# Electrostatic and Environmental Control of the Trion Fine Structure in Transition Metal Dichalcogenide Monolayers

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods: Three-Particle States in TMDC Monolayers

## 3. Results

## 4. Discussion

#### 4.1. Symmetry and Internal Structure of Trions

#### 4.2. Manipulating the Lowest Trion States

#### 4.3. The Role of Spin-Orbit and Exchange Interactions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ML | monolayers |

TMDC | transitional metal dichalcogenides |

2D | two-dimensional |

DFT | density functional theory |

GW | GW approximation |

## Appendix A. Computational Details: Single-Particle States

## Appendix B. Trion States

## Appendix C. Trion s_{z}τ = ±3/2 States

**Figure A1.**(

**a**) Trion energy splitting between the lowest ${s}^{z}\tau =\pm 3/2$ state and the lowest ${s}^{z}\tau =\pm 1/2$ states of the freestanding of MoS2 monolayer. For a wide range of doping, the fully dark ${s}^{z}\tau =\pm 3/2$ state will be the lowest trion state of the freestanding of MoS2 monolayer. (

**b**) Contributions of the Dirac single-particle band states in the vicinity of the K and $-K$ points to the lowest ${s}^{z}\tau =\pm 3/2$ trion state.

## References

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**Figure 1.**(

**a**) Schematics of the TMDC monolayer placed between two dielectrics that define environmental screening ${\epsilon}_{env}=({\epsilon}_{1}+{\epsilon}_{2})/2$. (

**b**–

**e**) Energy diagram of the transition energies of the three-particle states calculated in freestanding (${\epsilon}_{env}=1$) ML of MoS2 (

**b**), MoSe2 (

**c**), WS2 (

**d**), and WSe2 (

**e**) as a function of the Fermi level. Circle centers denote optical transition energies of trions. Colors and widths indicate their oscillator strengths (red) for bright states and (black) for dark ones. “Trion” and “exciton” denote trion and exciton states, depending on whether the second electron in the three-particle wavefunction is bound or not.

**Figure 2.**Contributions of Dirac single-particle band states in the vicinity of the K and $-K$ points to trions (panels (

**a**–

**d**)). A circle position points to the single-particle state and its radius gives weight to the exciton state; colors mark the spin projection ${s}^{z}$. The states T${}_{1}$ and T${}_{2}$ in panels (

**a**,

**b**) have $\tau {s}^{z}=1/2$, and the states T${}_{3}$ and T${}_{4}$ in panels (

**c**) and (

**d**) have $\tau {s}^{z}=-1/2$. Results are shown for ${E}_{F}=2.9$ meV.

**Figure 3.**Doping dependence of relative transition energies of trion states T${}_{1}$ and T${}_{2}$ (

**a**) and T${}_{3}$ and T${}_{4}$ (

**b**) and relative oscillator strength of trion states T${}_{1}$ and T${}_{2}$ (

**c**) and T${}_{3}$ and T${}_{4}$ (

**d**) calculated for a free-standing MoS2 ML. The transition energies in (

**a**,

**b**) are shown with respect to their average at each doping level.

**Figure 4.**Dependence of the oscillator strength ${S}_{{\mathrm{T}}_{1},{\mathrm{T}}_{2}}$ of the T${}_{1}$ and T${}_{2}$ trion states on doping. The color scale shows the ratio $({S}_{{\mathrm{T}}_{1}}-{S}_{{\mathrm{T}}_{2}})/({S}_{{\mathrm{T}}_{1}}+{S}_{{\mathrm{T}}_{2}})$.

**Table 1.**Model parameters for TMDC MLs: lattice constant a, effective mass m in units of free electron mass ${m}_{e}$, and spin-orbit couplings ${\lambda}_{c,v}$ are taken from Reference [61]. The layer thickness d and bulk dielectric constant ${\epsilon}_{bulk}$ are from Reference [64], and bandgap ${\mathsf{\Delta}}_{0}$ is from Reference [60] (see model (3)).

a [Å] | d [Å] | ${\mathit{\epsilon}}_{\mathbf{bulk}}$ | ${\Delta}_{0}$ [eV] | $\mathit{m}/{\mathit{m}}_{\mathit{e}}$ | ${\mathit{\lambda}}_{\mathit{c}}$ [meV] | ${\mathit{\lambda}}_{\mathit{v}}$ [meV] | |
---|---|---|---|---|---|---|---|

MoS2 | 3.185 | 6.12 | 16.3 | 2.087 | 0.520 | −1.41 | 74.60 |

MoSe2 | 3.319 | 6.54 | 17.9 | 1.817 | 0.608 | −10.45 | 93.25 |

WS2 | 3.180 | 6.14 | 14.6 | 2.250 | 0.351 | 15.72 | 213.46 |

WSe2 | 3.319 | 6.52 | 16.0 | 1.979 | 0.379 | 19.85 | 233.07 |

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**MDPI and ACS Style**

Zhumagulov, Y.V.; Vagov, A.; Gulevich, D.R.; Perebeinos, V.
Electrostatic and Environmental Control of the Trion Fine Structure in Transition Metal Dichalcogenide Monolayers. *Nanomaterials* **2022**, *12*, 3728.
https://doi.org/10.3390/nano12213728

**AMA Style**

Zhumagulov YV, Vagov A, Gulevich DR, Perebeinos V.
Electrostatic and Environmental Control of the Trion Fine Structure in Transition Metal Dichalcogenide Monolayers. *Nanomaterials*. 2022; 12(21):3728.
https://doi.org/10.3390/nano12213728

**Chicago/Turabian Style**

Zhumagulov, Yaroslav V., Alexei Vagov, Dmitry R. Gulevich, and Vasili Perebeinos.
2022. "Electrostatic and Environmental Control of the Trion Fine Structure in Transition Metal Dichalcogenide Monolayers" *Nanomaterials* 12, no. 21: 3728.
https://doi.org/10.3390/nano12213728