Advances in momentum-resolved EELS of phonons, excitons and plasmons in 2D materials and their heterostructures

TL;DR

The paper surveys momentum-resolved EELS (q-EELS) as a powerful, nanoscale probe of phonons, excitons, and plasmons in 2D materials and their heterostructures, anchored by the dielectric loss-function framework $\Gamma(q,\omega)=\mathrm{Im}[ -\epsilon^{-1}(q,\omega)]$ and the observable $I(\omega,q) \propto \frac{1}{q^{2}}\Gamma(q,\omega)$. It contrasts acquisition strategies (serial q-EELS with circular apertures versus $\omega$-$q$ mapping with rectangular slits) and discusses fundamental physics (the dual pump–probe role of the electron, nonlocal screening, and surface/relativistic contributions) and resolution limits ($\Delta q$ tied to convergence/collection angles) that shape data interpretation. The review then highlights key applications to plasmons, excitons, and phonons in 2D systems (including graphene, TMDCs, and hBN), emphasizing dispersions, anisotropies, exciton–polaritons, surface plasmon effects, and phonon polaritons, along with twist-angle–dependent phenomena in heterostructures. Finally, it outlines challenges and opportunities—signal limitations, quantification, cryogenic and in situ capabilities, tomographic/reciprocal-space mapping, and AI-driven data analysis—that are poised to expand q-EELS into a broadly applicable, multi-modal tool for uncovering emergent physics in low-dimensional materials.

Abstract

Functional nanomaterials, including 2D materials and their heterostructures are expected to impact fields ranging from catalysis, optoelectronics to nanophotonics. To realize their potential, novel experimental approaches need to be developed to characterize the combined materials and their components. Techniques using fast electrons, such as electron energy-loss spectroscopy (EELS), probe phenomena over an unrivaled energy range with high resolution. In addition, momentum-resolved EELS simultaneously records energy and momentum transfer to the sample and thus generates two-dimensional data sets for each beam position. This allows excitations that occur at large momentum transfer to be resolved, including those outside of the light cone and beyond the first Brillouin zone, all whilst retaining nanometer sized spatial selectivity. Such capabilities are particularly important when probing phonons, plasmons, excitons and their coupling in 2D materials and their heterostructures.

Figures (10)

• Figure 1: Overview of phenomena that are accessible to probe in the electron microscope using momentum-resolved EELS, including excitons in hexagonal boron nitride (hBN) and transition metal dichalcogenides (TMDCs), plasmons, phonons in graphene (G) and hBN as well as in TMDCs, and magnons. The vertical axis, “Detectability requirement”, denotes the relative difficulty of detecting each excitation using EELS. Smaller cross sections and/or weaker signals lead to a higher “Detectability requirement”.
• Figure 2: Setup of the electron microscope for q-EELS. a) Schematic showing the slit aperture technique (denoted here with EELS aperture), including the annular dark field (ADF) detector, the focusing quadrupole, and the magnetic prism in the spectrometer. b) A slit aperture is aligned to high symmetry directions, e.g. along $\Gamma \rightarrow K$ and $\Gamma \rightarrow M$ to generate energy-momentum ($\omega q$) maps. c) A circular aperture can also be used, referred to as serial q-EELS, to successively acquire q-EEL spectra to map out the loss spectra at specific momentum transfers.
• Figure 3: (a) Scattering geometry in momentum space. (b) Diffraction spot radius $\Delta q_\alpha$ as a function of convergence semi-angle $\alpha$, showing the intrinsic trade-off between momentum and spatial resolutions. Dashed lines indicate fractions of the Brillouin zone (BZ) of hBN along $\Gamma \rightarrow K$. (c) Momentum resolution specified as $q_{\beta^*} = (2\pi / \lambda) \sin(\beta^*)$ with $\beta^* = \sqrt{\beta^2 + \alpha^2}$. Examples of collection geometries from the literature are inserted from Hage et al.hage_nanoscale_2018, Senga et al.senga_position_2019, and Qi et al.qi_four-dimensional_2021. Note that practical resolution can be further limited by probe broadening and other instrumental factors.
• Figure 4: Delocalization length $d_{50}$ as a function of energy loss $\Delta E$ for two collection semi-angles, $\beta=3$ and $10$ mrad egerton2017scattering. Smaller collection angles (q-EELS regime) cut off localized scattering at lower energy losses, while larger angles (common STEM-EELS regime) extend the range. This illustrates how spatial resolution depends strongly on geometry. In practice, small, angle-limiting apertures in q-EELS can be used to traverse momentum space by isolating signal from delocalized small-angle scattering to more localized high-angle scattering.
• Figure 5: a) Experimentally measured dielectric function of WSe$_2$ in the optical regime li_dielectric_function_TMDCs_2014. b) Loss function including surface losses and relativistic effects computed from the dielectric function shown in (a) for different momentum transfers according to erni_relativistic_eels_2008. Each spectrum is assumed to be collected for an annular aperture spanning the angular range specified in the legend.