Size-Dependent Biexciton Quantum Yields and Carrier Dynamics of Quasi-

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1 Supporting Information Size-Dependent Biexciton Quantum Yields and Carrier Dynamics of Quasi- Two-Dimensional Core/Shell Nanoplatelets Xuedan Ma, Benjamin T. Diroll, Wooje Cho, Igor Fedin, Richard D. Schaller, Dmitri V. Talapin, Stephen K. Gray, Gary P. Wiederrecht, David J. Gosztola S1. Absorption and PL spectra of CdSe core NPLs Figure S1: Absorption and PL spectra of CdSe core NPLs. S2. Carrier wave function calculations of the NPLs The additional 2ML CdS shells in sample NPL5 can lead to carrier wave functions delocalized further into the additional shell layers mainly in the thickness direction because excitons in quasi-2d NPLs are strongly confined in this direction. To evaluate the influence of this carrier wave function delocalization on the NPL radiative lifetime, we calculate the electron/hole wave functions in the thickness direction using an effective mass model. 1 We assume that the thickness of the CdSe core is 1.2 nm, each CdS monolayer (ML) shell is 0.35 nm thick, the bulk band gaps of CdSe and CdS are 1.74 ev and 2.40 ev, the valence band offset between CdSe and CdS is

2 0.42 ev, the effective masses m e (CdSe) = 0.13 m 0, m hh (CdSe) = 0.90 m 0, m e (CdS) = 0.20 m 0, m hh (CdS) = 0.70 m 0, respectively. 2-6 The surrounding ligands are assumed to have an offset of 3.5 ev, and an effective mass of 1.0 m 0. Figure S2(a) below shows the calculated radial electron (top) and hole (bottom, inverted) wave functions R e,h (r) for the lowest energy states. It can be seen that coating the CdSe cores with CdS shells causes delocalization of the electron/hole wave functions into the shells. The radiative decay rate k r of a strongly confined system is directly proportional to the electron-hole wave function overlap integral Θ e-h, which is defined as Θ e-h = R e (r) R h (r)r 2 dr 2. Figure S2(b) shows the calculated shell-thickness dependent Θ e-h values. From 0ML to 2ML and eventually to 4ML CdS shells, the corresponding values of Θ e-h decreases from 0.91 to 0.88, and finally to Therefore, for sample NPL5, the additional two monolayer CdS shells cause only ~2.3% difference in the corresponding radiative lifetimes. Figure S2: (a) Calculated radial electron (top) and hole (bottom, inverted) wave functions R e,h (r) for the lowest energy states of a 1.2 nm thick CdSe core coated with different CdS shell thicknesses (core only: black; 2ML CdS: red; 4ML CdS: blue). (b) CdS shell-thickness dependent electron-hole wave function overlap integral Θ e-h values of 1.2 nm thick CdSe cores. S3. Determination of average absorbed photons per pulse N The absorption cross section of the NPL sample with the largest lateral size and shell thickness (NPL5, 33.4 ± 3.8 nm by 13.0 ± 2.8 nm, 4ML CdS) was assumed to be 1.4 x cm 2 according to Ref. 7 and 8. The laser wavelength and repetition rate are 400 nm and 1MHz, respectively. At the excitation power of ~ 1 nw and emission wavelength of 656 nm, we get from 9

3 C abs power N = photon energy spot size laser repetition rate that N ~0.14. Since the smaller NPLs have smaller absorption cross sections than that of sample NPL5 and the laser excitation power was kept below 1 nw during the study, we conclude that N was kept below ~0.14 for all the single NPL measurements. S4. Contribution of biexciton emission at low excitation powers A Poissonian absorption process with an average absorbed photons per pulse of N can be presented as: P(n, N ) = N n e N. n! The spectrally integrated single exciton and biexciton emission of a single NPL can be written as: 10 I 2X = Respectively. Therefore, I X = n=2 n=1 P(n, N )Q 1X = Q 1X (1 P(0, N )) P(n, N ) Q 2X = Q 2X (1 P(1, N ) P(0, N )) I 2X I X = Q 2X 1 P(1, N ) P(0, N ) Q 1X 1 P(0, N ) We define η = 1 P(1, N ) P(0, N ). Figure S3 shows N dependent η values. When N 0, the 1 P(0, N ) value of η, and consequently I 2X becomes negligible. In this experiment, N is kept below Therefore, η is below 0.1.

4 Figure S3: N dependent η values. S5. Pump-dependent g (2) (τ) measurements Figure S4: Pump-dependent g (2) (τ) traces at excitation powers corresponding to N = 0.14 (top curve) and N = (middle curve). No apparent change in the R value is observed from the two excitation powers, indicating our predication of N 1 is valid. The bottom curve is obtained by time gating the top curve at a gate delay time of 80 ns. The disappearance of the center peak suggests that only a single NPL is studied. S6. Size-dependent Q 2X /Q 1X values

5 PL intensity (a.u.) Figure S5: Figure 5(a) from the main text only showing the data points corresponding to samples NPL2 to NPL4. S7. Pump-power-dependent PL measurements 10 2 NPL1 NPL2 NPL NPL4 NPL <N> Figure S6: Pump-power-dependent PL intensities of NPL thin films. For clarity, all the curves are normalized to the PL intensities at N = 1. S8. Maximum PL intensity decay curves

6 Figure S7: PL decay curves of the small (blue) and large (black) NPLs mentioned in the main text. The decay curves are constructed from photons emitted during the maximum PL intensity periods. The red curves are single exponential fittings to the decay curves. The gray curve is the instrument response function. S9. PL intensity dependent lifetime analysis Figure S8. (a, e) PL timetraces of a small NPL from sample NPL1 (a) and a large NPL from sample NPL4 (e) with the background subtracted. The time bin is set to 100 ms. (b, f) PL intensity distributions extracted from (a) and (e). (c, g) PL intensity dependent decay lifetimes τ s determined by fitting the corresponding PL decay curves with stretched exponential functions. Here the y-axis is the corresponding bin center. Uncertainties in the fitted lifetime

7 values are plotted in orange. (d, h) g (2) (τ) traces of the small (d) and large (h) NPLs, respectively. The corresponding Q 2X /Q 1X values are 0.05 and 0.79, respectively. Figure S9. (a, e, i) PL timetraces of individual NPLs from sample NPL2 (a), NPL3 (e), and NPL5 (i) with the background subtracted. The time bin is set to 100 ms. (b, f, j) PL intensity distributions extracted from (a), (e), and (i). (c, g, k) PL intensity dependent decay lifetimes τ s determined by fitting the corresponding PL decay curves with stretched exponential functions. Here the y-axis is the corresponding bin center. Uncertainties in the fitted lifetime values are plotted in orange. (d, h, l) Decay rate distributions of the NPLs constructed from the stretched exponential fitting results in (c), (g), and (k). The color coding of the rate distribution curves are consistent with the colored circles in (c), (g), and (k). S10. Probability densities of on- and off-state duration times

8 Figure S10. Probability densities of on (left) and off (right) time intervals of the timetraces shown in Figure S8(a) and (e) of the Supporting Information plotted on a log-log scale. The red lines are linear fittings to the data. S11. Influence of defects on the effective exciton coherent motion area In principle, defects can serve as local traps of excitons. As a consequence, in the presence of defects, the effective exciton coherent motion area can be reduced and the effective radiative decay rate (k r ) can deviate from the electric dipole approximation, i.e., k r A c,eff with A c,eff being the effective exciton coherent motion area that is smaller than the real NPL size A. If we assume that defects are the sole reason for the deviation of radiative lifetime from the electric dipole approximation, in a simple assumption, we can define that the number of defects N defect is proportional to A A c,eff, i.e. N defect A A c,eff. From Figure 5(b), we observe that from NPL2 to NPL5, although there is an increase in the NPL size A, no significant change in the radiative lifetime is observed, indicating that k r and A c,eff remain mostly unchanged. Therefore, we can derive that the number of defects N defect increases with size, and this leads to an increase in the nonradiative decay rate k nr. Since Q 1x = k r /(k r +k nr ) and no significant change in k r is observed from Figure 5(b), we can derive that Q 1x should decrease with the NPL size. However, from Figure 5(c) we can clearly observe that this is not the case. Therefore, we can conclude that

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