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Plasmonic excitations in nanostructures
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Plasmonic excitations in nanostructures
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PLASMONIC EXCITATIONS IN NANOSTRUCTURES by Rodrigo Angelo Muniz 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 2011 Copyright 2011 Rodrigo Angelo Muniz Dedication This thesis is dedicated to my beloved wife K´ esia Sibery Tosta who always gave me support during the time of this work. ii Table of Contents Dedication ii List of Figures v Abstract vii Chapter 1: Introduction 1 1.1 Breakdown of Mie solution . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Non-local linear response theory . . . . . . . . . . . . . . . . . 5 1.1.2 Dielectric response of a metallic rod . . . . . . . . . . . . . . . 8 1.2 Beyond the Mie solution . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Localized plasmonic excitations in graphene . . . . . . . . . . 13 Chapter 2: Dielectric response of finite systems 16 2.1 Dielectric response of a diatomic molecule . . . . . . . . . . . . . . . . 18 2.2 Atomic chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Electronic states . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Self-consistent linear response . . . . . . . . . . . . . . . . . . 23 Chapter 3: Plasmonic excitations of atomic chains 26 3.1 Longitudinal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Transverse modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Angle of incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Number of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5 Parallel chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.6 Inter-atomic distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter 4: Polarization operator in real space 38 4.1 Non-interacting states . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 RPA treatment of the interaction . . . . . . . . . . . . . . . . . . . . . 39 4.3 Plasmonic modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 iii Chapter 5: Localized plasmons in graphene 45 5.1 Traits of plasmonic localization . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Local spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2.1 Spatial profile of plasmonic modes . . . . . . . . . . . . . . . . 49 5.2.2 Chemical potential dependence . . . . . . . . . . . . . . . . . 52 5.2.3 Impurity strength dependence . . . . . . . . . . . . . . . . . . 53 5.3 3D visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3.1 Zero doping with impurity . . . . . . . . . . . . . . . . . . . . 56 5.3.2 Finite doping with impurity . . . . . . . . . . . . . . . . . . . 58 Table 5.1 Dependence on the impurity potential and doping . . . . . 61 5.3.3 The effect of impurity size . . . . . . . . . . . . . . . . . . . . 61 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Chapter 6: Overlook 64 Bibliography 68 iv List of Figures 1.1 Induced energy for metallic rods. . . . . . . . . . . . . . . . . . . . . . 9 1.2 Plasmon modes of metallic rods. . . . . . . . . . . . . . . . . . . . . . 10 2.1 Dielectric response of a diatomic molecule. . . . . . . . . . . . . . . . 21 3.1 Longitudinal modes in atomic chains. . . . . . . . . . . . . . . . . . . 27 3.2 Transverse Modes in double chains. . . . . . . . . . . . . . . . . . . . 29 3.3 Dependence on the direction of the external electric field. . . . . . . . . 30 3.4 Dependence on the number of electrons in the atomic chain. . . . . . . 32 3.5 Two parallel chains connected by an extra atom. . . . . . . . . . . . . . 34 3.6 Dependence on the distance between neighbor atoms. . . . . . . . . . . 35 5.1 Histogram of induced charge in every site for every mode. . . . . . . . 46 5.2 Induced charge on a site near the impurity for each mode. . . . . . . . . 47 5.3 Single and two-particle spectral functions. . . . . . . . . . . . . . . . . 48 5.4 Spatial profiles of localized plasmons with different spectral strength. . 50 5.5 Charge density profiles of some plasmons in pristine graphene. . . . . . 50 5.6 Charge density profiles of some localized plasmons around the impurity. 51 5.7 Spectral function dependence on the chemical potential. . . . . . . . . . 52 5.8 Spectral function dependence on the impurity potential strength. . . . . 54 5.9 Spatial profile of modes for pristine graphene. . . . . . . . . . . . . . . 55 5.10 Spatial profile of modes around an impurity for no doping. . . . . . . . 56 v 5.11 Spatial profile of modes around an impurity for electron doping. . . . . 59 5.12 Spatial profile of modes around an impurity for hole doping. . . . . . . 60 5.13 Spatial profile of modes for different impurity sizes. . . . . . . . . . . . 62 vi Abstract The prevalent classical model of plasmonic calculations for nano-scale metallic clusters is based on the Mie solution. Which consists of solving Maxwell’s equations with the material being represented by a dielectric function on its spatial location. However, such a semi-empirical continuum description necessarily breaks down beyond a certain level of coarseness introduced by atomic length scales. Even the bulk based model used for the dielectric function fails by itself. This limitation of the Mie solution has been established by a quantum mechanical calculation with self-consistent treatment of the dielectric response. In order to understand better the plasmonic excitations at nearly atomic scale, we explored the collective electromagnetic response of atomic chains of various sizes and geometries, and we also computed plasmons in graphene in the presence of an impurity. For the atomic chains, we calculated the plasmonic resonances as a function of the system shape, direction of the external applied field, electron filling and atomic separa- tion. Their frequency, oscillator strength and spatial modulation of the induced charge density were analyzed. It was shown that longitudinal and transverse modes can be con- trolled in amplitude and frequency by the cluster size. It was also observed an abrupt dependence of the modes on the electronic filling. We also find that changes in atomic spacings have a very different impact on low-energy vs. high-energy modes. And it was seen that changing the position of a single atom in a nanostructure can completely alter vii its collective dielectric response. This strong sensitivity to small changes is the key to controlling the dielectric properties of atomic scale structures, and it can thus become the gateway to a new generation of quantum devices which effectively utilize quantum physics for new functionalities. For graphene it was shown that impurities induce the formation of nanoscale local- ized plasmonic excitations in graphene sheets. It was studied the dependence of these excitations on the magnitude and size of the impurity potential and electronic filling. It was shown that the impurity potential and doping can be used to tune the properties of nano-plasmonic excitations, demonstrating that graphene is an inherently plasmonic material. It was found that the chemical potential can be used to turn them on and off, but it does not affect their frequency. While their frequency and amplitude can be tuned by varying the strength of the impurity potential. The method employed for this calculation had not been seen before. In principle the results discussed can be tested experimentally by high-frequency optical probes or STM. These results showed that collective excitations in finite systems have properties different from their bulk correspondents. Since there is not a macroscopic number of electrons in the system, the variation of one single electron causes observable differ- ences. The localized resonant modes are very sensitive to even small variations in the system, for example the position of a single atom. This makes it difficult to establish general rules about the properties of collective excitations in atomic structures. On the other hand it also provides a vast range of possibilities that can be explored for achieving new functionalities. viii Chapter 1: Introduction One of the most fundamental questions regarding a quantum system is what are the states in which it can be found. This set of states is called the Hilbert space. Once the Hilbert space is determined it is important to determine what are the eigenstates of the hamiltonian, or the states with well defined energies. These states are then distinguished by their energies, and the next natural question is what are the ground(lowest energy) state and the excited ones. The study of the ground states allows one to identify different phases of the sys- tem, however the excitations are also crucial. In condensed matter physics the focus is usually on low temperatures, so most systems are described by low energy excita- tions(quasiparticles) above a well defined ground state. In such regime, the underlying constituents of the system, such as electrons, nuclei, etc. are forgotten and replaced by the operators corresponding to the quasiparticles. The interaction between the basic particles such as electrons, photons or phonons can lead to the formation of composite objects that behave as particles on their own right. Therefore some of these quasiparticles are collective, i.e. they involve a significant proportion of the electrons in the system all moving in a synchronized way. Which is one of the most interesting aspects of condensed matter systems, novel kinds of particles emerge from the elementary ones. These excitations can be quite exotic and have very useful characteristics for applications. 1 Collective excitations are well understood for macroscopic bulk systems but much remains to be explained for nanostructured materials. One important feature is that many usual symmetries like translational or rotational invariance are broken for nanos- tructures. And of course this has an impact on the properties of the emerging quasi- particles. There are even issues on the definition of what is actually collective for such small scales. The usual guideline of associating collective traits to macroscopic numbers looses sense. However, in the last decades there has been enormous progress in the production and manipulation of ever smaller systems approaching the atomic scale. Therefore it needs to be understood what are the properties of collective excitations(or their precursors) at this scale. This is also a very important technological problem. The exploration of those different finite size characteristics will be the only way to improve the efficiency of elec- tronic devices as the natural barrier of atomic scales are reached. Simply miniaturizing will not be possible anymore, and new devices will have to based on other paradigms. One of the simplest examples of collective excitations is the plasmon. The oscilla- tory motion of electrons interacting by the Coulomb repulsion in metals can be described in terms of charge density waves with their own physical properties. A simple way of envisioning that is by considering that the motion of water molecules can be described in terms of waves in the ocean. Since plasmons have a very short wavelength, they have the ability to concentrate light at the nano-scale regime, thereby beating the diffraction limit and opening the possibilities of vast technological applications. Below are some of the most relevant: • Plasmonic nanoclusters can magnify the electric field in their vicinity by sev- eral orders of magnitude and therefore provide a tremendous improvement of enhanced Raman scattering [45]. 2 • Plasmonic nanostructures added to a solar cell can improve its efficiency by increasing the cross section of the absorbing material without requiring a thicker layer of it, which makes the capture of the expelled electrons harder [3]. • Plasmonic waveguides can be used to substitute photons in the information trans- port extending photonics into the nanoscale world [49]. Plasmonics can also pro- vide a natural extension to semiconductor electronics devices which are very lossy in the nanoscale [10]. • In biology the potential applications include labeling and imaging, optical sensing and improvements of photo-therapy treatment of cancer, by doping plasmonic nanoclusters to the cancerous cells therefore increasing their light absorption [6]. The nanostructures employed in the above applications are still rather large and can mostly be still described by the same techniques developed for bulk systems with some adjustments. Nevertheless, at scales of 110nm the traditional approaches fail [32, 22], reinforcing the need of better physical understanding of the collective excitations in such systems. The theme of this thesis is the pursuit of this understanding specifically for plas- mons. It is also aimed at exploring the new degrees of freedom present in those broken- symmetry systems in order to provide novel applications unattainable with more restric- tive bulk structures. The rest of this chapter explains previous work in order to establish the break- down of the Mie solution for small particles, and then outlines further studies of the regime beyond it. The next chapters contain the methods used to compute the dielectric response of atomic structures, including collective modes, and analysis of their proper- ties. 3 1.1 Breakdown of Mie solution In this section it is explored a system whose description lies at the boundary between classical and quantum theory. There are of course many ways to approach this problem. Here, it is studied the interaction of classical light with small metal particles of cylin- drical shape. Specifically, it is considered a physical model that is capable of observing the transition from bulk material properties to nano-scale structures, for which quantum effects dominate. The prevalent classical model describing the interaction of visible and infrared elec- tromagnetic radiation with nano-scale metallic clusters is based on the Mie solution [39]. Which consists of solving Maxwell’s equations with the material being represented by a dielectric function on its spatial location. The dielectric function usually employed is the same as the one computed for bulk systems, extracted from approximations such as the Drude model, which explores the translational invariance clearly absent in small structures. This local continuum field method has been used to describe plasmon resonances in nanoparticles [67, 56, 35, 20]. However, such a semi-empirical continuum descrip- tion necessarily breaks down beyond a certain level of coarseness introduced by atomic length scales. Even the bulk based model used for the dielectric function fails by itself. Thus, this cannot be used to describe the interface between quantum and clas- sical macroscopic regimes. Moreover, extensions of Mie theory to inhomogeneous cluster shapes are commonly restricted to low-order harmonic expansions (e.g. elliptical distortions) and so do not exhaust the full realm of possible geometric configurations. In addition, near-field appli- cations, such as surface enhanced Raman scattering [45], are most naturally described using a real-space theory that includes the non-local electronic response of inhomoge- neous structures, again beyond the scope of Mie theory. 4 In the following sub-section it is described a microscopic approach that demon- strates the breakdown of this concept at atomic scales, whereas for large cluster sizes the classical predictions for the plasmon resonances are reproduced. Then it is shown for cylindrical particles of different diameters how this approach is useful in describ- ing the coexistence of excitations of quantum and classical character in inhomogeneous nanoscale structures, evidencing the transition between the two regimes. 1.1.1 Non-local linear response theory To capture the main single-particle and collective aspects of light-matter interaction in inhomogeneous nanoscale systems we adopt the linear response approximation [53, 58], valid for sufficiently low intensities of electromagnetic radiation. The following method was presented in Ref.[22, 34]. The starting point of this approach is the determination of the eigenenergiesE i and wave functions i (r) of the Hamiltonian for the electrons in the nanostructure. These can be obtained from several models of varying complexity, ranging from effective mass and tight-binding Hamiltonians to density functional theory and exact diagonalization or other sophisticated numerical solutions of Hubbard-type models. The following discus- sion is limited to the effective mass and the tight-binding model is considered in Chapter 2, noting that response optimization would require multiple evaluations ofE i and i (r). While desirable, the additional complexity of dealing with more sophisticated models would limit us to very small clusters and would thus make the optimization iterations numerically very expensive. The effective mass model starts from the Schr¨ odinger equation for noninteracting electrons with massm e and chargee moving in potentialV (r), given by H i (r) = ~ 2 2m e r 2 +V (r) i (r) =E i i (r) (1.1) 5 This hamiltonian is solved in conjunction with the Poisson equation which in turn determines the local potential due to the spatial distribution of the positive background charges. Using the jellium approximation, the resulting potential is implicitly given by r 2 V (r) = 4e(r) (1.2) where the density of the positive background charge(r) satisfies the condition of neu- trality inside the nanostructure so that Z (r)dr =N el (1.3) whereN el is the number of electrons. Once the electronic energy levels and wave functions have been obtained, it is pos- sible to calculate the dielectric susceptibility(r;r 0 ;!) using a real-space formulation of the random phase approximation [37, 19, 11] (r;r 0 ;!) = X i;j f(E i )f(E j ) E i E j !i i (r) i (r 0 ) j (r 0 ) j (r) (1.4) The induced charge density distribution function is then obtained by ind (r;!) = Z (r;r 0 ;!)( ind (r 0 ;!) + ext (r 0 ;!))dr 0 (1.5) where the induced potential is given by ind (r;!) = Z ind (r 0 ;!) jrr 0 j dr 0 (1.6) 6 One can avoid the large memory requirement to store (r;r 0 ;!) by calculating the induced charge density distribution iteratively via ind (r;!) = X i;j f(E i )f(E j ) E i E j !i i (r) j (r) Z i (r 0 ) tot (r 0 ;!) j (r 0 )dr 0 (1.7) with tot (r 0 ;!) = ind (r 0 ;!) + ext (r 0 ;!). The integrals can be evaluated numerically. Eqs. 2.31 and 2.32 are solved self-consistently by iterating ind (r;!) and ind (r;!). For sufficiently small clusters, this procedure typically converges in 3 to 8 steps when starting with ind (r;!) = 0, depending on the proximity to a resonance and on the value of the damping constant . A much better performance can be achieved when the initial ind (r;!) is taken as the solution of a previously solved nearby frequency. Upon its convergence, the frequency and spatial dependence of the induced electric field and the induced energy are obtained using E ind (r;!) =r ind (r;!) (1.8) and U ind (!) = Z jE ind (r;!)j 2 dr (1.9) The observed resonances in the induced energy and charge density distribution at certain driving frequencies of the applied electric field correspond to collective modes of the cluster. 7 1.1.2 Dielectric response of a metallic rod Turning now to metallic structures, described within the effective mass approach, we illustrate the application of inhomogeneous linear response to an infinitely long elliptic rod illuminated by an external field. Assume that the rod is aligned in thez direction and the electric field polarization is along the (1; 1; 0) direction. For the wave functions we assume periodic boundary conditions along thez direction. To quantify the response of arbitrary geometries to the applied field, we calculate the energy of the induced field, defined by W ind (!) = (1=2) Z jE ind (r;!)j 2 dr (1.10) In Fig. 1.1 the logarithm of the energy of the induced field (log 10 (W ind )) is displayed as a function of the applied external field photon energy~! and the characteristic system size R for the aspect ratio (a)a :b = 1 : 1:3 and (b)a :b = 1 : 2. For sufficiently large rod sizes, one clearly observes two plasmon resonances,! + and! , consistent with earlier predictions based on Mie theory [20]. Our method confirms that these resonances occur at ! + = ! p (b=(a +b)) 1=2 and ! = ! p (a=(a +b)) 1=2 , where ! p is the bulk plasmon frequency! p = p 4e 2 =m e . Because the scale in Fig. 1.1 is on a logarithmic scale, it is clear that the spectral intensity for large rod sizes is orders of magnitude greater compared to smaller rod sizes. Most importantly, it is evident from this figure that the classical picture of two well-defined resonances breaks down below a characteristic system size. For sufficiently small rod sizes, the two macroscopic resonances split into multi-level molecular excitations, with the overall spectral weight shifting toward lower energies. For the chosen parameters this transition occurs atR c 6:5L. BelowR c , the spectral intensity of the energy levels is reduced because fewer electrons participate in the individual resonances. 8 The physics determining the value ofR c may be illustrated by considering an infi- nite cylindrical rod of radius R and electron density . In the classical regime, the observed collective oscillations are only weakly damped, indicating that they are well separated from the quasi-continuum of single-particle excitations. This leads to the con- dition ! p R=v F 1, stating that the plasmon phase velocity is greater than the Fermi velocity v F of the electrons [52]. For r < R the electrons are trapped in a harmonic potential due to the uniform positive background [32] and the characteristic collective frequency is ! p = p 4e 2 =(2m e ). Estimating the Fermi velocity using a bulk value v F = (3 2 ) 1=3 ~=m e one obtains R R c = 1=6 3 1=3 p 2 ~ em 1=2 e 1=6 3:4 L for an elec- tron density = (1=480)L 3 , in reasonable, but only approximate, agreement with the calculated thresholdR c 6:5L shown in Fig.1.1. For system sizes below R c the dominant excitations are observed to shift toward lower energies as the positive background potential becomes increasingly anharmonic. Figure 1.1: Induced energy for metallic rods. (a) Logarithm of the energy of the induced electric field (log 10 (W ind )) (see text) in an elliptic metallic rod due to applied external fieldE ext along the (1; 1; 0) direction as a function of face surface size R and photon energy ~! of the external field in units of E 0 . Aspect ratio (semi-minor to semi-major axis)a : b = 1 : 1:3. TemperatureT = 0 K and carrier density = (1=480)L 3 , = 10 3 , and E 0 . (b) Same as (a) but with a :b = 1 : 2. Ref.[22]. 9 The harmonicity criterion for excitations can be expressed as 0 =R 1, where 0 = ~ m e! p 1=2 is the classical turning point for an electron in the ground state of the harmonic potential. This is equivalent to the conditionRR c = ( ~ 2 2mee 2 ) 1=4 , yieldingR 3L. The nature of the excitations changes as the characteristic sizeR crosses the quantum threshold R c . In Fig. 1.2, we show logarithm of the calculated energy of the induced field (log 10 (W ind )) as a function of external field frequency. As shown in Fig. 1.2(a), for relatively large rod sizes, the two distinct plasmon resonances labeled ! and ! + correspond to two orthogonal bipolar charge distributions, as indicated in the inset. The spatial orientations of these induced resonances are aligned with the semi-major and semi-minor axes, and do not depend on the direction of the incident field. In contrast, for system sizes less thanR c (Fig. 1.2(b)), the excitation spectrum consists of several lower-intensity modes, dominated by a low frequency resonance. However, such spatial Figure 1.2: Plasmon modes of metallic rods. (a) Logarithm of the energy of the induced electric field (log 10 (W ind )) in an elliptic rod of sizeR = 11L with aspect ratioa :b = 1 : 1:3 as a function of photon’s energy~! of the external field in units ofE 0 . Carrier density = (1=480)L 3 and = 10 3 E 0 . The direction of the external field is indicated by the arrow. Inset: induced charge density at the resonant frequencies! + and! . The boundary of the rod is shown using the solid line, and the dotted line shows the set of classical turning points, corresponding to the positive background potential. (b) Same as (a), but for sizeR = 3L. Ref. [22]. 10 characteristics of these modes are different from the classical limit, i.e. they arealigned with the incident field, indicating that Mie theory breaks down in the quantum regime. 1.2 Beyond the Mie solution The previous section showed that the semiclassical approach described is not applicable to very small nanostructures. But the quantum mechanical method developed for that analysis also hinted that the resonant modes of such systems may also be very different when compared to their macroscopic counterparts 1 . This means that new intuition needs to be developed for such scales, the usual properties of macroscopic plasmons might not hold and it is necessary to determine the new ones. In order to begin this process we explored the collective electromagnetic response in atomic chains of various sizes and geometries, aiming to understand and hence control their dielectric response [42]. The method was a fully quantum-mechanical description of the dielectric properties of the system, and the electronic energy levels and wave functions were calculated within the tight-binding model. Recent advances in nanoscience have created a vast number of experimentally acces- sible ways to configure atomic and molecular clusters into different geometries with strongly varying physical properties. Specifically, exquisite control of the shape and size of atomic and molecular clusters has made it now possible to investigate the collective electromagnetic response of ultra-small metal and semiconductor particles [32, 7, 43, 46]. The aim of this study was to model and examine plasmonic excitations in such structures, and thus to gain an understanding of the quantum-to-classical crossover of collective modes with increasing cluster size. There is obvious technological rel- evance to tunable collective modes in nanostructures. For example, surface plasmon 1 Macroscopic in this context also includes nanostructures of scales larger than100 nm. 11 resonances in metallic nanospheres and films have been found to be highly sensitive to nearby microscopic objects, and hence are currently investigated for potential sensing applications [26]. In this context, it is desirable to design customized nanostructures with specifically tailored resonance properties[40], and this study is intended to be a first step into this direction. It is natural to expect that in many cases the electromagnetic response of nanoclus- ters is considerably different from the bulk. In particular for very small clusters, the quantum properties of electrons confined in the structure need to be taken into account [5]. Moreover, unlike in the bulk, the coupling between single-particle excitations and collective modes can be very strongly affected by its system parameters. This sensitivity opens up excellent opportunities to optimize the dielectric response via tuning the clus- ter geometry and its electron filling. For example, by proper arrangement of atoms on a surface one can design nanostructures with controllable resonances in the near-infrared or visible frequency range [43, 46]. A possible application of such nanostructures is the creation of metamaterials with negative refractive index at a given frequency. Further- more, since geometry optimization of bulk resonators has demonstrated minimization of losses in metamaterials [57], it is also interesting to investigate the effect of the nanos- tructure shape on the loss function at a given resonance frequency. In order to explore the possible peculiarities in the dielectric response of small clus- ters, we investigated the formation of resonances in generic systems of finite conducting clusters, and examine how their frequency and spatial dielectric response depends on the system size and geometry. In particular, the non-locality of the dielectric response function in these structures is important, and will therefore be properly accounted for. A similar analysis for the case of smallmetallic nanostructures was performed recently, using an effective mass approximation [22, 12]. Here we focus on the opposite limit, namely we assume that electrons in the cluster can be effectively described using a 12 tight-binding model [59]. Because of the more localized nature of the electronic wave functions in this model the overall magnitude of the collective modes are expected to be strongly suppressed as compared to metallic clusters. The method and results found in that work are presented in the next two chapters. It was found that the system shape, the electron filling and the driving frequency of the external electric field strongly control the resonance properties of the collective exci- tations in the frequency and spatial domains. Furthermore, it was found that one can design spatially localized collective excitations by properly tailoring the nanostructure geometry. And we see that changing the position of a single atom in a nanostructure can completely alter its collective dielectric response. This strong sensitivity to small changes is the key to controlling the modes of ultra-small structures, and it can thus become the gateway to a new generation of quantum devices which effectively utilize quantum physics for new functionalities. 1.2.1 Localized plasmonic excitations in graphene Graphene is a single-layer atomically thin allotrope form of carbon, where carbon atoms are arranged in a honeycomb lattice. Ab-initio calculations have shown that graphene is a gapless semiconductor, for which the valence and conduction bands meet at the Fermi energy. Around this point the energy dispersion of the quasi-particle excitations has been found to be linear [63]. Its band structure has two independent cones which at half filling has low energy energy excitations described by the Dirac equation. Dirac fermions have a different dynamics compared to the usual fermions found in condensed matter systems, and combined to its single-atom thickness, graphene is a promising material for technological use [21]. Besides its unique electronic properties, graphene also has many related descendants such as nanoribbons, bilayers, nanotubes and spirals. Which arguably makes it the most manageable material known. Experimentally, the 13 system can be tuned into a metallic regime by adjusting the chemical potential using a gate voltage [47]. Due to these unique properties there is considerable current interest in the electronic properties of that material [13, 23, 30, 48, 25]. Scanning tunneling microscope(STM) experiments have shown that graphene is intrinsically disordered [9]. It has lattice defects, vacancies, ripples, dislocations, mag- netic impurities, etc. Particularly there are charge puddles caused by chemical adsorp- tion or molecules trapped between the sample and the substrate [70]. Previous studies of impurities have focused on understanding their effects on the ground state electronic properties of graphene [60, 31, 33, 51, 4, 65, 66, 9, 38, 62, 44, 18]. However, an equally important aspect is the study of the consequences of these charged impurities on the dielectric response for future electronic devices. There are some recent studies of plas- monic modes in graphene using the linearized band structure [68, 64, 27] and a full tight-binding band structure calculation [61]. The effects of a fully gapped band [55] and doping [29] have also been explored. But none of these works considered local- ized plasmons. A more recent work showed however that localized states can have a significant effect on tunneling [14]. These excitations have the ability to concentrate an electric field (visible photons) at the nano-scale regime, thereby beating the diffraction limit. The possibility of accomplishing this in a versatile material as graphene gives an opportunity to integrate plasmonics with other emerging nanotechnologies in multifunc- tional devices made of a single material. In order to explore the presence of such promising excitations in graphene, we did a work where plasmonic excitations were computed in the neighborhood of an impurity in a graphene sample [41]. It was used a non-local quantum mechanical model and a full analysis of the polar- ization operator was performed, allowing the extraction of all its poles. It was demon- strated that impurities induce the formation of nanoscale localized plasmonic modes, 14 and also that the chemical potential and impurity strength can be tuned to control target features of the localized modes, making graphene an intrinsic plasmonic material. Although there have been analytical calculations of plasmons in graphene, when plasmons are localized around impurities, the translational invariance of the lattice is lost and makes analytical calculations impractical. In this work it was used an approach different from the one employed for atomic chains described above. Here the denomina- tor of the polarization operator is diagonalized, providing full information about individ- ual plasmonic excitations, including spatial profiles and local spectral density of states. We studied the impurity effect as a function of the sign, strength, and size of the impu- rity potential, and the doping level of graphene. Due to this rich parameter space the computed data is multidimensional and very challenging to analyze. To overcome the problem of analyzing thousands of spatial patterns of nano-plasmonic modes, it was used a combined a quantitative and visual approach, requiring an advanced visualiza- tion tool for their efficient analysis, which can be achieved with parallel rendering. The interactive use of three-dimensional (3D) visualization allowed readily distinction of localized vs. delocalized plasmonic excitations, and also the identification of dipolar, quadrupolar and radial modes. The method is described in detail in Chapter 4 and the analysis of the results are shown in Chapter 5. 15 Chapter 2: Dielectric response of finite systems The usual way to compute the electric susceptibility is by going in momentum space. However this is not possible because the kind of systems being considered do not have translational invariance. Therefore the calculation must be done in the real space. This chapter begins by showing how the main formula of the method used is obtained. For a system with discrete spectrum one can write r = P ' (r)c and y r = P ' (r)c y (2.1) where ' (r) are the eigenstates of energy E = ~! . The electronic density can be also written as r = y r r = X ' (r)' (r)c y c (2.2) Therefore the electric susceptibility is (rt;r 0 t 0 ) = i (tt 0 )h[ r (t); r 0 (t 0 )]i = i P ' (r)' (r)' (r 0 )' (r 0 ) (tt 0 )e i(!! )t e i(! ! )t 0 Dh c y c ;c y c iE (2.3) 16 the commutator and expectation value are easily evaluated c y c c y c = c y c y c c = c y c c y c c y c h c y c ;c y c i = c y c c y c Dh c y c ;c y c iE = (n n ) (2.4) so (rt;r 0 t 0 ) = i P ' (r)' (r)' (r 0 )' (r 0 ) (tt 0 )e i(!! )t e i(! ! )t 0 (n n ) = i P ' (r)' (r)' (r 0 )' (r 0 ) (tt 0 )e i(!! )(tt 0 ) (n n ) (2.5) and using (tt 0 ) = Z 1 1 d! 2i e i!(tt 0 ) ! +i0 + (2.6) one gets (rt;r 0 t 0 ) = i P ' (r)' (r)' (r 0 )' (r 0 ) R 1 1 d! 2i e i ( !!+! )(tt 0 ) !+i0 + (n n ) = i R 1 1 d! 2i e i!(tt 0 ) !!+! +i0 + P ' (r)' (r)' (r 0 )' (r 0 ) (n n ) = R 1 1 d! 2 e i!(tt 0 ) P ' (r)' (r)' (r 0 )' (r 0 ) nn !!+! +i0 + (2.7) which gives (r;r 0 ;!) = X ' (r)' (r)' (r 0 )' (r 0 ) n n ! ! !i0 + (2.8) this formula will be used self-consistently with the electric potential in order to deter- mine the collective excitations. 17 In the next section this approach is illustrated by being applied to a two atom system [34] and the following section shows how it is employed for larger systems in [42]. 2.1 Dielectric response of a diatomic molecule Here it is shown explicitly how this approach works for a two-atom system, in which case the integrals can be performed analytically. For a diatomic system the tight-binding Hamiltonian is given by H = 0 @ t t 1 A (2.9) Diagonalizing this matrix yields the following eigenvalues and corresponding eigen- states: E 0 = t; j 0 i = j' a i +j' b i p 2 (2.10) E 1 = +t; j 1 i = j' a ij' b i p 2 (2.11) Herej' a i andj' b i are taken to be 1s-orbitals wave functions, centered at the atomsa andb respectively, ' a (r) = 1 p a 3 B e jrRaj a B (2.12) ' b (r) = 1 p a 3 B e jrR b j a B (2.13) 18 wherea B is an effective Bohr radius. Within the random phase approximation [19, 11], the susceptibility of the system only has two terms contributing to the sum over the states, (r;r 0 ;!) = f(E 1 )f(E 0 ) E 1 E 0 !i + f(E 0 )f(E 1 ) E 0 E 1 !i 0 (r) 0 (r 0 ) 1 (r 0 ) 1 (r) (2.14) Therefore the induced charge is given by ind (r;!) = 0 (r) 1 (r) 2(E 1 E 0 )(f(E 1 )f(E 0 )) (E 1 E 0 ) 2 (! +i ) 2 Z dr 0 0 (r 0 ) tot (r 0 ;!) 1 (r 0 ) (2.15) Let us now define 2Ef E (E) 2 ! 2 + 2 i2! Z dr 0 0 (r 0 ) tot (r 0 ;!) 1 (r 0 ) (2.16) where E =E 1 E 0 and f E =f(E 1 )f(E 0 ). Using the fact that fors-orbitals 0 (r) 1 (r) = (' a (r) +' b (r)) p 2 (' a (r)' b (r)) p 2 = ' 2 a (r)' 2 b (r) 2 (2.17) we can simply write ind (r;!) = 2 (' 2 a (r)' 2 b (r)) (2.18) which in turn allows the calculation of ind (r;!) via ind (r;!) = Z dr 0 ind (r 0 ;!) jrr 0 j = 2 Z dr 0 ' 2 a (r 0 )' 2 b (r 0 ) jrr 0 j (2.19) 19 Once this potential is calculated we are able to compute h 0 j tot j 1 i = Z dr 0 0 (r 0 ) tot (r 0 ;!) 1 (r 0 ) (2.20) and plug it back into Eq. 2.16 to obtain a linear equation for. Using elliptical coordinates withz a z b =R, this integral can be performed analyt- ically, yielding h 0 j tot j 1 i = ER 2 + 5 16a B 2R + ( 13 16a B + 3R 8a 2 B + 1 2R + R 2 12a 3 B )e 2R a B (2.21) where it is assumed that the external electric field of magnitudeE is applied along the z-direction, i.e. along the line connecting the two atoms. Plugging this result back into Eq. 2.16, one finds ER a 1 = 5 8a B 1 R + 13 8a B + 3R 4a 2 B + 1 R + R 2 6a 3 B e 2R a B E 2 ! 2 + 2 i2! Ef E (2.22) This enables us to determine a closed analytical form for the physical observables, i.e. we can express the induced charge and induced potential in terms ofr a =jrR a j andr a =jbR b j. Namely, ind (r;!) = 2a 3 B e 2ra a B e 2r b a B (2.23) ind (r;!) = 2 1 r a 1 a B + 1 r a e 2ra a B 1 r b + 1 a B + 1 r b e 2r b a B (2.24) 20 where the! dependency is contained in. The corresponding total induced energy is given by E ind (!) = 2 5 16a B 1 2R + 13 16a B + 3R 8a 2 B + 1 2R + R 2 12a 3 B e 2R a B (2.25) Figure 2.1: Dielectric response of a diatomic molecule. Dielectric response of a diatomic molecule. Left: induced charge density with the exter- nal electric field in the vertical direction, andR = 4a B . Right: Total induced energy as a function of! andR, with the tight-binding hopping parameter changing ast/R 3 . In Fig. 2.1, the calculated induced charge density and magnitude of the induced electric field are plotted as a function of frequency of the external electric field and the separation between the two atoms. This simple system only has a dipole resonance at the frequency which separates the bonding and anti-bonding energy eigenstates. As the distance between the atoms is increased, the tight-binding matrix element is expected to fall off rapidly. Here, we parametrize this effect with a generic power-law [24],t/ R 3 . Then the resonance frequency drops off with decreasing inter-atomic separation. 21 Furthermore, the magnitude of the induced electric field increases with inter-atomic distance, and hence with the cross-section of the diatomic “antenna” structure. 2.2 Atomic chains As it was explained in the first chapter, the interaction of electromagnetic radiation with nanoscale conducting clusters is conventionally described by semi-classical Mie theory [39]. This is a local, continuum-field model which uses empirical values of the linear optical response of the corresponding bulk material, and has been applied in nanoparti- cles to describe plasmon resonances [67, 56, 35, 20]. However, such a semi-empirical continuum description breaks down beyond a certain degree of roughness introduced by atomic length scales, and thus cannot be used to describe ultra-small systems. There- fore, we will use a self-consistent quantum-mechanical model which fully accounts for the non-locality of the dielectric response function [22]. Specifically, to identify the plasmonic modes in small clusters we calculate the total induced energy due to an applied external electric field with driving frequency!, and scan for the resonance peaks. The induced energy is determined within the non-local linear response approximation. 2.2.1 Electronic states To keep the computational complexity of this procedure at a minimum, we use a one- band tight-binding model to obtain the electronic energy levelsE i and wave functions i (r) as a linear combination of s-orbitals i (r) = X i;j ij '(rR j ); (2.26) 22 where'(rR j ) is the wave function of a s-orbital around an atom localized at position R j and ij are the coefficients of the eigenvector (with energyE i ) of the Hamiltonian, which has the matrix elements h'(rR i )jHj'(rR j )i = 8 > > > < > > > : i =j t i;j n:n: 0 otherwise (2.27) or in the second quantization language H 0 =t X <ij> c y i c j +c y j c i X i c y i c i (2.28) where<ij > means nearest neighbors,t is the tight-binding hopping parameter which determines the width of the electronic band by 4t and is the on-site potential that corresponds to the electronic energy at the center of the band. For all the results shown in the next chapter, it is set = 0 andt = 1, such that the energy levels are measured relative to the center of the band, and the energy scale is given by the hopping parameter. The Hamiltonian matrix is diagonalized using the Householder method to first obtain a tridiagonal matrix and then a QL algorithm for the final eigenvectors and eigenvalues [54]. 2.2.2 Self-consistent linear response Once the electronic wave functions have been obtained, it is possible to calculate the dielectric susceptibility(r;r 0 ;!) via (r;r 0 ;!) = X i;j f(E i )f(E j ) E i E j !i i (r) i (r 0 ) j (r 0 ) j (r): (2.29) 23 The induced charge density distribution function is then obtained by ind (r;!) = Z (r;r 0 ;!)( ind (r 0 ;!) + ext (r 0 ;!))dr 0 ; (2.30) where in turn the induced potential is given by ind (r;!) = Z ind (r 0 ;!) jrr 0 j dr 0 : (2.31) We avoid the large memory requirement to store(r;r 0 ;!) by calculating the induced charge density distribution iteratively via ind (r;!) = X i;j f(E i )f(E j ) E i E j !i i (r) j (r) Z i (r 0 ) tot (r 0 ;!) j (r 0 )dr 0 ; (2.32) with tot (r 0 ;!) = ind (r 0 ;!) + ext (r 0 ;!). The integrals are evaluated using a 4 th order formula obtained from a combination of Simpson’s Rule and Simpson’s 3/8 Rule. Eqs. 2.31 and 2.32 are solved self-consistently by iterating ind (r;!) and ind (r;!). This procedure typically converges in 3-8 steps when starting with ind (r;!) = 0, depending on the proximity to a resonance and on the value of the damping constant , which throughout this paper is chosen as = 0:08t. A much better performance can be achieved when the initial ind (r;!) is taken as the solution of a previously solved nearby frequency. Upon its convergence, the frequency and spatial dependence of the induced electric field and the induced energy are obtained using E ind (r;!) =r ind (r;!) (2.33) and U ind (!) = Z jE ind (r;!)j 2 dr (2.34) 24 The observed peaks in the induced energy allows the identification of the plasmonic modes, and the charge density distribution at certain driving frequencies of the applied electric field allows the analysis of spatial properties of each mode, such as localization. In the next chapter the results of [42] are presented. 25 Chapter 3: Plasmonic excitations of atomic chains The method explained in the previous chapter is applied to linear chains of atoms with the aim of developing basic physical intuition for systems in the atomic length scale. In order to accomplish that, it was studied the dependence of the modes on the direction of the external electric field, number of electrons in the system, geometry of the structure and the distance between the atoms. For each structure analyzed it is shown a plot of the total energy induced versus the frequency of the external electric field. The peaks in these graphs correspond to the collective resonance. The local induced charge density distribution is used for analyzing the characteristic spatial modulation of given plasmonic resonances. The energy scale is given in terms of the tight-binding hopping parametert, such that the frequency unit is t ~ , with~ = 1. 3.1 Longitudinal modes Let us first focus on the dielectric response function in linear chains of atoms, with the intent to identify the basic features of their collective excitations. Unless otherwise stated, the inter-atom spacing is fixed toa = 3r B , wherer B is the Bohr radius, and the 26 number of atoms in the chain is varied. The frequency dependence of the induced energy in such systems, exposed to a driving electric field along the chain direction, is shown in Fig. 3.1(a). It exhibits a series of resonances, which increase in number for chains with increasing length. As observed in the spatial charge density distribution, e.g. shown 0 1 2 3 4 5 (c) -1.3 -0.43 0.43 1.3 (b) -5.0 -1.7 1.7 5.0 (a) Log 10 (E TOT ) Frequency (t) N at = 2 N at = 3 N at = 4 N at = 5 N at = 6 Figure 3.1: Longitudinal modes in atomic chains. Longitudinal modes in atomic chains. (a) Decimal logarithm of the total induced energy (artificially offset) as a function of the frequency of an external electric field which is applied along the direction of the chain. The resonance peaks correspond to different modes. The arrows indicate the peaks for which the corresponding charge density profile are shown in (b) and (c). (b) Induced charge density distribution for the lowest energy mode at! = 0:73t in the 5-atom chain. (c) Induced charge density distribution for the highest-energy mode at! = 3:56t in the 6-atom chain. In (b) and (c) the arrows indicate the direction of the external applied electric field. 27 for the 5-atom chain in Fig. 3.1(b), the lowest peak corresponds to a dipole resonance. When increasing the system size N, there are more electronic levels available in the spectrum of the system and the spacing between them decreases, i.e., E n+1 E n = 4t sin 2N + 2 sin 2n + 2N + 2 / N1 1 N 2 : (3.1) Since the dipole resonance frequency is associated with the transition between the high- est occupied and the lowest vacant energy level, it also decreases for larger chains, as observed in Fig. 3.1(a). The resonances at higher frequencies correspond to higher har- monic charge density distributions. For example, in Fig. 3.1(c), we show the charge den- sity distribution corresponding to the highest frequency resonance of the 6-atom chain. In contrast to the dipole resonance, these modes show a rapidly oscillating charge den- sity distribution, and thus have the potential to provide spatial localization of collective excitations in more sophisticated structures. While an extension to much larger chains is numerically prohibitive within the current method, the finite-size scaling of the observed dielectric response of these clusters indicates that the frequency of the dominant low- energy plasmon mode scales as E/ 1=N 2 , consistent with the discussion above. 3.2 Transverse modes In order to study the transverse collective modes we apply an external electric field perpendicular to ladder structures made of coupled linear chains of atoms. 1 Fig. 3.2(a) shows that for every chain size there are two resonance peaks for the total induced energy, the higher energy is an end mode, as shown in Figs. 3.2(b) and (c) for the 3 and 5-atom double chains respectively, whereas the lower energy peak corresponds to a 1 Within this approach, at least two coupled chains are necessary to visualize charge redistributions along the transverse direction, since charge fluctuations within the orbitals are not accounted for. 28 0 1 2 3 4 5 6 (c) -0.090 -0.030 0.030 0.090 (b) -0.20 -0.067 0.067 0.20 (d) -1.3 -0.43 0.43 1.3 (a) Log 10 (E TOT ) Frequency (t) 3x2 4x2 5x2 Figure 3.2: Transverse Modes in double chains. Transverse Modes in coupled chain structures. (a) Logarithm of the total induced energy (artificially offset) as a function of the frequency of an external electric field applied transversely to the chain. The low energy mode is central, the analog of a bulk plasmon, and the high energy mode is located at the surface, the analog of a surface plasmon. The arrows indicate the peaks for which the corresponding charge density profiles are shown in the other insets. (b) Induced charge density distribution for the mode at! = 4:91t in the 3-atom double chain. (c) Induced charge density distribution for! = 5:41t in the 5-atom double chain. (d) Induced charge density distribution for ! = 1:93t in the 6-atom double chain. In (b), (c) and (d) the arrow indicates the direction of the external applied electric field. central mode, as displayed in Fig. 3.2(d) for the 6-atom double chain. It is also confirmed that as the length of the chain is increased, the central mode gets stronger relative to the end mode, which is the expected behavior for bulk versus surface excitations. These results are in agreement with the findings of Ref.[69]. 29 3.3 Angle of incidence 0 2 4 6 8 (c) -0.10 -0.033 0.033 0.10 (b) -0.11 -0.037 0.037 0.11 (a) Log 10 (E TOT ) Frequency (t) = 0 0 = 20 0 = 45 0 = 70 0 Figure 3.3: Dependence on the direction of the external electric field. Dependence on the direction of the applied electric field. (a) Logarithm of the total induced energy (artificially offset) as a function of frequency of external electric fields applied to a 4 6-rectangle at different incident angles. = 0°when the field is parallel to the 4-atom edge, and = 90°when it is parallel to the 6-atom edge. The arrows indicate the peaks for which the corresponding charge density profiles are shown in the other insets. (b) Induced charge distribution for = 0°and! = 3:15t. (c) Induced charge density distribution for = 90°and! = 2:15t. In (b) and (c) the arrow indicates the direction of the external applied electric field. Let us next examine what happens when the direction of the external electric field is varied. Fig. 3.3 shows the dielectric response of a 46-atom rectangular structure for different angles incidence directions of the applied field. When the field is parallel to one of the edges ( = 0°or = 90°), the response is essentially that of a single chain 30 with the same length, shown in Fig. 3.1(a). Also the induced spatial charge density modulations are analogous to those of the correspondent linear chain, which can be seen in Figs. 3.3(b) and (c). At intermediate angles the response is a superposition of the two above cases, changing gradually from one extremum to the other as the angle is changed. Notice for instance that as the angle increases, the peak at the same frequency of the 4-atom dipole resonance diminishes, while simultaneously another resonance is formed at the frequency of the dipole mode of a 6-atom chain when the angle is tuned from = 0° to = 90°. For = 0° there is only the peak at the frequency of the 4-atom chain dipole resonance, whereas for = 90° only the dipole peak corresponding to the 6-atom dipole frequency is present. The superposition of the responses from each direction is a consequence of the linear response approximation employed, since the response is a linear combination of those obtained from each direction component of the external field. 3.4 Number of electrons Next, let us analyze the dependence of the resonance modes on the number of electrons in the cluster. Fig. 3.4(a) shows significant changes in the response of a 9-atom chain with the external field applied along its direction. In particular, it is observed that the response is stronger when there are more electrons in the sample, a quite obvious fact since there are more particles contributing to the collective response. Moreover the res- onance frequencies of lower modes increase with the number of electrons, which can be understood as a consequence of the one-dimensional tight-binding density of states being smallest at the center of the band. Hence the energy levels around the Fermi energy are more sparse in the finite system, and therefore the excitations require larger frequencies at half-filling. The same does not hold for higher frequency modes since 31 these correspond to transitions between the lowest and highest levels for any number of electrons in the sample. Therefore these modes have the same frequency, independent of the electronic filling. Higher filling also allows the induced charge density to concen- trate closer to the boundaries of the structure, as a comparison between Figs. 3.4 (b) and (c) demonstrates. Fig. 3.4(b) shows that a 9-atom chain withN el = 1 electron has its induced charge density localized around the center of the chain. In contrast, Fig. 3.4(c) displays the induced charge density localized at the boundaries of the same structure 0 1 2 3 4 -6 -5 -4 -3 -2 -1 0 1 2 3 (c) -20 -6.7 6.7 20 (b) -0.80 -0.27 0.27 0.80 (a) Log 10 (E TOT ) Frequency (t) N el =1 N el =3 N el =5 N el =7 Figure 3.4: Dependence on the number of electrons in the atomic chain. Variation of the number of electrons. (a) Logarithm of the total induced energy as a function of the external electric field frequency. The number of electronsN el in a 9-atom chain is varied. The arrows indicate the modes whose charge density profile are shown in the other insets. (b) Induced charge density distribution forN el = 1 at ! = 0:36t. (c) Induced charge density distribution forN el = 9 at! = 0:53t. In (b) and (c) the arrow indicates the direction of the external applied electric field. 32 withN el = 9. This concentration closer to the surface happens because higher energy states have a stronger charge density modulation than the lower energy ones. Therefore the induced charge density is more localized for higher fillings, because at low fillings the excitations responsible for the induced charge density are between the more homo- geneous lower energy levels. This can be interpreted as a finite-size rendition of the fact that by increasing the electronic filling one obtains the classical response with all the induced charge density on the surface of the object. 3.5 Parallel chains Access to high energy states is very important for achieving spatial localization of the induced charge density, as the next example shows. In order to find a structure with spatially localized plasmons we consider two parallel 8-atom chains connected to each other by an extra atom at the center. When an external electric field is applied trans- versely to the chains, the electrons are stimulated to hop between them, but this is only realizable through the connection, therefore the plasmonic excitation is sharply local- ized around it. Fig. 3.5(a) shows the response of two 8-atom chains, Fig. 3.5(b) and (c) show respectively the induced charge density for the dipole and the highest modes. It is seen that the induced charge density of the lowest frequency mode is spread along the chains, whereas the high frequency plasmon more localized, since it corresponds to excitations to the highest energy state that has a large charge modulation as it was pointed out before. 3.6 Inter-atomic distance Let us finally analyze the dependence of the various dielectric response modes on the inter-atomic distance. Now We consider a fixed number of atoms in the chain but the 33 inter-atom distance is changed. The dipole moment of the chain is proportional to its length, and consequently also proportional to the distance between atoms. Hence one would naively expect that the strength of the dielectric response is strictly proportional to the atomic spacing. However, higher frequency modes require that the electrons are able to hop quickly along the chain in order to produce the fast charge oscillations of the mode. Hence the oscillator strength of the high frequency modes are suppressed for systems where electrons cannot move fast enough. In the tight-binding model, the 0 1 2 3 4 5 -3 -2 -1 0 1 (c) -0.75 -0.25 0.25 0.75 (b) -2.8 -0.93 0.93 2.8 (a) Log 10 Figure 3.5: Two parallel chains connected by an extra atom. Connection between two chains,N el = 1. (a) Logarithm of the total induced energy as a function of the frequency of an external electric field applied to two 8-atom chains with an extra atom connecting them at the center. The arrows indicate the peaks for which the corresponding charge density profiles are shown in the other insets. (b) Induced charge density distribution for! = 0:28t. (c) Induced charge density distribution for ! = 4:36t. In (b) and (c) the arrow indicates the direction of the external applied electric field. 34 hopping rate is determined by the hopping parametert and stems from the overlap of the atomic orbitals on different sites, which decreases with increasing spacing between atoms.[24] Therefore the high frequency modes are suppressed for chains with large inter-atom spacing, because the hopping is so weak that it overcomes the gain coming from a larger dipole moment. On the other hand the oscillator strength of the slow modes increases for larger spacings, because they do not require fast motion of electrons 0 2 4 6 -4 -3 -2 -1 0 1 2 3 Log 10 (E TOT ) Frequency (t 3 ) a = 2.5r B a = 3.0r B a = 3.5r B a = 4.0 Figure 3.6: Dependence on the distance between neighbor atoms. Variation of the distance between neighbor atomsa. Logarithm of the total induced energy as a function of the frequency of an external electric field applied to 7-atom chains with different spacing between atomsa in units of the Bohr radiusr B . The damping constant is kept constant for all the different atomic spacings. In this figure the frequency unit ist 3 the tight-binding hopping parameter fora = 3r B . The red lines connect corresponding peaks for systems with different inter-atom spacings, indicating that the oscillator strength of the dominant low-energy mode decreases with decreasing spacing, whereas the peaks of the higher-energy modes increase. 35 along the chain. In this case, the contribution from a larger dipole moment dominates over the suppression due to the smaller hopping rates. This fact is demonstrated in Fig. 3.6 where the response of 7-atom chains with different atomic spacings are shown. The tight-binding hopping parametert changes with the atomic spacinga, and here we considered a generic[24] power-law dependence t a 3 . Comparing the oscillator strength of the slowest mode for all the different spacings, we see thata = 4r B has the strongest response, whilea = 2:5r B has the weakest. On the other hand, for the fastest mode we see that the chain with spacinga = 2:5r B has the strongest response, while the chain witha = 4r B has such a small response that we can not see a peak because it is washed out by other peaks. 3.7 Summary In conclusion, we have analyzed the evolution of plasmonic resonances in small clus- ters as a function of the system shape, direction of the external applied field, electron filling and atomic separation. Using a fully quantum-mechanical, non-local response theory, we observe that longitudinal and transverse modes are very sensitive to these system parameters. This is reflected in their frequency, oscillator strength and the spa- tial modulation of the induced charge density. Specifically, we identify bulk and surface plasmonic excitations which can be controlled in amplitude and frequency by the cluster size. Furthermore, we observe a non-trivial filling dependence, which critically depends on the electronic level spacing in a given structure. We also find that changes in atomic spacings have a very different impact on low-energy vs. high-energy modes. And we see that changing the position of a single atom in a nanostructure can completely alter its collective dielectric response. This strong sensitivity to small changes is the key to controlling the modes of ultra-small structures, and it can thus become the gateway to a 36 new generation of quantum devices which effectively utilize quantum physics for new functionalities. 37 Chapter 4: Polarization operator in real space The method presented in Chapter 2 addressed the main limitations of the Mie solution: the homogeneous continuous description of the system, and the bulk based dielectric function. Using this technique, the identification of plasmons was accomplished by scanning the frequencies of the modes with strongest induced fields. However, it does not provide full information of all plasmon excitations supported by the system. In this chapter, the polarization operator is diagonalized, providing all its poles [41]. Thus complete information of the plasmon excitations is obtained, including the local spectral densities of states, which is a key quantity for comparison with STM experiments for instance. Therefore this method is also useful for understanding basic properties of collective excitations in graphene from a more fundamental point of view. 4.1 Non-interacting states The electronic structure of graphene is described in terms of a tight-binding model. The non-interacting single-particle hamiltonian includes the external potential. The direct Coulomb interaction is treated as a perturbation and the polarization operator is com- puted using the Random Phase Approximation (RPA). 38 We model the electronic structure of graphene using a one-band tight-binding Hamil- tonian, H 0 =t X <a;b> c y a c b +c y b c a + X a U a c y a c a X a c y a c a , (4.1) where t = 2:7eV is the hopping parameter and is the chemical potential. U 0 is the magnitude of the impurity potential,x 0 corresponds to its location, and denotes its spa- tial spread.U a is then the impurity potential felt at the other sitesx a 6=x 0 , parametrized by U a = U 0 exp jxax 0 j 2 2 2 , the range of the impurity potential will be only few lattice sites.H 0 can be represented as matrix taking the approximate form H 0 = 2 6 6 6 4 . . . U a1 t 0 . . . t U a t . . . 0 t U a+1 . . . 3 7 7 7 5 : (4.2) H 0 is diagonalized numerically, providing the eigenstates j 0 i and eigenvalues E 0 . Each state has an occupation number given by the Fermi function n 0 = [exp( E 0 k B T ) + 1] 1 . 4.2 RPA treatment of the interaction The direct Coulomb interaction is considered as a perturbation, H =H 0 +V =H 0 + X abmn V abmn c y a c y b c m c n ; (4.3) with V abmn = e 2 2 Z dx Z dx 0 ' a (x)' b (x 0 )' m (x 0 )' n (x) jxx 0 j ; (4.4) 39 where ' j (x) is the p z orbital at site j. If the overlap of p z orbitals at different sites is neglected ' a (x)' b (x) = ab ' b (x)' b (x) the interaction has the following non-zero elements V ab;ab =V aa;bb = V 0 jx a x b j : (4.5) In this case, the induced charge in the lattice basis is given by (x) = X ab ' a (x) ab ' b (x) = X b ' b (x) bb ' b (x). The linear response equation for the induced charge in this basis is ab (!) = X mn ab;mn (!) Ext mn (!). (4.6) Within the random phase approximation (RPA), the polarization operator is then obtained via (!) = 0 (!) 1V 0 (!) 1 (4.7) where 0 (!) is the polarization operator of the non-interacting system. In the basis of eigenstatesj 0 i, 0 (!) can be written as 0 ; (!) = P !m G 0 (! +i! m )G 0 (i! m ) = P !m (!+i!mE 0 ) (i!mE 0 b ) = n 0 n 0 E 0 E 0 ! (4.8) Defining the tensorial matrices n and E, which in the basis of eigenstates have elements n ; = n 0 n 0 E ; = E 0 E 0 (4.9) 40 and using the superindicesI = andJ = , which reshape a matrix into a vector in a column-wise fashion so while ;2 1;:::;n one has I 2 1;:::;n 2 , the above tensors can be written as diagonal matrices in the basis of eigenstates n = 2 6 6 6 4 n 0 n 0 1 0 . . . 0 n 0 n 0 0 . . . 0 n 0 n 0 +1 3 7 7 7 5 (4.10) and E = 2 6 6 6 4 E 0 E 0 1 0 . . . 0 E 0 E 0 0 . . . 0 E 0 E 0 +1 3 7 7 7 5 : (4.11) Then 0 (!) is also regarded as a tensorial matrix acting on the tensorial vectors ab , and thus can be represented as the product 0 (!) = n (E!I) 1 , (4.12) and finally, the polarization of the interacting system (!) can then be expressed as (!) = n (E!IV n) 1 . (4.13) 4.3 Plasmonic modes The collective excitations correspond to the poles of the polarization operator for the interacting system, therefore the charge density waves will have a spatial envelope = n such that (E!IV n) = 0. (4.14) 41 When the matrix E nV is diagonalized, the polarization has poles at ! = for each eigenvalue of E nV . Hence the modes correspond to charge density oscillations with amplitude I = X J n IJ J ; (4.15) such that (E!IV n) = 0: (4.16) The matrix E nV is diagonalized giving the eigenvalues! and eigenvectors I satisfying P J (E nV ) IJ J = ! I . The polarization has poles at ! = ! for each eigenvalue! of E nV . One can then find the spatial profile of the mode with energy ! through I = P J n IJ J . By going back to the lattice basis and neglecting the orbital overlap we get the local charge oscillation I = ! ab ' aa : (4.17) The amplitude of the induced charge density oscillation at site a is the vector a aa . Of course the induced charge will oscillate with frequency! , that is, a (t) = a exp(i! t). The retarded Green’s function R (!) = (! +i0 + ) gives the spectral density functionA (!) = 1 Im R (!), which is expressed as A (!) =n (E!IV n) . (4.18) Due to the delta function A ; (!) is non-zero only for plasmonic frequencies. The only spatial profiles it can display are those of the plasmons n ab;mn mn with mn satisfying (E!IV n) = 0. 42 In order to plot the plasmon density of states the representation (!! ) ! [(!!) 2 + 2 ] is used, where is chosen to be 0:05eV . Then the plasmon density of statestrA(!) can be written as trA (!) = X A (!) = X A (!! ) 2 + 2 (4.19) A will be referred to as the strength of the mode A = ( ) T n = X b ( ) T bb (n ) bb , (4.20) where n = is the charge profile of the mode. Using Eq. 4.20, the plasmon is considered to be localized when the major contribution to the sum comes from sites around the impurity. The discussion of our results in the next chapter is focused on such localized modes. The plasmon density of statestrA (!) can be accessed experimentally through scanning tunneling experiments that reveal either direct or inelastic tunneling signatures associated with plasmons or related lifetime effects [8]. For the numerical simulation a finite-size realization of the graphene lattice with 96 sites is considered. An impurity affects approximately one hexagon of the honeycomb lattice, as shown in Fig. 5.3(a). Periodic boundary conditions are used to eliminate the boundary modes. LAPACK routines are used for the numerical diagonalization ofH 0 and (!) [2]. Due to the overlap of other bands, the tight binding model is not accurate for energies larger than 4eV , therefore our results for plasmons are not accurate for energies larger than 8eV . For a lattice having N sites, the number of plasmonic modes scales as N 2 =4 = N=2N=2 for N=2 electron-hole pairs at half-filling. Therefore the number of data points for plasmon spatial profiles scales asN 3 =4, which becomes rapidly a big number 43 even for small lattice sizes. For this study it is analyzed the dependence of the modes on strength (magnitude ofU 0 ) and size (varying) of the impurity potential, and the doping level (changing) of the lattice, besides one of their most basic properties which is the frequency. Therefore the data that comes out of the calculation outlined above is not only large but also multi-dimensional. This poses some challenges on the visualization of the results and require more sophisticated methods of presenting the data in order to make it easier to extract the physically relevant features [17]. In this case it was used a parallel rendering tool [50], that allows 3D visualization of several data sets at once, with time efficient image processing. This permits the identification of features that could have been missed otherwise. The next chapter shows the results of the method presented here. 44 Chapter 5: Localized plasmons in graphene In this chapter the method previously described is applied to graphene in the presence of an impurity. The results show localized plasmonic modes around the impurity. An analysis of these excitations is presented based on the spectral function and spatial pro- file of the the induced charge modulations. We analyze the localization dependence on the chemical potential, size and strength of the impurity potential. It is found that the chemical potential determines the existence of localized nanoplasmons and the impurity potential strength can control the frequency of such modes. Besides it is also found that a combination of these two parameters can further enhance the localization of plasmonic excitations. These results can be tested by STM experiments. 5.1 Traits of plasmonic localization We begin by presenting in Fig. 5.1 the histograms of induced charge at each site of the lattice for every plasmon mode. This is done for pristine graphene and in the presence of the impurity. In both cases, the number of zero-induced charges is significantly larger than the number of nonzero induced charges, i.e, lattice sites that are unaffected. How- ever the number of zero-induced charges is slightly bigger in impure graphene than in 45 pristine graphene. The magnitude of induced charges decrease exponentially in impure graphene, while it has a Gaussian distribution in pristine graphene. The range of the magnitude of induced charges is slightly bigger for impure graphene. For localized plasmons, the induced charge in sites far from the impurity is suppressed while for sites nearby it have quite large induced charge. The effects off that in the histogram of induced charges is to increase the amount of very small and very large values while reducing amount of intermediate values. This is exactly what the figure shows. So it is possible to identify the localization of plasmons by the shape of the histogram of induced charges. Another trait of localization can be found by monitoring the induced charge at a site near the impurity as a function of energy. The calculation is performed with or without the presence of impurity. For the case with the impurity, the impurity potential strength isU 0 = 2:0t. In both cases the site chosen is the same which is the closest to 0 2 4 6 8 10 12 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0 2 4 6 8 10 12 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 (a) (b) log(N) log(N) Figure 5.1: Histogram of induced charge in every site for every mode. Histogram of induced charge in every site of the sample for every mode (a) pristine graphene and (b) graphene with a single impurity with a positive potentialU 0 = 1:0t. The distribution function of charges is symmetric with respect to the origin. The range of the magnitude of induced charges is increased in impure graphene. 46 -0.2 -0.1 0 0.1 0.2 0 1 2 3 4 5 6 -0.2 -0.1 0 0.1 0.2 0 1 2 3 4 5 6 (a) (b) !=t !=t Figure 5.2: Induced charge on a site near the impurity for each mode. Induced charges in (a) pristine and (b) impure graphene as a function of energy at a fixed nearest-neighbor lattice site from the impurity, marked by the arrow in Fig. 5.9(a). The result is for a positive impurity potentialU 0 = 2t. We see a larger range of induced charges in impure graphene. the impurity, however for pristine graphene every site is equivalent, so any site gives the result. The result is shown in Fig. 5.2, it is seen that the range of induced charges are bigger for the impure graphene. For localized modes the induced charge at that nearby site can be twice larger than what is seen in the modes of the pristine system. In addition, comparing Fig. 5.1(b) and Fig. 5.2(b) one can see that the range of induced charges in impure graphene increases with increasing impurity potential fromU 0 = 1:0t to 2:0t. 5.2 Local spectral function In Fig. 5.3(b) the single particle densities of states of pure and impurity doped graphene are shown. They feature two characteristic singularities around2:7eV and a V-shaped dip at the Fermi energy. Please note that there is a tail of states in the doped sys- tem beyond the regular band width and also some additional states around! = 0. In 47 Fig.5.3(c) the spectral density of all plasmon and the spectral density of localized plas- mons are shown for = 0 and U 0 = 3:7eV . The global density of plasmons can be understood in terms of the single particle density of states shown in Fig.5.3(b). There are very few single particle states available around ! = 0. Then there are two peaks Figure 5.3: Single and two-particle spectral functions. (a) Single impurity on the graphene lattice. Blue (dark gray) corresponds to 0eV and red (light gray) to 3:7eV . The black dot indicates the center of the impurityx 0 . (b) Single particle density of statesn(!) = 1 ImtrG 0 (!) = P (!E 0 ) for pure(U 0 = 0) and doped(U 0 = 3:7eV ) graphene, both with = 0. (c) Spectral density of plasmons in graphene with = 0 andU 0 = 3:7eV . “Global” corresponds to the density of statestrA(!) = P all A (!), and “Local” is the spectral density of plasmons localized around the impuritytrA(!) = P local A (!), here runs through localized modes only. The arrows indicate the modes shown in Fig.5.4. 48 around ! =2:7eV . And, at the extrema of the band the single particle density of states becomes small again. This implies a quadratically increasing spectral density at small!, a sharp increase around! = 2t, and a small impurity-dominated contribution around! = 16eV (see Fig. 5.3(c)). Localized modes occur throughout the available fre- quency spectrum. Non-local plasmonic modes are most abundant for 510eV , because of the larger phase space. The highest energy modes are all localized, i.e. the spec- tral density of local plasmons equals the total spectral density above 16eV . This is a consequence of the fact that the high energy plasmons are excitations from the lowest energy states, which tend to be uniformly spread throughout the lattice, to the tail of highest (localized) single-particle states, thus generating a very localized charge density profile. The high energy values for these localized plasmonic modes are set by the very localized nature of the impurity problem. Had the impurity potential range been much larger, on the scale of 10nm, one would expect those localized plasmonic modes with much smaller energy, on the scale of few eV . 5.2.1 Spatial profile of plasmonic modes In Figs. 5.4(a)-(d) the spatial profiles of some selected localized modes are shown. These are representatives for the diversity of localized modes. It is seen that some modes have strong dipole characteristics, whereas others have a strong quadrupole component. This point highlights the importance of resolving the spatial distribution of the induced charge for all possible modes. Previous methods would only be capable of detecting few modes with strong dipole moments [22, 12]. In order to easily extract information from the spatial profile of the modes it is impor- tant to have the capacity of rendering all of them at once. However, as mentioned in the previous chapter, there are about 2300 plasmon modes for a lattice withN = 96 sites. The processing of all these images was achieved by parallel rendering those modes 49 Figure 5.4: Spatial profiles of localized plasmons with different spectral strength. (a)-(d) Charge density profiles of some localized plasmons. Red (light gray) corresponds to negative charge and blue (dark gray) corresponds to positive charges. The plots show the induced charge (n ) bb for the mode on siteb. The frequencies ! and strengthsA = ( ) T n of each mode are also indicated. Figure 5.5: Charge density profiles of some plasmons in pristine graphene. The spatial distribution of the induced charges on every lattice site at given energy for pristine graphene. Only a few selected modes are shown. The energies w =!=t of the nanoplasmons label the panels. 50 with Paraview [50]. It is worth mentioning that using a single-processor computer it is impractical to extract the necessary information in real time (it takes several hours to resolve the spatial distribution for all energies). We extracted the energy-resolved spatial distribution of induced charges for pristine and impure graphene in order to analyze the effect of the impurity. But since there are many unique plasmon modes (characterized here by their unique induced charge spatial profiles), it is impossible to report all of them. It is shown only a few more selected plasmon modes in Fig. 5.5. Figure 5.6: Charge density profiles of some localized plasmons around the impurity. The spatial distribution of the induced charges on individual lattice sites at given energy near the impurity on graphene lattice. Only a few selected nano-plasmons are shown. The energies w =!=t of the plasmons label the panels. The localization of plasmons occurs at the nano-meter scale, hence the termnano-plasmons. Fig. 5.6 shows the energy-resolved spatial distribution of induced charges in impure graphene for a few selected localized nanoplasmons. There are several non-localized plasmons similar to those seen in pristine graphene as well. However pristine graphene does not present localized induced charge profiles. Besides modes having simple pat- terns of dipolar, quadrupolar, radial or triangular symmetry, some have highly non-trivial patterns like clumps of similar charges on nearest lattice sites. It should be emphasized 51 that the localization of these plasmonic modes occurs at the nano-meter scale focusing the electric field to a very small area. 5.2.2 Chemical potential dependence Figure 5.7: Spectral function dependence on the chemical potential. Dependence of the spectral densitytrA(!) of localized plasmon modes on the chemical potential and the frequency!. The impurity potential is kept fixed at U 0 = 3:7eV . The spectral density is shown via the color scale. The dependence of the intensity of the localized plasmons (for a fixed impurity potential) on the chemical potential is shown in Fig. 5.7. For larger chemical poten- tials, more modes are present at lower energies. This is a consequence of the occupa- tion of single particle states closer to those localized modes at higher energies. When the chemical potential has opposite sign with respect to the impurity potential, there is some spectral density at low energies. These modes stem from the extra single parti- cle states around ! = 0 in doped graphene, seen in Fig. 5.3(b). This shows that the 52 chemical potential, which can be experimentally controlled through a gate voltage, is an important tuning parameter for achieving certain target modes. A limiting aspect of this property though is that the frequency of the modes does not change very signifi- cantly. This happens because the chemical potential does not change the single particle states but only their occupation. In this sense, the chemical potential does not allow one to control the frequency of localized plasmons but only their existence. As it will be shown below the impurity potential U 0 changes the frequency of the localized modes more effectively. We emphasize that Fig. 5.7 shows localized plasmonic excitations. 1 5.2.3 Impurity strength dependence In order to change the frequency of modes, one can vary the impurity potential, because it actually affects the single particle states. Fig. 5.8 shows that for an appropriate change ofU 0 the frequency of some localized plasmons can be tuned by about 1eV . Note also that the impurity potential should be at least 2eV in order to obtain a significant intensity of localized modes, and some modes are strong only within a specific range of the impu- rity potential (see e.g. the features at! 6:0eV or! 7:5eV ). The impurity potential considerably changes the spectral intensity of localized plasmons and can therefore be used as a tuning parameter to achieve targeted spectral properties, especially when com- bined with variations of the chemical potential. This result is in agreement with the fact that the impurity changes the single particle density of states. [4, 65] Another striking feature displayed in this figure is the symmetry of the intensity of the modes with respect 1 There is no contradiction with the well known fact that the usual bulk plasma frequency for metals depends on the number of electrons in the system. In bulk plasmons almost every electron participate in the excitation while for a localized plasmon there is a definite amount of electrons involved. Adding an extra electron in the system directly affects a bulk plasmon mode while localized ones are only disturbed through the perturbative interaction. 53 to the sign of the impurity potential. This is a consequence of the single particle spec- trum being symmetric relative to = 0. Since the plasmon equally involves electrons and holes, its properties do not depend on the sign of the impurity potential. 5.3 3D visualization In Fig. 5.9(a) it is shown the geometry of the graphene lattice used for the calculations. The turquoise colored sphere, which sits at the center of a hexagonal cell, represents the position of the impurity. All results presented here are for a single impurity on a 2D lattice with periodic boundary conditions. Its properties are modeled by their effects on the on-site potentials at neighboring carbon sites. Unless otherwise specified, the spatial Figure 5.8: Spectral function dependence on the impurity potential strength. Dependence of the spectral densitytrA(!) for localized plasmons on the impurity potential magnitudeU 0 and on the frequency!. The chemical potential is kept fixed = 0. The spectral density is shown via the color scale. 54 extent of the impurity potential is restricted to the nearest neighbor carbon sites. The energy is measured in units of the hopping energyt = 2:7 eV . To study the plasmonic excitations in pristine graphene, we set the impurity poten- tial U 0 = 0 in our calculations. In Fig. 5.9(b) it is shown the distribution of induced charges for a small section of the graphene lattice as a function of energy. The Z-axis (here and in what follows) represents the energy ! of the plasmonic excitations. It is displayed the charge distribution only for a small section of the lattice, because this is the region of interest for a spatially small-sized impurity potential. The color and size of the glyphs represents the polarity and the magnitude of the induced charges, respec- tively. Fig. 5.9(b) will be used as a reference for comparisons with the system with an impurity. Figure 5.9: Spatial profile of modes for pristine graphene. (a)Left: 2D graphene honeycomb lattice with a single impurity (turquoise sphere) at the center of one of the hexagonal cells. (b)Right: We show the distribution of induced charges on some of the lattice sites of pristine graphene as a function of plasmon energy (third dimension or Z-axis, i.e., direction of skewers pointing out of the plane above each lattice site). The size and color of glyphs represent the magnitude and polarity of the induced charge, respectively. We show the charge distribution only in the region of interest near the impurity site. Here we set the impurity potentialU 0 = 0. 55 5.3.1 Zero doping with impurity Figure 5.10: Spatial profile of modes around an impurity for no doping. Distribution of the induced charges, on several lattice sites near the impurity, as a function of plasmon energy!. Results are shown for (a)Top left:U 0 =t = 0:47, (b)Top right: 0:79, (c)Bottom left: 1:42 and (d)Bottom right: 2:05, wheret = 2:7 eV is the hopping parameter, which is used as the unit of energy. The strengthU 0 of the impurity drastically modifies the local response of the lattice. The impurity strongly localizes the induced charges near the impurity site. It is studied the effect of a single impurity on the spatial distribution of the induced charges in the close vicinity of the impurity site for all plasmonic energies. First we discuss the effect of an impurity with a repelling (positive) potential. In Figure 5.10 56 we show results for the impurity potentialU 0 =t=0.47, 0.79, 1.42 and 2.05 respectively. Comparing Fig. 5.9(b) with Fig. 5.10(a), one can see that the region closest to the impu- rity site shows the largest change in induced charge distribution. As the impurity poten- tial is increased the change becomes more pronounced, see Figs. 5.10(b)-(d), as seen in the increase in the size of the 3D sphere glyphs used to represent the magnitude of the induced charges. More precisely, induced charges start increasingly to localize on neighboring sites of the impurity. These are the sites which are maximally affected by the impurity potential. The effect of an attractive impurity potential on the induced charge distribution is also investigated forU 0 =t =0:47,0:79,1:42 and2:05. Similar to the case of a positive impurity potential, the most pronounced change occurs on lattice sites in close proximity to the impurity. As the impurity potential is increased more and more induced charge is localized at these sites. For a given magnitude of the impurity potential, there is no difference in the induced charge distribution between the positive and negative impurity. To be more accurate about this visual inference, we compared the histogram(as illustrated in Section 5.1) of induced charges of all modes for positive and negative impurity potentials. Indeed, it is confirmed that the corresponding induced charge distributions are identical for attractive and repelling impurity potential at half filling. The exact same response of the graphene lattice for both positive and negative sin- gle impurity at zero doping is related to the fundamental particle-hole symmetry of graphene. The tight-binding calculation shows that the valence and conduction band meet at two nonequivalent points in the Brillouin zone, which are called Dirac points. At zero doping the valence band is completely filled and the conduction band is empty (half filling). The Fermi level lies at the Dirac point so valence and conductance bands are symmetric with respect to the Fermi level (particle-hole symmetric). The same response 57 of the lattice for positive and negative impurity potential originates from the particle- hole symmetric electronic band structure. We use this observation as a validity check for the numerical method implemented to calculate the plasmonic response in graphene. It will be seen later that once this symmetry is broken, by doping the system away from half filling (6= 0) for example, the response of the system will be different for positive and negative single impurity. 5.3.2 Finite doping with impurity The effect of doping is modeled by taking a nonzero value for the chemical potential . For electron doping the chemical potential moves above the Dirac point and lies in the conduction band, i.e., the chemical potential is positive. In Fig. 5.11(a)-(d) we show the effects of the impurity on the induced charge distributions for doping levels =t = 0:0, 0:75, 1:5 and 2:5, respectively. The strength of the impurity potential is U 0 = 3t. Comparing Fig. 5.11(a) with Figs. 5.11(b)-(d), one can see that as the level of electron doping is increased the localization of the induced charges at sites close to the impurity are enhanced. The magnitude of the induced charges on the remaining lattice sites is consequently suppressed. Moreover, the number of localized plasmon (nano- plasmons) increases as seen in the increased number of bigger 3D sphere glyphs along the energy axis. This effect is most clearly seen in Fig. 5.11. The origin for the increase in the number of nano-plasmons (in the presence of a positively charged impurity), due to electron doping, comes from a change in the elec- tronic and impurity states of graphene [41]. The impurity states lie at the band edge and electron doping brings the top of the filled graphene electronic states closer to the impurity states. This leads to an increase in the number of localized excitations near the impurity site. 58 When the system is hole doped, the chemical potential moves below the Dirac point and lies in the valence band. The effect is modeled by taking a negative value for the chemical potential . Fig. 5.12(a)-(d) shows the effects of the impurity on the charge distributions for the doping levels of=t = 0:0,0:75,1:5 and2:5 respectively. The Figure 5.11: Spatial profile of modes around an impurity for electron doping. Distribution of induced charges in graphene with positive impurity potentialU 0 = 3:0t and electron doping (a)Top left:=t = 0:0, (b)Top right: 0:75, (c)Bottom left: 1:5 and (d)Bottom right: 2:5. As the electron doping is increased the localization of the charges at sites close to the impurity site are enhanced, while it is decreased for lattice sites farther away. The number of localized plasmons (nano-plasmons) is increased as seen in the increasing number of bigger 3D sphere glyphs (charges) along the energy axis. 59 strength of the impurity potential is fixed at U 0 = 3:0t. Comparing Fig. 5.12(a) with the Figs. 5.12(b)-(d), one can see that as the hole doping is increased the localization of charges in close vicinity of the impurity is suppressed. The suppression of localization of nano-plasmons is caused by the increased separation between the top of the filled states of graphene and the impurity states. The different nano-plasmonic responses of Figure 5.12: Spatial profile of modes around an impurity for hole doping. Distribution of induced charges in graphene with positive impurity potentialU 0 = 3:0t and hole doping levels of (a)Top left:=t = 0:0, (b)Top right:0:75, (c)Bottom left: 1:5 and (d)Bottom right:2:5. As the hole doping is increased the localization of charges at sites close to the impurity is reduced. 60 the graphene lattice due to electron and hole doping originates from the breaking of particle hole-symmetry with doping away from half filling. impurity potential charge doping plasmonic localization positive electron enhanced negative hole enhanced positive hole suppressed negative electron suppressed Table 5.1: Dependence on the impurity potential and doping. Dependence of nano-plasmons in graphene on the sign of the impurity potential and doping (chemical potential). These results are listed in Table 5.1 to summarize the properties of nano-plasmonic localization with a fixed negative impurity potential but changing the chemical potential. It was found that in the hole (electron) doped case the negative impurity potential leads to an increase (decrease) of the localization of induced charges near the impurity. Thus a positive impurity potential for electron doping behaves similarly as a negative impurity potential for hole doping. Both of them enhance the localization around the impurity. In contrast, a positive impurity potential for hole doping and a negative impurity potential for electron doping suppresses localization of induced charges around the impurity. It is worth to notice that for a given magnitude of the chemical potential and impurity potential, flipping the sign of both of them gives identical nano-plasmonic response in energy and real space. 5.3.3 The effect of impurity size A spatially extended impurity is modeled by taking into account the effects of the impurity on lattice sites beyond nearest neighbor carbon sites. This is mathematically achieved by relaxing the decay of the impurity potential over distance by increasing the magnitude of=a 0 from 1:0 to 1:4, wherea 0 is the lattice constant. Here the impurity potential magnitude is fixed toU 0 = 2:0t. 61 In Figures 5.13(a)-(d), we show the effects of increased impurity size on the induced charge distributions. Comparing Fig. 5.13(a) with Figs. 5.13(b)-(d), one can see that as the size of the impurity potential is increased the localization of the induced charges in close vicinity of the impurity is increased (notice the increase in the range of the induced Figure 5.13: Spatial profile of modes for different impurity sizes. Distribution of induced charges, on several lattice sites near the impurity, as a function of energy!. Results are shown for increasing the spatial extent of the impurity potential by changing its Gaussian width (a)Top left:=a 0 = 1:0, (b)Top right: 1:2, (c)Bottom left: 1:3 and (d)Bottom right: 1:4, wherea 0 is the lattice constant. The impurity strength is fixedU 0 = 2:0t. Increasing the impurity size localizes more charges close to the impurity. It also enhances the weight of the induced charge on the next-nearest-neighbor lattice sites. 62 charge in the color legend of Fig. 5.13 compared to that of Fig. 5.10). Moreover, it also enhances the weight of the induced charges on the next-nearest-neighbor carbon sites (compared to the magnitude of the charges at the lattice site where the arrow points to). Here too, we find that more and more induced charges are localized closer to the impurity with an increase of the extent of the impurity. 5.4 Summary It was introduced an RPA approach which resolves the real space structure of plasmonic modes in graphene. This method was used to show that impurities induce the formation of nanoscale localized plasmonic excitations in graphene sheets. It was discussed the dependence of excitations on the model parameters such as the sign, magnitude, size of the impurity potential and number of electrons in graphene. The spatial profile of the modes was found to vary strongly with the particular resonance. It was shown that the impurity potential and doping can be used to tune the properties of nanoplasmonic excitations, demonstrating that graphene is an inherently plasmonic material. It was found that the chemical potential can be used to turn them on and off, but it does not affect their frequency. However, their frequency and amplitude can be tuned by varying the strength of the impurity potential. This theoretical study was a first step in exploring surface enhancement phenomena in graphene which may be useful for nanoscale technologies such as molecular sensing. In principle the results discussed can be tested experimentally by high-frequency optical probes or STM. However, in order to achieve such tests, calculations of extended or nanosized impurity clusters are very desirable as they would give plasmonic excitations spread over a larger size with smaller frequencies. 63 Chapter 6: Overlook The prevalent classical model of plasmonic calculations for nano-scale metallic clusters is based on the Mie solution. Which consists of solving Maxwell’s equations with the material being represented by a dielectric function on its spatial location. The dielectric function usually employed is computed for bulk systems, extracted from approximations such as the Drude model, which explores the translational invariance clearly absent in small structures. However, such a semi-empirical continuum description necessarily breaks down beyond a certain level of coarseness introduced by atomic length scales. Even the bulk based model used for the dielectric function fails by itself. This limitation of the Mie solution has been established in a previous work, where it was employed a quantum mechanical effective mass model with self-consistent calculation of the dielec- tric response. The inapplicability of the standard method of dielectric calculations for small sys- tems means that new intuition needs to be developed for such scales, the usual properties of macroscopic plasmons might not hold and it is necessary to determine the new ones. In order to begin this process we explored the collective electromagnetic response in atomic chains of various sizes and geometries, aiming to understand and hence control their dielectric response [42]. The method used for this calculation was similar to the one used for the demonstration of the breakdown of the Mie solution, but it had a tight binding model for the calculation of the electronic states. 64 Another work done was the calculation of plasmons in graphene in the presence of an impurity. The goal was to identify and control localized plasmons around the impurity [41]. The method employed for this calculation consisted of diagonalizing the denominator of the polarization operator in order to obtain all the resonances. This approach had not been seen before. For the atomic chains, we calculated the plasmonic resonances as a function of the system shape, direction of the external applied field, electron filling and atomic separa- tion. Their frequency, oscillator strength and spatial modulation of the induced charge density were analyzed. It was shown that longitudinal and transverse modes can be con- trolled in amplitude and frequency by the cluster size. It was also observed an abrupt dependence of the modes on the electronic filling. We also find that changes in atomic spacings have a very different impact on low-energy vs. high-energy modes. And it was seen that changing the position of a single atom in a nanostructure can completely alter its collective dielectric response. This strong sensitivity to small changes is the key to controlling the dielectric properties of atomic scale structures, and it can thus become the gateway to a new generation of quantum devices which effectively utilize quantum physics for new functionalities. For graphene it was shown that impurities induce the formation of nanoscale local- ized plasmonic excitations in graphene sheets. It was studied the dependence of these excitations on the magnitude and size of the impurity potential and electronic filling. It was shown that the impurity potential and doping can be used to tune the properties of nano-plasmonic excitations, demonstrating that graphene is an inherently plasmonic material. It was found that the chemical potential can be used to turn them on and off, but it does not affect their frequency. While their frequency and amplitude can be tuned by varying the strength of the impurity potential. In principle the results discussed can be tested experimentally by high-frequency optical probes or STM. However, in order 65 to achieve such tests, calculations of extended or nano-sized impurity clusters are very desirable as they would give plasmonic excitations spread over a larger size with smaller frequencies. These results showed that collective excitations in finite systems have properties different from their bulk correspondents. Since there is not a macroscopic number of electrons in the system, the variation of one single electron causes observable differ- ences. Those localized resonant modes are very sensitive to even small variations in the system, for example the position of a single atom. This makes it difficult to establish general rules about the properties of collective excitations in atomic structures. On the other hand it also provides a vast range of possibilities that can be explored for achieving new functionalities. By concentrating on low energy modes we can now simulate systems containing thousands of atoms. It is possible to study the effects of multiple impurities and how to arrange them in a way to achieve some target response. Besides, larger systems and lower frequencies is exactly what one needs for experimentally testing our results. In order to explore the full potential of graphene related systems for plasmonics, a more thorough analysis is necessary. Namely it should be considered different kinds of impurities such as lattice defects and adsorbed atoms, which can be magnetic or not and attached to a link or a site of the honeycomb lattice with different consequences [16]. As well as different structures such as nanotubes, nanoribbons whose edges behave as plas- monic waveguides [36], spirals that roll due to the Casimir force - an effect extremely sensitive to dielectric properties - and bilayers which shows exotic plasmons [28]. Our last results showed that localized plasmons have properties different from macroscopic plasmons. Therefore it will be interesting to do a scaling analysis in order to determine the emergence of macroscopic plasmons, in the same way that it is possible to identify precursors of phase transitions for finite systems. 66 With some minor adjustments it is also possible to employ the current method for studying systems presenting exotic excitations. One extension of our work in this direc- tion is to study systems with spin-orbit coupling. In this case collective excitations could have plasmonic and spin components. Therefore it could be possible to control plas- monic modes properties through an applied eternal magnetic field. 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Abstract (if available)
Abstract
The prevalent classical model of plasmonic calculations for nano-scale metallic clusters is based on the Mie solution. Which consists of solving Maxwell’s equations with the material being represented by a dielectric function on its spatial location. However, such a semi-empirical continuum description necessarily breaks down beyond a certain level of coarseness introduced by atomic length scales. Even the bulk based model used for the dielectric function fails by itself. This limitation of the Mie solution has been established by a quantum mechanical calculation with self-consistent treatment of the dielectric response. ❧ In order to understand better the plasmonic excitations at nearly atomic scale, we explored the collective electromagnetic response of atomic chains of various sizes and geometries, and we also computed plasmons in graphene in the presence of an impurity. ❧ For the atomic chains, we calculated the plasmonic resonances as a function of the system shape, direction of the external applied field, electron filling and atomic separation. ❧ Their frequency, oscillator strength and spatial modulation of the induced charge density were analyzed. It was shown that longitudinal and transverse modes can be controlled in amplitude and frequency by the cluster size. It was also observed an abrupt dependence of the modes on the electronic filling. We also find that changes in atomic spacings have a very different impact on low-energy vs. high-energy modes. And it was seen that changing the position of a single atom in a nanostructure can completely alter its collective dielectric response. This strong sensitivity to small changes is the key to controlling the dielectric properties of atomic scale structures, and it can thus become the gateway to a new generation of quantum devices which effectively utilize quantum physics for new functionalities. ❧ For graphene it was shown that impurities induce the formation of nanoscale localized plasmonic excitations in graphene sheets. It was studied the dependence of these excitations on the magnitude and size of the impurity potential and electronic filling. It was shown that the impurity potential and doping can be used to tune the properties of nano-plasmonic excitations, demonstrating that graphene is an inherently plasmonic material. It was found that the chemical potential can be used to turn them on and off, but it does not affect their frequency. While their frequency and amplitude can be tuned by varying the strength of the impurity potential. The method employed for this calculation had not been seen before. In principle the results discussed can be tested experimentally by high-frequency optical probes or STM. ❧ These results showed that collective excitations in finite systems have properties different from their bulk correspondents. Since there is not a macroscopic number of electrons in the system, the variation of one single electron causes observable differences. The localized resonant modes are very sensitive to even small variations in the system, for example the position of a single atom. This makes it difficult to establish general rules about the properties of collective excitations in atomic structures. On the other hand it also provides a vast range of possibilities that can be explored for achieving new functionalities.
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Muniz, Rodrigo Angelo
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Core Title
Plasmonic excitations in nanostructures
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College of Letters, Arts and Sciences
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Doctor of Philosophy
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Physics
Degree Conferral Date
2011-12
Publication Date
09/19/2011
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08/04/2011
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graphene,linear response,nanostructures,OAI-PMH Harvest,plasmonics
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linear response
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