Controlling multipolar surface plasmon excitation through the azimuthal phase structure of electron vortex beams

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(1)UNIVERSIDADE ESTADUAL DE CAMPINAS SISTEMA DE BIBLIOTECAS DA UNICAMP REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP. Versão do arquivo anexado / Version of attached file: Versão do Editor / Published Version Mais informações no site da editora / Further information on publisher's website: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.205418. DOI: 10.1103/PhysRevB.93.205418. Direitos autorais / Publisher's copyright statement: ©2016 by American Physical Society. All rights reserved.. DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo CEP 13083-970 – Campinas SP Fone: (19) 3521-6493 http://www.repositorio.unicamp.br.

(2) PHYSICAL REVIEW B 93, 205418 (2016). Controlling multipolar surface plasmon excitation through the azimuthal phase structure of electron vortex beams Daniel Ugarte1,2,* and Caterina Ducati1 1. Department of Materials Science and Metallurgy, University of Cambridge, Cambridge CB3 0FS, United Kingdom 2 Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas-UNICAMP, rua Sergio Buarque de Holanda 777, CEP 13083-859, Campinas - SP, Brazil (Received 9 September 2015; revised manuscript received 29 February 2016; published 11 May 2016) We have theoretically studied how the azimuthal phase structure of an electron vortex beam excites surface plasmons on metal particles of different geometries as observed in electron energy loss spectroscopy (EELS). We have developed a semiclassical approximation combining a ring-shaped beam and the dielectric formalism. Our results indicate that for the case of total orbital angular momentum transfer, we can manipulate surface plasmon multipole excitation and even attain an enhancement factor of several orders of magnitude. Since electron vortex beams interact with particles mostly through effects due to azimuthal symmetry, i.e., in the plane perpendicular to the electron beam, anisotropy information (longitudinal and transversal) of the sample may be derived in EELS studies by comparing nonvortex and vortex beam measurements. DOI: 10.1103/PhysRevB.93.205418. The exponential growth of nanoscience has stimulated an astonishing control of the chemical and physical properties of nanoparticles targeting technological issues in different fields, such as material science, catalysis, electronics, biology, medicine, etc. Transmission electron microscopy (TEM) methods have in parallel increased their already remarkable spatial resolution and analytical capabilities; in particular, the increase in energy resolution has allowed electron energy loss spectroscopy (EELS) to begin to probe optical transitions. This has rendered the nanometric spatial resolution study of electronic excitation of nanoparticles (NPs) a very active field of research, in particular addressing collective oscillation of valence electrons, called surface plasmons (SPs), which are strongly dependent on particle shape and electronic properties [1,2]. However, the comparison between electron and light-based spectroscopies is not easy, because electrons can excite modes without a clear dipole moment (dark modes) [1–3]. The recent generation of electron vortex beams (EVBs) [4–6], carrying orbital angular momentum (OAM), has stimulated theoretical and experimental studies targeting magnetic effects [7–11]. The interaction of vortex beams with helical systems (molecules, crystals, etc.) has also been explored [12,13]. In contrast, the azimuthal phase structure [14] has not yet been tested as a tool to manipulate the interaction with the in-plane rotational symmetry of the target. Light vortex beams have already been demonstrated to generate peculiar diffraction patterns when traversing apertures of different geometries or when interacting with resonant antennas [15,16]. These reports are very instructive because they associate the spatial distribution of the azimuthal phase in a vortex beam with the in-plane rotational symmetry of the target. We must note that the exploration of optical vortex beams is difficult due to their vanishing intensity at the beam center. For EVBs, this may be exploited by studying NPs located in the center of a donut beam and the electron beam traveling external. *. dmugarte@ifi.unicamp.br. 2469-9950/2016/93(20)/205418(9). to the particle [17–19]. The central zero intensity is not so restrictive, because of the long range Coulomb interaction between electron beam and sample. It is well known that surface plasmon modes in nanoparticles are mainly determined by morphology, so this represents an excellent case to study the coupling between the NP electronic excitations (surface plasmons) and a vortex beam azimuthal phase structure. Among the different approaches to calculate SP excitation by a high energy electron beam, the boundary element method (BEM), originally proposed by Garcia de Abajo et al. [20,21], is one of the most popular methodologies. Here we have theoretically studied the excitation of surface plasmons by EVBs as observed in EELS spectroscopy. To do so, we have developed a semiclassical approximation combining an azimuthal phase factor and the dielectric formalism based on the BEM method [20,21]. In particular, we have analyzed the case of maximum orbital OAM transfer where EVBs can be used to manipulate efficient plasmon multipole excitation.. I. THE BEM METHOD. This method derives the surface charge distribution σ (s) derived from the Poisson equation for a fast electron (described as φ ext ) traveling close to a dielectric homogeneous nanoparticle [described by a local dielectric response ε1 (ω), ω represents the energy] embedded in an uniform dielectric medium [ε2 (ω)]. A great advantage of BEM is that the procedure can be numerically implemented for particles of arbitrary shape by dividing the surface into small area elements [20,21]. It is important to emphasize that the surface charge distribution can be built by the weighted contribution of different interface surface modes σ i (s,ω), which are independent of energy and depend only on the shape of the considered nanoparticles [22,23]. These so-called geometrical eigenfunctions describe self-sustained surface charge oscillations modes (the SPs), which are obtained from an eigenvalue integral equation. Ouyang and Isaacson [24] have shown that the derived eigenvalue factors λi are real, and the σ i modes form a complete basis set that satisfy a (bi-)orthogonality property.. 205418-1. ©2016 American Physical Society.

(3) DANIEL UGARTE AND CATERINA DUCATI. PHYSICAL REVIEW B 93, 205418 (2016). The total surface charge density is the weighted sum over the modes [20,21] and the boundary potential can be expressed as a summatory of the surface charge mode contributions [20,21]: fi (ω) σ i (s) B , (1) φ (r,ω) = ds ε2 (ω) [(ω) − 2π λi ] |r − s| i 1 (ω) where (ω) = 2π εε22 (ω)+ε and fi (ω) = ds ds ns · (ω)−ε1 (ω) i. that can be used to extend the BEM formalism. For example, Mohammadi et al. [11] have considered an electron traveling in a helical trajectory [28] to generate an electron beam with angular momentum in order to study magnetic responses. Basically, an electron vortex beam (EVB) of order n showing an helical wave front can be described in cylindrical coordinates by the following wave function [14]:. ∗. (s ) ∇φ ext (s,ω) σ|s−s . | The energy loss may be calculated within a pseudostatic calculation, where the dissipation work results from the electric field due to the boundary potential evaluated at the electron position [EB (re )]. For a high energy electron, it is assumed that this electric field does no produce significant changes in the incident electron trajectory direction; then only the component parallel to the electron velocity is considered. Without loss of generality, we will consider an electron traveling along a rectilinear path re = r0 + vt, where v = (0,0, − |v|) and r0 is perpendicular to the velocity v. Also, we will consider that the medium around the particle is a vacuum [ε2 (ω) ≡ 1], then the energy loss results: ∞ EzB (re ) dz. (2) E = ρe −∞. Using the BEM derived boundary potential φ B and considering an electron beam external to the particle (no “Begrenzum effect” [20], hereafter we will always assume this configuration), the energy loss probability for the surface charge mode σi becomes 1 Im{−gi (ω)fi (ω)Ii (r0 ,ω)}, (3) (1 + λi )π 2 v ⊥ ωs where Ii (r0 ,ω) = ds σ i (s) e−i v K0 ( ω|r0v−s | ) and gi (ω) = 2/[ε1 (ω)(1 + λi ) + ε2 (ω)(1 − λi )]. The indexes ⊥ () represents the vector component perpendicular (parallel) to the electron trajectory, and K0 is the modified Bessel function of order zero. As the BEM method only evaluates the Coulomb interaction potential between two point charges [electron and the surface charge element σ (s)ds], the final energy loss probability merely involves K0 functions, which arise from the Fourier transform of a potential with a |r − s|−1 dependence [25,26]. For an external electron beam, the fi integrals can be further simplified using the second Green identity [27]: P i (r0 ,ω) =. fi (ω) = 2π (2 + λi ) Ii∗ (r0 ,ω).. (4). Introducing Eq. (4) into (3), we obtain a very instructive expression for the energy loss probability for a particular surface charge mode i: P i (r0 ,ω) =. 4 (2 + λi ) Im{−gi (ω)} Ii (r0 ,ω)2 , π v 2 (1 + λi ). (5). where the main energy dependence is contained in the gi (ω) factor, and geometrical aspects in the Ii integral [22,23]. II. SEMICLASSICAL MODELING OF AN EVB. In order to study the excitation of SP by electron vortex beams [4–6], we must find a semiclassical physical description. ψvortex = f (r) einϕ eikz z .. (6). Note that this wave function contains the conventional phase term associated with the wave propagation along kz and, in particular, an azimuthal phase term dependent on the angle ϕ which is spatially distributed (but fixed in time) in the plane perpendicular to the wave vector k. Another important characteristic of vortex beams is their central region with vanishing intensity [f (r = 0) = 0] [14]. Our aim is to find a semiclassical description of the EVB where we consider: (a) the azimuthal phase structure [14–16] and (b) an annular beam with zero intensity at the center [14]. To simulate the latter simple geometrical constraint, we may approximate the electron beam cross section [described by f (r)] as being an infinitively thin ring of radius R0 . At this point it is important to remind ourselves that the dielectric formulation to calculate energy loss is actually a two-step procedure. The first one considers how the incident electron beam induces a polarization boundary potential on the dielectric particle (φ B ). The second step considers the dissipative work realized the electric field (EB ) generated by the boundary potential on the incident charge itself. These two steps must now be implemented for an annular electron beam. In analogy to the numerical implementation in the BEM approach, we may discretize the ringlike beam as an ensemble of linear electron trajectories along the azimuthal angle ϕj , re,j (t) = r0,j + vt, where r0,j is perpendicular to the velocity v and |r0,j | = R0 . We will assume that the point charges ρj in all these parallel trajectories do not interact directly between them (the electron in ϕj will not be directly affected by the electron field from another electron in the annular beam). The interaction between electrons will be performed through the induced boundary potential. All electrons in the annular beam will contribute simultaneously to generate the (total) induced boundary potential during the first step of the calculation. In order to calculate the total energy loss probability (second step), we will consider the work performed on each electron of the ringlike beam by the (total) boundary potential. These calculations can be performed within the classical framework without any further approximation, yielding an energy loss probability that represents a kind of ϕ angular averaging, just adding the energy losses for all trajectories. In order to study physical effects associated with the azimuthal phase structure of an EVB, we must consider an additional semiclassical hypothesis that cannot be taken into account in a conventional dielectric calculation of surface plasmon excitation. This can be performed by associating an azimuthal phase term einϕj (n being an integer representing the incident EVB order) with the potential contribution associated with each electron trajectory (φjB ) (see Fig. 1). In these terms, the total boundary potential φTB is obtained through a first. 205418-2.

(4) CONTROLLING MULTIPOLAR SURFACE PLASMON . . .. PHYSICAL REVIEW B 93, 205418 (2016). (second step p). The total energy loss suffered by the annular beam and due to the geometrical mode i must include a second sum over all ρk charges around the ringlike beam: ∞ B,i i −ipϕk ρk e ET ,z (re,k ) dz. (10) ETot = −∞. k. For the sake of simplicity we will hereafter take ρk = 1 for all k. Applying further algebra, we can observe how the phase structure influences the calculations of the energy loss: ∞ B,i i ETot = e−ipϕk einϕj Ej,z (re,k )dz. (11) k. FIG. 1. Scheme of the configuration used to calculate SP excitation on NPs located at the center of a donut-cross-section electron vortex beam of order n.. summatory on the azimuthal angle ϕj : φTB (r,ω) = einϕj φjB (r,ω).. j. −∞. We can now derive the total energy loss probability suffered by the ringlike EVB by introducing the BEM result [Eqs. (3) and (4)] for a single electron trajectory in Eq. (11): 4(2 + λi ) i i∗ (ω) Sk,p (ω) , (12) Im −gi (ω) Sj,n 2 π v (1 + λi ). i where Sj,n (ω) = j einϕj Ii∗ (r0,j ,ω). This expression explicitly allows a purely geometrical interference between the azimuthal symmetries, the surface mode electric field (order n from the incident EVB) and the phase structure (order p) from the boundary electric field interaction. The vortex order p in our semiclassical description corresponds to absorbed angular momentum by the nanoparticles plasmon field; in a quantum mechanical description, a phase factor e−i(nf −ni )ϕ (ni and nf describe the initial and final states of the vortex beam order interacting with the NP) should appear when calculating the loss probability [12]. It is important to emphasize that this allows the evaluation of different OAM transfers, while at the same time using the classical BEM approach to calculate the dielectric potentials and energy losses for each trajectory k. A very instructive expression can be derived if we assume that the the incident and interaction vortex phase structures have the same order (p = n), or in other terms assuming a total OAM transfer to the NP (nf = 0). PTi ,n,p (ω) =. (7). j. Considering only the potential associated with a particular geometrical surface mode i, the summatory becomes einϕj φjB,i (r,ω). (8) φTB,i (r,ω) = j. Note that this assumption suggests that the potentials from regions positive and negative signs (for a position very close to the nanoparticle edge, this is approximatively associated with the sign of the neighboring surface charge densities) along the azimuthal angle may add constructively if the vortex azimuthal phase terms change in a similar way. This implies that if the EVB order n matches the azimuthal symmetry of the analyzed surface charge mode i, the vortex phase structure may generate a geometric constructive interference along the angle ϕ and consequently an enhancement of the induced total electric field for this surface charge mode. Taking the gradient, we derive the total electric field for the surface mode i due to the ring-shaped vortex beam: EB,i einϕj EB,i (9) T (r,ω) = j (r,ω). j. The second step of the dielectric formulation evaluates the interaction between the (total) electric field due to the (total) boundary potential acting on the different electrons j of the ringlike EVB beam. Keeping in mind that we will calculate the energy loss within the quasistatic approximation, the interaction between the electric field created by the charge j (EB,i j ) and the charge ρj itself should not depend explicitly on the azimuthal phase. In contrast, a phase term should appear when we consider the interaction between the electric field created by the electron j (EB,i j ) and another point charge ρk (j = k) of the annular beam. This can be obtained by considering that the interaction between the electric field and the point charge at re,k contains an additional phase term e−ipϕk (p being an integer). We have intentionally used a different index to better identify the phase structures associated with the boundary potential induction (first step n) and the dissipative work interaction. Pni (ω) =. i. 2 4(2 + λi ) Im{−gi (ω)} Sj,n (ω) . 2 π v (1 + λi ). (13). This expression is almost identical to the one derived for an individual electron trajectory [Eq. (5)]. Each azimuthal position of the discretized electron beam contributes independently to the energy loss through the Ii (r0,j ,ω) factors (integrals considering the interaction between the surface charge element and the incident point charge). Subsequently, these terms i are added taking into account the spatial phase factor (Sj,n summatory) and, then, the square modulus is evaluated. In this way, the azimuthal phase structure will directly interact with the azimuthal symmetry of SP modes described by σ i (s). We must emphasize that the experimental setup (needed to measure within the total OAM transfer condition) requires the installation of two vortex selection devices, one before and a second one after the interaction with the sample. This kind of measurement is experimentally possible, but this will be certainly very challenging considering that EVB generators and detectors are still far from ideal efficiency. However, this particular case allows certainly illuminating theoretical. 205418-3.

(5) DANIEL UGARTE AND CATERINA DUCATI. PHYSICAL REVIEW B 93, 205418 (2016). analysis of physical effects due to the interaction between the nanoparticle surface charge symmetry and the vortex beam of order; we hope that the results will stimulate experiments to study the interaction between surface plasmons and EVB. In the following sections we will consider this condition nf = 0 to analyze the energy losses associated with SP modes for particles of different morphologies (spheres, disks, and triangles) located at the center of the ring-shaped vortex beam and, also, for focused vortex beams where the plasmonic nanoparticles are located outside the donut beam. The study of more complex situations involving different OAM transfers. can be easily calculated using the approach developed here. Additional research work is in progress and will be described in forthcoming publications. Briefly we have implemented the nonretarded BEM methodology considering the phase effects in Python language, and the singular integrals were evaluated following the suggestion reported by Hohenester and Trügler [29]. For all calculations presented here, we have considered silver NPs immersed in vacuum (Ag dielectric properties were taken from experimental values [30]) using a high energy electron beam of energy 100 keV. We have analyzed rather low vortex orders associated with dipolar (D, n = 1), quadrupolar (Q, n = 2), and hexapolar (H, n = 3) phase structures. The ringlike beam without phase structure (obtained from the same expressions by making n = 0) is also very interesting because it represents the averaged energy loss spectrum around the NP and it provides a reference value for SP excitation probabilities in a standard EELS experiment. III. EXCITATION OF SURFACE PLASMONS BY EVB. As a first case study, we have addressed the SP excitation on spherical NPs. Although this highly symmetrical morphology does not seem suited to analyze azimuthal geometrical effect,. FIG. 2. Calculated energy loss probability for hybridized sphere SP modes (hl ) with well-defined azimuthal symmetry. A silver sphere of radius 5 nm located at the center of a vortex donut beam (8 nm in radius) has been considered; modes up to l = 6 were included. (a) Predicted spectra for different vortex order (n = 1–3); the intensity has been normalized in relation to the maximum signal predicted for the donut beam without azimuthal phase structure and they have been shifted vertically by 0.5 for an easily visualization. (b) In order to estimate potential symmetry-based enhancement effects, we compare the excitation of each hl surface mode for each vortex order (Pnl ) in relation to predicted probability of the same mode for a nonvortex beam (P0l , the values are indicated by black dots, the lines are just a guide for the eyes). This figure also contains the intensity evolution of intensity of the hl modes with vortex order normalized by the maximum of the spectra predicted for a donut beam without phase l ) in order to get an estimation of the experimental structure (Pmax,0 observability (indicated by white squares).. FIG. 3. Calculated energy loss probability for generation on a silver disk at the center of a EVB. (a) Predicted spectra for the nonvortex beam, where peaks associated with the first multipolar modes can be easily recognized (D, Q, H ). (b) When using vortex beams, just the SP mode matching vortex order is excited in the calculated spectra. Probabilities have been normalized using the maximum of the spectra predicted for the EVB without azimuthal phase structure.. 205418-4.

(6) CONTROLLING MULTIPOLAR SURFACE PLASMON . . .. PHYSICAL REVIEW B 93, 205418 (2016). the surface charge modes are the well-known spherical harmonics Ylm functions, which display well-defined polar and azimuthal dependence on the l and m indexes, respectively. It is also important to emphasize that spherical harmonics allow us many analytical calculations, as for example eigenvalues. [−1/(2l + 1)] and normalization factors [ (2l + 1)/(4π a 3 ), a is the sphere radius], which can be very useful to test numerical software implementations. If we analyze the multipolar contribution in a low loss EELS study of nanospheres, it is typically observed that the dipolar mode l = 1 is responsible for most of the energy loss and higher multipoles have a much lower excitation probability. The dipole mode is degenerated, presenting three different modes, m being ±1,0. Among these modes, the energy loss is due to Y10 , because the induced electric dipole is parallel to electron trajectory (here along the z axis). When considering a charge distribution which is rotationally symmetric around the z axis (e.g., Y10 ), the addition of the loss due to all trajectories of the ringlike EVB will cancel out due to the azimuthal phase factor. In contrast, for the in-plane dipolar modes (Y1±1 ) lying perpendicular to the electron trajectory, the charge distribution is even in z coordinate, then the total virtual work integral should be small for a single electron trajectory. However, a EVB with an azimuthal structure matching the surface charge distribution will display a geometrical resonance and an enhanced energy loss. This represents a very important property, because the use of EVBs allow the study of in-plane properties of the surface modes (perpendicular to the electron trajectory) which are difficult to gather in standard EELS studies. In analogy to hybridization of atomic orbitals (e.g., sp2 , sp3 , etc.), spherical harmonics can be combined to get a desired real surface charge hybrid mode showing well-defined azimuthal geometry. For example, we could generate an hybridized mode by the sum hl = [Yll + (−1)m Yl−l ]; this yields a SP mode that is real and with an azimuthal modulation cos(lϕ). These hl modes have a well-defined eigenvalue as λi depends exclusively on the l index and they represent an ideal case to analyze the effect of vortex beams on SP excitation. Our results show that only hybrid hl modes matching exactly the vortex order n are excited with an enhancement of several orders of magnitude, while all other modes display negligible intensity (see Fig. 2). The selectively excited hybrid mode intensities should be just 10–100 lower than the SP peak measured values using nonvortex beams. The calculations show that for each vortex beam the only excited hl surface mode matches exactly the vortex order n and all other modes display negligible contribution to the energy loss [this effect can be visualized in the progressive blue shift of the plasmon ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−. FIG. 4. EELS probability estimated on a silver nano triangle (side 13.6 nm, aspect ratio 10) at the center of a EVB of radius 10 nm. (a) Spectrum for the nonvortex beam, SP modes are observed at ∼2.3 (D) [and ∼2.9 (Q) eV]. (b) The use of EBV is not so effective to select/enhance the selected SP multipoles, except for. hexapolar modes. (c) Comparison between the maximum intensity of the different SP modes vs vortex order. Probabilities have been normalized using the maximum of the spectra predicted for the EVB without azimuthal phase structure. (d) Schematic diagram comparing the excitation of dipolar and quadrupolar modes when using a focused electron beam and a EVB (the contrast variation around the ring-shaped EVB represents pictorially the phase changing as a function of ϕ) (see text for explanations).. 205418-5.

(7) DANIEL UGARTE AND CATERINA DUCATI. PHYSICAL REVIEW B 93, 205418 (2016). FIG. 5. Comparison between the intensity of the surface modes when moving a triangular silver NP (side 13.6 nm, aspect-ratio 10) out of the EVB (radius 16 nm) center. The eccentricity is plotted as the horizontal axis, value 0 represents NP at the center, while value 1 corresponds to the particle edge at the ring center (see the schematic drawn at the upper region). Note that the weight of different modes changes significantly when the triangular particles are taken out of the donut beam, however no significant changes are observed for the n = 1 vortex beam.. peak for increasing vortex order, Fig. 2(a)]. Concerning the intensity, the use of vortex beams enhance the mode excitation by several orders of magnitude for the first vortex beams [attaining almost a 104 enhancement factor for the octupole l = 4, and diminishes for higher n values, Fig. 2(b)]. We attribute this decrease to the increasing role of the polar angle in the spherical harmonics and also the s dependent dephasing term inside the surface charge integrals Ii (r0 ,ω). Our theoretical model predicts that pure quadrupolar or hexapolar modes can be excited and their intensities should be just 1 or 2 orders of magnitude lower than the measured values using nonvortex beams [Fig. 2(b)]. High aspect-ratio NPs such as triangles or disks [31–34] show SP modes, which are very distinct on the energy scale. For disks, these modes show a well-defined geometrical charge distribution (e.g., D, Q, H , etc.) on the extended planar surfaces [34]. We have considered a silver disk of radius 5 nm at the center of a EVB of radius 8 nm; an aspect ratio equal to 10 (thickness 1 nm) was used for the calculations. This thickness is quite small and different from experimental studies (10–30 nm [31–34]); we have chosen geometrical parameters in order to calculate the energy loss probability neglecting retardation effects. We must keep mind that the geometrical modes depend on the morphology and aspect ratio, but not on the absolute size value, so we are confident of getting a reasonable prediction of spectral behavior in order to analyze experimental data. The calculated probabilities show again a clear resonance between vortex order and azimuthal symmetry of the disk SP modes. The SP peak intensity can be enhanced by several orders of magnitude [Fig. 3(b)]. The reinforcement factor goes from ∼1000 to ∼10 for vortex orders going from n = 1 to 3. Disk-shaped NPs show such strong enhancement due to their regular shape and, also, because our calculations have considered a perfect symmetry (NP located at the center of a ringlike beam). High aspect-ratio NP of triangular shape are a very interesting problem [31–33]. First, the particle morphology imposes several geometrical constraints (e.g., nonuniform impact parameter for a ring-shaped beam); second, although the relevant SP modes can be described as dipolar, quadrupolar, etc., the charge pattern displays just roughly a kind of rotational symmetry (see Fig. 4). Schmidt et al. [33] have shown that we should expect two dipolar (D1 , D2 ) and two quadrupolar modes (Q1 , Q2 ), however, a single mode arises with threefold (H) symmetry. Figures 4(a) and 4(b) compare energy loss probabilities expected for triangular particles located at the EVB center. The peak associated with dipolar modes (∼2.3 eV) is the strongest for vortex orders 1–2 and 4; quadrupolar modes (∼2.9 eV) gain importance for even vortex orders 2 and 4. The n = 3 EVB excites almost exclusively the H SP mode with intensity about ∼10 and 100 times higher than the D and Q modes, respectively; however the intensity is ∼1000 smaller than the dipolar mode excited with a donut beam without phase structure (representing a typical experiment). The reduced in-plane rotational symmetry of SP modes on triangular NPs renders much weaker the resonant coupling with the EVB phase structure. Spatially resolved EELS studies using narrow beams have shown that the excitation of D1 mode is localized close to the triangle apexes and the Q1 modes close. 205418-6.

(8) CONTROLLING MULTIPOLAR SURFACE PLASMON . . .. PHYSICAL REVIEW B 93, 205418 (2016). to triangle sides [31–33]. The stronger excitation of D1 and Q1 modes [charges are localized on a tip or the center of an edge of the NP, see Figs. 4(c) and 4(d)] is explained by the fact that focused electron beam experiments would generate a strong localization of the polarization charges close/around to beam position. But, in contrast to narrow electron beam experiments, modes such as D2 and Q2 gain importance when excited with an annular EVB around the NP because they display a somewhat better in-plane charge distribution azimuthal symmetry. For an isolated narrow beam mode D1 represents the best intuitive distribution of polarization charges, while for a ring-shaped EVB of order n = 1, both the D1 and D2 modes show equivalent geometrical configuration [see Fig. 4(d)]. The precedent results are very promising, because they clearly indicate a manipulation of multipole excitation by choosing the absorbed OAM to match the incident EVB order. However, we must keep in mind that these calculations have considered a perfect geometry (NP at the EVB center). Figure 5 shows the effect of moving the triangular NP out of the central position. As may be expected, the resonant effect is reduced as a function of eccentricity; this indicates that a selection of multipole modes must follow stringent experimental requirements. Surprisingly the movement of the triangle away from the center does not seem to influence strongly the multipole probabilities for an EVB n = 1. We must note that due to the morphology of the triangle, the distance between the particle edge and the ring-shaped beam (impact parameter effect) may induce effects that are not easy to predict. Intuitively, the scheme on the right in Fig. 4(d) may help us to understand why the spatial distributions of the phase structure seem to continue to produce strong azimuthal interference effect between the dipolar modes and the EVB n = 1. Although the eccentricity (e.g., moving the particle to the right or left), the phases for the upper and lower regions can still generate constructive interference. We must emphasize that eccentricity effects are stronger for the very symmetrical disklike NPs, but, as for triangular NPs, the dipolar mode (D) shows very little influence of eccentricity for an incident EVB n = 1. This is very positive, because a deeper understanding of a material electronic structure requires the comparison EELS and photon equivalent absorption spectroscopies [1,2], where dipolar selection rules apply.. IV. MAPPING OF SURFACE PLASMON USING A FOCUSED EVB. Our theoretical study has until now considered azimuthal phase effects on the excitation of surface plasmons on NPs located inside the donut beam. Another natural question arises: Does this azimuthal phase structure generate measurable effects outside the donut beam? Probably due to partial coherence effects the phase information should extend outside the donut to a certain distance (a coherence length). To analyze this situation we have calculated energy loss line profiles for a focused donut vortex beam (1 nm in diameter) scanning parallel to the side of a triangular metal particle (see Fig. 6). For a ringlike beam without a phase structure (this is obtained by making n = 0), the dipolar (D1 ) mode excitation is stronger for positions close to the triangles tips, while the Q1 mode. FIG. 6. Comparison of intensity profile of surface modes D1 and Q1 expected when a vortex beam n = 1 scans a line parallel to a triangular nanoparticle side (triangle side 13.6 nm, aspect/ratio 10, donut-beam radius 1 nm, and distance beam center to triangle edge 2 nm). The solid line describes a nonvortex ringlike beam (without azimuthal phase structure), where the loss profiles show a maxima of intensity close to triangle tips (indicated as dashed vertical lines) and side center for the D1 and Q1 modes, respectively [31–33]. In contrast for a EVB of order n = 1, the maximum of intensity profile is located in front of the nodes of the surface charge density (see text for explanations). We have included the threefold degenerate surface charge distribution for each mode.. is more intense close to the center of the sides, as already reported [31–33]. This tendency is reversed if we use a focused vortex beam with n = 1. Surprisingly, the intensity maxima occur when the EVB is closer to inversion nodes of the surface charge density; also, as an overall effect, this reduces the efficiency of the interaction (and the total spectrum intensity) by about three orders of magnitude. In fact, the different sign of charges at each side of the SP node is compensated by the vortex phase distribution (we have pictorially represented phase changes and SP mode charge density as a varying contrast pattern at the top of Fig. 6). This suggests that an EVB may be used to manipulate the sign of surface charges around it. For example, when we place a standard electron beam between two nanoparticles or nanorods, we stimulate modes where the region closest to the electron trajectory should be polarized with positive charges generating a symmetrical mode [35–38] for the coupled NP system. If we place a dipolar vortex beam (n = 1) between. 205418-7.

(9) DANIEL UGARTE AND CATERINA DUCATI. PHYSICAL REVIEW B 93, 205418 (2016). the particles the phase structure will enhance the excitation of antisymmetric configuration with a stronger dipole character around it. The regions closest to the vortex beam should have opposite charges on each particle. This asymmetrical mode is actually observed in optical absorption spectroscopy when the polarization is parallel to the nanoparticle/nanorod dipole axis [35–37] or on EELS studies if the beam is placed far from the interparticle region and close to the furthest NP/nanorod tip. In this way, this phenomenon may be exploited to induce dipolar character and modify SP dark mode excitation; a similar control of dark modes has already been proposed in optics using focused cylindrical vector beams [39]. Further work is in progress to analyze these kind of effects as the BEM-EVB approach developed here can be directly applied to study neighboring NPs (this approach does not contain any assumption about the connectivity of the considered surface [35,36]). Finally, by controlling the sign of charges in neighboring NPs, this effect can be used to modify interparticle forces and manipulate NP coalescence [40].. the rotational symmetry of the sample becomes in resonance with the phase distribution from the donut vortex beam. If the phase structure is generated by multipolar lenses (e.g., astigmatism [41]) a proper azimuthal angle alignment would be required to induce resonances; however, this also opens the opportunity to select different in-plane interaction directions. The ring-shaped description of EVBs used here can be easily adapted for other approaches such as discrete dipole approximation (DDA [42,43]) or extended to include retardation effects, etc. We expect that a more detailed full quantum mechanical description will allow us an exploitation of phase structures in electron microscopy based spectroscopies. As the absorbed OAM-NP interaction is dominated by symmetry effects in the plane perpendicular to the electron beam, anisotropy information (longitudinal and transversal) of the sample can be derived by comparing nonvortex and vortex beam measurements. This may also be extremely useful for studies associating tomography and low loss spectroscopy [38,44]. ACKNOWLEDGMENTS. V. SUMMARY. We have theoretically revealed that the azimuthal phase structure of ring-shaped electron vortex beams can be used to manipulate the excitation surface plasmon multipoles on nanoparticles by choosing the absorbed OAM to match the incident EVB order (nf = 0). 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