High-energy collective electronic excitations in layered transition-metal dichalcogenides

We characterize experimentally and theoretically the collective electronic excitations in two prototypical layered transition-metal dichalcogenides, ${\mathrm{NbSe}}_2$ and ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$. The energy- and momentum-dependent dynamical structure factor was measured by inelastic x-ray scattering (IXS) spectroscopy and simulated by time-dependent density-functional theory. We find good agreement between theory and experiment, provided that Nb semicore states are taken into account together with crystal local-field effects. Both materials have very similar spectra, characterized by two main plasmons at 9 and 23 eV, which we show to both have $\pi +\sigma$ character on the basis of a detailed analysis of the band structure. Finally, we discuss the role of the layer anisotropy in the dispersion of these plasmons.

Figure 1

Comparison between experimental (bold line) and theoretical (thin line) spectra of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ (a) and ${\mathrm{NbSe}}_2$ (b) evaluated at $\mathbf{Q}$ (in multiples of $2\pi /c$) along the $c$ axis with a broadening of 1.0 eV.

Figure 2

Comparison between the theoretical spectra (loss function) of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ at $\mathbf{Q}=0.77{\AA }^{-1}$ along the $c$ axis evaluated in different approximations. In this calculations we have used a broadening of 0.1 eV.

Figure 3

(a) Imaginary and (b) real part of the dielectric function and (c) loss function of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ at $\mathbf{Q}=0.77{\AA }^{-1}$ along the $c$ axis evaluated in RPA-NLF with and without semicore states with a broadening of 0.1 eV.

Figure 4

RPA loss function of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ evaluated at $\mathbf{Q}$ (in ${\AA }^{-1}$ units) along the $c$ axis with a broadening of 0.1 eV. The dashed red line is a guide for the eye.

Figure 5

Band structure and projected density of states of ${\mathrm{NbS}}_2$. The zero of the vertical axis indicates the Fermi level.

Figure 6

Same as Fig. 5 for ${\mathrm{NbSe}}_2$.

Figure 7

Real and imaginary part of the macroscopic dielectric function, ${\epsilon }_M={\epsilon }_1+i{\epsilon }_2$, of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ evaluated at $\mathbf{Q}=0.09{\AA }^{-1}$ with a broadening of 0.1 eV (a) neglecting and (b) taking into account LFE.

Figure 8

Comparison between the RPA spectra of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ evaluated for in-plane and out-of-plane momentum transfer $\mathbf{Q}=0.09{\AA }^{-1}$ with a broadening of 0.1 eV.

Figure 9

RPA spectra of ${\mathrm{Cu}}_{0.2}{\mathrm{NbS}}_2$ evaluated for in-plane $\mathbf{Q}$ (in ${\AA }^{-1}$ units) with a broadening of 0.1 eV. The dashed red line is a guide for the eye.