Band Structures of Plasmonic Polarons

Using state-of-the-art many-body calculations based on the “$GW$ plus cumulant” approach, we show that electron-plasmon interactions lead to the emergence of plasmonic polaron bands in the band structures of common semiconductors. Using silicon and group IV transition-metal dichalcogenide monolayers ($A{X}_2$ with $A=\mathrm{Mo}$,W and $X=\mathrm{S}$, Se) as prototypical examples, we demonstrate that these new bands are a general feature of systems characterized by well-defined plasmon resonances. We find that the energy versus momentum dispersion relations of these plasmonic structures closely follow the standard valence bands, although they appear broadened and blueshifted by the plasmon energy. Based on our results, we identify general criteria for observing plasmonic polaron bands in the angle-resolved photoelectron spectra of solids.

Figure 1

Angle-resolved spectral function of silicon on a logarithmic scale for wave vectors along the $\mathrm{\Gamma }\text{−}X$ high-symmetry line, evaluated using (a) the Sternheimer-$GW$ method ($\mathrm{S}GW$) and (b) the $\mathrm{S}GW$ plus cumulant ($\mathrm{S}GW+C_{\mathrm{A}\mathrm{H}\mathrm{K}}$) approach. (c) Measured x-ray photoemission spectrum of silicon (XPS) from Ref. [18]. The blue dashed lines indicate the three features discussed in the main text.

Figure 2

(a) Complete angle-resolved spectral function of monolayer ${\mathrm{MoS}}_2$ on a logarithmic scale for wave vectors along the $\mathrm{\Gamma }\text{−}M$ high-symmetry line, evaluated using the $GW+C_{\mathrm{AHK}}$ approach. (b,d) Contributions of plasmonic polarons to the spectral function in (a): $\pi$-plasmonic polarons (b) and $(\pi +\sigma )$-plasmonic polarons (d). (c) Quasiparticle band structure of monolayer ${\mathrm{MoS}}_2$ extracted from (a) (red solid lines), band structure of $\pi$-plasmonic polarons extracted from (b) (blue solid line), and band structure of $(\pi +\sigma )$-plasmonic polarons from (d) (yellow solid line). Energies are referenced to the valence band top.

Figure 3

Energy range of the quasiparticle bands (blue boxes) and the $\pi$-plasmonic polaron bands in monolayer ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, evaluated within the $GW+{\mathit{C}}_{\mathrm{AHK}}$ approach (yellow boxes). Bulk ${\mathrm{MoS}}_2$ is included for comparison (${\mathrm{MoS}}_2[\mathrm{b}]$). The topmost bands arise from the hybridization of transition-metal $d$ states and the chalcogenide $p$ states ($\mathrm{TM}\text{−}d/\mathrm{Ch}\text{−}p$); the low-energy ones arise from the chalcogenide $s$ states ($\mathrm{Ch}\text{−}s$). The plasma energies ${\omega }_{\pi }$ are given in eV.