Electron density dependent Zeeman splitting in WS$_{2}$ single layer

We report an experimental and theoretical study of the magnetic field dependence of the circular polarization $P$ and the Zeeman splitting $\Delta E_{z} $ of the photoluminescence (PL) due to recombination of charged excitons (X$^{-})$ formed because excess electrons are present. Results from three WS$_{2\, \, }$monolayers are reported. For one sample the circular polarization at $B \quad =$ 0 T is finite while $P$ is zero for the other two samples. The slope $\frac{dP}{dB}$ in all samples is similar ($\approx $2.4 {\%} per Tesla). In contrast, the Zeeman splittings are: -0.13 meV/T, -0.30 meV/T and -0.47 meV/T. The experimental results are compared with those from the microscopic tight binding model to evaluate spin, valley and orbital contribution to the Zeeman splitting. The effect of carrier density on $\Delta E_{z} $ is included in a Hartee-Fock picture assuming inhomogeneous carrier distribution. The electron droplets present are divided into spin-valley polarized and spin-valley unpolarized. The spin-valley polarized droplets contribute to Zeeman splitting and are responsible for the carrier density and sample dependent Zeeman splitting in WS$_{2}$.