The ultrafast onset of exciton formation in 2D semiconductors

The equilibrium and non-equilibrium optical properties of single-layer transition metal dichalcogenides (TMDs) are determined by strongly bound excitons. Exciton relaxation dynamics in TMDs have been extensively studied by time-domain optical spectroscopies. However, the formation dynamics of excitons following non-resonant photoexcitation of free electron-hole pairs have been challenging to directly probe because of their inherently fast timescales. Here, we use extremely short optical pulses to non-resonantly excite an electron-hole plasma and show the formation of two-dimensional excitons in single-layer MoS2 on the timescale of 30 fs via the induced changes to photo-absorption. These formation dynamics are significantly faster than in conventional 2D quantum wells and are attributed to the intense Coulombic interactions present in 2D TMDs. A theoretical model of a coherent polarization that dephases and relaxes to an incoherent exciton population reproduces the experimental dynamics on the sub-100-fs timescale and sheds light into the underlying mechanism of how the lowest-energy excitons, which are the most important for optoelectronic applications, form from higher-energy excitations. Importantly, a phonon-mediated exciton cascade from higher energy states to the ground excitonic state is found to be the rate-limiting process. These results set an ultimate timescale of the exciton formation in TMDs and elucidate the exceptionally fast physical mechanism behind this process.

The absorption spectrum in Supplementary Fig. 1a, is obtained from a reflective contrast measurements. The reflectivity from the sample (R M oS2 ) and the substrate (R sub ) are separately collected and the reflectivity contrast R contr is calculated as (R M oS 2 -R sub )/R sub .
Kramers-Kroening analysis [1] is used to fit the reflectivity contrast and to extract the absorption spectrum A=0.25 R contr (n s -1), where n s is the refractive index of the substrate.
The photoluminescence spectrum is measured at T=300 K with 2.5 eV excitation. The spectrum is dominated by the A exciton emission at ∼1.86eV ( Supplementary Fig. 1b).

Supplementary Note 2: Exciton population dynamics
As reported in the main text, the non-equilibrium optical response of TMDs is the result of the transient modification of the excitonic peaks induced by photo-excitation processes.
The physical processes can be categorized as many-body and population effects. by a careful line shape analysis of the exciton peak dynamics. An alternative way to single out population effect consists in integrating the ∆R/R map on a broad energy range around the excitonic resonances [2]. The pump-probe signal obtained by such a spectral integration does not depend on any energy shift and it is proportional to the exciton population density.
In Supplementary Fig. 2 we compare the ∆R/R dynamics measured at the maximum of the bleaching signal and the dynamics of the traces obtained by spectral integration around the exciton bleaching signal. For all the excitation energies, after renormalization, the traces almost overlap. This is a clear indication that the ∆R/R bleaching signals on resonance with the excitonic peaks are mainly determined by the population effect, while many-body effects play a minor role.

Supplementary Note 3: Exciton dynamics upon UV pump excitation
In Supplementary Figure 3, we reported the ∆R/R formation dynamics measured upon excitation with a pump pulse centered around 3.75 eV, i.e. almost 1 eV higher than the maximum value of the pump energy reported in the manuscript. In this experiment, the UV pump pulses are obtained by frequency up-conversion of a broadband NOPA as explained in details in Ref. [3]. These pulses are compressed down to 20 fs by a pair of prisms and temporally characterized by a two-dimensional spectral-shearing interferometry method The build up dynamics of A 1s and B 1s exciton transitions does not display any dependence on the pump and probe polarizations as reported in Supplementary Fig. 7 and on the excitation fluence as reported in Supplementary Fig. 8). -

Supplementary Note 8: Excitons in 1L-TMDs
The equilibrium optical response of the TMD is characterized by the microscopic interband polarization c † ξ,s,q v ξ,s,q , where c † ξ,s,q (v ξ,s,q ) generates a conduction (valence) band electron (hole) near the corner ξ = K, K of the first Brillouin zone with spin s = ↑, ↓ and two-dimensional in-plane wave vector q [7]. Solving the Wannier equation [8] accesses Coulomb correlated electron-hole excitations near the TMD band gaps: Here, µ λ = m e λ m h λ /(m e λ + m h λ ) denotes the reduced mass defined with respect to the effective masses of electrons (holes) in the conduction (valence) band m e λ (m h λ ) [9]. V q /ε q describes the screened Coulomb potential which includes the bare Coulomb potential V q accounting for finite thickness effects [10] and the non-local dielectric function ε q parameterized from density function theory calculations [11]. The analytical model ε q of the dielectric function characterizes the dielectric environment with a non-linear q-dependence. Due to the fact that excitons in TMDCs are stable over a wide range of excitation power and doping level [12,13], treating the Coulomb potential statically captures the investigated excitonic effects.
The evaluation of the Wannier equation (1) provides a set of bound and continuum solutions ϕ λ,q with energies λ [14,15]. The compound index λ comprises the high symmetry point ξ = K, K and spin s = ↑, ↓ of the involved electron and hole as well as the exciton state (bound and continuum states). This enables to work in a convenient excitonic basis including coherent exciton transitions P λ , biexcitons and exciton-exciton scattering states B η , as well as incoherent exciton populations N λ 1 ,λ 2 ,q . The exciton dephasing is chosen equivalent to the energetically lowest A 1s /B 1s state which has been adjusted to the experimentally measured absorption spectrum: γ Aν = γ A 1s = 39 meV/ , γ Bν = γ B 1s = 66 meV/ .