Electron energy losses and cathodoluminescence from complex plasmonic nanostructures: spectra, maps and radiation patterns from a generalized field propagator

We present a unified framework for the description of the interaction of fast electrons with complex nanostructures based on the Green dyadic method. We show that the computation of a generalized field propagator yields the electron energy losses and cathodoluminescence of nano-objects of arbitrary morphologies embedded in complex dielectric media. Spectra and maps for both penetrating and non-penetrating electron trajectories are provided. This numerical approach can be extended to describe complex experiments involving fast electrons and optically excited nanostructures.


Introduction
Energy losses from fast electrons have provided the first experimental evidence of surface plasmons (SP) in metallic films (Powell and Swan 1959) and nano-objects (Fujimoto et al 1967). Stimulated by both instrumental (monochromators, detectors) and methodological (signal deconvolution and processing) advances, electron energy loss spectroscopy (EELS) and cathodoluminescence (CL) have since demonstrated their unique potential in probing SPs of metallic nanostructures with nanometer spatial resolution (García de Abajo and Kociak 2008a, Sigle et al 2009, García de Abajo 2010, García de Abajo and Kociak 2008b, Alber et al 2012, Carbone et al 2009, Sigle et al 2009, Geuquet and Henrard 2010, Howie 2009, Myroshnychenko et al 2012, Scholl et al 2012, Bosman et al 2013. Besides the superior spatial resolution compared to optical techniques, the ability of fast electrons to probe the dark SP modes and provide simultaneously accurate information on the morphology and environment of the investigated nano-objects has given a central role to electron spectroscopies in experimental nanoplasmonics. Today, these techniques are used to investigate nanostructures of increasing complexity in which the particle morphology, the substrate or the interparticle interactions strongly influence their optical response. In return, the interpretation of these sophisticated experiments has stimulated the development of several simulation techniques based on the boundary element method (BEM) (García de Abajo et al 2002, Hohenester and Trügler 2012) or discrete dipole approximation (DDA) (Geuquet and Henrard 2010), for instance. The connection of the electron energy loss probability with the local density of electromagnetic modes of metallic particles has been analyzed in detail (García de Abajo and Kociak 2008a, Hohenester et al 2009, Boudarham and Kociak 2012. Recent advances in combined electron/optical spectroscopy techniques such as electron energy gain spectroscopy demonstrated in ultrafast transmission electron microscopes (TEM) or surface plasmon threedimensional (3D) imaging push forward the need and development of novel simulation techniques (Hohenester et al 2009, Nicoletti et al 2013, Hörl et al 2013, Rivacoba and Zabala 2014, Park and Zewail 2014. In this article, we extend the 3D Green dyadic method (3D-GDM) and show that several aspects of the interaction of fast electrons with nanostructures, namely electron energy losses and CL can be accurately described using a generalized field propagator. The GDM is a wellestablished technique and is particularly well-suited for the description of the optical properties of nanostructures of arbitrary geometries, either isolated or electromagnetically coupled to other nano-objects or located in the vicinity of planar interfaces (Girard andDereux 1996, Tripathy et al 2011). Besides the far-field and near-field optical properties, the GDM has been used to predict the temperature increase induced in plasmonic particles by an optical excitation , to describe scanning optical microscopy experiments (Teulle et al 2012) or to visualize the 3D charge distribution associated with surface plasmon excitations (Marty et al 2010). In this work, we extend the 3D-GDM to more complex scenarios involving fast electrons interacting with plasmonic nanostructures.
As depicted in figure 1, we first present the calculation of the energy losses of a fast electron impinging on a metallic nanostructure located on a substrate. The latter can be a simple homogeneous medium, but more complex cases, such as layered substrates, can also be considered. The electron is travelling along the (OZ) axis with a velocity = −v v e z and the substrate surface is the (OXY) plane. The metallic particle with permittivity ε ω ( ) m is lying on the substrate of dielectric constant ε sub and is surrounded by a medium of dielectric constant ε .

env
The moving electron creates both charge and current distributions that can be written e , respectively. These two spatiotemporal distributions are the sources of an electric field t E r ( , ) el that can be derived from Maxwellʼs equations (Jackson 1998 − v c 1 1 env 2 2 is the Lorentz contraction factor and cylindrical coordinates are used ( = + R z r e e R z ). For a 100 kV incident electron, ≈ v c 0.5 and γ = 1.2. K 0 and K 1 are modified Bessel functions of the second kind. Rigorously, equation 1 should be modified to account for the electron interaction with interfaces between two different media. As detailed in the appendix, these corrections can be calculated from the boundary conditions and remain small outside of a thin layer above and below the interface. In the appendix, we show that their influence on the computed energy losses is weak. In the following, we have discarded these corrections to minimize the required computational ressources and used the bulk expression ω E r ( , ) el in the EELS and CL simulations. Note that the electric field scattered by the nanostructures is calculated taking into account the substrate (see equations 10 and below). Only the incident field is approximated by its expression for an homogeneous medium (see also appendix). The contribution from Cherenkov radiation is not considered in the following.

Electron energy losses from a generalized field propagator
In this section, we show that it is possible to accurately compute the energy losses of a fast electron incident on any nanostructure from a generalized field propagator. The electric field associated with the electron induces a polarization distribution ω P r ( , ) inside the nanostructures. In turn, ω P r ( , ) generates a secondary electric field ω ′ E r ( , ) ind . The work accumulated by the electron against the Lorentz force decreases its kinetic energy by (García de is the electron energy loss probability per unit energy (García de Abajo 2010): Assuming a local response of the metal, the electric susceptibility χ ω ′ r ( , ) can be defined as: Then, the Fourier components ω E r ( , ) ind of the induced electric field are: is the self-consistent electric field in the nanostructure. This relation can be recast with a generalized propagator ω ′  r r ( , , ) that directly connects the electric field induced everywhere in the system to the electric field of the probing electron itself: in which the Green dyadic tensor ω ′  r r ( , , ) ensures self-consistency through: . For the latter, several options are available depending on the investigated situation (type of substrate, influence of retardation effects). In the most simple case, the non-retarded image dipole approximation yields a simple analytical expression of ω ″ S r r ( , , ) surf .
We first illustrate our method by considering a silver nanoprism located on a mica substrate as in (Nelayah et al 2007). In figure 2(a), we have computed the EELS of a 100 kV electron, 6 nm away from the apex of a gold nanoprism (position 1) or from the middle of an edge (position 2). The electron energy loss spectra display two clear features at 1.7 and 2.7 eV and a weaker shoulder around 3.4 eV. In figure 2(b), we have generated the EELS maps at these three energies. To do so, the position of the electron beam (X elec , Y elec ) is raster scanned over the nanoparticle, and the loss probability Γ E ( ) EELS loss is computed at each position for a fixed energy E loss . The computed maps show that the enhanced electron energy losses at 2.7 and 1.7 eV are due to the excitation of quadrupolar and dipolar SP modes, respectively. These three features have been detected and ascribed to different surface plasmon modes (Nelayah et al 2007). The excellent agreement with these experiments shows that our method can accurately reproduce both spectra and spatial distributions recorded in EELS experiments (Nelayah et al 2007).
As a second example, we now consider the interaction of fast electrons with a larger nanoprism. The gold nanoprism has an edge length of 950 nm and is located on a Si N 3 4 substrate (figure 2(c)). Similar nano-objects have been investigated by EELS, energy-filtered transmission electron microscopy (EFTEM) (Gu et al 2011), two-photon photoluminescence (Viarbitskaya et al 2013) and dark-field microscopy (Major et al 2013). The EELS spectrum is more complex and displays several features at 1, 1.4, 1.6 and 1.8 eV. We have mapped the electron energy loss probability at these four energies ( figure 2(d)). The EELS maps, in qualitative agreement with the experimental results of (Gu et al 2011), confirm that the obtained features are due to high order SP modes. The latter are similar to the Fabry-Pérot optical cavity modes. With increasing energy, higher order plasmon modes are excited in the metallic resonator, and more nodes are visible. It is worth emphasizing that the inner structure of these modes has not been observed in the experiments most likely because of the large thickness of the investigated nano-objects, which strongly decreases the electron transmission (Gu et al 2011).
3. Cathodoluminescence emission probability and radiation pattern from a generalized field propagator In this part, we show that using the same generalized propagator ω ′  r r ( , , ) of equation 9, another aspect of the interaction of fast electrons with nanostructures, namely, cathodoluminescence (CL), can be quantitatively reproduced. CL is widely used for the nanoscale exploration of the optical properties of semiconductor and metallic nanostructures (Zagonel et al 2010). It has recently been extended to measure in a scanning transmission electron microscope (STEM) the second order photon correlation of individual emitting centers in nanodiamonds (Tizei and Kociak 2013). The energy lost by an electron interacting with a metallic nanoparticle is either absorbed (and eventually converted into heat) or re-radiated in the far-field. In metallic nanostructures, the non-radiative inelastic decay channels are much faster than the radiative ones, and therefore light emission is dominated by electron-induced radiation emission; incoherent CL being much weaker (García de Abajo 2010). The coherent CL emission arises from the far-field radiation of the dipoles induced by the evanescent electric field associated with the moving charge. The total radiated energy ΔE CL can be computed from the flux of the Poynting vector (Myroshnychenko et al 2012): CL 0 2 The number of photons emitted per electron, unit solid angle and unit energy Γ Ω ω ( , ) CL is given by: CL 2 rad 2 In the latter expression, ω E r ( , ) rad is the Fourier transform of the total electric field radiated by the metallic nanostructure and k is the light wavevector in vacuum.
V rad The polarization distribution is obtained from the induced electric field using equations (1) and (8) and the susceptibility (6).
Figure 3(a) shows the CL spectrum collected using a mirror (NA = 0.8) placed above a gold nanowire located on a glass substrate. The electron probe passes 10 nm away from the nanowire end. Our calculations take into account the contribution from the glass substrate and the numerical aperture of the collecting mirror. Four features are clearly visible in figure 3(a) at 0.72, 1.12, 1.42 and 1.69 eV. To shed light on their origin, we have plotted the CL photon emission intensity as a function of the electron beam position at each resonance energy. Figures 3(b)-(e) shows that the spatial distribution of the CL emission probability has a standing wave profile similar to Fabry-Pérot cavity modes. These modes are the counterpart of the high order multipolar surface plasmon modes, evidenced on the EELS maps of figure 2(d). They have already been observed in silver and gold nanowires (Vesseur et al 2007, Gómez- (Coenen et al 2012). In our case, the SP resonances at 0.72, 1.12, 1.42 and 1.69 eV are ascribed respectively to the m = 2, 3, 4 and m = 5 surface plasmon resonances of the gold nanowire. A clear advantage of CL spectroscopy, besides its superior spatial resolution, is evidenced in figure 3(a) in which the extinction spectrum of the same nano-object is presented. Unless an asymmetric excitation (e.g. tilted beams) is used, optical spectroscopy techniques relying on plane-wave excitation are restricted to odd-order resonances (Dorfmüller et al 2010). Figure 3(a) clearly shows that the m = 2 and m = 4 resonances are not visible in the optical spectrum. The ability of CL spectroscopy to probe odd and even resonances is a consequence of the strong electric field gradients associated with fast electrons. Furthermore, it has recently been shown that surface plasmon modes of even parity give rise to symmetric Lorentzian lineshapes, whereas modes of odd parity display asymmetric line profiles in the optical extinction spectrum of gold nanorods under near-normal plane-wave illumination (Verellen et al 2014). The results of figure 3(a) suggest that this effect also occurs in CL experiments.
Interestingly, relation (13) gives access to the angular distribution of the emitted CL which is another signature of the excited resonances. To illustrate this point we have calculated the angular distribution of the CL from the gold nanowire excited by an electron beam focused either near the end of the nanowire (position 1, figure 3(f)) or at its center (position 2, figure 3(g)) and at the energy of the m = 4 resonance. The radiation patterns presented in figures 3(f)-(g) are the result of the interferences of the radiation emitted by the polarization distribution generated in the nano-object. For instance, figure 3(f) shows an asymmetric pattern because the electron excitation is shifted with respect to the center of the nanowire. Figure 3(g) bears close ressemblance with the radiation pattern reported for gold ridges and for the same m = 4 resonance using angle-resolved cathodoluminescence experiments (Coenen et al 2012).

Electron energy losses from nanostructures of arbitrary geometries in the presence of a substrate
The proposed method can also be applied to even more complex morphologies. To further illustrate this, we consider a randomly generated nanostructure consisting of 29 gold nanospheres (diameter D = 9 nm) partially or almost completely overlapping and located on a glass substrate ( figure 4(a)). Such a morphology mimicks the nanoporous metallic films recently investigated by combined STEM-EELS and high-angle annular dark-field microscopy (HAADF) (Bosman et al 2011, Losquin et al 2013. The latter studies have demonstrated that the peculiar optical properties of nanoporous metallic films originate from nanoscale variations of the number and spectral properties of the surface plasmon resonances supported by these complex objects with no clear correlation with their local morphology. Figures 4(b)-(d) shows the EELS probability maps computed at three different energies (1.5, 2 and 2.4 eV). As can be noticed, regions separated by few nanometers can have drastically different optical responses. Furthermore, as observed by (Bosman et al 2011), higher energy modes (2.4 eV) appear confined in smaller regions than lower energy modes (1.5 and 2 eV).

Conclusion
To summarize, we have presented a numerical framework that allows us to simulate, from a generalized propagator, the electron energy losses and CL caused by the interaction of a fast electron and a metallic nanostructure. This formalism accounts for both penetrating and nonpenetrating trajectories, and nanostructures of arbitrary morphologies, and rigorously takes into account the environment. Its numerical implementation is fast and does not rely on numerical libraries. An extension of this approach would be adequate for the computation of the optical forces exerted on metallic nanoparticles or to more complex experiments such as electron energy gain spectroscopy experiments (García de Abajo and Kociak 2008b, Yurtsever et al 2012, Talebi et al 2013, Park and Zewail 2014. We consider an electron traveling in a medium of dielectric constant ε 2 with a velocity = −v v e z ( > v 0) towards a substrate located in the < z 0 region and characterized by a dielectric constant ε 1 (see figure 1). When the fast electron crosses the interface between two media, expression 1 must be modified to take into account the influence of the charge distribution induced on the interface by the moving charge. It is possible to take into account these corrections by writing the electric field in medium i as the sum of the electric field of the electron in an infinite bulk medium of dielectric constant ε i , ω E r ( , ) i el, and the contribution from the surface The electron electric field in an infinite medium of dielectric constant ε 1 can be written as a superposition of plane waves with transverse magnetic polarization:  allows us to recover the continuity of the parallel electric field component. These results show that the influence of the interface contribution is weak outside of an ≈50 nm thick layer around the interface. It is furthermore possible to show that the relative importance of the corrections induced by the presence of the substrate is dependent upon the energy, as the low frequency Fourier components of the electric field are more strongly affected away from the interface.
To go further, we have computed the electron energy losses using the method proposed above, taking into account either the bulk expression ω E r ( , ) i el, for the electric field associated with the moving charge, or the corrected expression ω E r ( , ) i . Figure  These results of figure A2 show that the substrate-induced corrections to the incident Efield have a weak influence on the energy losses. This can be explained by the fact that the larger modification of the incident electric field induced by the interface contribution ω E r ( , ) i surf, is on the Z-component. On the contrary, along its trajectory, the electron probes the electric field radiated along the Z-component, which is radiated by dipoles oriented in-plane and is therefore not excited efficiently by the out-of-plane components of the electron electric field. Moreover, the nanosphere is only deposited on the vacuum/SiO 2 interface, and therefore its dielectric environment is mainly vacuum, which explains that no observable redshift is evidenced.