielectric
SiO2
kdiel = 3.9 [171, 172]
Ebr = 30 − 40[171]
Al2 O3
kdiel = 9[172]
Ebr = 6 − 8[171, 173, 174]
Al2 O3
kdiel = 10.3[174]
Ebr = 6 − 8[171, 173, 174]
HfO2
kdiel = 17[171]
Ebr = 13[171]
HfO2
kdiel = 18.7[174]
Ebr = 5.6[174]
HfSiO4
kdiel = 11[175]
Ebr = 10[175]
HfSiO4 (with SiO2 )
kdiel = 4.8 − 5.4[176]
Ebr = 10[175]
HfSiO4 (with HfO2 )
kdiel = 12.5 − 15.1[176]
Ebr = 10[175]

Nacc , d = 0.5

Nacc , d = 1

Nacc , d = 2.5

Nacc , d = 5

1.29 − 1.7

0.65 − 0.86

0.26 − 0.35

0.13 − 0.17

0.6 − 0.8

0.3 − 0.4

0.12 − 0.16

0.06 − 0.08

0.68 − 0.91

0.34 − 0.46

0.14 − 0.18

0.07 − 0.09

2.4

1.2

0.48

0.24

1.16

0.58

0.23

0.12

1.22

0.61

0.24

0.12

0.53 − 0.6

0.27 − 0.3

0.11 − 0.12

0.05 − 0.06

1.38 − 1.67

0.69 − 0.83

0.28 − 0.33

0.14 − 0.17

Table 5.1: Maximum achievable carrier concentration in ITO accumulation
layers for reported values of breakdown field and DC dielectric constants of
high-strength dielectrics. The breakdown voltage Ebr in given in MV/cm and
the carrier density Nacc in 1021 /cm3 . d is the accumulation layer thickness
given in nm.
is the electronic rest mass. Starting from the background carrier density Nb =
5 × 1020 /cm3 , we show in Fig. 5.2 the dielectric permittivity of ITO at visiblenear IR frequencies for increasing carrier concentration. We highlight that carrier
densities in the range 1020 − 1021 /cm3 yield an epsilon-near-zero (ENZ) response at
visible frequencies, with ITO transitioning from positive to negative values. In turn,
this induces topological transitions in the dispersion surface of the metamaterial, as
we discuss in what follows (see Figs. 5.6, 5.7, 5.8).
5.3

Tunable hyperbolic response at optical frequencies

Regarding the optical response of the metamaterial, we consider a toy model consisting of 20 nm of Ag (yellow layers in Fig. 5.1), separated by 15 nm of ITO (dark

88
5x1020/cm3

7.5x1020/cm3

1x1021/cm3

5x1021/cm3

2.5x1021/cm3

7.5x1021/cm3

Re(εITO)

(a) 10

ENZ
-10
-20
300

400

500

600

700

800

900

1000

900

1000

(b)

Im(εITO)

wavelength (in nm)
1.5
0.5
300

400

500

600

700

800

wavelength (in nm)
Figure 5.2: Complex dielectric permittivity of indium tin oxide at visible-near
IR frequencies. (a) Re(ITO ), (b) Im(ITO ).

blue and light blue colors for background and accumulation layers, respectively, in
Fig. 5.1). The two materials are electrically isolated from each other by 10 nm of
HfO2 dielectric layers (purple layers in Fig. 5.1). The thickness of HfO2 was chosen
to correspond to thicknesses routinely achievable in atomic layer deposition [173,
174]. The optical properties of Ag were taken from [75], while the Sellmeier equation with three poles was used for the refractive index of HfO2 [183]. Under applied
bias between the Ag and the ITO in the geometry of Fig. 5.1, we consider a 2.5
nm accumulation layer to be formed in the HfO2 -ITO interface [153, 161, 162]. In
Section 5.4 we perform a sensitivity analysis over the accumulation layer thickness
and maximum carrier density modulation in order to comprehensively assess the
tunability range for this design for various TCOs, transition-metal nitrides or degenerate semiconductors and high-k dielectrics. We model both the background ITO
and the accumulation layer using the Drude model with higher carrier concentration
compared to the background, as discussed above (see Eq. (5.2)).
The drastic change in the carrier concentration across the ITO background and accumulation layer interface yields tunable optical parameters. We use the parameter
retrieval method discussed in Chapter 2 and in [47] to calculate the effective opti-

89
5x1020/cm3

Re(εe,eff)

(a)
50

7.5x1020/cm3

1x1021/cm3

2.5x1021/cm3

5x1021/cm3

7.5x1021/cm3

ENP

50
300

400

500

600

700

800

900

1000

800

900

1000

wavelength (nm)

Im(εe,eff)

(b)
100
50
300

400

500

600

700

wavelength (nm)
Figure 5.3: Extraordinary dielectric permittivity of the TCO-based tunable
HMM depicted in Fig. 5.1. (a) Re(e,eff ), (b) Im(e,eff ). Legends correspond
to varying carrier concentration in the ITO accumulation layer, Nacc .

cal parameters o,eff , e,eff of this motif for increasing carrier concentration in the
accumulation layer of ITO. We note that, no high-index layers are incorporated in
the considered metamaterial, hence, following the results of Chapter 3, the magnetic permeability of this motif is approximately unity (µo,eff ≈ µe,eff ≈ 1), and the
effective magnetic parameters are not discussed further.
As the carrier concentration in the accumulation layer of ITO increases under applied
bias from the background value of 5 × 1020 /cm3 up to 7.5 × 1021 /cm3 , corresponding to experimentally reported carrier concentration changes [161], the Lorentzianshapped resonance in extraordinary permittivity e,eff blue-shifts as seen in Fig. 5.3.
This resonance arises form the coupling of the plasmonic modes supported on the
metal/dielectric interfaces to the bulk high-k modes of the HMM [33]. The epsilonnear-pole (ENP) wavelength of this dominant resonance blue-shifts under applied
bias by more than 60 nm, while remaining within the visible regime.
In contrast to e,eff that is largely tunable, o,eff does not exhibit significant tuning
with increasing bias, as shown in Fig. 5.4. The weak dependence of o,eff on Nacc is
expected; the ITO accumulation layers are relatively thin (2.5 nm) compared to the

(a) 10
Re(εo,eff)

-10
-20
-30
300

400

500

600

700

800

900

1000

800

900

1000

wavelength (nm)
(b) 5
Im(εo,eff)

300

400

500

600

700

wavelength (nm)
Figure 5.4: Ordinary dielectric permittivity of the TCO-based tunable HMM
depicted in Fig. 5.1. (a) Re(o,eff ), (b) Im(o,eff ). Color map similar to Fig.
5.3.
Ag layers (20 nm), the response of which dominates along the in-plane direction,
consistent with effective medium theories (see Section 2.2).
As discussed in Chapter 3, TM polarized bulk modes experience an extreme anisotropic
response when propagating in planar HMMs, as dictated from Eq. (3.2). Similar to naturally occurring uniaxial materials [1], we introduce here an effective
birefringence and dichroism for TM polarized modes, defined as Re( o,eff µo,eff ) −
Re( e,eff µo,eff ) and Im( o,eff µo,eff ) −Im( e,eff µo,eff ), respectively. These parameters are shown in Fig. 5.5 for increasing carrier concentration in the accumulation
layer of ITO. As expected, HMMs exhibit extreme anisotropies, which is manifest
in large birefringence values. These by far exceed the birefringence of the most
anisotropic natural materials like liquid crystals and other uniaxial inorganic crystals (c.f., Table 5.2). Notable is the broadband tunability of both birefringence and
dichroism across the whole visible spectrum.
The tunability of the effective parameters presented in Figs. 5.3, 5.4 has an effect
on the dispersion surface for TM polarized waves, as discussed in Chapter 3 (Eq.

91
(3.2)). This can be illustrated by defining a metamaterial figure of merit (FOM),
FOM=Re(kz,eff )/Im(kz,eff ). In Figs. 5.6 a, b, c we display this parameter for carrier
concentration Nacc = 5 × 1020 /cm3 = Nb , Nacc = 2.5 × 1021 /cm3 and Nacc = 7.5 ×
1021 /cm3 , respectively. Active modulation of the ITO accumulation layer yields
drastic changes in the metamaterial FOM. The “valleys” and “ridges” in Fig. 5.6
are a good criterion for identifying regions of enhanced and suppressed density of
optical states, respectively [71]. Namely, the ridge in Fig. 5.6a, at wavelengths
400 − 500 nm, indicates a band gap where propagation is forbidden. The large FOM
values away from the band gap regime emphasize the hyperbolic response of the
heterostructure, supporting large-k values. As the bias in increased, in other words
for increased carrier density, in Fig. 5.6b, an additional band gap is introduced at
larger wavelengths 700−800 nm, due to the secondary Lorentzian-shapped resonance
in e,eff , shown in Fig. 5.3 with green color. Further increase in the carrier density
or applied bias (Fig. 5.6c) leads to further spectral shift of the valleys and ridges.
We note that these figures of merit by far exceed previous generation metamaterials,
due to the intrinsic property of HMMs to support spectral regions of o,eff e,eff < 0
and, therefore, large wavenumbers.

Material
Birefringence
Al2 O2
0.08
LiNbO3
0.085
Rutile
0.287
Nematic liquid crystals 0.4
Table 5.2: Birefringence Re( o ) − Re( e ) of anisotropic dielectric media
The tunable optical parameters for this metamaterial motif also yield changes in
the isofrequency contours and topological transitions. In Fig. 5.7, we present the
complex isofrequency contours for TM bulk modes. Both kx and kz are normalized
to the free space wavenumber ko = ω/c. Application of DC bias across the HMM
unit cell yields drastic changes in both the shape and type of the dispersion surface.
Additionally, the surface area enclosed by the isofrequency contours, which is proportional to the total number of available optical states [41], changes significantly
with changes in ITO accumulation layer carrier concentration, yielding a route to
active control over the metamaterial optical density of states.
As can be seen from Fig. 5.7, for all the considered free-space wavelengths, the
topology of the dispersion surface changes with the ITO carrier density. The EFC
at wavelengths of 378 nm and 650 nm depict regions of type I and type II hyperbolic
dispersion, respectively. Hyperbolic dispersion of type I is a consequence of the
effective built-in dipole moment in the direction across the optical axis for oblique

92
5x1020/cm3

7.5x1020/cm3

1x1021/cm3

2.5x1021/cm3

5x1021/cm3

7.5x1021/cm3

Birefringence

(a) 5
-5
-10

300

400

500

600

700

800

900

1000

800

900

1000

wavelength (nm)

Dichroism

(b)

-5
-10

300

400

500

600

700

wavelength (nm)
Figure 5.5: (a) Birefringence defined as Re( o,eff µo,eff )−Re( e,eff µo,eff ), (b)
Dichroism defined as Im( o,eff µo,eff ) − Im( e,eff µo,eff ). Legends correspond
to varying carrier concentration in the ITO accumulation layer, Nacc .

incidence and TM polarization, yielding e,eff < 0. Type II hyperbolicity is the result
of the metallic response of the metamaterial along the in-plane direction (o,eff < 0)
and the dielectric response along the optical axis (e,eff > 0). Deviations of the
EFCs from perfect hyperbolic shapes are attributed to loss and spatial dispersion
(see Section 3.7).
For the wavelength of 520 nm, it is evident that the area enclosed by the real part of
the isofrequency surface decreases for increasing ITO carrier density. Additionally,
the losses increase with increasing ITO carrier density, as shown by the imaginary
parts of the EFCs. This wavelength falls inside a photonic band gap region for the
ITO carrier density Nacc = 7.5 × 1021 /cm3 . At this carrier density, both parameters
o,eff and e,eff are negative, leading to a forbidden band (see Eq. (3.2)). Hence, at
this wavelength regime, the metamaterial experiences a topological transition from
a hyperbolic state to a forbidden region with increasing carrier concentration. For
800 nm wavelength, a similar observation is made for the green curve corresponding
to ITO carrier density of Nacc = 2.5 × 1021 /cm3 . For this carrier concentration, the
metamaterial exhibits a band gap, as also indicated in Fig. 5.6b.

FOM

(a) 15

kx=0.2ko
kx=0.3ko
kx=3.4ko

5 10 /cm

kx=6.8ko

300

kx=9.4ko
kx=12.8ko

400

500

600

700

800

900

1000

FOM

(b) 10

kx=16ko
kx=20ko

2.5x1021/cm3
300

400

500

600

700

800

900

1000

600

700

800

900

1000

FOM

7.5x1021/cm3

300

400

500

wavelength (nm)
Figure 5.6: Tunable figure of merit defined as Re(kz )/Im(kz ) of the TCObased tunable HMM depicted in Fig. 5.1. Panels (a), (b), (c) correspond,
respectively, to Nacc = 5 × 1020 /cm3 = Nb , Nacc = 2.5 × 1021 /cm3 and Nacc =
7.5 × 1021 /cm3 .

To illustrate the effects of tunable optical parameters on the effective band structure, we show in Fig. 5.8 the change in the topology of the frequency-dependent
three-dimensional dispersion surface for the two extreme cases of charge carrier concentration in the ITO accumulation layer, in the lossless limit. Within the optical
regime, we see an effective omnidirectional band gap (noticeable in the band diagram along both the kx and kz axes in Figs. 5.8a, b, respectively), whose band edges
can be tuned by varying the applied bias. Additionally, we notice the appearance of
a new effective band gap and a new hyperbolic region, for larger accumulation layer
carrier concentration changes (Nacc = 7.5 × 1021 /cm3 in Figs. 5.8c, d). Hence, the
field-effect provides sufficient change of the conductive oxide permittivity to allow
for spectral shifting of the hyperbolic regions and band gaps of HMMs and even to
allow for active opening and closing of band gaps.
5.4

Sensitivity analysis

As shown in Eq. (5.1), the parameter that principally defines the tunability range
of the effective optical properties of our metamaterial is the product Nacc d, where

-1

-2
-6

-6
-6

Im(kz)

-0.5
-1
-6

-3

-3
-6

-6
-6

-4
-6

-2

0.5

800nm

-3

650nm

Re(kz)

-6

Re(kz)

Re(kz )

520nm

Re(kz)

Im(kz)

378nm

7.5x1021/cm3

2.5x1021/cm3

5x1020/cm3

-3

-6
-6

Figure 5.7: Complex isofrequency contours at different wavelengths for the
TCO-based tunable HMM depicted in Fig. 5.1. Blue, green and red colors correspond, respectively, to Nacc = 5 × 1020 /cm3 = Nb , Nacc = 2.5 × 1021 /cm3 and
Nacc = 7.5 × 1021 /cm3 . kx and kz are normalized to the free space wavenumber
ko = ω/c

Nacc is the maximum achievable carrier concentration in the accumulation layer of
the TCO before electrical breakdown occurs, and d is the Debye length or thickness of the accumulation layer. We perform a sensitivity analysis of the retrieved
extraordinary permittivity e,eff , separately as a function of Nacc and d, since e,eff
is the optical parameter most drastically affected by the field-effect (Fig. 5.3). Both
for Nacc and d, we consider values within the range of previous experimental reports
[153, 161, 162, 167–169]. We suggest that this approach can provide useful insight
here, and also for other semiconductors and TCOs or transition-metal nitrides, as
active constituent materials in field-effect tunable metamaterial realizations. In Fig.
5.9 we illustrate the tunability of the ENP region of the primary Lorentzian resonance (Fig. 5.3).

(b)

(ω in 1015 rad/s)

(a)

-1

(d)

(ω in 1015 rad/s)

kz (nm )

-1

kx (nm-1)

-1

kx (nm )
(c)

-1

kz (nm-1)

Figure 5.8: Three-dimensional tunable dispersion surface for the TCO-based
tunable HMM depicted in Fig. 5.1. (a) along the kx axis and (b) along the kz
axis for Nacc = 5 × 1020 /cm3 = Nb . (c) along the kx axis and (d) along the kz
axis for Nacc = 7.5 × 1021 /cm3 .

Both the carrier concentration Nacc and the Debye length d drastically impact the
spectral shift of e,eff . As expected, thicker accumulation layers (i.e. larger d) is
desirable as it introduces larger shifts of the ENP. Similarly, being able to gate
TCO materials at very high carrier densities is also desirable for increased optical
tuning. We emphasize here on the importance of the electrical quality of the oxide
layer, which is critical for achieving experimentally observable spectral tuning.

90
80
70

ENP shift (nm)

(b) 90

0.5 nm
1 nm
1.5 nm
2 nm
2.5 nm
3 nm
3.5 nm
4 nm
4.5 nm
5 nm

60
50

80
70

ENP shift (nm)

(a)

60
50
40

70
20

7.5x1020/cm3
1x1021/cm3
2.5x1021/cm3
5x1021/cm3
7.5x1021/cm3

carrier concentration (10 /cm )

Debye length (nm)

Figure 5.9: Effects of the electronic properties of ITO on the ENP wavelength
of e,eff . (a) Dependence of the ENP shift on the carrier concentration Nacc ,
for varying Debye lengths indicated with the legends. (b) Dependence of the
ENP shift on the Debye length, for varying maximum achievable carrier concentration in the accumulation layer before breakdown occurs.

5.5

Graphene as an active component

Graphene is a two-dimensional, single-atom thick semimetal, initially extracted from
graphite in 2004, by A. Geim and K. Novoselov [184]. From its two-dimensional
nature arises the intriguing property of exhibiting linear electronic dispersion of
the form E ∝ |~k| near its charge neutrality point (CNP), as shown in Fig. 5.10a.
This is in contrast to conventional metallic systems where the electronic dispersion
is parabolic (E ∝ |~k|2 ). At the Dirac point or CNP, the density of electronic
states vanishes, however it is possible to tune graphene Fermi’s level EF via carrier
injection, chemically or by electrostatic changes to the carrier concentration [5,
120, 121, 184]. Hence the conductance of graphene σ, and therefore its optical
properties can be actively modulated in motifs where graphene is gated against
another conducting material.
Another interesting aspect of graphene is its ability to guide extremely confined
surface plasmonic modes [120, 121], with confinement factors that exceed any three-

97
(a)

(b)

EF >0

CNP

ky
kx

Figure 5.10: (a) Graphene’s Dirac cone. Here E and kx , kz are the energy and
wavenumbers of electrons, respectively, not to be confused with the wavenumber k// in (b), which pertains to the incident electromagnetic field’s wavenumber. (b) Schematic of graphene-based tunable hyperbolic metamaterial unit
cell. Directions (o) and (e) refer to the ordinary and extraordinary axes.

dimensional, bulk plasmonic metal. In contrast to plasmonic metals or to the TCO
family discussed in the previous sections, graphene’s lower carrier density yields plasmonic properties in the IR range. Hence, it is an intriguing material to investigate
as a tunable metallic component in planar graphene/dielectric HMM configurations
(Fig. 5.10b).
In modeling graphene, we take its conductance at different values of EF from the
semi-analytical model provided in [185]. In order to account for two-dimensional
materials in the conventional transfer matrix approach discussed in Section 2.3 and
in Appendix A.1, we modify Maxwell’s boundary conditions for the magnetic field,
~ induced by the incident
to consider a non-zero surface current density J~ = σ E
~ graphene (or
~ [186]. This results in the transfer matrix for graphene M
electric field E

any two-dimensional material described via a sheet conductance σ), given by

~ graphene =

−4πσ/c 1

(5.3)

which is to be multiplied with the transfer matrix of Eq. (2.15), for multilayer

98
heterostructures involving both two-dimensional and conventional three-dimensional
materials.
5.6

Tunable hyperbolic response at infrared frequencies

We envisage multilayer configurations of alternating graphene/high-strength dielectric layers as depicted in Fig. 5.10b. Similar configurations have been considered in
[186] and in [187], however in previous reports, the dielectric layers were assumed
to be non-dispersive. We highlight that, typically, in the IR regime, dielectric materials exhibit phonon resonances as discussed in Chapter 4. For this reason, and in
order to examine a parameter space of experimentally reasonable configurations, we
assume here a typical polar dielectric response with Lorentzian-shapped resonances,
as discussed in Fig. 4.1. We particularly focus our studies in SiO2 dielectric layers,
due to the widespread use of SiO2 as an insulating material in graphene experiments
[5, 120]. As a result of accounting for phonon resonances, we observe interesting
ENZ regimes for the composite metamaterial, at IR frequencies. Such a response is
typically unfound in natural plasmonic metals, where the ENZ regime is physically
bound to UV-visible frequencies.
We perform transfer matrix calculations for the example system of Fig. 5.10b, for
SiO2 layers of thickness 100 nm, achievable either with electron-beam or thermal
evaporation, or growth on a Si substrate, for higher material quality. We consider
changes in graphene’s Fermi level EF from CNP, up to 1 eV. These can be induced by
gating the graphene mono-layers either against each other, as shown in Fig. 5.10b,
or against a common ground, which could be, for example, a substrate (such as
doped Si, which is commonly used for graphene experiments). The ordinary and
extraordinary permittivities of the metamaterial, o,eff and e,eff , at different EF , are
shown in Fig. 5.11. The effective parameters calculations were performed using the
parameter retrieval approach discussed in Chapter 2 and in [47].
Similar to the TCO-based metamaterial investigated above, the graphene-based
metamaterial studied here does not exhibit interesting magnetic properties (µo,eff ≈
µe,eff ≈ 1). This may seem counter-intuitive, as in Chapters 3 and 4 we emphasized that strong artificial magnetic effects require high- values to support strong
displacement currents J~disp [98] (see Section 3.3), and the polar dielectric medium
(SiO2 ) does exhibit high- near its phonon resonances (∼ 10µm and ∼ 20µm). However, as discussed in Section 3.3, for strong magnetic effects, the high- layers should
~ eff ∝ ~r × J~disp
be separated by some distance r so that a magnetic dipole moment M
is built. In the case examined here, the SiO2 layers are separated by graphene monolayer of negligible thickness, hence r = 0 and the magnetic effects are unimportant.

(a) 50

(b)

Re(εo,eff)

Im(εo,eff)

Re(εe,eff)

wavelength (μm)

(d)
40

-50

Im(εe,eff)

-50

SiO2
0 eV
0.5 eV
1 eV

wavelength (μm)

Figure 5.11: Complex effective dielectric permittivity of the graphene/SiO2
HMM depicted in Fig. 5.10, for different EF corresponding to different applied
bias V . (a) Re(o,eff ), (b) Im(o,eff ), (c) Re(e,eff ), (d) Im(e,eff ).

It is noteworthy that the extraordinary permittivity e,eff of the metamaterial coincides exactly with the permittivity of SiO2 , SiO2 , as shown in Figs. 5.11c, d, for all
considered gating levels or values of EF . This occurs due to the two-dimensional nature of graphene; electric fields with out-of-plane (z) components do not experience
the graphene’s response and, instead, feel a purely dielectric environment.
By contrast, the ordinary permittivity o,eff is largely tunable as depicted in Figs.
5.11a, b. At EF = 0, we obtain o,eff = e,eff = SiO2 . This result is expected; as
explained above, at EF = 0 eV, graphene has a vanishing density of electronic states,
hence it is not conductive and its effect on the optical properties of the metamaterial is negligible, along all coordinate directions. By increasing EF , we observe an
induced metallic response along the in-plane direction, i.e. for electric fields with a
component parallel to the metamaterial’s interfaces. Maximum tunability of o,eff
is observed at and below the phonon frequencies of the SiO2 . Particularly, we observe two induced and tunable ENZ regimes for o,eff , near the two phonons, i.e. at
wavelengths ∼ 10µm and ∼ 20µm. In Fig. 5.12 we provide an enlarged view of

100

40
SiO2

30
ENZ

Re(εo,eff)

ENZ
0 eV

10
-10
kz

-20
-30
-40

20
25
wavelength (μm)

1 eV

Figure 5.12: Re(o,eff ) of the graphene/SiO2 HMM depicted in Fig. 5.10, for
different EF corresponding to different applied bias V . The highlighted regions correspond to epsilon-near-zero regimes and topological transitions from
elliptical to hyperbolic dispersion.

Re(o,eff ).
With the reminder that e,eff = SiO2 for all considered values of EF , we infer from
Fig. 5.12 that this metamaterial exhibits topological transitions of its dispersion
surface from the elliptical state (o,eff e,eff > 0) to the hyperbolic state (o,eff < 0,
e,eff > 0), as higher gate bias V is applied, i.e. for increasing values of EF . These
transitions occur at wavelengths between the two SiO2 phonons and also for wavelengths above the 20µm phonon. In order to quantify the degree of hyperbolicity
and to estimate the range of observable changes in the metamaterial’s dispersion,
we consider the equifrequency contours for bulk, TM polarized propagating modes
(see Eq. (3.2)) at different wavelengths, as shown in Fig. 5.13.
From Fig. 5.13a, at small wavelengths where the tunability range is small as shown
in Fig. 5.12, the EFC does not change considerably for different values of EF .
Particularly, since o,eff is not largely tunable in this regime of wavelengths, we
have o,eff ≈ e,eff , hence the response is almost isotropic, as confirmed with the
circular shape of the EFCs in Fig. 5.13a. Small changes in the shape of the EFC

101

(a) 6

(b) 60
9.8μm
9.5μm

11μm

-2

-20

-4

-40

-6
-6

-4

-2

1 eV
16μm

-60
-60 -40 -20

0 eV
60

Figure 5.13: Isofrequency contours at different wavelengths for the
graphene/SiO2 HMM depicted in Fig. 5.10, for different EF corresponding
to different applied bias V . (a) Elliptical regime corresponding to wavelengths
at which o,eff e,eff > 0, (b) hyperbolic regime for wavelength 16µm, at which
o,eff e,eff < 0. kx and kz are normalized to the free space wavenumber ko = ω/c.

from circular to elliptical arise from small changes in o,eff with respect to e,eff . By
contrast to the small wavelength regime, between the two phonons of SiO2 (∼ 10µm
and ∼ 20µm), the response of this metamaterial changes dramatically with applied
bias or increased EF . This is shown with the EFC in Fig. 5.13b, for a wavelength
of 16µm. At small values of EF , o,eff ≈ e,eff and the response is isotropic. With
increasing EF , o,eff becomes negative, hence the EFC opens up into a hyperboloid.
Notably, the range of achievable wavenumbers kz is dramatically increased in the
hyperbolic regime, where we have kz ∼ 60ko .
A considerable impediment in practically utilizing the large wavenumbers supported
in traditional metallodielectric multilayer hyperbolic media -and, as a consequence,
their large density of optical states- is to find means of coupling into these highk states from free space (k// ≤ ko ) [71]. In the case of graphene-based HMMs,
however, this issue is easily circumvented, as even unpatterned graphene provides
access to wavenumbers as large as k// ∼ 20ko due to its plasmonic properties below
its plasma frequency [120, 121]. Importantly, by patterning graphene, we can access
wavenumbers as large as k// ∼ 100ko [120, 121], for which we display a patterned
top-most graphene layer in Fig. 5.10b. Hence, with graphene/dielectric multilayer
HMM configurations (Fig. 5.10b), it becomes possible to achieve confinement factors
for bulk propagating modes (apart from guided surface plasmonic modes) below the
diffraction limit.

102
5.7

Conclusion

In conclusion, we outlined methods for electronically tuning the response of planar
hyperbolic metamaterials at visible and IR frequencies, using transparent conductive
oxides and graphene as active components, respectively.
In the visible regime, we introduced a field-effect gating scheme for electrical modulation of the permittivity in transparent conductive oxide layers. By incorporating
a TCO into a typical metallodielectric metamaterial and altering the carrier density
in accumulation regions, we observe the opening and closing of optical omnidirectional band gaps controlled by applied bias, which corresponds to a tunable figure
of merit with values as high as 15 in the hyperbolic regime while vanishing at the
band gaps. The field-effect allows spectral tunability of the effective extraordinary
permittivity (e,eff ) along the optical axis (z direction). We predict blue shifts of
e,eff near its resonance by more than 60 nm in the visible regime. This also gives rise
to broadband tunability of the effective birefringence and dichroism, with potential
for novel photonic devices like tunable metallodielectric waveguides, optical sensors,
filters and polarizers. Such active control over the complex parameters of metamaterials is also essential for slow light media and holographic displays. Our sensitivity
analysis with respect to changes in the TCO carrier concentration and accumulation
layer thickness indicates the robustness of the field-effect as a tuning mechanism.
The straightforward fabrication of multilayer metamaterials by thin-film deposition
techniques suggests that the experimental realization of tunable field-effect metamaterials with active materials such as TCOs, transition-metal nitrides or degenerately
doped semiconductors is well within reach.
In the IR regime, we envisage graphene as a tunable element, integrated into planar heterostructures that allow carrier injection for optical tuning. Particularly, we
investigate heterostructures composed of alternating layers of graphene and highstrength polar dielectric materials. Via electrostatic gating in graphene layers, one
is able to actively alter the Fermi level EF , which in turn yields an overall tunable
response of the metamaterial. In contrast to the visible regime where the formation of accumulation and depletion regions along the optical axis (z) yields tunable
extraordinary permittivity, the two-dimensional nature of graphene yields optical
tuning in the direction parallel to the layers, i.e. for the ordinary permittivity o,eff .
By raising previous assumptions regarding the non-dispersive nature of the dielectric layers, we observe largely tunable values of o,eff near the phonon resonances of
the dielectric material. The tunability of the effective parameters extends across a
wide range of frequencies, from the IR to the microwave regime. The extraordinary
plasmonic properties of graphene allow large confinement factors and coupling into
very high-k modes (k// ∼ 100ko ), taking full advantage of the hyperbolic response

103
of planar metallodielectric heterostructures.
By combining the two concepts outlined above, i.e. (i) carrier injection in accumulation regions of TCOs, transition-metal nitrides or degenerately doped semiconductors via the field-effect and (ii) carrier injection in graphene via electrostatic gating,
one may envision gaining active control and tunable response along all coordinate
directions simultaneously, for frequencies ranging from the IR to the microwave
regime.

104
Chapter 6

ULTRA-LIGHT VAN DER WAALS HETEROSTRUCTURES
AS SUPERMETALS
“What could we do with layered structures with just the right layers? What would
the properties of materials be if we could really arrange the atoms the way we want
them? They would be very interesting to investigate theoretically.”
Richard P. Feynman, Plenty of Room at the Bottom, 1959
6.1

What makes a perfect reflector? Beyond noble metals

In this final chapter, we search for artificial matter that exhibits ultra-reflective
properties. Although the question of what makes a perfect reflector may seem
rather fundamental, the search for perfect reflection is motivated by technological
needs and applications. Particularly, nearly perfect reflection is a requirement in
designing compact waveguides that operate based on total-internal-reflection. It
is also a necessary functionality for state-of-the-art aerospace technology, where
the quality factor of a mirror is a crucial parameter. Perfectly reflecting materials
are also necessary for engineering emission, serving as back-reflectors in absorbing
systems for solar energy technologies [5, 188], as well as for cloacking macroscopic
objects, such as aircrafts.
So far, engineering reflection has been discussed in Chapter 1, in the context of photonic crystals and particularly Bragg mirrors. However, Bragg mirrors are limited
by narrow bandwidth as can be seen from Fig. 1.5. Aside from Bragg reflectors,
broadband perfect reflection requires high electrical conductance, in other words
low electrical resistivity, typically found in noble metals with large charge carrier
density. In Table 6.1, we list the twenty most conductive elemental metals, sorted
with increasing bulk room-temperature resistivity ρo,rt [189]. In the far-IR, certain
metals act as perfect electrical conductors (PECs). A PEC is formally defined as a
material with Re() → −∞, while Im() → 0 that reflects perfectly. Critical parameters for a PEC are the carrier relaxation time τ , which ought to be small for highly
conductive materials with few collisions and low loss (Im() → 0), and the Fermi
velocity vf , which needs to be large for carriers (electrons) with high mobility. From
Table 6.1 it is clear that gold (Au) and silver (Ag) outperform other metals and are
the two most widely used materials for mirroring systems in the mid-long-wave IR.

105
Element
Silver
Cooper
Gold
Aluminum
Calcium
Beryllium
Magnesium
Rhodium
Sodium
Iridium
Tungsten
Molybdenum
Zinc
Cobalt
Nickel
Potassium
Cadmium
Ruthenium
Indium
Osmium

ρo,rt (µΩ cm) vf (105 m/s)
τ (fs)
1.587
14.48
36.8
1.678
11.09
36.0
2.214
13.82
27.3
2.650
15.99
11.8
3.36
4.80
73.6
3.56
12.62
37.0/59.1
4.39
11.63
19.6/16.6
4.7
6.67
10.3
4.77
10.21
30.2
5.2
8.54
8.30
5.28
9.71
16.0
5.34
9.18
12.2
5.9
15.66
11.5/8.31
6.2
2.55
21.2/17.6
6.93
2.34
14.5
7.20
7.94
39.7
7.5
15.55
11.1/9.18
7.8
7.24
8.82/7.07
8.8
16.32
5.27/5.05
8.9
8.19
8.54/6.19

Table 6.1: List of the twenty most conductive elemental metals sorted with
increasing bulk room-temperature resistivity ρo,rt from [189]. For hexagonal
and tetragonal crystal structures (hexagonal closely packed and body-centered
tetragonal), the two listed values are for transport perpendicular and parallel
to the hexagonal/tetragonal axis.

However, in all conventional noble metals, the high density of states near the Fermi
level leads to a rather small relaxation time, which is on the order of tens of fs (see
Table 6.1). Furthermore, increased ohmic losses, expressed through the imaginary
part of the dielectric permittivity, Im(), further limit the performance of noble
metals as PECs. The degree of mobility of charge carriers can be measured in terms
of their effective mass, and noble metals typically exhibit a rather large effective
mass, approximately equal to the free electron mass [189]. Finally, we note that
conventional noble metals have high mass density. Decreasing mass density is highly
desirable, for example in space exploration devices and systems.
6.2

Graphene-based van der Waals heterostructures

In contrast to the aforementioned limitations of noble metals, the two-dimensional
nature of graphene and its unique linear electronic dispersion yield a much larger
relaxation time, due to its lower electronic density of states, as well as an effective

106
mass approaching zero, both of which enhance the carrier mobility in comparison
to metals [184]. Although graphene typically has lower carrier density than Ag and
Au, its important features of high mobility and tunable Fermi level, for example
through external bias or carrier injection, motivate us to seek for graphene-based
heterostructures that may surpass the photonic and plasmonic properties of noble
metals [18].
To circumvent the issue of low carrier density in graphene, we investigate a large
number of graphene sheets stacked in combination with other two-dimensional materials in layered arrangements, termed van der Waals (vdW) heterostructures [15,
190]. While the electronic and phononic properties of graphene-based heterostructures have been widely investigated, photonic research on these materials has previously focused on graphene layers separated by hundreds of nanometers to micrometers [186, 191–194]. In contrast, here [18] we focus on Angstrom (Å)-scale
photonics, where graphene and hexagonal boron nitride (hBN) mono-layers are at
their equilibrium separation (3.3 − 3.4 Å), as shown in Fig. 6.1a; We also investigate
doped graphite, where graphene sheets are at their equilibrium separation (3.3 − 3.4
Å); results for the free-standing graphene case with double the layer spacing are
shown throughout for comparison to highlight the effect of interlayer interactions.
These interlayer electronic interactions are accounted for using ab initio electronic
calculations [195] and make this Å-scale regime distinct from conventional photonic
crystals and metamaterials. We compare the performance of these heterostructures
as reflectors with silver and gold.
6.3

Computational approach and effective mass

For a layered arrangement composed of two-dimensional materials, the dominant inplane (ordinary) dielectric response is conveniently expressed through the effective
medium theory [185] as

∞// (ω) = 1 +

iσ(ω)
ωd

(6.1)

where d is the spacing between adjacent layers and σ(ω) is the sheet conductance of
the mono-layer, σ(ω) = σintra (ω) + σdirect (ω). The first term captures the intraband
response of free carriers in the material, and the second term describes the effect of
interband transitions and is evaluated using Fermi’s Golden rule. Both terms are
computed using previously developed density functional theory (DFT)-based ab initio predictions that are outlined in detail in Appendix A.3 [195–197]. We note that,
in computing the conductance of 2D materials stacked in vdW heterostructures, the

107

(a)

0.67 nm

Graphene/air

Graphene/hBN

(b)

102

(d)

10
-Re(ε)

τ (fs)

Graphite

103
102
101
100

(c)

0.3
0.2
0.1

(e)

Im(ε)

meff / me

101

103
102
101

0.2 0.4 0.6 0.8
∆ EF (eV)

100

10
20
30
wavelength (μm)

Figure 6.1: (a) Schematics of 2D materials, graphene-based vdW heterostructures and 3D noble metals. (b) Comparison of the average lifetime τ as a
function of the Fermi level in 2D materials, heterostructures and the best-case
3D metals: Ag and Au. (c) The effective mass parameter for the stack, as a
function of the EF , connects our analytical understanding of Å-scale metamaterials with ab initio calculations. In panels (d) and (e) we show the Re() and
Im() of graphene-based heterostructures in comparison with the permittivity of Au and Ag. Results for the heterostructures are shown at 0.2 eV Fermi
level, as this is the doping regime where they reflect maximally for wavelengths
above 10 µm.

108
effects of the environment and the electronic interactions between adjacent layers
are fully accounted for.
The dielectric permittivities of the vdW heterostructures examined here are presented in Figs. 6.1d, e in the limit of infinite number of layers, and compared with
the response of Ag and Au. We point out that results pertaining to Ag and Au
are also derived ab initio [196, 197] assuming perfect crystalline metals. At low frequencies, intraband transitions dominate and the response of the heterostructures
resembles a typical Drude material. In this limit of low frequencies we are, thus,
able to approximate the dielectric response of the vdW heterostructures with the
Drude model via

Drude = = 1 −

ωp2
ω 2 + iωγD

(6.2)

where ωp2 = n3D e2 /o meff,D , where o is the dielectric permittivity of free space,
γD
1 = τD is the Drude carrier relaxation time and n3D is the three-dimensional

carrier density of the heterostructure. meff,D is the Drude effective mass that we
seek to estimate.
Let us assume an arrangement of Ng graphene mono-layers separated by distance
d. The in-plane two-dimensional carrier density of graphene is given by n2D =
EF2 /(π~2 vF2 ) [193]. The three-dimensional carrier density of the heterostructure is
then given by

n3D = n2D

E2
Ng
Ng
= 2F 2
(Ng − 1)d π~ vF (Ng − 1)d

(6.3)

In the mid-far IR, where intraband transitions dominate, the conductance of graphene
can be approximated with the analytical expression [185, 193]

σ = σintra =

ie2 EF
π~2 (ω + iγg )

(6.4)

where γg− 1 = τg is the ab initio computed carrier relaxation time in graphene [195],
the computation of which is discussed in Appendix A.3. By equating Eqs. (6.1) and
(6.2) using the analysis above, we obtain

meff,D =

NG EF
NG − 1 vF2

(6.5)

and
τD = τg

(6.6)

109
Eqs. (6.5), (6.6) are useful in allowing us to build intuition about the response of
the vdW heterostructures as effective media with an effective mass that, in the large
number of layers, reduces to meff = Ev2F . Furthermore, unsurprisingly, from Eq. (6.6)

we derive that in the in-plane direction, in other words, for electromagnetic fields
incident normal to the layers, the carrier relaxation time of the three-dimensional
heterostructures is equal to the scattering time of the mono-layer itself.
In Figs. 6.1b, c we respectively show the parameters τ and meff of the graphenebased heterostructures as a function of the change in Fermi level ∆EF from the
neutral value (Dirac point). We highlight the extremely large τ ∼ 1 ps for undoped
graphene in air, which drops to ∼ 200 fs in graphite due to interlayer interactions.
In contrast, encapsulating graphene with boron nitride layers increases the undoped
τ even further to ∼ 2 ps, despite having the same spacing between adjacent monolayers as in graphite. With increasing EF and carrier density, τ drops rapidly. This
is a consequence of increasing the phase-space for electron-phonon scattering, which
in turn is due to increased electronic density of states near the Fermi level, g(EF ).
For comparison, we also show the emphab initio computed relaxation times for noble
metals, gold and silver, which are much smaller (∼ 30 and 40 fs respectively) and
mostly insensitive to EF since g(EF ) depends weakly on EF . We further note that
EF cannot be changed easily for these metals in experiments.
Regarding the effective mass for the vdW heterostructures, meff increases linearly
in the ideal case of perfect linear dispersion, with slight deviations due to band
structure effects (Fig. 6.1c). We highlight that meff for the vdW heterostructures is
consistently smaller than meff ≈ me of metals. Concluding, the graphene-based heterostructures have a much higher τ and a lower meff . These properties in graphenebased heterostructures compared to noble metals motivate us to investigate their
optical properties.
Although, from Figs. 6.1b, c, the maximum electronic scattering time and minimum
effective mass are seen at the charge neutrality point (∆EF ≈ 0 eV), graphene at 0
eV does not contain enough carriers to produce a strong metallic response. We find
that ∆EF ≈ 0.2 eV is the optimum doping level which provides the best tradeoff
between increasing carrier density and correspondingly decreasing scattering time.
Figs. 6.1d, e compare the real and imaginary parts of the in-plane permittivity
of the graphene heterostructures with those of noble metals at this carrier density,
in the infinite number of layers limit. The real parts of the permittivities of the
heterostructures in the IR are smaller than that of the noble metals by typically
two orders of magnitude. However, the ratio of imaginary to real parts is smaller
by an extra factor of 2 − 3. This smaller imaginary-to-real ratio compensates for

110
the lower plasma frequency and makes the graphene-based heterostructures better
reflectors in the IR due to reduced losses, as we discuss below. We point out here
that results pertaining to Ag and Au are also derived ab initio, assuming perfect
crystalline metals.
To realistically compare the utility of graphene vdW heterostructures as reflectors,
we transition from the infinite stacks of considered above to stacks of finite thickness. To do this, we perform electromagnetic transfer matrix calculations for layered
media [1]. Contrary to materials of finite thickness, we modify Maxwell’s boundary
conditions to account for a non-vanishing surface current density, arising from the
two-dimensional sheet conductance of the mono-layers, as discussed in Section 5.5
[186].
6.4

Shinier than gold and silver!

In Fig. 6.2 we show the reflectance of the vdW materials for the bulk (semi-infinite)
limit, as well as for finite stacks of 1000, 500, and 250 sheets and compare with
the Ag/Au slabs of the same mass per area, which is the relevant metric for ultralight reflectors. The reflectance of the vdW heterostructures surpasses that of noble
metals above a critical wavelength, λ > 10 µm for the semi-infinite case, λ > 15 µm
for 1000 layers, λ > 20 µm for 500 layers, and λ > 20 µm for 250 layers.
We note that silver and gold slabs of mass densities 19 µg/cm2 (that of 250 sheets)
correspond to thicknesses of 18.1 nm and 9.85 nm, respectively. Despite the highfidelity of state-of-the-art thin-film deposition systems, deposition of such ultra-thin
silver and gold films is still known to lead to island formation and grain-boundary
inhomogeneities that would further lessen the reflectance in realistic experimental
conditions. Fabrication of graphene-based structures entails challenges as well, for
example, issues with adhesion, buckling, and uniaxial tension, particularly when
integrated with traditional dielectric materials [198]. However, recent experimental advances in assembly of vdW graphene heterostructures provide a pathway to
uniform, large-scale components with potential for ultralight, long-IR mirrors [190,
199]. Furthermore, recent work has proposed Li-intercalated graphene layers (doped
graphite-Fig. 6.1) as a potential system to realize the structures discussed here [200,
201]. Finally we emphasize that the high reflectivity of graphene-based vdW heterostructures discussed here critically requires sufficient free carriers (for which we
selected ∆EF ≈ 0.2 eV). In contrast, undoped graphite with few free carriers exhibits
a much lower reflectance of ∼ 75% [202].

111
(a) Semi−infinite

Reflectance (%)

99.8
99.6
99.4

Graphene/air
Graphite
Graphene/hBN
Ag
Au

99.2
99.0

Reflectance (%)

(b) 1000 layers, 76 μg/cm2

99.5

(d) 250 layers, 19 μg/cm2

99.0
98.5
98.0
97.5

10
20
30
wavelength (μm)

40 0

20
30
40
wavelength (μm)

Figure 6.2: Normal incidence reflectance of graphene-air metamaterial, doped
graphite, graphene-hBN heterostructure (all doped at 0.2 eV) and comparison
to Ag and Au. The reflectance of graphene-based heterostructures surpasses
conventional noble metals in the mid-IR, above 10 µm. Panels correspond to
the same (per area) mass densities: (a): bulk, (b): 1000 layers, corresponding
to 72.4 nm of Ag and 39.4nm of Au. (c): 500 layers, corresponding to 36.2
nm of Ag and 19.7 nm of Au. (d): 250 layers, corresponding to 18.1 nm of
Ag and 9.85 nm of Au. Note that the silver and gold values are based on ab
initio results for perfect conditions and therefore are the theoretical limit for
these metals. Experimentally realized metallic thin films will compromise the
magnitude of reflectance.

6.5

Loss tangent and quality factors

In discussing the permittivities shown in Figs. 6.1d, e above, we noted that the
ratio of imaginary to real permittivity (i.e. the loss tangent tanδ) was smaller in the
vdW heterostructures than in noble metals, which resulted in lower losses. A metric
that directly illustrates the superior performance of graphene heterostructures as
plasmonic systems compared to noble metals is the material quality factor Q =
1/tanδ = Re()/Im() and its inverse, the material loss tangent, shown in Fig. 6.3a.
For the chosen doping level of 0.2 eV and for wavelengths λ > 15 µm, the vdW
heterostructures are expected to show an order of magnitude higher performance as
electromagnetic materials compared to noble metals, thereby making these materials

112
particularly suited for mid-long-wave IR applications. We note that the peaks in
Fig. 6.3a correspond to the respective plasma frequency for each material system, for
which Re() → 0; vdW heterostructures have much lower carrier density compared
to noble metals, hence their Q-factor peak is located approximately at λ ≈ 10 µm.
By contrast, noble metals that are highly doped have a plasma frequency in the
UV-visible spectrum, with Q-factor peaks in this regime (below λ ≈ 1 µm).

(a) 50

tanδ = |Im(ε) /Re (ε) |

Q = |Re(ε) /Im (ε) |

0.02

Graphene/air
Graphite
Graphene/hBN
Ag
Au

0.03
0.04
0.05
0.07
0.1
0.2
0.5

20
10

(b)

Qw=Re(β)/Im(β)

200
100

20
wavelength (μm)

Figure 6.3: (a) Material quality factor Q = Re()/Im() and loss tangent
(tanδ = Q−1 ) and (b) the plasmonic modal quality factor QW = Re(β)/Im(β)
for the symmetric mode in a metal/dielectric/metal waveguide, where the
‘metal’ is either a conventional noble metal or a vdW heterostructure and the
dielectric is 0.1 µm of air.

6.6

Plasmonic propagation in vdW heterostructures

Finally, we investigate the performance of these materials in an alternating metal/dielectric/metal
waveguide geometry. Such structures have been widely investigated in the visible
part of the spectrum using plasmonic metals, such as Ag and Au [8, 92, 135]. On the
basis of our results, we envision replacing the metal with a vdW heterostructure to
improve performance for mid-IR applications. For simplicity, we take the dielectric
to be 0.1 µm of air. We use our predicted dielectric functions in the formalism from

113
Alu et al. in [203] to evaluate the plasmonic in-plane wavenumber (β). Fig. 6.3b
compares the plasmonic quality factor QW = Re(β)/Im(β) for the symmetric mode
between vdW heterostructures and noble metals, while in Fig. 6.4 we explicitly
compare the corresponding dispersion relations. The plasmonic quality factor QW
estimates the propagation length in number of mode-wavelengths, which shows a
3 − 5-fold improvement for the vdW heterostructures compared to noble metals for
λ > 15 µm. In absolute terms, compared to propagation distances of 50 − 60 modewavelengths with use of Ag or Au in the mid-IR, a graphene/hBN- based waveguide
supports propagation distances that may exceed 200 mode-wavelengths.

(a)

Frequency, ν (THz)

(b)

60
50
40

Light line
Graphene/air
Graphite
Graphene/hBN
Ag
Au

30
20
10

-1
Re β (μm )

0.01 0.02 0.03
-1
Im β (μm )

Figure 6.4: Comparison of dispersion relations for the symmetric mode in
the metal/dielectric/metal waveguide considered in Fig. 6.3 (b). The larger
Re(β) with vdW heterostructures corresponds to smaller effective wavelength
and improved mode confinement. The smaller Im(β) for ν < 25 THz yields a
two-fold increase in the decay length of the mode.
Fig. 6.4 shows the dispersion relation of the symmetric waveguide mode, which
asymptotically approaches the surface plasma frequency ωsp [8, 92, 135], a typical
trend for any plasmonic mode. The modal wavenumber (Re(β)) reaches a maximum at ωsp and then returns towards the light line as losses increase and the mode
becomes leaky. This feature is observable in the mid-IR frequency range for all the
vdW heterostructures, with slightly different resonance frequencies ranging from 60
to 70 THz, while corresponding features for the noble metal waveguides will appear
above 500 THz in the UV-visible regime as discussed above. The larger in-plane
wavenumbers (Re(β)) of the modes in the vdW heterostructure-based waveguide
lead to shorter in-plane effective wavelengths and correspondingly higher mode confinement in the directions perpendicular to the waveguide’s interfaces. Furthermore,
the smaller imaginary parts (Im(β)) illustrate larger propagation distances, as dis-

114
cussed in Fig. 6.3b.
6.7

Conclusion

Using a combination of ab initio DFT methods and optical transfer matrix calculations we showed that the long electron relaxation times in graphene-based vdW
heterostructures lead to improved optical properties compared to noble metals, in
the mid-long-IR regime. In particular, we predicted an order of magnitude improvement in terms of their material performance as plasmonic metals. Furthermore, we
showed that graphene-based vdW heterostructures can surpass the reflectance limit
of Ag and Au [18].
This suggests the possibility of replacing current noble metal components in optoelectronic devices with vdW heterostructures, which can also be tuned in real time,
for example via external bias. Hence, our results hold promise for improved material
performance in active waveguiding systems, Salisbury screens for perfect absorption,
and engineering Purcell enhancements, among others. Furthermore, we highlight
that vdW heterostructures exhibit substantially reduced mass density compared to
noble metals, which constitutes them particularly relevant for aerospace applications, where mass density becomes an important figure of merit. The increased
carrier density in graphene-based vdW heterostructures required for unlocking their
low-loss plasmonic response in the mid-IR may be achieved by gating alternating
graphene layers separated by hBN. The doping of graphite, for example with Li
intercalation, is also an option for experimental realization of the concept proposed
here [200, 201].

115
Chapter 7

SUMMARY & OUTLOOK
“The science of today is the technology of tomorrow.”
Edward Teller
7.1

Summary

Research in artificial matter and light-matter interactions with metamaterials has
experienced rapid advancement within the last twenty years, mainly due to technological progress in nanofabrication. Three-dimensional laser lithography and electron beams allow one to write structures with precision in the nanometer scale.
Developed metamaterials have dramatically expanded the range of achievable electromagnetic properties and extraordinary values of dielectric permittivity and magnetic permeability have been reported, inaccessible with natural materials.
As miniaturization requirements increase, however, the structural complexity of
metamaterials must reduce. For this reason, within the last decade, research in
photonics has revisited an older problem; that of light propagation in planar, multilayer and unpatterned systems. It is noteworthy that phenomena initially thought
to be achievable only with complex three-dimensional arrangements are also supported, among other unprecedented effects, in planar metallodielectric multilayer
(hyperbolic) metamaterials, which is the subject of this thesis.
We started our investigation by developing an inverse problem solver; a parameter
retrieval approach that computes the macroscopic effective properties of any uniaxial metamaterial, and we studied planar metallodielectric multilayers; see Chapter
2 [47]. The purpose of introducing macroscopic effective parameters for metamaterials is to simplify the computation of wave propagation phenomena and to build
intuition for their principle of operation. Contrary to the long-standing perception
that in planar systems µ = 1, we found and experimentally confirmed in Chapter 3
that such systems support magnetic dipole moments and exhibit artificial magnetic
properties. We explicitly computed and verified experimentally extraordinary values of magnetic permeability ranging from positive, zero to negative values. These
findings simplify the ways we can harness optical artificial magnetism, using simple
thin-film deposition techniques without any lithography [12].

116
Until recently, most light-matter interactions were manipulated using the dielectric
properties of matter, rather than its magnetic properties that vanish at frequencies above the THz range. An important example is the surface plasmon polariton,
a surface wave that gave birth to the field of plasmonics, which occurs naturally
only for < 0 and for transverse magnetic polarization, due to lack of magnetic
properties in natural substances. We showed in Chapter 3 that, using artificially
magnetic multilayer metallodielectric metamaterials, magnetic plasmons can propagate simultaneously with typical SPPs, generalizing their properties for unpolarized
light.
We further extended and simplified these results by identifying material systems
where one obtains omnipolarization surface waves using high-permittivity materials, which are the best alternative to magnetic media that do not occur at high
frequencies, see Chapter 4. We investigated polar dielectrics near their phonon
energies at infrared frequencies and transition-metal dichalcogenides near their excitonic resonances at visible frequencies, in systems that support transverse electric
polarized waves simultaneously with their TE counterparts (i.e. hybrid plasmonic
waves).
Motivated by the possibility of creating actively tunable material systems, the response of which can be tuned dynamically, we transitioned in Chapter 5 to study active materials as components of planar metallodielectric metamaterials. We showed
that use of transparent conductive oxides in traditional field-effect metal/oxide/semiconductor
geometries can yield a tunable hyperbolic response at visible-near IR frequencies
[48]. Similar findings were reported with the use of graphene, a two-dimensional
semimetal that, when gated, alters its chemical potential and hence its optical response. As a component of a planar heterostructure, graphene can serve as a tunable
element, via external bias, for obtaining tunable metamaterial functionalities for frequencies ranging from the near-IR up to the microwave regime.
In Chapter 6 we studied heterostructures composed of purely two-dimensional materials. Motivated by the Dirac dispersion of electronic states in graphene that gives
practically zero effective mass, and by its very large electronic relaxation times compared to noble metals, we sought for graphene-based van der Waals heterostructures
that act as supermetals [18]. By comparing these material systems with noble metals, namely silver and gold, we obtain a 3- to 5-fold improvement in material quality
factors that may hold promise for future optical components with reflectivity that
surpasses the Ag and Au limit. The additional advantage of an order of magnitude
lower mass density of van der Waals heterostructures compared to noble metals,
these arrangements can be particularly useful for space exploration applications

117
and systems.
7.2

Outlook

Therefore, having developed knowledge in the following areas: (i) artificial magnetism in one-dimension, using planar structures, (ii) omnipolarization surface waves,
(iii) actively tunable response via external bias at visible and IR frequencies, and
(iv) modeling van der Waals heterostructures and inducing perfect electrical conductor response, we proceed by envisioning areas in which these ideas can be applied
and expanded. We further seek and comment on relevant applications where the
discussed concepts may be employed.
All-van der Waals perfect absorbers
Nanophotonic devices that emit radiation efficiently and selectively across the IR
range (700 nm-1 mm) hold promise for next generation energy technologies. Particularly, they allow electricity-free radiative cooling [204], the temperature reduction
of hot objects passively, through emission of electromagnetic radiation. This functionality is key for reducing energy consumption, for example lowering the required
cooling load in air-conditioned buildings, and consequently decreasing their sizable
negative impact on energy use. Furthermore, energy-harvesting devices, e.g. solar
cells, suffer from compromised performance when over-heated. Passive control of
their cooling will lower their power consumption and allow robust operation.
Kirchhoff’s law states that the far-field emission of a body equals its absorption.
Achieving perfect absorption in nanophotonics typically employs an absorbing multilayer system composed of different materials, terminated with a back reflector
able to redirect the otherwise-escaping radiation back into the structure [5]; see
Fig. 7.1a. Current state-of-the-art radiative cooling approaches use conventional
materials with individual layer thicknesses ranging between 13-80 nm [204]. For
the back-side reflector, typically silver and gold are utilized due to their PEC-like
response.
Based on our previous results on ultra-light perfect reflectors with van der Waals
graphene-based heterostructures (Chapter 6 and [18]), we envisage here all-van der
Waals-based perfect absorbers. Particularly, graphene-based heterostructures (including doped graphite and graphene/hexagonal boron nitride alternating monolayers) can comprise the back reflector with performance that surpasses Ag and
Au (see Fig. 6.2). Furthermore, previous results on other two-dimensional materials, for example black phosphorus, transition-metal dichalcogenides (molybdenum

118
(a) All-vdW perfect absorbers
700nm-2.5μm

(b) Artificial magnetism for controlling
light emission
magnetic dipole emitter (ion)

high-ε material
(TMDC, polar dielectric)

hybrid
vdW absorber

graphene/doped graphite

graphene-based
vdW back reflector

0.3-0.5nm

high-ε material

Memitter

Jd
-Jd

Meff

(d) Casimir forces in vdW hetero(c) Unpolarized plasmons for near-field
structures for nanoscale manipulation
heat transfer
HTE ETM
kTE

high-ε material
graphene/doped
graphite (ε<0)
high-ε material

kTM

high DOS
effective medium
(foams, aerosoles)

graphene-based
vdW reflectors

Figure 7.1: (a) All-van der Waals perfect absorbers, (b) Artificial magnetism
for controlling light emission, (c) Unpolarized plasmonics for near-field heat
transfer, (d) Casimir forces in van der Waals heterostructures.

and tungsten) [205] and others, suggest that the absorbing layer can also be completely replaced by two-dimensional materials, for realizing all-van der Waals-based
perfect absorbers. Replacing conventional absorbing materials with van der Waals
heterostructures for all of the structure’s components will dramatically decrease the
dimensions of perfect absorbing schemes down to the Angstrom scale, therefore allowing maximum light-matter interaction in minimum volume. Additionally, van
der Waals heterostructures have significantly reduced mass density compared to
conventional metals and dielectrics, therefore yielding ultra-light photonic devices.
See Fig. 7.1a.
Artificial magnetism for controlling light emission
Light emission from solid-state quantum emitters impacts a variety of applications
ranging from imaging fluorescence to performing simple quantum operations with
light-emitting particles. Controlling the environment of a quantum emitter allows

119
engineering its emission properties, such as directionality, lifetime and polarization.
Although light emission occurs due to both electric and magnetic dipoles, most
applications pertaining to luminescence and emission control are limited to electric
dipole transitions in ionic materials. This arises from the asymmetry in naturally
occurring substances that exhibit a wide range of dielectric properties, expressed
through the values of the permittivity values, , while at frequencies above the
THz range the magnetic permeability of most materials is strictly unity, µ = 1 (see
Chapter 3). Therefore, strong interactions between matter and emitted electric fields
can be achieved, whereas it is very challenging to observe strong optical interactions
with magnetic dipole emitters using conventional materials.
As discussed in Chapter 3, the absence of high-frequency natural magnetism has
motivated research in artificially magnetic metamaterials.

Split-ring resonators

and fishnet structures are currently the most widely reported magnetic elements
at frequencies beyond THz. However, the structural complexity of such motifs constitutes them challenging to realize experimentally at IR and optical frequencies
where light emission becomes relevant, requiring multiple lithographic steps with
nanoscale precision. By contrast, our results discussed in Chapter 3 suggest that
artificial magnetic response can be induced in planar multilayr arrangements, by
utilizing dielectric layers that support strong loop-like displacement currents and,
therefore, magnetic dipole moments [12].
We envision probing and enhancing light emission not only with electric dipoles but
also by utilizing the magnetic nature of light in experimentally reasonable, planar
geometries. Particularly, by engineering artificial magnetic dipoles to occur at frequencies where natural optical magnetic dipole transitions occur, for example in
lathanide ions such as trivalent erbium or europium [206], one can engineer the Purcell factor of magnetic dipole moments [207]. Furthermore, extending the concept
of artificial magnetism to two-dimensional materials and van der Waals heterostructures, one can envisage ultra-thin van der Waals heterostructures for controlling
light emission from electric and magnetic dipole emitters simultaneously, see Fig.
7.1b. Specifically, as discussed in Chapters 3 and 4, the key-requirement for inducing magnetic dipole moments in planar systems is very large permittivity . At
visible and IR frequencies there exists a plethora of such two-dimensional materials,
for example transition-metal dichalcogenides and polar dielectrics (c.f. hBN) near
their excitonic and phonon resonances, respectively. Utilizing such ultra-thin planar heterostructures for emission control can also be advantageous as the emitter
(ion)-heterostructure interaction occurs in a much smaller scale compared to bulky
conventional photonic multilayers, yielding improved emission control.

120
Unpolarized plasmonics for near-field heat transfer
A broad range of applications in medical technology, chemistry, energy, lasers, and
luminescence have advanced with the rise of plasmonics, which are surface-confined
electromagnetic modes at the interface between a metallic and a dielectric medium
(see Chapters 3, 4). Despite their well-known potential in achieving an extremely
high degree of confinement [92], plasmonics are limited to negative dielectric permittivity ( < 0) materials and to electromagnetic fields polarized with an electric
field normal to the direction of propagation, i.e. transverse magnetic fields. The
counterpart of a plasmon for transverse electric polarization, i.e. a magnetic plasmon, has not been reported due to the absence of naturally occurring materials with
negative magnetic permeability (µ < 0). Combining transverse magnetic and transverse electric surface waves in ultra-thin heterostructures will increase the impact
and relevance of plasmonics for technology. An important application is thermal
control, and particularly near-field heat transfer, a process enabling exchanges rates
beyond the Stefan-Boltzmann limit between objects separated by submicron distances, promising drastically improved throughput in thermophotovoltaics [208].
Heat transfer occurs via photon coupling between objects of different temperatures,
through plasmonic surface waves. The polarization dependence of surface plasmons
reduces their ability to tunnel photons to 50%, with the other half of the provided
thermal energy being wasted.
Utilizing our results in Chapter 3, suggesting that µ can be negative in metallodielectric heterostructures, and from our results in Chapter 4 where we explicitly
show transverse electric surface waves in materials with high-, we are motivated
to imagine applications of omnidirectional surface waves in near-field heat transfer
for energy applications. The relevant wavelength regime for solar applications is
from the visible spectrum to approximately 3 µm. In this regime, there exist a
large canvas of naturally occurring materials with high-, for example polar dielectric materials near their phonon energies that lie at IR frequencies. In the world
of two-dimensional materials, there are similar opportunities for magnetic surface
waves, using transition-metal dichalcogenides, with excitonic resonances that lie at
visible frequencies, and two-dimensional polar dielectric media, for example hexagonal boron nitride. Combining such materials with graphene [208], which supports
transverse magnetic polarized plasmons, we envisage phase-matched omnipolarization propagation of surface waves in van der Waals heterostructures. Achieving
excitation and detection of such waves can improve near-field heat transfer rates
and high-temperature (600-1200K) solid-state energy conversion, see Fig. 7.1c.

121
Casimir forces in graphene-based heterostructures for nanoscale
mechanical manipulation
Mechanical manipulation of nanoscale objects is key for nanomechanical devices
in sensing and biomedicine. Mechanical control at the single-atom-level can also
transform state-of-the-art fabrication techniques in two-dimensional materials to a
new era; particularly, although two-dimensional materials exfoliation and transferring have dramatically advanced, a challenge that impedes their integration in real
devices is graphene buckling, the phenomenon of a mono-layer of graphene experiencing uniaxial tension, leading to ripples in mono-layers that cannot remain truly
flat [198].
Inspired by the results of Chapter 6, and particularly the fact that graphene-based
van der Waals heterostructures perform as ultra-reflective metals in the mid-IR,
we envision their use in enhancing Casimir interactions. Practical issues in materials processing and nanofabrication, c.f. buckling, adhesion and friction, may
be resolved by actively controlling and mechanically manipulating one-atom-thick
two-dimensional materials using Casimir forces. The Casimir force between two
parallel uncharged conductors becomes dominant in the nanoscale, if the conductors exhibit a perfect electrical conductor response. Until now, noble metals have
been the most extensively investigated materials for Casimir effects. By contrast,
the Casimir energy between two graphene sheets is small and only measurable at
cryogenic temperatures [209]. However, conductors are heavy, which constitutes
them unsuitable for pursuing mechanical manipulation in the nanoscale, in contrast to graphene-based heterostructures (including graphite) that have an order of
magnitude lower mass density. Additionally, contrary to a single graphene sheets,
stacked arrangements of graphene mono-layers are ultra-reflective as we discussed in
Chapter 6. This constitutes them suitable candidates for Casimir interactions in the
nanoscale. We imagine configurations of parallel graphene-based heterostructures
for controlling their mutual Casimir interaction.
Apart from the material quality of the conductors, the magnitude of the Casimir
energy can be further controlled by engineering the electromagnetic environment
between the two conductors. Namely, the Casimir energy scales with the density
of optical states of the intermediate medium, which, in turn, scales with the refractive index. One may imagine using sparse arrays of high-refractive index foams or
aerosol-based effective media inside the cavity, for enhancing the Casimir interaction
as a result of (i) decreased weight load and (ii) increased number of available optical
modes, respectively, see Fig. 7.1d.

122
The aforementioned concepts and ideas are merely a small subset of the opportunities arising from our ability to design materials on-demand. It is fascinating that
today it is possible to selectively position two-dimensional mono-layers in configurations of stratified media with precision in the Angstrom-scale. This opens routes
for controlling the interaction between light and matter at the single-atom level,
where electronic and electromagnetic degrees of freedom compete. Complementary
to the classical notion of a metamaterial, the properties of which are determined by
its geometry, now it is possible to tailor light-matter interactions by simultaneously
engineering geometry (layer sequencing) as well as fundamental material properties
(electronic interactions between layers). The field of metamaterials is at a transition
point where novel physical phenomena are rapidly adopted into practical devices for
applications, and exciting times are ahead for a diverse array of technical disciplines
including mathematicians, physicists, material scientists, chemists, engineers, and
technologists.

123

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138
Appendix A

APPENDIX
A.1

Transfer Matrix Equations for an infinite bilayer heterostructure

For a detailed presentation of the transfer matrix approach, the reader in encouraged
to see [1]. In Section 2.3 we discussed the case of an unbound layered medium
composed of two materials with indices n1,2 (in the absence of magnetic properties
we have n1,2 = 1,2 ) and thicknesses l1,2 (see Fig. 2.2). We present below the
transfer matrix elements A, B, C, D, referring to Eqs. (2.17)-(2.18). For TE
polarization, we have:
1 kz,2 kz,1
ATE = e−ikz,1 l1 cos(kz,2 l2 ) − i
sin(kz,2 l2 )
2 kz,1 kz,2

1 kz,2 kz,1
BTE = eikz,1 l1 − i
sin(kz,2 l2 )
2 kz,1 kz,2

−ikz,1 l1

CTE = e

DTE = e

ikz,1 l1

1 kz,2 kz,1
sin(kz,2 l2 )
2 kz,1 kz,2

(A.1)

(A.2)

(A.3)

1 kz,2 kz,1
cos(kz,2 l2 ) + i
sin(kz,2 l2 )
2 kz,1 kz,2

(A.4)

Similarly, for TM polarization, the matrix elements are:
1 n2 kz,1 n21 kz,2
ATM = e−ikz,1 l1 cos(kz,2 l2 ) − i 22
sin(kz,2 l2 )
2 n1 kz,2 n22 kz,1

BTM = e

ikz,1 l1

CTM = e

DTM = e

ikz,1 l1

1 n2 kz,1 n21 kz,2
− i 22
sin(kz,2 l2 )
2 n1 kz,2 n22 kz,1

−ikz,1 l1

1 n22 kz,1 n21 kz,2
sin(kz,2 l2 )
2 n21 kz,2 n22 kz,1

(A.5)

(A.6)

1 n2 kz,1 n21 kz,2
cos(kz,2 l2 ) + i 22
sin(kz,2 l2 )
2 n1 kz,2 n22 kz,1

(A.7)

(A.8)

139
A.2

Counting optical states with the reflection pole method

In Section 3.8, and particularly in Fig. 3.11 we discussed the increase in the density of optical states (DOS) for multilayer configurations that, when homogenized,
exhibit the response o e < 0 (µo µe < 0) for TM (TE) polarization. Here, we compute the number of optical states for the example system discussed in Fig. 3.11.
We consider in-plane wavenumbers k// /ko ∈ [0, ndiel ] for radiation (bulk) modes,
while k// /ko > ndiel for surface waves, as discussed in detail in Section 4.3. Prior to
reading this section, the reader is encouraged to study the reflection pole method
(RPM) and its details (see Section 3.11 and [108]).
First, lets start with the physical picture of a multilayer system with complex reflection coefficients rTE , rTM , for the two linear polarizations. The analytical expression
of the density of states is [210, 211]

ρ(ω, d, ~k) =

3 k// 1 2
p (1+rTE e2ikz d )k12 +(1−rTM e2ikz d )kz2 +p2n (1+rTM e2ikz d )k//
2|p|2 k13 kz 2 p
(A.9)

where pp and pn are the parallel and perpendicular components of a dipole moment
located at distance d above the first planar interface, k1 is the wavenumber in the
incident medium and kz is the normal component of the wavenumer inside the
multilayer system. As explained in Section 3.8, the effective magnetic permeability
has an effect mainly in the TE polarization characteristics of planar metamaterials,
similar to the effective dielectric permittivity pertaining to TM properties. As can
be seen from the first term in Eq. A.9, in order for the TE polarization to contribute
to the density of optical states, the TE complex reflection coefficient must deviate
from the value of −1.
We perform analytical transfer matrix calculations and RPM calculations, together
with finite element simulations investigating the TE properties of a fifty-five layers
metamaterial consisting of silver and a dielectric material with refractive index ndiel ,
in the lossless limit, to simplify our analysis (similar to the one discussed in Figs.
3.11d, e). All finite element simulations were performed with the commercial package Comsol Multiphysics. We set the refractive index of the surrounding medium to
be nsur = 1.55, similar to Figs. 3.11d, e. Figs. A.1, A.2 show two cases: ndiel = 1.55
and ndiel = 4, respectively. The latter is the same as the case discussed in Fig. 3.11e.
In the case of ndiel = 1.55, no TE electromagnetic states couple to the multilayer
metamaterial and its response is reflective as shown in Fig. A.1a. This is confirmed
by analytical transfer matrix calculations of the TE reflectance (Fig. A.1d), which
is almost unity, while absorption is negligible (Fig. A.1d). Furthermore, as can be

140
seen in Fig. A.1c, Re(rTE ) barely deviates from the value of −1, indicating absence of TE polarization electromagnetic states, based on Eq. A.9. With use of the
arguments explained in Section 4.3 regarding the RPM, we present the parameter
dm11TE (λ)
dλ

in Figs. A.1f and g for radiation (bulk) and bounded (surface) modes,

respectively. No peaks are observed, confirming the absence of any TE states supported in the metamaterial in this frequency regime for the considered wavenumbers.
The wavenumbers accounted for in Figs. A.1f, g are chosen based on the Gaussian
beam in the finite element simulation shown in Fig. A.1a.
Furthermore, by homogenizing the layered metamaterial through [47], we obtain
negative dielectric permittivity o , while the magnetic permeabilities along both
coordinate directions µo and µe remain positive (Fig. A.1b). For these values of
o , µo and µe , the dispersion equation for bulk propagating modes (Eq. 3.3) has
no solutions, confirming further the absence of any optical states in this frequency
regime.
By contrast, increasing the dielectric index to ndiel = 4 leads to a drastically different
Re(rTE ) response, as shown in the simulation result of Supplementary Fig. A.2a.
The metamaterial absorbs most of the incident field, as confirmed through analytical
calculations showing enhanced TE absorption in Supplementary Fig. A.2e, while
the TE reflectance vanishes at resonant wavelengths (Fig. A.2d). As seen in Fig.
A.2c, now Re(rTE ) is drastically different from unity, indicating a TE contribution
to the density of states, based on Eq. (A.9). The density of states enhancement is
already obvious by the enhanced absorption, however we also show it explicitly in
Fig. A.2f; the reflection pole peaks represent states and their number is equal to the
number of bulk eigenmodes (or optical states) for the considered wavenumbers and
wavelengths. The number of TE bound (surface) states is also enhanced, as peaks
are also observed for wavenumbers k// /ko > nsur (Fig. A.2g). This response is also
interpreted in the effective homogeneous slab picture (with effective parameters o ,
µo and µe ); the TE response of the metamaterial is now hyperbolic as µo µe < 0
(Fig. A.2b). Similar to HMMs with o e < 0 supporting an enhanced density of TM
states [41, 76, 210, 211], we show an equivalent response for TE polarization due to
µo µe < 0.
A.3

Details on the electronic structure calculations in vdW heterostructures

Here we discuss the details for computing the electronic properties and sheet conductance of the graphene-based heterostructures discussed in Chapter 6. Both intraband and interband contributions to the sheet conductance discussed in Section
6.3 are extracted from the linearized Boltzmann equation using an ab initio collision

141
integral for the electron-phonon scattering processes [196]. The interband transitions term is evaluated using Fermi’s Golden rule for the imaginary part, and then
the Kramers-Kronig relation for the real part.
All these calculations use DFT predictions for the energies and matrix elements of
both the electrons and the phonons, hence automatically accounting for electronic
structure effects such as inter-layer interactions and response of electrons far from
the Dirac point, as well as scattering against both acoustic and optical phonons
including Umklapp and inter-valley processes. These calculations account for finite
temperature occupations of electrons and phonons, as well as for the coupling of
electrons with photons and phonons, thereby capturing all relevant radiative and
non-radiative processes that contribute to the optical response. However, in this
work, excitonic effects were neglected since the high free carrier density of the doped
materials we consider would strongly screen electron-hole interactions [212]. See
[196] for more details on the theoretical framework and references in [18].
For graphene and its heterostructures, changing the Fermi level EF changes the
equilibrium electron occupation factors in the Boltzmann equation as well as in
Fermi’s Golden rule. This affects both interband and intraband contributions to the
sheet conductance, and we account for this by explicitly evaluating them for several
different values of EF , ranging from the neutral (undoped) value to ∼ 1 eV above
it. This computational treatment assumes ideal doping and neglects losses due to
scattering by inactive dopants and defects introduced into the material; the effects
of such nonidealities will be specific to the experimental strategy for doping and are
not discussed here.

142
Re(Ey)
a.u.

-4

(b) 2.2

μ oμ e>0

-5
-5

Re(μo)
Re(μe)

1.8

x(μm)

-2.8

-2.85

-2.9

Re(rTE)

-2

Re(μ)

ETE

Re(εo)

z(μm)

(a) 4

-0.85
-0.9

-0.95

1.6
774

776

778

780

782

-2.95

-1

wavelength (nm)

(d)

-0.8

(e)

100

RTE %

99.996

0.01
0.008
0.006

99.994

0.004

99.992
99.99

wavelength (nm)

A TE / loss%

99.998

774 776 778 780 782

0.002

774 776 778 780 782

wavelength (nm)

774 776 778 780 782

wavelength (nm)

d(arg(m11TE (λ)))/d(λ)
a.u.

(f)

d(arg(m11TE (λ)))/d(λ)
a.u.

773

774

775

776

774

775

776

777

778

779

780

781

782

783

777

778

779

780

781

782

783

wavelength (nm)

(g)

773

wavelength (nm)

Figure A.1: (a) Simulation results for a fifty-five layers dielectric ndiel = 1.5:
50 nm/Ag: 20 nm multilayer metamaterial at 780 nm for TE polarization.
The surrounding medium has index nsur = 1.55. (b) Effective parameters for
the metamaterial in (a). (c) Re(rTE ) barely deviates from the value of −1. (d)
TE polarization reflectance RTE = |rTE |2 . (e) TE polarization absorption. (f)
reflection pole method for radiation (bulk) modes, (g) reflection pole method
for bound (surface) modes. Note 1: (c), (d), (e), (f), (g) are transfer matrix
analytical multilayer calculations for the multilayer described in (a). Note 2:
For (c), (d), (e), (f): Colors: from blue to red: k// from 1.45ko (blue) to 1.55ko
(red) - corresponding to the wavenumbers in the simulation of (a). Note 3:
For (g): Colors: from blue to red: k// from 1.55ko (blue) to 2ko (red).

143
(a)

ETE

z(μm)

-2

Re(Ey)
a.u.

μ oμ e<0

-4

(b) 0.4

Re(μo)
Re(μe)

(c)

Re(εo)

0.3

x(μm)

-0.2

Re(rTE)

-5

0.1

-0.8

-0.1

-10

-1

Re(μ)

0.2

774

776

778

780

782

wavelength (nm)

(d) 100

-0.6

(e)

774

776

778

780

782

774

776

778

780

782

wavelength (nm)

40
30

A TE / loss%

RTE %

-0.4

774

776

778

780

782

wavelength (nm)

d(arg(m11 TE (λ)))/d(λ)
a.u.

(f)

773

d(arg(m11 TE (λ)))/d(λ)
a.u.

(g)

773

774

775

776

774

775

776

777

778

779

780

781

782

783

777

778

779

780

781

782

783

wavelength (nm)

Figure A.2: (a) Simulation results for a fifty-five layers dielectric ndiel = 4: 50
nm/Ag: 20 nm multilayer metamaterial at 780 nm for TE polarization. The
surrounding medium has index nsur = 1.55. (b) Effective parameters for the
metamaterial in (a). (c) Re(rTE ) deviates from the value of −1 indicating
enhanced density of states. (d) TE polarization reflectance RTE = |rTE |2 . (e)
TE polarization absorption. (f) reflection pole method for radiation (bulk)
modes, (g) reflection pole method for bound (surface) modes. Note 1: (c),
(d), (e), (f), (g) are transfer matrix analytical multilayer calculations for the
multilayer described in (a). Note 2: For (c), (d), (e), (f): Colors: from blue to
red: k// from 1.45ko (blue) to 1.55ko (red) - corresponding to the wavenumbers
in the simulation of (a). Note 3: For (g): Colors: from blue to red: k// from
1.55ko (blue) to 2ko (red).