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[2024A00865G_Trilayer_Supplementary.pdf](https://mdr.nims.go.jp/filesets/cee3c1d4-6517-4a0f-9937-21db97396ec8/download)

## Creator

Liuxin Gu, Lifu Zhang, Ruihao Ni, Ming Xie, Dominik S. Wild, Suji Park, Houk Jang, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Mohammad Hafezi, You Zhou

## Rights

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s <a href="https://www.springernature.com/gp/open-science/policies/accepted-manuscript-terms">AM terms of use</a>, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1038/s41566-024-01434-x[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Giant optical nonlinearity of Fermi polarons in atomically thin semiconductors](https://mdr.nims.go.jp/datasets/57f71c49-c4c9-4aef-b703-5e5cce84d584)

## Fulltext

Summary of Content   Section 1. Analysis of Anti-Crossing between IXD and XA  Section 2. Doping-dependent photoluminescence map of trilayer WSe2  Section 3. Effect of optical pumping on the oscillator strength of XA+  Section 4. Optical nonlinearity under resonant excitation   Section 5. Effect of pumping photon energy on nonlinearity  Section 6. Optical nonlinearity under different powers in various devices  Section 7. Effect of CW vs. pulsed excitation  Section 8. Temperature dependence of XA+   Section 9. Estimation of XI density  Section 10. Effect of circular polarization on nonlinearity   Section 11. Electric-field tuning of Fermi polarons  Section 12. Comparison of nonlinearity with previous work   Reference     Section 1. Analysis of Anti-Crossing between IXD and XA    Figure S1 Anti-crossing between IXD and XA for various doping concentrations in device D2. a, Reflectance spectra (R/R0) as a function of the electric field in the intrinsic regime. b, Differential reflectance spectrum (d(R/R0)/dE) as a function of the electric field. The anti-crossing takes place at ~0.05 V/nm, consistent with device D1. c, d, Zoom-in view of the differential reflectance under electron (c) and hole(d) doped with an applied electric field. The white dashed lines represent the energies fitted with a two-level model. The fitted coupling strength W for electron and hole-doped side is around ~10 meV and comparable with that in intrinsic trilayer.  Figure S2 Analysis of anti-crossing between IXD and XA in device D1. a, Reflectance spectra (R/R0) as a function of the electric field near the anti-crossing region. b, Voltage derivative of reflectance spectra, d(R/R0)/dV. The white dashed lines in (a) and (b) represent the energies fitted with a two-level model. c, d, We study the anti-crossing between the IXD and XA based on a two-level system with a Hamiltonian:  𝐻 =  (𝐸1 𝑊𝑊 𝐸2) where E1 and E2 are the unperturbed energies of the IXD and XA, respectively, and W is the coupling strength. The new eigenvalues can be expressed as: 𝐸±  =  12(𝐸1 + 𝐸2) ±12√(𝐸1  −  𝐸2)2 + 4|𝑊|2 where 𝐸± correspond to the energies of the two branches. In (c), we extract the peak positions 𝐸± by fitting the reflectance spectra with the Lorentzian function. The error bars represent the variance of fitting peak energy. We then set E2 to be 1.715eV, which is the mean energy of XA, and keep it as a constant. E1 is calculated based on the IXD energy at zero electric fields and the stark shift. The stark shift slope 𝑘 is estimated to be -1.436 eV/V in this particular device. The fitted anti-crossing is shown in (d) with a fitting parameter of 𝑊 =  10 ± 2 𝑚𝑒𝑉.   Section 2. Doping-dependent photoluminescence map of trilayer WSe2   Figure S3 a, Doping-dependent photoluminescence of the trilayer WSe2 at Ez = 0 taken from D2 at 4 K. b, Degree of circular polarization (DOCP) of the XA+ and XA- . The bright emission in the range of 1.5~ 1.6 eV corresponds to the momentum indirect trion/Fermi polaron. In contrast, the higher energy emission around 1.7eV corresponds to the momentum direct (K-K) intralayer trion/Fermi polaron. Both charged excitons XI and XA exhibit a redshift with increasing doping density.     Section 3. Effect of optical pumping on the oscillator strength of XA+    Figure S4 The oscillator strength of XA+, extracted from the reflectance spectra of sample D3, remains almost unchanged under low pump power and begins to decrease with increasing excitation power, when the blueshift becomes obvious. Section 4. Optical nonlinearity under resonant excitation    Figure S5. a, Relative change in the reflectance induced by 1 𝜇W of resonant at resonant (718 to 730 nm) pulsed laser excitation under different doping. The color map is obtained by normalizing the reflectance change induced by the resonant excitation with respect to the reflectance without optical pumping,  ∆𝑅/𝑅 =  𝑅(1 𝜇𝑊)𝑅(0.1 𝜇𝑊)− 1. The pulse has ~100 ps duration with a 40 MHz repetition rate. b, c, Reflectance change induced by a pulsed laser excitation power of 1 𝜇W(b) and 3 𝜇W(c), as a function of electric field, under hole doping. Under a small electric field, XA+ shows a blueshift, but it begins to redshift under excitation at a higher electric field. With increasing power, this transition point shifts to a lower electric field.    Section 5. Effect of pumping photon energy on nonlinearity    Figure S6 (a, b) Blueshift of XA+ under laser excitation at different center wavelengths (~10 nm spectral width) with a fixed pumping power at (a) 16 𝜇𝑊 and (b) 66 𝜇𝑊. (c, d) The corresponding reflectance changes induced by optical pumping at different wavelengths show no significant wavelength dependence. (e, f) When exciting the system with photon energies below XA+, we did not observe significant blueshift (e), in contrast to higher energy excitation at 640nm (f). All data is acquired from device D3. We also note that the resonant excitation results in a much more pronounced blueshift (Fig. S7).   Section 6. Optical nonlinearity under different powers in various devices    Figure S7 Relative change in the reflectance induced by optical pumping as a function of doping for various devices under different excitation conditions. We notice that the reflectance change is smooth near 0V, particularly at low excitation power. (a) 10 𝜇W and high excitation power (b) 30 𝜇W with 635 nm CW laser pumping for Device D1. (c) 2.5 𝜇W and (d) 20 𝜇W 645nm pulsed laser excitation for device D1. (e) 10 𝜇W and (f) 20 𝜇W CW 635 nm laser pumping for Device D3.     Section 7. Effect of CW vs. pulsed excitation   Figure S8 Analysis of power-dependent blueshift of XA+ under CW laser (a, c) and white laser excitation (b, d). a,b, Extraction of interaction strength g from (a) CW laser and (b) pulsed laser pumping induced XA+ blueshift vs. exciton density for D1. The exciton density is calculated from pump flux based on 𝑛𝑋 = 𝑃𝛼𝜏/ℏ𝜔, where P is the pump power, 𝛼 is the absorption coefficient, 𝜏 is the lifetime of the Fermi polaron, ℏ𝜔 is the photon energy. The Fermi polaron lifetime is a few picoseconds, as measured in similar systems, and we use a value of 2 ps. The interaction strength is extracted from the linear fit of the 𝛥𝐸 − 𝑛𝑋 curve in the low exciton density regime. The larger g values under CW excitation could be related to the complex relaxation dynamics of the exciton populations and an overestimation of exciton density under pulsed excitation. We also fit the blue shift amount as ∆𝐸 =  𝑎 ∙ 𝑛𝑋(𝑏) over the entire data range. The fitting for CW laser and pulsed laser yields a coefficient of b as 0.52 with an R-square of 0.9516 for the CW laser and b of 0.32 with an R-square of 0.9317 for the pulsed laser, which shows a sublinear response for XA+ as polaron density. The hole doping density is kept at 8 x 1012 cm-2. The error bars represent the variance of fitting peak energy. c, d, The same data as shown in a, b, plotted on a semilog scale, to emphasize on the low power regime, with power as the x-axis. e, f, Power-dependent blueshift of XA+ under CW laser for e, device D1 and f, device D3. The error bars represent the variance of fitting peak energy.        Section 8. Temperature dependence of XA+    Figure S9 Temperature-dependent reflectance spectra of the trilayer under a constant doping density under zero electric field. In all cases, which include (a) intrinsic, (b) hole-doping, and (c) electron-doping, we observe strong redshift with increasing temperatures. Therefore, the observed nonlinearity, which corresponds to exciton blueshift, cannot be described as simple laser heating effects.                  Section 9. Estimation of XI density   Figure S10 Estimation of XI density as a function of pump power. a-c, Electric field-dependent PL map under different pump power (a) 3 𝜇w, (b) 50 𝜇w, (c)100 𝜇w at the trilayer region. At an electric field of 0.12 V/nm, a maximum blue shift in a value of 3.3 meV of the XI is observed. (f) XI exciton density inferred from the above XI blueshift under an applied electric field of 0.12 V/nm.                  Section 10.  Effect of circular polarization on Nonlinearity      Figure S11 Valley-polarized holes under resonant circularly polarized excitation. (a, b) Power-dependent blueshift of XA+ under resonant pumping with (a) 𝜎+/𝜎+ (pumping and probing K valley) and (b) 𝜎+/𝜎− configuration (pumping K, while probing K’ valley) when the sample is hole-doped. We observe a stronger blueshift of XA+ in (a). We observe a stronger blueshift of XA+ in (a). In particular, we observe a ~1.3 nm blueshift under 3 μW pump when the pump and probe are co-polarized and no obvious shift in the cross-polarized case. Further increasing the pumping power to 10 μW leads to a blueshift in the cross-polarized setup, albeit still being smaller than the co-polarized case, which suggests the holes are partially polarized in K vs. K’. The data is acquired from device D3. (c) Nonequilibrium hole accumulation in K and K’ valleys induced by selective valley pumping with a circularly polarized excitation. The resulting population imbalance between K and K’ causes different amounts of blueshift in XA+ nonlinearity between the two valleys.     Figure S12 Non-resonant excitation with circular polarized polarization. Different from the resonant excitation case (Fig. S12), under non-resonant circularly polarized pump (635 nm), we observe similar magnitude of blueshift XA+ under (a) 𝜎+/𝜎+and (b) 𝜎+/𝜎− configuration. This is likely due to the breakdown of valley-selective optical selection rules far from the band edge as well as fast depolarization of excitons and electrons during the relaxation process.    Section 10.  Electric-field tuning of Fermi polarons.   Figure S13 (a) Energy shift of XA- and XA+ with applied electric field with constant doping. The electron and hole doping densities are both kept at 4.9× 1012𝑐𝑚−2.The peak position is obtained by fitting the reflectance spectral with a Lorentzian model. The error bars represent the variance of fitting peak energy. b,c, Doping dependence of the intralayer Fermi polaron reflectance contrast R/R0 in trilayer with applied (a) 0.03 V/nm, (b) 0.05 V/nm electric field. The negative doping density represents hole hole-doped side. An obvious blueshift and broadening of XA+ is observed on the hole side with an increasing electric field, corresponding to the additional phase space filling due to the population transfer from Γ to K valleys. Such a shift is much weaker on the electron side.     Section 11.  Comparison of nonlinearity with previous work  Table S1  Here, we compare the amount of blueshift per pump power, which has important implications for low-power devices. Indeed, in our current work, much smaller pump power is needed to shift the absorption of excitons by a similar amount, in comparison with previous reports. This is related to the fact that the measured interaction strength g is much larger than previous reports and theoretical exchange/dipolar interactions. We also note that while one can observe a significant shift of interlayer excitons at relatively low power (Ref. [4, 5]), these species have negligible absorption because of long lifetime, as we discussed in the introduction of the paper.   This work* Ref. [1] Ref. [2] Ref. [3]  Ref. [4] Ref. [5] Species Fermi Polaron Hybridized Interlayer Exciton Interlayer Exciton 2s Exciton Polariton Interlayer Exciton Interlayer Exciton nx (cm-2) 109 ~4*1012 ~2*1010 ~6*109 1.2*1011 ~1012 ∆𝐸 (meV) 6 ~7 ~2.2 1-2 ~2 ~22 Power (𝜇W/𝜇m2) 37 3.25*106 1300 3*104 60 ~300 System Trilayer WSe2 Bilayer  MoS2 Bilayer MoS2 Monolayer WSe2 Bilayer WSe2 MoSe2/ hBN/ WSe2 Probe Absorption Absorption Absorption Absorption PL PL     References:  1. Datta, B. et al. Highly nonlinear dipolar exciton-polaritons in bilayer MoS2. Nat. Commun. 13, 1–7 (2022). 2. Louca, C. et al. Interspecies exciton interactions lead to enhanced nonlinearity of dipolar excitons and polaritons in MoS2 homobilayers. Nat. Commun. 14, 3818 (2023). 3. Gu, J. et al. Enhanced nonlinear interaction of polaritons via excitonic Rydberg states in monolayer WSe2. Nat. Commun. 12, 2269 (2021). 4. Wang, Z., Chiu, Y.-H., Honz, K., Mak, K. F. & Shan, J. Electrical Tuning of Interlayer Exciton Gases in WSe 2 Bilayers. Nano Lett. 18, 137–143 (2018). 5. Sun, Z. et al. Excitonic transport driven by repulsive dipolar interaction in a van der Waals heterostructure. Nat. Photonics 16, 79–85 (2022).