Vibrational electron energy loss spectroscopy in truncated dielectric slabs

Specially designed instrumentation for electron energy loss spectroscopy (EELS) in a scanning transmission electron microscope makes it possible to probe very low-loss excitations in matter with a focused electron beam. Here we study the nanoscale interaction of fast electrons with optical phonon modes in silica. In particular, we analyze the spatial dependence of EEL spectra in two geometrical arrangements: a free-standing truncated slab of silica and a slab with a junction between silica and silicon. In both cases, we identify different loss channels, involving polaritonic and nonpolaritonic contributions to the total electron energy loss, and we obtain the corresponding energy-filtered maps. Furthermore, we present a comparison of the theoretical simulations for a silica-silicon junction with experimental results, and we discuss the spatial resolution attainable from the energy-filtered map considering optical phonon excitations in a conventional experimental arrangement.

Figure 1

(a) Schematics of the considered geometry. The sample is a slab of thickness $d$ infinite in the $xy$ plane with a sharp interface in the plane $x=0$. We consider either a slab with a silica-silicon interface or a semi-infinite silica slab surrounded by vacuum. A 60-keV electron ($v=0.446c$) is moving in the positive $z$ direction at the distance $b$ from the silica-silicon or silica-vacuum interface at $x=0$. Negative values of $b$ stand for the beam inside silica. Considered limiting cases of (b) an infinite interface geometry and (c) an infinite silica slab.

Figure 2

(a) Real and imaginary parts of the dielectric function of silica ${\varepsilon }_{{\mathrm{SiO}}_2}$ together with the real part of the dielectric function of silicon (the imaginary part is four orders of magnitude smaller and thus negligible). (b) Bulk loss function of silica $\mathrm{Im}[-1/{\varepsilon }_{{\mathrm{SiO}}_2}]$ (dashed black curve) with peak energy at 154 meV, the silica-vacuum interface loss function $\mathrm{Im}[-1/({\varepsilon }_{{\mathrm{SiO}}_2}+1)]$ peaking at 142 meV (solid blue curve) and a much weaker silica-silicon interface loss function $\mathrm{Im}[-1/({\varepsilon }_{{\mathrm{SiO}}_2}+{\varepsilon }_{\mathrm{Si}})]$ with the maximum at 134 meV (dashed-dotted red curve). Vertical dashed lines mark the corresponding positions of the loss peaks. The gray shaded area in (b) emphasizes the region of bulk losses in ${\mathrm{SiO}}_2$.

Figure 3

(a,b) Integrands of Eq. (5) evaluated for a 60-keV electron beam passing through an infinite silica slab of thicknesses 10 and 100 nm, respectively. For better visualization, the magnitude of (a) was multiplied by 10. (c) The nonretarded [full lines, according to Eq. (5)] and retarded (dashed lines) loss probability divided by the slab thickness, evaluated for $d=10/100/1000$ nm as denoted in the plot. The momentum cutoff used in the calculation is $Q_{\mathrm{c}}=0.27{\AA }^{-1}$.

Figure 4

Numerically calculated momentum-resolved EEL probability $P(q_y,\omega )$ for a truncated silica slab (upper row) and a silica-silicon junction (lower row) for two impact parameters of the beam passing through silica ($b=-1000$ and $-50$ nm, respectively) and another impact parameter at the slab truncation or junction ($b=0$ nm). The slab thickness is $d=100$ nm. The intensity of the last two plots for the silica-silicon junction is multiplied by a factor of 2. The overlaying curves are analytically calculated maxima of momentum-dependent loss probabilities for the infinite silica slab (dashed green), the infinite silica-vacuum interface (solid blue), and the infinite silica-silicon interface (dashed-dotted red). Solid straight red lines denote the threshold for radiation losses.

Figure 5

Numerically calculated EEL spectra for an electron beam probing (a) a truncated silica slab and (b) a silica-silicon junction at varying impact parameter $b$ as displayed in the insets. $b$ is negative when the beam is inside silica and positive when it is placed either in vacuum or in silicon. The slab thickness is in both cases $d=100$ nm. Spectra are vertically shifted by a constant value for clarity.

Figure 6

Spatial dependence of EEL intensity at fixed energies calculated for a truncated silica slab (a) and at a silica-silicon junction (b). The energies correspond to the silica bulk loss peak (black circles), the coupled-surface loss for an infinite 100-nm-thick silica slab (green squares), the silica edge excitation [orange diamonds in (a)], and the nonpolaritonic Si-O-Si symmetric stretch signal [blue triangles in (a), multiplied by 3]. The spatial dependence of the silicon-silica junction excitation intensity is plotted in (b) (red diamonds). The peak intensities were obtained from spectra in Fig. 5. Dashed curves are fits of the decay for $b>0$ by $K_0(2\omega b/v)$ at 100 meV (light blue line, which traces well the calculated points—triangles), at coupled-edge loss energy (orange line, which deviates from the diamonds) and junction excitation energy (red line, which deviates from the diamonds). Graph (c) depicts the total intensity integrated over an energy window of 120–160 meV for the silica edge (dark blue circles) and silica-silicon slab junction (dark red squares). Symbols represent the calculated data, lines are guides to the eye. The pink dashed line shows the experimental result for the silica-silicon junction integrated over a 120–180 meV energy window.

Figure 7

Numerically calculated EEL spectra from Fig. 5 after convolution with a point spread function (PSF) represented by a Gaussian function of FWHM 20 meV mimicking the finite experimental resolution (full lines) and experimentally measured spectra (dashed lines, taken from Ref. [21]). The impact parameters as denoted next to the plot. The spectra are vertically shifted for clarity.