Atomically inspired $k \cdot p$ approach and valley Zeeman effect in transition metal dichalcogenide monolayers

We developed a six-band $k \cdot p$ model that describes the electronic states of monolayer transition metal dichalcogenides (TMDCs) in $K$-valleys. The set of parameters for the $k \cdot p$ model is uniquely determined by decomposing tight-binding (TB) models in the vicinity of $K^\pm$-points. First, we used TB models existing in literature to derive systematic parametrizations for different materials, including MoS$_2$, WS$_2$, MoSe$_2$ and WSe$_2$. Then, by using the derived six-band $k \cdot p$ Hamiltonian we calculated effective masses, Landau levels, and the effective exciton $g$-factor $g_{X^0}$ in different TMDCs. We showed that TB parameterizations existing in literature result in small absolute values of $g_{X^0}$, which are far from the experimentally measured $g_{X^0} \approx -4$. To further investigate this issue we derived two additional sets of $k \cdot p$ parameters by developing our own TB parameterizations based on simultaneous fitting of ab-initio calculated, within the density functional (DFT) and $GW$ approaches, energy dispersion and the value of $g_{X^0}$. We showed that the change in TB parameters, which only slightly affects the dispersion of higher conduction and deep valence bands, may result in a significant increase of $|g_{X^0}|$, yielding close-to-experiment values of $g_{X^0}$. Such a high parameter sensitivity of $g_{X^0}$ opens a way to further improvement of DFT and TB models.