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Design of graphene-based plasmonic optoelectronic devices

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Design of graphene-based plasmonic optoelectronic devices
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Chorsi, Hamid Taghizadeh ( author )
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Denver, Colo.
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University of Colorado Denver
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Graphene ( lcsh )
Optoelectronic devices ( lcsh )
Graphene ( fast )
Optoelectronic devices ( fast )
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theses ( marcgt )
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Once reckoned unfeasible to exist in nature, graphene, the first two-dimensional material ever found, has rocketed to stardom after its first isolation in 2004 by Nobel Laureates Andre K. Geim and Konstantin Novoselov. Graphene is a single atomic layer of sp2 hybridized carbon atoms arranged in a flat hexagonal honeycomb lattice. The astonishing properties of graphene originate from its inimitable smooth-sided conical band structure that converges to a single Dirac point. Graphene can be deemed as a novel platform for optical and electronic devices with the extraordinary capabilities of dynamically manipulating electromagnetic waves. One of the most fascinating properties of graphene is that it is a zero-gap semiconductor (because of its unique band structure in which Dirac Fermions are charge carriers) with very high electrical conductivity. Graphene possesses the highest thermal conductivity and the highest current density at room temperature (300K) ever measured (a million times that of copper). Graphene's capability to absorb approximately 2.3% of white light is also a unique property, especially considering that it is only 1 atom thick. Unlike conventional materials, the saturable absorption response of graphene is wavelength independent from UV to IR, mid-IR and even to THz frequencies. Graphene is the best transparent conductive film, the thinnest material known, and the strongest material discovered (about two hundred times stronger than steel), yet is highly mechanically flexible. Recently, it has been revealed both theoretically and experimentally that graphene supports surface plasmons (SPs), collective oscillations of free electrons at the interface between two media with dielectric constants of opposite sign, such as a metal and a dielectric. Graphene can be viewed as a strong contender for conventional plasmonic material due to the tunability of its optical conductivity (electrical and chemical doping) as well as its extreme confinement strength in the mid-infrared, both properties of which are unprecedented in metal-based plasmonics. Graphene has also exhibited some superiorities to noble metals such as supporting both transverse electric (TE) and transverse magnetic (TM) SPs with lower loss. This thesis focuses on designing novel optoelectronic devices based on graphene plasmonics. A novel type of metasurfaces will be presented and optoelectronic devices will be constructed based on the graphene plasmonics. A graphene-based plasmonic nanoribbon filter and an optical beam splitter has been proposed based on the designed metasurface and the electromagnetics of the wave propagation are numerically analyzed. Besides electronics perspective and applications, a near field scanning probe based on graphene plasmonics has been designed in this work that can be used for a variety of applications ranging from bio-imaging to nanofabrication.
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by Hamid Taghizadeh Chorsi.

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Full Text
DESIGN OF GRAPHENE-BASED PLASMONIC OPTOELECTRONIC DEVICES
by
HAMID TAGHIZADEH CHORSI B.S., Urmia University of Technology, 2012
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering
2016


2016
HAMID CHORSI ALL RIGHTS RESERVED
ii


This thesis for the Master of Science degree by Hamid Taghizadeh Chorsi has been approved for the Electrical Engineering Program by
Stephen Gedney, Chair Mark Golkowski
Yiming Deng


Chorsi, Hamid Taghizadeh (M.S., Electrical Engineering)
Design of Graphene-based Plasmonic Optoelectronic Devices Thesis Directed by Professor Stephen D. Gedney
ABSTRACT
Once reckoned unfeasible to exist in nature, graphene, the first two-dimensional material ever found, has rocketed to stardom after its first isolation in 2004 by Nobel Laureates Andre K. Geim and Konstantin Novoselov. Graphene is a single atomic layer of sp2 hybridized carbon atoms arranged in a flat hexagonal honeycomb lattice. The astonishing properties of graphene originate from its inimitable smooth-sided conical band structure that converges to a single Dirac point. Graphene can be deemed as a novel platform for optical and electronic devices with the extraordinary capabilities of dynamically manipulating electromagnetic waves. One of the most fascinating properties of graphene is that it is a zero-gap semiconductor (because of its unique band structure in which Dirac Fermions are charge carriers) with very high electrical conductivity. Graphene possesses the highest thermal conductivity and the highest current density at room temperature (300K) ever measured (a million times that of copper). Graphenes capability to absorb approximately 2.3% of white light is also a unique property, especially considering that it is only 1 atom thick. Unlike conventional materials, the saturable absorption response of graphene is wavelength independent from UV to IR, mid-IR and even to THz frequencies. Graphene is the best transparent conductive film, the thinnest material known, and the strongest material discovered (about two hundred times stronger than steel), yet is highly mechanically flexible. Recently, it has been revealed both theoretically and experimentally that graphene supports surface plasmons (SPs), collective oscillations of free electrons at the interface between two media with dielectric constants of opposite sign, such as a metal and a dielectric. Graphene can be viewed as a strong contender for conventional plasmonic material due to the tunability of its optical conductivity (electrical and chemical doping) as well as its extreme confinement strength in the mid-infrared, both properties of which are unprecedented in metal-based plasmonics.
IV


Graphene has also exhibited some superiorities to noble metals such as supporting both transverse electric (TE) and transverse magnetic (TM) SPs with lower loss. This thesis focuses on designing novel optoelectronic devices based on graphene plasmonics. A novel type of metasurfaces will be presented and optoelectronic devices will be constructed based on the graphene plasmonics. A graphene-based plasmonic nanoribbon filter and an optical beam splitter has been proposed based on the designed metasurface and the electromagnetics of the wave propagation are numerically analyzed. Besides electronics perspective and applications, a near field scanning probe based on graphene plasmonics has been designed in this work that can be used for a variety of applications ranging from bio-imaging to nanofabrication.
The form and content of this abstract are approved. I recommend its publication.
Approved: Stephen D. Gedney
v


ACKNOWLDEDGMENTS
I would like to express my deepest gratitude to my supervisor Professor Stephen Gedney for his generous support, detailed guidance and constant encouragement throughout my graduate studies at University of Colorado Denver. He was patient with questions and always available to answer them in a professional manner. His help and guidance are greatly appreciated.
I am particularly grateful to Professor Mark Golkowski for his guidance and support. It is his trust and motivation from beginning to end that enables me to focus my mind on my thesis regardless of challenges.
Specially, I would like to thank my parents, my brother and my sisters for their support. Without their love, dedication and sacrifice, none of this work can be accomplished.
Finally, I would like to express my special thanks to Shenandoah for her love and all the experience we have gone through together.
VI


TABLE OF CONTENTS
Chapter
1. Introduction......................................................................14
1.1. Motivation........................................................................14
1.2. Plasmonics........................................................................15
1.2.1. Localized surface plasmon resonances..........................................16
1.2.2. Surface plasmon polaritons (SPPs).............................................17
1.2.3. Bulk plasmons.................................................................18
1.3 Graphene and Graphene Plasmonics...................................................18
1.4. Scope of the Thesis...............................................................22
2. Theoretical Foundations...........................................................23
2.1. Introduction......................................................................23
2.2. Macroscopic Electromagnetics......................................................23
2.2.1. Maxwells equations...........................................................23
2.2.2. Wave equation.................................................................24
2.2.3. Boundary conditions: waves at an interface....................................25
2.2.4. Fresnel reflection and transmission coefficients..............................26
2.3. Surface Waves Electromagnetics....................................................28
2.4. Surface Plasmons Electromagnetics.................................................31
2.5. Transmission Line Approximation (TLA).............................................34
2.6. Graphene Plasmonics: Conductivity.................................................35
vii


2.7. Finite Element Method
36
3. Fabrication Methods..............................................................37
3.2. Nanopatterns on Microscale Structures...........................................38
3.2.1. Photolithography............................................................38
3.2.2. Electron beam lithography (EBL).............................................39
3.2.3. Focused-ion beam lithography (FIB)..........................................39
3.2.4. Scanning probe lithography (SPL)............................................40
3.2.5. Nanoimprint lithography (NIL)...............................................42
3.2.6. Nanosphere lithography (NSL)................................................44
3.2.7. Colloidal self-assembly.....................................................45
3.3. Plasmonic Patterns on Microscale Structures.....................................45
4. Tunable Graphene-Plasmonic Metasurface Optoelectronics...........................47
4.1. Introduction....................................................................47
4.2. Graphene Surface Plasmon Polariton Propagation..................................47
4.3. Graphene-Plasmonic Tunable Metasurface..........................................50
4.3. Plasmonic Filter................................................................52
4.4. Plasmonic Beam Splitter.........................................................58
5. Graphene-Plasmonic Apertureless Scanning Probes..................................64
5.1. Introduction....................................................................64
5.2. Simulation Results and Analysis.................................................65
viii
6. Conclusion and Outlook
71


References....................................................................73
Appendix A....................................................................74
Appendix B....................................................................75
IX


LIST OF FIGURES
Figure
1.1. Schematic representation of chip-scale device technologies.................................16
1.2. Illustration of the dipole polarizability of a spherical metal nanoparticle under the influence
of a plane wave.......................................................................17
1.3. Propagating surface plasmon polaritons (SPPs)....................................17
1.4. C-C bonds that form the hexagonal lattice of graphene............................19
1.5. (Left) the standard honeycomb lattice of graphene with basic lattice vectors. (Right) the
reciprocal lattice and its Brillouin zone.............................................20
1.6. Energy band structure of graphene for the whole first Brillouin zone.............20
2.1. Boundary surface.................................................................26
2.2. Fresnel reflection and transmission coefficients (a) TE, perpendicular, a.k.a, s-polarized (b) TM, parallel, a.k.a, p-polarized, (s polarization (senkrecht, aka TE or horizontal) has an E field that is perpendicular to the plane of incidence, p polarization (parallel aka TM or vertical) has an E field that is parallel to the plane of incidence), (s) and (p) stand for the German words
senkrecht (perpendicular) and parallel (parallel)............................................27
2.3. Reflections of TE and TM modes for n = 1.50 (with Brewsters angle (rTM = 0))..........28
2.4. Schematic of a surface wave propagating along a single metal-dielectric interface......30
2.5. Real and imaginary parts of the dielectric constant for gold in visible range of the
wavelength according to the Drude model......................................................32
2.6. Dispersion relation of SPPs.............................................................34
2.7. Surface plasmon waveguide...............................................................34
2.8. Surface conductivity of graphene calculated based on Kubo formula (a) real, (b)
imaginary....................................................................................36
x


3.1. Top-down and bottom-up nanofabrication approaches..................................37
3.2. Three primary exposure methods in photolithography.................................38
3.3. Diagram of an EBL instrument.......................................................39
3.4. (Left) Scanning probe microscopy (SPM) (Right) Atomic force microscopy (AFM).......41
3.5. (a) near-field scanning optical microscope (NSOM) instrument.......................42
3.6. Thermal and UV NIL.................................................................43
3.7. (left) Nanosphere lithography (NSL) process and (right) nanosphere optical lithography
(NSOL)...................................................................................44
3.8. Schematic of the prism-coupled plasmonic nanolithography process...................46
3.9. Schematic of a single metallic grating lithography..................................46
4.1. Surface plasmon wave propagation on graphene nanoribbon, (left) undoped-doped-undoped (UDU), (right) doped-undoped-doped (DUD). Results are electric field in y direction,
|Ey|.....................................................................................48
4.2. Graphene plasmonic propagation analysis, (up) input power at different ports, (middle)
Absolute value of the power density at 1.53 pm, and (bottom) at 1.42 pm.................49
4.3. Transmission (T) and absorption (A) spectra for graphene SPPs on a nanoribbon......50
4.4. permittivity of graphene at THz frequency for different Fermi levels (left) real, (right)
imaginary................................................................................51
4.5. (left) schematic presentation of the proposed graphene metasurface with S=600 nm and
w=50 nm. (right) different combinations of the square in a 3*3 structure based on the voltage gating (Fermi level); light green squares are undoped, dark blue doped...................51
4.6. schematic presentation of the proposed filter based on the proposed graphene
metasurface..............................................................................52
4.7. Electric field distribution for the designed graphene plasmonic filter, first configuration.
215THz (1.38 pm), 205THz (1.46 pm).......................................................53
XI


4.8. transmission spectrum for the first configuration presented in Figure 4.7...................54
4.9. transmission spectrum for the second configuration presented in Figure 4.10.................54
4.10. Electric field distribution for the designed graphene plasmonic filter, second
configuration....................................................................................55
4.11. Transmission spectrum for the third configuration presented in Figure 4.12................56
4.12. Electric field distribution for the designed graphene plasmonic filter, third
configuration.....................................................................................57
4.13. Electric field distribution for the designed graphene plasmonic filter, third
configuration.....................................................................................58
4.14. Beam splitter at 210 THz (1.42 pm), figures show the amplitude of the power density (|P|),
the background is doped graphene (1eV). Ports are undoped (as shown in the insect, light green regions are undoped). GSPPs propagate on the undoped region which forms DUD waveguide.........................................................................................59
4.15. Beam splitter, electric field intensity in x direction at 210 THz (1.42 pm),...............60
4.16. Beam splitter, electric field intensity in y direction at 210 THz (1.42 pm),...............61
Figure 4.17. Beam splitter power density for each scenario, (orders are the same as Figure
4.14).............................................................................................62
4.18. Beam splitter, transmission spectrum for each scenario (orders are the same for all the figures in this section)..........................................................................63
5.1. Artistic representation of the NSOM tip coated with a layer of graphene...................66
5.2. Refractive index of graphene calculated based on the Kubo formula, (right) Real, (left)
Imaginary.........................................................................................65
5.3. magnetic field |Bz| (a) type A (just Si02), (b) type B (Si02 covered with low refractive index
dielectric).......................................................................................67
5.4. Electric field and the calculated FWHM for the designed probes............................68
xii


5.5. FWHM with respect to the distance, (top) conventional apertureless (middle) Type A, (bottom) Type B.......................................................................
70


1. Introduction
1.1. Motivation
The second half of the twentieth century was revolutionized by the invention of the first transistor in 1947 at Bell Laboratories. Since then, semiconductor research has made giant leaps forward. Todays state of the art microprocessors exploit ultrafast silicon based transistors with dimensions on the order of 20 nm, and the length of the channel in a field effect transistor (FET) is less than 17 nm. Currently, progress toward further miniaturization of transistors is hampered by the fundamental quantum mechanical phenomena that occur at this scale, mainly the nature of charge transport in semiconductor materials.
A semiconductor is a material that partially conducts current using electrons and holes as charge carriers. By downscaling the channel size in transistors, electrons and holes enter the realm of quantum mechanics and assume relativistic behavior. In this realm, charge transport is not diffusive anymore as in macroscopic and mesoscopic transport: it is ballistic. This type of transport gives rise to several kinds of quantum forces, such as the Casmir force, the Van der Waals force, thermal effects, etc., which hinder the process towards smaller and ultimately faster transistors. On the other hand, interparticle and phonon interactions and ohmic losses slow down the speed of propagation of electron transport inside a crystalline structure like silicon.
Photonics is the science of light, which according to special relativity travels with the highest attainable speed in the universe. Photonics seeks to generate, control, and detect light waves and photons, which are particles of light. The question that immediately arises is: can we utilize photons to carry information? This question emerged a few decades ago and was answered in the late twentieth century with the development of fiber optics for broadband communication.
What most limits the integration of photonics and semiconductor electronics is their corresponding sizes as shown in Figure 1.1. Semiconductor electronic circuits can be fabricated
14


at dimensions below 50 nm. On the other hand, the dimensions of photonic devices are on the order of micrometers at best.
Silicon photonics can be referred to as a branch of photonics that combines photonics with the silicon platform of current electronic devices. Unfortunately, silicon does not find the same niche in optoelectronics due to the indirect nature of its band gap. Present-day photonics relies on compound semiconductors and their alloys. The diffraction limit is an optical effect which encumbers the progress toward the miniaturization of photonic devices by preventing localization of electromagnetic waves into nanoscale regions, scales much smaller than the wavelength of light in the material. Surface plasmon photonics, or plasmonics can provide a solution to this dilemma because plasmonics has both the capacity of photonics and the miniaturization of electronics.
1.2. Plasmonics
Plasmonics is the study of the interaction between the electromagnetic field and the free electrons in a metal. Moreover, it is the study of light at the nanometer scale. Plasmonics is capable of providing unique advantages in all areas of science and technology where the manipulation of light at the nanoscale is a prominent ingredient [1],
Plasmonics can squeeze light into dimensions far beyond the diffraction limit by coupling the light with the surface collective oscillation of free electrons at the interface of a metal and a dielectric. Plasmonics can be considered as a promising candidate for high-speed and high-density integrated circuits. It bridges microscale photonics and nanoscale electronics and offers similar speed of photonic devices and similar dimension of electronic devices.
A plasmon is a quantum for the collective oscillation of free electrons, as a photon is a quantum for light. The different types of plasmons that can be excited in metallic objects depend on their dimensions. In large three-dimensional metal structures, volume plasmons can exist in the bulk of the metal. At the interface between metals and dielectrics, propagating surface
15


plasmon polaritons (SPPs) can be excited. Low-dimensional metal structures such as nanoparticles maintain a wide variety of localized surface plasmon resonances (LSPRs).
Figure 1.1 illustrates the role of plasmonics with respect to other chip-scale device technologies. It shows how semiconductor electronics, photonics and plasmonics occupy well-defined domains on a graph of the operating speed and device dimensions. The dashed lines indicate physical/practical limitations of different technologies; semiconductor electronics in electronic processors tend to be limited in speed by thermal and interconnect delay time issues to about 10 GHz. Photonics is limited in its critical dimensions by the fundamental laws of diffraction. Plasmonics can serve as an effective bridge between similar-speed photonics and similar-size nano-electronics.
A
PHz
THz
GHz
MHz kHz
lOnm lOOnm lum 10nm 100|im 1mm Device Dimensions
. Schematic representation of chip-scale device technologies.
1.2.1. Localized surface plasmon resonances
Localized surface plasmon resonances (LSPRs) are confined, non-propagating surface plasmon waves and can be considered as the simplest type of plasmon wave. The free electron cloud of the nanostructure can be resonantly excited by polarized electromagnetic fields due to enhanced polarizabilities of the particles at certain frequencies as shown in Figure 1.2. These
Plasmonics
Photonics
Semiconductor
Electronics
The Past
TD
0
0
Cl
CO
0
Cl
O
Figure 1.1
16


enhanced polarizabilities give rise to strongly enhanced near fields in the proximity of the metal
Figure 1.2. Illustration of the dipole polarizability of a spherical metal nanoparticle under the
influence of a plane wave.
1.2.2. Surface plasmon polaritons (SPPs)
The solutions to Maxwells equations at the interface between two media with opposite real permittivity (e.g., metal and dielectric at THz), are propagating waves known as surface plasmon polaritons (SPPs). These collective oscillations of the free electrons due to polarized EM light in the metal constitute dispersive longitudinal waves that propagate along the interface and decay exponentially into both media.
17


1.2.3. Bulk plasmons
Volume plasmons are the most fundamental and intrinsic type of plasmon resonance that can be supported by a metal. These resonances occur at the plasma frequency of metals, which are transparent to radiation with higher frequencies and non-transparent to radiation with lower frequencies.
1.3 Graphene and Graphene Plasmonics
Graphene is a planar sheet of sp2-hybridized carbon atoms arranged in a two-dimensional (2D) hexagonal lattice [2], It is a zero-gap semiconductor with unprecedented electrical, optical, thermal, and mechanical properties. Due to its unique characteristics, it will play a crucial role in future optoelectronic technologies. The inimitable properties of graphene are mainly due to the band structure of the material, which consists of two bands touching each other at two nodes. The electronic spectrum around these two nodes is linear and can be approximated by Dirac cones. Graphene can be gated or doped, such that the Fermi energy can be freely tuned. Calculations of many physical properties of graphene demand deep knowledge of graphenes band structure and electron dispersion in the entire Brillouin zone.
The hexagonal lattice of graphene is characterized by two types of C-C bonds (cj,k),
which are formed by the four valence orbitals of carbon atom (2s, 2px, 2py, 2pz) as shown in Figure 1.4.
18


Figure 1.4. C-C bonds that form the hexagonal lattice of graphene
In graphene, the 2s and 2p electrons hybridize to form three sp2 orbitals and one p orbital. The natural tendency (that is, minimization of energy) is for the sp2 hybridized orbitals to arrange themselves at 120 angles from each other in a plane orthogonal to the p orbital. Between two hybridized carbon atoms, overlapping sp2 orbitals form a a bond while p orbitals form a second tt bond, accounting for graphenes 2D honeycomb lattice structure.
Figure 1.5 shows the standard lattice of graphene along with the unit cell. One can note two inequivalent sublattices which are labeled by A and B, which connect all states in one triangular sublattice.
19


Figure 1.5. (Left) the standard honeycomb lattice of graphene with basic lattice vectors. (Right)
the reciprocal lattice and its Brillouin zone.
In the Bravais lattice the primitive lattice vectors are given by
au =-^(3,V3)
(1.1)
where a is the nearest-neighbor C-C spacing (1.42 A). The reciprocal lattice vectors have the form
bl2= (1,V3).
a
(1.2)
Each point in one sublattice is coupled by three nearest-neighbor vectors to the other sublattice,
*=-(i,V3), *=-(1,-73), * =*-i,o).
(1.3)
The eigenenergies of graphene can be modeled by considering the overlap of free electrons. Considering very strong periodic potential (each electron is almost bound to a minimum), the tight binding approximation exploits the system Hamiltonian and electron wavefunctions of the atoms to find the eigenenergies. Suppose that the ground state of an
20


electron moving in the potential U(r) of an isolated atom is ¥k(j) = XC//^r-r/)
, (1.4)
where j sums over all the atoms. Considering the function in the Bloch form (wavefunction for a particle in a periodically-repeating environment) for a crystal of N atoms [3],
¥,. (r) AT172 £ exp(ik- r )cp(v- r )
J
two inequivalent sublattices A and B in graphene, we can split (1.5) to
(1.5)
{k\ H\k) = YJcsYJCj
s-A s'=A
++o-+>i>'
ik-S;
J= 1
{s | H | s')
(1.6)
Equation (1.6) has been derived based on the three nearest neighbors. By minimizing the energy with respect to the coefficients c, and taking the Hamiltonian, we can calculate the ground state of an electron as
H k) = t X e** = telk>a [1 + 2 e3^fl/2cos(kya S / 2)]
Z=1
We can now find the two eigenenergies as a function of k, [3],
E(k) = t ^3 + 2 cos(>/3 kya) + 4 cos(^- kya) cos(~ kxa) These energies are shown as a function of k in Figure (1.6).
(1.7)
(1.8)
21


Figure 1.6. Energy band structure of graphene for the whole first Brillouin zone.
5
1.4. Scope of the Thesis
This thesis investigates the concept, analysis, and design of novel optoelectronic devices based on graphene surface plasmons (GSPs) for nano-electronic, optoelectronic, imaging, and bio-sensing applications. Chapter one provides a brief introduction and review of the fundamentals of plasmonics and graphene palsmonics. In chapter two we will develop the theoretical foundations of the electromagnetic theory of surface plasmons and wave propagation in graphene nanoribbon. Designing nano and plasmonic structures requires a deep understanding of the fabrication processes used in constructing plasmonic nano structures. In chapter three, a brief review of the most state of the art plasmonic fabrication methods is presented. Chapter four will present a new type of graphene-plasmonic metasurface. Novel optoelectronic devices such as filters, benders, couplers, and power splitters are designed based on graphene nano-structures (nanowires and nano-slabs). Chapter five will present a novel design concept for an apertureless plasmonic probe for near-field scanning optical
22


microscopy (NSOM) based on GSPs. This is particularly important for bio-sensing and imaging applications, such as molecular and cellular analyses. Chapter six concludes the thesis and provides a roadmap for future work.
2. Theoretical Foundations
2.1. Introduction
Plasmonics explores how electromagnetic waves can be confined over dimensions on the order of or smaller than the wavelength. It is based on the interaction between electromagnetic radiation and conduction electrons. This chapter summarizes the most important theories and phenomena that form the basis for the study of plasmonic waves. Starting with a brief review of Maxwells equations, the electromagnetics of surface, bulk, and localized plasmons will be reviewed. The transmission line approximation (TLA) of SPP propagation along a plasmonic waveguide is also reviewed.
2.2. Macroscopic Electromagnetics 2.2.1. Maxwells equations
The interaction of metals with electromagnetic fields can be completely understood in a classical macroscopic framework based on Maxwells equations. In macroscopic electromagnetics the singular nature of charges and their associated currents is avoided by considering charge densities p and current densities J. In differential form and in SI units Maxwells equations can be written as:
VxE(iV) = Ot (2.1)
VxH(r,0 = ^^ + J(r,0, ot (2.2)
V*D(r, t) = p(r, t), (2.3)
V*B(r, 0 = 0, (2.4)
23


These equations link the four macroscopic fields D (the electric flux density), E (the electric
field), H (the magnetic field), and B (the magnetic flux density) with the external charge p and current J densities. The electromagnetic properties of the medium are commonly discussed in terms of the macroscopic polarization P and magnetization M according to:
D(r, t) = s{) E(r, t) + P(r, t), (2 5)
H(r ,t) = //01 B(r ,t) M(r ,t), (2.6)
Where e0 and p0 are the permittivity and the permeability of a vacuum, respectively. Since in this thesis we will only treat nonmagnetic media, we need not consider a magnetic response represented by M, but can limit our description to electric polarization effects. For a linear, isotropic, and nonmagnetic media, the constitutive relations can be written as:
D(r,t) = s0srE(r,t), (2.7)
B(r,t) = ju0jur H(r,t), (2.8)
er is called the dielectric constant or relative permittivity and pr= 1 the relative permeability of a nonmagnetic medium. The last important constitutive linear relationship we need to mention is
J(r,r) = cr E(r,r) (2.9)
in which a is conductivity.
2.2.2. Wave equation
The properties of waves (such as sound waves, lightwaves and water waves) can be described using the wave equation. To derive the wave equation, we take the curl of the first equation:
Vx VxE(r,r) = -VxB(r,4 dt (2.10)
-V2E(r, <) = -/! |-(VxH(r,0), dt (2.11)
Using Amperes law:
24


(2.12)
-V2 E(r,t) = -M^r + J(r, t)\
V2 E(r, t) = [is
dt dt
d2 E(r,Q
dt2
+ J(r,/)),
(2.13)
For a source-free region, the wave equation can be written as:
V:
dr
(2.14)
This is actually three equations, the x-, v- and z- vector components for the E field vector. For a plane wave moving in the x-direction this reduces to
d2E(r,t) _ d2 E(r,r) A
2 ^ ,2 U?
(2.15)
dxz dt2
The monochromatic solution (a simple set of complex traveling wave solutions) to this wave equation has the form:
E(x,0 = E0e2(M, (2-16)
Re\t
j{cot-kx)\ -a.x
= e x cos{a>t-/3.x),
attenuation propagating
(2.17)
where a quantifies attenuation, quantifies propagation, co = kc where c J- is the speed
4

co
of light, and = vphase
is the phase velocity. The relation of the phase velocity vphase versus the
frequency / is known as the dispersion relation.
2.2.3. Boundary conditions: waves at an interface
Material boundary conditions express the relationships of the electromagnetic vector fields (E, D. H. B) at the interface separating two different materials.
25


medium (2)
Figure 2.1. Boundary surface.
nx(Ej-E2) = 0 (2.18)
/7x(H1-H2) = J,, (2.19)
h (//j Hj//-, H2) 0 (2.20)
h-(jUlEl-Ju2E2) = ps (2.21)
The first equation states that the tangential electric fields are continuous across the interface. The second equation states that the tangential magnetic fields are discontinuous at the same location by an amount equal to the impressed surface electric current density.
2.2.4. Fresnel reflection and transmission coefficients
In this section, the fraction of a light wave reflected and transmitted by a planar interface between two media with different refractive indices is presented. Applying the boundary conditions to a uniform plane wave incident on a single planar interface leads to the Fresnel reflection and transmission coefficients.
26


Figure 2.2. Fresnel reflection and transmission coefficients (a) TE, perpendicular, a.k.a, s-polarized (b) TM, parallel, a.k.a, p-polarized, (s polarization (senkrecht, aka TE or horizontal) has an E field that is perpendicular to the plane of incidence, p polarization (parallel aka TM or vertical) has an E field that is parallel to the plane of incidence), (s) and (p) stand for the German words senkrecht (perpendicular) and parallel (parallel).
For the TE case, the tangential electric field is continuous across the boundary,
Vi{y = 0,t) + Vr{y = 0,t) = Vt{y = 0,t). (2.22)
The tangential magnetic field is continuous,
Yii(y = 0,t)cos6j+Y[r(y = 0,t)cos6r =Yit(y = 0,t)cos6t. (2.23)
Using B = E/u where u = c/n (n refractive index) and considering only the amplitude of the
waves (E0) at the boundary, a reflection coefficient
r =r Eo, ", cos#, -nt cos0t (2.24)
E0; nj cos Oj + 77, cos 0t
And transmission coefficient
E0, 2n; cos 0; (2.25)
E(1; 77, COS 0,+77, COS 0,
27


is derived. For the TM case, Fresnel reflection and transmission coefficients can be easily derived
as
E0;. nt cos 0t nt cos 0t (2.26)
Eo, n, cos 0, + n, cos 0t
Eo/ 2/7. cos 6>. (2.27)
t0, ni cos 0t + n, cos 0t
A plot of Fresnel reflection and transmission coefficients for n= 1.50 is shown in Figure 2.3.
Figure 2.3. Reflections of TE and TM modes for n = 1.50 (with Brewsters angle (riM = 0)).
At some angle, known as the critical angle, light traveling from a higher refractive index
medium to a lower refractive index medium will be refracted at 90. When the angle of incidence
exceeds the critical angle, there is no refracted light. All the incident light is reflected back into
the medium. The critical angle of incidence can be obtained for two media by
. w, (2.28)
sin 0C=, nx>n^ .
wi
2.3. Surface Waves Electromagnetics
Many condensed matter properties are governed by surface properties. Very often,
surface waves, which are defined as waves propagating along the interface between two media
and existing in both of them, play a key role. The allure of surface waves stems from the
28


confinement of energy to the close vicinity of the interface of the two partnering materials. Any change in the composition of either partnering material in that vicinity could alter-even eliminatethe surface wave. Zenneck in 1907 envisioned that the planar interface between air and ground supports the propagation of radiofrequency waves. The idea was later extended by Sommerfeld, and that surface wave has since become known as the Zenneck wave. The idea of the Zenneck wave was reapplied in the mid-20th century to the visible portion of the electromagnetic spectrum at the interface of a noble metal and a dielectric material which, initiating the concept of the surface plasmon polariton (SPP) wave. During the last three decades, two other types of surface electromagnetic waves have been discovered. The Dyakonov surface wave, proposed in 1988, travels along the planar interface of an isotropic dielectric material and a uniaxial dielectric material. Unlike either the Zenneck wave or the SPP wave, which require that one of the two partnering materials forming the interface to have a relative permittivity with a negative real part, the Dyakonov wave is a result of the difference in the crystallographic symmetries of the two partnering dielectric materials. Another type of surface electromagnetic wave is called Dyakonov-Tamm waves. This type of wave requires one of the two partnering mediums to be periodically nonhomogeneous normal to the planar interface. SPP waves guided by the interface of noble metals and dielectric media is the main concentration of this work. The next section will review the electromagnetics of the SPP waves. By following [4]let us investigate a planar interface between a metal and a dielectric.
29


Dielectric (e2)
y Metal
(/)
Figure 2.4. Schematic of a surface wave propagating along a single metal-dielectric interface.
For the metal £1= em at z<0, and £2= £d for the dielectric at z > 0. The wave equation in Helmholtz form has to be solved separately in each region

co2. fE(r,*)'
I B(r,r) J
-0
(2.29)
In general, Maxwells equations allow two sets of solutions with different polarizations TM (or p-polarized) and TE (or s-polarized) modes. Surface plasmon waves do not support a TE mode since the width of the waveguide is much smaller than the exciting wavelength. Considering the following boundary conditions
E,,-E,:= 0, (2.30)
= 0, (2.31)
That is, the parallel field component is continuous, whereas the perpendicular component is discontinuous. Solving (2.31) along with Gauss law the following relations for the wave vector can be obtained.
-K
7=1,2
(2.32)
30


Equation (2.32) shows the dispersion relation between the wave vector components and the
angular frequency.
2.4. Surface Plasmons Electromagnetics
We start by deriving the dielectric constant of metals. One of the simplest but nevertheless valuable models to describe the response of a metallic particle exposed to an electromagnetic field is the Drude-Sommerfeld model:
where e and me are the charge and effective mass of the free electrons, and Eo and oo are the amplitude and frequency of the applied electric field. The damping term yd is proportional to
yd = vP /1, where vF is the Fermi velocity and / is the electron mean free path between
scattering events. Equation (2.33) can be solved by r(t) r^l01t, which leads to the well-known dielectric function of Drude form
here o)p is the plasma frequency, and describes the ionic background in the metal (usually 3.7 for silver). If co is larger than cd the corresponding refractive index is a real quantity; on the other hand, iftyis smaller than cop, the refractive index is imaginary since sm is negative.
(2.33)
(2.34)
31


Figure 2.5. Real and imaginary parts of the dielectric constant for gold in visible range of the
wavelength according to the Drude model.
As shown in Figure 2.4, the simplest geometry supporting a SPP wave is a single, planar interface between a metal at y < 0, with a complex dielectric constant^, in which the real part
is negative (metals at THz region have negative real permittivity as shown in Figure 2.5: this is a very critical criterion for SPP waves since in this situation the wave can actually penetrate inside the metal), and a dielectric at y > 0, with a positive dielectric constant. The TM mode (Hz, Ex, and Ey) solutions are considered. Propagating waves can be described asE(.r,y,z) = E(v)ei/& in which /? = kx is called the propagation constant of the traveling wave. Considering Maxwells curl equations and knowing the propagation along the x-direction (8/dx = if3) and uniformity in the z-direction (8/dz = 0) and(<9/<9r = -f)the system of governing equations for the TM can be written as
^ . 1 8H, (2.35)
E, = z--------
cos()s oy
E,^H, . <2'36>
C'J£ri£
32


and the wave equation for the Hz component as
^ + (kle-p2) Hz=0. dy (2.37)
For the half-space in Figure 2.4, the TM wave can be written for upper and lower halves as
H, (2.38)
E,(y) = -;AJ-^e"V*=' COC(] £2 (2.39)
E (y) = A2 & e^e-^y y ^-V-2 for y>0, and (2.40)
Hz(y) = Aj eipxehy (2.41)
Ex (y) = i Aj kl e,fSxehy (0C{! (2.42)
COS0£-j for y<0. Due to boundary conditions and the wave equation for Hy, we have (2.43)
A A ^2 82 K ~ P ~KS\ -^1 7 O O O K s, k;:=f-kis2 (2.44)
Finally, the dispersion relation of SPPs propagating at the interface between the two half Spaces can be obtained as
p-\Isa V^l+^2 (2.45)
Figure 2.6 shows plots of (2.45). It can be seen from Figure 2.6 that for a given frequency, a free-space photon has less momentum than an SPP since they dont intersect. On the other hand, coupling medium such as a prism can match the photon momentum. SPPs coupling with a prism light with relative permittivity of 1.5 with silver is presented in Figure 2.6.
33


Figure 2.6. Dispersion relation of SPPs.
2.5. Transmission Line Approximation (TLA)
Transmission line approximation (TLA) is a fast and reliable analytical approach used to investigate for optical applications.
y t
Metal
0 ^ (£ Dielectric
Metal

Figure 2.7. Surface plasmon waveguide
34


The waveguide of width h (Figure 2.7) can replaced by a transmission line of
characteristic impedance [5]:
Z(h) =
/?(h)h77
(2.46)
k0s
where p (h) is the SPP propagation constant at wavelength A,
m=Uz,-^e s"
(2.47)
k0hsni
2.6. Graphene Plasmonics: Conductivity
The propagation of SPPs on graphene is strongly dependent on the Fermi level Ep (chemical potential /uc). This can be seen from the frequency-dependent conductivity profile for different values of the Fermi level, which has been obtained from the Kubo formula [6]
graphene () intra () + Ginter ()
2iei 2T
7rh(co + iT )
Winter O) = TT [0( 2^c ) ^111
ln[2cosh(^)]
i ,(a + 2jucy
]
(2.48)
(2.49)
(2.50)
4h" c' 27t (co-iicf
in which e is the electron charge, T is the temperature, h is the reduced Planks constant, co is the radian frequency, r is the relaxation time between collisions with the impurity ions, phonons, etc., ar\d0{a)-2juc) denotes a step function. Figure 2.8. shows the calculated conductivity for different values of Fermi level from 20 THz to 900 THz.
35


Figure 2.8. Surface conductivity of graphene calculated based on Kubo formula (a) real, (b)
imaginary.
2.7. Finite Element Method
Partial differential equations (PDEs) appear in the mathematical modelling of many physical phenomena in electromagnetics. The complexity of boundary conditions renders finding their solutions by purely analytical means (e.g. by Laplace and Fourier transform methods, or Mie series) either impossible or impracticable, and one has to resort to seeking numerical approximations to the unknown analytical solution.
The Finite element method (FEM) is essentially a method for defining basis functions on mesh elements and using those basis functions to discretize a PDE. With FEM, the unknown solution is expanded as a linear combination of basis functions, as in the case with the method of moments (MoM).
FEM can be applied to a vast variety of PDEs, as well as integral equations (lEs). In all these cases, the basic outline of the FEM is as follows:
(a) Create a mesh by dividing the simulation domain into elements. For 2D problems (as in the case of most simulations in this thesis) the elements are typically triangular.
36


(b) Define basis functions on the mesh that can accurately interpolate the field, and its derivative.
(c) Exploit the functional for the PDE in order to compute a set of matrix elements for the basis functions on one canonical mesh element.
(d) Assemble the element matrix entities into a global Rayleigh-Ritz matrix for the entire mesh.
(e) Generate a right-hand side vector from the known sources.
(f) Solve the linear system for the basis function weights.
(g) Apply post-processing to derive physical quantities.
3. Fabrication Methods
3.1. Introduction
All of the grand ambitions of plasmonics are necessarily dependent upon feasible fabrication methods. There are innumerable nanofabrication techniques with different performances, choice of which depends upon the materials, applications, and geometries of the desired structure. These techniques can be broadly classified into either a top-down or bottom-up approach. Top-down fabrication refers to methods where one commences with macroscopically dimensioned material and carves the nanostructure out of the larger structure. On the other hand, in the bottom-up approach, assembly begins with smaller units: positions of atoms or molecules are manipulated to piece together the nanostructure. The top-down and bottom-up approaches are schematically shown in Figure 3.1.
Bulk Material
Single Atoms or Elements
Figure 3.1. Top-down and bottom-up nanofabrication approaches.
37


The topic of Nanofabrication is far too vast to be covered in one chapter. The goal of this section of the thesis is simply to introduce the method and review the substantial body of literature on plasmonic fabrication.
3.2. Nanopatterns on Microscale Structures 3.2.1. Photolithography
In optical lithography, a mask or reticle is imaged onto a substrate which is painted with a thin layer of photoresist, a photosensitive polymer material. Focused photon energy causes chain-scission or cross-linking in the polymer. The mask pattern is then delineated into the photoresist after a development process. There are three primary exposure methods: contact, proximity, and projection, shown in Figure 3.2.
Proximity Contact Projection
Figure 3.2. Three primary exposure methods in photolithography.
In contact lithography, the photomask is brought into physical contact with the wafer and then exposed to light. Contact lithography offers high resolution, but mask damage and a resultant low yield make this process impractical in most production environments. In proximity lithography, a gap is placed between mask and wafer in the range of 10 to 30 micro meters. Although proximity lithography does not suffer from mask damage as in contact printing, its low resolution makes it unsuitable for sub-100-nm fabrication. In projection lithography, the image is
38


projected onto the wafer with the help of a system of lenses. In this case, the mask can be used several times, substantially reducing the mask per wafer cost.
3.2.2. Electron beam lithography (EBL)
Electron beam lithography (EBL) has evolved from scanning electron microscope (SEM) in early the 1960s by the introduction of an electron-sensitive polymer material, called polymethyl-methacrylate (PMMA). Figure 3.4. shows the diagram of an EBL instrument.
An electron gun is a device that generates and projects a beam of electrons onto a substrate. Electrons are first generated by cathodes or electron emitters, then accelerated and focused by electrostatic fields to obtain higher kinetic energy and shaped into an energetic beam. Finally, the guidance system, consisting of the electric and magnetic focusing coils and deflecting system, transmits the beam to a work point on the substrate.
voltage source cathode
focusing coil J
deflection coil A
vacuum
r~ electron beam
work piece
Figure 3.3. Diagram of an EBL instrument.
3.2.3. Focused-ion beam lithography (FIB)
A technique related to electron lithography is focused ion-beam lithography, commonly called FIB. FIB is based on the use of accelerated ions instead of electrons. If the wavelength of accelerated ions can be similar to that of accelerated electrons; therefore an atomic resolution is expected in the ideal case. The major difference lies in the mass of the ions that allows very efficient momentum transfer and therefore physical etching of almost any kind of material.
39


FIB lithography is similar to EBL, but provides more functionalities. Not only can focused ions can create a pattern on a resist, similar to EBL, but they are capable of locally removing away some parts of the structure by sputtering (subtractive lithography: a hydrogen ion is 1840 times heavier than an electron). FIB is capable of accurately depositing atoms with sub-1 Onm resolution (additive lithography).
3.2.4. Scanning probe lithography (SPL)
For low-cost nanoscale patterning technologies, scanning probe lithography (SPL) is definitely an alternative to expensive photon or charged beam techniques. Another problem with lithography using either photons or charged beams is that they always rely on a polymer material (photo-resist or electron-resist) as an imaging layer.
SPL, however, can be implemented with diverse mechanisms, such as a direct-write approach. SPL uses a scanning probe microscope device (a sharp tip) in close proximity to a sample to pattern nanometer-scale features on the sample. A scanning probe microscope (SPM) is an instrument that monitors the local interaction between a sharp tip (less than 100 nm in radius) and the sample to acquire physical, electrical, or chemical information about the surface with high spatial resolution. Today there are many different types of SPMs used for diverse applications ranging from biological probing to material science to semiconductor metrology.
Three major technologies within the SPM family are scanning tunneling microscopy (STM), atomic force microscopy (AFM), and near-field scanning optical microscopy (NSOM). The first SPM was the STM invented in 1981 by Binnig and Rohrer. As shown in Figure 3.4 (a), STM uses a sharpened conducting tip with a bias voltage applied between the tip and the target sample. When the tip is within the atomic range (about 1 nm) of the sample, electrons from the sample begin to tunnel through the gap to the tip or vice versa, depending on the sign of the bias voltage. The exponential dependence of the distance between the tip and the target gives
40


STM its remarkable sensitivity with subangstrom precision vertically and subnanometer resolution laterally.
The primary limitation of STM is that it can only be used to image conducting substrates. The AFM was developed to assuage this constraint. The AFM based techniques are less restrictive than that of STM because AFM can be conducted in a normal room environment and can be used to image any kind of materials.
photodiode
Figure 3.4. (Left) Scanning probe microscopy (SPM) (Right) Atomic force microscopy (AFM).
In AFM lithography, the interaction potential between the atoms of the end of the tip and the atoms of the target surface causes a localized force. This force is measured by the deflection of a laser beam which is focused on top the mechanical cantilever on which the tip is attached.
The third major SPM, in addition to STM and AFM, is the near-field scanning optical microscope (NSOM). The main idea here is to utilize the perturbations of the evanescent waves in the near-field of the sample due to the interaction between the tip and the sample surface, and convert it into propagating light that can be detected via photodetectors.
41


Figure 3.5. (a) near-field scanning optical microscope (NSOM) instrument.
Two main types of NSOM probes are aperture type NSOM and apertureless techniques. In the first case, a subwavelength size aperture on a scanning tip is used as an optical probe. This is usually an opening in a metal coating of either an optical fiber tip or of a cantilever. Spatial resolution in the aperture type SNOM is, in general, determined by the aperture diameter. Apertureless techniques are based on the near-field optical phenomena as well, but do not require passing the light through an aperture. For SPL, the quality of the tip is defined by its crystallinity, surface roughness, and radius of curvature. High-resolution lithographic tools such as focused ion and electron beam have been used to mill, sculpt, and grow sharp tips for high-resolution imaging purposes.
3.2.5. Nanoimprint lithography (NIL)
Nanoimprint lithography (NIL) is one of the most promising low-cost, high-throughput technologies for nanostructure fabrication. Its principle component is a patterned mold or stamp that is pressed onto the surface of a polymer, transferring its pattern. In 1995 [7], NIL was proposed and demonstrated as a technology for sub-50 nm nanopatterning. Depending on the type of polymer used, NIL can be done via a thermal or UV curing. Figure 3.10 shows the fabrication procedure of the nanoimprint technologies.
42


Thermal NIL
UV NIL
^artzniold
Polymer M substrate
X /
align mold and substrate
press mold and expose to UV
remove mold
etch residential layer
Figure 3.6. Thermal and UV NIL.
In thermal NIL, first a thermoplastic polymer (PMMA) is applied to a silicon substrate, then the complex is heated to above glass transition temperature (Tg), in which the polymer becomes viscous liquid. Next, the mold is pressed onto the surface with a high pressure. After the mold cavities are filled with molten PMMA, the complex is cooled below the glass transition temperature and the mold peeled off from the PMMA surface. Finally, the residue PMMA on the compressed areas is removed by anisotropic etching.
In order to overcome difficulties associated with thermal NIL, such as alignment errors and the time-consuming process of thermal transition, in 1999 Colburn et. el proposed another nanoimprint method based on the UV curing process. A UV NIL mold must be transparent to UV light, and quartz is a popular choice. Initially, the UV curable polymer is dispensed onto the substrate. The quartz mold is pressed onto the polymer surface with low pressure, then the polymer is exposed to UV to cure and solidify. After curing, the mold is released from the
43


substrate. More details of the physics and the choice of the material and resist can be found in
171
3.2.6. Nanosphere lithography (NSL)
To enhance the consistency of particle size and arrangement, in 1995 Van Duyne and colleagues proposed a unique and simple fabrication method for metal nanoparticles called nanosphere lithography (NSL). NSL utilizes tightly packed polystyrene spheres on a substrate surface as a masking layer. The method is schematically shown in Figure 3.7.
Polystyrene
nanospheres
Polystyrene
nanospheres
I Sputtering of Ag

i
Removal of nanospheres
I Removal of nanospheres
Figure 3.7. (left) Nanosphere lithography (NSL) process and (right) nanosphere optical
lithography (NSOL).
The first step in NSL is dropping polystyrene nanospheres on a pristine, pre-prepared glass substrate. The hexagonally close-packed (Fischer pattern) nanospheres create a crystal structure in which the gaps between the spheres form a regular array of dots. Next, the array is filled in with thermally evaporated silver. After the deposition, the polystyrene spheres are removed by agitating (sonicating) the entire substrate in either H2CI2 acid or absolute ethanol, and the product is an array of triangular dots. As an example, Figure 3.7 shows the triangular
44


nanoparticle shape after deposition by NSL. Nanosphere optical lithography (NSOL) utilizes polystyrene or silica nanospheres on a substrate surface as a lens array. UV light is then used to pattern the photoresist using light spots under the nanospheres.
3.2.1. Colloidal self-assembly
Colloidal self-assembly refers to self-assembly of particles or spheres with diameter from micrometers to nanometers in a liquid suspension. Colloidal self-assembly can be used in two forms for nanofabrication of sub-100 nm and plasmonic structures. The first way is to fabricate the nanostructures inside the liquid, which we will refer to here as colloidal synthesis. The second method, colloidal lithography, uses the fabricated nanostructures via colloidal synthesis as masks for other fabrication techniques, such as photolithography or nanoimprint lithography. Colloidal synthesis has been widely used for the fabrication of plasmonic nanoparticles. Examples include nanospheres, nanostars, nanorods, and nanoporous structures.
3.3. Plasmonic Patterns on Microscale Structures
So far, we have reviewed several nanofabrication techniques for the fabrication of sub-100 nm structures and plasmonic devices. All the aforementioned techniques perform at the interface between Nano and Micro scale, hence the motivation of naming the previous section, Nanopatterns on microscale structures. The physics of surface plasmons, mentioned in chapter two, is extremely interesting and heralds enchanting applications for nanofabrication. Recently, SPPs have been used to fabricate nanostructures, especially for patterning nanoscale structures.
The idea of prism-coupled plasmonic nanolithography originated from the excitation of SPP at the interface of a metal and dielectric via a prism. The main idea here is the use of evanescent waves (from the interaction of light with metal mask) to pattern photoresist. Figure 3.8, which is characterized by the Kretschmann configuration, shows the physical arrangement.
45


Metal
Figure 3.8. Schematic of the prism-coupled plasmonic nanolithography process.
An isosceles triangle is placed at the uppermost layer in order to excite the SPPs. The bottom surface of the prism is coated with a thin metal (silver) film and then brought into intimate contact with a thin photoresist coated on a substrate. When two mutually coherent TM (p-polarized) plane waves are incident on the base of the prism in the vicinity of the resonance angle, multiple counterparts of the SPPs arise everywhere on the interface. As a result, SPP interference fringes are formed in the photoresist.
Grating-coupled plasmonic nanolithography uses metallic grating masks along with appropriate structures to excite SPPs and pattern nanoscale features. As distinct from a Kretschmann scheme, the mask grating based scheme is much more compact.
Glass
Figure 3.9. Schematic of a single metallic grating lithography.
46


The schematic of plasmonic lithography configuration using metal mask is shown in Figure 3.9. It consists of a metal mask, which can be fabricated on a thin quartz glass by electron-beam lithography and lift off process. The mask is brought into intimate contact with a photoresist coated on a silica substrate. Normally incident light tunnels through the mask via SPP and reradiates in to the photoresist.
4. Tunable Graphene-Plasmonic Metasurface Optoelectronics
4.1. Introduction
Recently, graphene plasmonics has attracted extensive interest in several areas of application. The design of optical and plasmonic circuit devices such as beam benders, optical couplers, filters, and power splitters is an indispensable step towards next-generation optoelectronic devices. This chapter aims to present some of these components in terms of graphene plasmonics. Specially, a novel tunable graphene metasurface is presented. That is used to design plasmonic and optoelectronic devices. The nature of the GSPPs propagation on nano-ribbons is considered at the beginning of the chapter in order to elucidate later results. Then, a tunable plasmonic filter and a controllable plasmonic beam splitter are designed based on the proposed graphene metasurface and the EM wave propagation is investigated in detail.
4.2. Graphene Surface Plasmon Polariton Propagation
Propagation of graphene SPPs can be categorized into two groups: undoped-doped-undoped (UDU) and doped-undoped-doped (DUD). The two paradigms are presented in Figure 4.1.
47


Figure 4.1. Surface plasmon wave propagation on graphene nanoribbon, (left) undoped-doped-undoped (UDU), (right) doped-undoped-doped (DUD). Results are
electric field in y direction, |Ey|
propagation as shown in Figure 4.1. The UDU configuration has a stronger edge mode propagation than DUD; on the other hand, DUD has a more confined waveguide mode propagation.
In order to analyze the wave propagation on top of a graphene ribbon, we have considered a DUD configuration. The width of the waveguide is 600 nm. The input power is shown in Figure 4.2. Maximum power (7.07*108) is at a wavelength of 1.53 pm. The absolute value of the power density (|P|) at 1.53 pm and 1.42 pm is also shown Figure 4.2. The results show that after propagating 6 pm, power at port 2 is 44.8% of the input power (P1) at 1.53 pm and it is 12.2% of the input power (P1) at port 3 with respect to P1.
48


| P | at 1.53 pm
PI P2 P3
| P | at 1.42 pm
PI P2 P3
Figure 4.2. Graphene plasmonic propagation analysis, (up) input power at different ports, (middle) Absolute value of the power density at 1.53 pm, and (bottom) at 1.42
pm.
In order to understand the decay rate, the transmission and absorption spectra are calculated at P1 and P2 with respect to P1 and P2 and P3 with respect to each other based on the power ratio of the output to the input, T = Pout / Pjn.
49


Transmission
Figure 4.3. Transmission (T) and absorption (A) spectra for graphene SPPs on a
nanoribbon.
The main conclusion of this section is that power loss is a major characteristic of plasmonic structures. It should be also mentioned that in case of GSPPs, this loss is much less than that of metallic loss.
4.3. Graphene-Plasmonic Tunable Metasurface
Metasurfaces are 2D surfaces composed of subwavelength optical elements (such as nanoantennas) capable of manipulating light by changing the optical properties of the incident electromagnetic wave. Due to the surface nature of the SPPs, which occurs at the interface between two media with opposite-sign real permittivity, GSPPs can only occur at frequencies at which the permittivity of graphene is negative. The relation between permittivity of graphene and frequency for different Fermi levels is presented in Figure 4.4.
50


Figure 4.4. permittivity of graphene at THz frequency for different Fermi levels (left) real,
(right) imaginary.
The structure of the proposed graphene metasurface is a 3x3 grid of graphene surfaces as presented in Figure 4.5. The side of each square and the gap size are S=600 nm and w=50 nm, respectively. Each square (referred to as a switch) can be turned ON and OFF based on the Fermi level (chemical potential) and the background medium. That is, if the background is
doped, doping a square will close the switch (OFF) and undoping will open the switch (ON).
w
>*5*8*5* *2*2*2* !| £*S*2*i |2*2*2* *2*2*2*! *2*2*2* *2*2*2*! *2*2*2* i *2*2*2*! *2*2*2* 1 9m9m9m*. \m9m9m9m
!**2*2* I *2*2*2*! **2*2*2 Vg i*2*2*2 !*2*S*2< 2*2*2*2
m
k Si02 A
Si
Figure 4.5. (left) schematic presentation of the proposed graphene metasurface with S=600 nm
and w=50 nm. (right) different combinations of the square in a 3*3 structure based on the voltage
gating (Fermi level); light green squares are undoped, dark blue doped.
51


4.3. Plasmonic Filter
Filters of electromagnetic frequencies are of fundamental importance in the integrated signal processing circuit. Traditional metallic plasmonic filters exhibit good performance in the visible and near-infrared region. However, poor confinement of SPPs in the THz region limits their applications. GSPPs have high field confinement at THz, and thus graphene-based filters promise better performance. Furthermore, the tunability of graphene allows one to adjust the coupling parameters dynamically. In this section, the proposed graphene metasurface is used to design tunable plasmonic filters. The structure of the proposed filter is presented in Figure 4.6.
Figure 4.6. Schematic presentation of the proposed filter based on the proposed
graphene metasurface.
In Figure 4.6, Un = 4 pm and Lout= 3 pm. The three middle squares are undoped graphene (since we want the wave to propagate in the waveguide). Doping (voltage gating) any of the middle squares will close the waveguide. From this perspective, the simplest optical device that can be realized with the proposed metasurface is an optical switch that turns ON and OFF the nanoribbon.
The other squares in the upper layer and lower layers will be turned ON and OFF based on the applied voltage, to meet the requirements of the filter. The electric field of the first configuration that we have considered is presented in Figure 4.7 at 1.38 pm (mode 2) and 1.46 pm based on the transmission spectrum, (Model=1.55 pm).
52


Figure 4.7. Electric field distribution for the designed graphene plasmonic filter, first configuration. 215THz (1.38 pm), 205THz (1.46 pm).
53


Figure 4.8. transmission spectrum for the first configuration presented in Figure 4.7.
In the first configuration, only two squares are turned ON the (the top middle and bottom middle ones). As mentioned before, undoping a square in a doped background turns it ON. From Figure 4.8 it can be seen that the filter is a band-stop filter for mode 2=1.38 pm and mode 1=1.55 pm (or band pass at 1.46 pm). The second configuration can be easily realized by changing the Fermi level of the metasurface. In the second configuration we will turn on (undope) the side squares and will turn off the middle ones as shown in the Figure. Mode 2=1.38 pm and mode 1=1.55 pm. Middle is 1.48 pm.
Figure 4.9. transmission spectrum for the second configuration presented in Figure 4.10.
54


Figure 4.10. Electric field distribution for the designed graphene plasmonic filter, second configuration, 215THz (1.38 pm), 202THz (1.48 pm)
55


In the last configuration, all upper row and lower row squares are turned on. The transmission spectrum is presented in the Figure 4.11. Comparing Figures 4.8, 4.9, and 4.11, it can be seen that the type of filter has been changed from a band-stop filter with two dips to a high pass filter at the same operating frequency. This indeed shows the tunability and flexibility
of the proposed graphene based filter based on the proposed metasurface.
1
0.8
0
"to 0.6 co
E
CO
1 0.4
0.2
0
1.1 1.2 1.3 1.4 1.5 1.6 1.7
A(//m)
Figure 4.11. Transmission spectrum for the third configuration presented in Figure 4.12.
Electric field results in both x and v directions are presented for the third configuration at 1.51 pm (197 THz) and 1.24 pm (241 THz) in Figure 4.12.
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Figure 4.12. Electric field distribution for the designed graphene plasmonic filter, third
configuration.
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4.4. Plasmonic Beam Splitter
In order to demonstrate the applicability and versatility of the proposed metasurface, a tunable plasmonic beam splitter is designed and analyzed in this section. A doped-undoped-doped (DUD) configuration has been considered. The background medium is kept at 1eV chemical potential (which corresponds to carrier density of ns = 7.3274x1013c/?r2) and the
waveguides (buses or ports) are kept at undoped levels. The doping levels of the square metasurfaces is modified based on the desired output. In Figure 4.13 the proposed graphene metasurface is placed between four ports to be used as a power splitter.
Figure 4.13. Electric field distribution for the designed graphene plasmonic filter, third
configuration.
In Figure 4.13, gray areas are PML and Lp=4 pm, which is the same for port 2 and 3. The goal here is to design a beam splitter which is capable of changing its filtering properties
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based on the proposed metasurface and the tunability of graphene. Figure 4.14 presents the
power spectrum for the designed beam splitter.
Figure 4.14. Beam splitter at 210 THz (1.42 pm), figures show the amplitude of the power density (|P|), the background is doped graphene (1eV). Ports are undoped (as shown in the insect, light green regions are undoped). GSPPs propagate on the undoped region which forms
DUD waveguide.
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Figure 4.15. Beam splitter, electric field intensity in x direction at 210 THz (1.42 pm),
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Figure 4.16. Beam splitter, electric field intensity in y direction at 210 THz (1.42 pm),
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From the power density and electric field plots, the behavior of the structure as a beam splitter can be observed. Figure 4.17 shows the power density plots at ports 1,2 and 3 for each configuration.
A(/im) A(//m)
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Figure 4.18. Beam splitter, transmission spectrum for each scenario (orders are the same
for all the figures in this section).
From the results it can be clearly seen that the structure has a dynamic behavior and can be tuned based on the applied voltage, which changes the doping level of graphene. This is an obvious advantage of the proposed graphene metasurface compared to metallic plasmonic based structures, which are not tunable and cannot be changed after fabrication.
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5. Graphene-Plasmonic Apertureless Scanning Probes
5.1. Introduction
The resolution of conventional optical microscopy is governed by Rayleigh diffraction, which states that the resolution of an optical system depends on the wavelength and the numerical aperture of the lens. One important conclusion is that light will diffract when it propagates through a hole or a slit which is smaller than approximately half of its wavelength:
A A (5.1)
d =-------=------
InsmQ 2NA
where n is the refractive index and 8 is the angle of incidence.
Various endeavors have been pursued to overcome this fundamental optical limitation; in particular, near-field scanning optical microscopy (NSOM) shows great promise in overcoming the diffraction limit, by capturing the evanescent portion of the spatial spectrum of an image, before it rapidly decays away with distance from the object. The primary challenge for NSOM is providing highly localized electromagnetic energy near the tip of the scanning probe.
In this chapter, a novel approach towards designing high-throughput apertureless NSOM tips based on graphene-plasmonics is presented. Specifically, localized graphene plasmon waves are combined with nanofocusing of surface plasmon polaritons (SPPs) to squeeze the lateral surface plasmon waves into the apex of the tip. The near-field electromagnetic properties of the designed probes are characterized in detail and compared to the conventional apertureless and metallic-plasmonic probes. Results show the applicability and versatility of the graphene-plasmonic probe in engineering near-field scanning probes. The designed structure can have many applications in different areas of science, including nanosensing, light sources, optical imaging, quantum optics, and tip-enhanced Raman spectroscopy.
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5.2. Simulation Results and Analysis
Figure 5.1. Artistic representation of the NSOM tip coated with a layer of graphene.
In this thesis two NSOM probes are considered. Structure one (type A) consists of a Si02 tip with permittivity Ssi02= 2.1638 at 960 nm and a layer of graphene on the lateral surface. The second structure (type B) includes an extra layer of low refractive index dielectric on the lateral surface to provide stronger confinement.
The key factor for graphene-plasmonic excitation at this wavelength of operation (960 nm) is high doping (Fermi) level and appropriate angle of excitation. Optimal design parameters including optical and electrical properties of graphene and dimensions are obtained with a finite element method (FEM) solver.
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Figure 5.2 shows the calculated refractive index of graphene for different values of Fermi levels at the frequency range of 20 THz to 900 THz. Here, Graphene is assumed to have 1 nm thickness and is doped with Fermi level of 1 eV.
Figure 5.2. Refractive index of graphene calculated based on the Kubo formula, (right) Real,
(left) Imaginary.
In order to excite surface plasmon waves and focus them towards the probe tip, the lateral surface is excited with 960 nm light source. Figure 5.3 shows the magnetic field on the surface of the tip for type the A and type B probes.
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Figure 5.3. magnetic field |Bz| (a) type A (just Si02), (b) type B (Si02 covered with low refractive
index dielectric).
The surface plasmon waves are localized at the apex of the NSOM tip in both cases. Although type B has stronger confinement, it requires an extra dielectric layer that is 30 nm-thick dielectric on top of the probe. The electric field plots are presented in Figure 5.4. The electric energy density is calculated at a 20 nm distance from the tip. The designed graphene-plasmonic tips provide a higher electric field density near the tip. The electrical energy density of type B is the highest and has been augmented more than 106 times compared to an apertureless probe. For type B, the amplitude is one-tenth that of type A, but it exhibits lower full width-half maximum (FWHM), meaning a confinement of the near field to a smaller region.
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Figure 5.4. Electric field and the calculated FWHM for the designed probes.
The FWHM levels for a conventional apertureless tip, tip type A, and tip type B probes are calculated and presented in Figure 5.4. The calculated FWHMs of the three probes are: 63.64 nm (conventional apertureless probe), 27.88 nm (type A) and 24.97 nm (type B).
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In Fig. 5.5, the calculated FWHM of each probe corresponding to various tip-sample distances is shown. Within 10 and 30 nm from the probe tip, the FWHM of probe type A and B is less than 60 nm, and the two designed probes still hold a large energy density enhancement (over six orders of magnitude compared to the simple aperture probe).
69


Figure 5.5. FWHM with respect to the distance, (top) conventional apertureless (middle)
Type A, (bottom) Type B.
From Figure 5.5 and based on the calculated FWHM values, it can be inferred that Type B possesses the lowest FWHM, and it is thus capable of concentrating the electric field into smaller and more compact area. On the other hand, by increasing the distance from the tip, Type B loses its advantages. At 30 nm from the tip, Type A has lower FWHM. In most
70


NSOM devices the distance is usually less than 20 nm, so it may be concluded that Type B has better performance in terms of the power concentration per area.
In conclusion, we have shown that graphene surface plasmons provide excellent near-field enhancement for a NSOM tip compared to a conventional apertureless probe tip [8], Compared to other reported NSOM tips designed using metallic plasmonics, the proposed tip has smaller FWHM and can be used for various biological, imaging, and fabrication applications.
6. Conclusion and Outlook
In this thesis, the fundamental physics and theory of surface plasmon waves and graphene plasmonics have been reviewed and novel tunable optoelectronic devices were designed. A graphene-based plasmonic nanoribbon filter and an optical beam splitter were proposed based on the designed metasurface and the electromagnetics of the wave propagation were numerically analyzed. The filtering characteristics of the proposed filter were investigated according to different parameters such as the nanoribbon width, working frequency, and chemical potential. Besides the electronics properties of the graphene plasmonics, bio and nano-imaging aspect of it were also investigated in this work. An efficient and high throughput microscopy probe was proposed which can be used for a variety of applications.
For future work, it is desired to extend the fundamental knowledge about the solid state physics and graphene physics. Other applications of the proposed graphene metasurface have already been investigated. The proposed structure shows great performance when it is used as an optical coupler. Other applications need to be investigated. Supplementary numerical methods such as finite difference time domain can also be exploited to verify the results.
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The next step for the graphene-plasmonic microscopy tip, is to extend analysis to a 3D domain and simulate the performance of the tip and ultimately fabricate it and verify the physical performance.
72


REFERENCES
[1] H. T. Chorsi and S. D. Gedney, "Efficient high-order analysis of bowtie nanoantennas using the locally corrected Nyström method," Optics Express, vol. 23, pp. 31452-31459, 2015/11/30 2015.
[2] A. N. Grigorenko, M. Polini, and K. S. Novoselov, "Graphene plasmonics," Nat Photon, vol. 6, pp. 749-758, 11//print 2012.
[3] C. Kittel and D. F. Holcomb, "Introduction to Solid State Physics," American Journal of Physics, vol. 35, pp. 547-548, 1967.
[4] S. A. Maier, Plasmonics: fundamentals and applications'. Springer Science & Business Media, 2007.
[5] X. Li, J. Song, and J. X. J. Zhang, "Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub," Optics Communications, vol. 361, pp. 130-137, 2/15/2016.
[6] K. Ziegler, "Minimal conductivity of graphene: Nonuniversal values from the Kubo formula," Physical Review B, vol. 75, p. 233407, 06/15/2007.
[7] S. Y. Chou, P. R. Krauss, and P. J. Renstrom, "Nanoimprint lithography," Journal of Vacuum Science & Technology B, vol. 14, pp. 4129-4133, 1996.
[8] Y. Lee, A. Alu, and J. X. J. Zhang, "Efficient apertureless scanning probes using patterned plasmonic surfaces," Optics Express, vol. 19, pp. 25990-25999, 2011/12/19 2011.
73


APPENDIX A
MATLAB Script: Graphene Band Structure
clear all; close all; t=l ;
lattice=l.446; eps_enl=3; eps_en2=3;
K_vec_x=linspace(-2*pi/(lattice),2*pi/(lattice),100); K_vec_y=linspace(-2*pi/lattice,2*pi/lattice,100);
[K_meshx,K_meshy]=meshgrid(K_vec_x,K_vec_y);
energy_mesh=NaN([size(K_meshx,1),size(K_meshx,2),2]); for a=l: size(K_meshx,1)
energy_mesh(:,a,1)=(eps_enl+eps_en2)/2+sqrt(((eps_enl-eps_en2)A2)/4 + 4*tA2*( (cos(K_meshy(:,a)/2*lattice) .A2) + .
cos(sqrt(3)/2*K_meshx(:,a)*lattice).*cos(K_meshy(:,a)/2*lattic e) ) +1/4) ;
energy_mesh(:,a,2)=(eps_enl+eps_en2)/2-sqrt(((eps_enl-eps_en2)A2)/4 + 4*tA2*( (cos(K_meshy(:,a)/2*lattice) .A2) + .
cos(sqrt(3)/2*K_meshx(:,a)*lattice).*cos(K_meshy(:,a)/2*lattic
e) ) +1/4) ;
end
% set(handles.axes_mesh,'Color','w')
surf(K_meshx,K_meshy,real(energy_mesh(:,:,1)))
hold on
surf(K_meshx,K_meshy,real(energy_mesh(:,:,2))) shading interp
colormap (jet)
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APPENDIX B
MATLAB Script: Kubo Conductivity Formula
% Calculation of Conductivity of Graphene using Kubo Formula % clc
clear all; close all
j = sqrt(-1); e = 1.6e-19;
K_B = 1.3806503e-23; %Boltzmann constant T = 300; %Room temprature
hb = ( 6.626e-34)/(2*pi) ; % Dirac constant tau = O.le-12; %momentum relaxation time ga = 1/(2*tau); %scattering rate % gamma = 2*ta; % Gusynin J. Phys.: Cond. Mat. sigma_min = 6.085e-5;
vf = 10A6; %Fermi velocity engineering in graphene by
substrate modification Choongyu Hwang
w = pi*2el2*linspace(5,400,200);
m = length(w);
mu_c = 1.6e-19*[-0.5];
n = length(mu_c);
mu_ct = repmat(mu_c,m,1);
wt = repmat(transpose(w),1,n);
% Intraband term calculations sigma_d_intra = ...
-j*((eA2*K_B*T)./(pi*hbA2*(wt-j*2*ga))).*((mu_ct)/(K_B*T)... +2*log(exp(-mu_ct/(K_B*T))+1)); %intraband term
% Interband term calculations sigma_d_inter = zeros(m,n); eps = 1.6e-l9*linspace ( 0, 10, 600000) ; q = length(eps);
for i = 1:n muc = mu_c(i);
f_d_meps = 1./ (1 + exp( (-eps-muc)/(K_B*T) ) ) ; f_d_peps = 1./(1+exp((eps-muc)/(K_B*T))); for k = 1:m
sigma_d_inter(k,i) = ...
trapz(eps,-(j*eA2*(w(k)-j*2*ga)/(pi*hbA2))...
*(f_d_meps-f_d_peps) ./((w(k)-j*2*ga)A2-4*(eps/hb) .A2));
75


end
end
sigma_tot = sigma_d_inter+sigma_d_intra; % total conductivity
C = {Tkf, 'r', 'b',TcT, 'g', 'm', 'yT}7
pi = zeros(1,n);
ppl = zeros(1,n);
si = cell(1, n) ;
p2 = zeros(1,n);
pp2 = zeros(1,n);
s2 = cell (1, n) ;
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
figure(1)
% subplot(2,1,1) for i = 1:n hold on pi(i) = ...
plot(le-12*w/2/pi,(transpose(real((sigma_d_inter(:,i)))))/... (eA2/hb/4),'Color', C[2],'Linewidth',2);hold on ppl(i) = ...
plot(le-12*w/2/pi,(transpose(real((sigma_d_intra(:,i)))))/... (eA2/hb/4),'Color', ...
C [2],'Linewidth',2);hold on
s1[2] = sprintf('mu {c,%d} = %d meV',i,le3*mu_c(i)/(1.6e-l9)) end
ind = 1:n;
% axis square box on
xlabel('f (THZ)','fontsize',20,'fontweight','b'); ylabel('Re(\sigma)','fontsize',20,'fontweight','b'); legend(pi(ind), si{ind});
aao,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaQ.aaaaaaaao,aaaaaaa
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
figure(2)
% subplot(2,1,2)
% hold on for i = 1:n hold on p2 (i) = ...
plot(le-12*w/2/pi,-(transpose(imag((sigma_d_inter(:,i)))))/.. (eA2/hb/4),'Color', C[2],'Linewidth',2);hold on pp2(i) = ...
plot(le-12*w/2/pi,-(transpose(imag((sigma_d_intra(:,i)))))/.. (eA2/hb/4),'Color', ...
C [2],'Linewidth',2);hold on
s2[2] = sprintf('mu {c,%d} = %d meV',i,le3*mu_c(i)/(1.6e-l9) )
76


end
% axis square box on
xlabel('f (THZ)','fontsize',20,'fontweight' b ) ; ylabel('Im(\sigma)','fontsize',20,'fontweight','b');
% legend(p2(ind), s2{ind});
aaaaaaaaaaaaaaaaaaoaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
figure (3)
plot(le-12*w/2/pi,real(sigma_tot)) title('Real part of conductivity in S')
aaaaaaaaaaaaaaaaaaoaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
figure (4)
plot(le-12*w/2/pi,imag(sigma_tot))
title('Imaginary part of conductivity in S')
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Full Text

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DESIGN OF GRAPHENE BASED PLASMONIC OPTOELECTRONIC DEVICES by HAMID TAGHIZADEH CHORSI B.S., Urmia University of Technology, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2016

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ii 2016 HAMID CHORSI ALL RIGHTS RESERVED

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iii This thesis for the Master of Science degree by Hamid Taghizadeh Chorsi has been approved for the E lectrical Engineering Program by Stephen Gedney, Chair Mark Golkowski Yiming Deng July 31, 2016

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iv Chorsi, Hamid Taghizadeh (M.S., Electrical Engineering) Design of Graphene based Plasmonic Optoelectronic Devices Thesis Directed by Professor Stephen D. Gedney ABSTRACT Once reckoned unfeasible to exist in nature, graphene, the first two dimensional material ever found, has rocketed to stardom after its first isolation in 2004 by Nobel Laureates Andre K. Geim and Konstantin Novoselov. Graphene is a sin gle atomic layer of sp 2 hybridized carbon atoms arranged in a flat hexagonal honeycomb lattice. The astonishing properties of graphene originate from its inimitable smooth sided conical band structure that converges to a single Dirac point. Graphene can be deemed as a novel platform for optical and electronic devices with the extraordinary capabilities of dynamically manipulating electromagnetic waves. One of the most fascinating properties of graphene is that it is a zero gap semiconductor (because of its unique band structure in which Dirac Fermions are charge carriers) with very high electrical conductivity. Graphene possesses the highest thermal conductivity and the highest current density at room temperature (300K) ever measured (a million times that of absorb approximately 2.3% of white light is also a unique property, especially considering that it is only 1 atom thick. Unlike conventional materials, the saturable absorption response of graphene is wavelength independe nt from UV to IR, mid IR and even to THz frequencies. Graphene is the best transparent conductive film, the thinnest material known, and the strongest material discovered (about two hundred times stronger than steel) yet is highly mechanically flexible. R ecently, it has been revealed both theoretically and experimentally that graphene supports surface plasmons (SPs), collective oscillations of free electrons at the interface between two media with dielectric constants of opposite sign, such as a metal and a dielectric. Graphene can be viewed as a strong contender for conventional plasmonic material due to the tunability of its optical conductivity (electrical and chemical doping) as well as its extreme confinement strength in the mid infrared, both properti es of which are unprecedented in metal based plasmonics.

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v Graphene has also exhibited some superiorities to noble metals such as supporting both transverse electric (TE) and transverse magnetic (TM) SPs with lower loss. This thesis focus es on designing nove l optoelectronic devices based on graphene plasmonics. A novel type of metasurfaces will be presented and optoelectronic devices will be constructed based on the graphene plasmonics. A graphene based plasmonic nanoribbon filter and an optical beam splitter has been proposed based on the designed metasurface and the electromagnetics of the wave propagation are numerically analyzed. Besides electronics perspective and applications, a near field scanning probe based on graphene plasmonics has been designed in this work that can be used for a variety of applications ranging from bio imaging to nanofabrication. The form and content of this abstract are approved. I recommend its publication. Approved: Stephen D. Gedney

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vi ACKNOWLDEDGMENTS I would like to express my deepest gratitude to my supervisor Professor Stephen Gedney for his generous support, detailed guidance and constant encouragement throughout my graduate studies at University of Colorado Denver He was patient with questions and always available to answe r them in a professional manner. His help and guidance are greatly appreciated. I am particularly grateful to Professor Mark Golkowski for his guidance and support. It is his trust and motivation from beginning to end that enables me to focus my mind on my thesis regardless of challenges. Specially, I would like to thank my parents, my brother and my sisters for their support. Without their love, dedication and sacrifice, none of this work can be accomplished. Finally I would like to express my special tha nks to Shenandoah for her love and all the experience we have gone through together

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vii TABLE OF CONTENTS Chapter 1. Introduction ................................ ................................ ................................ ....................... 14 1. 1. Motivation ................................ ................................ ................................ ................................ ....... 14 1.2. Plasmonics ................................ ................................ ................................ ................................ .... 15 1.2.1. Localized surface plasmon resonances ................................ ................................ ............ 16 1.2.2. Surface plasmon polaritons (SPPs) ................................ ................................ ................... 17 1.2.3. Bulk plasmons ................................ ................................ ................................ ....................... 18 1.3 Graphene and Graphene Plasmonics ................................ ................................ ........................ 18 1.4. Scope of the Thesis ................................ ................................ ................................ ..................... 22 2. Theoretical Foundations ................................ ................................ ................................ ... 23 2.1. Introducti on ................................ ................................ ................................ ................................ .... 23 2.2. Macroscopic Electromagnetics ................................ ................................ ................................ ... 23 ................................ ................................ ................................ .............. 23 2.2.2. Wave equation ................................ ................................ ................................ ....................... 24 2.2.3. Boundary conditions: waves at an interface ................................ ................................ ..... 25 2.2.4. Fresnel reflection and transmission coef ficients ................................ ............................... 26 2.3. Surface Waves Electromagnetics ................................ ................................ .............................. 28 2.4. Surface Plasmons Electromagnetics ................................ ................................ ......................... 31 2.5. Transmission Line Approximation (TLA) ................................ ................................ ................... 34 2.6. Graphene Plasmonics: Conductivity ................................ ................................ .......................... 35

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viii 2.7. Finite Elem ent Method ................................ ................................ ................................ ................. 36 3. Fabrication Methods ................................ ................................ ................................ ......... 37 3.2. Nanopatterns on Microscale Structures ................................ ................................ .................... 38 3.2.1. Photolithography ................................ ................................ ................................ ................... 38 3.2.2. Electron beam lithography (EBL) ................................ ................................ ........................ 39 3.2.3. Focused ion beam lithography (FIB) ................................ ................................ .................. 39 3.2.4. Scanning probe lithography (SPL) ................................ ................................ ...................... 40 3.2.5. Nanoimprint lithography (NIL) ................................ ................................ ............................. 42 3.2.6. Nanosphere lithography (NSL) ................................ ................................ ............................ 44 3.2.7. Colloidal self assembly ................................ ................................ ................................ ......... 45 3.3. Plasmonic Patterns on Microscale Structures ................................ ................................ ......... 45 4. Tunable Graphene Plasmonic Metasurface Optoelectronics ................................ ........... 47 4.1. Introduction ................................ ................................ ................................ ................................ .... 47 4.2. Graphene Surface Plasmon Polariton Propagation ................................ ................................ 47 4.3. Graphene Plasmonic Tunable Metasurface ................................ ................................ ............. 50 4.3. Plasmonic Filter ................................ ................................ ................................ ............................ 52 4.4. Plasmonic Beam Splitter ................................ ................................ ................................ ............. 58 5. Graphene Plasmonic Apertureless Scanning Probes ................................ ....................... 64 5.1. Introduction ................................ ................................ ................................ ................................ .... 64 5.2. Simulation Results and Analysis ................................ ................................ ................................ 65 6. Conclusion and Outloo k ................................ ................................ ................................ .... 71

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ix R eferences ................................ ................................ ................................ ............................ 73 A ppendix A ................................ ................................ ................................ ............................ 74 A ppendix B ................................ ................................ ................................ ............................ 75

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x LIST OF FIGURES Figure 1.1. Schematic representation of chip scale device technologies 6 1.2. Illustration of the dipole polarizability of a spherical metal nanoparticle under the influence of a plane wave 7 1.3. Propagating surface plasmon polaritons (SPPs) 7 1.4. C C bonds that form the hexagonal lattice of graphene 9 1.5. (Left) the standard honeycomb lattice of graphene with basic lattice vectors. ( Right) the reciprocal lattice and its Brillouin zone 2 0 1.6. Energy band structure of graphene for the whole first Brillouin zone 2 0 2.1. Boundary surface 6 2.2. Fresnel reflection and tr ansmission coefficients (a) TE, perpendicular, a.k.a, s polarized (b) TM, parallel, a.k.a, p vertical) has an E field that is parallel to the plane of incidence), (s) and (p) stand for the German words senkrecht (perpendicular) and parallel (parallel) 7 e (rTM = 0)) ..2 8 2.4. Schematic of a surface wave propagating along a single metal dielectric int erface .. 3 0 2.5. Real and imaginary parts of the dielectric constant for gold in visible range of the wavelength according to the Drude model 3 2 2.6. Dispersion relation of SPPs 3 4 2.7. Surface plasmon waveguide ..3 4 2.8. Surface conductivity of graphene calculated based on Kubo formula (a) real, (b) imaginary 6

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xi 3.1. Top down and bottom up nanofabrication approaches 7 3.2. Three primary exposure methods in photolithography 8 3.3. Diagram of an EBL instrument 3 9 3.4. (Left) Scanning p robe microscopy (SPM) (Right) Atomic force microscopy (AFM) 4 1 3.5. (a) near field scanning optical microscope (NSOM) instrument 2 3.6. Thermal and UV NIL 3 3.7. (left) Nanosphere lithography (NSL) process a nd (right) nanosphere optical lithography (NSOL) 4 3.8. Schematic of the prism coupled plasmonic nanolithography process 6 3.9. Schematic of a single metallic grating lithography 6 4.1. S urface plasmon wave propagation on graphene nanoribbon. (left) undoped doped undoped (UDU), (right) doped undoped doped (DUD). Results are electric field in y direction, |Ey| 4 8 4.2. Graphene plasmonic propagation an alysis. (up) input power at different ports. (middle) Absolute value of the power density at 1.53 m, and (bottom) at 1.42 m 49 4.3. Transmission (T) and absorption (A) spectra for graphene SPPs on a nanoribbon 5 0 4.4. permittivity of graphen e at THz frequency for different Fermi levels (left) real, (right) imaginary 5 1 4.5. (left) schematic presentation of the proposed graphene metasurface with S=600 nm and w=50 nm. (right) different combinations of the squ are in a 3*3 structure based on the voltage gating (Fermi level); light green squares are undoped, dark blue doped 5 1 4.6. schematic presentation of the proposed filter based on the proposed graphene metasurface 2 4.7. Electric field distribution for the designed graphene plasmonic filter, first configuration. 215THz (1.38 m), 205THz (1.46 m) 5 3

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xii 4.8. transmission spectrum for the first configuration presented in Figure 4.7 .5 4 4.9. transmission spectrum for the second configuration presented in Figure 4.10 4 4.10. Electric field distribution for the designed graphene plasmonic filter, second configuration 5 4.11. Transmission spec trum for the third configuration presented in Figure 4.12 6 4.12. Electric field distribution for the designed graphene plasmonic filter, third configuration 7 4.13. Electric field distribution for the designed gr aphene plasmonic filter, third configuration 8 4.14. Beam splitter at 210 THz (1.42 m), figures show the amplitude of the power density (|P|), the background is doped graphene (1eV). Ports are undoped (as shown in the insect, light green regions are undoped). GSPPs propagate on the undoped region which forms DUD waveguide 59 4.15. Beam splitter, electric field intensity in x direction at 210 THz (1.42 m), 6 0 4.16. Beam splitt er, electric field intensity in y direction at 210 THz (1.42 m), 1 Figure 4.17. Beam splitter power density for each scenario. (orders are the same as Figure 4.14) 2 4.18. Beam splitter, transmission spectrum for each scenario (orders are the same for all the figures in this section) 3 5.1. Artistic representation of the NSOM tip coated with a layer of graphene 6 5.2. Refractive index of graphene calculated based on the Kubo formula. (right) Real, (left) Imaginary 5 5.3. magnetic field |Bz| (a) type A (just SiO2), (b) type B (SiO2 covered with low refractive index dielectric) 7 5.4. E lectric field and the calculated FWHM for the designed probes 6 8

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xiii 5.5. FWHM with respect to the distance. (top) conventional apertureless (middle) Type A, (bottom) Type B 7 0

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14 1. Introduction 1. 1. Motivation Th e second half of the twentieth century was revolutionized by the invention of the first transistor in 1947 at Bell Laboratories. Since then, semicon ductor research has made giant lea p s fast sil icon based transistors with dimensions on the order of 20 nm, and the length of the channel in a field effect transistor (FET) is less than 17 nm. Currently, progress toward further miniaturization of transistors is hampered b y the fundamental quantum mech anical phenomena that occur at this scale, mainly the nature of charge transport in semiconductor materials. A semiconductor is a material that partially conducts current using electrons and holes as charge carriers. By downscaling the channel size in tran sistors, electrons and holes enter the realm of quantum mechanics and assume relativistic behavior. In this realm, charge transport is not diffusive anymore as in macroscopic and mesoscopic transport : it is ballistic. This type of transport gives rise to s everal kinds of quantum forces, such as the Casmir force, the Van der W aals force, thermal effects, etc., which hinder the process towards smaller and ultimately faster transistors. On the other hand, interparticle and phonon interactions and ohmic losses slow down the speed of propagation of electron transport inside a crystalline structure like silicon. Photonics is the science of light, which according to special relativity travels with the highest attainable speed in the universe. Photonics seeks to gen erate, control, and detect light waves and photons, which are particles of light. The question that immediately a rise s is : c an we utilize photons to carry information? T his question emerged a few decades ago and was answered in the late twentieth century w ith the development of fiber optics for broadband communication. What most limits the integration of photonics and semiconductor electronics is their corresponding sizes as s h own in Figure 1.1. Semiconductor electronic circuits can be fabricated

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15 at dimensi ons below 50 nm. On the other hand, the dimension s of photonic devices are on the order of micrometer s at best. Silicon photonics can be referred to as a branch of photonics that combine s photonics with the silicon platform of current electronic devices. U nfortunately, silicon does not find the same niche in optoelectronics due to the indirect nature of its band gap. Present day photonics relies on compound semiconductors and their alloys. The d iffraction limit is an optical effect which encumbers the progr ess toward the miniaturization of photonic devices by preventing localization of electromagnetic waves into nanoscale regions scales much smaller than the provide a soluti on to this dilemma because plasmonics has both the capacity of photonics and the miniaturization of electronics. 1.2. Plasmonics Plasmonics is the study of the interaction between the electromagnetic field and the free electrons in a metal. Moreover, it is the study of light at the nanometer scale. Plasmonics is capable of providing unique advantages in all areas of science and technology where the manipulation of light at the nanoscale is a prominent ingredient [1] Plasmonics can squeeze light into dimensions far beyond the diffraction limit by coupling the light with the surface collective oscillation of f ree electrons at the interface of a metal and a dielectric. Plasmonics can be considered as a promising candidate for high speed and high density integrated circuits. It bridges microscale photonics and nanoscale electronics and offers similar speed of pho tonic devices and similar dimension of electronic devices. A plasmon is a quantum for the collective oscillation of free electrons, as a photon is a quantum for light. The different types of plasmons that can be excited in metallic objects depend on the ir dimensions. In large three dimensional metal structures volume plasmons can exist in the bulk of the metal. At the interface between metals and dielectrics propagating surface

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16 plasmon polaritons (SPPs) can be excited. Low dimensional metal structures such as nanoparticles maintain a wide variety of localized surface plasmon resonances (LSPRs). Figure 1.1 illustrates the role of plasmonics with respect to other chip scale device technologies. It shows how semiconductor electronics, photonics and plasmon ics occupy well defined domains on a graph of the operating speed and device dimensions. The dashed lines indicate physical/practical limitations of different technologies; semiconductor electronics in electronic processors tend to be limited in speed by t hermal and interconnect delay time issues to about 10 GHz. Photonics is limited in its critical dimensions by the fundamental laws of diffraction. Plasmonics can serve as an effective bridge between similar speed photonics and similar size nano electronics Figure 1 1. S chematic re presentation of chip scale device technologies. 1.2.1. Localized surface plasmon r esonances Localized surface plasmon resonances (LSPRs) are confined, non propagating surface plasmon w aves and can be considered as the simplest type of plasmon wave. The free electron cloud of the nanostructure can be resonantly excited by polarized electromagnetic fields due to enhanced polarizabilities of the particles at certain frequencies as shown in Figure 1.2 These Operating Speed kHz MHz GHz THz PHz 1 m 10nm 100nm 10 m 100 m 1mm Device Dimensions The Past Plasmonics Photonics Semiconductor Electronics

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17 enhanced polarizabilities give rise to strongly enhanced near fields in the proximity of the metal surface Figure 1. 2 Illustration of the dipole polarizability of a spherical metal nanoparticle under the influence of a plane wave 1.2.2. Surface plasmon p olaritons (SPPs) real permittivity ( e.g., metal and dielectric at THz), are propagating waves known as sur face plasmon polaritons (SPPs). These collective oscillations of the free electrons due to polarized EM light in the metal constitute dispersive longitudinal waves that propagate along the interface and decay exponentially into both media Figure 1 3 Propagating surface plasmon polaritons (SPPs)

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18 1.2.3. Bulk plasmons Volume plasmons are the most fundamental and intrinsic type of plasmon resonance that can be supported by a metal. These resonances occur at the plasma frequ ency of metals, which are transparent to radiation with higher frequencies and non transparent to radiation with lower frequencies. 1.3 Graphene and Graphene Plasmonics Graphene is a planar sheet of sp 2 hybridized carbon atoms arranged in a two dim ensional (2D) hexagonal lattice [2] It is a zero gap semiconductor with unprecedented electrical, optical, thermal and mechanical properties. Due to its unique characteristics i t will pla y a crucial role in future opto electron ic technologies. The inimitable properties of graphene are mainly due to the band structure of the material which consists of two bands touching each other at two nodes. The electronic spectrum around these two nodes is linear and can be approximated by D irac cones. Graphene can be gated or doped, such that the Fermi energy can be freely tuned. C alcul ations of many physical proper ties of graphene demand deep knowledge of electron dispersion in the entire Brillouin zone. The he xagonal lattice of graphene is characterized by two types of C C bonds ( ), which are formed by the four vale nce orbitals of carbon atom ( 2s, 2p x 2p y 2p z ) as shown in Figure 1.4.

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19 Figure 1. 4 C C bonds that form the hexagonal l attice of graphene In graphene the 2s and 2p electrons hybridize to form three sp 2 orbitals and one p orbital The natural tendency (that is, minimization of energy) is for the sp 2 hybridized orbitals to arrange themselves at 120 o angles from each other in a plane orthogonal to the p orbital. Between two hybridized carbon atoms, overlapping sp 2 orbitals form a bond while p orbitals form a second bond oneycomb lattice structure Figure 1.5 shows the standard lattice of gr aphene along with the unit cell One can note two inequivalent sublattices which are labeled by A and B w hi ch connect all states in one triangular sublattice.

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20 Figure 1.5. (Left) the standard honeycomb lattice of graphene with basic lattice vectors. (Right) the reciprocal lattice and its Brillouin zone. In the Bravais lattice the primitive la ttice vectors are given by ( 1 .1) where a is the nearest neighbor C C spacing ( ). The reciprocal lattice vectors have the form ( 1 2 ) E ach point in one sublattice is coupled b sublattice, ( 1 3 ) The eigenenergies of graphene can be modeled by considering the overlap of free electrons Con sidering very strong periodic potential ( each electron is almost bound to a minimum ), t he tight binding approximation exploits the system Hamiltonian and electron wavefunctions of the atoms to find the eigenenergies. Suppose that the ground state of an A B

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21 ele ctron moving in the potential U(r) of an isolated atom is Then the approximate wavefunction for one elect ron in the whole crystal is found by taki ng ( 1 4 ) wh ere j sums over all the atoms. C onsidering the function in the Bloch form ( wavefunction for a particle in a periodically repeating environment ) for a crystal of N atoms [3] ( 1 5 ) two inequivalent sublattices A and B in graphene, we can split (1.5) to ( 1 6 ) Equation (1.6) has been derived based on the three nearest neighbors By minimizing the energy with respect to the coefficients c and taking the Hamiltonian we can calculate the ground state of an electron as ( 1 7 ) We can now find the two eigenenergies as a function of k [3] ( 1 8 ) These energies are show n as a function of k in Figure (1.6).

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22 Figure 1.6 Energy band structure of graphene for the whole first Bril louin zone 1.4. Scope of the T hesis This thesis investigate s the concept, analysis, and design of novel o ptoelectronic devices based on graphene surface plasmons (GSPs) for nano electronic, optoelectronic, imaging, and bio sensing applications. Chapter o ne provide s a brief introduction and review of the fundamentals of plasmonics and g raphene palsmonics. In chapter two we will develop the theoretical foundations of the electromagnetic theory of surface plasmons and wave propagation in graphene nanoribbon Designing nano and plasmonic structures requires a deep understanding of the fabrication processes used in constructing plasmonic nano structures. In chapter three, a brief review of the most state of the art plasmonic fabrication methods is presented. Ch apter four will present a new type of graphene plasmonic metasurface. N ovel optoelectronic devices such as filters, benders, couplers and power splitter s are designed based on g raphene nano struct ures (nanowires and nano slabs). Chapter five will present a novel design concept for an apertureless plasmonic probe for near field scanning optical

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23 microscopy (NSOM) based on GSPs. This is particularly important for bio sensing and imaging applications, such as molecular and cellular analyses. Chapter six conclu des the thes is and provides a roadmap for future work. 2. T heoretical Foundations 2.1. Introduction Plasmonics explores how electromagnetic waves can be confined over dimensions on the order of or smaller than the wavelength. It is based on the interacti on between electromagnetic radiation and conduction electrons. This chapter summarizes the most important theories and phenomena that form the basis for the study of plasmonic waves. of surface, bulk, and localized plasmons will be review ed The t ransmission line approximation (TLA) of SPP propagation along a plasmonic waveguide is also reviewed. 2.2. Macroscopic Electromagnetics The interaction of metals with electromagnetic fields can be completely understood in a electromagnetics the singular nature of charges and their associated currents is avoided by considering charge densi ties and current densities J In differential form and in SI units (2.1) (2.2) (2.3) (2.4)

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24 These equations link the four macr oscopic fields D (the electric flux density ), E (the electric field), H (the magnetic field), and B (the magnetic flux density) with the external charge and current J densities. The electromagnetic properties of the medium are commonly discussed in terms of the macroscopic polarization P and magnetization M according to: (2.5) (2.6) Where 0 and 0 are the permittivity and the permeability of a vacuum, respectively. Since in this th esis we will only treat nonmagnetic media, we need not consider a magnetic response represented by M, but can limit our description to electric polarization effects. For a linear, isotropic and nonmagnetic media, the constitutive relations can be written as: (2.7) (2.8) r is called the dielectric constant or relative permittivity and r = 1 the relative permeability of a nonmagnetic medium. The last important constitutive linear relationship we need to mention is (2.9) in which is conductivity. 2.2.2. Wave equation The properties of waves (such as sound waves, light waves and water waves) can be described using the wave equation. To derive the wave equation, we take the curl of the first equation: (2.10) (2.11)

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25 (2.12) (2.13) For a source free region, the wave equation can be written as: (2.14) This is actually three equations, the x y and z vector components for the E field vector For a plane wave moving in the x direction this reduces to (2.15) The monochromatic solution (a simple set of complex traveling wave solutions) to this wave equation has the fo rm: (2.16) (2.17) where quantifies attenuation, quantifies propagation, where is the speed of light, and is the phase velocity. The relation of the phase velocity versus the frequency is known as the dispersion relation. 2.2.3. Boundary conditions: waves at an interfa ce Material b oundary conditions express the relationships of the electromagnetic vector fields (E, D, H, B) at the interface separating two different materials. attenuation propagating

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26 Figure 2.1. Boundary surface (2.18) (2.19) (2.20) (2.21) The first equation state s that the tangential electric fields are continuous across the interface. The second equation states that t he tangential magnetic fields are discont inuous at the same location by an amount equal to the impressed surface electric current density 2.2.4. Fresnel reflection and transmission coefficients In this section, the fraction of a light wave reflected and transmitted by a planar interface between two media with different refractive indices is presented Applying the boundary conditions to a uniform plane wave incident on a single planar interface leads to the Fresnel reflection and transmission coefficients.

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27 Figure 2.2. Fresnel reflection and tr ansmission coefficients (a) TE, perpendicular, a.k.a, s polarized (b) TM, parallel, a.k.a, p vertical) has an E field that is parallel to the plane of incidence), (s) and (p) stand for the German words senkrecht (perpendicular) and parallel (parallel). For the TE case, the tangential electric field is continuous across the boundary, (2.22) The tangential magnetic field is continuous (2.23) Using where (n refractive index) and considering only the amplitude of the waves ( ) at the boundary a reflection coefficient (2.24) A nd transmission coefficient (2.25)

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28 i s derived. For the TM case, Fresnel reflection and transmission coefficients can be eas ily deriv ed as (2.26) (2.27) A plot of Fresnel reflection and transmission coefficients for n = 1.50 is shown in Figure 2.3. Figure 2.3. Reflections of TE and TM modes for n = 1.50 ( with T M = 0)) At some angle, known as the critical angle, light traveling from a higher refractive index medium to a lower refractive index medium will be refracted at 90 o When the angle of incidence exceeds the critical angle, there is no refracted light. All the incident light is reflected back into the medium. The critical angle of incidence can be obtained for two media by (2.28) 2.3. Surface W aves Electromagnetics Many condensed matter properties are governed by surface proper ties. Very often, surface waves, which are defined as waves propagating along the interface between two media and existing in both of them, play a key role. The allure of surface waves stems from the

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29 confinement of energy t o the close vicinity of the inter face of the two partnering materials. Any change in the composition of either partnering material in that vicinity could alter even eliminate the surface wave. Zenneck in 1907 envisioned that the planar interface between air and ground supports the propaga tion of radiofrequency waves. The idea was later extended by Sommerfeld and that surface wave has since become known as the Zenneck wave. The idea of the Zenneck wave was reapplied in the mid 20th century to the visible portion of the electromagnetic spec trum at the interface of a noble metal and a dielectric material which initiating the concept of the surface plasmon polariton (SPP) wave. During the last three decades, two other types of surface electromagnetic waves have been discovered. The Dyakonov s urface wave proposed in 1988, travels along the planar interface of an isotropic dielectric material and a uniaxial dielectric material. Unlike either the Zenneck wave or the SPP wave, which require that one of the two partnering materials forming the int erface to have a relative permittivity with a negative real part, the Dyakonov wave is a result of the difference in the crystallographic symmetries of the two partnering dielectric materials. Another type of surface electromagnetic wave is called Dyakonov Tamm waves. This type of wave requires one of the two partnering mediums to be periodically nonhomogeneous normal to the planar interface. SPP waves guided by the interface of noble metals and dielectric media is the main concentration of this work. The n ext section will review the electromagnetics of the SPP waves. By following [4] let us investigate a planar interface between a metal and a dielectr ic.

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30 Figure 2.4. Schematic of a surface wave propagating along a single metal dielectric interf ace For the metal 1 = m at z <0, and 2 = d for the dielectric at z > 0. The wave equation in Helmholtz form has to be solved separately in each region (2.29) TM (or p polarized) and TE (or s polarized) modes. Surface plasmon waves do not support a TE mode since the width of the waveguide i s much smaller than the exciting wavelength. Considering the following boundary conditions (2.30) (2.31) That is, the parallel field component is continuous, whereas the perpendicular component is di can be obtained. (2.32)

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31 Equation (2.32) shows the dispersion relation between the wave vector components and the angular frequency. 2. 4. Surface P lasmons Electromagnetics We start by deriving the dielectric constant of metals. One of the simplest but nevertheless valuable models to describe the response of a metallic particle exposed to an electromagnetic field is the Drude Sommerfeld mo del: (2.33) where e and m e are the charge and effective mass of the free electrons, and E 0 amplitude and frequency of the applied electric field. The damping term is proportional to where is the Fermi velocity and l is the electron mean free path between scattering events. Equation (2.33) can be solved by which leads to the well known dielectric function of Drude form (2.34) here is the plasma frequency, and describes the ionic background in the metal (usually 3.7 for silver). If is larger than the corresponding refr active index is a real quantity; on the other hand if is smaller than the refractive index is imaginary since is negative.

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32 Figure 2.5. R eal and imaginary parts of the dielectric constant for gold in visible range of the wavelength acc ording to the Drude model As shown in Figure 2.4, the simplest geometry supporting a SPP wave is a single, planar interface between a metal at y < 0, with a complex dielectric constant in which the real part is negative (metals at THz region have negative real permittivity as shown in F igure 2.5: this is a very critical criterion for SPP waves since in this situation the wave can ac tually penetrate inside the metal), and a dielectric at y > 0, with a positive dielectric constant. The TM mode (H z E x and E y ) solutions are considered Propagating waves can be described as in which is called the propagation constant of the traveling wave curl equations and knowing the propagation along the x direction ( ) and uniformity in the z direction ( ) and the system of governing equations for the TM can be written as (2.35) (2.36)

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33 and the wave equation for the H z component as (2.37) For the half space in Figure 2 .4, the TM wave can be written for upper and lower halves as (2.38) (2.39) (2.40) f or y>0 and (2.41) (2.42) (2.43) f or y <0. Due to boundary conditions and the wave equation for H y we have (2.44) Finally, the dispersion relation of SPPs propagating at the interface between the two half Spaces can be obtained as (2.45) Figure 2.6 shows plots of (2.45). I t can be seen from Figure 2.6 that for a given frequency, a free space photon has less momentum than an SPP t intersect On the other hand, coupling medium such as a prism can match the photon momentum. SPPs c oupling with a prism light with relative permittivity of 1.5 with silver is presented in Figure 2.6

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34 Figure 2.6. Dispersion relation of SPPs 2.5. Trans mission L ine A pproximation (TLA) Transmission line approximation (TLA) is a fast and reliable analytical approach used to investigate for optical applications. Figure 2.7. Surface plasmon waveguide E x E y B z

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35 (Figure 2.7) can r eplaced by a transmission line of characteristic impedance [5] : (2.46) (2.47) 2.6. Graphene Plasmonics: Conductivity The propagation of SPPs on graphene is strongly dependent on the Fermi level (chemical potential ). This can be seen from the frequency dependent conductivity profile for different values of the Fermi level which has been obtained from the Kubo formula [6] (2.48) (2.49) (2.50) i n which e is the electron charge, T is the temperature, is the radian frequency, is the relaxation time between collisions with the impurity ion s, phonons, etc., and denotes a step function. Figure 2.8. shows the calculated conductivity for different values of Fermi level from 20 THz to 900 THz.

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36 Figure 2.8. Surface conductivity of graphene calculated based o n Kubo formula (a) real, (b) imaginary 2.7. F inite Element Method Partial differential equations (PDEs) appear in the mathematical modelling of many physical phenomena in e lectromagnetics. The complexity of boundary conditions renders finding their solut ions by purely analytical means (e.g. by Laplace and Fourier transform methods, or Mie series) either impossible or impracticable, and one has to resort to seeking numerical approximations to the unknown analytical solution. The Finite element method (FEM) is essentially a method for defining basis functions on mesh elements and using those basi s functions to discretize a PDE. With FEM, the unknown solution is expanded as a linear combination of basis functions, as in the case with the method of moments (Mo M). FE M can be applied to a vast variety of PDEs, as well as integral equations (IEs). In all these cases, the basic outline of the FEM is as follows: (a) Create a mesh by dividing the simulation domain into elements. For 2D problems (as in the case of most si mulations in this thesis) the elements are typically triangular.

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37 (b) Define basis functions on the mesh that can accurately interpolate the field, and its derivative (c) Exploit the functional for the PDE in order to compute a set of matrix elements for the bas is functions on one canonical mesh element. (d) Assemble the element matrix entities into a global Rayleigh Ritz matrix for the entire mesh. (e) Generate a right hand side vector from the known sources. (f) Solve the linear system for the basis function weights. (g) Apply post processing to derive physical quantities. 3. F abrication Methods 3.1. Introduction All of the grand ambitions of plasmonics are necessarily dependent upon feasible fabrication methods. There are innumerable nanofabrication techniques with different p erformances, choice of which depends upon the materials, applications, and geometries of the down fabrication refers to methods where one co mmences with macroscopically dimensioned material and carves the nanostructure out of the larger structure. On the other hand, in the bottom up approach, assembly begins with smaller units: positions of atoms or molecules are manipulated to piece together the nanostructure. The top down and bottom up approaches are schematically shown in Figure 3.1. Figure 3.1 Top down and bottom up nanofabrication approaches

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38 The topic of Nanofabrication is far too vast to be covered in one chapter. The goal of this sec tion of the thesis is simply to introduce the method and review the substantial body of literature on plasmonic fabrication. 3.2. Nanopatterns on Microscale S tructures 3.2.1. Photolithography In optical lithography, a mask or reticle is imaged onto a subst rate which is painted with a thin layer of photoresist, a photosensitive polymer material. Focused photon energy causes chain scission or cross linking in the polymer. The mask pattern is then delineated into the photoresist after a development process. Th ere are three primary exposure methods: contact, proximity, and projection, shown in Figure 3.2. Figure 3.2. Three primary exposure methods in photolithography In contact lithography, the photomask is brought into physical contact with the wafer and the n exposed to light. Contact lithography offers high resolution, but mask damage and a resultant low yield make this process impractical in most production environments. In proximity lithography, a gap is placed between mask and wafer in the range of 10 to 30 micro meters. Although proximity lithography does not suffer from mask damage as in contact printing, its low resolution makes it unsuitable for sub 100 nm fabrication. In projection lithography, the image is

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39 projected onto the wafer with the help of a system of lenses. In this case, the mask can be used several times, substantially reducing the mask per wafer cost. 3.2.2. E lectron beam l ithography (EBL) Electron beam lithography (EBL) has evolved from scanning electron microscope (SEM) in early the 1960 s by the introduction of an electron sensitive polymer material, called polymethyl methacrylate (PMMA). Figure 3.4. shows the diagram of an EBL instrument. An electron gun is a device that generates and projects a beam of electrons onto a substrate. Electr ons are first generated by cathodes or electron emitters then accelerated and focused by electrostatic fields to obtain higher kinetic energy and shaped into an energetic beam. Finally, the guidance system, consisting of the electric and magnetic focusing coils and deflecting system, transmits the beam to a work point on the substrate. Figure 3.3 Diagram of an EBL instrument. 3.2.3. F ocused ion b eam l ithography (FIB) A technique related to electron lithography is focused ion beam lithography, commonly c alled FIB. FIB is based on the use of accelerated ions instead of electrons. If the wavelength of accelerated ions can be similar t o that of accelerated electrons; therefore an atomic resolution is expected in the ideal case. The major difference lies in t he mass of the ions that allows very efficient momentum transfer and therefore physical etching of almost any kind of material.

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40 FIB lithography is similar to EBL, but provides more functionalities. Not only can focused ions can create a pattern on a resis t, similar to EBL, but they are capable of locally removing away some parts of the structure by sput tering (subtractive lithography: a hydrogen ion is 1840 times heavier than an electron). FIB is capable of accurately depositing atoms with sub 10nm resolut ion (additive lithography). 3.2.4. S canning probe l ithography (SPL) For low cost nanoscale patterning technologies, scanning probe lithography (SPL) is definitely an alternative to expensive photon or charged beam techniques. Another problem with lithograp hy using either photons or charged beams is that they always rely on a polymer material (photo resist or electron resist) as an imaging layer. SPL, however, can be implemented with diverse mechanisms, such as a direct write approach. SPL uses a scanning pr obe microscope device (a sharp tip) in close proximity to a sample to pattern nanometer scale features on the sample. A scanning probe microscope (SPM) is an instrument that monitors the local interaction between a sharp tip (less than 100 nm in radius) an d the sample to acquire physical, electrical, or chemical information about the surface with high spatial resolution. Today there are many different types of SPMs used for diverse applications ranging from biological probing to material science to semicond uctor metrology. Three major technologies within the SPM family are scanning tunneling microscopy (STM), atomic force microscopy (AFM), and near field scanning optical microscopy (NSOM). The first SPM was the STM invented in 1981 by Binnig an d Rohrer. As s hown in Figure 3.4 (a), STM uses a sharpened conducting tip with a bias voltage applied between the tip and the target sample. When the tip is within the atomic range (about 1 nm) of the sample, electrons from the sample begin to tunnel through the gap to the tip or vice versa, depending on the sign of the bias voltage. The exponential dependence of the distance between the tip and the target gives

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41 STM its remarkable sensitivity with subangstrom precision vertically and subnanometer resolution laterally. T he primary limitation of STM is that it can only be used to image conducting substrates. The AFM was developed to assuage this constraint. The AFM based techniques are less restrictive than that of STM because AFM can be conducted in a normal room environm ent and can be used to image any kind of materials. Figure 3.4. (Left) S cann ing probe microscopy (SPM) (Right) A tomic force microscopy (AFM) In AFM lithography, the interaction potential between the atoms of the end of the tip and the atoms of the targe t surface causes a localized force. This force is measured by the deflection of a laser beam which is focused on top the mechanical cantilever on which the tip is attached. The third major SPM, in addition to STM and AFM, is the near field scanning optical microscope (NSOM). The main idea here is to utilize the perturbations of the evanescent waves in the near field of the sample due to the interaction between the tip and the sample surface, and convert it into propagating light that can be detected via pho todetectors.

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42 Figure 3.5 (a) near field scanning optic al microscope (NSOM) instrument Two main types of NSOM probes are aperture type NSOM and apertureless techniques. In the first case, a subwavelength size aperture on a scanning tip is used as an opt ical probe. This is usually an opening in a metal coating of either an optical fiber tip or of a cantilever. Spatial resolution in the aperture type SNOM is, in general, determined by the aperture diameter. Apertureless techniques are based on the near fie ld optical phenomena as well, but do not require passing the light through an aperture. For SPL, the quality of the tip is defined by its crystallinity, surface roughness, and radius of curvature. High resolution lithographic tools such as focused ion and electron beam have been used to mill, sculpt, and grow sharp tips for high resolution imaging purposes 3.2.5. Nanoimprint lithography (NIL) Nanoimprint lithography (NIL) is one of the most promising low cost, high throughput technologies for nanostructure [7] NIL was proposed and demonstrated as a technology for sub 50 nm nanopatterning. Depending on the type of polymer used, NIL can be done via a thermal or UV curing. Figure 3.10 shows the fabrication procedure of the nanoimprint technologies.

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43 Figure 3.6 Thermal and UV NIL. I n thermal NIL, first a thermoplastic polymer (PMMA) is applied to a silicon substrate, then the complex is heated to above glass transition temperature (Tg), in which the polymer becomes viscous liquid. Next, the mold is pressed onto the surface with a hig h pressure. After the mold cavities are filled with molten PMMA, the complex is cooled below the glass transition temperature and the mold peeled off from the PMMA surface. Finally, the residue PMMA on the compressed areas is removed by anisotropic etching In order to overcome difficulties associated with thermal NIL, such as alignment errors and the time consuming process of thermal transition, in 1999 Colburn et. el proposed another nanoimprint method based on the UV curing process. A UV NIL mold must be transparent to UV light, and quartz is a popular choice. Initially, the UV curable polymer is dispensed onto the substrate. The quartz mold is pressed onto the polymer surface with low pressure, then the polymer is exposed to UV to cure and solidify. Afte r curing, the mold is released from the

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44 substrate. More details of the physics and the choice of the material and resist can be found in [7] 3.2.6. Nanosphere lithography (NSL) To enhance the consistency of particle size and arrangement, in 1995 Van Duyne and colleagues proposed a unique and simple fabrication method for metal nanoparticles called nanosphere lithography (NSL). NSL utilizes tightly packed polystyrene spheres on a substrate surface as a masking layer. The method is schematically shown in Figure 3.7. Figure 3.7. (left) Nanosphere lithography (NSL) process and (right) nanosphere optical lithography (NSOL). The first step in NSL is dropping polystyrene nanospheres on a pristine, pre prepared glass substrate. The hexagonally close packed (Fischer pattern) nanospheres create a crystal structure in which the gaps between the spheres form a regular array of dots. Next, the array is filled in with thermally evaporated si lver. After the deposition, the polystyrene spheres are removed by agitating (sonicating) the entire substrate in either H 2 Cl 2 acid or absolute ethanol, and the product is an array of triangular dots. As an example, Figure 3.7 shows the triangular

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45 nanopart icle shape after deposition by NSL. Nanosphere optical lithography (NSOL) utilizes polystyrene or silica nanospheres on a substrate surface as a lens array. UV light is then used to pattern the photoresist using light spots under the nanospheres. 3.2.7. Co lloidal self assembly Colloidal self assembly refers to self assembly of particles or spheres with diameter from micrometers to nanometers in a liquid suspension. Colloidal self assembly can be used in two forms for nanofabrication of sub 100 nm and plasmo nic structures. The first way is to fabricate the nanostructures inside the liquid, which we will refer to here as colloidal synthesis. The second method, colloidal lithography, uses the fabricated nanostructures via colloidal synthesis as masks for other fabrication techniques, such as photolithography or nanoimprint lithography. Colloidal synthesis has been widely used for the fabrication of plasmonic nanoparticles. Examples include nanospheres, nanostars, nanor ods, and nanoporous structures. 3.3. Plasmon ic Patterns on Microscale S tructures So far, we have reviewed several nanofabrication techniques for the fabrication of sub 100 nm structures and plasmonic devices. All the aforementioned techniques perform at the interface between Nano and Micro scale, he nce the motivation of naming the previous section chapter two, is extremely interesting and heralds enchanting applications for nanofabrication. Recently, SPPs have bee n used to fabricate nanostructures, especially for patterning nanoscale structures. The idea of prism coupled plasmonic nanolithography originated from the excitation of SPP at the interface of a metal and dielectric via a prism. The main idea here is the use of evanescent waves (from the interaction of light with metal mask) to pattern photoresist. Figure 3. 8 which is characterized by the Kretschmann configuration, shows the physical arrangement.

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46 Figure 3.8. Schematic of the prism coupled plasmonic nano lithography process An isosceles triangle is placed at the uppermost layer in order to excite the SPPs. The bottom surface of the prism is coated with a thin metal (silver) film and then brought into intimate contact with a thin photoresist coated on a su bstrate. When two mutually coherent TM (p polarized) plane waves are incident on the base of the prism in the vicinity of the resonance angle, multiple counterparts of the SPPs arise everywhere on the interface. As a result, SPP interference fringes are fo rmed in the photoresist. Grating coupled plasmonic nanolithography uses metallic grating masks along with appropriate structures to excite SPPs and pattern nanoscale features. As distinct from a Kretschmann scheme, the mask grating based scheme is much mo re compact. Figure 3.9 Schematic of a single metallic grating lithography.

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47 The schematic of plasmonic lithography configuration using metal mask is shown in Figure 3. 9 It consists of a metal mask, which can be fabricated on a thin quartz glass by ele ctron beam lithography and lift off process. The mask is brought into intimate contact with a photoresist coated on a silica substrate. Normally incident light tunnels through the mask via SPP and reradiates in to the photoresist. 4. Tunable Graphene Plasm onic Metasurface Optoelectronics 4.1. Introduction Recently, graphene plasmonics has attracted extensive interest in several areas of application. The d esign of optical and plasmonic circuit devices such as beam benders, optical couplers, filters, and powe r splitters is an indispensable step towards next generation optoelectronic devices This chapter aims to present some of these components in terms of graphene plasmonics. Specially a no vel tunable graphene metasurface is presented. That is use d to design plasmonic and optoelectronic devices. The nature of the GSPPs propagation on nano ribbons is considered at the beginning of the chapter in order to elucidate later result s Then, a tunable plasmonic filter and a controllable plasmonic beam splitter are de signed based on the proposed graphene metasurface and the EM wave propagation is investigated in detail. 4.2. Graphene Surface Plasmon Polariton Propagation Propagation of graphene SPPs can be categorized into two groups: undoped doped undoped (UDU) and doped undoped doped (DUD) The two paradigms are presented in F igure 4.1

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48 Graphene surface plasmons can support both edge mode and waveguide mode propagation as shown in Figure 4.1. The UDU configuration has a stronger edge mode propag ation than DUD; on the other hand, DUD has a more confined waveguide mode propagation. In order to analyze the wave propagation on top of a graphene ribbon, we have considered a DUD configuration. The width of the waveguide is 600 nm. The input power is s hown in Figure 4.2. Maximum power (7.07 10 8 ) is at a wavelength of 1.53 m. The absolute value of the power density (|P|) at 1.53 m and 1.42 m is also shown Figure 4.2 The results show that after propagating 6 m power at port 2 is 44.8 % of the input power (P1) at 1.53 m and it is 12.2 % of the input power (P1) at port 3 with respect to P1. Figure 4.1. Surface plasmon wave propagation on graphene nanoribbon. (left) undoped doped undoped (UDU), (right) doped undoped doped (DUD). Results are electric field in y direction, |Ey|

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49 I n order to understand the decay rate, the transmission and absorption spectra are calculated at P1 and P2 with respect to P1 and P2 and P3 with respect to each other based on the power ratio of the output to the input P1 P 2 P 3 | P | at 1.42 m | P | at 1.53 m Figure 4.2. Graphene plasmonic propagation analysis. ( up ) input power at different ports. (middle) A bsolute value of the power density at 1.53 m, and ( bottom ) at 1.42 m P 3 P 2 P1 | P | at 1.53 m

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50 The main conclusion of this section is that power loss is a major characteristic of plasmonic structures. It should be also mentioned that in case of GSPPs this loss is much less than that of metallic loss. 4.3. Graphene Plasmonic Tunable Metasurface Metasurfaces are 2D surfaces composed of subwavelength optical elements (such as nanoantennas) capable of manipulating light by changing the optical properties of the incident electromagnetic wave. Due to the surface nature of the SPPs, which occurs at the interface between two media with opposite sign real permittivity, GSPPs can only occur at frequencies at which the permittivity of graphene is negative. The relation between permittivity of graphene and frequency for different Fermi levels is presented in Figure 4.4. Figure 4.3. Transmission (T) and absorption (A) spect ra for graphene SPPs on a nanoribbon.

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51 The structure of the proposed graphene metasurface is a 33 grid of graphene surfaces as presented in Figure 4.5. The side of each square and the gap size are S=600 nm and w=50 nm respectively. Each square ( switch ) can be turned ON and OFF based on the Fermi level (chemical poten tial) and the background medium. That is if the backgr ound is doped, doping a square will close the switch (OFF) and undoping wi ll open the switch ( ON ) Figure 4.4. permittivity of graphene at THz frequency for different Fermi levels (left) real, (right) imaginary Figure 4.5. (left) schematic presentation of the proposed graph ene metasurface with S=600 nm and w=50 nm. (right) different combinations of the square in a 3*3 structure based on the voltage gating (Fermi level) ; light green squares are undoped dark blue doped

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52 4.3. Plasmonic Filter F ilters of electromagnetic frequencies are of fundamental importance in the integrated signal processing circuit. Traditiona l metallic plasmonic filters exhibit good performance in the visible and near infrared region. However, poor confinement of SPPs in the THz regi on limits their applications. GSPPs have high fiel d confinement at THz and thus graphene based filters promise better performance. Furthermore, the tunability of graphene allows one to adjust the coupling parameters dynamically. In this section, the proposed graphene metasurface is used to design tunable plasmonic filters. The structure of the proposed filter is pr esented in Figure 4.6. In Figure 4.6, L in = 4 m and L out = 3 m. The three middle squares are undoped graphene (since we want the wave to propagate in the waveguide). Doping (voltage gating) any of the middle squares will close the waveguide. From this perspective, the simplest optical device that can be realized with the proposed metasurface is an optical switch that turn s ON and OFF the nanoribbon. The other squares in the upper layer and lower layer s will be turned ON and OFF based on the applie d voltage to meet the requirements of the filter. The electric f i e l d of the first configuration that we have considered is presented in Figure 4.7 at 1.38 m (mode 2) and 1.46 m based on the transmission spectrum, (Mode1=1.55 m). P in P out L in L out Figure 4.6. S chematic presentation of the proposed filter based on the p roposed graphene metasurface.

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53 |Ex| f=215 THz |Ex| f=205 THz | Ey | f=215 THz f=205 THz | Ey | 0 1 Figure 4.7. Electric field distribution for the designed graphene plasmonic filter, first configuration. 215THz ( 1.38 m ), 205THz ( 1.46 m ).

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54 In the first configuration, only two squares are turned ON the (the top middle and bottom middle ones) As ment ioned before, undoping a square in a do ped background turns it ON From Figure 4.8 it can be seen that the filter is a band st op filter for m ode 2=1.38 m and mode 1=1.55 m (or band pass at 1.46 m ) The second configuration can be easily realized by changing the Fermi level of the metasurface. In the second configuration we will turn on (undope) the side squares and will turn o ff the middle ones as shown in the Figure. Mode 2=1.38 m and mode 1=1.55 m. Middle is 1.48 m. Mode 2 M ode 1 Figure 4.8. transmission spectrum for the first configuration presented in Figure 4.7 Figure 4.9. transmission spectrum for the second configuration presented in Figure 4.10

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55 f=215 THz |Ex| |Ex| f=202 THz |Ey| f=215 THz |Ey| f=202 THz Fig ure 4.10. Electric field distribution for the designed graphene plasmon ic filter, second configuration, 215THz ( 1.38 m ), 202THz ( 1.48 m )

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56 I n the last configuration, all upper row and lower row squares are turned on. The transmission spectrum is presented in the Figure 4.11. C omparing Figures 4.8, 4.9, and 4.11, i t can be seen that the type of filter has been changed from a band stop filter with two dips to a high pass filter at the same operating frequency. This indeed shows the tunability and flexibility of the proposed graphene based filter based on the proposed metasurface. Electric fi e l d results in both x and y directions are presented for the third configuration at 1.51 m (197 THz) and 1.24 m (241 THz) in Figure 4.12. Figure 4.11 T ransmission spectrum for the third configuration presented in Figure 4.12

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57 |Ex| |Ex| f=197 THz f= 241 THz |Ey| f=197 THz |Ey| f=241 THz Figure 4.12. Electric field distribution for the designed graphene plasmonic filter, third configuration.

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58 4.4. Plasmonic Beam Splitter In order to demonstrate the applicability and versatility of the proposed metasurface, a tunable plasmonic beam splitter is designed and analyzed in this section. A doped undoped doped (DUD) configuration has b een considered. The background medium is kept at 1eV chemical potential (which corresponds to carrier density of ) and the waveguides (buses or ports) are kept at undoped level s The doping level s of the square metasurfaces is modi fied based on the desired output. In Figure 4.13 the proposed graphene metasurface is placed between four ports to be used as a power splitter. In Figure 4.13, gray areas are PML and L p =4 m which is the same for port 2 and 3. The goal he re is to design a beam splitter which is capable of changing its filtering properties Figure 4.13. Electric field distribution for the designed graphene plasmonic filter, third configuration.

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59 based on the proposed metasurface and the tunability of graphene. Figure 4.14 presents the power spectrum for the designed beam splitter. Doped Graphene 1 0 Figure 4.14. Beam splitter at 210 THz (1.42 m) figures show the amplitude of the power density ( |P| ) the bac kground is doped graphene (1eV). P orts are undoped (as shown in the insect, l ight green regions are undoped). GSPPs propag ate on the undoped region which forms DUD waveguide.

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60 | Ex | |Ex| min |Ex| max | Ex | | Ex | | Ex | Figure 4.15. Beam splitter, electric field intensity in x direction at 210 THz (1.42 m)

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61 |E y | min |E y | max Figure 4.16. Beam splitter, electric field intensit y in y direction at 210 THz (1.42 m)

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62 From the power density and electric field plots, the behavior of the structure as a beam splitter can be observed Figure 4.17 shows the power density plots at ports 1,2 and 3 for each configuration Figure 4.17. Beam splitter power density for each scenario. (orders are the same as Figure 4.14 ).

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63 From the results it can be clearly seen that the structure has a dynamic behavior and can be tuned based on the applied voltage which change s the doping level of graphene. This is an obvious advantage o f the proposed graphene metasurface compared to metallic plasmonic based structures which are not tu nable and can not be changed after fabrication. Figure 4.18. Beam splitter, transmission spectrum for each scenario (orders are the same for all the figures in this section).

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64 5. Graphene Plasmonic Apertureless Scanning Probes 5.1. Introduction The resolution of conventional optical microscopy is governed by Rayleigh diffraction which states that the resolution of an optical system depends on the wavelength and the numerical aperture of the lens One important conclusion is that light will diffract when it propagates through a hole or a slit which is smaller than appro ximately half of its wavelength: (5.1) where n is the refractive index and is the angle of incidence Various endeavors have been pursued to overcome this fundamental optical limitation ; i n particular, near field scanning optical microscopy (NSOM) shows great promise in overcoming the diffraction limit, by capturing the evanescent portion of the spatial spectrum of an image, before it rapidly decays away with distance from the object. T he primary challenge for NSOM is providing highly localized electromagnetic energy near the tip of the scanning probe. In this chapter a novel approach towards design ing high throughput apertureless NSOM tips based on graphene plasmonics is presented Specif ically, localized graphene plasmon waves are combined with nanofocusing of surface plasmon polaritons (SPPs) to squeeze the lateral surface plasmon waves into the apex of the tip. The near field electromagnetic properties of the designed probes are charact erized in detail and compared to the conventional apertureless and metallic plasmonic probes. Results show the applicability and versatility of the graphene plasmonic probe in engineering near field scanning probes. The designed structure can have many app lications in different areas of science, including nanosensing, light sources, optical imaging, quantum optics, and tip enhanced Raman spectroscopy.

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65 5.2. Simulation R esults and Analysis Figure 5. 1 Artistic representation of the NSOM tip coated with a layer of graphene In this thesis two NSOM probes are considered Structure one (type A) consists of a SiO 2 tip with permittivity SiO2 = 2.1638 at 960 nm and a layer of graphene on the lateral surface. The second structure (type B) includes an extra layer of low refractive index dielectric on the lateral surface to provide stronger confinement. The key factor for graphene plasmonic excitation at this wavelength of operation (960 nm) is high doping (Fermi) level and appropriate angle of excitation. Optimal design parameters including optical and electrical properties of graphene and dimensions are obtained with a finite element metho d (FEM) solver.

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66 Figure 5.2 shows the calculated refractive index of graphene for different values of Fermi level s at the frequency range of 20 THz to 900 THz. Here, Graphene is assumed to have 1 nm thickness and is doped with Fermi level of 1 eV. In order to excite surface plasmon waves and focus them towards the probe tip the lateral surface is excited with 960 nm light source. Figure 5.3 shows the magnetic field on the surface of the tip for type the A and type B probes. Figure 5. 2 Refractive index of graphene calculated based on the Kubo formula (right) Real, (left) I maginary

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67 The s urface plasmon waves are localized at the apex of the NSOM tip in both cases Although type B has stro nger confinement it requires an extra dielectric layer that is 30 nm thick dielectric on top of the probe. The electric fi e l d plots are presented in Figu re 5.4. The electric energy density is calculated at a 20 nm distance from the tip. T he designed graphene plasmonic tips p rovide a higher electric field density near the tip. The e lectrical energy density of type B is the highest and has been augmented mor e than 10 6 times compared to an apertureless probe For type B the amplitude is one tenth tha t of type A, but it exhibits lower full width half maximum ( FWHM ) meaning a confinement of the near fi e l d to a smaller region. Figure 5. 3 magnetic field |B z| (a) type A (just SiO2), (b) type B (SiO2 covered with low refractive index dielectric)

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68 The FW HM levels for a conventional apertureless tip, tip type A and tip type B probes are calculated and presented in Figure 5.4. The calculated FWHMs of the three probes are: 63.64 nm (conventional apertureless probe), 27.88 nm (type A) and 24.97 nm (type B). | E | Ey | E x| | E x| | E x| | E y| | E y| | E y| FWHM=27.88 nm FWHM=24.97 nm FWHM=63.64 nm Figure 5.4. Electric field and the calculated FWHM for the designed pro bes.

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69 In Fig. 5.5, the calculated FWHM of each probe corresponding to various tip sample distances is shown. Within 10 and 30 nm from the probe tip, the FWHM of probe type A and B is less than 60 nm, and the two designed probes still hold a large energy density enhancement (over six orders of magnitude compared to the simple aperture probe). FWHM=41.1 FWHM= 72 3 1

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70 From Figure 5.5 and based on the calculated FWHM values, it can be inferred that Type B possesses the lowest FWHM, and it is t hus capable of concentrating the electric field into smaller and more compact area. On the other hand, by increasing the distance from the tip, Type B loses its advantages. At 30 nm from the tip, Type A has lower FWHM. In most FWHM= 23 0 1 nm FWHM= 29 22 nm FWHM= 20 08nm FWHM= 49 8 nm Figure 5.5. FWHM with respect to the distance. (top) conventional apertureless (middle) Type A, (bottom) Type B

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71 NSOM devices the distance is usually less than 20 nm so it may be concluded that Type B has better performance in terms of the power concentration per area. In conclusion, we have shown that graphene surface plasmon s provide excellent near field enhancement for a NSOM tip compared t o a conventional apertureless probe tip [8] Compared to other reported NSOM tips designed using metallic plasmonics, the proposed tip has smaller FWHM and can be used for various bio logical imaging, and fabrication applications. 6. Conclusion a nd Outlook In this thesis, the fundamental physics and theory of surface plasmon waves and graphene plasmonics have been review ed and novel tunable optoelectronic devices were designed. A graphene based plasmonic nanoribbon filter and an optical beam splitter were proposed based on the designed metasurface and the electromagnetics of the wave propagation were numer ically analyzed The filtering characteristics of the proposed filter were investigated according to different parameters such as the nanoribbon width, wo rking frequency, and chemical potential Besides the electronics properties of the graphene plasmonics, bio and nano imaging aspect of it were also investigated in this work. An efficient and high throughput microscopy probe was proposed which can be used for a variety of applications. For future wo r k, it is desired to extend the fundamental knowledge about the solid state physics and graphene physics. Other applications of the proposed graphene metasurface ha ve already been investigated. The proposed stru cture shows great performance when it is used as an optical coupler. O ther applications need to be investigated. Supplementary numerical methods such as finite difference time domain can also be exploited to verify the results.

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72 The next step for the graphe ne plasmonic microscopy tip, is to extend analysis to a 3D domain and simulate the performance of the tip and ultimately fabricate it and verify the physical performance.

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73 REFERENCES [1] H. T. Chorsi and S. D. Gedney, "Efficient high or der analysis of bowtie nanoantennas using the locally corrected Nyström method," Optics Express, vol. 23, pp. 31452 31459, 2015/11/30 2015. [2] A. N. Grigorenko, M. Polini, and K. S. Novoselov, "Graphene plasmonics," Nat Photon, vol. 6, pp. 749 758, 11//print 2012. [3] C. Kittel and D. F. Holcomb, "Introduction to Solid State Physics," American Journal of Physics, vol. 35, pp. 547 548, 1967. [4] S. A. Maier, Plasmonics: fundamentals and applications : Springer Science & Business Media, 2007. [5] X. Li J. Song, and J. X. J. Zhang, "Design of terahertz metal dielectric metal waveguide with microfluidic sensing stub," Optics Communications, vol. 361, pp. 130 137, 2/15/ 2016. [6] K. Ziegler, "Minimal conductivity of graphene: Nonuniversal values from the Kubo formula," Physical Review B, vol. 75, p. 233407, 06/15/ 2007. [7] S. Y. Chou, P. R. Krauss, and P. J. Renstrom, "Nanoimprint lithography," Journal of Vacuum Science & Technology B, vol. 14, pp. 4129 4133, 1996. [8] Y. Lee, A. Alu, and J. X. J. Zhang, "Efficient apertureless scanning probes using patterned plasmonic surfaces," Optics Express, vol. 19, pp. 25990 25999, 2011/12/19 2011.

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74 APPENDIX A MATLAB Script: Graphene Band Structure clear all ; close all ; t=1; lattice=1.446; eps_en1=3; eps_en2=3; K_vec_x=linspace( 2*pi/(lattice),2*pi/(lattice),100); K_vec_y=linspace( 2*pi/lattice,2*pi/lattice,100); [K_meshx,K_meshy]=meshgrid(K_vec_x,K_vec_y); energy_mesh=NaN([size(K_meshx,1),size(K_meshx,2),2]); for a=1: size(K_meshx,1) energy_mesh( :,a,1)=(eps_en1+eps_en2)/2+sqrt(((eps_en1 eps_en2)^2)/4+4*t^2*((cos(K_meshy(:,a)/2*lattice).^2)+ ... cos(sqrt(3)/2*K_meshx(:,a)*lattice).*cos(K_meshy(:,a)/2*lattic e))+1/4); energy_mesh(:,a,2)=(eps_en1+eps_en2)/2 sqrt(((eps_en1 eps_en2)^2)/4+4*t^ 2*((cos(K_meshy(:,a)/2*lattice).^2)+ ... cos(sqrt(3)/2*K_meshx(:,a)*lattice).*cos(K_meshy(:,a)/2*lattic e))+1/4); end % set(handles.axes_mesh,'Color','w') surf(K_meshx,K_meshy,real(energy_mesh(:,:,1))) hold on surf(K_meshx,K_meshy,real(energy_m esh(:,:,2))) shading interp colormap (jet)

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75 APPENDIX B MATLAB Script: Kubo Conductivity Formula % Calculation of Conductivity of Graphene using Kubo Formula % clc clear all ; close all j = sqrt( 1); e = 1.6e 19; K_B = 1.3806503e 23; %Boltzmann constant T = 300; %Room temprature hb = (6.626e 34)/(2*pi); % Dirac constant tau = 0.1e 12; %momentum relaxation time ga = 1/(2*tau); %scattering rate % gamma = 2*ta; % Gusynin J. Phys.: Cond. Mat. sigma_min = 6.085e 5; vf = 10^6; %Fermi velocity enginee ring in graphene by substrate modification Choongyu Hwang w = pi*2e12*linspace(5,400,200); m = length(w); mu_c = 1.6e 19*[ 0.5]; n = length(mu_c); mu_ct = repmat(mu_c,m,1); wt = repmat(transpose(w),1,n); % Intraband term calculations sigma_d_intra = ... j*((e^2*K_B*T)./(pi*hb^2*(wt j*2*ga))).*((mu_ct)/(K_B*T) ... +2*log(exp( mu_ct/(K_B*T))+1)); %intraband term % Interband term calculations sigma_d_inter = zeros(m,n); eps = 1.6e 19*linspace(0,10,600000); q = length(eps); for i = 1:n muc = mu_c(i) ; f_d_meps = 1./(1+exp(( eps muc)/(K_B*T))); f_d_peps = 1./(1+exp((eps muc)/(K_B*T))); for k = 1:m sigma_d_inter(k,i) = ... trapz(eps, (j*e^2*(w(k) j*2*ga)/(pi*hb^2)) ... *(f_d_meps f_d_peps)./((w(k) j*2*ga)^2 4*(eps/hb).^2));

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76 end end sigma_tot = sigma_d_in ter+sigma_d_intra; % total conductivity C = { 'k' 'r' 'b' 'c' 'g' 'm' 'y' }; p1 = zeros(1,n); pp1 = zeros(1,n); s1 = cell(1, n); p2 = zeros(1,n); pp2 = zeros(1,n); s2 = cell(1, n); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(1) % subplot(2,1,1) for i = 1:n hold on p1(i) = ... plot(1e 12*w/2/pi,(transpose(real((sigma_d_inter(:,i)))))/ ... (e^2/hb/4), 'Color' C [2] 'Linewidth' ,2);hold on pp1(i) = ... plot(1e 12*w/2/pi,(transpose(real(( sigma_d_intra(:,i)))))/ ... (e^2/hb/4), 'Color' ... C [2] 'Linewidth' ,2);hold on s1 [2] = sprintf( 'mu {c,%d} = %d meV' ,i,1e3*mu_c(i)/(1.6e 19)); end ind = 1:n; % axis square box on xlabel( 'f (THZ)' 'fontsize' ,20, 'fontweight' 'b' ); ylabel( 'Re( \ sigma)' 'fontsize' ,20, 'fontweight' 'b' ); legend(p1(ind), s1{ind}); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(2) % subplot(2,1,2) % hold on for i = 1:n hold on p2(i) = ... plot(1e 12*w/2/pi, (transpose(imag((sigma_d_inter(:,i)))))/ ... (e^2/hb/4), 'Color' C [2] 'Linewidth' ,2);hold on pp2(i) = ... plot(1e 12*w/2/pi, (transpose(imag((sigma_d_intr a(:,i)))))/ ... (e^2/hb/4), 'Color' ... C [2] 'Linewidth' ,2);hold on s2 [2] = sprintf( 'mu {c,%d} = %d meV' ,i,1e3*mu_c(i)/(1.6e 19));

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77 end % axis square box on xlabel( 'f (THZ)' 'fontsize' ,20, 'fontweight' 'b' ); ylabel( 'Im( \ sigma)' 'fontsize' ,20, 'fontweight' 'b' ); % legend(p2(ind), s2{ind}); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure (3) plot(1e 12*w/2/pi,real(sigma_tot)) title( 'Real p art of conductivity in S' ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure (4) plot(1e 12*w/2/pi,imag(sigma_tot)) title( 'Imaginary part of conductivity in S' )