Ultralow Auger-Assisted Interlayer Exciton Annihilation in WS2/WSe2 Moiré Heterobilayers

Transition metal dichalcogenide (TMD) heterobilayers have emerged as a promising platform for exploring solid-state quantum simulators and many-body quantum phenomena. Their type II band alignment, combined with the moiré superlattice, inevitably leads to nontrivial exciton interactions and dynamics. Here, we unveil the distinct Auger annihilation processes for delocalized interlayer excitons in WS2/WSe2 moiré heterobilayers. By fitting the characteristic efficiency droop and bimolecular recombination rate, we quantitatively determine an ultralow Auger coefficient of 1.3 × 10–5 cm2 s–1, which is >100-fold smaller than that of excitons in TMD monolayers. In addition, we reveal selective exciton upconversion into the WSe2 layer, which highlights the significance of intralayer electron Coulomb interactions in dictating the microscopic scattering pathways. The distinct Auger processes arising from spatial electron–hole separation have important implications for TMD heterobilayers while endowing interlayer excitons and their strongly correlated states with unique layer degrees of freedom.

and the curves are fitting results.In the experiment, the laser polarization and 2D crystals are left unaltered while the polarization angle of SHG is analyzed.Given that the angle between the crystal armchair direction and the SH field is two times the angle between the crystal armchair direction and the laser field, the twist angle  can thus be determined using the difference in polarization angle of the SHG signal between monolayers.The twist angle in this WS2/WSe2 heterobilayer is found to be  = 58°.(b) Reflection contrast spectra of WS2 monolayer, WSe2 monolayer, and WS2/WSe2 heterobilayer (hBL).The intralayer moiré excitons mX A WS 2 (mX A WSe 2 ) possess energies of 1.67-1.80eV (1.92-2.04eV), as previously reported [1].The emergence of multiple peaks demonstrates the formation of moiré minibands.(c) Raman characterization of the WS2/WSe2 heterobilayer, displaying individual features of the WS2 and WSe2 layers.Spectra have been shifted vertically for clarity.The presence of intralayer moiré excitons and interlayer phonon mode  2 1 demonstrates the high quality of the vdW interface.EQE demonstrates the absence of EQE droop for both X WS 2 (c) and X WSe 2 (d).These data show that exciton-exciton annihilation (EEA) is not the primary recombination channel in this excitation range.The reason behind is likely due to the much lower exciton population (  ) for the monolayer sample.As discussed in Note S6, under the same excitation conditions, the population difference between intralayer exciton and IX is estimated to be on the order of ~10 3 .If we take the Auger coefficient   of the monolayer sample as 10 −3 ~10 −1 cm 2 s −1 [6][7][8][9][10][11][12], the Auger recombination rate     2 is estimated to be 10 2 − 10 4 lower than the heterobilayer sample.This indicates that the EQE decrease for the monolayer sample is already beyond the current maximum power.

Note S1. The power-dependent PL intensity and EQE.
The ±0. .In this regard, the power-dependent   and EQE can be written as   ∝     ∝  0.5 ∝  0.5 and EQE ∝    ⁄ ∝  −0.5 , resulting in ±0.5 exponents for   and EQE.This rate equation analysis demonstrates that Auger-assisted IX annihilation is the dominant recombination mechanism at high power.

Note S2. Estimation of photoexcited IX density.
The photocarrier generation rate  can be calculated by:  = , where F is the photon flux density (or photon irradiance) of the incident laser, and A is the total absorbance of the WS2/WSe2 heterobilayer.The photon flux density F can be calculated by:  =  ( 2 ℏ) ⁄ , where P is the average power, ℏ = 2.33 eV is the photon energy and  2 is the spot size of the focused 532nm laser.The overall absorbance of the WS2/WSe2 heterobilayer is estimated to be  = 9.7% utilizing the transfer matrix method to solve electric field distributions in the multilayer structure.
The model takes into account the absorbance of TMD layers as well as the reflection/transmission losses of the multilayer structure.The transmission loss of the vacuum chamber window and the objective lens is also considered.Under continuous-wave laser excitation with an average power  = 1 μW , the generation rate is estimated to be  = 1.13 × 10 18 cm −2 s −1 .At the slope conversion point with  = 200 nW, we estimate a steady-state IX density of 8.4 × 10 10 cm −2 at  = 4K, which is significantly lower than the Mott transition point reported for TMD heterobilayers [13,14].

Note S3. Calculation of IX annihilation rate using Fermi's golden rule.
As shown in Fig. 2e  In the first fitting set shown in Fig. S6, we only change   while keeping the other coefficients fixed.This fitting set already gives an adequate fit to all experimental data (Figs.S6a-b) and captures the effects of thermally activated   (Fig. S6c).The fixed parameters are   = 2.5 × 10 5 s −1 , and   = 1.3 × 10 −5 cm 2 s −1 .Nonetheless, the second fitting set provided in the main text obviously fits all experimental data better (see Fig. 3 in the main text).It is worth noting that both fitting sets demonstrate that the Auger coefficient is on the order of 10 −5 cm 2 s −1 .On the other hand, we determine an efficient   = 13.4 ns −1 at room temperature, which is quite similar to the previously reported value [15].As a result, determining   for IXs at  ≥ 150K becomes problematic.For example, the time constant of IX annihilation is 20 ns for an IX density of around 10 12 cm −2 .At  ≥ 150K, however, the average nonradiative time   −1 is already less than 1 ns, which can cause a significant inaccuracy in determining   .As a result, our experimental findings show that nonradiative recombination must be considered when analyzing IX kinetics at high temperatures, especially with such a low Auger-assisted IX annihilation rate.We note that phonon-assisted recombination/relaxation of IXs also plays an important role in TMD moiré heterobilayers.However, since the phonon-assisted process is a low-order process, it cannot explain bimolecular recombination at a (    2 ) that depends quadratically on IX density, as shown by the characteristic EQE droop and density-dependent TRPL decay.We realize that, while this Auger recombination occurs under high-power/low-temperature conditions, the phononassisted process is reflected in the nonradiative recombination rate     , which dominates IX recombination at high temperatures.Additionally, we realize that the type of IXs may also play a role.Specifically, our results show Auger recombination of delocalized IXs, which behaves differently from moiré-trapped IXs.In the following, we compare their spectral characteristics and demonstrate delocalized IXs using diffusion measurements.
This distance is much larger than the moiré periodicity (~8 nm) and the spatial extent of an IX, indicating that diffusion must precede Auger recombination [6,10].As shown in Fig. S8c, spatially-resolved PL images reveal IX diffusion in the WS2/WSe2 heterobilayer, demonstrating that IXs are delocalized from the moiré potential.The distinct IX dynamics observed in the WS2/WSe2 heterobilayer indicate a rich moiré exciton phenomenon in TMD heterobilayers.We also realize that the difference between MoSe2/WSe2 and WS2/WSe2 heterobilayers may be related to moiré periodicity and specific excitation/material conditions [21][22][23][24], resulting in moiré-trapped IX phase for the former while delocalized IX phase for the latter.

Note S6. The negligible effect of intralayer excitons on IX recombination.
Here, we have included additional TRPL data to clarify the interaction between exciton and IX.
Below, we show the negligible effect of intralayer excitons on IX recombination, which is concluded from their significant difference in exciton lifetime and population on the order of ~10 3 .
This ensures that IX-IX interactions are responsible for nonradiative bimolecular recombination.
Figure S9 shows the TRPL traces of WS2 excitons, WSe2 excitons, and IXs in the WS2/WSe2 heterobilayer measured at  = 4K.The PL spectrum reveals various intralayer exciton complexes including: neutral exciton X 0 , charged exciton T, and other exciton complexes (Fig. S9a).It is observed that intralayer excitons exhibit much faster decays compared to IXs (Figs.S9b-d).The main decay lifetimes of WS2 excitons and WSe2 excitons are in the range of 0.2 − 0.5 ns (Figs. S9c-d and Table S1), which are significantly shorter than the IX lifetime (> 450 ns) and the time constant of IX-IX annihilation (~20 ns).This large lifetime difference therefore results in the negligible effect of intralayer excitons on IX recombination in TRPL measurements.On the other hand, in PL measurements using continuous-wave laser excitation, the lifetime difference translates into the difference in steady-state exciton population, i.e., yielding a huge population difference on the order of ~10 3 .In this context, our results demonstrate that IX recombination is primarily affected by IX-IX interactions.Finally, the interaction between IX and intralayer exciton in moiré superlattice is an important topic to explore.Following the above discussion, we realize that studying such interaction requires probing exciton dynamics on sub-nanosecond time scales, which is not feasible in current TRPL experiments due to limited time resolution (~0.45 ns).We anticipate that more advanced investigations, such as pump-probe measurements, will be needed to study these exciton interactions.

Figure S2 .
Figure S2.Double-exponential analysis of TRPL decay traces.(a) TRPL traces fitted with double-exponential functions.The dots are experimental data, and the curves are fitting results.(b) Power-dependent decay time constants  1 and  2 .(c) The relative intensity  at various excitation powers.In Fig. S2a, we present fits to the TRPL data using a double-exponential decay function: y() =  −  1 ⁄ + (1 − ) −  2 ⁄ , where  and (1 − ) denote the relative intensity of the fastdecay ( 1 ) and slow-decay ( 2 ) components.As shown in Fig. S2b, the time constant of the fastdecay component ( 1 ) decreases at high P, while that of the slow-decay component ( 2 ) remains nearly unchanged.As shown in Fig. S2c, the relative intensity  of the fast-decay component increases significantly as a function of P. Both behaviors indicate that the fast-decay (slow-decay) component mainly comes from the Auger-assisted annihilation (radiative recombination) of IXs.The density-dependent fast-decay component on short timescales is typical of bimolecular population decay caused by Auger-assisted IX annihilation.

Figure S4 .
Figure S4.TRPL trace fitting at high temperatures.(a) TRPL traces fitted with singleexponential functions.TRPL traces have been vertically shifted for clarity.(b) Instrument response function (IRF) of the time-correlated single photon counting system.The dots are experimental data, and the curves are fitting results.The IRF can be fitted by: y() = 0.79 − 0.45 ⁄

Figure S5 .
Figure S5.Power-dependent PL of monolayer WS2 and WSe2.(a-b) Power-dependent PL spectra of monolayer WS2 (a) and monolayer WSe2 (b) measured at T= 4K.It should be noted that multiple emission lines with energies lower than neutral excitons X0 are typical of exciton complexes in tungsten-based TMDs [2-5].The spectra in (a) with energies above 2.075 eV have been zoomed in by a factor of 10 to better visualize neutral excitons X0. (c-d) Power-dependent

Figure S6 .
Figure S6.The influence of temperature on interlayer exciton kinetics.(a-b) Density dependence of integrated PL intensity (a) and EQE (b) measured at elevated temperatures.The dots are experimental data, and the curves are fitting results.(c) Temperature dependence of   .In this fitting set, we only change   while keeping the other coefficients fixed.This fitting set already gives an adequate fit to all experimental data and captures the effects of thermally activated   .The most inaccurate fits are found at T= 50K and 100K, as shown by the blue dashed area.It should be noted that, while this fitting set provides an adequate fit, the second fitting set provided in the main text obviously gives a better fit to all experimental data (see Fig.3in the main text).

Figure S7 .
Figure S7.The influence of temperature on interlayer exciton kinetics.(a-b) Density dependence of integrated PL intensity (a) and EQE (b) measured at elevated temperatures.The dots are experimental data, and the curves are fitting results.

Figure S8 .
Figure S8.Interlayer excitons in the WS2/WSe2 heterobilayer.(a) Normalized PL spectra at  = 4K.As laser power increases, IX emission exhibits a nearly constant linewidth.(b) The integrated PL intensity and EQE of IX emission as a function of power.Note that PL efficiency droop spans nearly 4 orders of magnitude power range (blue region).(c) Spatially-resolved PL and laser images with P = 10 nW, where the data (dots) are fitted by Gaussian functions (lines).IX diffusion is evidenced by the wider PL profile beyond the laser spot.

Figure S9 .
Figure S9.TRPL of intralayer excitons in the WS2/WSe2 heterobilayer.(a) The PL spectrum at T= 4K shows emissions from WS2 excitons, WSe2 excitons, and IXs.The spectrum with energies above 1.53 eV has been zoomed in by a factor of 10.(b) The TRPL traces of WS2 neutral excitons and IXs in the WS2/WSe2 heterobilayer, showing a much faster decay of WS2 neutral excitons.TRPL traces of WS2 excitons (c) and WSe2 excitons (d), in which TRPL traces have been shifted vertically for clarity.TRPL traces have been deconvoluted from the IRF and fitted with single-exponential (double-exponential) functions for WS2 excitons (WSe2 excitons).The gray lines are experimental data, and the colored curves are fitting results.
5 exponents of   and EQE at high power are direct features of Auger-assisted IX annihilation.Because free carriers prefer form IXs at high levels of excitation, the IX kinetic model can thus be simplified as:    ⁄ =  −     −     −     2 , where   is the IX density,  is the photocarrier generation rate,   is the radiative recombination coefficient, and   is the Auger coefficient.Under steady-state excitation with a high IX density   , we have (    +     ) ≪     2 , which leads to  ≅     2

Note S4. Fitting of temperature-dependent IX kinetics.
in the main text, an Auger-assisted IX annihilation event is directly linked to one IX being recombined nonradiatively with energy absorbed by another IX to its excited state concurrently.In this case, the IX annihilation rate can be calculated using Fermi's golden rule between two interacting dipoles.The transition rate from an initial state |  ⟩ with total energy   to a final state |  ⟩ with total energy   is given by: Γ → =   −   ).The initial state is two IXs with energy   , while the final state is one IX with energy 2  , resulting in   −   = 0.The dipole-dipole interaction potential can be approximated as V ∝     3⁄ ,where   (  ) is the transition dipole moment of the energy "donor" ("acceptor") IX and  is the inter-excitonic distance.By assuming that the wavefunction can be separated as|  ⟩ = |   ⟩|   ⟩and |  ⟩ = | 0  ⟩|   ⟩, one can thus obtain the Auger-assisted IX annihilation rate as:Γ → =The proposed kinetic model accurately reproduces the entire temperature-dependent behavior of PL intensity and EQE.Two separate fitting sets are employed to examine the roles of   and   .
2 is proportional to the energy donor IX's radiative recombination rate, whereas the second matrix element |⟨   |  |   ⟩| 2 is proportional to the energy acceptor IX's absorption rate to the high-energy continuum state.Therefore, a weak oscillator strength (a long radiative lifetime) leads to a low Auger annihilation coefficient.This is the reason behind the ultralow Auger coefficient of IXs in TMD heterobilayers compared to excitons in TMD monolayers.

Table S1 . Lifetimes of WS2 excitons and WSe2 excitons in the WS2/WSe2 heterobilayer.
Notethat  1 may not be determined accurately since the overall time resolution of the TRPL system is ~0.45 ns.