Plasmonic response of graphene nanostructures

Universidad de Santiago de Compostela Departamento de Física Aplicada Plasmonic response of graphene nanostructures Memoria presentada por Iván Silveiro Flores para optar al grado de Doctor en Física Director: Prof. F. Javier García de Abajo Codirector: Dr. Sukosin Thongrattanasiri Santiago de Compostela, diciembre de 2015 The Institute of Photonic Sciences eXefg_fkfe`Zj%\j iii La presente memoria de tesis se realizó en el Nanophotonics Theory Group en: •Instituto de Química-Física “Rocasolano” (CSIC), Serrano 119, 28006; Madrid, España (octubre, 2011 – septiembre, 2013). •The Institute of Photonic Sciences (ICFO), Mediterranean Technology Park, Av. Carl Friedrich Gauss 3, 08860; Castelldefels (Barcelona), España (octubre, 2013 – diciembre, 2015). v Universidad de Santiago de Compostela Departamento de Física Aplicada D. Francisco Javier García de Abajo, Profesor de Investigación en el Institute of Photonic Sciences (ICFO), como director; y Dña. María Teresa Flores Arias, Profesora Titular de Universidad en el Dpto. de Física Aplicada de la Universidad de Santiago de Compostela, como tutora, CERTIFICAN: Que la memoria titulada “Plasmonic response of graphene nanostructures” realizada por Iván Silveiro Flores en el Instituto de Química-Física “Rocasolano” del CSIC en Madrid y el Institute of Photonic Sciences (ICFO) en Castelldefels (Barcelona) para el departamento de Física Aplicada de esta Universidad, ha sido revisada y está en disposición de ser depositada como Tesis Doctoral para la obtección del grado de Doctor en Física. Fdo.: F. Javier García de Abajo Fdo.: M. Teresa Flores Arias Director Tutora Fdo.: Iván Silveiro Flores Doctorando vii A mis padres y a mi hermano ix If you want to find the secrets of the universe, think in terms of energy, frequency, and vibration. Nikola Tesla List of Figures 1.1 Carbonallotropes............................. 16 1.2 Graphenelattice ............................. 19 1.3 Graphene band diagram . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 Density of states in graphene . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 Conductivity and dielectric function of graphene . . . . . . . . . . . . 28 1.6 Types of plasmons excited in graphene . . . . . . . . . . . . . . . . . 33 1.7 Reflection and transmission of graphene . . . . . . . . . . . . . . . . 35 1.8 Dispersion relation of graphene . . . . . . . . . . . . . . . . . . . . . 37 1.9 Localized surface plasmons (LSPs) in a graphene nanodisk . . . . . . 41 2.1 Plasmon wave function of graphene nanoribbons . . . . . . . . . . . . 49 2.2 LSPs in interacting pairs of graphene nanoribbons . . . . . . . . . . . 52 2.3 LSP energy in dimers and arrays of graphene nanoribbons . . . . . . 53 2.4 Absorbance of a bilayer array of graphene nanoribbons . . . . . . . . 54 2.5 LSPs in backgated graphene nanoribbons . . . . . . . . . . . . . . . . 59 2.6 LSPs in pairs of co-planar parallel graphene nanoribbons of opposite polarity .................................. 61 2.7 LSPs in individual graphene nanoribbons subject to a uniform externalelectricfield.............................. 63 2.8 Induced charge density of a graphene nanodisk under uniform doping and inhomogeneous doping by a uniform potential . . . . . . . . . . . 69 2.9 LSPs in a neutral graphene nanodisk exposed to a neighboring externalpointcharge.............................. 70 xvii xviii LIST OF FIGURES 2.10 Graphene-disk plasmon frequencies for three different doping configurations .................................. 71 2.11 Optical dispersion of periodically doped graphene . . . . . . . . . . . 73 2.12 Plasmonic bands in periodically doped graphene by point charges . . 77 2.13 Local density of optical states in periodically doped graphene by point charges................................... 80 3.1 Quantum LSPs in individual graphene nanoribbons . . . . . . . . . . 85 3.2 Quantum LSPs in interacting co-planar graphene nanoribbon dimers 86 3.3 Evolution of the LSP energy with the separation distance in co-planar armchair-edged dimers of graphene nanoribbons . . . . . . . . . . . . 87 3.4 Absorbance of a co-planar array of graphene nanoribbons with zigzag andarmchairedges............................ 89 4.1 Secondand third-harmonic generation in graphene . . . . . . . . . . 93 4.2 Linear and nonlinear plasmonic response of equilateral graphene nanotriangles .................................101 5.1 Surface-enhanced infrared absorption (SEIRA) spectroscopy with graphene plasmons. .................................105 5.2 Doping dependence in the absorption cross-section of molecules in SEIRAspectroscopy ...........................107 5.3 Molecular sensitivity of the doping-dependent frequency-integrated absorption in SEIRA spectroscopy . . . . . . . . . . . . . . . . . . . . 109 5.4 Surface-enhanced Raman scattering (SERS) with graphene plasmons. 111 A.1 Electrostatic potential over extended graphene by an external point charge ...................................119 C.1 Energy-level diagram of infrared absorption, Rayleigh scattering, and Ramanscattering.............................125 List of Tables 1.1 Analytical approximations for the parameters ξjand ηjof the electrostaticscalinglaw............................ 46 4.1 Polarizability unit conversion factors . . . . . . . . . . . . . . . . . . 92 xix List of Acronyms AC Armchair ac Alternating current ATR Attenuated total reflection BEM Boundary-element method BTE Boltzmann transport equation BZ Brillouin zone C-C Carbon-to-carbon CNT Carbon nanotube CVD Chemical vapor deposition dc Direct current DSDA Discrete surface-dipole approximation EELS Electron energy-loss spectroscopy e-h Electron-hole FWHM Full width at half maximum H-H Hydrogen-to-hydrogen LDOS Local density of optical states xxi xxii LIST OF ACRONYMS LSP Localized surface plasmon NIR Near infrared PWF Plasmon wave function RPA Random-phase approximation SEIRA Surface-enhanced infrared absorption SERS Surface-enhanced Raman scattering SHG Second-harmonic generation SM Supplementary material SPP Surface plasmon polariton STM Scanning tunneling microscope TE Transverse electric THG Third-harmonic generation TM Transverse magnetic UV Ultraviolet ZZ Zigzag Resumen El grafeno, considerado por muchos como el material del futuro, se define como una lámina plana constituida por átomos de carbono fuertemente entrelazados en una red bidimensional con forma de panal de abeja. Desde que en 2004 los físicos rusos K. Novoselov y A. Geim de la Universidad de Manchester (ambos galardonados con el Premio Nobel de Física en 2010) sintetizaran por primera vez láminas aisladas de grafeno mediante exfoliación mecánica, este revolucionario material ha despertado un gran interés debido a sus extraordinarias propiedades optoelectrónicas muy útiles para la nanofotónica (rama de la física que estudia la interacción de la luz con la materia en el rango del nanómetro, esto es, en distancias mil millones de veces más pequeñas que el metro). La naturaleza 2D de este alótropo de carbono en combinación con su singular estructura atómica, dan lugar a una poco corriente relación de dispersión lineal muy diferente a la típica parabólica de los metales nobles. Al producirse la interacción entre la luz y la materia (cuya longitud característica Dha de ser necesariamente más pequeña o del orden de la longitud de onda de la propia luz incidente, i.e.,D.λ), se genera una serie de interesantes fenómenos electromagnéticos entre los cuales se halla la excitación del objeto de estudio de esta tesis: el plasmón (i.e.,la oscilación colectiva de los electrones en metales nobles o grafeno). Debido al intenso confinamiento de estas oscilaciones eléctricas (en distancias por debajo del límite de difracción de la luz), los plasmones estimulan una fuerte interacción luz-materia e incrementan notablemente el campo eléctrico inducido. Es necesario destacar también que los plasmones son muy sensibles a la forma y tamaño de las nanoestructuras que los sustentan, al entorno dieléctrico y a la cantidad de electrones participando en la oscilación colectiva (relacionados éstos a su vez con la estructura de bandas electrónicas del material que los contiene). Un estricto control 1 2RESUMEN de estos parámetros es indispensable para poder conocer en profundidad la respuesta de cualquier nanoestructura capaz de sustentar plasmones. Centrémonos ahora en el elemento químico que conforma la estructura de grafeno: el carbono. Su isótopo más común, el 12 6C, posee 6 protones y 6 neutrones en el núcleo atómico, y 6 electrones moviéndose libremente alrededor de éste distribuidos en diferentes orbitales electrónicos. La configuración electrónica del átomo de carbono en el estado fundamental es 1s22s22p2, con dos electrones llenando por completo el orbital 1s, otros dos llenando el orbital 2s, y los dos electrones restantes ocupando distintos orbitales 2p. Sin embargo, cuando varios átomos de carbono están próximos entre sí, la interacción entre orbitales hace que sea más favorable energéticamente que un electrón del orbital 2s se excite hasta el orbital 2p desocupado, formándose de esta manera enlaces covalentes entre diferentes átomos vecinos. Por lo tanto, pasamos a tener cuatro estados cuánticos idénticos que pueden combinarse entre sí formando diferentes orbitales híbridos spi(con i= 1,2 o 3). La red atómica con forma de panal de abeja del grafeno es el resultado de la hibridación sp2entre un orbital s y dos orbitales p por cada átomo de carbono, a partir de la cual se forman enlaces covalentes σcon un ángulo característico de 120◦ entre átomos vecinos. Este enlace fuerte es el responsable de la extrema dureza de la red bidimensional de átomos de carbono, los cuales permanecen separados una distancia a0= 1.421 ◦ A. El orbital p (o también denominado π) que queda libre se orienta perpendicularmente a la lámina de átomos y puede formar un enlace débil con los orbitales de los carbonos de otras láminas mediante interacción de van der Waals. La singular estructura de bandas electrónicas del grafeno está producida por estos orbitales πy consta de dos bandas: una inferior, o también llamada banda de valencia, y una superior o de conducción. A bajas energías sus formas se asemejan a las de dos conos invertidos tocándose en un único punto. A este punto se le denomina comúnmente punto de Dirac (su posición exacta en el espacio de momentos se halla en el vértice del hexágono que conforma la primera zona de Brillouin del grafeno) y su importancia reside en que determina el nivel de Fermi del grafeno en estado neutro (también denominado grafeno pristino o grafeno sin dopar). En este estado de neutralidad, la banda de valencia está completamente llena con electrones π deslocalizados, mientras que la de conducción permanece vacía. En consecuencia, podemos tratar al grafeno en estado neutro como un semiconductor con una banda RESUMEN 3 prohibida de valor nulo, de manera que únicamente son posibles transiciones interbanda de pares electrón-hueco. La oscilación colectiva de los electrones deslocalizados πen grafeno da lugar a los llamados plasmones intrínsecos πcuyas energías oscilan entre los 4.5 y 7 eV, lo que los sitúa en la región ultravioleta del espectro electromagnético. Además, estos plasmones son muy poco ajustables lo que limita su relevancia en nanofotónica. Sin embargo, cuando agregamos electrones adicionales al grafeno, éstos comienzan a llenar estados desocupados en la banda de conducción hasta un cierto nivel que determina el nuevo valor del nivel de Fermi EF=~vF√πn, donde nes la densidad por unidad de área de estos electrones adicionales y vF≈c/300 su velocidad de desplazamiento (nótese que debido a la relación de dispersión lineal, los electrones adicionales son tratados como partículas sin masa). De este modo, una banda prohibida de anchura 2EFse abre y, además de las ya mencionadas transiciones inter-banda, ahora también son posibles transiciones intra-banda de pares electrón-hueco. El proceso de agregar nuevos electrones se conoce como dopado (más concretamente, este caso se denomina dopado tipo n) y, debido a la simetría de bandas, se produce el mismo efecto cuando electrones πson extraídos de la banda de valencia (i.e.,dopado tipo p, o mediante huecos en vez de electrones). El hecho de que el nivel de Fermi EF(también llamado nivel de dopado) sea ajustable con nasí como la peculiar relación de dispersión lineal, son propiedades únicas del grafeno que lo distinguen notablemente de los metales nobles estudiados habitualmente en nanofotónica. A diferencia del grafeno, nen los metales nobles apenas es ajustable y, además, cambios ostensibles en napenas afectan de manera significativa a las propiedades optoelectrónicas del metal. A las oscilaciones colectivas de los electrones adicionales en grafeno dopado se las conoce como plasmones extrínsecos o plasmones de Dirac. Sus frecuencias de resonancia abarcan desde los THz hasta el infrarrojo cercano y habitualmente se suelen dividir en dos subgrupos distintos dependiendo de sus propiedades de propagación: los polaritones del plasmón de superficie (SPPs, por su acrónimo en inglés) y los plasmones de superficie localizados (LSPs). Los primeros son modos electromagnéticos que se propagan en láminas de grafeno extendido situadas en la interfaz de separación entre dos medios dieléctricos. Paradójicamente, estos plasmones no pueden ser excitados directamente con luz externa debido a los diferentes vecto- 10 ABSTRACT shape of two inverted cones touching at one point. This point is the so-called Dirac point and marks the Fermi level in the neutral state. In this state, the valence band is completely filled with delocalized πelectrons, while the conduction band is empty. The absence of an optical gap between both bands permits one to treat graphene in this neutral state as a zero-energy bandgap semiconductor, where only electron-hole pair interband transitions can occur. The collective oscillation of the πelectrons gives rise to the so-called πintrinsic plasmons with energies &4.5eV, confining them in the ultraviolet region of the electromagnetic spectrum. Moreover, these plasmons present a low tunability, so their relevance for nanophotonics is limited. Interestingly, when extra electrons are added to graphene, they start filling unoccupied states in the conduction band up to a certain level that corresponds to the new shifted Fermi level EF=~vF√πn, where nis the density per unit area of these extra electrons, and vF≈c/300 their velocity. Therefore, an optical gap of width 2EFopens, and electron-hole pair intraband transitions are enabled in addition to interband. The process of adding new electrons is known as doping, and due to the symmetry of the bands, it produces the same effect as when removing πelectrons from the lower cone (i.e.,doping with holes instead of electrons). The tunability of the Fermi level with n, as well as the peculiar linear dispersion relation, are unique properties of graphene in comparison with typical noble metals. In contrast to graphene, noble metals do not show a salient tunability as significant changes in nbarely affect their global optoelectronic properties. The collective oscillations engaging extra electrons in doped graphene are known as Dirac plasmons, and their resonances embrace from THz to near infrared frequencies. They are commonly subdivided into two different subgroups depending on their propagating features: surface plasmon polaritons (SPPs) and localized surface plasmons (LSPs). The former are propagating electromagnetic modes sustained by extended graphene layers acting as an interface layer between two different dielectric media, and that cannot be directly excited by external light due to momentum mismatch. The latter are confined modes sustained by finite nanostructures that, in turn, can be effectively excited by light. Dirac plasmons present a bunch of appealing properties in comparison with those of noble metals (e.g.,they have longer lifetimes and higher quality factors, they show strong nonlinearities, and they possess the ability to boost the electric field enhancement by several orders of magnitude). ABSTRACT 11 Furthermore, SPPs display a much shorter wavelength than the incident light, which translates into a larger degree of field confinement. The control of these properties has spurred the development of new theoretical models able to predict the plasmonic response of graphene, promoting its application to optical detection, sensing, nonlinear optics, and light modulation. The optical dispersion of graphene Dirac plasmons is governed by the dynamics of the extra electrons in the conduction band. The simplest model capable of giving a fairly accurate description is the Drude model, where only intraband transitions at zero temperature are considered. Despite its simplicity, this model is considered a good approximation at photon energies well below the doping level. If this condition is not satisfied, a more elaborate and realistic model like the random-phase approximation shows up including the effects of a finite temperature and interband transitions. We present an extensive comparison between both models for different doping levels in the first chapter of this thesis. The theoretical electromagnetic modeling of graphene in the classical limit is based on the solution of macroscopic full-retarded Maxwell’s equations. Throughout this thesis, the exact solution of Maxwell’s equations is obtained by the boundaryelement method. Within this procedure, a system of surface-integral equations is evaluated at the boundaries of a non-magnetic, local (i.e.,independent of the parallel wave vector kkof the incident light), and linear (i.e., polarization responding linearly to the electric field of the incident light) graphene nanostructure with an arbitrary shape. After determining the boundary conditions, a rigorous numerical solution is accomplished through a finite discretization of the graphene boundaries, with the carbon layer modeled as a thin film, and with a denser surface grid placed near the film edges. Despite the versatility of this method, finding a solution can be a highly time-consuming process. However, for nanostructures with characteristic length Dmuch smaller than the incident light wavelength, we can safely work in the electrostatic limit (i.e.,non-retarded approach). Here, the interaction between light and graphene is regarded as instantaneous, and the problem reduces to solving the Poisson equation with the appropriate boundary conditions. Under these assumptions, we observe that the plasmonic response of graphene derived from electrostatics is in excellent agreement with the full numerical solution of Maxwell’s equations. Moreover, in the last section of the first chapter, we derive a useful analy- 12 ABSTRACT tical electrostatic scaling law that permits obtaining the resonance frequencies of the LSPs for any arbitrary shape by just knowing the size, doping level, and dielectric environment of the graphene nanostructure. Specifically, we show that the resonance frequencies of the LSPs fulfill approximately the expression ωp∝»EF/D, which already illustrates their strong tunability. In actual experiments, the doping of graphene is generally achieved via chemical methods or electrostatic gating. In the latter, a potential difference with respect to a backgate is applied to graphene, thus inducing a perpendicular electric field E0that is uniformly applied to one side of graphene, so that an induced doping density of electrons n=−E0/4πe is distributed to screen the field completely. For simplicity, LSPs are extensively studied in the literature assuming a uniform distribution of nover the graphene nanostructure. However, we find that nis actually distributed inhomogeneously with a profile depending on the specific geometrical configuration, thus affecting the plasmonic response of the carbon film. We analyze this behavior in the second chapter of this thesis, where we classically study different realistic doping configurations in nanoribbons, nanodisks, and extended graphene layers. We find that Dirac plasmons are sensitive to inhomogeneous doping distributions, and thus, the particular plasmonic response needs to be considered for the correct design of device applications. When the characteristic length of the nanostructure is of the order of the Fermi wavelength (i.e.,the de Broglie wavelength in the vicinity of the Fermi energy, which for graphene is λF=»4π/n =hvF/EF≈10.33nm when EF= 0.4eV; note that its tunability with ncontrasts with the nearly constant value in noble metals: λF≈0.52nm in gold), classical electromagnetism is no longer valid, and a quantummechanical approach for the description of the plasmonic response is necessary. In the third chapter of this thesis, we provide extensive quantum calculations (including nonlocal and edge effects, where we can distinguish between armchair or zigzag terminations) through a tight-binding model, in combination with the random-phase approximation. In particular, we notice that for narrow single and interacting nanoribbons, classical calculations disregard several physical effects that affect the plasmonic response of graphene. For example, when the doping level is lower than the resonance frequencies of the lowest-order dipolar LSPs, these are strongly quenched by zigzag edges and decay through the excitation of electronic zero-energy edge ABSTRACT 13 states. Besides, for armchair terminations, LSPs are also affected by nonlocalities leading to slight blueshifts and broadening. Furthermore, we note that nonlocal effects play an important role in interacting islands at short separations, which give rise to remarkable corrections in the resonance frequencies of the sustained LSPs. In fact, we conclude that graphene is a suitable platform for studying the quantum effects and nonlocalities on LSPs. The fourth chapter of this thesis is devoted to the study of the strong nonlinear plasmonic response of doped graphene. Specifically, we present a review of three of the most important nonlinear processes: second-harmonic generation, thirdharmonic generation, and Kerr effect. Moreover, starting from the Boltzmann transport equation, we extend the electrostatic scaling law formalism to nonlinear orders of the graphene conductivity and electric polarizability. We also provide a detailed comparison between the classical and quantum nonlinear plasmonic response of small equilateral graphene nanotriangles. We find that the classical approach underestimates nonlinearities except for low levels of doping. Our results reveal the crucial relevance of a comprehensive knowledge of nonlinear effects in graphene. Finally, in the fifth chapter, we show the outstanding potential of graphene LSPs to resolve the chemical identity of molecules. It is known that the identification of molecules usually involves the employment of inefficient sensing techniques including the use of costly spectrometers and laser sources. In order to avoid this, we present a new sensing mechanism that simply requires infrared lamps and doped graphene nanodisks. We prove that graphene LSPs allow the molecule to enhance greatly its capability to absorb or inelastically scatter impinging light changing its roto-vibrational energy. These absorption and scattering processes are the elementary principles of the spectroscopy techniques known as surface-enhanced infrared absorption and surface-enhanced Raman scattering, for which we obtain intensity enhancements of ∼103and ∼104, respectively. We claim that by just integrating the sensing signal over a broadband spectral range as a function of the graphene doping level EF, we can identify the chemical fingerprints of the molecule with an energy resolution given by the spectral width of the graphene LSPs. In conclusion, we consider that our results contribute to broadening the theoretical understanding of the plasmonic response of graphene under the usual assumption of uniform doping including the effects of nonlocalities and nonlinearities, and also 14 ABSTRACT under novel realistic inhomogeneous doping conditions. We believe that this thesis paves the way for future experimental studies of graphene-based nanodevices. Chapter 1 Introduction In this initial chapter, we discuss the general properties of graphene, the key material to which this thesis is devoted. In particular, we start by succinctly analyzing its history in the context of the discovery of different carbon allotropes (i.e.,diverse structural forms of carbon). We continue with a brief review on how to grow graphene experimentally, and then we study its singular band structure and multiple optoelectronic properties derived from its bidimensional character. For this purpose, we use a macroscopic classical approach for the electromagnetic description of graphene. Additionally, the quantum-mechanical model used in further chapters for a detailed microscopic study is also presented here. Finally, we analyze the main properties of the surface plasmons sustained by graphene. 1.1 History of graphene and other carbon allotropes The first carbon allotrope known in history was the 3D graphite [see Fig.1.1(c)]. It was discovered in England in the 16th century [1] and chiefly used in pencils. Although the functionality of pencils promptly spread all over the world, the ongoing term “graphite” was not conceived until 1789 by the geologist A. Werner, thus remarking its use for graphical reasons [2]. The atomic structure of graphite consists of stacked graphene layers and its utility for writing derives from the weak van der 15 16 CHAPTER 1. INTRODUCTION (a) (b) (c) Figure 1.1: Plots of different carbon allotropes where each sphere represents a carbon atom. (a) 0D molecule of C60, (b) 1D carbon nanotube, and (c) 3D graphite. Waals forces between the different sheets. In fact, after pressing a pencil against a sheet of paper, stacks of graphene are exfoliated from the graphite, and it is actually possible to find individual graphene layers adhered to the surface. Fullerenes are carbon molecules arranged in a spherical-like shape so that they can be considered as a 0D structure. The most representative fullerene structure is the C60 molecule also called “buckyball” [see Fig.1.1(a)]. This C60 molecule was first detected in 1985 [3], although its existence had been predicted previously [4]. Besides, fullerenes can be directly constructed from graphene with the replacement of some hexagons by pentagons creating positive curvature defects that result in a wrapped-up structure [5]. Single-walled carbon nanotubes (CNTs) were discovered in 1991 [6]. They present only hexagons wrapped into a seamless cylinder [see Fig.1.1(b)], so that they are regarded as 1D cylindrical molecules with a diameter in the order of the nanometer. However, due to the lack of tools for searching carbon flakes, we had to wait until 2004 for the milestone of the experimental isolation of the 2D carbon allotrope: graphene [7]. Graphene is a one-atom-thick monolayer of carbon atoms tightly arranged in a purely bidimensional honeycomb lattice [see Fig.1.2(a)]. The physicists K. Novoselov and A. Geim from Manchester University (both awarded the Physics Nobel Prize in 2010) showed that, by just rubbing graphite over a silica substrate in a process known as “mechanical exfoliation”, graphene could be readily detected by regular microscopy techniques [8]. Soon after, simultaneously with P. Kim from Columbia [9], they found evidence of the quantum Hall effect in graphene [10]. 1.2. SYNTHESIS OF GRAPHENE 17 1.2 Synthesis of graphene In this section, we review the main techniques so far developed for the synthesis of graphene. Some of them are feasible with modest means while others require advanced experimental equipment. Furthermore, the resulting samples do not present exactly the same properties. Specifically, the main synthesis techniques are: •Mechanical exfoliation: This is the most straightforward and, as mentioned in the previous section, the original technique used for the synthesis of graphene. Its principal advantages are that it is an easy way of producing graphene and that the resulting layers present high quality and great electrical properties. However, this technique presents a serious disadvantage: the distribution of the layers over the substrate is completely random and the subsequent identification of single layers of graphene is very time-consuming and difficult to scale up. •Epitaxial growth: This promising technique consists of exposing hexagonal-like silicon carbide substrates (SiC) to temperatures ∼1300◦Cso that the silicon atoms evaporate and the remaining carbon atoms form graphene [11]. Unfortunately, the charge distribution of the remaining graphene nanostructures is not always uniform. •Chemical vapor deposition (CVD): This is the most popular method to produce relatively high-quality graphene on a large scale. In this technique, disassociated carbon atoms in gas phase are accumulated on a substrate at a temperature of ∼1000◦C. The main problem with this technique is the complicated separation of graphene from the substrate once the system has cooled down. 1.3 Optoelectronic properties 1.3.1 sp2hybridization Carbon, the elementary basis of all the organic molecules, is the fundamental component of graphene. Its most common isotope, 12 6C, possesses 6 protons and 6 neutrons in the atomic nucleus, and 6 electrons moving freely in different orbitals. Thus, the electronic configuration of a carbon atom in the ground state is 1s22s22p2, with two electrons confined in the inner orbital 1s, other two electrons in 2s, and the two remaining allocated in the orbitals 2p. The energy difference between the 2s orbital 18 CHAPTER 1. INTRODUCTION and the three 2p orbitals (we name them as 2px,2py, and 2pz) is ∼4eV, hence in the ground state it is more favorable in terms of energy that two electrons fill the orbital 2s and the other two stay in distinct orbitals 2p. However, when various carbon atoms are in close proximity, it is more favorable to excite one electron from the 2s orbital to the remaining 2p empty orbital, thus forming covalent bonds between electrons of different atoms. Since the energy gain with the covalent bond is >4 eV, the system tends to stay in this excited state. Therefore, we have four identical quantum states |2si,|2pxi,|2pyi, and |2pzithat can combine resulting in different spi(i= 1,2, and 3) hybrid orbitals. Graphene presents sp2hybridization [i.e., the orbitals 2s,2px, and 2pyhybridize and combine among themselves forming a trigonal planar structure with an angle of 120◦between the carbon atoms as shown in Fig.1.2(a)]. Carbon atoms in graphene are separated a distance a0=1.421 ◦ Aand are strongly bonded between them by means of covalent σbonds, which are responsible for the strength of the planar carbon structure. The remaining 2pzor πorbital is oriented perpendicularly to the atomic plane and can bind covalently with other πorbitals of different atoms. Each carbon atom of graphene possesses one πorbital containing one πelectron, and then, due to the spin degeneracy (i.e.,gs= 2) the πorbitals are half populated. These πelectrons are delocalized and, as we explain in the next section, they form two bands: a lower-energy one (π, valence, or bonding band that is completely filled with electrons) and an upper-energy one (π∗, conduction, or anti-bonding band that is completely empty). 1.3.2 Graphene band structure The atomic structure of graphene is plotted in Fig.1.2(b) and can be understood as a triangular lattice with two atoms per unit cell (see shaded area) composed by two intersecting triangular Bravais sublattices. The lattice vectors are1 a1=a0 2Ä3,√3ä,a2=a0 2Ä3,−√3ä.(1.1) 1Gaussian units are used in all the equations throughout this thesis. 1.3. OPTOELECTRONIC PROPERTIES 19 (a) (b) (c) kx ky ΓK K´ b1 b2 a1 a2 δ1δ2 δ3 x y atomic structure real lattice reciprocal lattice a0 Figure 1.2: Bidimensional honeycomb lattice of graphene. (a) Sketch of the atomic structure of 2D graphene. (b) Triangular lattice of graphene formed by two intersecting triangular Bravais sublattices. The carbon atoms of each sublattice are represented by the green and orange dots, respectively. The lattice vectors are a1 and a2. The vectors δ1,δ2, and δ3connect nearest neighbor atoms. The distance between the carbon atoms is a0= 1.421 ◦ A. The shaded region is the area of the unit cell A0= 3√3a2 0/2. (c) Reciprocal lattice of graphene. The yellow hexagonal region represents the first Brillouin zone with center at Γwhile the brown and the grey represent the second and the third, respectively. The reciprocal lattice vectors (blue arrows) are b1and b2. The Dirac points are represented by red dots in the corners of the 1BZ and named as Kand K0. The resulting reciprocal lattice is shown in Fig.1.2(c) and presents vectors in the momentum space with coordinates b1=2π 3a0Ä1,√3ä,b2=2π 3a0Ä1,−√3ä,(1.2) that fulfill the condition ai·bj= 2πδij, where δij is the Kronecker delta. Furthermore, the vectors connecting the three nearest neighbor atoms in the real space are δ1=−a0(1,0),δ2=a0 2(1,√3),δ3=a0 2(1,−√3).(1.3) The first Brillouin zone (1BZ) of graphene corresponds to the yellow shaded hexagonal region with center at the Γpoint as detailed in Fig. 1.2(c). The vertex of this 1BZ are named as Kand K0points, and present the coordinates K=2π 3√3a0Ä√3,1ä, K0=2π 3√3a0Ä√3,−1ä.(1.4) 26 CHAPTER 1. INTRODUCTION that relates the interaction between the electromagnetic radiation and graphene: D(r, ω) = ε(r, ω)E(r, ω),B(r, ω) = H(r, ω),(1.12) where the frequency dependence of the media on the electromagnetic fields is encompassed now in the local dielectric function ε(r, ω)and an exp(−iωt)temporal dependence is always assumed. Throughout this thesis, the exact solution of full-retarded Maxwell’s equations is obtained using the boundary-element method (BEM) [27]. In BEM, a system of surface-integral equations is evaluated at the boundaries of geometries with arbitrary shapes. Once we determine the boundary conditions satisfied by the surface charges and currents, the system is numerically solved by discretizing the boundaries with a finite number of points. Nevertheless, when the graphene nanostructures present a size much smaller than the incident light wavelength (i.e., Dλ), their response can be described in the electrostatic limit (i.e.,non-retarded limit where we assume c→ ∞). The interaction between graphene and the external light is considered instantaneous, so that the temporal phase of the electromagnetic field is practically constant, and therefore, we can reduce our problem of finding the spatial field distribution to electrostatics [28]. The electric and magnetic fields are decoupled [i.e., ∇×E(r, ω)=0 and ∇ × H(r, ω)=0] and the solution of Maxwell’s equations, considering negligible external currents and charges, reduces to solving Poisson equation with the appropriate boundary conditions, ∇·ε(r, ω)∇Φ(r, ω) = −4πρind(R, ω)δ(z),(1.13) where ρind(R, ω) = −en(R, ω)is the 2D induced charge density in the graphene plane and δ(z)is the Dirac Delta function. From the previous equation we can directly get the expression of the electric field E(r, ω) = −∇Φ(r, ω), where we define r= (R, z)and assume that the graphene layer lies on the z= 0 plane. Moreover, the continuity equation relates the induced density and the surface current as ρind(R, ω) = −i ω∇·J(R, ω).(1.14) 1.4. ELECTROMAGNETIC MODELING OF GRAPHENE 27 This current can be also expressed as J(R, ω) = −σ(R, ω)∇Φ(R, ω), where σ(R, ω) is the in-plane graphene conductivity. As we will show in further sections, the solution of the former electrostatic expressions gives a suitably accurate optoelectronic response of the graphene nanostructures. In the local classical description of graphene, the dielectric function of the nanostructures is characterized by the relation ε(ω) = 1 + 4πiσ(ω) ωt ,(1.15) where, for simplicity, we assume that graphene is an isotropic and uniformly doped material (by doping we refer to the process of changing the Fermi energy of graphene). Moreover, the value used in the simulations for the graphene thickness needs to be well converged with t→0for finding a valid solution. Note that the nominal thickness of a one-atom-thick layer of graphene is tnom ≈0.334nm (i.e.,the interlayer separation of graphite [29]). The shape of the conductivity is characterized by the behavior of the free conduction electrons (or holes) sustained by graphene [see section(1.5.2)]. The simplest kinetic model that approximately describes the local dynamics of these electrons and its scattering processes is the Drude model [18]. Here, only intraband electron-hole (e-h) pair transitions at temperature T= 0 are considered [see section(1.5.2)], and the free electrons may decay through multiple channels (e.g.,collision with phonons, lattice defects or, more rarely, quenching with other electrons) with a phenomenological rate per unit time γ[this magnitude, also known as damping rate, comprises all the possible graphene loss channels, see section(1.5.3)]. The resulting graphene conductivity presents the form [30] σDrude(ω) = ie2 π~2 EF ω+ iγ.(1.16) Inserting this expression into Eq.(1.15), we can rewrite the dielectric function of graphene with a more recognizable expression according to the Drude model [26] εDrude(ω) = 1 −ÄωDrude bulk ä2 ω(ω+ iγ).(1.17) 28 CHAPTER 1. INTRODUCTION (a) 100 103 EF=0.4eV EF=0.1eV EF=1.0eV local RPA Drude 10-3 10-2 10-1 100 Energy, ω (eV) 10-3 { } 0 Re (ω) σ σ 0.1 1.0 2 0 { } 0 Im (ω) σ σ (b) (ω) ε -30 -15 0 15 Energy, ω (eV) 1 2 3 { } Re (ω) ε { } Im (ω) ε local-RPA bulk ω Drude bulk ω 1 Figure 1.5: Conductivity and dielectric function of doped graphene. (a) Real part of the conductivity of doped graphene as a function of the incident photon energy. We compare the Drude model (blue curves) with the local-RPA model at temperature T= 300K (red curves) for different doping energies EF. The inset shows the evolution of the imaginary part. The conductivity is normalized to that in pristine graphene σ0=e2/4~[31]. (b) Evolution of the dielectric function 1 + 4πiσ(ω)/ωt of a graphene slab with EF= 0.4eV and thickness t= 0.5nm. The purple vertical lines represent the bulk plasma frequency corresponding to local-RPA and Drude models, respectively. At these frequencies, graphene changes from a metallic to a dielectric-like behavior. The mobility here used is µ= 10000cm2/(V s). Here, ωDrude bulk = (2e/~)»EF/t is the Drude bulk plasma frequency that depends on the graphene Fermi energy and the thickness [e.g.,for EF= 0.4eV and t= 0.5nm –well converged value with tnom– we find ~ωDrude bulk ≈2.15eV, as shown within the purple vertical line on the right side of Fig. 1.5(b)]. Using Eq.(1.9), this bulk plasma frequency can be also expressed as ωDrude bulk =»4πne2/tm∗ e. The importance of ωDrude bulk relies on the fact that it determines how is the response displayed by graphene to external radiation: −For ω < ωDrude bulk , doped graphene shows a behavior similar to metals, so that conduction electrons (or holes) are capable of screening external radiation. The dielectric function satisfies Re{εDrude(ω)}<0. −For ω > ωDrude bulk , doped graphene behaves as a dielectric to external radiation, so that light can propagate through it. The dielectric function satisfies Re{εDrude(ω)}>0. 1.4. ELECTROMAGNETIC MODELING OF GRAPHENE 29 However, there are more elaborated models that take into account not only the scattering processes of the conduction electrons (or holes) and intraband transitions at T= 0, but also electronic interband transitions at finite temperatures. For example, within the random-phase approximation (RPA) [32, 33, 34, 35, 36] we can describe more realistically the conductivity of graphene [31] σRPA(kk, ω) = −iωχ(kk, ω), where χ(kk, ω)is the linear graphene susceptibility. In the local limit (kk→0), the local-RPA conductivity [31, 37] is expressed as follows σlocal−RPA(ω) = ie2 π~2−1 ω+ iγZ∞ −∞ d ||∂f ∂ +(/||) 1−42¿î~2(ω+ iγ)2óf ,(1.18) where fis the Fermi-Dirac electron distribution as a function of the energy [see Eq.(1.21)]. The first term inside the integral of Eq.(1.18) corresponds to the intraband transitions –dominant at photon energies roughly below the Fermi energy EF– and can be integrated analytically resulting in ie2E∗ F/~2(ω+ iγ), with E∗ F=EF+ 2kBTlnÄ1+e−EF/kBTä, thus converging to the Drude model at T= 0. The second term inside the integral of Eq.(1.18) corresponds to interband transitions and needs to be solved numerically. As depicted by the red curves in Fig. 1.5(a) for different Fermi energies, this term contributes a step function to the real part of σlocal−RPA(ω)when ≈2EF, which is smoothed by the inclusion of finite values of the temperature and damping. The effect of the interband transitions is also appreciable in the imaginary part of σlocal−RPA(ω)with a clear valley again at ≈2EF[see red curves of the inset of Fig.1.5(a)]. Moreover, we show in Fig.1.5(b) the evolution of the real and imaginary parts of the dielectric functions for the two local classical models under study. Although the Drude model (blue curves) can be considered a good approximation (particularly when EFfor EFsmaller than the optical phonon energy ∼0.2eV [38]), an in-depth analysis of the graphene conductivity including the interband transitions and finite temperature effects result in clear redshifts of the bulk plasma frequency. Additionally, since the choice of sign for time-dependence is exp(−iωt), we find in the whole frequency spectrum for both models that Im{ε(ω)}>0, so that we can assert that graphene is a lossy material. Finally, we want to mention that for graphene characteristic sizes larger than λF, 30 CHAPTER 1. INTRODUCTION the inclusion of nonlocal effects does not induce significant changes with respect to the local limit, and qualitatively similar results are observed [31]. The employed lifetime or relaxation time τ=γ−1of Fig. 1.5 is obtained using the impurity-limited approximation τ=µEF/ev2 F, where µis the direct current (dc) graphene electrical mobility [see section(1.3.4)]. For instance, in Fig.1.5(b) we use the parameters EF= 0.4eV and µ= 10000 cm2/(V s), thus the predicted lifetime in graphene is τ≈400fs (i.e.,~γ≈1.65meV). This value is substantially higher than τ≈10fs in gold [39]. In all the figures depicted in this introductory chapter, the damping rate is calculated using this dc procedure. 1.4.2 Quantum-mechanical description: Random-phase approximation The local classical descriptions of the electromagnetic response of graphene to external radiation presented in the previous section are no longer valid when the size of the graphene nanostructure Dfulfills D.λF. Hence, a microscopic quantummechanical procedure that takes also into account finite-size and edge effects, is necessary to characterize the optoelectronic response of graphene. In this thesis, we combine the RPA with a TB model similar to the one described in the section (1.3.2) [see Ref.[40] for a detailed description of the method used here]. However, in our case we only consider the hopping between nearest neighbors land l0, so that the TB Hamiltonian satisfies hl|H|l0i=−twith the already known hopping parameter t∼2.8eV. The diagonalization of this Hamiltonian permits finding the energy jof each single-electron state j. Once this is done, we are interested in finding the noninteracting susceptibility χ0 ll0(ω)using the RPA. This quantity relates the induced charge density ρind l(ω)with the total potential per unit area resulting from the sum of the external and induced potentials Φl(ω)=Φext l(ω)+Φind l(ω)as ρind l(ω) = Xl0χ0 ll0(ω)Φl0(ω),(1.19) where land l0run over the carbon sites of the graphene structure [41]. For simplicity, we have just considered the non-interacting susceptibility between electrons, otherwise finding a solution requires the use of many-body theory. By means of the 1.5. PLASMONS IN GRAPHENE 31 expansion coefficients ajl of each TB state Plajl |li, the non-interacting susceptibility yields χ0 ll0(ω) = −2e2Xjj0Äfj−fj0äajl a∗ jl0a∗ j0laj0l0 ~ω−j+j0+ i~/2τ,(1.20) where ~ωis the energy of the incident photon, τis the finite lifetime of the electronic excitations, and fj=1 1+e(j−EF)/kBT(1.21) is the Fermi-Dirac electron distribution of the state jat Fermi energy EFand temperature T. The factor 2 immediately after the equal sign in Eq.(1.20) stems from the spin degeneracy. Additionally, the total potential per unit area can be also expressed as Φl(ω)=Φext l(ω) + Xl0vll0(ω)ρind l0(ω),(1.22) where vll0(ω) = 1/rll0is the Coulomb interaction between sites land l0. Finally, combining Eqs.(1.19) and (1.22), we can get a self-consistent solution of the induced charge density given by ρind (ω) = î1−χ0(ω)·v(ω)ó−1·χ0(ω)·Φext (ω),(1.23) from which we can directly obtain the induced dipole moment [see section(1.5.3)] and the electric polarizability α(ω). A detailed study of the effects regarding this quantum-mechanical approach is fully illustrated for different graphene geometries in Chapters3 and 4. 1.5 Plasmons in graphene Graphene plasmons are collective surface oscillations of electrons with respect to the fixed positively-charged lattice of carbon nuclei. The oscillation occurs because the attractive binding exerted by the lattice is weak, so that electrons can move freely forming an electronic cloud reacting to external radiation. Plasmons sustained by graphene are generically divided into two different groups: intrinsic and extrinsic plasmons. The former are observable under any doping conditions –even in pristine graphene– and can be subdivided into two different subgroups: σand πplasmons. 32 CHAPTER 1. INTRODUCTION The latter, also named Dirac plasmons, only appear when graphene is extrinsically doped. In the next section, we briefly introduce the main properties of intrinsic graphene plasmons. Their extremely high resonance frequencies (&4.5eV) are in contrast to the suitable ones of Dirac plasmons (.3eV) which make them more interesting for nanophotonics opening a wide range of potential applications. Afterwards, we explain the main properties of these Dirac plasmons but always assuming a classical description for graphene. 1.5.1 σand πplasmons These plasmons arise respectively from the collective oscillation in the 2D plane of delocalized σand πelectrons [see section(1.3.2)]. They were first observed in fullerene [42, 43], and later in CNTs [44, 45] and extended graphene [46] via electron energy loss spectroscopy (EELS). This technique has been extensively employed for spectrally and spatially characterizing diverse plasmons in noble metals [47, 48, 49, 50, 51] and graphene [52, 53, 54]. Essentially, in EELS an electron beam passes near or through a plasmonic target and the electrons lose part of their energy by exciting plasmons. The resonance frequencies of πplasmons oscillate between 4.5and 7eV, while for σplasmons the range fluctuates from 14.5to 30eV, which confine both deeply in the UV regime. Moreover, they display a limited tunability, similarly to those in typical noble metals. Finally, we need to remark that at electron energies ∼5eV, vertical interband transitions between the πand π∗bands can occur for wave vectors situated between the Dirac points [see Fig.1.3(b)]. The resulting possible coupling of this mechanism with the πplasmon is known as plexciton [55]. In the rest of this thesis, we do not discuss these plasmons any more. 1.5.2 Dirac plasmons As explained in section(1.3.2), pristine graphene presents a peculiar linear dispersion relation around the Dirac points with a zero-energy optical gap between the completely filled valence band and the fully empty conduction band (i.e.,pristine graphene can be treated as a zero-gap semiconductor). Within these conditions, the 1.5. PLASMONS IN GRAPHENE 33 Energy (eV) 0 5 10 15 20 Intensity π plasmon σ plasmon Dirac plasmon Figure 1.6: Schematic representation of the different plasmons sustained by graphene. The blue region corresponds to the plasmons excited in pristine graphene, where we can distinguish between the πplasmons at energies ∼5eV, and the σplasmons at ∼15eV. For doped graphene, a new type of low-energy plasmons appear (red region), the so-called Dirac plasmons with energy .3eV. Dirac points mark exactly the Fermi level [see Fig.1.3(b)]. Besides, the only possible e-h excitation when graphene is illuminated under this null doping condition is an interband transition. These transitions are responsible for a nearly-constant absorption [30] of self-standing undoped extended graphene [56, 57] πα ≈2.3% (i.e.,conductivity σ0=e2/4~[31]), where α=e2/~c≈1/137.036 is the finestructure constant. When we add extra charge carriers (electrons) to the conduction band (i.e., ndoped graphene), an optical gap opens and the Fermi energy substantially shifts to EF=~vF√πn, where now nis the density of these extra electrons [see Fig.1.8(a) for an illustrative representation of the variation of EFwith the external doping]. Due to the symmetry of the bands, the doping can be also produced by the removal of valence electrons (i.e.,p-doped graphene), so that the Dirac massless particles responsible for the doping are now the holes in the valence band. The remarkable tunability of the Fermi level in graphene with the addition of external carriers is in contrast to the response of typical noble metals, which is rather unaffected by doping. Moreover, noble metals present a parabolic dispersion of their conduction band that electrons fill up to a certain level determined by EF, and there is no optical gap so that electrons below the Fermi level can move up to any of the unoccupied 34 CHAPTER 1. INTRODUCTION states above EF. However, significant changes in the electronic carrier density nin metals will only slightly affect their optoelectronic response. Furthermore, in metals nis expressed per unit of volume while in graphene it is per unit of area, so that a bigger quantity of electrons is required for inducing significant changes in EF. Experimentally, the injection of additional charge carriers (either electrons or holes) is usually achieved by electrostatic gating [58, 59] or with chemical methods [60, 61]. Within the former, a potential difference with respect to a backgate is applied to graphene, thus inducing a perpendicular dc electric field E0that is uniformly applied to one side of the carbon layer and screened by an induced doping charge carrier density n=−E0/4πe. Fermi levels as high as ∼1eV have been obtained using this technique [58, 59], which correspond to n∼75 ×1012 cm−2. Moreover, when light impinges on doped graphene, electrons from the conduction band can be excited up to unoccupied states. This can occur through the absorption of the energy and momentum of incident photons. Therefore, in addition to interband e-h transitions [i.e.,the green and blue arrows in Fig.1.8(a)], intraband e-h transition are also allowed [i.e., the magenta arrow in Fig.1.8(a)]. Within the opening of an optical gap of maximum width 2EF, apart from the e-h transitions, doped graphene can support surface plasmons without undergoing Landau damping [62]. These are the Dirac plasmons mentioned above –collective oscillations of the conduction electrons (or holes)– whose resonance frequency fall between the THz and the near infrared region (NIR) [35, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. Graphene Dirac plasmons present an impressive number of salient properties in comparison with those in typical noble metals: (i) stronger interaction with light; (ii) ability of producing a large field enhancement [26, 31] (>105in near-field intensities) upon external illumination; (iii) smaller wavelength in comparison to the external radiation which results in an extraordinary light and field confinement; (iv) longer lifetimes [38, 74]; and (v) stronger nonlinearities [75, 76, 77, 78, 79, 80, 81]. Furthermore, the spectral and spatial control via electrical gating over these collective oscillations have been intensively investigated in recent years both experimentally [64, 65, 66, 67, 68, 69, 71, 82, 83] and theoretically [31, 34, 35, 38, 30, 81, 84, 85, 86, 87, 88, 89, 90]. Owing to the former appealing properties, graphene Dirac plasmons can be currently found in a vast range of applications: light modulation [69, 71, 91], nonlinear optics [75, 76, 77, 78, 81], sensing [92, 93], 1.5. PLASMONS IN GRAPHENE 35 (a)1 (b) 1 0 E p polarizations polarization 2 ε 1 ε 1t r s s = + 0 E 1t r p p = − ref Eref E trans E trans E s r p r 2 ε 1 ε ( , k ω) σ ( , k ω) σ Figure 1.7: Reflection and transmission coefficients of impinging (a) spolarized and (b) ppolarized waves by a thin graphene layer. The graphene layer is characterized by its conductivity σ(kk, ω)and acts as an interface between two different media with dielectric functions ε1and ε2, respectively. The orientations of the incident (E0), reflected (Eref), and transmitted (Etrans) electric fields are depicted within blue arrows. and signal processing [63], among other feats. We focus now on the two different subgroups of Dirac plasmons sustained by doped graphene which we distinguish according to their propagating features: surface plasmon polaritons and localized surface plasmons. Surface plasmon polaritons in graphene The surface plasmon polaritons (SPPs) are electromagnetic modes traveling along an extended graphene layer acting as an interface between two dielectric media. They are transverse solutions [i.e.,∇·E(r, ω) = 0] of the macroscopic Maxwell’s equations described in Eq.(1.11) in the absence of external sources, and fulfill the condition of non-zero magnetic field in both surrounding dielectric media and graphene. Once the graphene interface –which we assume in the limit of a thin layer– is illuminated by a plane wave, we can easily obtain the relations between the reflected and transmitted electric fields at the interface through the Fresnel coefficients [26]. The expressions of these coefficients depend on the two different types of electromagnetic field polarizations. For spolarization [i.e.,TE modes: the incident light impinging from medium 1 is polarized with its electric field perpendicular to the 42 CHAPTER 1. INTRODUCTION 1.5.3 Optical losses Once excited, plasmons in graphene possess a finite lifetime γ−1before they dissipate either through radiative or non-radiative (inelastic) channels. The former are related with re-emission of photons [98], while the latter are associated with other path-ways [99] like coupling with phonons [38], collision with lattice defects [100], finite-size and edge effects [40, 101], or generation of hot e-h pairs [102]. Thus, we can split the decay rate into a radiative and non-radiative components as γ=γr+γnr. The radiative contribution is usually a factor ∼103smaller and is given by [98] ~γr(ω) = s(ε1+ε2) 2 4ω3 3c3|p(ω)|2,(1.32) where p(ω) = Rρind(R, ω)Rd2Ris the dipole moment associated with the graphene layer, and ε1|ε2the permittivities above|below graphene, respectively. Regarding the non-radiative decay channels, the coupling with phonons plays an important role above the threshold energy ∼0.2eV [38, 70]. Moreover, the edge effects also induce a remarkable increment of the damping due to the creation of electronic edge states [20]. There are two different types of edge terminations in the graphene nanostructures: armchair or zigzag [see right inset of Fig.3.1]. The electronic edge states are only present in zigzag terminations [20] and produce a strong plasmon quenching [40] when ~ωp&EF[101]. Under the same energy condition, for the armchair terminations the edge effects only induce a slight broadening of the plasmons [89, 101]. Finally, the decay of plasmons into e-h pairs because of Landau damping may occur when additional momentum is provided, and the plasmons fall inside the region of electronic interband transitions [see Fig.1.8(b)]. 1.5.4 Electrostatic scaling law In previous sections we have showed that for graphene nanostructures of size D (e.g.,width of ribbons, diameter of disks, lateral size of equilateral triangles. . . ) much smaller than the incident light wavelength, retardation effects can be neglected and we can safely work in the electrostatic limit. Thus, we can study the response of graphene in terms of an electrostatic potential Φ(r, ω). The self-consistent equation 1.5. PLASMONS IN GRAPHENE 43 of this potential created by a certain bidimensional charge density acting as an interface between two dielectric media with permittivities2ε1and ε2is [53] Φ(r, ω)=Φext(r, ω) + 2 (ε1+ε2)Zd2R0 |r−R0|ρind(R0, ω).(1.33) This expression is valid for any point r= (R, z), but since we restrict our analysis to the graphene sheet which lies in the plane z= 0, the self-consistent potential will only involve coordinate vectors R= (x, y). The expression of Eq.(1.33) is the sum of two contributions: the external perturbation [now rewritten as Φext(R, ω)], and the potential created by the charge induced in doped graphene under external illumination which is globally represented by the integral. We can relate this induced charge with the current by means of the continuity equation given in Eq.(1.14). Furthermore, if we assume a linear response of graphene, the current can be obtained from the multiplication of the local, in-plane graphene conductivity σ(R, ω)by the total electric field E(R, ω). Combining all the former elements, we can rearrange the self-consistent equation of the potential to obtain Φ(R, ω)=Φext(R, ω) + 2 (ε1+ε2) i ωZd2R0 |R−R0|∇R0·σ(R0, ω)∇R0Φ(R0, ω).(1.34) We need to remark that the abrupt change of the conductivity produced at the edge of the nanostructure produces a divergent contribution to the integral of Eq.(1.34). The numerical solution of this issue involves the implementation of a smoothing at the graphene edge [an example is described in section(2.3)] that barely affects the final result [90]. Owing to the complete lack of absolute length scales in electrostatics, we can introduce for simplicity the dimensionless coordinate vector ~ θ=R/D. Besides, we consider that the conductivity can be split into a position and a frequency dependent terms as σ(R, ω) = f(R)σ(ω). Under uniform doping, the occupation function f(R)takes the constant value 1 inside the graphene area and vanishes elsewhere. As we will see in the next chapter, this formalism can be easily applied to diverse inhomogeneous doping configurations by just modifying the spatial distribution of 2see Appendix A for the derivation of the factor 2/(ε1+ε2)regarding the dielectric environment 44 CHAPTER 1. INTRODUCTION f(R)[86, 90, 103]. Thus, implementing these elements in Eq. (1.34), we obtain Φ(~ θ, ω)=Φext(~ θ, ω) + η(ω)Zd2~ θ0 |~ θ−~ θ0|∇~ θ0·f(~ θ0)∇~ θ0Φ(~ θ0, ω),(1.35) where η(ω) = 2 (ε1+ε2) iσ(ω) ωD ,(1.36) is a dimensionless parameter that contains all the information about the full dependence of graphene on the frequency, temperature, size, dielectric environment, and doping level. Integrating by parts Eq.(1.35) and taking the in-plane gradient on both sides, we find ~ E(~ θ, ω) = ~ Eext(~ θ, ω) + η(ω)Zd2~ θ0M(~ θ, ~ θ0)·~ E(~ θ0, ω),(1.37) where ~ E(~ θ, ω) = qf(~ θ)E(~ θ, ω),(1.38) and M(~ θ, ~ θ0) = qf(~ θ)f(~ θ0)∇~ θ⊗∇~ θ01¿|~ θ−~ θ0|is a real and symmetric operator. This operator possesses a complete dimensionless set of real, negative eigenvalues 1/ηj, and eigenvectors ~ Ej(~ θ)that satisfy the eigenstate, orthogonality, and closure conditions given respectively by ηjZd2~ θ0M(~ θ, ~ θ0)·~ Ej(~ θ0) = ~ Ej(~ θ), Zd2~ θ~ Ej(~ θ)·~ Ej0(~ θ) = δjj0, Xj ~ Ej(~ θ)·~ Ej(~ θ0) = δ(~ θ−~ θ0)I2, (1.39) with I2being the 2×2unit matrix. Then, the solution to Eq.(1.37) can be transformed into ~ E(~ θ, ω) = Xj cj 1−η(ω)/ηj ~ Ej(~ θ),(1.40) where the expansion coefficients cj=Zd2~ θ~ Ej(~ θ)·~ Eext(~ θ, ω)(1.41) 1.5. PLASMONS IN GRAPHENE 45 hold the same units as the electric field. If we assume normal-incidence illumination (i.e.,kk= 0) with an associated external field uniformly distributed with amplitude E0and oriented along the direction ˆ xof the nanostructure [i.e., ~ Eext(~ θ) = qf(~ θ)E0ˆ x], we can get the expression of the polarizability along that direction α(ω)=(D3/E0)Rd2~ θ θxρind(~ θ, ω), where the induced density is ρind(~ θ, ω) = −iσ(ω) ωD ï∇~ θ·qf(~ θ)~ E(~ θ, ω)ò.(1.42) It is then convenient to insert Eq.(1.40) into this expression, then we finally find α(ω) = (ε1+ε2) 2D3Xj ξ2 j 1 η(ω)−1 ηj ,(1.43) where jruns over each eigenmode and ξ2 j=Zd2~ θqf(~ θ)Ej(θx) 2 ,(1.44) are real, positive, and dimensionless coefficients that only depend on the specific shape of the nanostructure considered. Interestingly, for the Drude model we find αDrude(ω)∝D5/2E1/2 F. The ξjand ηjfactors obey two useful sum rules [30] that enable estimating analytically their values: (i) in the weak coupling regime (i.e.,σ→0) upon application of the closure relation for ~ Ej(~ θ), we have Xjξ2 j=A D2,(1.45) where Ain the area of the graphene nanostructure; and (ii) in the limit of a perfect conductor (i.e.,σ→ ∞), we get −Xjηjξ2 j=2[α(0)/D3] (ε1+ε2).(1.46) In particular, for nanodisks we have [104] α(0)/D3= 1/6π, while for nanoribbons we find [105] α(0)/D3=L/16Dwhere L→ ∞ is the nanoribbon length. In Table1.1 we summarize the analytical values of these factors for both nanostructures 46 CHAPTER 1. INTRODUCTION Geometry −Pjηjξ2 jPjξ2 jPjηj disk [2/(ε1+ε2)](1/6π)π/4 [2/(ε1+ε2)] (−2/3π2) ribbon [2/(ε1+ε2)] (1/16) 1 [2/(ε1+ε2)](−1/16) Table 1.1: Analytical approximations for the parameters ξjand ηjof the electrostatic scaling law. Interestingly, for the nanostructures here considered, one finds that the first mode j= 1 (i.e., lowest-order dipolar mode) is dominant and owns most of the weight in the sums given in Eqs.(1.45) and (1.46) [30]. Additionally, we need to remark that the condition η(ωj) = ηjsets the plasmon frequency ωjof the corresponding eigenstate j. Since all the graphene properties are contained inside η(ω), these eigenvalues ηjonly depend on the geometrical shape of the graphene nanostructure and can be calculated once and for all in order to get the plasmon frequencies for any desired value of the temperature, size, dielectric environment, or doping level. In particular, using the Drude model [see Eq.(1.16)], we easily obtain [see AppendixB] ωDrude p≈ωDrude j−iγ/2,(1.47) with ωDrude j=e ~Ãñ2 ε1+ε2ôñ 1 −πηjôñEF Dô.(1.48) As an example, we take the conditions used in Fig.1.9 of EF= 0.4eV and D= 30nm of a uniformly doped, self-standing nanodisk. If we assume dominant the lowestorder dipolar plasmon, we find that the estimated LSP energy is ~ωDrude 1≈0.3eV, which is in good agreement with the numerical solution of full-retarded Maxwell’s equations depicted in Fig.1.9(a). The fact that the resonance frequencies of LSPs in graphene nanostructures obey the approximate behavior ωp∝»EF/D, illustrates their strong tunability and pave the way for plenty of potential applications to exploit the plasmonic properties of doped graphene. Chapter 2 Plasmons in multiple doping configurations In the previous chapter, we have presented a description of graphene plasmons sustained by extended layers or individual nanostructures under a uniform doping configuration. The plasmonic response of graphene under this doping scheme has been extensively described in the literature and plenty of potential applications have been developed up to date [30, 31, 63, 88]. It is well-known that Dirac plasmons arise in doped graphene with a resonance frequency proportional to n1/4, where nis the doping charge carrier density. However, we find that usually nis not uniformly distributed, and therefore, its profile depends on the particular geometrical configuration which modifies the plasmonic response of graphene. Besides, the interaction of neighboring nanostructures can also produce remarkable differences in the Dirac plasmons developed on each single island. Motivated by these facts, in this chapter we classically study the effects of different geometric, and inhomogeneous doping schemes exerted on LSPs (for convenience, during this chapter we will refer to them simply as “plasmons”) and SPPs. Using extensively the electrostatic scaling law described in section (1.5.4), we specifically discuss the following topics: −We start by analyzing the interaction between identical uniformly doped graph47 48 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS ene nanoribbons via a novel and easy-to-handle model based on the plasmon wave function (PWF) which presents a higher level of accuracy than other existing commonly-used models. −Then, we describe the properties of plasmons generated under multiple inhomogeneous doping configurations. Firstly, we analyze the plasmons sustained on nanoribbons with diverse non-uniform distributions of nover the surface. Secondly, we concentrate on plasmons supported by individual inhomogeneously doped graphene nanodisks. −Finally, we investigate SPPs created on periodically doped extended graphene which develop a plasmonic band structure. 2.1 Plasmons in interacting uniformly doped ribbons In this section, we describe the plasmonic response of uniformly doped graphene nanoribbons of width D. We focus on this geometry due to the availability of new synthesis methods with control down to the nanometer scale [106, 107, 108, 109]. For our purpose, we analytically derive a model based on the PWF together with the electrostatic scaling law formalism presented in section(1.5.4). We conveniently define the PWF associated with the electromagnetic mode jas ρj(θx) = ∂ ∂θx»f(θx)Ej(θx),(2.1) where we have assumed that the nanoribbon is normally illuminated with the electric field polarized along the transversal direction ˆ xwith amplitude E0. In Fig.2.1 we illustrate the interaction between two ribbons through the interaction of their PWFs. The ribbons are separated a center-to-center distance rand aligned along ˆ y (i.e.,they can lie in different zplanes) under the illumination conditions mentioned above. The impinging light induces a line dipole along ˆ x. The PWF profile at the plasmon frequency is also included displaying a uniform dipolar-like behavior along ˆ y. Using the definition of the PWF given in Eq.(2.1), the expansion coefficients of 2.1. PLASMONS IN INTERACTING UNIFORMLY DOPED RIBBONS 49 ll´ k r 0 E y x z Figure 2.1: Plasmon wave functions (PWFs) of two interacting graphene nanoribbons aligned along ˆ yunder normal-incidence illumination. We represent the PWF associated with the lowest-order dipolar mode of two graphene nanoribbons separated by a vector rll0and excited by a normal-incidence plane wave of transverse polarization. The shape of the PWF is obtained from the induced charge density at the frequency of this lowest-order plasmon (ρ1) and clearly shows a dipolar-like behavior going from positive (red color) to negative (blue color) values along the transversal direction of the ribbons. The figure is adapted from Ref.[110]. Eq.(1.41) can be now expressed as cj=−E0ξj, whereas the dimensionless coefficients of Eq.(1.44) remain ξ2 j=Zdθxθxρj(θx) 2 .(2.2) Remarkably, the PWF is proportional to the induced charge density of the plasmon mode j, whereas ξjcan be treated as the normalized dipole moment. Moreover, the PWF also fulfills the charge neutrality condition Rdθxρj(θx)=0. For simplicity, we study the response of identical ribbons within a spectral region dominated by their lowest-order transversal dipole mode (j= 1), neglecting other modes in what follows. Besides, we intentionally discard the xsubindex since we know that the geometric evolution of the PWF varies along the ˆ xdirection. Thus, we can express E(θ, ω)in terms of this lowest-order mode as E(θ, ω) = Xldl(ω)E1(θ−θl),(2.3) 50 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS where lruns over each different ribbon, and θlindicates the position of the central point of the nanoribbon. Inserting Eq. (2.3) into Eq.(1.37), and using the eigenstate and orthogonality conditions given in Eq. (1.39), we find the self-consistent equation dl(ω) = 1 1−η(ω)/η1ñ−E0ξ1+η(ω)Xl06=lMll0(θ, θ0)dl0(ω)ô,(2.4) for the expansion coefficients, where Mll0(θ, θ0) = ZdθZdθ0E1(θ−θl)M(θ, θ0)El0(θ0−θl0) =ZdθZdθ0ρ1(θ)ρ1(θ0) lnî(θ−θ0+θl−θl0)2+ (zl−zl0)2/W2ó, (2.5) describes the interaction between ribbons land l0. Notice that we have generalized Mll0(θ, θ0)in order to deal with ribbons that are located at different heights zl. Considering now the expression of the induced charge density given in Eq.(1.42), we can easily find the dipole moment induced in ribbon l pl(ω) = −η(ω)D2L dl(ω)ξ1.(2.6) Taking into account that the self-consistent interaction between the dipoles of different ribbons is [104] pl(ω) = α(ω)E0+α(ω)Xl06=lGll0pl0(ω),(2.7) where we identify Gll0=ñ1 D2Lξ2 1ôMll0(θ, θ0) = −[2/(ε1+ε2)] p2 1(ω)Zd2RZd2R0ρ1(R)ρ1(R0) |R−R0−rll0|(2.8) as the Green tensor [26, 111] for the electric field produced by a single dipole, which contains all the information about the interaction between nanoribbons. As we can observe, it is directly related with the PWF which is well approximated by the phenomenological analytical function [110] ρ1(θ)≈9θ(1 + e−5+20θ2 4√1−4θ2).(2.9) 2.1. PLASMONS IN INTERACTING UNIFORMLY DOPED RIBBONS 51 Moreover, p1(ω)is the dipole moment at the plasmon frequency, and α(ω) = [(ε1+ε2)/2]D2Lξ2 1¿[1/η(ω)−1/η1]is the individual nanoribbon polarizability in the single-mode approximation that presents a similar expression to Eq.(1.43). Remarkably, from our calculations we get that the fitting coefficients are η1≈ −0.069 and ξ1≈0.94, which are in good agreement with the purely analytical values predicted in section(1.5.4) Pjηj=−1/16 ≈ −0.063 and Pjξ2 j= 1. If we take now into account that the extinction cross-section is the sum of the expressions given in Eq.(1.30), for a system formed by interacting ribbons we get σext(ω) = 4πω cXll0Im®1 1/α(ω)−Gll0´,(2.10) where Gis a matrix of elements (1 −δll0)Gll0. Note that since we assume symmetry along the longitudinal direction of the nanoribbon with length L→ ∞, we in fact implement in Eq.(2.10) the polarizability per unit length α(ω)/L and GL, so that the resulting extinction cross-section in also given per unit length. We depict in Fig. 2.2 the extinction cross-section normalized to the nanoribbon area of two interacting self-standing graphene nanoribbons (l= 1,2). The dielectric response of the carbon layer is taken from the local-RPA model at temperature T= 300K [see Eq.(1.18)], where EF= 0.4eV is the Fermi level and ~γ= 20meV is the decay rate. The system displays a prominent dipolar plasmon that starts at the single-ribbon energy limit, corresponding to large separations between nanoribbons (blue curves) and undergoes dramatic redshifts (blueshifts) due to inter-ribbon attractive (repulsive) interaction in co-planar (stacked) arrangements, as illustrated in panel (a) [panel (b)]. The attractive (repulsive) character of this interaction is intuitively suggested from the induced charge distribution shown in Fig.2.1. We find that the results from the PWF model (dashed curves) agree well with the full numerical solution (solid curves) down to very small edge-to-edge separations. In contrast, the dipole-dipole model (dotted curves) only produces accurate results at large separations, and its failure at small distances is particularly severe for stacked ribbons. This dipole-dipole model treats each ribbon as a line dipole along the transversal direction [30], and thus, the plasmon of the system results from the interaction between different dipoles that are oriented parallel to the external radiation. In Fig.2.3 we plot the evolution of the plasmon frequency for interacting rib- 58 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS periodic array of surface dipoles with a small period acompared to the characteristic lengths of the structure, i.e., a=D¿ÄNdip −1ä, where Ndip is the number of dipoles across ˆ x. This formalism is known as discrete surface-dipole approximation (DSDA) [88], where we take the polarizability of each dipole such that the layer formed by a uniform lattice of dipoles has the same conductivity as a uniform layer of graphene. The sum over dipole elements along ˆ yis performed before a self-consistent solution is found, and therefore, the numerical problem reduces to solving a set of 2Ndip linear equations with 2Ndip variables (the dipole components along both ˆ xand ˆ y directions). In practice, convergence is achieved with a few hundred dipoles for the dimensions considered in this work. Here, we have modified this method by allowing each element to depend through EFon the spatial position along ˆ x. Finally, we just need to solve a linear eigensystem [86] that produces real eigenvalues η0 j(here the subindex jrefers to each eigenstate). Thus, using a similar formalism presented in Eq.(1.47), we express the plasmon frequencies as ωDrude p≈ω0 0¿»−η0 j−iγ/2, where ω0 0= (e/~)»h|EF|i/D defines a natural normalization frequency that we define for convenience. We show in Fig.2.5(b) the evolution of the average Fermi energy of the backgated nanoribbon normalized to E∞ Fas a function of the width-to-distance ratio D/d. In the limit Dd, which corresponds to the graphene nanostructure in close proximity to the backgate, h|EF|i converges smoothly to E∞ F. Besides, from the inset we observe that the spatial distribution of the Fermi energy is nearly uniform except in the edges where a sharp profile appears. In contrast, in the limit Dd, which corresponds to large separations between the nanoribbon and the backgate, the profile is determined by the interaction of the nanoribbon with a distant image. Remarkably, in this limit we find that h|EF|i/E∞ F∝»(d/D)¿»ln (d/D), and from the inset, we observe that the shape of the Fermi level is smooth and convergent. Finally, we find that the doping level diverges as ∝x−1/4with the distance to the nanoribbon edge x. In Fig.2.5(c) we plot the plasmon frequencies of the system ωDrude p, normalized to ω0 0, so that the ratio ωDrude p/ω0 0is a dimensionless number, independent of the specific width Dor doping level h|EF|i. As an example, for D= 100nm and h|EF|i = 0.6eV, we find ~ω0 0= 0.093eV and a dipolar plasmon energy when d= 10µm of ωDrude p≈ 0.19eV (i.e., λDrude p≈6.5µm). As we can observe from the solid curves, with this 2.2. PLASMONS IN INHOMOGENEOUSLY DOPED NANORIBBONS 59 quadrupole dipole backgated nanoribbon d backgate (V=0) x=0 x=D z x z=0 V (a) D/ D/d0.5 D/d=1 D/ D/ (b) 10.1 10 100 D/d F F E x E( ) 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 D/d=100D/d=0.1 Plasmon charge density x/D D/d=100 D/d=8.00 D/d=2.00 D/d=1.00 D/d=0.50 D/d=0.25 (c) 0.01 10.1 10 100 D/d 4 3 2 Drude p0 ω ω F F E E∞ 1 10 0.01 ´ Figure 2.5: Inhomogeneous electrostatic doping and plasmon modes of backgated graphene nanoribbons. (a) Sketch of the geometry under consideration where a single nanoribbon of width Dis placed at a potential Vover a backgate. The distance between the graphene and the backgate is d. (b) Average Fermi energy h|EF|i as a function of width-to-distance ratio D/d, normalized to the value E∞ F= ~vF»|V|/4ed obtained in the Ddlimit. The inset illustrates the EFdistribution along the nanoribbon, normalized to h|EF|i. (c) Frequency ωDrude pof the dipolar (orange curves) and quadrupolar (blue curves) plasmon modes, normalized to ω0 0= (e/~)»h|EF|i/D. The insets represent the induced charge density at the frequency of these plasmons (vertical axis) as a function of position across the nanoribbon (horizontal axis). The dashed curves indicate the Ddlimit. The figure is adapted from Ref.[86]. 60 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS normalization ωDrude p/ω0 0displays just a weak dependence on D/d for the dipolar and quadrupolar modes and is nearly constant when D < d. Moreover, the plasmon frequencies slightly differ from the ones corresponding to the uniform doping (dashed curves). The inset of Fig.2.5(c) shows the corresponding induced charge densities of the dipolar (orange curves) and quadrupolar (blue curves) modes for different D/d ratios, and obtained implementing the induced potential in Eq.(1.14) as ρind(x, ω) = [iσ(ω)/ω]∇2 xΦ(x, ω)via finite-difference derivation. We find that their profiles are only slightly affected by the change in doping profile relative to uniform doping (i.e.,the average level of doping is a dominant parameter, and the effect of edge divergences is only marginal). In conclusion, the plasmon frequencies and induced densities can be approximately described by assuming a uniform Fermi energy in backgated nanoribbons, thus supporting the validity of previous analyses for this configuration [31, 41, 85] although h|EF|i has to be appropriately scaled as plotted in Fig.2.5(b) to compensate the effect of finite D/d ratios. 2.2.2 Co-planar nanoribbon pairs at opposite potentials The second geometrical configuration here studied is illustrated in Fig.2.6(a). It consists of two co-planar parallel nanoribbons of opposite polarity. The graphene nanostructures present the same width Dand are separated by an edge-to-edge distance d. Here, the neighboring ribbons can act both as plasmonic structures and gates. We follow the same analysis described in the previous section for the backgated case in order to get the spatial distribution of the Fermi energy. In our current case, the potential created on the nanoribbon on the right is V=Zd/2+D d/2 dx0Z+∞ −∞ dy[−en(x0)]  1 »(x−x0)2+y2−1 »(x+x0)2+y2 .(2.23) Moreover, Eqs. (2.17) and (2.19)-(2.21) remain valid but Eq.(2.18) becomes F(θ, θ0) = 2 ln  θ0+θ θ0−θ.(2.24) 2.2. PLASMONS IN INHOMOGENEOUSLY DOPED NANORIBBONS 61 quadrupole dipole (a) D/ D/d0.5 D/d=1 D/ D/ 0 -co planar nanoribbons D dV −V z=0 z x x=0 -4 0-2 2 4 x/d 8 2 4 6 0D/d 8 D/d=3 D/d=1 D/d=0.2 F F E x E∞ ( ) Plasmon charge density gap gap 0.1 1 10 100 D/d 4 3 2 Drude p0 ω ω (c)(b) ´ Figure 2.6: Inhomogeneous electrostatic doping and plasmon modes of co-planar parallel graphene nanoribbons of opposite polarity. (a) Sketch of the geometry under consideration with two co-planar parallel nanoribbons of width D, separated a distance d, and set at opposite potentials −Vand V, respectively. (b) Fermi energy distribution across the graphene surface for different width-to-distance D/d ratios. The Fermi energy EFis normalized to the value E∞ F=~vF»|V|/4ed. (c) Frequency ωDrude pof the dipolar (orange curves) and quadrupolar (blue curves) plasmon modes, normalized to ω0 0= (e/~)»h|EF|i/D, as obtained from the Drude model [see Eq.(1.16)]. The solid (dashed) curves correspond to inhomogeneous (uniform) doping. The insets illustrate the induced charge density associated with both plasmon modes (vertical axis) as a function of position across the nanoribbon on the right (horizontal axis, with the position of the right side of the gap indicated by an arrow) for D/d = 0.2,1,3,and 10, respectively (curves evolving in the direction of the cambered arrows). The figure is adapted from Ref.[86]. 62 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS For symmetry, the density of doping charge carriers of the left nanoribbon is found as n(x+d/2) = −n(−x−d/2). Finally, the plasmon frequencies with this specific spatial distribution of the doping are obtained using again the DSDA approach. In Fig.2.6(b) we depict the doping profiles across the nanoribbons, which evolve from a shape similar to the one obtained for the single nanostructure shown in the inset of Fig.2.5(b) in the Ddlimit, towards a converged profile near the gap in the D/d → ∞ limit. From Fig.2.6(c) we observe that plasmons here greatly resemble those of neighboring uniformly doped nanoribbons for the same value of h|EF|i. Incidentally, plasmons in pairs of uniform nanoribbons had already been thoroughly investigated [41], including the redshift with decreasing d. 2.2.3 Individual nanoribbons under a uniform electric field The two inhomogeneous doping schemes considered so far for nanoribbons involve fabrication processes including contacts that allow to charge graphene electrically and could turn into structural defects. However, we can prevent these defects with an electrostatic doping through a uniform external electric field easily attainable experimentally by either distant gates or low-frequency radiation. For this reason, the last geometrical configuration studied here consists of a globally neutral nanoribbon exposed to an uniform external electric field E0oriented along its width D[see Fig.2.7(a)]. We follow the same procedure as in the two previous doping configurations in order to get the spatial distribution of the Fermi energy. If we take the nanoribbon to be placed at zero potential, we can write 0 = −E0x+Z+D/2 −D/2 dx0Z+∞ −∞ dy[−en(x0)] 1 »(x−x0)2+y2,(2.25) where the first term after the equal sign stays for the scalar potential produced by the external field. Now, using the normalization θ=x/D, the above equation reduces to E0θ=Z+1/2 −1/2 dθ0[−en(θ0d)]F(θ, θ0),(2.26) 2.2. PLASMONS IN INHOMOGENEOUSLY DOPED NANORIBBONS 63 (a)nanoribbon in an electric field x=−D/2 x=D/2 E0 0 ´ -0.4 0.0-0.2 0.2 0.4 x/D 3 1 2 0 E x E F F ( ) Plasmon charge density inhomo uniform (b) dipole 0 5 10 || k D || k D 15 0 5 10 15 5 3 2 Drude p0 ω ω 4 1(c) (d) inhomo dipole dipole uniform Figure 2.7: Inhomogeneous electrostatic doping and plasmon modes in individual graphene nanoribbons subject to a uniform external electric field E0. (a) Sketch of the geometry under consideration with a single nanoribbon of width Dexposed to a uniform external electric field. (b) Fermi energy distribution normalized to h|EF|i = 0.6~vF»E0/e where the dashed line shows corresponds to the uniform doping case [i.e.,EF(x) = h|EF|i]. The inset corresponds to the induced charge density associated with the dipolar plasmon mode excited when kk= 0 for inhomogeneous (solid curve) and uniform (dashed curve) doping conditions, respectively. Here, the plasmon frequency for inhomogeneous doping is ωDrude p≈1.45ω0 0, where ω0 0= (e/~)»h|EF|i/D. (c) Plasmon dispersion diagram representing the dependence of the density of optical states on frequency ωDrude pand wave vector parallel to the nanoribbon kk. (d) Same as panel (c) for uniform doping. The figure is adapted from Ref.[86]. 64 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS where F(θ, θ0) = −ln î(θ−θ0)2ó.(2.27) We use the discretization θi=−1/2+(i+ 1/2)/N to write an expression similar to Eq.(2.19), where Mii0is still given by Eq.(2.20) with κ= 1. Finally, the density of doping charge carriers is obtained from n(θid) = (−E0/e)Xi0îM−1óii0θi0.(2.28) We illustrate in Fig.2.7(b) the spatial distribution of the Fermi energy EF(x) = ~vF»π|n(x)|normalized to the average Fermi energy, which is found to be h|EF|i = 0.6~vF»E0/e. We observe that the doping profile (solid curves) exhibits a divergence at the edges again, and it vanishes at the center of the nanoribbon, where nchanges sign. The dashed line corresponds to the uniform doping. The resulting induced charge density of the dipolar plasmon for normal-incidence illumination (kk= 0) is plotted in the insets. For the inhomogeneous doping, it displays a large concentration of induced charges near the center of the ribbon, in contrast to the uniform doping case (dashed curve) that shows the typical dipolar-like behavior also observed in Fig.2.1(a). This inhomogeneous induced dipole-charge density concentration results from the vanishing of n, which can be understood as a thinning of the effective layer thickness, similar to the trapping of graphene plasmons at p-n junctions [119]. Although we have so far discussed plasmons that are invariant along the length of the nanoribbon (i.e.,as those excited by light impinging normally to graphene, kk= 0), we represent in Figs.2.7(c),(d) the full plasmon dispersion relation for a nanoribbon either under inhomogeneous doping produced by E0[Fig.2.7(c)], or uniform doping [Fig.2.7(d)]. The dispersion relations are rather different in both situations, with the inhomogeneously doped nanoribbon showing a denser set of modes, as well as higher localization of the lowest-energy plasmons for large kk[86]. Finally, we need to mention that the decay rate γobtained from the Drude model [see Eq.(1.16)] presents now a spatial dependence on the position through |EF(R)|. However, since the local contribution to losses is proportional to Re{σDrude(ω)}= (e2/π~2)|EF|γ¿(ω2+γ2)(i.e., independent of R), we conclude that the inhomogeneity of the decay rate is nonetheless translated into a uniform spatial distribution of 2.3. PLASMONS IN INHOMOGENEOUSLY DOPED NANODISKS 65 losses. We need to remark that, during this section, we have obtained the decay rate using the impurity-limited approximation for the constant dc graphene mobility µ= 10000cm2/(V s) [see last paragraph of section(1.4.1)]. 2.3 Plasmons in inhomogeneously doped nanodisks In this section, we change the geometry under study, and we focus on plasmons generated in self-standing graphene nanodisks with small diameter D. In addition to the typical uniform doping distribution, we discuss in particular two different inhomogeneous doping configurations: (i) charged nanodisks under a uniform potential where the additional amount of charge carriers is self-consistently distributed along the graphene surface, and (ii) neutral nanodisks exposed to a neighboring external point charge. Our work is based on the study of the electromagnetic monopolar (m= 0) and the lowest-order dipolar modes (m= 1) sustained by the nanodisks, whereas the potential evolves as Φ(R, ω) = Φ(R, ω)eimϕ, with the azimuthal angle ϕand the polar coordinates R= (R, ϕ)[see orange arrows in the lower inset of Fig. 2.8(b)]. In this section, the doping charge carriers density nis also obtained via an electrostatic boundary-element calculation, and the dielectric response of graphene is taken from the Drude model [see Eq.(1.16)]. 2.3.1 Disks under uniform potential doping We start by analyzing charged nanodisks where additional carriers are rearranged to produce a uniform potential V=ZR0<D/2 d2R0[−en(R0)] |R−R0|.(2.29) We numerically solve this integral equation by expanding the Coulomb interaction in terms of Legendre polynomials of order las [120] 66 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS 1 |R−R0|=X∞ l=0 Rl < Rl+1 > Pl(cos(ϕ−ϕ0)),(2.30) where R>= max{R, R0}and R<= min{R, R0}. Besides, changing the radial variable to θ= 2R/D, we can rewrite Eq.(2.29) as −2V eD =X∞ n=0 I2n,0 1 θ2n+1 Zθ 0 dθ0θ02n+1n(θ0) + θ2nZ1 θ dθ0 θ02nn(θ0) ,(2.31) where Il,m =Z+π −π dφcos(mφ)Pl(cos(φ)),(2.32) with φ=ϕ−ϕ0, and in particular, I2n,0=2π 16n [(2n)!]2 (n!)4, I2n+1,1=π 24n+1 (2n)!(2n+ 2)! n![(n+ 1)!]2, (2.33) with I2n+1,0=I2n,1= 0 for integers n. We solve the integral of Eq.(2.31) following a similar approach as in section(2.2), where a discretization through a set of N equally spaced points is used, so that θi= (i+ 1/2)κ, with i= 0, . . . , N −1and κ= 1/N. Finally, normalizing the density of charge carriers to n∞=−V/2πeD (i.e.,the density in each of the infinite plates of a capacitor with a voltage difference Vand a plate separation D/2), we find n(θi) n∞= 4πXN−1 i0=0 îM−1óii0,(2.34) with Mii0=                              X∞ n=0 I2n,0 In i0 θ2n+1 i , i0< i, X∞ n=0 I2n,0Ln i, i0=i, X∞ n=0 I2n,0Kn i0θ2n i, i0> i, (2.35) 2.3. PLASMONS IN INHOMOGENEOUSLY DOPED NANODISKS 67 where the coefficients In i0=Z(i0+1)/N i0/N θ02n+1dθ0, Ln i=1 θ2n+1 Z(i+1/2)/N i/N θ02n+1dθ0+θ2nZ(i+1)/N (i+1/2)/N dθ0 θ02n, Kn i0=Z(i0+1)/N i0/N dθ0 θ02n, (2.36) admit straightforward closed-form solutions. The charge carriers density profile obtained with this fine-element method agrees [90] with the purely analytical profile expected n(θ)/n∞= (4/π)/√1−θ2[121]. Moreover, the effective capacitance density C=Q/V from the total charge Q=Rρ(R)d2R= (πeD2/2) R1 0θ n(θ)dθ yields C= 0.33D[90], which is in good agreement with the analytical result C=D/π ≈0.32D[121]. In particular, this method converges for l∼50 terms in the Legendre polynomials, and N∼2000 discretization points along the radial distance of the disk. For the calculation of n, we can express n(R) = h(R)¿[π(D/2)2], where h(R) is a dimensionless envelope function related to the occupation function as follows f(R) = »h(R)¿h»h(R)i. In absence of external fields, considering (i) the azimuthal symmetry m= 0,1of the potential, (ii) the Drude model for the conductivity where σ(R, ω) = σ(ω)f(R), and (iii) the Coulomb expansion in Legendre polynomials presented in Eq.(2.30), we can express the electrostatic potential for the disk given in Eq.(1.34) as Φ(θ, ω) = η(ω)Z∞ 0 dθ0θ0 θ> gmÇθ< θ>åñΦ00f+ Φ0Çf0+f θ0å−m2f θ02Φô.(2.37) Here, we use that Φ0(R, ω) = ∂Φ(R, ω)/∂R, the parameter η(ω)is given by Eq. (1.36) with ε1=ε2= 1, and gm(µ) = X∞ l=0 µlIl,m,(2.38) with Il,m taken from Eq. (2.32). In order to get the plasmon frequencies, we discretize the self-consistent potential of Eq.(2.37) in the same manner as above for obtaining the charge carrier 74 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS 2.4.1 Periodic doping by point charges In this section, we particularly describe the periodic doping of self-standing extended graphene through a square array of point charges of either equal [see Fig.2.12(a)] or alternating [see Fig.2.12(b)] sign, placed at a certain distance dabove the extended graphene. Due to the periodicity, we can restrict our study to one unit cell with characteristic size a, and a similar scaling law to that presented in section(1.5.4) can be used here since λa. Assuming the Drude model [see Eq.(1.16)], the periodicity of the structure allows us to write the conductivity as Fourier series as σ(R, ω) = XGσG(ω) eiG·R,(2.42) where the σG(ω)coefficients are independent of the position. The potential in the real space can be expanded using the same formalism Φ(r, ω) = XG0ΦG0(ω) ei(G0+kk)·R−|G0+kk||z|,(2.43) where the dependence on kkcomes from the external potential. Implementing Eqs.(2.42) and (2.43) into Eq.(1.34), and considering that graphene is in the z= 0 plane, the self-consistent potential can be rewritten in the Fourier space as ΦG(ω)=Φext G(ω) + XG0 (G+kk)·(G0+kk) |G+kk|ñ−2πiσG−G0(ω) ωôΦG0(ω).(2.44) Before solving the former equation, we need to know the profile of the doping charge carrier density nunder our specific doping configurations. We proceed by (i) using the method of charge images [111], (ii) expressing the Coulomb potential as 1 r=Zd2Q (2π)2 2π QeiQ·R−Q|z|,(2.45) and (iii) performing the sum over charges via the identity 2.4. PLASMONS IN PERIODICALLY DOPED GRAPHENE 75 XieiQ·Ri=(2π)2 a2XGδ(Q−G),(2.46) where the sum on the left runs over 2D lattice sites Riand δis the Dirac delta function. We find that the screening charge carrier density per unit cell reduces to n(R) = |Q0/e| a2h(R),(2.47) where the envelope function h(R)is defined as h(R) = |Q0/e| a2XG00 eiG00·R−Gd î1−ei(G00 x+G00 y)a/2ó.(2.48) Here, we consider that the 2D origin R= 0 is taken below one of the positive charges. The factor 1inside the square bracket stays for the positive charge, and the complex exponential accounts the effect of the negative charge at the center of the unit cell. Since |EF(R)|=~vF»π|n(R)|, we conveniently define the occupation function f(R) = »|h(R)|=XG00 fG00 eiG00·R,(2.49) where the coefficients fG00 present the next expression fG00 =1 a2Zunit cell d2Re−iG00·R»|h(R)|.(2.50) Considering that G00 =G−G0, we can rewrite the second fraction of Eq. (2.44) as follows −2πiσG−G0(ω) ω=a 2π ω000 0 2 ω(ω+ iγ) fG−G0 f0 ,(2.51) where ω000 0= (e/~)»4πh|EF|i/a is the normalization frequency of the system used here. Moreover, we establish f0=h|EF|i/E∞ Fwith E∞ F=~vFqπ|Q0/e|¿a2(2.52) as the Fermi energy obtained in the equal-sign doping case in the limit da. Furthermore, we define the function ΨG(ω) = »|G+kk|ΦG(ω)so that Eq.(2.44) 76 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS reduces to a system of equations ΨG(ω) = Ψext G(ω) + ω000 0 2 ω(ω+ iγ)XG0NGG0ΨG0(ω),(2.53) where we have defined the real, symmetric, and dimensionless matrix NGG0=a 2πXG0 (G+kk)·(G0+kk) »|G+kk||G0+kk| fG−G0 f0 .(2.54) Using the same procedure as in the previous sections of this chapter, if we set Φext(ω)≈0, we need to solve an eigensystem [103] whose real eigenvalues η000 jpermit us to find the plasmonic bands, since ωDrude spp ≈ω000 0/»η000 j−iγ/2. We plot in Figs. 2.12(c),(d) the kk-resolved plasmonic bands for the equal-sign doping (panel (c)) and alternating-sign doping (panel (d)) along a representative path of the 1BZ [see lower inset in panel (d)]. We plot the bands for diverse values of the d/a ratio, which determines the amplitude of the periodic doping modulation. In panel (c), the plasmonic bands evolve from those of uniformly doped extended graphene (blue dashed curves) to match perpendicularly the edges of the 1BZ showing up band gaps (X and M points) for small values of d/a (e.g.,the green curves where d/a = 0.1). In panel (d), the two lowest plasmonic bands display a flat range that extends from X to M. Remarkably, under this doping configuration, the normalized SPP frequency ωDrude spp /ω000 0is independent of d/a, unless da. This behavior is produced by the vanishing of h|EF|i with increasing d/a due to sign cancellations in the doping potential [103]. In this limit, the doping charge carrier density n(x, y) = [4|Q0/e|/a2]{exp(−2πd/a)}[cos(2πx/a) + cos(2πy/a)] presents a perfect harmonic profile. Finally, we observe that higher-order bands are nearly dispersionless over the entire 1BZ. In Figs.2.12(e),(f) we depict the near-field intensity for selected SPP modes normalized to the maximum value of each case. The near-field is obtained from the potential components in the parallel wave vector space as E(r, ω) = −∇Φ(r, ω) = −∇XGZ1BZ d2kk (2π)2ei(G+kk)·R−|G+kk| |z|ΦG(ω),(2.55) for the z= 0 plane, where graphene lies. The multiple panels of Fig.2.12(e) represent 2.4. PLASMONS IN PERIODICALLY DOPED GRAPHENE 77 0 1 (a) (e) ( f ) B D A G H E F C (c) 8 d/a d/a = 0.2 d/a = 0.1 0.25 0.00 0.50 0.75 1.00 B D C A F G H 0.15 0.00 0.30 0.45 0.60 0.75 E (d) y x z (b) ΓX M Γ ΓX M kx ky ΓX M Γ |E|2 |Emax|2 0 Drude spp ω ω ´´´ 0.5 -0.5 -0.5 0.5 x/a y/a Figure 2.12: Plasmonic bands in periodically doped graphene by point charges. (a),(b) Sketches of the two different doping configurations considered. In panel (a), the doping is produced by a square array of equal-sign point charges Q0=e(red spheres) with periodicity a, and placed at a distance dover the graphene layer (note that panels (c) and (e) correspond to the equal-sign doping while panels (d) and (f) to alternating-sign doping). In panel (b), we alternate the sign by introducing negative charges Q0=−e(green spheres) in the center of each square lattice. In both schemes the graphene layer is assumed to be at z= 0 height and schematically represented by its Fermi level. (c),(d) Plasmonic bands along the 1BZ [see inset of panel (d)] for different values of the ratio d/a. The SPP frequency ωDrude spp is normalized to ω000 0= (e/~)»4πh|EF|i/a. (e),(f) Parallel component of the graphene near-field intensity at some representative SPP modes along the unit cell area (see plot of point C) for a ratio d/a = 0.1. The figure is adapted from Ref.[103]. 78 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS the unit cell area corresponding to the equal-sign doping. Here, we assume that a positive charge is placed over each corner of the plots. The SPP at A is a dipolar mode excited under normal incident light; B represents a plasmon moving along the ΓX direction; C is similar to B, but with a node along ΓM because it belongs to the second band and it has to be orthogonal to B; finally, D behaves similarly to B, but since it has a spatial period 2aalong ΓX, its intensity possesses maxima at the vertical sides of the unit cell. In the case of alternating-sign doping [see Fig.2.12(f)], a negative charge is added at the center of each plot. Here, the near-field intensities possess a strong localization in the regions of vanishing doping charge [i.e.,±x=±y= (m+ 1/2)a, with m being an integer]. This behavior is similar to the trapping of graphene plasmons in ribbons and disks already mentioned in sections(2.2.3) and (2.3.2). Therefore, we can interpret that the dispersionless response of the upper plasmonic bands is produced by the strong localization of the SPPs at the regions of vanishing n. In all these results, we have assumed that the decay rate is γ= 0.01ω000 0. 2.4.2 Local density of optical states If we place an optical emitter (e.g.,molecules and quantum dots) close to a graphene layer, the plasmonic response of the latter can be substantially modified, giving rise to changes in the radiative decay rate [131, 132] of the emitter. This phenomenon is quantified through a modulation in the local density of optical states (LDOS), which measures the intensity of the normalized photon modes as function of the position and energy [133, 134]. The radiative decay rate of an optical emitter placed at r0is [135] γr(ω) = (4π2ω|p(ω)|2/~)×LDOS(r0, ω), where p(ω)and ωare its transition dipole and emission frequency, respectively. Besides, the LDOS(r0, ω)is obtained from the electric field Eacting back at the position of the optical emitter as [26] LDOS(r0, ω) = s(ε1+ε2) 2îLDOS0(ω) + LDOSind(r0, ω)ó,(2.56) where ε1(ε2) is the permittivity above (below) graphene, LDOS0(ω) = ω2 3π2c3(2.57) 2.4. PLASMONS IN PERIODICALLY DOPED GRAPHENE 79 is the value in vacuum projected along a certain Cartesian direction, and LDOSind(r0, ω) = 1 2π2ω p2(ω)Im{E(r0, ω)·p∗(ω)}.(2.58) is the induced component. Thereby, as the plasmons of doped graphene trigger an enhancement of the induced electric field, large values of the LDOS can be achieved due to an increase in the interaction with nearby optical emitters. Remarkably, if we only consider graphene as the optical emitter itself, Eq.(2.58) is negligible [31], and thus, we recover the expression given in Eq. (1.32) for γr(ω). An alternative way of achieving high values of γr(ω)for optical emitters is via a periodic doping which leads to divergences in the LDOS associated with Van Hove singularities. Here, we study this possibility by considering the interaction of our periodically extended graphene layer with an external optical emitter placed at a height z0. Using a similar fashion as the total potential given in Eq. (2.55), we conveniently rewrite the expression of the external potential considering (i) Eq.(2.45), and (ii) Φext(r, ω) = −p(ω)·∇(1/|r−r0|), so that in the graphene plane we find Φext(R, ω) = XGZ1BZ d2kk (2π)2ei(G+kk)·RΦext G(ω),(2.59) where Φext G(ω) = −2π iÄG+kkä |G+kk|·p(ω) + sign(z0)pz(ω) e−i(G+kk)·R0−|G+kk| |z0|.(2.60) Furthermore, we can also set LDOSind =Z1BZ d2kkLDOSind(kk).(2.61) Thus, we solve separately each fixed kkcomponent so that Eq.(2.53) remains ΨG(ω) = "1−ω000 0 2 ω(ω+ iγ)NGG0#−1 Ψext G(ω) =XjîΨ† G,j(ω)·Ψext G(ω)ó 1−ω000 0 2/η000 j ω(ω+ iγ)  −1 ΨG,j(ω), (2.62) 80 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS (a) a2 a1 uniform 600 200 400 150 50 100 0.4 0.6 0.70.5 a 2 a 1 0 200 ( ) 3 0 LDOS LDOS ka  ×     0 ω ω ´´´ Drude spp 0.5 0.60.4 b 1 b 2 0 ω ω ´´´ Drude spp (b) y x z a1 a2 b2b1 (d) (c) b2 b1 uniform Figure 2.13: Enhancement of the local density of optical states (LDOS) in periodically doped graphene by point charges. (a),(b) LDOS under the two configurations of Figs.2.12(a),(b) for d/a = 0.1. The optical emitters (blue arrows) are placed in the plane of the doping charges and their orientations and positions are described in the insets. The black solid curves correspond to uniformly doped graphene. The vertical dashed lines indicate exactly the monopolar frequencies at the edge of the 1BZ where the band gaps open [e.g.,in panel (a), it corresponds to the point D of Fig.2.12(c)]. (c),(d) In-plane evolution of the LDOS for the peak frequencies of panels (a) and (b). The figure is adapted from Ref.[103]. where the sum in the second row extends over orthonormalized eigenstates of the matrix NGG0[i.e.,it fulfills the conditions displayed on Eq.(1.39)]. Inserting Eq.(2.55) into Eq.(2.58), and using Eq.(2.62), we finally find LDOSind(kk) = 1 4π3ω p2(ω)XjΨ† G,j(ω)·Ψext G(ω)2Im   ω000 0 2/η000 j ω000 0 2/η000 j−ω(ω+ iγ)  . (2.63) From this expression, we directly calculate the LDOS using Eq.(2.61) through a discretization of the 1BZ where a mesh of 200×200 points in kkis enough to achieve converged results. Fig.2.13 illustrates the evolution in the LDOS with the ratio ωDrude spp /ω000 0once we place an optical emitter close to the periodically doped structures presented in Figs.2.12(a),(b). Here, we assume that the optical emitter is suspended in the same plane as the doping charges (d/a = 0.1), and we study the different positions and orientations presented in the insets of Figs. 2.13(a),(b). For the equal-sign doping 2.5. CONCLUSIONS 81 [i.e.,panel (a)], we observe a divergence in the LDOS (see vertical black dashed line) associated with a Van Hove singularity at the saddle point where the first band crosses the X value of the parallel wave vector [i.e.,D point in Fig. 2.12(c)]. Remarkably, an even more pronounced singularity is related with the first flat band in the XM region with the alternating-sign doping as depicted in Fig.2.13(b). In comparison with the uniformly doped case (black solid curves) for the same level of h|EF|i, we find an enhancement of a factor ∼8in panel (a) and ∼64 in panel (b). In Figs.2.13(c),(d) we show the dependence of the LDOS on the lateral position that fully confirms the symmetry of the plasmonic bands contributing to the singularities: for equal-sign doping, the LDOS map [see Fig.2.13(c)] follows the near-field intensity at the saddle point D in Fig.2.12(e), which is dominated by components along x, just like the orientation of the optical emitter; besides, for alternating-sign doping, the LDOS divergence extends over the entire region of vanishing doping, following the same behavior of the SPP mode at the point H as observed in Fig.2.12(f). The graphene decay rate used here is again γ= 0.01ω000 0. 2.5 Conclusions In the first section of this chapter, we have introduced a new practical method to describe accurately the electrostatic plasmonic response of interacting uniformly doped graphene nanoribbons based on the plasmon wave function of an individual island (i.e.,magnitude proportional to the induced charge density associated with the plasmon). We have shown that our model constitutes a natural extension of the dipole-dipole interacting model but with a better degree of accuracy, especially at short distances between the edges of the nanostructures. Although we have limited our analysis to different arrangements of graphene nanoribbons (e.g., dimers, arrays, and bilayer arrays), our model can be easily adapted to complex 2D geometries of other 2D plasmonic materials and also incorporate more plasmon modes in each nanostructure. In the second section, we have investigated the classical effects of inhomogeneities in diverse doping distributions in graphene nanoribbons. In particular, we have focused on three different electrostatic doping configurations: backgated ribbons, co-planar ribbon pairs placed at opposite potentials, and individual ribbons subject 82 CHAPTER 2. PLASMONS IN MULTIPLE DOPING CONFIGURATIONS to a uniform electric field. In the first two cases, we have found that the plasmon frequencies and the induced charge densities are very similar to those of uniformly doped nanoribbons for the same value of the average Fermi energy h|EF|i. However, when a uniform electric field is used to dope graphene, an interesting scenario opens where plasmons present very different characteristics, e.g.,distinct plasmonic dispersion to the uniformly doped case and induced charge densities piling up near the center of the ribbon. This charge accumulation can offer an additional handle to engineer graphene plasmon modes. Our study is based on the interaction of multiple dipoles spread all over the ribbon area, and can be straightforwardly extended to other doping configurations of graphene nanoribbons or different geometries of even diverse 2D plasmonic materials which could be useful for the design of future optoelectronic devices. In section(2.3), we have continued with the classical study of electrostatic inhomogeneous doping, but focused on graphene nanodisks. We have found again that the plasmon frequency and the induced charge density for inhomogeneous doping under a uniform potential behave similarly as in the uniform doping case for the same value of h|EF|i. Conversely, different behavior is observed when we have neutral nanodisks exposed to a neighboring external charge. We have found regions inside the nanodisks with a huge concentration of induced charge density near to the points of vanishing doping, in a similar manner as the ribbons under uniform electric field described in section(2.2.3). Moreover, here the plasmon frequencies differ considerably from the uniform doping configuration, particularly for the monopolar mode and more remarkably when the distance between the external charge and the disk is smaller than the disk radius. Remarkably, we have shown that a single electron is sufficient to excite a plasmon so that this doping configuration could enable the optical sensing of atoms and ions. Finally, we have explored in section(2.4) a new way of doping electrostatically extended graphene through a periodic distribution of equally-signed or alternatelysigned point charges. Under these patterned conditions, we have observed that the radiative decay rate of an optical emitter close to graphene can be boosted via the divergences in the LDOS associated with Van Hove singularities of the plasmonic bands. The radiated energy is almost entirely converted into graphene SPPs, which are eventually dissipated inelastically as heat or by the creation of e-h pairs. Chapter 3 Quantum nonlocal effects in nanoribbons As introduced in section(1.4.2), the plasmonic response of doped graphene is influenced by nonlocal and quantum finite-size effects when the characteristic size of the island is .λF, and therefore, classical electromagnetism loses its validity. More specifically, upon the inclusion of a quantum-mechanical TB-RPA description, substantial differences in the LSP (renamed as “plasmon” for simplicity throughout this chapter) frequencies and broadening appear up to sizes of ∼10 nm in nanoribbons and ∼20nm in nanodisks [40]. Besides, the broadening of the resulting plasmon is very sensitive to the type of edge termination [i.e.,armchair (AC) or zigzag (ZZ); see right inset of Fig.3.1]. Following the mentioned in section(1.5.3), ZZ-edged nanostructures possess a band of zero-energy modes containing surface states populating the edge termination (i.e.,electronic edge states) [20]. Thus, when plasmons dissipate, if their energy is higher than the Fermi level [101], a strong quenching emerges due to coupling with these zero-energy edge states [40, 80, 84]. In this chapter, we study quantum nonlocal effects observable in the lowest-order dipolar plasmons sustained by individual and interacting graphene nanoribbons. For our purpose, we use the quantum TB-RPA model presented in section (1.4.2) and consider D= 6nm wide self-standing uniformly-doped nanoribbons at T= 300 K where we distinguish between AC or ZZ edge terminations. Besides, we compare our results with classical full-numerical calculations following the description given in 83 90 CHAPTER 3. QUANTUM NONLOCAL EFFECTS IN NANORIBBONS 3.4 Conclusions In this chapter we have shown that quantum nonlocal effects in narrow graphene nanoribbons exhibit the following general properties: (i) plasmons are quenched by ZZ edges, and they decay through excitation of electronic edge states, when the energy of the formers are above the Fermi level; (ii) quantum nonlocal effects (generally identified as blueshifts) increase with decreasing nanoribbon width, although a classical electromagnetic description produces reasonable results for individual nanoribbons of widths down to a few nanometers; (iii) nonlocal effects are important in interacting nanoribbons at short separations, leading to substantial blueshifts in the lowest-order dipolar plasmons of closely spaced dimers, in contrast to classical theory, which predicts a smooth convergence towards the double-width nanoribbon; (iv) remarkably, the removal of a single row of atoms produces a dramatic increase in the plasmon energy, observed both in dimers and in arrays; and (v) our realistic quantum-mechanical calculations yield plasmon energies that are pushed up to the NIR regime for nanoribbon widths of a few nanometers, similar to those that can be synthesized by chemical self-assembly [108]. Our results provide a solid theoretical background for understanding the plasmonic response of single and interacting graphene nanoribbons, where important quantum nonlocal effects must be considered in the design of potential device applications in the NIR regime. Chapter 4 Nonlinear optical effects Nonlinear optics studies the phenomena related with the nonlinear response of a material to an external electric field [137]. In the case of doped graphene nanostructures, due to its unique electronic band structure, a strong nonlinear optical response [75, 77, 138, 139, 140, 141, 142, 143, 144, 145, 146] can emerge exceeding that of conventional nonlinear materials. Moreover, nonlinearities can be further enhanced by interband e-h transitions, finite temperatures, and LSPs (renamed as “plasmons” for simplicity) [78, 80, 81, 147, 148, 149, 150, 151]. In previous chapters, all the results have been obtained under the assumption of a linear optical response. Now, we study in detail the nonlinear optical response of doped graphene nanostructures which can act as plasmon-driven nonlinear enhancers. We start our analysis by briefly explaining the properties of three of the main nonlinear processes: second-harmonic generation (SHG), third-harmonic generation (THG), and Kerr effect. Then, we concentrate on describing the classical nonlinear conductivities and polarizabilities of doped graphene through the Boltzmann transport equation (BTE), where only intraband transitions are considered [18, 75, 81, 144, 147, 152], and we compare our results with a quantum-mechanical approach that implements nonlocal, finite-size, and edge-termination effects in combination with interband transitions. Finally, we show that these features play an important role in the nonlinear response observable in the plasmons sustained on nanostructures with a characteristic size D∼10 −20nm. 91 92 CHAPTER 4. NONLINEAR OPTICAL EFFECTS 4.1 Description of nonlinear optical effects In doped graphene, the dipole moment per unit area or polarization P(ω)depends on the external ac field Eext(ω) = E0(e−iωt + eiωt)associated with an incident monochromatic plane wave at frequency ωas [137] P(ω) = χ(1)(ω)Eext(ω) + χ(2)(ω)Eext2(ω) + χ(3)(ω)Eext3(ω) + . . . ≡P(1)(ω) + P(2)(ω) + P(3)(ω) + . . . , (4.1) where the quantity χ(1)(ω)is known as the linear susceptibility, while χ(2)(ω), and χ(3)(ω)are the secondand third-order susceptibilities, respectively. Additionally, we name P(1)(ω) = χ(1)(ω)Eext(ω)as the linear polarization, which is related to conventional (linear) optics, with P(2)(ω) = χ(2)(ω)Eext2(ω)and P(3)(ω) = χ(3)(ω)Eext3(ω) as the secondand third-order nonlinear polarizations. If we define the areal density of carbon atoms in graphene as [80] nC= 3.8×1015 cm−2, we can express directly the nth-order polarizability as α(n)(ω) = nCχ(n)(ω). The conversion factors from esu (electrostatic units) to SI of α(n)(ω)are given in Table4.1. Remarkably, the nonlinear physical processes derived from P(2)(ω)are different to those arising from P(3)(ω). Moreover, second-order nonlinearities cannot occur in centrosymmetric nanostructures [137] (i.e.,systems with an inversion symmetry center). Finally, we need to quantize the effects of nonlinear processes in doped graphene. If we take χ(1)(ω)of the order of unity, we find [153] χ(2)(ω)≈10−5esu and χ(3)(ω)≈10−8esu, which are much higher than the predicted values in standard nonlinear materials [137] [χ(2)(ω)≈10−8esu and χ(3)(ω)≈10−15 esu]. Polarizability esu SI α(1)(ω) 1cm31.113 ×10−16 C2m2J−1 α(2)(ω) 1cm5statC−13.711 ×10−21 C3m3J−2 α(3)(ω) 1cm7statC−21.238 ×10−25 C4m4J−3 Table 4.1: Polarizability unit conversion factors (esu, electrostatic units). 4.1. DESCRIPTION OF NONLINEAR OPTICAL EFFECTS 93 2ω ω SHG 3ω ω ω THG (a) (b) (c) ω ω 2ω SHG ω 3ω THG ω ω Figure 4.1: Secondand third-harmonic generation by a graphene nanoisland. (a) Scheme of the geometry for the SHG (red arrows) and THG (green arrows). Here, a graphene nanoisland is illuminated by a high-intensity laser beam at frequency ω (yellow beam) and the resulting emitted harmonic waves are collected by the photon detector. (b),(c) Energy-level diagram describing the SHG [panel (b)] and THG processes [panel (c)], respectively. The solid horizontal line represents the ground state while the dashed lines correspond to virtual states. The figure is adapted from Ref.[81]. 4.1.1 Second-harmonic generation We start our general analysis of the nonlinear effects by qualitatively describing the second-harmonic generation process. According to Eq.(4.1), the modulus of the second-order polarization can be rewritten as P(2)(ω) = 2χ(2) 0(ω)E2 0+χ(2) 2ω(ω)E2 0Äe−2iωt + e2iωtä.(4.2) The first term after the equal sign corresponds to a contribution at zero frequency (detailed in the subindex of the susceptibility), which does not lead the re-emission of any photon. Thus, a static electric field may arise inside graphene. This process is known as “optical rectification”. On the other hand, the second term after the equal sign corresponds to a contribution at the double frequency of the incident light, so that a photon at the second-harmonic frequency 2ωcan be generated. The corresponding energy-level diagram is depicted in Fig.4.1(b). Here, two photons at frequency ωimpinging over a triangular graphene nanoisland [see Fig. 4.1(a)] are converted into a photon of frequency 2ωin a single quantum-mechanical process with two virtual states involved. Besides the generation of a second-harmonic, the main 94 CHAPTER 4. NONLINEAR OPTICAL EFFECTS re-emission of photons from graphene occurs at frequency ωdue to the dominant linear response. Interestingly, this nonlinear optical mechanism is commonly used in lasers for doubling the frequency of their fundamental mode such as in solid-state lasers (e.g.,Nd:YAG lasers operate at λ= 1064nm and by SHG we can obtain the typical green-colored emission at 532nm). 4.1.2 Third-harmonic generation and Kerr effect Now we focus on the full expression of the third-order polarization. For simplicity, in this case we consider a monochromatic external ac field Eext(ω) = E0cos(ωt). Therefore, we have P(3)(ω) = 1 4χ(3) 3ω(ω)E3 0cos(3ωt) + 3 4χ(3) ω(ω)E3 0cos(ωt).(4.3) Analogously to what it is detailed in section(4.1.1), the first term in Eq.(4.3) after the equal sign corresponds to a contribution at frequency 3ω. Namely, three photons at frequency ωare destroyed, and a third-harmonic is generated. Fig.4.3(c) describes the energy-level diagram of the process. On the other hand, the second term after the equal sign in Eq.(4.3) corresponds to a contribution at the same harmonic of the incident light. This nonlinear term contributes to the variation of the refractive index n=√εof the graphene nanostructure at high intensities. This process is commonly known as ac Kerr effect [154]. Hence, if we only consider the first-harmonic terms up to third-order, the graphene polarization from Eq.(4.1) at the incident frequency ωreads Pω(ω)≈ïχ(1) ω(ω) + 3 4χ(3) ω(ω)E2 0òE0cos(ωt).(4.4) If we take into account the relation given in Eq.(1.15) between the dielectric function εand the local graphene conductivity σ(ω) = −iωχ(ω), with χ(ω) = χ(1) ω(ω) + (3/4)χ(3) ω(ω)E2 0, we can find straightforwardly the relation n(ω)≈n0(ω) + 3π 2t n0(ω)χ(3) ω(ω)E2 0,(4.5) 4.2. CLASSICAL NONLINEAR OPTICAL CONDUCTIVITIES 95 where tis the graphene thickness and n0(ω) = q1+4πχ(1) ω(ω)/t is the linear refractive index. We observe that third-order nonlinearities introduce a linear dependence of the refractive index with the intensity of the external field. 4.2 Classical nonlinear optical conductivities In this section we present an easy to handle way to obtain the nonlinear conductivities in graphene within the framework of classical electrodynamics [see section(1.4.1)]. More specifically, we assume the electrostatic where the characteristic size of our nanostructure fulfills Dλ. We use a unified approach based upon the BTE valid for extended graphene which describes the evolution of the doping carrier distribution function f(n) k(R, t)of nth-order at position Rand momentum k. Within this theory kand Robey the classical equations of motion: ˙ R=vk= (1/~)∂k/∂k, and ~˙ k(R, t) = −eE(R, t), where E(R, t)is the sum of the external field Eext(t) and the induced field Eind(R, t)generated by the extra charge carriers [see section(1.5.2)]. We focus in energy scales .1eV, then the linearization of the graphene dispersion relation around the Dirac points given in Eq.(1.7) is fulfilled. Besides, we assume a temperature T= 0 and incident photon energies .2EF, so that interband e-h transitions can be neglected. Combining all the former conditions, the doping carrier dynamics are described by [153] ∂ ∂tf(n) k(R, t)−e ~E(R, t)·∇kf(n) k(R, t)±vFˆ k·∇Rf(n) k(R, t) = −1 τhf(n) k(R, t)−f(0) k(k)i (4.6) where the right-hand side corresponds to scattering mechanisms, f(0) k(k)is the Fermi-Dirac distribution [see Eq. (1.21)], and τis the graphene relaxation time. Eq.(4.6) fully describes the microscopic dynamics of the doping carriers on which other macroscopic quantities are strongly related. For example, the surface current in graphene upon illumination of the external ac field given in section(4.1) can be expressed as [81] J(n)(R, t) = −egvgsZd2k (2π)2vkf(n) k(R, t).(4.7) where gv=gs=2 are the valley and spin degeneracies. The nonlinear set of Eqs.(4.6) and (4.7) can be solved perturbatively in the Fourier space by expanding the wave 96 CHAPTER 4. NONLINEAR OPTICAL EFFECTS vector kin order to get the relation between J(n)(ω)and E(ω)(i.e.,the linear and nonlinear conductivities). Interestingly, at zeroth-order expansion in k, the firstorder or linear surface current remains J(1) ω(ω) = ie2 π~2 EF ω+ iτ−1E(ω),(4.8) from which the local linear conductivity directly arises coinciding with that from the Drude model [see Eq.(1.16)]. For the second-order current, this model does not allow us to consider the effect of the fundamental frequency. However, in order to get the expression for the second-harmonic we have that the zeroth-order expansion in kvanishes [81], so that the dominant term is the first-order term which corresponds to a nonlocal contribution (i.e.,the current depends on the electric field gradient). Thus, we can find a expression for the second-harmonic (see Ref.[81] for a detailed derivation) given by J(2) 2ω,i(ω) = σ(2) 2ω,ijkl(ω)Ej(ω)∇kEl(ω) =3ie3v2 F 8π~2(ω+ iτ−1)3ñ5 3δijδkl −δikδjl +1 3δilδjkôEj(ω)∇kEl(ω), (4.9) where ijkl denote in-plane vector indexes, and δij is the Kronecker delta. As a clarifying example, in the z= 0 graphene plane, for an ac electric field polarized along the i≡xdirection, we have J(2) 2ω,x(ω) = 3ie3v2 F 8π~2(ω+ iτ−1)3ñExÇ∂Ex ∂x +5 3 ∂Ey ∂y å+EyÇ1 3 ∂Ex ∂y −∂Ey ∂x åô,(4.10) where for simplicity we omit the frequency dependence of the electric field components. Similarly, we can calculate the contributions to the THG and Kerr effect. Here, the third-order currents are local (i.e.,the spatial variations in f(3) k(R, t)are negligible). In the case of third-harmonic current we have [153] J(3) 3ω(ω) = σ(3) 3ω(ω)E3(ω) = 3ie4v2 F 4π~2EF(ω+ iτ−1)(2ω+ iτ−1)(3ω+ iτ−1)E3(ω),(4.11) and for the Kerr effect, the current reads [153] 4.3. CLASSICAL NONLINEAR OPTICAL POLARIZABILITIES 97 J(3) ω(ω) = σ(3) ω(ω)E3(ω) = 9ie4v2 F 4π~2EF(ω+ iτ−1)(−ω+ iτ−1)(2ω+ iτ−1)E3(ω).(4.12) Remarkably, the two former equations slightly differ from a previous study [146]. In fact, by neglecting the scattering term in Eq.(4.6), and redefining the external ac field as Eext(t) = R∞ −∞ dωE0exp(−iωt +t/τ), we recover the purely intraband expressions given in Ref.[146]. 4.3 Classical nonlinear optical polarizabilities The versatility of the electrostatic scaling law presented in section(1.5.4) enables us to directly obtain simple expressions for the nonlinear polarizabilities similar to that in Eq.(1.43) for the linear polarizability. First, we generalize the continuity equation from Eq.(1.14) considering nonlinearities at order nfor the harmonic sas ρind sω (n)(R, ω) = −i sω∇·J(n) sω (R, ω).(4.13) We start by analyzing the second-order nonlinear polarizability for SHG processes. Here, we assume polarization along the in-plane direction iover a doped graphene nanostructure with size Dand by implementing Eq.(4.10) into Eq.(4.13), we have ρind 2ω,i (2)(ω) = −η(2) 2ω(ω)D∇i®ñ5 3δijδkl −δikδjl +1 3δilδjkôEj(ω)∇kEl(ω)´,(4.14) where we define, in an analogous manner to Eq.(1.36), the second-order parameter η(2) 2ω(ω) = i 2ωD 3ie3v2 F 8π~2(ω+ iτ−1)3.(4.15) If we now consider (i) the dimensionless coordinate θ=i/D , (ii) the change of variables given in Eq.(1.38), (iii) the approximation of dominant lowest-order dipolar mode j= 1 [see section. (1.5.4)], and (iv) Eqs.(2.3) and (2.4) for a single nanostruc- 98 CHAPTER 4. NONLINEAR OPTICAL EFFECTS ture (l= 1), we can express the second-order dipole moment as p(2) 2ω,θ(ω) = −η(2) 2ω(ω)D2d2 1(ω)ξ2,(4.16) where ξ2=1 D d2 1(ω1)Zid2R∇i®ñ5 3δijδkl −δikδjl +1 3δilδjkôEj(ω1)∇kEl(ω1)´,(4.17) is a dimensionless coefficient obtained at the frequency of the lowest-order mode ω1. Analogously to the fitting coefficients η1and ξ1presented in section(1.5.4), ξ2 only depends on the specific geometry considered and can be calculated once and for all specific sizes, doping level, and dielectric environment. Finally, implementing Eq.(4.16) into Eq.(4.1), we attain an electrostatic scaling-law for the second-order polarizability α(2) 2ω(ω) = −η(2) 2ω(ω)D2ξ2 1ξ2 [1 −η(ω)/η1]2,(4.18) where η(ω)is given by Eq.(1.36). Note that for nanoribbons, the second-order polarizability is per unit length, where L→ ∞ is the ribbon length, so that on the right-hand side of Eq.(4.18), the dependence with Dis linear. Furthermore, under our assumptions of negligible interband e-h transitions and T= 0, we have α2ω(2)(ω)∝D2E−1 F. Using the same formalism, at third-order the polarizability readily leads to α(3) sω (ω) = η(3) sω (ω)D3ξ3 1ξ3 [1 −η(ω)/η1]3,(4.19) where η(3) sω (ω) = iσ(3) sω (ω)/sω encloses the dependence on the nonlinear conductivity for s= 1 (Kerr) and s= 3 (THG), respectively. Moreover, we have αsω(3)(ω)∝ D5/2E−3/2 Fand the dimensionless coefficient ξ3=1 D2d3 1(ω1)Zid2R¶∇iE3 i(ω1)©.(4.20) 4.4. COMPARISON CLASSICAL-QUANTUM NONLINEAR EFFECTS 99 4.4 Comparison classical-quantum nonlinear effects In this section we compare the classical nonlinear response of nanostructured doped graphene using the formalism presented in sections(4.2) and (4.3) with a quantummechanical TB-RPA treatment similar to that given in section(1.4.2) and explained in detail elsewhere [153] that includes interband e-h transitions, in combination with finite-size, nonlocal, and edge effects at a temperature T= 300 K. Here, we focus our study on self-standing triangular graphene nanoislands with D∼10nm although more extended results for bigger sizes and even for other geometries like nanoribbons are detailed in Ref.[153]. The choice of a triangular shape permits us to use straightforwardly the electrostatic scaling law formalism for the classical linear, secondand third-order nonlinear polarizabilities [see Eqs.(1.43), (4.18), and (4.19)] and also to study islands containing either AC or ZZ edges [see Fig.4.2(b)]. Furthermore, we compare the contributions to the lowest-order dipolar mode sustained by the graphene island excited by normal-incident illumination with an electric ac field polarized as the yellow arrow in Fig.4.2(a). For the classical calculations, the fitting coefficients η1, ξ1, ξ2,and ξ3are computed once using the commercial finite-difference code COMSOLR . We assume a local dielectric function [see Eq.(1.15)] where the linear conductivity is obtained from the Drude model [see Eq.(1.16)] and the graphene thickness t= 0.5nm is well converged with respect to the t→0limit. For our triangular nanoislands we find η1≈ −0.093,ξ1≈0.541,ξ2≈0.117−i0.006, and ξ3≈0.916+i0.022. In the case of nanoribbons, we have [153] η1≈ −0.071 and ξ1≈0.951 for linear order [which are in good agreement with the purely analytical values presented in section (1.5.4) and also with those given in section(2.1)], and ξ3≈1.297 + i0.0362. Note that since the nanoribbon is a centrosymmetrical structure, α(2) 2ω(ω)=0. The spectra shown in Figs.4.2(c)-(f) display the evolution of the dipolar plasmon assuming either a linear [panel (c)] or nonlinear [panels (d)-(f)] optical response for different doping levels EFof the graphene nanoisland [see the color code in panel (c)]. The intrinsic damping rate is ~γ= 50meV. At first glance, we observe in these four panels that, unlike the case of nanoribbons shown in Fig. 3.1, for low doping the 106 CHAPTER 5. MOLECULAR SENSING WITH GRAPHENE [166] is fitted to a sum of Lorentzians as αmol(ω) = 1 ~2Xj κj ω2 j−ω(ω+ iγ),(5.1) where κjand ωjare fitting parameters (see complete data in Table 1 of Ref.[93]), and we assume a fixed bandwidth (intrinsic damping) ~γ= 0.7meV. We observe from Fig.5.1(c) that two of these Lorentzians (modes j= 2,3) dominate the spectral range at photon energies ~ωA≈0.087eV and ~ωB≈0.092eV, respectively. In any case, the polarizability is very small αmol ≈10−22 cm3. Upon normal-incidence illumination, the pyridine molecule placed at the position r0experiences a total enhanced local field Eloc(r0, ω) = E0+Eind(r0, ω) + Eself(r0, ω), given by the superposition of E0, the field induced by the graphene nanodisk Eind(r0, ω), and the self-induced field of the molecule Eself(r0, ω). The latter is negligibly small for most molecules, so that it is disregarded in what follows. Assuming dominant the absorption process over scattering made by the graphene nanodisk [see Fig.1.9(a)], the absorption cross-section obtained from the optical theorem [26] of the whole system formed by the molecule and graphene is given by the superposition σabs gr+mol(ω)≈4π kIm®[f0(ω) + f1(ω)] ˆn·E∗ 0 E2 0´.(5.2) Here, f0(ω)is the far-field amplitude only with graphene, f1(ω)is the far-field amplitude only with the molecule, and ˆnis the direction of the incident polarization. Taking into account that the dipole moment of the molecule is pmol(ω) = αmol(ω)Eloc(ω) and f1(ω) = k2pmol(ω), we find straightforwardly that the shift on the absorption cross-section of the molecule or SEIRA cross-section is ∆σabs(ω) = σabs gr+mol(ω)−σabs gr (ω) = 4πkIm{αmol(ω)} Eind(r0, ω) E0 2 ,(5.3) where we approximate the increase in the local field intensity as |Eind(r0, ω)/E0|2. We observe from Eq.(5.3) that the inclusion of graphene permits a linear enhancement of ∆σabs(ω)with the induced intensity, and this behavior is in turn more remarkable when the impinging radiation is on resonance with the plasmon frequency. 5.1. SEIRA 107 900 1200 h D EF (eV) 0.0 0.2 0.08 0.11 0.14 B Photon energy, ω (eV)  A (c) (a) cm-1 h = 1nm (b) 0.0 0.5 1.0 0 10 20 h (nm) max Area σ ∆ abs 0.2 1.0 0.6 0.04 0.00 plasmon E0 Area σ ∆ abs Figure 5.2: Doping dependence in the absorption cross-section of molecules in SEIRA spectroscopy. We represent in panel (c) the SEIRA cross-section normalized to the graphene disk area (∆σabs/Area) produced by the interaction of a layer of pyridine molecules with the graphene disk considered in Fig.5.1. The molecules are placed at a distance h= 1nm above the graphene plane [see panel (a)] with a density of one molecule per nm2and covering an area that extends well beyond the nanodisk edge. The lower-right inset in the panel (c) shows a zoom of a highphoton-energy region in which weaker pyridine resonances are observable. Panel (b) shows the hdependence of the absorption cross-section for photon and Fermi energies corresponding to the absolute maximum of ∆σabs/Area. The figure is adapted from Ref.[93]. 108 CHAPTER 5. MOLECULAR SENSING WITH GRAPHENE In Fig.5.1(d) we plot ∆σabs(ω)for a pyridine molecule placed at r0= (150,0,1) nm over the edge of the graphene nanodisk (z= 0 plane), with a conservative value of the mobility µ= 2000cm2/(V s). We observe that ∆σabs(ω)is clearly enhanced along the dipolar plasmon (green dashed) curve of the graphene nanodisk, which shifts in energy with varying doping level since ~ωp∝E1/2 F. Interestingly, ∆σabs(ω) experiencies an enhancement of ∼103with respect to the absorption cross-section of the isolated molecule. In practical applications, one is interested in sensing a small concentration of molecules placed over a range of distances from the graphene nanodisk. Here, we consider a circular monolayer of pyridine molecules with diameter Dmol = 450nm and a density of nmol = 1molecule/nm2separated a distance h= 1nm from graphene [see Fig.5.2(a)]. The change in the SEIRA cross-section is simply given now by ∆σabs(ω)=4πnmol kIm{αmol(ω)}ZS d2r0 Eind(r0, ω) E0 2 ,(5.4) where the integral is extended over the surface layer Sof pyridine molecules. The resulting values normalized to the graphene area are shown in Fig.5.2(c) as a function of the photon energies and EF. The important message from this plot is that there is a one-to-one correlation between the molecular resonant photon energies and EF at which the dipolar plasmon overlaps with those resonances. More precisely, this leads to peaks in ∆σabs(ω)at EF≈0.34 and 0.39eV, corresponding to the molecular resonances of energies ~ωA≈0.087eV and ~ωB≈0.092 eV, respectively. In Fig.5.2(b) we plot the maximum values of ∆σabs(ω)as a function of the separation distance h. We observe that the results are qualitatively similar over distances up to a few nanometers from the graphene, and therefore, the technique should be robust against the uncertainty in the exact location of the molecules, provided their separation is in the .10nm range. The above mentioned one-to-one correlation between molecular resonances and EFsuggests that it is possible to obtain spectral information by recording the absorption as a function of EF, rather than ~ω. Indeed, upon illumination by spectrally broad sources (e.g.,an infrared lamp), this graphene-based sensor device can discriminate resonant photon energies by examining the Fermi levels at which the measured (spectrally unresolved) absorbed power is peaked. The plasmons of the 5.1. SEIRA 109 0 20 40 60 σint (nm2 eV) EF (eV) 0.60.2 1.0 h = 1nm h = 2nm h = 4nm h = 8nm (a)104 100 EF (eV) 0.60.2 1.0 (b) µ=2000 cm2/Vs µ=5000 cm2/Vs µ=10000 cm2/Vs µ=20000 cm2/Vs 102 h = 16nm σint (nm2 eV) Figure 5.3: Molecular sensitivity of the doping-dependent frequency-integrated absorption in SEIRA spectroscopy. We analyze the integral over NIR photon energies (0−1eV) of the SEIRA cross-section (i.e.,σint =~R1 0dω∆σabs) produced by a layer of pyridine molecules as a function of EFfor (a) several molecule-graphene distances with fixed graphene mobility µ= 2000cm2/(V s), and (b) fixed distance h= 1nm and different values of µ. The figure is adapted from Ref.[93]. graphene nanodisk act as amplifiers of the incident light at gate-controlled photon energies. We illustrate this concept by calculating the integral over IR photon energies (0−1eV) of the SEIRA cross-section, σint =~R1 0dω∆σabs. In Fig.5.3(a) we plot the results for several molecule-layer/graphene distances which suggest that the sensor can perform similarly well up to a distance .10 nm. Additionally, we explore in Fig.5.3(b) a range of feasible graphene mobilities ranging from the conservative value that we use in Fig.5.1, to higher-quality graphene. Although the former is already capable of giving sufficient molecule-specific information to resolve the presence of pyridine molecules, we note that currently attainable high-quality graphene enables further discrimination of weak vibrational features (e.g.,it allows us to resolve the A and B resonances, which are separated by ∼5meV). We thus conclude that the spectral resolution of this spectrometer-free technique is limited by the intrinsic graphene plasmon damping ∼~τ−1. In all the previous results, we have assumed for simplicity that the target molecules are not adsorbed near the graphene nanodisk so that the optical properties of the latter do not change. A variation in EFdue to the molecules can be a serious problem that might limit the applicability of our proposed sensing technique as it adds an element of uncertainty in the determination of the graphene Dirac point [see sec- 110 CHAPTER 5. MOLECULAR SENSING WITH GRAPHENE tion(1.3.2)]. We anticipate several possible strategies to deal with this uncertainty: (i) the entire spectrum changes when moving EF, and therefore, it should be sufficient to resolve spectral distances associated with the molecular features; (ii) in many practical situations, one is interested in discriminating between a certain finite number of different detected molecules, then the proposed sensor can be calibrated for each of them; and (iii) the possible charge transfer between the molecule and graphene can be drastically reduced by the addition of a thin transparent insulating layer, which according to Fig.5.3(a) can have a thickness of several nanometers without causing a serious reduction in sensing capabilities. These strategies can be also extended to the SERS spectroscopy that we present below. 5.2 SERS This spectroscopy technique based on the Raman scattering [see AppendixC] was first used in 1974 [167] precisely with pyridine molecules. The authors observed reversible Raman-shifts originated from the direct adsorption of the molecules onto a layer of a plasmonic material (silver in this case) by applying an external potential. Raman scattering is a nonlinear second-order process proportional both to the incident light intensity and to the emission from the inelastically frequencyshifted transition dipole [168]. Unfortunately, owing to the non-resonant nature of this effect, the cross-sections of single Raman-active molecules are very small (∼10−26 cm2). Therefore, similarly to SEIRA, an enhancement in the local field intensity provided by a plasmonic layer is needed in order to increase the intensity of the Raman signal. In our case, we follow a similar procedure as in section (5.1), and we use the dipolar plasmon of the graphene nanodisk in order to amplify the Raman signal of a certain generic molecule at position r0. Remarkably, we supplement the system with a resonant silicon cavity [169] that amplifies the intensity of the incident pump light with energy ~ωpump = 0.422eV (i.e.,the emission line of an Er:YAG laser) [see Fig.5.4(a)]. The SERS enhancement with respect to the isolated molecule is given by EFSERS = Eind(r0, ωpump) E0 2 ptot(R, ω1) p0(ω1) 2 ,(5.5) 5.2. SERS 111 h d Silicon molecule ∆νR (cm-1) 1000 2000 3000 0.60.2 EF (eV) 1.0 104 SERS EF 10-2 101 1-1 0 2x/D -1 0 1 2z/D SERS EF 103 100 102 101 disk (a) (b) (c) E0 Figure 5.4: Surface-enhanced Raman scattering (SERS) with graphene plasmons. (a) Sketch of the system under consideration, consisting of a silicon sphere (diameter 1530nm, εSi = 12) placed at a distance d= 9nm above a graphene disk (diameter D= 300nm), which is in turn placed at a distance h= 1nm above a Raman active molecule. The system is illuminated with a 0.422eV light plane wave (i.e.,λ≈ 2.94µm) that is resonant with a mode of the sphere (i.e.,the sphere works as a nanofocuser, similarly to previous designs [169]). (b) SERS enhancement factor EFSERS (relative to an isolated molecule), as a function of Raman-shift ∆νRand graphene Fermi energy EFfor a molecule placed along the symmetry axis. (c) SERS enhancement as a function of the position of the molecule relative to the graphene disk for doping and Raman shift conditions corresponding to the open circle in panel (b). The figure is adapted from Ref.[93]. where ω1is the final frequency of the molecule after the Raman transition [see Eq.(C.2)]. Moreover, Eind(r0, ωpump)is the induced field at the incident light frequency, mainly controlled by the silicon cavity, whereas ptot(R, ω1) = p0(ω1) + pind(R, ω1)is the total superposition of the free-molecule Raman transition dipole moment p0(ω1), and the dipole induced by the molecule on the surrounding structure pind(R, ω1). For simplicity, in our calculation of Eind(r0, ωpump)we only consider the silicon cavity and neglect the graphene nanodisk. This happens because the Mie resonant quality factor of the silicon sphere is so high that the non-resonant effect of the graphene nanodisk can be neglected. Similarly, we obtain pind(R, ω1)as the dipole induced on the graphene nanodisk placed at Rwithout taking into account the silicon cavity. This is possible since the graphene nanodisk supports resonant plasmons at the Raman-shifted frequency ~ω1. Thus, as graphene appears on resonance with the final roto-vibrational energy of the molecule, we can neglect the effect of the silicon sphere. Furthermore, like in SEIRA, we neglect the self-induced 112 CHAPTER 5. MOLECULAR SENSING WITH GRAPHENE polarization of the molecule. In Fig.5.4(b) we plot the enhancement factor EFSERS as a function of the molecule Raman-shift ∆νRand EFfor the system depicted in Fig.5.4(a), with an edge-to-edge distance between the silicon sphere and the nanodisk d= 9nm, and a separation between the molecule and the nanodisk h= 1 nm. Under these conditions, the silicon cavity produces an enhancement of |Eind(r0, ωpump)/E0|2≈2200. The important message of this plot is that if we fix ∆νR, for each value of EFwe have a different value of EFSERS, and there is a specific one for which EFSERS is maximum (particularly here ∼104). Furthermore, the actual value strongly depends on the molecule position relative to the nanodisk [see Fig.5.4(c)], yielding again qualitatively similar performance for molecule-graphene distances in the .10nm region. 5.3 Conclusions In this chapter, we have presented a new infrared sensing strategy that avoids the use of costly and inefficient optical elements (e.g., spectrometers and laser sources) and simply involves infrared lamps and electrical doping of graphene through an externally-applied gate voltage. We have shown the ability of graphene nanodisks to resolve the chemical identity of adsorbed molecules from the measurement of broadband-integrated absorption and Raman scattering signals enhanced by the electrically tunable plasmons of this material. Their narrowness is sufficient to resolve the frequency of the molecular resonances in the integrated intensity as a function of the controlled doping level. This control is an important practical aspect of our proposed sensor. We contemplate a gating device in which a bottom gate is combined with a contact for the graphene. Electrical connectivity could be provided by a thin transparent insulating layer, as recently used to demonstrate active control of the plasmons sustained on the graphene nanodisk [71]. Alternatively, we expect similar results for graphene nanoribbons, whose plasmon frequencies and characteristics for transversal polarization are similar to those of the nanodisks, with the additional advantage that the former structures can be contacted in a region far from the active sensing area. Remarkably, the large confinement and induced intensity amplification associated with graphene plasmons leads to SEIRA and SERS intensity enhancements 5.3. CONCLUSIONS 113 reaching ∼103and ∼104, respectively, which foster the use of graphene to improve traditional sensing techniques based on spectrally resolved infrared absorption and Raman scattering. The roto-vibrational energy resolution of the proposed sensing scheme is determined by the spectral width of graphene plasmons (i.e., it is essentially limited by the plasmonic quality factor of the material) and can reach a few meV under currently realistic conditions. In summary, graphene plasmons provide a versatile platform for sensing, thus opening new possibilities for exploiting their large electro-optical tunability, and in particular, the realization of label-free chemical identification without the involvement of spectrometers and laser sources. Chapter 6 Conclusions In this thesis, we have realized an in-depth study of the plasmonic response of graphene under different novel conditions. We have focused on theoretical concepts that could be useful for the design of future graphene plasmonic devices. We thus proceed to summarize in this final chapter the overall conclusions of our work. In Chapter1, we have concisely introduced the general properties of graphene. We have started with a brief review of its history and the most common techniques of synthesis. Afterwards, we have studied its peculiar optoelectronic properties resulting from the singular linear band structure. Additionally, we have classically described the different types of surface Dirac plasmons that graphene, either within a finite geometry (LSPs) or as an extended layer (SPPs), can sustain under uniform doping conditions. Finally, we have presented a general electrostatic scaling law to find directly the classical LSP frequencies of a given graphene nanostructure as a function of its doping level, characteristic size, and dielectric environment. In Chapter2, we have studied the classical behavior of graphene Dirac plasmons under diverse geometric schemes, as well as under realistic inhomogeneous doping configurations. First, we have derived a novel method based on the plasmon wave function of an individual uniformly doped nanoribbon to study the interaction of multiple nanoislands. Remarkably, our model possesses a better degree of accuracy than the typical dipole-dipole interacting model. Later, we have shown how the inhomogeneities in the doping distribution affect the LSPs sustained on graphene nanoribbons, nanodisks, and the SPPs of extended layers. Specifically, we find 115 122 APPENDIX B. LSP RESONANCE FREQUENCY IN ELECTROSTATICS This expression can be reformulated as a second-order polynomial equation x= y(1 −a/x), with x≡ωDrude p,y≡ωDrude j, and a≡iγ/2. We can directly find the solution x=y 2"1 + s1−4a y#≈y 2ñ2−2a yô=y−a, (B.4) where we have used the approximation √1−κ≃1−κ/2, being κ= (4a/y)1. Therefore, after recovering the proper variables, we finally obtain the expression of the LSP frequency in the electrostatic regime shown in Eq.(1.47) ωDrude p≈ωDrude j−iγ/2.(B.5) Remarkably, for nanostructures with characteristic size comparable to the incident light wavelength, retardation effects become important and the final LSP frequency is redshifted [see Fig.1.9(a)]. Appendix C Infrared absorption and Raman scattering It is known from Quantum Mechanics that an electron bound to an atom possesses only certain energy levels corresponding to its electronic states. For example, for a hydrogen atom, the electronic energy levels are determined by [170] el n=−13.6/n2eV where n= 1,2,3. . . is the principal quantum number. The bound energy is expressed as a negative number because we need that much energy to unbind the electron from the hydrogen nucleus. Due to this quantization of the energy levels, for radiative electronic transitions the electron can only absorb photons with energies exactly matching the gap between two different electronic states. The energy of these photons typically falls in the visible and UV regimes. In the case of molecules, the scheme of energy levels is more complicated. In addition to electronic transitions, we need to consider that the constituting atoms can vibrate with respect to the “equilibrium distance”, and the whole molecule can rotate with respect to its mass center. Hence, the internal energy of the molecule int under the Born-Oppenheimer approximation [171] is given by the superposition of the electronic, vibrational, and rotational energies as int =el +vib +rot.(C.1) A schematic representation of the electronic, vibrational, and rotational energy levels 123 124 APPENDIX C. INFRARED ABSORPTION AND RAMAN SCATTERING is shown in Fig.C.1(a). Here, we observe that an electronic state of a generic molecule (purple curve) possess multiple vibrational energy levels (red horizontal lines), and among them, multiple rotational levels (green horizontal lines in the inset) show up with a much smaller difference of energy. Radiative vibrational transitions may occur by the absorption of photons in the NIR while, for rotational transitions, the required photon energies fall in the far IR and microwave regimes. Remarkably, in a vibrational transition, the molecule can also change its rotational energy level, giving rise to the roto-vibrational transition spectrum. Here, we focus on the vibrational transitions of a generic molecule. The way a molecule vibrates is called a vibrational mode. For molecules containing Natoms, there are 3N−5vibrational modes for linear samples, whereas for nonlinear ones 3N−6degrees of freedom arise. We know that when a photon interacts with a molecule with the appropriate energy, it can be absorbed so that the molecule gets excited to a higher vibrational energy. This process is called infrared absorption, and the schematic representation of the transition between the two vibrational energy levels involved is depicted within a vertical blue arrow in Fig.C.1(a). When we consider that after a certain relaxation time, the molecule decays back to a lower energy level emitting a second photon, then the process is called light scattering. Interestingly, this second photon does not need to be equally energetic as the initial photon. When the molecule returns to the original energy level from a virtual state, the emitted and absorbed photons present the same energy. We name this process as elastic or Rayleigh scattering, which strongly depends on the size of the target molecule and, as an example, it is the responsible of the blue color of the sky [172]. In Fig.C.1(b) we show the energy-level diagram of the process. On the other hand, if the molecule decays back to a roto-vibrational energy level different to the one it originated from, the emitted and absorbed photons may have different energy. In this case, we talk about inelastic or Raman scattering [172] which is typically very weak (∼1of 107photons). When the final roto-vibrational level of the molecule presents a higher energy than the initial level, this means that the emitted photon is shifted to a lower frequency (redshift); namely, it presents a Stokes Raman-shift [see blue arrows on the left in Fig.C.1(c)]. In the case of a final less energetic roto-vibrational energy level, the emitted photon is shifted to a higher frequency (blueshift) and presents anti-Stokes Raman-shift [see blue arrows 125 (a) Rayleigh scattering electronic state vibrational levels Energy rotational levels infrared absorption (b) Raman scattering (c) anti-StokesStokes Figure C.1: Energy-level diagram of infrared absorption, Rayleigh scattering, and Raman scattering. (a) Sketch of the infrared absorption of a photon by a generic molecule that undergoes a transition (blue vertical arrow) from a lower energetic vibrational energy level to a higher one (red horizontal lines). The purple curve corresponds to the electronic energy of the molecule while the inset represents the rotational energies (green horizontal lines) lying between each vibrational energy level. (b) Elastic or Rayleigh scattering of a molecule. The molecule is excited up to a virtual state (black dashed line) and decays back to the same vibrational energy level emitting a photon with the same energy as the incident photon. (c) Inelastic or Raman scattering of a molecule. Here, the molecule decays back to a vibrational level with higher (lower) energy than the original level thus producing Stokes (anti-Stokes) Raman scattering. on the right in Fig.C.1(c)]. Usually, in spectroscopy these shifts are expressed in cm−1using the relation ∆νR(cm−1) = ~(ωpump −ω1)×Ä8065.544cm−1eV−1ä,(C.2) where ~ωpump is the incident or pump photon energy (in eV), and ~ω1is the energy of the photon emitted by the molecule (also in eV). Thus, positive values of ∆νR correspond to Stokes shifts, and negative values to anti-Stokes ones. List of publications and contributions to conferences The research performed during the development of this thesis has led to the following publications and contributions to international conferences: Articles on which the thesis is based 1. J. D. Cox, I. Silveiro, and F. J. García de Abajo. Quantum effects in the nonlinear response of graphene plasmons. (submitted, September 2015). 2. M. T. Manzoni, I. Silveiro, F. J. García de Abajo, and D. E. Chang. Second-order quantum nonlinear optical processes in single graphene nanostructures and arrays. New J. Phys.,17(8), 083031 (2015). 3. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. Plasmon wave function of graphene nanoribbons. New J. Phys.,17(8), 083013 (2015). 4. A. Marini, I. Silveiro, and F. J. García de Abajo. Molecular sensing with tunable graphene plasmons. ACS Photonics,2(7), 876–882 (2015). 5. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. Quantum nonlocal effects in individual and interacting graphene nanoribbons. Light, Science and Applic.,4, e241 (2015). 6. I. Silveiro and F. J. García de Abajo. Plasmons in inhomogeneously doped neutral and charged graphene nanodisks. Appl. Phys. Lett.,104, 131103 (2014). 127 128 LIST OF PUBLICATIONS AND CONTRIBUTIONS TO CONFERENCES 7. I. Silveiro, A. Manjavacas, S. Thongrattanasiri, and F. J. García de Abajo. Plasmonic energy transfer in periodically doped graphene. New J. Phys.,15, 033042 (2013). 8. S. Thongrattanasiri, I. Silveiro, and F. J. García de Abajo. Plasmons in electrostatically doped graphene. Appl. Phys. Lett.,100, 201105 (2012). Contributions to international conferences Oral contributions 1. Graphene Optics Workshop 2015–Optics of Graphene and 2D Materials. Exeter, UK. Second-order nonlinearities in single graphene nano-structures and arrays. M. T. Manzoni, I. Silveiro, F. J. García de Abajo, and D. E. Chang. September, 2015. 2. SPIE Optics and Photonics. San Diego, California, USA. Infrared spectroscopy with tunable graphene plasmons. A. Marini, I. Silveiro, and F. J. García de Abajo. August, 2015. 3. 7th International Conference on Surface Plasmon Photonics (SPP7). Jerusalem, Israel. Plasmon wave function of graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. June, 2015. 4. ImagineNano 2015. Bilbao, Spain. Quantum nonlocal effects in individual and interacting graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. March, 2015. 5. Nanometa 2015. Seefeld in Tirol, Austria. Quantum nonlocal effects in individual and interacting graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. January, 2015. 6. SPIE Optics and Photonics. San Diego, California, USA. Plasmons in inhomogeneously doped neutral and charged graphene nanodisks. I. Silveiro and F. J. García de Abajo. August, 2014. 7. SPIE Optics and Photonics. San Diego, California, USA. Quantum nonlocal effects in individual and interacting graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. August, 2014. LIST OF PUBLICATIONS AND CONTRIBUTIONS TO CONFERENCES 129 Poster contributions 1. 7th International Conference on Surface Plasmon Photonics (SPP7). Jerusalem, Israel. Quantum nonlocal effects in individual and interacting graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. June, 2015. 2. 7th International Conference on Surface Plasmon Photonics (SPP7). Jerusalem, Israel. Molecular Sensing with Tunable Graphene Plasmons. A. Marini, I. Silveiro, and F. J. García de Abajo. June, 2015. 3. ImagineNano 2015. Bilbao, Spain. Quantum nonlocal effects in individual and interacting graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. March, 2015. 4. Nanometa 2015. Seefeld in Tirol, Austria. Plasmons in inhomogeneously doped neutral and charged graphene nanodisks. I. Silveiro and F. J. García de Abajo. January, 2015. 5. Nanometa 2015. Seefeld in Tirol, Austria. Plasmon wave function of graphene nanoribbons. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. January, 2015. 6. Nanolight Conference. Benasque, Huesca, Spain. Plasmons in inhomogeneously doped neutral and charged graphene nanodisks. I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo. March, 2014. 7. Graphene Nanophotonics. Benasque, Huesca, Spain. Plasmonic energy transfer in periodically doped graphene. I. Silveiro, A. Manjavacas, S. Thongrattanasiri, and F. J. García de Abajo. March, 2013. 8. International Topical Meeting on Nanophotonics and Metamaterials. Seefeld in Tirol, Austria. Plasmonic energy transfer in periodically doped graphene. I. Silveiro, A. Manjavacas, S. Thongrattanasiri, and F. J. García de Abajo. January, 2013. 9. 12th International Conference on Near-Field Optics, Nanophotonics and Related Techniques (NFO12). San Sebastián, Spain. 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