The absorption cross section of lead halide perovskite nanocrystals is important for understanding their photophysical properties, especially those depending on the density of photoexcited charge carriers. Despite its importance, there are large discrepancies among the reported absorption cross section values determined employing different methods. Here, we measured the absorption cross section of CsPbBr3 quantum dots (QDs) of varying sizes using elemental analysis and transient absorption (TA) saturation methods and compared with the previously reported values determined from elemental analysis and transient photoluminescence (PL) saturation methods. A careful comparison indicates that the reliable absorption cross section of lead halide perovskite QDs is obtained from both elemental analysis and TA saturation methods, while many previously reported values determined from the PL saturation method underestimate the absorption cross section.

The lead halide perovskite materials have been of great interest in recent years for photovoltaic1–4 and optoelectronic5–9 applications due to their optical, electronic, and transport properties surpassing those of many other semiconductor materials. Colloidal nanocrystals of these materials are also extensively studied as the superior alternative to many other semiconductor nanocrystals used for decades as the source of photons for their facile chemical tuning of bandgap and high luminescence quantum yield approaching unity in some cases.10,11 Furthermore, the controllable dimensionality (0D-2D) and the quantum confinement achievable in many lead halide perovskites render them highly attractive as a new family of quantum confined semiconductor nanocrystals applicable as the light harvester or emitter of photons.8,12,13 Low nonradiative loss of the photoexcited exciton and relatively large exciton Stokes shift in lead halide perovskite nanocrystals were considered beneficial for photonic applications such as lasing (or optical gain) with low threshold.8 

For such applications, information on the average number of photoexcited excitons (n) in the nanocrystal at the given excitation intensity is important, especially for n > 1.8,14–16 For instance, the radiative exciton recombination competes with a n-dependent nonradiative decay channel such as Auger relaxation, requiring the knowledge of n in order to understand such dynamics. Often, one estimates n from the absorption cross section (σ) and the excitation fluence assuming that σ is not very sensitive to the excitation density, which is a reasonable assumption at significantly above the bandgap with sufficiently large density of state. In principle, one can obtain the average absorption cross section of the nanocrystals from the elemental analysis (EA) in conjunction with the information of the size (volume), shape, and stoichiometric composition of the nanocrystal if the size and shape of the nanocrystals are sufficiently uniform and well-defined. Such a strategy has been successfully used for measuring the absorption cross section of various binary quantum dots (QDs).17–21 Another approach determining the average absorption cross section is analyzing the saturation of the transient absorption (TA) intensity at the peak of exciton absorption or transient photoluminescence (PL) intensity with increasing excitation fluence.22–24 Compared to the method based on elemental analysis, determination of the absorption cross section from the saturation of TA or PL intensity does not require information on the volume of the nanocrystals. This is an important advantage over the elemental analysis method, especially for the structures with anisotropic morphology where the determination of the volume is not always straightforward. It is worth mentioning that the absorption cross section of isolated single nanocrystals has also been measured via photothermal imaging of individual particles.25 Earlier studies in binary QDs have shown that both the elemental analysis and saturation approaches result in comparable absorption cross section values.20,22

In the case of lead halide perovskite QDs, the absorption cross sections reported so far are spread in a wide range differing by an order of magnitude depending on the method employed. Hens and co-workers employed the elemental analysis method to determine the size-dependent absorption cross section of CsPbBr3 QDs in the size range of 4–11 nm, which are similar to those of CdSe QDs of the same volume.15,17 On the other hand, several other studies that employed the transient PL intensity saturation method reported scattered values that are also an order of magnitude smaller than those of Hens and co-workers for QDs of similar volume.23,24 It is not yet clear why there is such a large discrepancy between the two measurements. However, an order of magnitude smaller value from the PL intensity saturation method is quite unusual since the QDs of many different materials exhibit absorption cross sections of the same order of magnitude for a given volume at the energies significantly above the bandgap.17,18,26

In this work, we measured the absorption cross section of CsPbBr3 QDs of varying sizes employing both elemental analysis and transient absorption (TA) saturation methods and compared with the previously reported data. From this comparative analysis, we obtained reliable absorption cross section values of CsPbBr3 QDs as a function of size in the strongly confined regime. Both the elemental analysis and TA saturation methods gave similar results that are also very close to the result of Hens and co-workers, suggesting the higher reliability of the absorption cross section determined from these two methods than the transient PL saturation method. The results from this study also show that the TA saturation method can reliably determine the absorption cross section of perovskite nanocrystals without knowledge of the volume, which will be particularly useful for nanocrystals with complex morphologies.

Monodisperse cube-shaped colloidal CsPbBr3 quantum dots (QDs) with edge lengths from 3.8 nm to 6.9 nm were synthesized employing the recently reported procedure, which utilized the thermodynamic equilibrium of halide for size control.12,27 These QDs are under the strong quantum confinement regime and exhibit strongly size-dependent exciton absorption energy, as shown in Fig. 1(a). The representative transmission electron microscope (TEM) images of the QDs of different sizes are also shown in Figs. 1(b)–1(f). The absorption spectra of CsPbBr3 QDs dispersed in hexane shown in Fig. 1(a) were obtained using a CCD spectrometer (Ocean Optics, QE65pro). TEM images of CsPbBr3 QDs were obtained on a FEI Tecnai G2 F20 ST field-emission TEM microscope. The average size (edge length) was found from extensive survey of the TEM images as described in previous literature, also allowing for a polynomial fit between size and bandgap.15,27

FIG. 1.

(a) Size-dependent absorption spectra and [(b)–(f)] TEM images of the CsPbBr3 quantum dots (QDs) used in this study. The size shown is the edge length of the QDs. Scale bar is 20 nm.

FIG. 1.

(a) Size-dependent absorption spectra and [(b)–(f)] TEM images of the CsPbBr3 quantum dots (QDs) used in this study. The size shown is the edge length of the QDs. Scale bar is 20 nm.

Close modal

Elemental composition of Cs and Pb was determined using inductively coupled plasma mass spectrometry (ICP-MS). The samples for the elemental analysis were prepared by first measuring the absorption spectra of the QD solution dispersed in hexane, which was subsequently dried and digested in 70% aqueous solution of nitric acid. Appropriate dilutions were made using 1% nitric acid. ICP-MS measurements were made with a NexION 300 ICP-MS instrument. The calibration curves used for the determination of the ion concentration of the samples were obtained using the standard solutions (Sigma Aldrich, 1000 mg/l stock) which were diluted to a known dilution factor. Indium and bismuth were used as internal standards for Cs and Pb, respectively.

Transient absorption measurements were performed on a home-built pump-probe transient absorption spectrometer. The 400 nm pump was generated by doubling the 800 nm output (80 fs, 3 kHz) of a titanium-sapphire amplifier (KM Laboratories) with a 300 μm-thick β-barium borate (BBO) crystal whose fluence was controlled by a pair of linear polarizer and a half waveplate.28 The white light supercontinuum probe was generated by focusing a few microjoules of 800 nm beam on a 1 mm-thick CaF2 window.28 Transient absorption data at a chosen probe wavelength were recorded using a pair of amplified Si photodiodes and a monochromator (Newport, Oriel Cornerstone 130), in conjunction with Boxcar gated integrators. QD samples dispersed in ∼30 ml of cyclohexane were circulated through a quartz flow cell with 2 mm pathlength to avoid any potential photodegradation. Cyclohexane was used for the TA measurement due to its slower evaporation rate compared to hexane allowing for easier control of the constant sample concentration during the prolonged measurement using the liquid flow cell. The full width at half maximum (FWHM) beam diameter of 400 nm pump beam was ∼230 µm, which was determined by using the razor blade method.29 The FWHM of the weaker probe beam (∼20 µm) was measured by directly imaging the probe beam on a CCD camera (DMK21BU04, The Imaging Source) at the sample position. The power of the 400 nm pump beam was measured with an Ophir Nova power meter and an Ophir 3A-P-SH-VI sensor. For the accurate measurement of the power of the 400 nm pump beam with minimal contamination from the 800 nm beam, two sets of dielectric mirrors selectively reflecting 400 nm and transmitting 800 nm were used after the BBO crystal. Any residual 800 nm light reaching to the power meter was corrected for by separately measuring the power of the leaking 800 nm beam in the absence of the BBO crystal in the beam path. Further details on the pump fluence determination are in the supplementary material.

The absorption cross section of the strongly confined CsPbBr3 QDs dispersed in hexane was determined employing two different methods in this work, i.e., elemental analysis and saturation of transient absorption (TA). The method based on the elemental analysis is effective when the volume and the elemental composition of the QDs are readily determined. Since the CsPbBr3 QDs synthesized in this study exhibit highly uniform size and morphology, the determination of the volume of the QD from the analysis of TEM images is relatively straightforward. The absorption cross section of the QDs was determined using Beer’s law from the measured absorption spectrum (Aλ) and molar concentration of QDs (cQD) in the sample solution determined from the elemental analysis and the volume of the QD. The absorption cross section (σλ) was calculated from the molar absorption coefficient (ϵλ) with the conversion factor of 2303/NA, where NA is Avogadro’s number,17 

Aλ=ελbcQD,σλ=ελ2303NA.
(1)

In this approach, the number of Pb ions (NPb) contained in the QD of volume (V = L3) was first determined by considering the cube-shaped QDs with the surface terminated with Br and passivated with the oleylammonium ion using the edge length (L) obtained from the TEM images (supplementary material, Table S1), cQD was calculated from NPb, and the molar concentration of Pb ions in the QD sample solution (cPb) was determined from the elemental analysis. Since the stoichiometric ratio Cs:Pb:Br in CsPbBr3 QDs deviates from 1:1:3 depending on the size and surface terminating element of the QD especially for Cs and Br, we chose Pb to determine the concentration of QDs that is considered less sensitive to the detailed surface structure.15,30,31

In Fig. 2, the absorption cross sections of CsPbBr3 QDs at 400 nm (σ400) determined using the method described above are plotted. For comparison, the result from the earlier work by Hens and co-workers,15 who employed the elemental analysis approach using a slightly different data analysis method, is also shown alongside σ400 determined by the TA saturation method as will be discussed later. σ400 of the QDs measured employing the elemental analysis approach from the two different studies (this work and Ref. 15) are similar to each other for the QDs of comparable sizes, indicating the consistency of this method in determining the absorption cross section of CsPbBr3 QDs. The small difference in the values of σ400 may partially reflect the difference in the uncertainty and distribution of the QD size in the samples used in these two studies. The absorption cross section determined from the elemental analysis depends directly on the number of atoms within the QD counted for each element, which varies with the size and shape of the nanocrystal and surface termination. Therefore, despite the general applicability of the elemental analysis-based method for the measurement of the absorption cross section, its accuracy depends strongly on the quality of information on the size and shape of the nanocrystal and its surface structure.

FIG. 2.

The absorption cross section of CsPbBr3 QDs of different sizes at 400 nm (σ400) determined from elemental analysis (EA) and transient absorption (TA) in this work. The error bar on the x-axis represents a maximum of 5% estimated dispersity in the edge length of the QDs. For comparison, the values reported in Ref. 15 from EA are also shown with the values in Refs. 23 and 24 from transient photoluminescence (PL) measurements.

FIG. 2.

The absorption cross section of CsPbBr3 QDs of different sizes at 400 nm (σ400) determined from elemental analysis (EA) and transient absorption (TA) in this work. The error bar on the x-axis represents a maximum of 5% estimated dispersity in the edge length of the QDs. For comparison, the values reported in Ref. 15 from EA are also shown with the values in Refs. 23 and 24 from transient photoluminescence (PL) measurements.

Close modal

The alternative methods determining the absorption cross section of the QDs that do not require such information rely on the saturation behavior of transient photoluminescence (PL) intensity or transient absorption (TA) bleach signal. In the case of transient PL intensity saturation, it should exhibit the saturation behavior described by the following equation, if the number of photons absorbed by the QD can be described by Poisson statistics:

IPL(tl)=a(1eN)=a(1eFph*σλ).
(2)

IPL(tl) is the transient PL intensity at a sufficiently long time tl after the excitation, at which the PL decay dynamics reflect the decay of only single excitons. N is the average number of photons absorbed per QD at the given excitation fluence. Fph and σλ are the photon fluence (in photons/cm2) and absorption cross section (in cm2) at the excitation wavelength, respectively. a is a normalization constant.8,23 For the lead halide perovskite nanocrystals, the majority of the reported absorption cross sections were determined by analyzing the PL intensity saturation including CsPbBr3, CsPbI3, and CsPbBrxI3-x.8,23,24Figure 2 includes the previously reported σ400 of CsPbBr3 QDs determined using the PL saturation method for comparison. The numerical values of σ400 from this study and the earlier studies are tabulated in the supplementary material (Table S2). Notably, the absorption cross section of CsPbBr3 QDs determined from the PL intensity saturation method is much smaller and scattered significantly between different measurements. In principle, both the elemental analysis and PL intensity saturation method should give comparable absorption cross sections. While the origin of such a large discrepancy existing in the literature is not clear, we note that PL saturation method can easily underestimate the absorption cross section when a spatially nonuniform excitation beam is used as will be discussed later.

Another way to determine the absorption cross section without the QD size information is to analyze the bleach signal at the peak of exciton absorption in the transient absorption (TA) data, which is methodologically similar to the PL intensity saturation method. Both have the same underlying assumption of the applicability of the Poisson distribution for the number of photons absorbed in the QDs, therefore sharing the similar equation relating the absorption cross section to the signal saturation curve.23 However, TA signal saturation has been much less frequently used than the PL saturation method for the perovskite QDs. In this study, taking advantage of recent progress in synthesis of highly uniform and stable QDs under prolonged TA measurement condition,27,28 we determined σ400 of CsPbBr3 QDs using the TA saturation method and compared with those determined by elemental analysis. For this purpose, the TA signal probed at the peak of the exciton absorption, ΔOD(t), was measured with 400 nm pump for each QD as a function of excitation fluence.

Figure 3(a) shows the normalized |ΔOD(t)|, |ΔOD(t)|norm, for the CsPbBr3 QDs with the edge length of L = 6.9 nm at varying pump fluences. The data in Fig. 3(a) are normalized to |ΔOD(200 ps)| at the saturating pump fluence. The pump fluence shown is the average pump fluence under the probe beam area, defined as the circular area with the diameter of full width at half maximum (FWHM) of the probe beam (∼20 µm). In this study, FWHM of the pump beam (∼230 µm) was much larger than that of the probe beam ensuring relatively homogeneous excitation fluence under the probed area. The data for the QDs of other sizes are in the supplementary material. The early-time dynamics of |ΔOD(t)| shown more clearly in Fig. 3(b) exhibit a rapidly decaying component on the time scale of several tens of picoseconds, which becomes more prominent as the pump fluence increases due to the Auger decay of multiple excitons. After the completion of the Auger decay, the slow decay occurs on the time scale of ∼5 ns reflecting the decaying of the remaining single exciton, which is nearly independent of the pump fluence.

FIG. 3.

(a) |ΔOD(t)|norm of CsPbBr3 QDs (L = 6.9 nm) probed at the peak of exciton absorption normalized to |ΔOD(200 ps)| at the saturating pump fluence. (b) Early-time dynamics showing the rapid decay from Auger recombination at the higher pump fluences more clearly.

FIG. 3.

(a) |ΔOD(t)|norm of CsPbBr3 QDs (L = 6.9 nm) probed at the peak of exciton absorption normalized to |ΔOD(200 ps)| at the saturating pump fluence. (b) Early-time dynamics showing the rapid decay from Auger recombination at the higher pump fluences more clearly.

Close modal

Figure 4 shows |ΔOD(200 ps)|norm for the QDs of four different sizes as a function of pump fluence extracted from |ΔOD(t)|norm such as shown in Fig. 3 for the analysis of their saturation behavior. The curves are then fit to a model as will be discussed shortly. Additional data for other sizes and fit to a model are in the supplementary material. We chose 200 ps as the sufficiently long pump-probe delay time (tl) that ensures the completion of Auger decay so that every photoexcited QD contributes equally to the TA signal regardless of the number of initially excited excitons. Since the dynamics of the slow decay component in the TA data are nearly independent of the fluence, the TA signal saturation curve is insensitive to the chosen tl as long as tl is sufficiently long.

FIG. 4.

OD(tl)|norm as a function of pump fluence for CsPbBr3 QDs of different sizes at tl = 200 ps. The curves are the best fit to Eq. (4).

FIG. 4.

OD(tl)|norm as a function of pump fluence for CsPbBr3 QDs of different sizes at tl = 200 ps. The curves are the best fit to Eq. (4).

Close modal

For the optically dilute samples, |ΔOD(tl)|norm should exhibit the pump photon fluence dependence of Eq. (3) similar to the case of PL intensity, where Fph is the photon fluence of the pump beam under the probe beam area,

|ΔODtl|norm=1eN=1eFph*σλ.
(3)

On the other hand, for the samples with significant absorbance at the pump wavelength, the attenuation of the pump beam through the sample that lowers the average photon fluence should be considered in the analysis of the saturation of |ΔOD(tl)|norm. For the sample solution in a cell of pathlength l exhibiting the absorbance of Al at the pump wavelength, the photon fluence at location x along the pathlength decays exponentially as FphAl,x,l,F0=F010Al(x/l), where F0 is the incident pump photon fluence impinging on the sample. This modifies the dependence of |ΔOD(tl)|norm on F0, as shown in Eq. (4),

|ΔODtl|norm=E1σλF010Al+E1σλF0Alln10+1,E1x=xeuudu.
(4)

While this correction is not necessary for the PL measurement that can be made easily with optically dilute samples, it is needed for TA measurements that often uses samples of higher absorption. In this study, where the absorbance of the sample at 400 nm was 0.2–0.5, using Eq. (3) underestimates the absorption cross section up to factor of ∼2. The curves in Fig. 4 are fits of the experimental data to Eq. (4). The quality of the fit in Fig. 4 is generally very good. While the fit is slightly worse in 6.9 nm QD than others, it may be due to variation of the quality of the sample rather than the failure of the Poisson statistics of photon absorption in this size. The derivation of Eq. (4) is in the supplementary material. σ400 of CsPbBr3 QDs of different sizes determined from the fit in Fig. 4 are also shown in Fig. 2 for comparison. σ400 determined using the TA saturation method is in good agreement with the result from the elemental analysis of this study and the earlier work by Hens and co-workers.15 The consistency of the values of σ400 from this comparison indicates that the reliable absorption cross section of CsPbBr3 QDs can be obtained from both the elemental analysis and TA saturation method. The wavelength-dependent absorption cross section (σλ) of CsPbBr3 QDs of several different sizes in the spectral range of 350–550 nm based on σ400 values determined from the analysis of TA saturation is plotted in Fig. S1 of the supplementary material.

It is puzzling that the reported absorption cross section of various lead halide perovskite nanocrystals determined using the transient PL saturation method is much smaller than those from the elemental analysis and TA saturation method. For instance, Ref. 23 reported σ400 of 1.0 × 10−15 cm2 for CsPbBr3 QDs with L = 5.9 nm (V = 205 nm3), which is more than an order of magnitude smaller than from this study as compared in Fig. 2. In the case of CsPbI3, QDs, σ400 in the range of 1–1.3 × 10−14 cm2 was reported for QDs of L = 11.2–11.4 nm,23,24,32 which is also more than an order of magnitude smaller than what we determined from the elemental analysis for the QDs of comparable size in a separate measurement (σ400 = ∼2.6 × 10−13 cm2). While the reason for such a discrepancy is uncertain, we note that the use of typical laser light with the Gaussian beam profile on the sample cell larger than the beam size can underestimate the absorption cross section when the entire PL signal is measured without spatial resolution. Because of the spatially varying PL saturation from the spatially inhomogeneous excitation intensity on the sample, the PL signal can appear to saturate more slowly when the entire PL signal is detected. For this reason, spatially resolved PL intensity measurement is required for an accurate analysis of the PL intensity vs excitation density when the excitation beam size is smaller than the size of the sample cell.33 

Now, we discuss the factors that can potentially affect the measurement of the absorption cross section via the elemental analysis method more significantly in lead halide perovskite nanocrystals compared to more common binary systems. For the elemental analysis method, the significant departure of the Cs:Pb:X stoichiometric ratio in QDs from the bulk value of 1:1:3 that depends on the size of the QD and on the reaction condition can complicate the determination of the absorption cross section. Because of the labile anion and the sensitivity of the crystal phase to the surrounding environment, lead halide perovskite nanocrystals are more susceptible to change its stoichiometric composition or structure in response to changes of ligand and solvent environment.31 For instance, the Cs/Pb ratio in the range of 0.62–1.5 that varies depending on the QD size, synthesis method, and aging of the sample was reported.9,14,15,27,31 A large deviation of the Cs/Pb ratio from 1 that varies with the thickness was also observed in 1D and 2D structures.13 The Br/Pb ratio is also shown to be size dependent, exhibiting increasing Br/Pb ratio with decreasing QD size in the Br-terminated QDs.27 In the case of CsPbCl3 QDs, postsynthesis self-anion exchange resulted in a significant increase of the absorption intensity with a concomitant increase of PL quantum yield without a noticeable change in the particle size, presumably by removing the existing Cl vacancies in the QDs.34 This contrasts to the case of CsPbBr3 QDs, where no change in absorption and PL intensity was observed upon self-anion exchange, which suggests much less halide vacancy in CsPbBr3 QDs than in CsPbCl3 QDs. Because of the large variation of the elemental composition that depends on various factors exemplified above, the accuracy of the elemental composition is particularly important in determination of the absorption cross section of lead halide perovskite nanocrystals via elemental analysis. When the uncertainty in volume and elemental composition of the nanocrystals is high, the TA saturation method could be more advantageous than the elemental analysis method.

The absorption cross section is an important spectroscopic parameter of the colloidal QDs, crucial to understand their photophysical properties that depend on the density of photoexcited excitons. The absorption cross section of the colloidal QDs has been measured in many different ways including elemental analysis and saturation of transient PL or transient absorption signal, which gave consistent results for various binary semiconductor QDs. However, there is a significant discrepancy in the reported absorption cross section values of lead halide perovskite QDs determined by using different methods, notably those determined by the transient PL intensity saturation method being much smaller than those from the elemental analysis. In this work, we made comparative analysis of the absorption cross section of CsPbBr3 QDs of varying sizes, employing elemental analysis and transient absorption (TA) saturation methods to establish more reliable absorption cross section values. The comparison of the results from this study and previous reports indicates that both the elemental analysis and TA saturation result in consistent absorption cross sections, which are of the same order of magnitude to those of II-VI QDs of similar volume. We concluded that the absorption cross section determined by elemental analysis and TA saturation is more accurate than those reported previously by using the transient PL saturation method that are smaller by an order of magnitude, although the reason for such large underestimation is not clear.

See supplementary material for further experimental details and derivation of Eq. (4)

This research was supported by the National Science Foundation (Grant No. CHE-1836538, O.H.-C.C.) and Institute for Basic Science (Grant No. IBS-R026-D1, D.H.S.). The authors thank the Elemental Analysis Laboratory and Microscopy and Imaging Center at Texas A&M University for ICP-MS and TEM measurements.

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Supplementary Material