Neutral and charged inter-valley biexcitons in monolayer MoSe2

In atomically thin transition metal dichalcogenides (TMDs), reduced dielectric screening of the Coulomb interaction leads to strongly correlated many-body states, including excitons and trions, that dominate the optical properties. Higher-order states, such as bound biexcitons, are possible but are difficult to identify unambiguously using linear optical spectroscopy methods. Here, we implement polarization-resolved two-dimensional coherent spectroscopy (2DCS) to unravel the complex optical response of monolayer MoSe2 and identify multiple higher-order correlated states. Decisive signatures of neutral and charged inter-valley biexcitons appear in cross-polarized two-dimensional spectra as distinct resonances with respective ∼20 and ∼5 meV binding energies—similar to recent calculations using variational and Monte Carlo methods. A theoretical model considering the valley-dependent optical selection rules reveals the quantum pathways that give rise to these states. Inter-valley biexcitons identified here, comprising of neutral and charged excitons from different valleys, offer new opportunities for developing ultrathin biexciton lasers and polarization-entangled photon sources.


Supplementary Figure 3:
Level scheme used to calculate the optical response for excitation with circularly polarized light. The nine-level energy scheme shows the possible excitation pathways using + (solid lines) and − (dashed-dotted lines) circularly polarized light. Red arrows mark the excitation of uncharged exciton states; blue arrows indicate the excitation of negatively charged exciton states with an additional electron (i.e., trions). The level scheme consists of the ground state as well as the singly and doubly excited states in manifolds e and f. Due to many-body interactions, the doubly excited states are shifted by Δ , Δ , and Δ with respect to the sum of the individual transitions they are built of (grey, dashed levels).

Supplementary Note 1: Eliminating Inhomogeneous Broadening using 2DCS
Optical 2DCS has been previously implemented to investigate biexcitons in semiconductor quantum wells and quantum dots. Using 2DCS, the biexciton can be isolated from resonances associated with other quasiparticles in the two-dimensional frequency plane as shown in Fig. 3 of the main text. Supplementary Figure 2 highlights this point. The top two panels depict the projection of the 2D spectrum onto the emission energy axis for co-circular (left) and cross-circular (right) polarization schemes. In this case, several, inhomogeneously broadened contributions to each peak spectrally overlap, which precludes independent analysis of the exciton, trion, and biexciton. This type of projection demonstrates the challenges associated with one-dimensional linear and nonlinear spectroscopy techniques for which inhomogeneity can mask the optical response even in samples with reasonable quality (i.e. small-to-moderate Stokes shifts and inhomogeneous broadening). By taking a slice in the 2D spectrum, shown in the bottom panels in Supplementary Figure 1, the effects of inhomogeneous broadening are eliminated and the exciton, neutral biexciton, and charged biexciton can be analyzed independently.

Supplementary Note 2: Spin-and valley-selective optical transitions in monolayer MoSe2
Within the effective mass approximation, a parabolic band structure is used around the optically Note that our model system considers a common ground state for both the exciton and trion.
The background charge carrier density of unbound electrons and holes is also present in the ground state 7 . An optical excitation creates an additional electron-hole pair, which can either form a neutral bound exciton state or capture a background carrier to form a bound trion state.
We show in Supplementary Figure 4 one specific pathway corresponding to the bound charged exciton state that appears in the 2D spectrum in Figs. 3b and 4b of the main text as peak XT b . The excitation pulse sequence and polarization scheme is the same as for the bound neutral biexciton; however, in the present case, the excitation and emission energies of the quantum pathway are and + − Δ , respectively.

Supplementary Note 3: Calculation of the rephasing one-quantum signal
In the following, the nonlinear optical response is calculated following Ref. [8]. In a four-wave mixing (FWM) experiment, a sequence of three pulses with defined polarization directions, envelopes ℰ , laser frequencies Ω , wave vectors j , and variable delay times is applied to the system (cf. Fig. 2b in the main text). Each pulse is centered at time : The signal depends on the delay times between the pulses (cf. Fig. 1 in the main text): The induced polarization s ( ) can be calculated using the third-order response function (3) : To extract the rephasing contribution, the heterodyne signal is detected along the phasematched direction s = − 1 + 2 + 3 . The corresponding rephasing 2D spectrum is composed Note that cross peaks in the spectra indicate coherent couplings between the excited states, since many-body interactions break the symmetry between the ↔ and ↔ transitions. As a consequence, the Liouville space pathways involving singly excited states (lower transitions) are not canceled by the pathway including the doubly excited states (upper transitions).
The possible optical transitions with their dipole strengths and energies are listed in Supplementary Table 1. The exciton states and ′ are denoted as , the trion states and ′ as , and the two mixed exciton-trion states ′ and ′ as , since they are of the same type (but can be selected using circularly polarized light). Note that in Figs. 3 and 4 of the main text, the two exciton-trion cross-peaks below and above the diagonal are labeled and , respectively, and are marked by a superscript ( and ) when they are interaction shifted.
Inhomogeneous broadening is included by averaging the signal functions for normally distributed values of the system resonances. We assume a perfect correlation of the transition energies during 1 and 3 : the changes in transition energy during 1 are directly correlated to the changes in transition energy during 3 . This is a reasonable assumption, since the (fixed) delay time 2 between the second and third pulse is vanishing and therefore no spectral diffusion processes will take place between 1 and 3 . The inhomogeneous broadening is set to inh = 4 meV. It leads to elongated peaks along the diagonal (ℏ 1 = −ℏ 3 ). The broadening of the cross-diagonal is determined by the homogeneous linewidth, which was set to hom = 2.0 meV for all transitions.
In order to consider laser bandwidth effects, we approximate the excitation laser spectrum by a Gaussian-shaped spectral distribution with the half width at half maximum laser = 30 meV and the central wavelength of the pulse c = 765 nm (coinciding with the trion resonance). All parameters were chosen in agreement with the experiment.

Supplementary Note 4: Discussion of the calculated spectra
The calculated rephasing one-quantum signals for two different polarization combinations are shown in Fig. 4 in the main text.  Table 1) along the emission energy axis. This value is nearly identical to the biexciton binding energy predicted from microscopic calculations 2,3 . The off-diagonal coupling peaks and are red-shifted by a few meV along the emission energy axis compared to the spectrum for co-circular excitation. This effect is directly connected to the ESA contributions involving doubly excited charged fiveparticle bound states ′ and ′ composed of an exciton in one and a trion in the other valley. The associated exciton-trion binding energy Δ = 5 meV is considerably smaller than the stronger bound biexciton state , which is in line with previous experimental and theoretical studies 3,9 . Since Δ is in the same order of magnitude as the homogeneous and inhomogeneous broadening, the un-shifted cross-diagonal and peaks stemming from the singly-excited state contribution (ESE pathways) spectrally overlap and interfere with the doubly-excited exciton-trion peaks from the ESA contribution. A similar interference between ESE and ESA pathways for the trion peak T occurs with an estimated trion-trion binding energy of Δ = 2 meV. Due to this small binding energy, the → transition energy is not well separated from the → transition and therefore they partly cancel each other out, which leads to a smaller overall amplitude of the superposed trion peak compared to the exciton peak . Note that this phenomenological model only provides an estimate of the trion-trion and exciton-trion binding energies, since effects such as doping concentrations, sample quality, excitation power, degree of circular polarization, higherorder effects beyond (3) , etc. may have appreciable impact on the spectral signatures and the bound quasi-particle states.