Layer-dependent anisotropic electronic structure of freestanding quasi-two-dimensional $\mathrm{Mo}{\mathrm{S}}_2$

The anisotropy of the electronic transition is a well-known characteristic of low-dimensional transition-metal dichalcogenides, but their layer-thickness dependence has not been properly investigated experimentally until now. Yet, it not only determines the optical properties of these low-dimensional materials, but also holds the key in revealing the underlying character of the electronic states involved. Here we used both angle-resolved electron energy-loss spectroscopy and spectral analysis of angle-integrated spectra to study the evolution of the anisotropic electronic transition involving the low-energy valence electrons in the freestanding $\mathrm{Mo}{\mathrm{S}}_2$ layers with different thicknesses. We are able to demonstrate that the well-known direct gap at 1.8 eV is only excited by the in-plane polarized field while the out-of-plane polarized optical gap is 2.4 ± 0.2 eV in monolayer $\mathrm{Mo}{\mathrm{S}}_2$. This contrasts with the much smaller anisotropic response found for the indirect gap in the few-layer $\mathrm{Mo}{\mathrm{S}}_2$ systems. In addition, we determined that the joint density of states associated with the indirect gap transition in the multilayer systems and the corresponding indirect transition in the monolayer case has a characteristic three-dimensional-like character. We attribute this to the soft-edge behavior of the confining potential and it is an important factor when considering the dynamical screening of the electric field at the relevant excitation energies. Our result provides a logical explanation for the large sensitivity of the indirect transition to thickness variation compared with that for the direct transition, in terms of quantum confinement effect.

Figure 1

The experimental setup and the resulting momentum-dependent spectra. (a) Inelastic electron scattering geometry in analogy with polarized optical measuring. The samples in the TEM are atomically thin $\mathrm{Mo}{\mathrm{S}}_2$ layers. EELS spectra with in-plane polarization $\left(\vec{\mathrm{q}}\perp c\right)$ dominate the larger-scattering-angle region (blue) and resemble the well-known optical excitation at normal incidence (right inset) onto atomic layers. Whereas, the out-of-plane polarized $\left(\vec{\mathrm{q}}//c\right)$ spectra reside in the small-scattering-angle region (red) in the momentum space, similar to the unexplored grazing-incidence case of optical wave (left inset). (b) Corresponding momentum-dependent spectra extracted from the $q\text{−}E$ diagram with $q_y$ along the Γ$M$ direction in (a).

Figure 2

(a) Angle-resolved low-loss spectra of monolayer $\mathrm{Mo}{\mathrm{S}}_2$ with $q_y$ in the Γ$K$ direction. The unit of $q_y$ is ${\AA }^{-1}$. The weak peak at 2.0 eV is due to the direct transition of $A,B$ excitons, and the strong peaks at 3.1 eV (α) and 4.5 eV (β) result from strong interband transitions (van Hove critical points). (b) Spectra of multilayer $\mathrm{Mo}{\mathrm{S}}_2$ with $q_y$ along the Γ$K$ direction. The δ absorption peak arises as $q_y$ increases.

Figure 3

Momentum-dependent spectra of monolayer $\mathrm{Mo}{\mathrm{S}}_2$ with $q_y$ along the Γ$M$ direction. The α peak still splits and an additional δ absorption peak appears as $q_y$ increases.

Figure 4

The fine structures of angle-integrated low-loss spectra revealing interband transitions in $\mathrm{Mo}{\mathrm{S}}_2$ with different thicknesses.

Figure 5

The relative change of $A,B$, α, and β peak (at 2, 3.1 and 4.5 eV) intensities with momentum transfer $q_y$. Peak intensity at $q_y$ is normalized by its counterpart at $q_y=0$. The experimental data is the case between the pure in-plane mode and the pure out-of-plane mode predicted by the calculation, illustrating that each spectrum should be a linear superposition of these two components.

Figure 6

(a), (b) The possible in-plane and out-of-plane spectral components in monolayer $\mathrm{Mo}{\mathrm{S}}_2$, based on the subtraction of two momentum-resolved spectra using suitable subtraction factor $c$. The optimal subtraction factor marked by red circles $(c=0.7$ for the in-plane component and 1.1 for the out-of-plane component) is based on the presence or absence of the physically realistic δ feature unique to the in-plane component. It is found that α peaks are split into α′, α″ in the in-plane component, while it is a single well-behaved peak in the out-of-plane component. Note that inappropriate subtraction will result in unphysically sharp spectral variation (pit) [29] in the difference spectrum, as shown by the arrows. (c), (d) The in-plane and out-of-plane spectra in multilayer $\mathrm{Mo}{\mathrm{S}}_2$, extracted from the original EELS data based on the same subtraction procedure.

Figure 7

Polarization dependence of near-gap EELS in atomically thin $\mathrm{Mo}{\mathrm{S}}_2$. (a) Absorption spectra of monolayer and multilayer $\mathrm{Mo}{\mathrm{S}}_2$ at in-plane $\left(\vec{\mathrm{q}}\perp c\right)$ and out-of-plane $\left(\vec{\mathrm{q}}//c\right)$ polarization. The black star (★) indicates the threshold energy (2.4 ± 0.2 eV) in out-of-plane polarization which is inaccessible by optical measurements. (b) Dispersion effect of the peak energy (peak position) of major absorption peaks in Fig. 2. All lines are drawn as a visual aid to the trend seen in the data.

Figure 8

Dimensionality analysis and quantum confinement effect of electronic transitions. (a) A closer look at band edge transition. Nonlinear fittings of near-gap fine structures give the indirect transition energy (Γ→$Q$ transition). (b) Schematic illustration of the band structure of monolayer $\mathrm{Mo}{\mathrm{S}}_2$ with global conduction band minimum (CBM) and valence band maximum (VBM) both at the $K$ point. The second VBM is located at the Γ point and the second CBM at the $Q$ point (almost midpoint of straight line Γ$K)$. For the sake of brevity, only one $Q$ point paraboloid is drawn (altogether six). The false color disks are the projection of parabolic dispersion to describe the positions of local VBM or CBM. (c) A summarized electronic transition energy between this work and other reports [13, 14, 34, 39]. Note the direct transition energies of different layers are extracted from the peak energy of $A,B$ exciton transitions. The black dashed circle highlights the indirect transition energy in the monolayer system which cannot be easily measured by optical pumping. Nonlinear fitting of indirect transition energies quantitatively confirms the quantum confinement effect.

Figure 9

(a) Low-loss EEL spectra from a fresh $\mathrm{Mo}{\mathrm{S}}_2$ monolayer (in red) and that after exposure to electron beam irradiation for 4 h at 60 kV. (b) Low-loss EEL spectra recorded from monolayer and multilayer $\mathrm{Mo}{\mathrm{S}}_2$ after extended electron beam irradiation.

Figure 10

Dielectric functions and their ratio $\left(a\right)$; the inset shows dielectric constants extracted from Ref. [34]. It shows small variation of “$a$” as a function of energy.