Vibrational Surface Electron-Energy-Loss Spectroscopy Probes Confined Surface-Phonon Modes

Recently, two reports [Krivanek et al. Nature (London) 514, 209 (2014), Lagos et al. Nature (London) 543, 529 (2017)] have demonstrated the amazing possibility to probe vibrational excitations from nanoparticles with a spatial resolution much smaller than the corresponding free-space phonon wavelength using electron-energy-loss spectroscopy (EELS). While Lagos et al. evidenced a strong spatial and spectral modulation of the EELS signal over a nanoparticle, Krivanek et al. did not. Here, we show that discrepancies among different EELS experiments as well as their relation to optical near- and far-field optical experiments [Dai et al. Science 343, 1125 (2014)] can be understood by introducing the concept of confined bright and dark surface phonon modes, whose density of states is probed by EELS. Such a concise formalism is the vibrational counterpart of the broadly used formalism for localized surface plasmons [Ouyang and Isaacson Philos. Mag. B 60, 481 (1989), García de Abajo and Aizpurua Phys. Rev. B 56, 15873 (1997), García de Abajo and Kociak Phys. Rev. Lett. 100, 106804 (2008), Boudarham and Kociak Phys. Rev. B 85, 245447 (2012)]; it makes it straightforward to predict or interpret phenomena already known for localized surface plasmons such as environment-related energy shifts or the possibility of 3D mapping of the related surface charge densities [Collins et al. ACS Photonics 2, 1628 (2015)].

Popular Summary

Phonons are vibration waves that propagate in solids. Their study and quantification is fundamental for understanding many material properties, from superconductivity to thermoelectricity. Numerous techniques have been developed for studying phonons, such as high-resolution electron-energy-loss spectroscopy (HREELS), in which electrons are sent toward an object of interest, and the energy of the phonons is deduced from that lost by the electrons. Recently, HREELS has been pushed to the nanometer scale. With the success of these experiments, researchers now want to determine what types of phonons are being measured and how to rationalize some of the observations. A similar problem once arose in HREELS experiments on electronic waves known as plasmons, experiments that were interpreted at the time using analogies with phonons. We show that, in a reversal of traditional analogies, current phonon experiments can now be interpreted using modern concepts for plasmons.

Most of the properties of surface plasmons in nano-objects can be understood by introducing a rigorous quasiclassic eigenmode decomposition, which we have adapted to the case of surface phonons. From this decomposition, all classical and linear optical properties of phonons can be deduced, such as optical cross sections, EELS probabilities, and the electromagnetic local density of states. Such a theory describes most of the recent observations, such as the energy of phonons in slabs of silica or the spatial modulation of surface phonons in cubes of magnesium oxide. It also permits researchers to bridge optical and EELS observations.

With the rise of new HREELS instrumentation able to probe phonons at the nanometer scale, this theory will help us to grasp the physics of forthcoming experiments, as well as predict new effects such as 3D mapping of surface phonons.

Figure 1

Analogy between surface phonons modes and surface plasmons modes. (a) Dispersion relation of the Fuchs-Kliewer modes for a slab of thickness $d$ (top) and a cylinder (bottom) of radius $r$ made up of MgO. The charge symmetry of the modes is sketched in the inset. For the cylinder, only the rotationally invariant mode branch is shown, as the other modes are essentially not dispersing [5]. Calculations have been performed in the quasistatic approximation (b) Same for SP modes in silver. (c) Dispersion relation for the cSPh of nanorods, reconstructed from a series of retarded simulation of nanorods of different lengths (diameter is 30 nm). The dotted line is the quasistatic dispersion relation for an infinite cylinder of the same diameter, showing the remarkable agreement between both approximations even for long lengths of rods. (d) Surface eigencharge distribution for cSPh of a nanorod, with the given mode orders and eigenvalues ${\lambda }_i$.

Figure 2

Optical cross sections, EELS, EMLDOS, and eigenpotentials for the cSPh in a nanorod of MgO. (a) Simulated optical cross sections for an incoming beam propagating perpendicular to the nanorod axis, and EELS spectrum for an electron beam located 10 nm away from one tip of the nanorod. All spectra have been shifted for clarity. Optical cross section scales are the same for extinction and absorption, and multiplied by $6\times {10}^4$ for scattering. The polarization of the electrical field is parallel to the nanorod axis, except for the dotted curve. The nanorod is 200 nm in length and 30 nm in diameter. (b) EELS maps for the four first modes of the nanorod. (c) Corresponding zEMLDOS maps taken at $z=10\mathrm{nm}$ from the surface of the rod. (d) Corresponding $z$-integrated eigenpotentials.

Figure 3

Optical extinction, absorption, and scattering cross sections for (a) A MgO nanoantenna and (b) a silver nanoantenna. Both antennas have the same size ($200\times 30\mathrm{nm}$). Note the absolute cross section values.

Figure 4

Dielectric environment effect. (a) Simulated EELS spectra for a cube of MgO (100-nm edge long) in vacuum, exhibiting a corner ($C$), an edge ($E$), and a face ($F$) mode depending on the beam position. (b) Simulated EELS spectra for a nanorod ($200\times 30\mathrm{nm}$) in vacuum (black) and embedded into a dielectric of refractive index equal to 1.4. The beam is positioned at 10 nm from the tip of the nanorod in both cases. (c) Same simulations as in (a), but for a cube deposited on a substrate of refractive index $n=2.3$. The former $C$, $E$, and $F$ mode split into two bands. The distal band is essentially consisting in a series of $C$, $E$, $F$ modes arising at almost the energy of the corresponding vacuum modes, while the proximal band is shifted towards the ${\omega }_{\mathrm{TO}}$ energy. Spectra corresponding to a given trajectory are indicated by their colors.

Figure 5

Modes symmetry for a cube in the quasistatic approximation. Values of ${\lambda }_i$ are given on top of the corresponding eigencharge distributions (red is minimum and blue maximum). (a)–(g) Corner modes and (h)–(l) edge modes. Corner modes have been separated with respect to their symmetries.