Experimental determination of the massive Dirac fermion model parameters for MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$

Monolayer MX$_2$ (M = Mo, W; X = S, Se) has drawn much attention recently for its possible application possibilities for optoelectronics, spintronics, and valleytronics. Its exotic optical and electronic properties include a direct band gap, circular polarization dependent optical transitions, and valence band (VB) spin band splitting at the $K$ and $-K$ points. These properties can be described within a minimal model, called the massive Dirac fermion model for which the parameters need to be experimentally determined. We propose that the parameters can be obtained from angle resolved photoemission (ARPES) data from bulk 2H-MX$_2$, instead of monolayer MX$_2$. Through tight binding calculations, we show how the electronic structure at high symmetry points evolves as the system changes from the monolayer to the three dimensional bulk 2H-MX$_2$ . We find vanishing $k_z$ dispersion and almost no change in the direct band gap at the $K$ and $-K$ points, in sharp contrast to the strong $k_z$ dispersion at the $\Gamma$ point. These facts allow us to extract the gap and spin band splitting at the $K$ point as well as the hopping energy from bulk ARPES data. We performed ARPES experiments on single crystals of MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$ at various photon energies and also with potassium evaporation. From the data, we determined the parameters for the massive Dirac fermion model for monolayer MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$.