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Theory optical excitation spectra and depolarization dynamics in bilayer WS$_2$ from viewpoint of excimers

We investigate the optical excitation spectra and the photoluminescence depolarization dynamics in bilayer WS$_2$. A different understanding of the optical excitation spectra in the recent photoluminescence experimentby Zhu {\em et al.} [arXiv:1403.6224] in bilayer WS$_2$ is proposed. In the experiment, four excitations (1.68, 1.93, 1.99 and 2.37 eV) are observed and identified to be indirect exciton for the $Γ$ valley, trion, A exciton and B exciton excitations, respectively, with the redshift for the A exciton energy measured to be 30$\sim$50 meV when the sample synthesized from monolayer to bilayer. According to our study, by considering there exist both the intra-layer and charge-transfer excitons in the bilayer WS$_2$, with inter-layer hopping of the hole, there exists excimer state composed by the superposition of the intra-layer and charge-transfer exciton states. Accordingly, we show that the four optical excitations in the bilayer WS$_2$ are the A charge-transfer exciton, ${\rm A}'$ excimer, ${\rm B}'$ excimer and B intra-layer exciton states, respectively, with the calculated resonance energies showing good agreement with the experiment. In our picture, the speculated indirect exciton, which involves a high-order phonon absorption/emission process, is not necessary. Furthermore, the binding energy for the excimer state is calculated to be 40 meV, providing reasonable explanation for the experimentally observed energy redshift of the A exciton. Based on the excimer states, we further derive the exchange interaction Hamiltonian. Then the photoluminescence depolarization dynamics due to the electron-hole exchange interaction is studied in the pump-probe setup by the kinetic spin Bloch equations. We find that ......