Exciton and trions binding energies in single-layer MoS$_2$: applications of the density-matrix time dependent density*

Exciton and trion binding energies of a single layer of MoS$_2$ are studied using a time- dependent density-functional theory formalism. Kohn-Sham orbitals of the initial state were obtained using ab initio electronic structure calculations based on density functional theory. Several types of exchange-correlation (XC) kernels are implemented in our code to compare their performance. As expected our results depend crucially on the XC kernels used. In particular, the exchange-only adiabatic local density approximation kernel results in the binding energy about 0.1 eV, which is smaller than those obtained using the GW theory approximation ($\sim$ 0.9 eV) [1]. We have generalized the approach on the case of trion excitations, which gives the trion binding energy $\sim$ 0.3eV when one used the LDA approximation. On the other hand, we demonstrate that the results for the experimental binding energies can be reproduced by using phenomenological local and long-range XC kernels. [1] T. Cheiwchanchamnangij and W. R. L. Lambrecht, Phys. Rev. B \textbf{85}, 205302 (2012).