Enabling valley selective exciton scattering in monolayer WSe2 through upconversion

Excitons, Coulomb bound electron–hole pairs, are composite bosons and their interactions in traditional semiconductors lead to condensation and light amplification. The much stronger Coulomb interaction in transition metal dichalcogenides such as WSe2 monolayers combined with the presence of the valley degree of freedom is expected to provide new opportunities for controlling excitonic effects. But so far the bosonic character of exciton scattering processes remains largely unexplored in these two-dimensional materials. Here we show that scattering between B-excitons and A-excitons preferably happens within the same valley in momentum space. This leads to power dependent, negative polarization of the hot B-exciton emission. We use a selective upconversion technique for efficient generation of B-excitons in the presence of resonantly excited A-excitons at lower energy; we also observe the excited A-excitons state 2s. Detuning of the continuous wave, low-power laser excitation outside the A-exciton resonance (with a full width at half maximum of 4 meV) results in vanishing upconversion signal.

In addition to the A:2s hot PL emission from upconversion we also observe anti-Stokes Raman scattering. This can be seen in Supplementary Fig. 2, where in addition to the A:2s a peak 133 meV above the laser energy is clearly visible in each spectrum, shifting as a function of the laser energy E L . This is the same data as in Supplementary Fig. 4e in the main text.
In Supplementary Fig. 3 we plot circularly co-and cross-polarized emission in black and red, respectively, of the A:2s upconversion PL. The circular polarization degree P c is plotted in blue. The polarization of the upconverted emission at the A:2s energy does not originate exclusively from the anti-stokes Raman process, the emission is globally polarized, not just at the Raman energy E L + 133 meV. The A:1s resonance is at 1.723 eV, so the top (bottom) panel shows excitation 3 meV above (2 meV below) resonance. The middle panel corresponds to resonant A:1s excitation. In Fig. 2d of the main text, we measure the emission of the states labelled A:2s and 3s as a function of temperature. In Supplementary Fig. 4 we plot the linewidth, energy separation and intensity ratio of the peaks, extracted from a double peak fit using a Lorentz function.

Supplementary Note 2 : Models
Here we develop a more formal model for the observed optical processes and their polarization selectivity based on the Supplementary Equations (1) - (8). One-photon absorption. In linear absorption one absorbed photon generates one exciton. Hence, for resonant excitation of the A:1s state the exciton occupancy N A:1s is directly proportional to the light intensity I.
In the linear regime the exciton generation rate can be conveniently presented as [2] Here A is the absorption coefficient of the TMD ML and R is the reflection coefficient of the sample,hω is the photon energy. Due to nonlinear effects, e.g., absorption saturation, or nonlinear (Auger) recombination of excitons the exciton occupancy can be sublinear function of the incident light intensity. The exciton occupancy where τ A is the lifetime of the A-exciton. The nonlinearities are either included in the of the main text (energy equals to 2hω, i.e. twice the laser energy) is proportional to the square of incident radiation intensity [3,4,5] In the studied situation the single photon energyhω equals to the A:1s-exciton energy.
Hence, the intermediate state for the two-photon process can be real. The two-photon excitation via real state (RS-2PA) can be viewed as a two-step process: creation of the A-exciton at the first step and transition of the A-exciton to the excited |f state via the second photon absorption. On the level of free electron-hole pairs the process is illustrated in Supplementary Fig. 5. In this process the generation rate of the excited excitons takes the form This dependence is weaker than I 2 due to possible saturation of linear absorption. Particu- Supplementary Table I where N ± A are the occupancies of σ ± polarized A:1s-excitons, τ B is the lifetime of B-excitons irrelated with relaxation towards the A-states, W describes the rate of the relaxation to A:1s-excitonic state. This description is simplified as we neglect spin/valley relaxation of excitons and, moreover, the exciton relaxation from B-to A-state can be with multiple steps, in which case a cascaded process may be relevant [8]. The circular polarization degree of B-excitons can be expressed, in the limit W τ B (1 + N ± A ) 1 as where