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Efficient generation of neutral and charged biexcitons in encapsulated WSe2 monolayers

Higher-order correlated excitonic states arise from the mutual interactions of excitons, which generally requires a significant exciton density and therefore high excitation levels. Here, we report the emergence of two biexcitons species, one neutral and one charged, in monolayer tungsten diselenide under moderate continuous-wave excitation. The efficient formation of biexcitons is facilitated by the long lifetime of the dark exciton state associated with a spin-forbidden transition, as well as improved sample quality from encapsulation between hexagonal boron nitride layers. From studies of the polarization and magnetic field dependence of the neutral biexciton, we conclude that this species is composed of a bright and a dark excitons residing in opposite valleys in momentum space. Our observations demonstrate that the distinctive features associated with biexciton states can be accessed at low light intensities and excitation densities.

The dark exciton related emission is identified by the radiation pattern of the PL, as imaged in the back focal plane of our collection objective. Using a Bertrand lens, we project the back focal plane onto the slit of an imaging spectrometer and thus measure the radiation pattern of each individual spectral peak [1]. Since the in-plane component of the dark exciton is negligible, it emits through coupling with the out-of-plane dipole of a higher-lying transition [2]. As a result, the dark exciton emits most strongly in the in-plane direction, and has a node in the normal direction, making it easily distinguishable from the emission patterns of any of the bright exciton species (Supplementary Fig. 1 Here, we consider the expected dependence of the biexciton emission on the exciton density, making use of the experimental intensity dependencies of the neutral and dark 2 excitons. We start from a system of rate equations that determine the densities of bright and dark exciton, as well as of the biexciton: where ρ X 0 , ρ X D and ρ XX are the densities and τ X 0 , τ X D and τ XX the lifetimes of X 0 , X D and XX and γ is the formation rate of the XX species, which we assume to be a constant.
P X 0 and P X D are the bright and dark exciton formation rates, which, in the steady-state low density regime, scale with the emission strength and therefore can be written as I α , where α is 1.22 and 1.03, respectively. We note that in these equations, we have neglected the possible role of the formation of dark biexcitons. In the low density regime, we do not expect this omission to be significant.
Solving for the steady-state density of the excitons, we obtain ρ X 0 = . The biexciton density is proportional to the product P X 0 P X D in the limit where ρ XX ρ X 0 τ XX τ X 0 . In this case, the biexciton emission is expected to scale with intensity as I α , α = 2.25. For higher biexciton densities, its power-dependence will flatten and become linearly proportional to the bright exciton formation rate, P X 0 . In our experiment, we are below the saturation regime, as the biexciton density increases nearly quadratically and its emission is weaker than that of the bright exciton.

Supplementary Note 3. Exciton collision and biexciton formation
Here we introduce a simple kinetic model to describe the formation of the observed biexciton. Biexcitons are created when two excitons collide. The formation rate is given accordingly by the collision frequency ν = zn a n b ,where n a,b are the (2D) densities of the constituent excitons. For thermalized excitons in 2D, we have z = σ ab · (πk B T )/(2µ ab ), where σ ab is the collision cross-section, k B T is the thermal energy, and µ ab = m a m b /(m a + m b ) ∼ 0.3m 0 is the reduced mass of the exciton. As the biexciton is composed of at least one bright exciton, the collision needs to occur within its lifetime τ X 0 . Given the radius of 3 1 nm in the exciton ground state [3], we estimate the collision cross-section as 2 nm. For the laser intensity at which we start to observe the emission from the biexciton feature, the bright exciton emission is 20 times stronger than the biexciton emission, indicating that n b /(ν · τ X 0 ) ∼ 20. This suggests the density of the other exciton species n a is about 370 µm −2 (for τ X 0 = 2 ps [4]). For our applied laser intensity I ∼ 10 Wcm −2 , this density level can be achieved for a long-lived dark exciton species.

Supplementary Note 4. Nonequilibrium distribution in the bright exciton pair
Under the strongest applied magnetic fields, the bright exciton pair manifests an inverted population distribution, i.e., the photoluminescence (PL) of the low energy branch (LEB) is weaker than of the low energy branch (HEB). Here, we discuss a mechanism that can lead to such a distribution.   Fig. 2). The deviation from the quadratic dependence of XX − was attributed to the partial equilibrium between the biexciton and the constituent exciton [9]. The slight superlinear power dependence of the trion is expected to have the same origin as that of the exciton in the intrinsic regime, which has been attributed to partial filling of defect states at low laser excitation. As the defects are gradually filled, the loss of excitons/trions to the defect states is reduced, thus giving rise to a superlinear power dependence of the emission intensity.

Supplementary Note 6. Polarization resolved biexciton emission
We also measured the polarization dependence of the biexciton emission (Fig. 3). Clearly the XX and XX − emission in Fig. 3a and 3b have the same circular polarization as the excitation, confirming their intrinsic nature. In addition, we observe that circular polarization of the photoluminescence is present over the entire spectrum, except for the emission of the dark exciton, which cannot support circular polarization with its out-of-plane transition dipole moment. We find that the degree of circular polarization is enhanced in the n doped regime and suppressed in the p doped regime (Fig. 3 b & c). No linear polarization is observed in either the XX or XX − peak.