Inelastic electron scattering by intra- and interband plasmons in rhenium trioxide, tungsten trioxide, and some tungsten bronzes

Energy-loss spectra were measured by backscattering 500-eV electrons from films of W${\mathrm{O}}_3$ grown on single-crystal W(100), and from single crystals of ${\mathrm{Na}}_x$ W${\mathrm{O}}_3$ ($0.35<x<\mathrm{}0.86\mathrm{}$) and Re${\mathrm{O}}_3$ which were cleaved in ultrahigh vacuum. The spectrum of a crystal with $x=0.61\mathrm{}$ was nearly identical to the surface loss function $\mathrm{Im}[-{(\epsilon +1)}^{-1\mathrm{}}]$ calculated from $\epsilon$ measured by optical reflectivity on a crystal with $x=0.65\mathrm{}$. For $x=0.61\mathrm{}$, two plasmons are observed, ${\omega }_{-}=1.90\mathrm{}$ eV, and ${\omega }_{+}=6.5\mathrm{}$ eV. ${\omega }_{-}$ scales approximately as $x^{\frac{1}{2}}$ and is assigned to the conduction-electron resonance. The plasmon lifetime ${\tau }_p=1.9\mathrm{}\times {10}^{-15\mathrm{}}$ sec determined from the observed linewidth agrees well with a value for the conduction-electron relaxation time ${\tau }_e=2.1\mathrm{}\times {10}^{-15\mathrm{}}$ sec we obtained from previously reported reflectivity data, but is shorter than a value ${\tau }_c=6.7\mathrm{}\times {10}^{-15\mathrm{}}$ sec we estimated from previously reported dc conductivity data. The higher-energy plasmon ${\omega }_{+}$ is assigned to a screened longitudinal resonance of an interband excitation near 5 eV. ${\omega }_{+}$ is essentially independent of $x$ for $0\le x<\mathrm{}0.9\mathrm{}$. By fitting $\epsilon$ between 0 and 10 eV with a model dielectric function, we found a value for the mean effective conduction electron mass $m^{*}=0.80\mathrm{} m_e$, the interband contribution to the static dielectric constant ${\epsilon }_s=4.35\mathrm{}$, and values for parameters characterizing the interband resonance. Taking these latter parameters to be independent of $x$, we calculated the bulk and surface loss functions for different $x$ values. These calculations indicate that mode hybridization is unimportant for $x<\mathrm{}1\mathrm{}$ in the bronzes. The effects of damping or joint density-of-states width, screening, and mode interference on the plasmon spectra are discussed. It is shown that Re