Quantum Dot Self‐Assembly Enables Low‐Threshold Lasing

Abstract Perovskite quantum dots (QDs) are of interest for solution‐processed lasers; however, their short Auger lifetime has limited lasing operation principally to the femtosecond temporal regime the photoexcitation levels to achieve optical gain threshold are up to two orders of magnitude higher in the nanosecond regime than in the femtosecond. Here the authors report QD superlattices in which the gain medium facilitates excitonic delocalization to decrease Auger recombination and in which the macroscopic dimensions of the structures provide the optical feedback required for lasing. The authors develope a self‐assembly strategy that relies on sodiumd—an assembly director that passivates the surface of the QDs and induces self‐assembly to form ordered three‐dimensional cubic structures. A density functional theory model that accounts for the attraction forces between QDs allows to explain self‐assembly and superlattice formation. Compared to conventional organic‐ligand‐passivated QDs, sodium enables higher attractive forces, ultimately leading to the formation of micron‐length scale structures and the optical faceting required for feedback. Simultaneously, the decreased inter‐dot distance enabled by the new ligand enhances exciton delocalization among QDs, as demonstrated by the dynamically red‐shifted photoluminescence. These structures function as the lasing cavity and the gain medium, enabling nanosecond‐sustained lasing with a threshold of 25 µJ cm–2.

-PL spectra of isolated dots and SLs   Table S1 -ASE and lasing characteristics of perovskite quantum dots Note S1 -Balance between quantum confinement and Auger recombination Note S2 -Carrier transport with inorganic ligands Note S3 -Carrier dynamics under femtosecond and nanosecond optical excitation  Under femtosecond-pulsed excitation, superlattices with inorganic ligands and superlattices with organic ligands show traces of optical gain (ASE and lasing). The threshold for optical gain is higher in SLs (lasing at P th =11 μJ.cm -2 , Figure S5b) in comparison to QDs with organic ligands (ASE at P th =4.1 μJ.cm -2 , Figure S3) presumably because of factors such as the lower oscillator strength (Note S3).
However, once using nanosecond-pulsed excitation, SLs formed with inorganic ligands sustain lasing at low thresholds. On the other hand, SLs formed with organic ligands do not show any traces of optical gain. This suggests that non-radiative processes that are not triggered in the femtosecond time-scale but are triggered in the nanosecond regime, such as Auger recombination, quench lasing/ASE in SLs with organic ligands but not in SLs with inorganic ligands.  Excitonic occupation (Fig. 3e) was determined based on the estimated absorption cross-section.
The average QD size is determined to be of 6.9 nm (Fig. S4a), consistent with the synthesis conditions used (see Methods). Based on previously reported cross-section values of CsPbBr 3 QDs at 335 nm 1 and the average diameter of 6.9 nm, the absorption cross-section is estimated to be around 5.3x10 -14 cm 2 (Fig. S4b). The absorption spectrum of CsPbBr 3 is used to scale the estimated absorption cross-section from 335 nm to 325 nm (Fig S4c), the excitation wavelength used during the power-dependence analysis shown in Fig. 3e. The average excitonic occupancy is determined based on the relation shown below.      Assuming an optical resonator consisting of two sets of mirrors where light is reflected with normal incidence on both mirrors, we have the following resonance relation: Where λ is the wavelength of light, n is the refractive index of the medium and l is an integer.
The refractive index of the SLs should be of around 1.9-2. The cavity length of SLs is of around 2 (size distribution provided in Figure S13). The lasing wavelength we have obtained in Figure 4 is around 536 nm. This leads us to the cavity mode of l ~ 15. For cavity mode l = 14 and l = 16 (the modes that resonate at the closest wavelengths to 536 nm), we obtain λ l=14 =571 nm and λ l=16 =500 nm. Both modes are outside of the gain bandwidth of the laser, which is limited by the gain medium and it is of 7-15 nm judging by the ASE width (Fig. S3). The single lasing peak is tunable from around 527 nm to 544 nm (within the gain bandwidth) by slightly varying the cavity length. It is important to note that we do not have such precise synthetic controlthe synthesis leads to SLs with a range of sizes. The exciton binding energy was evaluated by the temperature dependence of the integrated PL signal and fitted by the following Arrhenius equation 5,6 : where I 0 is the integrated PL intensity at 0 K, A is a constant, E b the binding energy and kb the Boltzmann constant. Since the PL intensity flats out for lower temperatures, I 0 at 0 K is extrapolated from the 120K-180K where intensity remains constant.
This method can be used when assuming that depopulation processes are dominated by thermal dissociation and radiative spontaneous emission processes 5 . The decrease in PL with increasing temperature is due to the increase in thermal dissociation rate of excitons at high temperatures 5 . This is a reasonable assumption considering the high PLQY (> 80%) of the material system at the low fluence (<N> < 1) used in this analysis.
Auger recombination is associated with high exciton binding energy because of enhanced Coulomb electron-hole interaction. A higher binding energy leads to less uniformly distributed carriers in space, therefore increasing the multi-body interaction probability (two electrons and one hole at the same position) and accelerating Auger recombination 6,7 . In one-dimensionally confined materials, the Auger recombination rate is proportional to the third power of binding energy 6,8 . Therefore, reducing the binding energy leads to a decrease in Auger recombination in quantum-confined systems 6 . Figure S11 shows that the binding energy is decreased by 30 % from the uncoupled QDs to SLs. This is due to the decreased inter-dot distance (as shown in Fig. 1 and Fig. 2, inter-dot distance is reduced from 10 nm to 3 nm in a SL), which enables exciton delocalization and leads to decreased Auger recombination rates.
It is important to note that decreasing E b also decreases first-order exciton recombination due to trap-assisted nonradiative recombination 6 . However, lasing at the nanosecond regime is not limited by trap-assisted recombination but by Auger recombination; therefore, decreasing Auger recombination will improve lasing operation.    To assess the properties of multiexciton states, we excite the uncoupled QDs and SLs with 80 fs pulses (1 kHz repetition rate) using a wavelength of 400 nm. Figure S15 shows an example of time-resolved PL obtained using a streak camera (Hamamatsu C10910). At low excitation fluence, time-resolved PL shows the typical single-exciton recombination lifetime of tens-of-ns (as shown in Figure 3c). When increasing the excitation fluence, a much faster time component (tens-of-ps) emerges. This is a signature of the generation of multiple excitons which decay via Auger recombination 10 .
Uncoupled QDs show a much stronger Auger recombination rate in comparison to SLs ( Figure  S15). Fitting yields a fast lifetime component of approximately 72 ps for uncoupled QDs and 135 ps for SLs. We attribute the improvement to the delocalization of charge carriers in the SL structure.
We note that both uncoupled QDs and SLs show traces of optical gain (ASE in uncoupled QDs [ Figure S3] and lasing in SLs [ Figure S5]) under femtosecond excitation. The fluence used for this analysis was chosen by increasing the excitation power until traces of a faster time component emerged. This was sufficient for direct comparison. Using higher fluences -approaching that of optical gainmay reduce lifetime due to accelerated radiative emission due to stimulated emission. We note that although the analysis was conducted with the same fluence for SLs and uncoupled QDs (and therefore, similar <N>), QDs are closer to ASE threshold (0.3P th ) than SLs (0.1P th ). The small differences between the fluence used for QDs and SLs arise from limitations controlling the laser power.

Note S3
We compare the carrier dynamics of the SLs under excitation with femtosecond (40 fs, 10 kHz) and nanosecond (1.1 ns, 15 kHz) optical excitation. The analysis here provided follows Qin, J. et al. 15 .
Under femtosecond-pulsed excitation, the pulse time (τ pulse ) is much shorter than the carrier lifetime (τ pulse << τ carrier ). As such, the carrier recombination rate is negligible compared with the carrier generation rate (G) and the carrier density is attributed only to the carrier generation.
The carrier density (n) is therefore calculated based on Equation 1: Where R is the reflectance (R~10%), α is the absorption coefficient (5x10 4 cm -1 ), d is the thickness of the SLs (d~2 μm; due to the large size of the SLs, we use the penetration depth, L = 1/ α), I fs is the laser fluence (J.cm -2 ) of the excitation at wavelength λ (400 nm), h is Plank's constant, and c is the speed of light. This relation is applicable for femtosecond and picosecond laser excitation.
In Figure S15a, we calculated the carrier density under femtosecond pulsed excitation using Equation (1) The carrier density at optical gain threshold using Equation 1 (assuming the carrier recombination rate is negligible compared with the carrier generation rate), as shown in Figure  S15, is around 6.1x10 17 cm -3 , similar to that of perovskite lasers (approximately 10 18 cm -3 carriers 15 ).
For pulse excitation comparable with the carrier lifetime (τ pulse ~ τ carrier ), the carrier density should be derived by considering both the carrier injection and carrier recombination simultaneously. The longer pulse can be regarded as quasi-continuous-wave excitation due to similar dynamics as in continuous-wave excitation 15 . Equilibrium is realized when G is equal to the carrier recombination rate, therefore (Equation 2): Where P ns is peak power density. The carrier lifetime is obtained by: Where a, b and c correspond to the first-order, second-order and third-order recombination rate coefficients. The first-order term contains radiative recombinationwe extract it from PL lifetime at low carrier densities (Figure 3c, approximately 20 ns, in good agreement with previous reports 9 ). The second-order term contains free electron-hole recombination. The thirdorder term contains Auger recombination. The total carrier lifetime is no longer than any of the three individual lifetimes. At the high fluences required for perovskite lasing (~10 18 cm -3 ) 15 , the carrier lifetime is dominated by Auger lifetime (τ carrier ~ τ c ). Figure S15b compares the carrier density for a carrier lifetime of 40 ps, 400 ps and 4000 ps.
As shown in Figure S15b, the carrier density at threshold with femtosecond laser excitation (6.1x10 17 cm -3 ) is matched by the carrier density at threshold with nanosecond laser excitation for a Auger lifetime of 100-400 ps.
The analysis presented here does not substitute direct measurement of Auger lifetime using transient absorption and photoluminescent techniques. However, as noted in the main text, due to focusing and signal limitations, we could not characterize Auger recombination using transient absorption. We could not probe a single superlattice directly; rather, we were limited to probing a larger area of the substrate. Thus, any signal obtained from the SLs overlapped with a stronger signal from the surrounding uncoupled QDs.