Electron energy loss spectroscopy
Popularized in the 1990’s, electron energy loss spectroscopy (EELS) consists in using a beam of fast-moving electrons which energy is known, to scan a device and/or a material. The non-zero probability of each electron to interact with the structure produces a measurable energy loss, thus providing information on the structure. In particular, various plasmonic resonances can be investigated when the electron beam passes close to the sample.
In order to compute numerically the electron energy loss spectrum from a metallic nanostructure using a time-domain approach, we follow the procedure proposed by C. Matyssek et al. in a DGTD framework. An electron travels at speed v along a trajectory re(t) colinear to the z axis (e.g. re(t) = r0 + v.t.ez). The field generated in vacuum by the moving electron is knwon analytically for each spatial position r(t) for all time t. The electron’s trajectory brushes past an aluminum nanosphere with a minimal distance b, which is typically of a few nanometers. In return, the scattered field radiated by the excited plasmons acts back on the electron, slightly lowering its kinetic energy. These losses are extremely low (at least a thousand times) compared to the total energy of the electron. Hence, a rather good approximation, known as no-recoil approximation, consists in neglecting the induced slow-down in the loss computation. The classical way of expressing the energy lost by the electron is to express it as a frequency-dependent loss probability P(ω), which represents the probability of an electron to lose an energy equal to hpω where hp is the Planck constant.
The incident field (i.e. the above-mentioned generated field) is singular at the electron location (e.g. for r = re). To avoid this particular problem in practice, this incident field is imposed at a certain distance from the electron’s trajectory, on a TF/SF (Ttal Field/Scattered Field) interface. To do so, a cylindrical surface of sufficient length enclosing electron’s trajectory is defined in the computational domain. To avoid the singular field at the top and bottom edges of the cylinder, the TF/SF surface is closed inside-out, so the TF region is not convex. One might note that this technique is only valid if the electron beam does not travel throught the material. In this latter case, using the TF/SF interface method can only lead to approximate results, since a portion of the scatterer is excluded from the total field region.
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Mesh setup for a metallic sphere EELS spectrum computation. The gray cells correspond to the PML, and the red ones to the metallic sphere. The green triangles define the TF/SF interface, which is closed inside-out thanks to a cylinder connecting the upper and lower faces. The z extension of the TF/SF box is voluntarily reduced for clarity.
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| v = 0.1c0 | v = 0.2c0 |
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| v = 0.3c0 |
EELS spectra of a single aluminium nanosphere for various electron velocities. A DGDT-P4 method is used in conjunction with curvilinear elements for the sphere.
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| The electron’s trajectory is oriented from bottom to top | The field induced by the electron in the vicinity of the sphere excites a surface plasmon | While the electron moves away, the plasmon continues to resonate |
Ez field map during an EELS experiment. The gray cells correspond to the SF cells, in which the field is not represented. In this case the electron velocity is v = 0.2c0. For the three views, the field values are arbitrarily scaled to [−1, 1].