# Theory of Excitons in Atomically Thin Semiconductors: Tight-Binding Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}, representative of a wide class of materials, and review our minimal tight-binding model, which reproduces low-energy physics around the Fermi level and, at the same time, allows for the understanding of their electronic structure. Next, we describe how electron-hole pair excitations from the mean-field-level ground state are constructed. The electron–electron interactions mix the electron-hole pair excitations, resulting in excitonic wave functions and energies obtained by solving the Bethe–Salpeter equation. This is enabled by the efficient computation of the Coulomb matrix elements optimized for two-dimensional crystals. Next, we discuss non-local screening in various geometries usually used in experiments. We conclude with a discussion of the fine structure and excited excitonic spectra. In particular, we discuss the effect of band nesting on the exciton fine structure; Coulomb interactions; and the topology of the wave functions, screening and dielectric environment. Finally, we follow by adding another layer and discuss excitons in heterostructures built from two-dimensional semiconductors.

## 1. General Overview

#### 1.1. 2D van der Waals Materials

#### 1.2. Optical Properties of TMDs and Their Heterostructures

#### 1.3. Review of Theory of Correlated Optical Excitations

_{0}W

_{0}+ BSE), almost identical to the experimental, is calculated for MoS${}_{2}$ in vacuum. The similarity of these values is coincidental and stems from the cancellation of self-energy and excitonic renormalizations, which are true even when the dielectric environment (vacuum, SiO${}_{2}$, hBN) is vastly different. Recent GW calculations of MoS${}_{2}$ on hBN [232] help to rationalize this, showing the actual position of the GW-normalized “free particle” band gap.

## 2. Electronic Structure of MX${}_{2}$ Semiconductors

#### 2.1. Ab Initio Insight into Electronic Structure

#### 2.2. Minimal Tight-Binding Hamiltonian

## 3. Tight-Binding Theory of Optical Excitations

#### 3.1. Bethe–Salpeter Equation

#### 3.2. Coulomb Matrix Elements

#### 3.3. Screening of Coulomb Interactions

## 4. Mechanisms of the Renormalization of the X Spectrum

## 5. Summary

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TMDs | Transition metal dichalcogenides |

X | Exciton |

hBN | Hexagonal boron nitride |

BSE | Bethe–Salpeter equation |

VB | Valence band |

CB | Conduction band |

DFT | Density functional theory |

SOC | Spin-orbit coupling |

TB | Tight-binding |

BZ | Brillouin zone |

EMA | Effective mass approximation |

mDF | Massive Dirac fermion |

R.-K. | Rytova–Keldysh |

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**Figure 1.**Schematic structure of spin-split bands in monolayer MX${}_{2}$$+K$/$-K$ valleys. Respective A/B excitonic transitions are shown as dashed arrows. Additional solid arrows denote different contributions to the $+K$ and $-K$ valleys’ responses to magnetic fields. The solid blue and red arrows show the bare electron Zeeman contribution in magnetic field $\overrightarrow{B}=(0,0,{B}_{z})$; the green—the atomic orbital Landé contribution and the black ones—the valley Zeeman contribution.

**Figure 2.**(

**a**) Top view of the structure of MX${}_{2}$ in the 2H phase: metal atoms are denoted by blue dots, and chalcogen by red ones. (

**b**) Side view of MX${}_{2}$, showing that the atoms are organized into three layers, central metal and two chalcogen, with structural constants parametrized by ${d}_{\Vert}$ and ${d}_{\perp}$.

**Figure 3.**Color-mapped localization of a given k-resolved eigenenergy on Mo and S${}_{2}$ spheres and symmetry of eigenvalues across the Brillouin zone. Circles (crossed rectangles) denote symmetric (anti-symmetric) orbitals with respect to the metal plane.

**Figure 4.**

**Left**: TB dispersion obtained after optimizing the SK parameters to reproduce all even DFT bands.

**Right**: TB dispersion optimized to reproduce the transition energy between the VB and CB. We note that the former reproduces the VB very well, while the latter one reproduces the CB very well, especially on the $K-\Gamma $ line.

**Figure 5.**(

**a**) Spin texture of the CB, showing the spin orientation of the lowest CB spin-split band. (

**b**) Corresponding VB spin texture.

**Figure 6.**Comparison between dispersion models along the $K-\Gamma $ line. DFT dispersion is denoted by black circles, TB by blue rectangles, massive Dirac fermion by green diamonds, and parabolic (effective mass) model by red triangles. The corresponding connecting lines are shown as a guide to the eye.

**Figure 7.**(

**a**) Choice of $+K$ valley on the whole BZ. (

**b**) Construction of the valley around a single K point.

**Figure 8.**(

**a**) The single electron–single hole picture (exciton in effective mass approximation) in which interaction creates a spectrum of bound states. (

**b**) Exact picture where the “hole” is created by exciting the electron from the filled ground state in the VB. The exciton is then constructed as a coherent superposition of all possible excitations for a given center-of-mass momentum ${\overrightarrow{Q}}_{CM}$ interacting via the Coulomb interaction.

**Figure 9.**Graphical representation of two types of interaction between an electron and a hole: (

**a**) direct process (

**b**) exchange process.

**Figure 10.**(

**a**) Schematic picture of the dielectric environment of the MoS${}_{2}$ monolayer on the SiO${}_{2}$ substrate. (

**b**) Slab model of the MoSe${}_{2}$/WSe${}_{2}$ heterostructure. Each of the MX${}_{2}$ layers with the width ${d}_{1,2}$, respectively, is described by the dielectric constant ${\epsilon}_{{l}_{1},{l}_{2}}$.

**Figure 11.**The first two shells of the excitonic spectrum with full tight-binding direct interaction form factors. (

**a**) Effect of the form factors compared to ${F}^{D}=1$ approximation. (

**b**) Effect of different static screening on the spectrum. On both (

**a**,

**b**), tight-binding energies of electron and hole are used.

**Figure 12.**Exciton fine structure for the MoS${}_{2}$ monolayer (

**a**) on the SiO${}_{2}$ substrate and (

**b**) encapsulated with hBN, in a full TB model with complex electron–hole interaction included. Results restricted to the first three shells. The topological splitting of $2p$, $3p$, and $3d$ states in the excitonic spectrum of the MoS${}_{2}$ layer is presented.

**Figure 13.**Interlayer exciton fine structure for the MoSe${}_{2}$/WSe${}_{2}$ heterostructure encapsulated with hBN in the EMA, restricted to $1s$, $2p$, and $2s$ states. Interlayer A/B/$\tilde{A}$/$\tilde{B}$ exciton types have been distinguished, where A denotes the transition WSe${}_{2}$–MoSe${}_{2}$ spin-up, B transition MoSe${}_{2}$–WSe${}_{2}$ spin-down, $\tilde{A}$ transition MoSe${}_{2}$–WSe${}_{2}$ spin-up, and $\tilde{B}$ transition WSe${}_{2}$–MoSe${}_{2}$ spin-down, respectively.

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Bieniek, M.; Sadecka, K.; Szulakowska, L.; Hawrylak, P.
Theory of Excitons in Atomically Thin Semiconductors: Tight-Binding Approach. *Nanomaterials* **2022**, *12*, 1582.
https://doi.org/10.3390/nano12091582

**AMA Style**

Bieniek M, Sadecka K, Szulakowska L, Hawrylak P.
Theory of Excitons in Atomically Thin Semiconductors: Tight-Binding Approach. *Nanomaterials*. 2022; 12(9):1582.
https://doi.org/10.3390/nano12091582

**Chicago/Turabian Style**

Bieniek, Maciej, Katarzyna Sadecka, Ludmiła Szulakowska, and Paweł Hawrylak.
2022. "Theory of Excitons in Atomically Thin Semiconductors: Tight-Binding Approach" *Nanomaterials* 12, no. 9: 1582.
https://doi.org/10.3390/nano12091582