Relativistic electron energy loss spectroscopy of solid and core-shell nanowires

Near-field optical spectroscopy with nanometer spatial resolution using transition radiation from a focused, relativistic electron beam is demonstrated for Ge nanowires. The excitation of the waveguide modes is absent in theoretical treatments that neglect retardation because the coupling of the swift electron to the optical modes of the nanowire is a consequence of its relativistic nature. Here, relativistic energy loss probabilities for aloof electron trajectories near a perpendicularly oriented single nanowire are analytically calculated using a local dielectric theory. Multimode and zero-mode Ge nanowires are experimentally measured in this geometry with a $\sim 2\text{-nm}$-size electron probe. Observations fall in excellent agreement with the retarded calculations. Optical eigenmodes are identified from comparison of the experimental electron energy loss spectra to theoretical dispersion maps. Adding a Drude-type metallic shell to a dielectric nanowire provides surface-plasmon modes in the same energy range as the dielectric waveguide, leading to strongly coupled hybrid modes that can be efficiently excited by the incident electron. Such results demonstrate a powerful solution for optical studies of nanosystems with nanometer spatial resolution over a broadband energy range.

Figure 1

(Color online) Geometry of the experiment. The core-shell nanowire extends along the $y$ axis and consists of an inner core of radius $a$, described by the dielectric function ${ϵ}_a$, while the outer shell extends to radius $c$, described by ${ϵ}_c$. The aloof electron travels parallel to the $z$ axis and passes closest to the wire at a distance $b$ from the center of the wire.

Figure 2

(a) Scanning electron microscopy (SEM) image of Ge wires grown on a Si substrate. (b) High-angle annular dark field STEM images of individual Ge nanowires dispersed on perforated ${\text{SiN}}_x$ films.

Figure 3

(Color online) Experimental and simulated EELS for a 286-nm-diameter nanowire. A fitted background model (dashed curve) is added to the theory (short dashed curve) resulting in the solid curve which agrees well with the experiment (dotted curve).

Figure 4

(Color online) Real and imaginary components of the dielectric function for Ge. The imaginary component decreases to zero below 2 eV indicating negligible optical absorption in this regime.

Figure 5

(Color online) Background-subtracted EEL for 143- and 23.5-nm-radius (or 286- and 47-nm-diameter) wires and the corresponding theory. The fitted background model is subtracted from the raw spectrum. (a) The first three theoretical peaks (dashed curve) in the 286-nm-diameter wire are resolved by experiment (solid curve). (b) The absence of the peaks is shown for the 47-nm-diameter wire for both theory (dashed curve) and experiment (solid curve).

Figure 6

(Color online) (a) Loss-dispersion map generated from $S\left(g,\omega \right)$ for 23.5- and 143-nm-radius (or 47- and 286-nm-diameter) wires. (b) The corresponding nonretarded loss-dispersion map. Intensity is multiplied by 10 for similar contrast. The impact parameter in all cases is 3 nm away from the surface.

Figure 7

(Color online) The energy loss probability as a function of diameter. Larger diameters are shown to support a higher density of peaks with improved definition. The increase in intensity above 4.44 eV is indicative of absorption due to the Ge interband transition. The intensities are normalized.

Figure 8

(Color online) (a) Loss-dispersion maps for a solid 140-nm-diameter Au nanowire and for a Au shell consisting of inner and outer diameters of 130 and 140 nm, respectively. In the case of the shell, the first three modes are due to the lower-energy (or symmetric) branches of the surface plasmonic modes and can be identified from bottom to top as $m=0$, $m=1$, and $m=2$ modes. (b) EELS for different shell thicknesses, $t$, with the outer diameter fixed at 140 nm. The splitting of the peak originates from the $m=1$ and $m=2$ symmetric modes. Intensity is enhanced for thinner shells as the coupling strength between the inner and outer surfaces increase.

Figure 9

(Color online) EELS for a Au shell consisting of inner and outer diameters of 160 and 200 nm, respectively. Different dielectric cores result in the shifting and splitting of energies depending on their dielectric constant. Inset shows only the real part of the dielectric constants for ${\text{SiO}}_2$ and GaN as the imaginary component is small in the energy range of consideration $\left(1<E<3.5\text{eV}\right)$.

Figure 10

(Color online) The loss-dispersion maps for the same system considered in Fig. 9. The gradual redshift of the symmetric modes occurs for higher dielectric constants.

Figure 11

(Color online) Loss-dispersion maps for a GaN core, a Drude-model Au shell, and the combined system. Only the $m=1$ mode is considered for illustration purposes. The core diameter is 160 nm and the outer-shell diameter is 170 nm. The impact parameter in all cases is 1 nm away from the outer shell (or 6 nm away from the core). The ${\text{HE}}_{11}$ mode apparent in the isolated GaN core converges to the antisymmetric mode from the Drude-type Au shell.