Figure 1

(Color online) Atlas of spatially resolved spectroscopy techniques. They are organized according to their space and energy resolution (see Nomenclature for a list of acronyms). The relation between wavelength and energy is represented by dashed curves for photons and electrons. The universal electron inelastic mean free path (EIMFP) is given as a function of electron energy. The skin depth of Ag is calculated from optical data compiled by 349.Reuse & Permissions

Figure 2

(Color online) Schematic representation of various types of electron microscope equipped to perform spectroscopy. (a) Scanning transmission electron microscopes (STEMs) allow analysis of the energy distribution of electrons transmitted through a thin specimen ($\lesssim 100\mathrm{nm}$ thick), on which the incident beam is focused down to a $<0.1\mathrm{nm}$ spot. Large-momentum transfers are collected through annular dark-field detectors (ADFs). Cathodoluminescence (CL) light detection is also possible on a STEM. (b) Scanning electron microscopes (SEMs) collect secondary electrons (SEs) to form images or CL emission to perform spectroscopy, with spatial resolution down to $\sim 2\mathrm{nm}$. (c) Photoemission electron microscopes (PEEMs) can render spectroscopic information on the photoexcitation process and electron-hole $\left(e\text{−}h\right)$ dynamics with $\lesssim 50\mathrm{nm}$ resolution.Reuse & Permissions

Figure 3

(Color online) Schematic representation of some excitation processes triggered in a solid by a swift passing electron. We emphasize their connection to measured signals in electron microscopes: CL, secondary electron emission (SEE), and electron energy-loss spectroscopy (EELS). The electron can launch propagating surface plasmon polaritons (SPPs; see Sec. 5D). It can also produce surface plasmons localized in nanoparticles or in surface features (Sec. 6). Localized surface plasmons typically decay by coupling to radiation, thus giving rise to contribution (1) to CL (Sec. 4A1). Electronic excitations in the sample (e.g., $e\text{−}h$ pairs) can decay radiatively back to the initial state [coherent CL emission (2), automatically included in the random-phase approximation (RPA) dielectric function as a bubble diagram (273); see Sec. 4A1] or to a different excited state (incoherent CL emission; Sec. 4A2). In metals, secondary electronic excitations constitute the dominant decay channel, producing Auger electrons (AEs), sometimes above the vacuum level, so that they contribute to the detected SEE. Direct excitation of electrons from the target is also possible, producing true SEs.Reuse & Permissions

Figure 4

(Color online) Typical electron energy-loss spectrum. We can identify a narrow zero-loss peak (ZLP; $\mathrm{FWHM}\approx 0.2\mathrm{eV}$), collective modes in the valence loss region (e.g., the $3.7\mathrm{eV}$ plasmon of Ag in the inset), and much weaker core excitations at higher lost energies. The spectrum has been calculated for $200\mathrm{keV}$ electrons traversing a $20\mathrm{nm}$ Ag film using optical data for Ag (349) and assuming a collection angle of $5\mathrm{mrad}$.Reuse & Permissions

Figure 5

(Color online) Wave-vector–frequency diagram in a polaritonic material. The diagram shows the light dispersion relation $q^2=k^2ϵ$ described by Eq. (A3) (thick solid curves) and its intersection with allowed transfers coming from an electron moving with velocity $\mathbf{v}$ (shaded region). Point C signals the frequency threshold of Cherenkov radiation (CR) emission.Reuse & Permissions

Figure 6

(Color online) Evanescent character of the electromagnetic field produced by a fast electron. Transverse-spatial-direction dependence of the $\exp (-i\omega t)$ contribution to the electromagnetic field set up by an electron moving in vacuum with velocity $v=0.7c$ ($\gamma \approx 1.4$ and kinetic energy $\approx 200\mathrm{keV}$) along the positive $z$ axis. The only nonvanishing components ($E_R$, $E_z$, and $H_{\phi }$) decay exponentially at large distance $R$ from the trajectory. The inset shows the orientation of these components relative to the electron velocity vector. The small-$R$ limit is dominated by the $1∕R$ divergence of $E_R$ and $H_{\phi }$.Reuse & Permissions

Figure 7

(Color online) Schematic representation of the Coulomb interaction between a swift electron and a specimen, showing the elements involved in Eq. (14), including the target electron-density operator $\widehat{\rho }$ at point ${\mathbf{r}}^{\prime }$.Reuse & Permissions

Figure 8

(Color online) Bulk excitations in aluminum. (a) The conduction band of Al is well described by a degenerate free-electron gas of Fermi wave vector $k_F=1.75{\mathrm{\AA }}^{-1}$. The $e\text{−}h$ pair excitations supported by this material are contained in the shaded region within the plotted wave-vector–energy $\left(q\text{−}\hslash \omega \right)$ diagram. The volume plasmon dispersion relation measured by 42 (symbols) is compared to the longitudinal plasmon-pole approximation $\omega ={[{\omega }_p^2+{(\hslash ∕m_e)}^2({\beta }^2{q}^2+q^4∕4)]}^{1∕2}$ (thick solid curve) derived from the condition ${ϵ}_{\mathrm{lon}}(q,\omega )=0$, as obtained from Eq. (A7) with $\beta ={(3∕5)}^{1∕2}{k}_F$ and ${ϵ}_b=1$. (b) Logarithmic-scale representation of some of the curves in (a) compared to the light line $(\omega =qc)$ and the transverse bulk plasmon dispersion relation $(\omega ={[{\omega }_p^2+q^2{c}^2]}^{1∕2})$. The latter, which is obtained by inserting Eq. (A8) into $q^2=k^2{ϵ}_{\mathrm{tr}}$, gives rise to light emission in thin films irradiated by electrons (138; 482).Reuse & Permissions

Figure 9

(Color online) Planar interface separating two media of dielectric functions ${ϵ}_1$ and ${ϵ}_2$, respectively. An electron is shown moving parallel to the boundary inside medium 1 (left), with impact parameter $b$ relative to the interface. Another electron is represented crossing the interface at normal incidence (right).Reuse & Permissions

Figure 10

First observation of surface plasmons. They were detected in the energy-loss distribution of $2020\mathrm{eV}$ electrons specularly reflected on an Al surface under 45° incidence. Adapted from 374.Reuse & Permissions

Figure 11

(Color online) Surface plasmon dispersion relation in Ag (left) and excitation of SPPs by swift electrons (right). The SPPs saturate at Ritchie’s nonretarded frequency for large parallel wave vector and follow the light line $\omega =q_{\parallel }c$ in the low-energy limit. An electron moving with velocity $\mathbf{v}$ parallel to the surface preferentially excites plasmons of frequency and wave vector related by $\omega \lesssim q_{\parallel }v$. This condition is indicated by solid dots in the dispersion relation for the two electron energies considered on the right. Faster electrons generate plasmons of lower $q_{\parallel }$ and longer wavelength. The impact parameter is $10\mathrm{nm}$ in the field plots. The dielectric function of Ag is taken from 349.Reuse & Permissions

Figure 12

(Color online) Experimental determination of the SPP dispersion relation in a thin film. Symbols: Measured data obtained from energy and angle distributions of fast electrons traversing a partially oxidized Al thin film. Continuous curves: SPP branches calculated from optical data. Aadapted from 359.Reuse & Permissions

Figure 13

(Color online) Notation used for homogeneous spheres and cylinders of radius $a$, with an electron passing at a distance $b$ from the center.Reuse & Permissions

Figure 14

(Color online) Plasmon losses and spatial resolution in Al spheres. (a) The loss probability is calculated for two different particle sizes, exhibiting several multipole plasmons that redshift as the particle radius increases. Solid curves: Result of Eq. (33) using local response. Dashed curves: Same equation with nonlocal scattering coefficients (see Appendix ). Dotted curves: Prediction of Eq. (34) (dipole approximation). The Al is described by the dielectric function of Eq. (A1) with $\hslash {\omega }_p=15\mathrm{eV}$ and $\hslash \eta =0.5\mathrm{eV}$. (b) Impact-parameter dependence of the dipolar mode maximum. (c) Apparent radius of the dipole excitation in an Al sphere, modeled within local response and defined as the impact parameter $b_{1∕2}$ for which the EELS probability of a $200\mathrm{keV}$ electron is half the maximum as a function of $b$. The dipole-energy dependence on particle size (right scale) is taken into account. The $v\gamma ∕2\omega$ distance of exponential decay in ${\Gamma }_{\mathrm{EELS},\mathrm{dip}}$ [Eq. (34)] is shown for comparison.Reuse & Permissions

Figure 15

(Color online) Scheme of parametrization used in the boundary element method.Reuse & Permissions

Figure 16

(Color online) Mapping plasmon modes in an individual Ag triangular nanoprism through EELS. The triangles were supported on mica. (a) Loss spectra acquired at different positions of the electron beam over the particle (see inset). (b) Parameter-free calculated loss spectra, given in probability per incident electron per eV of energy-loss range. (c)–(e) Simulated energy-filtered images at the positions of absolute maxima in the spectra A–C, respectively. (f)–(h) Measured energy-filtered images. Adapted from 128Reuse & Permissions

Figure 17

(Color online) Electron-beam-induced plasmon generation in gold nanoparticle chains. (a) Scheme of the structure and geometry under consideration. (b) Induced electric field $\left|E^{\mathrm{ind}}(x,\omega )\right|$ (in frequency space $\omega$) produced by the electron along the points of the $\overline{AB}$ segment [see double arrow in (a), $\Vert \widehat{\mathbf{x}}]$. (c) Same as (b) in the time domain $\left|E^{\mathrm{ind}}(x,t)\right|$ [see Eq. (1)]. (d) Fourier transform of $E^{\mathrm{ind}}(x,\omega )$ with respect to $x$, showing the dispersion relation of plasmons excited by the electron and propagating along the chain. The velocity of the launched pulse is found to be $\sim 0.75c$ [dashed line in (c)], in good agreement with the group velocity at the maximum of excitation probability along the guided-mode dispersion curve [dashed line in (d)]. From 69.Reuse & Permissions

Figure 18

(Color online) Energy loss of an electron passing near a composite material. (a) Energy-loss spectra for an electron moving parallel to the [100] direction of the (100) surface of a simple-cubic crystal formed by six layers of aluminum spheres in vacuum. The lattice period is $5\mathrm{nm}$. We consider two different filling fractions as shown by labels. The electron is moving at a distance of $1\mathrm{nm}$ from the sphere surfaces with velocity $v=0.4c$. The dashed curves are obtained by neglecting the interaction among spheres, whereas the solid curves correspond to the full solution of Maxwell’s equations using a layer KKR approach. (b) Surface loss function for the same crystal as in (a) using the effective dielectric function obtained from Maxwell-Garnett theory [broken curves, Eq. (43)] and from the layer KKR method (solid curves).Reuse & Permissions

Figure 19

(Color online) Probing plasmons in a single-wall carbon nanotube via EELS. The experiment shows prominent $\pi$ and $\sigma$ plasmons that are relatively well described both by the discrete-dipole approximation (DDA) and by a dielectric approach. The experimental curve and the dielectric theory are from 437. The DDA calculation performed here for a $16\times 16$ nanotube is based on the dielectric function of graphite, taken from 350.Reuse & Permissions

Figure 20

(Color online) Gap determination using EELS. (a) and (b) Direct and indirect gaps. (c) and (d) Loss probability near direct and indirect gaps. Adapted from 381. (e) Example of simultaneous fit of indirect and direct gaps in GaN for $E_{\mathrm{g}}^{\mathrm{i}}=2\mathrm{eV}$ and $E_{\mathrm{g}}^{\mathrm{d}}=3.2\mathrm{eV}$, adapted from 265.Reuse & Permissions

Figure 21

(Color online) Some of the processes taking place during energy transfer from a swift electron to a sample. (a) Excitation from the specimen ground state $g$ to an excited state $e$, and subsequent radiative decay of the latter (see Sec. 4A1). (b) Excitation followed by incoherent radiative decay to an intermediate state $i$ (see Sec. 4A2). (c) Excitation followed by nonradiative decay.Reuse & Permissions

Figure 22

(Color online) Intuitive explanation of transition radiation. (a) An electron cancels its image charge when it crosses a metal surface. (b) This cancellation creates an effective dipole capable of emitting light and generating plasmons. (c) The resulting transition radiation has a dipolar angular pattern (solid curve) similar to that arising from a dipole normal to the surface (dashed curve) and vanishing for grazing emission, in contrast to the pattern of the dipole in the absence of the surface (dotted curve). The angular pattern is calculated for $200\mathrm{keV}$ electrons impinging on a planar Au surface and for $600\mathrm{nm}$ light wavelength.Reuse & Permissions

Figure 23

(Color online) Excitation of SPPs and TR by electron bombardment of a metal surface. (a) Spectral dependence of the TR and SPP emission probabilities on electron beam bombardment under normal incidence. A $200\mathrm{keV}$ electron impinges on a planar metal surface, giving rise to TR [dashed curves, calculated from Eq. (56)] and SPPs [solid curves, obtained from Eq. (71)]. The metal dielectric function has been taken from 224. The probability is given per unit of light wavelength range. (b) Dependence of the TR and SPP emission probabilities in gold on electron kinetic energy for two different free-space light wavelengths, $\lambda =800$ and $1500\mathrm{nm}$. (c) Measured (symbols) and simulated (dashed curve) SPP emission probability spectra for normally incident $30\mathrm{keV}$ electrons in gold. The SPP emission rate produced by an oscillating dipole placed at the surface is shown for comparison (solid curve), normalized to the spectral integral of TR and far-field dipole radiation. (d) Computed TR spectrum for $30\mathrm{keV}$ electrons normally impinging on a gold surface. Notice that the probability is normalized per unit wavelength, so it corresponds to $(\omega /\lambda )\Gamma \left(\omega \right)$ with $\omega =2\pi c/\lambda$. (c), (d) From 255.Reuse & Permissions

Figure 24

(Color online) Tutorial description of the Cherenkov effect in a photonic crystal. A charge moving in a homogeneous dielectric with velocity below the Cherenkov threshold $v<c∕n$ (a) cannot transfer energy and momentum [shaded region in (b)] overlapping the light dispersion line [upper line in (b)]. However, a periodic modulation of the index of refraction (c) allows light to undergo umklapp processes [i.e., absorption or emission of lattice momentum, as shown in (d)], so that band folding over the first Brillouin zone of the resulting photonic crystal (e) produces mixing and gap openings in the photonic band structure (f). The charge can now couple to propagating optical modes.Reuse & Permissions

Figure 25

(Color online) Experimental observation of CR losses in a photonic crystal. A $1.4\text{−}\mu \mathrm{m}$-thick self-standing porous alumina film, as shown in the TEM image of (a), is used to measure the energy loss experienced by electrons moving inside one of the holes (b). The measured energy-loss spectrum is represented in (c) for 120 and $200\mathrm{keV}$ electrons (symbols) alongside theoretical calculations based on full solution of Maxwell’s equations for an electron moving inside the 2D photonic crystal (dashed and solid curves, respectively). The theory has been convoluted with a measured ZLP of $0.86\mathrm{eV}$ FWHM. Adapted from 170.Reuse & Permissions

Figure 26

(Color online) Losses in aloof trajectories due to diffraction radiation. Measured and simulated energy-loss spectra shown for $200\mathrm{keV}$ electrons passing just outside alumina spherical particles. Symbols: Measured data taken from 1. Thick curves: Retarded EELS calculations using Eq. (33). Dotted curves: Nonretarded EELS calculation obtained from Eq. (31). Thin solid curve: CL calculation for the larger sphere using Eq. (58). The computed probability is given per eV of energy-loss range. The dielectric function of alumina is taken from 349.Reuse & Permissions

Figure 27

(Color online) Smith-Purcell effect in particle chains. Energy-loss probability and Smith-Purcell radiation emission for an electron moving parallel to an infinite periodic string of aligned Al nanoparticles (see upper inset). The probability is normalized per sphere and per eV of energy-loss range. The loss probability for an individual sphere is also shown for comparison. Adapted from 156.Reuse & Permissions

Figure 28

(Color online) Plasmon mapping of single metallic nanoparticles using CL. (a) Measured and calculated CL spectra for a $122\mathrm{nm}$ Ag nanoparticle under grazing incidence, showing dipole $(l=1)$ and quadrupole $(l=2)$ plasmons. The dielectric function of Ag is taken from 224. (b) Energy-filtered CL scan corresponding to the dipole-mode maximum of the spectra shown in (a). The collected light is passing through a polarizer with the orientation shown by the arrow. (c) Symbols: Measured dispersion of multipole plasmons with particle size. Curves: Mie theory. (d) Plasmon width obtained from Mie theory. Adapted from 513.Reuse & Permissions

Figure 29

(Color online) Comparison of EELS and CL intensities in a gold nanodisk. Two different electron beam trajectories are considered (see inset) separated $10\mathrm{nm}$ from the disk in both cases. The disk edges have been rounded with a curvature radius of $3\mathrm{nm}$. These results have been obtained using the retarded BEM with the dielectric function of gold from 224.Reuse & Permissions

Figure 30

(Color online) Momentum transfer from an electron to a spherical alumina particle. (a) Sketch of a $200\mathrm{keV}$ electron passing at a distance of $10\mathrm{nm}$ from the surface of an alumina particle. (b) Linear-momentum transfer, quantified by the velocity change of the particle along directions parallel and perpendicular to the beam and normalized per incident electron. Solid curves: Full electromagnetic calculation. Dashed curves: Dipole approximation [Eq. (62), with the prescription of Eq. (36) for $\alpha$]. (c) Change in rotational velocity produced by angular momentum transfer from a single electron: Full electromagnetic simulation (solid curve) compared to the dipole approximation [dashed curve, Eq. (63)]. From 159.Reuse & Permissions

Figure 31

(Color online) Electron self-interaction mediated by a metallic nanoparticle. Loss probability according to Eq. (65) for two coherent electron beams passing close to opposite sides of a $20\mathrm{nm}$ gold sphere (see inset). The gold is described with the dielectric function of 224. The electron energy is $200\mathrm{keV}$.Reuse & Permissions

Figure 32

(Color online) Pioneering evidence of energy gain after transmission of fast electrons through a thin foil. Energy-gain events involve room-temperature phonon absorption by the electron and are strongly reduced at lower temperature. Adapted from 53.Reuse & Permissions

Figure 33

(Color online) Schematic description of electron energy-gain spectroscopy (EEGS). (a) An electron passes near a nanostructure (a nanoparticle in the illustration) that is illuminated by light of frequency $\omega$. (b) A regular EELS spectrum acquired in a scanning transmission electron microscope (STEM) shows a gain peak on the left-hand side with respect to the zero-loss peak, corresponding to electrons that have absorbed one photon out of the external light. (c) The electron energy-gain probability for light of frequency $\omega$ is proportional to the area of the energy-gain peak. The gain process is assisted by localized excitations of the nanostructure. They supply the necessary momentum to couple evanescent scattered light to the electron, so that the EEGS probability exhibits features as a function of photon frequency mimicking the excitation modes of the sample.Reuse & Permissions

Figure 34

(Color online) Spatial visualization of plasmon launching. (a) The magnetic induction $\mathbf{B}(\mathbf{r},\omega )$ set up by an electron impinging on a semi-infinite silver sample [see Eq. (55)] is represented as a function of distance to the trajectory along the surface and is resolved in energy components $\hslash \omega$. Plasmon generation leads to oscillations of increasing wavelength with decreasing plasmon energy, in agreement with the dispersion relation of Fig. 11. The induction is calculated at a distance of $10\mathrm{nm}$ from the surface. (b) Full spatial dependence of the $\hslash \omega =1.5\mathrm{eV}$ component of the magnetic induction under the same conditions as in (a). The left inset shows a logarithmic scale overview stretching $1.2\mathrm{mm}$ along the surface, in which SPP and TR components are clearly separated. The dielectric function of Ag is taken from 224.Reuse & Permissions

Figure 35

(Color online) Probing nonlocal effects through EELS in metal nanoparticle dimers. Loss probability of a $200\mathrm{keV}$ electron passing near dimers formed by spherical gold particles of $20\mathrm{nm}$ in diameter. The loss spectra are dominated by the longitudinal dipole mode. Local (dashed curves) and nonlocal (solid curves) calculations are compared for several separations between the particle surfaces $d$. Consecutive curves are vertically offset by $0.003{\mathrm{eV}}^{-1}$ for clarity. The electron trajectory is perpendicular to the dimer axis and located as indicated in the upper right inset. The lower left inset shows the $\hslash \omega =2.26\mathrm{eV}$ component of the total-field intensity in logarithmic scale for $d=0.6\mathrm{nm}$ calculated from the BEM; the field diverges along the electron trajectory (vertical bright line) and it has a local maximum in the dimer-gap region.Reuse & Permissions

Figure 36

(Color online) Comparison between EELS, CL, and optical extinction spectroscopy. (a) Electron energy-loss spectrum (EELS) of the dumbbell gold nanoparticle shown in the dark-field TEM image inset. (b) Plasmon maps obtained from the intensity of electrons that have lost an energy corresponding to the dipolar longitudinal mode $\left(1.7\mathrm{eV}\right)$ and the dipolar transverse mode $\left(2.4\mathrm{eV}\right)$. (c) Calculated EELS probability (left scale) and extinction cross section (right scale) of a nanoparticle mimicking the dumbbell of (a). The cross section is obtained for external electric field orientations both parallel and perpendicular to the particle axis in order to excite longitudinal $\left(l\right)$ and transversal $\left(t\right)$ plasmons, respectively. The EELS spectra correspond to positions of the electron beam as shown in (d) by circles. The longitudinal mode is redshifted in the experiment, as compared to the calculation, presumably due to the effect of a mica substrate, which is not accounted for here. (d) Calculated EELS plasmon maps, with the particle contour shown as a shaded area. Notice that the experimental images of (b) exhibit depletion of electron intensity for trajectories passing through the particle due to strong elastic scattering of the electrons with gold atoms. (e) Near-field plots for longitudinal and transversal polarizations of the incident electric field under external optical illumination. (f) Comparison of EELS and CL probabilities (solid and dashed curves, respectively) with the optical extinction cross section ${\sigma }_{\mathrm{ext}}$ (dash-dotted curve) for a metal sphere described by the dielectric function of Eq. (A1). The damping rate is $\eta =0.04{\omega }_p$ (similar to Al), the sphere radius is one-tenth of the free-space wavelength at frequency ${\omega }_p$, and the impact parameter is $b=1.1a$. The experimental data of (a) and (b) are taken from 57. The calculations of (c)–(e) are adapted from 319.Reuse & Permissions