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1 doi:.8/nature Supplementary Figure Temperature dependent PL decay. Hydrostatically (a, c) and biaxially (b, d) strained CQDs show very similar temperature dependence. It should be noted that the lowest-energy excitons in CQDs are dark due to spin-selection rules, but the nearest bright states are only mev away and photoluminescence comes from these thermally populated bright states. Biaxial strain does not affect the electron-hole exchange interaction strength, and the darkbright splitting remains small (Fig. a-b and Extended Data Fig. d). Furthermore, it is known that the dark-bright states splitting results in slower PL decays at low temperature in shell-free dots. However, the trend in thick-shell dots is opposite due to the efficient negative charging of the dots in the absence of molecular oxygen in the environment. Trion s lowest excitonic state is bright, resulting in faster PL decay at low temperature. Thus, temperature-dependent PL decay is shown to be similar in the two types of CQDs. The insets show the TEM images of two types CQD.

2 doi:.8/nature a c M CdS VASP CdSe VASP M A A b d CdS QNANO M CdSe QNANO M A A e g CdSe Hydrostatic - - CdSe Biaxial in XY Z fixed f h CdS Hydrostatic - - CdS Biaxial in XY Z fixed Supplementary Figure Comparison of CdSe and CdS band structures computed with VASP and fitted with QNANO, DFT-calculated dependences of band energies vs. strain. Tight-binding reproduces the behavior exactly.

3 doi:.8/nature a. Hydrostatically strained CQDs b. Biaxially strained CQDs A c J/cm - 7 J/cm - 7 J/cm - 7 J/cm - 87 J/cm - 7 J/cm - 77 J/cm - 87 J/cm - 97 J/cm J/cm A d A + A. A + A Supplementary Figure Optical gain threshold measurements. Delta A and delta A+A of hydrostatically (a, b) and biaxially (c, d) strained CQD solution, respectively. The optical densities were carefully adjusted to. with mm light path length to ensure they absorb the same number of photons. Photoexciting energy was selected as.8 ev (7 nm) to excite core absorption to minimize the possible differences in absorption cross section. Comparable PL red shift after shell growth (Fig. a and b) between hydrostatically and biaxially strained dots suggests that electron confinement, and thus core absorption cross section, are similar, allowing for comparing the thresholds. The instantaneous total absorption was collected 7 ps after pulsed excitation, and this provides the sum of ground state absorption and bleaching. At a certain power, the absorption turns negative, giving evidence of optical gain. There is no excited state absorption on the red side of the first exciton peak; it is harder to ascertain this for higher energy photons but they become less relevant for achieving optical gain. By comparing the PL (Fig. ) and ASE peak positions (Extended Data Fig. 7), we verified that biexciton binding energies in biaxially strained dots remain comparable to control dots (sub- mev), insufficient to affect noticeably the gain threshold (Simulation Methods and Extended Data Fig. de).

4 doi:.8/nature Supplementary Figure SEM cross section, and AFM images of CQD films. a, Crosssectional SEM of hydrostatically strained CQD film, showing a thickness of nm. b, AFM image and height trace of hydrostatically strained CQDs. c, Cross-sectional SEM of hydrostatically strained CQD film, showing a nm film thickness. d, AFM image and height trace of biaxially strained CQDs. Supplementary Figure PC-DFB substrate and Cl - exchanged biaxially strained CQDs used for CW lasing. a, AFM image of the PC-DFB device, showing MgF pillars on top of a single crystal MgF substrate. The y-axis has been shifted to make the pillar height relative to the substrate surface at nm. Dark lines are scratches inherent in the substrate. The insets show height trace of ~ nm pillars in the grating. b, Multiple exciton lifetimes of biaxially strained CQDs before and after Cl - exchange: they have quite similar decay behavior with given photoexcitation intensity (.8 ev, 7 nm, 7 µj/cm ).

5 doi:.8/nature Supplementary Table Theoretical simulations parameters. Anion-cation tight-binding parameters Elastic constants and tight binding strain parameters Parameter CdSe CdS Parameter CdSe CdS lattice_constant_a.98 A.7 A E_s lattice_constant_c 7.9 A.7 A E_p.7.87 c E_pz.7.9 c.9. E_sstar.8.88 c..89 E_d Delta_so.9.7 Alpha (bond stretching) Beta (bond bending) T_ss N_ss.9877.E-8 T_sp N_sp.8.9 T_ps..9 N_ps.8.9 T_pp_sigma.77.8 N_pp_sigma.98.9 T_pp_pi N_pp_pi. 7.9 T_ssstar.9E N_ssstar T_sstars N_sstars T_psstar.7.8E-8 N_psstar.7.9 T_sstarp 8.E-7.8 N_sstarp.7.9 T_sstarsstar N_sstarsstar T_sd N_sd.7. T_ds E-7 N_ds.7. T_pd_sigma N_pd_sigma..8 T_dp_sigma N_dp_sigma..8 T_pd_pi.87. N_pd_pi

6 doi:.8/nature T_dp_pi.9.9 N_dp_pi.999. T_sstard -.7E N_sstard T_dsstar.7.7E-7 N_dsstar T_dd_sigma N_dd_sigma T_dd_pi N_dd_pi T_dd_delta N_dd_delta E_strain_shift 7 7. C_diag C_ss.7.89 C_sp..97 C_ps..99 C_pp C_ssstar C_sstars C_psstar.E-.9 C_sstarp C_sstarsstar C_sd.E-.89 C_ds C_pd.99.9E-8 C_dp E-8 C_sstard.77.9 C_dsstar.E-.8 C_dd 7.E-.9E-8 References

7 doi:.8/nature Javaux, C. et al. Thermal Activation of Non-Radiative Auger Recombination in Charged Colloidal Nanocrystals. Nat. Nanotechnol. 8, (). Califano, M., Franceschetti, A. & Zunger, A. Temperature Dependence of Excitonic Radiative Decay in CdSe Quantum Dots: The Role of Surface Hole Traps. Nano Lett., - (). 7