Comparative analysis of plasmon modes in layered Lindhard metals and strange metals

The enigmatic strange metal remains one of the central unsolved problems of 21st century science. Understanding this phase of matter requires knowledge of the momentum- and energy-resolved dynamic charge susceptibility, $\chi(q,\omega)$, especially at finite momentum. Inelastic electron scattering (EELS), performed in either transmission (T-EELS) or reflection (R-EELS) geometries, is a powerful probe of $\chi(q,\omega)$. For the prototypical strange metal Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$, T-EELS, R-EELS, and infrared (IR) spectroscopy agree at $q \sim 0$, all revealing a highly damped plasmon near 1 eV. At larger $q$, however, EELS results show unresolved discrepancies. Since IR data are highly reproducible, it is advantageous to use IR data to calculate what the expected EELS response should be at modest $q$. Building on prior R-EELS work [J. Chen \textit{et al.}, Phys. Rev. B. \textbf{109}, 045108 (2024)], we extend this approach to T-EELS for finite stacks of metallic layers, comparing a "textbook" Lindhard metal to a strange metal. In the Lindhard case, the low-$q$ response is dominated by long-lived, standing wave plasmon modes arising from interlayer Coulomb coupling, with in-plane dispersions that resemble the well-known Fetter modes of layered metals. This behavior depends only on the geometry and the long-ranged nature of the Coulomb interaction, and is largely insensitive to layer details. At larger $q$, the response reflects the microscopic properties of individual layers. For the strange metal, calculations based on IR data predict a highly damped plasmon with weak dispersion and no distinct surface mode. While our results match IR and R-EELS at low $q$, they do not reproduce any published EELS spectra at large $q$, highlighting unresolved discrepancies that demand further experimental investigation.