Strongly Coupled Plasmon and Phonon Polaritons as Seen by Photon and Electron Probes

The ability to control and modify infrared excitations in condensed matter is of both fundamental and applied interest. Here we explore a system supporting low-energy excitations, in particular, mid-infrared localized plasmon modes and phonon polaritons that are tuned to be strongly coupled. We study the coupled modes by using far-field infrared spectroscopy, state-of-the-art monochromated electron energy-loss spectroscopy, numerical simulations, and analytical modeling. We demonstrate that the electron probe facilitates a precise characterization of polaritons constituting the coupled system, and enables an active control over the coupling and the resulting sample response both in frequency and space. Although far-field optical spectra can be substantially different from near-field electron energy-loss spectra, we show that a direct comparison is possible via postprocessing and right positioning of the electron beam. The resulting spectra allow us to evaluate the key parameters of the coupled system, such as the coupling strength, which we demonstrate to be probe independent. Our work establishes a rigorous description of the spectral features observed in light- and localized electron-based spectroscopies, which can be extended to the analysis of analogous optical systems with applications in heat management and electromagnetic field concentration or nanofocusing.

Figure 1

(a) Numerically calculated EEL (green) and optical absorption (red) spectra of an infinitely extended ${\mathrm{Si}\mathrm{O}}_2$ thin film with a thickness of $t=40$ nm. The dielectric response of ${\mathrm{Si}\mathrm{O}}_2$ is modeled using the complex dielectric response obtained from experimental measurements in Ref. [38]. All the spectra are probe-position invariant. Vertical dashed lines denote energies of the TO and LO phonons in ${\mathrm{Si}\mathrm{O}}_2$. The inset shows charge-symmetric (left) and charge-antisymmetric (right) SPhP modes. (b) Spectral response of a gold (dielectric response taken from Ref. [39]) rectangular plasmonic antenna with length $L=3\text{μ}\mathrm{m}$, width $w=400$ nm, and height $h=25$ nm on top of a substrate of thickness $t=40$ nm with a constant dielectric response (characterized by a relative dielectric constant ${ϵ}_{\mathrm{sub}}=1.8$). Absorption (red line) cross section is calculated for excitation by a plane wave impinging at normal incidence along the $z$ axis with a linear polarization aligned along the long antenna’s axis ($x$ axis). EEL spectra are obtained for an electron beam placed 10 nm outside the antenna’s corner (solid green line) and at the antenna’s center (dashed green line), as shown in the schematics. The electron-beam energy is 60 keV in both (a) and (b). (c) The $z$ component of the total electric field emerging in the excitation of the ${\mathrm{Si}\mathrm{O}}_2$ film by focused electrons (trajectories shown by green arrows) at two energies around the peak in the EEL spectrum in (a). (d) The $z$ component of the electric field confirming the excitation of the dipolar plasmon by the plane wave (top red framed plot) and the electron beam placed close to the antenna side (middle green solid framed plot). When the electron beam passes close to the antenna center, it can only weakly excite a higher-order plasmon mode (bottom dashed green framed plot). All plots are obtained at energy $150$ meV, approximately corresponding to the peaks in (b), and are extracted at the central plane $y=0$ while the electron beam is passing in the plane $y_b=d/2+10$ nm (10 nm from the antenna’s shorter edge). In (c), (d), we show total field in the electron-beam excitation (green frames) and induced field for the plane-wave excitation (red frame). Real part of the field is shown in panels (c) while imaginary part is plotted in (d).

Figure 2

Comparison of electron and light spectra for an approximately optimally coupled antenna-substrate system (individual system components have the same geometry and dimensions as in Fig. 1). (a) Experimentally measured FTIR spectrum corresponding to $(1-T_{\mathrm{rel}})$, where $T_{\mathrm{rel}}$ is the transmission (orange line) for light polarized along the long antenna axis, divided by a reference transmission spectrum on a plain SiO₂ layer. Numerically calculated absorption (red) and scattering (blue) cross-section spectra for an entire system are shown for comparison. (b) Experimental versus calculated electron spectra (thick versus thin lines) obtained for the 60-keV beam, which is placed just next to the antenna and scanned along the antenna’s long axis. Colors approximately correspond to the electron positions as marked schematically in the inset. (c) Selected EEL spectra from (b) after subtraction of the (reference) spectrum recorded at the center of the antenna, which is dominated by uncoupled SPhPs and LO phonon excitation in ${\mathrm{Si}\mathrm{O}}_2$ [black lines in (b)]. For clarity, subsequent spectra in (b) and (c) are vertically shifted by a constant offset. The vertical dashed lines denote energies of TO and LO phonon modes in ${\mathrm{Si}\mathrm{O}}_2$. (d) The $z$ component of the electric field for plane-wave and electron excitation (red and blue framed plots, respectively) at different energies as denoted above. The electron beam is placed at the side of the antenna or close to its center (solid versus dashed framed plots; electron trajectories are represented by green lines). All plots are extracted at the central plane $y=0$ while the electron beam is passing in the plane $y_b=w/2+10$ nm (10 nm from the antenna’s shorter edge). We show total field in the electron-beam excitation (green frame) and induced field for the plane-wave excitation (red frames).

Figure 3

Coupled system response as a function of the plasmonic antenna length. (a),(b) Optical scattering and (c),(d) reference-subtracted EEL spectra (see Fig. 2 for details). Simulated pseudodispersions in (a) and (c) are obtained via the transformation $k_{\mathrm{LSP}}=\pi /L$, where $L$ is the antenna length and $k_{\mathrm{LSP}}$ is the effective wave vector of the dipolar LSP. Gray dashed lines trace energies of the uncoupled LSPs and PhPs, whereas the colored symbols denote energies of the new hybrid modes characterized by eigenvalues obtained from fitting of spectra and solution of $|\mathbf{M}|=0$. Examples of numerically calculated, fitted, and experimental spectra (thin, dashed, and thick curves, respectively) for three selected lengths (2.4, 3.0, 3.6 $\text{μ}\mathrm{m}$) are shown in (b) and (d) [denoted by vertical colored dashed lines in (a) and (c)]. All EEL spectra are obtained for a 60-keV electron beam placed close to the corner of the antenna [violet line in Fig. 2]. Experimental measurements of EELS and FTIR are performed on the same sample.

Figure 4

Coupling parameters compared to their damping extracted from the fitted optical spectra in Fig. 3 (fitting reference-subtracted EEL spectra provides values with a difference within about $1$ meV). The criterion $2g_i>{\gamma }_{\mathrm{LSP}}+{\gamma }_{{\mathrm{SPhP}}_i}$ establishes the strong coupling.

Figure 5

(a) Comparison of theoretically calculated absorption ($A$, red line), reflection ($R$, blue line), and $1-$ transmission ($1-T$, orange line) spectra for the plane-wave excitation and the geometry considered in Fig. 2. (b) Comparison of “relative” transmission spectra ($1-T_{\mathrm{rel}}$) obtained by dividing the total transmission spectra by a reference on a plain ${\mathrm{Si}\mathrm{O}}_2$ layer. We show calculated and experimentally measured FTIR spectra (green versus orange lines) together with the calculated scattering cross section (dark blue line).

Figure 6

Comparison of theoretically calculated EEL spectra after convolution with a Gaussian function of 14 meV FWHM (thin lines) with experimentally measured spectra (thick lines) for the varying beam position as shown in the inset and three antenna’s lengths.