Effect of spin-orbit interaction on the optical spectra of single-layer, double-layer, and bulk MoS${}_2$

We present converged ab initio calculations of the optical absorption spectra of single-layer, double-layer, and bulk MoS${}_2$. Both the quasiparticle-energy calculations (on the level of the GW approximation ) and the calculation of the absorption spectra (on the level of the Bethe-Salpeter equation) explicitly include spin-orbit coupling, using the full spinorial Kohn-Sham wave functions as input. Without excitonic effects, the absorption spectra would have the form of a step function, corresponding to the joint density of states of a parabolic band dispersion in two dimensions. This profile is deformed by a pronounced bound excitonic peak below the continuum onset. The peak is split by spin-orbit interaction in the case of single-layer and (mostly) by interlayer interaction in the case of double-layer and bulk MoS${}_2$. The resulting absorption spectra are thus very similar in the three cases, but the interpretation of the spectra is different. Differences in the spectra can be seen in the shape of the absorption spectra at 3 eV where the spectra of the single and double layers are dominated by a strongly bound exciton.

Figure 1

From left to right: Band structure of MoS${}_2$ single layer, double layer, and bulk in the LDA (thin dashed lines) and in the $G_0{W}_0$ approximation (thick continuous lines).

Figure 2

(a)–(c) Optical spectra for single-layer, double-layer, and bulk MoS${}_2$, obtained with the BSE (solid lines) and RPA (dashed lines). LDA and GW band gaps at $K$ are marked with vertical dashed lines, the absorption threshold is marked with a vertical solid line. Note that the wiggles above 2.5 eV in panel (c) are due to finite $\mathbf{k}$-point sampling. (d)–(f) Experimental absorption spectra (Refs. 3 and 4) in comparison to the calculations (solid lines, shifted by about $-0.2$ eV).

Figure 3

Exciton wave functions: (a) top view of bound exciton $A$ of single layer, (b) top view of bound exciton at 3.0 eV of single layer, (c) lateral view of the exciton $A$ of the single layer, (d) lateral view of the exciton $A$ in bulk. In all cases, the hole has been placed on the Mo atom in the center of the figure. Images were realized with the software XCrySDen (Ref. 38).

Figure 4

Left axis: Bethe-Salpeter spectra for several $\mathbf{k}$-point grids. Right axis: Exciton binding energy ($E_b$) as a function of the number of irreducible $\mathbf{k}$ points. Note the correspondence in the colors of the spectra and $E_b$.