# On the Exciton Fine-Structure of Transition-Metal Dichalcogenides Mono-Layers

## TL;DR

This paper investigates the exciton fine-structure in transition-metal dichalcogenide monolayers, analyzing how spin-orbit coupling and electron-hole exchange interactions influence exciton energy levels and state mixing.

## Contribution

It introduces a model Hamiltonian framework incorporating spin-orbit effects and exchange interactions to analyze exciton fine-structure in these materials.

## Key findings

- Electron-hole exchange causes energy shifts in exciton states.
- Spin-triplet states are mixed due to exchange interactions.
- The model explains optical activity differences among exciton states.

## Abstract

In order to discuss the exciton fine-structure of transition-metal dichalcogenides mono-layers, excitons are first defined in the subspace of electron- and hole states, including the lowest conduction band (LCB) and the uppermost valence band (UVB). Both bands are spin degenerate at the Gamma-point. All other states are neglected. The resulting exciton states are analyzed in the framework of an invariant expansion of a model Hamiltonian: The spin-orbit coupling in the conduction- and valence band is simulated by introducing a fictive magnetic field, giving rise to a splitting of the electron- and hole states outside the $\Gamma$-point. Then the electron-hole exchange-interaction is introduced into the exciton Hamiltonian. It is due to the fact that electron and hole are indistinguishable particles in the exciton problem. In D3h crystal symmetry this electron-hole exchange-interaction has two different contributions: While a first term accounts for an energy re-normalization of all exciton states, a second term does not influence the optical active (spin-singlet) states but affects only the optical inactive (spin-triplet) states, which become mixed in-between the different exciton series.