University of Southern California Dissertations and Theses

Plasmonic excitations in quantum materials: topological insulators and metallic monolayers on dielectric substrates

(USC Thesis Other)

Plasmonic excitations in quantum materials: topological insulators and metallic monolayers on dielectric substrates

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content Plasmonic Excitations in Quantum Materials: Topological Insulators and Metallic Monolayers on Dielectric Substrates by Zhihao Jiang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) December 2021 Copyright 2021 Zhihao Jiang More Is Different. - P. W. Anderson ii Dedication Dedicated to my parents who always tell me to keep confident and do not be afraid of difficulties. iii Acknowledgements Pursuing a Physics PhD degree is a longtime and challenging experience. It would not be possible for me to carry out any work in this thesis without the guidance and support from numerous people to whom I owe a debt of gratitude. Firstly, I am most grateful to my supervisor, Prof. Stephan Haas, for guiding me throughout all the research work I have done since my first year of the PhD program. Instead of telling me what to do, he always motivates me to have independent thinking and to explore my own ideas, which eventually trains me to be an independent researcher. He is always kind and considerate. I should say that my pleasant research experience with him is the main reason that motivates me to continue staying in the academic world. I also want to thank him for often bringing me into discussions with different expert people, and for his generous support for my visiting study in the Radboud University. Besides my supervisor, I am also very grateful to other committee members of my defense. They are Prof. Aiichiro Nakano, Prof. Wei Wu, Prof. Eli Levenson-Falk and Prof. Lorenzo Campos Venuti. Their insightful comments on my thesis and defense presentation are very helpful. I also appreciate questions and criticism they posed that have widened my knowledge scope and encouraged me to think deeper. I also want to thank Prof. Robin Shakeshaft for being a committee member of my PhD qualification exam. iv I have my sincere thanks to Dr. Malte Rösner who has been more than generous to collaborate with me and teach me a lot of things. I would in my heart respect him as my second research advisor for tons of invaluable help he has given me. From him I learned screened Coulomb in- teraction, random phase approximation and properties of real two-dimensional materials which constitute main ingredients for this thesis. I appreciate his patience. I also like his academic rigor. I want to thank Prof. Wei Wu, again, and Dr. Boxiang Song who are my experimental collab- orators. From them I learned practical experimental techniques such as sample fabrication and spectrum measurements besides my theoretical study. Working with them greatly enlarges my knowledge scope. I want to thank more collaborators and colleagues. Dr. Roelof Groenewald has given me a lot of help with improving my Python programming skills and with using the high-performance super-computing resource. Henning Schlömer is an expert on topological physics. I benefit a lot from discussions with him. We have studied together the plasmons in 1D and 2D topological insulators. Yuling Guan is my collaborator for investigating stability of plasmons in perturbed topological insulators. It is happy to work with him and also fun to chat with him about life. I have learned a lot from Dr. Gautam Rai about the Aubry-André-Harper model and quasicrystals. Dr. Zhengzhi (Jack) Ma has shared with me his previous works on the Su-Shrieffer-Heeger model which are very helpful. I enjoy talking with my Greek friends Dr. George Courcoubetis and Dr. Georgios Styliaris who always come with cool stories about bio-physics and quantum information science. I am also grateful to Joseph Vandiver and Gökhan Esirgen for being the leaders of my teaching assistant work. I appreciate the opportunites they provided. To me, teaching is more than just v a way of earning a financial support. It is a very valuable experience which has improved my communicating skills and made me love to share knowledge. Moreover, I am deeply thankful to all my teachers for their effects on educating me, starting from the primary school to the graduate school. They do not only teach me knowledge, but also affect my sense of values. They make me love science and seek the truth. I will always remember the encouragements and spiritual supports that my many sincere friends have given me when I am in a hard time. Most importantly, I will never forget my families, especially my parents and my loved girl- friend Cong Liu. Without their endless love and unconditional support, I will not be where I am today. Finally, I would like to say "thank you" to myself for never giving up. I am proud of myself for still possessing an innocent love for science and truth after having experienced many aspects of life. vi TableofContents Dedication iii Acknowledgements iv ListofFigures ix Abstract xi Chapter1: Introduction 1 Chapter2: ElectronsinQuantumMaterials: Many-BodyDescriptionandTopology 4 2.1 Many-Body Description of Condensed Matter Systems . . . . . . . . . . . . . . . 5 2.1.1 General Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Born-Oppenheimer Approximation and the Electronic Hamiltonian . . . . 6 2.1.3 Second Quantization Representation for Quantum Many-Body Systems . 7 2.2 Random Phase Approximation and Plasmons . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Density-Density Correlation Function . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Screened Coulomb Interaction and Dielectric Function . . . . . . . . . . . 14 2.2.3 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.4 Excitation Spectra of Electrons: Particle-Hole Pairs and Plasmons . . . . . 17 2.2.5 RPA Dielectric Function in the Real-Space Representation . . . . . . . . . 21 2.3 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 One-Dimensional Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . . 30 2.3.2 Generalized Aubry-André-Harper Model . . . . . . . . . . . . . . . . . . . 33 2.3.3 Two-Dimensional Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . 38 Chapter3: PlasmonsinTopologicalInsulators 40 3.1 Momentum-Space Plasmons in the Su-Schrieffer-Heeger Model . . . . . . . . . . 41 3.1.1 General Method for Band Electrons . . . . . . . . . . . . . . . . . . . . . . 41 3.1.2 Momentum-Space Dispersions . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.3 An Ordinary Two-band Insulator . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Real-Space Plasmons in an Open Su-Schrieffer-Heeger Model . . . . . . . . . . . . 66 3.2.1 Electron Energy Loss Spectrum and Localized Plasmons . . . . . . . . . . 67 3.2.2 Topological Origin of Localized Plasmons . . . . . . . . . . . . . . . . . . 69 vii 3.2.3 Mirror SSH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.4 Stability of Topological Plasmons Against Disorder . . . . . . . . . . . . . 74 3.2.5 Effects of Coulomb Interactions on Localized Plasmons . . . . . . . . . . . 79 3.2.6 Excitation Spectra under External Driving Field . . . . . . . . . . . . . . . 83 3.3 Plasmons in Two-Dimensional Topological Insulators . . . . . . . . . . . . . . . . 85 Chapter4: CoulombEngineeringonTwo-DimensionalPlasmons 88 4.1 Generic Feature of Plasmons in 2D materials . . . . . . . . . . . . . . . . . . . . . 90 4.1.1 Dynamical Screened Coulomb Interaction . . . . . . . . . . . . . . . . . . 91 4.1.2 Substrate Controlled 2D Plasmons . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Structured Dielectric Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.1 Evaluating the Screened Coulomb Interaction . . . . . . . . . . . . . . . . 101 4.2.2 Real-Space 2D Plasmons in Heterogeneous Screening Environment . . . . 105 4.2.3 A quantitative Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 Plasmonic Waveguide from Coulomb Engineering . . . . . . . . . . . . . . . . . . 110 4.3.1 Waveguide Imprinted by Dielectric Structure . . . . . . . . . . . . . . . . 111 4.3.2 Materials Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter5: ConclusionsandOutlooks 115 References 119 OwnPublications 130 viii ListofFigures 2.1 Feynmandiagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 ScreenedCoulombinteractioninelectrongas. . . . . . . . . . . . . . . . . . 14 2.3 Particle-holeexcitationsandplasmons. . . . . . . . . . . . . . . . . . . . . . 19 2.4 ElectronicstructuresofSu-Schrieffer-Heegermodelanditsmirrorvariant. 30 2.5 Contourstracedout bytherelativephases ˜ ϕ 2 intheSSH model, ask is takenaroundaloopinthefirstBrillouinZone. . . . . . . . . . . . . . . . . . 35 2.6 Two-dimensionalSu-Schrieffer-Heegermodel. . . . . . . . . . . . . . . . . . 38 3.1 One-dimensionaltight-bindingmodeloftwoorbitals. . . . . . . . . . . . . 42 3.2 OverlapfunctionsintheSSHmodelfordifferentbasisatomseparations. . 48 3.3 Momentum-spaceplasmonsinSSHmodel. . . . . . . . . . . . . . . . . . . . 54 3.4 TuningtheSSHmodelbetweendimerizedandanti-dimerizedregimes. . . 61 3.5 Illustrationoftrappedatomsinone-dimensionalperiodicpotentialwells. 63 3.6 Plasmonsinanordinarytwo-bandinsulator. . . . . . . . . . . . . . . . . . . 65 3.7 Real-spaceplasmonsinSu-Schrieffer-Heegermodelwithopenboundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.8 EELSdecompositionintheSSHmodel. . . . . . . . . . . . . . . . . . . . . . . 69 3.9 Real-spacePlasmonsinmirrorSSHmodel. . . . . . . . . . . . . . . . . . . . 73 3.10 StabilityoftheinterfaceplasmoninthemirrorSSHmodel. . . . . . . . . . 75 ix 3.11 Effectoftheglobalscreeningenvironmentontheplasmonspectrum. . . . 80 3.12 Removal of topological plasmon degeneracy and enhancement of plasmonlocalizationwithlocallyvaryingCoulombinteraction. . . . . . . 82 3.13 ExcitationspectraofSSHmodelunderdifferentexternalfield. . . . . . . . 83 3.14 ExcitationspectraofmirrorSSHmodelunderdifferentexternalfield. . . 85 3.15 Real-spaceplasmonsinthetwo-dimensionalSu-Schrieffer-Heegermodel. 86 4.1 Substratecontrolledplasmonpatternsinatomicallythinmetals. . . . . . . 89 4.2 Internalscreeningmodelin2Dmaterial. . . . . . . . . . . . . . . . . . . . . . 91 4.3 Substratecontrolledtwo-dimensionalplasmons. . . . . . . . . . . . . . . . . 96 4.4 Real-spacechargedensitymodulationsoftwotypicalplasmonmodes. . . 99 4.5 Imagechargemodelofa2Dmaterialembeddedinadielectricstructure withasingleverticalinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.6 Heterogeneous plasmon patterns from spatially structured dielectric environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.7 Impactofthedielectriccontrasttotheplasmonicpatterns. . . . . . . . . . 109 4.8 Plasmonicwaveguidesfromspatiallystructureddielectricenvironments. 111 x Abstract Correlation and topology are probably the two most intriguing properties of electrons. While in many situations they are investigated independently, since many topological properties of elec- tronic systems can be well described within the framework of single-particle physics, in some context, they can be playing an important role together. In recent years, there has been increased interest in the elementary excitations of topological electronic systems, such as plasmons, exci- tons and magnons. It is interesting to study whether these excitations inherit any topological signatures from the constituent electronic states, and how this would depend on the electron- electron Coulomb interactions in the system. In the first part of this thesis, we study plasmonic excitations in topological insulators. We fo- cus on the Su-Schrieffer-Heeger (SSH) model in both one-dimensional (1D) and two-dimensional (2D) lattices. We use a fully quantum mechanical approach to evaluate the dielectric function in both momentum space and real space within the random phase approximation, from which we extract the electron energy loss spectra (EELS) that characterize the plasmonic excitations in the system. In momentum space, we observe different branches of plasmon dispersions that can be tuned by varying the inter-atomic separation between basis atoms. In real space, we observed plasmonic excitations that are localized on the boundaries of the 1D SSH and 2D SSH models, xi when they are in topologically non-trivial phases. These localized edge plasmons serve as topo- logical signature of the underlying model in electronic collective excitations. They can be detected in experiments. These localized plasmons are shown to originate from the constituent topolog- ical electronic edge states and meanwhile to be affected by the bulk electronic states which are providing the background screening. We investigate the stability of the localized modes specifi- cally on a topological interface of connecting two topologically distinct 1D SSH models against disorder. We find that both the excitation energy and the real-space localization of the localized plasmon mode are less stable than those of the single-electron topological state when the disorder becomes significant. In the second part of this thesis, we discuss 2D plasmons in layered materials on dielectric substrates. Coulomb interactions play an essential role in atomically-thin materials. On one hand, they are strong and long-ranged in layered systems due to the lack of environmental screening. On the other hand, they can be efficiently tuned by means of surrounding dielectric materials. Thus all physical properties which decisively depend on the exact structure of the electronic in- teractions can be in principle efficiently controlled and manipulated from the outside via Coulomb engineering. This concept has already been applied successfully to semiconductors for tuning the bandgap and exciton binding energies. We show that this concept can also be used to metallic systems to engineer plasmonic excitations. Specifically, we demonstrate that a spatially struc- tured dielectric environment can be used to non-invasively confine plasmons in an unperturbed homogeneous metallic 2D system by modifications of its many-body interactions. This observa- tion motivates us to propose a conceptually novel way to design 2D plasmonic waveguides via Coulomb engineering. Our numerical results indicate that plasmons can be confined to a 10-nm scale. In contrast to conventional functionalization mechanisms, this scheme relies on a purely xii many-body concept and does not involve any direct modifications to the active material itself. We discuss proper materials choices of both the active plasmonic 2D layer and the screening dielectric substrate for our proposal to be experimentally feasible. xiii Chapter1 Introduction Physical properties of a condensed matter system can be studied by probing the system with an external perturbation and then measuring its response. For example, in order to study optical properties of a material, one can conduct spectroscopic experiments by injecting beams of pho- tons, electrons or other particles to perturb the sample, and measure the corresponding spectra, such as absorption, photoemission, electron-energy loss, etc [1]. In these experiments, electrons interact with the external perturbing potential and are excited. Moreover, due to the internal Coulomb correlation between electrons, the measured spectra reflect many-body properties of the electronic system, i.e., the spectra of quasi-particles. The general theoretical formalism describing these probing-responding phenomena is called the response theory. The measurable observables and the perturbing potentials are related by re- sponse functions. They are, in fact, retarded Green functions evaluated in many-body electronic states. In this thesis, we will particularly focus on plasmonic excitations in quantum materi- als. These are, phenomenologically, electron density oscillations induced by an external driving electromagnetic field. The response function is the charge susceptibility, which is in general dynamical and spatially non-local. Experimentally, plasmonic excitations can be characterized 1 by the electron energy loss spectrum (EELS), which can be effectively calculated from the di- electric response function. In Chapter 2, we will review the relevant many-body concepts and techniques that are needed for quantitatively studying plasmonic excitations. These include the density-density correlation function (whose retarded version is just the susceptibility function), the screened Coulomb interaction, the random phase approximation and the dielectric response function. After this discussion, we are well armed to calculate the plasmonic excitations in dif- ferent quantum systems. The first quantum systems on which we will be focusing are topological insulators (TIs). TIs have been studied extensively in the last few decades, starting with the experimental discovery of the integer quantum Hall effect [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. They are manifested in systems with gapped bulk states and symmetry protected, conducting edge states. The electronic topological properties in these systems have already been well characterized in great detail using non-interacting band theory. Recently, increasing attention has been focused on the effects of many-body interactions in these systems. For instance, researchers have started to investigate the collective excitations in TIs, such as plasmons [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], excitons [34, 35, 36, 37], and magnons [38, 39, 40]. The work to be presented in this thesis mainly focuses on plasmonic excitations in 1D and 2D TIs. In the last section of Chapter 2, we briefly review models of TIs to be considered, by discussing their single-electron topology. These include the 1D Su-Schrieffer-Heeger (SSH) model, the generalized Aubry-André-Harper model, and the 2D SSH model. In Chapter 3, we give detailed descriptions of plasmonic excitations in these models by evaluating their electron energy loss spectra in both momentum space and real space. In both spaces we observe topological signatures in these collective excitations, indicating 2 that the non-trivial topology of the single-electron states have a clear impact on their collective excitations. In Chapter 4, we present our work on 2D plasmonic excitations in layered metallic materi- als embedded in dielectric environments. Plasmons are collective excitations rendered by dy- namical screened Coulomb interactions. Coulomb interactions in 2D materials are susceptible to their screening environment. Therefore, all physical properties which decisively depend on the electron-electron Coulomb interaction can, in principle, be efficiently controlled and manipulated from the outside. To understand how this can be realized for 2D plasmons, we first study the generic features of 2D plasmons in a homogeneous 2D material embedded in a homogeneous di- electric background by carefully considering all involved screening channels in Section 4.1. Then, in Section 4.2, we show that spatially confined plasmons can be induced in a 2D homogeneous material purely by a structured dielectric environment. Based on this observation, we propose a conceptually novel way to design plasmonic waveguides by dielectric engineering in Section 4.3. That is, the propagating plasmonic waves in the 2D material are guided by the imprint of an external dielectric structure. We will summarize our conclusions and offer an outlook in Chapter 5. 3 Chapter2 ElectronsinQuantumMaterials: Many-BodyDescription andTopology In this chapter, we give a general review of the background theories and methods that are rel- evant to the specific research topics to be presented in later chapters of this thesis. Broadly speaking, we are interested in many-body properties of electrons in interacting condensed mat- ter systems. In Section 2.1, we introduce this general problem, along with some approximations and description formalism. We will then focus on plasmonic excitations in Section 2.2. There we will discuss the key relevant ingredients including the density-density correlation function, the screened Coulomb interaction, the random phase approximation, the dielectric response function and the electron energy loss spectrum. Besides correlations, topology is another fascinating property of electrons. Topological insu- lators are the main focus discussed in the first part of this thesis. We will briefly introduce them in Section 2.3. Although many topological properties of electrons can already be well described within single-electron band theory, we are particularly interested in thecollectivemodes of these materials. 4 2.1 Many-BodyDescriptionofCondensedMatterSystems Many-body condensed matter systems have been one of the most interesting and fruitful research areas that attracts extensive interest from various communities of physicists, chemists and ma- terial scientists. They are studied in order to understand the properties of existing materials, as well as to design new, functional materials, or meta-materials. In this section, we give a short overview of the many-body condensed matter problem. The general Hamiltonian describing in- teracting electrons and ions is introduced in Section 2.1.1. This is followed by a discussion of the Born-Oppenheimer approximation which decomposes the problem into an electronic part and an ionic part, discussed in Section 2.1.2. In Section 2.1.3, we briefly introduce the second quantization formalism, which is a very useful representation to describe many-body physics. 2.1.1 GeneralHamiltonian Many-body condensed matter systems consist of a huge number (on the order of ∼ 10 23 ) of interacting electrons and ions. The general description can be summarized by the Hamiltonian H = X i p 2 i 2m e + 1 2 X i̸=j e 2 |r i − r j | + X I P 2 I 2M I + 1 2 X I̸=J Z I Z J e 2 |R I − R J | + X i,I V I (r i − R I ) (2.1) where m e is the mass of an electron, and M I is the math of the ion I. The capital letters R andP denote the positions and momenta of ions, whereas the lower-case lettersr andp denote the positions and momenta of the electrons. e is the charge of an electron andZ I is the atomic number of the ionI. The first two terms in Eqn. 2.1 are the kinetic energy of the electrons and their mutual Coulomb interactions. The third and fourth terms represent the kinetic energy of 5 the ions and the mutual Coulomb interactions between the ions. The last term accounts for the interactions between the electrons and the ions. Despite the clear intuition behind this Hamilto- nian, the problem itself is extremely difficult to solve, which is essentially due to the interaction terms that make the problem much more challenging than single-particle physics. Even for a numerical solution, due to the large number of particles, the computational complexity is way beyond the capacity of any existing supercomputer. Therefore, in order to proceed, it is necessary to introduce appropriate approximations. 2.1.2 Born-OppenheimerApproximationandtheElectronicHamiltonian A typical first-step approximation for the above Hamiltonian is the so called Born-Oppenheimer (BO) approximation. Physically, the BO approximation assumes that electrons and ions can be treated separately based on the fact that they have very different masses ( M I ∼ 10 3 m e ). The elec- trons are assumed to follow the motion of ions instantaneously. Therefore, when studying elec- trons, we can neglect the kinetic energy of ions and regard electrons to be in a background of ef- fectivelyfixedions at positionsR 1 ,R 2 , ..., etc. These fixed ions provide a static external potential for the electrons. The separated electronic Hamiltonian is given by H e = X i " p 2 i 2m e + X I V I (r i − R I ) # + 1 2 X i̸=j e 2 |r i − r j | (2.2) where the ionic positions{R I } are now just parameters. Throughout this thesis we will not consider ionic properties, but we focus on the electronic Hamiltonian 2.2. We can see that the first term is just a simple summation of independent elec- trons in an external potential (arising from ions), which is in principle not hard to solve. The 6 second term, describing electron-electron Coulomb interactions, is essential here. On one hand, it drastically complicates the problem for which an exact solution has not been achieved so far. For a practical calculation, people have introduced many types of techniques and approxima- tions according to the goal of problem. For example, the Hartree-Fock mean-field approximation and the density functional theory (with approximated exchange-correlation functionals) are ex- tensively used for evaluating ground state properties. For excitation properties and dynamical responses, general approaches include time-dependent density functional theory and many-body perturbation theory based on Green functions [1]. On the other hand, electron-electron Coulomb interactions cause a multitude of fascinating properties and phenomena that would otherwise not exist, such as correlated insulating states, magnetism, high-temperature superconductivity, quasiparticles, collective excitations and etc. In this thesis, we focus on plasmonic excitation in electronic systems. These are collective oscillations of the electron density arising from long-range Coulomb interactions. The quanta of these oscillations are called plasmons. We will study plasmons by evaluating the microscopic di- electric response function using the random phase approximation (RPA). These will be discussed in detail in Section 2.2. 2.1.3 SecondQuantizationRepresentationforQuantumMany-BodySystems Here, we give a brief introduction to the second quantization formalism. This is a clever repre- sentation which is especially useful for describing quantum many-body systems of indistinguish- able particles. The traditional formalism of describing many-body quantum states by many-body wave functions with proper (anti-)symmetrization becomes very involved and intracable when 7 the number of particles becomes large. In the second quantization formalism, instead of specify- ing the quantum state of each individual particle, we specify the occupation number of particles in each quantum stateα , such that anN-particle state can be simply represented as |Ψ N ⟩=|n α n β ···⟩ (2.3) wheren α is the number of particles in stateα ,n β is the number of particles in stateβ , etc. And in total, N = n α +n β +··· . This is a very compact representation. The state|Ψ N ⟩ lives in a Hilbert space called Fock space. The Fock space in general does not have a restriction on the total number of particles. Furthermore, we introduce creation operators c † and annihilation operators c (also called ladder operators) which add or remove particles, or exchange two particles, when combined. Depending on whether these manipulations are done on bosons or fermions, the resulting states fulfill the corresponding symmetry or anti-symmetry requirement. This is conveniently captured in simple commutation relations of these ladder operators, {c α ,c † β }=δ αβ , for fermions, (2.4) h b α ,b † β i =δ αβ , for bosons. (2.5) We focus on electronic systems in this thesis. We can express its Hamiltonian 2.2 using these creation and annihilation operators. Here, we simply give the most general result, H =− X mn X σ t mn c † mσ c nσ + X mnpq X σσ ′ U mnpq c † mσ c † nσ ′ c pσ ′c qσ . (2.6) 8 The first term contains the one-body (independent-particle) operator expressed in the second quantization representation. It includes the kinetic energy of electrons and the potential energy of electrons in an external field. The second term contains two-particle (interacting) operators. This accounts for mutual Coulomb interactions between electrons. This general Hamiltonian 2.6 can be simplified to specific, well-known models such as the independent-particle tight-binding model and the Hubbard model. In our work, we will be using the second quantization electronic Hamiltonian very often. 2.2 RandomPhaseApproximationandPlasmons In this section, we introduce the main concepts and methods that will be used later for studying plasmons. Plasmons are electron density oscillations. We first discuss the density-density correla- tion function in Section 2.2.1, which describes the response of the electron density to an external perturbing potential. Due to correlations, electrons can polarize their environment, such that their effective interactions with other electrons are screened. The screened Coulomb interaction is discussed in Section 2.2.2, together with the dielectric function which quantitatively describes the screening effect. In Section 2.2.3, we introduce the random phase approximation which is a well-known approximation for calculating the dielectric function. In Section 2.2.4, we discuss the excitation spectra of electrons that can be extracted from the dielectric function. These in- clude particle-hole excitations and plasmonic excitations. Finally, in Section 2.2.5, we talk about the implementation of the random phase approximation in real space. This method will be used extensively in almost all chapters presented in this thesis. 9 2.2.1 Density-DensityCorrelationFunction In this section, we introduce the density-density correlation function which describes the re- sponse of electron density to an external perturbing potential. It will be useful for understanding excitation properties of electronic systems, including their plasmonic excitations. By definition, the density-density correlation functionχ is a two-particle Green function which, in momentum space, is given by [41] χ (q,t)=i⟨Φ GS |T[ρ (q,t)ρ (− q,0)]|Φ GS ⟩ (2.7) where|Φ GS ⟩ is the ground state of the interacting system,T is the time ordering operator, and ρ (q,t) is the density operator ρ (q,t)= X k,σ c † k,σ (t)c k+q,σ (t). (2.8) Its retarded version is also known as the susceptibility function, because it describes the charge density fluctuation induced by the external perturbing potential, i.e. ρ ind (q,t) = R dt ′ χ (q,t− t ′ )V ext (q,t ′ ) ∗ . The difficulty in solving χ using the defining equation 2.7 comes from the unknown ground state|Φ GS ⟩ of the interacting many-body system. In the many-body perturbation formalism, one starts from the non-interacting ground state|ϕ (0) GS ⟩ at the remote part and "adiabatically switches ∗ In this scenario, aΘ -function will appear in the definition of χ (q,t− t ′ ) due to causality. 10 (a) (b) (d) = + + ⋯ (c) (e) = + + ⋯ RPA (f) RPA = + + ⋯ RPA Figure 2.1: Feynmandiagrams. (a) Two fermion loops. (b) Particle-hole bubble diagram. (c) An example of a reducible diagram. (d) Diagrammatic representation of Dyson series for full inter- acting Coulomb interaction. (e) Diagrammatic representation of Dyson series for RPA Coulomb interaction. (f) Diagrammatic representation of Dyson series for RPA susceptibility function. on" the interaction via theS matrix to bring the ground state into the fully interacting version |ϕ GS ⟩ at the present time. This leads to χ (q,t)=i ⟨Φ (0) GS |T[ρ (q,t)ρ (− q,0)S(∞,−∞ )]|Φ (0) GS ⟩ ⟨Φ (0) GS |S(∞,−∞ )|Φ (0) GS ⟩ . (2.9) Both the numerator and the denominator can be evaluated by diagrammatic perturbation expan- sion. Due to convenient cancellations it turns out we only need to evaluate the numerator with connected diagrams. 11 Let us first evaluate the density-density correlation function in the non-interacting ground state, which we denote asχ 0 (q,t) here. Below we use⟨···⟩ 0 (⟨···⟩ ) to represent the expectation taken in the non-interacting (interacting) ground state. We can write χ 0 (q,t)=i⟨T[ρ (q,t)ρ (− q,0)]⟩ 0 =i X k,σ X k ′ ,σ ′ ⟨T[c † k,σ (t)c k+q,σ (t)c † k ′ ,σ ′ (0)c k ′ − q,σ ′(0)]⟩ 0 . (2.10) To evaluate the four-operator expectation inside the summation, we can use Wick’s theorem to form contractions. Each annihilation (creation) operator has to be paired to a creation (annihi- lation) operator, so there are only two non-vanishing ways of pairing. The first way is to pair c † k,σ (t) withc k+q,σ (t) andc † k ′ ,σ ′ (0) withc k ′ − q,σ ′(0). This would force the momentum transferq to be zero, leading to a Feynman diagram of two independent fermion loops, see Fig. 2.1(a). Each fermion loop just gives the electron density. So the total contribution of this diagram is a constant that is proportional to N 2 where N is the number of electrons. This term does not induce any non-local correlation (q = 0) and thus will be ignored from now on. The second way is to pair c † k,σ (t) with c k ′ − q,σ (0) and c † k ′ ,σ ′ (0) with c k+q,σ (t), leading to a Feynman diagram of a particle- hole bubble in Fig. 2.1(b). In this way, χ 0 (q,t) is also called the polarization function which is denoted asΠ 0 (q,t), since it describes the propagation of particle-hole pairs. In this thesis, we use χ 0 and Π 0 interchangeably to mean the same thing - the non-interacting polarization function. 12 This particle-hole bubble diagram can be evaluated by a simple product of the non-interacting electron’s Green function and the non-interacting hole’s Green function. Therefore, we have Π 0 (q,t)=2i X k G 0 (k+q,t)G 0 (k,− t) (2.11) where the factor 2 corresponds to the spin degeneracy (of electrons). We analyze it in the fre- quency domain by doing the Fourier transformation Π 0 (q,ω)= Z dte iωt Π 0 (q,t). (2.12) The Fourier transformation the the productG 0 (k+q,t)G 0 (k,− t) is the just convolution of the Fourier-transferred Green functions in the frequency domain, namely Z dte iωt G 0 (k+q,t)G 0 (k,− t) = 1 2π FT[G 0 (k+q,t)]∗ FT[G 0 (k,− t)] = 1 2π G 0 (k+q,ω)∗ G 0 (k,− ω) = 1 2π Z dω ′ G 0 (k+q,ω ′ )G 0 (k,ω ′ − ω) (2.13) Therefore, we get the non-interacting polarization functionΠ 0 (q,ω) as Π 0 (q,ω)= i π X k Z dω ′ G 0 (k+q,ω ′ )G 0 (k,ω ′ − ω). (2.14) For the full interacting polarization function Π , we need to consider contributions from all high-order diagrams including internal interaction line(s). The particle-hole bubble diagram 13 - hole - “dressed” electron Figure 2.2: ScreenedCoulombinteractioninelectrongas. An electron can repel other elec- trons due to the Coulomb repulsion, creating a hole area nearby. Its interactions with other electrons are effectively screened by the hole cloud. shown in Fig. 2.1(b) does not have an internal interaction line and is therefore regarded as the zero-th order contribution. Practically, for each order, we just need to consider summing over irreducible diagrams that can not be split into two parts by cutting an interaction line, leading to so-calledirreduciblepolarizationfunction. An example ofreducible diagram is shown in Fig. 2.1(c). This scheme is analogous to the irreducible self-energy used for constructing Dyson series of the full interacting Green function. 2.2.2 ScreenedCoulombInteractionandDielectricFunction We discuss the screened Coulomb interactions between electrons and introduce the dielectric function in this section. It is well-known that the Coulomb interaction between two point charges q 1 andq 2 separated by a distancer in vacuum is simply given by V c (r)= q 1 q 2 r . (2.15) 14 In the case of electrons in a true material (or in a many-body quantum system), the effective interaction between them is more complicated due to the polarizable environment. In an intuitive physical picture in Fig. 2.2, an electron located at the positionr can repel other electrons around it due to the Coulomb repulsion and create a hole area nearby. Another electron, located atr ′ , does not see a “bare" electron atr, but instead an electron dressed by a “hole cloud." This “hole cloud" screens the interaction between these two electrons atr andr ′ , leading to a weakened effective interaction. This effective interaction is called the screened Coulomb interaction, in contrast to the bare Coulomb interaction in Eqn. 2.15. The screened Coulomb interaction can be evaluated through the Dyson equation using the bare Coulomb interaction and the polarization function Π . The diagrammatic representation is shown in Fig. 2.1(d), which is analogous to evaluating the interacting Green function using the non-interacting one with the self-energy. We usually denote the screened Coulomb interaction byW and it is given by W =V c +V c Π V c +V c Π V c Π V c +··· =V c +V c Π W (2.16) =⇒ W = V c 1− V c Π . (2.17) We can see that the screened Coulomb interaction is renormalized from the bare one by a factor (1− V c Π) − 1 due to the polarizability Π . This factor is defined as the dielectric function, ε = 1− V c Π , which can be used to quantitatively describe the screening effect. As a simple example in classical electromagnetism, the Coulomb interaction between two charges in an uniform medium with the dielectric constantε m is screened by a factor1/ε m . In general, the screening effect in a quantum material can be much more complicated than a simple constant. It can be dynamical and 15 spatially non-local, characterized byε(rt,r ′ t ′ ). The explicit formula depends on the microscopic structure of the system. In this thesis, we will generally consider the dynamical and non-local dielectric function using the microscopic quantum mechanical treatment. Specifically, if the time evolution is homogeneous, we can analyseε(ω,r,r ′ ) by Fourier transformingε(rt,r ′ t ′ ) into the frequency domain. This is done throughout the thesis. Moreover, if the system has translational invariance, we can further transform the dielectric function into the momentum space and obtain ε(ω,q), otherwise a real-space treatment will be needed. In this thesis, we will consider both the momentum-space dielectric function and the real-space dielectric function depending on the systems we considered. 2.2.3 RandomPhaseApproximation We introduce the random phase approximation in this section, which will be used for all calcula- tions of dielectric functions throughout the thesis. As we can see from the Eqn. 2.17, in order to calculate the full screened Coulomb interaction, we need the full irreducible polarization func- tionΠ which in principle includes all polarization diagrams on all orders of interaction. This is practically very hard to solve. One simple approximation is that we only keep one type of the diagram, the zero-th order electron-hole bubble diagram [Fig. 2.1(b), corresponding to Π 0 ] and consider interactions between them to an infinite order. This is called the random phase approxi- mation (RPA), which can be diagrammatically represented in Fig. 2.1(e). Within RPA, the screened 16 Coulomb interaction can be evaluated by simply knowing the bare Coulomb interactionV c and the non-interacting polarization functionΠ 0 as W RPA = V c 1− V c Π 0 . (2.18) We can also evaluate the RPA charge susceptibility function. This is also done by just keeping the simple electron-hole bubble diagram in the irreducible polarization, as illustrated in Fig. 2.1(f). The final result is given by a Dyson series as χ RPA =Π 0 +Π 0 V c Π 0 +Π 0 V c Π 0 V c Π 0 +··· =Π 0 +Π 0 V c χ RPA (2.19) =⇒ χ RPA = Π 0 1− Π 0 V c . (2.20) The RPA dielectric function is correspondingly defined as the denominator of Eqn 2.18: ε RPA =1− V c Π 0 . (2.21) 2.2.4 ExcitationSpectraofElectrons: Particle-HolePairsandPlasmons In the following we evaluate the non-interacting polarization function and the RPA dielectric function explicitly, from which we can study excitation properties of the many-body system. We will use the momentum-space representation here to illustrate the main ideas, assuming full translational invariance of the system. Namely, we are considering homogeneous electron gas. The full real-space treatment of an open system will be provided in the next section (Section 2.2.5), 17 and the treatment for band electrons will be discussed in detail in the next chapter (Chapter 3) when we consider two-band topological insulators. We first recall that the non-interacting electronic Green function is G 0 (k,ω)= 1 ω− ξ k +iδ k (2.22) whereξ k is the energy measured relative to the chemical potential andδ k = δ sgn(ξ k ) is a small broadening which is positive (negative) forξ k > 0 (ξ k < 0). Then, by evaluating the integral in Eqn. 2.14 for the retarded propagator, we get the non-interacting polarization function Π 0 (q,ω)=2 X k f k (1− f k+q ) ϵ k+q − ϵ k − ω− iδ (2.23) where ϵ k is the energy dispersion, f k is the Fermi-Dirac distribution function and δ = 0 + is a small positive broadening. We will always consider the zero temperature limit, so the Fermi- Dirac functionf k is the Heaviside step function which equals 0 (1) ifϵ k is above (below) the Fermi level. Before going to the dielectric function, we first study the non-interacting polarization function by specifically investigating its imaginary part † . The imaginary part ofΠ 0 (q,ω) is proportional to the spectral functionS 0 (q,ω). Using the identity 1 X +iδ =P.V 1 X − iπδ (X), (2.24) † The real part leads to the Lindhard function that will not be considered in this thesis. 18 + Fermi sea (a) particle-hole continuum 1D 3D 2D (b) Figure 2.3: Particle-holeexcitationsandplasmons. (a) Illustrate of a particle-hole excitation. (b) Plasmon dispersions for 1D, 2D and 3D electron gas. The shaded area is the particle-hole continuum where plasmons are damped. we get the spectral function as S 0 (q,ω)∝2 X k f k (1− f k+q )δ (ϵ k+q − ϵ k − ω). (2.25) The spectral function describes particle-hole excitations in the system, which will be explained here. For the aboveS 0 (q,ω) to be non-vanishing, we needf k =1, indicating a occupied state (a particle) at the energyϵ k . We also needf k+q =0, indicating an empty state (a hole) at the energy ϵ k+q . For the ground state of a Fermi sea, this excitation is depicted in Fig. 2.3(a). The excitation happens at the frequency ω which conserves the energy as is enforced by the delta function δ (ϵ k+q − ϵ k − ω). These are real excitations. These excitations extend over a continuous region in the spectrum which we call the particle-hole continuum. As they do not have a well-defined energy-momentum relation, they are considered as incoherent excitations. 19 We now proceed to the dielectric function with the Coulomb interaction taken into account on the RPA level. We have ε RPA (q,ω)=1− V c (q)Π 0 (q,ω). (2.26) We will always consider the dielectric function within RPA in this thesis, so we omit the subscript “RPA" for the simplification of notation. Physically, ε(q,ω) describes the total potential gener- ated in the system responding to an external perturbing potential:V tot (q,ω)=V ext (q,ω)/ε(q,ω). Similar to the classical definition, we consider ε(q,ω) = 0 to be the condition for plasma oscil- lations whose quanta are called plasmons. These are collective excitations of the electron density mediated by the Coulomb interaction which live outside the particle-hole continuum. In contrast to particle-hole excitations, plasmons usually have a well-defined dispersion relation in the ω− q space. Quantitatively, it is more convenient to study plasmonic excitations by evaluating the negative imaginary part of the inverse dielectric function− Im h 1 ε(q,ω) i . This is in fact proportional to the electron energy loss spectrum (EELS) which can be measured in experiments. Using the Eqn. 2.26 with some algebra, we can get EELS(q,ω)=− Im 1 1− V c (q)Π 0 (q,ω) = V c (q)ImΠ 0 (q,ω) [1− V c ReΠ 0 (q,ω)] 2 +[V c (q)ImΠ 0 (q,ω)] 2 . (2.27) 20 From 2.27, we can see that EELS inherits the information of particle-hole excitations which are in- dicated by singularities inImΠ 0 (q,ω). Moreover, there are other excitations outside the particle- hole continuum whereImΠ 0 (q,ω) goes to zero, when1− V C ReΠ 0 (q,ω) also approaches 0. These are plasmonic excitations where the EELS becomes singular outside the particle-hole continuum. We note that ImΠ 0 (q,ω) = 0 and 1− V C ReΠ 0 (q,ω) = 0 are equivalent to ε(q,ω) = 0, the defining condition of plasmonic excitations. Plasmon dispersion relations are usually sensitively dependent on the dimensionality of the system. This is due to the different forms of the Coulomb interaction V c (q) for the momentum transfer q in different dimensionalities. In Fig. 2.3, we sketch the plasmon dispersions for one- , two-, and three-dimensional spaces. When the plasmon dispersion enters into the particle- hole continuum, plasmons become unstable and decay into particle-hole pairs. This is usually called Landau damping. From this simple sketching picture, we can already see some merits of 2D plasmons. Comparing to 1D case, they live higher above the Landau-damping regime and are therefore less damped. Comparing to 3D case, 2D materials are more tunable via screening environment. 2D plasmons will be our intensively focused topic in Chapter 4. 2.2.5 RPADielectricFunctionintheReal-SpaceRepresentation In this section we derive the RPA dielectric function in the real-space representation, namely, ε(ω,r,r ′ ) depending onr andr ′ . This can be used to study the response in non-translational- invariant systems, such as heterojunctions, quasicrystals, fractals, Moiré patterns, or any arbitrar- ily shaped structures. This can also be used to study systems which have important boundary properties, such as topological edge states. 21 We use a self-consistent-field (SCF) approach to derive the real-space RPA dielectric function ε(ω,r,r ′ ), which has a more clear physical picture [42]. We consider the electronic system being perturbed by an external potentialδV ext (r ′ ,t). This induces charge density fluctuation which is given by δρ ind (r,t)= Z dt ′ dr ′ χ (r,r ′ ,t− t ′ )δV ext (r ′ ,t ′ ). (2.28) The total potentialδV tot (r,t) seen by electrons is the summation of the external potentialδV ext (r,t) and the induced potentialδV ind (r,t) due to the induced chargeδρ ind (r,t). It is in this sense that we call this approach the SCF approach. The induced potential δV ind (r,t), in general, can be very complicated including the contributions from classical Hartree potential and the exchange- correlation potential. Here, we only keep the Hartree potential and neglect the exchange-correlation part, which corresponds to the random phase approximation in this context. For the Hartree part, we can write the induced potential as δV ind (r,t)= Z V c (r− r ′ )ρ ind (r ′ ,t)dr ′ . (2.29) 22 The total potential is therefore given by δV tot (r,t)=δV ext (r,t)+δV ind (r,t) (2.30) =δV ext (r,t)+ Z V c (r− r ′ )δρ ind (r ′ ,t)dr ′ (2.31) =δV ext (r,t)+ Z V c (r− r ′ ) Z χ (r ′ ,r ′′ ,t− t ′ )δV ext (r ′′ ,t ′ )dt ′ dr ′′ dr ′ (2.32) = Z δ (r− r ′′ ,t− t ′ )+ Z V c (r− r ′ ) Z χ (r ′ ,r ′′ ,t− t ′ )dr ′ δV ext (r ′′ ,t ′ )dt ′ dr ′′ , (2.33) where we have used the relation 2.28 when going from the step 2.31 to 2.32. The dielectric re- sponse function is defined as the total potential generated by the external potential in this self- consistent manner: δV tot (r,t)= Z dt ′ dr ′ ε − 1 (r,r ′ ,t− t ′ )δV ext (r ′ ,t ′ ). (2.34) By comparing 2.33 and 2.34, we obtain the real-space RPA dielectric response function as ε − 1 (r,r ′ ,t− t ′ )=δ (r− r ′ ,t− t ′ )+ Z V c (r− r ′′ ) Z χ (r ′′ ,r ′ ,t− t ′ )dr ′′ . (2.35) We generally consider the response in the frequency domain by Fourier transforming the time dependence. So, we can write ε − 1 (r,r ′ ,ω)=δ (r− r ′ )+ Z V c (r− r ′′ ) Z χ (r ′′ ,r ′ ,ω)dr ′′ . (2.36) 23 In the above formula we have the full susceptibility functionχ . We can find its relation with the non-interacting susceptibilityχ 0 within the RPA framework. In fact,χ 0 is defined as the response of the electron density to the total potential: δρ ind (r,ω)= Z dr ′ χ 0 (r,r ′ ,ω)δV tot (r ′ ,ω). (2.37) Using the relations 2.30 and 2.28, we get δρ ind (r,ω)= Z dr ′ χ 0 (r,r ′ ,ω)+ Z dr ′′ χ 0 (r,r ′′ ,ω) Z dr ′′′ V c (r ′′ − r ′′′ )χ (r ′′′ ,r ′ ,ω) V ext (r ′ ,ω), (2.38) from which we extract the relation betweenχ 0 andχ as χ (r,r ′ ,ω)=χ 0 (r,r ′ ,ω)+ Z dr ′′′ Z dr ′′ χ 0 (r,r ′′ ,ω)V c (r ′′ − r ′′′ )χ (r ′′′ ,r ′ ,ω). (2.39) This is the Dyson seriesχ =χ 0 +χ 0 V c χ which we have seen before and which is now written in the real-space representation. We now give the non-interacting polarization function in the real space. It is χ 0 (r,r ′ ,ω)=2 X i,j f(E i )− f(E j ) E i − E j − ω− iδ ϕ i (r) ∗ ϕ i (r ′ )ϕ j (r ′ ) ∗ ϕ j (r) (2.40) where E i and ϕ i (r) are the eigenenergy and the real-space wavefunction of the i-th electronic orbital. f(·) is again the Fermi function andδ is a finite broadening. 24 In this thesis, we always consider real-space systems on a discrete latticeR n . Therefore, it is more convenient to use the atomic basis formed by atomic orbitals on each site. In this discrete basis, all response functions are expressed as matrices. We will give all formulas explicitly here. The non-interacting polarization matrix is calculated by [χ 0 (ω)] ab =2 X i,j f(E i )− f(E j ) E i − E j − ω− iδ ϕ ∗ ia ϕ ib ϕ ∗ jb ϕ ja . (2.41) Herea andb label the atomic sitesR a andR b . ϕ ia is the component of thei-th electronic eigen- state on the atoma, i.e.ϕ ia =⟨ϕ (r− R a )|ϕ ( r)⟩ withϕ (r− R a ) being the atomic orbital centered on the siteR a . In our calculations, we will obtain these components from empirical tight-binding models. Furthermore, the RPA susceptibility function written in 2.39 now takes the matrix form as χ (ω)=χ 0 (ω)+χ 0 (ω)V c χ (ω) =⇒ χ (ω)=[I− χ 0 (ω)V c ] − 1 χ 0 (ω). (2.42) The RPA dielectric function given in 2.36 has the matrix representation as ε − 1 (ω)= I+V c χ (ω) (2.43) = I+V c [I− χ 0 (ω)V c ] − 1 χ 0 (ω) (2.44) = I+V c [I+χ 0 (ω)V c +χ 0 (ω)V c χ 0 (ω)V c +··· ]χ 0 (ω) (2.45) = I+V c χ 0 (ω)+V c χ 0 (ω)V c χ 0 (ω)+··· (2.46) = 1 I− V c χ 0 (ω) (2.47) 25 where from 2.43 to 2.44 we have used the relation 2.42. Finally we get the RPA dielectric matrix in the atomic basis as ε(ω)= I− V c χ 0 (ω) (2.48) For completeness, we also need the real-space Coulomb interaction V c to be expressed in the atomic basis. The Coulomb matrix is given by [V c ] ab =          e 2 /(ε env |r a − r b |) ifa̸=b, U 0 /ε env ifa=b. (2.49) ε env is the dielectric constant of the background environment. For the on-site Coulomb interac- tion, we use the parameterU 0 . We will now explain how to extract plasmonic excitations from the real-space RPA dielectric matrix given in the Eqn. 2.48. This is not very straightforward because the dielectric function is now a matrix, unlike the scalar functionε(q,ω) in the momentum space. In this case, we define the electron energy loss spectrum (EELS) using the eigenvalues ofε(ω). For each single frequency ω, we calculate the eigenvalues{ϵ n (ω),n = 1,2,...} of the matrixε(ω). We define the “leading order" EELS as EELS(ω)=max n − Im 1 ϵ n (ω) . (2.50) That is, we select the eigenvalue among{ϵ n (ω)} which maximizes− Im h 1 ϵ n(ω) i for each fre- quency. After doing this over the entire frequency range, we get a smooth spectrum EELS(ω) 26 which are peaked at plasmon frequencies ‡ . For any identified plasmon mode ω p , we can further obtain its real-space charge modulation pattern by utilizing the eigenvector of the dielectric func- tion. Suppose that for the freqencyω p , the selected eigenvalue ofε(ω) for calculating EELS(ω p ) is indexed asϵ max (ω). Let us denote the corresponding eigenvector byξ max (ω), we can then get the induced charge-density distribution pattern of the plasmon modeω p in the atomic basis as ρ (ω p )=χ 0 (ω p )ξ max (ω p ). (2.51) Noting that what we have solved is an eigen-problem, it is therefore not appropriate to interpret ρ (ω) as the real physical charge density. Instead, it simply gives a qualitative pattern. Neverthe- less, this is already very advantageous as it can provide us visualization of a plasmon mode in the real space. Specifically, we determine whether a plasmon mode is an extended mode, a confined mode or a localized mode. Similarly, we can define the second EELS by selecting the second maximum of− Im h 1 ϵ n(ω) i at each frequencyω. In most cases, the leading order EELS is enough for our analysis. However, when a plasmon mode atω p is a degenerate mode due to the symmetry of the underlying model structure, the second EELS will also show a peak atω p . In fact, for a two-fold degenerate plasmon mode, the leading order EELS and the second EELS coincide, indicating a degenerate subspace furnished by degenerate eigenvectors of the dielectric matrixε(ω p ). In general, we can define the third, forth, and all orders of EELS in the same manner. In the following discussions, “EELS" always means the “leading order EELS." When we need to consider other orders, we will specify them. ‡ More strictly, we also need the real part of eigenvalueϵ n (ω) to vanish at the plasmon frequency. 27 The plasmon modes identified from the EELS are eigenmodes of the system. However, EELS does not provide more information about how to experimentally excite a specific mode. When a specific external driving field is applied, some plasmon modes can be excited and become bright, while other modes are not excited, i.e. they are dark modes. The actual excitation spectrum sensitively depends on the character of the external perturbing field, especially its polarization direction. Microscopically, for a mode to be excited, the external perturbing potential must induce a non-vanishing transition matrix between the corresponding initial and final states. To show the excitation spectrum with respect to a certain external perturbing potentialV ext (ω), we can utilize the RPA interacting susceptibility matrixχ 0 (ω) calculated in 2.42 to evaluate the induced charge density in the atomic basis as ρ ind (ω)=χ (ω)V ext (ω). (2.52) A smoother induced charge distribution functionρ ind (r,ω) in the continuous real space can be obtained by transformingρ ind (ω) from the atomic basis representation to the continuousr-space representation, using atomic orbitals on each site. The transformation is given by ρ ind (r,ω)= X a [ρ ind (ω)] a ϕ (r− R a ) (2.53) with ϕ (r− R a ) the atomic orbital centered at the site R a . The induced charge generates the induced potential V ind (r,ω) = R dr ′ ρ ind (r ′ ,ω)/|r− r ′ |, leading to spatially distributed induced electric field E ind (r,ω) = −∇ V ind (r,ω). The corresponding excitation spectrum is finally ob- tained from the frequency-dependent induced energyU ind (ω)= R |E ind (r,ω)| 2 dr by integrating 28 the energy density over the space [43]. Similar to the EELS, the induced energy spectrumU ind (ω) is also peaked at plasmon frequencies. In contrast to the EELS(ω) and the qualitative charge den- sity pattern obtained in 2.51, U ind (ω) and ρ ind (r,ω) depend on the actually applied electromag- netic field V ext (r,ω) and result in quantitative induced charge densities, induced electric fields and induced energies, allowing for direct comparisons to experiments. 2.3 TopologicalInsulators Topology has been playing an increasingly important role in modern theory of electrons. This can be dated back to the experimental observation of quantized Hall conductivity in a quantum Hall system. Soon after this, it is realized that the quantized conductivity can be characterized by an integer, called the Chern number, which is a topological invariant. The integer quantum Hall system is known as the first topological insulator (TI) called Chern insulator. In general, TIs are characterized by a fundamentally non-trivial electronic structure, which has an insulating bulk spectrum but conducting edge states on the surface. Moreover, these edge states are symme- try protected and are robust against disorder. In this thesis, we will study plasmonic excitations in TIs. In this section, we briefly introduce the relevant models that will be considered later in Chapter 3. They are one-dimensional Su-Schrieffer-Heeger model (Section 2.3.1) and its straight- forward generalization to the Aubry-André-Harper model (Section 2.3.2), and two-dimensional Su-Schrieffer-Heeger model (Section 2.3.3). 29 0 20 40 60 80 100 n -2 -1 0 1 2 Eigenenergy (eV) trivial SSH topological SSH x ψ s 1 x s 2 s 1 s 2 0 20 40 60 80 100 n -2 -1 0 1 2 Eigenenergy (eV) m-SSH with strong interface m-SSH with weak interface x ψ s 3 x s 4 x s 5 x s 6 s 3 s 4 s 5 s 6 (a) (c) (b) (d) (e) (f) (g) (h) (i) (j) Figure 2.4:ElectronicstructuresofSu-Schrieffer-Heeger(SSH)modelanditsmirrorvari- ant. (a) Illustration of the real-space SSH model on a bipartite tight-binding chain. (b) Corre- sponding energy levels of the SSH model in a 100-site open chain. We observe two zero-energy edge states (s 1 ands 2 ) in the topological phase, whereas no zero-energy edge states are present in the trivial phase. (c-d) Wave functions of the zero-energy statess 1 ands 2 , showing localization at the edges. (e) Illustration of the mirrored SSH model with inversion at the center of the chain. (f) Corresponding energy levels of the m-SSH model on a 103-site open chain. We observe localized zero-energy interface and edge states (s 3− 6 ), depending on the interface and edge properties of the chain. (g-j) Wave functions of the localized interface states 3 and localized interface and edge statess 4,5,6 . 2.3.1 One-DimensionalSu-Schrieffer-HeegerModel We introduce the one-dimensional (1D) Su-Schrieffer-Heeger (SSH) model in this section and briefly discuss its topology. The 1D SSH model is a bi-particle chain with alternating hopping parameters, as illustrated in Fig. 2.4(a). The Hamiltonian can be written as H =t N X m=1 (c † m,B c m,A + H.c.)+t ′ N− 1 X m=1 (c † m+1,A c m,B + H.c.), (2.54) whereN is the number of unit cells, andA andB label the two basis atoms within a unit cellm. Respectively,t andt ′ describe intra- and inter-cell hopping parameters. With periodic boundary 30 conditions, the Hamiltonian can be transformed into the momentum space. For each momentum k, we have [44] H(k)=     0 h ∗ (k) h(k) 0     , (2.55) whereh(k)=h x (k)+ih y (k),h x (k)=Re(t)+|t ′ |cos[ka+arg(t ′ )],h y (k)=− Im(t)+|t ′ |sin[ka+ arg(t ′ )], and a is the lattice spacing. The bulk topological invariant of the SSH model is the winding numberW, which can be evaluated by the formula W = 1 2πi Z π − π dk d dk ln[h(k)]. (2.56) We choose hopping parameterst andt ′ to be real. Fort>t ′ we obtainW =0, and the system is correspondingly in the trivial phase, whereas fort < t ′ , the winding number isW = 1, and the system is in the topologically non-trivial phase. The phase transition occurs att = t ′ , i.e. where the bulk band gap closes. The bulk-boundary correspondence implies that the topology of the SSH model can also be recognized by the number of pairs of zero-energy edge statesN p-es in the case of open boundary conditions. Zero-energy edge states always come in pairs due to the chiral symmetry of the SSH model. Here, the chiral operator is the Pauli matrixσ z such that σ z H(k)σ z =− H(k). (2.57) 31 This leads to the result that, whenE(k) is an eigenenergy ofH(k) with the eigenvector|Ψ( k)⟩, then − E(k) is also an eigenenergy of H(k) whose eigenvector is simply given by σ z |Ψ( k)⟩. These two states are called chiral partners. In fact, in the SSH model, the number of zero-energy edge state pairs is equal to the winding number defined from the bulk, namely, N p-es =W. In Fig. 2.4(b) we show the energy spectrum of the trivial (t = 1.25 eV > 0.75 eV = t ′ ) and topological (t=0.75eV <1.25eV =t ′ ) SSH models of100 sites. The electronic structure in the trivial phase corresponds to a gapped particle-hole symmetric insulator withN p-es =0, whereas in the topologically non-trivial phase we observe N p-es = 1, i.e. we find a pair of degenerate zero-energy electronic states, denoted as s 1 and s 2 in Fig. 2.4(b), in the middle of the band gap. Moreover, their wave functions are localized at the edges of the chain, as shown in Fig. 2.4(c) and Fig. 2.4(d), which confirms that they are edge states. In this case, these two zero-energy edge states are chiral partners of each other. Gapless edge mode(s) can also appear on the interface connecting two topologically distinct insulators. Intuitively, this is because the topological invariant changes when passing over the interface. The interface therefore must close the gap. We can construct a variant of the SSH model by reflecting the simple SSH chain at one edge site. This new model is mirror-symmetric, with an interface in the center [Fig. 2.4(e)], which we call the mirror-SSH (m-SSH) model. We will also study plasmonic excitations in the m-SSH model later, so we also briefly discuss its electronic structure here. The interface connecting two topologically distinct SSH chains supports an additional localized zero-energy state. This kind of topological zero-energy mode was first found by Jackiw and Rebbi [45], and is called the Jackiw-Rebbi mid-gap state. The character of this interface state depends on the property of the interface. Here, by our construction, the interface hopping can be either strongt>t ′ or weakt<t ′ , and the interface and two edges have 32 the same hopping parameter. In Fig. 2.4(f) we show the energy spectrum of the m-SSH model with103 sites for both scenarios. In both cases, we observe zero-energy states in the center of the band gap. Fort>t ′ , there is only one zero-energy state [s 3 in Fig. 2.4(f)], which is localized at the interface [Fig. 2.4(g)]. Fort < t ′ , there are three zero-energy states [s 4 , s 5 ands 6 in Fig. 2.4(f)], which are localized at the interface and at the edges of chain [Fig. 2.4(h-j)] § . 2.3.2 GeneralizedAubry-André-HarperModel The 1D SSH model can be easily generalized to the so-called Aubry-André-Harper (AAH) model which can contain more than two basis atoms in a unit cell. The most general AAH Hamiltonian ofN sites can be written as [46] H = N− 1 X n=1 (t n c † n+1 c n + H.c.)+ N X n=1 v i c † n c n (2.58) where t n =t[1+λ cos(2πbn +ϕ t )], (2.59) v n =vcos(2πdn +ϕ v ). (2.60) § The number of edge and interface states can also be understood in terms of valence-bond decorations corre- sponding to the different states. For the simple SSH model in the topologically trivial state [blue in Fig. 2.4(b)], the resulting valence bond solid connects every site with a neighbor, resulting in a perfect product state of valence bonds. In contrast, in the valence-bond decoration corresponding to the topological phase [red in Fig. 2.4(b)], the two outer bonds remain uncoupled, i.e. dangling bonds, leading to the observeds 1 ands 2 mid-gap zero-energy states. For the m-SSH model with neighboring two strong bonds at the central mirror interface [blue in Fig. 2.4(d)], the resulting three-site strongly coupled object is a non-bonding state, whereas the remaining states are valence bonds. This re- sults in a localized state at the interface. Finally, in the opposite decoration of the m-SSH chain [red in Fig. 2.4(d)], there are dangling bonds at each end of the chain as well as at the mirror interface, resulting in localized zero-energy states at each of these positions. 33 AAH model has many different versions. If v = 0 and t,λ ̸= 0, the modulation is only on the hopping parameters, leading to the off-diagonal AAH model. In contrast, ifv̸= 0 andλ = 0, we get the diagonal AAH model with the onsite energy modulation. The two modulations can exist simultaneously for non-zerov andλ . Moreover, depending on the modulation periodb andd, the model can be commensurate or incommensurate, or even become a quasi-crystal. Generally, for the commensurate AAH model, with increased number of basis atoms in the unit cell, the band structure and edge states become more complicated, depending specifically on how model parameters are modulated. Here, we want to report a general approach which can characterize the band topology and edge states in 1D multi-orbital tight-binding models using simple diagrams. We consider a 1D periodic super-lattice withM orbitals in each super-unit-cell. In the AAH model these orbitals are theT basis atoms. By choosing periodic boundary conditions and Fourier transforming the Hamiltonian with respect to the external degrees of the freedom, we can reduce it to lower-dimensional ˆ H(k), which lives in the internal Hilbert space, spanned by the basis of M orbitals for each momentum k in the first Brillouin zone (BZ). The corresponding energy spectraE n (k) can be obtained by diagonalizing ˆ H(k), and the associated eigenstates|n,k⟩ can be written in the basis ofM orbitals. Below we focus on a specific band, omitting the band index n. Generally, we can write |k⟩= M X a=1 C a (k)|a⟩. (2.61) 34 Figure 2.5: Contourstracedoutbytherelativephases ˜ ϕ 2 intheSSHmodel,ask istaken aroundaloopinthefirstBrillouinZone. (a) shows a complete circle, corresponding to a non- trivial Berry connection atϕ =0. Similarly, (b) shows another example for a fully enclosed area at ϕ =0.4π . In contrast, (c) and (d) show cases without fully enclosed areas, yielding topologically trivial Berry phases,γ =0. 35 The Berry connection is defined as A(k) = i⟨k|∂ k |k⟩, and the Berry phase is calculated via the integralγ = H A(k)dk. Assuming that theM orbitals are orthogonal, we can then calculate the Berry connection from A(k)= X a C ∗ a (k)C ′ a (k), (2.62) and the Berry phase from γ =i I A(k)dk =i X a I C ∗ a (k)C ′ a (k)dk. (2.63) The coefficients C a (k) are complex functions ofk, which can be written asC a (k)=r a (k)e iϕ a(k) , wherer a (k) andϕ a (k) are real functions. Using these two functions we can representC ∗ a (k)C ′ a (k) as C ∗ a (k)C ′ a (k)=r a e − iϕ a [r ′ a e iϕ a +r a e iϕ a iϕ ′ a ] (2.64) =r a r ′ a +ir 2 a ϕ ′ a . Sinceγ is real, we just need to keep the second term in the above expression (there is an extrai outside the integral and summation). The Berry phase is then given by γ =− X a I r 2 a dϕ a =− X a S a . (2.65) HereS a has the geometric interpretation that it is the area swept by the head of the coefficient C a (k) in the complex plane, whenk goes around the first BZ in a loop. 36 A few words about the gauge. The Berry phase is gauge independent, but during the calcula- tion we often need to specify a (usually smooth) gauge to perform our calculations, i.e., taking derivatives. In our approach, this gauge degree of freedom is that we can specify any arbitrary phase for one orbital, say ϕ a , and then all the phases for the other orbitals, ϕ b for b ̸= a, are determined accordingly. So the global phase is not important, but the relative phases between different orbitals are! We can not give arbitrary phases to all orbitals. We therefore go slightly beyond Eq. 2.65 and choose the phase of the first orbital always to be zero,ϕ 1 (k) = 0 for allk. Then, the other phases are determined accordingly and should now be regarded as fixed relative phases with respect to the first orbital. Denoting the relative phases as ˜ ϕ a ( ˜ ϕ 1 ≡ 1), we arrive at our final expression for the Berry phase, γ =− X a̸=1 I r 2 a d ˜ ϕ a =− X a̸=1 ˜ S a . (2.66) A big advantage of using therelativephase is that it is usually a smooth function, even in numer- ical calculations, as it is not arbitrary. The computer may assign arbitrary global phase for each state|k⟩, but the relative phase is always a fixed function of k. In the special case wherer a =1/ √ M for alla, as it is for example the case in the Su-Schrieffer- Heeger (SSH) model, the formula above can be further simplified, i.e., γ =− 1 M X a̸=1 I d ˜ ϕ a =− 1 M X a̸=1 2πv a , (2.67) wherev a is the number of vortices. In the SSH model (a special case of AAH model), it is mapped to the winding number. Fig. 2.5 shows the contours of the relative phases ˜ ϕ 2 in the SSH model, 37 A B D C (a) (b) Figure 2.6: Two-dimensionalSu-Schrieffer-Heegermodel. (a) Real-space illustration of the 2D SSH model. There are four basis atoms in a unit cell. The intra and inter-cell hopping param- eters aret andt ′ in bothx- andy-direction. withk going around a loop in the first BZ. We can see in topological cases (the phase ϕ =0,0.4π ) the relative phase completes a circle, whereas in the other cases it just traces an arc, which does not contribute to a closed area ˜ S 2 , and therefore yields no contribution to the Berry phase. For a general topologically non-trivial band, the relative phase will wind the origin an interger number, sayingn, of times. In this case, the open boundary model will haven edge states coming from this band. Thus, by calculating the relative phase winding, we can predict the edge states properties for a generic model. 2.3.3 Two-DimensionalSu-Schrieffer-HeegerModel In this last small section of the chapter, we briefly introduce the two-dimensional (2D) version of the SSH model. More details of this model is given by [47]. The 2D SSH model is simply a square-lattice model with alternating hopping parameters in both thex- and they-direction, as illustrated in Fig. 2.6(a). There are four basis atoms, A, B, C and D in a unit cell. The intra and inter-cell hopping parameters aret andt ′ . 38 Unlike other 2D TIs, the 2D SSH model has zero Berry curvature everywhere in the Brillouin zone due to the coexisting time-reversal symmetry and inversion symmetry. So it is not a Chern insulator. It is also not a quantum spin Hall insulator. The topological phases of the model can be characterized by the so-called 2D Zak phase [47] which is given by γ = Z BZ dkTr[⟨ψ |i∇ k |ψ ⟩]=          (0,0) ift>t ′ , (π,π ) ift<t ′ . (2.68) Like other TIs, the 2D SSH model can also have mid-gap edge states on the boundaries when the bulk is in a topologically non-trivial phase. These edge states can exist along the four-side boundary for a fully finite-sized open model or along the upper and lower edges of a nano-ribbon structure. The corresponding electronic structures are shown in Fig. 2.6(b). In the next chapter, we will also study plasmonic excitations in the 2D SSH model. 39 Chapter3 PlasmonsinTopologicalInsulators In this chapter, we start to investigate plasmonic excitations in topological insulators in detail, using both the momentum-space approach and the real-space approach. We will first study plasmons in the 1D Su-Schrieffer-Heeger (SSH) model in the momentum space, given in Section 3.1. Specifically, in Section 3.1.1, we will introduce the general framework for calculating dielectric function and plasmonic excitations in multi-band electronic models. We emphasize the importance of basis choices for a consistent and reasonable calculation. Then in Section 3.1.2, we will analyse detailed behavior of plasmon dispersions of the SSH model in the momentum space. We provide both analytical and numerical calculations. Lastly, in Section 3.1.3, we perform the same calculation to an ordinary atomic insulator, showing quite different results from those observed in the SSH model. Then we study plasmons in the 1D SSH model in the real space, given in Section 3.2. We first show the observation of localized plasmons in topologically non-trivial SSH model from the elec- tron energy loss spectrum (EELS) in Section 3.2.1, followed by a discussion on their topological origin in Section 3.2.2. In Section 3.2.3 we briefly show localized interface plasmons observed in the mirror SSH (mSSH) model. The stability of the interface plasmons is then investigated in 40 detail in Section 3.2.4. In Section 3.2.5 we discuss effects of Coulomb interactions on plasmons in the SSH model and the mSSH model, which clearly distinguish these collective excitations from their single-electron counterparts. In Section 3.2.6, we show excitation spectra of the these mod- els under different external perturbing potentials. This illustrate whether an eigenmode indicated in EELS can be experimentally excited by a specific external field or not. Finally, in Section 3.3, we will show our calculations for plasmons in the 2D SSH model. We will consider a finite-sized sample with open boundaries. A fully real-space approach is applied. Similarly, we also observe edge plasmons on the boundary of the sample. However, on the 1D boundary of the 2D sample, these edge plasmons are propagating modes showing a nice disper- sion relation. 3.1 Momentum-SpacePlasmonsintheSu-Schrieffer-Heeger Model 3.1.1 GeneralMethodforBandElectrons In this section, we discuss a general method, based on the random phase approximation, of cal- culating plasmonic excitations in band electrons from the tight-binding approach, followed by specific analysis of the SSH Model in the next section. The common feature of the model is it has an external translational invariant degree of freedom over the unit cell, and meanwhile, non- translational degrees of freedom with a unit cell. Therefore the treatment is neither fully in the real space nor fully in the momentum space, but in a combined version. In the electronic structure theory this is nothing but the energy spectrum forms bands in the first Brillouin zone (1BZ) due 41 A B 1 2 … … A B Figure 3.1: One-dimensional tight-binding model of two orbitals. In each unit cell, there are two distinct atoms A and B which can in principle host different shapes of atomic orbitals. The lattice spacing isa and the inter-cell basis sepatation isδ . to the internal degrees. In plasmonic calculations, more careful attentions may be need when we evaluate the polarization function and especially Coulomb interaction among internal degrees of freedom. The non-triviality comes from the basis choice, which will not affect the final physical results, but definitely affect intermediate steps of calculations both analytically and numerically. To illustrate the method, let us consider a simple one-dimensional (1D) lattice with a basis, whose primitive cell consists of two distinct atoms, A and B, which can host different orbitals ϕ A andϕ B , as depicted in Fig. 3.1. The method can be straightforwardly generalized to any dimen- sionality with more than two basis orbitals in a primitive cell. In this 1D bi-particle lattice, we set the lattice spacing to bea and the separation between two atoms in the primitive cell to beδ . This simple structure can be parameterized to describe some well-known two-band models, such as the SSH model [48] and the Rice-Mele model [49]. 42 It is firstly important to set up a basis with which we express the Hamiltonian matrix and then proceed to calculate response functions. Following the standard tight-binding approach, we form a linear combination of the atomic orbitals (LCAO) for each basis atom [42], ψ A/B k (r)= 1 √ N X n e ik·R A/B n ϕ A/B (r− R A/B n ), (3.1) whereR α n is the position of atom α (α ∈ {A, B}) in the n-th unit cell, andk is the momentum vector in the first Brillouin zone (1BZ). From Fig. 3.1 it is clear that R B n − R A n =δ = δ ˆ x. These LCAOs serve as basis functions in which we can expand the electronic states and write down the Hamiltonian matrix. Due to the translational invariance over the unit cell, the Hamiltonian can be written for each momentum k in the 1BZ, where the Hilbert space is spanned by the basis functions ψ A k (r) and ψ B k (r). We consider hoppings only between nearest neighbors of atomic sites, as indicated byt 1 andt 2 in Fig. 3.1. Hence, the Hamiltonian matrix is given by H(k)=     V A t 1 e ik·δ +t 2 e − ik·(a− δ ) t ∗ 1 e − ik·δ +t ∗ 2 e ik·(a− δ ) V B     (3.2) whereV A andV B are on-site energies of the basis atom A and the basis atom B. In principle,V A , V B ,t 1 andt 2 can be obtained from first-principles calculations or determined from experiments. Here in our model simulation, we treat them as tunable parameters which can be chosen at will, in order to analyze plasmonic properties in various parameter regimes. By diagonalizingH(k), we get the electronic eigenenergiesE n (k) and corresponding eigenstatesΦ n (k). In the current case there are totally two bands ofE n (k) and each eigenstateΦ n (k) has two components on two LCAO basis. 43 We can further evaluate the non-interacting polarization functionΠ 0 . Recalling that in Chap- ter 2 we have discussed the full momentum-space approach and the real-space approach, here we will combine the both, namely, the momentum-space description is applied over the external degree of freedom while the real-space description is kept for the internal degrees of freedom (LCAOsψ A andψ B , not original atomic orbitalsϕ A andϕ B ). We therefore have a two-by-two ma- trix form ofΠ 0 (q,ω) for eachq andω, whose matrix elements in the LCAO basis are explicitly given by [Π 0 (q,ω)] αβ =2 X k,n,n ′ f n ′ ,k+q − f n,k E n ′ ,k+q − E n,k − ω− i0 + × Φ ∗ n,k,α Φ n ′ ,k+q,α Φ ∗ n ′ ,k+q,β Φ n,k,β . (3.3) We use Greek letters here to label the basis components. Φ n,k,α is simply the component of the state Φ n (k) on the LCAO orbital of the sublattice α . Similarly, we can then write the Coulomb interaction V(q) in the same basis, which is, in principle, also a two-by-two matrix for each momentumq. The RPA dielectric function naturally follows as ε(q,ω)=I− V(q)Π 0 (q,ω), (3.4) which is, again, also a two-by-two matrix. At this point we want to state and then explicitly prove that in the basis constructed from the LCAOs all the Coulomb matrix elements are identical. This is to say thatV αβ (q)≡ V(q), which do not really depend on the sub-lattice indices as labeled. This allows us to avoid the matrix product calculation in Eqn. 3.4 and just use a scalar Coulomb interaction function V(q) multiplied by a scalar polarization function with a slightly modified form. We want emphasize that this is often 44 only true in the LCAO basis. In many literatures and books, the scalar forms are used but without specifying the basis requirement. Surely, the final results of the physical spectrum do not depend on the basis choice, however, the evaluating process, especially the numerical implementation needs this careful attention on basis consistency. In other words, as long as we are not using the LCAO basis (which is very common), then matrix production must be performed and numerically programmed. In fact, this is because the explicit form of electronic eigenstates will depend on basis choice although eigenenergies do not, so the polarization matrix elements evaluated by Eqn. 3.3 also depend on the basis choice. Let us start to prove that in the basis constructed from the LCAOs all the Coulomb matrix elements are identical. We will prove for the general case that is not limited to two internal orbitals. The LCAO for a basis atom located atτ α is constructed as ψ α k (r)= 1 √ N X n e ik·(R α n +τ α ) ϕ (r− R α n − τ α )= e ik·τ α √ N X n e ik·R α n ϕ (r− R α n − τ α ). (3.5) We can evaluate Coulomb interaction matrix elements in this basis. Generally, we have V αβ (k)=⟨ψ α k (r)|V|ψ β k (r)⟩ = e − ik·τ α e ik·τ β N X n,n ′ e − ik·(Rn− R n ′) ⟨ϕ (r− R α n − τ α )|V|ϕ (r− R β n ′ − τ β )⟩ = e − ik·τ α e ik·τ β N X n,n ′ e − ik·(Rn− R n ′) e 2 |R n − R n ′ +τ α − τ β | . (3.6) 45 From this general formula, we can see that, for diagonal terms,α =β , we have V αα (k)=⟨ψ α k (r)|V|ψ α k (r)⟩ = 1 N X n,n ′ e − ik·(Rn− R n ′) e 2 |R n − R n ′| =FT e 2 R (3.7) For off-diagonal terms that α ̸=β , lettingδ τ =τ α − τ β , we get V αβ (k)=⟨ψ α k (r)|V|ψ β k (r)⟩ = e − ik·δ τ N X n,n ′ e − ik·(Rn− R n ′) e 2 |R n − R n ′ +δ τ | =e − ik·δ τ e ik·δ τ FT e 2 R =FT e 2 R . (3.8) In the last line we have used the shifting theorem of Fourier transformation. Clearly, we can see that the diagonal elements and off-diagonal elements are all equal, i.e. V αβ (k) ≡ V(k). In this situation, when we proceed to evaluate the matrix production ofV(q) andΠ 0 (q,ω), we will get [V(q)Π 0 ] αβ = X γ [V(q)] αγ [Π 0 ] γβ =V(q) X γ [Π 0 ] γβ =V(q) ˜ Π β 0 . (3.9) 46 So the matrix structure will look like V(q)Π 0 =V(q)             ˜ Π α 0 ˜ Π β 0 ˜ Π γ 0 ··· ˜ Π α 0 ˜ Π β 0 ˜ Π γ 0 ··· ˜ Π α 0 ˜ Π β 0 ˜ Π γ 0 ··· . . . . . . . . . . . .             . (3.10) The only non-zero eigenvalue of this matrix isλ =V(q)( ˜ Π α 0 + ˜ Π β 0 + ˜ Π γ 0 +··· ). Denoting now a scalar polarization function byΠ 0 (q,ω)= ˜ Π α 0 + ˜ Π β 0 + ˜ Π γ 0 +··· , we finally arrive at a scalar RPA dielectric function, ε(q,ω)=1− V(q)Π 0 (q,ω). (3.11) In fact,Π 0 (q,ω) can now be evaluated by a very compact formula which is given by Π 0 (q,ω)=2 X k,n,n ′ (f n ′ ,k+q − f n,k ) E n ′ ,k+q − E n,k − ω− i0 + ×|⟨ Φ n,k |Φ n ′ ,k+q ⟩| 2 . (3.12) For the following analysis, we define a quantity called overlap function that O nn ′(k,k+q)≡|⟨ Φ n,k |Φ n ′ ,k+q ⟩| 2 . (3.13) 47 -0.5 0 0.5 k/ (a.1) a = 2, = 0.1 (a.2) a = 2, = 0.5 (a.3) a = 2, = 1 (a.4) a = 2, = 1.5 (a.5) a = 2, = 1.9 0 0.5 1 q/ -0.5 0 0.5 k/ (b.1) 0 0.5 1 q/ (b.2) 0 0.5 1 q/ (b.3) 0 0.5 1 q/ (b.4) 0 0.5 1 q/ (b.5) 0 1 Figure 3.2: OverlapfunctionsintheSSHmodelfordifferentbasisatomseparations. δ is the inter-cell atomic separation. Hoppings parameterst 1 andt 2 are set to be0.5 eV and1.5 eV. (a.1-a.5) Intra-band overlap functions. (b.1-b.5) Inter-band overlap functions. The form of 3.12 is seen in many literatures and the such defined overlap function is also called band coherence factor sometimes [50]. For the dielectric function we can get the electron energy loss spectrum (EELS) by evaluating the negative imaginary part of its inverse: EELS(q,ω)=− Im 1 ε(q,ω) . (3.14) By tracking the singularities of the EELS outside the particle-hole (p-h) continuum, we can iden- tify plasmon dispersion(s). Note that the overlap functions in Eq. 3.13 fully capture the informa- tion of internal orbitals, such as the separation parameter δ , which is the reason for it to be a central quantity investigated in upcoming sections. 3.1.2 Momentum-SpaceDispersions Let us now consider the one-dimensional (1D) Su-Schrieffer-Heeger (SSH) model parameterized by setting unequal hopping matrix elementst 1 ̸=t 2 while keeping zero on-site energies for both 48 basis atoms V A = V B = 0. The model has two electronic bands separated by an energy gap of E g = 2|t 1 − t 2 |. The topology of this model has already been discussed in Chapter 2. Using the eigenstates calculated in the LCAO basis, the overlap functions take a simple explicit form as O nn ′(k,k+q)=          1 2 + 1 2 cos(φ k,q ) ifn=n ′ , 1 2 − 1 2 cos(φ k,q ) ifn̸=n ′ , (3.15) where φ k,q =qδ +arg(z k,q ) =qδ +arg t 1 +t 2 e − ika− iqa t 1 +t 2 e − ika . (3.16) We have replaced vectorsk andq byk andq ranging over[− π a , π a ) because of the 1D space. From Eqns 3.15 and 3.16, we can see that the overlap functionO nn ′(k,k+q) is closely related to the topology of the SSH model via the phase φ k,q . As we recall from Section 2.3.1, the two topologically distinct phases of the SSH model corresponding to the cases oft 1 >t 2 andt 1 <t 2 . We show the variation of the second term in the phase φ k,q for two limiting cases: (1) when t 1 /t 2 → 0, arg(z k,q )→− qa; and (2) whent 2 /t 1 → 0, arg(z k,q )→ 0. Furthermore, the overlap function depends on the real-space configuration of the internal degrees of freedom, i.e., δ , via the first term in the phase φ k,q [see Eqn. 3.16]. The overlap function is a factor in the polariza- tion function 3.12, which ultimately determines the dielectric response function and plasmonic excitations. Thus, we expect that plasmon dispersions are tunable viaδ or viat 1 /t 2 . 49 We first study the overlap function before evaluating the plasmon dispersions. We fix hopping parameters to bet 1 =0.5 eV<t 2 =1.5 eV, while changingδ , the real-space separation between two basis atoms. In Fig. 3.2, we show the overlap functions O nn ′(k,k +q) for both intra-band transitions [n = n ′ , Figs. 3.2(a.1)-(a.5)] as well as inter-band transitions [n ̸= n ′ , Figs. 3.2(b.1)- (b.5)] for different intra-cell atom separation δ . It can be clearly seen how the intra-cell separation δ affects the overlap function. For large δ , the intra-band overlap is close to1 [Fig.3.2(a.5)] over the entire(k,q) parameter space, whereas the inter-band overlap function is close to0 [Fig.3.2(b.5)]. From Eqn. 3.15, it follows that there is a complementarity of the intra- and inter-band overlap functions in the SSH model, namely, they add up to 1. Therefore, when the intra-band (inter-band) overlap is enhanced, the inter-band (intra-band) overlap is suppressed. This complementarity is generally true for multi-band models in any dimensionality, which can be proved by using the completeness relation of electronic eigenstates, i.e. for any bandn, X n ′ O nn ′(k,k+q)= X n ′ |⟨Φ n,k |Φ n ′ ,k+q ⟩| 2 =1. (3.17) We specify two different regimes depending on parameters δ , t 1 and t 2 . As we see from Fig. 3.1, the hopping parameter over the distanceδ ist 1 . In the natural scenario, whenδ is small, t 1 is large. We call this the dimerized (D) regime. Figs 3.2(a.4-a.5) and Figs 3.2(b.4-b.5) belong to the D regime. In the opposite, when both δ and t 1 are small at the same time, we call it the anti-dimerized (AD) regime. This is unnatural and may be realized artificially. Figs 3.2(a.1-a.2) and Figs 3.2(b.1-b.2) belong to the AD regime. Later we will shown that the D regime and AD regime support enhanced low-energy plasmons and high-energy plasmons, respectively, due to enhanced intra-band overlap function and inter-band overlap function. 50 We now proceed to calculate plasmon dispersions in the SSH model and explore effects of overlap function on it. We set the Fermi level to be E F = − 1.5 eV to partially fill the low- energy band E 0,k while leaving the high-energy band E 1,k is empty, such that both intra-band and inter-band transition channels are open, leading to gapless low-energy plasmons and gapped high-energy plasmons. Before showing the numerical results of plasmon dispersions, we analyse the low-energy plasmons in detail semi-analytically and explicitly show the effects of inter-band and intra-band overlap functions. The low-energy plasmons arise from the intra-band polarization within the metallic band E 0,k , subject to the inter-band dielectric screening. Too see this, we re-write the full dielectric function as ε(q,ω)=1− V(q)[Π intra 0 (q,ω)+Π inter 0 (q,ω)] =ε inter (q,ω)[1− W(q,ω)Π intra 0 (q,ω)]. (3.18) Here ε inter (q,ω) is the inter-band dielectric function which is evaluated as ε inter (q,ω) = 1− V(q)χ inter 0 (q,ω). W(q,ω) = [ε inter (q,ω)] − 1 V(q) is the screened Coulomb interaction. Plasmon frequencies solve the equation ε(q,ω p ) = 0. From 3.18, we can see that this is equivalent to consider the metallic band (intra-band) polarization under the screened Coulomb interaction W(q,ω). The solutions of the plasmons satisfy 1− W(q,ω p )ReΠ intra 0 (q,ω p )=0 (3.19) 51 outside the particle-hole continuum. The intra-band polarization function can be evaluated di- rectly using the formula 3.12 withn=n ′ =0: Π intra 0 (q,ω)≃ k F X k=− k F 4(E 0,k+q − E 0,k )O 00 (k,k+q) (ω+i0 + ) 2 − (E 0,k+q − E 0,k ) 2 . (3.20) O 00 (k,k+q) represents the intra-band overlap function, as illustrated above in Figs. 3.2(a.1)-(a.5). For the screened Coulomb interaction W(q,ω), we need to evaluate the inter-band dielectric function. For the frequency ω in the range of low-energy plasmons, we can approximately use the static screening from inter-band transitions, namelyW(q,ω)≈ W(q,ω =0)=[ε inter (q,ω = 0)] − 1 V(q). The static inter-band dielectric function for the SSH model takes a simplified form ε inter (q)≃1− 4V(q) k F X k=− k F O 01 (k,k+q) E 0,k +E 0,k+q , (3.21) where O 01 (k,q) is the inter-band overlap function which can be evaluated from Eq. 3.15 and is plotted in Figs. 3.2(b.1)-(b.5). In deriving Eq. 3.21, we made use of the chiral symmetry of the energy bands,E 0,k =− E 1,k . Finally, for the bare Coulomb interactionV(q) in the 1D momentum space, it needs more care because the Fourier transformation integral diverges. Here we actually consider a quasi-one-dimensional system with a small cross-sectionl 2 (l ≪ chainlength). We useV(q) = 2e 2 K 0 (ql)/a referred from [51], whereK 0 (·) is the zeroth modified Bessel function of the second kind. In our simulations,l is taken to be half of the lattice spacinga. 52 We further proceed to derive explicit effects of the overlap functions on plasmon frequencies ω p . First, as we are interested in the plasmon solutions|ω p | 2 ≫ (E 0,k+q − E 0,k ) 2 [52], where we simplify the real part of the intra-band polarization function to Re[Π intra 0 (q,ω p )]≈ 4 ω 2 p Z k F k F − q dkg(k)O 00 (k,k+q)× (E 0,k+q − E 0,k ). (3.22) Here,g(k) is the density of states. For the half-filled low energy band, we make a linear approx- imation to the energy difference around k F , E 0,k+q − E 0,k ≈ ∂ k E 0,k | k F q =v F q, (3.23) which is adequate for an appreciable range ofq (|q|⪅0.4 − 1 ). This leads to Re[Π intra 0 (q,ω p )]≈ 4qv F ω 2 p Z k F k F − q dkg(k)O 00 (k,k+q). (3.24) Putting this back into Eq. 3.19, we find that the plasmon frequencies are approximately given by ω 2 p ≈ 4qW(q)v F Z k F k F − q dkg(k)O 00 (k,k+q). (3.25) From this expression, we recognize an explicit dependence of the low-energy plasmonsω p on the Fermi velocity v F , the screened Coulomb interaction W(q) which is negatively correlated with the inter-band overlap function, and an integral of the intra-band overlap function over a range q around k F . Thus, the overlap functions affect plasmons through at least two aspects. First, the screened Coulomb interactionW(q) depends on the inter-band overlap function. When the 53 0 0.5 1 q/ (Å 1 ) 0 1 2 3 4 5 6 7 (eV) = 0.1 = 0.5 = 1.0 = 1.5 = 1.9 Figure 3.3: Momentum-space plasmons in SSH model. Dispersion relations for different intra-cell atomic separationsδ are given. Hopping parameters are the same to those in Fig. 3.2. The dark areas represent the particle-hole continuum. inter-band overlap function is large, the screening effect is stronger and the Coulomb interaction W(q) is further reduced, which in turn leads to a softening of the plasmon dispersion. This corresponds to the smallδ limit (the AD regime) for intermediateq [for example, see Figs. 3.2(b.1) and (b.2)]. In contrast, for largeδ (the D regime), the inter-band screening effect is weak due to the reduced inter-band overlap function [Figs. 3.2(b.4) and (b.5)]. Second, the plasmon frequency is proportional to the integral of the intra-band overlap function, R k F k F − q dk g(k)O 00 (k,k + q), around the Fermi vector k F . Thus, in general, for larger intra-band overlaps, the low-energy plasmon dispersion is enhanced to lie higher above the Landau damping region. We expect the high-energy branch of plasmons follow the opposite trend due to the complementary of inter- and intra-band overlap functions. 54 We are now ready to show numerical results for the plasmon dispersions of the SSH model for different basis atom separations δ , by showing the EELS in Fig. 3.3. As expected, we observe that changing δ affects both the low-energy and the high-energy plasmons due to the changed overlap functions. Generally, as one branch becomes more prominent, the other branch is sup- pressed, which can be attributed to the complementarity of the intra- and inter-band overlap functions as mentioned before. For largerδ , the low-energy plasmon branch lies further outside the intra-band electron-hole continuum and therefore remains undamped for a larger range of momentaq. In this regime, we can see that the intra-band overlap function is large over the en- tire(k,q) space, whereas the inter-band overlap function is almost zero globally [see Figs. 3.2(a.5) and (b.5)], indicating negligible inter-band screening effects on the low-energy plasmons. In con- trast, the high-energy plasmon branch becomes more prominent with gradually decreasing δ . And meanwhile, the low-energy plasmon branch is softened and moves closer to the electron- hole continuum. This is due to the enhanced inter-band overlap function and reduced intra-band overlap function [see Figs. 3.2(a.1) and (b.1)]. Thus, we can tune the parameterδ to selectively en- hance the low-energy or the high-energy plasmon branch and suppress the other one, supposed thatt 1 andt 2 are kept fixed. It is noticeable from Fig. 3.3 that the effect vanishes for small momenta q. This matches our observation of the overlap functions in Fig. 3.2 that the most vulnerable part is the intermediate q-regime. Here we provide a quantitative analysis on this by performing a smallq expansion of 55 the polarization functions. To do this, we reformulate the non-interacting polarization function Π 0 (q,ω) of the SSH model in the form Π 0 (q,ω)=2 X k,n,n ′ f n,k O nn ′(k,k+q)T nn ′(ω,k,q), (3.26) where T nn ′(ω,k,q)= 2(E n ′ ,k+q − E n,k ) (ω+i0 + ) 2 − (E n ′ ,k+q − E n,k ) 2 . (3.27) For the above derivation, we used the following properties: (1) change of variables,k+q→− k, justified by the periodicity (in k) of the sum in Eq. 3.26; (2) the time reversal (TR) symmetry, lead- ing toE n,k =E n,− k andf n,k =f n,− k ; (3) the overlap function obeyO nn ′(k,k+q)=O n ′ n (k+q,k), which is easily seen from Eq. 3.13. By separating the summed terms into the overlap function O nn ′(k,k + q) and the T-function T nn ′(ω,k,q), we can perform small-q expansion for each of them. For the intra-band polarization function, we setn=n ′ =0 and get Π intra 0 (q,ω)=2 k F X k=− k F O 00 (k,k+q)T 00 (ω,k,q). (3.28) To obtain the small-q behavior of the intra-band overlap functionO 00 (k,k+q)=(1+cosφ k,q )/2, we first expand cosφ k,q as cosφ k,q =1− q 2 2 (∂ q φ k,q ) 2 | q=0 − q 3 2 [∂ q φ k,q ∂ 2 q φ k,q ]| q=0 +o(q 4 ). (3.29) 56 There is no linear term in q. Thus, for small q, the overlap functions have negligible effects on the dispersion. The cubic term appears because φ k,q =qδ +arg(t 1 +t 2 e − ika− iqa )− arg(t 1 +t 2 e − ika ) (3.30) is neither even nor odd in q. The derivatives in the expansion coefficients can be calculated explicitly. They are ∂ q φ k,q | q=0 =δ +∂ q arg(t 1 +t 2 e − ika− iqa )| q=0 =δ − a t 2 2 +t 1 t 2 cos(ka) t 2 1 +t 2 2 +2t 1 t 2 cos(ka) ≜α δ (k) (3.31) and ∂ 2 q φ k,q | q=0 =∂ k α δ (k)= a 2 t 1 t 2 sin(ka)(t 2 1 − t 2 2 ) [t 2 1 +t 2 2 +2t 1 t 2 cos(ka)] 2 ≜β (k). (3.32) α δ (k) is an even function ink and depends onδ , whereasβ (k) is an odd function ink and does not depend onδ . Finally, the intra-band overlap function, up to third order inq, is given by O 00 (k,k+q)=1− α 2 δ (k)q 2 4 − α δ (k)β (k)q 3 4 +o(q 4 ). (3.33) For the small-q expansion of T 0,0 (ω,k,q), since we are interested in plasmon solutions where ω≫ (E n ′ ,k+q − E n,k ), we can write the real part approximately as T 00 (ω,k,q)≈ 2 ω 2 q∂ k E 0,k + q 2 2 ∂ 2 k E 0,k + q 3 3! ∂ 3 k E 0,k + q 4 4! ∂ 4 k E 0,k +··· . (3.34) 57 Combining 3.33 and 3.34, we can obtain the small-q behavior ofΠ intra 0 (q,ω). The summation over k goes from− k F to k F . The linear term vanishes because ∂ k E 0,k is an odd function in k. The coefficient (apart from 4 ω 2 ) for the second order term,q 2 , is C 2 = k F X k=− k F 1 2 ∂ 2 k E 0,k . (3.35) For the free electron gas with a quadratic dispersion, this reduces to the familiar result which is the inverse of the electron mass (apart from a constant factor). We can see that C 2 does not depend on the overlap function and δ , which means the intra-band polarization function is not affected by δ even to the second order ofq. For theq 3 term, the coefficient C 3 = k F X k=− k F 1 3! ∂ 3 k E 0,k − 1 4 α 2 δ (k)∂ k E 0,k (3.36) also vanishes because the summed term is odd ink. The fourth order coefficient is C 4 (δ )= k F X k=− k F 1 4! ∂ 4 k E 0,k − 1 8 α 2 δ (k)∂ 2 k E 0,k − 1 4 α δ (k)β (k)∂ k E 0,k . (3.37) This is non-vanishing because the summed term is even in k, and it implicitly depends on δ throughα δ (k). So, for the intra-band polarization function, the overlap function becomes relevant from the fourth order term, i.e. q 4 -term. We stop here by neglecting higher orders and write Π intra 0 (q,ω) as Π intra 0 (q,ω)= 4 ω 2 [C 2 q 2 +C 4 (δ )q 4 +o(q 5 )]. (3.38) 58 For the inter-band polarization function, we setn=0 andn ′ =1, which gives Π inter 0 (q,ω)=2 k F X k=− k F O 01 (k,k+q)T 01 (ω,k,q). (3.39) The smallq expansion forO 01 (k,k+q) can be simply obtained by1− O 00 (k,k+q), which starts with theq 2 -term. ForT 01 (ω,k,q), we also use its static limit withω = 0, which turns out to be a good approximation for low-energy plasmons. Due to the chiral symmetry of the model, we haveE 1,k =− E 0,k . T 01 (k,q) can then be expanded as T 01 (k,q)≈ 1 E 0,k 1− 1 2E 0,k (E 0,k+q − E 0,k )+ 1 4E 2 0,k (E 0,k+q − E 0,k ) 2 +··· . (3.40) Combining O 01 (k,k +q) and T 01 (k,q), we observe that there is also no linear term in q in the inter-band polarizationΠ inter 0 (q). The lowest order inq is the quadratic term whose coefficient is ˜ C 2 (δ )= 1 2 k F X k=− k F α 2 δ (k) E 0,k , (3.41) which implicitly depends onδ viaα δ (k). Unlike the intra-band polarization, the inter-band po- larization already starts to be relevant to the overlap function from the second orderq 2 . Hence, we write Π inter 0 (q)= ˜ C 2 (δ )q 2 +o(q 3 ). (3.42) 59 The inter-band screening is therefore ε inter (q)=1− V(q) ˜ C 2 (δ )q 2 . (3.43) Because ˜ C 2 is negative (E 0,k is negative), the inter-band screening is typically larger than 1, which means that the screened Coulomb interactionW(q) = [ε inter (q)] − 1 V(q) is weakened. Typically, for small q, tuning the overlap functions firstly affects the inter-band screening (from q 2 ) and then the intra-band polarization (fromq 4 ). The approximated solutions for low-energy plasmons at smallq can be determined by setting 1− W(q)Π intra 0 (q,ω p )=0, (3.44) which yields ω 2 p ≈ 4V(q)[C 2 q 2 +C 4 (δ )q 4 ] 1− V(q) ˜ C 2 (δ )q 2 . (3.45) Theδ -independent part is simply obtained by ignoring ˜ C 2 (δ ) and the last two terms inC 4 (δ ) [see Eq. 3.37]. We plot thisδ -independent part as a white dashed line in Fig. 3.3. We can see it nicely follows the trend of all low-energy plasmon dispersion curves for smallq. Asq gets larger, they start to deviate. At this point we want to emphasize that we are not aiming to distinguishing between the topologically trivial and non-trivial phases of the SSH model based on the bulk behavior of the overlap function and the corresponding plasmon dispersions. In fact, this is also not achievable because the underlying topology of the model is not characterized by the bulk spectrum. This is 60 0 0.5 1 /a 0 1 2 t 2 /t 1 AD AD D D -1 0 1 Figure 3.4: TuningtheSSHmodelbetweendimerizedandanti-dimerizedregimes. Phase factorcos(φ k,q ) in Eq. 3.15 atk =0 andq =π/ 2 for different δ/a andt 2 /t 1 is shown. The vertical axis t 2 /t 1 can be extended to infinity without making any essential difference to the plot. Red indicates the dimerized (D) regime, and blue denotes the anti-dimerized (AD) regime. Tuning from D to AD (or AD to D) can be done by either changingδ , i.e., along a horizontal path, or by changing hopping parameterst 1 andt 2 , i.e., along a vertical path. Whenδ/a =0.5, switchingt 1 andt 2 makes no difference. 61 similar to the single-particle topology where the topological invariant is not evaluated from the momentum-space energy spectrum, but instead from the energy eigenstates. It seems to imply that we may be able distinguish different topological phases of the model using plasmonic eigen- states. As we will soon see in Section 3.2, this is indeed the case when we consider the open SSH model using the real-space treatment. Here, the essential distinction is actually the D and AD scenarios [30] as we defined before. In our discussion above, we tune the model from D to AD by decreasingδ fromδ >a/ 2 toδ t 2 = 0.5 eV show the same behavior as in Fig. 3.2 fort 1 = 0.5 eV < t 2 = 1.5 eV, withδ replaced bya− δ . In Fig. 3.4, we summarize these two ways of tuning between D and AD regimes. Specifically, when δ = a/2, switchingt 1 andt 2 has no effect on the bulk properties of the model. However, when the model is cut into a finite chain with open boundaries, switching t 1 andt 2 fundamentally changes the electronic as well as the plasmonic properties on the system boundaries [53, 29, 30]. We close this small section with some considerations on possible experimental realizations of the D and AD regimes. In the above discussion, we freely tune the inter-atomic separationδ and hopping parameters t 1 and t 2 independently. In the context of solid materials, this is obviously 62 (a) V(x) x t 1 t 2 E > E < a (b) t 1 t 2 E < E > a Figure 3.5: Illustration of trapped atoms in one-dimensional periodic potential wells. Here, the separation between basis atoms,δ , is held fixed, and we consider two different regimes: (a) strong intra-basis tunneling,t 1 >t 2 , and (b) strong inter-basis tunneling,t 2 >t 1 . not realistic, as the hopping matrix element between two sites typically increases when the inter- particle separation is reduced, due to more wavefunction overlap between neighboring sites. For example, this is the case when a Peierls instability occurs, whereby the phonon-induced structural dimerization of the underlying lattice leads to an increased hopping matrix element between neighboring atoms within the same basis and a decreased inter-basis hopping. In this case, for example ift 1 = 0.5 eV andt 2 = 1.5 eV, it impliesδ > a − δ , i.e., the large-δ (D) regime appears to be the more natural in the solid state context. For this situation we found above that the low-energy plasmons are dominant, with small inter-band screening effect. However, here we consider an alternative, synthetic approach to realize this as well as the opposite (AD) situation, i.e., small inter-particle separation (δ . Meanwhile, the potential barrier between atom B and atom A in the next unit cell has a relatively larger widtha− δ (a is the lattice spacing which is always fixed) with a very small heightE < . If we properly tune the barrier heightsE > andE < , it is in principle possible to have a smaller tunneling at the smaller separation. To illustrate this, we perform a semi-quantitative estimate by assuming the potential barriers are square barriers. Then, the tunneling probability through a barrier of widthd and heightE h is proportional toe − √ E h d . In order to havet 1 < t 2 , we requiree − √ E>δ <e − √ E<(a− δ ) , which leads to the condition E> E< > (a− δ ) 2 δ 2 . 3.1.3 AnOrdinaryTwo-bandInsulator Although we mentioned before that we are not aiming to distinguishing topological phases of the underlying electronic system using the bulk plasmon dispersions, it does not mean that the topology of the model is totally irrelavent to its plasmonic excitations. In fact, we notice that the selective tunability of high-energy and low-energy plasmons viaδ or viat 1 /t 2 is particulary remarkable in systems with topologically non-trivial band structures. We have not fully figured out the exact reason behind this, but we have at least verified this for the 1D SSH model just discussed, as well as several two-dimensional (2D) topological insulators [30]. Actually, for the 1D SSH model, we can explicitly see the interplay between the topology and the intra-cell atomic separation from the phase factor given in Eq. 3.16. 64 Figure 3.6: Plasmonsinanordinarytwo-bandinsulator. (a) Plasmon dispersions in an ordi- nary insulator for different choices of the separation between basis atoms, δ . The dark areas rep- resent the particle-hole continuum. (b)-(k) Overlap functions for the various parameters shown in (a). 65 As a benchmark, we will show that the same tuning technique does not apply to a purely ordinary insulator. The ordinary insulator is parametrized byt 1 =t 2 =1.0 eV andV A =− V B = 1.0 eV. This parameter set generates the same bandwidth and energy gap as the SSH model discussed above. So, they have exactly same energy band structure. In this ordinary insulator, the energy gap is opened up by unequal on-site energies of the two atoms in the unit cell. We show the plasmon dispersions for different δ in Fig. 3.6(a). Unlike in the SSH model, there is no significant tuning effects by varing δ . Accordingly, the overlap functions also do not show a clear trend by tuningδ [see Figs. 3.6(b)-(k)]. 3.2 Real-Space Plasmons in an Open Su-Schrieffer-Heeger Model In this section we study the real-space dielectric response in an open-ended Su-Schrieffer-Heeger (SSH) Model using the real-space RPA approach introduced in Section 2.2. As we have discussed before, the topologically non-trivial open SSH model has zero-energy mid-gap states living in the band gap. These are topologically protected edge states that are exponentially localized at the boundaries of the system. We will see that they have clear impact on plasmons excited in the real space. 66 3.2.1 ElectronEnergyLossSpectrumandLocalizedPlasmons We consider the real-space Hamiltonian of the SSH model which is given by ˆ H =t 1 N X m=1 (|m,B⟩⟨m,A|+h.c.)+t 2 N− 1 X m=1 (|m+1,A⟩⟨m,B|+h.c.). (3.46) It describes a one-dimensional chain with alternating hopping parameters t 1 and t 2 . We use A and B to label the two basis atoms in a unit-cell, and N is total number of unit-cells. In the following calculation, we set N = 50, indicating a total number of 100 sites, and consider two topologically distinct phases corresponding tot 1 =1.25ev>t 2 =0.75eV (the trivial phase) and t 1 = 0.75eV < t 2 = 1.25eV (the topologically non-trivial phase). The spacing between any two sites is set to bea=2 Åuniformly. The Fermi level is set to beE F =0eV, corresponding to half- filling. We calculate the electron energy loss spectra (EELS) which are shown in Fig. 3.7(a). In both phases, we find a low-energy plasmon continuum (LPC) and a high-energy plasmon continuum (HPC), which are separated by a plasmonic gap. These two continua result from the two internal degrees of freedom (sub-lattices A and B) in the unit cell. The excitations in these continua consist of bulk plasmons with charge oscillations extended over the entire chain. However, the real-space modulations of plasmons in LPC and HPC are of different types. Plasmons in LPC have inter-unit-cell charge modulations, showing long oscillation wavelengths, while plasmons in HPC have intra-unit-cell charge modulations, showing a short oscillation wavelengths. In Figs. 3.7(b) and (c) we show examples of real-space plasmon modes in LPC and HPC. This also results in the spectrum gap separating them. The gap is, however, also controlled by the strength of Coulomb interactions in the system. If we increase the global dielectric constant of the medium, 67 3 4 5 6 7 8 ω (eV) EELS(ω) topological SSH trivial SSH LPC HPC LPC (a) (b) (c) (d) (e) X X HPC p 1 p 2 p 1,2 Figure 3.7:Real-spaceplasmonsinSu-Schrieffer-Heegermodelwithopenboundarycon- ditions. (a) Electron Energy Loss Spectrum (EELS) of the topologically trivial (blue) and non- trivial (red) SSH models on an open-ended 100-site chain. (b) Typical charge modulation of a bulk plasmon in the lower plasmon continuum (LPC). (c) Typical charge modulation of a bulk plasmon in the higher plasmon continuum (HPC). (d) and (e) are the charge modulations of the two-fold degenerate localized plasmon, indicated by the red arrow (p 1,2 ) in (a), which is only ob- served in the topological phase. the gap will shrink. Meanwhile, we notice that the lower bound of the LPC is entirely inherited from the single-particle band gap. which is expected. In the topologically non-trivial phase, besides the two plasmon continua, we find an addi- tional excitation at ω ≈ 4.63 eV which lives in the plasmonic gap, indicated by the red arrow in Fig. 3.7(a). This non-dispersive plasmon mode has a very similar localization character as the single-electron edge states we observed in Fig. 2.4. Namely, this mode is highly localized at the ends of the SSH chain. Meanwhile, it is also a two-fold degenerate mode whose real-space charge-distribution pattern can be either even or odd, as illustrated in Fig. 3.7(d) and Fig. 3.7(e), respectively. This resemblance motivates us to seek a deeper understanding of the origin of these 68 2 4 6 8 ω (eV) EELS(ω) EELS with χ 0 full EELS with χ 0 bulk EELS with χ 0 topo (a) (b) (c) X X p 1,2 p 2 p 1 Figure 3.8: EELSdecompositionintheSSHmodel. (a) EELS of the 100-site open-ended topo- logical SSH chain, calculated usingΠ full 0 ,Π bulk 0 andΠ topo 0 respectively. (b) and (c) are the charge modulations of the two-fold degenerate mode at ω ≈ 4.07 eV (indicated by the red arrow) ob- served in the topological EELS. localized plasmonic modes which specifically observed in the topologically non-trivial SSH chain. This will be discussed in detailed in the coming section. 3.2.2 TopologicalOriginofLocalizedPlasmons We now dig into the topological origin of the localized plasmons observed in the previous section. To do this, we decompose the full polarization function into different parts and investigate their contributions individually. This is somehow similar to the idea of constraint RPA. Here, we will decompose the fullΠ 0 into two parts by separating the summation in Eqn. 2.41 as follows: X i,j ... | {z } ⇒Π full 0 = X i∈TS X j/ ∈TS ··· + X i/ ∈TS X j∈TS ... | {z } ⇒Π topo 0 + X i,j/ ∈TS ... | {z } ⇒Π bulk 0 , (3.47) 69 where TS is the set of topological zero-energy states. We call the two terms on the right hand side the topological part of the polarization,Π topo 0 , and the bulk part of the polarization,Π topo 0 . It is worth to clarify thatΠ topo 0 still includes information of bulk states because it actually sums over transitions between a topological electronic state and a bulk electronic state.Π topo 0 , however, describes only transitions between bulk electronic states and is therefore a purely bulk property. Due to the degeneracy of the two zero-energy edge states, there is no transition between the them, so that we can ignore the term P i,j∈TS in the above decomposition ∗ . With such defined Π topo 0 and Π bulk 0 , we can proceed to calculate the dielectric function correspondingly, and finally derive the EELS which we call EELS topo (ω) and EELS bulk (ω). Doing this, we can delineate the bulk and edge states contributions to the plasmon spectrum. It should be clarified that this is purely a math- ematical decomposition, whereas in real experimental measurement we should always get the full contribution which is EELS full (ω). In Fig. 3.8(a), we show al spectra EELS topo (ω), EELS bulk (ω) and EELS full (ω). As we see, EELS bulk (ω) only reproduces extended bulk plasmonic excitations forming LPC and HPC, with no sign of the non-dispersive and localized plasmon mode, as we observed before in EELS full (ω) around the frequency of ω ≈ 4.63 eV. In contrast, EELS topo (ω) indicates a few isolated sharp peaks. By examining the real-space charge modulations of these modes, we find that they are indeed strongly localized modes living on the ends of the SSH chain. For example, we show the plasmon mode atω≈ 4.07eV (indicated by the arrow) in Figs. 3.8(b) and (c). It also has two-fold degeneracy which resembles the results we saw before in Figs. 3.7(d) and (e) where the full polarization was considered. However, we notice that the edge modes we obtained from pureΠ topo 0 (ω) are more localized that those obtained from the full polarization that ∗ This is different from higher-dimensional spaces. There the topological edge states form a dispersive band, allow intra-band transitions between topological electronic states. 70 differs by the contribution Π bulk 0 (ω). Meanwhile, the precise excitation energies are also different: ω ≈ 4.07 eV versus ω ≈ 4.63 eV. Thus, we conclude that the localized plasmons we observed are originated fromΠ topo 0 (ω). At the same time, Π bulk 0 (ω) affects these localized plasmons as a background screening, shifting their excitation energies and modifying their localization lengths. These localized plasmon modes can be regarded as collective topological signatures of the under- lying electronic model. To understand the above observations deeper, we focus on the matrix structure ofΠ topo 0 and provide a semi-analytical analysis. The matrix element ofΠ topo 0 can be evaluated using Eqn 3.47 and Eqn. 2.41, which can be simplified to the form Π topo 0 (ω) ab =P ab S ab (ω) (3.48) with P ab =⟨ϕ a |P|ϕ b ⟩= X i∈TS ⟨ϕ a |ψ i ⟩⟨ψ i |ϕ b ⟩ (3.49) and S ab (ω)= X i/ ∈TS E i [2f(E i )− 1] E 2 i − (ω+iγ ) 2 ψ ia ψ ib , (3.50) assuming that the tight-binding wave functions are real and all topological states have zero en- ergy and occupation number 0.5. P is the projection operator to the space of topological zero- energy states. Each element ofΠ topo 0 is thus a product of the projection operator to the space of topological zero-energy states P ab and S ab (ω) which is a sum (integral) over all bulk electronic 71 states. The topological electronic states ψ i∈TS (r) are strongly localized at the edges, so that we find non-zero elements of P ab only fora andb close to the edges. S ab (ω) does not vary abruptly with a and b since the bulk states extend across the entire chain. S ab (ω) thus does not indicate any localization character. The resultingΠ topo 0 matrix is very sparse with entries only close to the upper-left and lower-right corners (where indicesa andb are close to the ends of the chain). The corresponding matrix representation of the dielectric functionε topo (ω) = I− V c Π topo 0 also inherits this “corner localization" character from Π topo 0 (ω). It is in this sense that these local- ized plasmonic excitations indeed originate from the localized topological electronic edge states, which is why we refer to them as topological plasmons hereafter. It is important to notice that there is still a bulk-related component inΠ topo 0 fromS ab (ω), as defined by Eq. 3.50. 3.2.3 MirrorSSHModel We also study the plasmonic excitations in the mirror SSH (mSSH) model. The general phenom- ena are similar to those observed in the SSH model, so here we just report the results of our nu- merical calculations briefly. As we recall from Section 2.3, the mSSH model is simply constructed by reflecting the simple SSH chain at one edge site. The mSSH model is inversion symmetric with respect to the mirror interface localized at the middle of the chain. The model can hold localized edge states as well as interface states depending on the parameter choices oft 1 andt 2 . Fig. 3.9(a) shows the EELS(ω) of the m-SSH model with strong (t 1 = 1.25,t 2 = 0.75) and weak (t 1 = 0.75,t 2 = 1.25) interface hopping parameters. In both cases, we again observe a LPC, a HPC as well as additional localized plasmonic excitations living in the gap, labeled by p 1,2,3,4 . This is similar to the case of the SSH model. However, unlike the SSH model, the mSSH 72 3 4 5 6 7 8 ω (eV) EELS(ω) m-SSH with strong interface hopping m-SSH with weak interface hopping LPC HPC (a) (b) (c) (d) (e) X X p 3,4 p 3 p 4 p 2 p 2 p 1 p 1 Figure 3.9: Real-space Plasmons in mirror SSH model. (a) EELS of the m-SSH model on an open-ended 103-site chain with strong (blue) and weak (red) hopping at the central mirror interface. (b) and (c) show the charge modulation of the localized interface plasmon in the m- SSH model with strong and weak interface hopping, indicated by the blue (p 1 ) and green (p 2 ) arrows in (a). (d) and (e) are the charge modulations of the two-fold degenerate localized edge plasmon, indicated by the red arrow (p 3,4 ) in (a), observed in the m-SSH model with weak interface hopping. 73 model also holds interface plasmons due to the mirror structure located in the middle of chain. In the strong interface case, there is just one additional modep 1 atω 1 ≈ 6.0eV, with a dipole- like charge distribution localized at the interface, as shown in Fig. 3.9(b). In the weak interface case, we find two additional plasmonic excitations, p 2 and p 3,4 , at ω 2 ≈ 5.04 eV and ω 3,4 ≈ 4.63eV, respectively.p 2 is an interface mode [Fig. 3.9(c)], but with different symmetry compared to the dipole-like mode shown in Fig. 3.9(b). p 3 and p 4 are two-fold degenerate edge modes whose real-space charge-distribution pattern can either be even [Fig. 3.9(d)] or odd [Fig. 3.9(e)]. They are essentially the same as the localized edge plasmons observed in the simple SSH model because both, the open-ended topological SSH chain and the open-ended m-SSH chain with weak interface hoppings, have weakly linked edges (t 1 = 0.75), leading to dangling bonds at the two ends of the open chain. 3.2.4 StabilityofTopologicalPlasmonsAgainstDisorder It is natural to ask that whether these topology-originated localized plasmons are robust against disorder. As we know, the single-electron topology is symmetry protected. The electronic edge states are robust against external perturbations as long as they do not close the band gap. Plas- mon, however, is an elementary excitation of different character. It is collective and correlated (due to the RPA Coulomb interaction considered here). We have seen in previous sections that although the localized edge plasmons are originated from topological single-electron edge states, bulk electronic states still contribute throughΠ bulk 0 , shifting their excitation energy and spread- ing out their spatial localization. We therefore expect that the localized topological plasmons are generally less stable than their constituent topological single-particle states. 74 -2 -1 0 1 2 E (eV) DOS unperturbed system a = 0.05 eV a = 0.10 eV a = 0.15 eV a = 0.20 eV 5 5.5 6 6.5 7 ω (eV) EELS full (ω) 4 4.2 4.4 4.6 4.8 5 ω (eV) EELS topo (ω) 0 0.05 0.1 0.15 0.2 a (eV) 0 2 4 6 R-STD (10 -2 ) EELS full EELS topo p-IPR full p-IPR topo 0 0.05 0.1 0.15 0.2 a (eV) 2 3 4 5 IPR (10 -1 ) p-IPR full p-IPR topo e-IPR unperturbed systems fit (a) (b) (d) (c) (e) Figure 3.10: Stability of the interface plasmon in the mirror SSH model. (a) Averaged single-particle density of states over 500 realizations with different strengths of hopping disorder a. (b) and (c) are averaged EELS around the interface topological plasmon over 500 realizations calculated fromΠ full 0 andΠ topo 0 , respectively. (d) Relative standard deviation (R-STD) of the ex- citation energies and real-space charge modulations of the topological plasmon modes shown in (b) and (c) as functions ofa. (e) Averaged inverse participation ratio of the topological electronic state (e-IPR) and of the interface plasmon mode calculated fromΠ full/topo 0 as functions ofa. Here we numerically study the stability of localized plasmons against hopping disorder, i.e., the off-diagonal disorder in the Hamiltonian matrix. We focus on the localized interface plasmon mode atω≈ 6.0 eV observed in the m-SSH model with strong interface hoppings [p 1 in Fig. 3.9]. There are several reasons for choosing to focus on this mode: 1) the mode in non-degenerate, which simplifies the problem; 2) the mode is deeply located in the plasmonic gap between LPC and HPC and is easy to be tracked when the disorder is turned on. The hopping disorder is modeled by a uniformly distributed noiseδt ∼ U(− a,a) on top of the hopping matrix elements t 1 andt 2 . The strength of the perturbation is limited to a certain range, i.e.,a <|t− t ′ |/2, such that it will not induce any topological transition by reversing the order oft andt ′ . We consider 500 realizations for eacha, ranging from0.025eV to0.2eV. 75 We study the effects of the disorder on the single-particle energies and wave functions as well as the plasmonic excitations energies and corresponding charge-modulation patterns. We want to compare them. Specifically, in the energy picture, we will compare the stability of plasmonic EELS to the stability of electronic density of states (eDOS). In the real-space picture, we will characterize the localization of plasmon charge modulation and the electronic wavefunction by introducing the inverse-participation ratios (IPR). The IPR of a normalizedN-dimensional vector ψ is calculated viaIPR = P N i |ψ i | 4 , wherei is the index of the components. An increased IPR indicates a stronger localized distribution, while a decreased IPR implies delocalization. We will calculate the electronic IPR (e-IPR) from the eigenvector of the Hamiltonian and the plasmonic IPR (p-IPR) from the eigenvector of the dielectric matrix derived fromΠ full/topo 0 (p-IPR full/topo ). Fig. 3.10(a) shows the unperturbed and (averaged) perturbed electronic DOS. The bulk elec- tronic states are affected by the perturbation, whereas the topological electronic state at zero energy is unchanged. Fig. 3.10(b) shows the averaged EELS around the focused interface mode, which broadens when disorder is introduced. The excitation energy of this localized plasmon is thus less stable against the hopping noise than the topological single-particle edge state. To quantify this behaviour, we plot the relative standard deviation of the plasmonic excitation en- ergyω p (R-STD † ) as a function ofa in Fig. 3.10(d) (blue dots). From this we clearly see thatω p gets more and more uncertain with increasing disorder. As discussed in the previous section, we can separate the full polarization Π full 0 = Π topo 0 +Π bulk 0 into a “topological” and a “bulk” part. From this we can calculate EELS topo (ω) [Fig. 3.10(c)] and the corresponding R-STD [Fig. 3.10(d), red squares]. The latter shows a very similar trend of decreasing stability as discussed before. We † The R-STD is defined by the square-root of the second central moment of the averaged EELS, divided by the unperturbed excitation energy. 76 can thus conclude that this instability of the plasmon excitation energy is due to bulk electronic states insideΠ topo 0 [Eq. (3.48)]. The bulk polarization functionΠ bulk 0 does not drastically affect the stability of the excitation energy, although it shifts the excitation energy. The stability of real-space localization of the interface plasmon shows a different and more complicated story. We evaluate the IPR of the wavefunction of the single-electron interface state (e-IPR) and the IPR of the real-space charge modulation of the interface plasmon mode. For the latter one, we evaluate it in two ways: using only the topological part of the polarizationΠ topo 0 (p-IPR topo ) and using the full polarizationΠ full 0 (p-IPR full ). We show all results in Fig. 3.10(e). As expected, the e-IPR does not vary much for all noise levels, indicating that the wavefunction of the single-electron interface state is stable. This is not surprising since the interface connects two topologically distinct SSH insulators and is therefore forced to hold a gapless state. The noise does not change the topological character of the two sides of the interface, i.e., it does not induce a gap closure. For the real-space plasmonic state, the phenomenon becomes more interesting. If we focus on p-IPR full [red squares in Fig. 3.10(e)], it is obviously affected by the hopping noise a. When the noise level increases, p-IPR full starts to decrease, indicating a trend of delocalization. We can fit the decay of localization by an exponential function as p-IPR full =Ae − a 2 /δ (3.51) withA≈ 0.5 andδ ≈ 1 (eV) 2 . Asa 2 is proportional to the variance of the uniform distribution U(− a,a), so p-IPR full decays exponentially with the variance of the perturbation. On the other hand, if we focus on p-IPR topo [blue triangles in Fig. 3.10(e)], we can see that it is not varied by the noise. Thus, in contrast to the instability of the plasmon excitation energy which is due to the 77 bulk admisture inΠ topo 0 , the delocalization of the real-space plasmon mode is attributed to the noised bulk polarization functionΠ bulk 0 , as can be seen by comparing the full and the topological p-IPR in Fig. 3.10(e). In other words, the disorder-induced delocalization tendency of the interface plasmon is driven by the perturbed bulk screening properties. The observations can be concluded as follows. The noise inΠ topo 0 broadens the EELS but does not affect the localization of the interface plasmon mode, whereas the noise in Π bulk 0 delocalizes the mode in the real space but it does not further broaden the spectrum very much. In fact, this is not too complicated to understand and we will provide a semi-analytical explanation below. First of all, we realize that, mathematically, the conclusion roughly means that the noise inΠ topo 0 affects its eigen-values but not eigen-states (so the energy of the mode is unstable but the distribution of the mode is quite stable). This would be true if the perturbed ˜ Π topo 0 and the unperturbed Π topo 0 just differ by a scaling factor. Below we will see that this approximately holds. Going back the decomposition of the full polarization function and the definition of Π topo 0 , we construct the perturbed dielectric function as ˜ ε topo = I− V c ˜ Π topo 0 (3.52) in which the perturbed ˜ Π topo 0 is constructed from the matrices ˜ P and ˜ S(ω) [see Eqn. 3.50]. The only quantity within this expression which is modulated by the applied perturbation is ˜ S(ω), which can be approximated by ˜ S(ω) ≈ c(a)S(ω) because the noise is uniform over the entire real-space chain. c(a) is a scaling factor depending on the perturbation strengtha. Thus ˜ ε topo is affected just by a simple scaling factor which does not change its eigenvector, but its eigenvalue. 78 Correspondingly, the excitation energy [EELS topo (ω)] is affected by the perturbation, but the real- space localization [p-IPR topo ] is not. For the full quantities, i.e., p-IPR full and EELS full (ω) derived from the perturbed˜ ε= I− V c ˜ Π full 0 , this line of argumentation does not hold anymore. Here, ˜ Π full 0 cannot be approximated as a scaled version ofχ full 0 , so that both eigenvectors and eigenvalues are affected by the perturbation. To quantify whether the plasmonic excitation energy or its charge- localization is more affected by the disorder, we also plot the R-STD of the p-IPRs in Fig. 3.10(d). From this we find that the topological plasmon localization is less affected than its excitation energy for perturbation strengths ofa≲0.15eV only. Finally, it is important to note thatδ ≈ 1 (eV) 2 in Eqn. 3.51 is a rather large value compared to the energy scale of practical perturbation. In fact, √ δ is here of the order of the band gap (1.0 eV in our numerical simulation), which is likely beyond the limit of experimentally achiev- able perturbation levels. Therefore, we want to conclude that the real-space localization of the topologically-originated interface plasmon mode in the m-SSH model is rather stable when sub- jected to sufficiently small hopping noise, while its excitation energy will be renormalized. 3.2.5 EffectsofCoulombInteractionsonLocalizedPlasmons Despite many similarities observed between the electronic picture and the plasmonic picture, the plasmon excitations are fundamentally different single-electron physics. In this section, we will see that the plasmonic excitation is strongly affected by the Coulomb interactions in the system. We first investigate the effects of globally varied Coulomb interactions. The simplest way to do this is to change the background dielectric constantε b , which will reduce the Coulomb interactions in the system globally by a factor 1 ε b . This will shift the excitation energies of all 79 EELS(ω) ε b = 1.0 ε b = 1.1 ε b = 1.2 ε b = 1.3 3 4 5 6 7 8 ω (eV) EELS(ω) SSH model m-SSH model (a) (b) Figure 3.11:Effectoftheglobalscreeningenvironmentontheplasmonspectrum. Shown are the EELS of (a) the open-ended 100-site topological SSH chain and (b) the open-ended 103-site m-SSH chain with strong interface, subject to different background dielectric constant ε b . plasmons including topological ones. We demonstrate this by showing the EELS(ω) of the SSH and m-SSH models in their topological non-trivial phase for varyingε b in Fig. 3.11. The excitation spectra shift continuously to lower energies with increasingε b , which is also reported in previous studies. In more detail, we notice that the high-energy modes shift stronger than the low-energy modes. As a result, the plasmonic gap between LPC and HPC shrinks with increasing background dielectric screening, which was mentioned before. The Coulomb interaction can also be controlled to break the symmetry and remove some degeneracies when going from the single-electron spectrum to the plasmon spectrum. This will depend specifically on the form of the Coulomb interaction as a function of the real-space co- ordinate. We recall that in the topological phase of the SSH model, the topological plasmon at ω≈ 4.63 eV is a two-fold degenerate mode whose real-space pattern can be evenly or oddly lo- calized at the ends of the chain [Figs. 3.7(d) and (e)]. The single-particle topological zero-energy 80 edge states are also two-fold degenerate. In this case, the degeneracy of single-particle topologi- cal states is fully inherited by the topological plasmons due to equivalent local Coulomb environ- ments at the right and left edges of the chain. Things are slightly differen in the m-SSH model with weak interface hopping. There we obtain one interface plasmon mode at ω ≈ 5.04 eV in addition to two two-fold degenerateedgeplasmonmodes atω≈ 4.63eV. The constituent single- particle topological zero-energy states, on the other hand, are three-fold degenerate, with wave functions localized at the mirror interface and at the two ends of the chain. In this case, the de- generacy of the constituent single-particle states is only partly inherited to the derived plasmon modes and split into “1+2.” The interface mode can now be distinguished from the edge mode in the EELS due to its different local Coulomb environment, i.e., the interface site is exposed to the Coulomb interaction from the left and right part of the system, while the edges just feel one tail of the Coulomb interaction. To verify the above argument, we consider breaking the Coulomb environment equivalence in the SSH model by using different local Coulomb potentials on the left and right edges, i.e. U R 0 ̸=U L 0 . Figs. 3.12(a) and (b) show the resulting EELS(ω) withU R 0 =U L 0 ,1.2U L 0 ,1.5U L 0 ,2U L 0 . ForU L 0 ̸=U R 0 [Fig. 3.12(b)] the two-fold degenerate edge plasmon is split into two non-degenerate modes: the left mode (p L 1,2,3 ) and the right mode (p R 1,2,3 ) with different excitation energies. By in- creasing the difference between U L 0 andU R 0 we also increase the energy difference between the two modes. With increasing on-site Coulomb interactionU R 0 , the charge distribution at the right edge gets more localized in contrast to the mode localized at the left edge, which, for instance, can be seen by comparing the modep L 3 [Fig. 3.12(c)] and the modep R 3 [Fig. 3.12(d)]. Such a manip- ulation of the topological plasmons by changes to the local Coulomb interactions is remarkable, since it does not affect other excitations in the system and can be used as a tuning knob solely 81 EELS(ω) U 0 R = U 0 L 3 4 5 6 ω (eV) EELS(ω) U 0 R = 1.2U 0 L U 0 R = 1.5U 0 L U 0 R = 2.0U 0 L (a) (b) (c) (d) X X p L 1,2,3 p R 2 p R 3 p R 1 p R 3 p L 3 Figure 3.12: Removal of topological plasmon degeneracy and enhancement of plasmon localizationwithlocallyvaryingCoulombinteraction. (a) EELS of the SSH model with sym- metric on-site Coulomb interactionU R 0 =U L 0 , showing a two-fold degenerate plasmon mode. (b) EELS of SSH models with asymmetric on-site Coulomb interactionU R 0 = 1.2U L 0 ,1.5U L 0 ,2.0U L 0 . In each case, the degenerate mode in (a) splits into a left edge mode (p L 1,2,3 , indicated by the solid arrow) and a right edge mode (p R 1,2,3 , indicated by dashed arrows). (c) and (d) are real space charge modulations of modesp L 3 andp R 3 , showing that the right edge mode with larger on-site Coulomb interaction is more localized. 82 Log(U ind ) linear external potential 0 2 4 6 8 ω (eV) Log(U ind ) centered quadratic external potential X X (a) (b) (c) (d) p 1 p 1 p 2 p 2 Figure 3.13: Excitation spectra of SSH model under different external field. Induced en- ergy spectra of 100-site open-ended topological SSH chain, subject to a (a) linear and to a (b) centered quadratic external electrical potential, are shown. The two-fold degenerate topological edge plasmon excitation is indicated by a red dashed line. (c) and (d) show charge-distribution patterns of this mode corresponding to the linear and the centered quadratic external potentials. for the topological plasmon edge mode. In contrast, global changes to the Coulomb interactions as just discussed before will affect all plasmons simultaneously. 3.2.6 ExcitationSpectraunderExternalDrivingField While the EELS shows all possible plasmonic excitations of a system, it does not yield any in- formation about the excitations generated by specific external electromagnetic fields. In reality the symmetry of the external electromagnetic field will strongly influence which modes will and can be excited. Specifically, it depends on whether Π 0 (ω)ϕ ext (ω) is zero or not [see Eqs. 2.52 83 and 2.42]. We therefore turn to the induced energy spectrumU ind (ω), which renders the realistic response to specific applied external electromagnetic fields. To this end we focus on the topological plasmons in the SSH and m-SSH models subject to linear and centered quadratic external electrical potentials. The linear potential (LP) is an odd function, whereas the centered quadratic potential (CQP) is an even function in real space. Figs. 3.13(a) and (b) show the induced energy spectra of the SSH model subject to both potentials. The topological plasmon mode atω≈ 4.63eV, as shown in Fig. 3.7(a), also appears in the induced energy spectrum for each of these cases (the red dashed line). Fig. 3.13(c) and (d) depict the cor- responding real-space charge distributions which inherit the symmetry of the external potential. In more detail, we find an odd real-space charge distribution by using the LP [Fig. 3.13(c)] which resembles the one shown in Fig. 3.7(e) and an even distribution by using the CQP [Fig. 3.13(d)] as found in Fig. 3.7(d). Therefore, we can excite either one of these two degenerate modes by choosing certain external potentials. Figs. 3.14(a) and (b) show the induced energy spectra of the m-SSH model with strong interface hopping, also subject to LP and CQP. The EELS of this model [Fig. 3.9(a)] has a plasmon mode atω ≈ 6.00eV with a dipole-like real-space charge-distribution pattern observed in Fig. 3.9(b). Thus, it can be excited by LP but not by CQP as verified in Fig. 3.14. The symmetry of the external perturbation has thus a strong effect to the topological plas- monic excitations while the rest of the spectrum is unchanged and is therefore well-suited to study these special and highly localized states. 84 Log(U ind ) linear external potential 0 2 4 6 8 ω (eV) Log(Uind) centered quadratic external potential (a) (b) P Figure 3.14:ExcitationspectraofmirrorSSHmodelunderdifferentexternalfield. Induced energy spectra of 103-site open-ended m-SSH chain with strong interface hoppings, subject to a (a) linear and to a (b) centered quadratic external electrical potential, are shown. The topological interface plasmon excitation is indicated by a red dashed line. 3.3 PlasmonsinTwo-DimensionalTopologicalInsulators In the last section of this chapter, we discuss our work on plasmons in 2D topological insulators. Specifically, we will focus on the 2D SSH model here. The topology and single-electron energy structure have already be discussed in Section 2.3.3. Here, we will evaluate plasmonic excitations in this model in the case of open boundary conditions. We perform a fully real-space RPA calculation on a finite 2D SSH model of 30× 30 sites. We set the Fermi level to be in the upper bulk band gap, as shown in the inset of Fig. 3.15 (a). When the model is in the topologically trivial phase (the 2D Zak phase isγ = (0,0)), this bulk band gap is empty. Therefore, only inter-band transitions between bulk states can happen. However, when the model is in the topologically non-trivial phase (the 2D Zak phase isγ = (π,π )), this bulk band gap will be filled by edge states that live on the boundaries. In this case, intra-band transitions between egde states dominant the low-energy physics. Thus, the above setup can 85 Figure 3.15:Real-spaceplasmonsinthetwo-dimensionalSu-Schrieffer-Heegermodel . (a) EELS of the 2D SSH model of30× 30 in both the trivial phase and the topologically non-trivial phase. The Fermi level lies in the upper bulk band gap of the model. In the trivial phase we only obverse extended bulk plasmons living in a gapped, high-energy spectrum. In the topologically non-trivial phase, we observe localized edge plasmons at low energies that are induced by intra- band transtions between topological electronic edge states. (b) Edge plasmon dispersion read off from the real space calculation for the 2D SSH model of different sizes and shapes [ 30× 30 (stars), 40× 20 (crosses),50× 50 (triangles)], fitted to the 1D SSH bulk plasmon dispersion evaluated in the momentum space. 86 nicely decouple the contributions coming from the bulk and mid-gap topological edge states in different topological phases of the model. The EELS of the 2D SSH model for both the trivial phase and the topologically non-trivial phase are calculated and shown in the middle of Fig. 3.15 (a). In the trivial phase, we only observe a gapped spectrum consisted of extended bulk modesω≳4.5 eV. Several typical modes are plotted in the lower row of Fig. 3.15 (a). In the topologically non- trivial phase, we observed a gapless spectrum with low-energy excitations arising from intra-band transition between topological edge states, forω≲2.5eV. To characterize these plasmons in the real-space, we plot their charge distribution patterns in the upper row of Fig. 3.15 (a). These plasmon modes are localized around the boundaries of the sample, i.e. they are topologically originated edge plasmons. Moreover, these modes show a nice energy-momentum dispersion. By analyzing the wavelength of the plasmon waves appearing in the localized plasmonic spectrum, we extract their dispersion ω(q) for various system shapes and sizes. We find that they can be well fitted to the RPA bulk plasmon dispersion of the 1D SSH model in the momentum space with identical hopping and environmental parameters by properly tuning the internal screening in the quasi-1D Coulomb interaction. Specifically, for the quasi-1D momentum-space Coulomb interaction of the formV(q)=2e 2 K 0 (lq)/ε b , the fitting can be optimized over the effective cross- sectionl. In our calculation, we find the optimal l is close to the lattice spacingδ . Interestingly, this optimal cross-section isindependent of the 2D sample size and shape, as illustrated in Fig. 3.15 (d). This indicates that these localized edge plasmons are largely decoupled from the bulk part of the system, such that the bulk screening effects do not influence their excitation character. This is a clear distinction from the 1D case we investigated above. 87 Chapter4 CoulombEngineeringonTwo-DimensionalPlasmons Plasmons in two-dimensional (2D) materials is a fascinating research field attracting extensive research interests from both scientists and engineers. There are basically two reasons for this. On one hand, plasmons in a 2D material have intrinsic advantages. They display an enhanced low- energy dispersion curve that is highly detached from the Landau damping regime, which makes them intrinsically low-damped and long-living. Due to the dimensionality reduction, plasmons in 2D materials are also more confined than surface plasmons on the surface of three-dimensional (3D) metallic bulk. On the other hand, plasmons in 2D materials are tunable from the external environment, for example, by applying a dielectric substrate. This makes 2D plasmonic devices particularly important and useful in technological applications. The tunability of 2D plasmons by the screening environment is essentially due to the tunable Coulomb interactions within the 2D material. This is intuitively clear since electric field lines connecting two charged particlesq 1 andq 2 on a 2D plane pass over the entire 3D space. Because the Coulomb interaction is long-ranged, it can be easily affected by the background dielectric materials located below and/or above it. This idea of tuning many-body electronic properties within a layered material by modifying Coulomb interactions from the outside is calledCoulomb 88 engineering. In fact, such a concept has already been quite extensively applied to layered semi- conductors [54, 55, 56, 57, 58, 59, 60] as well as to homogeneously embedded Mott insulators [61]. It can, for example, renormalize semiconductor band gap or exciton binding energies. (a) (b) (c) (d) Figure 4.1: Substrate controlled plasmon patterns in atomically thin metals. (a) Unsup- ported and (b)-(d) dielectrically supported metallic layer with a typical plasmonic charge distri- bution. (c),(d) Structured dielectric substrates induce spatially patterned plasmonic excitations. In this section, we apply the concept of Coulomb engineering to activemetallicsystems where possibly strong internal and external screening channels compete. Specifically, we will consider homogeneous as well as horizontally staggered dielectric environments applied to the layered material. Here, we simply illustrate this idea using Fig. 4.1. More detailed calculations and analy- sis will be provided in following sections accordingly. Fig. 4.1(a) depicts plasmonic excitations in a sample of unsupported metallic layer, whereas Fig. 4.1(b) shows modified excitations when the layer is put on a dielectric substrate. Both cases are spatially homogeneous in the 2D plane. In Fig. 4.1(c) we show a heterojunction structure of the dielectric substrate supporting the layered 89 material. In this case, plasmons excited in the layered material is not spatially homogeneous any more, but is imprinted by the structured dielectric environment. Fig. 4.1(d) follows the same idea by using a staggered dielectric substrate with two parallel interfaces. This motives us to design a plasmonic waveguide by properly choosing dielectric constants in different sectors. In following sections, we will show step-by-step evolving of the above idea. 4.1 GenericFeatureofPlasmonsin2Dmaterials In this section, we investigate plasmons in a metallic layer embedded in a homogeneous dielectric background. This also includes the unsupported case where the background is just vacuum. For a quantitatively description, we consider the plasmonic excitations in layered metals using a single-orbital generalized Hubbard model of the form ˆ H =− t X σ,<ij> ˆ c † iσ ˆ c jσ + X σ,i U ii n iσ n i¯σ + X σ,σ ′ ,i>j U ij n iσ n jσ ′ (4.1) where ˆ c † iσ (ˆ c iσ ) creates (annihilates) an electron of spin σ at the site i, n iσ are the correspond- ing occupation operators, t = 1 eV is the nearest neighbour hopping, and U ij is the non-local Coulomb interaction. We use a square lattice of fixed lattice constant a 0 =3 Å and set the Fermi energy toE F =− 3 eV, and treat the model within the random phase approximation. Previously in Chapter 2, we have briefly discussed RPA plasmons in the two-dimensional electron gas (2DEG) model which show a square-root like dispersion relation. In the real 2D materials, electrons form bands due to discrete crystal structure. The screening effects get more complicated. Moreover, real 2D materials are usually deposited on dielectric substrates which 90 induces another external screening channel. Therefore, plasmon dispersions will deviate from the square-root shape. We will discuss all these in detail in following sub-sections, starting from the dynamical screened Coulomb interaction in a real 2D material. 4.1.1 DynamicalScreenedCoulombInteraction ℎ/2 − ℎ/2 0 ε m ε en v 1 ε en v2 Figure 4.2: Internalscreeningmodelin2Dmaterial.. The layered material along thez = 0 plane is modeled by a dielectric slab of heighth with dielectric constantε m . To realistically describe the collective excitations of a generic layered metal in terms of the above model, it is essential to properly account for all involved screening channels, which is considered in the dynamical screened Coulomb interactionW(q,ω) given by W(q,ω)= v q 1− v q Π( q,ω) = v q ε(q,ω) (4.2) within the random phase approximation (RPA). Here Π( q,ω), ε(q,ω), and v q = 2πe 2 q are the total polarization, the dielectric function, and the 2D bare Coulomb interaction, respectively. q 91 is the in-plane momentum transfer. The full polarization functionΠ( q,ω) involves all screening channels. Generically, we consider there are three main contributions: Π( q,ω)=Π M (q,ω)+Π inter (q,ω)+Π env (q,ω). (4.3) The first term Π M (q,ω) represents the intra-band polarization due to electronic transitions within the metallic band of the material. The second term,Π inter (q,ω), is the inter-band polarization con- tributed by electronic transitions between different bands. These two are both intrinsic screening effects inside the layered material itself [62, 63, 52]. On top of them, we have the third screening channelΠ env (q,ω) coming from the external environment, such as substrates and/or capping lay- ers. From here we can clearly see that, in order for the external dielectric engineering to be really effective, we Π env to be at least comparable to the material-intrinsic screening. If the material- intrinsic screening is too large, then the external dielectric screening only plays a negligible role. This condition is more easily to be satisfied in a semiconductor or an insulator because there is no metallic screeningΠ M , so dielectric engineering has already been applied to those categories of materials. For metallic materials, dielectric engineering is less applicable so far, but should still be possible for suitably chosen materials. In our calculations, we actually split the full polarization function into two parts,Π M andΠ res , that are evaluated separately. The first part Π M is just the metallic screening explained above, which is modeled by the above introduced Hamiltonian and calculated using a fully quantum- mechanical RPA approach. The second term Π res , which we call the residual screening here, combines contributions from the inter-band screening within the material and the environmental screening from the external background. It is effectively treated as a semiconductor screening, 92 which can be analytically approximated by a classical electrostatic ∗ model of a dielectric slab of effective height h and intrinsic dielectric constant ε m embedded in a environmental screening background (Fig. 4.2). Let’s consider two charged particlesq 1 andq 2 located on thez = 0 plane with an inter-particle separationr. Clearly, the interaction between them is screened both by the dielectric slab ε m (internal material screening) and the dielectric background ε env1/env2 (external environmental screening). This screened Coulomb interaction in the real space can be easily calculated analytically by iterated image-charge method. The image charges (empty circles) are generated by the real charge (solid circle) due to the two parallel horizontal interfaces. There is in principle an infinite number of image charges along the z-direction. Their charges and positions are determined byε m ,ε env1 ,ε env2 andh. The fully screened Coulomb interactionU(r) is given by the sum of all of these contributions. For a simple case ofε env1 =ε env2 =ε b ,U(r) is given by U(r)= e 2 ε m r +2 ∞ X n=1 e 2 β n b ε m z n (r) , (4.4) where β b = (ε m − ε b )/(ε m +ε b ) and z n (r) = p r 2 +(nh) 2 . The first term on the right hand side of Eq. (4.4) is the contribution from the real (source) charge, whereas the second term results from the image charges. By Fourier transformingU(r) into the momentum space, we get U(q)= v q (1+2˜ εe − qh + ˜ ε 2 e − 2qh ) ε m (1− ˜ ε 2 e − 2qh ) (4.5) ∗ For the low-energy gapless plasmons considered here, it is appropriate to neglect the frequency dependence in the residual polarization using the static screening approximation. 93 in which ˜ ε = (ε m − ε sub )/(ε m +ε sub ). From this, we extract the effective non-local screening in the momentum space as ε Res (q)=ε m 1− ˜ ε 2 e − 2qh 1+2˜ εe − qh + ˜ ε 2 e − 2qh . (4.6) It is worth mentioning that the two parametersε m andh are determined from the intrinsic prop- erties of the 2D material and can be calculated from first-principles [64, 54, 63]. Here, we are not aiming to focusing on a specific material with high precision. Instead, we want to describe the generic feature. Nevertheless, we still want our parameter sets to be realistic, i.e., to be close to true 2D materials. Therefore, numerically, we set h = 5.76 Å and ε m = 10, similar to the situation in transition mental dichalcogenide monolayers [55]. ε env1 andε env2 , on the other hand, are tunable environmental dielectric constants in our calculation. To see the variation ofε Res (q) as a function ofq more explicitly, we plot the functionε Res (q) for two different choices of parameters in Fig. 4.3. In one choice, ε env1 =ε env2 =1.0, correspond- ing to the free standing case, while in the other choice ε env1 = ε env2 = 9.0, indicating that the layered material is embedded in a dielectric background. We can see that for small momentaq, the screening function are drastically different for the free standing case and embedded case, indi- cating a possible promising energy regime for dielectric engineering. For large momentaq, they both converge to the limit of the material-intrinsic dielectric constantε m . This is also physically expected. Because largeq effectively corresponds to small r in the real space, which means the the two charged particles have a small separation and therefore a more short-ranged interaction. In this situation, the environmental dielectric screening has negligible effect. We will always 94 consider the substrate and the capping layer to have thesame dielectric structure to simplify our analysis and numerical calculations. From now on, we will denoteε env1 =ε env2 =ε sub . 4.1.2 SubstrateControlled2DPlasmons As we properly accounted for the inter-band screening effect and the external dielectric screening inε Res (q), we can now proceed to evaluate the full dielectric response function. We rewrite the fully screened Coulomb interaction utilizing the residual screeningε Res (q), which ultimately reads W(q,ω)= U q 1− U q Π M (q,ω) = U q ε M (q,ω) , (4.7) whereU q = v(q) ε Res (q) . The full dielectric function is thus given by the productε(q,ω)=ε M (q,ω)ε Res (q). This allows us to define the electron energy loss spectrum (EELS) as EELS(q,ω)∝− Im 1 ε(q,ω) . (4.8) According to the implicit definition of plasmonic excitations that ε(q,ω p (q)) = 0, the EELS is maximized along the plasmonic dispersionω p (q). By plotting EELS(q,ω), We can extractω p (q) along a path in the momentum space, along with the metallic polarization function Π M (q,ω) representing the intra-band Landau damping regime. Our model parameters are chosen to ap- proximately reproduce the plasmonic energy scales of metallic transition metal dichalcogenides [63, 65, 62] and doped hexagonal boron nitride [66]. We first consider the free-standing case where ε sub = 1. In Fig. 4.3 (d), we can see that, even without any supporting dielectric substrate, the plasmonic dispersion deviates quickly from the 95 ℎ (b) Figure 4.3: Substrate controlled two-dimensional plasmons. (a) Effective residual screen- ing function ε Res (q) for different homogeneous dielectric substrates. The dashed lines indicate the slopes of the function at q = 0. (b) Illustration of plasmon excitation in a layered mate- rial embedded in a homogeneous dielectric environment. (c) Electron energy loss spectra of the two-dimensional metallic monolayer shown in (b), with different choices of the environmental dielectric constant. (d) Plasmon dispersions in momentum space aroundΓ . The grey area repre- sents the electron-hole continuum. 96 generic √ q-like dispersion known for purely two-dimensional electron gas (2DEG) model. This deviation is a consequence of broken continuous translational symmetry in a lattice structure, such that the non-local residual screening ε Res (q) arises from the inter-band transitions within the layered metal [62, 63, 52]. For this free-standing case, its long and short wavelength limits are ε Res (q = 0) = 1 andε Res (q≫ 1) = ε m , respectively. Hence, in the long wavelength limit the ef- fective intrinsic inter-band screening is negligible [67]. The interaction is “effectively" in the vac- uum andω p (q)∝ √ q holds. For larger momenta, however,ε Res (q) increases successively, which suppresses ω p (q) and eventually pushes it into the particle-hole continuum. Importantly, the corresponding flattening of the plasmonic dispersion takes place at momenta which are clearly detached from the particle-hole continuum, so that Landau damping and thus plasmonic losses are drastically reduced in layered metals at small and intermediate momenta [52, 66, 62, 63]. The dispersion flattening, in turn, leads to a prominent enhancement of the plasmonic spectral func- tion at intermediate frequencies, as it is clearly visible in the EELS(ω), shown in Fig. 4.3 (c) (green shaded region). This behavior of the dispersion relation observed in our simple model is qualitatively a generic feature for 2D plasmons in real, layered materials. More precise first-principles calculations have generally confirmed this for a variety of 2D materials. Specifically, they can be metallic, semi- metallic, and doped semiconducting monolayer materials including graphene [68, 69, 65, 67], Li-intercalated hexagonal boron nitride [66], doped black phosphorus [70, 71], metallic T- [72, 73] and 2H-phase [62, 52] transition metal dichalcognides, as well as in their doped 2H-phase semiconducting counterparts [63]. The details of the dispersion flattening and the corresponding energy of the enhanced spectral weight in the full electron energy loss spectrum of these materials 97 are highly material specific and vary between a few 100 meV to about 1 to 2 eV but share the common origin explained above. In the next, we investigate the effects of environmental screening on the plasmons excited in the free-standing 2D material. We sandwich the 2D layer between a substrate and a capping layer of the same dielectric constant. The configuration is illustrated as in Fig. 4.3 (b). We vary the environmental screening by changing the dielectric constant ε sub , i.e., using different substrate materials. The calculated plasmonic dispersion relations for ε sub = 3 and ε sub = 9 are shown in Fig. 4.3 (b). It is clear that the plasmonic dispersion relations are drastically affected by the dielectric environment, especially for the long-wavelength (smallq) part. This is consistent with our previous analysis on the residual screening function ε Res (q). More quantitatively, the long- wavelength limit of the effective residual screening is changed to ε Res (q =0)=ε sub , compared to ε Res (q =0)=1 in the free-standing case. This leads to a decreased Coulomb interactionU q , and subsequently to a suppressed plasmonic dispersion stays closer to the Landau damping regime. In the full EELS(ω) shown in Fig. 4.3 (c), we correspondingly find a decreased and broadened max- imum at intermediate frequencies, whereas the remainder of the plasmonic spectrum is largely unaffected. Therefore, we have found that this intermediate frequency range in the plasmonic spectral function is most susceptible to changes in the environmental screening of the layered material. It is also the most promising energy regime where the dielectric engineering can have a significant effect on the 2D plasmons. We focus on this intermediate energy range and study more details of the plasmons ex- cited there in the real space. We pick two frequencies in this energy range: ω ≈ 0.77 eV and ω≈ 1.08 eV. For each frequency, we calculate the plasmon modes for different choices of the di- electric environmentε sub and plot their real-space charge density patterns, shown in Figs. 4.4 (a-f). 98 Figure 4.4: Real-space charge density modulations of two typical plasmon modes. At ω≈ 0.77eV andω≈ 1.08eV, plotted for three different dielectric environments. Generically, for all cases, we observe extended modes over the entire sample with checkerboard- like patterns oscillating along thex andy directions. For a higher excitation frequency, the mode also has a larger momentum q which in the real space corresponding to a shorter oscillation wavelength. All modes are spatially homogeneous because the layered material as well as the screening environment is homogeneous. However, for a fixed excitation frequency, the charge oscillation patterns shown are quite different for different ε sub . With increased environmental screening, the plasmon wavelength decreases. This nicely matches the plasmonic dispersion re- lations we calculated before in Fig. 4.3(d). There, we can see, for a fixed excitation frequency ω (corresponding to a horizontal line in the plot of EELS(q,ω)), the corresponding momentum q is larger for increased environmental dielectric constant. In other words, we can also say that the same type of mode (characterized by the momentumq) is red-shifted, due to the suppressed 99 dispersion curve. This demonstrates how spatial patterns of 2D plasmons can be controlled by means of dielectric substrates. Here, we just simply change the dielectric constant of the environ- ment globally, without altering its structure. The effect is also uniform in the entire 2D space. In the following, we will apply a structured dielectric environment to the 2D material and explore its effects on the spatial distribution of plasmonic excitations. 4.2 StructuredDielectricEnvironment The simplest structured dielectric environment is a heterojunction consisting of two different dielectric materialsε L sub andε R sub on two sides of a contacting interface, as illustrated in Fig. 4.6 (a). The heterogeneous environment clearly breaks the translational invariance in the x-direction, therefore we will use the real-space representation to calculate the plasmonic excitations, utilizing a supercell consisting of80× 80 sites (237Å× 237Å for the sample scale). This is in practical broken into two parts in our calculation, which is worth to be clarified here. The first part is calculating the real-space screened Coulomb interactionU ab between any two lattice pointsa andb on the z = 0 plane. This is done using a full real-space approach. This interaction is, however, hardly to be calculated exactly using a simple way. Below, we will introduce an approximated way to evaluate it. The second part is calculating the metallic polarization functionΠ M also on the lattice. For this part, we first calculate Π M in the momentum space that is reciprocal to the real-space lattice. This is because the 2D material itself is still spatially homogeneous and translationally invariant. Once we obtainedΠ M in the reciprocal space, we can transform it back to the real-space lattice, obtaining[Π M ] ab , by Fourier transformation. This saves an enormous amount of time for numerical calculation comparing to the direct real-space approach whose time complexity scale 100 asN 4 (N is the total number of lattice sites). WithU andΠ M evaluated over the real-space grid, we can finally obtain the dielectric matrix over the lattice as [ε] ab =I− X c U ac [Π M ] cb . (4.9) As in the previous section, we will start by discussing the screened Coulomb interaction first. 4.2.1 EvaluatingtheScreenedCoulombInteraction The 2D layered material, modeled by a 80× 80 square lattice, is now sandwiched in a hetero- junction dielectric background as show in Fig. 4.6(a). To write down the real-space screened Coulomb interaction matrix over the lattice, we need to be able to evaluate the interactionU(r) between any two points on thez =0 plane. By our knowledge, there is no simple (and even “not- simple") analytical way to get an exact solution. The numerical solution is possibly achievable by solving the Poisson equation iteratively. However, the singularity due to the crossing interfaces and the requirement for proper infinite-distance boundary conditions would make the numerical approach hard and expensive. Here we propose a method to solve this problem approximately, based on the image-charge technique. This has not been reported before, so we will describe the method in detail here. Let us first review a simpler case where the dielectric slab is absent, which is shown in Fig. 4.5(b). The whole space has only one vertical interface separating spaces ofε sub1 andε sub2 . In this scenario, the Coulomb potential (interaction) can be solved exactly using the image-charge method. Suppose that a charge (electron) e is placed on the xz-plane at (− d,0) [black dot in 101 Fig. 4.5 (b),x = 0 represents the interface], it will induce bound charges on the dielectric inter- face. The effect of the induced charges can be accounted by properly introduced image charges. For a pointr located in the left half space (r x <0), the total Coulomb potential is given by the real charge e and an image charge α 1 e located at the position (d,0). So the point-point interaction between two charges is U(r)= e 2 ε sub 1 1 r q + α 1 r 1 . (4.10) wherer q = |r+dˆ x| is the distance fromr to the real charge andr 1 = |r− dˆ x| is the distance fromr to the image charge at (d,0). For a pointr located in the right half space (r x > 0), the total Coulomb potential is, on the other band, given by the real chargee and an image chargeα 2 e both located at the position(− d,0) † , leading to U(r)= e 2 ε sub 2 1 r q + α 2 r 2 . (4.11) in whichr 2 =r q . By solving the boundary conditions on the interface, we get the coefficients α 1 = ε sub 1 − ε sub 2 ε sub 1 +ε sub 2 and α 2 = ε sub 2 − ε sub 1 ε sub 1 +ε sub 1 . (4.12) We then consider another simpler case in which the vertical dielectric interface is absent. This has already been discussed before (Fig. 4.2). The interaction between two points on the z = 0 † They are often regarded together as one single image charge introduced to solve the Coulomb potential in the right half space, as discussed in standard textbooks. Here we want to write the formula in a more symmetric manner. For each sub-space, we will write the Coulomb interaction to be the real-charge contribution plus a correction from an image charge. 102 ε m ε sub2 ε sub 1 ℎ/2 − ℎ/2 0 − (a) − ε sub 1 ε sub2 (b) ε m ε sub2 ε sub 1 ℎ/2 − ℎ/2 0 − ≈ − � ε 1 0 � ε 2 0 � ε 1 1 � ε 2 1 ℎ � ε 1 2 � ε 2 2 2ℎ + + + ⋯ (c) Figure 4.5: Imagechargemodelofa2Dmaterialembeddedinadielectricstructurewith asingleverticalinterface. 103 plane can also be solvedexactly by a infinite series of image charges which is explicitly given by Eqn (4.4). Here we slightly re-formulate this equation and write it as U(r)=e ∞ X n=−∞ e z n (r)˜ ε b,n . (4.13) In this way, we can interpret U(r) as the Coulomb interaction between an electron at r with a series of (electron) point charges positioned at z n which are each embedded in different homo- geneous dielectric backgrounds ˜ ε b,n = ε m /β |n| b . And we know that, with the image charges, the two horizontal interfaces can be effectively removed. Now, to model the effect of the additional vertical dielectric interface (between ε sub1 andε sub2 ), we just need to introduce vertical image charges for each of these iterated horizontal image charge as described in Eqs. (4.10, 4.13) and illustrated in Fig. 4.5 (c). The full approximate interactions can thus be written as a summation of the form U(r)= X n U n (r), (4.14) where each U n (r) is evaluated as in the simple image-charge method introduced before, how- ever, with adjusted parameters. Specifically we need to replace ε sub 1 andε sub 2 by ˜ ε 1,n and ˜ ε 2,n to evaluate α 1,n and α 2,n , and replace r q by z n,q (r) = q r 2 q +(nh) 2 , r 1 by z n,1 (r) = p r 2 1 +(nh) 2 andr 2 byz n,2 (r)= p r 2 2 +(nh) 2 . Altogether we get U(r)=          P n e 2 ˜ ε 1,n 1 zn,q(r) + α 1,n z n,1 (r) , ifr x <0, P n e 2 ˜ ε 2,n 2 zn,q(r) + α 2,n z n,2 (r) , ifr x >0. (4.15) 104 4.2.2 Real-Space2DPlasmonsinHeterogeneousScreeningEnvironment We are now ready to calculate the dielectric function on the real-space lattice. The screened Coulomb matrixU is calculated by the method introduced above. The non-interacting metallic- band polarization, evaluated in the momentum space, is given by Π M (q,ω)= 1 Ω BZ X σ k f(k)− f(k+q) E(k)− E(k+q)+ω+iγ (4.16) withE(k) being the non-interacting single-particle metallic-band dispersion atk,f(k) the cor- responding Fermi function, andγ = 0.02 eV a finite broadening. The real-space representation in the atomic basis can then be obtained via an inverse Fourier transformation [74], [Π M (ω)] ab = 1 N X q Π 0 (q,ω)e iq·(Ra− R b ) . (4.17) Here, the vectors R a/b are defined on the real-space lattice. This two-step calculation greatly improves the computational efficiency. Finally, we obtain the real-space dielectric function as a matrix in the atomic basis via [ε(ω)] ab =δ ab − X c U(R a − R c )[Π M (ω)] cb . (4.18) 105 To obtain the EELS and further extract plasmonic excitations, we go through the exactly same approach as introduced in Section 2.2 and apply to the real-space SSH model in Section 3.2. The plasmonic excitations are identified from a spectral decomposition of the dielectric matrix, ε(ω)= X n ε n (ω)|ϕ n (ω)⟩⟨ϕ n (ω)|, (4.19) by selecting the “leading” dielectric eigenvalueε max (ω) which maximizes the electron energy loss spectrum EELS(ω)∝− Im[1/ε n (ω)] together with its eigenvector|ϕ max (ω)⟩ for each frequency. The real-space charge modulation of a plasmon mode at the frequencyω p can be obtained from |ρ ind (ω p )⟩=Π 0 (ω p )|ϕ tot ⟩=Π 0 (ω p )|ϕ max (ω p )⟩ (4.20) [75]. We note that by restricting the continuous position coordinates r and r ′ to the discrete lattice positionsR a andR b we effectively neglect local-field effects [74]. Let us show the numerical results for a heterogeneous dielectric environment consisting of ε L sub =1 andε R sub =9, as depicted in Fig. 4.6 (a). Using the real-space approach introduced above, we obtain the resulting heterogeneous EELS shown in Fig. 4.6 (c). In the same figure we also show the two homogeneous cases withε sub =1 and withε sub =9 for comparison. By comparing the heterogeneous spectrum to the unsupported one, we observe that the prominent spectral maximum at intermediate excitation energies (green shaded) is still present, but suppressed. This energy range is again most susceptible to the external environment screening. For excitation energies below0.5 eV we see that the homogeneous EELS in the unsupported situation vanishes. 106 This is a finite-size effect of the supercell calculation which suppresses all plasmons with wave- lengths larger than the supercell size, c.f. Fig. 4.3, which can also be seen for the supported case. The heterogeneous spectral function thus clearly inherits characteristics from both homogeneous limits. More interesting phenomena are observed in the real-space charge modulations of plasmons in the heterogeneous case. They are no longer homogeneous due to the applied dielectric struc- ture. At a small excitation energy, i.e.ω≈ 0.41 eV, the plasmon is mostly localized in theε R sub =9 region, as seen in Fig. 4.6 (d) ‡ . At intermediate excitation energies of the green shaded area, they prominently reside in theε L sub =1 area, as observed in Figs. 4.6 (e) (ω≈ 0.92 eV), (f) (ω≈ 1.18 eV) and (g) (ω≈ 1.29 eV). This clearly illustrates how heterogeneous plasmonic patterns can be externally and non- invasively induced inhomogeneous layered materials via spatially structured dielectric substrates. The substrate induced heterogeneous plasmonic patterns behave, however, slightly differently compared to their homogeneous counterparts depicted in Fig. 4.4. For intermediate excitation en- ergies, the confined plasmon modes in the subspace of ε L sub =1 is are now quasi-one-dimensional and propagating along the direction parallel to the dielectric interface. The mode patterns evolve with increasing energy [Figs. 4.6(e-g)]. The dispersion relation of these quasi-one-dimensional modes is linear as shown in Fig. 4.6 (b). Additionally, there are some spurious, strongly damped, plasmonic excitations present in theε R sub region. Interestingly, we notice that the fully heterogeneous EELS [black line in Fig. 4.6 (c)] can be approximately reconstructed by taking a simple average over the homogeneous data (blue and ‡ This has been explained before as due to the finite-size effect, which leads to the vanished spectrum of ε sub =1 and the dominant spectrum ofε sub =1 forω <0.5 eV. 107 Figure 4.6: Heterogeneous plasmon patterns from spatially structured dielectric envi- ronments. (a) Sketch of the layered metal embedded in a heterogeous dielectric environment. (b) Dispersion relation of the plasmon mode in the left subspace with ε L sub = 1.0 and ε R sub = 9 (black dots) together with the dispersion from Fig. 4.3 (c) forε sub = 1. (c) Total EELS (black) for ε L sub = 1 and ε R sub = 9 together with homogeneous EELS (red and blue). (d-g) Typical plasmon modes at different frequencies. 108 red lines). From this we understand that theε L sub = 1 environment barely affects the low-energy plasmonic excitations confined in the ε R sub = 9 area, which does not hold vice versa. I.e. the ε R sub = 9 slightly damps the plasmonic excitations on the ε L sub = 1 side. However, except from this, these two patterns behave largely independently on the other side. 4.2.3 AquantitativeDescription 0.0 0.5 1.0 1.5 2.0 (eV) 0 5 10 15 EELS( ) (a) L, R = 1, 2 L, R = 1, 4 L, R = 1, 6 L, R = 1, 9 0 50 100 X X ind (A.U.) (f) 2 4 6 9 R 20 30 40 50 60 70 80 weight(%) (g) W L W R X Y (b) L = 1 R = 2 = 1.21 eV X (c) L = 1 R = 4 = 1.19 eV X (d) L = 1 R = 6 = 1.19 eV X (e) L = 1 R = 9 = 1.18 eV Figure 4.7: Impact of the dielectric contrast to the plasmonic patterns. (a) Total EELS for a different dielectric contrasts. (b-e) Corresponding real-space patterns for ω ≈ 1.2 eV. (f) Nor- malizedx− component of the charge distributions of these plasmon modes. (g) Variation of the charge weight in the left (W L ) and right (W R ) subspaces. We proceed with a quantitative analysis of these confined excitations by investigating their dependence on the dielectric contrast ratioε R sub /ε L sub in the substrate. To this end, we fix the left subspace to beε L sub =1 and vary the right subspace by changingε R sub to be 2, 4, 6 and 9. The EELS for all these combinations are calculated and plotted in Fig. 4.7 (a). We observe that increasing the contrast affects the EELS in a non-trivial way. At small dielectric contrasts, the full heterogeneous 109 EELS resembles a homogeneous one, with just one broad maximum at intermediate excitation energies, which is reflected in the rather spread-out plasmonic excitation pattern depicted in Fig. 4.7 (b). Upon increasing the substrate dielectric contrast, we find an increasing localization of the plasmonic pattern on theε L sub side, with decreasing weight on theε R sub side forω≈ 1.2 eV, as depicted in Figs. 4.7 (c-e). We can furthermore quantify this increasing localization with the help of the normalized charge distribution function, ˜ ρ ind (x), obtained by integrating the absolute value of the real space charge distribution function over the y-component [see Fig. 4.7 (f)]. By additionally integrating ˜ ρ ind (x) over the left and right regions we can define sub-space weights W L/R , which we show in Fig. 4.7 (g). We observe that up to nearly 80% of the charge can be confined within the left region for the maximum dielectric contrast considered here. As a result the plasmonic excitation in the ε L sub area gets relatively brighter. However, increasing ε R sub also damps the excitation in the ε L sub region. Therefore, achieving an optimal dielectric contrast will be a trade-off between spatial contrast and brightness of the plasmons on the active side. We note that theε L sub -confined quasi-one-dimensional plasmon wavelength is not affected by ε R sub . 4.3 PlasmonicWaveguidefromCoulombEngineering The possibility to non-invasively induce spatially patterned plasmonic excitations in a homoge- neous layered material by means of dielectric interfaces in the dielectric environment motivates us to propose a new class of plasmonic waveguides that utilizes two parallel vertical dielectric interfaces in the environment. So far, 2D plasmonic devices have been manufactured either by strongly invasive processes, such as structuring the activate material itself [76, 77, 78, 79, 80], by changes to the single-particle properties of the plasmon-hosting system [81, 82, 83, 84], or by 110 creating metallic heterostructures [85]. Comparing to them, our plasmonic waveguide design is based on a fundamentally different concept, i.e. we are engineering the many-bodyproperty from the outside without touching the active material. In this section, we will first show our numerical calculations on this design. Then we will discuss on material choices for a feasible experimental realization. 4.3.1 WaveguideImprintedbyDielectricStructure Figure 4.8: Plasmonicwaveguidesfromspatiallystructureddielectricenvironments. (a) Illustration of the plasmonic waveguide design. (b) Dependence of the EELS on the core widthd. (c-f) Real space charge modulations of typical plasmon modes atω≈ 1.0 eV. The plasmonic waveguide considered here is constructed by sandwiching the homogeneous layered material in a staggered dielectric environment of three subspaces and two interfaces, as 111 illustrated in Fig. 4.8 (a). This is supposed to confine well-defined quasi-one-dimensional plas- mons in the central region if the substrate’s dielectric constant there is smaller than in the outer substrate regions. We now verify this proposal by considering a system withε L sub =ε R sub =9 and ε C sub =1. We will vary the widthd of the central substrate area, in order to see to what extent we can confine the mode using this design. Not surprisingly, there will be a trade-off between the mode confinement and the mode brightness. In Fig. 4.8 (a) we show the EELS for various d. Starting from large d, we recover the pre- viously discussed maximum at intermediate excitation energies. Upon decreasing d, this maxi- mum diminishes until it vanishes below d ≲ 60 Å (20 unit cells). This behavior becomes clear by examining the corresponding real space patterns shown in Figs. 4.8 (c-f). Fig. 4.8 (c) reveals that the plasmon mode is now indeed spatially confined in the central substrate area and propa- gates parallel to the substrate dielectric interfaces. Thus, the maximum in EELS(ω) results from the low-dielectric substrate region, and the correspondingly confined plasmonic excitation there. This concept of designing the plasmonic waveguide by purely engineering the external screening structure has been proved to be realizable. Upon decreasingd, we observe in Figs. 4.8 (d-f) that the plasmonic waveguide behavior persists, but with gradually decreasing contrast to theε L/R sub areas, until it nearly vanishes for the smallestd shown in Fig. 4.8 (f). Thus, the increasing environmental screening from the increasingε L/R sub regions gradually damps the confined excitation in the center of the device until no spatial structure is visible anymore. Nevertheless, a clear confinement can be achieved for waveguides with widths down to about30 unit cells, which is on the order of90 Å here. Like in the case of the dielectric contrast, there is a trade-off between the field confinement and loss in the optimization of these plasmonic waveguides [86, 87]. This is a demonstration 112 of a novel type of plasmonic waveguides, based on Coulomb-engineered homogeneous layered metallic materials. 4.3.2 MaterialsChoices We briefly discuss a general guidance for materials choices here, in order to make the above plas- monic waveguide more likely to be experimentally feasible. Generally, this includes choosing an optimal active layered material that hosts plasmonic excitations, and suitable dielectric materials forming a structured screening environment. We discuss both here. ActiveMaterialCandidates. Our proposal depends strongly on the active material itself, i.e. not all 2D metals will be equally suitable. Specifically, the Coulomb interaction between the electrons in the active metallic layer should be very sensitive to environmental screening. This is the case if all layer-internal polarization channels (intra- and inter-band) are rather small, in other words: materials with a small density of states at the Fermi level (or with a small effective mass) and with a metallic (low-energy plasmon hosting) band which is energetically well separated from all other valence and conduction bands. Both of these properties are satisfied in graphene: it hosts low-loss plasmons [88, 69, 78, 89, 90], and the Coulomb interactions have been experimentally shown to be rather susceptible to external polarization [67, 91, 85]. Alternatively, slightly doped semi-conducting layered materials, such as electron or hole doped MoS 2 or WS 2 [63] or electron doped PtX 2 (with X∈{S,Se,Te}) [92, 93] or InSe [94] could be suitable candidates due to reduced intra-band polarization (small effective electron / hole masses) as well as metallic3R-NbS 2 and1T -AlCl 2 due to their low internal plasmonic losses [65]. In con- trast hole doped PtX 2 or InSe as well as conventional metallic 2H-phase TMDCs, such as TaS 2 113 or VS 2 , are likely less preferable due their enhanced density of states (due to rather flat upmost valence bands) [95, 92, 93] and/or due to the close vicinity of the low-energy plasmon-hosting band to other fully occupied / empty bands which both increases the internal screening [65]. Additionally, layered materials with strongly anisotropic screening properties, such as doped black phosphorus or T-phase WTe 2 , have the potential to add another interesting degree of free- dom to the proposed plasmonic waveguides: due to their strongly anisotropic plasmonic disper- sions [70, 71, 96, 73, 72] we expect the direction in which the plasmon dispersion is higher in energy to be more prone to the environmental screening due to the reduced effective mass in the corresponding direction. If this direction is parallel to the dielectric interfaces in the substrate the plasmonic confinement might be enhanced or achieved with reduced dielectric contrast. Other- wise, in case these two directions are perpendicular to each other the dielectric contrast in the environment might need to be strongly enhanced. Structured Environments. For the creation of structured environments, as needed for the pro- posed waveguides, we envision laterally grown and vertically cut lithographic structures, twisted layered materials [97, 98, 99], or novel fractionalized 2D system [100, 101, 102] to be possible routes to pursue. As discussed above, the dielectric contrast in these structures should be rather high. Furthermore, it will be interesting to create environmental screening structures with more than just one or two dielectric interfaces. One could, for example, imagine periodically patterned substrates for optimal light-matter coupling, two-dimensional dielectric structures that either create plasmonic checkerboard patterns or plasmonic quantum dots, or non-linear plasmonic waveguides [103]. 114 Chapter5 ConclusionsandOutlooks Using a fully quantum-mechanical atomic-scale random phase approximation approach, we have studied plasmonic excitations in two kinds of quantum materials - topological insulators and layered metals. In this chapter, we conclude our results and provide an outlook for future research directions. Conclusions. The first part of this thesis was devoted to plasmonic excitations in topological insulators. We find that the topology of electrons has clear impacts on their collective excita- tions, i.e., plasmons. By analyzing the electron energy loss spectra of the one-dimensional (1D) Su-Schrieffer-Heeger (SSH) model, the 1D mirrored SSH model and the 2D SSH model, we ob- serve plasmons locally excited on the boundaries of all these materials, but only when they are in topologically non-trivial phases. Thus, these localized edge plasmons should be regarded as collective topological features of the underlying model, which can be experimentally detected. Edge plasmons observed in the 1D SSH model and the 2D SSH model are slightly different. In 1D, edge plasmons are spatially localized, energetically non-dispersive, zero-dimensional (0D) modes. These modes can be very confined and bright, which look similar to quantum dots, but they are more easily affected by the noise in the bulk part of the sample, because the polarization 115 transition channels include admixture of bulk electronic states. In 2D, edge plasmons are also confined to the boundary, but they are propagating along the boundary, showing a characteristic dispersion of 1D plasmons. These modes are quite immune to the bulk screening effect because they are induced by transitions purely among topological electronic edge states that form the low- energy conducting band. For these different characters, we expect that the 0D edge plasmons in the 1D topological insulator may be used to enhance the local electromagnetic field for Raman and fluorescence spectroscopic measurements, while the propagating 1D edge plasmons on the boundaries of the 2D topological insulators may be used to transmit light and carry information. In the second part, we studied 2D plasmons excited in metallic monolayers that are tunable from the outside by Coulomb engineering. We find that 2D plasmons in layered materials are very susceptible to the screening environment, i.e. dielectric substrates. By using uniform dielec- tric substrates of different dielectric constants, the plasmonic spectral density and damping are drastically different. Moreover, we find that using a structured dielectric substrates can induce spatially inhomogeneous plasmons in perfectly uniform 2D materials. Specifically, we find that on a heterogeneous dielectric substrate, 2D plasmons will be confined to the side with a smaller dielectric constant and less screened Coulomb interaction. According to our quantitative calcu- lation, for a heterogeneous substrate consisted of dielectric materialsε L = 1 andε R = 9 on two sides, nearly80% of induced charges are confined in the left subarea. Based on this idea, we pro- pose a way to design a 2D plasmonic waveguide using stripe-structured dielectric environment. In contrast to previous plasmonic waveguide fabricated by either shaping or gating the plasmonic host materials, our approach only relies on a passive pre-structured environment, to which the active layer needs to be exposed. 116 Outlook. There are many fascinating directions to be explored in the context of collective excitations in topological insulators. One direction we want to further explore is whether we can use the momentum-space plasmon dispersion to detect the bulk topological index, for example, the Chern number of the Haldane model. We are also working on plasmons in a quantum spin Hall insulator. In all our previous work, electron spin is just accounted by a simple degeneracy factor 2. By removing the spin degeneracy somehow (for example by using a Zeeman field), we want to explore “spin-polarized" plasmons by investigating the EELS for each copy of electron spin. However, to do this, we may need to consider exchange interaction above the RPA Coulomb interaction we have used so far. We want to see whether plasmons can be used to carry the spin information. Moreover, we are also interested in exploring other types of collective excitations, for example, magnons, in topological systems. Another interesting context is the interplay between topology and correlation. It has recently been observed that the strongly correlated magic-angle twisted bilayer graphene shows an quan- tum anomalous Hall effect at 3/4 filling [104]. It has also been reported that very strong corre- lations can destroy the topology [105]. In all our calculations, we have considered the Coulomb interaction on the RPA level and within the linear response framework. It would be interest- ing to explore interactions in topological systems beyond RPA, although it would be numerically demanding. Our calculations on Coulomb engineered 2D plasmons suggest that it is feasible to externally functionalize a homogeneous 2D metallic layer by means of structured dielectric environments, thus creating new plasmonic wave guides using existing experimental techniques and available layered materials. This concept can be generalized to a variety of other many-body properties rendered by dynamical screened Coulomb interactions, including many-body excitations, such as 117 magnons, and many-body instabilities, such as superconductivity or magnetism. The proposed plasmonic wave guide in this thesis is therefore just one possible example of a more general concept for Coulomb-engineering of many-body properties in metallic layered materials with fascinating further applications. 118 Bibliography [1] Giovanni Onida, Lucia Reining, and Angel Rubio. “Electronic excitations: density-functional versus many-body Green’s-function approaches”. In: Reviews of Modern Physics 74.2 (June 2002), p. 601.doi: 10.1103/RevModPhys.74.601. [2] K. V. Klitzing, G. Dorda, and M. Pepper. “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance”. In: Physical Review Letters 6 (Aug. 1980), pp. 494–497.issn: 0031-9007.doi: 10.1103/PhysRevLett.45.494. [3] D. C. Tsui, H. L. Stormer, and A. C. Gossard. “Two-Dimensional Magnetotransport in the Extreme Quantum Limit”. In: Physical Review Letters 48.22 (May 1982), pp. 1559–1562. issn: 00319007.doi: 10.1103/PhysRevLett.48.1559. [4] C. L. Kane and E. J. Mele. “Z2 topological order and the quantum spin hall effect”. In: Physical Review Letters 95.14 (2005), p. 146802.issn: 00319007.doi: 10.1103/PhysRevLett.95.146802. [5] C. L. Kane and E. J. Mele. “Quantum Spin hall effect in graphene”. In: Physical Review Letters 95.22 (2005), p. 226801.issn: 00319007.doi: 10.1103/PhysRevLett.95.226801. [6] Yuanbo Zhang, Yan Wen Tan, Horst L. Stormer, and Philip Kim. “Experimental observation of the quantum Hall effect and Berry’s phase in graphene”. In: Nature 438.7065 (2005), pp. 201–204.issn: 00280836.doi: 10.1038/nature04235. [7] B. Andrei Bernevig and Shou Cheng Zhang. “Quantum spin hall effect”. In: Physical Review Letters 96.10 (2006), p. 106802.issn: 00319007.doi: 10.1103/PhysRevLett.96.106802. [8] B. Andrei Bernevig, Taylor L. Hughes, and Shou Cheng Zhang. “Quantum spin hall effect and topological phase transition in HgTe quantum wells”. In: Science 314.5806 (2006), pp. 1757–1761.issn: 00368075.doi: 10.1126/science.1133734. 119 [9] Markus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao Liang Qi, and Shou Cheng Zhang. “Quantum spin hall insulator state in HgTe quantum wells”. In: Science 318.5851 (2007), pp. 766–770.issn: 00368075.doi: 10.1126/science.1148047. [10] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim. “Room-temperature quantum hall effect in graphene”. In: Science 315.5817 (2007), pp. 1379–1379.issn: 00368075.doi: 10.1126/science.1137201. [11] Liang Fu, C. L. Kane, and E. J. Mele. “Topological insulators in three dimensions”. In: Physical Review Letters 98.10 (2007), p. 106803.issn: 00319007.doi: 10.1103/PhysRevLett.98.106803. [12] Liang Fu and C. L. Kane. “Topological insulators with inversion symmetry”. In: Physical Review B 76.4 (2007), p. 045302.issn: 10980121.doi: 10.1103/PhysRevB.76.045302. [13] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan. “A topological Dirac insulator in a quantum spin Hall phase”. In: Nature 452.7190 (2008), pp. 970–974. issn: 14764687.doi: 10.1038/nature06843. [14] Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K. Mo, X. L. Qi, H. J. Zhang, P. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen. “Experimental realization of a three-dimensional topological insulator, Bi2Te3”. In: Science 325.5937 (2009), pp. 178–181.issn: 00368075.doi: 10.1126/science.1173034. [15] K. Kuroda, M. Ye, A. Kimura, S. V. Eremeev, E. E. Krasovskii, E. V. Chulkov, Y. Ueda, K. Miyamoto, T. Okuda, K. Shimada, H. Namatame, and M. Taniguchi. “Experimental realization of a three-dimensional topological insulator phase in ternary chalcogenide TlBiSe2”. In: Physical Review Letters 105.14 (2010), p. 146801.issn: 00319007.doi: 10.1103/PhysRevLett.105.146801. [16] Yan Gong, Jingwen Guo, Jiaheng Li, Kejing Zhu, Menghan Liao, Xiaozhi Liu, Qinghua Zhang, Lin Gu, Lin Tang, Xiao Feng, Ding Zhang, Wei Li, Canli Song, Lili Wang, Pu Yu, Xi Chen, Yayu Wang, Hong Yao, Wenhui Duan, Yong Xu, Shou Cheng Zhang, Xucun Ma, Qi Kun Xue, and Ke He. “Experimental Realization of an Intrinsic Magnetic Topological Insulator”. In: Chinese Physics Letters 36.7 (2019), p. 076801.issn: 17413540.doi: 10.1088/0256-307X/36/7/076801. [17] Ryo Noguchi, T. Takahashi, K. Kuroda, M. Ochi, T. Shirasawa, M. Sakano, C. Bareille, M. Nakayama, M. D. Watson, K. Yaji, A. Harasawa, H. Iwasawa, P. Dudin, T. K. Kim, M. Hoesch, V. Kandyba, A. Giampietri, A. Barinov, S. Shin, R. Arita, T. Sasagawa, and Takeshi Kondo. “A weak topological insulator state in quasi-one-dimensional bismuth iodide”. In: Nature 566.7745 (2019), pp. 518–522.issn: 14764687.doi: 10.1038/s41586-019-0927-7. 120 [18] M. Z. Hasan and C. L. Kane. “Colloquium: Topological insulators”. In: Reviews of Modern Physics 82.4 (2010), p. 3045.issn: 00346861.doi: 10.1103/RevModPhys.82.3045. [19] Xiao Liang Qi and Shou Cheng Zhang. “Topological insulators and superconductors”. In: Reviews of Modern Physics 83.4 (2011), p. 1057.issn: 00346861.doi: 10.1103/RevModPhys.83.1057. eprint: 1008.2026. [20] Andreas Karch. “Surface plasmons and topological insulators”. In: Physical Review B 83.24 (2011), p. 245432.issn: 10980121.doi: 10.1103/PhysRevB.83.245432. [21] Yoshinori Okada and Vidya Madhavan. “Topological insulators: plasmons at the surface”. In: Nature nanotechnology 8.8 (2013), pp. 541–542.issn: 17483395.doi: 10.1038/nnano.2013.157. [22] Robert Schütky, Christian Ertler, Andreas Trügler, and Ulrich Hohenester. “Surface plasmons in doped topological insulators”. In: Physical Review B 88.19 (2013), p. 195311. issn: 10980121.doi: 10.1103/PhysRevB.88.195311. [23] Dmitry K. Efimkin, Yurii E. Lozovik, and Alexey A. Sokolik. “Collective excitations on a surface of topological insulator”. In: Nanoscale Research Letters 7.1 (2012), p. 163.issn: 1556276X.doi: 10.1186/1556-276X-7-163. [24] Jian Yuan, Weiliang Ma, Lei Zhang, Yao Lu, Meng Zhao, Hongli Guo, Jin Zhao, Wenzhi Yu, Yupeng Zhang, Kai Zhang, Hui Ying Hoh, Xiaofeng Li, Kian Ping Loh, Shaojuan Li, Cheng Wei Qiu, and Qiaoliang Bao. “Infrared Nanoimaging Reveals the Surface Metallic Plasmons in Topological Insulator”. In: ACS Photonics 4.12 (2017), pp. 3055–3062.issn: 23304022.doi: 10.1021/acsphotonics.7b00568. [25] Alexander M. Dubrovkin, Giorgio Adamo, Jun Yin, Lan Wang, Cesare Soci, Qi Jie Wang, and Nikolay I. Zheludev. “Visible Range Plasmonic Modes on Topological Insulator Nanostructures”. In: Advanced Optical Materials 5.3 (2017), p. 1600768.issn: 21951071. doi: 10.1002/adom.201600768. [26] Dafei Jin, Thomas Christensen, Marin Soljačić, Nicholas X. Fang, Ling Lu, and Xiang Zhang. “Infrared Topological Plasmons in Graphene”. In: Physical Review Letters 118.24 (2017), p. 245301.issn: 10797114.doi: 10.1103/PhysRevLett.118.245301. [27] P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi. “Observation of Dirac plasmons in a topological insulator”. In: Nature Nanotechnology 8.8 (2013), pp. 556–560.issn: 17483395. doi: 10.1038/nnano.2013.134. [28] Prabhu K. Venuthurumilli, Xiaolei Wen, Vasudevan Iyer, Yong P. Chen, and Xianfan Xu. “Near-Field Imaging of Surface Plasmons from the Bulk and Surface State of Topological Insulator Bi2Te2Se”. In: ACS Photonics 6.10 (2019), pp. 2492–2498.issn: 23304022.doi: 10.1021/acsphotonics.9b00814. 121 [29] Zhihao Jiang, Malte Rösner, Roelof E. Groenewald, and Stephan Haas. “Localized plasmons in topological insulators”. In: Physical Review B 101.4 (2020), p. 045106.issn: 24699969.doi: 10.1103/PhysRevB.101.045106. [30] Henning Schlömer, Zhihao Jiang, and Stephan Haas. “Plasmons in two-dimensional topological insulators”. In: Physical Review B 103.11 (Mar. 2021), p. 115116.issn: 24699969.doi: 10.1103/PhysRevB.103.115116. [31] Yi Ping Lai, I. Tan Lin, Kuang Hsiung Wu, and Jia Ming Liu. “Plasmonics in topological insulators”. In: Nanomaterials and Nanotechnology 4 (2014), p. 13.issn: 18479804.doi: 10.5772/58558. [32] Theresa P. Ginley and Stephanie Law. “Coupled Dirac Plasmons in Topological Insulators”. In: Advanced Optical Materials 6.13 (2018), p. 1800113.issn: 21951071.doi: 10.1002/adom.201800113. [33] Junjie Qi, Haiwen Liu, and X. C. Xie. “Surface plasmon polaritons in topological insulators”. In: Physical Review B 89.15 (2014), p. 155420.issn: 1550235X.doi: 10.1103/PhysRevB.89.155420. [34] S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C.H. Liew, M. Segev, C. Schneider, and S. Höfling. “Exciton-polariton topological insulator”. In: Nature 562.7728 (2018), pp. 552–556.issn: 14764687.doi: 10.1038/s41586-018-0601-5. [35] Johannes Knolle and Nigel R. Cooper. “Excitons in topological Kondo insulators: Theory of thermodynamic and transport anomalies in SmB6”. In: Physical Review Letters 118.9 (Mar. 2017), p. 096604.issn: 10797114.doi: 10.1103/PhysRevLett.118.096604. [36] Yaroslav V. Kartashov and Dmitry V. Skryabin. “Bistable Topological Insulator with Exciton-Polaritons”. In: Physical Review Letters 119.25 (2017), p. 253904.issn: 10797114. doi: 10.1103/PhysRevLett.119.253904. [37] H. H. Kung, A. P. Goyal, D. L. Maslov, X. Wang, A. Lee, A. F. Kemper, S. W. Cheong, and G. Blumberg. “Observation of chiral surface excitons in a topological insulator Bi 2 Se 3”. In: Proceedings of the National Academy of Sciences of the United States of America 116.10 (2019), pp. 4006–4011.issn: 10916490.doi: 10.1073/pnas.1813514116. [38] Andreas Rückriegel, Arne Brataas, and Rembert A. Duine. “Bulk and edge spin transport in topological magnon insulators”. In: Physical Review B 97.8 (2018), p. 081106.issn: 24699969.doi: 10.1103/PhysRevB.97.081106. [39] M. Malki and G. S. Uhrig. “Topological magnon bands for magnonics”. In: Physical Review B 99.17 (2019), p. 174412.issn: 24699969.doi: 10.1103/PhysRevB.99.174412. 122 [40] Alexander Mook, Jürgen Henk, and Ingrid Mertig. “Edge states in topological magnon insulators”. In: Physical Review B 90.2 (2014), p. 024412.issn: 1550235X.doi: 10.1103/PhysRevB.90.024412. [41] Michael El-Batanouny. Advanced Quantum Condensed Matter Physics: One-Body, Many-Body, and Topological Perspectives. Cambridge University Press, 2020. [42] Marvin L. Cohen and Steven G. Louie. Fundamentals of Condensed Matter Physics. 2016. doi: 10.1017/cbo9781139031783. [43] Rodrigo A. Muniz, Stephan Haas, A. F. J. Levi, and Ilya Grigorenko. “Plasmonic excitations in tight-binding nanostructures”. In: Physical Review B 80.4 (July 2009), p. 045413.issn: 1098-0121.doi: 10.1103/PhysRevB.80.045413. [44] János K Asbóth, László Oroszlány, and András Pályi. “A short course on topological insulators”. In: Lecture notes in physics 919 (2016). [45] Roman Jackiw and Cláudio Rebbi. “Solitons with fermion number1/2”. In: Physical Review D 13.12 (1976), p. 3398. [46] Sriram Ganeshan, Kai Sun, and S. Das Sarma. “Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models”. In: Physical Review Letters 110.18 (May 2013), p. 180403.doi: 10.1103/PhysRevLett.110.180403. [47] Feng Liu and Katsunori Wakabayashi. “Novel topological phase with a zero berry curvature”. In: Physical review letters 118.7 (2017), p. 076803. [48] W. P. Su, J. R. Schrieffer, and A. J. Heeger. “Solitons in polyacetylene”. In: Physical Review Letters 42.25 (1979), p. 1698.issn: 00319007.doi: 10.1103/PhysRevLett.42.1698. [49] M. J. Rice and E. J. Mele. “Elementary excitations of a linearly conjugated diatomic polymer”. In: Physical Review Letters 49.19 (1982), p. 1455.issn: 00319007.doi: 10.1103/PhysRevLett.49.1455. [50] Cyprian Lewandowski and Leonid Levitov. “Intrinsically undamped plasmon modes in narrow electron bands”. In: Proceedings of the National Academy of Sciences of the United States of America 116.42 (2019), pp. 20869–20874.issn: 10916490.doi: 10.1073/pnas.1909069116. [51] Bm Santoyo. “Plasmons in three, two and one dimension”. In: Revista Mexicana de Fisica 39.4 (1993), pp. 640–65.url: https://rmf.smf.mx/ojs/rmf/article/download/2356/2324. [52] Felipe H. da Jornada, Lede Xian, Angel Rubio, and Steven G. Louie. “Universal slow plasmons and giant field enhancement in atomically thin quasi-two-dimensional metals”. In: Nature Communications 11.1 (2020), pp. 1–10.issn: 20411723.doi: 10.1038/s41467-020-14826-8. 123 [53] János K Asbóth, László Oroszlány, and András Pályi. “A short course on topological insulators”. In: Lecture notes in physics 919 (2016), pp. 997–1000.url: https://link.springer.com/book/10.1007%5C%2F978-3-319-25607-8. [54] M. Rösner, C. Steinke, M. Lorke, C. Gies, F. Jahnke, and T. O. Wehling. “Two-Dimensional Heterojunctions from Nonlocal Manipulations of the Interactions”. In: Nano Letters 16.4 (Apr. 2016), pp. 2322–2327.issn: 15306992.doi: 10.1021/acs.nanolett.5b05009. [55] C. Steinke, T. O. Wehling, and M. Rösner. “Coulomb-engineered heterojunctions and dynamical screening in transition metal dichalcogenide monolayers”. In: Physical Review B 102.11 (Sept. 2020), p. 115111.issn: 24699969.doi: 10.1103/PhysRevB.102.115111. [56] Archana Raja, Andrey Chaves, Jaeeun Yu, Ghidewon Arefe, Heather M. Hill, Albert F. Rigosi, Timothy C. Berkelbach, Philipp Nagler, Christian Schüller, Tobias Korn, Colin Nuckolls, James Hone, Louis E. Brus, Tony F. Heinz, David R. Reichman, and Alexey Chernikov. “Coulomb engineering of the bandgap and excitons in two-dimensional materials”. In: Nature Communications 8.1 (May 2017), pp. 1–7.issn: 20411723.doi: 10.1038/ncomms15251. [57] Matthias Florian, Malte Hartmann, Alexander Steinhoff, Julian Klein, Alexander W. Holleitner, Jonathan J. Finley, Tim O. Wehling, Michael Kaniber, and Christopher Gies. “The Dielectric Impact of Layer Distances on Exciton and Trion Binding Energies in van der Waals Heterostructures”. In: Nano Letters 18.4 (Apr. 2018), pp. 2725–2732.issn: 15306992.doi: 10.1021/acs.nanolett.8b00840. [58] Philipp Steinleitner, Philipp Merkl, Alexander Graf, Philipp Nagler, Kenji Watanabe, Takashi Taniguchi, Jonas Zipfel, Christian Schüller, Tobias Korn, Alexey Chernikov, Samuel Brem, Malte Selig, Gunnar Berghäuser, Ermin Malic, and Rupert Huber. “Dielectric Engineering of Electronic Correlations in a van der Waals Heterostructure”. In: Nano Letters 18.2 (Feb. 2018), pp. 1402–1409.issn: 15306992.doi: 10.1021/acs.nanolett.7b05132. [59] M. Iqbal Bakti Utama, Hans Kleemann, Wenyu Zhao, Chin Shen Ong, Felipe H. da Jornada, Diana Y. Qiu, Hui Cai, Han Li, Rai Kou, Sihan Zhao, Sheng Wang, Kenji Watanabe, Takashi Taniguchi, Sefaattin Tongay, Alex Zettl, Steven G. Louie, and Feng Wang. “A dielectric-defined lateral heterojunction in a monolayer semiconductor”. In: Nature Electronics 2.2 (Feb. 2019), pp. 60–65.issn: 25201131.doi: 10.1038/s41928-019-0207-4. [60] Yuto Kajino, Kenichi Oto, and Yasuhiro Yamada. “Modification of Optical Properties in Monolayer WS2 on Dielectric Substrates by Coulomb Engineering”. In: Journal of Physical Chemistry C 123.22 (June 2019), pp. 14097–14102.issn: 19327455.doi: 10.1021/acs.jpcc.9b04514. 124 [61] Erik G. C. P. van Loon, Malte Schüler, Daniel Springer, Giorgio Sangiovanni, Jan M. Tomczak, and Tim O. Wehling. “Coulomb Engineering of two-dimensional Mott materials”. In: arXiv (Jan. 2020), p. 2001.01735.url: http://arxiv.org/abs/2001.01735. [62] Kirsten Andersen and Kristian S. Thygesen. “Plasmons in metallic monolayer and bilayer transition metal dichalcogenides”. In: Physical Review B 88.15 (Oct. 2013), p. 155128.issn: 10980121.doi: 10.1103/PhysRevB.88.155128. [63] R. E. Groenewald, M. Rösner, G. Schönhoff, S. Haas, and T. O. Wehling. “Valley plasmonics in transition metal dichalcogenides”. In: Physical Review B 93.20 (May 2016), p. 205145.issn: 24699969.doi: 10.1103/PhysRevB.93.205145. [64] M. Rösner, E. Şaşlolu, C. Friedrich, S. Blügel, and T. O. Wehling. “Wannier function approach to realistic Coulomb interactions in layered materials and heterostructures”. In: Physical Review B 92.8 (Aug. 2015), p. 085102.issn: 1550235X.doi: 10.1103/PhysRevB.92.085102. [65] Kirsten Andersen, Simone Latini, and Kristian S. Thygesen. “Dielectric Genome of van der Waals Heterostructures”. In: Nano Letters 15.7 (July 2015), pp. 4616–4621.issn: 15306992.doi: 10.1021/acs.nanolett.5b01251. [66] Ivor Lončarić, Zoran Rukelj, Vyacheslav M. Silkin, and Vito Despoja. “Strong two-dimensional plasmon in Li-intercalated hexagonal boron-nitride film with low damping”. In: npj 2D Materials and Applications 2.1 (Dec. 2018), p. 33.issn: 23977132. doi: 10.1038/s41699-018-0078-y. [67] Mark B. Lundeberg, Yuanda Gao, Reza Asgari, Cheng Tan, Ben Van Duppen, Marta Autore, Pablo Alonso-González, Achim Woessner, Kenji Watanabe, Takashi Taniguchi, Rainer Hillenbrand, James Hone, Marco Polini, and Frank H.L. Koppens. “Tuning quantum nonlocal effects in graphene plasmonics”. In: Science 357.6347 (July 2017), pp. 187–191.issn: 10959203.doi: 10.1126/science.aan2735. [68] Yu Liu, R. F. Willis, K. V. Emtsev, and Th Seyller. “Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets”. In: Physical Review B 78.20 (Nov. 2008), p. 201403.issn: 10980121.doi: 10.1103/PhysRevB.78.201403. [69] Tony Low and Phaedon Avouris. “Graphene plasmonics for terahertz to mid-infrared applications”. In: ACS Nano 8.2 (Feb. 2014), pp. 1086–1101.issn: 19360851.doi: 10.1021/nn406627u. [70] Tony Low, Rafael Roldán, Han Wang, Fengnian Xia, Phaedon Avouris, Luis Martín Moreno, and Francisco Guinea. “Plasmons and Screening in Monolayer and Multilayer Black Phosphorus”. In: Physical Review Letters 113.10 (Sept. 4, 2014), p. 106802.doi: 10.1103/PhysRevLett.113.106802. 125 [71] D. A. Prishchenko, V. G. Mazurenko, M. I. Katsnelson, and A. N. Rudenko. “Coulomb interactions and screening effects in few-layer black phosphorus: a tight-binding consideration beyond the long-wavelength limit”. In: 2D Materials 4.2 (Apr. 3, 2017), p. 025064.issn: 2053-1583.doi: 10.1088/2053-1583/aa676b. (Visited on 04/14/2021). [72] Chong Wang, Shenyang Huang, Qiaoxia Xing, Yuangang Xie, Chaoyu Song, Fanjie Wang, and Hugen Yan. “Van der Waals thin films of WTe 2 for natural hyperbolic plasmonic surfaces”. In: Nature Communications 11.1 (Mar. 3, 2020), p. 1158.issn: 2041-1723.doi: 10.1038/s41467-020-15001-9. (Visited on 04/14/2021). [73] Chong Wang, Yangye Sun, Shenyang Huang, Qiaoxia Xing, Guowei Zhang, Chaoyu Song, Fanjie Wang, Yuangang Xie, Yuchen Lei, Zhengzong Sun, and Hugen Yan. “Tunable Plasmons in Large-Area ${\mathrm{W}\mathrm{Te}}_{2}$ Thin Films”. In: Physical Review Applied 15.1 (Jan. 7, 2021), p. 014010.doi: 10.1103/PhysRevApplied.15.014010. (Visited on 04/14/2021). [74] M. Van Schilfgaarde and M. I. Katsnelson. “First-principles theory of nonlocal screening in graphene”. In: Physical Review B 83.8 (Feb. 2011), p. 081409.issn: 10980121.doi: 10.1103/PhysRevB.83.081409. [75] Ali Fahimniya, Cyprian Lewandowski, and Leonid Levitov. “Dipole-active collective excitations in moiré flat bands”. In: arXiv (Nov. 2020). arXiv: 2011.02982.url: http://arxiv.org/abs/2011.02982. [76] A. Yu Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno. “Edge and waveguide terahertz surface plasmon modes in graphene microribbons”. In: Physical Review B 84.16 (Oct. 2011), p. 161407.issn: 10980121.doi: 10.1103/PhysRevB.84.161407. [77] Shengjun Yuan, Fengping Jin, Rafael Roldán, Antti Pekka Jauho, and M. I. Katsnelson. “Screening and collective modes in disordered graphene antidot lattices”. In: Physical Review B 88.19 (Nov. 2013), p. 195401.issn: 10980121.doi: 10.1103/PhysRevB.88.195401. [78] Johan Christensen, Alejandro Manjavacas, Sukosin Thongrattanasiri, Frank H.L. Koppens, and F. Javier García De Abajo. “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons”. In: ACS Nano 6.1 (Jan. 2012), pp. 431–440.issn: 19360851.doi: 10.1021/nn2037626. [79] Jin Tae Kim and Sung-Yool Choi. “Graphene-based plasmonic waveguides for photonic integrated circuits”. In: Optics Express 19.24 (Nov. 2011), p. 24557.issn: 1094-4087.doi: 10.1364/oe.19.024557. [80] Sukosin Thongrattanasiri, Frank H.L. Koppens, and F. Javier García De Abajo. “Complete optical absorption in periodically patterned graphene”. In: Physical Review Letters 108.4 (Jan. 2012), p. 047401.issn: 00319007.doi: 10.1103/PhysRevLett.108.047401. 126 [81] Ashkan Vakil and Nader Engheta. “Transformation optics using graphene”. In: Science 332.6035 (June 2011), pp. 1291–1294.issn: 00368075.doi: 10.1126/science.1202691. [82] Ming Liu, Xiaobo Yin, Erick Ulin-Avila, Baisong Geng, Thomas Zentgraf, Long Ju, Feng Wang, and Xiang Zhang. “A graphene-based broadband optical modulator”. In: Nature 474.7349 (June 2011), pp. 64–67.issn: 00280836.doi: 10.1038/nature10067. [83] D. A. Prishchenko, V. G. Mazurenko, M. I. Katsnelson, and A. N. Rudenko. “Gate-tunable infrared plasmons in electron-doped single-layer antimony”. In: Physical Review B 98.20 (Nov. 2018), p. 201401.issn: 24699969.doi: 10.1103/PhysRevB.98.201401. [84] Guus Slotman, Alexander Rudenko, Edo Van Veen, Mikhail I. Katsnelson, Rafael Roldán, and Shengjun Yuan. “Plasmon spectrum of single-layer antimonene”. In: Physical Review B 98.15 (Oct. 2018), p. 155411.issn: 24699969.doi: 10.1103/PhysRevB.98.155411. [85] David Alcaraz Iranzo, Sébastien Nanot, Eduardo J.C. Dias, Itai Epstein, Cheng Peng, Dmitri K. Efetov, Mark B. Lundeberg, Romain Parret, Johann Osmond, Jin Yong Hong, Jing Kong, Dirk R. Englund, Nuno M.R. Peres, and Frank H.L. Koppens. “Probing the ultimate plasmon confinement limits with a van der Waals heterostructure”. In: Science 360.6386 (Apr. 2018), pp. 291–295.issn: 10959203.doi: 10.1126/science.aar8438. [86] Stephan J.P. Kress, Felipe V. Antolinez, Patrizia Richner, Sriharsha V. Jayanti, David K. Kim, Ferry Prins, Andreas Riedinger, Maximilian P.C. Fischer, Stefan Meyer, Kevin M. McPeak, Dimos Poulikakos, and David J. Norris. “Wedge Waveguides and Resonators for Quantum Plasmonics”. In: Nano Letters 15.9 (Sept. 2015), pp. 6267–6275. issn: 15306992.doi: 10.1021/acs.nanolett.5b03051. [87] M. S. Tame, K. R. McEnery, Ş K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim. “Quantum plasmonics”. In: Nature Physics 9.6 (June 2013), pp. 329–340.issn: 17452481.doi: 10.1038/nphys2615. [88] Marinko Jablan, Hrvoje Buljan, and Marin Soljačić. “Plasmonics in graphene at infrared frequencies”. In: Physical Review B 80.24 (Dec. 2009), p. 245435.issn: 10980121.doi: 10.1103/PhysRevB.80.245435. [89] A. N. Grigorenko, M. Polini, and K. S. Novoselov. “Graphene plasmonics”. In: Nature Photonics 6.11 (Nov. 2012), pp. 749–758.issn: 17494885.doi: 10.1038/nphoton.2012.262. [90] Xiaoguang Luo, Teng Qiu, Weibing Lu, and Zhenhua Ni. “Plasmons in graphene: Recent progress and applications”. In: Materials Science and Engineering R: Reports 74.11 (Nov. 2013), pp. 351–376.issn: 0927796X.doi: 10.1016/j.mser.2013.09.001. 127 [91] M. Kim, S. G. Xu, A. I. Berdyugin, A. Principi, S. Slizovskiy, N. Xin, P. Kumaravadivel, W. Kuang, M. Hamer, R. Krishna Kumar, R. V. Gorbachev, K. Watanabe, T. Taniguchi, I. V. Grigorieva, V. I. Fal’ko, M. Polini, and A. K. Geim. “Control of electron-electron interaction in graphene by proximity screenings”. In: Nature Communications 11.1 (Dec. 2020), pp. 1–6.issn: 20411723.doi: 10.1038/s41467-020-15829-1. [92] Rovi Angelo B. Villaos, Christian P. Crisostomo, Zhi-Quan Huang, Shin-Ming Huang, Allan Abraham B. Padama, Marvin A. Albao, Hsin Lin, and Feng-Chuan Chuang. “Thickness dependent electronic properties of Pt dichalcogenides”. In: npj 2D Materials and Applications 3.1 (Jan. 16, 2019), pp. 1–8.issn: 2397-7132.doi: 10.1038/s41699-018-0085-z. (Visited on 04/14/2021). [93] Ali Obies Muhsen Almayyali, Bahjat B. Kadhim, and Hamad Rahman Jappor. “Tunable electronic and optical properties of 2D PtS2/MoS2 van der Waals heterostructure”. In: Physica E: Low-dimensional Systems and Nanostructures 118 (Apr. 1, 2020), p. 113866. issn: 1386-9477.doi: 10.1016/j.physe.2019.113866. (Visited on 04/14/2021). [94] Yi-min Ding, Jun-jie Shi, Congxin Xia, Min Zhang, Juan Du, Pu Huang, Meng Wu, Hui Wang, Yu-lang Cen, and Shu-hang Pan. “Enhancement of hole mobility in InSe monolayer via an InSe and black phosphorus heterostructure”. In: Nanoscale 9.38 (Oct. 5, 2017), pp. 14682–14689.issn: 2040-3372.doi: 10.1039/C7NR02725G. (Visited on 04/14/2021). [95] A. V. Lugovskoi, M. I. Katsnelson, and A. N. Rudenko. “Strong Electron-Phonon Coupling and its Influence on the Transport and Optical Properties of Hole-Doped Single-Layer InSe”. In: Physical Review Letters 123.17 (Oct. 23, 2019), p. 176401.doi: 10.1103/PhysRevLett.123.176401. (Visited on 04/14/2021). [96] Brian Kiraly, Elze J. Knol, Klara Volckaert, Deepnarayan Biswas, Alexander N. Rudenko, Danil A. Prishchenko, Vladimir G. Mazurenko, Mikhail I. Katsnelson, Philip Hofmann, Daniel Wegner, and Alexander A. Khajetoorians. “Anisotropic Two-Dimensional Screening at the Surface of Black Phosphorus”. In: Physical Review Letters 123.21 (Nov. 21, 2019), p. 216403.doi: 10.1103/PhysRevLett.123.216403. (Visited on 04/14/2021). [97] Rafi Bistritzer and Allan H. MacDonald. “Moiré bands in twisted double-layer graphene”. In: Proceedings of the National Academy of Sciences of the United States of America 108.30 (July 2011), pp. 12233–12237.issn: 00278424.doi: 10.1073/pnas.1108174108. [98] Eva Y. Andrei and Allan H. MacDonald. “Graphene bilayers with a twist”. In: Nature Materials 19.12 (Dec. 2020), pp. 1265–1275.issn: 14764660.doi: 10.1038/s41563-020-00840-0. 128 [99] Astrid Weston, Yichao Zou, Vladimir Enaldiev, Alex Summerfield, Nicholas Clark, Viktor Zólyomi, Abigail Graham, Celal Yelgel, Samuel Magorrian, Mingwei Zhou, Johanna Zultak, David Hopkinson, Alexei Barinov, Thomas H. Bointon, Andrey Kretinin, Neil R. Wilson, Peter H. Beton, Vladimir I. Fal’ko, Sarah J. Haigh, and Roman Gorbachev. “Atomic reconstruction in twisted bilayers of transition metal dichalcogenides”. In: Nature Nanotechnology 15.7 (July 2020), pp. 592–597.issn: 17483395.doi: 10.1038/s41565-020-0682-9. [100] Tom Westerhout, Edo Van Veen, Mikhail I. Katsnelson, and Shengjun Yuan. “Plasmon confinement in fractal quantum systems”. In: Physical Review B 97.20 (May 2018), p. 205434.issn: 24699969.doi: 10.1103/PhysRevB.97.205434. [101] Ting He, Zhen Wang, Fang Zhong, Hehai Fang, Peng Wang, and Weida Hu. “Etching Techniques in 2D Materials”. In: Advanced Materials Technologies 4.8 (Aug. 2019), p. 1900064.issn: 2365-709X.doi: 10.1002/admt.201900064. [102] Francesco De Nicola, Nikhil Santh Puthiya Purayil, Vaidotas Miˆ seikis, Davide Spirito, Andrea Tomadin, Camilla Coletti, Marco Polini, Roman Krahne, and Vittorio Pellegrini. “Graphene Plasmonic Fractal Metamaterials for Broadband Photodetectors”. In:Scientific Reports 10.1 (Dec. 2020), pp. 17–19.issn: 20452322.doi: 10.1038/s41598-020-63099-0. [103] Prasana K. Sahoo, Shahriar Memaran, Yan Xin, Luis Balicas, and Humberto R. Gutiérrez. “One-pot growth of two-dimensional lateral heterostructures via sequential edge-epitaxy”. In: Nature 553.7686 (Jan. 2018), pp. 63–67.issn: 14764687.doi: 10.1038/nature25155. [104] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young. “Intrinsic quantized anomalous Hall effect in a moiré heterostructure”. In: Science 367.6480 (Feb. 2020), pp. 900–903.doi: 10.1126/SCIENCE.AAY5533. [105] Ari M. Turner, Erez Berg, and Ady Stern. “Gapping fragile topological bands by interactions”. In: (Apr. 2021). arXiv: 2104.09528.url: https://arxiv.org/abs/2104.09528v1. 129 OwnPublications 1. Zhihao Jiang, Stephan Haas and Malte Rösner. "Plasmonic waveguides from Coulomb- engineered two-dimensional metals." 2D Materials8 (2021), 035037. 2. Yuling Guan, Zhihao Jiang and Stephan Haas. "Control of Plasmons in Topological In- sulators via Local Perturbations." arXiv:2102.02347 (2021). (Accepted to Physical Review B) 3. ZhihaoJiang, Henning Schlömer and Stephan Haas. "Control of Plasmons in Doped Topo- logical Insulators via Basis Atoms." Physical Review B104 (2021), 045135 4. Henning Schlömer,ZhihaoJiang and Stephan Haas. "Plasmons in Two-Dimensional Topo- logical Insulators." Physical Review B103 (2021), 115116. 5. Boxiang Song,ZhihaoJiang ∗ , Zerui Liu, Yunxiang Wang, Fanxin Liu, Stephen B. Cronin, Hao Yang, Deming Meng, Buyun Chen, Pan Hu, Adam M. Schwartzberg, Stefano Cabrini, Stephan Haas, and Wei Wu. "Probing the Mechanisms of Strong Fluorescence Enhancement in Plasmonic Nanogaps with Sub-nanometer Precision." ACS Nano14 (2020), 14769-14778. 6. Zhihao Jiang, Malte Rösner, Roelof E. Groenewald, and Stephan Haas. "Localized plas- mons in topological insulators." Physical Review B101 (2020), 045106. ∗ Boxiang Song and Zhihao Jiang contribute equally. 130

Abstract (if available)

Abstract Correlation and topology are probably the two most intriguing properties of electrons. While in many situations they are investigated independently, since many topological properties of electronic systems can be well described within the framework of single-particle physics, in some context, they can be playing an important role together. In recent years, there has been increased interest in the elementary excitations of topological electronic systems, such as plasmons, excitons and magnons. It is interesting to study whether these excitations inherit any topological signatures from the constituent electronic states, and how this would depend on the electron-electron Coulomb interactions in the system. ❧ In the first part of this thesis, we study plasmonic excitations in topological insulators. We focus on the Su-Schrieffer-Heeger (SSH) model in both one-dimensional (1D) and two-dimensional (2D) lattices. We use a fully quantum mechanical approach to evaluate the dielectric function in both momentum space and real space within the random phase approximation, from which we extract the electron energy loss spectra (EELS) that characterize the plasmonic excitations in the system. In momentum space, we observe different branches of plasmon dispersions that can be tuned by varying the inter-atomic separation between basis atoms. In real space, we observed plasmonic excitations that are localized on the boundaries of the 1D SSH and 2D SSH models, when they are in topologically non-trivial phases. These localized edge plasmons serve as topological signature of the underlying model in electronic collective excitations. They can be detected in experiments. These localized plasmons are shown to originate from the constituent topological electronic edge states and meanwhile to be affected by the bulk electronic states which are providing the background screening. We investigate the stability of the localized modes specifically on a topological interface of connecting two topologically distinct 1D SSH models against disorder. We find that both the excitation energy and the real-space localization of the localized plasmon mode are less stable than those of the single-electron topological state when the disorder becomes significant. ❧ In the second part of this thesis, we discuss 2D plasmons in layered materials on dielectric substrates. Coulomb interactions play an essential role in atomically-thin materials. On one hand, they are strong and long-ranged in layered systems due to the lack of environmental screening. On the other hand, they can be efficiently tuned by means of surrounding dielectric materials. Thus all physical properties which decisively depend on the exact structure of the electronic interactions can be in principle efficiently controlled and manipulated from the outside via Coulomb engineering. This concept has already been applied successfully to semiconductors for tuning the bandgap and exciton binding energies. We show that this concept can also be used to metallic systems to engineer plasmonic excitations. Specifically, we demonstrate that a spatially structured dielectric environment can be used to non-invasively confine plasmons in an unperturbed homogeneous metallic 2D system by modifications of its many-body interactions. This observation motivates us to propose a conceptually novel way to design 2D plasmonic waveguides via Coulomb engineering. Our numerical results indicate that plasmons can be confined to a 10-nm scale. In contrast to conventional functionalization mechanisms, this scheme relies on a purely many-body concept and does not involve any direct modifications to the active material itself. We discuss proper materials choices of both the active plasmonic 2D layer and the screening dielectric substrate for our proposal to be experimentally feasible.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Plasmons in quantum materials

PDF

Topological protection of quantum coherence in a dissipative, disordered environment

PDF

Plasmonic excitations in nanostructures

PDF

Coulomb interactions and superconductivity in low dimensional materials

PDF

Hot electron driven photocatalysis and plasmonic sensing using metallic gratings

PDF

Out-of-equilibrium dynamics of inhomogeneous quantum systems

PDF

Signatures of topology in a quasicrystal: a case study of the non-interacting and superconducting Fibonacci chain

PDF

Disordered quantum spin chains with long-range antiferromagnetic interactions

PDF

Superconductivity in low-dimensional topological systems

PDF

Healing of defects in random antiferromagnetic spin chains

PDF

Quantum information techniques in condensed matter: quantum equilibration, entanglement typicality, detection of topological order

PDF

Destructive decomposition of quantum measurements and continuous error detection and suppression using two-body local interactions

PDF

Fabrication and application of plasmonic nanostructures: a story of nano-fingers

PDF

Quantum computation by transport: development and potential implementations

PDF

High throughput computational framework for synthesis and accelerated discovery of dielectric polymer materials using polarizable reactive molecular dynamics and graph neural networks

PDF

Surface plasmon resonant enhancement of photocatalysis

PDF

Phase diagram of disordered quantum antiferromagnets

PDF

Light Emission from Carbon Nanotubes and Two-Dimensional Materials

PDF

Entanglement parity effects in quantum spin chains

PDF

Multi-scale quantum dynamics and machine learning for next generation energy applications

Asset Metadata

Creator Jiang, Zhihao (author)

Core Title Plasmonic excitations in quantum materials: topological insulators and metallic monolayers on dielectric substrates

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2021-12

Publication Date 11/22/2021

Defense Date 09/29/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag dielectric screening,OAI-PMH Harvest,plasmons,random phase approximation,topological insulators,two-dimensional materials

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan (committee chair), Levenson-Falk, Eli (committee member), Nakano, Aiichiro (committee member), Venuti, Lorenzo Campos (committee member), Wu, Wei (committee member)

Creator Email jzh9261@outlook.com,zhihaoji@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC17239131

Unique identifier UC17239131

Legacy Identifier etd-JiangZhiha-10256

Document Type Dissertation

Format application/pdf (imt)

Rights Jiang, Zhihao

Type texts

Source 20211124-wayne-usctheses-batch-900-nissen (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu