Spectral Information of EG-2D Ag Plasmonic Nanostructures Revealed by Analyzing Trajectory of Hyperspectral sSNOM Mapping in Argand Space
Authors: Li Xinyi , Dong Chengye , Maniyara Rinu Abraham , Turker Furkan , Robinson Joshua , Rotkin Slava V. 
Published: August 2024

It is well known that optical microscopies, such as confocal Raman or PL spectroscopy, provide information regarding the optical properties of 2DM or modulation of the latter in their vdWHs, still, with a spatial resolution limited to the wavelength of excitation light. Since the size of the plasmonic nanostructures is usually order of magnitude smaller than the excitation wavelength, or even smaller, it is not feasible to directly map the field confinement with such optical microscopies. Scattering Scanning Near-Field Optical Microscopy, also known as “sSNOM”, is an AFM based imaging technique, which can not only directly probe the local optical surface impedance of the 2DMs or their vdWHs, but also capable of high-resolution mapping of optical response. This suggests it to be an ideal tool for visualizing the localized field of surface modes and associated charge distribution in plasmonic nanostructures, as well as for revealing other light-matter coupling phenomena at nanometer-scale resolution (~ 10 nm).
Unlike confocal Raman or PL spectroscopy, it is not straight-forward for sSNOM to obtain spectral information. It is accomplished by one of two approaches: either by analysis of a quantitative sSNOM data, such as a signal intensity or the periodicity of standing wave fringes, measured at series of excitation wavelength – i.e., via a hyperspectral mapping of the region of interest. In another approach (nano-FTIR) a broad-band excitation is utilized to take single-point spectra, repeated at every pixel over the region of interest. Both approaches proved to provide consistent results though correlation of the resulting spectra with specific materials parameters is a tough problem due to complex mechanisms of near-field response.
For either approach, proper referencing/normalization of the demodulated near-field signal has to be done before starting data analysis. This is due to the near-field signals are influenced by tuning of setup and may vary from wavelength to wavelength. Most commonly, near-field signal from the substrate (typically, Si or SiO2) or from the electrodes (Au or Pt) is used for referencing/normalization, anticipating that it either does not contribute to signal dispersion or the contribution is additive and can be subtracted by referencing. Such “trivial background” referencing procedure cannot be proved if the background signal itself is non-uniform vs. spatial coordinates and/or wavelength. Typical examples include: wave diffraction patterns, where local intensity of the pixel is a function of the excitation even in the absence of dispersion; or the case of the substrate’s “background” being a resonance strongly coupled to the sample signal.
Most success was demonstrated in spectroscopic near-field analysis of standing waves – the method which is self-calibrated without any reference. Indeed, the periodicity of a standing wave pattern (the distance between wave nodes and peaks) is independent of the (non-calibrated) intensity of the sSNOM signal itself. Plotting periodicity vs. excitation wavelength gives directly the dispersion relation of surface modes. Said that, in order to get the periodicity data, analysis should be performed over a region of several wavelengths (of the surface mode) to have a few distinct nodes and peaks, which undermines the idea of sub-diffractional imaging of nanomaterials. Furthermore, if the polariton wavelength is comparable with the size of plasmonic nanostructure one can barely observe a single wavelength period. If the size is even smaller, instead of forming interference fringe patterns, the polariton wave will be confined within or in the vicinity of the nanostructures and appear as “hot spot”, not a wave. The period and, therefore, dispersion relation cannot be derived. Similarly, in highly disordered samples, the sSNOM map shows a complicated diffraction pattern which contains only short fragments of the waves, much less than a full period, obstructing dispersion analysis as well.
To overcome this challenge, we developed an analytic approach for sSNOM hyperspectral mapping to reveal the local value of surface wave propagation constant and map it with substantially sub-wavelength resolution. It applies to the surface polaritons that are confined by the plasmonic nanostructures which have sub-diffraction size compared to the polariton wavelength, which is typically much shorter than the laser excitation wavelength. The proposed method is based on eikonal representation of the surface wave. By tracing the phase change of the wave and analyzing the trajectory of phasors (Argand or Nyquist plot of the sSNOM data) in complex space demodulated near-field signal allows natural referencing/normalization at each laser excitation wavelength. Thus, the hyperspectral sequence of the demodulated near-field signal yields the map of a propagation constant of surface polaritons, demonstrated below for (epitaxial graphene) EG-Ag plasmonic nanostructures with sub-wavelength sizes, lithographically fabricated on SiC substrate.