Unusually large exciton binding energy in multilayered 2H-MoTe2

Although large exciton binding energies of typically 0.6–1.0 eV are observed for monolayer transition metal dichalcogenides (TMDs) owing to strong Coulomb interaction, multilayered TMDs yield relatively low exciton binding energies owing to increased dielectric screening. Recently, the ideal carrier-multiplication threshold energy of twice the bandgap has been realized in multilayered semiconducting 2H-MoTe2 with a conversion efficiency of 99%, which suggests strong Coulomb interaction. However, the origin of strong Coulomb interaction in multilayered 2H-MoTe2, including the exciton binding energy, has not been elucidated to date. In this study, unusually large exciton binding energy is observed through optical spectroscopy conducted on CVD-grown 2H-MoTe2. To extract exciton binding energy, the optical conductivity is fitted using the Lorentz model to describe the exciton peaks and the Tauc–Lorentz model to describe the indirect and direct bandgaps. The exciton binding energy of 4 nm thick multilayered 2H-MoTe2 is approximately 300 meV, which is unusually large by one order of magnitude when compared with other multilayered TMD semiconductors such as 2H-MoS2 or 2H-MoSe2. This finding is interpreted in terms of small exciton radius based on the 2D Rydberg model. The exciton radius of multilayered 2H-MoTe2 resembles that of monolayer 2H-MoTe2, whereas those of multilayered 2H-MoS2 and 2H-MoSe2 are large when compared with monolayer 2H-MoS2 and 2H-MoSe2. From the large exciton binding energy in multilayered 2H-MoTe2, it is expected to realize the future applications such as room-temperature and high-temperature polariton lasing.

Our analysis consists of two fitting processes: One is for obtaining the optical conductivity from the measured transmittance spectra by using the transfer matrix method and the other is for obtaining the characteristic absorption modes including exciton modes, indirect and direct bandgaps from the optical conductivity by using the Lorentz and Tauc-Lorentz models. The followings are detailed descriptions for those two separated fitting processes.

Optical conductivity obtained from measured transmittance using transfer matrix method
We used the transfer matrix method 1,2 to obtain the optical properties of each layer from the measured optical spectra of a thin multilayered sample. The information on the optical properties (or constants) of each layer is contained in the transfer matrix 1 . Because all possible the multi-reflections at all the interfaces are included in the transfer matrix method, reliable optical properties (or constants) of each layer can be obtained using this method. Detailed theoretical description of the transfer matrix method can be found in previous literature 1-3 . In our case, each sample consists of two layers: Thin transition metal dichalcogenide (TMD) film and quartz substrate. To obtain the optical conductivity of the thin film and substrate, one should measure, at least, two set of transmittance spectra: Bare substrate and TMD film/substrate (see Figure 1d). We analyzed the measured transmittance spectra of bare substrate and each sample (TMD film/substrate) by using the transfer matrix method. First, we fitted the transmittance spectrum of bare quartz substrate (see Figure 1d) and obtained the fitting parameters for the substrate layer. Then, we fitted the transmittance spectrum of the TMD film/substrate using the fitting parameters of the substrate layer and additional fitting parameters for the TMD film layer (see Figure 1d). We note that, for the fitting to the transmittance of TMD/substrate, we fixed the fitting parameters of the substrate and adjusted the fitting parameters for the TMD layer. Therefore, in this way, we obtained two sets of fitting parameters for the two layers, separately. To achieve a good-quality fit to the transmittance spectrum, we used many Lorentz modes and employed a least-squares procedure to adjust the fitting parameters of the Lorentz modes for each layer (see Table 1 in Ref. [3]). Using the resulting fitting parameters of the TMD layer, we obtained the optical conductivity of TMD film (as shown in Figure 2a); usually, the obtained optical conductivity is a smooth curve. We note that the Lorentz modes used here should be discriminated from the Lorentz modes used for fitting the optical conductivity with the Lorentz and Tauc-Lorentz models. Here, we used many Lorentz modes to achieve as good quality of fits as we can (see Figure 1d).

Fitting procedure of optical conductivity using Lorentz and Tauc-Lorentz models
In Figure 2c, we show the optical conductivity data obtained from the measured transmittance spectra using the transfer matrix method and the corresponding fits obtained by using the two models: Lorentz and Tauc-Lorentz (TL) models. The Lorentz model can be used to describe symmetric exciton absorptions (electron-hole pairs), whereas the TL model can be used to describe asymmetric direct and indirect bandgap absorptions. To obtain a reliable fit to the optical conductivity by using the Lorentz and TL model, the following several factors are considered. The absorption modes in this analysis were based on the theoretically calculated band structure of multilayer 2H-MoTe2, of which schematic is shown in Figure 2b.
The positions and widths of Lorentz modes for A and B (A' and B') excitons are unambiguously determined from the measured spectra. We assumed that the strengths ( i ) of the Lorentz modes of the A and B (A' and B') excitons are similar to each other because they originate from a single exciton level. Based on the schematic band structure of multilayer 2H-MoTe2, we constrained that the exciton levels at the K-point and Γ-point are lower than the conduction bands at the K-point and Γ-point, respectively. In the TL mode, there are two positional parameters of the bandgap ( g,j ) and the absorption maximum ( j ). We also constrained that j of the indirect transition is smaller than g,j of the K-point direct transition. Also, j of the K-point direct transition is smaller than g,j of the Γ-point direct transition. Beyond 2 eV, the fitting is further improved by adding two unknown Lorentz components (X and X'). We note that X and X' are not new; they were observed and denoted as  and C in previous literature 4 . The energy difference between the conduction band minimum and the exciton level is defined as the exciton binding energy. We fitted the optical conductivities with Lorentz and Tauc-Lorentz modes at various temperatures between 8 and 350 K as shown Figure S2, S3, and S4.
Interestingly, there is a narrow energy region that we could not fit well; the region is in between A and B excitons (see Fig. 2c). To get a better fit in this region, we need to add an additional peak, as shown in Figure S6. This additional peak can be assigned as the 2s state of the A exciton. The strength of the peak increases as the film thickness increases. In

S2. Real part of optical conductivity and Lorentz and TL model fits for 2 nm thick 2H-
MoTe2 at various temperatures.

S3. Real part of optical conductivity and Lorentz and TL model fits for 4 nm thick 2H-
MoTe2 at various temperatures.

S5. Real part of optical conductivity and fit with four Lorentz and three TL modes in 2H-
MoTe2. Figure S5. Fits of real part of optical conductivity with four Lorentz and three TL modes for 2, 4, and 10 nm thick 2H-MoTe2 at 8 K. The second derivative spectrum clearly shows a peak ( peak) located between A and B excitons. The insets and panels on the right show the optical conductivity data and fits including the peak.