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In order to discuss the exciton fine-structure of transition-metaldichalcogenides mono-layers, excitons are first defined in the subspace ofelectron- and hole states, including the lowest conduction band (LCB) and theuppermost valence band (UVB). Both bands are spin degenerate at theGamma-point. All other states are neglected. The resulting exciton states areanalyzed in the framework of an invariant expansion of a model Hamiltonian: Thespin-orbit coupling in the conduction- and valence band is simulated byintroducing a fictive magnetic field, giving rise to a splitting of theelectron- and hole states outside the $\Gamma$-point. Then the electron-holeexchange-interaction is introduced into the exciton Hamiltonian. It is due tothe fact that electron and hole are indistinguishable particles in the excitonproblem. In D3h crystal symmetry this electron-hole exchange-interaction hastwo different contributions: While a first term accounts for an energyre-normalization of all exciton states, a second term does not influence theoptical active (spin-singlet) states but affects only the optical inactive(spin-triplet) states, which become mixed in-between the different excitonseries.

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