Optical and electronic properties of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ under pressure: Revealing the spin-polarized nature of bulk electronic bands

Monolayers of transition-metal dichalcogenide semiconductors present spin-valley locked electronic bands, a property with applications in valleytronics and spintronics that is usually believed to be absent in their centrosymmetric (as the bilayer or bulk) counterparts. Here we show that bulk $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ hides a spin-polarized nature of states determining its direct band gap, with the spin sequence of valence and conduction bands expected for its single layer. This relevant finding is attained by investigating the behavior of the binding energy of $A$ and $B$ excitons under high pressure, by means of absorption measurements and density-functional-theory calculations. These results raise an unusual situation in which bright and dark exciton degeneracy is naturally broken in a centrosymmetric material. Additionally, the phonon-assisted scattering process of excitons has been studied by analyzing the pressure dependence of the linewidth of discrete excitons observed at the absorption coefficient edge of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$. Also, the pressure dependence of the indirect optical transitions of bulk $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ has been analyzed by absorption measurements and density-functional-theory calculations. These results reflect a progressive closure of the indirect band gap as pressure increases, indicating that metallization of bulk $\mathrm{Mo}{\mathrm{S}}_2$ may occur at pressures higher than 26 GPa.

Figure 1

(a) Fundamental absorption edge of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ measured under different applied hydrostatic pressures. Evolution with pressure of the identified $A,B$, and $C$ features has been indicated by small solid bars. The inset shows an example of the result of fitting Eq. (2) to the absorption coefficient measured at the selected pressure of 5.07 GPa (red solid line). Each one of the contributions to the fitting curve has been included in the inset (blue solid lines), these corresponding to transitions involving states from the highest (I) and the next (II) valence bands at the $K$ point, as well as these attributable to transitions located around the midpoint of the Γ$K$ high-symmetry direction (III). (b) Pressure dependence of the energies of the $A$ and $B$ band-to-band direct transitions $\left(E_{0\mathrm{A};\mathrm{B}}\right)$ and the ground $A_1$ and $B_1$ excitonic transitions $\left(E_{1A;B}\right)$, respectively, as extracted from the fitting of Eq. (2) to the experimental results showed in (a). Black and grey symbols represent values obtained in experiments in which hydrostatic pressure increases or decreases, respectively. Error intervals, as extracted from the fitting of Eq. (2) to the experimental data in Fig. 1, are smaller than the dimensions of the data points. The pressure dependence of the $A$ and $B$ band-to-band direct transitions, those obtained by DFT calculations, have been included. (c) Pressure dependence of the total band splitting $\mathrm{\Delta }=E_{0\mathrm{B}}-E_{0\mathrm{A}}$ at the $K$ point of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, as obtained from the experimental results collected in (b) (grey and black symbols). Error intervals have been propagated from data shown in (b). Open circles represent the pressure dependence of ${\mathrm{\Delta }}_{\mathrm{v}}$, as obtained by DFT calculations. Since, a priori, optical direct transitions may occur from the two highest valence bands to each one of the two lowest conduction bands at the $K$ point, the pressure dependence of Δ may be expected to evolve between ${\mathrm{\Delta }}_{\mathrm{v}}+\left|{\mathrm{\Delta }}_{\mathrm{c}}\right|$ [as would correspond to the $\mathrm{Mo}{\mathrm{S}}_2$-like configuration illustrated in Fig. 3] and ${\mathrm{\Delta }}_{\mathrm{v}}-\left|{\mathrm{\Delta }}_{\mathrm{c}}\right|$ [i.e., as the $\mathrm{WS}{\mathrm{e}}_2$-like configuration illustrated in Fig. 3]. The range of Δ comprised between these two cases has been indicated by a light-red area and vertical errorlike bars, as obtained by DFT. For comparison purposes, the pressure dependence of the difference $E_{1\mathrm{B}}-E_{1\mathrm{A}}$ has been included in this plot, as extracted from the data shown in (b). Dashed lines are linear fits to the experimental data.

Figure 2

(a) Electronic band structure of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ calculated by DFT along the Γ$K$ and the KH high-symmetry directions of the Brillouin zone (also shown). Calculations have been performed at ambient conditions (left), 4 GPa (center) and 8 GPa (right). The top and bottom panels show zooms of the corresponding electronic conduction and valence bands located in the vicinity of the direct band gap of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, respectively. (b) Illustration of the spin-polarized nature of the electronic bands at the $K(K$′) point of each single layer composing the unit cell of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ (left side). The characteristics of these bands remain basically unaltered when two layers stack together due to their strongly localized nature, giving rise to energy and spin degenerate bands in the centrosymmetric material that retain spin information from layer of provenience (right side).

Figure 3

(a) Schematic description of the two options of allowed $A$ and $B$ direct optical transitions, those following a $\mathrm{WS}{\mathrm{e}}_2$-like or a $\mathrm{Mo}{\mathrm{S}}_2$-like spin-order configuration of a ML. (b) Pressure dependence of the BE of ground $A$ and $B$ excitons of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, as extracted from the results shown in Fig. 1 by using that $\mathrm{B}{\mathrm{E}}_{1\mathrm{A};\mathrm{B}}=E_{0\mathrm{A};\mathrm{B}}-E_{1\mathrm{A};\mathrm{B}}$ (black and grey circles). Error intervals have been propagated from data shown in Fig. 1. The pressure dependence of the BE of the ground $A$ and $B$ excitons, as calculated in the frame of the Gerlach-Pollmann model, has been included. These calculations have been performed for each one of the two possible sets of optical transitions that have been considered in (a). The color code used to plot the results of these calculations is the same as that used for the background of illustrations shown in (a), i.e., orange for $\mathrm{WS}{\mathrm{e}}_2$-like excitons and green for $\mathrm{Mo}{\mathrm{S}}_2$-like excitons. (c) Proposed scheme of the bright and dark distribution of $A$ and $B$ excitons in $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, as obtained from the results shown in (a). (d) Illustration of the different radiative properties of eventual single-photon emitters in 2D TMDs based on W and Mo transition metals, provided that spin information of extended states remains unaltered when giving rise to impurity-related localized states.

Figure 4

Pressure dependence of the ${\mathrm{\Gamma }}_1$ half width of the first excitonic peaks $(A_1$ and $B_1$ excitons) of the fundamental absorption edge of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$. Dashed lines are fitting curves obtained by means of Eq. (6). Although these curves have been plotted in the whole 0–8-GPa range, the fitting processes have been performed in the range of pressure comprised between 0 and 6.5 GPa $(A_1$ exciton) and 0–4 GPa $(B_1$ exciton), to conduct further discussion.

Figure 5

Pressure dependence of high-energy excitonic transitions giving rise to the $C$-labeled feature identified in the curves showed in Fig. 1. Error intervals have been obtained by fitting Eq. (2) to the experimental data in Fig. 1. The pressure dependence of the band gap at the $Q$ point has been included, as obtained from DFT band-structure calculations [Fig. 2]. Solid lines are linear fitting curves.

Figure 6

(a) Square root of the absorption edge measured in $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ samples 11 μm thick (bottom spectra) and 1.2 μm thick (top spectra) for pressures ranging from ambient conditions up to 26.1 GPa. Solid lines are linear fittings of ${\alpha }^{1/2}$ and extrapolated to $\alpha =0$ to estimate the indirect band gap at each pressure. (b) Pressure dependence of the indirect band gap of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, as extracted from (a) (black circles) and as obtained by DFT calculations (open circles). Solid lines are linear fitting curves.

Figure 7

Electronic band structure of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ calculated by DFT along the Γ$K$ and the KH high-symmetry directions at 12 GPa (left), 17 GPa (center), and 22 GPa (right). Note the progressive closure of the indirect band gap as pressure increases, indicated by red dashed lines.

Figure 8

(a) Pressure dependence of the effective mass of each one (see inset) of the two valence (VA and VB) and conduction bands $(C1$ and $C2$) of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ in the vicinity of the $K$ point, as obtained by parabolic fitting of their respective band dispersion calculated by DFT along the Γ$K\left(m_{\perp }^{*}\right)$ and KH $\left(m_{\parallel }^{*}\right)$ high-symmetry directions. The $m_{\parallel }^{*}$ of the $C2$ conduction band is not shown since it exceeds, by more than one order of magnitude, the scale range comprised by this plot, for pressures lower than 8 GPa. (b),(c) Pressure dependence of the exciton reduced mass obtained for each one of the optical transitions that may occur between the $A$ and $B$ valence bands and the $C1$ and $C2$ conduction bands, respectively. For the $B$ transitions, ${\mu }_{\mathrm{B}\parallel }$ values are much higher than these of ${\mu }_{\mathrm{B}\perp }$, justifying their 2D nature.

Figure 9

(a) Dispersion of the valence and conduction bands of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ in the vicinity of the $K$ point along the Γ$K$ and KH high-symmetry directions, as calculated by DFT. Grey shaded area represents the band gap. (b) Localization degree of states of the each one of the bands shown in (a) within a ML of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, as obtained by DFT calculations performed along the ΓKH directions. (c) Spin projection in the $z$ direction (i.e., perpendicular to the layers) calculated for conduction- and valence-band states located along the ΓKH directions. This spin projection has been calculated for one of the two MLs of the unit cell of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$. (d) Momentum dependence, along the ΓKH directions, of the spin-layer polarization within a particular ML of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, as obtained by DFT. In all plots, bands have been labeled as in the inset of Fig. 8. Vertical dashed lines indicate the position of the $K$ point.

Figure 10

Calculated real and imaginary parts of the dielectric function in the parallel (a) and perpendicular (b) to the $c$-axis directions, as obtained by DFT calculations in the independent particle approximation for bulk $\mathrm{Mo}{\mathrm{S}}_2$ under ambient conditions and under pressures of 4 and 8 GPa.

Figure 11

Pressure dependence of ${\varepsilon }_{\perp }\left(0\right)$ and ${\varepsilon }_{\parallel }\left(0\right)$, as obtained by DFT calculations in the local-field approximation.

Figure 12

Dispersion of the real part of the $n_{\perp }\left(\omega \right)$ of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ measured under different applied pressures. Solid lines are fitting curves to the experimental $n_{\perp }\left(\omega \right)$ dispersion curves as obtained by using Eq. (D1). The inset shows the pressure dependence of ${\varepsilon }_{\perp }\left(0\right)$ as obtained from these fittings. The pressure dependence of the ${\varepsilon }_{\perp }\left(0\right)$ obtained by DFT has been also included in the inset, for comparison purposes.