A model for optical gain in colloidal nanoplatelets† †Electronic supplementary information (ESI) available: Synthesis procedure, additional transient absorption spectra and kinetics, fitting procedures, and fitting parameters. See DOI: 10.1039/c7sc04294a

Optical gain in CdSe nanoplatelets is shown to be independent on their lateral size and can be explained by a new optical gain model for 2D nanoplatelets.

. TEM images of (a) NPLa, (b) NPLb, and (c) NPLc. The length and width of all samples are determined by the solid lines marked in lower panels of (a) for NPLa, right panel of (b) for NPLb, and lower panel of (c) for NPLc.
The length, width, and area of all CdSe colloidal NPL samples are measured according to Figure S1 (NPLa to NPLc) and Figure 1a (NPLd). The distribution of NPL length and width is shown in Figure S2. The average length, width, and area with the errors are listed in Table S1.

Amplified Spontaneous Emission (ASE) measurement: all ASE measurements
were conducted at room temperature in air. The NPL film was pumped with the same 400 nm pulses described in TA set-up. The pump beam was focused into a stripe along the NPL films by a cylindrical lens. The length of the stripe was determined to be the same as the beam size before focusing (2.2 mm) and the width of the stripe was determined to be 32 m using the knife-edge technique. The emission was detected at the edge of NPL films along the stripe direction by a USB spectrometer (HR2000+, Ocean Optics). Figure S3. Static PL spectra of all CdSe NPL samples. NPLc4.

S6. Amplified spontaneous emission measurements
As another way of characterizing potential laser performance of NPLs, we also studied how ASE threshold depends on the optical density of the sample. For this study, three NPLc films, named Film1 to Film3 with increasing absorbance ( Figure S5a), were prepared by spin-coating NPLc solution with increasing concentrations on a glass substrate.
ASE measurement was carried out using the same 400 nm pulse excitation that was also used for the TA measurement. The pump beam was focused into a stripe along the film plane using a cylindrical lens. The emission was detected at the film edge along the stripe direction. The emission spectra of Film1 to Film3 as a function of fluence are shown in Figure S5b to S5d, respectively. The emission spectra can be assigned to ASE, consistent with previous reports. 2, 3 Unlike previous reports, no single exciton band edge emission, which is centered at ~518 nm, was observed in the ASE measurement. This likely indicates that single exciton band edge emission intensity is too small to be observed under our experimental conditions. As shown in Figure

S7. Optical gain mechanisms and threshold model
The absorption coefficient ( ) at OG energy E OG of the NPL ensemble, following Eq. 3 of the main text, is given by:  Table S2. We plotted m th (N s ) as a function of N s in Figure S6   The average number of photons in NPLs can be related to pump fluence I: where ℎ is pump photon energy, N A is Avogadro constant, is molar absorption coefficient per unit NPL volume, z is NPL thickness, A Z[ is NPL lateral area, L is light path of cuvette (1 mm), and C m is molar concentration of NPL solution. From Eq. S3, we can obtain the pump fluence at OG threshold, i.e. the OG threshold I th : where OD = zA Z[ C W L is optical density at pump wavelength. Substituting in m th (N s )/N s ~0.49, Eq. S4 can be rewritten as: We used Eq. S5 to fit the optical density dependent OG threshold and ASE threshold as shown in Figure 3c and 4d, respectively. The best fit gave the fitting parameter, B, as 88.9 J/cm 2 in Figure 3c and 89.7 J/cm 2 in Figure 4d.
Eq. S2 also predicts how optical gain increase with m. Shown in Figure S7a  single exciton remained at long delay time (t L , 800-1000 ps). Therefore, the A exciton bleach at t L (800-1000 ps) is proportional to the percentage of excited NPLs: 1-P 0 (m).
Following the same analyzing methods as reported in our previous work, 5 we define the normalized TA signal at t L as: where λ is A exciton wavelength (~512 nm). These normalized TA signals represent the probability of finding excited NPLs in the solution sample. At high excitation intensities, when all NPLs were excited, ∆S , t R approached one, from which the scaling factor S( ) was determined. According to Eq. S5, m is proportional to pump fluence: m=CI, and C is the photon encountering cross-section, which is proportional to lateral area (A QW ).
Therefore, Eq. S8 became: As shown in Figure S7c to f, fitting ∆S 512nm, t R as a function of pump fluence I to Eq. S10 yields the value of parameter C of 0.023, 0.031, 0.040, and 0.045 cm 2 /µJ for NPLa to d, respectively. Parameter C as a function of lateral area is plotted in Figure S7g, which confirms its linear increase with lateral area.
Our experimental OG amplitude as a function of m of all NPL samples were fitted to Eq. S3 with N s as the fitting parameter. The fitting results is plotted in Figure S7h. This plot agrees well with the measured OG saturation process and average exciton number at threshold (m th ) with different N s .