# The Key Role of Non-Local Screening in the Environment-Insensitive Exciton Fine Structures of Transition-Metal Dichalcogenide Monolayers

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## Abstract

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## 1. Introduction

## 2. Computational and Experimental Methods

#### 2.1. First-Principles Calculations and Bethe–Salpeter Equation (BSE) for Exciton Spectra

#### 2.2. Sample Fabrication and Experimental Photoluminescence (PL) Measurement

## 3. Results and Discussion

#### 3.1. Quasi-Particle Band Structures of the WSe${}_{2}$ Monolayer

#### 3.2. Theory of Exciton Fine Structures of TMD-MLs under Dielectric Screenings

#### 3.3. Exciton Fine Structures in the Approximation of Local Screening

#### 3.4. Exciton Fine Structure under Non-Local Dielectric Screening

#### 3.5. Dielectric Environmental Dependencies of Exciton Fine Structures

#### 3.6. Non-Linear Correlation between the BX-DX Splitting and Exciton-Binding Energy under Non-Local Screening

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Valley− and spin−resolved quasi-particle band structures of WSe${}_{2}$-ML calculated by using the HSE06 exchange-correlation functionals in the DFT [62]. The blue (orange) color stands for the spin-down (spin-up) Bloch states. Left inset: the first Brillouin zone of WSe${}_{2}$-ML, with the indication of high-symmetry points. Middle inset: the close-up of the lowest conduction bands around the K valley, with the spin-splitting, ${\mathsf{\Delta}}_{c}=19.6$ meV. (

**b**) Schematics of the intra-valley BX and SFDX, and the inter-valley MFDX and MF-SF-DX states, with the valence holes located at the K valley. (

**c**) Schematics of BX and DX in a TMD-ML under a dielectric environment. In the approximation of local screening, the dielectric screening of the TMD-embedded layered system is simply characterized by a fixed dielectric constant ${\epsilon}_{eff}$, which is the same for both BX and DX. Because of the heavier reduced mass of DX, the Bohr radius (binding energy) of DX is significantly smaller (larger) than the lighter BX. (

**d**) Beyond the approximation of local screening, the non-local screenings for the BX and DX in the TMD-embedded layered system are described by the $\mathit{q}$-dependent dielectric function, $\epsilon \left(\mathit{q}\right)$, leading to the unequal effective screening for the BX and DX. The non-local screenings reduce the differences between the Bohr radii and binding energies of the BX and DX.

**Figure 2.**Comparison of the measured fine structure spectrum of PL at $T=10$ K (top panel) and BSE-calculated exciton fine structure (bottom panel) of hBN-encapsulated WSe${}_{2}$-ML. In the top panel, the measured PL spectrum is indicated by the blue profile. In the PL spectrum, the fitted peak by the red (black) profile is attributed to the BX (SFDX) state [46]. The BSE-calculated energy splitting of 42.7 meV between BX and SFDX states is in quantitative agreement with the fine structure of the measured 41.8 meV in the PL spectrum.

**Figure 3.**The BSE−calculated low-lying exciton fine structures of free-standing WSe${}_{2}$-ML (

**a**) with fixed ${\epsilon}_{eff}=4.39$ in the approximation of local screening and (

**b**) with the $\mathit{q}$-dependent non-local dielectric function, $\epsilon \left(\mathit{q}\right)$ (see the inset at the bottom), beyond the approximation of local screening. (i–iii) in (

**a**,

**b**): The exciton fine structure spectra calculated by solving the modified or exact BSE that take into account (i) only the kinetic energies, (ii) the kinetic energies and the direct Coulomb interaction, and (iii) all of the kinetic energies and both the direct and exchange Coulomb interactions, respectively. From the evolution of the spectra from (i), (ii), and (iii), the e-h direct interaction is the dictating mechanism of the BX-SFDX splitting. Comparing (

**a**,

**b**), the non-local screening is shown to make the overall shrinkage of the fine structure splittings between the BX and various DXs.

**Figure 4.**(

**a**) The WSe${}_{2}$-ML−embedded dielectric environments considered in this work. In cases (I)–(VII), all of the substrate and capping layers are considered semi-infinitely thick, except for the hBN layer, whose thickness is set at 10 nm. (

**b**) The calculated q-dependent dielectric functions for dielectric systems (I)–(VII). (

**c**) The calculated relative energy levels of the BX, MFDX, and SFDX states of the WSe${}_{2}$-ML under the dielectric screenings of (I)–(VII), indicated by filled triangles (MFDX) and circles (SFDX) in different colors, in the approximation of local screening with the fitted dielectric constants, ${\epsilon}_{eff}$. Dark dashed and gray solid curves are for eye-guiding; (

**d**) is the same as (

**c**), but under the non-local dielectric screening described by $\epsilon \left(q\right)$. (

**e**) [(

**f**)]: The log–log plot for the BX-DX splittings, ${\Delta}_{B,D}^{X}={E}_{B}^{X}-{E}_{D}^{X}$, given by (

**c**) [(

**d**)], fitted by the power law of ${\Delta}_{B,D}^{X}\propto {\epsilon}_{eff}^{-n}$ with the exponent n. The white-filled symbols in (

**e**) are the calculated splittings, disregarding ${\Delta}_{c}$. Taking into account the non-local screening, the exponent is decreased to the small values of $n\sim 0.58,0.46$, indicating the weak environmental dependence of the BX-DX splitting.

**Figure 5.**The non-linear correlation between the BX-SFDX splittings (${\Delta}_{B,SF}^{X}$) and binding energies of BX (${E}_{B}^{b}$) of WSe${}_{2}$-ML, changing the dielectric environments from (I) to (VII) in Figure 4a. The colored circles were obtained by solving DFT-based BSE for the WSe${}_{2}$-MLs in the multi-layered dielectric structures with full consideration of the non-local dielectric functions. The gray dashed line is the predicted linear ${\Delta}_{B,SF}^{X}$−${E}_{B}^{b}$ by the conventional hydrogen model in the approximation of local screening. The red and blue solid lines are the non-linear correlations simulated by the extended hydrogen model with the expanded non-local dielectric functions in the first- and second-order approximations of $\epsilon \left(q\right)$, respectively. The sub-linear correlation between ${\Delta}_{B,SF}^{X}$ and ${E}_{B}^{b}$ manifests the insensitive environmental dependence of the exciton fine structures as the consequence of non-local dielectric screening.

**Table 1.**Effective masses of the conduction electrons and valence holes in the K- or Q-valleys, the reduced masses of the bright exciton (${\mu}_{B}$), and that of the spin-forbidden dark exciton (${\mu}_{SF}$) in the free electron mass ${m}_{0}$ unit. The method for determining the effective mass is detailed in the supporting information (See Figure S6 and Table S4 for the fitting results).

Effective Masses of Carriers (${\mathit{m}}_{0}$) | Reduced Masses of Excitons (${\mathit{m}}_{0}$) | ||||
---|---|---|---|---|---|

${m}_{{v}_{1},K}$ | ${m}_{{c}_{1},K}$ | ${m}_{{c}_{2},K}$ | ${m}_{{c}_{1},Q}$ | ${\mu}_{B}$ | ${\mu}_{SF}$ |

0.32 | 0.40 | 0.29 | 0.51 | 0.15 | 0.18 |

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## Share and Cite

**MDPI and ACS Style**

Li, W.-H.; Lin, J.-D.; Lo, P.-Y.; Peng, G.-H.; Hei, C.-Y.; Chen, S.-Y.; Cheng, S.-J.
The Key Role of Non-Local Screening in the Environment-Insensitive Exciton Fine Structures of Transition-Metal Dichalcogenide Monolayers. *Nanomaterials* **2023**, *13*, 1739.
https://doi.org/10.3390/nano13111739

**AMA Style**

Li W-H, Lin J-D, Lo P-Y, Peng G-H, Hei C-Y, Chen S-Y, Cheng S-J.
The Key Role of Non-Local Screening in the Environment-Insensitive Exciton Fine Structures of Transition-Metal Dichalcogenide Monolayers. *Nanomaterials*. 2023; 13(11):1739.
https://doi.org/10.3390/nano13111739

**Chicago/Turabian Style**

Li, Wei-Hua, Jhen-Dong Lin, Ping-Yuan Lo, Guan-Hao Peng, Ching-Yu Hei, Shao-Yu Chen, and Shun-Jen Cheng.
2023. "The Key Role of Non-Local Screening in the Environment-Insensitive Exciton Fine Structures of Transition-Metal Dichalcogenide Monolayers" *Nanomaterials* 13, no. 11: 1739.
https://doi.org/10.3390/nano13111739