Magnetic brightening, large valley Zeeman splitting, and dynamics of long-lived A and B dark excitonic states in monolayer WS2

We theoretically investigate the magnetic brightening, valley Zeeman splitting, and valley dynamics of long-lived A and B dark excitons (A-DEs and B-DEs) in monolayer ${\mathrm{WS}}_{2}$, subjected to tilted magnetic fields. Biexciton states and indirect intervalley excitons, with the latter composed of a hole in the $K$ valley and an electron in the $\mathrm{\ensuremath{\Lambda}}$ valley, are also taken into account. We reveal that both in-plane and out-of-plane field components greatly affect the magnetic brightening process due to correlations among different excitonic quasiparticles but in distinct roles. Specifically, the in-plane component primarily enables the dark-state brightening, while the perpendicular one not only modifies the bright-dark splitting inside a given valley, but also lifts the valley degeneracy of either bright or dark excitonic states. Moreover, in the presence of a tilted magnetic field with a nonzero in-plane component, we observe that both A-DEs and B-DEs can be brightened, with the latter requiring a much stronger field for a certain intensity of light emission. Remarkably, we find that the valley splitting of A-DEs is around twice as large as that of A bright excitons but vanishes for B-DEs, as a result of the combined effect of the spin and orbital magnetic moments on the energy shifts of the two valleys. In addition, as the tilt angle of the field with respect to the normal direction of ${\mathrm{WS}}_{2}$ sheet increases from ${0}^{\ensuremath{\circ}}$ to ${180}^{\ensuremath{\circ}}$, the bright-dark splittings for A and B excitons exhibit an opposite behavior, but both feature an ``X''-like pattern when the field is oriented in-plane. Finally, it is observed that the temperature greatly affects the light emission of bright and brightened dark states, while the corresponding temperature dependence is distinct for A and B excitons.