Momentum- and frequency-resolved collective electronic excitations in solids: insights from spectroscopy and first-principles calculations

TL;DR

This paper surveys momentum- and frequency-resolved collective electronic excitations in solids, emphasizing a unified dielectric-response framework based on $\varepsilon({\mathbf q},\omega)$ and its inverse. It highlights spectral-band-structure (SBS) representations and the emerging multipole–Padé approximant (MPA($\mathbf q$)) framework as compact, ab initio tools to interpret plasmonic, excitonic, and mixed modes across metals, semiconductors, and low-dimensional systems. The review covers experimental probes (EELS and IXS), data reconstruction, and the quantitative linking of experiment with first-principles calculations (RPA, TD-DFT, GW+BSE), including best practices for reproducible analysis. Open challenges include improved post-processing, beyond-RPA physics, accurate linewidth extraction, and data-driven approaches to automate mode identification, all aiming toward predictive, material-specific dielectric-function analyses for next-generation spectroscopy.

Abstract

Collective electronic excitations, including plasmons, excitons, and intra- and interband transitions, play a central role in determining the dynamic screening, optical response, and energy transport properties of materials. Recent advances in momentum- and frequency-resolved spectroscopies, such as electron energy-loss spectroscopy (EELS) and inelastic x-ray scattering (IXS), together with progress in first-principles many-body perturbation theory (MBPT) calculations, now allow collective excitations to be mapped with considerable precision across the Brillouin zone. This topical review surveys current developments in the representation and interpretation of both experimental and theoretical dielectric-response spectra. Particular emphasis is placed on recent ways of representing spectral band structures (SBS) of the direct and inverse dielectric functions, such as analytical approaches based on multipole-Padé approximants in momentum and frequency (MPA($\q$)), which provide a combined band-like description of the dispersion of the main collective excitations. We discuss how features observed in metals, semiconductors, and low dimensional systems reflect the interplay between electronic structure, screening strength, and local-field effects, and how post-processing procedures can improve the quantitative comparison between experiment and theory. Finally, we provide perspectives on open challenges and potential developments in quantitative dielectric-function analyses.

Figures (4)

• Figure 1: Relation between first-principles calculations at different levels of the many-body theory and the observables measured with EELS/IXS experiments. Conceptual flow linking the microscopic inverse dielectric matrix in Eq. (12), the macroscopic dielectric function in Eq. (13), and the energy-loss function in Eq. (14). In Eq. (12), $\chi$ denotes the fully interacting (reducible) polarizability, and the dielectric matrix is restricted to the longitudinal response. Transverse electromagnetic effects are discussed separately in the context of polaritons.
• Figure 2: Examples of momentum and frequency dependent response functions of different materials. 1. Figure adapted from Ref. Alkauskas2013PRB (permission granted by license No. RNP/26/JAN/100835) corresponding to the loss function of bulk Cu obtained from REELS measurements. 2. Color-map representation of the bulk ZnO data from Ref. Leon2024ZnO (permission granted by license No. RNP/26/JAN/100838) corresponding to direct and inverse loss functions, $\mathrm{Im}[-\varepsilon^{-1}(\omega,{\mathbf q})]$ and $\mathrm{Im}[\varepsilon(\omega,{\mathbf q})]$, computed at the BSE@PBE$+U$ level theort and compared to EELS measurements along the $M \Gamma$ direction. 3. Figure adapted from Ref. Fugallo2015PRB (permission granted by license No. RNP/26/JAN/100836) corresponding to the dynamical structural factor of hexagonal BN monolayer computed with GW and BSE and compared with IXS measurements. The main spectral features at low energies are indicated for Cu and ZnO.
• Figure 3: Color-map representation of the energy- and momentum-dependent $\mathrm{Im}[\varepsilon({\mathbf q},\omega)]$ (a) and its inverse $\mathrm{Im}[\varepsilon^{-1}({\mathbf q},\omega)]$ (b) along a high-symmetry ${\mathbf q}$-path in the Brillouin zone, replotted from the RPA data of Ref. Leon2024ZnO (permission granted by license No. RNP/26/JAN/100838). Experimental absorption edges are included where available.
• Figure 4: (left panel) Comparison of the computed RPA loss function of Cu in the optical limit with EELS, REELS measurements, and reference IPA results. The data is taken from Ref. Leon2025metals. (middle panel) Spectral $Y (\omega, {\mathbf q})$ band structure of Cu computed from the numerical RPA results along the $X \Gamma N$${\mathbf q}$-path. (right panels) Spectral $Y ({\mathbf q}, \omega)$ band structure of Cu in the $\Gamma N$ direction reconstructed with MPA(${\mathbf q}$) with a number of $n_Y = 12$ poles.