Mapping of the dark exciton landscape in transition metal dichalcogenides

Transition metal dichalcogenides (TMDs) exhibit a remarkable exciton physics including bright and optically forbidden dark excitonic states. Here, we show how dark excitons can be experimentally revealed by probing the intraexcitonic $1s-2p$ transition. Distinguishing the optical response shortly after the excitation and after the exciton thermalization allows us to reveal the relative position of bright and dark excitons. We find both in theory and experiment a clear blueshift in the optical response demonstrating the transition of bright exciton populations into lower-lying dark excitonic states.

Figure 1

Theoretically predicted blueshift. (a) Schematic illustration of the modeled experiment, where first, a pump pulse induces a coherent exciton density in the $1s$ state, which is then probed with a weak pulse matching the energy of the intraexcitonic $1s-2p$ transition. We can distinguish two scenarios, where the energetically lowest state is either dark or bright. (b) For very short pump-probe delay times (coherent limit), the coherent exciton density has not decayed yet and dominates the linear optical response. (c) For longer delay times, incoherent excitons have already been formed and have thermalized through exciton-phonon coupling. In this incoherent limit, the optical response is dominated by the lowest-lying excitonic states, where the population of incoherent excitons is the largest. (d) Theoretically predicted absorption spectrum for ${\mathrm{WSe}}_2$ in the coherent (yellow area) and incoherent (blue area) regime. The first reflects the population of bright $K-K$ excitons, while the second results from the energetically lowest dark $K-\mathrm{\Lambda }$ excitons.

Figure 2

Experimentally measured blueshift. Resonance energy of the intraexcitonic $1s-2p$ transition in a monolayer ${\mathrm{WSe}}_2$ on a diamond substrate as a function of the pump-probe delay time $t_{\text{pp}}$ after resonant interband optical injection of coherent A $1s$ excitons. We observe a clear blueshift of roughly 30 meV after a few ps. The error bars represent the 95% confidence interval of the resonance energy. The inset shows the real part of the pump-induced midinfrared conductivity $\mathrm{\Delta }{\sigma }_1$ (corresponding to the optical absorption) as a function of the photon energy for two characteristic pump delay times. Here, colored spheres correspond to experimental data, while shaded areas show the result of the Lorentz-oscillator model [22]. The gray dots in the main figure show another experiment on monolayer ${\mathrm{WSe}}_2$ encapsulated by hexagonal boron nitride (hBN) demonstrating a clear blueshift of the $1s-2p$ transition [27].

Figure 3

Comparison of different TMDs. Theoretical prediction of the optical response to an infrared red probe pulse for monolayer (a) $\mathrm{WSe}2$, (b) ${\mathrm{MoSe}}_2$, (c) ${\mathrm{WS}}_2$, and (d) ${\mathrm{MoS}}_2$, distinguishing the coherent (orange) and incoherent (blue) contribution including bright $K-K$ as well as dark $K-K^{(\prime )}, K-{\mathrm{\Lambda }}^{(\prime )}$, and $\mathrm{\Gamma }-K^{(\prime )}$ excitons. TMDs exhibiting energetically lower-lying dark excitonic states are characterized by a clear blueshift of the incoherent contribution. (a), (c) In W-based TMDs, the equilibrium distribution of excitonic occupation favors the low-lying $K-{\mathrm{\Lambda }}^{(\prime )}$ (solid blue lines) and $K-K^{(\prime )}$ states (dashed lines). (b) In contrast, the major contribution in ${\mathrm{MoSe}}_2$ stems from the bright $K-K$ exciton, revealing that in this material dark states are energetically higher. (d) In ${\mathrm{MoS}}_2$, the lowest states are the dark $\mathrm{\Gamma }-K^{(\prime )}$ excitons. Note that there is no fitting involved in this figure and the spectral position of all excitonic resonances has been microscopically calculated based on the solution of the Wannier equation.