Theory of electron energy loss near plasmonic wires, nanorods, and cones

We present an analytical description of the electron energy loss near plasmonic nanostructures with cylindrical symmetry as a three-step process: (i) electron-induced excitation of surface plasmon modes, (ii) their propagation and reflection, and (iii) the backaction on the electrons. The model incorporates relativistic effects, retardation, and light emission, and can treat arbitrary tilted electron trajectories. Convincing agreement with recent experimental data is demonstrated.

Figure 1

Sketch of a straight infinite cylinder (in vacuum) traversed by an electron at an angle $\alpha$. A value of $\alpha =0$ corresponds to a trajectory perpendicular to the axis of the cylinder, and $\alpha =\pi /2$ to a parallel trajectory. Vector $\mathbf{b}$ defines the position of the plane of incidence, and its orientation is defined by $\mathbf{v}$ and $\widehat{\mathbf{z}}$.

Figure 2

(a) Energy-loss probability distribution (thick blue line) and contributions from different azimuthal modes (thin lines) for a 200 keV electron passing near an infinite gold cylinder with impact parameter $b_{\perp }=520$ nm (cylinder of $500\mathrm{nm}$ radius) perpendicularly to the wire (at $\alpha =0$). (b) The same for a tilted electron trajectory at $\alpha =0.47\pi$. The results have been obtained using the dielectric function of gold from Johnson and Christy [85].

Figure 3

(a)–(c) Energy-loss probability distribution for a 200 keV electron passing near an infinite gold cylinder as a function of the wire radius. The impact parameter of the electron is $b_{\perp }=R+20$ nm. (d)–(g) The radial component of the induced electric field, $\text{Re}\left[E_{\rho }^{\text{ind}}(\mathbf{r},\omega )exp(-i\omega t)\right]$, is plotted in the transverse plane at energy $\omega =2$ eV with a spatially dependent time variable defined at every point as $t=y/v$, where $v$ is the electron velocity. The dashed white lines depict the corresponding electron trajectories. (h)–(k) The $y$ component of the induced electric field, $\text{Re}\left[E_y^{\text{ind}}(\mathbf{r},\omega )exp(-i\omega t)\right]$, is analogously plotted.

Figure 4

Absolute value of the $T$-matrix element $T_{q,0}^{22}$ evaluated as a function of $q$ and $\omega$ for a gold cylinder of 15 nm radius. The red and blue lines illustrate the complex propagation constant $q_{\left|\right|}$, whose real and imaginary parts determine the pole location of the $T$ matrix in the complex $q$ plane. The dashed white line indicates the light line. The results have been obtained using the dielectric function of gold from Johnson and Christy [85].

Figure 5

(a) Energy- and position-dependent loss probability distribution evaluated along the conical surface at distance $b$ from the apex (see inset). (b),(c) The position-dependent loss probability distribution at $\omega =1.0$ and $0.75$ eV. The circles illustrate the experimental data from Ref. [51].

Figure 6

(a) Energy- and position-dependent loss probability distribution evaluated along a 700-nm-long gold nanorod of 13 nm radius (see inset). (b) The energy-loss probability distribution summed over $b\in (0,L)$ (blue line) and corresponding results obtained by means of the boundary element method [64] (black line). (c)–(e) The position-dependent loss probability distributions evaluated at selected loss energies (blue lines) and results of the boundary element method (circles). The results have been obtained using the dielectric function of gold from Johnson and Christy [85].

Figure 7

Energy-loss probability distribution (blue line) for a 460-nm-long silver nanorod of 13 nm radius averaged over $b\in (0,L)$, as shown in the inset. The dashed line illustrates the contribution from the TM $m=0$ mode. $n$ indicates the resonance number from Eq. (30). The results have been obtained using the dielectric function of silver from Ref. [92] and the dielectric function of amorphous carbon from Ref. [93]. (Compare Fig. 1 from Ref. [46].)