The hydrodynamic approach for plasmonics in graphene

Pedro Alexandre Ferreira Passos The hydrodynamic approach for plasmonics in graphene October 2022 UMinho | 2022 Pedro Passos The hydrodynamic approach for plasmonics in graphene Universidade do Minho Escola de Ciências Universidade do Minho Escola de Ciências Pedro Alexandre Ferreira Passos The hydrodynamic approach for plasmonics in graphene Master Thesis Master in Physics Work developed under the supervision of: Dr. Nuno Miguel Machado Reis Peres October 2022 COPYRIGHT AND TERMS OF USE OF THIS WORK BY A THIRD PARTY This is academic work that can be used by third parties as long as internationally accepted rules and good practices regarding copyright and related rights are respected. Accordingly, this work may be used under the license provided below. If the user needs permission to make use of the work under conditions not provided for in the indicated licensing, they should contact the author through the RepositoriUM of Universidade do Minho. License granted to the users of this work Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International CC BY-NC-SA 4.0 https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en iv Acknowledgements The development of this thesis was a regarding and fulfilling experience. It is the culmination of all the hard work and the effort that i put in my degree and master’s in physics taken in University of Minho. The five years that i spend on this institution allowed me to have personal growth and became more aware of how exciting and stimulating science can be. The memories and experiences that i retrieve along the way, were only possible due to the numerous people who offer their unconditional support and that i would not be able to overcome the challenges that were imposed on me. To all these people it is my wish to leave a mention here. Firstly, i would like to thank my supervisor Dr Nuno Peres for giving me the tools necessary to accomplish this thesis, for its availability and readiness to answer my questions, helping me to overcome the difficulties. Also for showing me how research in science is done, and at this point i must mention FCT for granting me with a research grant on graphene, whose project lead to this thesis. Next, i must thank all my friend and colleagues from outside and inside my course for their support and for always be there for me. In particular, to my fellow course friends: Patricia, Mélina, Pinheiro, Mónica and Rui, with whom i spend many days studying. Without you this would not have half the fun. Finally, my biggest thanks goes to my family, whom i dedicate this thesis to. To my cousin Claudia for the english revision and to my brother Tiago for helping me with the design for some figures in this thesis. Not forgetting the rest of my family, you were the inspiration and my strength to overcome many obstacles that came in my way during these years. To all of you i am forever grateful. v STATEMENT OF INTEGRITY I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the Universidade do Minho. ,(Pl ae) (D ate ) (Pedro Alexandre Ferreira Passos) vi Abstract The hydrodynamic approach for plasmonics in graphene The hydrodynamic model approach to plasmonics is based on the simultaneous solution of Euler’s equation, Poisson’s equation, and the continuity equation. The quantum mechanical effects enter the model via the statistical pressure induced by the gas of electrons. It is also a form of including the effect of nonlocality. The aim of this thesis is to study the dispersion relation of the surface plasmon-polaritons and the plasmonic wakes created by an external potential, when a graphene sheet is in the vicinity of a metal. It is known they disperse linearly with the wave vector, therefore are of acoustic nature. This problem has been studied for normal plasmons, but the study for acoustic plasmons is missing. In the first part of this thesis, the hydrodynamic model will be used to solve some electrostatic boundary-value problems in planar geometry, that will give the linear dispersion of the SPPs in graphene near a semi-infinite and finite nonlocal metal. The study for the metals, gold and titanium, showed that the nonlocal effects are more visible in titanium, due to its intrinsic proprieties, such as plasmon frequency and background permittivity. The dielectric separation between graphene and metal also enhances the nonlocal effects. The decrease of the dielectric thickness increases the nonlocality. Regarding the finite metal, the results show that the increase of the metal thickness results in a higher energy of the surface plasmon-polaritons in graphene. In this case, the dispersion is also linear in the wavenumber 𝑘. The second part encompasses the study of the induced potential in graphene, due to an external charge moving parallel to graphene in the y-direction at an height 𝑧0. When graphene is in the vicinity of a dielectric an oscillatory V-shaped pattern was perceived, and the dependence of the angle on the Froude number (or dependence on the velocity of the external charge) provided two different regions, a constant angle region for low Froude numbers where the wake angle takes the value of 21◦. This is similar to the Kelvin region, where the angle takes the constant value of 19.47◦. A transition for a Mach region occurs for a plasmonic Froude number of 2.2, where the decrease of the angle happens for higher velocities following the law 1/𝑣. When a local metal is added to the system, the oscillatory behavior vanishes and a more continuous V-shaped wake appears in graphene. In this case, the angles follow a quadratic polynomial law, where these decrease with the increasing Froude number. Studying the phase velocity and the dispersion for the classical water wakes and the plasmonic wakes it is possible to see two limiting cases for 𝜅=−1and 𝜅=0, which correspond to pure gravity waves in deep water and gravity waves in shallow water, respectively. In such manner, it is possible to make an analogy between gravity waves and the plasmonic waves in graphene. Keywords: Nonlocality, SPPs, Graphene, Plasmonic wakes, Kelvin region, Mach region vii Resumo A abordagem do modelo hidrodinâmico à plasmônica é baseada na solução simultânea da equação de Euler, da equação de Poisson e da equação de continuidade. Os efeitos da mecânica quântica entram no modelo através da pressão estatística induzida pelo gás de eletrões. É também uma forma de incluir o efeito da não-localidade. O objetivo desta tese é estudar a relação de dispersão e as perturbações plasmônicas criadas por um potencial externo, quando o grafeno está nas proximidades de um metal. É sabido que os plasmões dispersam linearmente com o vetor de onda, portanto possuem uma natureza acústica. Este problema foi estudado para plasmões normais, no entanto falta o estudo para plasmões acústicos. Na primeira parte desta tese, o modelo hidrodinâmico será usado para resolver alguns problemas eletrostáticos de valor de fronteira, que darão a dispersão linear dos SPPs quando o grafeno está próximo a um metal não local semi-finito e finito. O estudo para os metais ouro e titânio mostraram que os efeitos não locais são mais visíveis no titânio, devido às suas propriedades intrínsecas, como frequência plasmônica e permissividade de fundo. A separação dielétrica entre o grafeno e o metal, também potencializa os efeitos não locais. A diminuição da espessura do dielétrico, aumenta a não-localidade. Em relação ao metal finito, os resultados mostram que o aumento da espessura do metal leva ao aumento da energia dos plasmões-polaritões de superfície no grafeno. Neste caso, a dispersão também é linear no vetor de onda 𝑘. A segunda parte abrange o estudo do potencial induzido no grafeno, devido a uma carga externa movendo-se paralelamente ao grafeno na direção dos y’s, a uma altura 𝑧0. Quando o grafeno está na vizinhança de um dielétrico, foi visto um padrão oscilatório em forma de ”V”, e a dependência do ângulo no número de Froude (ou dependência da velocidade da carga externa) mostra duas regiões diferentes, a primeira de ângulo constante para números de Froude baixos onde o ângulo do cone é constante e de valor 21◦. Sendo semelhante à região de Kelvin, onde o ângulo constante toma o valor de 19.47◦. Observa-se uma transição para a região de Mach ocorre para um número de Froude plasmónico de 2.2, onde a diminuição do ângulo ocorre para velocidades mais altas seguindo a lei 1/𝑣. Quando um metal local é adicionado ao sistema, o comportamento oscilatório é quebrado e uma onda em forma de ”V”contínua aparece no grafeno. Neste caso, os ângulos seguem uma lei polinomial quadrática, onde os ângulos diminuem com o aumento do número de Froude. Estudando a velocidade de fase e a dispersão para as ondas clássicas na água e as ondas plasmônicas é possível ver dois casos limites para 𝜅=−1 e𝜅=0, que correspondem a ondas gravíticas puras em águas profundas e ondas gravíticas em águas rasas, respectivamente. Desta forma, é possível fazer uma analogia entre as ondas gravitacionais e as ondas plasmônicas no grafeno. Palavras-chave: Não-localidade, SPPs, Grafeno, Ondas plasmónicas, Região de Kelvin, Região de Mach viii Contents List of Figures xi List of Tables xix Acronyms xxi 1 Introduction 1 1.1 General Topics in Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 GraphenePlasmonics............................... 5 1.3 Nonlocal theory: a hydrodynamic approach . . . . . . . . . . . . . . . . . . . . 9 2 An Hydrodynamic Model 11 2.1 Boltzman and Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Hydrodynamic Model of a 3D electron gas in the electrostatic regime . . . . . . . . 13 2.2.1 2D electron gas and graphene . . . . . . . . . . . . . . . . . . . . . . 16 2.3 BoundaryConditions ............................... 18 3 Plasmonic Nanostructures 20 3.1 Electrostatic configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.1 Metal-dielectric interface . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.2 PlanarSlab ............................... 24 3.1.3 2Delectrongas ............................. 30 3.1.4 Graphene ................................ 30 3.2 Graphene-Metal Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Hydrodynamic Model in the Presence of External Potentials 44 4.1 Motion Parallel to the Graphene Sheet . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Graphene in the vicinity of a dielectric . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Graphene in the vicinity of a local metal . . . . . . . . . . . . . . . . . . . . . . 53 5 Plasmonic Kelvin and Mach wakes 61 ix LIST OF FIGURES 5.4 Relation dispersion of water waves for the exact (dashed red curve) and for the limiting cases (𝜅=0,−1,1/3) given by equations (5.6), (5.4) and (5.8), respectively. The exact solution is given by equation (5.3). The left figure shows the exact dispersion (red dashed curve) for a water depth of ℎ=5mm, which for smaller wavenumbers follows the dispersion of gravity waves in shallow water. For the right figure, we have a depth of ℎ=1m, and the dispersion behaves as a pure gravity wave in an infinite water depth. The surface water-air tension admitted was 𝜎=72 mN/m [33] and the density of water 𝜌=1000 kg/m3. The gravity acceleration is the known 9.8m/s2. ...................... 68 5.5 Wake pattern of the induced electrostatic potential, in units of Φ0, given by equation (4.35), when graphene is in the vicinity of a dielectric. The top figures show the density plots for a particle moving at a height 𝑧0=1𝜇m above graphene. The bottom ones represent the potential created by the traveling particle at 𝑧0=0.1𝜇m. In this figure, the oscillatory behavior of the induced potential is visible. Note the presence of the transverse and divergent waves in the pattern, which makes this case very similar to the ship wakes generated in the water. The velocity of the moving particle is given by 𝑣=0.1𝑐and a Fermi energy of 𝐸𝐹=0.17 eV. .................................... 71 5.6 Wake pattern of the induced electrostatic potential, in units of Φ0, given by equation (4.35) for different Froude numbers: 𝐹𝑝𝑙 =2.7(left) and 𝐹𝑝𝑙 =4.5(right), which corresponds to an external particle velocity (𝑧0=1𝜇m) of 𝑣=0.3𝑐and 𝑣=0.5𝑐, respectively. Qualitatively is possible to see a decrease in the wake angle for higher Froude numbers and the dominance of the divergent waves in the wake pattern. The Fermi energy is of 𝐸𝐹=0.17 eV. . . . . 72 5.7 Graphical representation of half angle of the plasmonic cone for graphene in the vicinity of a dielectric for 𝜖𝑑=1(vacuum), given by table 1. There is a transition from a Kelvin-type wake to a Mach-type wake at 𝐹𝑝𝑙 ∼2.2as the Froude number increases. The other parameters were: 𝐸𝐹=0.17 eV and 𝑧0=1𝜇m. The dashed black line corresponds to the angle given by 𝜃=1/(𝛿𝐹𝑝𝑙 )and 𝛿=0.020, which only holds for angles in the Mach region. The dashed blue line is the Kelvin angle given by Kelvin’s theory (𝜃𝐾=19.47◦). The dashed grey line is a fit performed to the angles enclosed in the Kelvin region, with an angle value of 𝜃𝑎𝑝𝑝 =21◦. The transition between the two regions occurs at the interception of the dashed blackandgreylines. ................................. 73 xvi LIST OF FIGURES 5.8 Wake pattern of the induced electrostatic potential, in units of Φ0, given by equation (4.63), when graphene is in the vicinity of a local metal. The top figures show the density plots for a distance between metal and graphene of 𝑑=0.01 𝜇m. The bottom ones represent the potential created in graphene for a dielectric thickness of 𝑑=0.1𝜇m. It is possible to understand the continuous nature of the potential in all plots and the nonexistence of transverse and divergent waves. The wake angles are smaller for a higher dielectric constant 𝜖𝑑and a smaller separation of metal/graphene 𝑑. The velocity of the moving particle is 𝑣=0.1𝑐, the Fermi energy is of 𝐸𝐹=0.17 eV and the external particle is moving at a height of 𝑧0=1𝜇m above graphene. The chosen metal was titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.2. ...................................... 75 5.9 Wake pattern of the induced electrostatic potential, in units of Φ0, given by equation (4.63) for different Froude numbers: 𝐹𝑝𝑙 =2.7(left) and 𝐹𝑝𝑙 =4.5(right), which corresponds to an external particle velocity (𝑧0=1𝜇m) of 𝑣=0.3𝑐and 𝑣=0.5𝑐, respectively. Qualitatively it is possible to see a decrease in the wake angle for higher Froude numbers. The parameters are: Fermi energy of 𝐸𝐹=0.17 eV, dielectric thickness of 𝑑=0.01 𝜇m in vacuum (𝜖𝑑=1). The chosen metal was titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.2. ............. 76 5.10 Graphical representation for the apparent angle 𝜃𝑎𝑝𝑝, given by table 2, of the plasmonic cone for graphene in the vicinity of a local metal for 𝑧0=1𝜇m. In this case, there is no Kelvin or Mach region in the whole domain. The angles follow a quadratic law given by 1/𝛿0+1/𝛿1𝐹𝑝𝑙 +1/𝛿2𝐹2 𝑝𝑙 , with 𝛿0=0.26,𝛿1=0.10 and 𝛿2=1.30. The parameters considered are: 𝐸𝐹=0.17 eV, 𝑑=0.01 𝜇m and 𝜖𝑑=1. For titanium: 𝜔𝑝=2.80eV and 𝜖∞=2.20....................................... 77 5.11 Graphical representation for the apparent angle𝜃𝑎𝑝𝑝, given by table 3, of the plasmonic cone for graphene in the vicinity of a local metal for 𝑧0=0.1𝜇m. In this case, the only theory that describes the angle behavior is the Mach theory, 𝜃=1/(𝛿𝐹𝑝𝑙 ), where 𝛿=0.027 . The parameters considered are: 𝐸𝐹=0.17 eV, 𝑑=0.01 𝜇m and 𝜖𝑑=1. For titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.20.............................. 78 5.12 Representation of the configurations in the classical and plasmonic problems. In (a) the figure shows a perturbation moving in the y-direction in the water, which has a height ℎregarding the bottom land. The plasmonic configuration (b) represents graphene in the vicinity of a dielectric, with an external particle moving with velocity 𝑣𝑦at a height 𝑧0from the graphene sheet. Configuration (c) is characterized by graphene in the vicinity of a local metal with a dielectric separation of thickness 𝑑. Once again, we have an external charge moving with velocity 𝑣𝑦at a height 𝑧0fromgraphene......................... 79 xvii LIST OF FIGURES 5.13 Wake pattern for the plasmonic (left) and classical (right) cases. Both patterns show similar structures, typical of the limiting case 𝜅=−1, which are the oscillatory behavior and the presence of transverse and divergent waves. The bottom figures show disturbances speeds higher than the top ones, which bring a decrease on the wake angle. The case of graphene in the vicinity of a dielectric is similar to water wakes in deep water. For the plasmonic case the parameters are: 𝐸𝐹=0.17 eV and 𝑧0=0.1𝜇m. Right figures retrieved from [25]. . 80 5.14 Graphical representation of the wake pattern in graphene near a metal (left) and the water wake pattern (right) for the limiting case 𝜅=−0.1. This corresponds to water wakes in shallow water. Both shapes show us a continuous behavior in the central cone. In the water wake there are waves forming in the outer region of the pattern which is analogous to the bluest region in the plasmonic wake. For the plasmonic wake the parameters are: 𝑣=0.1𝑐, 𝐸𝐹=0.17 eV, 𝑑=0.01 𝜇m and 𝑧0=1𝜇m. The chosen metal was titanium. The right figureretrievedfrom[25]................................ 81 5.15 Graphical representation of the wake pattern in graphene near a metal for 𝑑≪𝑧0(left) and 𝑑≫𝑧0(right). For the limiting case 𝜅=0we are in the shallow water regime, where 𝑑=0.01 𝜇m and 𝑧0=1𝜇m. When we increase the dielectric thickness, there is a regime inversion and the oscillatory behavior is recovered, because we now have 𝑑=1𝜇m and 𝑧0=0.1𝜇m and this case is analogous to water wakes in deep water. The parameters are: 𝑣=0.1𝑐and 𝐸𝐹=0.17 eV. The chosen metal was titanium. . . . . . . . . . . . . . 82 xviii List of Tables 1 Apparent half wake angle 𝜃𝑎𝑝𝑝 (in degrees) of the cone produced by the moving plasmons in graphene induced by an external particle with parallel motion at a height 𝑧0=1𝜇m. The Fermi energy of the electron gas is 𝑒𝐹=0.17 eV. The last line of the table gives the Froude number for this problem obtained to the expression 𝐹𝑝𝑙 =p𝑣2/𝑧0𝑎........... 73 2 Apparent half wake angle 𝜃𝑎𝑝𝑝 (in degrees) of the cone produced by the moving plasmons in graphene induced by an external particle with parallel motion at a height 𝑧0=1𝜇m. The Fermi energy of the electron gas is of 𝐸𝐹=0.17 eV and the dielectric thickness of 𝑑=0.01 𝜇m for vacuum (𝜖𝑑=1). The last line of the table gives the Froude number for this problem obtained to the expression 𝐹𝑝𝑙 =p𝑣2/𝑧0𝑎................. 77 3 Apparent half wake angle 𝜃𝑎𝑝𝑝 (in degrees) of the cone produced by the moving plasmons in graphene induced by an external particle with parallel motion at a height 𝑧0=0.1𝜇m. The Fermi energy of the electron gas is of 𝑒𝐹=0.17 eV and the dielectric thickness of 𝑑=0.01 𝜇m for vacuum (𝜖𝑑=1). The last line of the table gives the Froude number for this problem obtained to the expression 𝐹𝑝𝑙 =p𝑣2/𝑧0𝑎................. 77 4 Numerical Value for the integrals in equation (4.63), for different velocities. The parameters were: 𝑑=0.01 𝜇m and 𝑧0=1𝜇m and 𝜃′=𝜋. Note that for every velocity the potential can be seen as Φ𝑖𝑛 (r, 𝑧, 𝑡)=2𝐼2approximately. For every 𝑟the values for 𝐼1and 𝐼2are very close to each other and we can consider 𝐼1+𝐼2=2𝐼2in a good approximation. . . . . . 99 5 Numerical Value for the integrals in equation (4.63), for different velocities. The parameters were: 𝑑=0.01 𝜇m and 𝑧0=0.1𝜇m and 𝜃′=𝜋. Note that for every velocity the potential can be seen as Φ𝑖𝑛 (r, 𝑧, 𝑡)=2𝐼2approximately. For every 𝑟the values for 𝐼1and 𝐼2are very close to each other and we can consider 𝐼1+𝐼2=2𝐼2in a good approximation. . . . . . 100 6 Numerical Value for the integrals in equation (4.63), for different velocities. The parameters were: 𝑑=0.1𝜇m and 𝑧0=1𝜇m and 𝜃′=𝜋. Note that for every velocity the potential can be seen as Φ𝑖𝑛 (r, 𝑧, 𝑡)=2𝐼2approximately. For every 𝑟the values for 𝐼1+𝐼2=2𝐼2are very close to each other and we can consider 𝐼1=𝐼2in a good approximation. . . . . . . . 101 xix LIST OF TABLES 7 Numerical Value for the integrals in equation (4.63), for different velocities. The parameters were: 𝑑=0.01 𝜇m, 𝑧0=1𝜇m, 𝜃′=3(for velocities of 0.05𝑐and 0.1𝑐) and 𝜃′=3.1 (for velocities of 0.3𝑐and 0.8𝑐). Note that for every velocity the potential can be seen as Φ𝑖𝑛 (r, 𝑧, 𝑡)=2𝐼2approximately. For every 𝑟the values for 𝐼1and 𝐼2are very close to each other and we can consider 𝐼1+𝐼2=2𝐼2in a good approximation. . . . . . . . . . . . 102 xx Acronyms 2D 2-Dimensional 3D 3-Dimensional BVPs Boundary Value Problems GHz Gigahertz HDM Hydrodynamic Model LED Light Emitting Diode LRA Local Response Approximation SPPs Surface Plasmon-Polaritons THz Terahertz xxi Chapter 1 Introduction The study of plasmonic and nanoplasmonic in noble metals and semiconductors has become of the uttermost importance in the latest years. Because of its remarkable electromagnetic, optical, and thermodynamic proprieties, the study of Surface Plasmon-Polaritons (SPPs) and bulk plasmons have led to the discovery of groundbreaking technology in many areas of science, overcoming many problems and limitations of non-plasmonic devices in use. Smaller sizes and higher response speeds will characterize these new plasmonic technologies. Graphene and plasmonics comes together to develop a new field of physics known as graphene plasmonics, due to the important proprieties of graphene, such as its 2Dimensional (2D) nature, graphene is an excellent material to study the plasmonic effects in tighter regions. This means that, in graphene, the surface plasmon-polaritons are confined to atomic-size structures where the nonlocal effects cannot be ignored. The Hydrodynamic Model (HDM) came to life as a new way to study electron gases and graphene in the light of the nonlocal effects. In this thesis, we will analyze the hydrodynamic model and its application to graphene by perceiving it as a nearly perfect fluid, in order to have a hydrodynamic approach to plasmonics in graphene. The thesis will be structured in the following way: in chapter 1 we will introduce the basics of plasmonic and graphene plasmonic, as well as the instances of importance of nonlocal effects which play an important role in the optical and electromagnetic proprieties of plasmons, with the usage of the hydrodynamic model. In chapter 2, we will perform a mathematical derivation of the HDM for 3-Dimensional (3D) and 2D electron gases in the electrostatic regime. In chapter 3, we will use the hydrodynamic model as a tool to derive the dispersion relation of some electrostatic problems in planar geometry. In chapter 4 we will derive the expressions for the induced potential in graphene when an external particle is moving parallel to it. The study will cover two cases: graphene in the vicinity of a dielectric and graphene in the vicinity of a nonlocal metal. Finally, in chapter 5, we will discuss about classical Kelvin and Mach waves and provide an analogy between water wakes and graphene potential wakes created by the moving external particle, in order to recognize 1 CHAPTER 1. INTRODUCTION the differences and similarities between the two different phenomena. 1.1 General Topics in Plasmonics In the latest years, plasmonics and nanoplasmonics have been interesting and sovereign fields of research in physics and nanophotonics. This field of research allows us to see how electromagnetic fields can be confined to scales no greater than their wavelength dimensions. These light-matter interactions lead to optical proprieties, giving rise to the main fields of investigation, known as nanophotonics. In recent years, the emergence of faster, smaller, and more efficient electronic devices brought a new perspective and interest in plasmonics and nanophotonics. This has led to the investigation and building of groundbreaking technology, such as electronic devices, terahertz detectors [1, 2], signal processing devices, plasmonic integrated circuits [3] and some technological applications in other areas of science [4]. Although the scientific enthusiasm for plasmonics is relatively new, the unknown discovery of plasma effects can be attributed to Wood in 1902, due to an effect, which later became known as Wood anomalies [5]. At the time, these anomalies were considered to be one of the most fascinating phenomena in optics. Wood anomalies are essential rapid variations in the intensity of the light diffracted spectrum. We can observe this abnormal phenomenon through the spectrum of visible light diffracted by a metal grating. Even though it was a remarkable observation at the time, Wood could not find an explanation for the phenomena. Further work was developed with the aim of explaining these anomalies. Lord Rayleigh, who stated that the existence of anomalies occurs at a wavelength that corresponds to the transition to a higher order spectrum, conducted the first experiments on Wood’s anomalies. Rayleigh’s study was of limited interpretation and the first reappraisal to his study was made by Strong showing that the metal influences the position of the anomalies [6]. Since the location of the transition is a consequence of the geometry of the grating, the metal does not influence it. So Rayleigh study was in need of a revision. Figure 1.1: Representation of an evanescent surface plasmon wave at an interface between a metal and a dielectric. The wave propagates along the x-axis, without loss of generality. 2 1.1. GENERAL TOPICS IN PLASMONICS Only a few years later, with the subsequent work of Fano, was a theoretical breakthrough on Wood anomalies provided. Fano observed that the origin of the anomalies was due to the excitation of evanescent surface waves propagating along the metal grating, meaning along the surface of the material [6]. This was the first step towards a plasmonic description of the phenomena, although further research (see [6, 7] for more details) was necessary in order to relate these anomalous excitations with the surface plasmons. Surface plasmons are localized excitations of an electron gas that propagates as waves along an interface of two materials (usually a metal and a dielectric), decaying exponentially away from the interface (see figure 1.1). It can also be seen as polarization oscillations of metallic nanoparticles. In a way as plasmons are collective excitation of electrons, metals are the perfect materials to observe these phenomena. From a solid-state physics point of view, metals are known as good conductors. Thus, once we apply an external field to a metallic system, the electrons with energies near the Fermi level can move freely in the material. In such case, we can conclude that the metal is a free electron gas bounded to a certain geometry. Of course, the electrons are moving in a charge ion background where the only interactions are the instantaneous electron-ion collisions. To get the most basic electromagnetic and optical responses for a metal, Drude came up with a simple model, having in mind the particularity of considering the metal as a free electron gas. This model later became known as the Drude model and is stated in the following way 𝜖(𝜔)=1−𝜔2 𝑝 𝜔2+𝑖Γ𝜔(1.1) where 𝜔𝑝=p𝑛𝑒2/𝑚𝜖0is the plasma frequency of the electron gas and Γis a phenomenological parameter that counts for the scattering rate of electrons with the ionic background. The plasma frequency comes from applying an electric field to a thin metallic film that embodies 𝑛free electrons per unit volume. This will exert a force on the film, which can be quantified using Newton’s second law of motion for a single electron with mass 𝑚and an electric charge 𝑒. Recalling that surface plasmons are visible in the interface between two materials, classically they are possible if one of the two materials has a real and negative permittivity 𝜖, such as the Drude model provides. The other material is usually taken to be a dielectric, the most simple being vacuum with real and positive permittivity. For an ideal electron gas, this is, for Γ=0, the Drude permittivity is negative for frequencies smaller than the plasma frequency 𝜔𝑝. As such, one characteristic of the surface plasmons is that the frequency for surface waves to exist within the interface of the material is taken to be at 𝜔<𝜔𝑝. To find these surface modes, one must find a solution to Maxwell’s equations, which give rise to longitudinal waves that propagate along the interface between the two media but are exponentially damped away from the surface. However, the plasmons do not exist alone, and that is a very interesting point. We know that the usual study of metallic structures is accompanied by a dielectric, which forms a certain interface. The electrons in the metal are excited to form plasmons while simultaneously in the dielectric an electromagnetic wave couples to a polarization excitation in the material, which are known as polaritons. In that case, there is a more general description of the plasma phenomena, of the evanescent waves that occur in the interface between the two materials. It is common to describe them as surface plasmon-polaritons (SPPs). The dispersion for the plasmons in 3 CHAPTER 1. INTRODUCTION the wavelength of the radiation of the free space, and the condition𝑘/𝑘𝐹>1, tells that the nonlocal effects have an important place in the electromagnetic response of the system, which can change the results predicted by the local model [15, 16]. In [17] it is shown that graphene-metal plasmons (graphene in the vicinity of a metal) have energies in the mid-IR range of the electromagnetic spectrum with a wavenumber of 200 𝜇m−1, which puts graphene in the strong nonlocal regime because the nonlocal condition is 𝑘𝑐/𝑘𝐹𝑣𝐹∼100. This means that the conductivity of graphene must depend on the wavenumber 𝜔and on the wavevector kto build up the proper nonlocal effects. However, plasmonic structures, even when small, possess a large number of atoms to consider. In that order, it was necessary to find an alternative calculation to the nonlocal Drude dielectric and graphene conductivity functions because of their time-consuming and computational burden. With a need for an alternative approach to these calculations, a nonlocal model, also known as the hydrodynamic model it came to light. To explain the experimental results, Bloch and Jensen introduced a theory that explains the behavior of an electron gas when in the presence of external fields [15]. The hydrodynamic model allows the inclusion of nonlocal effects by considering a term that corresponds to the statistical pressure (where quantum effects are contained) on the position of a particle. The Coulomb interactions arise by coupling Euler’s equation with Maxwell equations of electromagnetism. The hydrodynamic model can be applied to a 2D electron gas, which is considered a perfect fluid. However, nonlocal plasmonic effects in the THz and mid-IR spectral range are not possible for noble metals due to the poor spatial confinement at these frequencies. In that order, graphene emerges as a new material to study these effects in this specific spectral range. The massless Dirac electrons in graphene behave as a nearly perfect fluid, whose electronic motion is given by the Navier-Stokes equation, from which Euler’s equation follows. This nonlocal model can characterize the conductivity of graphene and its plasmonic properties. In the following sections, we will give the mathematical details of the hydrodynamic model and equations that describes this model. In the lastest years, the hydrodynamic model has become the reference to study the nonlocal effects of metallic interfaces and metallic nanostructures. 10 Chapter 2 An Hydrodynamic Model The Hydrodynamic model came about as a quantum nonlocal theory to describe the plasmonic properties of thin metallic structures, whereas the local-response theory failed to provide a coherent description of plasmonics in metals [1]. This nonlocal theory allows treating the electrons in a metal as a degenerate electron gas. The hydrodynamic model is built on three principal equations: Euler’s equation (or equation of drift motion), the Poisson equation, and the continuity equation. In this chapter, we will see that the Boltzmann equation is the starting point to derive the hydrodynamic equations and, with the proper boundary conditions for a specific system, these equations will stand useful in providing the plasmonic properties of metallic nanostructures and nanoparticles. Moreover, it is provided the hydrodynamic information for 2-dimensional systems such as the degenerate 2D electron gas and a graphene layer. 2.1 Boltzman and Euler Equations The Hydrodynamic Model has its foundations in Boltzmann and Euler’s equations of fluid dynamics. The study of these equations will provide us with some insight and knowledge into how this model works and will allow us to reach a set of hydrodynamic equations for an electron gas in the electrostatic regime. This electron gas can be characterized by a distribution function 𝑓(r,v)𝑑r𝑑v, which gives us the number of electrons in the gas having position and velocity centered at rand vwhen we take into account, respectively, the small volume 𝑑rand the small velocity range 𝑑v. Considering the case for conservative forces F=−∇Φ, where Φis the potential energy, the time evolution of the distribution function 𝑓(r,v) is given by the Boltzmann equation 𝜕𝑓 𝜕𝑡 +v·𝜕𝑓 𝜕r+g·𝜕𝑓 𝜕v=0(2.1) 11 CHAPTER 2. AN HYDRODYNAMIC MODEL where grepresents the external forces per unit mass. Equation (2.1) is also called the collisionless Boltzmann equation. To account for collisions of the electrons in the gas, it is necessary to add a term of the form Γ(𝑡)to the right-hand side of this equation. However for our purposes, the collision term can be neglected. The position of the particles as a function of time can be obtained, simply by integrating (2.1) on the coordinate v. In that order let us take the zero moment of the Boltzmann equation. ∫𝑑v𝜕𝑓 (r,v) 𝜕𝑡 +v·𝜕𝑓 (r,v) 𝜕r+g·𝜕𝑓 (r,v) 𝜕v=0(2.2) By introducing the mass density (assuming the same mass m for all particles in the gas) and the velocity moment, respectively, using the relations 𝜌(r)=∫𝑑v𝑓(r,v)𝑚(2.3) ⟨𝑣𝑖⟩=1 𝜌(r)∫𝑑v𝑓(r,v)𝑚𝑣𝑖(2.4) then equation (2.2) can be expressed in the form 𝜌(r) 𝜕𝑡 +𝜕 𝜕r· [𝜌(r)⟨v⟩] =0(2.5) which is the continuity equation that amounts to the conservation of mass. Note that we have used the divergence theorem in the last term of equation (2.2), which provides the identity ∫𝑉 𝑑v𝜕𝑓 (r,v) 𝜕v=∫𝑆∞ 𝑓(r,v)𝑑SV=0(2.6) where 𝑓(r,v)=0 over a surface at infinity, 𝑆∞. Let us now consider the first moment of Boltzman equation ∫𝑑vv 𝜕𝑓 (r,v) 𝜕𝑡 +v·𝜕𝑓 (r,v) 𝜕r+g·𝜕𝑓 (r,v) 𝜕v=0(2.7) This last equation can be simplified by recalling the expressions of mass density (eq. 2.3) and velocity moment (eq. 2.4) as 𝜕 𝜕𝑡 [𝜌(r)⟨v⟩] +Õ 𝑖 𝜕 𝜕𝑥𝑖[𝜌(r)⟨v𝑣𝑖⟩] −g𝜌(r)=0(2.8) where the third term was obtained with integration by parts. This last equation is called the momentum equation and, together with the continuity equation, the Euler equation naturally appears, as we will show. First, it is necessary to introduce the tensor 𝜏2 𝑖𝑗 =⟨𝑣𝑖𝑣𝑗⟩ − ⟨𝑣𝑖⟩⟨𝑣𝑗⟩. The term ⟨𝑣𝑖𝑣𝑗⟩is the second moment of the velocity and can be written in an integral form as. ⟨𝑣𝑖𝑣𝑗⟩=1 𝜌(r)∫𝑑v𝑓(r,v)𝑚𝑣𝑖𝑣𝑗(2.9) 12 2.2. HYDRODYNAMIC MODEL OF A 3D ELECTRON GAS IN THE ELECTROSTATIC REGIME by defining the tensor in this way, it will be possible to write the term with the second momentum in equation (2.8) in terms of products of first moments. Subtracting from the momentum equation [eq. (2.8)] the continuity equation we obtain, for each component 𝜌(r)𝜕⟨𝑣𝑗⟩ 𝜕𝑡 +𝜌(r)Õ 𝑖⟨𝑣𝑖⟩𝜕⟨𝑣𝑗⟩ 𝜕𝑥𝑖 =𝑔𝑗𝜌(r) −Õ 𝑖 𝜕[𝜌(r)𝜏2 𝑖 𝑗 ] 𝜕𝑥𝑖 (2.10) or in vectorial terms 𝜌(r)𝜕⟨v⟩ 𝜕𝑡 +𝜌(r)(⟨v⟩ ·∇)⟨v⟩=g𝜌(r) −∇𝑃(2.11) This last equation is Euler’s equation of fluid dynamics and it is the starting point for the hydrodynamic model of plasmons in metals and graphene. Quantum mechanical effects enter this model via the statistical pressure 𝑃induced by the electron gas. In addition this pressure is a function of position due to charge inhomogeneity. Since the statistical pressure depends on the position-dependent charge density, this model is a good approach to describing nonlocal plasmonic effects in metals and graphene. 2.2 Hydrodynamic Model of a 3D electron gas in the electrostatic regime The Hydrodynamic model is a nonlocal theory described by three equations: Euler’s equation (the equation of motion), the continuity equation, and the Poisson equation. Since this model describes the behavior of an electron gas when in the presence of external fields, it is natural to think that the hydrodynamic model combines Euler’s equation of motion [eq. (2.11)] with Maxwell’s equations of electromagnetism. These equations, when properly linearized, will be capable of providing a nonlocal description of plasmonics, such as the dispersion of the surface plasmon-polaritons and the potentials created in these structures. The validity of the hydrodynamic model holds for different nanostructure geometries when provided with specific boundary conditions for each case. For that, Euler’s equation can be written in terms of the density of particles in the gas 𝑛(r)where the mass density is 𝜌(r)=𝑚𝑛(r) 𝑚𝑛(r)𝜕v 𝜕𝑡 +𝑚𝑛(r)(v· ∇)v=g𝑚𝑛(r) −∇𝑃(2.12) Other physical phenomena, such as electron-phonon and electron-electron interactions, can be introduced to the study, but only if we consider scattering effects. Then we add the term 𝑚𝑛(r)v/𝜏in the left-hand side of the previous equation, where 𝜏is the relaxation time that takes into consideration the momentum nonconservation of the electrons in the gas. Let us consider now that this gas is being perturbed by an external electric field E. Since grepresents the external forces per unit mass, we have g=−𝑒E/𝑚, where −𝑒is the electron charge. In the electrostatic regime we write E=−∇𝜙(r), so that the external force can be expressed in terms of the electrostatic potential 𝜙, which leads to 13 CHAPTER 2. AN HYDRODYNAMIC MODEL 𝑚𝑛(r)𝜕v 𝜕𝑡 +𝑚𝑛(r)(v· ∇)v=𝑒𝑛(r)∇𝜙(r) −∇𝑃(2.13) The statistical pressure stored in the kinetic energy of the electron gas gives the simplest estimate for the pressure. In that case, for a 3D Fermi gas, the determination of the pressure requires the calculation of the kinetic energy. 𝐾𝑔=2𝑉∫𝑑k (2𝜋)3 ℏ2𝑘2 2𝑚𝜃(𝑘𝐹−𝑘)=𝑉 5𝜋2 ℏ2𝑘5 𝐹 2𝑚(2.14) where𝑉is the volume of the system and 𝑘𝐹the Fermi wavenumber. The integral can be done in spherical coordinates, 𝑑k=𝑘sin 𝜃𝑑𝑘𝑑𝜃𝑑𝜑, with the azimutal angle integrated into 0≤𝜑≤2𝜋and the polar angle with integration limits in 0≤𝜃≤𝜋. The Heaviside function states that the upper limit in the integral in 𝑑𝑘 must be 𝑘𝐹, so the wavenumber of the particles in the electron gas can never be greater than the Fermi wavenumber. On the other hand, the total number of particles reads 𝑁𝑒=2𝑉∫𝑑k (2𝜋)3𝜃(𝑘𝐹−𝑘)=𝑉 3𝜋2𝑘3 𝐹(2.15) we solved this last integral in the same way as the integral in equation (2.14). For an electron gas, we can write the Fermi wavenumber in terms of 𝑁𝑒and 𝑉as 𝑘𝐹=3𝜋2𝑁𝑒 𝑉1/3 (2.16) Then, the kinetic energy is rewritten as 𝐾𝑔=1 5𝜋2 ℏ2 2𝑚3𝜋2𝑁𝑒5/3𝑉−2/3(2.17) from where it follows the statistical pressure as 𝑃=−𝜕𝐾𝑔 𝜕𝑉 =𝛾𝑛5/3(2.18) where 𝑛(r)is the electron density which is defined as the number of particles per unit volume in the electron gas, and it is related to this last equation of state with 𝛾=(1/5)(3𝜋2)2/3(ℏ2/𝑚). The gradient of the pressure is given by ∇𝑃=5 3𝛾𝑛2/3∇𝑛(r)(2.19) Finally its possible to describe the hydrodynamic model in the electrostatic limit with the Euler’s equation [eq. (2.20a)], the Poisson’s equation [eq. (2.20b)] and the continuity equation [eq. (2.20c)] as 14 2.2. HYDRODYNAMIC MODEL OF A 3D ELECTRON GAS IN THE ELECTROSTATIC REGIME 𝜕v 𝜕𝑡 + (v·∇)v=𝑒 𝑚∇𝜙(r) − 5 3𝛾1 𝑚𝑛1/3∇𝑛(r)(2.20a) ∇2𝜙=−𝑒 𝜖0[𝑛+−𝑛(r)] (2.20b) 0=𝜕𝑛(r) 𝜕𝑡 +∇· [𝑛(r)v](2.20c) we have 𝑛+as the ionic background charge density neutralizing the electron gas and 𝜖0as the vacuum permittivity. To find a solution to these hydrodynamic equations, it becomes necessary to proceed to their linearization since equation (2.20a) in its form includes nonlinear effects that are unnecessary to the case in study, in which the applied electric field is a weak perturbation, and linear terms are enough to describe the potentials and the electronic densities of the electron gas. By the usual linearization method, we get: 𝑛(r, 𝑡)=𝑛0+𝑛1(r, 𝑡) + 𝑛2(r, 𝑡) + ..., and 𝜙(r, 𝑡)=𝜙0(r) + 𝜙1(r, 𝑡) + 𝜙2(r, 𝑡) + ..., the zeroth-order elements correspond to the electronic density and the static potential in the equilibrium state (static), where the 𝑛0is assumed to be constant throughout the homogeneous and infinite electron gas. The linear case is implicit in the first-order perturbation terms and, assuming 𝑛0≫𝑛1≫𝑛2, we can argue that the linear order is enough to describe the system. Note the velocity vis intrinsic of first order (linear), there is no current flowing in the static system so 𝑣0=0. Which explains the nonlinear character of the hydrodynamic equations and the necessity to carry out a linearization of the same. With this in mind, we write the linear hydrodynamic model as: 𝜕v 𝜕𝑡 =𝑒 𝑚∇𝜙1(r) − 5 3𝛾1 𝑚𝑛1/3 0∇𝑛1(r)(2.21) ∇2𝜙1(r)=𝑒 𝜖0 𝑛1(r)(2.22) 0=𝜕𝑛1(r) 𝜕𝑡 +𝑛0∇·v(2.23) Taking the divergence of the linearized Euler’s equation multiplied by 𝑛0, it becomes more intuitive to use the Poisson’s equation and the continuity equation to write Euler’s equation as 𝛽2∇2−𝜕2 𝜕𝑡2−𝜔2 𝑝𝑛1(r)=0(2.24) where 𝛽is a parameter that characterizes the nonlocality of the theory and is defined as 𝛽2= ℏ2𝑘2 𝐹/3𝑚2= 𝑣2 𝐹/3in our model. However, we can consider other sources of nonlocality that take our hydrodynamic model more realistic, and in that case, the value of the nonlocal parameter will slightly change ([17],[18],[19]). In the electrostatic regime, the value of 𝛽obtained in equation (2.24) is considered for low frequencies. To obtain the local response of the system, nothing is easier than taking 𝛽→0. Physically, the nonlocal parameter tells us the speed of propagation of density disturbances in the electron gas. The wave equation (2.24) also gives the bulk plasmon frequency 𝜔𝑝=p𝑒2𝑛0/𝑚𝜖0(see chapter 1.1) and is independent of 15 CHAPTER 2. AN HYDRODYNAMIC MODEL the velocity of light c, but rather depends on the Fermi velocity 𝑣𝐹of the electron gas. For an infinite and homogeneous medium the electronic density is given as a plane wave solution 𝑛1(r, 𝑡)=𝐴𝑒𝑖(k·r−𝜔𝑡). With equation (2.24) the dispersion admits solutions of the form 𝜔𝑏𝑢𝑙𝑘 =q𝜔2 𝑝+𝛽2𝑘2(2.25) Once 𝑛1(r)is determined, the potential follows from equation (2.22), and the velocity from equation (2.23). 2.2.1 2D electron gas and graphene In the latter section, we derived the linearized hydrodynamic equations for a 3-dimensional electron gas. However, it is possible to reproduce the same equations but for 2-dimensional configurations. This will be important because we want to study some structures where graphene will play an important role. For 2D structures, the position vector is r=(r∥, 𝑧), with r∥being the 2D position vector in the plane of the 2D electron gas or in graphene, this is for 𝑧=0, while the configurations are chosen to be perpendicular to the 𝑧-direction. The starting point is once again Euler’s equation 𝑚𝑛(r∥)𝜕v 𝜕𝑡 +𝑚𝑛(r∥)(v· ∇)v=𝑒𝑛(r∥)∇𝜙(r∥, 𝑧 =0) −𝛿(𝑧)∇𝑃(2.26) There are some differences between the electron gas and the layer of graphene, so let us calculate the gradient of the pressure for each case. Note that the calculations are similar to the 3-dimensional electron gas. 2D electron gas For a 2D electron gas the kinetic energy reads 𝐾𝑔=2𝐴∫𝑘𝑑𝑘𝑑𝜃 (2𝜋)2 ℏ2𝑘2 2𝑚𝜃(𝑘𝐹−𝑘)=𝐴 𝜋 ℏ2𝑘4 𝐹 8𝑚(2.27) where 𝐴is the area of the system. The total number of electrons in the gas is given by 𝑁𝑒=2𝐴∫𝑘𝑑𝑘𝑑𝜃 (2𝜋)2𝜃(𝑘𝐹−𝑘)=𝐴 2𝜋𝑘2 𝐹(2.28) we can then write the Fermi momentum in terms of the 2D homogeneous electronic density 𝑛0=𝑁𝑒/𝐴 as 𝑘𝐹=√2𝜋𝑛0(2.29) It follows that the statistical pressure is given by 𝑃=−𝜕𝐾𝑔 𝜕𝐴 =𝛾2𝐷𝑛2 0(2.30) 16 2.2. HYDRODYNAMIC MODEL OF A 3D ELECTRON GAS IN THE ELECTROSTATIC REGIME where 𝛾2𝐷= ℏ2𝜋/2𝑚. The calculations were made to a homogeneous 2D electron gas, however; we are interested in the pressure of the inhomogeneous gas. In that case, for obtaining the gradient of the pressure, we promote 𝑛0to a position-dependent on 2D electronic density, that is, 𝑛0→𝑛2𝐷(r∥). With this assumption, the gradient of the pressure as the form ∇𝑃=2𝛾2𝐷𝑛2𝐷(r∥)∇𝑛2𝐷(r∥)(2.31) We now linearize the hydrodynamic equations by assuming 𝑛0≫𝑛1, then 𝜙(r∥) ≈ 𝜙0(r∥) +𝜙1(r∥) and 𝑛(r∥) ≈ 𝑛0+𝑛1(r∥). This leads to the linearized 2-dimensional hydrodynamic equations (Euler’s equation, Poisson equation and continuity equation) 𝜕v 𝜕𝑡 =𝑒 𝑚∇𝜙1(r,0) − 2𝛾2𝐷 𝑚∇𝑛1(r∥)(2.32a) ∇2𝜙1(r)=𝑒 𝜖0 𝛿(𝑧)𝑛1(r∥,0)(2.32b) 0=𝜕𝑛1(r∥) 𝜕𝑡 +𝑛0∇·v(2.32c) where we divided Euler’s equation [eq. (2.32a)] by the mass of the electron in the gas 𝑚and by the homogeneous electronic density. Graphene It is natural to think that to obtain the hydrodynamic equations for graphene we could just change the mass of the electrons in the 2D electron gas by the effective mass of the electron in graphene 𝑚𝑔= ℏ𝑘𝐹/𝑣𝐹. However the previous analysis is not entirely consistent for graphene, as the pressure has a different functional form from the 2D electron gas, due to the linear dispersion of the electrons in graphene [17]. Since we are dealing with a 2D material, the method of linearization is the same as the 2D electron gas. Also, the Fermi momentum in graphene is given by 𝑘𝐹=√𝜋𝑛0, which follows from the calculation of the number of particles in the graphene layer. It presented the kinetic energy as 𝐾𝑔=4𝐴∫𝑘𝑑𝑘𝑑𝜃 (2𝜋)2𝑣𝐹ℏ𝑘=𝑣𝐹ℏ2 3𝜋𝜋3/2𝑁3/2 𝑒𝐴−1/2(2.33) where 𝐴is the area of the graphene layer, 𝑣𝐹is the Fermi velocity of the electron in graphene and 𝑁𝑒is the total number of electrons. In a similar way to the 2D electron gas and assuming a position-dependent electronic density, we get the gradient of the pressure in the form ∇𝑃=𝑣𝐹ℏ1 2q𝜋𝑛(r∥)∇𝑛(r∥)(2.34) At last the linearized momentum equation (Euler’s equation) in real space can be written in terms of the effective mass for graphene 𝑚𝑔= ℏ𝑘𝐹/𝑣𝐹as 17 CHAPTER 2. AN HYDRODYNAMIC MODEL 𝜕v 𝜕𝑡 =𝑒𝑣𝐹 ℏ𝑘𝐹∇𝜙1(r∥,0) − 𝑣2 𝐹 2𝑛0∇𝑛1(r∥)(2.35) The remaining equations are the Poisson’s equation and the continuity equation given by (2.32b) and (2.32c), respectively. 2.3 Boundary Conditions In the latter sections, we described the behavior of an electron gas in an infinite system at the lights of the hydrodynamic model. Now let us recall equation (2.24), this is the wave equation that can be solved for different geometries by writing the Laplacian in an appropriate system of coordinates. This is a physics problem known as Boundary Value Problems (BVPs) and it makes necessary to provide some boundary conditions at the surface of the electron gas to account for the proper solutions for the problems. In the electrostatic approximation, let us consider an interface 𝑆with free surface charge density 𝜎separating two media with different relative permittivity and a vector nnormal to that surface. The first boundary condition follows from the continuity of the potential at the separation of the two media 𝜙(𝑚𝑒𝑑𝑖𝑢𝑚 1) 1(r, 𝑡)𝑆 =𝜙(𝑚𝑒𝑑𝑖𝑢𝑚 2) 1(r, 𝑡)𝑆 (2.36) The second boundary condition follows from the continuity of the electric displacement vector component perpendicular to the interface, and since E=∇𝜙the condition can be expressed as the continuity of the normal derivative of the electrostatic potential at the interface 𝑆 𝜖1 𝜕𝜙(𝑚𝑒𝑑𝑖𝑢𝑚 1) 1(r, 𝑡) 𝜕𝑛 𝑆 =𝜖2 𝜕𝜙(𝑚𝑒𝑑𝑖𝑢𝑚 2) 1(r, 𝑡) 𝜕𝑛 𝑆 (2.37) The hydrodynamic model also provides an extra conditions due to the fact that the current flowing, perpendicular to the interface 𝑆, must vanish. In other words, the normal component of the electron velocity field 𝑣is zero at 𝑆 𝑛0v·ˆ𝑛=0(2.38) this last condition can be written in a more interesting way. What condition (2.38) tell us, is that the normal component of the velocity to the surface is zero, and consequently so does the time derivative of the velocity. Then, by using equation (2.21), the last boundary condition follows as 𝜖0𝜔2 𝑝 𝑒𝑛0 𝜕𝜙1 𝜕𝑛 𝑆 =𝛽2 𝑛0 𝜕𝑛1 𝜕𝑛 𝑆 (2.39) In this way the hydrodynamic equations can be solved with the constrains provided by the boundary conditions given by equations (2.36), (2.37) and (2.39). Now let us consider that medium 2 is a metallic system, in this case, the metal dielectric function is given by the Drude model (see chapter 1.1) 18 2.3. BOUNDARY CONDITIONS 𝜖(𝜔)=𝜖∞−𝜔2 𝑝 𝜔2(2.40) where 𝜖∞is a background permittivity that accounts for all the effects that are not provided by the free electrons, such as high-frequency interband transitions. Note that in equation (2.40) we are not considering damping effects (Γ=0). In the hydrodynamic model, it already held all the charges accountable within the theory, so in this case, for a nonlocal metal the boundary condition that states the continuity of the normal derivative of the potential [eq. (2.37)] is changed as 𝜖1 𝜕𝜙(𝑚𝑒𝑑𝑖𝑢𝑚 1) 1(r, 𝑡) 𝜕𝑛 𝑆 =𝜖∞ 𝜕𝜙(𝑚𝑒𝑡𝑎𝑙) 1(r, 𝑡) 𝜕𝑛 𝑆 (2.41) If we are in the presence of local metal, it is only necessary to substitute 𝜖∞for 𝜖(𝜔)in the last equation, and that is because the Drude model provides a local Drude dielectric function. Whithin the hydrodynamic equations, we are now in a good position to study the optical properties of some electrostatic configurations and see how the nonlocal effects are present in the electrostatic systems. 19 CHAPTER 3. PLASMONIC NANOSTRUCTURES 𝐷1(𝑧)=𝐷1𝑒𝑘𝑧 (3.18) 𝑀(𝑧)=𝐵1cosh(𝑘𝑧) +𝐶1cosh(𝛼𝑧) +𝐵2sinh(𝑘𝑧) +𝐶1sinh(𝛼𝑧)(3.19) 𝐷2(𝑧)=𝐷2𝑒−𝑘𝑧 (3.20) Just like the metal-dielectric interface, the potential outside the metal must decay as |𝑧| → ∞. Inside the metal, we write the potential as a linear combination of sinh and cosh functions because we have a finite metal and two surfaces at 𝑧=±𝑎. We could also write the solution as exponential functions [20]. The coefficient for 𝑀(𝑧)remains valid if 𝐶1=𝑒 𝜖0 𝑛0 𝛼2−𝑘2𝐴1(3.21) 𝐶2=𝑒 𝜖0 𝑛0 𝛼2−𝑘2𝐴2(3.22) To determine the dispersion for this system we first apply the boundary conditions (2.36) and (2.37) to the surface of the slab at 𝑧=±𝑎. 𝐷1=hsinh(𝑘𝑎)cosh(𝛼𝑎) − 𝛼 𝑘cosh(𝑘𝑎)sinh(𝛼𝑎)i𝐶1 +h𝛼 𝑘sinh(𝑘𝑎)cosh(𝛼𝑎) −cosh(𝑘𝑎)sinh(𝛼𝑎)i𝐶2(3.23) 𝐷2=hsinh(𝑘𝑎)cosh(𝛼𝑎) − 𝛼 𝑘cosh(𝑘𝑎)sinh(𝛼𝑎)i𝐶1 −h𝛼 𝑘sinh(𝑘𝑎)cosh(𝛼𝑎) −cosh(𝑘𝑎)sinh(𝛼𝑎)i𝐶2(3.24) 𝐵1=−hcosh(𝛼𝑎) + 𝛼 𝑘sinh(𝛼𝑎)i𝑒−𝑘𝑎𝐶1(3.25) 𝐵2=−hcosh(𝛼𝑎) + 𝛼 𝑘sinh(𝛼𝑎)i𝑒−𝑘𝑎𝐶2(3.26) Finally, the dispersion relation for the plasmons for this system within the nonlocal hydrodynamic model is obtained by applying the last boundary condition (2.39) 𝛼sinh(𝛼𝑎)"𝜔2 𝜔2 𝑝−1 21−𝑒−2𝑘𝑎1+𝑘 𝛼tanh(𝛼𝑎)#𝐴1 ±𝛼cosh(𝛼𝑎)"𝜔2 𝜔2 𝑝−1 21+𝑒−2𝑘𝑎1+𝑘 𝛼coth(𝛼𝑎)#𝐴2=0(3.27) 26 3.1. ELECTROSTATIC CONFIGURATIONS The finite metal slab contains two types of solutions for the potential: even and odd modes. For an even mode 𝐴2=0and Ω2=1 21−𝑒−2𝑥1+𝑥 𝑦tanh(𝑦)(3.28) On the contrary, for an odd mode 𝐴1=0and the dispersion equation reads Ω2=1 21+𝑒−2𝑥1+𝑥 𝑦coth(𝑦)(3.29) To simplify the expressions, we defined some dimensionless variables: Ω=𝜔/𝜔𝑝;𝑥=𝑘𝑎 and 𝜁= 𝑎𝜔𝑝/𝛽. In this case, 𝜁is the nonlocal parameter, but different from 𝛽this parameter defines the local case for 𝜁→ ∞, which is also called the non-dispersive limit [18]. Let’s also define the variable 𝑦= 𝛼𝑎 =p𝑥2−𝜁2(Ω2−1)which determines the surface plasmon-polaritons modes, for real 𝑦, and the bulk plasmons modes for imaginary 𝑦. We can find the plasmon dispersion for the even and odd modes by solving the transcendental equations (3.28) and (3.29). 5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 2.5 3.0 5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (a) Even Modes Odd Modes (b) Figure 3.4: Dispersion curves in a finite metal slab with thickness 2𝑎for 𝜁=10. (a) Even modes given by equation (3.28). (b) Odd modes given by equation (3.29). The curve with 𝑛=0corresponds to the surface mode, while the curves with 𝑛>0corresponds to bulk modes. The dashed curve is the dispersion for the infinite medium given by equation (2.25). Let’s now study the dispersion curves presented in figure 3.4. For a fixed wavenumber 𝑘there is more than one root to the transcendental equations. In that order, we found all the solutions falling in the frequency interval 0⪕Ω⪕3. We presented the solutions for the even (a) and the odd (b) modes. Depicted with the dispersion relations is also computed the dispersion for an infinite homogeneous medium given by equation (2.25). The first difference between the even and odd modes comes from the frequency at 𝑘𝑎 =0. For this frequency, the lowest curves with 𝑛=0, which corresponds to surface modes (𝛼real), are different from zero in the odd modes dispersion. Actually the surface plasmon dispersion at 𝑘𝑎 =0 takes the value of 𝜔∼𝜔𝑝. For higher values of the wavenumber, the surface mode tends to disperse similarly to the infinite medium. 27 CHAPTER 3. PLASMONIC NANOSTRUCTURES To study the nonlocal effects, we provide it in figure 3.5 the dispersion relation for the surface modes and the first volume mode (𝑛=1) for different values of 𝜁. Our analysis shows the differences already known for the surface modes. For the even surface mode, all the curves have a null frequency at 𝑘𝑎 =0 while for the odd surface modes the dispersion takes 𝜔∼𝜔𝑝at 𝑘𝑎 =0, no matter the value of the nonlocal parameter 𝜁. The volume modes are very similar to the even and odd modes. However, when the nonlocal parameter increases, the curves converge to the non-dispersive case where 𝜁→ ∞ (or 𝛽→0), this is the local behavior of the plasmon dispersion. Actually, for this limit, the volume modes disperse to Ω=1. For the surface modes, the even and odd curves start to behave, respectively, as Ω2=1 21−𝑒−2𝑥(3.30) Ω2=1 21+𝑒−2𝑥(3.31) This equations corresponds to the dashed curves in figure 3.5. 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1/√280 40 20 10 Surface Even Modes Surface Odd Modes 5 10 15 20 25 30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 40 80 Volume Even Mode n = 1 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1/√2 10 80 40 20 0 5 10 15 20 25 30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Volume Odd Mode n = 1 10 80 40 20 Figure 3.5: Dispersion curves in the slab for different values of the parameter 𝜁=𝑎𝜔𝑝/𝛽(10,20,40,80). it is represented the surface modes and the volume mode 𝑛=1for the even (left) and odd (right) cases. The dashed curve corresponds to the non-dispersive model (𝜁=∞). Consider now the induced electrostatic potential 𝜙1(𝑧)in the various regions given by the equation system (3.16). In figure 3.6, we show the potential profile for the even and odd modes for different values 28 3.1. ELECTROSTATIC CONFIGURATIONS of 𝜁. Each figure shows the potential for the three regions in the study. For 𝑧/𝑎<−1and 𝑧/𝑎>−1, we have the electrostatic potential in vacuum. For the metal region (−1<𝑧/𝑎<1) the potential is delimited by the grey vertical lines. To draw the surface mode potential, we normalize them to a specific 𝑧/𝑎point. For the even mode, we normalize the potential at 𝑧=0and for the odd surface mode at 𝑧=0.5𝑎. This normalization became necessary to have a better comparison between the different curves. For the potential in the volume modes, the normalization did not show necessary. It is also interesting to note that the odd modes possess a node at the center of the slab, which does not occur in the even modes. Once again, as the nonlocal parameter increases, the potential tends to the non-dispersive limit (dashed line). Figure 3.6: Electrostatic potential 𝜙1(𝑧)in a slab for various values of 𝜁(10,20,40,80). To plot the surface mode we took 𝑘𝑎 =2and for the volume modes 𝑘𝑎 =5. The dashed line corresponds to the non-dispersive model (𝜁→ ∞). 29 CHAPTER 3. PLASMONIC NANOSTRUCTURES 3.1.3 2D electron gas To start the study of 2D materials, let us consider the simple case of a 2D electron gas surrounded by vacuum with his interface perpendicular to the 𝑧axis and located at 𝑧=0. We now consider the potential in the momentum space 𝜙1(k, 𝑧, 𝜔), assuming 𝜙1(k, 𝑧, 𝜔)=𝐴𝑒−𝑘𝑧 for 𝑧>0and 𝜙1(k, 𝑧, 𝜔)=𝐴𝑒𝑘𝑧 for 𝑧<0(note the potentials satisfy Poisson’s equation (3.3c) in vacuum). The coefficients A and B need to be determined by the boundary conditions (2.36) and (3.1). Which gives the relations 𝐴=𝐵(3.32) −𝑘(𝐴+𝐵)=𝑒 𝜖0 𝑛1(k, 𝜔)(3.33) which provides 𝐴=−𝑒 2𝑘𝜖0 𝑛1(k, 𝜔)(3.34) With these last new relations, equation (3.3a) gives an expression for the velocity v(k, 𝜔)written in terms of the electronic density𝑛1(k, 𝜔). At last, the continuity equation (3.3b) with the expression for the velocity gives 𝜔2=𝑛0𝑒2 2𝜖0𝑚𝑘+𝜋𝑛0 ℏ2𝑘2 𝑚2(3.35) which is the well-known expression for the surface plasmons in a 2D electron gas [21]: 𝜔∝𝑘at high wavenumber and 𝜔∝√𝑘at small wavenumbers. Here ℏis the Planck’s constant an 𝑐the velocity of light. For low frequencies, the result for the dispersion is only given by the first term, since for realistic parameters 𝑘<𝑘𝐹. We can write the expression in a simpler and more intuitive form by considering the electronic density 𝑛0=𝑘2 𝐹/2𝜋and the mass of the electron gas as 𝑚= ℏ𝑘𝐹/𝑣𝐹, where 𝑣𝐹is the Fermi velocity. With these definitions, the dispersion for the 2D electrons becomes, 𝜔2=𝑒2𝑘𝐹 4𝜋𝜖0ℏ𝑘+𝑣2 𝐹 2𝑘2(3.36) In [21] it is seen the dispersion given by equation (3.36). The plasmon dispersion obtained is proportional to √𝑘and the SPPs are seen in the range of frequencies of Gigahertz (GHz). 3.1.4 Graphene Obtaining the dispersion of the surface plasmon in graphene is a lot similar to the calculations made for the 2D electron gas. We consider a graphene sheet (2-dimensional) in vacuum located at 𝑧=0and with the normal vector parallel to the𝑧axis. Of course the potential is the same as before: 𝜙1(k, 𝑧, 𝜔)=𝐴𝑒−𝑘𝑧 for 𝑧>0and 𝜙1(k, 𝑧, 𝜔)=𝐴𝑒𝑘𝑧 for 𝑧<0. From the boundary conditions follows equations (3.32) and (3.33), but in this case 𝑛1(k, 𝜔)is the electronic density for graphene. Recalling that 30 3.2. GRAPHENE-METAL INTERFACE 𝜙1(k,0, 𝜔)=−𝑒 2𝑘𝜖0 𝑛1(k, 𝜔)(3.37) Once more, we apply the potential in Euler’s equation to obtain a relation between the velocity and the electronic density 𝑛1(k, 𝜔). Then the continuity equation provides the expression for the dispersion ℏ2𝜔2=2𝛼𝐹𝑆𝐸𝐹ℏ𝑐𝑘 +𝑣2 𝐹ℏ2 2𝑘2≈2𝛼𝐹𝑆 𝐸𝐹ℏ𝑐𝑘 (3.38) with the approximate result valid for realistic parameters (𝑘<𝑘𝐹). The last term in equation (3.38) is the hydrodynamic correction to the dispersion relation of plasmons in graphene. Recall the Fermi momentum for graphene 𝑘𝐹=√𝜋𝑛0and the Fermi energy 𝐸𝐹= ℏ𝑘𝐹𝑣𝐹. The fine structure constant is given by 𝛼𝐹𝑆 =𝑒2/4𝜋𝜖0ℏ𝑐. In figure 3.7 is represented the dispersion of the surface plasmons in graphene given by equation (3.38) for different values of the Fermi energy. The energy of the plasmons in graphene it is in the order of 0−80 meV as seen in (a), which corresponds to a frequency plasmon in the 0−15 terahertz region (b). As expected, the plasmons have a dispersion proportional to √𝑘for low momentum frequencies, while for larger 𝑘the curves tend to a linear dispersion. In [20] is stated that the validity of the hydrodynamic model is limited to the frequency values of 𝑘𝑣𝐹≪𝜔≪𝑘𝐹𝑣𝐹. The typical value for the homogeneous electronic density in graphene is in the order of 𝑛0⩽10−13 cm−2, and the corresponding Fermi energy must respect the condition 𝐸𝐹⩽0.4eV. This will allow us to study the plasmonic properties of graphene in the THz to a mid-infrared range of frequencies, which is confirmed by figure 3.7 (b). 0.000 0.002 0.004 0.006 0.008 0.010 0 5 10 15 20 (a) (b) 0.000 0.002 0.004 0.006 0.008 0.010 0 20 40 60 80 100 Figure 3.7: Dispersion relation [eq. (3.38)] for a single layer of graphene in vacuum for different values of the Fermi energies 𝐸𝐹=(0.2,0.3,0.4)eV and a typical Fermi velocity of 𝑣𝐹=𝑐/300. (a) Dispersion in terms of energy of the surface plasmons in the graphene layer. (b) Dispersion of plasmons in graphene regarding frequencies. Plasmonic effects are found in the THz to mid-infrared range of frequencies. 3.2 Graphene-Metal Interface The electrostatic boundary-value problems solved in the latter sections are already known by literature [17, 18, 20, 21]. Our attention must go now to the problem of the generation of plasmons in a graphene 31 CHAPTER 3. PLASMONIC NANOSTRUCTURES sheet when it is in the vicinity of a metal. At a first approach, this section will focus on the study of the nonlocal effects in the dispersion of the surface plasmons and the potentials created in graphene for two different planar configurations: first we will consider a semi-infinite nonlocal metal slab and then we generalize the results for a finite nonlocal metal slab. To simplify the algebra, graphene will always be located at 𝑧=0in the next problems. Graphene in the vicinity of a semi-infinite metal Let’s then consider a planar configuration perpendicular to the z-axis and infinite for the x and y axis, as we can see in figure 3.8. The layered heterostructure is embedded in vacuum (𝜖𝑑=1). For this geometry, we have graphene separated from metal through a dielectric of thickness𝑑. The metal occupies the region 𝑧<−𝑑while the dielectric is placed at −𝑑<𝑧<0. Dielectric εd Z Metal ε(ω) 0 -d Graphene Figure 3.8: Schematic representation of graphene in the vicinity of a semi-infinite metal. Between the graphene sheet and the metal exists a dielectric of thickness 𝑑and dielectric constant 𝜖𝑑. For this planar geometry the electronic density and the potentials are given, respectively, by 𝑛1(r, 𝑡)=            0if 𝑧>0 0if −𝑑<𝑧<0 𝑛(𝑧)𝑒𝑖(k·r∥−𝜔𝑡)if 𝑧<−𝑑 (3.39) 𝜙1(r, 𝑡)=            𝑀(𝑧)𝑒𝑖(k·r∥−𝜔𝑡)if 𝑧<−𝑑 𝐷(𝑧)𝑒𝑖(k·r∥−𝜔𝑡)if −𝑑<𝑧<0 𝐺(𝑧)𝑒𝑖(k·r∥−𝜔𝑡)if 𝑧>0 (3.40) 32 3.2. GRAPHENE-METAL INTERFACE Note that the electronic density for graphene is given by: 𝑛1(r, 𝑡)=𝛿(𝑧)𝑛1(r∥, 𝑡), however, for now, we are interested in the study of the potential for 𝑧>0(in vacuum) where the electronic density is zero, since there are no charge carriers. Graphene density will play an important role in the boundary conditions as already seen in section 3.1.4. The term 𝑛(𝑧)in the electronic density of the metal (𝑧<−𝑑) can be determined by recalling the differential equation (3.5) and the parameter 𝛼=q𝑘2+ (𝜔2 𝑝−𝜔2)/𝛽2[eq. (3.6)]. Of course, the local metal is rescued when we take 𝛽→0. The differential equation as a solution of the form 𝑛(𝑧)=𝑛0𝐴𝑒𝛼𝑧 (3.41) For the potential, we solve Poisson’s equation [eq. (2.22)] inside the metal, where the electronic density is given by the last equation, and we solve the Laplace equation for the other regions. This provides 𝑀(𝑧)=𝐵𝑒𝛼𝑧 +𝐶𝑒𝑘𝑧 (3.42) 𝐷(𝑧)=𝐷𝑒−𝑘𝑧 +𝐹𝑒𝑘𝑧 (3.43) 𝐺(𝑧)=𝐺𝑒−𝑘𝑧 (3.44) In this case, the boundary conditions must apply to the surfaces at 𝑧=0(graphene/dielectric) and 𝑧=−𝑑(metal/dielectric). For the graphene-dielectric interface, the boundary conditions are given by equations (2.36) and (3.1), this last condition, we already know that due to the polarization of graphene, the normal derivative of the potential at the interface becomes discontinuous. For the metal-dielectric interface, we solve equations (2.36) and (2.37) and for the metal, the hydrodynamic model provide us with equation (2.39). Contrary to the problem of the metal-dielectric interface and the planar slab, we will consider 𝜖∞≠1. In this section, we will compare the strength of the nonlocal effects in two distinct metals: gold (Au) and titanium (Ti). In both cases, the background permittivity is different from 1. Within this thesis, we will only consider these two metals, even though this study is not restricted to them and we can extend the theory to other types of metals. Solving the BCs equations is essential to determine the coefficients in the potentials. In latter electrostatic problems solved, until now, the equations were quite simple to solve through algebraic methods. However, in this case we are presented with a certain level of complexity, but luckily we have five coefficients for five boundary conditions and so we can use matricial methods to solve the linear system of equations. This is, it is only necessary to confirm the equality 𝑀·𝑉=𝑁, where𝑉is the coefficient matrix 𝑉=[𝐵,𝐶, 𝐷, 𝐹,𝐺]and 𝑁the matrix of the free-coefficient terms 𝑁=[0,0,0,(𝑒/𝜖0)𝑛1(k, 𝜔),0]. The matrix M is given by 33 CHAPTER 3. PLASMONIC NANOSTRUCTURES 𝑀= 𝑒−𝛼𝑑 𝑒−𝑘𝑑 −𝑒𝑘𝑑 −𝑒−𝑘𝑑 0 𝜖∞𝛼𝑒−𝛼𝑑 𝜖∞𝑘𝑒−𝑘𝑑 𝜖𝑑𝑘𝑒𝑘𝑑 −𝜖𝑑𝑘𝑒−𝑘𝑑 0 0 0 1 1 −1 0 0 𝜖𝑑𝑘−𝜖𝑑𝑘−𝑘 (𝜔2/𝜔2 𝑝)(𝛼/𝑘)𝑒(𝑘−𝛼)𝑑1 0 0 0  The solution for all coefficients and consequently the potentials can be consulted in appendix A. For our purposes, we only need the potential in graphene to determine the dispersion of the surface plasmons in it. By that, in graphene (at 𝑧=0) we have 𝜙1(k,0, 𝜔)=𝐺(0)=𝑒 𝜖0 𝑛1(k, 𝜔)[𝐷(𝑘, 𝜔) +𝐹(𝑘, 𝜔)] (3.45) with the auxiliary function 𝐷(𝑘, 𝜔)and 𝐹(𝑘, 𝜔), respectively 𝐷(𝑘, 𝜔)=−𝜖𝑑(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜔2 𝑝−𝜔2)𝜖∞ 𝑘[𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞] (3.46) 𝐹(𝑘, 𝜔)=−𝑒2𝑑𝑘 (𝜖𝑑(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜔2−𝜔2 𝑝)𝜖∞) 𝑘[𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞] (3.47) Finally, the potencial (3.45) goes into the Euler’s equation (3.3a) for graphene, and the dispersion relation follows from the continuity equation (3.3b) ℏ2𝜔2="−4𝛼𝐹𝑆 ℏ𝑐𝐸𝐹(𝐷(𝑘, 𝜔) +𝐹(𝑘, 𝜔)) + ℏ2𝑣2 𝐹 2#𝑘2(3.48) Figure 3.9 and 3.10 represent the energy spectrum of the surface plasmon-polaritons, given by the last equation when a graphene layer is in the vicinity of a nonlocal semi-infinite metal. The spectrum is shown for two different values of the separation between the metal and graphene, this is the dielectric thickness d. As already said, we consider two different metals, gold (Au) and titanium (Ti). The nonlocal parameter 𝛽in our model is given by 𝛽=𝑣𝐹/√3, which for gold and titanium take the values 0.0027𝑐and 0.0035𝑐, respectively. Gold is described by the plasma frequency 𝜔𝑝=8.84 eV and by the background permittivity 𝜖∞=9.84. While for titanium, the parameters are: 𝜔𝑝=2.80 eV and 𝜖∞=2.2[15]. In both figures, we study the impact of the nonlocal effects probed by the metal in the spectrum of the surface plasmons in graphene and we compare the respective curves with the spectrum for the local case (𝛽→0) For gold and titanium, a linear character describes the spectrum of the surface plasmons. This means that the SPPs dispersion for graphene in a vicinity of a semi-infinite metal is proportional to the wavenumber, 𝜔∝𝑘. This is in contrast with the spectrum obtained for graphene in vacuum (figure 34 3.2. GRAPHENE-METAL INTERFACE 3.7), where the plasmons disperse by following the law √𝑘. Looking now at the differences between the spectrum of the two metals, it is visible that the strength of the nonlocal effects is more enhanced in the titanium figures, where the dispersion curves has a higher separation when regarding the local spectrum. Figure 3.9: Spectrum of the graphene surface plasmon-polaritons given by equation (3.48) when in a graphene-metal interface, where the metal in case was gold. To improve the visualization of the nonlocal effects it was necessary approximate the spectrum, so note that it does not start at 𝑘=0. The dielectric chosen was silicon dioxide, 𝜖𝑑=3.9and with thickness 𝑑=10 nm and 𝑑=5nm. The nonlocal effects are studied for the nonlocal parameter 𝛽=0.0027𝑐. We also computed the local spectrum (𝛽→0). The other parameters are: 𝜔𝑝=8.84 eV, 𝜖∞=9.84 and 𝐸𝐹=0.17 eV. Figure 3.10: Spectrum of the graphene surface plasmon-polaritons given by equation (3.48) when in a graphene-metal interface, where the metal in case was titanium. The dielectric chosen was silicon dioxide, 𝜖𝑑=3.9and with thickness 𝑑=10 nm and 𝑑=5nm. The nonlocal effects are studied for the nonlocal parameter 𝛽=0.0035𝑐. We also computed the local spectrum (𝛽→0). The other parameters are: 𝜔𝑝=2.80 eV, 𝜖∞=2.20 and 𝐸𝐹=0.17 eV. The nonlocality is more distinguishable in titanium due to their smaller plasma frequency 𝜔𝑝and background permittivity 𝜖∞. Actually, we have 𝜔𝑝,𝑇𝑖 /𝜔𝑝,𝐴𝑢 ≃0.32 and 𝜖∞,𝑇𝑖/𝜖∞,𝐴𝑢 ≃0.22. These two factors are crucial for the presence of nonlocal effects in the system. Furthermore, in our case we are not considering relaxation frequencies (𝛾=0), however, this variable also plays an important role in the study of nonlocal effects, more information about this parameter can be found in ref. [15]. All this leads to a 35 CHAPTER 3. PLASMONIC NANOSTRUCTURES Finally, within this expression we apply the limit𝑑→0to obtain 1/2𝑘and the dispersion for the graphene SPPs in vacuum is given by ℏ2𝜔2=2𝛼𝐹𝑆ℏ𝑐𝐸𝐹𝑘+ ℏ2𝑣2 𝐹 2𝑘2(3.61) which is the exact expression obtained for a graphene sheet in vacuum given by equation (3.38). This shows us that the plasmon dispersion obtained when graphene is near a finite metal provide the expected expressions for graphene near a local semi-infinite metal [eq. (3.59) ] and a graphene sheet in vacuum [eq. (3.61)] when taking the appropriate limits, respectively. In figure 3.17 is shown the dispersion for this two limits altogether with the dispersion for graphene in the vicinity of a semi-infinite local metal given by equation (3.48), which is expected to be the same as the limit spectrum 𝜖∞→ ∞. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 10 20 30 40 Figure 3.17: Spectrum limits of equation (3.57) that corresponds to graphene near a finite metal. For 𝜖∞→ ∞ we obtain the spectrum of a local semi-infinite metal, that corresponds to the semi-infinite local metal (dashed green line). The dielectric chosen was silicon dioxide (𝜖𝑑2=3.9) and a thickness of 𝑑=10 nm. The metal chosen was titanium. For the limit 𝑎→0and 𝑑→0we obtain the problem of graphene in a dielectric, which in this case we choose to be vacuum (𝜖𝑑1=𝜖𝑑2=1) and this spectrum agrees with the spectrum of graphene in vacuum represented in figure (3.7). The Fermi energy has a value of 𝐸𝐹=0.17 eV. At last, we represent the potentials in figure 3.18 for titanium (𝛽=0.0035𝑐) with a thickness of 𝑎=10 nm. The potentials were drawn by fixing 𝑘=3𝜇m−1and for a dielectric thickness of 𝑑=10 nm and 𝑑=5nm. It is for 𝑑=5nm that we see a clearer distinction between the local and nonlocal potentials, because as we decrease the dielectric thickness, the nonlocal effects become more enhanced due to the higher spatial confinement of the fields. Once again, there is a continuity for the potential in the various regions. 42 3.2. GRAPHENE-METAL INTERFACE Figure 3.18: Electrostatic potential for titanium (Ti) while fixing 𝑘=3𝜇m−1and 𝑎=10 nm The dashed curve corresponds to the local model (𝛽→0). The thickness of the dielectric 2 is 𝑑=10 nm (left) and 𝑑=5nm (right). The Fermi energy takes a value of 𝐸𝐹=0.17 eV, the dielectric constants are 𝜖𝑑1=1 and 𝜖𝑑2=3.9. For titanium the parameters are: 𝜔𝑝=2.80 and 𝜖∞=2.2. 43 Chapter 4 Hydrodynamic Model in the Presence of External Potentials In this chapter, we will extend the hydrodynamic model in the electrostatic approximation to describe the plasmonic effects created by the potential of an external charge in graphene. In the latter chapter, the intrinsic potentials of the mediums within the hydrodynamic theory provided the characteristic dispersion curves for the surface plasmons in graphene. Now that we know the typical energy at which the plasmons occur in graphene, it will be interesting to add an external moving charge to the system to study the outcome of the potential in graphene. In this thesis, we will study the potential induced in graphene by a particle moving parallel to it. Of course, the motion perpendicular to the graphene sheet can also be achieved [17] (or any other motion, for that matter). Furthermore, we derive a simple expression for the induced potential 𝜙𝑖𝑛 (r∥, 𝑧, 𝜔)due to a moving charge 𝑍𝑒. The following external charge densities describe this moving charge, regarding the two motion cases 𝜌𝑒𝑥 (r∥, 𝑧, 𝑡)=𝑍𝑒𝛿 (𝑥)𝛿(𝑦)𝛿(𝑧−𝑣𝑡)(4.1) 𝜌𝑒𝑥 (r∥, 𝑧, 𝑡)=𝑍𝑒𝛿 (𝑥)𝛿(𝑦−𝑣𝑡)𝛿(𝑧−𝑧0)(4.2) where the particle is moving at speed 𝑣. Note that delta functions define the direction of the motion. In equation (4.1), the particle has a perpendicular motion to the graphene plane (piercing it). On the other hand, equation (4.2) describes the parallel motion to the graphene plane at a height 𝑧=𝑧0, where we also considered that the charge moves in the direction of the y-axis. We will determine the potential through equation (3.3c) having into account the presence of external charges. This equation can be solved with the Green´s function method and the potentials will be given in the momentum space. Therefore, it is necessary to perform a Fourier transforms in rand t of the charge densities, which gives 44 4.1. MOTION PARALLEL TO THE GRAPHENE SHEET 𝜌𝑒𝑥 (k, 𝑧, 𝜔)=𝑍𝑒 ∫𝑑𝑡𝑑r∥𝑒−𝑖(k·r∥−𝜔𝑡)𝛿(𝑥)𝛿(𝑦)𝛿(𝑧−𝑣𝑡)=𝑍𝑒 𝑣𝑒𝑖𝜔𝑧/𝑣(4.3) 𝜌𝑒𝑥 (k, 𝑧, 𝜔)=𝑍𝑒 ∫𝑑𝑡𝑑r∥𝑒−𝑖(k·r∥−𝜔𝑡)𝛿(𝑥)𝛿(𝑦−𝑣𝑡)𝛿(𝑧−𝑧0)=2𝜋𝑍𝑒𝛿(𝑧−𝑧0)𝛿(𝜔−𝑘𝑦𝑣)(4.4) As stated we are interested in the motion parallel to the graphene sheet, thus in the next sections we will use equation (4.4) to obtain the induced potential in the graphene sheet for two configurations: graphene in the vicinity of a dielectric and graphene in the vicinity of local metal. The parallel motion of the external charge will induce a dragging of the surface plasmons in graphene, which will lead to the formation of plasmonic wakes on the surface of graphene. First, we will have to reconstruct the hydrodynamic model to account for external charges. The methods to derive the expressions for the external and induced potentials were first deduced by Fetter in [23]. 4.1 Motion Parallel to the Graphene Sheet The additional presence of external electrostatic forces acting on the electron gas must have a description in the hydrodynamic model. This adds extra terms to Euler’s and Poisson’s equation. The continuity equation remains unchanged. With this external potential, we get 𝜕v 𝜕𝑡 =𝑒𝑣𝐹 ℏ𝑘𝐹∇𝜙1(r∥,0) +𝜙𝑒𝑥 (r∥,0)−𝑣2 𝐹 2𝑛0∇𝑛1(r∥)(4.5a) ∇2[𝜙1(r) +𝜙𝑒𝑥 (r)]=−𝑒 𝜖0 𝜌𝑒𝑥 (r) + 𝑒 𝜖0 𝛿(𝑧)𝑛1(r∥,0)(4.5b) where 𝜙𝑒𝑥 (r∥,0)is the external potential acting on the electron gas due to the external forces, while 𝜌𝑒𝑥 (r) is the volume density of external charges. Note that we write the last equations for the parameters in graphene. To recall, 𝑘𝐹and 𝑣𝐹are the Fermi momentum and the Fermi velocity, respectively. 𝑛0=𝑘2 𝐹/𝜋 is the homogeneous electronic density for graphene. Of course, we can also do it for the 2D electron gas by using equation (2.32a) instead. Moreover, we are interested in the electronic response of graphene towards an external force. Performing a Fourier transform of the quantities in equations (4.5a) and (4.5b), and we obtain −𝑖𝜔v(k, 𝜔)=𝑒𝑣𝐹 ℏ𝑘𝐹 𝑖k𝜙(k,0, 𝜔) − 𝑣2 𝐹 2𝑛0 𝑖k𝑛1(k, 𝜔)(4.6a) 𝜕2 𝜕𝑧2−𝑘2𝜙(k, 𝑧, 𝜔)=−1 𝜖0 𝜌𝑒𝑥 (k, 𝑧, 𝜔) + 𝑒 𝜖0 𝛿(𝑧)𝑛1(k, 𝜔)(4.6b) In this equation we defined the total potential as 𝜙(k, 𝑧, 𝜔)=𝜙1(k, 𝑧, 𝜔)+𝜙𝑒𝑥 (k, 𝑧, 𝜔), and 𝜌𝑒𝑥 (k, 𝑧, 𝜔) is the volume density of external charges. The continuity equation remains the same as equation (3.3b). 45 CHAPTER 4. HYDRODYNAMIC MODEL IN THE PRESENCE OF EXTERNAL POTENTIALS To solve equation (4.6b) we use Green’s function method to obtain the potential. The free space Green’s function is defined as [24] 𝜕2 𝜕𝑧2−𝑘2𝑔(k, 𝑧 −𝑧′, 𝜔)=−𝛿(𝑧−𝑧′)(4.7) where it follows the general solution for the potential 𝜙(k, 𝑧, 𝜔)=1 𝜖0∫𝑑𝑧′𝑔(k, 𝑧 −𝑧′, 𝜔)𝜌(k, 𝑧′, 𝜔)(4.8) where the total volume density of charge is given by 𝜌(k, 𝑧, 𝜔)=𝜌𝑒𝑥 (k, 𝑧, 𝜔) −𝑒𝛿(𝑧)𝑛1(k, 𝜔)(4.9) The solution for equation (4.7) gives the usual function (check with appendix B) 𝑔(k, 𝑧 −𝑧′, 𝜔)=1 2𝑘𝑒−𝑘|𝑧−𝑧′|(4.10) then the total potential in equation (4.8) can be rewritten with the charge densities provided in equation (4.9) as 𝜙(k, 𝑧, 𝜔)=∫𝑑𝑧′1 2𝑘𝜖0 𝑒−𝑘|𝑧−𝑧′|𝜌𝑒𝑥 (k, 𝑧′, 𝜔) − 𝑒 2𝑘𝜖0 𝑒−𝑘|𝑧|𝑛1(k, 𝜔)(4.11) which can be thought as 𝜙(k, 𝑧, 𝜔)=Φ𝑒𝑥 (k, 𝑧, 𝜔) − Φ1(k, 𝑧, 𝜔). We now replace this result in the hydrodynamic equation (4.6a), obtaining 𝜔v(k, 𝜔)=𝑒𝑣𝐹 ℏ𝑘𝐹 k[Φ1(k,0, 𝜔) −Φ𝑒𝑥 (k,0, 𝜔)]+𝑣2 𝐹 2𝑛0 k𝑛1(k, 𝜔)(4.12) With this last equation, we obtain the velocity of the electron gas which we plug into the continuity equation. By that, we obtain an expression for 𝑛1(k, 𝜔)that depends on the external charges. This will allow us to write the induced potential in terms of 𝜌𝑒𝑥 (k, 𝑧, 𝜔). In this way, the induced and external potentials are given, respectively, by Φ𝑖𝑛 (k, 𝑧, 𝜔)=−Φ1(k, 𝑧, 𝜔)=𝑛0𝑒2𝑣𝐹 4𝜖2 0ℏ𝑘𝐹 𝑒−𝑘|𝑧| 𝜔2−𝜔2 𝑠𝑝𝑝 ∫𝑑𝑧′𝑒−𝑘|𝑧′|𝜌𝑒𝑥 (k, 𝑧′, 𝜔)(4.13) Φ𝑒𝑥 (k, 𝑧, 𝜔)=∫𝑑𝑧′1 2𝑘𝜖0 𝑒−𝑘|𝑧−𝑧′|𝜌𝑒𝑥 (k, 𝑧′, 𝜔)(4.14) where 𝜔𝑠𝑝𝑝 is the known dispersion for the surface plasmon-polaritons in graphene. Note the potentials are a function of the external volume density of charges 𝜌(k, 𝑧′, 𝜔)in the medium. In that case, we just plug equation (4.4) for the parallel motion, to obtain the potential induced in graphene by the external charges 46 4.2. GRAPHENE IN THE VICINITY OF A DIELECTRIC Φ𝑖𝑛 (k, 𝑧, 𝜔)=2𝜋𝑍 𝑛0𝑒3𝑣𝐹 4𝜖2 0ℏ𝑘𝐹 𝑒−𝑘(|𝑧|+𝑧0) 𝜔2−𝜔2 𝑠𝑝𝑝 𝛿(𝜔−𝑘𝑦𝑣)(4.15) Φ𝑒𝑥 (k, 𝑧, 𝜔)=2𝜋𝑍𝑒 2𝑘𝜖0 𝑒−𝑘|𝑧−𝑧0|𝛿(𝜔−𝑘𝑦𝑣)(4.16) To obtain the last potentials in real space, we only need to perform an inverse Fourier transform, as will be done in the next sections. 4.2 Graphene in the vicinity of a dielectric When graphene is in the vicinity of a dielectric (𝜖𝑑), the calculations are similar to graphene in vacuum. However, in such case, the Green function must differ from that of the free space [eq. (4.10)], moreover, the method to get the potential remains the same. To determine the potentials when a charge is moving parallel to a graphene sheet in the vicinity of a dielectric, it is necessary to first calculate the Green’s function in graphene. By that, we have graphene at 𝑧=0and the dielectric with dielectric constant 𝜖𝑑at 𝑧<0as seen in figure 4.1. Consider that we have vacuum at 𝑧>0. Dielectric εd Z 0 z' Graphene I II III Figure 4.1: Graphene in the vicinity of a dielectric configuration. Graphene is at 𝑧=0, while the dielectric is present at 𝑧<0. Vacuum (𝜖𝑑=1) extends for 𝑧>0. We want to find the Green function at an arbitrary 𝑧′. Once again, the Green function is defined in equation (4.7). Where we consider an exponential behavior for these functions 𝑔(𝑧, 𝑧′)=            𝑔𝐼(𝑧, 𝑧′)=𝐴(𝑧′)𝑒−𝑘𝑧 if 𝑧>𝑧′ 𝑔𝐼𝐼 (𝑧, 𝑧′)=𝐶1(𝑧′)𝑒−𝑘𝑧 +𝐶2(𝑧′)𝑒𝑘𝑧 if 0<𝑧<𝑧′ 𝑔𝐼𝐼𝐼(𝑧, 𝑧′)=𝐵(𝑧′)𝑒𝑘𝑧 if 𝑧<0 (4.17) 47 CHAPTER 4. HYDRODYNAMIC MODEL IN THE PRESENCE OF EXTERNAL POTENTIALS The Green functions in equation (4.17) must, at all cases, decay as |𝑧| → ∞. In particular, 𝑔𝐼(𝑧, 𝑧′) must decay as 𝑧tends to positive values of infinity, while 𝑔𝐼𝐼𝐼(𝑧, 𝑧′)decays for negative values of infinity. Of course, this is like the calculations done in chapter 3 for the potentials, and as in that case, it is also necessary to apply some boundary conditions to the different regions, as to obtain the coefficients in the Green function. These conditions are just the continuity of the Green function and the discontinuity of the normal derivative to the function at the boundaries (the mathematical formulation is clear in appendix B). Because the Green function is like a point charge potential, only at 𝑧=𝑧′are the derivatives not continuous. We consider all the remaining charges of the configuration when we calculate the potential with equation (4.8). Solving the linear system of the equation provided by the boundary conditions we get for the Green functions 𝑔(𝑧, 𝑧′)=            𝑔𝐼(𝑧, 𝑧′)=𝑒−𝑘(𝑧−𝑧′) 2𝑘+𝑟𝑑𝑒−𝑘(𝑧+𝑧′) 2𝑘if 𝑧>𝑧′ 𝑔𝐼𝐼 (𝑧, 𝑧′)=𝑟𝑑𝑒−𝑘(𝑧+𝑧′) 2𝑘+𝑒𝑘(𝑧+𝑧′) 2𝑘if 0<𝑧<𝑧′ 𝑔𝐼𝐼𝐼(𝑧, 𝑧′)=𝑡𝑑𝑒𝑘(𝑧−𝑧′) 2𝑘if 𝑧<0 (4.18) By thinking of the Green functions as propagating waves through the mediums, we can express these with the ”reflection”(𝑟𝑑) and ”transmission”(𝑡𝑑) coefficients as 𝑟𝑑=1−𝜖𝑑 1+𝜖𝑑(4.19) 𝑡𝑑=2 1+𝜖𝑑(4.20) We now want the potential created due to the parallel motion of an external particle (at 𝑧>0) to the graphene sheet. Since graphene is at 𝑧=0we want the Green function for this region, so we also admit 𝑧′=0. Here 𝑔𝐼𝐼 (𝑧, 𝑧′)vanishes, because this region cease to exist and the induced potential in graphene is given by 𝑔𝐼(𝑧, 𝑧′)and 𝑔𝐼𝐼𝐼 (𝑧, 𝑧′). However, if we want the external potential, then we have to consider 0<𝑧<𝑧′and we must make use of the Green function 𝑔𝐼𝐼 (𝑧, 𝑧′). Equation (4.8) demonstrates the total potential and the external and induced potentials are, respectively Φ𝑒𝑥 (k, 𝑧, 𝜔)=1 𝜖0∫𝑑𝑧′𝑒𝑘(𝑧−𝑧′) 2𝑘+𝑟𝑑 𝑒−𝑘(𝑧+𝑧′) 2𝑘𝜌𝑒𝑥 (k, 𝑧′, 𝜔)(4.21) Φ𝑖𝑛 (k, 𝑧, 𝜔)=−Φ1(k, 𝑧, 𝜔)=−𝑒 𝜖0 𝑡𝑑 𝑒−𝑘|𝑧| 2𝑘𝑛1(k, 𝜔)(4.22) Notice that 𝜖𝑑=1then the ”reflection”coefficient is 𝑟𝑑=0, and the ”transmission”coefficient obtained is 𝑡𝑑=1, and the potentials in the last equations are reduced to the expressions in equations (4.13) and (4.14). Using the results for the potentials in the hydrodynamic equation (4.12), we obtain a relation for the velocity which, once again, we put in the continuity equation (3.3b) which get us the following equation 48 4.2. GRAPHENE IN THE VICINITY OF A DIELECTRIC 0=−𝜔2𝑛1(k, 𝜔) +𝑛0𝑘2"−𝑒𝑣𝐹 ℏ𝑘𝐹 Φ𝑒𝑥 (k,0, 𝜔) + 𝑒2𝑣𝐹 𝑘(1+𝜖𝑑)𝜖0ℏ𝑘𝐹+𝑣2 𝐹 2𝑛0!𝑛1(k, 𝜔)#(4.23) where the term in curve brackets multiplied by 𝑛0𝑘2is the dispersion of the surface plasmon-polariton 𝜔𝑠𝑝𝑝, for graphene in this configuration. Recall that we can write the dispersion in terms of the Fermi energy 𝐸𝐹= ℏ𝑘𝐹𝑣𝐹and the fine structure constant 𝛼𝐹𝑆 =𝑒2/4𝜋𝜖0ℏ𝑐, by considering the homogeneous electronic density as 𝑛0=𝑘2 𝐹/𝜋. So the relation for the SPPs in graphene is given by 𝜔2 𝑠𝑝𝑝 =4𝛼𝐹𝑆𝐸𝐹ℏ𝑐 (1+𝜖𝑑)ℏ2𝑘+𝑣2 𝐹 2𝑘2(4.24) Note that if we take 𝜖𝑑=1we have graphene in vacuum and the dispersion obtained through this method is the same as the one obtained in equation (3.38). Also, resorting to this last equation, we can get the inhomogeneous electronic density 𝑛1(k, 𝜔)through equation (4.23) 𝑛1(k, 𝜔)=−(2𝛼𝐹𝑆𝐸𝐹ℏ𝑐/𝑒) ℏ2𝜔2−ℏ2𝜔2 𝑠𝑝𝑝 𝑘∫𝑑𝑧′𝑡𝑑𝑒−𝑘𝑧′𝜌𝑒𝑥 (k, 𝑧′, 𝜔)(4.25) Since the induced potential in graphene depends on 𝑛1(k, 𝜔)we can write it as Φ𝑖𝑛 (k, 𝑧, 𝜔)=1 𝜖0 𝛼𝐹𝑆𝐸𝐹ℏ𝑐𝑡2 𝑑𝑒−𝑘|𝑧| ℏ2𝜔2−ℏ2𝜔𝑠𝑝𝑝 ∫𝑑𝑧′𝑒−𝑘𝑧′𝜌𝑒𝑥 (k, 𝑧′, 𝜔)(4.26) Now we want to study the parallel motion to the graphene sheet, where the volume density of external charge (4.4) provides Φ𝑖𝑛 (k, 𝑧, 𝜔)=2𝜋𝑣2Φ0 𝑡2 𝑑𝑒−𝑘(|𝑧|+𝑧0) 𝜔2−𝜔2 𝑠𝑝𝑝 𝛿(𝜔−𝑘𝑦𝑣)(4.27) where the prefactor Φ0=𝑍𝑒 𝜖0 𝛼𝐹𝑆 𝐸𝐹ℏ𝑐 ℏ2𝑣2has units of electric potential and we defined it to make the potentials easier to read. Note that the delta function implies that the particle disperses with a frequency given by 𝜔=𝑘𝑦𝑣. Fourier transforming to real space and time equation (4.27), we obtain for Φ𝑖𝑛 (r, 𝑧, 𝑡) Φ𝑖𝑛 (r, 𝑧, 𝑡)=𝑣2Φ0∫∞ −∞ 𝑑𝜔𝑑k (2𝜋)2 𝑡2 𝑑𝑒−𝑘(|𝑧|+𝑧0)𝑒𝑖(r∥·k−𝜔𝑡) 𝜔2−𝜔2 𝑠𝑝𝑝 𝛿(𝜔−𝑘𝑦𝑣)(4.28) This last equation finally gives us the potential induced in the surface of the graphene sheet. However, the expression as such, still needs to be resolved, so as to provide us with results for the problems in question. We will work out a method to calculate the integral in a more convenient form. The calculation method is based on a mathematical description given to us by [17], where the Sokhotski-Plemelj formula will allow us to divide the integral into two distinct new integrals: a principal value integral and an imaginary integral. Before we start the calculation of the integral, let us compute the phase velocity 𝑣𝑝=𝜔/𝑘of the plasmons in graphene 49 CHAPTER 4. HYDRODYNAMIC MODEL IN THE PRESENCE OF EXTERNAL POTENTIALS 𝑣𝑝=√𝑎𝑘 𝑘=𝜔 𝑎𝜅(4.29) with 𝜅=−1. Thus, plasmons in graphene behave as gravity waves [25], with an acceleration 𝑎. The argument of the 𝛿-function in equation (4.28) implies that 𝜔=𝑘𝑣 cos 𝜃, which also gives us the following condition cos 𝜃=𝜔 𝑣𝑘 =𝑣𝑝 𝑣≤1(4.30) by that, the velocity 𝑣of the charge must be larger than 𝑣𝑝. Using 𝜔=√𝑎𝑘, we must also have 𝑎 𝑣2≤𝑘(4.31) This means that the 𝑘integral has domain 𝑘∈ [𝑎/𝑣2,∞[. Now that we have defined the domain of integration of the potential integral in equation (4.28), let us write it in polar coordinates for simplicity Φ𝑖𝑛 (r∥, 𝑧, 𝑡)=𝑣2𝑡2 𝑑Φ0∫2𝜋 0 𝑑𝜃 2𝜋∫∞ 0 𝑘𝑑𝑘 2𝜋 𝑒−𝑘(|𝑧|+𝑧0)𝑒𝑖𝑘[𝑟cos(𝜃−𝜃′)−𝑣𝑡 cos 𝜃] (𝑘𝑣 cos 𝜃)2+𝑖sgn(cos 𝜃)𝜂−𝑎𝑘 (4.32) In this last integral we made some important considerations, where the polar angle 𝜃is the angle that the vector kmakes with the y-axis. While the angle 𝜃′is the polar angle of r∥, and by that end the scalar product k·ris given by 𝑘𝑟 cos(𝜃−𝜃′). Note that the delta function implies 𝜔=𝑘𝑦𝑣, but the y-component of kcan be written as 𝑘𝑦=𝑘cos 𝜃. To treat the dispersion of the surface plasmon-polaritons in graphene [eq. (4.24)], we know that for real parameters (𝑘≪𝑘𝐹) the hydrodynamic term can be neglected. So we get 𝜔2 𝑠𝑝𝑝 =𝑎𝑘, where 𝑎=4𝛼𝐹𝑆 𝐸𝐹ℏ𝑐 (1+𝜖𝑑)ℏ2is a parameter with units of acceleration and characterizes the 2D electron gas in graphene. Finally, to solve the integral in the complex plane we get the term 𝑖sgn(cos 𝜃)𝜂, where 𝜂→0and sgn(cos 𝜃)is the sign function for cosine. Equation (4.29) can be resolved by using the Sokhotski-Plemelj theorem Φ𝑖𝑛 (r∥, 𝑧, 𝑡)=𝑡2 𝑑Φ0∫2𝜋 0 𝑑𝜃 2𝜋⨏∞ 0 𝑑𝑘 2𝜋 𝑒−𝑘𝜇𝑒𝑖𝑘𝜈 𝑘cos2𝜃−𝑎/𝑣2 −𝑖𝜋𝑡2 𝑑Φ0∫2𝜋 0 𝑑𝜃 2𝜋∫∞ 0 𝑑𝑘 2𝜋𝑒−𝑘𝜇𝑒𝑖𝑘𝜈 sgn(cos 𝜃)𝛿(𝑘cos2𝜃−𝑎/𝑣2)(4.33) where the symbol ⨏stands for the principal value of the integral. The parameters 𝜇and 𝜈are given by 𝜇=|𝑧| +𝑧0(4.34) 𝜈=[𝑟cos(𝜃−𝜃′) −𝑣𝑡 cos 𝜃](4.35) Now let’s perform a variable change of the form 𝑘cos2𝜃−𝑎/𝑣2=𝜅, then the last integral can be written as 50 4.2. GRAPHENE IN THE VICINITY OF A DIELECTRIC Φ𝑖𝑛 (r∥, 𝑧, 𝑡)=Φ0𝑡2 𝑑∫2𝜋 0 𝑑𝜃 (2𝜋)2 𝑒−𝑓(𝜃) cos2𝜃⨏∞ −𝑎/𝑣2 𝑑𝜅 𝑒−𝑣2𝑓(𝜃)𝜅/𝑎 𝜅 −𝑖𝜋𝑡2 𝑑Φ0∫2𝜋 0 𝑑𝜃 (2𝜋)2 𝑒−𝑓(𝜃) cos2𝜃sgn(cos 𝜃)(4.36) In this case the sign function constrains the limits of integration, that is, if the cosine function is positive the sign function takes the value of +1within the integration limits [−𝜋/2, 𝜋/2], while for negative values of cos 𝜃the sign function takes −1in the range of integration [𝜋/2,3𝜋/2]. We also defined the function 𝑓(𝜃)as 𝑓(𝜃)=𝑎 cos2𝜃𝑣2[𝜇−𝑖𝜈(𝜃)] (4.37) with 𝜇and 𝜈defined in equations (4.34) and (4.35), respectively. The principal value of the integral in 𝜅 is a known special function, called the exponential integral function Ei(𝑥). So we write the integral as Figure 4.2: Top: Dispersion relation of the graphene SPPs, when in the vicinity of a dielectric given by equation (4.24) in terms of energy (left) and frequency (right). The green curve corresponds to the particle dispersion that follows the law 𝜔=𝑘𝑣, along the direction of the moving particle (y-axis). Note that the velocity of the moving particle 𝑣gives the slope of the spectrum. Bottom: Induced electrostatic potential along the direction of propagation of the moving particle. The parameters are: an electronic density of 105𝜇m−2which gives a Fermi energy of 𝐸𝐹=0.37 eV ,𝑣=0.1𝑐,𝑧0=0.01 𝜇m and 𝑡=0s. The plots show the curves for two distinct values of the dielectric constant, 𝜖𝑑=1and 𝜖𝑑=2. 51 CHAPTER 4. HYDRODYNAMIC MODEL IN THE PRESENCE OF EXTERNAL POTENTIALS Having in mind the sign term, the polar integral can be written in two parts 2∫𝜋/2 0 𝑑𝜃𝑔(𝑘, 𝜃) 4 Õ 𝑖=1 𝛿(𝜃−𝜃𝑖) |2𝑣2𝑘2cos 𝜃𝑖sin 𝜃𝑖|−2∫𝜋 𝜋/2 𝑑𝜃𝑔(𝑘, 𝜃) 4 Õ 𝑖=1 𝛿(𝜃−𝜃𝑖) |2𝑣2𝑘2cos 𝜃𝑖sin 𝜃𝑖|(4.60) where we have used the following mathematical property 𝛿(𝑓(𝜃)) −→ 𝑁 Õ 𝑖=1 𝛿(𝜃−𝜃𝑖)1 |𝑓′(𝜃𝑖)|(4.61) The factor of two appears in each integral because we are only dealing with cosine functions, and in that case, the sign function will provide that for the region of integration where cos 𝜃has the same sign, the value of the integral will be the same. At last, we need to look at the angles 𝜃𝑖to see which one falls in the angle interval of integration. The integral is zero for the angles 𝜃𝑖that are not in the range of the limits of integration. In that case figure 7, shows us that for 𝜃∈0,𝜋 2only 𝜃1contributes for the integral and for 𝜃∈𝜋 2, 𝜋only 𝜃2gives a solution different from zero. This means that the solution of the polar integral is 𝑔(𝑘, 𝜃1) |𝑣2𝑘2cos 𝜃1sin 𝜃1|−𝑔(𝑘, 𝜃2) |𝑣2𝑘2cos 𝜃2sin 𝜃2|(4.62) 0 50 100 150 200 250 1 2 3 4 5 π 2π/3 π/2 θ1 θ2 θ3 θ4 Figure 4.6: Contribution of the polar angles for each interval of the polar integral (4.59). The black lines represented in the figures are the polar angles 𝜃𝑖=(𝜃1, 𝜃2,𝜃3, 𝜃4)=𝜃∈ (arccos(𝜔𝑆𝑃𝑃 𝑘𝑣 ), 𝜋 − arccos(𝜔𝑆𝑃𝑃 𝑘𝑣 ), 𝜋 +arccos(𝜔𝑆𝑃𝑃 𝑘𝑣 ),2𝜋−arccos(𝜔𝑆𝑃𝑃 𝑘𝑣 )). Only the angles 𝜃1and 𝜃2are in the range of the limits of integration, however, the factor of 2 in the integral covers the polar angles of the upper branch (𝜃3and 𝜃4). The parameters are the same as in figure 4.5. Finally, the induced potential when graphene is in the vicinity of a local metal is given by 58 4.3. GRAPHENE IN THE VICINITY OF A LOCAL METAL Φ𝑖𝑛 (r, 𝑧, 𝑡)=𝑣2𝑡2 𝑑Φ0∫2𝜋 0 𝑑𝜃 2𝜋⨏∞ 0 𝑘𝑑𝑘 2𝜋 𝑒−𝑘𝛽𝑒𝑖𝑘𝛾 (𝑘𝑣 cos 𝜃)2−𝜔2 𝑠𝑝𝑝 𝑟𝐷(𝑘𝑣 cos 𝜃) +𝑒2𝑘𝑑 𝑟𝑟𝐷(𝑘𝑣 cos 𝜃) +𝑒2𝑘𝑑  −𝑖𝑣2𝑡2 𝑑Φ0∫∞ 0 𝑑𝑘 4𝜋𝑔(𝑘,𝜃1) |𝑣2𝑘2cos 𝜃1sin 𝜃1|−𝑔(𝑘, 𝜃2) |𝑣2𝑘2cos 𝜃2sin 𝜃2|(4.63) Figure 4.7: Induced electrostatic potential along the direction of propagation of the moving particle given by equation (4.63). The parameters are: an electronic density of 105𝜇𝑚−2which gives a Fermi energy of 𝐸𝐹=0.37eV ,𝑣=0.1𝑐,𝑧0=1𝜇m ,𝑡=0s and for a dielectric thickness of 𝑑=0.01 𝜇m. The plots show the curves for two distinct values of the dielectric constant, 𝜖𝑑=1and 𝜖𝑑=2. Note that in this problem the potential is continuous rather than oscillatory, like in the graphene in the vicinity of a dielectric problem. Contrary to the case of the dielectric, the potential of graphene in the vicinity of a local metal is not oscillatory, but rather presents a peak and then goes from zero to higher values of 𝑟. This means that the metal breaks the oscillatory behavior of the potential, as seen in figure 4.7. However, this happens because the dispersion relation of the SPPs plays an important role in the potential’s behavior. Let us recall the dispersion for graphene in the vicinity of a dielectric (fig. 4.2), in this case, the spectrum is proportional to √𝑘and the potential is oscillatory, but when we add metal to the configuration, the dispersion changes to a behavior proportional to 𝑘and the potential ceases to present an oscillatory nature. Moreover, when we increase the thickness 𝑑of the dielectric, the dispersion tends to √𝑘and the oscillations in the graphene surface are recovered. This argument was also confirmed by [25] where the differences of waves propagating on the surface of the water are directly related to the dispersion of those waves in the media. The transversal and longitudinal cuts for the induced potential are represented in figure 4.8. The longitudinal representation shows us that the presence of the metal altogether with a small enough separation between the metal and graphene, suffices for the potential to behave more continuously, rather than 59 CHAPTER 4. HYDRODYNAMIC MODEL IN THE PRESENCE OF EXTERNAL POTENTIALS Figure 4.8: Induced potential in graphene given by equation (4.63). The transversal cut (left) was done for 𝑦0=−15 𝜇m and the longitudinal cut (right) for 𝑥0=0𝜇m. In this case, the longitudinal representation shows the break of the oscillatory behavior. The potentials were obtained with the parameters: 𝐸𝐹= 0.17 eV, 𝑣=0.1𝑐,𝑧0=1𝜇m and for a dielectric thickness of 𝑑=0.01 𝜇m with a dielectric constant of 𝜖𝑑=1. The metal chosen was titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.20. present an oscillatory nature. In appendix 3, we show that the integrals in equation (4.63) have approximately the same numerical value and we can use only the second integral multiplied by a factor of two to compute the induced potential. This becomes necessary, because the second integral is computationally faster to evaluate than the first integral. 60 Chapter 5 Plasmonic Kelvin and Mach wakes In the previous chapter, we deduced the expressions for the induced potential in graphene created by an external particle moving parallel to it, with constant velocity 𝑣(along the y-axis). The presence of this external particle will give rise to plasmonic wakes on the surface of graphene. Our aim for this chapter is to study the shape and the proprieties of this type of wakes. In [17] the same study was conducted for graphene in vacuum, and it was uncovered that the external particle creates a V-shape pattern on the surface of graphene. By that it is possible to make an analogy with the waves created on the surface of a liquid, having been disturbed by a moving object at a uniform pace (like the external particle for graphene). The half-angle that encloses this V-shape wake is commonly known to be of arcsin(1/3)=19.47◦, which means that the angle of the wake remains constant, at any speed of the moving object. Since this angle was first proved by Lord Kelvin, this type of waves are known as Kelvin waves. However, recent studies have suggested that the angle decreases with the increase of the velocity of the moving perturbation. The measure of the angles for different velocities of the moving perturbation will let us know, where the Kelvin region ends and where another one begins. It was found that for higher velocities the angle decreases with the law 1/𝑣, which is indicative of the Mach type waves [25–28]. We are now interested to see, if these studies for the classical Kelvin and Mach waves can be transferred to the plasmonic wakes created on the surface of graphene, for the configurations of the previous chapter. Furthermore, we will study the half-angle of the V-shaped potential, in order to make a parallelism between the Kelvin and Mach wakes in graphene. Firstly, within this chapter, we will discuss about classical Kelvin and Mach waves, so as to have a starting point in the study of this type of wakes in graphene. In particular, the plasmonic wakes will be analyzed in two cases: graphene in the vicinity of a dielectric and graphene in the vicinity of a local metal (the same configurations in chapter 4). 61 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES 5.1 Classical Kelvin and Mach wakes Let us consider a plane-free surface of steady liquid in equilibrium and in a gravitational field. This field is characterized by the gravitational field strength 𝑔. To the surface of the liquid, a moving external disturbance can be applied in order to disturb the state of equilibrium of the liquid at a given point. The effects of this perturbation will then extend to the entire surface and a specific pattern will appear on the surface of the liquid. This pattern depends on the shape and on the motion of the object, which is perturbing the surface. Since we wish to study the parallel motion of an external perturbation in graphene, it is quite clear that we have to analyze and understand the parallel motion on the surface of a liquid, where the moving object has a rectilinear motion. For simplicity, let us consider the surface of the water. When a perturbation is applied, this will create water wakes, which will propagate through all the free surface. Various studies describe this type of wakes by looking at the waves produced by a moving ship or by a duck swimming in a pond with velocity 𝑣[28]. It is not only interesting to study this physical phenomenon in water, but it is also a fairly suitable method to probe the surface of soft systems, where some usual techniques, such as electron microscopy, impose some constraints. In the paper [29] it is precisely shown that, a moving tip with speed 𝑣parallel to the surface of the soft system will induce a local distortion, which will dissipate in the liquid surface through the form of waves. The study was also made for a point-like particle, such as an electron or a proton moving at a nanoscale distance from the surface, similar to our plasmonic problem [30]. A moving ship with velocity 𝑣traveling along a calm water surface creates a V-shaped wave moving in an opposite direction from that of the moving ship. These waves have some properties that make them a very interesting object of study. One of the theories that explains the propagation and creation of this type of wake pattern is the Kelvin theory, which studies Kelvin type waves. However, some studies suggest that this theory only works for small velocities of the ship. This implies that as the velocity increases, the waves cannot be described by the Kelvin theory but by another one called Mach theory, which studies Mach type of waves. We will now analyze these two theories in order to understand when we can use them and how the waves are affected by the velocity of the moving perturbation. Note that to create these types of waves we could choose a different perturbation other than the ship, as long as the perturbation was able to create waves on the surface of the water. The majority of the studies use the waves created by ships because it is easy to obtain real satellite images (see figure 5.1) from these waves as well as distinguish the different structures in the ship wakes depending on different velocities of the ship. Furthermore, we will define an important parameter that describes the drag (also related to the velocity) of the wave. This parameter is the Froude number, which is adimensional 𝐹=𝑣 p𝑔𝐿 (5.1) where 𝑔is the gravitational acceleration and 𝐿is the hull length of the ship. The Froude number will help us interpret the moment when the transition between a Kelvin wave and the Mach wave occurs, for what 62 5.1. CLASSICAL KELVIN AND MACH WAKES Froude number, and consequently for what velocity the waves change their behavior. A Kelvin wave pattern is composed of two types of waves: transverse waves and divergent waves. Transverse waves are waves that appear in the middle of the ship’s wake and are aligned perpendicular to the ship’s course, following its crests. Divergent waves belong in the outer region, on the sides of the wave pattern, where the wave behaves differently from those in the middle region (figure 5.1 right). It was observed in [28] that the amplitudes of the transverse and divergent waves depend directly on the Froude number (5.1). This implies that, as the Froude number increases (increasing velocities), the amplitude of the divergent wave dominates the wake pattern, and the amplitude of the transverse wave becomes increasingly smaller, looking like it vanishes for higher Froude numbers. Therefore, for high Froude numbers, we are expecting a wave pattern dominated by divergent waves. Another propriety of this type of wave is the wake angle, which is the opening angle of the wake cone. Kelvin theory suggests that the half-angle that encloses the Kelvin wave is constant, independent of the velocity of the ship (or any other perturbation), and takes the value 𝜃𝐾=arctan 1 √8≈19.47◦(5.2) Figure 5.1: (Left) Real image of ship wake created on the surface of the water by a ship traveling with velocity 𝑣.𝛼is the wake angle defined from the slope of the dashed yellow line, which gives the maximum amplitude of the wake. The wake angle is half of the angle of the wave produced by the ship for a Froude number 𝐹=0.15. Figure adapted from [27]. (Right) Wave pattern of a moving ship at 𝐹=0.26. There are some wave structures represented in the figure, such as the wave crest and the wave trough that correspond to the maximum and minimum amplitudes of the wave, respectively, and the wavelength 𝐿𝑤. It is also possible to see that half of the cone angle is the Kelvin angle of about 19.47◦(total angle of 38.93◦). Figure retrieved from [31]. However, some recent observations and calculations have shown that for fast-moving ships, the angle of the wave decreases with velocity [27]. The decrease in the angle can be explained by assuming that a ship cannot generate a wavelength greater than its hull length. However, this hypothesis is still not well established and is open to questioning. Another explanation for the decrease of the wake angle resides in the fact that the wave pattern is defined by the location of its highest peaks, which means that for 63 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES fast-moving ships, the highest peaks don’t lie on the outermost divergent waves anymore, resulting in a smaller apparent angle. This also happens, because when the Froude number increases, so does the velocity and the ship enters the planning regime and moves away less mass of water, which results in a smaller apparent angle. In figure 5.2, various waves produced by the ship for different Froude numbers are represented. We see that for higher Froude numbers the highest peaks of the wakes no longer lie on the outermost divergent waves, which means that the angle of the ship wakes decreases as the velocity of the ship increases. Figure 5.2: Plan view of the ship wakes produced for different Froude numbers. The solid red dots represent the highest peaks, and the black line is the best fit for those points. Clearly, for higher Froude numbers, the highest peaks do not lie anymore in the outermost divergent waves. For an easy comparison, the figures are scaled as ˜𝑦=𝑦/𝐹2and ˜𝑥=𝑥/𝐹2. This figure was retrieved from [26]. This clearly demonstrates a transition from the Kelvin regime to a different regime. In figure 5.3 we see that for higher velocities, the angle of the cone follows the law 𝐹−1(or 1/𝑣) which is consistent with the Mach regime, where the wake angles follow this specific law. For smaller Froude numbers, the Kelvin regime behavior is clearly perceived, as the wake angles are constant and their numerical value is very close to the Kelvin angle (accounting for the error, the numerical angle interval is within the Kelvin angle). Therefore, it is clear that a transition between these two regimes mirrors a change of behavior for a specific 64 5.1. CLASSICAL KELVIN AND MACH WAKES Froude number. From figure 5.3 the Froude numbers in the range 0.8<𝐹<3are in the Kelvin region, where the angles are constant and slightly below the Kelvin angle. For Froude numbers bigger than 4there is a transition to the Mach region where the angles decrease by the law 𝐹−1. This is consistent with figure 5.2 (c) and (d), where half of the cone angle decreases with the increase of the Froude number. Figure 5.3: Apparent wake angles 𝜃𝑎𝑝𝑝 in degrees as a function of the Froude number. The blue line is the best fit to the angles for the Mach region as it follows the law 𝐹−1. The black dashed line is Kelvin’s angle 19.47◦. The green dot-dashed curve is the root mean squared error, scaled by 𝐹2. Figure retrieved from [26]. It is important to notice that the description of such waves presented in this chapter was assumed to be linear and the perturbation on the water was created by a single source (in this case a ship). However, in [26] is also studied the problem for nonlinear solutions as well for other kinds of sources, like source doublets. In the nonlinear regime, the apparent wake angle increase and becomes greater than the Kelvin angle. In the following sections of this chapter, we will provide a mathematical description for this type of wakes, starting with the study of the dispersion of water waves and then proceeding to get an expression for the surface displacement brought by the moving ship. 5.1.1 Dispersion of water waves The complexity of studying the formation and the dispersion of water waves, in particular those caused by a moving disturbance, is well known, due to external and internal factors. Such factors, like the surface tension and the depth of the water, can lead to complicating dispersion for the waves we wish to study. Not to mention that the study of ships in open air can be rather difficult due to winds and perturbations 65 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES of the water, which can cause turbulence and, in that way, interfere with the wake pattern produced by a moving ship. However, let us consider the perfect conditions for the observation of these wakes. In that case, the dispersion relation between angular frequency and the wave number 𝑘is given by [25, 32] 𝜔2(𝑘)=𝑔𝑘 +𝜎𝑘3 𝜌tanh(𝑘ℎ)(5.3) where 𝑔is the gravitational field strength, 𝜎the surface tension, 𝜌the density of the water, and ℎthe water depth. Equation (5.3) provides us with waves of different characteristics, regarding the parameters chosen: if the surface tension is null on the interface air-water (𝜎=0), we are in the presence of pure gravity waves. While for 𝜎≠0the capillary gravity waves dominate the wave pattern. For pure gravity waves short compared to the depth of the water, that is ℎ→ ∞, we get the limit limℎ→∞ tanh(𝑘ℎ)=1and the dispersion relation (5.3) reduces to 𝜔2(𝑘)=𝑔𝑘 (5.4) The phase velocity can now be taken through 𝑣𝑝=𝜔/𝑘, which gives 𝑣𝑝=r𝑔 𝑘=𝑔 𝜔=𝜔 𝑔−1 (5.5) This is a lot similar to the phase velocity of plasmons in graphene in the vicinity of a dielectric (section 4.2), obtained in equation (4.37). That is why we consider that plasmons behave as gravity waves in graphene with acceleration given by 𝑎=4𝛼𝐹𝑆 𝐸𝐹ℏ𝑐 (1+𝜖𝑑)ℏ2. Where 𝛼𝐹𝑆 is the fine structure constant, 𝐸𝐹is the Fermi energy, and 𝜖𝑑is the dielectric constant. For pure gravity waves with wavelengths greater than the water depth, we have tanh(𝑘ℎ) ≈ 𝑘ℎ, we get for the dispersion relation 𝜔2=𝑔ℎ𝑘2(5.6) and the phase velocity 𝑣𝑝is given by 𝑣𝑝=p𝑔ℎ =p𝑔ℎ𝜔0(5.7) Once again, we can make an analogy with the phase velocity of graphene in the vicinity of a local metal [equation (4.51)]. Where the phase velocity is equal to the group velocity 𝑣𝑔=𝜕𝜔/𝜕𝑘. In the problem described in the last chapter, the phase velocity was given by 𝑣0, which we obtained as the slope of the plasmonic spectrum in figure 4.5. The slope can be taken as 𝜕𝜔/𝜕𝑘 so we infer that the phase velocity is equal to the group velocity for gravity waves in shallow water depth and for the graphene in the vicinity of a local metal. For gravity waves of any kind, the waves created are V-shaped waves that stretch behind the moving object. With this study, we can already see, that this kind of wake pattern will appear on the surface of graphene when an external particle is moving parallel to it, as we will analyze in the next section. For capillary gravity waves (𝜎≠0) in deep water, the dispersion (5.3) gets the form 66 5.1. CLASSICAL KELVIN AND MACH WAKES 𝜔2=𝜎𝑘3 𝜌(5.8) where we neglect 𝑔𝑘 in comparison to the capillary term. Of course, this gives a phase velocity of 𝑣𝑝=s𝜎𝑘 𝜌=𝜔𝜎 𝜌1/3 (5.9) If we define the limiting cases for 𝜅=0,−1,1/3then, the general phase velocity for water waves can be written as 𝑣𝑝=𝑣0𝜔 𝜔0𝜅 (5.10) where 𝑣0and 𝜔0are constants: for pure gravity waves in deep water (𝜅=−1) we get 𝑣0𝜔0=𝑔, for gravity waves in shallow water (𝜅=0) the constants are 𝑣0=p𝑔ℎ and finally, for capillary waves (𝜅=1/3) the phase velocity gives 𝑣3 0/𝜔0=𝜎/𝜌. The authors of [25] also found an expression to calculate the Kelvin angle, which is based on the geometry of the wakes and the group velocity. For the wake half-angle, it was discovered 𝜃𝐾=arctan 1 2r1 𝜅(𝜅−1)(5.11) What this expression tells us, is that the kelvin angle depends on the type of wake that we are studying. For 𝜅=−1the angle is of 𝜃𝐾=19.47◦, which is the angle found by Kelvin. In opposite, for 𝜅=0the Kelvin angle occurs at 𝜃𝐾=90◦. However, for this last limit, the pattern wakes only develops at longer times and becomes less evident, further losing its meaning. The dispersion relation for the limiting cases (𝜅=0,−1,1/3), altogether with the exact solution [eq. (5.3)] can be seen in figure 5.4. It shows us that for pure gravity waves in infinite water depth (𝜅=−1), the spectrum follows the law √𝑘, just like graphene in the vicinity of a dielectric. For gravity waves in shallow water (𝜅=0), the waves are linear in the wavenumber, similar to when we add a local metal to a graphene configuration. This tells us we can make an analogy between these waves and the plasmonic cases studied in chapter 4. 5.1.2 Surface displacement A mathematical approach is necessary to have a better understanding of the V-shape wave pattern generated by a moving perturbation. For that, we will determine the shape of the surface displacement generated by a pressure field 𝑝(𝑥,𝑦). Let us also consider a moving perturbation traveling in 𝑥-direction with velocity 𝑣. The free equilibrium surface is the plane 𝑥𝑦 and the z-axis is oriented perpendicular to the surface. We want to find the displacement of the free 𝑥𝑦 surface, 𝑧=𝜁(𝑥,𝑦, 𝑡), from the equilibrium position. Assuming an irrotational fluid motion [29], the velocity can be expressed as a velocity potential 67 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES region, the angle of the plasmonic wake pattern is almost constant and with a value of 𝜃𝑎𝑝𝑝 =21◦(grey line), which is slightly above the Kelvin angle of 𝜃𝐾=19.47◦(blue line) produced in water waves. For 𝐹𝑝𝑙 ≫1the wake angle starts to change behavior and the angles start to decrease by following the law 1/𝐹𝑝𝑙 , which indicates a Mach-type region. In that order, we study the angles by producing a fitting function of the type 𝜃𝑎𝑝𝑝 ≈1 𝛿r𝑧0𝑎 𝑣2=1 𝛿 1 𝐹𝑝𝑙 (5.28) that gives the apparent wake angles as the Froude number increases. A transition from the Kelvin region to the Mach region is clear at 𝐹𝑝𝑙 ∼2.2(red line), where 𝛿is a real constant that was found to be the order of 𝛿=0.020 for 𝜖𝑑=1(vacuum). These results are consistent with the study of the formation of plasmonic wakes in graphene provided by [17]. 5.2.2 Graphene in the vicinity of a local metal For graphene in the vicinity of a local metal (configuration 4.4) the oscillatory behavior of the plasmons in graphene vanishes and a rather continuous nature will become more distinct. We already studied this phenomenon in chapter 4.3. Equation (4.63) gave the results in figure 4.7, which precisely shows the continuous nature of the potential. In a similar manner to the problem of graphene in the vicinity of a dielectric we show, in figure 5.8, a plan view of the plasmonic wakes in graphene in the presence of a local metal for different dielectric thickness 𝑑and different dielectric constants 𝜖𝑑. From the density plots, it is possible to infer the cease of the oscillatory behavior of the potential in graphene but is replaced by a somewhat continuous behavior in the cone. Note that in this case there are not any transverse and divergent waves in the pattern. All these differences come from the presence of the metal, which is chosen to be titanium. If the distance between the metal and the graphene sheet (in other words, the dielectric thickness 𝑑) increases, it is expected the appearance of the oscillatory nature of the plasmons in graphene, due to the weak effect that the metal will provide to the configuration. However, for the dielectric thickness chosen (𝑑=0.01 𝜇m and 𝑑=0.1𝜇m)the effects of the metal are quite evident as seen in figure 5.8. We also observe that the wake angle decreases with the increase of the dielectric thickness 𝜖𝑑, just like in the case of graphene in the vicinity of a dielectric. The dielectric thickness 𝑑, also plays an important role in the variation of the wake angle, in such a way that for 𝑑=0.01 𝜇m the cone angle is smaller than for 𝑑=0.1𝜇m. This means that the wake angle is greater for a higher separation between the metal and graphene. To study the V-shaped pattern of the potential created in graphene, in figure 5.9, it is shown the wake pattern for two different velocities of the external particle. Similar to graphene near a dielectric, the wake angle decreases with the increase of the velocity of the particle (and with the increase of the Froude number). However, the continuous nature of the potential remains, even for higher particle speeds. It remains to be seen if, in such case, we can depict a Kelvin and Mach region by measuring the half angle 74 5.2. KELVIN AND MACH WAKES IN GRAPHENE of the V-shaped pattern. In tables 2 and 3, the apparent wake angles for different velocities, and an external particle height of 𝑧0=1𝜇m and 𝑧0=0.1𝜇m are respectively presented. Figure 5.8: Wake pattern of the induced electrostatic potential, in units of Φ0, given by equation (4.63), when graphene is in the vicinity of a local metal. The top figures show the density plots for a distance between metal and graphene of 𝑑=0.01 𝜇m. The bottom ones represent the potential created in graphene for a dielectric thickness of 𝑑=0.1𝜇m. It is possible to understand the continuous nature of the potential in all plots and the nonexistence of transverse and divergent waves. The wake angles are smaller for a higher dielectric constant 𝜖𝑑and a smaller separation of metal/graphene 𝑑. The velocity of the moving particle is 𝑣=0.1𝑐, the Fermi energy is of 𝐸𝐹=0.17 eV and the external particle is moving at a height of 𝑧0=1𝜇m above graphene. The chosen metal was titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.2. In figure 5.10 is represented how the half wake angle of the potential cone created by an external particle moving parallel to graphene at a distance 𝑧0=1𝜇m from it, changes with the plasmonic Froude number. Contrary to the dielectric case, there is not a Kelvin region and a Mach region visible in the whole domain. However, following an inverse quadratic law where the best fit to the angle points has to do the law 1/𝛿0+1/𝛿1𝐹𝑝𝑙 +1/𝛿2𝐹2 𝑝𝑙 , or in other words, the half angle of the wake pattern in graphene near a metal can be taken as 𝜃𝑎𝑝𝑝 ≈3.87 +9.92 𝐹𝑝𝑙 +0.77 𝐹2 𝑝𝑙 (5.29) 75 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES where 𝛿0=0.26,𝛿1=0.10 and 𝛿2=1.30. In this case there is no transition for a different angle behavior, since there are no Kelvin and Mach regions. Also, from equation (5.11) the Kelvin angle was expected to be at 𝜃𝐾=𝜋/2, because in this case 𝜅=0, which means that the angle increases for lower Froude numbers. Of course, for lower Froude numbers the linear and quadratic terms have a bigger contribution to the angle and as such their value increases. For higher Froude numbers the quadratic term has a small contribution to the angle and equation (5.29) is dominated by linear and the constant terms. This behavior was also perceived in classical water waves in [35]. In this study, a disturbance with a certain length 𝐿and a width 𝐵was considered, as for axisymmetric disturbances the aspect ratio parameter 𝐴=𝐿/𝐵must be one. The Kelvin and Mach regions are reached, which means the angle starts to behave consistently, and for higher speeds it scales as 𝜃𝑎𝑝𝑝 ∝𝐹−1 𝑝𝑙 . In the case where 𝐴≫1, we find ourselves in the presence of nonaxisymmetric disturbances and angle scales as 𝜃𝑎𝑝𝑝 ∝𝐹−2 𝑝𝑙 , which show the quadratic nature of some water wakes. For figure 5.11 the apparent angles of the wake pattern in graphene for a particle traveling at a height 𝑧0=0.1𝜇m above graphene are drawn. In this case, the whole domain can be described by the law 1/𝑣[equation (5.28]. Once again, the Kelvin region ceases to exist and the angles are contained within the Mach region, where the angle is given, once again, by equation (5.28), with 𝛿=0.027. As the velocity of the external particle increases, and consequently the Froude number, the apparent wake angle 𝜃𝑎𝑝𝑝 decreases, such as in the dielectric case. Figure 5.9: Wake pattern of the induced electrostatic potential, in units of Φ0, given by equation (4.63) for different Froude numbers: 𝐹𝑝𝑙 =2.7(left) and 𝐹𝑝𝑙 =4.5(right), which corresponds to an external particle velocity (𝑧0=1𝜇m) of 𝑣=0.3𝑐and 𝑣=0.5𝑐, respectively. Qualitatively it is possible to see a decrease in the wake angle for higher Froude numbers. The parameters are: Fermi energy of 𝐸𝐹=0.17 eV, dielectric thickness of 𝑑=0.01 𝜇m in vacuum (𝜖𝑑=1). The chosen metal was titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.2. 76 5.3. CLASSICAL WAKES AND PLASMONIC WAKES Table 2: Apparent half wake angle 𝜃𝑎𝑝𝑝 (in degrees) of the cone produced by the moving plasmons in graphene induced by an external particle with parallel motion at a height 𝑧0=1𝜇m. The Fermi energy of the electron gas is of 𝐸𝐹=0.17 eV and the dielectric thickness of 𝑑=0.01 𝜇m for vacuum (𝜖𝑑=1). The last line of the table gives the Froude number for this problem obtained to the expression 𝐹𝑝𝑙 =p𝑣2/𝑧0𝑎. 𝑣/𝑐0.04 0.055 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 𝜃𝑎𝑝𝑝 37.3 27.1 21.8 19.4 17.5 16.0 9.2 7.2 6.6 6.1 5.8 5.6 5.5 5.4 𝐹𝑝𝑙 0.36 0.49 0.63 0.72 0.81 0.89 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 0.5 1 2 5 5 10 20 50 Figure 5.10: Graphical representation for the apparent angle 𝜃𝑎𝑝𝑝, given by table 2, of the plasmonic cone for graphene in the vicinity of a local metal for 𝑧0=1𝜇m. In this case, there is no Kelvin or Mach region in the whole domain. The angles follow a quadratic law given by 1/𝛿0+1/𝛿1𝐹𝑝𝑙 +1/𝛿2𝐹2 𝑝𝑙 , with 𝛿0=0.26, 𝛿1=0.10 and 𝛿2=1.30. The parameters considered are: 𝐸𝐹=0.17 eV, 𝑑=0.01 𝜇m and 𝜖𝑑=1. For titanium: 𝜔𝑝=2.80eV and 𝜖∞=2.20. Table 3: Apparent half wake angle 𝜃𝑎𝑝𝑝 (in degrees) of the cone produced by the moving plasmons in graphene induced by an external particle with parallel motion at a height 𝑧0=0.1𝜇m. The Fermi energy of the electron gas is of𝑒𝐹=0.17 eV and the dielectric thickness of𝑑=0.01 𝜇m for vacuum (𝜖𝑑=1). The last line of the table gives the Froude number for this problem obtained to the expression 𝐹𝑝𝑙 =p𝑣2/𝑧0𝑎. 𝑣/𝑐0.03 0.05 0.07 0.09 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 𝜃𝑎𝑝𝑝 44.8 26.6 19.1 14.5 13.0 8.6 6.4 4.3 3.2 2.6 2.0 1.8 1.6 𝐹𝑝𝑙 0.84 1.4 2.0 2.5 2.8 4.2 5.6 8.4 11.3 14.0 16.9 19.7 22.5 5.3 Classical Wakes and Plasmonic Wakes To finalize the study of the plasmonic wakes in graphene it is necessary to analyze the V-shape pattern and compare it to the wakes produced in water. Even more, we proceed to see the major similarities and differences between the classical and plasmonic wakes. This will allow us to understand the influence of 77 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES 1 2 5 10 20 2 5 10 20 50 Figure 5.11: Graphical representation for the apparent angle 𝜃𝑎𝑝𝑝, given by table 3, of the plasmonic cone for graphene in the vicinity of a local metal for 𝑧0=0.1𝜇m. In this case, the only theory that describes the angle behavior is the Mach theory, 𝜃=1/(𝛿𝐹𝑝𝑙 ), where 𝛿=0.027 . The parameters considered are: 𝐸𝐹=0.17 eV, 𝑑=0.01 𝜇m and 𝜖𝑑=1. For titanium: 𝜔𝑝=2.80 eV and 𝜖∞=2.20. the dispersion relation and the problem parameters in the shape of this type of wakes. In figure 5.12, it is shown a 3D representation for the problems we want to compare. The first one is the classical wakes that originated on the surface of the water by a moving source traveling in the 𝑦-direction. For water waves, the parameters in question are the gravity acceleration, 𝑔, the height of the water, ℎ, and the velocity of the source producing the waves. The plasmonic configurations are the same as the ones studied in chapter 4. For graphene in the vicinity of a semi-infinite dielectric (configuration (b)), we have the acceleration of the electron gas 𝑎=4𝛼𝐹𝑆 𝐸𝐹ℏ𝑐 (1+𝜖𝑑)ℏ2, in graphene. The moving charge is placed at a height 𝑧0and moves with velocity 𝑣𝑦. When graphene is in the vicinity of a local metal (configuration (c)) we add an extra parameter to the problem, which is the dielectric thickness 𝑑. In section 5.1.1 we study the dispersion relation of water waves, where it was shown that the phase velocity is related to the angular frequency as 𝑣𝑝∝𝜔𝜅. For pure gravity waves in deep water the dispersion is proportional to √𝑘[equation 5.4], which means that the phase velocity is proportional to the inverse of the angular frequency, 𝑣𝑝∝𝜔−1, which means that 𝜅=−1. This provides an analogy with the electrostatic problem of graphene in the vicinity of a dielectric, where the dispersion relation is given by equation (4.24) and like pure gravity waves the dispersion is proportional to √𝑘and the phase velocity is proportional to the inverse of𝜔. As such, we can think of the plasmonic wake pattern as being similar to the wake pattern of pure gravity waves in deep water. Let us consider figure 5.13, where the calculated wakes for pure gravity waves and the plasmonic wakes for graphene in the vicinity of a dielectric are calculated. Both waves have the similarity of having 𝜅=−1, which in other words, means that both wakes follow the law 𝑣𝑝∝𝜔−1. The left 3D density plots show us the wakes created by an external particle moving parallel 78 5.3. CLASSICAL WAKES AND PLASMONIC WAKES Figure 5.12: Representation of the configurations in the classical and plasmonic problems. In (a) the figure shows a perturbation moving in the y-direction in the water, which has a height ℎregarding the bottom land. The plasmonic configuration (b) represents graphene in the vicinity of a dielectric, with an external particle moving with velocity 𝑣𝑦at a height 𝑧0from the graphene sheet. Configuration (c) is characterized by graphene in the vicinity of a local metal with a dielectric separation of thickness 𝑑. Once again, we have an external charge moving with velocity 𝑣𝑦at a height 𝑧0from graphene. to graphene at a height 𝑧0=0.1𝜇m, and the right density plots are the calculated pattern for pure gravity waves in deep water [25]. In the top panel, the wakes are generated by disturbances moving slower than those from the bottom panels. The clear similarities are given by the decrease of the V-shape pattern angle as the velocity increases and there are present transverse and divergent waves in both cases. This indicates that once we have graphene in the vicinity of a dielectric, an analogy can be made with pure gravity waves in shallow water. Here the semi-infinite dielectric plays the part of the deep water, where ℎ→ ∞. Let us now take a look at when graphene is in the vicinity of a local metal. In this case, the dispersion is linear in the wavenumber 𝑘, which means that the phase velocity is constant and equal to the group velocity of the waves. The same happens for gravity waves in shallow water [equation (5.6)], that is, for gravity waves with wavelengths greater than the water depth. So when the dispersion of the waves is linear in 𝑘, then this limiting case is characterized by having 𝜅=0, for both problems. In figure 5.14, the classical wake pattern for 𝜅close to zero and the analogous problem for the plasmonic wakes are shown. The classical wakes were computed with 𝜅=−0.1. This choice is due to the impossibility of having 𝜅=0. These type of waves always show features for 𝜅>0, because very short capillary waves cannot be avoided even when reducing the surface tension 𝜎to low values. However, this value of 𝜅is very close to 79 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES Figure 5.13: Wake pattern for the plasmonic (left) and classical (right) cases. Both patterns show similar structures, typical of the limiting case 𝜅=−1, which are the oscillatory behavior and the presence of transverse and divergent waves. The bottom figures show disturbances speeds higher than the top ones, which bring a decrease on the wake angle. The case of graphene in the vicinity of a dielectric is similar to water wakes in deep water. For the plasmonic case the parameters are: 𝐸𝐹=0.17 eV and 𝑧0=0.1𝜇m. Right figures retrieved from [25]. zero and will allow us to study the wakes in this regime. This means that we have a parameter 0<𝛿≪1 which changes the dispersion relation to 𝜔∝𝑘1−𝛿and, consequently the phase velocity takes the form 𝑣𝑝=𝜔/𝑘∝𝑘−𝛿. In this case 𝛿=−𝜅=0.1. For the plasmonic case, we are expecting the same to happen. However, let us study two different regimes: the first regime for 𝑑≪𝑧0corresponds to the case of shallow water, while the second regime 𝑑≫𝑧0corresponds to the deep water case. This is the same as saying that when the dielectric separation between metal and graphene increases then the influence of the metal in the system begins to decrease and we recover the oscillatory behavior of the potential and consequently the limiting case of 𝜅=−1, which is analogous to the case of pure gravity waves in deep water. The similarities between the two wake patterns are visible in the continuous wake that the perturbations create on the surface of the water and on the surface of graphene, which means that when 80 5.3. CLASSICAL WAKES AND PLASMONIC WAKES graphene is in the vicinity of a local metal and 𝑑≪𝑧0the dispersion is linear in the wavenumber and the plasmonic configuration can be seen as waves in shallow water. In the classical case the depth is controlled by the height, ℎof the water, while for the plasmonic case what controls the shallow nature is the dielectric thickness, 𝑑, and the distance of the external particle and graphene, 𝑧0. For the regime of 𝑑≫𝑧0we retrieve the oscillatory plasmonic shape in graphene, which implies the same behavior as pure gravity waves in deep water, as seen in figure 5.15 where we compare the two different regimes. Figure 5.14: Graphical representation of the wake pattern in graphene near a metal (left) and the water wake pattern (right) for the limiting case 𝜅=−0.1. This corresponds to water wakes in shallow water. Both shapes show us a continuous behavior in the central cone. In the water wake there are waves forming in the outer region of the pattern which is analogous to the bluest region in the plasmonic wake. For the plasmonic wake the parameters are: 𝑣=0.1𝑐,𝐸𝐹=0.17 eV, 𝑑=0.01 𝜇m and 𝑧0=1𝜇m. The chosen metal was titanium. The right figure retrieved from [25]. From this chapter, a connection between the shape pattern of classical water waves and the plasmonic wakes propagating in graphene and created by an external charge moving parallel to it is visible. This reinforces the argument that the dispersion of the water waves and the surface plasmon-polaritons in graphene play an essential role in the shape of the wake pattern produced on the surface of the mediums. In such manner, the hydrodynamic model can be seen as a way to understand not only the nonlocal effects in materials at the nanoscale but also provide a connection between the classical and plasmonic world, which can lead to very interesting results, such as the ones obtained in this thesis. Finally, these results can lead to a more profound understanding of the optical and electromagnetic proprieties of plasmonic nanostructures, specifically how nonlocal graphene behave in the presence of external potentials and how we can use these results to boost the search for faster and more compact nanotechnology. 81 CHAPTER 5. PLASMONIC KELVIN AND MACH WAKES Figure 5.15: Graphical representation of the wake pattern in graphene near a metal for 𝑑≪𝑧0(left) and 𝑑≫𝑧0(right). For the limiting case 𝜅=0we are in the shallow water regime, where 𝑑=0.01 𝜇m and 𝑧0=1𝜇m. When we increase the dielectric thickness, there is a regime inversion and the oscillatory behavior is recovered, because we now have 𝑑=1𝜇m and 𝑧0=0.1𝜇m and this case is analogous to water wakes in deep water. The parameters are: 𝑣=0.1𝑐and 𝐸𝐹=0.17 eV. The chosen metal was titanium. 82 C h a p t e r 6 Conclusion The hydrodynamic model is a nonlocal theory that couples the Euler equation with Maxwell’s equations of electromagnetism, and it is constituted by the following equations: Euler’s equation, Poisson equation, and the continuity equation. This model is generalized for an electron gas (regarding the spatial dimension). However, it is possible to consider various geometries composed of metals, semiconductors, and dielectric materials where the model remains valid if supported by specific boundary conditions for each geometry. The main goal of this thesis was to make use the hydrodynamic model to analyze the plasmonic nonlocal effect in graphene, considering an electrostatic approximation for the electron gas. Various planar geometries in the form of slabs oriented perpendicular to the z-axis and infinite in all the other directions were studied. Through a first approach, the nonlocal effects in the spectrum of the surface plasmonpolaritons and on the electrostatic potentials in some problems, such as a semi-infinite and finite metal slab embedded in vacuum were analyzed. For a graphene sheet in vacuum and a 2D electron gas, it was recovered the known dispersion of the SPPs, which are proportional to the square root of the wavenumber 𝑘. Literature was already aware of these problems. The additional problems came with graphene in the vicinity of a nonlocal metal. When graphene is in the vicinity of a semi-infinite metal, the dispersion of the SPPs presents a linear nature, rather than a 𝜔∝√𝑘behavior like graphene in vacuum. As graphene is moved away from the metal, that is, the dielectric separation increases, it was verified that the SPPs show higher energies from a higher dielectric thickness. Therefore, the further graphene is from the metal, the greater the growth in terms of surface plasmon energy. However, when increasing the dielectric separation, the nonlocal metal will start to behave more like a local metal. For graphene near a finite metal with thickness 𝑎, it was seen a linear dispersion for different values of the thickness 𝑎. Moreover, the energy of the plasmons decrease with the decrease of the metal slab thickness, which is a counter intuitive result, because as the metal thickness decrease the spectrum of the SPPs should be the one obtained for graphene near a semi-infinite metal. However, when we take the limit of the background permittivity 83 APPENDIX A. PLASMONIC ELECTROSTATIC CALCULATIONS 𝐷+𝐹=𝐺(A.7) −𝑘𝐺 −𝜖𝑑[−𝑘𝐷 +𝑘𝐹]=𝑒 𝜖0 𝑛2𝐷,1(k, 𝜔)(A.8) Recall that 𝛼is a nonlocal parameter given by equation (3.6). The hydrodynamic model provides an extra boundary conditions, that states the null current of the electron gas at the normal direction of the interface. In mathematical terms this means 𝜖0𝜔2 𝑝 𝑒𝑛0 𝜕𝜙(𝑚𝑒𝑡𝑎𝑙) 1 𝜕𝑧 =𝛽2 𝑛0 𝜕𝑛(𝑚𝑒𝑡𝑎𝑙) 1 𝜕𝑧 (A.9) where 𝜔𝑝is the plasma frequency, 𝑛0the homogeneous electronic density and 𝛽the nonlocal parameter. The first order electronic density 𝑛1is given by equation (3.41). This boundary condition give rise to the expression 𝜔2 𝑝h𝛼𝐵𝑒−𝛼𝑑 +𝑘𝐶𝑒−𝑘𝑑 i+ (𝜔2−𝜔2 𝑝)𝛼𝐵𝑒−𝛼𝑑 =0(A.10) Dividing this last equation by 𝜔2 𝑝we obtain 𝐶=−𝜔2 𝜔2 𝑝 𝛼 𝑘𝑒(𝑘−𝛼)𝑑𝐵(A.11) At last, we derived the expressions that allow us to construct the coefficient matrix 𝑀presented in chapter 3.2. The solution for all coefficients is then given by 𝐵(𝑘, 𝜔)= 2𝜖𝑑𝜔2 𝑝𝑒𝑑(𝑘+𝛼) 𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞ (A.12) 𝐶(𝑘, 𝜔)=−2𝜖𝑑𝛼𝑒2𝑑𝑘𝜔2 𝑘[𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞] (A.13) 𝐷(𝑘, 𝜔)=−𝜖𝑑(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜔2 𝑝−𝜔2)𝜖∞ 𝑘[𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞] (A.14) 𝐹(𝑘, 𝜔)=−𝑒2𝑑𝑘 (𝜖𝑑(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜔2−𝜔2 𝑝)𝜖∞) 𝑘[𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞] (A.15) 90 𝐺(𝑘, 𝜔)=−(1+𝑒2𝑑𝑘)(𝛼𝜔2−𝑘𝜔2 𝑝)𝜖𝑑+𝛼(−1+𝑒2𝑑𝑘)(𝜔2−𝜔2 𝑝)𝜖∞ 𝑘[𝜖𝑑(1−𝜖𝑑+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝛼𝜔2−𝑘𝜔2 𝑝) +𝛼(𝜖𝑑−1+𝑒2𝑑𝑘 (1+𝜖𝑑))(𝜔2−𝜔2 𝑝)𝜖∞] (A.16) Then the potentials in the momentum space take the form 𝜙1(k, 𝑧, 𝜔)=            𝑒 𝜖0𝑛1(k, 𝜔)[𝐵(𝑘, 𝜔)𝑒𝛼𝑧 +𝐶(𝑘, 𝜔)𝑒𝑘𝑧]if 𝑧<−𝑑 𝑒 𝜖0𝑛1(k, 𝜔)[𝐷(𝑘, 𝜔)𝑒−𝑘𝑧 +𝐹(𝑘, 𝜔)𝑒𝑘𝑧]if −𝑑<𝑧<0 𝑒 𝜖0𝑛1(k, 𝜔)[𝐷(𝑘, 𝜔) +𝐹(𝑘, 𝜔)]𝑒−𝑘𝑧 if 𝑧>0 (A.17) Now let us take a look at the problem of graphene in the vicinity of a finite metal of thickness 𝑎, which is given by configuration 3.13. In this case, the boundary condition to apply take the form 𝜙𝑚𝑒𝑡𝑎𝑙 1(−𝑑−𝑎)=𝜙𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 1 1(−𝑑−𝑎)(A.18) 𝜖∞ 𝜕𝜙𝑚𝑒𝑡𝑎𝑙 1(−𝑑−𝑎) 𝜕𝑧 =𝜖𝑑1 𝜕𝜙𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 1 1(−𝑑−𝑎) 𝜕𝑧 (A.19) 𝜙𝑚𝑒𝑡𝑎𝑙 1(−𝑑)=𝜙𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 2 1(−𝑑)(A.20) 𝜖∞ 𝜕𝜙𝑚𝑒𝑡𝑎𝑙 1(−𝑑) 𝜕𝑧 =𝜖𝑑2 𝜕𝜙𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 2 1(−𝑑) 𝜕𝑧 (A.21) 𝜙𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 2 1(0)=𝜙𝑔𝑟𝑎𝑝ℎ𝑒𝑛𝑒 1(0)(A.22) 𝜕𝜙𝑔𝑟𝑎𝑝ℎ𝑒𝑛𝑒 1(0) 𝜕𝑧 −𝜖𝑑2 𝜕𝜙𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 2 1(0) 𝜕𝑧 =𝑒 𝜖0 𝑛1(k, 𝜔)(A.23) The hydrodynamic model provides the boundary condition described in equation (A.9), but in this case it has to be solves in the boundaries at 𝑧=−(𝑑+𝑎)and 𝑧=−𝑑. By using the potentials expressed in equations (3.52), (3.53), (3.54) and (3.55) in these boundary conditions it is found the coefficient expressions 𝐵1cosh(𝛼(𝑑+𝑎))+𝐶1cosh(𝑘(𝑑+𝑎))−𝐵2sinh(𝛼(𝑑+𝑎))−𝐶2sinh(𝑘(𝑑+𝑎)) =𝐷𝑒−𝑘(𝑑+𝑎)(A.24) 𝜖∞[−𝐵1𝛼sinh(𝛼(𝑑+𝑎)) −𝐶1𝑘sinh(𝑘(𝑑+𝑎)) +𝐵2𝛼cosh(𝛼(𝑑+𝑎)) +𝐶2𝑘cosh(𝑘(𝑑+𝑎))] =𝜖𝑑1𝑘𝐷𝑒−𝑘(𝑑+𝑎)(A.25) 91 APPENDIX A. PLASMONIC ELECTROSTATIC CALCULATIONS 𝐵1cosh(𝛼𝑑) +𝐶1cosh(𝑘𝑑) −𝐵2sinh(𝛼𝑑) −𝐶2sinh(𝑘𝑑)=𝐹cosh(𝑘𝑑) −𝐸sinh(𝑘𝑑)(A.26) 𝜖∞[−𝐵1𝛼sinh(𝛼𝑑) −𝐶1𝑘sinh(𝑘𝑑) +𝐵2𝛼cosh(𝛼𝑑) +𝐶2𝑘cosh(𝑘𝑑)] =𝜖𝑑2(−𝐹𝑘 sinh(𝑘𝑑) +𝐸𝑘 cosh(𝑘𝑑)) (A.27) 𝐹=𝐺(A.28) −𝐺𝑘 −𝜖𝑑2𝐸𝑘 =𝑒 𝜖0 𝑛2𝐷,1(k, 𝜔)(A.29) 𝜔2 𝜔2 𝑝 𝛼[𝐵2cosh(𝛼(𝑑+𝑎)) −𝐵1sinh(𝛼(𝑑+𝑎))] +𝑘[𝐶2cosh(𝑘(𝑑+𝑎)) −𝐶1sinh(𝑘(𝑑+𝑎))]=0(A.30) 𝜔2 𝜔2 𝑝 𝛼[𝐵2cosh(𝛼𝑑) −𝐵1sinh(𝛼𝑑)]+𝑘[𝐶2cosh(𝑘𝑑) −𝐶1sinh(𝑘𝑑)]=0(A.31) In a similar way as graphene in the vicinity of a semi-infinite metal, due to the complexity of the latter linear equation, it is necessary to solve the linear system of equations through matricial methods. In that order, we solve equation 𝑀·𝑉=𝑁to obtain the coefficient matrix 𝑉=[𝐵1, 𝐵2,𝐶1,𝐶2, 𝐷, 𝐸, 𝐹,𝐺], with 𝑁 the matrix of the free-coefficient terms 𝑁=[0,0,0,0,0,(𝑒/𝜖0)𝑛1(k, 𝜔),0,0]. The matrix M is given by 𝑀=  cosh(𝛼(𝑑+𝑎)) −sinh(𝛼(𝑑+𝑎)) cosh(𝑘(𝑑+𝑎)) −𝜖∞𝛼sinh(𝛼(𝑑+𝑎)) 𝜖∞𝛼cosh(𝛼(𝑑+𝑎)) −𝜖∞𝑘sinh(𝑘(𝑑+𝑎)) cosh(𝛼𝑑) −sinh(𝛼𝑑)cosh(𝑘𝑑) −𝜖∞𝛼sinh(𝛼𝑑)𝜖∞𝛼cosh(𝛼𝑑) −𝜖∞𝑘sinh(𝑘𝑑) 0 0 0 0 0 0 −𝜔2 𝜔2 𝑝𝛼sinh(𝛼(𝑑+𝑎)) 𝜔2 𝜔2 𝑝𝛼cosh(𝛼(𝑑+𝑎)) −𝑘sinh(𝑘(𝑑+𝑎)) −𝜔2 𝜔2 𝑝𝛼sinh(𝛼𝑑)𝜔2 𝜔2 𝑝𝛼cosh(𝛼𝑑) −𝑘sinh(𝑘𝑑) . . . 92 . . . −sinh(𝑘(𝑑+𝑎)) −𝑒−𝑘(𝑑+𝑎)0 0 0 𝜖∞𝑘cosh(𝑘(𝑑+𝑎)) −𝜖𝑑1𝑘𝑒−𝑘(𝑑+𝑎)0 0 0 −sinh(𝑘𝑑)0 sinh(𝑘𝑑) −cosh(𝑘𝑑)0 𝜖∞𝑘cosh(𝑘𝑑)0−𝜖𝑑2𝑘cosh(𝑘𝑑)𝜖𝑑2𝑘sinh(𝑘𝑑)0 0 0 0 1 −1 0 0 −𝜖𝑑2𝑘0−𝑘 𝑘cosh(𝑘(𝑑+𝑎)) 0 0 0 0 𝑘cosh(𝑘𝑑)0 0 0 0  By solving the linear system of equation with matrix M and matrix 𝑁, the solution for all the coefficients follow from matrix𝑉. The potential in graphene (at 𝑧=0) is in this way given by equation (3.56), with the auxiliary functions given by 𝐺1(𝑘, 𝜔)=𝜖𝑑1𝜖𝑑2cosh(𝑘𝑑)[ −2𝑘𝛼𝜔2𝜔2 𝑝(−1+cosh(𝑘𝑎)cosh(𝛼𝑎)) +𝛼2𝜔4+𝑘2𝜔4 𝑝sinh(𝑘𝑎)sinh(𝛼𝑎)i(A.32) 𝐺2(𝑘, 𝜔)=𝛼𝜖∞(𝜔2−𝜔2 𝑝)h𝜖𝑑2cosh(𝑘𝑑) +𝜖𝑑1sinh(𝑘𝑑)−𝑘𝜔2 𝑝cosh(𝛼𝑎)sinh(𝑘𝑎) +𝛼𝜔2cosh(𝑘𝑎)sinh(𝛼𝑎)+𝛼𝜖∞(𝜔2−𝜔2 𝑝)sinh(𝑘𝑎)sinh(𝑘𝑑)sinh(𝛼𝑎)i(A.33) 𝑇1(𝑘, 𝜔)=𝜖𝑑1𝜖𝑑2𝑘(cosh(𝑘𝑑) +𝜖𝑑2sinh(𝑘𝑑)) −2𝑘𝛼𝜔2𝜔2 𝑝(−1+cosh(𝑘𝑎)cosh(𝛼𝑎))+𝛼2𝜔4+𝑘2𝜔4 𝑝sinh(𝑘𝑎)sinh(𝛼𝑎)(A.34) 𝑇2(𝑘, 𝜔)= 𝛼𝜖∞𝑘(𝜔2−𝜔2 𝑝)h−((1+𝜖𝑑1)𝜖𝑑2cosh(𝑘𝑑)+(𝜖𝑑1+𝜖2 𝑑2)sinh(𝑘𝑑)) 𝑘𝜔2 𝑝cosh(𝛼𝑎)sinh(𝑘𝑎) −𝛼𝜔2cosh(𝑘𝑎)sinh(𝛼𝑎)+𝛼𝜖∞(𝜔2−𝜔2 𝑝)sinh(𝑘𝑎)sinh(𝛼𝑎)(𝜖𝑑2cosh(𝑘𝑑) +sinh(𝑘𝑑))i (A.35) By this way, the matricial method provides the solution for the coefficients in the electrostatic potential, necessary to determine the spectrum of the surface plasmon-polaritons in graphene, like is done explicitly in chapter 3.2. 93 Appendix B Green’s Functions The Green’s function method is a simpler mathematical way to solve differential equations that contains an inhomogeneous term or in other words a source term, just like Poisson equation. In this thesis, we made use of the Green’s functions to solve the differential equation (4.6b), where the left-hand side of this equation is the source term that has a contribution from the external charges and the electrostatic charges intrinsic to the materials. In this appendix we will show the equations and the boundary conditions that give rise the solution provided by equations (4.10), (4.18) and (4.40), that correspond to the Green’s functions for graphene in vacuum, graphene near a dielectric and graphene near a local metal. Graphene in the vicinity of a dielectric To start let us see the case of graphene in a dielectric, with constant permittivity 𝜖𝑑and the dielectric present at 𝑧<0as seen in figure 4.1. To find the solution of equation (4.6b) we need to impose boundary conditions to the Green’s function based on the geometry of the problem. The boundary conditions are given by 𝑔𝐼(𝑧′, 𝑧′)=𝑔𝐼𝐼 (𝑧′, 𝑧′)(B.1) 𝜕𝑔𝐼(𝑧′, 𝑧′) 𝜕𝑧 −𝜕𝑔𝐼𝐼 (𝑧′, 𝑧′) 𝜕𝑧 =−1(B.2) 𝑔𝐼𝐼 (0, 𝑧′)=𝑔𝐼𝐼𝐼 (0, 𝑧′)(B.3) 𝜕𝑔𝐼𝐼 (0, 𝑧′) 𝜕𝑧 =𝜖𝑑 𝜕𝑔𝐼𝐼𝐼 (0, 𝑧′) 𝜕𝑧 (B.4) 94 The oscillatory form of the Green’s functions is shown in equation (4.17), by substitution in these last boundary conditions we obtain the following explicit set of equations 𝐴(𝑧′)𝑒−𝑘𝑧′=𝐶1(𝑧′)𝑒−𝑘𝑧′+𝐶2(𝑧′)𝑒𝑘𝑧′(B.5) 𝑘𝐴(𝑧′)𝑒−𝑘𝑧′−𝑘𝐶1(𝑧′)𝑒−𝑘𝑧′+𝑘𝐶2(𝑧′)𝑒𝑘𝑧′=1(B.6) 𝐶1(𝑧′) +𝐶2(𝑧′)=𝐵(𝑧′)(B.7) −𝑘𝐶1(𝑧′) +𝑘𝐶2(𝑧′)=1 𝜖𝑑 𝑘𝐵(𝑧′)(B.8) Solving this linear system of equation, we obtain the following coefficients 𝐴(𝑧′)=𝑒𝑘𝑧′ 2𝑘+1−𝜖𝑑 1+𝜖𝑑𝑒−𝑘𝑧′ 2𝑘(B.9) 𝐵(𝑧′)=1 (1+𝜖𝑑)𝑘𝑒−𝑘𝑧′(B.10) 𝐶1(𝑧′)=1−𝜖𝑑 1+𝜖𝑑1 2𝑘𝑒−𝑘𝑧′(B.11) 𝐶2(𝑧′)=1 2𝑘𝑒−𝑘𝑧′(B.12) This lead to the solution for the Green’s function, when graphene is in the vicinity of a dielectric as seen in equation (4.18). To obtain the expression for a graphene sheet in vacuum, we simply compute 𝜖𝑑=1 in the last expressions. This will lead to 𝑔𝐼𝐼 (𝑧, 𝑧′)=𝑔𝐼𝐼𝐼 (𝑧, 𝑧′), and the Green function in this case can be expressed like equation (4.10). Where the modulus assures the different solution for the two distinct regions (𝑧>𝑧′and 𝑧<𝑧′) Graphene in the vicinity of a local metal Now for the graphene near a local metal the procedure is similar, but in this case we will have a new boundary conditions at the interface metal/dielectric, that correspond to the regions 𝐼𝑉 and 𝐼𝐼𝐼, respectively. Just like the configuration in figure 4.4. For this geometry the boundary conditions are 𝑔𝐼(𝑧′, 𝑧′)=𝑔𝐼𝐼 (𝑧′, 𝑧′)(B.13) 𝜕𝑔𝐼(𝑧′, 𝑧′) 𝜕𝑧 −𝜕𝑔𝐼𝐼 (𝑧′, 𝑧′) 𝜕𝑧 =−1(B.14) 95 APPENDIX B. GREEN’S FUNCTIONS 𝑔𝐼𝐼 (0, 𝑧′)=𝑔𝐼𝐼𝐼 (0, 𝑧′)(B.15) 𝜕𝑔𝐼𝐼 (0, 𝑧′) 𝜕𝑧 =𝜖𝑑 𝜕𝑔𝐼𝐼𝐼 (0, 𝑧′) 𝜕𝑧 (B.16) 𝑔𝐼𝐼𝐼(−𝑑, 𝑧′)=𝑔𝐼𝑉 (−𝑑, 𝑧′)(B.17) 𝜖𝑑 𝜕𝑔𝐼𝐼𝐼 (−𝑑, 𝑧′) 𝜕𝑧 =𝜖(𝜔)𝜕𝑔𝐼𝑉 (−𝑑, 𝑧′) 𝜕𝑧 (B.18) where 𝜖(𝜔)is the Drude dielectric function given by equation (2.40). Solving the boundary conditions with the Green functions presented in equation (4.39), we have 𝐴(𝑧′)𝑒−𝑘𝑧′=𝐶1(𝑧′)𝑒−𝑘𝑧′+𝐶2(𝑧′)𝑒𝑘𝑧′(B.19) 𝑘𝐴(𝑧′)𝑒−𝑘𝑧′−𝑘𝐶1(𝑧′)𝑒−𝑘𝑧′+𝑘𝐶2(𝑧′)𝑒𝑘𝑧′=1(B.20) 𝐶1(𝑧′) +𝐶2(𝑧′)=𝐵1(𝑧′) +𝐵2(𝑧′)(B.21) −𝐶1(𝑧′) +𝐶2(𝑧′)=𝜖𝑑(−𝐵1(𝑧′) +𝐵2(𝑧′))(B.22) 𝐵1(𝑧′)𝑒𝑘𝑑 +𝐵2(𝑧′)𝑒−𝑘𝑑 =𝐷(𝑧′)𝑒−𝑘𝑑 (B.23) 𝜖𝑑−𝐵1(𝑧′)𝑒𝑘𝑑 +𝐵2(𝑧′)𝑒−𝑘𝑑 =𝜖(𝜔)𝐷(𝑧′)𝑒−𝑘𝑑 (B.24) The coefficients in the last equations are obtained by solving the linear system through matricial methods in a similar way that was solved in chapter 3 and in appendix A, for the electrostatic problems of graphene near a semi-infinite and finite nonlocal metal, respectively. This leads to 𝐴(𝑧′)=𝑒−𝑘𝑧′ 2𝑘𝑟𝐷+𝑟𝑑𝑒2𝑘𝑑 𝑟𝑑𝑟𝐷+𝑒2𝑘𝑑 +𝑒2𝑘𝑧′(B.25) 𝐵1(𝑧′)=𝑒−𝑘𝑧′ 2𝑘𝑡𝑑𝑟𝐷 𝑟𝑑𝑟𝐷+𝑒2𝑘𝑑 (B.26) 𝐵2(𝑧′)=𝑒−𝑘𝑧′ 2𝑘𝑡𝑑𝑒2𝑘𝑑 𝑟𝑑𝑟𝐷+𝑒2𝑘𝑑 (B.27) 𝐶1(𝑧′)=𝑒−𝑘𝑧′ 2𝑘𝑟𝐷+𝑟𝑑𝑒2𝑘𝑑 𝑟𝑑𝑟𝐷+𝑒2𝑘𝑑 (B.28) 96 𝐶2(𝑧′)=𝑒−𝑘𝑧′ 2𝑘(B.29) 𝐷(𝑧′)=𝑒−𝑘𝑧′ 2𝑘𝑡𝑑𝑡𝐷𝑒2𝑘𝑑 𝑟𝑑𝑟𝐷+𝑒2𝑘𝑑 (B.30) These coefficients give rise to the Green’s functions admitted in equation (4.40). The Green function method provides, in this way the solution for Poisson’s equation. This leads to the determination of the external charge potential and the potential induced in graphene by that external charge, as is stated in chapter 4. 97 Appendix C Graphene/Metal Induced Potential Integral When graphene is in the vicinity of a local metal, the potential induced in the graphene sheet by an external charge moving with velocity 𝑣in the y-direction can be computationally burden, due to the numerical solution of the integral. The potential integral in equation (4.63) can be separated into the integrals 𝐼1and 𝐼2, with 𝐼1=𝑣2𝑡2 𝑑∫2𝜋 0 𝑑𝜃 2𝜋⨏∞ 0 𝑘𝑑𝑘 2𝜋 𝑒−𝑘𝛽𝑒𝑖𝑘𝛾 (𝑘𝑣 cos 𝜃)2−𝜔2 𝑠𝑝𝑝 𝑟𝐷(𝑘𝑣 cos 𝜃) +𝑒2𝑘𝑑 𝑟𝑟𝐷(𝑘𝑣 cos 𝜃) +𝑒2𝑘𝑑 (C.1) 𝐼2=−𝑖𝑣2𝑡2 𝑑∫∞ 0 𝑑𝑘 4𝜋𝑔(𝑘,𝜃1) |𝑣2𝑘2cos 𝜃1sin 𝜃1|−𝑔(𝑘, 𝜃2) |𝑣2𝑘2cos 𝜃2sin 𝜃2|(C.2) Computing these integrals separately, show us that 𝐼1is more time consuming than 𝐼2. Luckily, both potential have the same form, which means that present a peak of potential and then become continuous. Regarding their numerical value, it was seen that both integral are approximately equal to each other, this is |𝐼1|=|𝐼2|. Since their value is equal and 𝐼2is computationally faster than 𝐼1we can obtain the induced potential only by using the second integral, this is Φ𝑖𝑛 (r, 𝑧, 𝑡)=2𝐼2(C.3) To make our point clearer, in the following tables is computed the numerical values for the integrals 𝐼1 and 𝐼2, for different values of the external charge velocity 𝑣, for various values of the polar coordinate 𝑟, different dielectric thickness 𝑑and external charge height 𝑧0. In table 7 it is also computed the integrals for different angles of 𝜃′. To recall this variable is the polar angle of r(see chapter 4.2). The tables show us that equation (C.3) is a good approximation to the induced potential where the ratio between the two integrals show that the values for 𝐼1and 𝐼2are very close to each other and we can consider 𝐼1+𝐼2=2𝐼2 in a good approximation. In all cases the dielectric constant chosen was 𝜖𝑑=1(vacuum). 98 Table 4: Numerical Value for the integrals in equation (4.63), for different velocities. The parameters were: 𝑑=0.01 𝜇m and 𝑧0=1𝜇m and 𝜃′=𝜋. Note that for every velocity the potential can be seen as Φ𝑖𝑛 (r, 𝑧, 𝑡)=2𝐼2approximately. For every 𝑟the values for 𝐼1and 𝐼2are very close to each other and we can consider 𝐼1+𝐼2=2𝐼2in a good approximation. 𝑣/𝑐 𝑟 [𝜇m] |𝐼1| |𝐼2||𝐼1| |𝐼2||𝐼1|+|𝐼2| 2|𝐼2| 1 0.0019 0.0029 0.66 0.83 3 0.0033 0.0038 0.87 0.93 0.05 6 0.0023 0.0026 0.89 0.94 10 0.0015 0.0017 0.88 0.94 12 0.0013 0.0014 0.93 0.96 15 0.0011 0.0012 0.93 0.96 1 0.0021 0.0031 0.68 0.84 2 0.0047 0.0054 0.87 0.94 0.1 5 0.0070 0.0073 0.96 0.98 8 0.0061 0.0063 0.97 0.98 10 0.0053 0.0055 0.96 0.98 15 0.0039 0.0041 0.95 0.98 1 0.0022 0.0031 0.71 0.85 2 0.0055 0.0062 0.88 0.94 0.3 4 0.0112 0.0116 0.97 0.98 8 0.0186 0.0188 0.99 0.99 10 0.0204 0.0205 ∼1∼1 15 0.0214 0.0215 ∼1∼1 4 0.0118 0.0122 0.97 0.98 8 0.0222 0.0224 0.99 ∼1 0.5 15 0.0328 0.0329 ∼1∼1 20 0.0355 0.0356 ∼1∼1 25 0.0356 0.0357 ∼1∼1 30 0.0345 0.345 ∼1∼1 5 0.0151 0.0155 0.97 0.99 10 0.0294 0.0296 0.99 ∼1 0.9 20 0.0507 0.0508 ∼1∼1 30 0.0614 0.0615 ∼1∼1 40 0.0645 0.0645 ∼1∼1 50 0.0632 0.0632 ∼1∼1 99