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

Abstract: 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.