Valley plasmonics in transition metal dichalcogenides

The rich phenomenology of plasmonic excitations in the dichalcogenides is analyzed as a function of doping. The many-body polarization, the dielectric response function, and electron energy loss spectra are calculated using an ab initio based model involving material-realistic Coulomb interactions, band structure, and spin-orbit coupling. Focusing on the representative case of ${\mathrm{MoS}}_2$, a plethora of plasmon bands are observed, originating from scattering processes within and between the conduction or valence band valleys. We discuss the resulting square-root and linear collective modes, arising from long-range versus short-range screening of the Coulomb potential. We show that the multiorbital nature of the bands and spin-orbit coupling strongly affects intervalley scattering processes by gapping certain two-particle modes at large momentum transfer.

Figure 1

Sketch of the Fermi surfaces in hole- (left) and electron-doped (right) monolayer ${\mathrm{MoS}}_2$. The predominant orbital-character weight of each surface is indicated by red ($d_{z^2}$) and blue ($d_{xy}$ and $d_{x^2-y^2}$) fillings. Points of high symmetry are indicated by different markers.

Figure 2

Real and imaginary parts of the polarization functions ($d_{xy}/d_{xy}$ channel) and EELS spectra for hole-doped ${\mathrm{MoS}}_2$ without (top row) and with (bottom row) spin-orbit coupling. The insets in (a) and (d) illustrate the Fermi surface pockets around $K$ and $K^{\prime }$.

Figure 3

Real parts of the polarization functions for $d_{z^2}/d_{z^2}$ scattering at low and high electron doping concentration without SOC. The insets illustrate the Fermi surfaces.

Figure 4

Real parts of the polarization functions for $d_{xy}/d_{xy}$ scattering at elevated electron doping concentration ($K$ and $\mathrm{\Sigma }$ valleys are partially occupied) without and with SOC. The insets illustrate the Fermi surfaces.

Figure 5

Band structure of ${\mathrm{MoS}}_2$ with spin-orbit coupling (top). The two lowest conduction bands and highest valence band are shown. The doping levels discussed in the main text are shown with horizontal lines: green (“hole doping”) $E_f=-0.15\mathrm{eV}$, blue (“low electron doping”) $E_f=2.52\mathrm{eV}$, and cyan (“high electron doping”) $E_f=2.67\mathrm{eV}$. The lower panel shows a sketch of the splitting due to SOC at $K$ and $K^{\prime }$ in the valence band and at $\mathrm{\Sigma }$ and ${\mathrm{\Sigma }}^{\prime }$ in the lowest conduction band.

Figure 6

(a) Plasmon dispersions from our ab initio based model (dots) in comparison with analytic $\mathbf{k}·\mathbf{p}$ models by Scholz et al. (blue) [30] and Kechedzhi and Abergel (red) [31]. In the data by Kechedzhi and Abergel both spin components and their coupling are included, while the data of Scholz et al. includes a single spin component only. Data shown as red (blue) dots results from the ab initio model by using a simple constantly screened Coulomb interaction and the full (spin resolved) polarization function. The data shown as green dots arise from the complete ab initio model. (b) Plasmon dispersions for undoped ${\mathrm{NbS}}_2$ from (right) full ab initio calculations from Ref. [43] and (left) EELS data obtained from our ab initio based model.