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Using Bulk-Like Nanocrystals to Probe Intrinsic Optical Gain Characteristics of Inorganic Lead Halide Perovskites Pieter Geiregat, Jorick Maes, Kai Chen, Emile Drijvers, Jonathan De Roo, Justin M Hodgkiss, and Zeger Hens ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.8b05092 • Publication Date (Web): 20 Sep 2018 Downloaded from http://pubs.acs.org on September 21, 2018

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Using Bulk-Like Nanocrystals to Probe Intrinsic Optical Gain Characteristics of Inorganic Lead Halide Perovskites Pieter Geiregat,∗,†,‡ Jorick Maes,†,‡ Kai Chen,¶,§,k Emile Drijvers,†,‡ Jonathan De Roo,⊥ Justin M. Hodgkiss,¶,§,k and Zeger Hens†,‡ †Physics and Chemistry of Nanostructures group, Department of Chemistry, Ghent University, Ghent, 9000, Belgium ‡Center for Nano and Biophotonics, Ghent University, Ghent, 9000, Belgium ¶The MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington, 6012, New Zealand §School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington, 6012, New Zealand kThe Dodd-Walls Centre for Photonic and Quantum Technologies, Wellington, 6012, New Zealand ⊥Department of Chemistry, Columbia University, New York City, 10025, USA E-mail: [email protected]

Abstract Following the introduction of perovskites for photovoltaic solar energy conversion, the use of these materials as a general purpose opto-electronic material for displays, lighting and lasing has been explored. However, while reports on stimulated emission and lasing by perovskites show great promise, a comprehensive quantification of their

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optical gain characteristics is lacking. Here, we measure gain coefficients, clarify the gain mechanism, and explore the gain dynamics of colloidal CsPbBr3 nanocrystals by deploying a unique combination of broadband transient absorption and ultrafast fluorescence spectroscopy. Opposite from current literature, we show that optical gain in such nanocrystals is supported by stimulated emission from free carriers, and not from excitons or bi-excitons. Importantly, we demonstrate that the concomitant gain coefficients and thresholds agree with literature results reported for perovksite thin films. Finally, we show that, even in the case of fully inorganic lead halide perovskites, a cooling bottleneck hampers the development of net stimulated emission at high excitation density. Based on these results, we propose that bulk-like colloidal nanocrystals in general offer a unique testbed to quantify optical gain of novel photonic materials and in particular for lead halide perovskites.

Keywords perovskite, nanocrystal, exciton, optical gain, broadband pump-probe spectroscopy A large variety of lead halide perovskites can now be synthesized as bulk materials, continuous thin films 1 and nanocrystals (NCs). 2,3 Although the initial interest in these materials involved their use as light-absorber in photovoltaic solar cells, 4–6 several studies demonstrated that perovskites can be rightly seen as multi-purpose solution-processable semiconductors for opto-electronics. 7–9 Here, an application that stands out is the use of perovskite (nano)materials as an optical gain material, 10–12 that can be used across the visible and nearinfrared spectrum. Studies on optical gain and lasing by perovskites typically make use of amplified spontaneous emission (ASE) to quantify gain properties. However, the reported net modal gain ranges from a few 10 cm−1 up to 450 cm−1 at optical pump thresholds varying from from 1 to 100 µJ/cm2 . 7,13,14 Clearly, this large spread on even the most fundamental gain characteristics of lead halide perovskites hampers a reliable quantification of their ultimate performance as a gain material, and it complicates the understanding of the

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underlying gain mechanism. The first difficulty to reliably quantify optical gain follows from the use of ASE as an analysis method. The main ASE observables, such as the threshold at which amplification is first observed and the concomitant gain coefficient, strongly depend on the measurement conditions. Spurious light scattering, limited optical mode confinement, or overall sample preparation and crystallinity, can affect measurement results and prevent the sound quantification of gain characteristics. Moreover, ASE-based studies will typically yield a threshold for net stimulated emission that is a gross quantity, such as the energy density per pulse of the optical pump laser. While such external characteristics are more easily measured and have practical relevance, understanding the mechanisms that underly stimulated emission requires insight in intrinsic properties such as the threshold carrier density. An alternative approach was followed by Sutherland et al., who used pump-probe spectroscopy to study and quantify optical gain development in high quality solid films of methylammonium lead iodide (MAPbI3 ). 1 Although pump-probe spectra can be far more insightful than ASE-spectra for the quantitative analysis of optical gain, a proper quantification and interpretation of the gain characteristics is hampered also in this case by the strong transient photo-refractive effects in thin films, which can mask bleach and optical gain features. 15,16 A second problem for the quantitative analysis of optical gain by perovskites is material quality. Perovskite thin films can have limited stability in ambient, and often feature ill-defined grain boundaries, composition variations, high surface roughness, or a low photoluminescence quantum yield. 17,18 An alternative could involve the study of optical gain using colloidal perovskite NCs as a model system. Pioneered by the work of Protesescu et al., CsPbX3 (X = Cl, Br, I) NCs have quickly become an attractive alternative to solid and continuous thin films, 6,19,20 and similar colloidal synthesis methods have been developed for the synthesis of MAPbX3 . 21 As demonstrated by several studies on CdSe-based quantum dots (QDs), pump-probe spectroscopy is ideally suited to quantify the optical gain characteristics of such colloidal NCs since they can be prepared as scatter-free dispersions with a

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known absorbance spectrum. In the case of perovskite NCs, colloidal dispersions can be airstable and feature photoluminescence quantum yields (PLQYs) approaching unity. Hence, by analyzing such dispersions, all the issues that complicate quantitative pump-probe spectroscopy in the case of thin films can be overcome, 2,22,23 and material gain coefficients of up to 450 cm−1 have been reported under femtosecond and nanosecond optical pumping. 24 What is unclear, however, is the agreement or not between the optical gain characteristics of perovskite NCs and bulk perovskites. Most strikingly, optical gain in colloidal CsPbBr3 NCs is typically discussed in terms of the state-filling framework that was originally developed for CdSe-based QDs, even if the CsPbBr3 NCs investigated show little size quantization. Building on the above argumentation, we present a comprehensive study of optical gain development in CsPbBr3 NCs, with sizes exceeding the exciton Bohr radius. Using quantitative broadband transient absorption and ultrafast photoluminescence spectroscopy, we show that optical gain develops at carrier densities matching the threshold expected for free-carrier mediated gain in bulk perovskites and we demonstrate that the material gain can be as high as 2000 cm−1 . Opposite from recent reports, 24 these findings indicate that optical gain in perovskite NCs is comparable to that in the corresponding bulk material, and involves stimulated emission by free carriers rather than excitons. In fact, we find that a optical gain is counteracted by a lingering excitonic absorption, which limits the gain bandwidth and lowers the material gain. In addition, despite indications that fully inorganic systems do not suffer from hot carrier cooling, 25 we observe a cooling bottleneck at high carrier density that strongly hampers optical gain development. We conclude that high quality bulk NCs are an advantageous model system to study the intrinsic photo-physics of solution processable semiconductors by pump-probe spectroscopy, where measurement artifacts that come with thin film studies or result from poor material quality can be avoided.

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Figure 1: (a) Transmission Electron Microscopy (TEM) image of the 12.7 nm CsPbBr3 NCs used in this work, see also Supporting Information S1. (b) Intrinsic absorption coefficient and photoluminescence spectrum for 12.7 nm CsPbBr3 NCs. Inset: photoluminescence decay at the PL peak wavelength, together with a single exponential fit. (c) Decomposition of the absorption spectrum using Elliot’s theory of Wannier Excitons (see Supporting Information S2) showing the exciton feature red-shifted from the continuum absorption. The latter deviates from the classical square-root dependence due to a small Coulomb enhancement near the band edge. (d) Sizing curve of the exciton feature based on TEM analysis. The vertical dashed line indicates the Bohr diameter dB , as calculated by Protesescu et al. 2

Results Material Characteristics of CsPbBr3 Nanocrystals We focus in this work on dispersions of CsPbBr3 perovskite NCs with a cubic shape and an edge length of 12.7 nm, a size that corresponds to the regime of weak confinement. A transmission electron microscopy (TEM) analysis shows that the such NC ensembles have a size dispersion of ≈ 10 %, see Figure 1a and Supporting Information S1 for a quantitative analysis. We obtained such a low size dispersion thanks to the specific size selective precipitation used during the purification of the crude reaction product (see Methods Section). Figure 1b displays the absorbance and photoluminescence spectra of a typical 12.7 5


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nm CsPbBr3 NC dispersion in hexane. The photoluminescence decay is shown in the inset of Figure 1, where a 4.6 ns decay time is extracted by means of a single exponential fit. The quantum yield of these NCs is determined using an integrating sphere and typically exceeds 65%, a number attesting to the excellent material quality. As can be seen in Figure 1b, the band-edge absorption exhibits an initial peak followed by a steadily increasing absorbance. These features closely resemble the absorption edge of hybrid organic-inorganic perovskites such as MAPbX3 , where several reports have decomposed the absorbance spectrum using Elliot’s theory of Wannier excitons, see Supporting Information S2. 26,27 An example of such a decomposition for 12.7 nm NCs is shown in Figure 1c. A clear exciton component – with a full width at half maximum (FWHM) of 41 meV – and a free carrier absorption profile can be distinguished. This decomposition yields an exciton binding energy of 36 ±2 meV, a number comparable to literature reports on MAPbBr3 and quite close to the value of 40 meV calculated by Protesescu and coworkers. 2 By repeating this spectral decomposition for batches of NCs with a different average size, we obtain a plot of the exciton energy as function of the NC size (see Figure 1d), which indicates that the exciton energy exhibits only a weak size-dependence in the case of 12.7 nm CsPbBr3 NCs. This is indeed to be expected as the Bohr radius of CsPbBr3 is only 3.5 nm, 2 such that 12.7 nm NCs are in the weak confinement regime. We also notice that the free carrier absorbance is somewhat enhanced near the band edge and deviates from the square root dependence, which is attributed to Coulomb interactions (i.e. Sommerfeld enhancement). 28

Quantifying optical gain We studied the development of optical gain by means of quantitative femtosecond transient absorption (TA) and photoluminescence spectroscopy. In both experiments, a short 110 fs pump pulse was used to photo-excite a NC dispersion and thereby create electron-hole pairs. Using excitation at either 400 (3.1 eV) or 500 nm (2.48 eV), we varied the excess energy relative to the band edges (see Methods Section). In TA, we used an equally short yet 6


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broadband probe pulse to track the change of absorbance as a function of both time delay t and probe wavelength λ. As such, we obtained time-wavelength maps of the differential absorbance ∆A(λ, t), from which the actual sample absorbance A = ∆A + A0 was calculated by means of the linear absorbance spectrum A0 . Note that TA on colloidal NC dispersions does not suffer from transient refractive effects which can complicate data-analysis. 15 Also, thanks to the sample preparation resulting in an aggregate-free nanocolloid of 12.7 nm NCs, the A0 spectrum does not show any scattering since individual NCs are much smaller than the relevant wavelengths (400-600 nm). The combined absence of scattering and transient refractive effects enabled us to make quantitative claims on optical gain thresholds and magnitudes. In the case of colloidal NCs, a quantitative TA study therefore provides the most optimal material characteristics, i.e., the maximum attainable material gain and the minimal carrier density threshold for transparency. Such characteristics offer direct insight into the gain mechanism since the measurements are not obscured by external factors such as cavity or film loss, a common issue when evaluating gain performance through, e.g., variable-stripe length or ASE methods. Quantities that can be obtained by means of gain spectroscopy through TA include the threshold excitation density for net gain, the material gain and the gain spectrum. To obtain the threshold excitation density, we calculate the average number hN i of absorbed photons per NC as the product of the photon flux Jph and the absorption cross section of a single NC σλ . To determine this cross section, we rely on recently published absorption coefficients, 29 from which we obtain a cross section σ400 of 1.4 × 10−13 cm2 at 400 nm for 12.7 nm NCs. A useful metric to compare our results to bulk systems is the carrier density n, which we can obtain as the ratio hN i/VN C of the average NC occupation and the NC volume. In the case of 12.7 nm NC, this implies that a situation where hN i = 1 corresponds to a density of 4.9 × 1017 cm−3 . Using these definitions, we can define the gain threshold in terms of the average occupation or the carrier density at which optical gain first occurs. Note that this value can depend on the excitation wavelength.

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Recently, the intrinsic absorption coefficient µi of CsPbBr3 NCs was published by Maes et al. (see also Methods), where a size-independent figure was obtained in particular at 335 nm. 29 By means of µi,335 , any absorbance spectrum can be rescaled to yield the spectrum of the intrinsic absorption coefficient, which is the absorption coefficient a NC composite would have if the NC volume fraction were 1. When A or, equivalently, µi turns negative after photo-excitation, the dispersion will exhibit net stimulated emission, for which −µi = gi can be identified as the material gain. Here, we will use this material gain to quantify optical gain by CsPbBr3 NCs since this number provides a basis to compare optical gain by nanocrystalline and bulk CsPbBr3 . Moreover, from the material gain, the gain of any NC composite in a given optical configuration can be obtained as the product of gi , the volume fraction f of NCs in that composite and the modal confinement. 30 Note that both the material gain and the mode confinement are usually strongly wavelength dependent.

Transient Absorption with Non-Resonant and Resonant Pumping. We first studied the development of optical gain using optical pumping at 400 nm, a wavelength that was also used by Yakunin et al. in their ASE study. 24 Figure 2a shows the 2D map of the differential intrinsic absorption coefficient ∆µi for pump powers leading to a carrier density n = 1.4 × 1019 cm−3 or, concomitantly, hN i = 28. At early times, a strong photo-induced absorption is observed at wavelengths longer than the CsPbBr3 NC bandgap, which can be attributed to band gap renormalization (BGR). A similar effect upon nonresonant excitation was observed in bulk MAPbI3 films by Price and coworkers. 15 Carrier cooling turns this photo-induced absorption within 1 − 2 ps into a bleach feature as the renormalized band-edge states fill up. Figure 2b displays near band-gap spectral cuts at 3 ps – a time delay corresponding to the maximal bleach – of the transient intrinsic absorption coefficient µi = µi,0 + ∆µi , measured for increasing fluence. A time delay of 3 ps also avoids excessive carrier loss as non-radiative relaxation occurs on a multiple nanosecond timescale, see Supporting Information S1.3. The 8


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Figure 2: Gain development using a 400 nm (3.1 eV) pump. (a) 2D map of the differential intrinsic absorption coefficient ∆µi (cm−1 ). The initial carrier density is 1.4 × 1019 cm−3 . (b) Spectral cuts of the transient intrinsic absorption coefficient µi = µi,0 + ∆µi at 3 ps for different carrier densities. Inset: zoom on the long wavelength part, showing that µi turns negative with increasing carrier density. (c) Material gain spectra gi and linear absorption (solid black line) for different carrier densities showing a transition from net absorption to net gain. (d) The material gain as function of carrier density at 530 and 540 nm. A gain threshold of 1.6 × 1018 cm−3 is extracted. arrows in the figure highlight both the pronounced reduction of the exciton absorbance and the increase of the short wavelength absorbance. Remarkably, even for the most extreme occupations used here, which reach up to hN i = 100 electron-hole pairs per NC, no full saturation or inversion of the exciton line is observed. Zooming in on the long wavelength tail, we observe a region where µi turns negative, tantamount to optical gain. Figure 2c plots the corresponding material gain gi = −µi together with the linear intrinsic absorption coefficient µi,0 at different excitation densities. Clearly, the gain spectrum is red-shifted relative the to absorbance spectrum, showing net optical gain at wavelengths where linear absorption is almost or completely absent. Optical gain peaks between 530 and 540 nm, wavelengths that correspond well to literature ASE reports. 24 The dashed lines in Figure 2c

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represent the material gain measured when the excitation density exceeds 2 × 1019 cm−3 . Remarkably, rather than saturating at a fixed time delay, we find that the material gain drops at these higher excitation densities. Figure 2d represents the material gain at 530 nm, the wavelength where the maximum gain was measured, and at 540 nm, where net gain is observed first, as a function of the photo-generated carrier density. It follows that transparency is achieved at a threshold density of 1.6 × 1018 cm−3 , which corresponds to an average occupation of 3.3 electron-hole pairs per NCs. Also here, the drop of the material gain with increasing excitation density is clearly visible. A complication coming with optical pumping at 400 nm (3.1 eV) is that each excitation has an excess energy of about 650 meV relative to the band-edge states that must be dissipated upon charge-carrier cooling. Bearing in mind the plethora of observations of slow carrier cooling in hybrid organic-inorganic perovskites, we therefore extended our investigation to near-resonant excitation at 500 nm. Figure 3a shows again a 2D map of the differential intrinsic absorption coefficient. As compared to photo-excitation at 400 nm (Figure 2a), pumping at 500 nm results in some pronounced differences. First, no photoinduced absorption appears at early times. While this does not mean BGR is absent, it implies that carriers immediately occupy the shifted energy levels as expected for resonant excitation. The similar bleach bandwidth suggests that mainly carrier density, not initial carrier energy/temperature dictates the BGR. Figure 3b shows the intrinsic absorption coefficient µi 3 ps after photo-excitation. Again, a decrease of the band edge absorbance and an increase of absorption at shorter wavelengths is observed. Zooming in on the long wavelength tail (inset, Figure 3b), we observe also in this case a region where µi turns negative. Figure 3c represents the resulting material gain spectra. While these spectra have the same spectral width as for 400 nm photo-excitation, the maximum material gain is substantially higher and reaches up to 2000 cm−1 for the highest excitation density, see Figure 3c. In addition, rather than dropping, it appears that the gain coefficient saturates with increasing excitation density. On the other hand, the

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Figure 3: Gain development using a 500 nm pump (a) 2D map of the differential absorbance ∆µi (cm−1 ) after photo-excitation creating an initial carrier density of 1.2 × 1019 cm−3 . We observe a distinctly different behavior from pumping at 400 nm (see Figure 2). No strong photo-induced band gap renormalization is observed as the renormalized states are immediately occupied by charge carriers. (b) Near-band edge spectral cuts at 3 ps for different carrier densities. (c) Material gain spectra gi (cm−1 ) and linear absorption (solid black line) for different carrier densities showing a transition from net absorption to net gain. (d) The gain coefficient as function of carrier density at the peak wavelength (530 nm) and the first wavelength showing net gain (540 nm). threshold excitation density of 1.8 × 1018 cm−3 is comparable to that obtained for 400 nm photo-excitation. We note that in this case, the reduction of the absorption cross-section at 500 nm following photo-excitation has been taken into account to calculate the carrier densities.

Mechanism of Optical Gain in CsPbBr3 Nanocrystals In earlier work, 24 the occurrence of optical gain in colloidal CsPbBr3 NCs in the 10-12 nm size range was interpreted in terms of a biexciton-exciton transition that exhibits net stimulated emission due to the filling of discrete single-electron states, not unlike CdSe-based QDs. 31 A

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Figure 4: Excitons and free carriers (a) Carrier density nthres required for net optical gain, as measured for various sizes of CsPbBr3 NCs under weak confinement. The horizontal blue line indicates the estimated Mott transition (see Supporting Information S6) and the red shaded region indicates the range of gain thresholds for the free carrier mechanism based on calculations of the effective mass from density functional theory in the generalized gradient approximation (GGA) with or without considering van der Waals (vdW) interactions. (b) Lingering exciton absorption (solid red line) extracted from the non-linear absorbance (black line) 3 ps after photo-excitation with 400 nm and a carrier density of 1.4×1019 cm−3 . The free carrier profile (dashed red) is obtained from numerical modeling, see Supporting Information S5. fingerprint characteristic of this QD gain model is that the gain threshold, expressed as the average number hN i of electron-hole pairs per NC at which transparency is reached, is size independent and close to unity. 32 This implies that the threshold charge carrier density n should increase as 1/V when the dimensions of the NCs are reduced. Such a trend contrasts with bulk semiconductors, where gain is reached at a fixed threshold density nf c of free carriers as given by: 33 nf c ≈ 1.5 ×

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Here, mr and M are the reduced and total exciton mass, respectively, whereas T is the carrier temperature. Figure 4a displays the threshold density nthres = 1.6 × 1018 cm−3 we deduced for 12.7 nm CsPbBr3 NCs by means of transient absorption spectroscopy at a delay of 3 ps. In addition, the figure shows nthres obtained using a similar analysis on 11.7, 8.6 and 6.7 nm NCs (see Supporting Information S1). Opposite from the discrete state-filling model, we find that transparency is reached in these different samples at largely the same carrier density

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of ∼ 1.6 × 1018 cm−3 . Note that this density corresponds to a similar threshold energy flux of about 10 µJ/cm2 as reported in literature. 24 In the case of 12.7 nm NCs, the threshold density corresponds to an average exciton occupation of hN i = 3.2, a number that is larger than expected based on a discrete state-filling model. Next to experimental data, Figure 4a also represents the threshold density for plasma gain at 300 K as predicted by Eq 1, where we use effective masses calculated by Yettapu et al. by means of density functional theory in the generalized gradient approximation (GGA) with or without considering van der Waals (vdW) interactions. 34 Interestingly, we find that the experimental threshold density falls in between these estimated thresholds for plasma gain in bulk CsPbBr3 . A second fingerprint of the QD gain model that applied to CdSe-based QDs is the progressive bleach of the exciton absorption with increasing exciton occupation of hN i, followed by the complete reversal of the intrinsic absorption spectrum to yield the saturated material gain spectrum. 31 Figure 4b represents a horizontal slice of the 2D map in Figure 3a on an energy scale, which was recorded after a pump pulse that creates on average 30 electron-hole pairs per NC, i.e. a density of 1.4×1019 cm−3 . One sees that even under such conditions, the exciton absorption is not fully quenched and remains superimposed on a smooth background which changes from net gain in the 2.3–2.35 eV region, to absorption at higher energies. Given the size-independent threshold density and the persistent exciton absorption, we can rule out an interpretation of optical gain in 12.7 nm CsPbBr3 NCs in terms of the QD gain model. The agreement between the experimental threshold density and the theoretical threshold for free carrier gain in bulk perovskites rather points towards a similarity between gain in 12.7 CsPbBr3 NCs and bulk CsPbBr3 . In excitonic materials with a large band-edge density of states, net stimulated emission can either result from exciton-related transitions, such as the recombination of excitonic molecules, exciton-exciton scattering or exciton-free carrier scattering, or from free-carrier recombination. 28 Several authors have argued that optical excitation in Pb-based perovskites leads to the formation of an electron-hole plasma. 35,36 Both the width op the material gain spectrum, which exceeds the exciton binding energy

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by a factor of 5, and the agreement between the experimental threshold density and the expected threshold nf c for plasma gain in perovskites indicate that also in the case of 12.7 nm CsPbBr3 NCs, optical gain results from free-carrier recombination. This interpretation is supported by the work of Yettapu et al., who showed by THz spectroscopy that photo-excited CsPbBr3 NCs of similar sizes exhibit a real photoconductivity, 34 a distinctive characteristic of free carriers. If optical gain in 12.7 nm CsPbBr3 NCs can indeed be attributed to plasma gain, it should be possible to simulate the nonlinear absorption spectrum measured shortly after optical pumping (see Figure 4b). To do so, we implemented a bulk state-filling model in which the spectral broadening and the band-gap renormalization are the only free parameters (see Supporting Information S5). 37 As shown in Figure 4, such an approach describes very well the background of the nonlinear absorption, and its evolution from net absorbance at higher photon energy to net gain at lower photon energy. We obtained a best fit using a moderate BGR of 34 meV and a broadening of 2Γ0 , where Γ0 = 41 meV is the broadening used to fit the linear absorption profile. A similar increase in broadening was also observed by Yang, ? measured on bulk perovskite films. 26 As shown in Figure 4b, subtracting this free carrier gain profile from the net non-linear absorbance (black line) yields the lingering exciton absorption, which in this case amounts to ca. 10 % of the exciton absorption of an unexcited sample. Similar observations were made by Yang et al. in the case of bulk MAPbI3 , 26 where they found a moderate reduction of the exciton oscillator strength by a factor of 2 at carrier densities comparable to the gain threshold density we report here. This persistent exciton absorption is remarkable since one would expect screening by free carriers to fully quench the exciton oscillator strength. 38 This remaining exciton absorption indicates that excitons can still be created through photoexcitation in CsPbBr3 NCs containing a dense electron-hole plasma. However, we expect that such excitons quickly dissociate since the threshold density for optical gain is already higher than the so-called Mott density nM ott (see Figure 4a), which is the carrier density

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Figure 5: (a-b) Net gain maps for pumping at (a) 400 nm (a) and (b) 500 nm at low density showing the gain lifetime is ca. 100 ps across the entire gain bandwidth. (c) Kinetics of the gain buildup at 530 nm for pumping at 400 nm. Until a carrier density of about 1019 cm−3 (red traces), the gain builds up in a few ps. Beyond this carrier density (red traces), the maximum gain saturates and the buildup time (indicated by vertical markers) substantially increases reaching up to 15 ps for the highest carrier densities used here. (d) Kinetics of gain buildup for pumping at 500 nm where gain builds up instantaneously and reaches a maximum of 2100 cm−1 . We observe a clear drop due to non-radiative Auger recombination on a timescale comparable to the gain buildup under non-resonant excitation, i.e. close to 20 ps. Clearly, the non-radiative recombination during slow carrier cooling limits the maximum gain under non-resonant photo-excitation. at which carrier-carrier interactions are screened to the point where excitons dissociate (see Supporting Information S6). 28

Decay of Photo-Excited Carriers Further evidence that the optical gain mechanism in 12.7 nm CsPbBr3 NCs is highly similar to that in bulk CsPbBr3 comes from the decay of the photo-excited carriers. Figures 5a and 5b show 2D maps of the material gain gi after a carrier density of ≈ 1.4 × 1019 cm−3 was created by photo-excitation with 400 and 500 nm pump light, respectively. In both figures, we have only color-coded combinations of delay and wavelength that correspond to optical 15


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gain. This representation highlights the delayed buildup of net gain in the case of 400 nm excitation as compared to the almost instantaneous appearance of optical gain in the case of 500 nm excitation. Moreover, the gain spectrum is considerably broader in the latter case, and net gain persists longer, up to 100 ps. By means of absorption transients at a fixed wavelength (530 nm, see Figure 5c), we can assess in more detail the fluence dependence of the buildup and decay of optical gain. In the case of 400 nm excitation, optical gain appears after about 2 ps as long as n remains below 1019 cm−3 . However, when the carrier density approaches 1019 cm−3 , the maximum material gain saturates and a further increase of the carrier density delays the gain buildup and lowers the maximum material gain attained during the transient. At the highest carrier densities used (7.5 × 1019 cm−3 ), already 15 ps are needed before the material gain attains its maximum value. In the same way, the time span during which optical gain persists initially increases when the carrier density is raised, but eventually levels of at 80 ps for the highest densities used. In the case that a pump wavelength of 500 nm is used, we find that optical gain builds up instantaneously and reaches almost directly its maximum value, regardless of the carrier density. As compared to pumping at 400 nm, the maximum material gain is significantly higher – up to 2000 cm−1 for the highest carrier density – and lasts for 100 ps. In addition, we observe a rapid decay in the first 20 ps, which is absent in the case of 400 nm pumping. For fluences near and above the gain threshold, the differential absorption transient can be fitted using a two-body recombination model, in which the decay of the differential absorption – and thus of the carrier density – is proportional to the square of the carrier density n: dn = −k2 n2 dt

(2)

As shown in Supporting Information S4, this yields a second order recombination rate constant k2 = 2.2 ± 0.4 × 10−9 cm3 s−1 , which matches very well with the work of Manser et al. on carrier dynamics in bulk MAPbI3 . 35 At the transparency density of 1.6 × 1018 cm−3 , this 16


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Figure 6: Carrier cooling after excitation at 400 nm (a) 2D map of differential absorbance creating a large initial carrier density of 2.5 × 1019 cm−3 . (b) Spectral cuts on an energy scale at different time delays extracted from (a) showing a tailing to higher energies. The inset shows a fit to the Boltzmann distribution (see main text) which allows us to extract an effective temperature for the electronic gas. (c) Temperature of the electronic system versus delay time for different initial carrier densities. We observe that all traces evolve to room temperature (horizontal dashed line) yet at the highest densities the carrier temperature remains for several ps above even 1000K. (d) PL map after photo-excitation with 400 nm pulses creating a carrier density of 4.5 × 1019 cm−3 . (e) Spectral cuts from (d) at various increasing time delays (f) Temperature from PL analysis (red markers) compared to TA (black markers) analysis. gives rise to a two-body recombination time of τ2 of 280 ps. Following Manser et al., we interpret this quadratic carrier loss rate as reflecting geminate electron-hole pair recombination – and not Auger recombination as is common for multiple electron-hole pairs in colloidal QDs – which would indicate that the development of optical gain is initially not limited by non-radiative processes.

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Slow Charge-Carrier Cooling in CsPbBr3 Nanocrystals Figure 5c shows that optical pumping at a large excess energy and high carrier density counteracts the buildup and saturation of the optical gain. Looking at the ∆A-map recorded under such pumping conditions (400 nm, n = 2.5 × 1019 cm−3 ) represented in Figure 6a) makes clear that this suppression of optical gain concurs with the appearance of a pronounced high energy tailing of the band-edge bleach. This bleach tail builds up instantaneously and requires several tens of picoseconds to narrow down to the band edge. Since TA effectively probes the occupation of energy levels, this high energy bleach can be interpreted as a slow relaxation of the initially hot carriers, by which we mean carriers with an excess energy relative to the band edge, towards the band edge. Usually, this process is fast in bulk semiconductors due to rapid emission of phonons, i.e. heat. 28 In order to understand and quantify the apparent cooling bottleneck, we analyzed in more detail the temperature of the photo-generated charge carriers as a function of the initial carrier density and the excess energy provided per carrier. To do so, we assumed that the time-dependent high energy bleach reflects a Boltzmann occupation of energy levels (see Figure 6b). This approach is commonly used in literature 15,26 and applies when (a) rapid internal thermalization of the electron/hole gas leads to a Fermi-Dirac distribution characterized by a unique temperature, (b) photon energies are substantially above the respective electron or hole quasi-Fermi level, (c) the electron and hole gas have equal temperature, and (d) cooling does not induce a significant change in the (spectral shape) of the absorption spectrum. Assuming that these conditions are met when using photon energies exceeding 2.55 eV, we can write the high energy bleach as:

∆A(¯ hω) ∝ e−

E−Ef kTc

(3)

This proportionality enables us to extract an effective carrier temperature Tc from the high

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energy tail of the band-edge bleach, which we represent as a function of the delay time in Figure 6c for various carrier densities after photo-excitation at 400 nm. At the lowest carrier density analysed (3.4 × 1018 cm−3 ), the initially elevated carrier temperature exhibits a rapid drop, and reaches room temperature (horizontal dashed line) within ∼ 2 ps. We assign this efficient equilibration of the charge carriers with the crystal lattice to the rapid emission of optical phonons mediated through Fröhlich coupling, a common observation for polar semiconductors. 28,39 Upon increasing the initial carrier density, one sees that the first stage of rapid cooling – labeled as A Figure 6c – persists but thermal equilibrium between the free carrier gas and the room-temperature environment is only attained after a second stage – labeled as B – of slow cooling. At the highest densities, for example, the temperature is still at 1500 K after 2 ps and remains above 500 K for up to 50 ps. Clearly, this second regime points towards a cooling bottleneck, similar to observations made on other types of perovskite materials. 25,40 This pronounced reduction of the cooling rate is typically assigned to a rapid thermalization of hot carriers and optical phonons. This cooling mechanism brings carriers to the band edges when the density of carriers is low, such that the dissipation of the excess energy does not significantly increase the phonon occupation or, equivalently, the phonon temperature. At higher carrier densities, however, the increasing energy dissipation by the cooling charge-carrier gas will make that it equilbrates with the phonon gas at a temperature well above room temperature. This leads to a situation where hot carriers can no longer dissipate energy by emit phonons, effectively forcing them to remain at high temperature. A more detailed analysis shows that phonon emission lifetimes increase from a few tens of fs to over 10 ps at the highest densities used here, see Supporting Information S7. We note that apart from overpopulation of the phonon modes, also a decreased coupling to these modes can cause a cooling bottleneck, e.g. due to carrier induced screening of the Fröhlich coupling. 39 However, this effect should, in a first approximation, be independent of the initial excess energy per carrier and only depend on the absolute carrier density. We do

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observe an effect of the initial carrier energy by varying the pump from 400 nm to 330 nm for an equal absolute carrier density, as is shown in Supporting Information S7, hence ruling out this screening mechanism as the main contributor to the observed cooling bottleneck. To further support the analysis of the high energy bleach tail in terms of a Boltzmann occupation of energy levels as determined by the carrier temperature, we used ultrafast luminescence spectroscopy as an alternative analysis method, see Methods. Since a transient absorption signal may be masked by photo-induced absorption, the interpretation of transient fluorescence can be more straightforward. Figure 6d shows the luminescence spectrum of a dispersion of 12.7 nm CsPbBr3 NCs that was pumped at 400 nm excitation with 110 fs pulses that each created a carrier density of 4.5 × 1019 cm−3 . Also here, a high energy tail shows up that we analyzed in a similar fashion as for the TA analysis. Figure 6e represents the thus obtained decay of the carrier temperature, which shows the same behavior as observed from TA (see Supporting Information S7). A detailed comparison of the temperature extracted from TA and PL at equal carrier density, however, points towards interesting differences. Whereas the temperatures in stage B largely coincide, TA yields a significantly higher initial temperature in stage A than PL. Since photoluminescence involves a transition between two energy levels that are occupied by an electron and a hole, whereas an absorbance bleach only requires that one of both levels is occupied, this discrepancy possibly indicates a difference in the cooling rate for electrons and holes at high carrier density. Finally, we note that the luminescence also shows a strong broadening towards longer wavelengths, matching the gain bandwidths observed earlier. This indicates that these perovskite materials also show strong spontaneous emission in the region where net optical gain can be achieved. One should note that the ability to amplify an incoming probe beam did not guarantee this. The combination of both efficient amplification and spontaneous emission sets the scene for efficient lasing action, as observed in literature by several groups. 24 Importantly, the observed cooling bottleneck accounts for the observations made earlier in this work on the slow gain buildup and limited maximal gain under strong non-resonant

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excitation. Indeed, when cooling slows down, carriers take longer to reach the energy levels close the band gap where population inversion is attained and the buildup of optical gain is delayed. Moreover, fast non-radiative recombination will eliminate a substantial part of the carrier population during this prolonged cooling stage. The concomitant reduction of the maximum gain can be clearly noticed in Figures 5c and 5d, where the measured maximum material gain is reduced from close to 2000 cm−1 attained within 0.2 ps for near-resonant pumping to 1000 cm−1 after 10 ps for non-resonant pumping. Note that this latter figure matches well the maximum value found for resonant excitation at the same time delay of ca. 10 ps at maximum carrier density (see Figure 5d).

Discussion Since TA spectroscopy provides the material gain spectrum, a direct comparison of the gain characteristics determined here, such as bandwidth, magnitude and lifetime, with literature data is possible. For example, Yakunin et al. reported that optical pumping using 400 nm light of films of CsPbBr3 NCs results in amplified spontaneous emission (ASE) in the wavelength range of 535–540 nm. 24 Clearly, this finding matches perfectly the gain band shown in Figure 2b. In addition, these authors obtained modal gain coefficients of 450 cm−1 through variable stripe length measurements. To compare these figures with our results, the material gain reported in Figure 2b should be rescaled by the NC volume fraction in the film, which amounts to ca. 60% for close-packed cubes with organic ligands. This results in a gain coefficient of 720 cm−1 for unity optical mode confinement. Absent any losses, this would imply a modal confinement factor of 0.6 in the work of Yakunin et al., a reasonable figure for thick NC films. We have shown that weakly confined perovskite NCs show the gain thresholds expected for their bulk counterparts, and we have attributed optical gain in such NCs to plasma gain. Recent work by Ghoh et al. also indicated similarities between nanocrystals and bulk

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perovskites, yet their study was limited to low density regimes without any proper quantification. 16 These results indicate that bulk-like NCs may constitute an attractive model system to analyze optical gain – or opto-electronic characteristics in general – in the corresponding bulk material. Clearly, the fact that we measure similar carrier loss rates in the case of CsPbBr3 NCs as reported for bulk CsPbBr3 supports this point further. Especially in view of transient absorption spectroscopy, colloidal NCs make for an interesting test bed. The quality of the NCs makes that carrier trapping is absent and the lack of transient refractive (sub-gap) effects and/or strong scattering allows for a direct quantification of the main gain characteristics. As a result, the material gain obtained using TA spectroscopy constitutes an upper limit of what a material can attain under conditions where residual losses are negligibe, and only corrections for local field effects are needed to translate this figure to the corresponding bulk material. In the case of CsPbBr3 NC, we estimate the corresponding local field factor at 0.44 (see Supporting Information S3), which makes that a material gain of 2000cm−1 for CsPbBr3 NCs corresponds to 4500 cm−1 for bulk CsPbBr3 . Such a figure compares well with a study on films of MAPbI3 by Sutherland et al., where they found a material gain of up to 3000 cm−1 . 7 Building on the idea that weakly confined perovskite NCs can be used to study the behavior bulk perovskites, we can also use the results of this work to predict the performance of perovskites for applications in stringent high carrier density regimes, such as those required for lasing. Such regimes have not been studied thoroughly in literature as the main focus of the community has been on developing these materials for solar energy conversion, which is a low carrier density application. Based on our results, we expect that a carrier cooling bottleneck will occur in fully inorganic perovskites at high carrier densities, which deteriorates the optical gain performance of these materials. This conclusion seems at odds with many reports claiming that the organic cations play a major role in slowing down carrier cooling in perovskites. 25,40 What we find, however, is that the occurrence or not of a cooling bottleneck strongly depends on the carrier density regime. At low density, no significant bottleneck for cooling exists. This agrees with the conclusion

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of Zhu et al., who found no hot carrier luminescence in bulk CsPbBr3 . 25 When increasing the density, however, we enter a different regime where carrier cooling bottlenecks are not uncommon for polar semiconductors. Indeed, similar observations were made in GaAs. 39 To confirm the occurrence of the cooling bottleneck in bulk perovskites, further studies should compare organic-inorganic and fully inorganic perovskites on a systematic basis, keeping the excess energy and photo-generated density as separate variables. A first attempt was given by Li et al. 41 who analyzed similarly sized QDs as this work. A comparison, see Supporting Information S7, shows that the organic cation can lead to even stronger cooling bottleneck issues.

Conclusions In summary, we have shown that weakly confined nanocrystals are an excellent test bed for evaluating the intrinsic properties of their bulk counterparts. In particular, we have shown that combining perovskite NCs with quantitative ultrafast spectroscopy presents an excellent platform to study intrinsic optical gain properties of the perovskite material family. We have shown that net optical gain in perovskite materials occurs via a classical free carrier mechanism, with a limited role for excitons. In fact, a lingering exciton-related absorption even counteracts the development of broadband optical gain. Finally, we have shown that for off-resonant photo-excitation a cooling bottleneck at high carrier density manifests, which will both limit the maximum achievable gain and slow down gain buildup, thereby forcing it to compete with non-radiative processes.

Methods Materials Rather than developing a new synthesis method for inorganic CsPbBr3 nanocrystals (NCs), we focus here on an approach reported by De Roo et al. 23 and Maes et al. 29 to perform a post-synthesis treatment to obtain monodisperse batches. We refer to the Sup23


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porting Information S1, for more details on this protocol.

Quantitative Pump-Probe Spectroscopy Samples were excited using 110 femtosecond pump pulses with varying wavelength created from the 800 nm fundamental (Spitfire Ace, Spectra Physics) through non-linear conversion in an OPA (Light Conversion, TOPAS). Probe pulses were generated in a thin CaF2 crystal using the 800 nm fundamental. The pulses were delayed relative to the pump using a delay stage with maximum delay of 5 nanoseconds. The probe spectrum in our experiments covers the UV-VIS window from 350 nm up to 700 nm. CsPbBr3 NCs were dispersed in an optically transparent solvent (n-hexane) and continuously stirred to avoid charging or photo-degradation. The quantum yield before and after the measurements was comparable at a typical value of ≈ 65%, measured using an integrating sphere, see also Supporting Information S1. The average number of photo-generated charge pairs at time zero, noted as hN i, can be calculated from the photon flux Jph , the cuvette length L and the nanocrystal absorption cross section at the pump wavelength σλp : hN i = Jph × σλp ×

−α

1−e 0,λp α0,λp L

L

. The photon flux is

calculated from the beam area, obtained through a Thorlabs CCD beam profiler, and defined as Abeam = 2π×σx σy where σi is the standard deviation in the i = x, y direction. We can write the cross section at a given wavelength as the product of the intrinsic absorption coefficient at that wavelength and the volume of the NC: 42 σλ = µi,0 (λ)×VN C . The intrinsic absorption coefficient µi,0 (λ) was determined in earlier work through a combination of elemental analysis and TEM, where 29 Maes et al. showed that µi,0 is a size-independent quantity with a value of: µi,0 (335nm) = 1.59 ± 0.05 × 105 cm−1 . Ultrafast Photoluminescence Spectroscopy For (ultrafast) photoluminescence (PL) spectroscopy, samples were measured using a cuvette with a 1 mm optical path length and an optical density of 0.1 at the exciton absorption peak to avoid strong re-absorption. Samples were not stirred, but translated along 1 axis to avoid photo-charging. The detection of the broadband PL on femtosecond timescales was made possible by using a newly developed 24


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transient grating technique by Chen et al. 43 A Ti:Sapphire amplifier system (Spectra-Physics Spitfire Ace) operating at 3 kHz generating 110 fs pulses was split into two parts. One part was frequency doubled to 400 nm using a BBO crystal and focused to a 70 µm spot on the sample. The PL is collimated using an off-axis parabolic mirror and refocused on a polished slice of fused silica. The second part, ca. 40 µJ, of the 800 nm output was split using a 50:50 beam splitter creating two gate beams that are focused on the fused silica with a crossing angle of 8 degrees. The instantaneous grating generated by the interfering gate beams create an instantaneous gate which is used to temporally resolve the decay over a broad wavelength range. The scatter of the pump beam was suppressed using a 430 nm long-pass filter and the pump polarization was set a magic angle relative to the PL collection. Spectra were averaged over 15000 shots for every time delay.

Acknowledgement PG acknowledges the FWO-Vlaanderen for a postdoctoral fellowship. ZH acknowledges support by the European Commission via the Marie-Sklodowska Curie action Phonsi (H2020MSCA-ITN-642656), by the Research Foundation Flanders (project 17006602) and by Ghent University (GOA no. 01G01513). JMH and KC acknowledge support from the Marsden Fund.

Supporting Information Available The Supporting Information contains a detailed overview of the various nanocrystals used in this work and more in-depth analysis of the transient absorption/fluorescence data and gain modeling.

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References 1. Sutherland, B. R.; Hoogland, S.; Adachi, M. M.; Kanjanaboos, P.; Wong, C. T. O.; McDowell, J. J.; Xu, J.; Voznyy, O.; Ning, Z.; Houtepen, A. J.; Sargent, E. H. Perovskite Thin Films via Atomic Layer Deposition. Adv. Mater. 2015, 27, 53–58. 2. Protesescu, L.; Yakunin, S.; Bodnarchuk, M. I.; Krieg, F.; Caputo, R.; Hendon, C. H.; Yang, R. X.; Walsh, A.; Kovalenko, M. V. Nanocrystals of Cesium Lead Halide Perovskites (CsPbX3, X = Cl, Br, and I): Novel Optoelectronic Materials Showing Bright Emission with Wide Color Gamut. Nano Lett. 2015, 15, 3692–3696. 3. Dong, Y.; Qiao, T.; Kim, D.; Parobek, D.; Rossi, D.; Son, D. H. Precise Control of Quantum Confinement in Cesium Lead Halide Perovskite Quantum Dots via Thermodynamic Equilibrium. Nano Lett. 2018, 18, 3716–3722. 4. Zhou, H.; Chen, Q.; Li, G.; Luo, S.; Song, T.; Duan, H.; Hong, Z.; You, J.; Liu, Y.; Yang, Y. Interface Engineering of Highly Efficient Perovskite Solar Cells. Science 2014, 542 . 5. Brenner, T. M.; Egger, D. A.; Kronik, L.; Hodes, G.; Cahen, D. Hybrid Organic—Inorganic Perovskites:

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