Electronic structure, spin-orbit coupling, and interlayer interaction in bulk ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$

We present in-depth measurements of the electronic band structure of the transition-metal dichalcogenides (TMDs) ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$ using angle-resolved photoemission spectroscopy, with focus on the energy splittings in their valence bands at the $K$ point of the Brillouin zone. Experimental results are interpreted in terms of our parallel first-principles computations. We find that interlayer interaction only weakly contributes to the splitting in bulk ${\mathrm{WS}}_2$, resolving previous debates on its relative strength. We additionally find that across a range of TMDs, the band gap generally decreases with increasing magnitude of the valence-band splitting, molecular mass, or ratio of the out-of-plane to in-plane lattice constant. Our results provide an important reference for future studies of electronic properties of ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$ and their applications in spintronics and valleytronics devices.

Figure 1

Constant-energy maps of the electronic band structure of bulk (a) ${\mathrm{MoS}}_2$ and (b) ${\mathrm{WS}}_2$ taken at photon energies of 50 and 90 eV, respectively. The maps were taken at and below the energy corresponding to their respective valence-band maximum (VBM) at $\overline{K}$. The dashed white line indicates the two-dimensional (reduced in the $k_z$ dimension, as shown in the inset) hexagonal Brillouin zone. The evolution of the valence band with an increase in binding energy about the $\overline{\mathrm{\Gamma }},\overline{K}$, and $\overline{K^{\prime }}$ points can be observed for both samples. An energy splitting at the $\overline{K}$ and $\overline{K^{\prime }}$ points can be observed in the ${\mathrm{MoS}}_2$ maps, most easily at 170 or 260 meV below the VBM (white arrows). The splitting for ${\mathrm{WS}}_2$ is not observed 170 meV below the VBM but can be observed 540 meV below the VBM (white arrows). Trigonal warping of the valence band about the $\overline{K}$ and $\overline{K^{\prime }}$ points can be observed for both samples.

Figure 2

(a) Photon energy dependence from 52 to 120 eV and theoretical calculations of the electronic band structure of bulk ${\mathrm{WS}}_2$ along the $\overline{\mathrm{\Gamma }}$–$\overline{K}$ high-symmetry direction. The horizontal red (yellow) dashed lines correspond to the calculated location of bulk ${\mathrm{WS}}_2$ bands at the $\mathrm{\Gamma }$ point $(A$ point). Observed intensity shifts of the valence band near the $\overline{\mathrm{\Gamma }}$ point at different binding energies are evidence of strong $k_z$ dispersion. (b) Energy distribution curves (EDCs) (solid black lines) taken along $k_x=k_y=0{\text{Å}}^{-1}$ of the respective maps in (a) (e.g., along the white dashed line in the 52 eV map). The vertical red (yellow) dashed lines correspond to the calculated location of bulk ${\mathrm{WS}}_2$ bands at the $\mathrm{\Gamma }$ point $(A$ point). The peak in intensity (denoted by the tick marks) shifts periodically between higher and lower binding energies as a result of the strong $k_z$ dispersion. Peaks in the 70 and 110 eV EDCs as well as the top of the feature in the 90 eV EDC correspond to the theoretical band maximum at $A$.

Figure 3

High-resolution maps of the electronic band structure of bulk (a) ${\mathrm{MoS}}_2$ and (b) ${\mathrm{WS}}_2$ along the $\overline{\mathrm{\Gamma }}$–$\overline{K}$ high-symmetry direction with theoretical calculations overlaid as dashed blue lines. Spin-orbit coupling and interlayer-interaction-induced splitting in the valence band near the $\overline{K}$ point are fully resolved for both samples, as indicated by the white arrows and the respective EDCs (upper insets) taken along the upper white dashed lines. (Here the ${\mathrm{WS}}_2$ EDC was integrated in momentum between the two white tick marks.) The splitting is found to be approximately $170\pm 2$ meV for ${\mathrm{MoS}}_2$ and $425\pm 18$ meV for ${\mathrm{WS}}_2$. An additional splitting in a higher-binding-energy ${\mathrm{WS}}_2$ valence band is observed within the white dashed rectangle in (b). The lower inset presents an EDC taken along the vertical white dashed line corresponding to the energy and momentum location of the maximum splitting of $196\pm 22$ meV. (Here the EDC was integrated in momentum between the two white tick marks and smoothed with a Savitzky-Golay filter.) The calculated band structure is found to be in excellent agreement with the experimental band structure for both samples. The discrepancy in the location of the valence band near the $\overline{\mathrm{\Gamma }}$ point for bulk ${\mathrm{MoS}}_2$ can be attributed to its strong $k_z$ dispersion.

Figure 4

(a) Magnitude of valence-band splitting at the $\overline{K}$ point for bulk ${\mathrm{MoS}}_2$ [11, 31], ${\mathrm{MoSe}}_2$ [31, 34], ${\mathrm{MoTe}}_2$ [31, 35], ${\mathrm{WS}}_2$ [8], and ${\mathrm{WSe}}_2$ [36, 37]. (b) Magnitude of the splitting vs band gap $E_G$ for bulk ${\mathrm{MoS}}_2$ [31, 38, 39, 40], ${\mathrm{MoSe}}_2$ [31, 38, 39, 40, 41], ${\mathrm{MoTe}}_2$ [31, 40], ${\mathrm{WS}}_2$ [38, 39, 42], and ${\mathrm{WSe}}_2$ [38, 39, 41, 43]. Ratio of out-of-plane to in-plane lattice constant $\frac{c}{a}$ [31, 44] vs (c) magnitude of valence-band splitting and (d) band gap. The dashed lines are guides to the eye indicating the different trends for compounds containing molybdenum (red) and those containing tungsten (blue). The trends are further subdivided between experimental (circles) and theoretical (triangles) data. For all plots, solid (open) symbols represent data from this (other) work(s).