First-principles ultrafast exciton dynamics and time-domain spectroscopies: Dark-exciton mediated valley depolarization in monolayer ${\mathrm{WSe}}_2$

Calculations combining first-principles electron-phonon ($e$-ph) interactions with the Boltzmann equation enable studies of ultrafast carrier and phonon dynamics. However, in materials with weak Coulomb screening, electrons and holes form bound excitons so their scattering processes become correlated, posing additional challenges for modeling nonequilibrium physics. Here we show calculations of ultrafast exciton dynamics and related time-domain spectroscopies using ab initio exciton-phonon (ex-ph) interactions together with an excitonic Boltzmann equation. Starting from the nonequilibrium exciton populations, we develop simulations of time-domain absorption and photoemission spectra that take into account electron-hole correlations. We use this method to study monolayer ${\mathrm{WSe}}_2$, where our calculations predict subpicosecond timescales for exciton relaxation and valley depolarization and reveal the key role of intermediate dark excitons. The approach introduced in this paper enables a quantitative description of nonequilibrium dynamics and ultrafast spectroscopies in materials with strongly bound excitons.

Figure 1

(a) Exciton energy versus momentum dispersion in monolayer ${\mathrm{WSe}}_2$, with minima at M and Q. The two lowest bright excitons are the third and fourth states at $\mathrm{\Gamma }$, with energy $E_0=1.665$ eV indicated with a blue dot. (b) Electronic band structure showing the transitions that make up dark excitons with $\mathbf{Q}=(\mathrm{K},\mathrm{Q},\mathrm{M})$ (left) and the corresponding electron-hole pairs shown schematically in the electronic BZ (right).

Figure 2

Computed PL linewidth from ex-ph interactions in monolayer ${\mathrm{WSe}}_2$, shown as a function of temperature and compared with experiment [42]. The inset shows two main ex-ph scattering processes for bright exciton relaxation, $\mathrm{\Gamma }$ to M and $\mathrm{\Gamma }$ to K. The computed results are shifted upward by 7 meV to match experiment.

Figure 3

Snapshots of exciton relaxation in monolayer ${\mathrm{WSe}}_2$ at 300 K, shown at times $t=30$, 200, and 400 fs from top to bottom. We map the exciton populations onto the exciton band structure, with a spot radius proportional to the logarithm of the exciton populations, $log\left(F_{n\mathbf{Q}}\right)$. Next to each panel, we give the exciton energy distribution, $F\left(E\right)={\sum }_{n\mathbf{Q}}{F}_{n\mathbf{Q}}\delta (E-E_{n\mathbf{Q}})$ (in arbitrary units, shown with magenta solid lines).

Figure 4

(a) Schematic of exciton valley depolarization involving intermediate dark excitons. The bright A excitons at $\mathrm{\Gamma }$, corresponding to electron-hole pairs in the electronic K and ${\mathrm{K}}^{\prime }$ valleys, can scatter into each other in a process mediated by dark excitons. (b) Exciton populations in the electronic K and ${\mathrm{K}}^{\prime }$ valleys following a simulated circularly polarized 50 fs Gaussian pump pulse at 300 K temperature. (c) Corresponding ratio of the K and ${\mathrm{K}}^{\prime }$ valley populations after the pump pulse. The depolarization times extracted from the simulations are ${\tau }_{\mathrm{d}}=185$ fs at 77 K and ${\tau }_{\mathrm{d}}=66$ fs at 300 K.

Figure 5

Simulated tr-ARPES spectra at times $t=30,200$, and 400 fs, colored according to the computed tr-ARPES signal $I(k,\omega ;t)$ in Eq. (5) and plotted as a function of electron crystal momentum.

Figure 6

(a) Schematic of our simulated pump-probe transient absorption settings. (b) Transient absorption spectra of monolayer ${\mathrm{WSe}}_2$ pumped with a right-handed circularly polarized (${\sigma }^{+}$) incident pulse. We plot separately the transient absorption spectra up to 300 fs for probes with ${\sigma }^{+}$ (left panel) and ${\sigma }^{-}$ (right panel) polarizations.