Trion fine structure and coupled spin–valley dynamics in monolayer tungsten disulfide

Monolayer transition-metal dichalcogenides have recently emerged as possible candidates for valleytronic applications, as the spin and valley pseudospin are directly coupled and stabilized by a large spin splitting. The optical properties of these two-dimensional crystals are dominated by tightly bound electron–hole pairs (excitons) and more complex quasiparticles such as charged excitons (trions). Here we investigate monolayer WS2 samples via photoluminescence and time-resolved Kerr rotation. In photoluminescence and in energy-dependent Kerr rotation measurements, we are able to resolve two different trion states, which we interpret as intravalley and intervalley trions. Using time-resolved Kerr rotation, we observe a rapid initial valley polarization decay for the A exciton and the trion states. Subsequently, we observe a crossover towards exciton–exciton interaction-related dynamics, consistent with the formation and decay of optically dark A excitons. By contrast, resonant excitation of the B exciton transition leads to a very slow decay of the Kerr signal.


A. Supplementary Note 1: Calculation of background carrier concentration in our samples
To extract the background carrier density, we utilize the fact that in the TMD monolayers, the exciton-trion splitting ∆XT observed in optical spectroscopy sensitively depends on the chemical potential (Fermi energy E F ) [1]. This is due to the fact that for resonant creation (probed in absorption-related experiments like Kerr spectroscopy) or annihilation (probed in PL experiments) of a trion, energy and momentum conservation have to be taken into account [2]. We can extract this splitting directly from our PL and energy-dependent Kerr spectra. Here, in order to be able to compare our results directly to literature data, we neglect the fact that we are able to observe a ne structure splitting of the trion in our measurements, and extract the energetic position of the trion by (a) using a single Gaussian to t its spectral position in the PL spectrum and (b) calculating the weighted average of the energetic positions of the two trion resonances in the t to the energy-dependent Kerr spectrum. We nd an exciton-trion separation of 31 meV (PL data) and 31.5 meV (Kerr data). Following the reasoning presented by Chernikov et al. [2], we can calculate the Fermi energy according to Here, ∆ 0 is the trion binding energy in the limit of zero background carrier density. Using the value of ∆ 0 = 23 ± 1 meV [2], we nd that E F = 12 ± 3 meV. For a two-dimensional electron gas, carrier density and Fermi energy are related by Here, g V = 2 is the valley degeneracy factor. Using a conduction-band mass of m * = 0.
In order to describe the reectance spectrum, we need to utilize a transfer matrix method approach [4] to account for multiple reections at the interfaces of our SiO 2 /Si substrate.
By assuming three excitonic resonances, we are able to get excellent agreement between our data and our model. We nd the following energetic positions for the excitonic resonances: C. Supplementary Note 3: Substrate-induced eects on energy-dependent Kerr rotation spectrum As shown in supplementary Note 2, we are able to account for the eects of multiple reections due to the silicon/SiO 2 substrate by means of a transfer matrix approach. We apply this approach to model the energy dependence of the Kerr rotation angle as well, following the procedure described here [5]. For this, we need to introduce another variable in addition to the oscillator strength of the individual resonances, namely the valley polarization of each resonance, which is created by the circularly polarized pump in the experiment.
Additionally, we need to consider that the spectral positions of the individual resonances extracted from the white-light reectance measurement may need to be shifted slightly to account for small dierences between the ake used in that measurement and in the Kerr rotation measurements. Supplementary Figure 3 shows the results of a t to the Kerr rotation data which takes these considerations into account, using three resonances, a trion, a neutral A exciton and a neutral B exciton. To match the resonances observed in the Kerr rotation data, we red-shifted the spectral positions for the A exciton and its trion by 10 meV, and that of the B exciton by 20 meV. As the gure shows, this t is in good agreement with the experimental data in the range of the neutral A and B excitons. However, it fails to describe the feature we associate with the trions. To improve our t, we utilize two trion resonances corresponding to the two trion species observed in our PL experiments. For this, we assume that the antisymmetric Lorentzians for the two trion species have opposite signs of β, but comparable damping rates Γ . As shown in Supplementary Figure 4, this t captures all the salient features of the experimental data. It also allows us to extract the energetic splitting between the two trion species, which we nd to be in good agreement with the value found in our PL experiments.

D. Supplementary Note 4: Time-dependence of exciton Kerr resonance
To investigate possible bandgap renormalization eects as a function of exciton density, we look more closely at our energy-dependent Kerr rotation spectra. Supplementary Figure 5 compares energy-dependent Kerr rotation data for dierent time delays, extracted from the dataset shown in Figure 2(a) of the main manuscript. We clearly see that the spectral positions of the resonances do not change as function of delay time. We note that in the time interval from ∆t = 0.5 ps to ∆t = 10 ps, the Kerr signal decays by almost one order of magnitude, and a substantial part of this decay is due to photocarrier recombination, so that the exciton density is also signicantly reduced from its initial value. If there were excitondensity-dependent bandgap renormalization eects in the density range that we create, we The spectrum is dominated by the biexciton feature marked XX, as expected for high carrier densities, the charged exciton states are visible as a high-energy shoulder of the XX state, and a relatively weak neutral A exciton peak is also observed. In the spectral range where the B exciton resonance is observed in our Kerr spectroscopy and white-light reectance experiments, no PL emission is observable, as the inset shows.

A. Photoluminescence
For the PL measurements shown in Supplementary Figure 6, we utilize a picosecond pulsed diode laser system (Picoquant LDH-P-FA-355) emitting at 355 nm in conjunction with a self-built confocal microscope system. The laser was coupled into a 100x objective and focussed to a spot size of about 1 µm on the sample surface. The PL was collected by the same objective and recorded using a single-grating spectrometer equipped with a CCD sensor. A low-pass lter with an onset at a wavelength of 400 nm was used in front of the spectrometer slit to suppress stray light from the laser. The sample was mounted in a small He-ow cryostat.

B. White-light reectance
For white-light reectance measurements, the output of an incandescent lamp was focussed onto a pinhole, and an achromatic lens was used to collimate the emission from the pinhole. This was coupled into a self-built microscope setup with a 100x objective lens, so that the white light could be focussed down to a spot size of about 4 µm on the sample surface. The sample was mounted in a small He-ow cryostat. The reected light was collected with the same objective lens and coupled into a spectrometer, where it was detected with a